MODELING EMERGING SOLAR CELL MATERIALS AND DEVICES
By
Non Thongprong

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics – Doctor of Philosophy
2017

ABSTRACT
MODELING EMERGING SOLAR CELL MATERIALS AND DEVICES
By
Non Thongprong
Organic photovoltaics (OPVs) and perovskite solar cells are emerging classes of solar cell
that are promising for clean energy alternatives to fossil fuels. Understanding fundamental
physics of these materials is crucial for improving their energy conversion efficiencies and
promoting them to practical applications.
Current density-voltage (JV) curves; which are important indicators of OPV efficiency,
have direct connections to many fundamental properties of solar cells. They can be described
by the Shockley diode equation, resulting in fitting parameters; series and parallel resistance
(Rs and Rp ), diode saturation current (J0 ) and ideality factor (n). However, the Shockley
equation was developed specifically for inorganic p-n junction diodes, so it lacks physical
meanings when it is applied to OPVs. Hence, the purposes of this work are to understand
the fundamental physics of OPVs and to develop new diode equations in the same form as
the Shockley equation that are based on OPV physics.
We develop a numerical drift-diffusion simulation model to study bilayer OPVs, which
will be called the drift-diffusion for bilayer interface (DD-BI) model. The model solves
Poisson, drift-diffusion and current-continuity equations self-consistently for charge densities
and potential profiles of a bilayer device with an organic heterojunction interface described
by the GWWF model. We also derive new diode equations that have JV curves consistent
with the DD-BI model and thus will be called self-consistent diode (SCD) equations.
Using the DD-BI and the SCD model allows us to understand working principles of bilayer
OPVs and physical definitions of the Shockley parameters. Due to low carrier mobilities in
OPVs, space charge accumulation is common especially near the interface and electrodes.
Hence, quasi-Fermi levels (i.e. chemical potentials), which depend on charge densities, are
modified around the interface, resulting in a splitting of quasi-Fermi levels that works as

a driving potential for the heterojunction diode. This brings about the meaning of Rs as
the resistance that gives rise to the diode voltage equal to the interface quasi-Fermi level
splitting instead of the voltage between the electrodes. Quasi-Fermi levels that drop near
the electrodes because of unmatched electrode work functions or due to charge injection can
also increase Rs . Furthermore, we are able to study dissociation and recombination rates of
bound charge pairs across the interface (i.e. polaron pairs or PPs) and arrive at the physical
meaning of Rp as recombination resistance of PPs. In the dark, PP density is very low, so
Rp is possibly caused by a tunneling leakage current at the interface. Ideality factors are
parameters that depend on the split of quasi-Fermi levels and the ratio of recombination rate
to recombination rate at equilibrium. Even though they are related to trap characteristics
as normally understood, their relations are complicated and careful interpretations of fitted
ideality factors are needed. Our models are successfully applied to actual devices, and useful
physics can be deduced, for example differences between the Shockley parameters under dark
and illumination conditions.
Another purpose of this thesis is to study electronic properties of CsSnBr3 perovskite
and processes of growing the perovskite film using an epitaxy technique. Calculation results
using density functional theory reveal that a CsSnBr3 film that is grown on a NaCl(100)
substrate can undergo a phase transition to CsSn2 Br5 , which is a wide-bandgap semiconductor material. Actual mechanisms of the transition and the interface between CsSnBr3
and CsSn2 Br5 are interesting for future studies.

ACKNOWLEDGEMENTS

Every time I look back to the past of my life, I am surprised by how far I have journeyed
the academic path and how much I have succeeded. I would not be here if I had not had
support from the following persons.
First of all, I would like to thank my advisor, professor Duxbury, for understanding
me, correcting my mistakes, and supporting me academically and financially. I would like to
thank all members in my guidance committee: professor Pengpeng Zhang for letting me work
in her lab for a semester; professor Richard Lunt for being a great collaborator; professor
Norman Birge and professor Mark Dykman for being very good teachers in the courses I
took and guiding my PhD goals. I would like to thank all professors in the department and
the department of Physics and Astronomy for being a great place to study for PhD.
I would like express great gratitude to the Development and Promotion of Science and
Technology Talents Project (DPST), which is a program supported by the Royal Thai Government, for recruiting me into this program and for funding me since high school.
I would also like to thank Lili Wang, who is a postdoc in Professor Lunt’s group, and
Peggy Young, who works with Dr Lunt as well. It has been pleasure to work with you two.
Thank you, Gift Kanokkorn Pimcharoen, for giving me the prototype of the numerical model.
I would like to mention David Miller and my Thai friend; Thanaphong Phongpreecha, who
give me advice about doing DFT simulations in VASP. Thank you all my friends; Dan Olds,
Jenny Portman, Bin, Xukun, Brian, Connor and Joe Williams, who make the fourth floor
of BPS building a wonderful place to work in.
Lastly, I could not say thanks enough to my mom and dad, who have been raising and
taking care of me even when we are thousands of miles away. I would also like to thank
all Thai friends, Thai Student Association and Office of Educational Affairs at Royal Thai
Embassy in Washington DC for helping me through the years at MSU.

iv

TABLE OF CONTENTS

LIST OF TABLES

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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
CHAPTER 1 BACKGROUND AND MOTIVATION . . . . . . . . . . . . . . . . .
1.1 Global Warming and the Need for Solar Energy Technologies . . . . . . . . .
1.2 Introduction to Organic Photovoltaics . . . . . . . . . . . . . . . . . . . . . .
1.3 My Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Shockley Equation and Equivalent Electrical Circuit Model . . . . . . . . . .
1.5 Efficiency parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 GWWF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Shockley-Read-Hall Recombination . . . . . . . . . . . . . . . . . . . . . . .
1.8 Shockley-Read-Hall Recombination with Exponential Trap Distributions . .
1.9 Dissociation and Recombination Rates of Polaron Pairs . . . . . . . . . . . .
1.10 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2 ANALYTICAL MODEL WITH SPACE CHARGES
2.1 Self-Consistence Ideal Diode (SCD) Equations . . . . . . .
2.1.1 The f-f Recombination . . . . . . . . . . . . . . . .
2.1.2 The f-t Recombination . . . . . . . . . . . . . . . .
2.1.3 The f-f and f-t Recombination . . . . . . . . . . . .
2.1.4 The SRH Recombination . . . . . . . . . . . . . . .
2.2 Diode Voltage, Series and Parallel Resistances . . . . . . .
2.3 Definition of Illumination Conditions . . . . . . . . . . . .
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 3 NUMERICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 the Drift-Diffusion model for Bilayer Interfaces (DD-BI model) . . . . . . .
3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4 PHYSICS OF OPVS . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 The Differences Between the Current Models and the GWWF Model . . . .
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Strategies to Improve the GWWF Model . . . . . . . . . . . . . . . .
4.2 Effects of Carrier Mobilities . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Space charge effects and quasi-Fermi levels splitting . . . . . . . . . .
4.2.2 Effect of mobility on space charges . . . . . . . . . . . . . . . . . . .
4.2.3 Dissociation and recombination of polaron pairs: The parallel resistance

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4.4

4.5
4.6
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4.2.4 Effect of mobility on the series resistance . . . . . . . . . .
Effects of Injection Barriers . . . . . . . . . . . . . . . . . . . . .
4.3.1 JV Characteristic Response with Varied Injection Barriers
4.3.2 Reduced Built-in Potential from High Injection Barriers . .
4.3.3 Unaffected Interface Densities: Unaffected Voc . . . . . . .
Metal/Organic Semiconductor Contact . . . . . . . . . . . . . . .
4.4.1 Mott-Schottky Model With Vacuum Level Shift . . . . . .
4.4.2 Integer Charge Transfer (ICT) Model . . . . . . . . . . . .
4.4.3 Organic/Metal Interfaces Under Illumination . . . . . . . .
4.4.4 Organic/Metal Interfaces In the Dark . . . . . . . . . . . .
4.4.5 Components of Series Resistance . . . . . . . . . . . . . .
4.4.6 Choosing the Anode and the Cathode . . . . . . . . . . . .
Ideality Factor and Trap-assisted Recombination . . . . . . . . . .
Comparison between the SRH and the f-t Recombination . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 5 INTERPRETATION OF SHOCKLEY PARAMETERS : DARK VERSUS LIGHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Organic Salt Photovoltaic Devices . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Effects of the Shockley Parameters on Dark JV curves . . . . . . . . . . . . .
5.3 Effects of the Shockley Parameters on illuminated JV curves . . . . . . . . .
5.4 Parallel Resistance in the Dark . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Trap-to-trap Recombination as the Dark Leakage Current . . . . . .
5.4.2 Simulating the CyTRIS:C60 Device . . . . . . . . . . . . . . . . . . .
5.4.3 Including a Constant Dark Parallel Resistance in the DD-BI and the
SCD models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Complete Equivalent Electrical Circuit Model . . . . . . . . . . . . . . . . .
5.5.1 Total Parallel Resistance under Illumination Conditions . . . . . . . .
5.6 Shockley Parameters: Dark Versus Light . . . . . . . . . . . . . . . . . . . .
5.6.1 Diode Saturation Current (J0 ) . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Series Resistance (Rs ) . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Ideality Factor (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Parallel Resistance (Rp ) . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.5 Errors of Using Dark and Illumination Shockley Parameters Interchangeably . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125
126

CHAPTER 6 AB INITIO CALCULATION OF AN EPITAXIAL PEROVSKITE . .
6.1 Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Cubic Perovskite . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 CsSn2 Br5 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Band Structure and Density of States Calculations of CsSnBr3 and CsSn2 Br5
6.2.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 The PBE Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2.1 CsSnBr3 Bands and DOS . . . . . . . . . . . . . . . . . . .

128
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vi

100
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6.3
6.4
6.5
6.6
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6.2.2.2 CsSn2 Br5 Bands and DOS . . . . . . . . . . . . . . . . . .
6.2.3 The HSE06 Calculations . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Summary and Discussion of Band Gaps . . . . . . . . . . . . . . . .
Adsorption of the Precursor Atoms Onto the NaCl(100) Surface . . . . . .
Effects of the NaCl Substrate on the CsSnBr3 band structure . . . . . . . .
CsSnBr3 Film Growing Processes: Layer by Layer Deposition . . . . . . .
Useful Research Directions : Heteroepitaxy, Critical Thickness and Phase
Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 7 CONCLUSIONS AND OUTLOOKS . . . . . . . . . . . . . . . . . . 154
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

vii

LIST OF TABLES

Table 4.1: Summary of parameters used in the comparison between the GWWF
model and the models developed in this study. . . . . . . . . . . . . . . .

48

Table 4.2: Four cases of GWWF calculations with improved values of δD , δA , FI
and Rs to include space charge effects. . . . . . . . . . . . . . . . . . . . .

54

Table 4.3: Summary of parameters used in simulations of P3HT:PCBM bilayer solar
cells, with hole mobility as the independent variable . . . . . . . . . . . .

57

Table 4.4: Parameters indicating power conversion efficiency; short circuit current
(Jsc ), open circuit voltage (Voc ), and fill factor (F F ), of the P3HT:PCBM
bilayer solar cells with different P3HT hole mobility, obtained from figure
4.6(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Table 4.5: Summary of parameters used to study effects of cathode injection barriers φc 69
Table 4.6: Summary of parameters used to study effects of hole characteristic trap
temperature (Tt,D ) in a P3HT:PCBM bilayer OPV. . . . . . . . . . . . .

89

Table 4.7: Dependence of device parameters; Jsc , Voc and FF, on hole characteristic
trap temperature (Tt,D .) . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Table 4.8: Summary of the simulation parameters used in the comparison between
the SRH recombination (Eq. 1.43) and the f-t recombination (Eq. 1.8). .

94

Table 4.9: Comparisons of the Jsc , Voc and F F values between the devices having
RSRH (Eq. 1.43) and Rf-t (Eq. 1.8) as the interface recombination. The
values are measured from the JV curves in Fig. 4.33, and the devices are
described by Table 4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Table 5.1: Summary of parameters used to study effects of the trap-to-trap recombination (Rt-t ) in terms of the effective mobility µtunn in the CyTRIS:C60
device[1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Table 6.1: Summary of the band gap values and lattice parameters of CsSnBr3 and
CsSn2 Br5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Table 6.2: Resulting adsorption energies (Eads ) and perpendicular distances in the
study of adsorption of Cs, Sn and Br precursor atoms on the NaCl(100)
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

viii

LIST OF FIGURES

Figure 1.1: Energy band diagram and working principles of typical OPVs showing
several important steps: (1) exciton generation; (2) exciton diffusion;
(3) forming of a charge transfer exciton or a polaron pair (PP); (4)
dissociation of PP; (5) charge transport and collection. Adapted from
[2, 3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 1.2: Equivalent electrical circuit with single diode for a solar cell . . . . . . .

11

Figure 1.3: Equivalent electrical circuit with double diode for a solar cell . . . . . . .

12

Figure 1.4: Example of JV characteristic and output power versus voltage of a solar
cell with an illustration of squareness of the JV curve. . . . . . . . . . . .

13

Figure 1.5: Energy level diagram and possible interactions occurring at the heterointerface between the active layers of a bilayer organic solar cell as
proposed by GWWF. Adapted from[4] . . . . . . . . . . . . . . . . . . .

14

Figure 1.6: Diagram showing the four processes constituting the total ShockleyRead-Hall recombination: electron capture rate (rnc ); electron emission
rate (rne ); hole capture rate (rpc ); and hole emission rate (rpe ) . . . . . .

19

Figure 1.7: Energy band diagram of an organic semiconductor showing exponential
tail state distributions from the LUMO (gn (E)) and from the HOMO
(gp (E)) that are filled with trapped electrons and holes up to Ef n and
Ef p respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 1.8: Total SRH recombination at the D/A interface as a result of net capture and emission of the interface charges (nI and pI ) by the interface
donor trapped holes and the interface acceptor trapped electrons having
energies Etp and Etn respectively. . . . . . . . . . . . . . . . . . . . . . .

24

Figure 2.1: Schematic flow of the constant exciton current (qJX ) through the PP
processes at the interface resulting in the photo-current qJX ηP P d and
the recombined exciton current (qJX (1 − ηP P d )) that is flowing through Rp . 35
Figure 3.1: Flowchart of the numerical DD-BI model. . . . . . . . . . . . . . . . . .

38

Figure 4.1: Comparison of JV characteristics generated by the DD-BI model (symbol), the SCD equation in Equation 2.14 (solid line), the GWWF model
with δD = 0.9 and δA = 0.1 (dashed line) and the GWWF model with
δD = 0.5 and δA = 0.5 (dashed and dotted line). . . . . . . . . . . . . . .

49

ix

Figure 4.2: Comparison of JV characteristics generated by the DD-BI model (symbol), the SCD equation in Equation 2.14 (solid line), the GWWF model
with δD = 0.9 and δA = 0.1 (dashed line) and the GWWF model with
δD = 0.5 and δA = 0.5 (dashed and dotted line) for the devices with
φa /φc equal to (a) 0.1eV/0.3eV, (b) 0.2eV/0.3eV and (c) 0.3eV/0.3eV. . .

49

Figure 4.3: (a) Interface hole (pI ) and electron (nI ) densities versus the applied
voltage calculated by the DD-BI model (filled symbols) and the equations used by GWWF (Eqs. 1.12 and 1.13)(open symbols) that have
fractions of potential dropped δD and δA shown in figure (b) as their
arguments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 4.4: Graphs of the interface electric field strengths (FI ), polaron pair dissociation rates and probabilities (kP P d and ηP P d ) of the device with
φa /φc = 0.1eV /0.6eV calculated by the DD-BI models (black), the GWWF
model with δD /δA = 0.9/0.1 (red) and the GWWF model with δD and
δA as a function of Va derived from the DD-BI model(green). . . . . . . .

52

Figure 4.5: Comparisons of JV characteristics calculated by the DD-BI model, the
SCD model and the four cases of modified GWWF calculations that
are summarized in Table 4.2, in the devices with φa /φc equal to (a)
0.1eV/0.6eV, (2) 0.0eV/0.0eV and (3) 0.3eV/0.3eV. The values of the
fitted Rs that are used in case 4 are written in each plot. . . . . . . . . .

55

Figure 4.6: (a) J-V curves under illumination conditions of P3HT:PCBM bilayer
photovoltaic cells and (b) ln|J| − V curves of the same devices under
dark conditions. Parameters used in these simulations are in table 4.3. .

57

Figure 4.7: (a) A comparison of hole (red lines) and electron (black lines) densities
at Va = 0V between dark (or equilibrium) conditions (sold lines) and illuminated conditions (dashed line), of the device with µp = 10−6 m2 /V s.
(b) Energy level diagram of the same device under both conditions including equilibrium (dashed line), quasi electron Fermi (green line) and
hole (blue line) energy levels. . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 4.8: (a) Densities of electron (black) and hole (red) within the devices with
different donor hole mobility, under Va = 0V . Equilibrium charge densities are almost invariant with µp and plotted in the solid lines. (b) Energy level diagram showing different quasi-Fermi levels deviation/splitting
in the devices. The top LUMO and bottom HOMO bands are almost
invariant with the mobility. . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 4.9: Polaron pair dissociation rate (kP P d ), recombination rate (kP P r ) and
dissociation probability/efficiency (ηP P d ), as a function of applied voltage. 63

x

Figure 4.10: Plots of (a) kP P d , (b) kP P r , and (c) ηP P d as a function of voltage in
the devices with different hole mobility. . . . . . . . . . . . . . . . . . . .

64

Figure 4.11: Parallel resistances (Rp ) as a function of applied voltage calculated by
equation 2.28 from the devices having different donor µp . . . . . . . . . .

65

Figure 4.12: Energy diagram showing LUMO level (top solid line) and HOMO level
(bottom solid line), including electron and hole quasi-Fermi levels (red
dashed and blue dotted-dashed line respectively) of the devices with (a)
µp = 10−6 m2 /V s and (b) µp = 10−9 m2 /V s at Va = 0.4V . . . . . . . . .

66

Figure 4.13: Plot of series resistance (Rs ) functions of the devices with different hole
mobility in donor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Figure 4.14: Plots of the diode voltage (Vd = Ef n,I −Ef p,I ) and total change of quasiFermi energies from the electrodes to the interface (Vd ), i.e. Va − Vd ,
with respect Va . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 4.15: (a) JV characteristics of devices under influence of increasing cathode
work function (injection barrier), and (b) their extracted Jsc , Voc , and FF. 69
Figure 4.16: Potential profiles (ψ(x)) of the devices with different φc at applied voltage (a) 0V and (b) 0.5V. . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Figure 4.17: Carrier density profiles (n(x) and p(x)) showing only majority carriers,
i.e. holes in the donor and electrons in the acceptor, of the devices with
φc = 0.0eV, 0.2eV and 0.4eV at the same applied voltage of 0.5V. . . . .

71

Figure 4.18: Plotting as a function of φc of (a) interface electron and hole densities
with their multiplication and (b) open circuit voltages extracted from
the DD-BI model simulations (same as in Fig. 4.15b), compared with a
calculated one using equation 4.12. . . . . . . . . . . . . . . . . . . . . .

73

Figure 4.19: Illustration of a Mott-Schottky contact between a metal with work finction Φm and an n-type semiconductor with chemical potential Φs , where
Φm > Φs , forming a blocking contact and downward band bending. . . .

74

Figure 4.20: Illustration of an ohmic contact between a metal with work finction Φm
and a semiconductor with Fermi level Φs , where Φm < Φs , forming an
ohmic contact and upward band bending. . . . . . . . . . . . . . . . . .

75

Figure 4.21: Integer Charge Transfer (ICT) model describing spontaneous charge
tunneling at a metal/organic semiconductor interface of the case (a)
Φm > EICT + , (b) EICT − < Φm < EICT + and (c) Φm < EICT − . . . . .

77

xi

Figure 4.22: A simple diagram showing the bilayer structure and some important
positions at the anode and the cathode organic/metal and the heterojunction interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Figure 4.23: Band diagram illustrating energy levels of the devices with φa = 0.2eV
and φc =0.0eV, 0.3eV and 0.6eV under illumination. . . . . . . . . . . . .

79

Figure 4.24: Plots of vacuum level shift (∆), voltage loss (Vloss ) and acceptor electron
quasi-Fermi level (Ef n,L−1 ) at the cathode relative to the cathode work
function (W Fc ) as a function of φc . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.25: Band diagram illustrating energy levels of the devices with φc =0.0eV,
0.3eV and 0.6eV in the dark. . . . . . . . . . . . . . . . . . . . . . . . . .

81

Figure 4.26: Diagram showing possible deviations of hole quasi-Fermi level at the anode contact and bulk donor and electron quasi-Fermi level at the cathode
contact and bulk acceptor, that can be represented by a component of
series resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Figure 4.27: Plots of series resistance as a function of voltage under influences of
changing cathode injection barrier from 0.0eV to 0.6eV with fixed anode
injection barrier at 0.2eV. . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Figure 4.28: Plots of the component series resistances Rsa , RsD , RsA and Rsc , and
total Rs being the sum of the four components as a function of applied
voltage of the devices with different cathode injection barrier. . . . . . .

84

Figure 4.29: Ideality factors nD and nA in the symmetric device with characteristic
trap temperatures of electron (Tt,A ) and hole (Tt,D ) equal to 1000K. . .

88

Figure 4.30: JV characteristic responses of the devices with different hole characteristic trap temperature (Tt,D ) but fixed electron characteristic trap
temperature (Tt,A ) at 1000K. . . . . . . . . . . . . . . . . . . . . . . . .

88

Figure 4.31: Plots of the ideality factors (a) nD and (b) nA as a function of voltage,
calculated using equations 2.17 and 2.18 respectively, under the influences of varying Tt,D . The inset shows the interface electron and hole
densities in dependence on voltage when Tt,D = 1200K. . . . . . . . . . .

90

Figure 4.32: Plots of JV components, namely, diode currents from free-electron to
free-hole recombination (f-f), free-electron to trapped-hole recombination (fn-tp) and free-hole to trapped-electron recombination (fp-tn), and
the photo-current (qJX ηP P d ), of the devices with fixed Tt,A = 1000K
and Tt,D = 1200K, 800K, and 400K. . . . . . . . . . . . . . . . . . . . .

91

xii

Figure 4.33: Comparisons between the illuminated JV curves calculated using Rf-t
(Eq. 1.8) and RSRH (Eq. 1.43) of the devices with Tt,D and Tt,A equal
to a) 1000K and b) 450K. . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 4.34: a) Total RSRH and Rf-t rates and b) their separate rates from different
mechanisms; RSRH = RSRH,D + RSRH,A and Rf-t = Rrec,n + Rrec,p ,
of the devices with µn /µp = 100 and Tt,D(A) = 1000K. . . . . . . . . . .

96

Figure 5.1: a) Molecular structures of the heptamethine cation (Cy+ ) and b) its
−
−
counter anions: (from left to right) I− ; PF−
6 ; SbF6 ; TRIS ; and
TPFB− , that are used as the donor in the device with the architecture shown in c). From Ref. [1] . . . . . . . . . . . . . . . . . . . . . . . 101
Figure 5.2: JV curves under a) nominal 1 sun illumination and b) dark condition
from measurements (symbols) and fitting with the generalized Shockley
equation (Eq. 1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Figure 5.3: Effects of the Shockley parameters a) Rp , b) J0 , c) n and d) Rs on
the shapes of dark JV curves calculated by the conventional Shockley
equation (Eq. 1.3) using each parameter as a constant. Control variables
are summarized at the top of each plot. . . . . . . . . . . . . . . . . . . . 103
Figure 5.4: Plots of the total dark currents (J), the diode currents (Jdiode ) and the
leakage currents (Jleak ) where J = Jdiode + Jleak as a function of voltage
for the cases of a) Rp = 3 × 105 and b) 3 × 101 Ωm2 . . . . . . . . . . . . 104
Figure 5.5: A figure showing the three regions in a dark current, namely, the leakage
region, the diode region, and the Ohmic region, including how the shape
in each region is affected by the Shockley parameters. adapted from [5] . 106
Figure 5.6: Effects of the Shockley parameters a) Rp , b) J0 , c) n and d) Rs on the
shapes of illuminated JV curves calculated by the conventional Shockley
equation (Eq. 1.3) using each parameter as a constant. Control variables
are summarized at the top of each plot. . . . . . . . . . . . . . . . . . . . 107
Figure 5.7: (a) Energy band diagram and the thickness of each layer in the CyTRIS:C60
device. (b) Fermi-level pinning, vacuum level shifts, and possible hole
injecting mechanism at the ITO/MoO3 /CyTRIS interfaces ( adapted
from [6]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Figure 5.8: Comparisons between the simulated JV curves of the CyTRIS:C60 device and the experimental data under a) illumination and b) dark conditions113
Figure 5.9: Plots of the currents originated from RSRH (JSRH ) and Rtt (Jtun )
defined in Eqs. 5.3 and 5.4 and the total dark currents J = JSRH + Jtun
for the devices with µtun equals to a) 1.0 × 10−11 and b) 5.0 × 10−10 m2 /V s.114

xiii

Figure 5.10: Plot of the parallel resistance Rp_tun calculated by Eq. 5.5 as a function
of applied voltage in the device with µtun = 5.0 × 10− 10m2 /V s . . . . . . 115
Figure 5.11: Comparisons of JV curves of the CyTRIS:C60 device under (a) illumination and (b) dark conditions between the experimental results (circle
symbol), the DD-BI and the SCD results (lines with a symbol) and
the Shockley equation’s results (Eq. 1.3) (solid lines) using the fitted
parameters from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Figure 5.12: A schematic showing the bilayer OPV architecture with each important
physical mechanism being represented by an electrical circuit component. All the electrical components form the complete EEC model. . . . 118
Figure 5.13: A comparison between Rs , Rp_P P , Rp_tun and total device resistance
calculated by Eqs. 2.27, 2.28, 5.5 and [∂J/∂Va ]−1 respectively. The
device is from Section 5.4 with µtun = 5 × 10−10 m2 /V s and under illumination conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Figure 5.14: JV curves of the CyTRIS:C60 (black lines with circles) and the P3HT:PCBM
(red lines with squares) devices under (a) illumination and (b) dark conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 5.15: Plots of J0 as a function of Va extracted from (a) the CyTRIS:C60 and
(b) P3HT:PCBM devices ( Figs. 5.14) under dark (black solid lines)
and illumination (red dashed line) conditions. . . . . . . . . . . . . . . . 122
Figure 5.16: Comparisons between illumination and dark conditions of (a) kP P d (Va ),
kP P r (Va ), (b) FI (Va ) and ηP P d = kP P d /(kP P d +kP P r ) of the CyTRIS:C60
device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 5.17: Comparisons between illumination (red dashed lines) and dark (black
solid lines) series resistances of (a) the CyTRIS:C60 and (b) P3HT:PCBM
devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 5.18: Ideality factor functions calculated by Eq. 2.23 of (a) the CyTRIS:C60
and (b) the P3HT:PCBM devices under dark and illumination conditions. (inset) zoomed portion of figure (a) for Va > 0V . . . . . . . . . . 124
Figure 5.19: Plots of parallel resistances under illumination (Rp_P P ) of the CyTRIS:C60
(black with circles) and the P3HT:PCBM (red with squares) devices calculated using Eq. 2.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 6.1: (a) Structure of a cubic perovskite showing octahedra of BX6 and the
cation A filling the cuboctahedral void. (b) and (c) Two representations
of the perovskite unit cell with the labels of the Cs, Sn, and Br ions in
figure (b) for the case of CsSnBr3 . . . . . . . . . . . . . . . . . . . . . . . 130

xiv

Figure 6.2: a) Tetragonal unit cell of CsSn2 Br5 with the definitions of the basis
vectors and b) its projection onto the b/c plane. c) Projection onto the
a/b plane of the [Sn2 Br5 ]n−
n polymeric sheet. Gray, purple and brown
spheres represent Cs, Sn and Br atoms, respectively. . . . . . . . . . . . . 132
Figure 6.3: Triclinic unit cell of CsSn2 Br5 that is projected into a) the a/c plane,
b) the b/c plane and c) the a/b plane. Gray, purple and brown spheres
represent Cs, Sn and Br atoms, respectively. . . . . . . . . . . . . . . . . 133
Figure 6.4: The first brillouin zones of (a) a cubic, (b) a triclinic and (c) a tetragonal
unit cell with a path along the edges of the irreducible brillouin zones.
adapted from [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure 6.5: Ground state energy (E0 ) convergence tests of the cubic phase as a
function of (a) cut off energy (Ecut ) using a Γ-centered k-point grid
9x9x9 and (b) k-point grid of NxNxN where N is the x-axis value with
Ecut = 275eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 6.6: Ground state energy (E0 ) convergence tests of the optimized triclinic
unit cell of CsSn2 Br5 as a function of (a) cut off energy (Ecut ) using a
Γ-centered k-point grid 13x13x12 and (b) Γ-centered k-point grids with
a fixed Ecut = 300eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Figure 6.7: Electronic band structure and projected density of states the cubic
CsSnBr3 structure calculated using the PBE functional. . . . . . . . . . 138
Figure 6.8: Zoomed in projected density of states around the VBM and the CBM
of the cubic CsSnBr3 calculated using the PBE functional. . . . . . . . . 139
Figure 6.9: Electronic band structure and projected density of states of CsSn2 Br5
calculated using the optimized triclinic unit cell and the PBE functional.

139

Figure 6.10: Electronic band structure and projected density of states of CsSn2 Br5
calculated using the optimized tetragonal unit cell and the PBE functional.140
Figure 6.11: Calculated band structures of a) the cubic CsSnBr3 and b) the triclinic
CsSn2 Br5 using the HSE06 functional. . . . . . . . . . . . . . . . . . . . 141
Figure 6.12: (a) The unit cell used for the infinite NaCl(100) surface under 3D periodic boundary conditions. The larger yellow spheres represent Na
atoms, and the smaller green spheres represent Cl atoms. (b) Top view
of a single-layer NaCl(100) surface. . . . . . . . . . . . . . . . . . . . . . 144
Figure 6.13: Structure of the CsSnBr3 crystal bonded to the NaCl(100) surface by
the Sn2+ Cl− ionic bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

xv

Figure 6.14: Unit cells of the cubic CsSnBr3 perovskite films having the thicknesses
of 1.7 nm that are used to study effects of the NaCl substrate. . . . . . . 146
Figure 6.15: Band structure and PDOS functions of the CsSnBr3 film that is attached
to the NaCl substrate, calculated using the PBE functional. . . . . . . . 147
Figure 6.16: Band structure and PDOS functions of the CsSnBr3 film without the
NaCl substrate, calculated using the PBE functional. . . . . . . . . . . . 148
Figure 6.17: Simulation results of depositing atomic layers of CsSnBr3 onto the NaCl
substrate. Na, Cl, Cs, Sn, Br atoms are represented by yellow, green,
gray, purple and brown spheres, respectively. . . . . . . . . . . . . . . . . 149
Figure 6.18: Simulation results showing film deposition processes after the third fullcover SnBr layer. Na, Cl, Cs, Sn, Br atoms are represented by yellow,
green, gray, purple and brown spheres, respectively. . . . . . . . . . . . . 151

xvi

KEY TO ABBREVIATIONS

OPVs Organic Photovoltaics
GWWF the model proposed by Giebink, Wiederrecht, Wasielewski and Forrest
JV current density-voltage characteristics
EEC the equivalent electrical circuit model
MPP maximum power point
eq equilibrium conditions (i.e. no light and applied voltage)
D/A interface the interface between the donor and the acceptor layers
DD-BI the drift-diffusion for bilayer interfaces model
SCD the self-consistent diode equations
HOMO, LUMO the highest occupied molecular orbital and the lowest unoccupied molecular orbital
PP polaron pair at the D/A interface (i.e. charge-transfer exciton)
CT charge-transfer exciton at the D/A interface (i.e. polaron pair)
f-f recombination between free electrons and free holes at the D/A interface
fn-tp recombination between free electrons and trapped holes at the D/A interface
fp-tn recombination between free holes and trapped electrons at the D/A interface
t-t recombination between trapped electrons and trapped holes at the D/A interface
SRH the Shockley-Read-Hall recombination
Jsc , Voc short-circuit current and open-circuit voltage
Jmax , Vmax output current density and voltage at the MPP
Jd , Jdiode diode current density
Jleak leakage current density
Jleak,dark leakage current density in the dark
Jph photo-current density
Jtun current density from tunneling processes
Va the applied voltage (equal to W Fc - W Fa )
xvii

Vd the diode voltage
Vt the thermal voltage
J0 the diode saturation current in the Shockley equation
Js0 the diode saturation current from the f-f recombination
JsD , JsA the diode saturation currents arising from donor trapped holes (the fn-tp recombination) and acceptor trapped electrons (the fp-tn recombination)
Rs total series resistance and series resistance in the Shockley equation
Rsa , RsD , RsA , Rsc series resistances caused by the anode contact, the bulk donor, the
bulk acceptor and the cathode contact
Rp total parallel resistance and parallel resistance in the Shockley equation
Rp_tun recombination resistance of tunneling current density
Rp_P P recombination resistance of the recombined exciton current density
Rp,dark , Rp,illu parallel resistances under dark and illumination conditions
n ideality factor
nD , nA ideality factors arising from donor trapped holes (the fn-tp recombination) and
acceptor trapped electrons (the fp-tn recombination)
JX exciton current density reaching the D/A interface
a0 the width of the D/A interface
relative dielectric permittivity
Eb polaron pair binding energy
ζ polaron pair density at the D/A interface
FI electric field strength at the D/A interface
kP P d , kP P r dissociation and recombination rates of interface polaron pairs
ηP P d polaron pair dissociation probability
R free carrier recombination at the D/A interface
∆EHL interface energy gap
krec recombination coefficient of the f-f recombination
krec,n , krec,p recombination coefficients of the fn-tp and the fp-tn recombinations
ni intrinsic free carrier density
xviii

nI , pI free electron and hole densities at the D/A interface
nIt , pIt trapped electron and hole densities at the D/A interface
nc , pc free electron and hole densities at the cathode and the anode contacts
δD , δA fractions of potential dropped across the donor and the acceptor
Vbi built-in potential
W Fa , W Fc work functions of the anode and the cathode
φa , φc injection barriers at the anode and the cathode
NLU M O , NHOM O densities of states at the acceptor LUMO and the donor HOMO band
edges
Ef equilibrium Fermi energy
Ef n , Ef p quasi-Fermi energies of electrons and holes
Ef n,I , Ef p,I quasi-Fermi energies of electrons and holes at the D/A interface
HD , HA trap densities of states at the donor HOMO and the acceptor LUMO band edges
Tt,D , Tt,A characteristic trap temperatures of donor trapped hole and acceptor trapped
electron distributions
lD , lA characteristic trap parameters of donor trapped hole (lD = Tt,D /T ) and acceptor
trapped electron (lA = Tt,A /T ) distributions
Et,D , Et,A characteristic trap energies of donor trapped hole (Et,D = kB Tt,D ) and acceptor
trapped electron (Et,A = kB Tt,A ) distributions
Nt trap density in the SRH recombination
Cn , Cp capture rates of electrons and holes in the SRH recombination
µn , µp electron and hole mobilities
n(x), p(x) electron and hole densities as a function of position (i.e. electron and hole density
profiles)
ψ(x) electrostatic potential as a function of position (i.e. potential profile)

xix

CHAPTER 1
BACKGROUND AND MOTIVATION

1.1

Global Warming and the Need for Solar Energy Technologies

In the book Sustainable Energy — Without the Hot Air [8], David JC MacKay points out
that there are three main reasons why we have to rethink about our uses of fossil fuels: (1)
fossil fuels are limited; (2) we need energy security; and (3) burning fossil fuels causes climate
change. Energy security is defined as the ability to access energy resources at an affordable
price. Points (1) and (2) may cause economic and political problems, but more importantly,
climate change and global warming, caused by by-products of burning fossil fuels, may result
in uninhabitable regions and unhealthy climate and environment across the planet.
According to the International Energy Agency[9], in 2014, oil, coal and natural gas; which
are the three primary fossil fuels, have their shares at 39.9%, 11.4% and 15.1%, respectively,
of the direct global energy consumption of 109,612 terawatt-hours (TWh), where watt-hour
is a unit of energy equal to 3.6 × 103 Joules. These percentages do not include the additional
use of oil, coal or natural gas in producing electricity. Thus, fossil fuels are the primary
energy resources consumed by humans. A statistical prediction based on the ratios of world
consumption to reserved fossil fuels in 2006 [10] predicts that oil, coal and gas will last
about 40, 200 and 70 years, respectively, meaning that we will approximately run out of
these resources in 2046, 2206 and 2076. Even though advanced extraction techniques like
hydraulic fracturing [11] and discoveries of new crude oil locations and new forms of resources
like methane hydrate on the seabed [12] can extend the supply time hundreds of years, the
resources will be depleted eventually.
Almost all of the utilizations of fossil fuels involve burning the resources to get heat, and
then using the heat to do work. For example, we burn natural gas to warm our houses, burn
oil to run the heat engines in our cars, and burn coals to boil water to produce electricity in
power plants. These processes are not the most efficient or favorable as we lose energy during
1

heat transfers, carbon dioxide (CO2 ) gas; which is a greenhouse gas, is emitted as a product
of combustion reactions (e.g. CH4 + 2O2 → CO2 + 2H2 O). Naturally, CO2 gas is a part of
atmospheric gases that absorbs and re-emits infrared parts of sunlight so that Earth is livable.
Introducing extra CO2 content to the atmosphere from human activities undoubtedly results
in higher global temperature. There are clear evidences that CO2 concentration in the air
has increased at an accelerating rate after the industrial revolution [8, 13]. According to the
global temperature data by the National Aeronautics and Space Administration (NASA)
[14], the year 2016 is the warmest year since 1880, having 0.99◦ C higher temperature than
the 1951-1980 average. At this rate, global average temperature is predicted to increase
by 2-6◦ C within a century; which will result in a sea level rise between 0.5 to 1.5 meters
[15]. Due to the problems associated with using fossil fuels, sustainable, renewable and clean
energies are necessary.
The amount of sunlight energy hitting Earth in two hours is more than an annual total
global energy consumption from all resources combined [16]. This means that, theoretically,
we could cut the uses all energies and replace them with solar energy alone. In a qualitative
view, assuming that a solar panel can harvest about 20% of solar irradiance (i.e. solar energy
per unit time and unit area), whose average is around 300 Wm−2 [17] and there are 12 hours
of daylight, the panel can produce a yearly energy of
20% × 300W m−2 × 60seconds × 60minutes × 12hours × 365days
= 262.8kW hm−2 . (1.1)
3.6 × 106 J/KW h
In order to have as much energy as the 2014 total global assumption, we need to install
109, 612 × 109 KW h
= 4.17 × 1011 m2 = 4.17 × 105 km2
262.8KW hm−2

(1.2)

or 645km x 645km of solar panels, which is only about 0.28% of Earth’s land area[18]. This
futuristic idea can be practical if solar panels are geographically distributed over available
areas, like roof surface area [19]. For example, if half of the population in Spain, where
average irradiation is 1,600 kW h/m2 [20, 21], installed 14m2 [19] of the 20%-efficiency solar
panels on their roofs, 42% of total electricity consumption (i.e. 249 Twh in 2014 [9]) could
2

be replaced by solar energy; which is equivalent to reduction of fossil fuel usage by 9 million
tonnes (i.e. 1 Million Tonnes of Oil Equivalent (MTOE) is 11.63 TWh).
Solar cell technologies have been developed since the mid-20th century, and now we are
on the verge of the third generation of solar cells. The first generation solar cells; which are
still the most efficient and dominate the solar cell market, are made of silicon (Si). They are
classified by the type of silicon used; which can be monocrystalline Si (up to 25% efficiency),
polycrytalline Si (up to 20% efficiency) and amorphous Si (less than 10% efficiency) [22].
The second generation of solar cells is based on thin-film technologies. Three main types of
solar cells in this generation are thin-film amorphous Si, cadmium telluride (CdTe) (up to
18% efficiency [22]) and copper indium gallium diselenide (CIGS) (up to 16% efficiency[22]).
However, solar cells in this generation are not as popular as the first generation’s ones, and
share less than 1% in the solar panel market [23]. The third generation solar cells, which
are at the focal point of this thesis and the research community, are based on solutionprocessed materials, where promising candidates include organic polymer:fullerene, organic
hybrid polymer, dye-sensitized and perovskite solar cells [24, 25].
Apart from the impressive efficiencies and lifetimes of silicon solar cells, pure silicon;
which is expensive and energy intensive, is needed[26]. For each kilogram of pure silicon,
electrical energy ranging from 15 KWh to 90 KWh is required; which means, in terms of
energy profits, silicon-based solar cells may take up to two years to generate the same amount
of electricity that was spent in producing them [26].
Organic photovoltaics or solar cells (OPVs or OSCs) are promising for the next generation solar cells because of the following advantages over inorganic solar cell technologies.
First, they can be manufactured using all-solution processes at low temperature and standard pressure. These include roll-to-roll printing techniques[27]. Second, they can be light,
flexible, transparent, integrable to other products, cheaper to manufacture than inorganic
solar cells, have short energy payback time and less environmental impact[28]. Results from a
cost analysis indicate that current OPV technologies have price per produced energy at 1.06-

3

8.90$ per watt-peak (Wp ), and can be as low as 0.05-0.64$ per Wp on an industrial scale[27].
Watt-peak is defined as the nominal power of a solar cell, measured under standard Test
Conditions; light intensity of 1KW/m2 , at latitude 42◦ N (airmass 1.5) and temperature
25◦ C. These prices per Wp are comparable to those of silicon solar cells[29]. Energy payback
time (EPBT) of OPV cells can be as low as 0.2 years, which is better than many renewable
energies, such as, wind (EPBT = 0.39 years), polycrystalline and monocrystalline Si PVs
(EPBT = 1.65-4.14 years), hydropower (EPBT = 0.5 years) and biomass (EPBT = 5-10
years)[30]. Furthermore, OPVs are among the cleanest sustainable energies because they
produce only 0.01-0.06 kg of CO2 per kW of produced electricity, which is comparable to
wind and hydropower energies, and less than 10% of CO2 amount emitted by natural gas[30].
We can see that OPVs have a lot of potential to be the next generation solar cells, or even
applied at a global scale to replace/reduce the uses of fossil fuels. However, low efficiencies
(less than 5% under actual operation) and short lifetimes are two main drawbacks of OPVs
that make them stay on laboratory shelves, instead of store shelves. These are the reasons
why we have to study and understand the physics of OPVS.

