HIGH TEMPERATURE HALL EFFECT MEASUREMENT SYSTEM DESIGN,
MEASUREMENT AND ANALYSIS
By
Isil Berkun

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Doctor of Philosophy
2015

ABSTRACT
HIGH TEMPERATURE HALL EFFECT MEASUREMENT SYSTEM
DESIGN, MEASUREMENT AND ANALYSIS
By
Isil Berkun
A reliable knowledge of the transport properties of semiconductor materials is essential
for the development and understanding of a number of electronic devices. In this thesis,
the work on developing a Hall Effect measurement system with software based data acquisition and control for a temperature range of 300K-700K will be described. A system was
developed for high temperature measurements of materials including single crystal diamond,
poly-crystalline diamond, and thermoelectric compounds. An added capability for monitoring the current versus voltage behavior of the contacts was used for studying the influence of
ohmic and non-ohmic contacts on Hall Effect measurements. The system has been primarily used for testing the transport properties of boron-doped single crystal diamond (SCD)
deposited in a microwave plasma-assisted chemical vapor deposition (MPCVD) reactor [1].
Diamond has several outstanding properties that are of high interest for its development as
an electronic material. These include a relatively wide band gap of 5.5 (eV), high thermal
conductivity, high mobility, high saturation velocity, and a high breakdown voltage. For a
temperature range of 300K-700K, IV curves, Hall mobilities and carrier concentrations are
shown. Temperature dependent Hall effect measurements have shown carrier concentrations
from below 1017 cm−3 to approximately 1021 cm−3 with mobilities ranging from 763(cm2 /V s)
to 0.15(cm2 /V s) respectively. Simulation results have shown the effects of single and mixed
carrier models, activation energies, effective mass and doping concentrations. These studies have been helpful in the development of single crystal diamond for diode applications.

Reference materials of Ge and GaAs were used to test the Hall Effect system. The system
was also used to characterize polycrystalline diamond deposited on glass for electrochemical applications, and M g2 (Si, Sn) compounds which are promising candidates of low-cost,
light weight and non-toxic thermoelectric materials made from abundant elements and are
suited for power generation application in the intermediate temperature range of (600 K 800 K). In this work the thermoelectric materials were synthesized by a solid-state reaction using a molten-salt sealing method. The ingots produced were then powder processed,
followed by pulsed electric sintering (PECS) densification. A set of M g2.08 Si0.4−x Sn0.6 Sbx
(0 ≤ x ≤ 0.072) compounds were investigated and a peak ZT of 1.50 was obtained at 716 K
in M g2.08 Si0.364 Sn0.6 Sb0.036 [2]. The high ZT value is related to a high electrical conductivity
in these samples, which are possibly caused by a magnesium deficiency in the final product. Analysis of the measured results using LabVIEW and MATLAB developed programs
showed good agreement with expected results and gave insight on mixed carrier dopant
concentrations.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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

vii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . .
1.1 Why Diamond? . . . . . . . . . . . . . . . . . . . . . .
1.2 Electrical Transport Characterization . . . . . . . . . .
1.2.0.1 Hall Effect Systems . . . . . . . . . . .
1.2.0.2 Hall Effect Measurements on Diamond
1.2.1 Ohmic Contacts . . . . . . . . . . . . . . . . . .
1.3 Hall Effect Measurement Methodology . . . . . . . . .
1.3.1 Van der Pauw Method . . . . . . . . . . . . . .
1.3.2 Hall Bar Method . . . . . . . . . . . . . . . . .
1.3.3 Quantitative Mobility Spectrum Analysis . . . .
1.4 This Work . . . . . . . . . . . . . . . . . . . . . . . . .

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2
2
4
4
5
13
17
17
20
22
24

Chapter 2 Temperature Dependent Hall Effect System
2.1 System Setup and Design . . . . . . . . . . . . . . . . .
2.1.1 System Overview . . . . . . . . . . . . . . . . . .
2.1.2 System Components . . . . . . . . . . . . . . . .
2.1.3 System Connections . . . . . . . . . . . . . . . .
2.2 Sample Mounting Procedure . . . . . . . . . . . . . . . .
2.3 PID Control . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Proportional Control . . . . . . . . . . . . . . . .
2.3.2 Proportional + Derivative Control . . . . . . . .
2.3.3 Proportional + Integral+ Derivative Control . . .
2.3.4 PI Control in This System . . . . . . . . . . . . .
2.4 Software Control . . . . . . . . . . . . . . . . . . . . . .

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26
26
26
29
31
35
38
39
39
40
41
44

Chapter 3 Measurement Results . . . . . . . . . . . . .
3.1 Reference Materials . . . . . . . . . . . . . . . . . . . .
3.1.1 p-Type GaAs . . . . . . . . . . . . . . . . . . .
3.1.2 n-Type GaAs . . . . . . . . . . . . . . . . . . .
3.1.3 p-Type Ge . . . . . . . . . . . . . . . . . . . . .
3.2 Single Crystal Diamond . . . . . . . . . . . . . . . . .
3.2.1 Influence of Ohmic contacts . . . . . . . . . . .
3.2.2 Effect of Applied Current to Hall Measurements

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46
47
47
51
54
57
58
62

iv

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3.3

Thermoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Carrier Concentration and Hall Mobility . . . . . . . . . . . . . . . .

Chapter 4 Analysis and Discussion . . . . . .
4.1 Band Theory . . . . . . . . . . . . . . . . .
4.2 Metal-Semiconductor Contacts . . . . . . . .
4.3 Characterization of Silicon and Germanium
4.3.1 p-Type Ge . . . . . . . . . . . . . . .
4.4 Characterization of Diamond Materials . . .
4.5 Physical Analysis Results . . . . . . . . . . .
4.6 Software Tools for Analysis . . . . . . . . . .

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65
66

. 69
. 69
. 72
. 84
. 88
. 90
. 100
. 102

Chapter 5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 107
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
BIBLIOGRAPHY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

v

LIST OF TABLES

Table 1.1

Electrical properties of common wide bandgap semiconductors compared to silicon [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Concentration, activation energy an Hall mobility of seven boron
doped homoepitaxial diamond, Borst et al. [4] . . . . . . . . . . . .

7

The mobility and activation energy values for room temperature on
low boron doped samples [5]. . . . . . . . . . . . . . . . . . . . . . .

12

Table 2.1

PID Manual Tuning [6] . . . . . . . . . . . . . . . . . . . . . . . . .

41

Table 2.2

PI Control Summary Table . . . . . . . . . . . . . . . . . . . . . . .

43

Table 1.2

Table 1.3

vi

LIST OF FIGURES

Figure 1.1

Mounted optical grade polycrystalline diamond window, ready for use
in high power CO2 laser optics [7] . . . . . . . . . . . . . . . . . . .

3

Room temperature resistivity as a function of doping concentration.
[8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 1.3

Van der Pauw configuration . . . . . . . . . . . . . . . . . . . . . .

18

Figure 1.4

Hall bar configuration . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figure 2.1

Hall mobility data comparison between Ohmic and non-Ohmic contacts for a moderately doped single crystal diamond sample (SND02)

27

Figure 2.2

Temperature Stability Effects . . . . . . . . . . . . . . . . . . . . . .

28

Figure 2.3

Hall Effect System Flow Chart . . . . . . . . . . . . . . . . . . . . .

28

Figure 2.4

Hall Effect System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 2.5

Hall Effect System Magnet Components . . . . . . . . . . . . . . . .

31

Figure 2.6

Stable Connections . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Figure 2.7

Van der Pauw Sample Connections . . . . . . . . . . . . . . . . . . .

34

Figure 2.8

In Line Sample Connections . . . . . . . . . . . . . . . . . . . . . .

35

Figure 2.9

Hall Effect System Sample Stage . . . . . . . . . . . . . . . . . . . .

37

Figure 2.10

Effect of Proportional Gain to Temperature Stabilization . . . . . .

42

Figure 2.11

Temperature vs time representation for the PID control . . . . . . .

44

Figure 3.1

Temperature dependent IV curve for p-type GaAs without contact
deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Temperature dependent IV curve for p-type GaAs with contact deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 1.2

Figure 3.2

vii

Figure 3.3

Temperature dependent resistivity for p-type GaAs

. . . . . . . . .

49

Figure 3.4

Temperature dependent mobility for p-type GaAs . . . . . . . . . .

50

Figure 3.5

Temperature dependent carrier concentration for p-type GaAs . . .

51

Figure 3.6

Temperature dependent IV curve for n-type GaAs with deposited
metal contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Temperature dependent mobility comparison for n-type GaAs with
contact deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Temperature dependent concentration for n-type GaAs contact deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Figure 3.9

IV curve for p-type Ge . . . . . . . . . . . . . . . . . . . . . . . . .

54

Figure 3.10

Temperature dependent resistivity for p-type Ge . . . . . . . . . . .

55

Figure 3.11

Temperature dependent carrier concentration for p-type Ge . . . . .

56

Figure 3.12

Temperature dependent mobility for p-type Ge . . . . . . . . . . . .

56

Figure 3.13

Current vs. voltage comparison for Ohmic and non-Ohmic contacts
for a moderately doped single crystal diamond (SND02) . . . . . . .

58

Hall mobility data comparison between Ohmic and non-Ohmic contacts for a moderately doped single crystal diamond sample (SND02)

59

Figure 3.15

IV measurements for a moderately doped sample (SND02) . . . . .

60

Figure 3.16

Temperature dependent concentration measurements for a moderately doped sample (SND02) with comparison to data reported by
Borst, et al[9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Figure 3.7

Figure 3.8

Figure 3.14

Figure 3.17

Temperature dependent mobility measurements for a moderately doped
sample (SND02) with comparison to data reported by Borst, et al.[9] 61

Figure 3.18

Effect of applied current level to concentration measurements for a
moderately doped sample (SND02) . . . . . . . . . . . . . . . . . .