1.2

Introduction to Organic Photovoltaics

Photovoltaic (PV) cells, or solar cells, are semiconductor devices that generate a current and
voltage from the energy of incident photons. The current and the voltage are therefore the
most important parameters that need to be maximized. They depend on each other and
are often plotted as a current density-voltage (JV) characteristic curve. Having an analytic
equation to describe the JV curves is undoubtedly necessary for effective design of modules
consisting of many thousands of solar cells.
JV curves are usually diode-like, so the most ubiquitous JV equation is composed of the
Shockley p-n junction diode term[31], a photo-current term, and a leakage term. Series and
parallel resistances (Rs and Rp ) are usually included to represent physical loss mechanisms
in PV cells[17, 32]. The resulting equation is generally referred to as the Shockley equation.
4

It can meaningfully describe p-n junction PV cells, which are p-n junction diodes both when
the light is absent and present.
When new types of PV cell emerge, the Shockley equation and the working principles
of p-n junction PV cells are also applied to them, for example, organic photovoltaic (OPV)
cells[33, 34, 35, 36, 37, 38], dye sensitized solar cells (DSSCs)[39], perovskite solar cells[40, 41],
and organic salt solar cells[1], because they all exhibit diode-like behavior similar to the p-n
junction. Despite the fact that the Shockley equation fits their JV curves, their working
principles and structures are different from the p-n junction PV cells[42, 43]. This has led to
a debate on whether the Shockley equation is universal; which is still open, even in 2015[44].
Diode saturation current (J0 ), Rs , Rp and the ideality factor (n) are important Shockley
parameters that are usually fitted and extracted by experimentalists to show their devices’
physical processes. Rs is interpreted as the total resistance of the bulk layers and contacts
[35, 45, 46, 5], while Rp is understood to represent leakage current[32] that is caused by
either pinholes[5] or direct contact between the electrodes (e.g. anode and cathode) and
the active layers[47]. The ideality factor n is an indicator of the free-carrier recombination
type. In the ideal case with n = 1, free charges recombine via bimolecular recombination[44].
n exactly equal to 2[48] in the p-n junction picture arises from the Shockley-Read-Hall
recombination[49, 50] where free charges recombine with trapped charges that act as recombination centers. In reality, any values of n can be extracted, and n > 1 is interpreted as an
evidence of trap-assisted recombination[51, 52].
Thin-film OPV cells are the main focus in this study. A typical energy band diagram and
working principles of OPVs are illustrated in Figure 1.1[2, 3]. The active layer, which is the
layer that adsorbs photons and produces free carriers, is made of an electron-donor and an
electron-acceptor material. They are often called donor and acceptor for short. The donor
material can be a macrocyclic compound[38, 53], a polymer[54, 55], etc., and the acceptor is
usually a fullerene derivative[38, 54]. The first OPV cell was invented by Tang in 1986[53] in
a form of planar heterojunction, or bilayer, structure where the donor and the acceptor are

5

deposited separately into two layers. Later, the idea of blending the donor and the acceptor
in the bulk heterojunction (BHJ) structure [56, 57, 58] was introduced and gained popularity
among researchers. Recently, attention has been drawn back to the planar heterojunction
because it leads to a superior OPV morphology[59, 60, 61], and novel PV technologies like
peroveskite solar cells[62, 63, 64] also adopted it. Hence, the bilayer structure is at the focal
point in this study.

Figure 1.1: Energy band diagram and working principles of typical OPVs showing several
important steps: (1) exciton generation; (2) exciton diffusion; (3) forming of a charge transfer
exciton or a polaron pair (PP); (4) dissociation of PP; (5) charge transport and collection.
Adapted from [2, 3].
As illustrated in Fig. 1.1, typical working principles of OPVs are as follows[2, 3]. First,
incident photons excite electrons in the donor molecules from the highest occupied molecular
orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). However, because the
dielectric permittivities are typically low in organic semiconductors, the excited electrons are
coulombically bound to electron holes (or holes) in the HOMO, resulting in tightly bound
electron-hole pairs; or excitons, that have a large binding energy of around 0.3 - 1.0 eV[65].
Excitons in organic materials are usually confined within the molecular radius and thus are
categorized as Frenkel excitons. The exciton generation processes are step (1) in Figure
6

1.1. These excitons are electrically neutral and diffuse around via hopping mechanisms
that are caused by the exciton concentration gradient. Typical exciton duffusion length in
organic materials is around 1-10 nm[2]. If the excitons can diffuse to the donor-acceptor
(D/A) interface as shown in step (2) in Figure 1.1, the excited electrons undergo charge
transfer under the influence of the LUMO energy offset; ELU M O,D − ELU M O,A , to a more
favorable energy level in the acceptor, as labeled by process (3). Electronic relaxations of the
donor holes and geometric relaxations of the donor molecules occur, resulting in self-localized
states called hole polarons[66]. The same relaxations apply to the acceptor electrons and
the acceptor molecules, leading to electron polarons. After the charge transfer in step (3),
electron polarons and hole polarons are still bound together across the D/A interface, and
a pair of electron and hole polarons is defined as a polaron pair (PP) or a charge-transfer
(CT) exciton. The coulombic bond in a PP can be broken by an external electric field
(shown step (4)), resulting in a free electron in the acceptor and a free hole in the donor. In
the final step, these free carriers are transported under influences of an applied electric field
(drift) and charge concentration gradients (diffusion) to the electrodes, so they are collected
and utilized. Accordingly, the key mechanism that distinguishes OPV cells from others is
dissociation of PPs into free carriers[67, 68].
The paper by Giebink, Wiederrecht, Wasielewski and Forrest (GWWF) [4] was probably
the first and one of very few papers trying to derive PV equations from the physics of
CT states. The resulting equations are in the same form as the Shockley equation but
written in terms of CT dissociation and recombination rates. This work ties the opposite
ends of solar cells together and validates the applicability of the Shockley equation to OPV
cells. There are, however, some improvements that can be made to the GWWF model to
make it more accurate. For example, GWWF calculated the interface charge densities using
fractions of potential dropped across the donor and the acceptor, which is strictly valid
only when the sum of drift and diffusion currents is zero[69]. The interface electric field
which is very important in determining the dissociation of CT excitons is also inaccurate

7

because it is calculated from the interface charge densities. Additionally, the series resistance,
which is an intrinsic property depending on charge transport and contacts, is missing in
the GWWF derivation due to some assumptions. Even though it was added manually
later so the equations fitted with experimental results better, its physical meaning was still
absent. Another OPV characteristic that is hardly included in any OPV equations is the
accumulation of charges due to poor transport properties[44], which are referred to as space
charge effects.
Numerical simulation solving the Poisson equation and the current continuity equations
self-consistently can bring about the correct interface charge densities and the inclusion
of space charge effects. It is a widely accepted method used to study OPV cells[70, 71,
72, 73, 74]. Barker et al. [69] developed a numerical model for bilayer OPV cells that
include interface processes similar to the GWWF model but with the space charge effects
included. It is able to reproduce OPV behaviors such as dependence of Voc on incident light
intensity. Nonetheless, it needs some modifications that make it more suitable for OPV cells.
For instance, Langevin-like bimolecular recombination[75, 76] that was used for free carrier
recombination can be replaced by a trap-assisted recombination[4, 77]. Most importantly, it
would be more helpful to have an analytic JV equation to go with the numerical simulation.
In this study, a numerical simulation for thin-film bilayer OPV cells is developed and
utilized; and new JV equations in the same form as the Shockley equation with quasi-Fermi
energies are developed. They will be called the Drift-Diffusion model for Bilayer Interfaces
(DD-BI model) and the Self-Consistent Ideal Diode (SCD) models/equations respectively.
The DD-BI model; which has space charge effects included, can calculate the right interface
charge densities, the right interface electric field and the right D/A CT rates. Using these
quantities as a function of the PV voltage in the SCD equations not only makes them able
to reproduce the DD-BI’s JV curves, but also able to provide physical meanings of the
Shockley parameters (e.g Rs , Rp , n). This is the proof that the Shockley form equations
are appropriate to describe OPV JV curves[44]. However, each parameter in the equations

8

is not constant in general but a function of applied voltage as well as other parameters in
the model.

1.3

My Contributions

There are two parts in this thesis. The first part is simulating bilayer OPVs using a computer program and models that we develop ourselves, and the second part is calculating
electronic properties of a perovskite and related materials using density functional theory
with a commercial software.
For the first part, as stated earlier, the Shockley equation that was developed for inorganic
p-n junctions can fit OPV JV curves regardless of their different natures, so I started this
part with curiosity of origins of the Shockley equation and its parameters in terms of organic
semiconductor physics. I took a Fortran code of 1D drift-diffusion simulation that was written
by Kanokkorn Pimcharoen; another PhD candidate in our group, and included the bilayer
interface as suggested in the GWWF model; which is a Shockley equation model for OPVs,
resulting in the DD-BI model in Chapter 3. However, JV results from the DD-BI model
do not agree with GWWF diode equations, so I made comparisons between the two models
and found that the GWWF model does not treat many OPV properties correctly, especially
interface densities, space charge effects and series resistance (see Section 4.1). Thus, I derived
new diode equations for OPVs using charge densities as a function of quasi-Fermi levels,
resulting in the SCD equations in Chapter 2; which are always consistent with the DD-BI
model. Using the DD-BI and SCD models allows me to understand principles of bilayer OPVs
in terms of quasi-Fermi levels and define actual meanings of all the Shockley parameters for
OPVs. Many characters of OPVs are studied (see Chapter 4) and new interesting results are
observed. For example, injection barriers smaller than about 0.2eV do not affect JV curves,
and increasing injection barriers almost do not affect open-circuit voltage. I can show by
using the DD-BI and the SCD models that small injection barriers result in quasi-Fermi level
pinning, so they do not affect JV curves, and interface densities; which mainly determine
9

Voc , are not significantly affected by increasing injection barriers. I can also show that
ideality factor functions that I derived are actually general forms of a famous ideality factor
equation but provide more details about trap characteristics. In Chapter 5, I include more
mechanisms governing leakage current in the dark and thus arrive at a complete diode model
from OPVs. My models can fit and explain actual devices and provide more understanding
of the Shockley parameters both under illumination and dark conditions.
My contributions in the second part of the thesis are that I use the program VASP to
calculate band structures and band gaps of CsSnBr3 and CsSn2 Br5 in DFT levels in order
to confirm experimental results of growing a CsSnBr3 film on a NaCl(100) substrate using a
vapor phase heteroepitaxy technique. Another purpose; which is still in progress, is to find
the atomic configuration at the interface between CsSnBr3 and CsSn2 Br5 in order to verify
that CsSnBr3 can transform into CsSn2 Br5 during the growth process.

1.4

Shockley Equation and Equivalent Electrical Circuit Model

Most inorganic solar cells nowadays are based on p-n junction architectures that have been
developed and studied since the 1950s[78]. For this reason, the ideal diode equation of a
p-n junction, which was developed by Shockley[31], has always been applied to solar cells to
describe their current-voltage (JV) response. The generalized form of the Shockley equation
that includes photogenerated current (Jph ), series resistance (Rs ), and parallel resistance
(Rp ) is given by
J = J0 exp

Va − JRs
q(Va − JRs )
− 1 − Jph +
,
nkB T
Rp

(1.3)

where J0 is the saturation current of minority carriers, q is the elementary charge, Va is
applied voltage as well as the voltage that a solar cell produces at a corresponding J, n
is the ideality factor, kB is the Boltzmann constant, and T is temperature[32, 17]. Rs is
a resistance from contacts and bulk materials, and Rp comes from leaking current at the
p-n junction that can be represented by a shunt path. This conventional definition of Rp
10

applies to OPVs only in the dark because Rp under illumination conditions is dominated by
recombination resistance of interface PPs (See Section 2.2 and 5.5). The ideality factor n is an
indicator of charge transport/recombination mechanisms. Conventionally, the recombination
current gives rise to n = 2 while the current from charge diffusion has n = 1[48]. There are
three terms on the right hand side of Eq. 1.3, i.e. diode current(Jd ), photocurrent, and
leakage current(Jleak ), from left to right. Each of these terms can be represented by an
electrical component in the equivalent electrical circuit (EEC) model, shown in Fig. 1.2.

Figure 1.2: Equivalent electrical circuit with single diode for a solar cell
However, the Shockley equation and the EEC model sometimes cannot accurately fit
experimental data. A double diode model was introduced to capture both recombination
and diffusion currents[79], with the Shockley equation modified accordingly to
J = J01 exp

q(Va − JRs )
Va − JRS
q(Va − JRs )
− 1 + J02 exp
− 1 − Jph +
, (1.4)
n1 kB T
n2 kB T
Rp

where the first diode term represents diffusion current having n1 = 1 and the second term
represents recombination current having n2 = 2[79, 48]. Later, it was shown that n1 = 1
and n2 = 2 are not always true, and both of them should also be fitting parameters to get a
better fit to actual devices[80]. Figure 1.3 depicts the EEC model for the double diode case.

11

Figure 1.3: Equivalent electrical circuit with double diode for a solar cell

1.5

Efficiency parameters

One of the simplest ways to measure and display how well a solar cell can convert sunlight
energy into electrical energy is to use efficiency parameters associated with a JV characteristic
curve. If a solar cell is connected to an external load, the relation between current and voltage
that it can deliver is often plotted as the lower part of Figure 1.4, where negative current is
defined as a current flowing out of the anode under illumination. Load or external resistance
will determine the current and voltage. If the cell produces large current, the produced
voltage has to be small, and vice versa. The point where maximum current is achieved is
the point where the voltage is zero and is called short circuit (sc) conditions. The current
here is short circuit current (Jsc ). On the other hand, the cell will have maximum voltage
if the current is zero. This condition is called open circuit (oc), having the voltage of open
circuit voltage (Voc ).
Different points on the JV curve correspond to different output power defined by P ower =
J × V (see upper part of Fig. 1.4). There is one point where the output power is maximum.
This is the point where solar cells operate, and it is called maximum power point (MPP).
The associated voltage and current are Vmax and Jmax , respectively. Efficiency of a solar
cell is exhibited by power conversion efficiency (PCE) which is the ratio of electrical output
of the solar cell at the MPP to the power of incident sunlight. The power of sunlight per

12

Figure 1.4: Example of JV characteristic and output power versus voltage of a solar cell with
an illustration of squareness of the JV curve.
unit area; irradiance, is G with the unit W att/m2 . PCE is given by
P CE =

Jmax Vmax
.
G

(1.5)

However, it is more useful to correlate the MPP with Jsc and Voc so that the shape of the JV
curve can be roughly presented. Another parameter, named Fill Factor (FF), is introduced
and defined by F F = Jmax Vmax /Jsc Voc . According to the lower part of Fig. 1.4, FF is the
ratio of the area of the inner red rectangle to the area of the outer red rectangle. FF is then
squareness of JV characteristics. Consequently, it is more general to calculate PCE with the
following form.
P CE =

Jsc Voc F F
G
13

(1.6)

Ultimately, larger Jsc , Voc and FF, give higher PCE and a better solar cell. It is the primary
goal of researchers to develop solar cell technology so that these parameters are as high as
possible.

1.6

GWWF Model

Giebink, Wiederrecht, Wasielewski and Forrest (GWWF) proposed a model for organic heterojunctions that includes important interface processes and derived ideal diode equations
from the model[4]. Below is an overview of this model.

Figure 1.5: Energy level diagram and possible interactions occurring at the heterointerface
between the active layers of a bilayer organic solar cell as proposed by GWWF. Adapted
from[4]
The GWWF model can be depicted by Fig. 1.5. Photoexcited donors produce bound
excited electron-hole pairs, or excitons[81], that can diffuse to the donor-acceptor (D/A)
interface with a current density JX . The electrons transfer from the lowest unoccupied
molecular orbital (LUMO) of the donor to the LUMO of the acceptor, but they are still

14

coulombically bound with the donor holes[56]. This forms paired charges called charge
transfer (CT) excitons or polaron pairs (PPs). The D/A interface is a0 thick, where a0 is the
PP binding length. The PPs, having total density of ζ: can either recombine to ground states
with a rate kP P r , the polaron pair recombination rate[82]; or dissociate to be free charge
carriers with a rate kP P d , the Onsager-Bruan electric field assisted dissociation rate[83].
Two more interactions affect ζ. One is thermally activated PP from the ground states with
a rate kP P r ζeq . The other is recombination of free charges at the interface, denoted by R
in this work. Competition between total recombination rate and total dissociation rate will
result in the current density (J) flowing outward or inward at the interface, and it is this
current that will be calculated.
Two possible mechanisms of the recombination (R) are considered. The first one is
free-carrier to free-carrier Langevin bimolecular recombination[75, 76]. It is denoted by f-f
recombination, which follows the same notation as in Ref.[84]. The rate is given by
Rf-f = krec nI pI ,

(1.7)

where krec is the Langevin bimolecular recombination coefficient, and nI (pI ) is the interface
free electron (hole) density. krec is given by krec = q(µn + µp )/ 0 , where µn (µp ) is the
electron (hole) mobility,

is the dielectric constant, and 0 is dielectric permittivity of free

space. The second recombination mechanism is trapped-assisted recombination which is a
recombination between free-carriers and trapped-charges (f-t). Its expression is given by
Rf-t = krec,n nI pIt + krec,p pI nIt ,

(1.8)

where krec,n is the recombination coefficient of free-electrons to trapped-holes (fn-tp), krec,p
is the recombination coefficient of free-holes to trapped-electrons (fp-tn), and nIt (pIt ) is the
interface trapped electron (hole) density.
Assuming steady state conditions of ζ, nI and pI , the following rate equations are obtained.
JX
− kP P r (ζ − ζeq ) − kP P d ζ + R = 0.
a0
15

(1.9)

kP P d ζ − R +

J
= 0.
qa0

(1.10)

Solving these equations leads to an expression for the current density J as
J = qa0 (1 − ηP P d ) R −

kP P d
kP P d,eq

Req − qJX ηP P d ,

(1.11)

where ηP P d is PP dissociation probability expressed as ηP P d = kP P d /(kP P d + kP P r )[83, 85],
kP P d,eq is the equilibrium polaron pair dissociation rate, and Req is the interface free carrier
recombination rate under equilibrium. Equilibrium here refers to when there is no light; JX =
0, and no applied voltage; Va = 0. In order to have an explicit equation for J, expressions for
R in Eq. 1.7 or Eq. 1.8 and for the interface densities nI and pI are needed. GWWF used
the following relations to relate nI and pI with cathode electron density (nc ) and anode hole
density (pc ).
nI = nc exp

δA q(Va − Vbi )
.
kB T

(1.12)

pI = pc exp

δD q(Va − Vbi )
,
kB T

(1.13)

where δA and δD are fractions of potential dropped across donor and acceptor respectively,
and Vbi is the built-in potential given by
Vbi = W Fa − W Fc = ∆EHL − φa − φc ,

(1.14)

where W Fa and W Fc are anode and cathode work function respectively, ∆EHL is the
interface energy gap defined by ∆EHL = ELU M O,A − EHOM O,D , and φc and φa are cathode
and anode injection barriers, respectively. Equations 1.12 and 1.13 are strictly valid only
when current density vanishes, i.e. at around the open circuit conditions; Va = Voc [4, 69],
but they are applied to all Va in the GWWF model. Contact charge densities nc and pc can
be approximated by
nc = NLU M O exp(−φc /kB T ),

(1.15)

pc = NHOM O exp(−φa /kB T ),

(1.16)

and

16

where NLU M O is the density of states of electrons at the acceptor LUMO band edge, and
NHOM O is the hole density of states at the highest occupied molecular orbital (HOMO)
band edge of donor.
In the case of f-f recombination, GWWF derived an ideal diode equation for organic
heterojunctions that is given by
J = Js0 exp

q(Va − JRs )
− 1 − qJX ηP P d ,
kB T

(1.17)

where Js0 is the diode saturation current. It is given by
Js0 = qa0 krec NHOM O NLU M O (1 − ηP P d )exp(−∆EHL /kB T ),

(1.18)

where series resistance Rs is manually added here. This equation resembles the Shockley
ideal equation for solar cells that was derived from the physics of inorganic p-n junction in
1949[31].
In the case of trap-assisted recombination, expressions for trapped charge densities nIt
and pIt have to be formulated first. However, it is important to introduce the physics of
quasi-Fermi levels prior to that. Under equilibrium conditions, the total chemical potential
of an electron in a substance is the Fermi level Ef . In semiconductors, the Fermi level
designates the amount of free electrons in the LUMO bands and free holes in the HOMO
bands through the relations;
n = NLU M O exp

−(ELU M O − Ef )
,
kB T

(1.19)

p = NHOM O exp

−(Ef − EHOM O )
.
kB T

(1.20)

and

However, once a semiconductor is disturbed from equilibrium by either illumination or applied voltage, electrons from the HOMO will be excited to the LUMO and holes will be
produced in the HOMO. These two excited free charge densities cannot be described by just
one Fermi level as in Eq. 1.19 and Eq. 1.20, but two quasi-Fermi levels are needed, namely,
the electron quasi-Fermi level (Ef n ) and the hole quasi-Fermi level (Ef p )[17]. In simple
17

words, the electron and hole densities change as they have their own Fermi levels, which are
given by
n = NLU M O exp

−(ELU M O − Ef n )
,
kB T

(1.21)

p = NHOM O exp

−(Ef p − EHOM O )
.
kB T

(1.22)

and

Assuming an exponential trap distribution with characteristic trapped electron energy
Et,A = kB Tt,A and characteristic trapped hole energy Et,D = kB Tt,D , where Tt,A and Tt,D
are corresponding characteristic trap temperatures, trapped electron and hole densities can
be approximated by

and

(1/lA )
−(ELU M O − Ef n )
n
,
≈ HA
nt ≈ HA exp
kB Tt,A
NLU M O

(1.23)

(1/lD )
−(Ef p − EHOM O )
p
pt ≈ HD exp
≈ HD
,
kB Tt,D
NHOM O

(1.24)

where HD and HA are the total trap densities in donor and acceptor respectively, and
lA = Tt,A /T and lD = Tt,D /T . These equations are actually true only at T = 0K, where
the injected or photo-generated charges are trapped and fill up the DOS up to the quasiFermi levels[86]. However, they can be applicable to any temperature if T < Tt,D and Tt,A ,
and if Fermi-Dirac distributions of free electrons and holes are established and n < nt and
p < pt [86]. More details about the exponential trap distribution can be found in Section 1.8.
As a result, the ideal diode equation with trap-assisted recombination in the GWWF
model is given by
J = JsD exp

q(Va − JRs )
q(Va − JRs )
− 1 + JsA exp
− 1 − qJX ηP P d ,
nD kB T
nA kB T

(1.25)

where JsD and JsA are diode saturation currents, and nD and nA are ideality factors. They
are a function that have δD and δA as one of the arguments. Detailed expressions for JsD ,
JsA , nD , and nA in the GWWF model can be found in the original work[4].

18

1.7

Shockley-Read-Hall Recombination

In addition to the trap-assisted recombination in Eq. 1.8, the Shockley-Read-Hall (SRH) recombination [49, 50] is a widely accepted recombination theory in disordered materials. Even
though it was originally derived using inorganic semiconductor physics, it can be extended
to organic devices[87, 88, 89].
The integral part of the SRH recombination is that defects in a disordered solid can capture (or trap) free electrons or holes from transport levels. They can then either be emitted
(or detrapped) back to the transport levels or act as recombination centers, recombining
with other free carriers. The defects or traps are localized states having trap energy (Et )
lying within the forbidden energy gap

Figure 1.6: Diagram showing the four processes constituting the total Shockley-Read-Hall
recombination: electron capture rate (rnc ); electron emission rate (rne ); hole capture rate
(rpc ); and hole emission rate (rpe )
The derivation of the SRH recombination equation is done by applying the detailed
balance principle to the competing trapping and detrapping rates of both electrons and
holes. Figure 1.6 shows the four rates associated with a single trap energy level at Et .
For Nt as the density of trap states, the probability that the states are occupied by trapped
electrons ft ; the rate at which free electrons (with density n) are captured by the unoccupied

19

traps is
rnc = Cn nNt (1 − ft ),

(1.26)

where Cn is the electron capture coefficient. The occupied traps can also emit the trapped
electrons back to the LUMO bands by the rate
rne = en Nt ft ,

(1.27)

where en is the electron emission coefficient. At the equilibrium conditions where rnc = rne ,
a relation between Cn and en is established and leads to
rne = Cn n1 Nt ft ,

(1.28)

where n1 is the number of electrons in the LUMO bands when the electron Fermi level equals
to Et [49], i.e.
n1 = NLU M O exp

Et − ELU M O
.
kB T

(1.29)

When the semiconductor is not at equilibrium, the total electron capture rate is
Unc = rnc − rne = Cn nNt (1 − ft ) − Cn n1 Nt ft .

(1.30)

Similarly, following the same considerations gives rise to the hole capture and emission
rates that can be expressed by
rpc = Cp pNt ft ,

(1.31)

rpe = Cp p1 Nt (1 − ft ),

(1.32)

and

respectively. p1 is the number of holes in the HOMO bands when Ef = Et , so
p1 = NHOM O exp

EHOM O − Et
.
kB T

(1.33)

Consequently, the net hole capture rate is
Upc = rpc − rpe = Cp pNt ft − Cp p1 Nt (1 − ft ).
20

(1.34)

Under steady-state conditions where n and p are invariant with time, the rate Unc must
equal Upc which also equal the total SRH recombination rate (RSRH ). Solving Unc = Upc
for ft in terms of the other variables, the famous RSRH rate is obtained and given by
RSRH =

Cn Cp Nt (np − n1 p1 )
.
Cn (n + n1 ) + Cp (p + p1 )

(1.35)

From Equations 1.29 and 1.33,
n1 p1

= NLU M O NHOM O exp((EHOM O − ELU M O )/kB T )
= NLU M O NHOM O exp(−Eg /kB T ) = n2i ,

(1.36)

where Eg = ELU M O −EHOM O is the semiconductor energy gap and ni is the intrinsic carrier
density.
The values of the capture rates Cn and Cp have to be specified. Here we use the assumption that stationary traps and free carriers recombine under the influence of their mutual
coulomb field, so the rates can be described by the Langevin bimolecular recombination[51,
77], i.e.
µp

= krec,p ,

(1.37)

qµn

= krec,n .

(1.38)

Cp = q
and
Cn =

We can see that many aspects of the SRH recombination are similar to the f-t recombination
in Eq. 1.8. However, the two theories are actually different, and the differences will be
discussed thoroughly in Section 4.6.

1.8

Shockley-Read-Hall Recombination with Exponential Trap Distributions

In the previous section, trap states are represented by only a single trap energy level at
Et . In reality, there are many trap energy levels distributed in the band gap, so the trap
states need to be described by a trap density of states function. Gaussian and exponential
21

distributions are the two most acknowledged trap distribution functions[90], and because of
their shapes in the gap, traps are frequently referred to as tail states. In this section, the
SRH recombination is extended to include the exponential trap distribution by integrating
over electron trap energy Etn in acceptor and hole trap energy Etp in donor.

Figure 1.7: Energy band diagram of an organic semiconductor showing exponential tail
state distributions from the LUMO (gn (E)) and from the HOMO (gp (E)) that are filled
with trapped electrons and holes up to Ef n and Ef p respectively.
Consider the exponential tail states in the band gap in Fig. 1.7 of a semiconductor that
is disturbed form equilibrium so that electron and hole densities are described by their own
quasi-Fermi levels Ef n and Ef p . Non-intrinsic electrons and holes that are introduced to
the system have to fill up the traps to the energies Ef n and Ef p . An exponential DOS of
trapped electrons per unit energy range Etn can be given by [86, 91]
gn (Etn ) =

HA
Etn − ELU M O
exp
,
kB Tt,A
kB Tt,A

(1.39)

where HA and Tt,A are the total trapped electron density at the LUMO edge and the
characteristic trapped electron temperature, respectively. They are the same as in Eq. 1.23
in the GWWF model section. Similarly, an exponential DOS function of trapped holes is

22

given by
gp (Etp ) =

EHOM O − Etp
HD
exp
,
kB Tt,D
kB Tt,D

(1.40)

where HD and Tt,D have the same definitions as in Eq. 1.24. Assuming that traps occupy
the states according to a Fermi-Dirac distribution function f (Etn,p ) = 1/[1 + exp((Etn,p −
Ef n,p )/kB T )], trapped electron and hole densities (nt and pt ) can be calculated from Eqs.
1.39 and 1.40 by
nt =

Ef n
−∞

gn (Etn )f (Etn )dEtn ≈ HA

Ef n − ELU M O
kB Tt,A
exp
,
kB Tt,A − kB T
kB Tt,A

(1.41)

and
pt =

∞
Ef p

gp (Etp )[1 − f (Etp )]dEtp ≈ HD

EHOM O − Ef p
kB Tt,D
exp
,
kB Tt,D − kB T
kB Tt,D

(1.42)

as similarly proved by Ref. [92]. Generally, for T < Tt,D(A) , n < nt and p < pt , Equations
1.41 and 1.42 can be approximated by Equations 1.23 and 1.24 respectively.
Each trap level in the DOS functions gn (ET n ) and gp (Etp ) serves as a trapping/detrapping
or recombination center as described in the SRH theory. Trapped hole density is much higher
than trapped electron density in a donor and vice versa for trapped electron density in an
acceptor; so that the interface carriers nI and pI are most probably captured and emitted
by donor trapped holes at Etp and acceptor trapped electrons at Etn as illustrated by Figure
1.8. Consequently, the total SRH recombination rate at the D/A interface is calculated by
the integration[51, 93];
RSRH =
+

gn (Etn )Cn Cp (nI pI −n2i,A )
Ef n
EHOM O,A Cn [n +n (Etn )]+Cp [p +p (Etn )] dEtn
I
1A
I 1A
2
g
(E
)C
C
(n
p
ELU M O,D
p tp n p I I −ni,D )
dEtp ,
Ef p
Cn nI +n1D (Etp ) +Cp pI +p1D (Etp )

(1.43)
where n1D(A) and p1D(A) are n1 and p1 as Eqs. 1.29 and 1.33 in the donor(acceptor), and
n2i,A(D) = NLU M O,A(D) NHOM O,A(D) exp(−Eg,A(D) /kB T ), where Eg,A(D) is the intrinsic
energy gap of the acceptor(donor). For simplicity, NLU M O = NLU M O,A = NLU M O,D and
NHOM O = NHOM O,A = NHOM O,D are used throughout this study.
23

Figure 1.8: Total SRH recombination at the D/A interface as a result of net capture and
emission of the interface charges (nI and pI ) by the interface donor trapped holes and the
interface acceptor trapped electrons having energies Etp and Etn respectively.

1.9

Dissociation and Recombination Rates of Polaron Pairs

Interface polaron pairs, or the lowest lying charge transfer state, can either dissociate or
recombine. Both of these rates are a function of the interface electric field (FI ) as used in
the GWWF model and in our analytical and numerical models. First, for the field that points
out of the interface (i.e. dissociating field), the dissociation rate (kP P d ) is described by the
Bruan model[83]. It can be expressed by a dissociation rate at zero field (kP P d,0 ) multiplied
by a field dependent term suggested by Onsager[94]. The resulting rate is sometimes called
Onsager-Bruan (OB) dissociation rate, which is given by
√
J1 [2 −2b]
kP P d = kP P d,0 √
,
−2b

(1.44)

2 T 2 , where
where J1 is the first order Bessel function of the first kind and b = q 3 FI /8π kB

is spatially averaged dielectric constant. The rate (kP P d,0 ) is derived from dissociation of

24

an ion pair accompanied by consideration of detailed balance of the CT state. It is given by
kP P d,0 =

3
krec exp(−Eb /kB T ),
4πa30

(1.45)

where krec is Langevin bimolecular rate constant and Eb is polaron pair binding energy
described by
Eb =

q2
.
4π a0

(1.46)

Secondly, when the field points to the interface and suppresses the dissociation, an additional barrier is added to the PP binding energy, and kP P d is averaged over the half sphere
at the D/A interface[4]. The mathematical expression is given by
kP P d =

1 π/2 3
−Eb + FI rc cosθ
krec exp
dθ,
3
π −π/2 4πa0
Vt

(1.47)

where FI rc cosθ is the additioanl barrier, rc = q 2 /(4π kB T ) is the Onsager radius, and θ is
the angel between the PP separation length vector and the D/A interface normal vector.
Another fate of the PPs is that they recombine with the rate kP P r . According to paper
II of GWWF [82], experimental data allowed them to formulate a field dependent kP P r that
is given by
kP P r = kP P r0 exp(

−qFI a0
),
kB T

(1.48)

where kP P r0 is PP recombination in zero field. Note that the exponential term has opposite
sign from the original paper because here the positive field is defined as the field that helps
charge separation.

1.10

Overview

The contents in this thesis are organized as follows. In Chapter 1, introductions to solar
cells, organic solar cells, the Shockley equation and its parameters, typical solar cell JV
curves and the GWWF model are provided. Physics of charge distributions, trap densities
and charge recombination processes are explained. The chapter ends with the interface
polaron pair rates that are used in the GWWF model and our models.
25

In Chapter 2, we present our analytical models, namely the self-consistent diode (SCD)
models, using different types of free carrier recombination. Physical definitions of saturated
diode currents, ideality factors, series and parallel resistances, the diode voltage are formulated and explained.
In Chapter 3, the computational drift-diffusion for bilayer interfaces (DD-BI) model
that is developed in this dissertation is presented. Basics of the drift-diffusion model and
how to solve the equations in the model computationally are explained.
Chapter 4 focuses on using the SCD and the DD-BI models to elucidate fundamental
physics of bilayer OPVs. This chapter starts with advantages of the DD-BI and the SCD
models when they deal with space charge effects, interface properties, series and parallel
resistances as compared with the GWWF model. We show how charge densities distribute as
a function of a distance from an electrode and how these distributions affect quasi-Fermi levels
(chemical potentials) as a function of the distance. Effects of carrier mobilities on the charge
distributions and quasi-Fermi levels are explained and connected to physical definitions of
measurable series and parallel resistances. After that, physics regarding the electrodes’ work
functions, e.g. injection barriers and organic-semiconductor/metal contacts, are studied.
Relations between ideality factors and trap characteristic energies are presented, and remarks
on interpretations of fitted ideality factors are given. Finally, the SRH recombination is
compared with its reduced forms.
Chapter 5 copes with applications of our models on actual bilayer devices through
use of the Shockley parameters Rs , Rp , J0 and n to explain the shapes of JV curves under
illumination and dark conditions. Additional leakage current and Rp in the dark are proposed
and compared with those under illumination conditions. The Shockley parameters as a
function of applied voltage in the presence and absence of light are compared. Finally,
remarks on relations between fitted Shockley parameters and physical properties are given.
In Chapter 6, we show ab initio calculations of growing a CsSnBr3 perovskite film;
which is a promising material for bilayer photovoltaics, using a heteroepitaxy technique.

26

Band structures and densities of states of CsSnBr3 and CsSn2 Br5 are simulated using density
functional theory. Processes of growing the epitaxial CsSnBr3 film are studied, and possible
future works regarding a phase transition from CsSnBr3 to CsSn2 Br5 are suggested.

27

CHAPTER 2
ANALYTICAL MODEL WITH SPACE CHARGES

2.1

Self-Consistence Ideal Diode (SCD) Equations

The interface densities in the GWWF model (Equations 1.12 and 1.13) are valid only at the
open circuit conditions and do not take space charge effects into account. More accurate
forms of the ideal diode equation with a better representation of space charge effects can be
derived using more accurate interface densities calculations. In order to do this, Equations
1.12 and 1.13 are replaced by Equations 1.21 and 1.22, respectively. The latter have space
charge effects included in terms of quasi-Fermi levels.
Here, we use the intermediate equation of J (Eq. 1.11) in the GWWF model as a general
form of JV equation. It can be rewritten as
J = qa0 (1 − ηP P d )Req

k
R
− P P d − qJX ηP P d .
Req kP P d,eq

(2.1)

This equation is an exact solution for any recombination types, based on the detailed balance
conditions of the GWWF interface rates. With the approximation kP P d /kP P d,eq ≈ 1[4] and
R usually depends on the voltage exponentially, we will show in this section that it is in
the same form as the Shockley equation with the term qa0 (1 − ηP P d )Req equivalent to the
saturation diode current (J0 ) and qJX ηP P d containing information about the leakage photocurrent.
In this part, ideal diode equations with space charge effects (or SCD equations) for
bilayer solar cells with the f-f recombination (Eq. 1.7) and the f-t recombination (Eq. 1.8)
are derived analytically. They will be compared and used cooperatively with the numerical
DD-BI model summarized in Chapter 3, to elucidate the OPV physics in Chapter 4

28

2.1.1

The f-f Recombination

For the f-f recombination processes, R is given by (from Eqs. 1.7, 1.21, 1.22)
Rf-f = krec NLU M O NHOM O exp −

Ef n,I − Ef p,I
ELU M O,A,I − EHOM O,D,I
exp
,
kB T
kB T
(2.2)

where ELU M O,A,I − EHOM O,D,I is the interface acceptor LUMO energy minus the interface
donor HOMO energy. It is identical to the interface energy gap ∆EHL in the GWWF
model and can be used interchangeably. At the equilibrium conditions where JX = 0 and
Va = 0, electron and hole quasi-Fermi levels do not split from the intrinsic Fermi level and
Ef n = Ef p = Ef . For this reason, recombination rate at equilibrium is given by
Rf-f,eq = krec NLU M O NHOM O exp −

ELU M O,A,I − EHOM O,D,I
.
kB T

(2.3)

Substituting Rf-f and Rf-f,eq into Eq. 2.1 yields a new ideal diode equation;
J=

qa0 (1 − ηP P d ) krec NLU M O NHOM O exp −
× exp

Ef n,I − Ef p,I
kB T

−

kP P d
kP P d,eq

ELU M O,A,I − EHOM O,D,I
kB T

− qJX ηP P d .