63

Effect of applied current to Hall mobility measurements for a moderately doped sample (SND02) . . . . . . . . . . . . . . . . . . . . . .

64

Figure 3.19

viii

Figure 3.20

Effect of applied current to resistivity measurements for a moderately
doped sample (SND02) . . . . . . . . . . . . . . . . . . . . . . . . .

65

Carrier concentration versus temperature for nominal compositions
of M g2.08 Si0.4−X Sn0.6 SbX . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 3.22

Hall mobilities for M g2.08 Si0.4−X Sn0.6 SbX . . . . . . . . . . . . . . .

68

Figure 4.1

Intrinsic conduction. (a) Density of levels plotted for two separated
bands separated by energy gap, g . A hole in the valance band is
formed when energy not less than g is supplied to the system. The
electron from the valance band is liberated in the conduction band.
(b) Impurity levels supply electrons to the conduction band.[10] . . .

70

Energy band diagrams of metal-semiconductor contacts. Metal and
semiconductor (a) in separated systems, and (b) connected into one
system. As the gap δ (c) is reduced and (d) becomes zero. [11]. . . .

75

Electron affinity of diamond (χ) and work function (φ) for different
doping and orientation [12] . . . . . . . . . . . . . . . . . . . . . . .

77

Dependence of specific contact resistanceon doping concentration (and
E00 ), barrier height, and temperature. TE, TFE and FE are indicated
[11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.5

Sample image for transmission line measurements (TLM). . . . . . .

80

Figure 4.6

Transmission line measurements (TLM).

. . . . . . . . . . . . . . .

81

Figure 4.7

A zone diagram of an ohmic contact with a) low barrier height and
b) high doping level [13]. . . . . . . . . . . . . . . . . . . . . . . . .

81

Low boron doped diamond (LB04) selective doping process steps.
a)Low boron doped surface b)High boron doped surface c) Surface
with Al mask before etching d) Surface with Al mask after etching.

82

Shadow mask alignment prior to Ti/Au deposition on boron doped
diamond sample (LB04) . . . . . . . . . . . . . . . . . . . . . . . .

83

IV curves at various temperatures (300K to 700K) for boron doped
diamond sample (LB04). . . . . . . . . . . . . . . . . . . . . . . . .

84

Hall coefficient data for Ge [10] . . . . . . . . . . . . . . . . . . . . .

85

Figure 3.21

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

ix

Figure 4.12

Resistivity data for Ge [10] . . . . . . . . . . . . . . . . . . . . . . .

87

Figure 4.13

Concentration vs temperature fit for p-type germanium . . . . . . .

89

Figure 4.14

Conductivity vs temperature fit for p-type germanium . . . . . . . .

90

Figure 4.15

Temperature dependent IV curves for SND26. (a)Temperature range
of 300K-700K (b)Effect of annealing at 300K. . . . . . . . . . . . . .

91

Figure 4.16

Concentration vs temperature fit for SND26 . . . . . . . . . . . . .

92

Figure 4.17

Activation energy for SND26 . . . . . . . . . . . . . . . . . . . . . .

93

Figure 4.18

Fitting vs experimental results using single carrier (dashed line) and
mixed carrier (straight line) model for SND26 . . . . . . . . . . . . .

93

Figure 4.19

Fitting with varied Na for SND26 . . . . . . . . . . . . . . . . . . .

94

Figure 4.20

Fitting with varied Ea for SND26 . . . . . . . . . . . . . . . . . . .

94

Figure 4.21

Fitting with varied Nd for SND26 . . . . . . . . . . . . . . . . . . .

95

Figure 4.22

Fitting with varied Na for SND26 . . . . . . . . . . . . . . . . . . .

95

Figure 4.23

Fitting with varied mp/mo for SND26 . . . . . . . . . . . . . . . . .

96

Figure 4.24

Mobility analysis, simulation and experimental values for SND25 . .

97

Figure 4.25

Activation energy for SND25 . . . . . . . . . . . . . . . . . . . . . .

97

Figure 4.26

Activation energy for SND33 . . . . . . . . . . . . . . . . . . . . . .

98

Figure 4.27

Activation energy for SND34 . . . . . . . . . . . . . . . . . . . . . .

98

Figure 4.28

Temperature dependent mobility . . . . . . . . . . . . . . . . . . . .

99

Figure 4.29

Temperature dependent concentration . . . . . . . . . . . . . . . . .

99

Figure 4.30

Concentration vs temperature fit for SND25 . . . . . . . . . . . . . 100

Figure 4.31

Concentration vs temperature fit for SND34 . . . . . . . . . . . . . 101

Figure 4.32

SIMS measurements for diamond samples [14] . . . . . . . . . . . . 102

Figure 4.33

Concentration vs temperature fit for LB04 . . . . . . . . . . . . . . 102
x

Figure 4.34

Left: All the postivite and negative charges are shown. Right: Zoomed
in image where the neutrality equation is satisfied. . . . . . . . . . . 105

xi

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my advisor, Prof. Timothy Hogan, who
leads by example, for his outstanding guidance, consistent encouragement and being my role
model both professionally and personally. Special thanks go to Prof. Timothy Grotjohn,
Prof. Donnie Reinhard, Prof. Jes Asmussen and Prof. Greg Swain for being my committee
members and their excellent support and guidance throughout my PhD studies. Also, I
would like to thank my colleagues in the Electronic Materials and Pulsed Laser Deposition
and Diamond Electronics Labs.
My gratitude extends to my parents, Hayriye and Ismail Berkun, for their support and
encouragement. Also I would like to thank to my best friend and grandma Ayhan Berkun,
my family members Funda Berkun and Alparslan Berkun. Their prayers for me during
the course of this study have brought many blessings into my life. Special gratitude and
appreciation goes to my yoga family in East Lansing. Their friendship and community
have provided me support and encouragement. Last but not least, I would like to extend
my sincere gratitude to my friends Will Cicola, Karyn Coward, Kazuko Fuchi, Sedat Gur,
Letizia Tozzini, Ami Blue, Kitty Cox, Cagri Ozcaglar, Kathleen Donahue and many others.

1

Chapter 1
Introduction

1.1

Why Diamond?

Diamond has a broad range of applications incuding but not limited to power electronics, micro - electromechanical systems (MEMS), biomedical sensors, optics, thermal management,
electrochemical electrodes and lasers. A few examples for diamond’s common application
areas are as follows:
Diamond is a versatile material. It is commonly used in improving the mechanical performance of industrial parts, processed by different methods including physical vapor deposition (PVD). Due to the extraordinary electrochemical properties of diamond including
relatively low background current which results to improved signal-to-backround (SBR) and
signal-to-noise (SNR) ratios, diamond has a high application potential in both aqueous and
non-aqueous media and holds microstructural and morphological stability at relatively high
temperatures (e.g. 180◦ C) [15]. Diamond has a high thermal conductivity (approximately
four times higher than copper), thus diamond is a practical material for thermal applications. The chemical and biological inertness of diamond allows for its use in severe chemical
and radioactive environments that destroys other materials and helps it find practical applications in biosensor area [16]. Diamond has a highly symmetrical lattice as well as very
strong bonding, thus, diamonds cubic symmetry minimizes the impact of multiple crystalline
orientations in the film on performance which makes it a useful material for optics as seen
2

Bandgap (eV )
Maximum Electric Field (M V /cm)
Electron Mobility (cm2 /V s)
Hole Mobility (cm2 /V s)
Thermal Conductivity (W/mK)
Relative Dielectric Constant

Si
GaAs 4H-SiC 6H-SiC GaN Diamond
1.12 1.424
3.26
3.03
3.295
5.47
0.41 0.48
2.4
2.4
5
20
1450 8600
900
400
2000
4500
480
130
120
100
200
3800
1.5
0.46
4.9
4.9
1.3
20
11.7 12.9
10
9.7
5.35
5.7

Table 1.1 Electrical properties of common wide bandgap semiconductors compared to silicon
[3]
in Fig. 1.1. Diamond is transparent at a wide range of wavelengths from the mid-ultraviolet
through the microwave portion of the spectrum and beyond [7].

Figure 1.1 Mounted optical grade polycrystalline diamond window, ready for use in high
power CO2 laser optics [7]

Due to the properties of diamond which are described above it is a high interest material
for various applications. Diamond is compared with other common semiconductors in Table.
1.1. This work mainly focuses on applications of diamond in electronic devices, taking
advantage of it’s higher power, higher operating voltages and higher operational temperatures
due to wide band gap.
Commonly used growth techniques for making synthetic diamond include the high pressure high temperature technique (HPHT) [17], hot filament chemical vapor deposition (HFCVD),
3

and microwave plasma chemical vapor deposition (MPCVD) [18]. The HPHT method provides growth of bulk crystals which are used to grow thin single crystal films on the top
surface in this work, while the CVD-method is used for homoepitaxial growth of single crystal films on single crystal substrates. Thus, the crystal quality of films highly depends on the
quality of substrate. Characterization of MPCVD grown p (mainly boron doped) or n type
(mainly phosphorus doped) single crystal diamond films (SCD) on top of HPHT substrates
for power electronic applications will be the main focus of this thesis. An essential need in
the study of boron or phosphorus doped diamond semiconductors is the reliable and accurate determination of the critical electronic transport properties, such as the temperature
dependent conductivity, carrier concentration, and carrier mobility of the deposited film. A
detailed description on the measurement of these critical properties for a temperature range
of 300K - 700K are provided in the following sections of this thesis.