(2.4)

With the same assumption in the GWWF model that kP P d /kP P d,eq ≈ 1[4], it is reduced to
J = Js0 exp

Ef n,I − Ef p,I
kB T

− 1 − qJX ηP P d ,

(2.5)

where the diode saturation current is given by
Js0

= qa0 (1 − ηP P d ) krec NLU M O NHOM O exp −

ELU M O,A,I − EHOM O,D,I
kB T

= qa0 (1 − ηP P d )krec nI,eq pI,eq
= qa0 (1 − ηP P d )Rf-f,eq

(2.6)

The ideal diode equation (Eq. 2.5) is similar to the one in the GWWF model except the
diode voltage that equals to Ef n,I − Ef p,I instead of Va − JRs .

29

2.1.2

The f-t Recombination

For the f-t recombination, Eq. 1.8 is written in two separate terms as
R = Rfn-tp + Rfp-tn = krec,n nI pIt + krec,p pI nIt

(2.7)

for simplicity of derivation. Use this form of R to rewrite Equation 1.11 as
J=
=
+

qa0 (1 − ηP P d ) Rfn-tp + Rfp-tn −

kP P d
kP P d,eq

Rfn-tp,eq + Rfp-tn,eq

− qJX ηP P d

Rfn-tp
k
− PPd
Rfn-tp,eq kP P d,eq
Rfp-tn
k
− P P d − qJX ηP P d .
qa0 (1 − ηP P d )Rfp-tn,eq
Rfp-tn,eq kP P d,eq
qa0 (1 − ηP P d )Rfn-tp,eq

(2.8)

This equation has two diode terms having different physical origins that are in the same
form as the general JV equation (Eq. 2.1). Using the interface charge densities in Eqs. 1.21
and 1.22 along with the trapped charge densities in terms of quasi-Fermi levels in Eqs. 1.23
and 1.24, the electron capture rate by trapped holes is
Rfn-tp = krec,n NLU M O HD exp −ELU M O,A,I + EHOM O,D,I /lD + Ef n,I − Ef p,I /lD ,
(2.9)
and the free hole capture rate by trapped electrons is
Rfp-tn = krec,p NHOM O HA exp −ELU M O,A,I /lA + EHOM O,D,I + Ef n,I /lA − Ef p,I .
(2.10)
Here the characteristic trap energy Et,D(A) is written as lD(A) kB T and E = E/kB T . Under
the equilibrium conditions, these two equations are given by
Rfn-tp,eq = krec,n NLU M O HD exp −ELU M O,A,I + EHOM O,D,I /lD + (1 − 1/lD )Ef

(2.11)

and
Rfp-tn,eq = krec,p NHOM O HA exp −ELU M O,A,I /lA + EHOM O,D,I + (1/lA − 1)Ef . (2.12)

30

Using Eqs. 2.9 - 2.12 in 2.8 yields the ideal diode equation with space charges;


qa0 (1 − ηP P d )krec,n NLU M O HD exp −ELU M O,A,I +

J=

× exp (Ef n,I − Ef p,I ) 1 +

qa0 (1 − ηP P d )krec,p NHOM O HA exp 
× exp (Ef n,I − Ef p,I ) 1 +

−

lD

(Ef p,I − Ef )(1 − 1/lD )
Ef n,I − Ef p,I


+

EHOM O,D,I

−ELU M O,A,I
lA

−



1
+ 1−
Ef 
lD

kP P d
kP P d,eq


1
+ EHOM O,D,I +
− 1 Ef 
lA

(Ef n,I − Ef )(1/lA − 1)
Ef n,I − Ef p,I

−

kP P d
kP P d,eq
(2.13)

qJX ηP P d .

It can be written in a simple form as,
J = JsD exp

Ef n,I − Ef p,I
nD kB T

− 1 + JsA exp

Ef n,I − Ef p,I
nA kB T

− 1 − qJX ηP P d , (2.14)

where JsD and JsA are respectively saturation currents associated with the fn-tp and the
fp-tn processes, which are given by


= qa0 (1 − ηP P d )krec,n NLU M O HD exp −ELU M O,A,I +

JsD

EHOM O,D,I
lD



1
+ 1−
Ef 
lD

= qa0 (1 − ηP P d )krec,n nI,eq pIt,eq
= qa0 (1 − ηP P d )Rfn-tp,eq ,

(2.15)

and


JsA

= qa0 (1 − ηP P d )krec,p NHOM O HA exp 

−ELU M O,A,I
lA



1
+ EHOM O,D,I +
− 1 Ef 
lA

= qa0 (1 − ηP P d )krec,p pI,eq nIt,eq
= qa0 (1 − ηP P d )Rfp-tn,eq ,

(2.16)

and nD and nA are ideality factors associated with the two recombination processes formulated as

(Ef p,I − Ef,I )(1 − 1/lD ) −1
nD = 1 +
Ef n,I − Ef p,I

(2.17)

(Ef n,I − Ef,I )(1/lA − 1) −1
.
Ef n,I − Ef p,I

(2.18)

and
nA = 1 +

31

2.1.3

The f-f and f-t Recombination

In the previous cases, nI and pI are designated to recombine either with other free carriers
with the rate krec or with trapped charges with the rate krec,n(p) . However, it is possible
that the f-f and the f-t recombination processes coexist, leading to the total recombination
rate of
Rf-f + f-t = krec nI pI + krec,n nI pIt + krec,p pI nIt .

(2.19)

The ideal diode equation is then given,
J=

Ef n,I − Ef p,I
Ef n,I − Ef p,I
− 1 + JsD exp
kB T
nD kB T
Ef n,I − Ef p,I
+JsA exp
− 1 − qJX ηP P d ,
nA kB T

Js0 exp

−1
(2.20)

where all the parameters are explained above. This equation is similar to the double diode
EEC model in section 1.4. There are multiple diode currents where one of them has the
ideality factor of unity (n = 1), and the diode term from trap-assisted recombination is split
into two distinct processes; fn-tp and fp-tn.
We should emphasize that the derived ideal equations in Eqs. 2.5, 2.14 and 2.20 come
explicitly from fundamental physics. Each term has mathematical expression that has physical meaning. They do not have any extra terms for the resistances Rs and Rp because these
are intrinsic properties and already included in the Fermi-Dirac charge distributions (Eqs.
1.21 and 1.22) and the PP rates. Unlike many other theoretical works that view Rs and Rp
as extrinsic or fitting parameters, we instead extract them and their physical meanings from
the equations. This is shown analytically in the next section and numerically in Chapter 4.

2.1.4

The SRH Recombination

For the SRH recombination in the integral from (Eq. 1.43), an explicit SCD equation like
Eq. 2.5 for the f-f recombination or Eq. 2.14 for the f-t recombination cannot be written.
However, we can use the general form of current density proposed by GWWF (Eq. 2.1) to

32

obtain an implicit SCD equation for the SRH process; which is
J = qa0 (1 − ηP P d )RSRH,eq

RSRH
k
− P P d − qJX ηP P d ,
RSRH,eq kP P d,eq

(2.21)

where RSRH,eq is the total SRH recombination rate under equilibrium conditions. Assuming that the term RSRH /RSRH,eq is in the same form as in the f-f and the f-t cases, i.e.
RSRH /RSRH,eq = exp (Ef n,I − Ef p,I )/nSRH kB T , where nSRH is the ideality factor from
the SRH recombination, Equation 2.21 can be reduced to the generalized Shockley form;
J = Js,SRH exp

Ef n,I − Ef p,I
nSRH kB T

− 1 − qJX ηP P d ,

(2.22)

where Js,SRH = qa0 (1 − ηP P d )RSRH,eq is the diode saturation current, and nSRH is given
by
nSRH =

2.2

Ef n,I − Ef p,I
R

.

kB T ln R SRH
SRH,eq

(2.23)

Diode Voltage, Series and Parallel Resistances

As seen in the generalized Shockley equation and the EEC model (see section 1.4), series (Rs )
and parallel (Rp ) resistance are as important as Jsc , Voc , and F F as efficiency parameters.
Experimentalists usually fit the Shockley equation (Eq. 1.3) with their JV data to extract
Rs and Rp . Conventionally, Rs represents total resistance of all bulk materials and contacts,
and Rp is interpreted as leakage current at the interface due to imperfect contact. Smaller
Rs is preferable as well as higher Rp . In order to understand the physical meaning of Rs
and Rp in the OPV point of view, the diode voltage (Vd ) has to be discussed first.
The diode equation is usually given by
Jd = J0 (exp(Vd /nVt ) − 1),

(2.24)

where Jd is a current flowing through the diode, Vd is diode voltage and Vt = kB T /q is
thermal voltage. Vd is defined as a voltage right across the junction or interface, that drives
the current Jd through it. In the case of p-n junctions, it is originally and sometimes called
33

junction voltage[31]. It is the applied voltage (Va ) across anode and cathode reduced by the
total series resistance loss;
Vd = Va − JRs .

(2.25)

Comparing this to equation 2.20 for organic heterojunctions, we have
Vd = Ef n,I − Ef p,I .

(2.26)

Hence, what drives a current through the D/A interface is the quasi-Fermi level splitting,
which equals the difference between the interface electron quasi-Fermi level in acceptor
(Ef n,I ) and the interface hole quasi-Fermi level in donor (Ef p,I ). It should be noted that,
by saying driving a current through the interface, holes in the donor and electrons in the
acceptor are driven to the interface to recombine, resulting in a hole current in the donor and
an electron current in the acceptor that are equal to each other. The splitting Ef n,I − Ef p,I
actually promotes interface free carrier recombination (R), and hence both holes and electrons are driven to the interface. Because the bilayer device is unipolar, i.e. current density
is conducted by holes in the donor and by electrons in the acceptor, higher recombination R
implies that more current is flowing in the device.
The relations for Vd lead to Rs that is given by
Rs =

Va − (Ef n,I − Ef p,I ) (W Fc − Ef n,I ) + (Ef p,I − W Fa )
=
,
J
J

(2.27)

where the second form is obtained by using Va = W Fc − W Fa where W Fc and W Fa are the
work functions of cathode and anode, respectively. According to the first form, one can view
Rs as a resistance that reduces the voltage difference Va between the anode and cathode to
become the actual diode voltage Ef n,I − Ef p,I at the interface. More precisely, according to
the second form, Rs causes total deviation of energy levels from W Fa to Ef p,I and from W Fc
to Ef n,I . It will be shown later that quasi-Fermi levels can be reduced by charge transfer
at the contacts and bulk materials due to space charges. This agrees with the conventional
meaning of Rs .

34

For Rp , by writing the term qJX ηP P d in both the GWWF model and Eq. 2.20 as
qJX + qJX (1 − ηP P d ) to represent a constant photo-generated and a leakage current, a
simple comparison between this and the generalized Shockley equation (Eq. 1.3) results in
Rp =

Ef n,I − Ef p,I
Vd
=
.
qJX (1 − ηP P d ) qJX (1 − ηP P d )

(2.28)

Figure 2.1 is an illustration of the following Rp explanation. The term qJX can be interpreted
as a constant supply of PPs as if the interface is connected to a reservoir of incoming PPs.
The term (1−ηP P d ) = kP P r /(kP P r +kP P d ) is the probability of geminate PP recombination
via kP P r . Thus, qJX (1 − ηP P d ) is a recombination current that is like a fictitious leakage
flowing backward against the photocurrent. Due to that the interface has a voltage of
Vd = Ef n,I − Ef p,I , the parallel resistance is then a fictitious resistance that controls the
recombination current.
Ef n,I − Ef p,I
qJX

qJX (1 − ηP P d )

PPs

qJX ηP P d

Rp

Figure 2.1: Schematic flow of the constant exciton current (qJX ) through the PP processes
at the interface resulting in the photo-current qJX ηP P d and the recombined exciton current
(qJX (1 − ηP P d )) that is flowing through Rp .

2.3

Definition of Illumination Conditions

Solar cells are usually tested under standard test conditions (stc) where the solar cell temperature is kept at 25◦ , solar irradiance is 1000 W/m2 and the testing location is at latitude
42◦ N. The latter is usually referred to as air-mass 1.5 (AM1.5), which is defined as the ratio
of optical path length to the Sun to optical path length to the Sun when the Sun is directly
above, i.e. AM1.5 = 1/sin(42◦ ) = cosec(42◦ ) = 1.5.
35

There are many factors affecting the amount of excitons that are created by exposing a
solar cell to sunlight. These include the light reflectivity, the light adsorptivity, refractive
index and exciton generation rate per unit volume as a function of the depth from the surface
of the solar cell. In this study, these factors are not taken into account, and the illumination
conditions are represented by a constant exciton current JX which is a fitting parameter and
approximately equal to the short circuit current (i.e. Jsc ≈ qJX ).

2.4

Conclusions

In this chapter, we show derivations of our analytic JV equations; the self-consistent diode
(SCD) equations. The derivations are similar to those in the GWWF model, but instead
of using the problematic interface charge relations in Equations 1.12 and 1.13 (see Section
4.1), we use the ones that obey quasi Fermi-Dirac statistics in Equations 1.21 and 1.22.
For the f-f recombination, Equations 2.4, 2.5 and 2.6 are the SCD equations. For the ft recombination, the SCD equation is shown in Equation 2.14, and the definitions of the
saturated diode currents and ideality factors are given by Equations 2.15, 2.16, 2.17 and
2.18. If the recombination does not have an explicit expression (e.g. the SRH recombination),
Equations 2.22 and 2.23 are general SCD equations that can be applied for any recombination
types.
Additionally, physical meanings of series and parallel resistances are obtained. Based
on the SCD equations, series resistance is the resistance that results in the diode voltage of
Ef n,I −Ef p,I (i.e. Va −JRs = Ef n,I −Ef p,I ), and parallel resistance under illumination conditions is the recombination resistance for the recombined exciton current (qJX (1 − ηP P d )).

36

CHAPTER 3
NUMERICAL MODEL

3.1

the Drift-Diffusion model for Bilayer Interfaces (DD-BI model)

The DD-BI model is a combination of the numerical drift-diffusion model and the GWWF
interface model. The drift-diffusion model is a computational iterative scheme, comprising solving the Poisson equation, the current continuity equations, and the drift-diffusion
equations. This model has been widely applied to many semiconductor applications[69, 70,
71, 72, 73, 74] with the main purpose of including space charge effects. Because of the
very small thickness-to-surface area ratio of OPV films, one-dimensional modeling along the
thickness dimension is often sufficient to represent an OPV device. The Poisson equation in
one dimension is given by
q
∂2
ψ(x) = [n(x) − p(x)],
2
∂x

(3.1)

where ψ(x) is electrostatic potential at a distance x from the anode. The time-independent
current continuity equations are given by
∂
Jn (x) = −qU (x),
∂x
∂
Jp (x) = qU (x),
∂x

(3.2)

where Jn(p) is electron (hole) current density and U (x) is total free carrier generation rate,
which is equal to kP P d ζ − R. These equations are normally coupled with the following
drift-diffusion equations.
∂
∂
ψ + qµn Vt n,
∂x
∂x
∂
∂
Jp = −qpµp ψ − qµp Vt p,
∂x
∂x

Jn = −qnµn

(3.3)

where the argument x is dropped for convenience. One can couple Eqs. 3.2 with Eq. 3.3 by
finding derivatives of Eq. 3.3 with respect to x and comparing to the charge generation rate
37

±qU (x). Current density equations containing both drift and diffusion terms in Eq. 3.3 are
also used in calculations of total current density (J), which is the sum of Jn and Jp .
In computer programming, the continuous distance x is discretized into a mesh. In this
study, the zeroth mesh point is at the anode and the Lth mesh point is at the cathode.
Adjacent x nodes are separated by the mesh width (δx) of around 1-2 nm. The total film
thickness is, hence, Lδx. The D/A interface is between the point lth and (l + 1)th . For
simplicity, the mesh width is assigned to be equal to the interface width, which is 1.5nm
according to the GWWF model.

Figure 3.1: Flowchart of the numerical DD-BI model.
The DD-BI model is implemented into a computer code using the flowchart as shown in
38

Fig. 3.1. It is developed based on the Gummel scheme[95] and a work by Koster et al.[70].
Firstly, initial and boundary conditions of charge densities and potential profile are set using
the following conditions.
p0 = NHOM O . pL = NHOM O exp(−(ELU M O,A − EHOM O,A )/Vt ).

(3.4)

nL = NLU M O . n0 = NLU M O exp(−(ELU M O,D − EHOM O,D )/Vt ).

(3.5)

ψL − ψ0 = Vbi − Va .

(3.6)

The subscripts refer to the nth mesh points and Vbi is built-in potential defined by
Vbi = W Fa − W Fc = ∆EHL − φa − φc .

(3.7)

The built-in potential is a potential that helps move free carriers from the interface to the
electrodes where they can be extracted to an external circuit. For initial values of ni , pi ,
and ψi where i does not belong to one of the above boundary conditions, data from a preconverged calculation with similar Va can be used. If they are not available, the smallest
value in Eq. 3.4 and Eq. 3.5 is a good initial value for any ni and pi . For ψi , a linear
function from ψ0 to ψL is also a good guess for the initial potential.
After setting the initial conditions, the Poisson equation in Eq. 3.1 is solved numerically.
In order to do this, it has to be discretized. First, consider the following Taylor series
expansions.
δx3 (3)
δx4 (4)
δx2
ψ (x) +
ψ (x) +
ψ (x) + ...
2
6
24
δx2
δx3 (3)
δx4 (4)
ψ(x − δx) = ψ(x) − δxψ (x) +
ψ (x) −
ψ (x) +
ψ (x) + ...
2
6
24
ψ(x + δx) = ψ(x) + δxψ (x) +

(3.8)
(3.9)

One can have an approximation for ∂ 2 ψ/∂x2 by adding the two Taylor expansions.
ψ(x − δx) − 2ψ(x) + ψ(x + δx)
∂2
ψ=
+ O(δx4 ),
2
∂x
δx2

(3.10)

where O(δx4 ) is an error proportional to δx4 , which is very small compared to the other
terms. Consequently, the discrete form of the Poisson equation is obtained as
ψi−1 − 2ψi + ψi+1 q
= [ni − pi ],
δx2
39

(3.11)

for i = 1, 2, ..., L − 1. However, numerical difference between ni , pi and ψi can be problematic
and causes instability in numerical calculations. This can be alleviated by scaling all the
variables so they are in comparable orders of magnitude. Here, the scaled variables are
ψi = ψi /Vt , ni = ni /N , and pi = pi /N where N ≈ NLU M O or NHOM O . Please note that
the prime marks here indicate numerical scaling, not the first derivative. Ultimately, the
solvable Poisson equation is given by
ψi−1 − 2ψi + ψi+1 =

qN δx2
[ni − pi ].
Vt

(3.12)

If trapped charges are taken into account, ni and pi are total densities including the trapped
charges. There are L − 1 equations of Eq. 3.12 from the 1st to L − 1th node, which can be
written in a matrix form as

−2


 1



 0


 .
 ..




0

1
−2
1

···

0
1

0 ···



0   ψ1 

0 ···

0

−2 1 · · ·
..
.

0
..
.

0

0

1





ψ2 



ψ3 
=

.. 
. 



ψL−1
















−2









n1 − p1

  ψ0 
  
  
 
n2 − p2 
  0 
  
  
 
n3 − p3 
− 0 ,
  
  . 
..
  .. 
.
  
  
  
ψL
nL−1 − pL−1







2
qN δx 

Vt 







(3.13)

or
Aψ ψ = bψ ,

(3.14)

where Aψ is a tridiagonal matrix. Here Poisson’s equation in Eq. 3.14 is solved for a
potential profile ψ from known charge profiles n and p. This can be done by taking the
matrix inverse to obtain ψ = A−1
ψ bψ . The mathematics theorem that is used for inverting
a tridiagonal matrix in this study was proposed by R.A. Usmani[96]. This step is repeated
until the highest percentage change of ψi is less than a small number ( 1 ≈ 1.0 × 10−18 or
1.0 × 10−16 ). This is usually easy to converge to 1 within only one repeat.
New converged values, or updated, ψi provide a new electrostatic field at the interface
(FI ) by
FI = Fl+1/2 = −(ψl+1 − ψl )/a0 .
40

(3.15)

This new field will update the interface rates in the GWWF model. Three rates, kP P d , kP P r
and R, are taken into consideration. Let’s define the positive field as the field pointing from
acceptor to donor; i.e. the field that promotes charge dissociation and extraction, and vice
versa for negative field. Equation 1.44, written in terms of a Bessel function expansion is
kP P d =

3
b2 b3
b4
k
exp(−E
/k
T
)
1
+
b
+
+
+
,
rec
b B
3 18 180
4πa30

(3.16)

which is useful to calculate the kP P d rate under a positive field. If FI is negative, Eq. 1.47
is used instead. The rate kP P r expressed in Eq. 1.48 is also straightforwardly updated. For
the free carrier recombination R, Eqs. 1.7 and/or 1.8 or 1.43 are applied, depending on the
recombination type at the interface. After that, the polaron pair density (ζ) is accordingly
updated using detailed balance conditions suggested in the GWWF model. It is simply given
by
J /a + kP P r ζeq + R
.
ζ= X 0
kP P r + kP P d

(3.17)

Ultimately, net generation rate of free carriers is updated using the relation Ul = Ul+1 =
kP P d ζ − R.
The current continuity equations coupled with the drift diffusion equations are solved in
the following step using new ψi and Ul(l+1) as one of the arguments. However, according to
Scharfetter and Gummel[97], when the potential difference between two adjacent mesh points
exceeds 2Vt , numerical instability may occur and the standard finite difference approach
is unable to solve the problem. They suggested a method to mitigate this problem by
considering charge densities as a function of x in the form n(x) = a + becx which finally leads
to more stable discretized current density equations;
qVt N µp,i+1
(B(−δψi+1 )pi+1 − B(δψi+1 )pi ),
δx
qVt N µn,i+1
Jn,i+1/2 =
(B(δψi+1 )ni+1 − B(−δψi+1 )ni ),
δx

Jp,i+1/2 = −

(3.18)
(3.19)

where δψi+1 = ψi+1 − ψi and B() is the Bernouli function defined by
B(x) =

x
.
exp(x) − 1
41

(3.20)

Any energy offsets (e.g. ELU M O,A − ELU M O,D for an electron at the interface) and barriers (e.g. injection barriers φa and φc ) are simply added to δψi at the corresponding ith
point. The discrete current density relations are now solvable by the standard central finite
difference, i.e.
∂Jn(p),i
∂x

≈

Jn(p),i+1/2 − Jn(p),i−1/2
δx

,

(3.21)

and results in the final formulas to be solved iteratively as
qVt N
µn,i+1 B(δψi+1 )ni+1 − (µn,i+1 B(−δψi+1 ) + µn,i B(δψi ))ni + µn,i B(−δψi )ni−1 = qUi ,
δx2
qVt N
−
µp,i+1 B(−δψi+1 )pi+1 − (µp,i+1 B(δψi+1 ) + µp,i B(−δψi ))pi + µp,i B(δψi )pi−1 = qUi .
δx2

−

(3.22)
Similarly to Poisson equation, they form matrix equations
An n = bn

(3.23)

Ap p = bp ,

(3.24)

and

respectively, where

42



−(µn,2 B(−δψ2 ) + µn,1 B(δψ1 ))
µn,2 B(δψ2 )



µn,2 B(−δψ2 )
−(µn,3 B(−δψ3 ) + µn,2 B(δψ2 ))

An = 
.

..


0
···






Ap = 






0

···

µn,3 B(δψ3 )

···
..
.

µn,L−1 B(−δψL−1 ) −(µn,L B(−δψL ) + µn,L−1 B(δψL−1 ))

−(µp,2 B(δψ2 ) + µp,1 B(−δψ1 ))

µp,2 B(−δψ2 )

0

···

µp,2 B(δψ2 )
..
.

−(µp,3 B(δψ3 ) + µp,2 B(−δψ2 ))

µp,3 B(−δψ3 )

···
..
.

0

···



n
 1 


 n 
 2 
n =  . ,
 . 
 . 


nL−1



µp,L−1 B(δψL−1 ) −(µp,L B(δψL ) + µp,L−1 B(−δψL−1 ))





U1
µn,1 B(−δψ1 )n0

 


 





2
U
0
δx  2  

bn = −
 . −
,
.

Vt N  .  
.
.
.

 


 

UL−1
µn,L B(δψL )nL





p
 1 


 p 
 2 
p =  . ,
 . 
 . 


pL−1








0


0

,



(3.25)

0


0

,



(3.26)


U
µ B(δψ1 )p0
 1   p,1


 





2
U
0
δx  2  

bp = −
 . −
.
.

Vt N  .  
.
.
.

 


 

UL−1
µp,L B(−δψL )pL

(3.27)

43

The matrices An and Ap are tridiagonal and can be solved with the same method as
used for solving the Poisson equation.
The algorithm is an iterative scheme. Starting from guessed charge density profiles [n]i
and [p]i , Poisson’s equation yields a new potential profile [ψ]i , that is inserted in the current
continuity equations to be solved for new charge density profiles, which starts another loop
of the scheme. These will be repeated until the percentage change of [n]i and [p]i are very
small, which is normally around the significant decimal digits of the numerical format used in
the program. Double-precision floating-point format, having about 15-17 significant decimal
digits, is used in this study, so the convergence condition is
max

|ni,new − ni,old | + |pi,new − pi,old |
ni,old + pi,old

10−14 ,

(3.28)

where the subscripts old and new refer to values from the previous and current loop, respectively. After the simulation is converged, total current density J is calculated by summing
Eq. 3.18 and Eq. 3.19, at any mesh point i.
It should be noted that, even using the Scharfetter-Gummel method[97], numerical instability is still a problem. This problem is alleviated by setting charge densities for the next
iteration with mixed values from the old densities, e.g., ni,new = wni,new + (1 − w)ni,old ,
where w is the mixing parameter. It is normally less than 0.1 or as small as 10−5 in hardto-converge circumstances such as at high Va , where numerical instability is severe. Here,
the DD-BI model works well up to Va a little higher than Voc , after which convergence is
difficult.

3.2

Conclusions

In this chapter, we present our numerical model; the drift-diffusion for bilayer interface (DDBI) model, which is an extension of the GWWF but interface properties, such as interface
densities, interface electric field, dissociation and recombination rates of polaron pairs, are
solved self-consistently using an iterative numerical method instead of an approximation.
44

The advantages of the DD-BI model are better treatments of space charge effects and more
correct interface properties.
The algorithm of the model is illustrated in Figure 3.1. The central part of the program
is to solve the Poisson equation (Eq. 3.12) and the drift-diffusion equations coupled with
current continuity equations (Eq. 3.22) iteratively until the charge density profiles (n(x) and
p(x)) converge. The equations are written in tridiagonal matrices and can be solved using
a mathematical theorem to invert the tridiagonal matrices. Contributions of the GWWF
model are the interface rates and polar pair density in Eq. 3.17. Once a simulation converges,
Equations 3.18 and 3.19 are used to calculate total current density, resulting in a J − Va
characteristic.

45

CHAPTER 4
PHYSICS OF OPVS
Fundamental physics of OPVs is studied in this chapter using the DD-BI and the SCD models
that are described in Chapter 3 and 2, respectively. We start this chapter by comparing our
models with the GWWF model to show that better treatments of interface densities, space
charge effects, series resistance and interface electric field are included in our models. In
Section 4.2, effects of carrier mobilities are explained in terms of spatial charge distributions,
quasi-Fermi levels, interface rates, series and parallel resistances. In Section 4.3 and 4.4, we
explain effects of injection barriers using our models in conjunction with the ICT and MottSchottkey models. Understanding how quasi-Fermi levels drop across injection barriers leads
to better understanding of series resistance. In Section 4.5, relations between ideality factors
and trap characteristics are studied. Lastly, in Section 4.6, we show differences between the
SRH recombination and the f-t recombination which is understood as a reduced form the
SRH recombination.

4.1

4.1.1

The Differences Between the Current Models and the GWWF
Model
Introduction

Interface charge densities (nI and pI ) are very crucial in describing the D/A interface because
they determine the interface recombination, R, in Eqs. 1.7, 1.8, 1.35 and 1.43 and the
interface electric field (FI ); which is an important contributer to the rates kP P d (Eqs. 1.44
and 1.47), kP P r (Eq. 1.48) and, most importantly, the polaron pair dissociation efficiency,
ηP P d = kP P d /(kP P d + kP P r ). The densities nI and pI are obtained from solving the charge
transport equations (Eqs. 3.1, 3.2 and 3.3), using electrode charge densities (nC and pC )
and the terminal voltage (Vbi − Va ) as boundary conditions. However, because of the low
mobilities in OPVs, spatial charge accumulation (i.e. space charge effect) is common and
46

complicates the charge transport, so the equations have to be solved self-consistently.
The GWWF model; which is not a self-consistent model, uses Eqs. 1.12 and 1.13, that
require voltage drops across donor (δD ) and acceptor (δA ) as inputs, to estimate nI , pI
and FI . As we will show in this section, Eqs. 1.12 and 1.13 are invalid and result in
misinterpretations in the following aspects: (i) Interface densities and rates; (ii) Space charge
effects; (iii) Series resistance (Rs ); (iv) Injection barriers (φa and φc ); (v) δD and δA ; (vi)
Traps in the bulk and at the interface.
In contrast, the DD-BI model; which is the computational model developed in this study,
is based on the numerical self-consistent drift-diffusion simulation and the D/A interface
described by the GWWF model. The purpose of combining these models is to provide a
self-consistent extension of the GWWF model. The SCD equations which are analytic forms
of the DD-BI model also have the self-consistency if each parameter is an actual function of
Va , extracted from the DD-BI model. Accordingly, advantages of the DD-BI and the SCD
models are that they treat the aspects (i)-(vi) self-consistently. This part of thesis shows the
differences between the DD-BI/SCD model and the GWWF model.

4.1.2

Simulation Details

The models are compared by generating JV characteristics based on the same set of parameters; which are taken from Table II of the GWWF paper, and are summarized in Table
4.1. It should be remarked that the injection barriers (φa and φc ) cancel out in the GWWF
derivation, so no information about them or their effects can be found in the paper. From
the interfacial energy gap (∆EHL ) of 1.2eV and the built-in potential (Vbi ) of 0.5eV, the
summation of φc and φa has to be 0.7eV according to Vbi = ∆EHL − φc − φa . For this reason, three sets of φa and φc are analyzed: (1) φa = 0.1eV and φc = 0.6eV ; (2) φa = 0.2eV and
φc = 0.5eV ; (3) φa = 0.3eV and φc = 0.4eV . However, these high injection barriers are not
desirable for efficiency OPVs, so we include three more cases of lower φa /φc : (1) 0.1eV/0.3eV;
(2) 0.2eV/0.3eV; and (3) 0.3eV/0.3eV. Another remark is that in the GWWF model, series

47

resistance (Rs ) is inserted manually (Rs = 1 × 10−4 Ωm2 ). In the DD-BI model, Rs is an
intrinsic property, arising from the mobilities of the donor and acceptor and the contacts at
the anode and the cathode (more discussion can be found in Section 4.4.5). Hence, Rs is not
added to the DD-BI model simulations, but is instead a result of the analysis. Additionally,
the DD-BI model solves drift-diffusion equations for the interface charge densities, but the
GWWF model uses Equations 1.12 and 1.13 with empirical values of the fractions of potential dropped δD = 0.9 and δA = 0.1. We find that for devices with the same thicknesses
and mobilities in donor and acceptor, δD = δA = 0.5 is more reasonable, so we include this
value into the simulations. The final remark is that the interface electric field is obtained
from Poisson equation’s solutions in the DD-BI and the SCD models, but in the GWWF
JV calculations it is calculated using the method summarized at the end of the GWWF
paper[4], based on pC and pI that are predicted by Eqs. 1.16 and 1.13 .
Parameter
Donor and acceptor thickness
µp = µn
φa /φc
Rs †
δD /δA †
Tt,A = Tt,D
kP P r
NHOM O /NLU M O
HD /HA
JX
a0
∆EHL
Vbi
recombination type
†

Value
40nm.
−7
10 m2 /V s
Figure 4.1: 0.1eV/0.6eV; 0.2eV/0.5eV; 0.3 eV/0.4eV,
Figure 4.2: 0.1eV/0.3eV; 0.2eV/0.3eV; 0.3 eV/0.3eV.
1 × 10−4 Ωm2
0.9/0.1 and 0.5/0.5
30
1000K
−1
(1µs) (constant)
1E27 m−3
1E24 m−3
4.3E20 m−2 s−1
1.5nm
1.2eV
0.5eV
the f-t recombination (Eq. 1.8)

Rs , δD , and δA are used in the GWWF model only.

Table 4.1: Summary of parameters used in the comparison between the GWWF model and
the models developed in this study.

48

4.1.3

Results and Discussion
50
φa(eV) / φc(eV)
0.1 / 0.6
0.2 / 0.5
0.3 / 0.4

0

-2

J (A m )

25

-25
-50
-75

-0.4

-0.2

0

0.2

Voltage (V)

0.4

0.6

Figure 4.1: Comparison of JV characteristics generated by the DD-BI model (symbol), the
SCD equation in Equation 2.14 (solid line), the GWWF model with δD = 0.9 and δA = 0.1
(dashed line) and the GWWF model with δD = 0.5 and δA = 0.5 (dashed and dotted line).
50

-2

J (A m )

25
0

DD-BI
SCD
GWWF δD=0.9, δA=0.1
GWWF δD=0.5, δA=0.5

DD-BI
SCD
GWWF δD=0.9, δA=0.1
GWWF δD=0.5, δA=0.5

DD-BI
SCD
GWWF δD=0.9, δA=0.1
GWWF δD=0.5, δA=0.5

-25
-50
-75 -0.4 -0.2

0 0.2 0.4
Voltage (V)

(a) φa /φc = 0.1eV /0.3eV

0.6

-0.4 -0.2

0 0.2 0.4
Voltage (V)

0.6

(b) φa /φc = 0.2eV /0.3eV

-0.4 -0.2

0 0.2 0.4
Voltage (V)

0.6

(c) φa /φc = 0.3eV /0.3eV

Figure 4.2: Comparison of JV characteristics generated by the DD-BI model (symbol), the
SCD equation in Equation 2.14 (solid line), the GWWF model with δD = 0.9 and δA = 0.1
(dashed line) and the GWWF model with δD = 0.5 and δA = 0.5 (dashed and dotted line)
for the devices with φa /φc equal to (a) 0.1eV/0.3eV, (b) 0.2eV/0.3eV and (c) 0.3eV/0.3eV.
The resulting JV curves are compared in Figures 4.1 and 4.2 for the two sets of φa and
φc . The GWWF JV curves both with δD = 0.9 and δA = 0.1 (dashed lines) and δD = 0.5
and δA = 0.5 (dashed and dotted lines) are completely different from the DD-BI (symbols)
and the SCD (solid lines) curves that are consistent with each other. The differences can be
categorized into two aspects: (1) the currents at Va > Voc in the GWWF results are much
higher than the DD-BI and the SCD cases; and (2) the DD-BI and the SCD models show
49

severe s-shaped behaviors if the injection barriers are too high or reduced FFs if the injection
barriers are moderate, while the GWWF results show high FFs in most cases.
The prime variables that give rise to the discrepancies between the GWWF model and
the DD-BI/SCD model are the interface charge densities (nI and pI ). Equations 1.12 and
1.13, which are used in the GWWF model, have δD and δA as one of their arguments, where
δD +δA = 1 is assumed. However, there are several disadvantages to making this assumption.
First, as stated in the GWWF paper, Eqs. 1.12 and 1.13 are valid only when J = 0
(open circuit conditions). Using the equations at any other Va results in nI and pI that are
inaccurate and do not include space charge effects. Comparisons of nI and pI calculated
by Eqs. 1.12 and 1.13 and the DD-BI model in the device with φa = 0.1eV and φc = 0.6eV
are given in Figure 4.3(a). The DD-BI self-consistent values of δD and δA are functions of
applied voltage as shown in Figure 4.3(b). We can see that, even with the actual functions
of δD (Va ) and δA (Va ), the GWWF Equations 1.12 and 1.13 fail to predict the interface
densities except for pI in the small voltage range at high Va . One important feature that is
shown in the DD-BI results is that nI and pI do not change with Va in the small and reverse
bias regimes. This indicates accumulation of charge in space which is common for OPVs due
to the low mobilities and the low dielectric permittivity. The accumulation is often referred
to as space charge accumulation or space charge effect; which is not included in the GWWF
model. It is interesting that δD and δA can change dramatically over the whole range of
voltage. They diverge when Va = Vbi , and may have negative values when the direction of
the averaged electric field in an active layer is switched due to space charge effects. The
topic will be discussed in Section 4.3.2.
Inaccurate values of nI and pI bring about an erroneous interface recombination, interface
electric field, kP P d and, most importantly, ηP P d in the GWWF calculation. The interface
electric field (FI ) is one of the most important parameters in determining the rate kP P d as
seen in Eqs. 1.44 and 1.47. In Figures. 4.1 and 4.2, kP P r is used as a constant, but kP P r
as a function of FI (Eq. 1.48) is more accurate to describe the PP recombination and often

50

-3

nI,pI (m )

10

24

2
20

10
10

δD(Va), δA(Va)

10

16

12

10

pI DD-BI
nI DD-BI
pI Eq. 1.11
nI Eq. 1.10

8

-0.5

Va(V)

0

0

δD
δA

-1

4

10 -1

1

0.5

(a) Interface charge densities

-0.5

0
Va (V)

0.5

(b) Fractions of potential dropped

Figure 4.3: (a) Interface hole (pI ) and electron (nI ) densities versus the applied voltage
calculated by the DD-BI model (filled symbols) and the equations used by GWWF (Eqs.
1.12 and 1.13)(open symbols) that have fractions of potential dropped δD and δA shown in
figure (b) as their arguments.
used[82]. We can see that ηP P d = kP P d /(kP P d + kP P r ), which has a major impact on fill
factor through the photo-current density (qJX ηP P d ), is directly affected by FI . The DD-BI
model solves the Poisson equation for a potential profile (ψ(x)), so FI can be calculated
by FI = −∂ψ(x)/∂x at the interface. On the other hand, because the GWWF model does
not use any iterative numerical scheme to solve for ψ(x), FI has to be approximated using
a method that relates FI with differences between the electrode charge densities (nc and
pc ) and the interface charge densities (nI and pI ). However, Eqs. 1.12 and 1.13 greatly
underestimate nI and pI because of the lack of treatment of the space charge accumulation
at the interface. Thus, the GWWF model overestimates FI , kP P d and ηP P d (FI ), resulting
in a JV curve with high FF and efficiency. Figure 4.4 shows FI , kP P d and ηP P d as a
function of Va in the device with φa /φc = 0.1eV /0.6eV from three different models: the
DD-BI model; the GWWF model with δD /δA = 0.9/0.1; and the GWWF model with actual
values of δD (Va ) and δA (Va ) from the DD-BI results. Strengths of the FI fields predicted
by the GWWF method are always higher than the DD-BI results, regardless of the choices
of δD and δA . This leads to kP P d that can be much higher than it should be and ηP P d that
is close to a hundred percent (i.e. almost all of the excitons are dissociated and turned into

51

the photo-current). We note the similarities between the ηP P d graphs and the shapes (fill

0

-2×10
-3×10
-4×10

DD-BI
GWWF δD/δA = 0.9/0.1
GWWF actual δD, δA

8

7

-1

-1×10

10

kPPd (s )

Interface electric field (V/m)

factors) of the JV Curves in Fig. 4.1.

7

DD-BI
GWWF δD/δA = 0.9/0.1
GWWF actual δD, δA

7

7

-0.4

-0.2

0

Va (V)

0.2

10
10

0.4

7

6

10

(a) Interface electric field strength

5

-0.4

-0.2

0

Va (V)

0.2

0.4

(b) Polaron pair dissociation rate

ηPPd

1

DD-BI
GWWF δD/δA = 0.9/0.1
GWWF actual δD, δA

0.1 -0.4

-0.2

0

Va (V)

0.2

0.4

(c) Polaron pair dissociation probability

Figure 4.4: Graphs of the interface electric field strengths (FI ), polaron pair dissociation
rates and probabilities (kP P d and ηP P d ) of the device with φa /φc = 0.1eV /0.6eV calculated
by the DD-BI models (black), the GWWF model with δD /δA = 0.9/0.1 (red) and the GWWF
model with δD and δA as a function of Va derived from the DD-BI model(green).
Thirdly, Rs , which is an intrinsic property associated charge densities and quasi-Fermi
levels, is removed by the assumption δD + δA = 1. The product of nI and pI is given by
nI pI = nc pc exp

q(Va − Vbi )
−q∆EHL
qVa
≈ NLU M O NHOM O exp
exp
,
kB T
kB T
kB T

(4.1)

or
qVa
nI pI ≈ n2i exp
,
kB T

52

(4.2)

where nc and pc are charge densities at the electrodes given by Equations 1.15 and 1.16, and
ni = [NLU M O NHOM O exp(−q∆EHL /kB T )]1/2 is intrinsic charge density. However, this is
not true in general because when the device is under illumination or bias, the electron and
hole densities will change according to their quasi-Fermi levels, so that[17]
nI pI = n2i exp (Ef n,I − Ef p,I )/kB T .

(4.3)

The voltage in the exponential term of nI pI is the diode voltage; Vd = Va − JRs , as shown in
Chapter 2. The assumption δD + δA = 1 removes Rs from the model, so it has to be included
manually at the end of the GWWF derivation.
Furthermore, the summation δD + δA = 1 + FI a0 /(Vbi − Va ) instead of one as assumed
in the GWWF model where a0 is the interface width. By definition in the GWWF model,
δD (δA ) is based on the potential drop from the anode(cathode) to the edge of donor(acceptor)
at the interface. This leaves the potential drop across the interface itself out, hence a term
of magnitude of FI a0 is omitted. Since FI is of the order of 106 − 107 V /m as shown in Fig.
4.4 and a0 ≈ 10−9 m, the error of using δD + δA = 1 could be as high as 10% assuming that
Vbi − Va is of the order of 0.1V. In addition, δD and δA change drastically from one system
to another and can take many values. They also depend on the cathode work function[98],
donor/acceptor thicknesses, the device voltage[99], and the mobilities.
For the conclusions of this section, we now go back to the two aspects of the JV differences
that were raised earlier. For aspect (1), the steep JV curves calculated by the GWWF method
are the absence of Rs which is most relevant at high voltage[32]. In-depth discussion about
the effects of Rs on JV curves can be found in Sections 5.2 and 5.3. For aspect (2), the
s-shaped JV characteristics are mainly caused by the accumulation of space charge[100, 101]
that can reduce FI and ηP P d , and is not included in the GWWF model. It will be showed
in Section 4.3.2 that reduced built-in potential from high injection barriers can also cause sshaped JV behaviors, which are observed experimentally when high work function electrodes
are used[102].

53

4.1.4

Strategies to Improve the GWWF Model

In order to prove the points about FI and Rs in the previous section, in this section we
would like to show that GWWF JV curves can be consistent with DD-BI and SCD curves if
space charge effects are included, by using more reasonable values of δD , δA , FI and Rs . It
is also interesting to see the conditions in which the GWWF model is a good approximation.
Here, we simulate devices having the same parameters as in Table 4.1, and with the
injection barriers φa /φc = 0.1eV /0.6eV , 0.0eV /0.0eV and 0.3eV /0.3eV . The latter two cases
of the injection barriers represent symmetric devices with Ohmic contacts and contacts with
small Schottky barriers, respectively. Four cases of modified GWWF calculations, which are
summarized in Table 4.2, are calculated and compared to the DD-BI and the SCD results.
From case 1 to case 4, we correct the parameters’ values one at a time, to clearly show effects
of each parameter. In case 1, we use the original values of δD , δA , FI and Rs as suggested by
GWWF. According to the results shown in Fig. 4.3(b), δA = δD = 0.5 is more appropriate
if the thicknesses of the donor and the acceptor are equal, so we start using this value from
case 2. Up to this point, FI is still calculated from the approximation method in the GWWF
paper that is based on δD and Eq. 1.13. Accordingly, in case 3, we use FI as a function of
Va that is derived from the DD-BI simulations. Finally, in case 4, a fitted Rs is used instead
of 1 × 10−4 Ωm2
case
1
2
3
4

δD , δA
0.9, 0.1
0.5, 0.5
0.5, 0.5
0.5, 0.5

FI
GWWF method
GWWF method
DD-BI results
DD-BI results

Rs
1 × 10−4 Ωm2
1 × 10−4 Ωm2
1 × 10−4 Ωm2
Fitted

Table 4.2: Four cases of GWWF calculations with improved values of δD , δA , FI and Rs to
include space charge effects.
The calculated GWWF, DD-BI and SCD JV curves are shown and compared in Fig. 4.5.
Some interesting remarks about the results are as follows. Firstly, the SCD and the DD-BI
results are always consistent with each other. Secondly, the original GWWF calculations

54

50

50
-2

0

SCD
DD-BI
GWWF case 1
GWWF case 2
GWWF case 3
2
GWWF case 4:Rs=3E-4 Ωm

75

J (A m )

-2

J (A m )

25

100

SCD
DD-BI
GWWF case 1
GWWF case 2
GWWF case 3
2
GWWF case 4:Rs=1E-2 Ωm

-25

25
0
-25

-50
-75

-50

-0.4

-0.2

0

Voltage (V)

0.2

-75

0.4

-0.4

(a) φa /φc = 0.1eV /0.6eV

-0.2

0

0.2

Voltage (V)

0.4

0.6

(b) φa /φc = 0.0eV /0.0eV

100
75

-2

J (A m )

50
25

SCD
DD-BI
GWWF case 1
GWWF case 2
GWWF case 3
2
GWWF case 4:Rs=2E-3 Ωm

0
-25
-50
-75

-0.4

-0.2

0

0.2

Voltage (V)

0.4

0.6

(c) φa /φc = 0.3eV /0.3eV

Figure 4.5: Comparisons of JV characteristics calculated by the DD-BI model, the SCD
model and the four cases of modified GWWF calculations that are summarized in Table 4.2,
in the devices with φa /φc equal to (a) 0.1eV/0.6eV, (2) 0.0eV/0.0eV and (3) 0.3eV/0.3eV.
The values of the fitted Rs that are used in case 4 are written in each plot.
(case 1) are not consistent with the drift-diffusion DD-BI results in any φa /φc cases and also
result in the curves with the highest FFs. As discussed earlier, this is because the GWWF
method overestimates FI , kP P d and ηP P d from the underestimated nI and pI . Thirdly,
using δD = δA = 0.5 alone does not give rise to a better JV prediction because Equations
1.12 and 1.13 that have δD and δA as arguments are invalid to calculate nI , pI , and the
resulting FI . However, using δD = δA = 0.5 concurrently with the right FI function from
the DD-BI results drastically improves the GWWF results, as we can see from the case-3
graphs. This is another indication of how important the right FI is in predicting JV curves.
Finally, the case-3 GWWF graphs still have inconsistent currents at high Va that have to be

55

corrected by using the more appropriate values of Rs .
In conclusion, we proved that a correct function FI (Va ) and a fitted Rs are needed in the
GWWF model in order to make it a reasonable model. Moreover, even with the invalidities
of Eqs. 1.12, 1.13 and the assumptions regarding δD and δA , δD = δA = 0.5 is the best set
up if the interface sits in the middle of the active layer. All of the physical quantities, such
as spatial charge densities (n(x) and p(x)), potential profile (ψ(x)) and electric field (F (x)),
are highly correlated with each other according to the laws of Electrostatics. Hence, it is
best to avoid using the problematic approximations (Eqs. 1.12, 1.13, δD and δA ) altogether.

4.2

Effects of Carrier Mobilities

Free carrier mobilities are clearly one of the most important factors limiting OPV efficiency.
It is commonly known that they directly affect charge transport, so the higher they are the
better. Somehow, it is less known in what way they affect other less obvious properties like
charge recombination, space charge accumulation, charge-transfer state recombination and
dissociation, etc. This leads to the purpose of this section; studying effects of free carrier
mobilities.
One of the benchmark materials that have been extensively studied is poly-3-hexyl thiophene (P3HT):phenyl-C61-butyric acid methyl ester (PCBM) donor-acceptor system. Parameters from these materials will be mainly used in this section. In practice, the "true"
bilayer structure is difficult to achieve in this system[60, 61] due to fast interpenetration of
the two materials even at low temperature. Hole mobility in P3HT intrinsically depends
on its regioregularity (order of repeating monomers) and molecular weight of the P3HT
solution, which was shown to be around 10−4 − 10−5 cm2 /V s[103]. Consequently, hole mobility in P3HT within this range and above will be studied to develop a detailed mechanistic
picture of the benefits of higher mobility.
Table 4.3 summarizes important parameters used in the simulations. Simulated JV curves
both under illumination and dark conditions are shown in figure 4.6. Efficiency parameters
56

Parameter
µp
µn
Donor/acceptor thickness
φa /φc

Value

10−6 − 10−9 m2 /V s

2.0 × 10−7 m2 /V s[104]
50nm/22nm [59]
0.2 eV[105], 0.2 eV
3.9 0
58 meV[103]
1µs−1
-4.9eV/-3.0eV
-6.1eV/-3.7eV
1E27 m−3
1E24 m−3
4.3E20 m−2 s−1
1.0nm
Bimoleular + trap-assisted

Et,A = Et,D
kP P r,0
Donor ELU M O /EHOM O
Acceptor ELU M O /EHOM O
NHOM O /NLU M O
HD /HA
JX
a0 = δx
recombination type

Table 4.3: Summary of parameters used in simulations of P3HT:PCBM bilayer solar cells,
with hole mobility as the independent variable
(a) Light

-2

Jlight (A m )

60
40

-1 -1

10

2

-1 -1

10

2

-1 -1

2

-1 -1

µp = 1E-7 m V s
µp = 1E-8 m V s

20
0

2

µp = 1E-6 m V s

-2

80

µp = 1E-9 m V s
-0.4

-0.2

0

0.2

Jdark (A m )

100

0.4

-20

0.6

0.8

(b) Dark

0

-2

10

-4

10

-6

10

10

-40

10

-60

10

Voltage (V)

2

-8

-10
-12

-0.4

-0.2

0

0.2

0.4

Voltage (V)

0.6

0.8

Figure 4.6: (a) J-V curves under illumination conditions of P3HT:PCBM bilayer photovoltaic
cells and (b) ln|J| − V curves of the same devices under dark conditions. Parameters used
in these simulations are in table 4.3.
from the illuminated cells are listed in table 4.4. As expected, energy conversion efficiency
is reduced with the reduced hole mobility. Short circuit current and fill factor are affected
heavily while Voc is surprisingly left intact. At the open circuit point, slopes of the graphs
are very steep in the high mobility devices and become less steep as the mobility is lowered,
especially with µp = 10−9 m2 /V s. The slope of a JV curve at open circuit voltage is often a
good indication of series resistance (Rs )[32]. This agrees with the results and can be simply
57

understood by decreased donor electrical conductivity (σ = qpµp ). However, deeper understanding the mechanism behind the slope change reveals more insights. On the other hand,
slope at short circuit conditions is connected to the parallel resistance (Rp )[32]. Leakage current is the cause of Rp in the conventional picture, whereas polaron pair (or charge-transfer
state) recombination is more likely the reason in the OPV context under illumination. The
least slope as occurs in the high µp devices is preferred at this point. There are a lot of details
regarding Rs and Rp that will be further discussed in this section and Section 5.5. There
are, however, a couple most important and fundamental aspects that have to be understood
first; space charge effects, quasi-Fermi levels and polaron pair dissociation.
µp
2
[m V −1 s−1 ]
1E-6
1E-7
1E-8
1E-9

Jsc
[Am−2 ]
63.03
49.99
43.62
35.65

Voc
[V ]
0.59
0.58
0.58
0.58

FF
[%]
58.19
38.46
32.44
22.68

Table 4.4: Parameters indicating power conversion efficiency; short circuit current (Jsc ),
open circuit voltage (Voc ), and fill factor (F F ), of the P3HT:PCBM bilayer solar cells with
different P3HT hole mobility, obtained from figure 4.6(a).

4.2.1

Space charge effects and quasi-Fermi levels splitting

This part is devoted to explain space charge accumulation and quasi-Fermi levels splitting at
the D/A interface, which causes the driving potential of the diode current. Heterojunctions
of organic donor-acceptor are similar, but not the same, as the inorganic p-n junctions[42].
Quasi-Fermi levels splitting is also found in a p-n junction, but the regions near the junction
are depletion regions due to charge transfer across the interface. This does not occur in
a D/A interface because charge densities are much lower, and there are barriers from the
donor and acceptor energy levels differences. Therefore, it is crucial to understand the D/A
heterojunction from the organic point of view itself.
Figure 4.7 shows a comparison of charge densities and energy levels of the device with

58

2
24

LUMO

18

10

12

10

(a)

Energy (eV)

-3

Charge densities (m )

10

Donor
Acceptor

6

10

0

10

-6

10

-12

n(x) Dark (Equilibrium)
p(x) Dark (Equilibrium)
n(x) Illuminated
p(x) Illuminated
2e-08

4e-08

(b)
Donor

0
HOMO

Acceptor

4e-08

6e-08

-1

10

0

1

Efn
Efp
Ef,eq

6e-08

8e-08

-2
0

Position (m)

2e-08

Position (m)

8e-08

Figure 4.7: (a) A comparison of hole (red lines) and electron (black lines) densities at
Va = 0V between dark (or equilibrium) conditions (sold lines) and illuminated conditions
(dashed line), of the device with µp = 10−6 m2 /V s. (b) Energy level diagram of the same
device under both conditions including equilibrium (dashed line), quasi electron Fermi (green
line) and hole (blue line) energy levels.
µp = 10−6 m2 /V s under dark and illumination conditions at Va = 0V . The former conditions
will be referred to as equilibrium conditions. Under these conditions, holes and electrons,
that are injected from the anode and the cathode, diffuse until they reach the interface
where they are blocked by a barrier (e.g. ELU M O − ELU M O ). The quasi-Fermi energies
D
A
of electron and hole (dashed line in Fig. 4.7(b)) are equal and equilibrated throughout the
device when there is no illumination. Equilibrated , or aligned, Fermi levels (Ef,eq ) are
necessary otherwise currents would flow as;[17]
Jn (r) = µn n∇Ef n

(4.4)

Jp (r) = µp p∇Ef p ,

(4.5)

, but it is obvious that there is no net current at the equilibrium conditions. The LUMO
and HOMO bands are almost linear from the anode to the cathode as shown in Fig. 4.7(b),
due to the linear potential. Hence ELU M O − Ef n and Ef p − EHOM O are also a linear
function of position. As a result, according to equations 1.21 and 1.22, charge densities
spread exponentially from the injection sources (see Figure 4.7(a)).
In the case of illuminated conditions, the D/A interface is now another source of charge
59

24

(a)

2

Reducing µp

(b)

-3

Carrier densities (m )

10

1

Energy (eV)

12

10

0

2

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

µp = 1E-6 m V s

10

µp = 1E-7 m V s
10

LUMO

0

HOMO
Reduced µp

-1

µp = 1E-8 m V s

-12

Reduced µp

µp = 1E-9 m V s
Equilibrium at any µp

0

2e-08

4e-08

6e-08

8e-08

Position (m)

-2
0

2e-08

4e-08

6e-08

8e-08

Position (m)

Figure 4.8: (a) Densities of electron (black) and hole (red) within the devices with different
donor hole mobility, under Va = 0V . Equilibrium charge densities are almost invariant with
µp and plotted in the solid lines. (b) Energy level diagram showing different quasi-Fermi
levels deviation/splitting in the devices. The top LUMO and bottom HOMO bands are
almost invariant with the mobility.
from the dissociation of polaron pairs. Interface charges move toward the electrodes under the
influences of the built-in potential or bias (drift) and diffusion, until steady state conditions
(time-independent) of charges and any interface rates are reached. There are now two regions
of charge distribution, namely an exponential decay region near the electrodes (e.g. the first
20 nm near both the donor and acceptor) and a constant charge region near the interface (e.g.
from 20-50 nm in the donor and a few nanometers in the acceptor). According to equations
1.21 and 1.22, in the constant charge regions, donor EHOM O and Ef p are parallel, as well
as acceptor ELU M O and Ef n . This makes the quasi-Fermi levels Ef n and Ef p split and
deviate from Ef,eq , giving rise to the quasi-Fermi level difference Ef n,I − Ef p,I at the D/A
interface, which is the actual diode voltage.

4.2.2

Effect of mobility on space charges

The origins of quasi-Fermi level splitting have been discussed in the previous part. This part
will be about how hole mobility affects space charge accumulation and energy splitting at
the interface.
Figure 4.8 shows how charge densities and energy levels are affected by varying the hole
60

mobility. First of all, under equilibrium conditions for µp = 10−6 to 10−8 m2 /V s, equilibrium
charge densities and energy levels are almost invariant, but for µp = 10−9 m2 /V s there is a
small change caused by band bending. It is not shown in the figure and will be discussed
in detail later under the band-bending section (Section 4.4). Therefore, solid red and black
lines in Fig. 4.8(a) respectively represent equilibrium hole and electron densities in all cases
of µp . When there are incident photons, the two distinct regions, i.e. the exponential charge
distribution and the constant charge distribution, can be seen in all cases (see figure 4.8(a)).
The impacts of lower µp is that dissociated free carriers are more difficult to move and
collected at the electrodes, so they accumulate more and raise the amount of charge in the
constant charge distribution region. This is normally viewed as reduced charge transport
and collection coefficient caused by low mobility. The raised hole density in the constant
density region decreases the hole quasi-Fermi level since holes are unoccupied electron states,
so electrons can be added or removed using lower energies. Then, Ef p is closer, but still
parallel, to EHOM O (see Eq. 1.22), resulting in more splitting of the quasi-Fermi levels
Ef n,I − Ef p,I at the interface as can be seen in Fig. 4.8(b).
The D/A heterojunction which behaves like a diode is biased with more diode voltage Vd =
Ef n,I − Ef p,I from the lowered mobility. According to ideal equations in Shockley equation
form (for example Eqs. 1.3 and 2.20), the diode conducts more current and subtracts more
from the photocurrent, so the reduction of photocurrent is observed in Fig. 4.6(a). Physically
speaking, lowered mobility worsens charge transport so there is more accumulation near the
interface. This forces dissociated charges from polaron pairs to recombine back to polaron
pairs, and finally are lost by polaron pair recombination (kP P r ζ).

4.2.3

Dissociation and recombination of polaron pairs: The parallel resistance

Polaron pair (PP) recombination and dissociation are vital parts of OPV physics. Let’s first
see how their rates vary as a function of applied voltage in Figure 4.9.
The rate kP P d is the polaron pair dissociation rate, governing the amount of polaron

61

pairs to be dissociated into free carriers (kP P d ζ); while kP P r is the rate for recombining
PP to the ground state, leading to loss of carriers. Introduction to kP P d and kP P r can
be found in section 1.9. Maximized kP P d and minimized kP P r are undoubtedly primary
goals of OPV research. They depend on interface electric field (FI ), but oppositely and
kP P d is more sensitive to it. It changes more than one order of magnitude as seen in
Fig. 4.9. Within the reverse or small bias regimes, built-in potential provides large FI
that points from acceptor to donor, breaking the coulombic bond of a PP, hence promoting
dissociation and also pulling carriers away from the interface. This makes kP P d high and
kP P r low. As the voltage increases into the working solar cell regime, the applied voltage
counters the built-in potential and lessens FI or even reverses its direction. Charges are
driven to the interface and recombine, reducing kP P d and enlarging kP P r . The combination,
ηP P d = kP P d /(kP P d + kP P r ) is a very important parameter, capturing the competition of
the two PP rates. One might consider the D/A interface as an exciton current separator,
splitting the incoming exciton current JX into usable photocurrent qJX ηP P d and wasted
current qJX (1 − ηP P d) leaking to ground states. In Fig. 4.9, ηP P d is close to one at low
voltage because of the built-in potential. In this device, the maximum power point is at
0.47V, where only about 60% of exciton flux becomes free charge flux.
Figure 4.10 shows how the three parameters (kP P d , kP P r , ηP P d ) react to a lowered hole
mobility. The dissociation rate, kP P d , drops rapidly with decreasing hole mobility; however,
from µp = 10−8 to 10−9 m2 /V s and at high Va , kP P d almost does not change. Surprisingly,
kP P r is not influenced by the mobility from µp = 10−6 down to 10−8 m2 /V s, but it is when
µp = 10−9 m2 /V s. Nonetheless, the increment at this mobility is small as compared to the
reduction of kP P d (notice that the scale of kP P r is linear, not logarithmic). In terms of
PP rates, OPV efficiency is thus impacted from poor charge transport through diminished
kP P d , but less severely from elevated kP P r . Noticeably, since kP P r is weakly affected by
µp , the ηP P d plots in figure 4.10(c) follow the trends of kP P d . Reduced ηP P d greatly affects
the harvestable exciton flux. It was discussed already that only 60% of JX is harvestable

62

1

ηPPd

7

0.8

-1

kPPr [s ], kPPd [s ]

10

kPPd

ηPPd

0.6

-1

kPPr
0.4
6

10

0.2

-0.4

-0.2

0

0.2

0.4

0.6

0
0.8

Voltage (V)
Figure 4.9: Polaron pair dissociation rate (kP P d ), recombination rate (kP P r ) and dissociation probability/efficiency (ηP P d ), as a function of applied voltage.
for µp = 10−6 m2 /V s at maximum power point. For µp = 10−9 m2 /V s, the MPP point is at
0.25V, having only about 30% of excitons leading to free carriers coming out of the interface.
Note that, further losses occur from free carrier recombination.
Parallel resistance (Rp ), based on the theory in section 2.2, mostly depends ηP P d . In the
conventional interpretation, Rp is due to leakage current from one electrode to the other[5],
which can be caused by manufacturing defects like pinholes. We will show in Chapter 5
that an OPV cell without actual shorting electrodes can have leakage currents both under
illumination and dark conditions, but these leakage currents are more likely recombination
currents at the D/A interface. Illumination Rp is related to recombinations of interface
polaron-pairs, and dark Rp is possibly from trap-related tunneling processes. Rp is usually
extracted at Va = 0V or lower, and at this voltage, the total photocurrent qJX ηP P d is higher
than the diode current many orders of magnitude. The slope of a graph, which is understood
as Rp−1 [32], is accordingly the slope of ηP P d (dηP P d /dV ).
Figure 4.11 is a comparison plot of Rp as a function of Va showing the effects of donor
hole mobility. They are calculated using Equation 2.28;
Rp =

Ef n,I − Ef p,I
.
qJX (1 − ηP P d )
63

(4.6)

8

10

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

1.2×10

6

µp = 1E-8 m V s

7

1.0×10

6

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

µp = 1E-7 m V s

6

µp = 1E-8 m V s
µp = 1E-9 m V s

-1

µp = 1E-9 m V s

2

µp = 1E-6 m V s

(b)

µp = 1E-7 m V s

-1

kPPd [s ]

(a)
10

2

µp = 1E-6 m V s

kPPr [s ]

10

5

8.0×10

5

6.0×10
5

10

-0.4

-0.2

0

0.2

0.4

Voltage (V)

0.8

0.6

-0.4 -0.2

0

0.2

0.4

Voltage (V)

0.6

0.8

1

(c)
0.8

ηPPd

0.6

0.4

2

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

µp = 1E-6 m V s
µp = 1E-7 m V s

0.2

µp = 1E-8 m V s
µp = 1E-9 m V s

0

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)

Figure 4.10: Plots of (a) kP P d , (b) kP P r , and (c) ηP P d as a function of voltage in the devices
with different hole mobility.
In the Va < 0V range where Rp is most in effect, Rp is reduced with the same trend as
kP P d and ηP P d in figure 4.10 (a) and (c). Lower PP dissociation caused by the decreased
µp makes the "virtual" leakage current of qJx (1 − ηP P d ) higher and, hence, Rp smaller. In
the DD-BI model, Rp is thus predominantly due to PP recombination, and it is inversely
related to it. A low recombination rate leads to large Rp .

4.2.4

Effect of mobility on the series resistance

Another resistance that is more directly related to mobilities is series resistance (Rs ). Since
the applied voltage is the difference between anode and cathode work functions; Va = W Fc −
W Fa , then the relation Vd = Va − JRs = Ef n,I − Ef p,I leads to equation 2.27, i.e.
Rs =

(W Fc − Ef n,I ) + (Ef p,I − W Fa )
.
J

64

(4.7)

2

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

2

Parallel resistance (Ωm )

µp = 1E-6 m V s
-1

µp = 1E-7 m V s

10

µp = 1E-8 m V s
µp = 1E-9 m V s

10

-2

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)
Figure 4.11: Parallel resistances (Rp ) as a function of applied voltage calculated by equation
2.28 from the devices having different donor µp .
Typically, if the injection barriers are not higher than about 0.2eV, Ef n near the cathode
(Ef n,c ) will be pinned to W Fc and Ef p near the anode (Ef p,a ) will be also pinned to W Fa
because they are in the Fermi level pinning regime[106]. This will be discussed further in
the injection barrier section (Section 4.4). Hence, one can write Rs as
Rs =

(Ef n,c − Ef n,I ) + (Ef p,I − Ef p,a )
.
J

(4.8)

Consequently, Rs causes Ef n,I and Ef p,I at the interface to be changed from Ef n,c and
Ef p,a at the cathode and anode respectively.
Energy levels at Va = 0.4V of the devices with µp = 10−6 m2 /V s and 10−9 m2 /V s are
chosen as examples and shown in figure 4.12. It can be seen that ELU M O , EHOM O and
Ef n are not much different between the two cases, but Ef p in the bulk donor moves more
from the anode level in the device with µp = 10−9 m2 /V s than the other. This shifting
is caused by more accumulation of space charge as seen in figure 4.8(a), due to the lower
mobility of this device.
Finally, Rs as a function of applied voltage of the different µp devices are plotted in
figure 4.13, calculated using equation 2.27. The high Va region is where Rs is usually most
relevant and experimentally extracted. For µp = 10−6 − 10−7 m2 /V s, Rs values here agree
65

2

2
Efn
Efp

(a)

(b)
1

Energy (eV)

Energy (eV)

1
0
-1
-6

2

µp = 10 m /Vs
-2
-3
0

0
-1
-9

-2
2e-08

4e-08

Position (m)

6e-08

8e-08

-3
0

2

µp = 10 m /Vs

2e-08

4e-08

Position (m)

6e-08

8e-08

Figure 4.12: Energy diagram showing LUMO level (top solid line) and HOMO level (bottom
solid line), including electron and hole quasi-Fermi levels (red dashed and blue dotted-dashed
line respectively) of the devices with (a) µp = 10−6 m2 /V s and (b) µp = 10−9 m2 /V s at
Va = 0.4V
with typical experimentally extracted values of 1Ωcm2 or 10−4 Ωm2 . It is also increased
by orders of magnitude as µp decreases. One interesting thing is that Rs (Va ) itself varies
considerably (two orders of magnitude) in the high mobility devices, but it changes much less
(factors of five) in the lowest mobility device. This raises the question of when the constant
value Rs Shockley equation can be applicable, which will be discussed in Chapter 5.
-1

2

Series resistance (Ωm )

10

10

-2

10

-3

2

-1 -1

2

-1 -1

2

-1 -1

2

-1 -1

µp = 1E-6 m V s
µp = 1E-7 m V s

10

-4

10

µp = 1E-8 m V s
µp = 1E-9 m V s

-5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)
Figure 4.13: Plot of series resistance (Rs ) functions of the devices with different hole mobility
in donor.

66

The reason why Rs (Va ) values decrease at high Va in high mobility devices is as follows.
From Eq. 2.27, Rs = Va − (Ef n,I − Ef p,I ) /J, where J is the total current, which is negative
in the fourth quadrant of JV plot, and Ef n,I − Ef p,I is the diode voltage (Vd ). Consider Vd
and Va − Vd of the device with µp = 10−6 m/ V s in Figure 4.14. The term Va − Vd indicates
how much quasi-Fermi levels change over the bulk layers to the actual voltage biasing the
interface (Vd ). For Va < 0.4V , Vd shows little variation with Va because of space charge
accumulation, so Va − Vd is almost a linear function of Va and highly negative. At Va = 0.4V ,
a lot of charges are injected so the interface quasi-Fermi levels start to align with the electrode
values. This makes Vd ≈ Va and Va − Vd approaching zero. As a result, Rs that depends on
Va − Vd decreases at high Va . This can be simply viewed as, at high Va , there are more free
carriers to conduct currents so the electrical resistivity of the device decreases. On the other
hand, Rs (Va ) of the device with µp = 10−9 m2 /V s does not follow the same trend as the
others because of the stronger degree of space charge effects and the s-shaped JV behavior.
0.8

Vd (V), Va - Vd (V)

0.6
0.4
0.2
0.0
-0.2
-0.4

Vd
Va - Vd
Va

-0.6
-0.8
-0.4

-0.2

0

0.2

0.4

0.6

Va (V)

Figure 4.14: Plots of the diode voltage (Vd = Ef n,I − Ef p,I ) and total change of quasi-Fermi
energies from the electrodes to the interface (Vd ), i.e. Va − Vd , with respect Va

67

4.3

Effects of Injection Barriers

Injection barriers at the anode (φa ) and the cathode (φc ) are defined by W Fa − EHOM O,D
and ELU M O,A − W Fc , where W Fa and W Fc are the anode and the cathode work function,
respectively. In this study, only positive values of φa and φc are taken into consideration. A
stack of indium tin oxide (ITO) and poly(3,4-ethylenedioxythiophene) polystyrene sulfonate
(PDOT:PSS) is ubiquitously used as the transparent hole extracting layer (anode) for the
P3HT:PCBM system [59, 107, 108, 109]. In some cases, ITO/MoOX and other material
systems are used as the anode[1, 110]. On the other hand, Ca, Mg, Al LiF/Al, and Au
are metallic elements usually chosen to be the cathode[111]. The question why some materials/elements perform better than the others have been partially answered by considering
alignment of energy levels[66, 110], but a complete understanding including space charge effects, charge recombination, interface polaron pair recombination, etc. has not been achieved.
In order to make the right choices of anode and cathode that optimize the efficiency, effects
of injection barriers on OPV are of importance to understand.

4.3.1

JV Characteristic Response with Varied Injection Barriers

JV responses of the devices with parameters summarized in table 4.5 under illumination
conditions are simulated and showed in figure 4.15a, as well as their efficiency parameters
in figure 4.15b. The active layer is P3HT/PCBM bilayer. The anode is ITO/PDOT:PSS,
having the injection barrier (φa ) around 0.2eV.[105]. Cathode work function serves as the
independent variable in this study, and is varied from 0.0 to 0.4eV; resulting in many interesting responses of the JV curves. First, photocurrent is reduced with the increasing
φc , giving rise to a lower FF and the s-shape JV behavior which are main reasons for the
degraded efficiency. As seen in figure 4.15b, FF is only at 0.48 when φc = 0.4eV , which is
reduced by 26% relative to when φc = 0eV , while Jsc is reduced by less than 4%. Second, Voc
is affected very slightly by the φc . Third and most importantly, φc

0.2eV does not have

noticeable effects on the JV curves. Before investigating any of these behaviors, it should be
68

noted that these s-shape JV characteristics with unchanged Voc being induced by changing
cathode work function are commonly observed in experiments[102, 112, 113].
Parameter
µp
µn
Donor/acceptor thickness
φa
φc

Value
1.0 × 10−8 m2 /V s
2.0 × 10−7 m2 /V s[104]
40nm/40nm
0.2 eV[105]
0.0, 0.1, 0.2, 0.3, 0.35, 0.4 eV
3.9 0
58 meV[103]
(20µs)−1
-4.9eV/-3.0eV
-6.1eV/-3.7eV
1028 /1026 m−3
1024 m−3
4.3 × 1020 m−2 s−1
1.0nm
Bimoleular + trap-assisted

Et,A = Et,D
kP P r,0
Donor ELU M O /EHOM O
Acceptor ELU M O /EHOM O
NHOM O /NLU M O
HD = HA
JX
a0 = δx
recombination type

Table 4.5: Summary of parameters used to study effects of cathode injection barriers φc

0

-0.4

-0.2

0

-64

0.2

0.4

0.6

0.8

-65
-66

0.55

-66

-JSC

-66

0.50

-50
0

-2

0.60

VOC
FF

-JSC (A m )

50

φa/φc = 0.2/0.00eV
φa/φc = 0.2/0.10eV
φa/φc = 0.2/0.20eV
φa/φc = 0.2/0.30eV
φa/φc = 0.2/0.35eV
φa/φc = 0.2/0.40eV

VOC (V), FF (%)

-2

J (A m )

100

-64

0.65

0.1

0.2

0.3

-67
0.4

φc (eV)

Voltage (V)
(a) JV characteristics

(b) Efficiency parameters

Figure 4.15: (a) JV characteristics of devices under influence of increasing cathode work
function (injection barrier), and (b) their extracted Jsc , Voc , and FF.

69

0.5

1

(a) Va = 0V

0.4
0.2
0
0

Interface

0.6

0.4

Potential (V)

Potential (V)

0.8

(b) Va = 0.5V

φc = 0.00eV
φc = 0.10eV
φc = 0.20eV
φc = 0.30eV
φc = 0.35eV
φc = 0.40eV

0.3
0.2
0.1

2e-08

4e-08

6e-08

0
0

8e-08

Position (x)

2e-08

4e-08

6e-08

8e-08

Position (x)

Figure 4.16: Potential profiles (ψ(x)) of the devices with different φc at applied voltage (a)
0V and (b) 0.5V.
4.3.2

Reduced Built-in Potential from High Injection Barriers

The most obvious reason for the reduced photocurrent is that the built-in potential is lessened
by higher φc (i.e. work functions). Built-in potential (Vbi ) is integrated into an OPV from
the difference of the electrodes’ work function. It is often given by Vbi = W Fc − W Fa ,
which is also Vbi = ∆EHL − φa − φc . This potential difference helps pulling photo-generated
holes and electrons from the D/A interface to be collected at the anode and the cathode
respectively. High work function materials are consequently chosen for the anode, and low
work function materials are used as cathodes. Figure set 4.16 shows potential profiles at (a)
Va = 0V and (b) Va = 0.5V , under illumination. This potential (ψ(x)) drives free carriers
along its gradient resulting in the drift currents; qp(x)µp (x)E for hole and qn(x)µp (x)E
for electron, where E = −∂ψ(x)/∂x is the electric field. In the figures, the anode and the
cathode are at x = 0m and x = 80nm respectively. Non-linear ψ(x) can be seen at the
anode for any φc and at the cathode for φc = 0.0eV . This is caused by band bending from
spontaneous charge transfer at the electrodes, which will be discussed later. Total potential
difference between the anode and the cathode is W Fc − W Fa (i.e. Vbi ). Hence, increasing
φc from 0.1eV to 0.4eV linearly reduces Vbi and results in smaller electric field E = −∂ψ/∂x
from the smaller slopes of potential profiles. With less help from the built-in field, less

70

drift photocurrents can reach the electrodes. Situations at Va = 0.5V in figure 4.16(b) are
more obvious and severe. The potential profiles are more curved by accumulation of photogenerated charges at the interface, further reducing electric field and drift currents out of
the interface. For φc ≥ 0.3eV , Vbi is low enough so that the direction of the acceptor electric
field is switched, driving free charges back to the interface. This is confirmed in Figure 4.17
where majority electron and hole densities are plotted as a function of position, in the devices
with φc = 0.0, 0.2, and 0.4 eV. In the case of φc = 0.0eV, electron density is high near the
cathode due to high drift current as explained earlier and significant carrier injection from
the cathode. Electron density near the contact can be approximated by Eq. 1.15 stating
that nc = NLU M O exp(−φc /kB T ). Conversely, when φc = 0.4eV , even though n(x) near the
cathode is very sparse because of the high φc , free electrons from PP dissociation do not
diffuse to the cathode because they are impeded by the inverted electric field, i.e. backward
drift current wins over the diffusion current. Thus, dissociated electrons are harder to extract
and cause the reduction of photo-current, due to increased free carrier recombination.
n(x) : φc = 0.0eV
p(x) : φc = 0.0eV
n(x) : φc = 0.2eV
p(x) : φc = 0.2eV
n(x) : φc = 0.4eV
p(x) : φc = 0.4eV

27

-3

Charge densities (m )

10

24

10

21

10

Acceptor

Donor
18

10 0

2e-08

4e-08

6e-08

8e-08

Position (m)
Figure 4.17: Carrier density profiles (n(x) and p(x)) showing only majority carriers, i.e.
holes in the donor and electrons in the acceptor, of the devices with φc = 0.0eV, 0.2eV and
0.4eV at the same applied voltage of 0.5V.

71

4.3.3

Unaffected Interface Densities: Unaffected Voc

From the lack of driving potential at high φc , interface charge densities are not changed
much compared to the contact densities (see figure 4.17). This leads to the origins of another
character seen in the JV curves in figure 4.15a, namely the unaffected Voc .
Open circuit voltage is related to the quasi-Fermi levels through[114];
qVoc = (Ef n − Ef p )|Va =Voc .