1.2
1.2.0.1

Electrical Transport Characterization
Hall Effect Systems

In order to accurately characterize the electrical transport properties of semiconductors,
temperature dependent Hall effect systems have been built by several research groups and
reported in the literature, such as systems for thermoelectrics materials where the Van der
Pauw method is used with pressure contacts [19], systems for refractory materials having
high charge carrier concentrations; generally with low mobilities [20], some systems include
a circuit which combines the best features of both ac and dc apparatus where errors due
to thermal emf’s and the Ettingshausen effect are eliminated [21], measurement systems for
ferromagnetic semiconductors [22], systems for low-mobility samples at high temperatures
4

[23], systems that are capable of measuring eight samples simultaneously [24], systems for
low electron mobility materials such as copper [25]. A temperature dependent Hall Effect
measurement system with software based data acquisition and control was built and tested
as a part of this PhD work [1].This system alters from the previously reported systems by
adding several control components including a software based PID temperature controller,
temperature dependent I-V measurements for monitoring the annealing effects on contact
resistances under vacuum environment, and the capability to accommodate samples for both
van der Pauw and inline geometrical configurations. Electronic transport measurements are
shown for boron-doped single crystal diamond (SCD) films deposited in a microwave plasmaassisted chemical vapor deposition (MPCVD) reactor. The influence of ohmic contacts and
PID temperature control accuracy and stability are studied. For a temperature range of
300K - 700K IV curves, Hall mobilities and carrier concentrations are presented as shown in
the following sections in more detail.

1.2.0.2

Hall Effect Measurements on Diamond

Electrical transport studies for diamond in literature consists of several types of diamond
samples including HPHT grown bulk single crystal diamond and MPCVD grown diamond
thin films. Temperature dependent Hall effect measurements have been reported by several groups. These studies report three types of conduction mechanisms along with three
activation energies.

σ = σ1 ∗ exp(−Ea1 /kT ) + σ2 ∗ exp(−Ea2 /kT ) + σ3 ∗ exp(−Ea3 /kT )

(1.1)

In Eq. 1.1, Ea1 is the activation energy for valance band conduction due to acceptor

5

ionization, Ea2 is an activation energy for impurity band conduction due to the motion of
carriers over neutral acceptors and Ea3 is an activation energy for hopping conduction due

Hopping
conduction

105

p(Ω c m)

103

Metallic conduction

to the transition of holes from occupied to unoccupied acceptor sites.

101
Valence band
conduction

10-1

10-3
1016

1017

1018

1019

1020

1021

[Β] (cm-3)

Figure 1.2 Room temperature resistivity as a function of doping concentration. [8]

The resistivities of the diamond films vs. the boron doping are plotted in Fig. 1.2 by
Lagrange et al.[8]. According to the report, the resistivity of an undoped film was around
1013 Ωcm. Considering the results derived from the I(T ) curves, depending on the doping,
three conduction mechanisms are observed at room temperature. For [B]< 1019 cm3 , the
conduction is mainly dominated by the free holes within the valence band. As the resistivity
decreases linearly with [B], the decrease in mobility is relatively low in this region. Between
1019 and 3∗1020 cm3 , the hopping conduction dominates, and the resistivity decreases strongly
with [B]. Above 3 ∗ 1020 cm3 , progression of resistivity is not as strong due to the metallic
behaviour of the conduction.
Borst et al. fabricated and characterized boron doped homoepitaxial diamond films using

6

Sample #
#1
#2
#3
#4
#5
#6
#7

Concentration (cm−3 )
7 − 15x1017
–
1.2x1019
4 − 5x1019
1.2x1020
3x1020
3 − 9x1018

Activation energy (eV )
0.34
0.33
0.27
0.16
0
0
0.30

µ ∼ TLx - THy
TL0.8 - TH−2.4
TL1.1 - TH−2.3
TL1.6 - TH−1.8
TL1.3 - TH−1.5
T 2.8
Almost linear
TH1.3 - TH2.2

µmin - µmax (cm2 /V s)
5x10−6 - 250
3.7x10−4 - 170
0.11 - 55
0.55 - 16
0-4
3.5 - 4.5
2.2x10−2 - 100

Table 1.2 Concentration, activation energy an Hall mobility of seven boron doped homoepitaxial diamond, Borst et al. [4]
microwave plasma chemical vapor deposition (MPCVD). Substrate types were type IIa for
samples #1 and #7 and type Ib for all the other samples. Concentration and mobility data
are summarized in Table. 1.2 for the temperature range of 100 to 1300K [4].
They have reported 3 main conduction mechanisms taking place: 1. Valance band conduction. 2. Impurity band conduction (sample #5) and 3. Hopping conduction (samples
#2, #5 and #7). Sample # 6 is degenerated and showed metallic conduction.
In all other samples, valance band conduction was observed above 300K. Above 773K,
the resistance of the type Ib sample becomes comparable and even lower than the resistance
of the Hall bars which results as a misguiding raise of the conductivity. As a consequence, it
has been reported that only samples # 1 and #7 were reliable in the high temperature range
and showed the predicted transition from the exponential increase of the freezeout region to
the exhaustion or saturation region.
Hopping conduction can also be present at room temperature in boron doped diamond
due to the wide bandgap of diamond and large ionization energy of boron. In this case,
nitrogen impurities and other unidentified defects contributed as donors with concentration
nD and gave their electron to the boron acceptors of concentration. Tunneling may occur in
the form of nearest neighbor hopping or variable range hopping by thermal activation.
Another parameter which influences diamond electronic device performances is the elec7

tron and hole mobility. The empirical model used to fit the experimental Hall effect measurements reported in literature has been implemented into Silvaco finite element based software
via a built-in C language interpreter, in order to take into account the doping concentration and temperature dependence of the mobility [26]. An empirical mobility model they
developed shows good agreement with the experiments over a wide range of doping concentration, from NA = 1014 cm3 , where the mobility is equal to 2016cm2 /V s, to NA = 1021 cm3
where the mobility falls down to unity. For the phosphorus donor, the mobility is maximum
µmax = 1030cm2 /V s,for Nd = 1014 cm−3 , and close to unity when Nd = 1020 cm−3 .
During the determination of the electrical transport properties of the materials, defects
and the shape of the sample have a critical effect. Electrical properties in the temperature
range of 180 − 800K with the influence of defects and geometry for high quality bulk single
crystal IIb-type diamonds grown by the HPHT method are extensively studied by Bormashov
et al. by measuring the defect free region from the actual sample [27]. They did not
observe a significant change in resistivity, however observed a significant difference in the
Hall effect data. The non-rectangular shaped sample showed a lower carrier mobility due to
the inhomogenous acceptor and donor concentrations.
Borsmashov et al. determined the actual concentration of acceptors (NA , boron) and
compensating donors (ND , nitrogen) by the regression analysis of the temperature dependencies of Hall effect data using the least squares method. They varied both concentration
values of the theoretical model of conductivity where Fermi distribution of electrons and
holes and the neutrality equation is used, until minumum deviation from the experimental
data was achieved. Tau-approximation fit for collusion integral has been used and four scattering mechanisms of carriers were taken into account: by ionized and neutral impurities,
and optical and acoustic phonons. At low boron conditions, acceptor concentration decreases
8

and compensation ratio increases (for mixtures containing less than 0.001% boron).
The maximum mobility achieved was in the range of 2000 − 2200cm2 /V s at T = 300K
and more than 7000cm2 /V s at T = 180K was found from the mixture which contained
0.0005 to 0.002% boron. As a result of low impurity concentration and low compensation
ratio, carrier scattering is not as strong on ionized and non-ionized impurities. Thus, the
dominating mechanism is the phonon (lattice) scattering of holes where scattering by neutral
and ionized impurity atoms is minimal. In this particular case, the mobility of holes does
not depend on the impurity concentration.
For the heavily doped sample, Hall carrier mobility was shown to be very low in the
whole temperature range. In addition to the deviation from the valance band conductivity
to hopping conduction, defects play a significant role in the electrical transport properties
of samples.
Pernot et al. combined theoretical and experimental studies for Hall hole mobility of
homoepitaxial boron-doped diamonds in the temperature range of 100K to 900K where the
maximum hole mobility was reported to be around 2000cm2 /V s [28]. Four hole scattering
mechanisms have been considered: ionized impurity scattering, neutral impurity scattering,
acoustic phonon scattering, and nonpolar optical phonon scattering. Scattering mechanisms
and their anisotropy as well as the anisotropy of the valance band energy surface are the
three main factors that contribute to the combine Hall scattering factor, rH which is used
to compare the theoretical to experimental data.
(i)In order to study the temperature dependency of the mobility in low boron doped
diamond, they looked at four samples with fixed acceptor concentration NA = 1016 cm−3 and
donor concentrations ranging from ND = 0 to 1014 , 1015 , 5x1015 cm−3 . They have observed
the temperature dependence of the four samples are very similar in the low temperature
9

range (150-300K) with a joint effect of ionized impurity scattering and acoustic phonon scattering (below 200K). However, the high temperature range (larger than 300K) is dominated
by phonon acoustical and optical phonons. The acoustical phonon scattering limited the
mobility between 200K and 300K.
For the sample with NA = 1016 cm−3 and ND = 1015 cm−3 , the neutral impurity contribution is considerably small in the whole temperature range. This negligible contribution
was due to the low density of neutral boron atoms. Between 100K and 200K, ionized impurity scattering dominates due to the ionized compensated centers. Between 200K and
400K, acoustic phonnon scattering dominates which results in a temperature dependency of
mobility as T −3/2 . For the high temperature range, optical phonon scattering dominates.
Mobility reported at 700K is 100cm2 /V s [28]. Effective mass of light holes is low, as a result,
they exhibit the highest mobility of the three bands. Effective mass of the spin-orbit holes
is in between the light and heavy holes, thus, they have a mobility close but reasonably
higher than the total mobility. Most influencing holes are the the heavy holes due to the
high density of holes in this band.
(ii)In order to study the temperature dependency of the mobility in highly boron doped
diamond, they have investigated several different samples. The moderately doped sample
with NA = 2.5x1017 cm−3 and ND = 1015 cm−3 , low temperature mobility is controlled mainly
by acoustic scattering, the impurity and neutral impurity scattering mechanisms does not
have a significant effect. For the sample with NA = 2x1018 cm−3 and ND = 3x1015 cm−3 ,
the most important scattering mechanism between 200 and 400K is the neutral impurity
scattering, which gives a mobility of 1000cm2 /V s at room temperature with an activation
energy of 0.37eV for boron. This corresponds to an ionization ratio of p/NA lower than
a few % up to 400K indicating that a large majority of the boron acceptors are neutral.
10