(4.9)

This agrees with the diode voltage derived in this study that Vd = Ef n,I − Ef p,I = Va − JRs .
Under open circuit conditions, the fact that J = 0 results in Va = Voc = Ef n,I − Ef p,I = Vd .
At this point, Ef n in acceptor and Ef p in donor are invariant with the position, being aligned
from the electrodes to the interface. In order to approximate Voc , consider the expressions
for charge densities in equations 1.21 and 1.22. Interface quasi-Fermi levels can be obtained
by taking a logarithm;
nI
,
Ef n,I = ELU M O,A,I + kB T ln
NLU M O
pI
Ef p,I = EHOM O,D,I − kB T ln
.
NHOM O

(4.10)
(4.11)

The subscripts D, A, and I refer to the donor, the acceptor and the interface respectively.
Substitute these into the Voc equation.
k T
NLU M O NHOM O
Voc = ∆EHL − B ln
,
q
nI pI

(4.12)

where ∆EHL = ELU M O,A,I − EHOM O,D,I is interface energy gap in eV. All the variables
in Eq. 4.12 are independent of φc , except nI pI . The larger nI pI is, the closer Ef n,I is to
ELU M O,A,I , and Ef p,I to EHOM O,D,I , leading to larger Voc . Figure 4.18(a) shows plots of
nI and pI at Voc as a function of φc . Even with the reversed acceptor electric field at higher
φc , nI only decreases slightly from diffusion out of the interface. Interface holes then face less
recombination from the lesser electrons and accumulate more. With one density goes down
and the other goes up, nI pI does not change much as φc increases (notice that the nI pI scale
72

is linear), so is the Voc . Figure 4.18(b) shows a comparison between Voc obtained from the JV
curves in Fig. 4.15a and approximated by Eq. 4.12 using nI pI from Fig. 4.18(a). It should
be stressed that the y-scale is very narrow. Both graphs indicate that Voc changes less than
20mV while φc changes by 0.4eV. The approximated Voc is close and agrees with the extracted
one with a small constant error arising from using ∆EHL = 1.2eV as a constant. In reality,
there is band bending at the interface[66] making ∆EHL = ELU M O,A,I − EHOM O,D,I .
23

10

10

45

0.655

(b)
nIpI

0.650

10

22

nI
pI

10

0.1

0.645
0.640

21

0

VOC (V)

-6

nIpI (m )

-3

nI and pI (m )

(a)

0.2

0.3

10
0.4

44

0.635
0

φc (eV)

Numerical Simulation
VOC = Efn,I - Efp,I

0.1

0.2

φc (eV)

0.3

0.4

Figure 4.18: Plotting as a function of φc of (a) interface electron and hole densities with their
multiplication and (b) open circuit voltages extracted from the DD-BI model simulations
(same as in Fig. 4.15b), compared with a calculated one using equation 4.12.
In conclusion, nI pI is not affected significantly by the reduced built-in field resulting from
an increased φc . Open circuit voltage which entirely depends on nI pI is also not greatly
affected. Optimizing OPVs by using matching electrode work functions (small injection
barriers) does not improve Voc but significantly improves FF and Jsc .

4.4

Metal/Organic Semiconductor Contact

The question why φc

0.2eV does not have any noticeable effects on the JV curves will

be investigated in this part. The main role is played by the metal/organic semiconductor
contact at the cathode where a lot of physics is going on. This section will be for possible
explanations of the nature of the contact, starting from Mott-Schottky model to Integer
Charge Transfer Model. Band bending, the intrinsic property that has been mentioned
73

many times, will also be discussed. The simulation results will be compared and explained
with these established theories.

4.4.1

Mott-Schottky Model With Vacuum Level Shift

The original Mott-Schottky (MS) model was proposed in the 1940s in two separate papers by
N. F. Mott and W. Schottky. The model was developed for metal/inorganic semiconductor
contacts, but had been applied, without serious verification, to the field of organic semiconductor during its boom period[115]. Until around in 2000, there were many experimental
evidences of an interface dipole that induces a vacuum level shift, so the model was slightly
modified by including it.

Figure 4.19: Illustration of a Mott-Schottky contact between a metal with work finction
Φm and an n-type semiconductor with chemical potential Φs , where Φm > Φs , forming a
blocking contact and downward band bending.
Figure 4.19 is a schematic conclusion of one example of the MS contacts. A metal with
the work function Φm is brought into contact with an n-type inorganic semiconductor with
the Fermi level (chemical potential) Φs , where Φm > Φs . After the contact, Fermi level
has to be equilibrated at the interface and aligned throughout the bulk semiconductor. To

74

achieve this, electrons have to flow from the semiconductor (higher energy) to the metal
(lower energy), giving rise to energy levels bending, or band bending. In the original MS
model, interface vacuum level alignment is assumed. However, this does not hold in most
organic/metal interfaces due to an interface dipole modifying vacuum level, so a vacuum
level shift (∆) was added to the model[115]. Potential drop from the band bending (Vbb )
is |Φm − Φs |, and total potential loss (Vloss ) including the vacuum shift is Vbb + ∆. In this
example, a Mott-Schottky (blocking) barrier is formed, making the contact rectifying. In
the opposite circumstances shown in figure 4.20 where Φm < Φs , electrons flow from the
metal to the semiconductor, inducing upward band bending instead of downward. There is
no Mott-Schottky barrier and the contact is ohmic (non-rectifying). Possible vacuum level
shift is also included in the figure.

Figure 4.20: Illustration of an ohmic contact between a metal with work finction Φm and a
semiconductor with Fermi level Φs , where Φm < Φs , forming an ohmic contact and upward
band bending.

75

4.4.2

Integer Charge Transfer (ICT) Model

Integer Charge Transfer (ICT) model has gained more popularity than the MS model recently as it is more applicable and able to describe organic/metal interfaces, and it has been
confirmed by experiments [106]. There are many models describing different types of organic/metal interfaces, but the ICT model is suitable for interfaces of a solution processed
(spin-coating, ink-jet printing) polymer coating on oxide passivated metal[66]. This type of
interface is the most common in OPVs due to the ease of the solution process.
Organic π-conjugated molecules and polymers have a soft geometric structure so that
when they receive or give away an electron, both electronic and geometric states are relaxed[66].
Hence, free holes and electrons in an organic polymer are considered as polarons. Their local
polaronic states can have energies within the forbidden energy gap[66]. Based on the ICT
model, EICT + is defined as an energy loss when an electron is transferred out of the organic
material. This level lies just above the HOMO band edge, within the energy gap. Similarly,
EICT − is an amount of energy gained by the organic semiconductor if there is an electron
transferred into it. This level is below and close the LUMO edge.
When a substrate with a work function Φm is brought into contact with an organic
semiconductor, three possible cases as shown in figure 4.21 may occur.
(a) If Φm > EICT + , electron polarons in the EICT + states tunnel through the interface to
the substrate, creating hole polarons and reducing total energy of the organic semiconductor.
This continues until EICT + is equilibrated with the substrate work function, usually referred
to as Fermi level pinning. There is a potential dipole from the unbalanced charges, producing
a shift of vacuum level (∆) at the interface, equal to Φm − EICT + .
(b) If EICT − < Φm < EICT + , spontaneous charge transfer does not occur due to unfavorable energy loss and gain of the organic material[66]. Once taken into contact, all energy
levels are the same as when they are separated, making their vacuum level aligned. This
situation is consequently called Vacuum Level Alignment in the ICT model.
(c) If Φm < EICT − , electrons from the substrate will spontaneously tunnel to the organic

76

(a)

(b)

(c)

Figure 4.21: Integer Charge Transfer (ICT) model describing spontaneous charge tunneling
at a metal/organic semiconductor interface of the case (a) Φm > EICT + , (b) EICT − < Φm <
EICT + and (c) Φm < EICT − .

77

layer, raising its EICT − and all other energy levels. Similar to the case (a), this continues
until the EICT − level equal the substrate work function, so the Fermi level is pinned. Upward
vacuum level shift; ∆ = EICT − − Φm , also occurs at the interface.

4.4.3

Organic/Metal Interfaces Under Illumination

It was found that the computational results have distinct properties between under illumination and the dark. Thus, a cathode organic/metal interface under illumination is investigated
first in this part. The cathode injection barrier is taken to be to 0.6eV to clearly show its
influence. Other parameters are the same as in table 4.5. Analysis of φc effects will involve
referring to important spatial positions in the device. Figure 4.22 is a simple diagram of the
bilayer structure and some important positions (nodes). As stated in the computational theory chapter (Chapter 3), the position in the direction of film thickness is divided into nodes
(mesh). The anode is at the node 0; the donor is from the 1st to the lth node. The acceptor
at the vicinity of the hetero-junction is at the l + 1th node. Most importantly, the node L − 1
is the acceptor node next to the organic/metal interface, and the Lth node is the cathode.
Based on the MS and the ICT models, the vacuum level shift given by ∆ = ψL−1 − ψL .
Fermi level pinning can be checked by comparing Ef n,L−1 − W Fc with zero. If it is zero,
then the Fermi level, or more correctly quasi-Fermi level, is pinned. If there is band bending
in the acceptor bulk, it will be Vbb = ψl+1 − ψL−1 . Lastly, total potential drop from the
cathode to the hetero-interface is Vbb + ∆ = ψl+1 − ψL .

Figure 4.22: A simple diagram showing the bilayer structure and some important positions
at the anode and the cathode organic/metal and the hetero-junction interfaces.

78

Energy (eV)

2
1
0
-1

EHOMO
ELUMO
Efn
Efp

-20

2e-08

4e-08

Position (m)

6e-08

8e-08 0

2e-08

4e-08

Position (m)

(a) φc = 0.0eV

6e-08

8e-08 0

2e-08

4e-08

Position (m)

(b) φc = 0.3eV

6e-08

8e-08

(c) φc = 0.6eV

Figure 4.23: Band diagram illustrating energy levels of the devices with φa = 0.2eV and φc
=0.0eV, 0.3eV and 0.6eV under illumination.
Figure 4.23 shows energy band diagrams from anode to cathode of the devices with φc =
0.0eV, 0.3eV and 0.6eV under flat band conditions and illumination. Flat band conditions
are when the applied voltage cancels out the built-in potential so the bands are flat and drift
currents vanish. However, there is unavoidable band bending that makes completely flat
bands impossible in some cases. The applied voltages to achieve flat bands in figure 4.23 are
0.594V, 0.509V and 0.214V respectively. Total potential drop from the cathode to the D/A
interface under flat band conditions will be called voltage loss (Vloss ) in donor. The name
is taken from the fact that Vbi is lost due to Vbb and ∆, producing less built-in electric field.
Figure 4.24 shows ∆ at the cathode, Vloss and Ef n,L−1 − W Fc as a function of φc .

∆ and Vloss (V)

0.25
0.20

Efn,L-1 - WFc

Vloss
∆

0.15

0.10

0.15
0.10

0.05
0.05
0.00
0

0.1

0.2

0.3

0.4

0.5

Efn,L-1 - WFc (eV)

0.20

0.00
0.6

φc (eV)
Figure 4.24: Plots of vacuum level shift (∆), voltage loss (Vloss ) and acceptor electron quasiFermi level (Ef n,L−1 ) at the cathode relative to the cathode work function (W Fc ) as a
function of φc .
79

According to figures 4.23 and 4.24, some conclusions can be drawn as follows. Firstly, Two
regimes of φc , i.e. Fermi level pinning (with vacuum level shift) and vacuum level alignment
(with unpinned Fermi level) as suggested by the ICT model, are observed. The black and
blue curves in figure 4.24 clearly exhibit these cases. At low φc , Ef n,L−1 is close to W Fc
and moves along with it when it is varied. Vacuum level shift is also visible in this range.
However at φc ≈ 0.3eV , Ef n,L−1 starts to unpin from W FC when it moves down further
into the acceptor energy gap. At the same time, ∆ approaches and remains zero. This
gives a rough prediction that the EICT − is around 0.3eV. Secondly, unlike the ICT model
stating that band bending should not occur[66], band banding is clearly seen in both figures
4.24 and 4.23(a). Band bending is caused by accumulation of space charges and should be
inevitable in OPVs because of the low mobilities. It is usually observed experimentally, for
example, Ishii et al. [115] saw band banding in a C60 film on a Copper substrate by using
Kelvin probe measurements. However, there is only upward band bending when φc is low,
but no downward one to be found when φc is high, as predicted by the MS model. At high
φc , band banding and vacuum level shift are less significant (see figure 4.24) and the bands
seem to be more flat (see figure 4.23(b) and (c)). Not much change of the bands can be
observed when increases φc from 0.3eV to 0.6eV. The positions of Ef n and Ef p in the bulk
as well as their split at the D/A interface also do not alter notably. These are other proofs
of the unchanged Voc that was discussed previously. Thirdly, vacuum level shift is not as
high as expected by the MS and ICT models. They predict that the shift should be about
|0.3 − φc | V, taken into account that EICT − ≈ 0.3V . However, it is just tens of mV from
the simulation results. However, when include band bending into the shift leading to Vloss ,
it is in an agreeable range as predicted by the ICT and MS models. This discrepancy may
arise from different assumptions and the way each model approaches the problem. The MS
model was originally developed for inorganic/metal interfaces, but can work reasonably well
with organic interfaces, if small modifications are included. The ICT model was empirically
developed from a lot of photoelectron spectroscopy data. Some theoretical works at the

80

molecular level might be needed to verify and correct the model. The DD-BI model is
based on more fundamental Physics like Poisson equation, but better organic descriptions,
for example at the organic/metal interfaces, are needed. Nevertheless, they all work well to
describe OPVs to some extent.

4.4.4

Organic/Metal Interfaces In the Dark

Energy (eV)

2
1
0
-1

EHOMO
ELUMO
Efn
Efp

-20

2e-08

4e-08

Position (m)

6e-08

(a) φc = 0.0eV

8e-08 0

2e-08

4e-08

Position (m)

6e-08

(b) φc = 0.3eV

8e-08 0

2e-08

4e-08

Position (m)

6e-08

8e-08

(c) φc = 0.6eV

Figure 4.25: Band diagram illustrating energy levels of the devices with φc =0.0eV, 0.3eV
and 0.6eV in the dark.
Figure 4.25 shows energy band diagrams in the dark under flat band conditions of the
devices with φc =0.0eV, 0.3eV and 0.6eV. The flat band voltages are 0.609V, 0.536V and
0.255V respectively. These are slightly different from the one under illumination because
there is no space charge accumulation at the D/A interface in the dark.
The most important trait of the interface in the dark is that the electron quasi-Fermi
level in the donor is always pinned at the cathode, regardless of φc . Figures 4.23(c) and
4.25(c) illustrate this. Figure 4.25(a), (b) and (c) show Ef n,L−1 being exactly at 0.0eV,
0.3eV and 0.6eV respectively below the energy of the Lth node. There are no separated φc
regimes like in the devices under illumination within the ICT model. The results are more
agreeable, but not entirely, with the MS model where Fermi level pinning is always assumed.
Band bending and vacuum level shift similar to the illumination cases in figure 4.24 are also
expressed under the dark.

81

4.4.5

Components of Series Resistance

Series resistances (Rs ) was discussed in section 4.2.4 without any effects of the organic/metal
interface at the cathode or the anode, so only the bulk donor and acceptor contributed to
the Rs . It is commonly known that not only the bulk parts, but also any contacts are
factors influencing Rs . There is an experiment by Y. Shen et al.[46], studying effects of
film thickness on the series resistance. The results show that Rs does come from the two
contributions, with about 17% from the contacts and 83% from the bulk.
Since Rs is controlled by changes in the quasi-Fermi levels, as explained in section 4.2.4,
it is important to consider four possible causes of changes in th quasi-Fermi energies: (1)
anode contact; (2) bulk donor; (3) bulk acceptor; and (4) cathode contact. These give rise
to four components of Rs that will be named as Rsa , RsD , RsA and Rsc respectively. See
figure 4.26 for an illustration. At the anode or the cathode contact, if the injection barrier is
too high so that the quasi-Fermi level is not pinned, there will be a drop of the quasi-Fermi
energies. This can be represented by the contact series resistances Rsa and Rsc . To view this
from another angle, high injection barriers bring about a Mott-Schottky blocking barrier,
which resists charge injection/extraction. For the bulk donor and acceptor, poor charge
transport (mobility) gives rise to lower conductivity and reduces/changes the quasi-Fermi
levels in the regions around the D/A interface. These cause the donor and acceptor to have
the resistance RsD and RsA respectively. Since this topic was discussed quite expansively
in section 4.2.4, effects of injection barriers will only be focused on. Similar to total Rs
equation in Eq. 2.27, each Rs component is simply given by
Rs,i−j =

Ef n(p),j − Ef n(p),i
Ji−j

,

(4.13)

where Rs,i−j and Ji−j are series resistance and total current density between the node ith
and i + 1th , respectively.
Figure 4.27 shows how Rs changes with increasing φc , under illumination. For φc = 0.0ev
to 0.2eV, the change of Rs is negligible. The graph for φc = 0.1eV sits on top and entirely
covers the graph of φc =0.0eV. Since mobilities are fixed, the Rs change is only caused by
82

Figure 4.26: Diagram showing possible deviations of hole quasi-Fermi level at the anode
contact and bulk donor and electron quasi-Fermi level at the cathode contact and bulk
acceptor, that can be represented by a component of series resistance.
the cathode and anode interfaces. However, in this range of φc , quasi-Fermi level pinning
occurs and there is no drop of the Ef n at the cathode, so there is no contribution of Rsc at
all. Then total Rs is not affected by φc and/or φa that are in Fermi level pinning regime.
Starting at φc = 0.3eV, Rs at high voltage begins to increase. As stated earlier, Rs matters
more at the high-voltage than low-voltage regimes. Thus, this Rs increment affects the OPV
efficiency as seen in figure 4.15a, even though Rs values at low voltage are the same.
Change of Rs from φc = 0.2eV to 0.6eV is interesting to look further how those four
components of Rs play their role. Figure 4.28 shows plots of Rsa , RsD , RsA and Rsc as a
function of voltage of the devices with φc =0.2eV, 0.4eV and 0.6eV. They are calculated by
equation 4.13. Anode injection barrier is fixed at 0.2eV. Firstly, it can be seen that Rsa is
negligibly small and not changed at all. This is because of the unchanged φa and pinning
of Ef p at the anode. Secondly, bulk donor resistance RsD is also not affected noticeably by
the increased φc , due to the fixed φa and donor hole density that is not disturbed much.
Thirdly, RsD is high and composes most of the Rs when both φa and φc equal to 0.2eV.
This is because of the lower hole mobility in the donor as compared to electron mobility.

83

10

0

2

Rs (Ωm )

Increasing φc
-2

10

φc = 0.0eV
φc = 0.1eV
φc = 0.2eV
φc = 0.3eV
φc = 0.4eV
φc = 0.5eV
φc = 0.6eV

-4

10

-0.4

-0.2

0

0.2

0.4

Voltage (V)

0.8

0.6

Figure 4.27: Plots of series resistance as a function of voltage under influences of changing
cathode injection barrier from 0.0eV to 0.6eV with fixed anode injection barrier at 0.2eV.
0

10

0

10

0

2

Components of Rs (Ωm )

10

-2

-2

Rsa
RsD
RsA
Rsc
Rs,total

-4

10

-6

10

10

-2

10

10

-4

-0.2

0

-4

10

10

-6

-6

10

-8

-0.4

10

0.2

0.4

Voltage (V)

(a) φc = 0.2eV

0.6

0.8 10

10

-8

-0.4

-0.2

0

0.2

0.4

Voltage (V)

(b) φc = 0.4eV

0.6

0.8 10

-8

-0.4

-0.2

0

0.2

Voltage (V)

0.4

(c) φc = 0.6eV

Figure 4.28: Plots of the component series resistances Rsa , RsD , RsA and Rsc , and total Rs
being the sum of the four components as a function of applied voltage of the devices with
different cathode injection barrier.
However, higher φc greatly reduces electron density in the acceptor, making Ef n deviation
in the acceptor much more significant. This raises RsA so that it becomes a major part of
the total Rs . Fourthly Rsc increases its importance as φc increases. From φc = 0.2eV where
Ef n is pinned with W Fc and Rsc is negligible to φc =0.4eV, the barrier φc has entered the
vacuum level alignment regime, dropping Ef n,L−1 a little from W Fc . The resistance Rsc
increases many orders of magnitude, but is still insignificant compared to the bulk resistances
RsD and RsA . At φc = 0.6eV, the drop of Ef n,L−1 − W Fc becomes substantial as seen in
figure 4.24, so that Rsc is more than the donor resistance RsD and considerably contributes

84

to the total Rs .
In conclusion, if both φc and φa barriers are in the Fermi level pinning regime, i.e. less
than about 0.3eV, contacts resistances Rsa and Rsc are negligible. Series resistance depends
on the bulk layer having smaller carrier mobility. Increasing any injection barrier over this
limit augments the resistant components associated with it. One is its contact, and the other
is the charge density it injects into the bulk. For the case of φc , it increases RsA because less
electrons can be injected into the acceptor and Rsc because the Fermi level is not pinned
but dropped across the organic/metal interface.

4.4.6

Choosing the Anode and the Cathode

Effects of injection barriers have been discussed from many standpoints such as built-in
potential, band bending, organic/metal interfaces and contact series resistances. The most
important question that remains unanswered is: What are the right values of the anode and
the cathode work functions that maximize the OPV efficiency?
Speaking in terms of built-in potential; Vbi = |W Fa − W Fc |, so maximizing this by using
the highest W Fa and the lowest W Fc seems to be a right strategy. However, this is not true
if W Fa is higher than the donor HOMO level and/or W Fc is lower than acceptor LUMO
level. All energies here are measured relative to the vacuum level. Effects of negative φa
and φc that are beyond the scope of this study was investigated experimentally by Y.S.
Eo et al.[111]. They studied effects of cathode work function in a bulk hetero-junction
P3HT:PCBM OPV using PDOT:PSS as the anode. Following materials were used as the
cathode: Ca (W Fc = 2.89eV), Mg (W Fc = 3.66eV), Al (W Fc = 4.20eV), LiF/Al (W Fc =
4.30eV) and Au (W Fc = 5.10eV). With the PCBM LUMO level around 4.3eV, Ca, Mg and
Al having a negative φc resulted in worse OPV efficiency than LiF/Al that has about zero
φc , with descending efficiency from Al to Mg to Ca. The worst efficiency belonged to gold
that has φc ≈ 0.8eV . Hence, small positive φa and φc are preferred values for high efficiency.
There is, however, room that φc and φa can vary around without affecting the efficiency.

85

If φc is between ELU M O,A and acceptor EICT − and φa is between EHOM O,D and donor
EICT + , charge transfer and Fermi level pinning will make φa and φc irrelevant because the
injection barriers are countered/compensated by band bending and vacuum level shift. The
PCBM EICT − is about 0.3eV according to this study. Using φa and φc in these ranges also
benefits the Rs by minimizing contact resistances.

4.5

Ideality Factor and Trap-assisted Recombination

The ideality factor (n), or quality factor when specifically used with p-n junction diodes in old
literature [116], is another fitting parameter in the Shockley equation which is as important as
the other parameters in the model. Despite the fact that it is just a single number, typically
ranging around 1-2, it is a useful indicator of the recombination mechanism in microscopic
levels. n equal to 1 is commonly understood as an indication of bimolecular recombination,
while n > 1 indicates that trap-assisted recombination dominates [52]. In this part, the
origins of n will be discussed. Their equations developed in this study (equations 2.17 and
2.18) will be showed to be generalized forms of famous expressions for n. Effects of trap
states on the ideality factors will be investigated. Later, the origins of n = 1 and errors from
interpreting a fitted n will be discussed.
Recombination of free carriers and trapped charges brings about a value of n larger
than one. Following is a well acknowledged analytic theory that connects n to the trap
charateristic[116, 52, 51]. Let nf be the free carrier density, where it could belong to either
electrons or holes. Trapped charge density (nt ) can be approximated by the exponential trap
distribution through[52, 51]
k T /EU

nt ∝ nfB

,

(4.14)

where EU is the characteristic Urbach tail energy. Assuming that the hole and the electron densities are close to each other, then the trap-assisted recombination depends on
1+kB T /EU
.

nf

Note that Eq. 4.14 is the same as the trapped charge densities in Equations 1.23

and 1.24 used in the GWWF model and this work, but the characteristic trapped electron and
86

hole energies are expressed separately through Et,A = kB Tt,A and Et,D = kB Tt,D respectively,
instead of EU . Using the fact that the diode current originates from the recombination[116],
the relation
1 kB T
+
n=
2 2EU

−1

(4.15)

is obtained[116, 52, 51]. Thus an extracted n can be used to calculate the Urbach tail energy
EU .
In this study, there are two ideality factors; nD in Eq. 2.17 from the trapped-hole to
free-electron recombination (fn-tp) and nA in Eq. 2.18 from the trapped-electron to freehole recombination (fp-tn). They are indeed general forms of Eq. 4.15 if total symmetry is
assumed. It means that, for example: µn = µp ; donor and acceptor thicknesses are equal;
both injection barriers are the same. nD is rewritten here for convenience
(Ef p,I − Ef,I )(1 − 1/lD ) −1
,
nD = 1 +
Ef n,I − Ef p,I

(4.16)

where, lD = Tt,D /T . In the symmetric case, Ef p,I and Ef n,I split equally from the equilibrium Ef,I under influences of bias or illumination and hence the magnitude of Ef p,I − Ef,I
is half of Ef n,I − Ef p,I . Note that Ef,I is higher than Ef p,I . Thus,
nD = 1 −

(1 − 1/lD )
2

−1

=

T
1
+
2 2Tt,D

−1

=

k T
1
+ B
2 2Et,D

−1

,

(4.17)

where Et,D = Et,A = EU and nD = nA = n.
To prove the claim in the previous paragraph computationally, a symmetric device with
only trap-assisted recombination was simulated with the following parameters: the thicknesses were 40 nm for both donor and acceptor; φa = φc = 0.2eV ; µp = µn = 2 × 10−7 m2 /V s;
= 3.5 0 ; most importantly, Tt,D = Tt,A = 1000K. nD and nA calculated by Equations 2.17
and 2.18 are plotted in Figure 4.29. Please notice the very fine scale on the y-axis. They
are very close to each other, with small variations at high voltage. Using nD = nA ≈ 1.5384
to calculate the characteristic trap energy in Equations 4.15, EU = 86.16meV is obtained,
which exactly is kB Tt,D or kB Tt,A when Tt,D = Tt,A = 1000K.

87

Ideality factors

1.53850
nD
nA

1.53845

-0.4 -0.2

0

0.2

0.4

Voltage (V)

0.6

0.8

Figure 4.29: Ideality factors nD and nA in the symmetric device with characteristic trap
temperatures of electron (Tt,A ) and hole (Tt,D ) equal to 1000K.
Knowing that the nD and nA equations are general forms of n in the established theory,
studying their responses to the characteristic trap energies in general is the next step. Devices
with different Tt,D varying from 400K to 1200K but the same Tt,A at 1000K were simulated.
Other parameters are listed in Table 4.6. Their JV curves are plotted in Figure 4.30, and their
efficiency are summarized in Table 4.7. According to the results, Jsc shows little variation
across the Tt,D change. Voc improves around 50mV from the diminishing trapped holes.
FF is also slightly improved. Interestingly, there is not much change of the efficiency from
reducing Tt,D from 800K to 400K. nD and nA in dependence on voltage are also plotted in
Figure 4.31.
600

-2

J (A m )

500
400
300
200

Tt,D = 1200K
Tt,D = 1000K
Tt,D = 800K
Tt,D = 600K
Tt,D = 400K

100
0
-100
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Voltage (V)

Figure 4.30: JV characteristic responses of the devices with different hole characteristic trap
temperature (Tt,D ) but fixed electron characteristic trap temperature (Tt,A ) at 1000K.
88

Parameter
µp
µn
Donor/acceptor thickness
φa /φc

Value

2.0 × 10−8 m2 /V s[103]

Tt,D
Tt,A
kP P r,0
Donor ELU M O /EHOM O
Acceptor ELU M O /EHOM O
NHOM O /NLU M O
HD /HA
JX
a0 = δx
recombination type

2.0 × 10−7 m2 /V s[104]
50nm/22nm [59]
0.2 eV[105], 0.0 eV
3.5 0
400K-1200K
1000K
(20µs)−1
-4.9eV/-3.0eV
-6.1eV/-3.7eV
1E27 m−3
1E24 m−3
4.3E20 m−2 s−1
1.0nm
Bimoleular + trap-assisted

Table 4.6: Summary of parameters used to study effects of hole characteristic trap temperature (Tt,D ) in a P3HT:PCBM bilayer OPV.
Tt,D
[K]
1200
1000
800
600
400

Et,D
[meV ]
103.4
86.2
69.0
51.7
34.5

Tt,A
[K]
1000
1000
1000
1000
1000

Et,A
[meV ]
86.2
86.2
86.2
86.2
86.2

Jsc
[Am−2 ]
63.326
66.542
66.677
66.741
66.757

Voc
[V ]
0.575
0.597
0.613
0.622
0.625

FF
[%]
56.527
57.535
58.275
58.692
58.800

Table 4.7: Dependence of device parameters; Jsc , Voc and FF, on hole characteristic trap
temperature (Tt,D .)
There are some interesting characters of nD and nA in Figure 4.31 that are worth discussing. First, nD and nA are not constant, but a function of voltage and many other
parameters that impact the charge densities. However, higher characteristic trap temperatures/energies make the corresponding ideality factors more susceptible to the variations.
For instance, nD at Tt,D = 1200K changes considerably from 1.7 at low voltage to 1.4 at
Va = 0.8V , but nD at Tt,D = 400K is more or less constant around 1.15 across the whole
range of voltage. The reasons behind the variations of nD and nA are the interface free

89

2.0

Increasing Tt,D

1.7

nA

nD

1.8

-3

Tt,D = 1200K
Tt,D = 1000K
Tt,D = 800K
Tt,D = 600K
Tt,D = 400K

Interface density (m )

2.2

1.6

1.6

24

10

pI
nI

23

10

22

10

21

10

20

10

19

10

-0.4 -0.2 0

0.2 0.4 0.6 0.8

Voltage (V)

1.4
1.5
1.2
1.0 -0.4 -0.2

0

0.2

0.4

Voltage (V)

0.6

0.8

1.4 -0.4 -0.2

(a) nD (Va )

0

0.2

0.4

Voltage (V)

0.6

0.8

(b) nA (Va )

Figure 4.31: Plots of the ideality factors (a) nD and (b) nA as a function of voltage, calculated
using equations 2.17 and 2.18 respectively, under the influences of varying Tt,D . The inset
shows the interface electron and hole densities in dependence on voltage when Tt,D = 1200K.
charge densities. Let’s rewrite nD and nA in Equations 2.17 and 2.18 by
nD = 1 −

|Ef p,I − Ef,I |(1 − 1/lD ) −1
,
Ef n,I − Ef p,I

nA = 1 −

|Ef n,I − Ef,I |(1 − 1/lA ) −1
.
Ef n,I − Ef p,I
(4.18)

From these equations, when either electron or hole has substantially higher density than the
other, its quasi-Fermi level splits from the equilibrium level farther than the other, making
the terms |Ef p,I − Ef,I | and |Ef n,I − Ef,I | more different. Thus, nD and nA will be affected
oppositely. Let’s consider the case of Tt,D = 1200K for example. The inset of Figure 4.31(b)
shows the interface electron (nI ) and hole (pI ) densities as a function of voltage. Starting
at about 0.4V, nI rises many orders of magnitude because of the higher injection through
the cathode. This shifts Ef n,I farther from Ef,I . Hence, the term |Ef n,I − Ef,I | becomes
greater than |Ef p,I − Ef,I |, so nA is increased and nD is decreased.
Secondly, nD does decrease with the decreased Tt,D , suggesting that ideality factors
can be used as indicators of the recombination process. To elucidate this more clearly, see
Figure 4.32 where each component of the total current densities of the devices with Tt,D =
1200K, 800K, and 400K is plotted separately. Each curve is calculated by the corresponding
term in Equation 2.20. It should be stressed that Eq. 2.20 is the analytic solution that
can precisely reproduce numerical JV results. In the plots, there are three diode currents
90

stemming from three possible recombination processes: (i) free-electron to free-hole (f-f); (ii)
free-electron to trapped-hole (fn-tp); (iii) free-hole to trapped-electron (fp-tn). The term
qJX ηP P d is the photo-current including the polaron pair recombination. At Tt,D = 1200K,
JD,f n−tp dominates over the entire graph due to the high concentration of trapped holes.
The total current density can be approximated by J ≈ JsD [exp((Ef n,I − Ef p,I )/nD kB T ) −
1] − qJX ηP P d , which is in the form of the Shockley equation. If this graph is fitted to the
Shockley equation, the fitted n should be correlated to nD and contain information about
Et,D . In the case of Tt,D = 400K, the importance of fn-tp recombination is abated greatly,
so that JD,f n−tp is now negligible. JD,f p−tn increases only slightly as there are less trapped
electrons due to their higher mobility. Accordingly, nA will not greatly influence the fitted

-2

Current densities (A m )

n although it is as high as 1.6.
500
400
300
200

Jtotal : Numerical & analytic
JD, f-f
JD, fn-tp
JD, fp-tn
qJXηPPd

100
0
-1000

0.2

0.4

Voltage (V)

(a) Tt,D = 1200K

0.6

0

0.2

0.4

Voltage (V)

0.6

(b) Tt,D = 800K

0

0.2

0.4

Voltage (V)

0.6

(c) Tt,D = 400K

Figure 4.32: Plots of JV components, namely, diode currents from free-electron to freehole recombination (f-f), free-electron to trapped-hole recombination (fn-tp) and free-hole
to trapped-electron recombination (fp-tn), and the photo-current (qJX ηP P d ), of the devices
with fixed Tt,A = 1000K and Tt,D = 1200K, 800K, and 400K.
The device with Tt,D = 800K brings us to the third point, which is about complexity
and errors of interpreting fitted n. In reality there is not just one dominant diode current
similar to the cases of Tt,D = 1200K and 400K, but a couple of them, all contributing to the
total diode current. For Tt,D = 800K, or equivalently Et,D = 69meV , which is in the typical
range of the characteristic trap energy found in OPVs [103, 52, 84, 117], both JD,f −f and

91

JD,f n−tp are comparable. Ignoring JD,f p−tn , the total current density is approximately
J ≈ JsD [exp((Ef n,I − Ef p,I )/nD kB T ) − 1] + Js0 [exp((Ef n,I − Ef p,I )/kB T ) − 1] − qJX ηP P d ,
where JsD , Js0 , and nD are functions of the voltage. Fitting this graph to the Shockley
equation;
J = J0 [exp(Va − JRs )/nkB T ) − 1] − Jph + (Va − JRs )/Rp ,
where J0 , Rs , Rp and n are constant, will result in considerable error. Note that different
methods of extracting parameters give values of the parameters that can differ by orders of
magnitude [118, 119]. One might fit the device with Tt,D = 800K and get n ≈ 1, leading to
the conclusion that trap states were irrelevant and free charges recombine via bimolecular
recombination, which is not true. Ultimately, fitting JV curves to the Shockley equation
using the same fitting method are useful for comparison purposes among the devices, but
may offer wrong physics if the extracted numbers are used in quantitative analysis, unless
the fitted devices are in some specific conditions like the cases of Tt,D = 1200K or 400K.
If a device is dominated by only one recombination mechanism, such as the device with
Tt,D = 1200K where the recombination between free electrons and trapped holes dominates
and in the device with Tt,D = 400K where trapped holes are not significant and the total
recombination is governed by Rf-f , fitting to the standard Shockley equation using one ideality
factor can be useful. For example, if the donor layer of a bilayer device is an amorphous
polymer and the acceptor is crystalline C60 , recombination between trapped holes in the
donor and free electrons in the acceptor is likely the dominant process and results in only
one dominant diode current. One ideality factor is sufficient to describe its JV curves and
provides information about the recombination.
Finally, is it possible that there are devices with n = 1, exactly? Theoretically yes but
practically no. A device with only bimolecular recombination can be simulated and has the
analytic expression (Eq. 2.5) with n = 1 to describe it. However, all other parameters have
to be correct functions of the voltage. Without delicate measuring tools, these functions
are unknown in reality. With literature evidence that bimolecular recombination usually
92

overestimates recombination rate and recombination coefficient [76], extracting n = 1 from
an OPV and concluding that bimolecular recombination is dominant can be misleading.

4.6

Comparison between the SRH and the f-t Recombination

The terms n1 , p1 and n1 p1 in the SRH recombination (Eq. 1.35) are almost always assumed
to be negligible in the literature[77, 87, 88]. n1 and p1 are the densities of electrons and
holes in the LUMO and the HOMO when their Fermi levels sit at Et respectively (see Eqs.
1.29 and 1.33)[49], and n1 p1 = n2i . The SRH recombination equation is then reduced to
RSRH =

Cn Cp Nt nI pI
.
C n nI + C p pI

Kuik et al.[77] show that if the limiting cases Cn

Cp or Cp

(4.19)
Cn are applied, the equation

further reduces to
RSRH = Cp Nt pI ,

(4.20)

RSRH = Cn Nt nI .

(4.21)

or

Each of these equations is exactly one of the terms in the f-t recombination in Eq. 1.8. This
brings about questions: Is the f-t recombination a simplified model of the SRH recombination? Are there any relations between the two models? These questions are answered in this
section by the DD-BI model’s simulations.
The simulations are carried out using the parameters summarized in Table 4.8. We choose
three different sets of the mobilities, i.e. µn /µp = 1, 10 and 100 where µn = 1 × 10−7 m2 /V s,
and Tt,D = Tt,A = 1000K and 450K to represent devices with deep and shallow tail states,
respectively. One group of the devices has the SRH recombination in the integral form (Eq.
1.43) as the interface free-carrier recombination, while the other has the f-t recombination
(Eq. 1.8). The same capture rates as described in Eqs. 1.37 and 1.38 are used.
Figures 4.33a and 4.33b show comparisons between the simulated JV curves under illumination conditions for Tt,D = Tt,A = 1000K and 450K respectively. Table 4.9 summarizes
93

Parameter
µp
µn
Donor/acceptor thickness
φa /φc

Value
1.0 × 10−8 m2 /V s, 1.0 × 10−9 m2 /V s
1.0 × 10−7 m2 /V s
40nm/40nm
0.20 eV/0.20 eV
3.0 0
450K, 1000K
Eq. 1.48 with kP P r,0 = 10µs−1
-4.9eV/-3.0eV
-6.1eV/-3.7eV
1.0E27 m−3
1.0E24 m−3
4.3E20 m−2 s−1
1.5nm
Shockley-Read-Hall or f-t

1.0 × 10−7 m2 /V s,

Tt,D = Tt,A
kP P r
Donor ELU M O /EHOM O
Acceptor ELU M O /EHOM O
NHOM O /NLU M O
HD /HA
JX
a0 = δx
recombination type

Table 4.8: Summary of the simulation parameters used in the comparison between the SRH
recombination (Eq. 1.43) and the f-t recombination (Eq. 1.8).
the devices’ Jsc , Voc and FF values. There are a couple of interesting points that can be
concluded as follows.
40

-20
-40

-2

-2

0

20

J (Am )

20

J (Am )

40

SRH µn/µp = 1
f-t µn/µp = 1
SRH µn/µp = 10
f-t µn/µp = 10
SRH µn/µp = 100
f-t µn/µp = 100

-20
-40
-60

-60
-80

0

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)
(a) Tt,D = Tt,A = 1000K

-80

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)
(b) Tt,D = Tt,A = 450K

Figure 4.33: Comparisons between the illuminated JV curves calculated using Rf-t (Eq. 1.8)
and RSRH (Eq. 1.43) of the devices with Tt,D and Tt,A equal to a) 1000K and b) 450K.
First, the two recombination types create similar illuminated JV curves in the reverse
and the small bias regimes, but the f-t recombination results in a higher diode current at high
bias. As we discussed earlier that the diode current is caused by the interface free-carrier
94

Tt,D , Tt,A
[K]
1000

µn
2
[m /V s]

µp
2
[m /V s]

Recombination

1 × 10−7 m2 /V s

1 × 10−9 m2 /V s

1000

1 × 10−7 m2 /V s

1 × 10−8 m2 /V s

1000

1 × 10−7 m2 /V s

1 × 10−7 m2 /V s

450

1 × 10−7 m2 /V s

1 × 10−9 m2 /V s

450

1 × 10−7 m2 /V s

1 × 10−8 m2 /V s

450

1 × 10−7 m2 /V s

1 × 10−7 m2 /V s

SRH
f-t
SRH
f-t
SRH
f-t
SRH
f-t
SRH
f-t
SRH
f-t

Jsc
[Am−2 ]
57.84
57.16
64.82
64.72
67.11
67.08
64.96
64.95
66.76
66.76
67.77
67.77

Voc
[V ]
0.7225
0.6045
0.6599
0.6045
0.6273
0.6050
0.8167
0.7210
0.7658
0.7210
0.7401
0.7212

FF
[%]
23.80
25.97
43.38
45.55
58.78
59.29
34.91
39.09
52.38
55.55
63.64
65.22

Table 4.9: Comparisons of the Jsc , Voc and F F values between the devices having RSRH
(Eq. 1.43) and Rf-t (Eq. 1.8) as the interface recombination. The values are measured from
the JV curves in Fig. 4.33, and the devices are described by Table 4.8.
recombination, all the plots indicate that Rf-t is higher than RSRH when Cn = krec,n and
Cp = krec,p . This agrees with Figure 4.34a that shows the total recombination rates as a
function of voltage of the device with µp = 10−9 m2 /V s, µn = 10−7 ms /V s and Tt,D = Tt,A =
1000k. The rate Rf-t is about 2-3 times as high as the rate RSRH for the whole voltage
range.
Secondly, the terms n1 , p1 and n1 p1 in the SRH equation (Eqs. 1.35 or 1.43) can actually
be ignored. The simulation results for all six devices show that n1 p1 = n2i has no effects on
the JV curves both under illumination and dark conditions, and the terms n1 and p1 cause
only negligible changes on the curves (not shown here). Hence, the reduced SRH equation
(Eq. 4.19) is valid. We would like to emphasize that this is not caused by n1 < n or p1 < p
because n1 and p1 can be more than or comparable to n and p at some values of Et , but
by that the thermal emissions of the trapped charges are negligible if Et is between Ef n
and Ef p [87]. Only small regions of trap states close to Ef n or Ef p can thermally emit the
trapped charges to the transport levels.

95

30

30

10

10

-1

Recombination (s )

total RSRH
total Rf-t

-1

Recombination (s )

10

29

10

10

28

27

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)

10

29

10

10

10

RSRH,D
RSRH,A
Rrec,n
Rrec,p

28

27

26

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)

(a) Total rates

(b) Separate rates

Figure 4.34: a) Total RSRH and Rf-t rates and b) their separate rates from different mechanisms; RSRH = RSRH,D + RSRH,A and Rf-t = Rrec,n + Rrec,p , of the devices with
µn /µp = 100 and Tt,D(A) = 1000K.
Thirdly, the f-t recombination is not a simplified form of the SRH recombination but a
different model. There are two main reasons. First, the assumptions Cn

Cp and Cp

Cn

are not always valid since µn and µp at the D/A interface are not necessarily very different.
Second, even if they are significantly different, RSRH does not reduce to Rf-t because it has
an extra process related to the unoccupied trap states that Rf-t does not have. Based on the
SRH theory, not only the occupied trap states (trapped electrons) can capture free holes,
but the unoccupied trap states can also capture free electrons. This is shown in Figure 4.34b
where separate rates of the total RSRH and the total Rf-t of the device with µn /µp = 100
and Tt,D(A) = 1000K in Fig. 4.34a are plotted. RSRH,A and RSRH,D are the SRH rates
originated from the donor and the acceptor trap states (the first and the second terms of
Eq. 1.43), respectively. Rrec,n = krec,n nI pIt , and Rrec,p = krec,p pI nIt . Note that RSRH,A
has physical meanings similar to Rrec,p , so RSRH,A is expected to reduce to Rrec,p when
Cn

Cp , and vice versa for RSRH,D and Rrec,n . According to the results, Rrec,p is greatly

reduced at high voltage because pI becomes depleted from its exceedingly recombination
with the injected electrons. However, RSRH,A does not decrease from the lack of pI because
the unoccupied traps can still capture the interface electrons. A similar situation happens

96

to RSRH,D and thus explains the differences between RSRH,D and Rrec,n .
Surprisingly, even with the different fundamental physics of RSRH and Rf-t , the total
rates differ by just a factor of 2-3. According to Table 4.9 and Fig. 4.33b, the calculated JV
curves can be very close for Va < Voc especially in the devices with more balanced mobilities
and narrow tail states.
We have shown in this section that RSRH and Rf-t are different by their natures, but their
values are not far apart when Cn = krec,n and Cp = krec,p . However, they are considerably
different at Voc , so using the reduced forms RSRH = Cp Nt pI or RSRH = Cn Nt nI are not
recommended at this voltage. Additionally, the terms n1 , p1 and n1 p1 in the SRH rate (Eq.
1.35) are negligible due to the irrelevant thermal charge emission processes.

4.7

Conclusions

In this chapter, fundamental physics of bilayer OPVs are studied using the DD-BI and the
SCD models. The purpose is to gain more insights and make connections between the
fundamental physics and measurable Shockley parameters; Rs , Rp and n.
In Section 4.1, comparisons between our models and the GWWF model are made. The
advantages of the DD-BI and the SCD models are that they solve for spatial charge densities
and potential profile self-consistently, so they have better treatments of interface charge
densities, interface electric field, space charge effects and series resistance. On the other
hand, the GWWF model uses Equations 1.12 and 1.13 to estimate the interface electric field
which is an important argument in kP P d and ηP P d . However, Equations 1.12 and 1.13 do
not have accumulation of space charges and series resistance included, so the GWWF model
tends to overestimate the interface electric field, kP P d and ηP P d , resulting in overestimated
device efficiencies.
In Section 4.2, we present simulation results regarding effects of carrier mobilities on device efficiencies, charge transport and quasi-Fermi levels. First, we show that there are
two regions of spatial charge distribution: (1) exponential decay regions near the elec97

trodes; (2) constant density regions near the D/A interface, which are caused by space
charge accumulations due to low carrier mobilities. The constant charge regions give rise
to changes in quasi-Fermi levels and split the quasi-Fermi levels at the interface; which is
the diode voltage (Vd = Ef n,I − Ef p,I ), and bring about the definition of series resistance
(Ef n,I − Ef p,I = Va − JRs ). In terms of interface rates, we show that mobilities effect kP P d
the most, resulting in a reduction of dissociation efficiency (ηP P d ), which can be represented
by an increased parallel resistance.
In Section 4.3, effects injection barriers are presented. Increased injection barriers reduce
the device efficiency through reductions of fill factor and short-circuit current, but opencircuit voltage almost does not change. This is because interface charge densities; which
are directly related to open-circuit voltage, are not significantly affected by the increased
injection barriers.
One interesting feature that is observed in Section 4.3 is that injection barriers smaller
than about 0.3eV do not effect JV curves noticeably. In Section 4.4, we focus on this aspect
by considering organic semiconductor/metal contacts at the electrodes. The results show
that, under illumination, physics at the contacts are similar to that in the ICT model, i.e.
small injection barriers result in charge transfer and quasi-Fermi level pinning, but large
injection barriers result in significant drops of quasi-Fermi levels over the barriers. These
explain the JV behaviors caused by small injection barriers. However, under dark conditions, the contacts are more likely explained by the Mott-Schottkey model because charge
transfer and quasi-Fermi level pinning always occur. Ultimately, the knowledge of organic
semiconductor/metal contacts allows us to understand and formulate each component of
series resistance explicitly.
In Section 4.5, we study origins of ideality factors and show that our expressions of ideality
factors in Equation 2.17 and 2.18 are general forms of a famous ideality factor expression
[51]. We also show that ideality factor functions change accordingly with characteristic trap
energies and can be used as indicators of trap-assisted recombination.

98

In Section 4.6, the f-t recombination, which is understood as a reduced form of the SRH
recombination, is compared with the SRH recombination using exponential trap distributions. The results show that their recombination rates are comparable at applied voltage
below Voc , but they are actually different models and result in different JV curves especially
in terms of Voc ’s. This is because the f-t model only considers recombination with occupied
trap states, but the SRH model also includes capturing of free carriers by unoccupied trap
states.

99

CHAPTER 5
INTERPRETATION OF SHOCKLEY PARAMETERS : DARK VERSUS
LIGHT
Despite the fact that each property of a solar cell needs to be described by a complicated
function, using a set of constant Shockley parameters; J0 , Rs , Rp and n, is still one of the
most preferable methods for describing a solar cell device and its JV curves because it is
simple, yet sufficiently descriptive. However, the parameters are always used interchangeably
between illumination and dark conditions, despite the different physics of the two conditions.
A question then arises: are there any errors associated with using only one set of the parameters interchangeably? In order to answer this question, we simulate and compare functions
of the Shockley parameters under both conditions using our models, but first we have to
include another mechanism for leakage current and Rp in the dark because our illumination Rp model in Eq. 2.28, which is the recombination resistance of the recombined exciton
current; qJX (1 − ηP P d ), does not apply when JX = 0 in the dark.
This chapter starts with an introduction of a new type of solar cell from our collaborators;
organic salt solar cells. We then explain effects of the Shockley parameters on JV curves
under illumination and dark conditions using the standard Shockley equation (Eq. 1.3) with
constant parameters. After proposing possible leakage mechanisms, we apply the models
to one of the organic salt solar cells and the P3HT:PCBM device in Chapter 4, and then
discuss the differences of dark and illumination Shockley parameters.

5.1

Organic Salt Photovoltaic Devices

One particular bilayer system that is interesting because of its transparency and ability to
utilize near-infrared parts of the light spectrum is the organic salt solar cell [1, 120]. Our
collaborators in Department of Chemical Engineering and Materials Science, Michigan State
University, are focusing on this system, so their work in Ref.[1] will be used and analyzed in

100

this chapter.
The architecture of the device is a bilayer donor-acceptor so that the DD-BI and the SCD
models are applicable. The donor is composed of a heptamethine or cyanine cation (Cy+ )
and a small molecule anion. The molecular structures are shown in Fig. 5.1. There are many
−
choices of the anion, and PF−
6 , SbF6 , ∆-tris(tetrachloro-1,2 benzenediolato)phosphate(V)

(∆-TRISPHAT− or TRIS− ) and Tetrakis(pentafluorophenyl)borate (TPFB− ) are used in
Ref. [1]. The acceptor is a C60 film. A stack of indium tin oxide (ITO) and MoO3 serves as
the anode and Ag is the cathode.

Figure 5.1: a) Molecular structures of the heptamethine cation (Cy+ ) and b) its counter
−
−
−
anions: (from left to right) I− ; PF−
6 ; SbF6 ; TRIS ; and TPFB , that are used as the donor
in the device with the architecture shown in c). From Ref. [1]
Figure 5.2 shows JV characteristics, measured under nominal 1 sun and dark conditions.
Solar cell efficiency under illumination is improving from CyPF6 to CySbF6 to CyTRIS and
to CyTPFB. According to the paper[1], there are several possible reasons for the improvement
including: (1) the donor HOMO and LUMO levels are tuned so they are more favorable,
(2) the interface size is increased, abating polaron pair recombination, (3) exciton diffusion
length increases and (4) trap states are not very dominating. The CyTRIS and especially
the CyTPFB devices also show kink-shaped behaviors in their dark JV curves. Because the
HOMO and LUMO levels of the CyTRIS:C60 device (see supplementary materials of Ref.
101

Figure 5.2: JV curves under a) nominal 1 sun illumination and b) dark condition from
measurements (symbols) and fitting with the generalized Shockley equation (Eq. 1.3)
[1]) are suitable for simulating with the DD-BI and the SCD models, the device will be fitted,
so the Shockley parameters are extracted and analyzed.

5.2

Effects of the Shockley Parameters on Dark JV curves

In the upcoming sections, the analyses of dark currents require discussions in terms of the
Shockley parameters, namely, J0 , Rs , Rp and n, and how they affect the diode and the
leakage currents. In addition to their physical meanings that are discussed in detail in
Chapters 2 and 4, simple understanding of their effects on the shapes of dark currents is
important for analyzing experimental JV data. This section presents how the shapes of dark
currents are changed by those parameters using graphical methods.
Figure 5.3 shows impacts of the Shockley parameters on dark currents’ shapes calculated
by the conventional Shockley equation in Eq. 1.3. The values of the parameters are typical
for the cyanine organic salt solar cells[1].
102

-2

2

2

J0 = 1E-7 Am , Rs = 9E-4 Ωm , n = 1.5, T = 300 K

Reducing Rp

2

Rp = 3E6 Ωm

2

Rp = 3E5 Ωm

-2

10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10-0.5

4

J (A m )

-2

J (A m )

3

2

Rp = 3E4 Ωm

2

Rp = 3E3 Ωm

2

Rp = 3E2 Ωm

2

Rp = 3E1 Ωm

2

Rp = 3E0 Ωm

0

0.5

1

1.5

Voltage (V)

10
3
10
2
10
1
10
Increasing J0
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10-0.5
0

(a) Effects of Rp
-2

2

5

-2

J (A m )

-2

J (A m )

Increasing n

n = 1.00
n = 1.25
n = 1.50
n = 1.75
n = 2.00
n = 2.25
0.5

Voltage (V)

1

0.5

Voltage (V)

-2

2

4

0

-2

J0 = 1E-8 Am
-2
J0 = 1E-7 Am
-2
J0 = 1E-6 Am
-2
J0 = 1E-5 Am
-2
J0 = 1E-4 Am
-2
J0 = 1E-3 Am

1.5

(b) Effects of J0

J0 = 1E-7 Am , Rs = 9E-4 Ωm , Rp = 3E3 Ωm , T = 300 K

10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10-0.5

2

Rs = 9E-4 Ωm , Rp = 3E2 Ωm , n = 1.5, T = 300 K

1

1.5

(c) Effects of n

2

J0 = 1E-7 Am , Rp = 3E2 Ωm , n = 1.5, T = 300 K

10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10-0.5

2

Rs = 1E-5 Ω m
2
Rs = 1E-4 Ω m
2
Rs = 1E-3 Ω m
2
Rs = 1E-2 Ω m
2
Rs = 1E-1 Ω m
2
Rs = 1.0 Ω m
Increasing Rs

0

0.5

Voltage (V)

1

1.5

(d) Effects of Rs

Figure 5.3: Effects of the Shockley parameters a) Rp , b) J0 , c) n and d) Rs on the shapes
of dark JV curves calculated by the conventional Shockley equation (Eq. 1.3) using each
parameter as a constant. Control variables are summarized at the top of each plot.
In order to understand the Rp effects in Figure 5.3a where Rp varies from 3 × 106 Ωm2
(high Rp ) to 3 Ωm2 (low Rp ), consider the total current (J) as a summation of the diode
current; Jdiode = J0 [exp(q(Va − JRs )/kB T ) − 1], and the leakage current; Jleak = (Va −
JRs )/Rp . They are plotted in Fig. 5.4 for the cases of Rp = 3 × 105 and 3 × 101 Ωm2
as examples. Because Jdiode exponentially depends on the voltage while Jleak is a linear
function of it, Jleak is comparable to Jdiode only when the voltage is low or reversed. Lowering Rp increases Jleak . Since Jdiode is not affected by Rp , reducing Rp is like vertically

103

lifting the Jleak graph up while keeping the Jdiode graph unchanged. This increases the total
current at low and reverse voltage where Jleak dominates, so that the overall JV curve is
kink-shaped.
10

3

2

10

1

10

0

10

10
10

J
Jdiode
Jleak

1

10

0

10

-1

10

-3

-4

10
10

-2

-2

10
10

J (A m )

-2

J (A m )

-1

10

10
10

-2

10
10

-5

10

-7

-3

-4

10

-6

-5

-6
-7

10

10

-8

10-0.5

3

2

-8

0

0.5

1

1.5

10-0.5

Voltage (V)

0

0.5

1

1.5

Voltage (V)

(a) Rp = 3 × 105 Ωm2 , low leakage current

(b) Rp = 3 × 101 Ωm2 , high leakage current

Figure 5.4: Plots of the total dark currents (J), the diode currents (Jdiode ) and the leakage
currents (Jleak ) where J = Jdiode + Jleak as a function of voltage for the cases of a) Rp =
3 × 105 and b) 3 × 101 Ωm2
For the effects of J0 , consider Figure 5.3b where J0 varies from 1×10−3 to 1×10−8 Am−2 .
Since J0 is the prefactor of the diode term, it simply determines the magnitude Jdiode but
does not change its slope or the leakage current. Graphically, J0 can be viewed as a shifting
parameter of Jdiode . Increasing J0 is like shifting the diode current up along the y-axis
and toward the origin of the x-axis. If Jdiode at low voltage is high enough, it becomes the
dominant current, so the kink-shaped behavior caused by Rp cannot be seen.
The ideality factor n is another parameter that affects Jdiode , but in a different way.
Figure 5.3c shows dark JV curves having different n values ranging from 1 to 2.25. Assuming
that the diode voltage; Vd = Va − JRs , is a linear function of Va , so Vd = aVa + b, where a
and b are constant. The slope of the logarithmic diode current can be approximated by
∂ln(Jdiode )/∂Va ≈ ∂ln[J0 exp(qVd /nkB T )]/∂Va ≈ qa/nkB T . Hence, n−1 is associated with
the slope of dark current in the region where Jdiode

Jleak and dominates the curve. n

can be graphically viewed as a tilting parameter of Jdiode . As seen in Fig. 5.3c, the dark

104

JV curves are tilted downward by the higher values of n, while the reverse currents or the
kink-shaped characteristics are less affected
It is clear from Fig. 5.3d that Rs is very important at high voltage. In order to understand
this, consider the Shockley equation in Eq. 1.3 along with its equivalent circuit in Fig. 1.2.
At low or reverse bias, the diode is not conductive, so it can be excluded from the circuit.
Rp is then connected to Rs in series, which is negligible due to Rp
voltage, the diode is conductive, so Jdiode

Rs . At high applied

Jleak . The shunt path with Rp that conducts

Jleak can be neglected. The circuit is reduced to the diode connected in series with Rs . If
the diode is fully conducting, the entire circuit is reduced to just the biased Rs . Hence,
Ohm’s law applies and the dark current is J ≈ Va /Rs . In a graphical sense, Rs is most
important after the Jdiode dominating regime, by vertically lifting/lowering the total dark
current curves.
A dark JV curve can have up to three regions with different slopes[5]. For example, Fig.
5.4b shows different slopes in the ranges: Va < 0.5V ; 0.5V < Va < 1.0V ; and Va > 1.0V .
According to the discussions above, the slope at Va < 0.5V belongs to Jleak . The slope at
0.5V < Va < 1.0V is governed by Jdiode . For Va > 1.0V , the current obeys Ohm’s law, so
that the slope is [∂J/∂Va ]−1 ≈ Rs .
In conclusion, a dark JV curve can be divided into the leakage region, the diode region,
and the Ohmic region, as illustrated by Figure 5.5. In the leakage region, the curve is lifted
or lowered by Rp , causing the kink-shaped characteristic if Rp is low enough. In the diode
region, the magnitude of the graph is controlled by J0 , and the slope is controlled by n. J0
and n can be graphically viewed as shifting and tilting parameters, respectively. Finally, Rs
affects dark currents at very high Va by lowering or raising the currents according to Ohm’s
law.

105

Figure 5.5: A figure showing the three regions in a dark current, namely, the leakage region,
the diode region, and the Ohmic region, including how the shape in each region is affected
by the Shockley parameters. adapted from [5]

5.3

Effects of the Shockley Parameters on illuminated JV curves

JV curves under illumination can also provide basic information about the Shockley parameters and physical properties, but not as clearly and informatively as dark JV curves. Figure
5.6 shows the effects of the parameters Rp , J0 , n and Rs on illuminated JV curves calculated
by the conventional Shockley equation (Eq. 1.3). The values of the control variables are the
same as in Section 5.2 where the parameters’ effects are studied under dark conditions. An
additional constant photo-current term (Jph ) equal to 70Am−2 is added in all the plots to
represent the photogeneration processes, so J = Jdiode + Jleak − Jph .
There are a couple of interesting points about Rp in Fig. 5.6a. First, Rp under illumination has to be very low compared to Rp under dark in order to fit this data to the Shockley
model. The dark Rp values in Fig. 5.3a are typical for organic solar cells, which are about
tens to hundreds Ωm2 [1, 121, 122], but Rp under illumination in Fig. 5.6a does not show
any effects unless it is lower than 10−2 Ωm2 (i.e. about four orders of magnitude lower than
those in the dark). This topic will be examined thoroughly in Sections 5.4 and 5.5. The

106

-2

400

2

600

J0 = 1E-7 Am

2

500

J0 = 1E-6 Am

2

400

J0 = 1E-5 Am

300

J0 = 1E-4 Am

Rp = 3E-3 Ωm
Rp = 6E-3 Ωm

-2

-2

J (A m )

600

2

700

J (A m )

800

2

Rs = 9E-4 Ωm , Rp = 3E2 Ωm , n = 1.5, T = 300 K

2

J0 = 1E-7 Am , Rs = 9E-4 Ωm , n = 1.5, T = 300 K

Rp = 3E-2 Ωm

2

Rp = 3E-1 Ωm , and higher

200

-2
-2
-2
-2

200
100

0

0
-200
-0.5

0

0.5

1

-100
-0.5

1.5

Voltage (V)
(a) Effects of Rp

0

0.5

(b) Effects of J0
-2

-2

2

J0 = 1E-7 Am , Rs = 9E-4 Ωm , Rp = 3E3 Ωm , T = 300 K

600

400
-2

400

2

500

n = 1.00
n = 1.25
n = 1.50
n = 1.75
n = 2.00

J (A m )

-2

J (A m )

500

300
200

300
200
100

100

Rs = 1.0E-4 Ωm

2

= 1.0E-3 Ωm
2
= 2.5E-3 Ωm
2
= 5.0E-3 Ωm
2
= 7.5E-3 Ωm
2
= 1.0E-2 Ωm
Increasing Rs

0

0
-100
-0.5

2

J0 = 1E-7 Am , Rp = 3E2 Ωm , n = 1.25, T = 300 K

2

700
600

1

Voltage (V)

0

0.5

Voltage (V)

1

1.5

(c) Effects of n

-100
-0.5

0

0.5

1

Voltage (V)
(d) Effects of Rs

Figure 5.6: Effects of the Shockley parameters a) Rp , b) J0 , c) n and d) Rs on the shapes
of illuminated JV curves calculated by the conventional Shockley equation (Eq. 1.3) using
each parameter as a constant. Control variables are summarized at the top of each plot.
conclusion is that parallel resistances under dark and illumination conditions arise from two
different physical processes; the dark Rp is connected to trap-related tunneling processes
which are insignificant compared to the illuminated Rp that virtually depicts the resistivity
of the recombined exciton current (see Sections 2.2 and 4.2.3). Secondly, the slope at around
the short circuit point is affected the most by Rp , as shown in Fig. 5.6a. This can be a
helpful guide in determining properties of an actual device or possible ranges of parameters.
For example, if the slope at Va = 0V is very flat, one could conclude that the rate kP P r is

107

not very high, and thus the PP dissociation probability (ηP P d ) should approach a hundred
percent. kP P r can be set at around (1-100 µs)−1 in this case. On the other hand, if the
slope is very steep, kP P r can be as high as (10-100 ns)−1 , making the PP recombination
the main loss mechanism.
According to Figs. 5.6b and 5.6c, effects of J0 and n are not easily distinguishable, in
contrast to the dark cases where n controls the slope of the diode current and J0 shifts the
current toward the graph origin. We can only deduce that higher J0 and lower n increase
the diode current, so it responds at lower voltage. More information about n and J0 requires
a fitting.
Finally, Rs is the resistance that reduces the slope around Voc and higher, which accordingly reduces fill factor. The reasons why Rs affects at high voltage are discussed in the
previous section. The behavior shown in Fig. 5.6d agrees with the numerical results in Figs.
4.6a and 4.15a where Rs is changed by changing the mobility and the contact component of
Rs , respectively.
In summary, even though the reactions of illuminated JV curves to the changes of the
Shockley parameters are not as clear as for dark JV curves’, the slopes at Va = 0V and at
Voc which are affected by Rp and Rs can be simple clues of related physical parameters and
properties, such as, PP recombination rate, mobility and injection barriers.

5.4

Parallel Resistance in the Dark

The experimental dark JV curves of the CyTRIS and CyTPFB devices in Fig. 5.2 are
particularly interesting because they clearly show the kink-shaped behavior. According to
Section 5.2, the kink-shaped trait is caused by low J0 and low Rp , so that the total current
in the small and reverse bias regime is dominated by the leakage current. However, the
mechanisms governing the leakage current and Rp under dark conditions have not been
included in the models yet. According to the Rp expression (Eq. 2.28), Rp is a recombination
due to a fraction of the current (qJX (1 − ηP P d )) that is lost at the D/A interface. This
108

definition of Rp does not apply to devices under dark conditions because it predicts that
there is no leakage current when JX = 0.
In the following sections, mechanisms for the dark leakage current are proposed and
added to the numerical models. Numerical JV curves containing these mechanisms will be
compared to the experimental results for the CyTRIS case.

5.4.1

Trap-to-trap Recombination as the Dark Leakage Current

Charge tunneling and trap states have been demonstrated to be the primary cause of the
leakage current in inorganic semiconductor devices[123, 124, 125, 126, 127]. If there are
trap states in the forbidden energy gap, charges from one side of the semiconductor do
not have to tunnel through the whole thickness but to the trap states that can capture
the tunneling charges and emit them to the other side of the semiconductor. This is the
well established trap-assisted tunneling (TAT) theory[125, 126, 127], which is necessary in
reproducing the low-bias JV curves in computational simulations[128, 129], where the leakage
current dominates. Recently, the TAT model was used to describe the tunneling leakage
current in a thin organic layer[130], implying that trap states and tunneling processes play
important roles in the organic devices’ leakage current.
In addition to the f-f and the f-t recombination mechanisms that are discussed extensively
in Chapters 2 and 4, recombination between localized or trap states can be taken into
consideration[84, 87]. Though some studies suggest that this recombination should not
happen[87], we will show that it results in a dark leakage current that has similar behaviors
to those of the dark leakage current caused by Rp in the Shockley model. The recombination
also brings about an extra parallel resistance in the SCD equations that has values close to
experimentally extracted Rp values from dark JV curves.
Here, we model the recombination between localized states in a similar way to that in
Ref.[84]. We consider the tunneling and recombination of trapped charges through the D/A
interface and that the traps have an effective mobility µtun . Using the same notation as in

109

Ref.[84], the trap-to-trap (t-t) recombination can be modeled as a bimolecular recombination
between nIt and pIt with the mobility µtun ;
Rt-t =

qµtun
0

nIt pIt ,

(5.1)

where µtun is used as a constant and a fitting parameter, for simplicity.

5.4.2

Simulating the CyTRIS:C60 Device

We choose to simulate the CyTRIS:C60 device (Fig. 5.1) to study the effects of the t-t
recombination and dark Rp . Figure 5.7a shows the energy band diagram of each layer and
the device architecture; which is indium tin oxide (ITO) (100nm)/MoO3 (10nm)/CyTRIS
(12 nm)/ C60 (40 nm)/bathocuproine (BCP) (7.5 nm)/Ag (100 nm). The CyTRIS and C60
layers are the active layers that convert photons into free electrons and holes. MoO3 and
BCP work as the protective inter-layers between the active layers and the electrodes ( i.e.
ITO and Ag).
MoO3 has deep-lying electronic states; the conduction band minimum and the valence
band maximum are at 6.7eV and 9.68eV below the vacuum level respectively [6]. Including
a thin MoO3 layer helps improving hole injection from anode[6]. The mechanism, which is
illustrated in Fig. 5.7b, is that charge transfer between ITO and MoO3 gives rise to the
Fermi-level pining between the two layers and a vacuum level shift (∆1 ), making the MoO3
a strongly n-type semiconductor. Once an ITO/MoO3 layer is brought into contact with
a donor (CyTRIS in this case), instead of injecting holes directly from the ITO, electrons
are injected from the donor HOMO to the MoO3 LUMO, leaving holes in the donor HOMO
bands[6]. As a result, an ohmic contact and another vacuum level shift (∆2 ) occur at the
interface between ITO/MoO3 and CyTRIS, making the quasi-Fermi level aligned in all three
layers. Based on Section 4.4, we model the ITO/MoO3 /CyTRIS contacts as an injecting
contact with a small injection barrier.
On the other hand, a thin BCP layer is usually deposited between the electron acceptor
(C60 ) and the metal cathode to prevent thermal damage during the cathode deposition and
110

(b) Fermi-level pinning at ITO/
MoO3 / CyTRIS interfaces

(a) Energy band diagram

Figure 5.7: (a) Energy band diagram and the thickness of each layer in the CyTRIS:C60
device. (b) Fermi-level pinning, vacuum level shifts, and possible hole injecting mechanism
at the ITO/MoO3 /CyTRIS interfaces ( adapted from [6]).
to serve as an exciton blocking layer[131]. Based on the energy band diagrams in Fig. 5.7a,
electrons cannot transport through the LUMO of BCP because of the high barrier between
C60 and BCP. However, chemical interaction between BCP and the metal cathode results
in induced transport levels and trap states in the BCP gap that electrons can transport
through [132]. For simplicity, we model the BCP/Ag layer as an injecting cathode with the
work function close to that of Ag.
Simulations for the CyTRIS:C60 device are performed with the parameters listed in
Table 5.1. µn and µp are the estimated mobilities of electrons in C60 and holes in CyTRIS,
respectively. µtun is empirically chosen to be around 10−9 to 10−11 m2 /V s. The dielectric
constant is estimated from the real (n ) and imaginary (k) parts of the refractive indices
shown in the supplementary material of Ref. [1], using the relation [133]
= 0 (n 2 − k 2 ).

111

(5.2)

The total recombination rate is a summation of the SRH recombination with the exponential
trap distributions summarized in Section 1.8 and Rt-t in Eq. 5.1.
Parameter
µp
µn
µtun
Donor/acceptor thickness
φa /φc
Tt,D = Tt,A
kP P r,0
Donor ELU M O /EHOM O
Acceptor ELU M O /EHOM O
NHOM O /NLU M O
HD /HA
JX
a0 = δx
recombination type

Value
1.0 × 10−7 m2 /V s
1.0 × 10−6 m2 /V s
0, 10−9 − ×10−11 m2 /V s
12nm/40nm[1]
0.20 eV/0.25 eV
4.2 0
1200K
5 × 105 s−1
-4.9eV/-3.6eV [1]
-6.6eV/-4.0eV [1]
3.2E24 m−3
2.0E23 m−3
3.5E20 m−2 s−1
1.0nm
Shockley-Read-Hall + trap-to-trap

Table 5.1: Summary of parameters used to study effects of the trap-to-trap recombination
(Rt-t ) in terms of the effective mobility µtunn in the CyTRIS:C60 device[1].
The simulation results under illumination and dark conditions are respectively plotted in
Figs. 5.8a and 5.8b including the experimental data for comparisons. For the illumination
cases, µtun from 10−9 to 10−11 m2 /V s does not have any noticeable effects on the JV curves,
and they agree well with the experimental one for the whole range of Va . For the dark cases,
the simulated curves in the range Va > 0.3V , i.e. the diode regime, are consistent with the
experimental result, with µtun ≈ 5 × 10−10 m2 /V s giving a reasonable fit at lower Va . The
curves in the reverse bias, however, are quite erroneous as they are flatter and lower than the
measured values. Based on the discussions in section 5.2, negative- and low-voltage current
is governed by the dark leakage current which is separable from the diode current (see Fig.
5.4). Since the diode term contains all the physical parameters, and the graphs agree well in
the diode dominating regime, the parameters in Table 5.1 are good approximations for the
actual ones.
Including Rt-t influences JV curves in identical ways to including the dark Rp . This can
112

3

10

150

0

10

2

50

Increasing µtun
Experimental

-2

J (A m )

Simulated for µtun = 1E-11 to 1E-9 m /Vs
Experimental

-2

J (A m )

100

-3

2

10

µtun = 1E-9 m /Vs
2

µtun = 5E-10 m /Vs
2

-6

µtun = 1E-10 m /Vs

10

0

2

µtun = 1E-11 m /Vs
no Rt-t

-9

10

-50
-0.5

0

0

-0.5

0.5

0.5

1

Voltage (V)

Voltage (V)

(a) Illumination

(b) Dark

Figure 5.8: Comparisons between the simulated JV curves of the CyTRIS:C60 device and
the experimental data under a) illumination and b) dark conditions
be shown by the following explanations. Firstly, Rt-t graphically affects the illuminated and
dark JV curves in Figs. 5.8a and 5.8b analogously to those in Figs. 5.6a and 5.3a, i.e. it
does not affect the illuminated JV curves but affects the low- and negative-voltage parts of
the dark currents.
Secondly, an extra current term originated from Rt-t has similar properties to the leakage
current. Because we use the integral form of RSRH (Eq. 1.43), in order to separate currents
from different origins, the general GWWF JV relation in the analytic form (Eq. 1.11) has
to be used instead of Eq. 3.3. Using R = RSRH + Rt-t in Eq. 1.11 yields
J=

=

k
RSRH
− PPd
RSRH,eq kP P d,eq
k
+qa0 (1 − ηP P d ) Rt-t − P P d Rt-t,eq − qJX ηP P d
kP P d,eq
qa0 (1 − ηP P d )RSRH,eq

JSRH + Jtun − qJX ηP P d ,

(5.3)

where
Jtun = qa0 (1 − ηP P d )(Rt-t −

kP P d
kP P d,eq

Rt-t,eq )

(5.4)

is the tunneling current form Rt-t , and each parameter is a function of Va and taken from
numerical simulations. This equation results in exactly the same currents as numerically
calculated by Eq. 3.3. Figure 5.9 shows plots of J, JSRH and Jtun of the devices with µtun =
113

1×10−11 and 5×10−10 m2 /V s under dark conditions. The diode-like JSRH is invariant upon
changing µtun and equals to the total current curve when Rt-t is excluded. The shapes of
Jtun in both graphs are also unchanged but shifted up by the higher µtun . These behaviors
are similar to Fig. 5.4 when Jleak is lifted by reducing the dark Rp .
2

2

10

0

10

-2

0

10
-2

J (A m )

-2

J (A m )

10
10

10

JSRH
Jtun
J = JSRH + Jt-t

-4
-6

10

-2
-4
-6

10

10

-8

-8

10

10

-10

10

10

-10

-0.4

-0.2

0
0.2
0.4
Voltage (V)

0.6

0.8

(a) µtun = 1.0 × 10−11 m2 /V s

10

-0.4

-0.2

0
0.2
0.4
Voltage (V)

0.6

0.8

(b) µtun = 5.0 × 10−10 m2 /V s

Figure 5.9: Plots of the currents originated from RSRH (JSRH ) and Rtt (Jtun ) defined in
Eqs. 5.3 and 5.4 and the total dark currents J = JSRH + Jtun for the devices with µtun
equals to a) 1.0 × 10−11 and b) 5.0 × 10−10 m2 /V s.
Thirdly, if the current Jtun is considered as the dark leakage current flowing through a
parallel resistance Rp_tun , the Rp_tun value will be in accordance with the experimental
one. Using the same definition of Rp as in Section 2.2, i.e. that it is biased by the interface
voltage difference Ef n,I − Ef p,I yields
Rp_tun =

Ef n,I − Ef p,I
.
Jtun

(5.5)

A plot of Rp_tun as a function of voltage in the device with µtun = 5 × 10−10 Ωm2 is shown
in Figure 5.10. In the range 0.0V < Va < 0.3V where the leakage current Jtun caused by
Rp_tun dominates, the resistance values are around hundreds to a thousand of Ωm2 which
agree well with the extracted value of 1,920 Ωm2 from the actual device.[1]
Therefore, it can be concluded that the trap-to-trap recombination can describe the
trap-related tunneling processes that cause the dark leakage current. Rt-t gives rise to Jtun

114

4

10

3

2

Rp_tun (Ω m )

10

2

10

1

10

0

10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage (V)
Figure 5.10: Plot of the parallel resistance Rp_tun calculated by Eq. 5.5 as a function of
applied voltage in the device with µtun = 5.0 × 10− 10m2 /V s
that behaves similarly to the dark leakage current and Rp_tun that has values close to the
typically measured parallel resistance in the dark.
However, Rt-t and µtun oversimplify the actual tunneling processes, and cannot reproduce reverse-bias dark JV curves. More sophisticated theories have to be applied. For
example, using the Miller-Abrahams rate[91] or a more correct form of charge hopping rate
between localized states[134] in the master equation[135] to derive the mobility across the
D/A interface is one possibility. The second possibility is to consider charge tunneling
through the whole donor layer which is only 12-nanometer thick, by using the TAT current and energy-level-dependent tunneling coefficient[130]. Additionally, one could use the
electric-field-dependent mobility[136] in the modified SRH recombination to include the TAT
processes[126, 127]. These topics are useful directions for future work.

5.4.3

Including a Constant Dark Parallel Resistance in the DD-BI and the SCD
models.