For the most highly doped sample with NA = 1.7x1019 cm−3 and ND = 5x1015 cm−3 , the
low temperature mobility is completely controlled by neutral impurity mode up to 600K
which corresponds to a mobility value of 200cm2 /V s. For all the three highly boron doped
samples described above, the low temperature mobility decreases due to hopping. From
valance band conduction to hopping conduction transition temperature is 150K, 200K and
223K respectively (from low to high doped sample).
Pernot et al. summarized the mobility vs temperature relationship as follows: at 300K,
lattice scattering (acoustic and optical) at low doped materials up to 1017 cm−3 . When
acceptor concentration NA is larger than 1017 cm−3 neutral impurity scattering is important,
when NA is larger than 3x1018 cm−3 , neutral impurity scattering is dominating. Ionized
impurity scattering is reported to be not as important in the whole temperature range due
to the neglection of the compensation and the high ionization energy of boron compared to
the ratio of the ionized boron. At 500K, lattice scattering is reported to be important up to
NA = 1018 cm−3 . In the case of highly doped diamond such as acceptor concentrations higher
than 1018 cm−3 , neutral impurity mode becomes important and for acceptor concentrations
higher than 2x1019 cm−3 , neutral impurity mode dominates and ionized impurities are not
important [28].
Volpe et al. studied the mobility for low boron doped samples where the acceptor density
of the samples has been estimated by capacitance voltage, C(V), measurements around
1016 cm3 with a maximum mobility value of 1870cm2 /V s at 292 K. Table 1.3 summarizes the
mobility and activation energy values for room temperature [5].
Sample #1 was measured to have a mobility significantly higher than the other samples.
This is unexpected since the total dopant concentration is comparable to that of sample #2.
The activation energy for sample #1, however, is significantly lower than for samples #2
11

Sample #
#1
#2
#3

NA (cm−3 ) ND (cm−3 )
1.5x1016
4.7x1015
16
1.7x10
1.5x1015
2.2x1016
2.5x1014

Ea (eV ) µ at 292K (cm2 /V s)
0.295
1870
0.380
850
0.380
870

Table 1.3 The mobility and activation energy values for room temperature on low boron
doped samples [5].
and #3. An underestimation of the ionization energy would result in an overestimation of
the donor density since the two parameters are strongly correlated. Essentially, the ionized
impurity scattering should not be the dominant scattering mechanism for donor densities
lower than 5x1015 cm3 , and the limitation of the mobility at room temperature suggested to
be due to the crystalline quality of the layer, mostly related to the growth conditions [5].
It is common that more than one scattering mechanism is present in diamond. Time
of flight (TOF) measurements are one of the few methods to identify the scattering mechanisms involved. CVD diamond samples are compared with data obtained on commercial
samples, literature and theoretical data for IIa diamond in order to report dominant scattering mechacnisms for different temperature regimes by Nesladek et al. [29]. TOF tecnique
was also used by Isberg et al. in order to determine the low-field drift mobility values on
thick diamond plates for p-i junction diode studies [30]. The boron concentration of their
sample was reported to be 1x1020 cm−3 .
Deposition conditions have a tremendous effect on surface morphology and electrical
properties of boron-doped diamond films synthesized by MPCVD. Effects of trimethylboron
on diamond (100) to electronic transport properties of the substrate are presented by Tsubota et al. [31]. They also reported the effects of hydrogen plasma treatment to the surface
smoothness and elecrical properties of the boron-doped diamond films. Barjon et al. optimized the growth conditions for MPCVD grown boron doped diamond samples with boron

12

concentrations ranging from 1x1018 cm−3 to 3x1019 cm−3 and reported the reduction of compensation effects [32]. The effect of substrate temperature between 850 − 950◦ C and growth
rate on the doping efficiency of single crystal boron doped diamond was recently reported by
Demlow et al. where heavily boron doped samples with a substrate temperature of 950◦ C
showed fewer defects and higher doping efficiency than the samples grown at 850◦ C for boron
concentrations ranging from 8.9x1018 cm−3 to 5.8x1019 cm−3 [14].

1.2.1

Ohmic Contacts

Ohmic contacts are essential for accurate Hall effect measurements. The use of silver paste
as a contact to diamond yields a high contact resistance.
Das et al. reviewed the processing of metal contacts such as Al or Au on CVD films which
showed highly resistive ohmic or nominally asymmetric behavior, where these contacts can
be used for forming rectifying contacts on synthetic and natural semiconducting diamond
crystals, and also this work pointed out the ohmic contact formation by using metals such
as Ti, Ta and Mo with a layer of Au on top, followed by annealing [33].
Transition-metal/Au bilayer metallization have been studied by several groups. Moazed
et al. used transition metals including Ti, Ta, Mo and Ni with a thickness of 100 − 150˚
A and
a 1500˚
A layer of Au on top, followed by annealling at 885C ◦ for 8-16min in an H2 ambient
to form ohmic contacts [34]. Volpe et al. used Ti/Pt/Au pads for low boron doped samples
followed by annealing at 750C ◦ during 30 min to obtain reliable ohmic contacts [5]. Ohmic
contacts made out of Ti/Al/Au were deposited by PVD followed by annealing in argon at
600◦ C for the boron doped side of the p-i diode as reported by Isberg et al. [30].
Experimental contact studies have been reported from the same group by using Ti/Au
contacts on diamond by B ion implantation where contact resistance was on the order of
13

10−6 Ω.cm2 [35]. They also fabricated similar contacts on polycrystalline films as well as
a natural IIb crystal [36] and reported the electrical characterization of the diamond films
including Hall effect measurements and other characterization techniques such as secondary
ion mass spectroscopy (SIMS) [37].
Thin films of Mo, Mo/Au and Mo/Ni/Au were also used as contact metals on diamond by
a thermally activated solid state reaction process which produced adherent ohmic contacts
after annealing at 950◦ C by Moazed et al. [34].
Barjon et al. reported the resistivity of MPCVD grown boron doped diamond samples
where Ti/Au contacts were fabricated followed by annealing at 450C ◦ for 30 min under Ar
in order to obtain an ohmic contact [32].
Using a similar approach, Hewett et al. tested the contacts including Ti, Mo, Al, Ni, Au,
Pt and Pd and reported contact resistance as low as 2x10−5 Ωcm2 on single crystalline films
with boron concentrations of the order of 1020 cm−3 . They have studied a transmission line
model (TLM) for their contact resistance measurements as well as surface roughness measurements to verify the formation of a carbide for Ta/Au, Ti/Au and Mo/Au contacts[38].
On heavily doped diamond samples with a concentration value of 1019 cm−3 , Geis observed that ohmic contacts can be obtained with most evaporated metals without any additional processing with contact resistance of 10−6 Ωcm2 [39]. Braunstein et al. used boron
implantation-doping of diamond. They have reported conductivity and Hall measurements
showing results as a typical heavily doped semiconductor. Ohmic contacts were achieved by
using sharp tungsten probes [40].
An annealed Ti/Pt/Au trilayer metallization is a commonly used procedure as reported
by Hoff et al. for a naturally occurring type IIb diamond crystal where the Pt served
successfully as a barrier to Ti diffusion into the Au capping layer [41], by Bormashov et al.
14

in the Van der Pauw configuration [27] and by Chen et al. on boron-doped homoepitaxial
diamond films with a reported contact resistance of 1.3x105 Ωcm2 [42].
Mo/Au contacts deposited on a type II b diamond crystal also reported to show ohmic
behaviour with no indication of deterioration or degradation of performance [43].
In normal conditions Au contacts alone on top of boron doped (100) diamond would
form rectifying contacts, however Chu et al. showed the fabrication of ohmic contacts by
exposure to an argon sputtering prior to the deposition of Au onto semiconducting samples.
They also observed that the presputtering step improved the adhesion of the Au film which
is commonly a big challange to achieve without a transition metal between diamond and Au
[44].
Indium has been used in the past years on semiconducting diamond but due to its low
melting point it is not suitable for high temperature environments. Indium has been used
successfully to make an ohmic contact on semiconducting diamond crystals where higher
temperature regimes are not studied [39].
In diamond diode studies, high quality contacts are essential. Twitchen et al. showed an
abstract demonstration of a 2.5kV diamond diode provided by electrical measurements using
a circular gold Schottky contact, with an area of 1 > mm2 ,on large area free standing single
crystal diamond consisting of a thin high purity layer 1 < 1013 [B]/cm3 on a thicker heavily
boron-doped 1 > 1019 [B]/cm3 substrate with an ohmic back contact. The diode structures
were fabricated using a microwave-assisted chemical vapor deposition process [45].
A Si/SiC/diamond multilayer structure has been investigated on natural type-IIb diamond by Fang et al [46]. They deposited 500˚
A of Si by ion-beam mixing using Kr+ at
240 keV and substrate temperature of 700◦ C. Following the ion beam mixing, an annealing
step was processed at 1200◦ C for an hour in forming gas. Transmission line measurements
15

(TLM) showed measurements showed specific contact resistance of 5x10−3 Ωcm2 for samples
without ion mixing and 1x10−3 Ωcm2 for samples with ion mixing. They have also compared
Ta/Au contacts on top of diamond with a specific contact resistivity of 1x10−3 Ωcm where it
is reported to be non-uniform compared to the Si/diamond contacts.
Geis et al. have established ohmic contacts to type-IIb diamond using surface modification induced by argon-fluoride (ArF) excimer laser radiation with pulses of 193nm. It is
reported that the modified layer was stable at 1800C ◦ and was approximately 40 − 60cm
thick which was composed of diamond and graphite-like material with a minumum contact
resistance of (4 − 10)x10−4 Ωcm2 [47].
Sandhu was able to form ohmic contacts at room temperature to formerly implanted
layers in insulating diamond by a selected-area implantation method [48].
Selective doping is a critical method especially for more lightly doped diamond samples
in order to obtain ohmic contacs. Borst et al. reported electrical transport characteristics for
MPCVD grown B, P, Li and Na doped diamond films which were selectively grown as Hall
bars by using a sputtered SiO2 mask on the diamond substrates. Electrical metal contacts
were electron-beam evaporated and annealed at high temperatures [9]. In another paper,
Borst et al. have used a selective growth techique with a Hall bar geometry using a sputtered
SiO2 mask followed by bonding the gold wires to the Mo/Pt/Au contacts [4].
Prins et al. used implantation of boron ions to form low resistance ohmic contacts on
a type IIb diamond, followed by annealing. Top ion-damaged layer was then removed by
etching in a hot acid solution. The remaining higly doped thin surface was then processed
with contact deposition [49].

16

1.3

Hall Effect Measurement Methodology

Resistivity and Hall effect measurements are taken according to the ASTM standard F76-08
[50] which includes best practices for measuring Hall effect for two different sample geometries
of van der Pauw and Hall bar configurations as described in 1.3.1 and 1.3.2 respectively.
The Hall effect system includes the following components:
• Electromagnet (up to 1T field strength)
• Sample stage (cryostat with a high temperature stage)
• Current meter to provide current through the sample
• Voltmeter for measuring conductivity and Hall voltages
• Power supplies for the electromagnet and for temperature control of the sample stage
• A computer controlled switching system for controlling magnetic field direction and
voltage measurements
• A temperature detector and heater

1.3.1

Van der Pauw Method

Van der Pauw specimens, can have various different geometries according to the standards
[50]. Most commonly used one for this work is the square geometry shown in Fig. 1.3.

17

2

1

③

③

③

③

4

3

Figure 1.3 Van der Pauw configuration

Contact evaluation is required for measurement quality, ideally linear current voltage
characteristics for each contact pair should be observed to be able to obtain reliable measurements. For this work, IV curves for all contact pairs have been tested for different
temperature values between 300K to 700K. When the desired temperature stability is maintained, measurements start with resistivity, by applying current and measuring voltage as
shown in Eq. 1.2 and 1.3 where the notation VAB,CD refers to the potential difference VC −VD
measured between contacts C and D when current enters Contact A and and exits Contact B.
The constant 1.331 ∼ π/4 ln(2), units are in volts for voltages, V, centimeters for thickness,
t, amperes for current, I, and the geometrical factors, fA and fB are function of resistance
ratio, more details can be found in the ASTM standard F76-08[50].

1.1331fA t
[V21,34 − V12,34 + V32,41 − V23,41 ] Ω.cm
I
1.1331fB t
ρB =
[V43,12 − V34,12 + V14,23 − V41,23 ] Ω.cm
I
ρA =

(1.2)
(1.3)

If ρA is not equal to ρB within ± % 10, the specimen is inhomogenous and a more uniform
specimen is required. Average resistivity ρav is shown in Eq. 1.4.

18

ρavg =

ρA + ρB
Ω.cm
2

(1.4)

After the resistivity calculations, magnetic field (+B is out of the sample) is introduced
in both directions to calculate the Hall coefficient as shown in Eq. 1.5 and Eq. 1.6. Magnetic
field and current directions are flipped to eliminate non-Hall contributions.

2.5 × 107 t
[V31,42 (+B) − V13,42 (+B) + V13,42 (−B) − V31,42 (−B)] cm3 .C −3
BI
2.5 × 107 t
[V42,13 (+B) − V24,13 (+B) + V24,13 (−B) − V42,13 (−B)] cm3 .C −3
=
BI

RHC =

(1.5)

RHD

(1.6)

Similar to resistivity, If RHC is not equal to RHD within ± % 10, the specimen is inhomogenous and a more uniform specimen is required. Average Hall coefficient RHav is shown
in Eq. 1.7. Units are in volts for voltages, V , centimeters for thickness, t, amperes for current,
I, and Gauss for magnetic field, B.

RHav =

RHC + RHD
cm3 .C −3
2

(1.7)

Hall mobility can be calculated using the average Hall coefficient, RHav and resistivity, ρav
as shown in Eq. 1.8.
µH =

|RHav |
cm.V −1 .s−1
ρav

(1.8)

For n-type samples, Hall coefficient is negative whereas, for p-type samples Hall coefficient
is positive. Carrier concentration for a p-type semiconductor can be calculated using the
average Hall coefficient as shown in Eq. 1.9.

19

p=

1.3.2

1
RHavg × q

cm−3

(1.9)

Hall Bar Method

Hall bar samples can have different type of geometries [50]. The thickness of the sample
needs to be uniform to ± %1 and should be less than 0.1cm. Length of the specimen must
be between 1cm to 1.5cm. Length to width ratio should be ideally greater than 5, however,
4 is the minimum ratio value that would be acceptable for measurements.
Similar to the Van der Pauw samples, Hall bar samples also should have linear current
voltage characteristics. To evaluate the contacts, every contact pair should be separately
tested for different temperature points. When the desired temperature stability is maintained, measurements start with resistivity, by applying current and measuring voltage as
shown in Eq. 1.10 and 1.11 for Hall bar sample with 6 contacts where W is the width, d1
and d2 are the distances between the contacts in cm as shown in Fig. 1.4.

Figure 1.4 Hall bar configuration

20

Wt
[V12,46 − V21,46 ] Ω.cm
2Id1
Wt
[V12,35 − V21,35 ] Ω.cm
ρB =
2Id2
ρA =

(1.10)
(1.11)

If ρA is not equal to ρB within ± % 10, the specimen is inhomogenous and a more uniform
specimen is required. Average resistivity ρav is shown in Eq. 1.12.

ρavg =

ρA + ρB
Ω.cm
2

(1.12)

Once resistivity is calculated, magnetic field (+B is out of the sample) is introduced in
both directions to calculate the Hall coefficient as shown in Eq. 1.13 and Eq. 1.14.

2.5 × 107 t
[V12,65 (+B) − V21,65 (+B) + V21,65 (−B) − V21,65 (−B)] cm3 .C −3 (1.13)
BI
2.5 × 107 t
=
[V12,43 (+B) − V21,43 (+B) + V21,43 (−B) − V12,43 (−B)] cm3 .C −3 (1.14)
BI

RHA =
RHB

Similar to resistivity and hall coefficient in VdP samples shown in 1.3.1, If RHA is not
equal to RHB within ± % 10, the specimen is inhomogenous and a more uniform specimen is
required. Average Hall coefficient RHav is shown in Eq. 1.15. Units are in volts for voltages,V,
centimeters for thickness, t, amperes for current, I, and Gauss for magnetic field, B.

RHav =

RHA + RHB
cm3 .C −1
2

(1.15)

Hall mobility can be calculated using the average Hall coefficient, RHav and resistivity, ρav

21

as shown in Eq. 1.16.
µH =

|RHav |
cm.V −1 .s−1
ρav

(1.16)

Carrier concentration for a n-type semiconductor can be calculated using the average
Hall coefficient as shown in Eq. 1.17.

n=

1.3.3

1
RHavg × q

cm−3

(1.17)

Quantitative Mobility Spectrum Analysis

Quantitative Mobility Spectrum Analysis (QMSA) is a novel method for characterization of
diamond films with mixed carrier conduction.
Conventional single magnetic field Hall characterization is not capable of providing a
precise determination of the electronic transport properties in materials with mixed carrier
conduction. When magnetotransport experiments are performed as a function of magnetic
field, one can measure the data to obtain densities and mobilities for each carrier present as
shown in the literature [51, 52, 53, 54].
QMSA is the most effective tool to characterize mixed conduction devices. It is possible
to decide the number of carriers to fit the data and QMSA is capable of showing how well data
fits that parameter without any prior assumption. When systematic temperature analysis
are included in the study, more details can be obtained [55].
In the magnetic field dependent Hall Effect measurement, current is applied from terminal
1 to 2 on the Hall bar configuration as shown in Fig. 1.4 and a magnetic field, Bz , is applied
perpendicular to the device. By measuring the voltage V35 between terminal 3 and 5, the
resistivity ρ(Ω.Cm) is calculated by Eq. 1.18.

22

ρ=

Ex
V35 /d2
=
Jx
I12 /(W.t)

(1.18)

The Hall voltage, Vy , is measured between terminals 3 and 4, and the Hall coefficient
RH (cm3 C −1 ) can be calculated by Eq. 1.19.

RH =

Ey
Vy /dy
dz Vy
=
=
Bz Jx
Bz Ix /(W.t)
Bz Ix

(1.19)

The carrier concentration n(cm−3 ) and mobility µ are easily obtained from Eq. 1.20 and
1.21.

n=

1
eRH

(1.20)

RH
ρ

(1.21)

µ=−

The conductivity tensor σij (Ω−1 cm−1 ) can be calculated from Eq. 1.22 and 1.23.

σxx (B) =

qnµ
1 + (µB)2

(1.22)

σxy (B) =

qnµ2 B
1 + (µB)2

(1.23)

The sheet resistivity can still be measured when film thickness is unknown as shown in Eq.
1.24.

ρs =

dy Vx
dx Ix
23

Ω
sq

(1.24)

Sheet resistivity can lead to sheet carrier concentration ns (cm−2 ). Conductivity tensor
can be shown as Eq. 1.25 and 1.26.

σxx (Bj ) =

enµ
1 + (µBj )2

(1.25)

σxy (Bj ) =

enµ2 Bj
1 + (µBj )2

(1.26)

where e is the electron charge, n is the sheet carrier concentration, Bj is the variable
magnetic field, and µ is the electron mobility. For the case of more than one carriers, each
carrier can have different electron mobilities.