In this part, we propose another method of including a leakage current under dark conditions
(Jleak,dark ) to the DD-BI and the SCD models. Here, we use the Shockley form (Eq. 1.3)

115

of the leakage current;
Jleak,dark =

Va − JRs Ef n,I − Ef p,I
=
,
Rp,dark
Rp,dark

(5.6)

where the parallel resistance Rp,dark is manually included.
Including Eq. 5.6 in the SCD equations (Eqs. 2.5, 2.14, or 2.22) is straightforward. For
the case of the SRH recombination, the SCD equation (Eq. 2.22) becomes
J = Js,SRH exp

Ef n,I − Ef p,I
nSRH kB T

− 1 − qJX ηP P d +

Ef n,I − Ef p,I
.
Rp,dark

(5.7)

To include the dark leakage current in the self-consistent DD-BI model, some modifications
have to be made to the model. First, the total current J in the GWWF interface model
(Fig. 1.5) has to include Jleak,dark by simply replacing J with J − Jleak,dark . According to
the rate equation 1.10, the interface free carrier generation rate (U) is modified to
U = kP P d ζ − R −

Ef n,I − Ef p,I
Jleak,dark
= kP P d ζ − R −
qa0
qa0 Rp,dark

(5.8)

to cover the loss from the leakage current. Hence, Ef n,I − Ef p,I has to be updated after
solving the Poisson equation in the computer algorithm (Fig. 3.1). This can be done by
using Equations 1.19 and 1.20 to express Ef n,I − Ef p,I in terms of the interface densities
(nI and pI ), i.e.
Ef n,I − Ef p,I = ∆EHL − FI a0 + Vt ln

nI pI
,
NLU M O NHOM O

(5.9)

where the potential drop across the D/A interface (FI a0 ) is included.
The CyTRIS:C60 device is simulated, using the same parameters as in Table 5.1 except
that the t-t recombination is replaced by the leakage processes through the constant Rp,dark ,
which is chosen to be 80 to 8 × 104 Ωm2 . Figure 5.11 shows the simulation results, and for
comparison purposes, JV curves calculated by the generalized Shockley equation (Eq. 1.3)
using fitted Shockley parameters from Ref. [1]: J0 = 1.3 × 10−6 A/m2 ; Rs = 9 × 10−4 Ωm2 ;
Rp = 1920 Ωm2 ; Jph ≈ Jsc = 48A/m2 ; and n = 1.34 and 1.36 under illumination and dark
conditions respectively, are also included. Note that these parameters are extracted from
116

2

10

160

Experimental

120

DD-BI, SCD with
2
Rp,dark = 8E1 to 8E4 Ω m

80

0

10
-2

J (Am )

-2

J (Am )

200

Fitted Shockley equation
from Ref.[1]

40

10
10

Rp,dark = 8E1 Ω m
Rp,dark = 8E2 Ω m

-4

Rp,dark = 8E3 Ω m

2
2
2
2

0

Rp,dark = 8E4 Ω m
Fitted Shockley eq.
from Ref.[1]

-6

10

-40
-80

-2

Experimental
Reducing Rp,dark

-8

-0.4

-0.2

0

0.2

0.4

Va (V)

0.6

0.8

1

10

-0.4 -0.2

(a) Illumination

0

0.2

0.4

Va (V)

0.6

0.8

1

1.2

(b) Dark

Figure 5.11: Comparisons of JV curves of the CyTRIS:C60 device under (a) illumination
and (b) dark conditions between the experimental results (circle symbol), the DD-BI and
the SCD results (lines with a symbol) and the Shockley equation’s results (Eq. 1.3) (solid
lines) using the fitted parameters from Ref. [1].
the measured dark curve, but they are used to describe the device both under dark and
illumination conditions.
Following points can be deduced from the comparisons. First, the graphs calculated by
the SRH SCD equation with Rp,dark included are consistent with the DD-BI simulations
in all the cases of Rp,dark . The DD-BI and the SCD JV curves respond to the change of
Rp,dark similarly to the Shockley equation results in Sections 5.2 and 5.3. Second, according
to the results in the dark, the DD-BI/SCD curve with Rp,dark ≈ 800Ωm2 fits reasonably
well with the experimental one, and is very close to the standard Shockley curve using the
parameters from Ref. [1]. However, both plots do not agree well with the experimental result
in the reverse bias regime, where the actual slope is steeper. This implies that a more correct
description of Jleak,dark is needed. Thirdly, the Shockley equation using only the dark Rp
cannot reproduce the illumination JV curve because the leakage mechanisms under the two
conditions are different. Two Rp values are needed, and the one under illumination has to be
much lower than the one in the dark in order to have a leaking photo-current and a sloping
JV at low Va . This topic and the errors of using only one set of Shockley parameters under
both conditions will be discussed in the upcoming sections.
117

5.5

Complete Equivalent Electrical Circuit Model

With more insights of Rp in the dark and the physical definitions of the Shockley parameters
in Chapter 2 and 4, we now have a complete picture of the equivalent electrical circuit model
(Complete EEC) in Fig. 5.12.

Figure 5.12: A schematic showing the bilayer OPV architecture with each important physical
mechanism being represented by an electrical circuit component. All the electrical components form the complete EEC model.
First of all, we would like to emphasize that each electrical component does represent
an actual physical mechanism. Starting from the top of Fig. 5.12, series resistance portrays
the drops of quasi-Fermi levels caused by the anode contact, bulk donor, bulk acceptor
and the cathode contact, so there are four Rs components (See Section 4.4.5). There are
three main processes at the D/A interface, from left to right: (1) photo-generation processes
enclosed by the gray box; (2) free-carrier recombination pathways with a diode representing
each pathway; (3) the leakage in the dark which is possibly caused by the trap-related
118

tunneling. For process (1), the current source (the circle symbol with an arrow) produces
a constant current of qJX , but qJX (1 − ηP P d ) recombines as if it leaks through a virtual
parallel resistance (Rp_P P ) so that only qJX ηP P d comes out as the voltage-dependent photocurrent. If the device is under dark conditions, the whole gray box disappears, leaving only
process (3) governing the dark leakage current. For process (2), if the recombination is
RSRH , one diode suffices. If RSRH is separated into Rfn-tp and Rfp-tn , there have to be
two diodes, but they may not conduct currents equally. An extra diode is included if the
f-f recombination is taken into account. Lastly, the clear evidences of dark leakage current
(Jleak,dark ) that are discussed in Section 5.4 bring about another parallel resistance which
is denoted by Rp,dark (or Rp,tun in Eq. 5.5 if the leakage mechanism is described by traprelated tunneling processes).
A complete general JV equation in the same form as the Shockley equation can be written
based on the Complete EEC model;
J

=

Jdiode,i + Jleak + Jleak,dark − qJX
i

=

Jsi exp
i

Ef n,I − Ef p,I
ni kB T

−1 +

Ef n,I − Ef p,I Ef n,I − Ef p,I
+
− qJX (, 5.10)
Rp_P P
Rp,dark

where i runs over the free-carrier recombination types (i.e. SRH or fn-tp + fp-tn and/or f-f),
Jleak = qJX (1 − ηP P d ) is the PP leakage current, Jleak,dark is the dark leakage current that
is usually smaller than Jleak , and Jsi is the diode saturated current for the recombination
type i (Ri ). If an analytic form for Ri is known, Jsi can be written explicitly, for instance,
Eq. 2.6 for Rf-f , Eq. 2.15 for Rfn-tp and Eq. 2.16 for Rfp-tn . If it is not known, Jsi can
always be written in the general form as
Jsi = qa0 (1 − ηP P d )Ri,eq ,

(5.11)

according to Eq. 2.1. ni is the ideality factor associated with the recombination Ri . For the
f-f recombination, n = 1. nD in Eq. 2.17 and nA in Eq. 2.18 are the ideality factors for the
fn-tp and the fp-tn recombinations respectively. If the SRH recombination is used, Eq. 2.23
can be used to calculate the SRH ideality factor function.
119

5.5.1

Total Parallel Resistance under Illumination Conditions

We can see that there are two parallel resistances (Rp_P P and Rp,dark ) if the device is under
illumination conditions. In this section, the total parallel resistance is explored.
Figure 5.13 shows Rs , Rp_P P , Rp,dark = Rp_tun as a function of applied voltage calculated by Eqs. 2.27, 2.28 and 5.5 as well as the total resistance R = [∂J/∂Va ]−1 , using
µtun = 5 × 10−10 Ωm2 from Section 5.4. According to the results, Rp_P P is much lower than
Rp_tun , so that the total parallel resistance under illumination is
Rp,illu =

1
Rp_pp

+

−1

1
Rp_tun

(5.12)

≈ Rp_P P ,

i.e. Rp,dark (or Rp_tun ) and the possible leakage processes in Section 5.4 can be ignored
altogether under illumination conditions, unless the light intensity is low enough so that
the leakage photo-current (qJX (1 − ηP P s )) is comparable to Jleak,dark [137]. The values of
Rp_P P in the reverse- and small-bias regimes are around 0.1 Ωm2 which agrees with the
total Rp under illumination predicted by the Shockley equation in Fig. 5.6a. The graph
of R = [∂J/∂Va ]−1 also points to the same conclusion. It converges to Rp_P P at highly
negative voltage where Rp dominates.

Rs
Rp_PP
Rp_tun

2

Resistances (Ω m )

100

[dJ/dVa]

1

-1

0.01

0.0001

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Voltage : Va (V)

Figure 5.13: A comparison between Rs , Rp_P P , Rp_tun and total device resistance calculated by Eqs. 2.27, 2.28, 5.5 and [∂J/∂Va ]−1 respectively. The device is from Section 5.4
with µtun = 5 × 10−10 m2 /V s and under illumination conditions

120

5.6

Shockley Parameters: Dark Versus Light

We have seen in Sections 4.2 and 4.4 that physics of OPVs under dark and illumination conditions are different because there are many more free carriers under illumination conditions
than in the dark. Nevertheless, only one set of extracted Shockley parameters; J0 , n, Rs and
Rp , is usually reported and used to describe JV curves under both conditions. In this part,
we compare the differences between the Shockley parameters under the two conditions, and
discuss errors of using the parameters interchangeably.
Here, we choose to simulate the CyTRIS:C60 device using the parameters shown in Table
5.1 with a constant Rp,dark = 800Ωm2 as proposed in Section 5.4.3, and the P3HT:PCBM
device using the same parameters as in Table 4.5, but the recombination is replaced by
the SRH recombination (Eq. 1.43) and a constant Rp,dark = 100Ωm2 is included. We use
this value of Rp,dark because it results in Jleak,dark that has similar magnitude to those in
literature [138, 139]. Simulating the two devices provides us conclusions that can be applied
to a variety of OPV systems.
3

100

10

2

CyTRIS:C60
P3HT:PCBM

75

1

10

50

0

10
-2

J (Am )

-2

J (Am )

CyTRIS:C60
P3HT:PCBM

10

25
0

-1

10

-2

10

-3

10

-25

-4

10

-50
-75

-5

10

-6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(a) Illumination

10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(b) Dark

Figure 5.14: JV curves of the CyTRIS:C60 (black lines with circles) and the P3HT:PCBM
(red lines with squares) devices under (a) illumination and (b) dark conditions.
Figure 5.14 shows JV curves under illumination and dark conditions of the CyTRIS:C60
and P3HT:PCBM devices. The following are comparisons of J0 , Rs , Rp and n as a function
of Va that are extracted from the figure.

121

5.6.1

Diode Saturation Current (J0 )

J0 functions that are extracted from dark and illumination JV curves are not significantly
different and can be used interchangeably. This is shown in Figure 5.15 where the J0 (Va )
functions of the two devices are plotted. The reason is that J0 which is expressed through
Eq. 5.11 only depends on q, a0 , ηP P d and Req (recombination rate in equilibrium) that do
not change between the two conditions.
-8

10

-6

Dark
Illumination

J0 (Am )

Dark
Illumination
-2

-2

J0 (Am )

10

10

-7

-8

10

-9

10

-10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Va (V)

10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Va (V)

(a) CyTRIS:C60

(b) P3HT:PCBM

Figure 5.15: Plots of J0 as a function of Va extracted from (a) the CyTRIS:C60 and (b)
P3HT:PCBM devices ( Figs. 5.14) under dark (black solid lines) and illumination (red
dashed line) conditions.
Figures 5.16a and 5.16b show kP P d , kP P r , interface electric field (FI ) and ηP P d as a
function of Va of the CyTRIS:C60 device both under dark and illumination conditions. Even
though free carrier densities under illumination increase considerably from photo-generation
processes (see Fig. 4.7), the device potential profile (ψ(x)) still mainly depends on the
voltage difference between the anode and the cathode (Vbi − Va ), and band bending near
the electrodes. ψ(x) and the resulting FI in Fig. 5.16b are slightly altered by the photogenerated charges, so kP P d , kP P r and ηP P d = kP P d /(kP P d + kP P r ) that depend primarily
on FI do not change significantly. As a result, J0 = qa0 (1 − ηP P d )Req hardly changes when
light is introduced.

122

1

8

10

Dark
Illumination

6

5×10

5

7×10

0.8

6

-1×10

5

-2×10

-0.4

-0.2

0

0.2

0.4

0.6

0.4

7

5

5

Dark
Illumination

7

3×10
10

0.6

7

5

4×10

10

6

-5×10

ηPPd

5

5×10

FI (v/m)

-1

kPPr (s )

6×10

10

-1

kPPd (s )

0

5

7

-2×10

2×10

-0.4

-0.2

0

0.2

0.4

0.6

0.2

Va (V)

Va (V)

(a) kP P d and kP P r

(b) FI and ηP P d

Figure 5.16: Comparisons between illumination and dark conditions of (a) kP P d (Va ),
kP P r (Va ), (b) FI (Va ) and ηP P d = kP P d /(kP P d + kP P r ) of the CyTRIS:C60 device.
5.6.2

Series Resistance (Rs )

Figure 5.17 shows graphs of Rs versus Va of the two devices under consideration, in the
presence and the absence of light. In order to describe a JV curve correctly, Rs has to
be a function of Va , but due to limitations of finding Rs (Va ), a constant Rs is normally
extracted, using Rs = [∂J/∂Va ]−1
Voc [32, 140]. Accordingly, we compare Rs (Va ) around Voc
in Figure 5.17, where the CyTRIS:C60 and the P3HT:PCBM devices have Voc = 0.60V and
0.74V , respectively.
10
10

10

3

Dark
Illumination

2

10
10

1

2

Rs (Ωm )

2

Rs (Ωm )

10
10

0

-1

10

Dark
Illumination

3

2

1

10
10

0

-1

-2

10

-3

10

10

-2

10

-3

-4

10

4

-0.4

-0.2

0

0.2

0.4

0.6

10

0.8

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Va (V)

Va (V)

(a) CyTRIS:C60

(b) P3HT:PCBM

Figure 5.17: Comparisons between illumination (red dashed lines) and dark (black solid
lines) series resistances of (a) the CyTRIS:C60 and (b) P3HT:PCBM devices.
Based on the data, Rs (Voc ) values of the CyTRIS:C60 device are 9.32 × 10−4 Ωm2 and
123

8.12 × 10−4 Ωm2 under dark and illumination conditions, respectively. For P3HT:PCBM,
dark and illumination Rs (Voc ) are 1.49×10−3 Ωm2 and 1.33×10−3 Ωm2 , respectively. Hence,
Rs (Voc ) values under the two conditions are about 12-14% different in these two devices.
These agree with the Rs values shown in Ref. [138], where Rs values from six different
devices are different around 10-50% between under dark and illumination conditions.

5.6.3

Ideality Factor (n)

According to Section 4.5, ideality factor is related to the split of quasi-Fermi levels at the
interface (Ef n,I − Ef p,I ) and trap characteristic temperatures (Tt,D and Tt,A ). In general
form, it is expressed by Equation 2.23, which also depends on Ef n,I − Ef p,I and R(Va )/Req
where R(Va ) is recombination rate at Va and Req is the equilibrium recombination rate. Due
to that R(Va ), Ef n,I and Ef p,I under illumination and dark conditions are different because
of the different amounts of charges, ideality factor values under the two conditions are also
different, but the degree of the difference varies from system to system.
1.5

1.5

4

1.4

1.4

1.3

1.20

0.2

0.4

Va (V)

0.6

0.8

ideality factor

ideality factor

ideality factor

8

6

2

0

Dark
Illumination

Dark
Illumination

1.3
1.2
1.1

-0.4

-0.2

0

0.2

0.4

0.6

1.0

0.8

0.2

0.4

0.6

0.8

Va (V)

Va (V)

(a) CyTRIS:C60

(b) P3HT:PCBM

Figure 5.18: Ideality factor functions calculated by Eq. 2.23 of (a) the CyTRIS:C60 and (b)
the P3HT:PCBM devices under dark and illumination conditions. (inset) zoomed portion
of figure (a) for Va > 0V
Figure 5.18 shows comparisons of dark and illumination ideality factor functions calculated using Eq. 2.23 of the two devices. For the P3HT:PCBM graph, we only show
the positive bias regime because the dark n(Va ) function diverge around V = 0V from
124

ln(R(Va )/Req ) ≈ 0 in the denominator of Eq. 2.23. For the CyTRIS:C60 case, the dark
and illumination n(Va ) functions are close to each other (i.e. less than 10% different) in
the positive bias regime, which is where n relevant the most according to Sections 5.2 and
5.3. This is in accordance with the results from Ref. [1] where extracted ideality factors
from dark curves are close to those calculated by light intensity dependent Voc data. On
the other hand, the n(Va ) functions of the P3HT:PCBM device are significantly different
because of the term Ef n,I − Ef p,I (not shown here). Ef n,I − Ef p,I is related to the interface
charge densities and can be affected by many parameters, for example, dielectric constant,
mobilities, and density of states near the HOMO or LUMO edges.

5.6.4

Parallel Resistance (Rp )

Unlike the other three parameters whose values under dark and illumination conditions can
be close or different by a small factor, Rp values under illumination and dark conditions
are totally different because they arise from different origins. Figure 5.19 shows plots of
illumination Rp functions (Rp_P P from Section 5.5), which are recombination resistances
of the recombined exciton currents; qJX (1 − ηP P d ). Their values are lower than Rp,dark ;
Rp,dark = 800Ωm2 for CyTRIS:C60 and 100Ωm2 for P3HT:PCBM, by at least two orders of
magnitude. Similar results are observed experimentally in Ref. [138] where Rp values under
dark are higher than the ones under illumination by 10 − 105 times.

5.6.5

Errors of Using Dark and Illumination Shockley Parameters Interchangeably

According to the results of the four Shockley parameters, J0 is the only parameter that can
be used interchangeably between dark and illumination conditions. Rp is on the opposite
end of the spectrum because its values under dark and illumination are different by orders
of magnitude. Extracted Rp values from dark JV curves must not be applied to fit other
Shockley parameters or derive any physical properties from JV curves under illumination.

125

10

0

2

Rp (Ωm )

CyTRIS:C60
P3HT:PCBM

-1

10

-2

10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Va (V)

Figure 5.19: Plots of parallel resistances under illumination (Rp_P P ) of the CyTRIS:C60
(black with circles) and the P3HT:PCBM (red with squares) devices calculated using Eq.
2.28.
For Rs and n, their differences between dark and illumination conditions are not very high,
and vary from system to system. However, using extracted values from one conditions with
the other conditions can cause errors and wrong interpretations of the results. For example,
if a dark Rs value, which is slightly higher than the illumination value (see Fig. 5.17), is
applied to describe an illumination curve, the diode voltage (Va − JRs ) is underestimated.
In order to maintain the right diode current (J0 [exp((Va − JRs )/nkB T ) − 1]), n has to be
reduced from the actual value to compensate the error caused by Rs . Therefore, it is best to
use two sets of the shockley parameters to describe JV curves under illumination and dark
conditions.

5.7

Conclusions

In this chapter, we focus on applying the DD-BI and SCD models and the theories of OPVs
that are discussed in the previous chapters to describe an actual bilayer device in terms of
Shockley parameters; Rs , Rp , J0 and n. The device is a cyanine organic salt solar cell with
TRIS counterion (CyTRIS:C60 ), which has an interesting kink-shaped JV behavior under
dark conditions. We start this chapter with studying the effects of the Shockley parameters
on JV curves under dark (Section 5.2) and illumination (Section 5.3) conditions. The results

126

show that the kink-shaped JV behavior is caused by a leakage current in the dark due to
low Rp and low J0 .
In addition to the illumination leakage current and Rp in Section 2.2, we include a leakage
current and Rp in the dark in our models by using the trap-to-trap (t-t) recombination and
the standard Shockley form of leakage current with a constant Rp . Both methods result in
a dark leakage current that can bring about the kink-shaped JV behavior. However, leakage
currents from the two models do not fit perfectly with experimental results especially in the
reverse bias regime, so better descriptions of dark leakage current in bilayer OPVs are still
needed.
With the dark Rp processes, we now have a complete electrical equivalent circuit (EEC)
model that includes physical processes of bilayer OPVS, having a dark Rp resistance connected to the illumination Rp resistance in parallel. In Section 5.6, we use the DD-BI, the
SCD and the complete EEC to study differences of the Shockley parameters’ values under
dark and illumination conditions and possible errors of using the parameters interchangeably.
According to the results, only J0 can be used interchangeably, and Rs and n values under
the two conditions can be significantly different, but most importantly dark and illumination
Rp values are different by many of orders of magnitude. Hence, it is best to use two sets of
the parameters to avoid errors and misinterpretations of fitting results.

127

CHAPTER 6
AB INITIO CALCULATION OF AN EPITAXIAL PEROVSKITE
Since the first photovoltaic application in 2009[141], perovskite solar cells have attracted
tremendous attraction from the research community, and this discovery is among the top
ten Breakthrough of the Year, ranked by the Science magazine in 2013. A type of perovskite
material of particular interest is caesium tin halide (CsSnX3 where X = Cl, Br or I). Even
though they have been known since the 1980s[142, 143], the availability of atomistic calculations and advanced computer technologies have just allowed us recently to understand them
fundamentally.
A research group at MSU lead by professor Richard R. Lunt is now concentrating on
synthesizing CsSnBr3 using an epitaxy technique. It allows an epitaxial CsSnBr3 film to
grow on the (100) surface of a NaCl substrate, from gaseous precursor of CsBr:SnBr2 . The
CsSnBr3 crystal usually maintains cubic phase at room temperature[142], but the film undergoes a phase transition during the growth process. This is possibly caused by the induced
compressive strain from lattice mismatch (i.e. a = 5.64 Å for NaCl and a = 5.804 Å[144]
for CsSnBr3 ) or chemical interactions between the gaseous precursors. Different ratios of
the precursors also affect the phase transition. For example, for CsBr:SnBr2 at 1:1 ratio,
the first several layers are in the cubic phase and then another unknown phase forms in the
thick film limit. For the ratio of 2:1, the structure maintains a cubic structure regardless of
the thickness of the film.
Based on ultraviolet-visible (UV-Vis) spectrophotometry measurements, the cubic phase
of CsSnBr3 has an energy gap of around 1.8eV and the unknown phase in the thick film limit
has a gap of around 3.4eV. X-ray photoelectron (XPS) spectroscopy results show that the
actual ratio of Sn:Br in the unknown phase is about 1:1 to 1:2, instead of 1:3, and the amount
of Cs ion is less than that of CsSnBr3 . With this information and measured X-ray diffraction
(XRD) patterns, the unknown phase/structure could be cesium pentabromodistannate (II)

128

or CsSn2 Br5 [145]. Hence, the objective of the study described in this chapter is to calculate
properties of the two compounds using Ab initio calculations and compare the simulation
results with experimental results, so the identity of CsSn2 Br5 can be confirmed. Mechanisms
related to the film growth and the phase transition are also considered.
All calculations in this chapter are performed using the Vienna Ab initio Simulation
Package (VASP), which is a quantum solid-state simulator. The method implemented in the
package is density functional theory (DFT). Necessary background of the related quantum
mechanics and the DFT are summarized in the appendix. The contents of this chapter start
with an introduction to perovskites and cesium pentabromodistannate (II) in Section 6.1.
Simulated electronic band structures are described and compared in Section 6.2. In Section
6.3, adsorption of the precursor atoms onto the (100) NaCl surface is investigated to find
the lowest energy atomic configuration of the CsSnBr3 /NaCl interface. Section 6.4 then
shows effects of the NaCl substrate on the band structures. Section 6.5 shows simulations
of the CsSnBr3 film growing processes by depositing the structure layer by layer. Finally, in
Section 6.6, possible explanations behind the phase transition are suggested.

6.1
6.1.1

Crystal Structures
Cubic Perovskite

The chemical formula of perovskite materials is ABX3 where its stoichiometry can be I-VVI3 , II-IV-VI3 , III-III-VI3 and I-II-VII3 [146]. For the I-II-VII3 family that CsSnBr3 belongs
to, A is a cation with 1+ charge and 12 nearest neighbors (NN), B is a 2+ cation with 6
NN, and X is a 1- halide anion. The structure and the unit cells are shown in Figure 6.1.
There exist more complex structures of perovskite materials which are beyond the scope of
this study.
The structure under investigation consists of ions B and X forming BX6 octahedra
throughout the bulk, and the A cations occupy the cuboctahedral voids[147, 148]. There are

129

(a) Crystal Structure

(b) Unit cell type A

(c) unit cell type B

Figure 6.1: (a) Structure of a cubic perovskite showing octahedra of BX6 and the cation A
filling the cuboctahedral void. (b) and (c) Two representations of the perovskite unit cell
with the labels of the Cs, Sn, and Br ions in figure (b) for the case of CsSnBr3 .
two representations of the unit cell, shown in Figures 6.1b and 6.1c. There are many possible
choices of the ions A, B, and X, however a perovskite material will be stable only if its geo√
metric tolerance factor (t) is close to one; t = (rA +rX )/ 2(rB +rX ), where ri is the effective
radii of the ion i[148]. The factor t is derived from the ratio between the lattice constants
√
(a) calculated from the AB plane (i.e. the middle plane in Fig. 6.1b); a1 = 2(rA + rX )
and from the AX2 plane (i.e. the bottommost plane in Fig. 6.1b); a2 = 2(rB + rX ), so
t = a1 /a2 . For a list of the effective radii, see the paper by Shanon[149]. The BX6 octahedra
can be considered as a scaffold with the cations A supporting it. Accordingly, the size of the
cation A is quite specific and large alkali metals like caesium suit the position. Too small A
cation results in t < 1 and octahedral distortion results. Alkali metal halide perovskites (e.g.

130

CsSnBr3 ) should have 0.813 < t < 1.107 [150] in order to be stable. Examples of possibly
stable perovskites are CsCuX3 , KCuX3 , KCoX3 , KMnX3 , etc., where X = Cl, Br, I.

6.1.2

CsSn2 Br5 Structure

The crystal structure of cesium pentabromodistannate (II) (CsSn2 Br5 ) is shown in Figure
6.2, which is proposed by Abrahams et al. in their paper [145]. The following is a brief
description of the structure that is summarized from the paper.
The structure is composed by two-dimensional sheets of polymeric anion [Sn2 Br5 ]n−
n and
inter-layer planes of Cs cations. Its tetragonal unit cell is shown in Figure 6.2a along with
the definitions of the basis vectors. Projection of the unit cell onto the b/c plane is shown in
Figure 6.2b, and Figure 6.2c shows the 2D [Sn2 Br5 ]n−
n sheet when it is viewed along the c
direction. The polymeric sheet is composed of four-sided rings of 4 Sn atoms and 4 Br atoms,
with each side shared with other rings. Each Sn atom forms 4 bonds with 4 neighboring Br
atoms (note that there are Br atoms right under the dangling Br atoms in Fig. 6.2c) and
has a pair of non-bonding valence electrons, so the bond shape at each Sn site is pyramidal.
Figure 6.3 shows another way of choosing CsSn2 Br5 unit cell that has different c basis
vector from that in Figure 6.2, but the same a and b basis vectors. Instead of including 2
sheets of [Sn2 Br5 ]n−
n and 2 planes of Cs into the tetragonal unit cell in Fig. 6.2, the unit
cell in Fig. 6.3, which is triclinic, has only one sheet of [Sn2 Br5 ]n−
n and one Cs plane. Thus,
the triclinic unit cell is more economic than the tetragonal unit cell.
One unit cell can be transformed into another by using a transformation matrix in the
relation;



a

b

c

= a b c

P11


P
 21


P31



P12 P13 
P22
P32



P23 
,


P33

(6.1)

where basis vectors a, b and c are transformed into new basis vectors a’, b’ and c’ by the
transformation matrix P. In the case of CsSn2 Br5 , the transformation matrices to transform

131

(a) CsSn2 Br5 Crystal

(b) Projection onto the b/c
plane

(c) Sn2 Br5 plane

Figure 6.2: a) Tetragonal unit cell of CsSn2 Br5 with the definitions of the basis vectors and
b) its projection onto the b/c plane. c) Projection onto the a/b plane of the [Sn2 Br5 ]n−
n
polymeric sheet. Gray, purple and brown spheres represent Cs, Sn and Br atoms, respectively.
from the triclinic unit cell to the tetragonal unit cell and vice versa are given by;


1


P=
0







0 1
1



1
,



and P

0 0 2

1


=
0





0 −0.5
1

0 0

respectively.

132



−0.5
,



0.5

(6.2)

(a) Projection onto the a/c plane

(b) Projection onto the b/c plane

(c) Projection onto the a/b plane

Figure 6.3: Triclinic unit cell of CsSn2 Br5 that is projected into a) the a/c plane, b) the b/c
plane and c) the a/b plane. Gray, purple and brown spheres represent Cs, Sn and Br atoms,
respectively.

6.2

Band Structure and Density of States Calculations of CsSnBr3
and CsSn2 Br5

In this part, we calculate electronic band structures and densities of states of CsSnBr3 and
CsSn2 Br5 at the DFT level using the program VASP. We use the Perdew-Burke-Ernzerhof
(PBE) functional [151]; which belongs to the class of generalized gradient approximations
(GGA), and the Heyd-Scuseria-Ernzerhof (HSE06) functional [152]; which is a screened
hybrid approximation for the exchange-correlation functional. In the case of CsSn2 Br5 ,
we use the triclinic unit cell primarily because of the smaller number of atoms, but the
tetragonal unit cell is also calculated for comparison purposes. First of all, we perform
structural relaxations on the unit cells that are described in the previous section using the

133

PBE functional to find the optimized unit cells. The optimized cubic unit cell of CsSnBr3
has the lattice constant of 5.888Å, and the optimized triclinic unit cell of CsSn2 Br5 has the
following parameters; a = 8.6177A, b = 8.6190A, c = 10.1028A, α = 115.28◦ , β = 115.25◦
and γ = 90◦ . Computational details are as follows.

6.2.1

Computational Details

The structural optimizations are performed in two steps; rough and fine optimizations. During the rough optimizations, we use a conjugate gradient algorithm (IBRION = 2) to relax
the initial structures until total force on each ion is less than 0.3 eV/Å (EDIFFG = -0.3).,
and then the structures are further optimized using a quasi-Newton algorithm (IBRION =
1) until the total force is less than 0.01 eV/Å.
For band structure calculations, the first Brillouin zones of cubic, triclinic and tetragonal
unit cells are illustrated in Figure 6.4. For cubic band structures, the path along the high
symmetry k-points is chosen as Γ-X-M-R-Γ. The paths for the cases of triclinic or tetragonal
systems follow the suggestions by Setyawan and Curtarolo[7] which are X-Γ-Y|L-Γ-Z|N-ΓM|R-Γ and Γ-X-M-Γ-Z-R-A-Z|X-R|M-A respectively. Details of the high symmetry k vectors
can be found in the paper [7] as well.

(a) Cubic

(b) Triclinic

(c) Tetragonal

Figure 6.4: The first brillouin zones of (a) a cubic, (b) a triclinic and (c) a tetragonal unit
cell with a path along the edges of the irreducible brillouin zones. adapted from [7]

134

Ground state energy : E0 (eV)

Ground state energy : E0 (eV)

-15.840

-15.845

-15.850

-15.855
200

250

300

350

400

450

500

Cut-off energy : Ecut (eV)

-15.800
-15.810
-15.820
-15.830
-15.840
-15.850
-15.860

5

10

15

20

k-point grid (NxNxN)

(a) Ecut test with fixed 9x9x9 k-point grid

(b) K-points test with fixed Ecut = 275eV

Figure 6.5: Ground state energy (E0 ) convergence tests of the cubic phase as a function
of (a) cut off energy (Ecut ) using a Γ-centered k-point grid 9x9x9 and (b) k-point grid of
NxNxN where N is the x-axis value with Ecut = 275eV.
Convergence tests of the ground state energy (E0 ) due to the choices of k-point grid and
cut-off energy (Ecut ) of the plane wave basis set are shown in Figure 6.5 for the optimized
cubic phase. The tests are based on the PBE functional. Throughout this chapter, the
Gaussian smearing (ISMEAR = 0) with the width of 0.05eV (SIGMA = 0.05) are used,
unless stated otherwise. The precision of the calculation is set to 10−5 eV/atom (EDIFF
= 1E-5), although EDIFF = 1E-4 is sufficient according to the VASP manual. The graph
shows that Ecut = 300eV, or 275eV if the computational time is needed to be reduced, are
sufficient and result in an error of about 1-2 meV. These Ecut ’s are considerably higher than
the default value set by VASP at 220eV, based on the Cs_sv PAW potential. Change of E0
caused by k-point grids finer than 8x8x8 is not clearly observed, so it is chosen.
Figure 6.6 shows the E0 convergence tests of the optimized CsSn2 Br5 triclinic unit cell.
Figure 6.6a shows that E0 (Ecut ) has considerable fluctuation for Ecut < 300eV , so Ecut =
300eV; which brings about an error of about 1-2meV as compared to that of Ecut = 350eV, is
chosen. According to Fig. 6.6b where the major scale separation of the E0 axis is 0.01meV,
k-point grids finer than 6x6x4 result in an negligible improvement of the accuracy. However,
if a symmetric k-point grid (e.g. 6x6x6) is used, the program will show an error message

135

Ground state energy : E0 (eV)

-50.625
-50.630
-50.635
-50.640
-50.645
-50.650

300

250

350

Cut-off energy : Ecut (eV)

(a) Ecut test with a fixed 13x13x12
k-point grid

(b) K-points test with a fixed Ecut = 300eV

Figure 6.6: Ground state energy (E0 ) convergence tests of the optimized triclinic unit cell of
CsSn2 Br5 as a function of (a) cut off energy (Ecut ) using a Γ-centered k-point grid 13x13x12
and (b) Γ-centered k-point grids with a fixed Ecut = 300eV.
about unmatched k-points in the irreducible Brillouin zone (IBZ) to the lattice type. This
is because of the unequal lengths of the reciprocal lattice vectors; which are determined by
b1 = 2π

a 2 × a3
,
a1 · (a2 × a3 )

b2 = 2π

a3 × a 1
a2 · (a3 × a1 )

and

b3 = 2π

a1 × a 2
,
a3 · (a1 × a2 )

(6.3)
−1

where ai is the Bravais lattice vector. For the optimized triclinic unit cell, |b1 | = 0.131Å
−1

|b2 | = 0.131Å

−1

and |b3 | = 0.124Å

,

, i.e. |bi | is approximately inversely proportional to |ai |.

In order to have an equally-spaced k-point grid in the IBZ, the grid has to be roughly a1 x 1b
x 1c . In this case, the grid 6x6x4 does not result in the error message and is sufficient.
The projected augmented-wave (PAW) potentials[153, 154]; which are the default methods in VASP, are used to replace the complex potentials of the core electrons. For Cs atoms,
semi-core s electrons have to be treated as valence electrons, so the potential file Cs_sv is
used.

136

6.2.2
6.2.2.1

The PBE Calculations
CsSnBr3 Bands and DOS

Figure 6.7 shows the calculated band structure and projected density of states (PDOS) of
the cubic phase (a = 5.888 Å) CsSnBr3 , using the PBE functional. The PBE functional is
known to underestimate band gaps[155], and this character happens here as well. A direct
gap of 0.639 eV occurs at the R point (0.5 0.5 0.5). Figure 6.8 shows the magnified PDOS
around the valence band maximum (VBM) and the conduction band minimum (CBM). In
order to clearly understand effects of atomic orbitals on the bands, consider the electron
orbitals that are treated as valence orbitals in the following PAW potential files:
the Cs_sv PAW potential : 5s2 5p6 6s1;
the Sn PAW potential : 5s2 5p2;
the Br PAW potential : 4s2 4p5.
Energies in the following analysis are referenced to the Fermi level which is set to zero. The
dominant band in the energy range -8.5 to -6 eV is mainly from the Sn 5s orbitals; while
from energies -5 to -2 eV, the bands are dominated by the Br 4p states. Near the VBM, the
bands are a mixture of the Sn 5s and the Br 4p states. According to Huang and Lambrecht
[156], these VBM bands are from anti-bonding interactions of the Sn 5s and Br 4p. Above
the gap, the Sn 5p orbitals mainly comprise the CBM bands, until around the energy 2 to 4
eV, the bands are mixed between the Sn 5p, the Br 4p and the Cs semi-core d sates. Over
4eV, the bands of these highly excited states belong to the Cs semi-core d sates. Except for
the gap, these results agree well with those of Huang and Lambrecht[156] where the more
accurate GW method is employed.

6.2.2.2

CsSn2 Br5 Bands and DOS

The band structure and the PDOS of CsSn2 Br5 are shown in Figure 6.9. The VBM and
the CBM are at the Z point (-0.5 0.0 0.5) and M point (0.0 0.0 0.5), respectively, resulting

137

Figure 6.7: Electronic band structure and projected density of states the cubic CsSnBr3
structure calculated using the PBE functional.
in an indirect band gap of 2.467eV. However, the bands near the VBM are quite flat and
the energies at the symmetry points at the edges of the Brillouin zone (i.e. X, Y, L, Z, N,
M and R) are very close to each other. In the CBM bands, the energy at Γ point is also
very close to the CBM at M point. The flat VBM bands indicate that effective hole masses
(m∗h = h
¯2 (∂ 2 Ek /∂k 2 )−1 ) are high; possibly resulting in a low hole mobility; which is not
preferable in photovoltaic applications.
Atomic orbital characteristics of the bands are as follows. At -7 eV, the flat bands and
the corresponding high DOS peak belong to the localized, full-shell semi-core Cs 5p states.
The peaks at around -6eV are contributed by the Sn 5s states. From the energy about -3.5
to -0.5 eV, the bands are mainly from to the Br 4p orbitals. Near the VBM, the bands are
due to a mixture of Sn 5s and Br 4p states; which are similar to the case of CsSnBr3 , and a
small contribution from the Sn 5p orbitals. The bands near the CBM are mainly composed
of the Sn 5p states, and approximately over 4eV, the bands are from both the Sn 5p and Br
138

Density (states/eV)

3
Total DOS
Sn (s)
Sn (p)
Br (p)
Br (d)

2

1

0

-2

-1

0

1

2

Energy (eV)

Figure 6.8: Zoomed in projected density of states around the VBM and the CBM of the
cubic CsSnBr3 calculated using the PBE functional.

Figure 6.9: Electronic band structure and projected density of states of CsSn2 Br5 calculated
using the optimized triclinic unit cell and the PBE functional.
4p orbitals.
Figure 6.10 is a plot of band structure and PDOS functions of CsSn2 Br5 calculated using
the optimized tetragonal unit cell; whose unit cell parameters are a = 8.591Å, b = 8.591Å
139

Figure 6.10: Electronic band structure and projected density of states of CsSn2 Br5 calculated
using the optimized tetragonal unit cell and the PBE functional.
and c = 16.133Å, and the PBE functional. The contributions of atomic orbitals as a function
of energy that are shown in the graph agree well with those of the optimized triclinic unit
cell in Fig. 6.9. The band structure also shows similar flat VBM bands and an indirect band
gap of 2.430 eV, which is close to the value of 2.467 eV in the triclinic case.

6.2.3

The HSE06 Calculations

The HSE06[152] functional is employed for the band gap calculations as an effort to improve
the accuracy. It is considerably more accurate than the local density approximations (LDA)
or generalized gradient approximations (GGA) (e.g. the PBE functional) and is even more
accurate than the single-shot G0 W0 calculation in binary compounds of GaAs, InAs, GaN
and InN[157]

140

(a) HSE06 band structure of the cubic CsSnBr3

(b) HSE06 band structure of the triclinic CsSn2 Br5

Figure 6.11: Calculated band structures of a) the cubic CsSnBr3 and b) the triclinic
CsSn2 Br5 using the HSE06 functional.
Figure 6.11 represents band structures of the cubic CsSnBr3 and CsSn2 Br5 with the
optimized triclinic unit cell, calculated using the HSE06 hybrid functional. For the CsSn2 Br5 ,
we use the path L-Γ-Z|N-Γ-M instead of the full path as shown in the PBE calculation to
reduce the computational time. The shapes of the band structures are not significantly

141

different from the PBE results, but the gaps are broadened to 1.112eV for CsSnBr3 and
3.287 eV for CsSn2 Br5 .