1.4

This Work

In this work boron doped single crystal diamond (SCD) films were deposited on high pressure
high temperature SCD substrates in a microwave plasma-assisted chemical vapor deposition
(MPACVD) bell-jar reactor. Dopant concentrations ranged from below 1017 cm−3 to over
1020 cm−3 [56].
A high temperature Hall effect system which operates in the temperature range of 300K
to 700K was designed and built as a part of this thesis. System capabilities not only include
temperature dependent resistivity, carrier concentration and mobility experiments but also
include the effects of influences such as temperature stability and non-ohmic contacts on the
electronic transport measurements [1].
Temperature dependent Hall effect measurements for p-type single crystal diamond have
shown carrier concentrations from below 1017 cm−3 to approximately 1021 cm−3 with mobili-

24

ties ranging from 763(cm2 /V s) to 0.15(cm2 /V s) respectively.
This work is motivated by the investigation of diamond diodes with breakdown voltages
of around 1200V , thus an extensive characterization of diamond films are essential. In this
work, the developed Hall effect systems was designed with an open architecture which allows
us to analyze all of the individual signals measured for both of Van der Pauw and Hall bar
configurations from 300K to 700K. Compared to some existing systems, this measurement
system includes the capability to accommodate both Van der Pauw and Hall bar (or inline)
configurations in order to avoid electrical problems associated with the misalignment of
the contacts, distances between the contacts are recorded and taken into account when
calculating the Hall voltage according to ASTM standards [50]. Sample geometries are
flexible and are not predefined in order to fit to the certain sample holders. This system has
a broad range for low voltage measurements for metals as well as resistive materials.

25

Chapter 2
Temperature Dependent Hall Effect
System

2.1
2.1.1

System Setup and Design
System Overview

Temperature dependent Hall effect system is an essential measurement tool for characterizing the temperature dependent transport properties of semiconductors such as resistivity,
mobility and concentration. It is also critical for defining the type of mobile charges in a
given semiconductor, either as positive or negative. Depending on the type of the charge
carriers, a semiconductor can be defined as p or n type.
A new temperature dependent Hall effect system is presented to characterize the properties of new materials, primarily doped diamond films and thermoelectrics. This system is
fully software automated, thus, all the inputs can be given to the computer independent of
the user’s physical location, i.e. a mobile phone access is sufficient to run the data acquisition
and measurement system control and to check on the status of the measurements.
Heaters are hooked up to the stage and PID control is embedded to the software. Temperature control of the sample stage is accomplished by proportional-integral-derivative (PID)
control of heater cartridges embedded in the oxygen free high conductivity (OFHC) copper

26

sample stage. Measurements of heating up from 300K to 700K with 25K steps takes less
than 7h. Measurements are also taken when cooling back down to room temperature to be
able to observe the repeatability of the measurement. The system is capable of computer
controlled temperature dependent IV analysis to evaluate the quality of contacts, usually
before the Hall voltage measurement since ohmic contacts are needed for accurate results.
Temperature dependent I-V measurements also provide insightful information on the influence of annealing and help gain understanding of the impact of non-ohmic contacts on the
Hall measurements. Fig. 2.1 shows Hall mobility for a moderately doped diamond sample
with ohmic and non-ohmic contacts with comparable values (in the same order of magnitude)
but significantly more scatter in the data for the non-ohmic contacts.
250
non-Ohmic contacts
Ohmic contacts

2

Hall Mobility (cm /Vs)

200

150

100

50

0
300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 2.1 Hall mobility data comparison between Ohmic and non-Ohmic contacts for a
moderately doped single crystal diamond sample (SND02)

The number of voltage and temperature readings taken can be controlled by the user,
saving the average values and standard deviations in the measurement file for evaluation of
repeatability and quality of the measurement.
27

19

-3

Conecentration (cm )

5x10

19

T not stable
T stable

4x10

19

3x10

19

2x10

19

1x10

0
300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 2.2 Temperature Stability Effects

Diamond was used as a reference sample to investigate the influence of temperature
stability on the measured values of concentration as shown in Fig. 2.2. The influence of such
non-stable conditions is depicted by the red circular symbols and leads to increased scatter
in the measurements.
In this system, as seen in the flowchart, Fig. 2.3, after temperature stability is achieved,
the resistivity and Hall coefficient measurements are taken, then the temperature is measured
again for comparison. If the temperature has changed more than 2K over the course of the
measurements, it is considered not stable and the measurement is repeated. Due to the
efficient PID control in this system, same temperature repetitions have not been observed.

Enter Inputs

Set Temperature

PID Control

Measure Resistivity

Measure Hall

Figure 2.3 Hall Effect System Flow Chart

28

Final Temperature Check

2.1.2

System Components

Figure 2.4 Hall Effect System

As seen in the Fig. 2.4, system utilizes a Janis continuous flow cryostat onto which a high
temperature measurement stage has been added with heat shields, nichrome heaters, and a
copper sample stage. Instruments used include: a source meter (Keithley 2400), a voltmeter
(Keithley 2002), a switch system (Keithley 7002), total of three power supplies where two
29

of them connected to the nichrome heaters (KLP 150 and KLP 300) and the other one
connected to the magnet (KLP 75).
Keithley’s Series 2400 Source Measure Unit, which can practically be used as a voltage
source, a current source, a voltage meter, a current meter, and an ohmmeter, is added as
a source meter in this configuration due to the power source characteristics such as low
noise, precision, and readback. The multimeter capabilities include but not limited to high
repeatability and low noise. It has 0.012% accuracy with 51/2 digit resolution. The source
voltage and measured voltage range from 5µV to 210V and 1µV to 211V , respectively;
the source current and measured current range from 50pA to 1.05A and 10pA to 1.055A,
respectively. Measurable resistance ranges from 100µΩ to 211M Ω and the maximum source
power is 22W . The Keithley 2400 is connected to the computer through a GPIB connection
where 520 readings per second can be delivered.
Keithley’s Series 2002 Voltage Measure Unit, among other functions, can measure DC
voltage, AC voltage, DC current, AC current, two and four-wire resistance, frequency (voltage and current), and temperature (resistance temperature devices or thermocouples).
In this system, Keithley 2002 has been used to measure voltage and temperature. The
measured voltage and the measured current range from 200mV to 1000V and 200µA to
2mA respectively. For temperature measurements with E type thermocouple, there are
two possible methods. First one is to read the temperature directly from the meter where
temperature range from −200◦ C to −1000◦ C with ±0.5◦ C accuracy and second one is to
convert the voltage readings into temperature.
The Model 7002 Switch System has been used for van der Pauw samples (VdP). One
independent card slot is dedicated to direct the contact pairs to the current source and voltage
meter. Use of the 7002 switching system allows for fewer measurement instruments and more
30

measurement connection configurations to the sample to more accurately measure sample
uniformity and to average measurements in different configurations in order to eliminate
spurious non-Hall contributions to the measurement.
Magnetic field is provided by GMW magnet system 5403 as seen with the magnetic field
sensor in Fig. 2.5. Magnet is powered by the power supply, KLP 75 and provides a magnetic
field of 0.7 Tesla.

Figure 2.5 Hall Effect System Magnet Components

2.1.3

System Connections

The cryostat chamber is Janis ST100 which includes a 10 pin connector, labeled as connector
A, there are two separate connectors to connector A, one for samples with VdP (Van der
Pauw) geometry and one for Hall bar (in-line) geometry. Cryostat also includes a separate
19 pin connector, labeled as B, which stays connected during the measurement independent
from the sample type. The schematic for the 19 pin connector can be seen in Fig. 2.6.
31

A
B
C
D
E
F
G
H
J
K
L
M
N
P
U
V

Wiring see from outside of the connector

M A B
P C

L N
K U
J T

V

R D
S

E

H G F

Anode (+) of Si diode on cryostat
Anode (+) of Si diode on cryostat
Cathode (-) of Si diode on cryostat
Cathode (-) of Si diode on cryostat
Power (+) to HT stage heater #1
Power (-) to HT stage heater #1
Power (+) to HT stage heater #2
Power (-) to HT stage heater #2
Power (+) to cryostat heater in epoxy
Power (-) to cryostat heater in epoxy
Type E - TC #1 (+) chromel
Type E - TC #1 (-) constantan
Type E - TC #2 (+) chromel
Type E - TC #2 (-) constantan
Type E - TC #3 (+) chromel
Type E - TC #3 (-) constantan

Figure 2.6 Stable Connections

Connector B is connected at all times during the measurement, independent from the
sample type, either VdP or in-line geometry. Through the connector B, two nichrome stage
heaters, each are sandwiched between the aluminum nitride plates, are connected to two
Kepco power supplies. The nichrome heaters are the main part of the PID control system
which is custom designed and software automated and will be explained in further detail
in the following sections. Three E type Omega thermocouples are connected to the Keitley
2002 rear card, channel one and two. The third thermocouple is currently not active. The
main reason for using E type thermocouples, which are combinations of chromel (nickelchromium) and constantan (copper-nickel), is the low sensitivity to the magnetic field. Ten
readings of the thermocouple voltage are averaged and saved to the data file along with the
standard deviation.
For VdP samples, the connection schematic can be seen in Fig. 2.7. After Physical Vapor
32

Deposition (PVD) of metal contacts, silver paste was used to connect 50µm diameter gold
wires to the contacts. Several pastes were investigated for electrical connection to the samples
including a water based silver paste, an alcohol based silver paste, and a carbon paste. The
water based silver paste was found to easily detach from the sample during the measurements,
and the carbon paste was challenging to apply, required significant preparation time, had a
shorter shelf-life, and was difficult to remove from the sample after the measurement. The
alcohol based silver paste was found to be a good choice for ease of mounting, long shelf-life,
ease of removal, and good repeatability in the measured data through several temperature
cycles of data collection.
As part of the research for this dissertation, software was developed in the LabVIEW
programming environment for system control and data acquisition. The measurement steps
were programmed to follow the ASTM standard [50] for Hall effect measurements. The software was developed such that individual subroutines could be run independently allowing
for open access to various measured values and open access testing of the samples. This is
a critical feature of the system that empowers the user to troubleshoot measurement difficulties, further investigate unexpected results, and rapidly reconfigure the software control
to develop new measurements. All voltage and current readings are averaged after taking
ten measurements along with the standard deviations and the user is able to change the
settings through the graphical user interface (or front panel) of the program. As a custom
built open access system, user can be able to change the settings through the front panel of
the program.