6.2.4

Summary and Discussion of Band Gaps

Table 6.1 summarizes the values of the band gaps and the optimized lattice parameters used
in the band calculations.
material
CsSnBr3
CsSn2 Br5
(triclinic)
CsSn2 Br5
(tetragonal)

lattice parameters
a = 5.888 Å
a = 8.618 Å, b = 8.619 Å, c = 10.10 Å,
α = 115.28◦ , β = 115.25◦ , γ = 90◦
a = 8.591 Å, b = 8.591 Å, c = 16.133 Å,

PBE gap (eV)
0.639
2.467

HSE06 gap (eV)
1.112
3.287

2.430

-

Table 6.1: Summary of the band gap values and lattice parameters of CsSnBr3 and CsSn2 Br5 .
For CsSnBr3 , an actual compound is stable in a cubic phase at room temperature [142]
with a lattice constant of 5.804 Å [144], showing a black color and has an energy gap of 1.8eV
[143]. According to the simulation results, the optimized lattice constant is 5.888Å, and the
band gaps are 0.639eV and 1.112eV when the PBE and the HSE06 functionals are used
respectively. The underestimated PBE gap is unsurprising and expected because the PBE
functional is commonly known to underestimate band gaps by around 1-2 eV or even 3 eV
in some wide-gap semiconductors or insulators[158, 159]. In this case, the underestimation
is around 1.2eV. For the HSE06 functional, the result is considerably improved but still
underestimates the gap. The improvement comes from the fact that the HSE06 functional
includes an exact Hartree-Fock exchange energy in short-range coulomb interactions but
screens coulomb interactions in the long-range limit.
For the case of CsSn2 Br5 , if the same underestimation is assumed, the estimated gap
should be around 3.6 - 3.8 eV according to the simulation results of the two functionals.
These wide gap values agree with the colorlessness of the film and are in reasonable agreement
with the experimental band gap value of 3.3eV.
142

In conclusion, it is likely that the unknown phase in the thick-film limit is CsSn2 Br5 based
on the band calculations. This agrees with Ref. [145], which states that the CsSnBr3 and
CsSn2 Br5 compounds usually forms simultaneously during sample preparation of CsSn2 Br5 .

6.3

Adsorption of the Precursor Atoms Onto the NaCl(100) Surface

The purpose of the study described in this section is to find the atomic configuration at
the interface between the NaCl(100) surface and the cubic CsSnBr3 . This can be done by
studying adsorption of the precursor atoms during the film deposition. The situation is that
there is a clean surface of NaCl(100) and free atoms of Cs, Sn, and Br, suspended over it.
The VASP package can be used to perform atomic relaxations to see if the free atoms attach
to the surface. Considering the adsorption energy (Eads ) also helps to determine the most
probable absorption configuration. Eads is a released energy as a result of an adsorption
process, which is given by
Eads = Esurf ace+atom − (Esurf ace + Eatom ),

(6.4)

where Esurf ace , Eatom and Esurf ace+atom are the total energies of the clean surface, the free
single atom and the surface with the adsorbed atom respectively. They can be calculated
from the relaxed structures using VASP.
The calculation procedure is as follows. First, Esurf ace is calculated using the unit
cell of the NaCl(100) surface shown in Figure 6.12, using the lattice parameter of 5.64 Å,
which is the experimental value. Since VASP is implemented with 3D periodic boundary
conditions, large vacuum (>15Å in this simulation) has to be included to represent the
surface. Second, an atom of Cs, Sn or Br is placed near the surface, and then relaxed to the
lowest energy position. It is relaxed freely in any direction, while the surface is kept frozen.
With the hypothesis that the atom should be adsorbed on top of a surface atom, there are
six possibilities to put the atoms on, which are listed in Table 6.2. Finally, the total energies
of a single Cs, Sn and Br atom are calculated and used in the Eads equation.
143

All the calculations are based on the DFT with the PBE[151] version of the GGA functional. A k-point grid of 8x8x2 is used in calculations involving the surface, and 4x4x4
k-points are used in the single-atom calculations. A cut-of energy of 340eV is used. During
the relaxations, the two-step relaxation method as described in Section 6.2.1 is carried out.

(a) Unit cell of the
NaCl(100) surface

(b) Top view of a single-layer NaCl
(100)

Figure 6.12: (a) The unit cell used for the infinite NaCl(100) surface under 3D periodic
boundary conditions. The larger yellow spheres represent Na atoms, and the smaller green
spheres represent Cl atoms. (b) Top view of a single-layer NaCl(100) surface.
The simulation results are shown in Table 6.2 in terms of Eads and the perpendicular
distances between the relaxed atoms and the surface. The most probable case is that a Sn
atom is adsorbed onto a Cl surface atom. Its adsorption energy is much higher than that
of the other atoms. Its distance of 2.7 Å suggests formation of a chemical bonding. For the
other cases, the lower Eads and the higher distances indicate non-bonding, especially in the
Br cases where Br atoms are repelled from the surface.
Based on these results, it is hypothesized that the CsSnBr3 crystal is attached to the
NaCl(100) surface by a Sn-Cl bond. This is natural as the bottom-most Br ion in each SnBr6
octahedron is replaced by the surface Cl ion. The structure is shown in Figure 6.13. It is
further confirmed by VASP calculations that this is stable. The chemical bonding in the

144

Adsorbing site
Cl
Na

Adsorbed atom
Cs
Sn
Br
Cs
Sn
Br

Eads (eV)
-0.1712
-0.5790
-0.1192
-0.0945
-0.0827
-0.1239

Distance from the surface (Å)
3.639
2.717
6.741
4.392
3.532
6.089

Table 6.2: Resulting adsorption energies (Eads ) and perpendicular distances in the study of
adsorption of Cs, Sn and Br precursor atoms on the NaCl(100) surface.

Figure 6.13: Structure of the CsSnBr3 crystal bonded to the NaCl(100) surface by the
Sn2+ Cl− ionic bonds.
metal halide octahedral network is considered to be mixed ionic/covalent interactions, so the
standard ionic model (i.e. Sn2+ + Br− ) is applicable[146]. The NaCl crystal is also formed
by ionic bonding of Na+ and Cl− . Hence, ionic bonding in the SnBr5 Cl surface octahedra
is likely possible.

6.4

Effects of the NaCl Substrate on the CsSnBr3 band structure

In the experiments, properties of the films, e.g. band gaps and crystal structures, are measured on samples that are still attached to the NaCl substrate. Hence, it is interesting to
examine if the NaCl substrate has any effects on the films. In this part, we calculate and
145

compare band structures and PDOS functions of two cubic CsSnBr3 films; one is attached
to the NaCl substrate and the other is not.

(a) Attached
to the NaCl
substrate

(b) Without
the NaCl
substrate

Figure 6.14: Unit cells of the cubic CsSnBr3 perovskite films having the thicknesses of 1.7
nm that are used to study effects of the NaCl substrate.
The atomic configuration result from the previous section is used in this study. Figure
6.14 shows the unit cells of the two films. The structures are relaxed using the PBE functional
but the NaCl substrate is kept frozen similarly to that in the previous section. The perovskite
films are composed of three layers of the CsSnBr3 cubic unit cell, resulting in the film
thicknesses of 1.7 nm. Both unit cells are tetragonal unit cells with the same parameters;
a = 5.64Å, b = 5.64Å and c = 40.64Å. The DFT calculation procedures that are described
in Section 6.2.1 are applied to these systems, using Ecut of 300eV and k-point grids of 9x9x1
for band structure calculations and 17x17x2 for density of states calculations.
The band structure and DOS calculation results are shown in Figures 6.15 and 6.16.

146

There are higher number of non-degenerate bands as compared to the bulk result in Figure
6.7 because the unit cell is a super cell with distorted atomic positions that are caused by
relaxations near the vacuum and the NaCl surfaces. The differences between the results of
the films with and without the substrate can be categorized as follows.

Figure 6.15: Band structure and PDOS functions of the CsSnBr3 film that is attached to
the NaCl substrate, calculated using the PBE functional.
First, the NaCl substrate gives rise to additional bands around the energies -5 to -3 eV
and around 4eV, with respect to the Fermi level at 0eV. The extra bands at the negative
energies are contributed by the Cl 3p orbitals. The origins of the extra bands at the energy
4eV are not clear as the PDOS plot indicates that they arise from increased DOS’s of Cs
and Br. This may be caused by coulomb interactions at the interface and more insights are
needed. Contributions from Na DOS can be seen around these bands, but they are negligibly
small.
The extra bands do not noticeably change the original bands of CsSnBr3 and hence lead
to a change of the band gap by only around 10meV, i.e. the energy gap of the film with the
NaCl substrate is 0.633eV and the gap of the film without the substrate is 0.645eV. This is
147

Figure 6.16: Band structure and PDOS functions of the CsSnBr3 film without the NaCl
substrate, calculated using the PBE functional.
because NaCl crystals are formed by ionic bonding of full-shell Na+ and CL− ions, resulting
in an insulating crystal with a very wide band gap of 8.97eV [160]. Hence, there should be no
electron transfer or electron sharing at the CsSnBr3 /NaCl interface, so electronic structures
of the CsSnBr3 film are almost unchanged.
We can see that the effects caused by the NaCl substrate are not very significant, even
when the CsSnBr3 film is very thin. In the thick-film limit where the film is around 40 nm,
properties of the samples should be of those the bulk perovskite and the substrate effects
can be neglected.

6.5

CsSnBr3 Film Growing Processes: Layer by Layer Deposition

Now we have the most probable atomic configuration at the CsSnBr3 /NaCl interface as
shown in Figure 6.13, which has the unit cell as shown in Figure 6.14. This part of thesis
shows the dynamics of the film growth processes based on that configuration, by simulating

148

depositions of the CsSnBr3 film; one atomic layer after another.
For the simulations, we use the two-step structural optimization that is described in
Section 6.2.1 to relax the film structures until the total force acting on each atom is less
than 0.01 eV/Å. The unit cells are similar to that in Fig. 6.14, including a large vacuum
space over the film. We allow all atoms to relax including Na and Cl atoms in the substrate,
but the unit cells’ shape and volume are fixed (ISIF = 2 in VASP) to represent the induced
strain as a result of the lattice mismatch between the cubic CsSnBr3 crystal and the NaCl
substrate. Ecut of 340eV and k-point grid of 8x8x2 are used in all the simulations.

Figure 6.17: Simulation results of depositing atomic layers of CsSnBr3 onto the NaCl substrate. Na, Cl, Cs, Sn, Br atoms are represented by yellow, green, gray, purple and brown
spheres, respectively.
Figure 6.17 shows the simulations results. For the first layer, a Sn atom is put on top of
the Cl atom in the middle of the cell, and then Br atoms are placed over Na atoms at the
edges of the unit cell. This first layer will result in the complete cubic CsSnBr3 structure as
shown in Figures 6.13 and 6.14. Note that the Cl atoms at the corners of the unit cell are
not covered by Sn atoms because these spaces will form cuboctahedral voids for Cs atoms.
149

Based on the simulation, this first SnBr2 layer attaches to the substrate and is stable.
For the second layer; which is an atomic layer of Cs and Br atoms, a Br atom is placed
on top of the middle Sn atom of the first layer and forms a bond. Cs atoms are placed at
the corners of the unit cell to fill the cuboctahedral voids. The relaxed structure; which is a
sheet of SnBr5 Cl octrahedra attached to the NaCl substrate, is stable.
A surprising result is observed in the third layer, which is an atomic SnBr2 layer similar
to the first layer. The structural relaxation does not converge to the required force even at
the 250th ionic step, while the other relaxations usually converge within 50 ionic steps. The
introduced layer is repelled from the film as shown in the Figure. However, if the third and
the fourth layers are deposited simultaneously as shown by the rightmost structure in Fig.
6.17. The structure is stable, and the convergence of the relaxation is obtained. The most
probable explanation is that, in cubic CsSnBr3 perovskites, Sn and Br atoms bond with each
other and form the octahedron scaffold. If the octahedra are incomplete (e.g. the third layer
situation), the bonds’ strength may be less than those of the complete one (e.g. the third
and the fourth layers together), so a coulomb repulsion dominates and repels the layer.
Figure 6.18 shows another situation that can occur for the third layer. Instead of depositing the same SnBr2 layer as the first layer that covers only half of the Cl atoms, we
deposit a SnBr layer that cover all atoms of the layer beneath it. This full-cover SnBr layer
results in a surprising outcome, as it attaches and is stable. However, if other layers can
attach on top of this layer, the structure will not cubic CsSnBr3 anymore. We try a couple
of atomic configurations for the fourth layer after the full-cover SnBr layer (see Figure 6.18
for example), but they do not form a bond or attach to the third SnBr layer.
In conclusion, we simulate the film growing processes in this section and arrive at two
interesting results. First, the deposited film has to form complete octahedra SnBr6 layers in
order to from cubic perovskite, or, second, other more energetically favorable layers like the
full-cover SnBr layer in the third layer may form and cause a phase transition.

150

Figure 6.18: Simulation results showing film deposition processes after the third full-cover
SnBr layer. Na, Cl, Cs, Sn, Br atoms are represented by yellow, green, gray, purple and
brown spheres, respectively.

6.6

Useful Research Directions : Heteroepitaxy, Critical Thickness and Phase Transition

Heteroepitaxy, which is the technique of growing a crystal film of one material on a crystal
substrate of another material used in this study, usually leads to a lattice mismatch between
the film and the substrate and a mismatch (or misfit) strain. The misfit strain (f ) is defined
by [161]
f=

as − ae
,
ae

(6.5)

where as and ae are the lattice constants of the substrate and the epitaxial film, respectively.
The strain can be a tensile strain(as > a0 ; f > 0) or a compressive strain (as < a0 ; f < 0)
[161]. In the present study, the relaxed lattice constant of CsSnBr3 in the cubic phase is
5.888Å, and NaCl has the lattice constant of 5.64Å, resulting in a misfit strain of -4.21%.
This strain is not very high, so the first several layers of the epitaxial CsSnBr3 film are
151

pseudomorphic, i.e. they match their in-plane lattice constant with that of the substrate
and adjust their out-of-plane lattice constant accordingly. The pseudomorphic layers can be
grown until the film thickness reaches a critical thickness where atom dislocations (or misfit
dislocations) occur to reduce the strain energy [161], after which the film lattice approaches
its natural states. In a similar perovskite system [162], an epitaxial film of LaAlO3 is grown
on a SrTiO3 (100) substrate, resulting in a lattice mismatch of -3.17% and a critical thickness
of 3.8 nm.
Based on the physics of heteroepitaxy and the results we find in Section 6.5, there seems
to be a connection between the misfit strain and the transition from CsSnBr3 to CsSn2 Br5 .
Studying this relation is crucial to understanding physics of the phase transition and the
CsSnB3 /CsSn2 Br5 interface. Here, we suggest the following steps for future work. First,
total and surface energies of a CsSnBr3 /NaCl sample as a function of the CsSnBr3 thickness
can be calculated using DFT to find the critical thickness, and then this thickness is compared
to experimental thicknesses at which transitions occur. Secondly, the interface between
CsSnBr3 and CsSn2 Br5 can be studied directly by first finding surface configurations that
bring about the lowest energies in the two materials. The two surfaces are then taken into
contact, and the total energy can be calculated. With information of crystal orientations
from experiments, the correct interface configuration can be predicted.

6.7

Conclusions

In this chapter, we present our numerical studies regarding growing of cubic CsSnBr3 peroveskite films on NaCl(100) substrates using a vapor phase epitaxy technique. Based on
experimental measurements of band gap and lattice constants, the hypothesis is that the cubic CsSnBr3 perovskite undergoes a phase transition to CsSn2 Br5 . We test this hypothesis
by simulating electronic band structures using DFT with the PBE and the HSE06 functionals. The results indicate that CsSn2 Br5 has an energy gap of around 3.6 - 3.8eV, which
agrees with the experimental band gaps and its transparency. Therefore, it is likely that the
152

hypothesis is true.
We also study interface between the NaCl substrate and CsSnBr3 . Considering adsorption energies of the precursor atoms reveals that CsSnBr3 attaches to the substrate through
bondings between Sn atoms and surface Cl atoms. This configuration is then used to study
effects of the substrate on electronic properties of the film. The results show that the effects
are not significant and can be negligible in the thick-film limit.
Finally, the film growing processes are studied by simulating depositions of CsSnBr3
atomic layers onto the NaCl substrate, one layer at a time. Structural relaxation calculations
confirm that several layers of the cubic CsSnBr3 perovskite can be stably grown on the
NaCl(100) surface, but the film may be unstable if the topmost layer does not form complete
SnBr6 octahedra. This may be the origin of the phase transition and can be a topic of future
studies.

153

CHAPTER 7
CONCLUSIONS AND OUTLOOKS

7.1

Conclusions

Working principles of bilayer organic photovoltaics (OPVs) and electronic properties of an
epitaxial CsSnBr3 perovskite film are the focuses of this dissertation.
For bilayer OPVs, we developed the DD-BI model, which is a numerical model in the
form of a computer software, to simulate fundamental mechanisms of bilayer OPVs and
the SCD equations, which are analytic equations that can describe results from the DD-BI
model. These models are based on the GWWF model, which is a diode equation model
bilayer OPVs. We consider three types of carrier recombination: free-carrier to free-carrier
(f-f) (or bimolecular) recombination; free-carrier to trapped-charge (f-t) recombination; and
the Shockley-Read-Hall (SRH) recombination. The SCD equations are in the same form
as the Shockley diode equation, which is universally used to fit current-voltage (JV) curves
of solar cells resulting in fitting parameters: series and parallel resistances (Rs and Rp );
ideality factor (n); and diode saturation current (J0 ). Together with short-circuit current
(Jsc ), open-circuit voltage (Voc ) and fill factor (FF), these parameters are figures of merit of
solar cells.
The DD-BI and the SCD models were shown to be extensions of the GWWF model to
include space charge effects, correct physical definitions of Rs and self-consistent interface
properties. A drawback of the GWWF model is that it estimates interface charge densities
using relations in terms of empirical potential drops across the donor and the acceptor that
do not include space charge effects and cancel out the definition of Rs in the derivation. The
GWWF model does not show space charge accumulation around the interface and results
in overestimations of interface electric field and polaron pair dissociation rate and efficiency
(kP P d and ηP P d ).
In Section 4.2, we studied influences of carrier mobilities on space charge accumulation
154

and quasi-Fermi levels (chemical potentials). We find that space charge accumulation around
the interface raises electron quasi-Fermi energy (Ef n,I ) in the accepter and reduces hole
quasi-Fermi energy (Ef p,I ) in the donor, resulting in a split of quasi-Fermi energies (Ef n,I −
Ef p,I ) at the interface. This split is a driving potential of the interface recombination and
can be defined as the diode voltage (Vd ); which equals to Va − JRs . Hence, the definition of
Rs is obtained as the total change of quasi-Fermi energies from the electrodes to the interface
divided by the total current density. The degrees of charge accumulation and quasi-Fermi
energy splitting increase with reducing mobilities, giving rise to greater values of Rs . Parallel
resistance under illumination conditions is also studied. It is a recombination resistance of
the recombined exciton current (qJX (1 − ηP P d )). Higher accumulation of charge caused
by low mobilities results in a higher recombination rate and higher qJX (1 − ηP P d ), which is
represented by lower recombination resistance (Rp ) values. This agrees with the conventional
picture of Rp as a shunt path for a leakage current, but qJX (1 − ηP P d ) is the actual leakage
current of bilayer OPVs under illumination.
In Sections 4.3 and 4.4 we studied effects of injection barriers, which are defined as
the differences between the electrode work functions and the donor HOMO energy and the
acceptor LUMO energy. High injection barriers cause s-shaped JV behaviors, resulting in
lower Jsc and FF values, but they do not significantly affect Voc as interface charge densities
and interface quasi-Fermi levels are not significantly affected. Interestingly, injection barriers
smaller than about 0.3 eV do not have any noticeable effects on JV curves. The reason
behind this is quasi-Fermi level pinning due to charge transfer between the active layers and
the electrodes, which can be explained by the integer charge transfer (ICT) model. If the
cathode work function is between the acceptor LUMO level and the acceptor Ef n , charge
transfer from the cathode to Ef n occurs and the quasi-Fermi level is pinned, and vice versa
for the anode work function that sits between the donor HOMO level and the donor Ef p . If
the electrodes’ work functions are out of these ranges, charge transfer does not occur, and
quasi-Fermi levels drop across the contacts, giving rise to extra components of Rs .

155

In Section 4.5, relations between ideality factors and trap characteristics are studied.
In general, ideality factors depend on Ef n,I − Ef p,I and the ratio of recombination rate
to recombination rate at equilibrium. In terms of interface quasi-Fermi energies, ideality
factors indicate deviations of Ef n,I and Ef p,I from the equilibrium Fermi energy. These
mechanisms can result in ideality factor values of around 1-2 which can be used as rough
indicators of trap characteristics. However, actual relations between ideality factors and
traps are complicated, and interpretations of fitted ideality factors have to be careful. For
example, a total current may be composed of many currents from different trap-assisted
recombination types, each of which has a different ideality factor and may not be the same
as that of the total current, so deducing a trap characteristic energy (Urbach tail energy)
value from the fitted ideality factor is likely misleading. Furthermore, we show that our
ideality factor equations are general forms of a famous definition of ideality factor.
In Section 4.6, the SRH recombination is compared with the f-t recombination, which
is understood as a reduced form of the SRH recombination. According to the results, even
though their recombination rates are comparable at voltage lower than Voc , they are actually
different processes, because the SRH recombination includes capturing of free carriers by
unoccupied trap states into account, while the f-t recombination does not. The differences
are clearer at voltage higher than Voc when many more free carriers are injected. The f-t
recombination rate reduces because traps cannot capture the injected charges, but the SRH
recombination increases as the injected charges can fill empty trap states.
In Chapter 5, we apply the DD-BI and the SCD models to describe an experimental
JV curve of a cyanine organic salt solar cell. The JV curve has an interesting kink-shaped
behavior under dark conditions that is caused by a leakage current. However, the definition
of leakage current under illumination conditions as qJX (1 − ηP P d ) does not apply in the
dark where JX = 0, so we include another mechanism for the dark leakage current. Two
processes are proposed: (1) recombination between localized states; (2) using an empirical
dark Rp as a constant in the standard Shockley form. The two processes are in reasonable

156

agreement with the experimental results in the positive voltage regime. Hence, the total Rp
value is the total resistance of dark Rp and illumination Rp that are connected in parallel.
Under illumination conditions, illumination Rp dominates the total Rp because illumination
Rp values are much lower than dark Rp values. In the dark, dark Rp is the only process
that governs interface leakage current, so it becomes dominant. We also study differences
between Shockley parameters under illumination and dark conditions. The results show
that only J0 can be used interchangeably between the two conditions because it depends
on intrinsic properties that are not affected by illumination. On the other hand, Rs , n and
most importantly Rp between the two conditions can be different by orders of magnitude.
Using extracted parameters from one condition to describe devices under the other conditions
results in misinterpretations and errors.
In Chapter 6 we focus on electronic properties of an epitaxial CsSnBr3 film that is
grown on a NaCl(100) substrate using a vapor phase heteroepitaxy technique. CsSnBr3 is
stable in a black cubic phase at room temperature and has a band gap of around 1.8 eV.
Growing a CsSnBr3 film using the technique results in a colorless film with a band gap of
around 3.3 - 3.5 eV in the thick-film limit, which is hypothesized to be CsSn2 Br5 . Band
structure calculations using DFT show that CsSn2 Br5 is a wide band gap semiconductor
with an agreeable estimated band gap with the experimental result, and thus the film is
likely CsSn2 Br5 . We also study atomic configurations at the interface between the NaCl
substrate and CsSnBr3 and effects of the NaCl substrate on the growth film. The result is
that the NaCl substrate does not significantly affect electronic properties of the grown film.
Finally, we consider the growth process of a CsSnBr3 film and find that a phase transition
to CsSn2 Br5 is possible if the topmost layer of the CsSnBr3 film does not form a complete
octahedron structure of SnBr6 .

157

7.2

Outlook

For OPVs, there are many possibilities for future research. First, a more accurate description
of dark leakage current is needed. We suggest including a trap-assisted tunneling process
for the leakage current. Second, at the present, the research community tends to go toward
perovskite solar cells for the next generation of solar cell. Many efficient perovskite solar
cells are based on bilayer structures. Hence, it is interesting to adapt the DD-BI and the
SCD models for bilayer perovskite solar cells. Additionally, due to the fact that the true
bilayer organic interface is difficult to achieve because of fast interpenetration between the
donor and the acceptor, our models can be modified to have a bulk-heterojunction region
around the interface for better descriptions of bilayer OPVs.
For CsSnBr3 perovskite, the interface between CsSnBr3 and CsSn2 Br5 and the conditions
under which the phase transition occurs are interesting directions. CsSn2 Br5 is not suitable
for photovoltaic applications because of the wide band gap. Hence, understanding the transition conditions is important to determine the film growth conditions that result in a more
perfect cubic CsSnBr3 crystal. Additionally, according to the Shockley-Queisser limits, the
optimal band gap for single-junction solar cells is around 1.3 eV. Tuning the CBM and the
VBM to optimize band gap of CsSnBr3 using a doping process is an interesting study and
can be done using DFT simulations.

158

APPENDIX

159

APPENDIX
QUANTUM MECHANICS BACKGROUND FOR AB INITIO
CALCULATIONS
This appendix provides an overview of quantum mechanics necessary to understand some
important basics of ab inito calculations. Although the Schrodinger equation can accurately
describe atomic systems, solving the equation for an exact solution is difficult in practice, as
stated by Dirac four years after the advent of the Schrodinger equation:
"The underlying physical laws necessary for the mathematical theory of a large
part of physics and the whole of chemistry are thus completely known, and the
difficulty is only that the exact application of these laws leads to equations much
too complicated to be soluble. It therefore becomes desirable that approximate
practical methods of applying quantum mechanics should be developed, which can
lead to an explanation of the main features of complex atomic systems without
too much computation".
— P. A. M. Dirac, Proc. R. Soc. Lond. A 123, 714 (1929)
Hence, approximations are needed to deduce useful information from the equation. One of
the most preferred approximations, which is used in this study, is density functional theory
(DFT). The main principle of DFT is that total energy of a system is treated as a functional
of charge density instead of wave function.
In order to introduce to DFT, the Schrodinger equation for many-body systems, the
Born - Oppenheimer approximation, the Hartree-Fock (HF) approximation and the KohnSham DFT have to be introduced first. The latter can be considered as the basis of current
atomistic DFT simulations. Beyond this overview, in-depth discussions of the theories can
be found in quantum chemistry books and review papers, for example, the book by David
Sholl and Janice Steckel [163], which is an introduction to the HF theory, and the book by
Szabo and Ostlund [164]. The narrative review paper Nobel Lecture by W. Kohn [165] who
160

is the founder of modern DFT is also recommended. For more thorough reviews, see the
papers by Jones and Gunnarsson[166] and by Geerlings et al.[167].

A.1

The Schrodinger Equation and the Born-Oppenheimer Approximation

The primary goal of computational quantum calculations is to have the most accurate solution for the time-independent, non-relativistic Schrodinger equation of many-body systems.
Consider a system of M nuclei with masses MA and charges ZA at positions RA and N
electrons with spin coordinates ωi at positions ri . Let xi = {ri , ωi } denote the set of the four
coordinates necessary to describe the electron i. The Schrodinger equation can be written
as
ˆ
HΨ(x
1 , · · · , xn , R1 , · · · , R1 ) = EΨ(x1 , · · · , xn , R1 , · · · , R1 ),

(A.1)

where the Hamiltonian is
M M Z Z
N
N N 1
N M Z
1 M 1 2
A B
A
ˆ = −1
+
∇2i −
∇A −
+
.
H
2 i=1
2 A=1 MA
r
r
R
AB
i=1 A=1 iA i=1 j>i ij A=1 B>A

(A.2)

The Hamiltonian is written in atomic units where the electron mass (me ), the elementary
charge (q), the factor 4π 0 and the reduced Planck’s constant (¯
h) are set to 1. There are five
energy terms in the Hamiltonian. The first two terms are kinetic energies of the electrons
and the nuclei. The third term represents attractive electrostatic potentials between the
electrons and the nuclei. The fourth and fifth terms are repulsive electron-electron and
nucleus-nucleus electrostatic potentials, respectively.
According to the Born-Oppenheimer approximation, nuclei are much heavier and move
much slower than electrons, so nuclei can be considered stationary. As a result, the nuclear
kinetic energy in the second term can be neglected, and the total nucleus-nucleus repulsive
potential can be considered as a constant; Enuc =

M
A=1

ZA ZB
M
B>A RAB ,

which can be sepa-

rated from the Hamiltonian. The remaining terms depend only on electrons’ positions, and

161

ˆ el ) as
we can write the electron Hamiltonian (H
N
N M Z
N N 1
A
ˆ el = − 1
H
∇2i −
= Tˆ + VˆN e + Vˆee .
+
2 i=1
r
r
ij
iA
i=1 A=1
i=1 i<j

(A.3)

Thus, the total energy of the Schrodinger equation in Eq. A.1 is reduced to Etotal = Eel +
Enuc , where Eel is total electron energy from the relation;
ˆ el Ψel = Eel Ψel ,
H

(A.4)

where Ψel is the electronic wave function.

A.2

Hartree-Fock Approximation

As a result of the Born-Oppenheimer approximation, the electron wave function Ψel ; which
has to be antisymmetric, is sought for. One way to describe the antisymmetric ground state
wave function (Ψel,0 ) is using the Slater determinant;
χ1 (x1 )

χ2 (x1 )

···

χN (x1 )

1 χ1 (x2 )
Ψel,0 = √
..
N!
.

χ2 (x2 )
..
.

···

χN (x2 )
,
..
.

(A.5)

χ1 (xN ) χ2 (xN ) · · · χN (xN )
where χi (xj ) are orthogonal spin orbitals, which are a product of spatial orbitals ψi (rj ) and
a spin function; α(ω) (spin up) or β(ω) (spin down). This formulation of Ψel,0 satisfies the
antisymmetry because an interchange between any two sets of the spatial-spin coordinates
x results in −Ψel,0 . Note that the Pauli exclusion principle, which does not allow two
electrons to have exactly the same set of x, is already included in the antisymmetry. Using
the Hamiltonian in Equation A.3 and the variational principle, a trial function of the Slater
determinant can be minimized so that Ψel,0 is obtained.
The Hartree-Fock approximation states that there exists a set of orthogonal spin orbitals
that minimizes the energy, and hence the ground state Hartree-Fock wave function is found
162

(i.e. ΨHF = Ψel,0 ). The Hamiltonian in Eq. A.3 can be written as a summation;
ˆ el =
H

N

M Z
N 1
1
A
(− ∇2i −
).
+
2
r
r
i=1
A=1 iA i<j ij

(A.6)

Using this form of the Hamiltonian and the orthogonality of spin orbitals; χ∗i (x)χj (x)dx =
ˆ el |ΨHF >. This results in the
δij , the electron energy can be calculated; Eel =< ΨHF |H
Hartree-Fock differential equations;
fˆχi = i χi ,

i = 1, 2, · · · , N

(A.7)

where fˆ is the Fock operator which is expressed by
M

1
ZA
fˆ = − ∇2i −
+ VHF (xi ),
2
r
A=1 iA

(A.8)

where VHF (xi ) is the Hartree-Fock potential which is given by
VHF (xi ) =

N

ˆ j (xi )).
(Jˆj (xi ) − K

(A.9)

j=i

The operator Jˆj (xi ), which is called the Coulomb operator, represents a repulsive potential
perceived by an electron at xi due to the distribution of other electrons at xj . The operator
−1 dx . The operator K
ˆ j (xi ), which is called the exchange
is given by Jˆj (x1 ) = |χj (x2 )|2 r12
2

term, has no meaning in the classical sense but arises mathematically from the Hartreeˆ j (x1 )χi (x1 ) =
Fock approximation. When it operates onto a spin orbital, it is defined by K
−1 χ (x )dx χ (x ).
χ∗j (x2 )r12
2 j 1
i 2

The essence of the Hartree-Fock approximation is that instead of dealing with the manybody Schrodinger equation, each electron can be treated independently under averaged influences of the nuclei (the 2nd term in Equation A.8) and other electrons (through VHF ).
In other words, the atomic system is reduced to non-interacting electron systems in a mean
potential field, which can be solved self-consistently.

A.3

Density Functional Theory

The theory that kindled the idea behind DFT was the Thomas-Fermi (TF) model of electron
gas, which results in an expression for total kinetic energy of electrons in terms of spatial
163

electron density (n(r)). Hohenberg and Kohn (HK) then derived the first theorem of DFT
with the purpose of using n(r) instead of wave function Ψ in solving the Schrodinger equation. The theorem states that the ground-state Hamiltonian and energy of the Schrodinger
equation are unique functionals of electron density[168]. Two keywords that have to be understood here are "unique" and "functional". The former means that it is not possible for
different systems with different Hamiltonians to have the same ground state n(r). Hence,
ˆ el and Ψel . For the latter, an energy
a ground state electron density can be used to find H
functional, which is usually written as E[n(r)], takes the entire vector field n(r) as an input
and gives a scalar field of energy as an output.
In the same paper by Hohenberg and Kohn[168], they also show that E[n(r)] obeys the
variational principle, so minimizing E[˜
n(r)] where n
˜ (r) is any trial electron density will yield
the ground state energy:
E0 = min E[˜
n(r)].
n
˜ (r)

(A.10)

Consequently, the energy functional can be written similarly to the Hamiltonian in Equation
A.3 as
E[n(r)] = T [n(r)] + Eee [n(r)] + EN e [n(r)] = FHK [n(r)] + EN e [n(r)].

(A.11)

The kinetic energy functional (T [n(r)]) and the electron-electron potential functional (Eee [n(r)])
are usually combined together as the Hohenberg-Kohn functional FHK [n(r)]. The term
EN e [n(r)] is the nucleus-electron potential energy functional that can be expressed through
EN e [n(r)] =

n(r)VN e (r)dr,

(A.12)

where the nucleus-electron potential VN e (r) is sometimes called external potential (Vext (r)).
Note that E[n(r)] in Eq. A.11 is still exact. FHK [n(r)] is a universal functional that is
absolutely invariant from system to system. If it was known, DFT could solve any problems
for the exact ground state energies. Unfortunately, its expression is not known exactly.
Eee [n(r)] can be written in terms of the classical coulomb interaction (J[n(r)]) and a non-

164

classical coulomb interaction (Encl [n(r)]) as;
Eee [n(r)] = J[n(r)] + Encl [n(r)],

A.4

where J[n(r)] =

1
2

n(r)n(r )
drdr .
|r − r |

(A.13)

Kohn-Sham Equations

Inspired by the non-interacting electron system in the Hartree-Fock formalism, Kohn and
his post-doc L. Sham overcame the unavailability of FHK [n(r)] by introducing the fictitious
Kohn-Sham (KS) system[169]. It is a system of non-interacting particles that can be described by the Kohn-Sham orbitals (φi ) and obeys the following single-particle Schrodinger
equation;
ˆ KS φi (r) = − 1 ∇2 + VKS (r) φi (r) = i φi (r),
h
2

(A.14)

ˆ KS is the single-particle Kohn-Sham Hamiltonian, VKS (r) is the Kohn-Sham potenwhere h
tial and i is the orbital energy of the KS orbital φi (r). With the following constraints of
total energy and particle density;
E=

N
i,

n(r) =

N

|φi (r)|2 ,

(A.15)

i=1

i=1

where E and n(r) are the same as those in the original electron system, one can solve the
Kohn-Sham equations for the ground state properties.
The question is: What should the KS potential be? First, the HK functional (FHK [n(r)])
is rewritten as
FHK [n(r)] = TS [n(r)] + J[n(r)] + EXC [n(r)],

(A.16)

where TS [n(r)] is the Kohn-Sham kinetic energy, which is given by
Ts [n(r)] =

N

1
< φi | ∇2 |φi >,
2
i=1

(A.17)

J[n(r)] is the classical Coulomb interaction in Eq. A.13, and EXC [n(r)] is the exchangecorrelation functional. The Exchange energy refers a reduction of Coulomb energy from
spatial separations of electrons with the same spin, and the correlation energy refers to
165

the energy difference between the actual many-body system and the one that is calculated
by the Hartree-Fock approximation[170]. EXC [n(r)] also contains all unknown or additional
functionals.
Using the new FHK [n(r)] in the Euler-Lagrange equation[167];
VN e (r) +

δFHK [n(r)]
δT [n(r)]
= VKS (r) + S
,
δn(r)
δn(r)

(A.18)

we have
VKS (r) = VN e (r) +

n(r )
dr + VXC (r),
|r − r |

(A.19)

where the exchange correlation potential (VXC (r)) is a functional derivative in the form
VXC (r) =

δEXC [n(r)]
.
δn(r)

(A.20)

With VKS (r) in Eq. A.19, Equations A.14 become the Kohn-Sham equations.

A.5

Exchange-Correlation Functionals

Based on the previous discussion about the Kohn-Sham DFT, the exchange-correlation (XC)
term in Eq. A.20 is the only term that has no explicit expression and has to be approximated.
There are many classes and many functionals within a class. The Perdew-Burke-Ernzerhof
(PBE) functional [151], which belongs to the generalized gradient approximation (GGA)
class, is used mainly in this work. It does not require a lot of computational resources but
provides moderate accuracy. However, it usually underestimates energy gaps. The HSE06
function[152, 171], which is a hybrid XC functional, is also included in this work. The
following are brief explanations of these functionals.

A.5.1

The PBE functional

The local density approximations (LDA) are the simplest approximations for the XC functional. The principle is that electrons are considered as a homogeneous electron gas with density n(r). Assuming that the homogeneous XC energy per electron is XC (r) = hom
XC [n(r)],
166

the XC energy functional is given by
LDA [n(r)] =
EXC

n(r) XC (r)dr.

(A.21)

Even though it is based on local approximations [170] and is not highly accurate, it is
overwhelmingly successful and is the basis of more sophisticated functionals.
Including electron spins in the LDA yields the local spin density approximations (LSDA),
and the XC functional is given by
LSDA [n (r), n (r)] =
EXC
↑
↓

n(r) XC (n↑ , n↓ )dr.

(A.22)

The generalized gradient approximations (GGA) take another step further than the
LSDA by including inhomogeneities of n(r) about r in terms of gradients of the spin density. Perdew-Burke-Ernzerhof[151] approach this approximation by adding a correction term
H(n↑ , n↓ , ∇n) to the homogeneous exchange functional homo
(n↑ , n↓ ), which finally results
X
in
P BE [n (r), n (r)] =
EXC
↑
↓

n(r) homo
[n(r)]FXC [n(r), ∇n(r)]dr,
X

(A.23)

where FXC is an enchantment factor.

A.5.2

Hybrid Functionals

As an effort to increase the accuracy beyond the GGA approximations, a certain amount
of Hartree-Fock (HF) exact exchange energy is included, resulting in a hybrid functional.
One example of hybrid functionals is the PBE0 functional [172], which treats the exchangecorrelation energy (Exc ) as
P BE0 = 1 E HF + 3 E P BE + E P BE ,
Exc
c
4 x
4 x

(A.24)

where the exchange energy is a mixture of PBE and HF exchange energies in a ratio of 3:1,
but the correlation energy is taken to be the PBE result. This full-range (or global) hybrid
functional is usually expensive and difficult to converge, and often overestimates band gaps
of materials [173].
167

Heyd-Scuseria-Ernzerhof (HSE) [152, 171] propose a hybrid functional that includes effects of screened Coulomb potential by separating the Coulomb operator into short-range
HSE06 ) is given by
(SR) and long-range (LR) parts. The resulting XC energy (Exc
HSE06 = 1 E HF,SR (w) + 3 E P BE,SR (w) + E P BE,LR + E P BE ,
Exc
x
c
4 x
4 x

(A.25)

where w is an adjustable parameter determining the range of the short-range part. According to the equation, the short-range part of the exchange energy is a combination of HF
and PBE energies similar to that of the PBE0 functional, but the long-range part and the
correlation energy are from PBE results. If the parameter w is set to zero, the short-range
part diminishes and the HSE06 functional reduces to the PBE0 functional. The HSE06
functional predicts band gaps more accurately than the global PBE0 functional but uses
significantly less computational resources [173].

168

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