33

Wiring see from outside
of the connector A

VdP Sample
A

A VdP 1 #A

D

A

H

B VdP 2 #B
K C

G J
F
B

B

E

C

D

C VdP 3 #C
D VdP 4 #D

Keithley 7002 Matrix Card
D C B A
1

Keithley 2002 - Rear- H
Keithley 2002 - Rear- L

2
3

Keithley 2400 - Front- H
Keithley 2400 - Front- L

4

Figure 2.7 Van der Pauw Sample Connections

For the in-line samples, connection schematic can be seen in Fig. 2.8. After metal
deposition of the contacts, 6 or 8 connections are made to the sample with 50µm thick gold
wires attached using silver paste. All voltage and current readings are the average of ten
measurements and recorded to a file along with the standard deviations.

34

A Inline V1 #A
Wiring see from outside
of the connector A
In-Line Sample
A

B

C Inline V3 #C
C

E

H
D

H

G

B Inline V2 #B

A

G J

F

F

E

B

D Inline I1 #D

K C

E Inline I2 #E

D

F Inline V4 #F
G Inline V5 #G
H Inline V6 #H

Keithley 2002 Rear Card
Channel 4 5 6 7 8 9 10
High

A B H G A B C

Low

B C G F H G F

Keithley 2400 Rear
High
D
Low

E

Figure 2.8 In Line Sample Connections

2.2

Sample Mounting Procedure

Both evaporation and sputtering of the metal contacts have been investigated. Both deposition techniques resulted in ohmic contacts after annealing. The e-beam evaporation system
used allows for up to four different deposition materials in a single run. The sputtering system accommodates two sputtering targets and the total runtime is shorter due to the higher
deposition rate and the less stringent vacuum requirements prior to starting the deposition.
Thus, sputtering was more frequency used in this research.
Metals for ohmic contact fabrication are chosen by the electron affinity of the material and
the work function of the metal which will be further discussed in Chapter 4. For the boron
doped single or poly crystalline p-type diamond films, ohmic contacts are commonly formed
by a Ti and Au layered structure, with thicknesses of 30nm and 100 − 300nm respectively.

35

For phosphorus doped single crystal n-type diamond films, in addition to Ti/Au, Al
contacts are also widely used.
For thermoelectric or heavily doped samples, silver paste is sufficient to obtain ohmic
contacts. Thus, no additional metal deposition is necessary.
The sample stage (Fig. 2.9) is made out of oxygen free high conductivity (OFHC)
copper. For uniform temperature, a cylindrical copper shield surrounds the sample stage
and consists of two copper cylinders - one inside the other with a thin sheet of thermal
insulation between them. A screw at the end of the shield hold it in place. A thin plate
of aluminum nitride electrically isolates the sample from the copper stage while providing
good thermal connection and high temperature compatibility. Sample to be measured is
then placed on an aluminum nitrite plate and wiring process is performed using silver paste
under the microscope. The 50µm diameter gold wires are attached to the sample with silver
paste which is allowed to dry for approximately 30 minutes. Care is taken during sample
mounting to adjust the Au wires so there is no shorting of the wires to the copper shield,
copper stage, or to other wires.

36

Figure 2.9 Hall Effect System Sample Stage

Once the sample is mounted into the system, the cryostat is carried from the microscope
back to the Hall effect system. The sample orientation should be made perpendicular to the
magnetic field. This orientation can be found by running some of the subroutines developed
and monitoring the Hall voltage while adjusting the rotation of the cryostat relative to the
magnetic field. When the Hall voltage reaches a maximum value, the proper orientation is
established. Rough pump is turned on followed by the turbo pump. After this process, all of
the measurements steps can be also operated by remote access, i.e. laptop or a smart phone.
Once the required vacuum level is reached, a current versus voltage measurement sequence
is initiated to study the effects of annealing on the ohmic nature of the contacts. Due to the
temperature dependent IV measurement capabilities of the system, annealing effects can be
simultaneously monitored starting from room temperature up to 750K. The system can also
by cycled in temperature a number of times (defined by the user) so repeatability studies can
also easily be done. The switching system allows the user to monitor the IV measurements
for 6 different contact pairs, including the neighbor and diagonal contacts which can be
37

chosen for VdP samples through the LabVIEW software. Thus, one can compare different
contact pairs and confirm that they are sufficient to start the resistivity and Hall effect
measurements at temperatures ranging from 300K to 750K.

2.3

PID Control

The influence of temperature stabilization was shown in Fig. 2.2 and the importance of
temperature stabilization is highlighted in the ASTM F76 Hall effect measurement standard.
A program was therefore developed for temperature control and was based on the Ziegler
Nichols method [57]. PID control is a form of feedback to obtain robustness. Instability is a
risk unless the PID loop is properly tuned.
As Loo [58] explained in her Ph.D thesis, PID includes three components: Proportional
(P) - also known as Gain, Integral (I) - also known as Automatic Reset or Reset, and
Derivative (D) - also known as Rate.
The algorithm is commonly used in the following forms:
• Proportional only (P)
• Proportional and Integral (PI)
• Proportional, Integral and Derivative (PID)
• Proportional and Derivative (PD)
In a PID control system, response of the controller to an error, the degree to which the
controller overshoots the set point and the degree of the system oscillations are the main
parameters to take into account in order to build an efficient temperature control [59].

38

2.3.1

Proportional Control

The proportional term generates an output value that is proportional to the present error
value. The proportional response can be tuned by multiplying the error by a constant, P ,
called the proportional gain constant.
P applies power, W , to the heater in proportion to the difference in actual temperature
and the set-point. In terms of performance, proportional gain is preferable to an On-Off
type controller.

W = P x(Ts − T0 )

(2.1)

Ts is the set temperature point and T0 is the actual temperature of the system. In
the presence of a large gain, the system responds faster to changes, however also becomes
progressively under damped and eventually unstable. Contrarily, a small gain results in a
relatively smaller output response to a large input error and a less sensitive controller.

2.3.2

Proportional + Derivative Control

High proportional control, P , can cause instability and overshoot problems. In order to
eliminate this, a derivative gain or damping constant, D, can be added to the control which
slows the rate of change of the controller output. In order to calculate the derivative of the
process error, slope of the error over time (rate of change) is determined and multiplied by
the derivative gain D.
In proportional and derivative, P D, control, a D term is added proportional to the
time-derivative of the error signal.

39

W = P x (Ts − T0 ) + Dx

d
(Ts − T0 )
dt

(2.2)

D can be adjusted to achieve an optimally damped response. Too high damping will
cause the system to be slow to stabilize, on the other hand, too low damping will result in
overshoot.

2.3.3

Proportional + Integral+ Derivative Control

P D control can overcome the overshoot problems, however, accelerating the movement of
the process towards set point and eliminating the residual steady-state error can be achieved
by adding the integral term, I, to the function which is proportional to both the magnitude
of the error and the duration of the error. The integral in P ID controller is the total of
the instantaneous errors over time and gives the accumulated offset which should have been
corrected previously. The accumulated error is then multiplied by the integral gain, I, and
added to controller output [60]. Due to fact that the integral term responds to accumulated
errors from the past, it can cause the present value to overshoot the set point value. To
prevent this condition from occurring in our work, we have introduced error reinitialize
functions to the P ID control software in LabVIEW environment.

W = P x (Ts − T0 ) + Dx

d
(Ts − T0 ) + Ix
dt

(Ts − T0 )dt

(2.3)

D term has not been used in our system, however it is commonly used to reduce the
magnitude of the overshoot produced by the integral component and to improve the combined
controller process stability. I term is used to change the heater power until the time averaged
value of the temperature is zero.
40

For manual tuning, first I and D values are set to zero. Then the P is increased until
periodic oscillations are observed, then P should be set to approximately half of that value
for a quarter amplitude decay response. Then I is increased until any offset is corrected
in sufficient time for the process. Too high I values will cause poor temperature stability.
Finally increase D if required, until the loop is acceptably quick to reach its reference after
a load disturbance. Too high D values will cause overshoot in the temperature. A fast P ID
loop tuning usually overshoots slightly to reach the set point more quickly, then stabilizes at
the set temperature [61]. Effects of increasing a parameter independently are summarized
in Table 2.1.
Parameter

Rise time

Overshoot

Settling time

Steady-state error

Stability

P

Decrease

Increase

Small change

Decrease

Degrade

I

Decrease

Increase

Increase

Eliminate

Degrade

D

Minor change

Decrease

Decrease

No effect in theory

Improve if small

Table 2.1 PID Manual Tuning [6]

2.3.4

PI Control in This System

For this system a custom made temperature controller is designed using two nichrome heaters
and two power supplies controlled by LabVIEW program. Maximum allowed current for the
power supplies are set to 2.5A.
In order to determine the optimum PI parameters, for fixed temperatures over the measurement range, between 300K-700K, the proportional gain was incrementally increased until
stable temperature oscillations were observed as seen in Fig. 2.10.

41

316

314

314

Temperature(K)

Temperature (K)

316

312

310

308

306
0

312

310

308

100

200

300

400

500

306
0

600

100

300

400

500

600

500

600

(b) P=4, I=0

316

316

314

314

Temperature (K)

Temperature (K)

(a) P=2, I=0

312

310

312

310

308

308

306
0

200

Time (sec)

Time (sec)

100

200

300

400

500

600

306
0

Time (sec)

100

200

300

400

Time (sec)

(c) P=8, I=0

(d) P=16, I=0

Figure 2.10 Effect of Proportional Gain to Temperature Stabilization

The proportional gain was then decreased to 0.45 of the value for sustained oscillations,
and the period of these oscillations was used to determine the integral value as 1.2 times the
proportional gain over the period of oscillation. This procedure was follwed by fine tuning
the PI parameters in LabVIEW. The derivative gain was set equal to zero as summarized in
Table 2.2.

42

Control Type
P
PI

Kp

Ki

POSC

-

0.45POSC

1.2P/T

Table 2.2 PI Control Summary Table

After the fully automated P ID control software program is established. Temperature
control consists of the following steps to achieve stabilization:
• Read the gain, P and the integral coefficients from the table which is created by
optimizing and fine tuning the parameters for that temperature range.
• Save the data and execute the program when the following three conditions are met
(i)Actual temperature is in the set temperature limits (ii) Actual temperature remains
in the set temperature limits for 300 seconds (iii) Standard deviations of the last 1000
temperature readings are lower than 0.2.
This procedure gave satisfactory results over the temperature range of 300 − 750K with
temperature variation less than 2K within the 7 minute long Hall effect and resistivity
measurement.
In Fig. 2.11, PID control efficiency for this system can be seen between the temperature
range of 300K to 500K with 50K temperature steps. Less than 5min is sufficient in order
to reach to a stable temperature point.

43

420

360

400

350

Temperature (K)

Temperature (K)

370

340
330
320

360
340
320

310
300

380

300
0

50

100 150 200 250 300 350 400

0

100 200 300 400 500 600 700 800

Time (Sec)

Time (Sec)

(a) 300K − 350K

(b) 300K − 400K
500

460
440

450

Temperature (K)

Tempeature (K)

420
400
380
360

400

350

340
320
300

300
0

200

400

600

800

1000

1200

0

200 400 600 800 1000 1200 1400 1600

Time (Sec)

Time (Sec)

(c) 300K − 450K

(d) 300K − 500K

Figure 2.11 Temperature vs time representation for the PID control

2.4

Software Control

Except for the vacuum pumps, all of the instruments in the system are controlled by software
developed in LabVIEW. The communication between the instruments and the computer is
through GPIB address cables. Expected input values are the sample geometry, number of
voltage and current readings to average, applied current value, temperature range with step
44

size and location of the measurement data file to be saved. Data is saved in three .txt files,
one includes a summary data of resistivity, carrier concentration and mobility for all the
temperature values and the other two data files include all the details of the measurement
including the standard deviations for resistivity and Hall measurements, respectively. For
samples with Van der Pauw geometry, in addition to the input values given above, thickness
of the sample is a user input to the program. For samples with in-line (Hall bar) geometry, additional inputs are thickness and the width of the sample and distance between the
contacts.
Once the user inputs are entered into the user interface of the LabVIEW program, the
automated measurement starts running and it typically takes about 7 hours for resistivity
and Hall effect measurements from 300K to 700K with 25K steps. During the measurement,
the system can be remotely accessed through a smart phone, tablet or a laptop independent
of the user’s physical location. Measurements start with the P ID control program to achieve
a stable temperature point, once the set temperature point is achieved and stabilized, resistivity measurements are taken and saved into the data sheet, followed by the Hall effect
measurements. After all of the measurements are taken at a certain temperature point, temperature is measured again and if it is less than ±2K compared to the start temperature, the
system moves to the next temperature set point. If the temperature difference is more than
±2K, the same temperature point is repeated and saved, until the ASTM conditions are
met. Currently, in this system, due to the efficient P ID control system built, temperature
set point repetitions have not been observed.

45

Chapter 3
Measurement Results
Hall effect measurement is a very important technique for understanding the electrical conduction properties of semiconductors such as carrier density, electrical resistivity, and the
mobility of carriers. It is also critical for defining the type of mobile charges in a given
semiconductor, either as positive or negative, depending on the type of the charge carriers, a
semiconductor can be defined as p or n type. This temperature dependent Hall effect system
has been tested between 296-700K and for both in-line and van der Pauw samples. Resistivity and Hall measurements of n and p type GaAs and Ge materials with known properties
has been measured as reference materials.
This system is fully software automated, thus, all the inputs can be given to the computer
independent from user’s physical location, i.e. a mobile phone access is sufficient to run the
whole experiment. The typical time for measurements over the 300K to 700K temperature
range with 25K steps takes less than 7 hours. Measurements are also taken when cooling
back down to room temperature to be able to observe the repeatability of the measurement.
Temperature dependent IV measurements also provide critical information for observing
the annealing effects specifically when the contacts are deposited on the sample with physical
vapor deposition (PVD), e-beam or sputtering methods, thus, unexpected measurement
results due to contacts can be easily detected and solved.

46

3.1
3.1.1

Reference Materials
p-Type GaAs

Before deposition of metal contacts, silver paste has been used to make contacts to the
material. Current versus voltage (or I-V) measurements as seen in Fig. 3.1 show ohmic
behavior only at 700K when no metal contacts have been deposited to the sample. This
measurement process to 700K also anneals the contacts. After annealing, the sample is cooled
back to 300K and another I-V measurement is taken for comparison to the pre-annealing
data. The after annealing data are labeled as 300KAA in Fig. 3.1 which shows that the I-V
behavior changed after annealing, but remained non-ohmic.
10
300K
300K_AA
400K
500K
600K
700K

Curent (mA)

5

0

-5

-10

-3

-2

-1

0

1

2

3

Voltage (V)

Figure 3.1 Temperature dependent IV curve for p-type GaAs without contact deposition

Resistivity and Hall effect measurements were taken on this p-type GaAs sample with
the silver paste contacts in order to investigate the influence of non-ohmic contacts on the
measured data.
47

In order to compare the effects of metal contacts custom made e-beam target (80wt%
Ag, 10wt% In and 10wt% Zn) was used to deposit metal contacts on the four corners of the
VdP sample. As Fig. 3.2 shows, the contacts initially showed non-ohmic behavior, however
the I-V measurements became linear above approximately 500K and remained ohmic after
annealing and cooling back to room temperature.
10
300K
300K_AA
400K
500K
600K
700K

Current (mA)

5

0

-5

-10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Voltage (V)

Figure 3.2 Temperature dependent IV curve for p-type GaAs with contact deposition

After achieving ohmic contacts, resistivity and Hall measurements were taken with different applied current levels to investigate the effect of applied current as seen in Fig.3.3, 3.4
and Fig. 3.5. No significant dependence on the applied current level could be seen as would
be expected for low resistance contacts and high uniformity in the sample.

48

0.1

0.1
300K-700K
700K-300K

300K-700K
700K-300K

Resistivity (Ohm.Cm)

0.09

0.08
0.07
0.06
0.05

0.08
0.07
0.06
0.05

0.04

0.04

0.03
300 350 400 450 500 550 600 650 700

0.03
300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) I=20mA

(b) I=15mA

0.1
300K-700K
700K-300K

0.09

Resistivity (Ohm.cm)

Resistivity (Ohm.cm)

0.09

0.08
0.07
0.06
0.05
0.04
0.03
300 350 400 450 500 550 600 650 700

Temperature (K)

(c) I=10mA

Figure 3.3 Temperature dependent resistivity for p-type GaAs

49

120

120

300K-700K
700K-300K

300K-700K
700K-300K

100

2

Hall Mobility (cm /Vs)

80

60

80

60

40

40

20
300 350 400 450 500 550 600 650 700

20
300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) I=20mA

(b) I=15mA

120
300K-700K
700K-300K
100

2

Mobility (cm /Vs)

2

Mobility (cm /Vs)

100

80

60

40

20
300 350 400 450 500 550 600 650 700

Temperature (K)

(c) I=10mA

Figure 3.4 Temperature dependent mobility for p-type GaAs

50

18

2.1x10

18

18

2x10

18

1.9x10

18

1.8x10

18

1.7x10

18

1.6x10

18

1.5x10

18

-3

Concentration (cm )

-3

Concentration (cm )

2x10

18

1.9x10

18

1.8x10

18

1.7x10

18

1.6x10

18

1.5x10

2.1x10

300K-700K
700K-300K

300 350 400 450 500 550 600 650 700

300K-700K
700K-300K

300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) I=20mA

(b) I=15mA
18

2.1x10

300K-700K
700K-300K

18

-3

Concentration (cm )

2x10

18

1.9x10

18

1.8x10

18

1.7x10

18

1.6x10

18

1.5x10

300 350 400 450 500 550 600 650 700

Temperature (K)

(c) I=10mA

Figure 3.5 Temperature dependent carrier concentration for p-type GaAs

3.1.2

n-Type GaAs

To test the measurement system, both p-type and n-type samples of GaAs were measured
as reference samples. In addition, the influence of temperature stability was studied to see
the effects of PID temperature control. Both van der Pauw (VdP) and Hall bar sample
geometries were investigated with the n-type GaAs samples. Metal contacts to the samples
were deposited by e-beam evaporation of target material made of (90wt% Ag, 5wt% In and

51

10wt% Ge). was used to deposit metal contacts on the four corners of the VdP and two ends
of the Hall bar sample. The typical measurement procedures begin with current vs. voltage
behavior of various contact pairs. This is repeated at various temperatures to investigate
the annealing effects on the contacts. The IV measurements for contacts from one edge to
the other of a Hall bar shaped sample of gallium arsenide show non-ohmic behavior until
650K above which linear IV behavior is found. After this annealing step to 700K, the IV
behavior remained Ohmic even down to 300K as seen in Fig. 3.6.
10
300K
300K_AA
400K
500K
600K
700K

Current (mA)

5

0

-5

-10
-15

-10

-5

0

5

10

15

Voltage (V)

Figure 3.6 Temperature dependent IV curve for n-type GaAs with deposited metal contacts

Mobility data collected is in a good agreement with two different sample configurations,
(VdP and Hall bar) and literature values [62] as seen in Fig. 3.7.
Temperature dependent carrier concentration can be seen from 300K-700K and 700K300K in Fig.3.8 where ND = 6.3x1016 cm−3 at near room temperature.
To test the system accuracy we compared our results to those shown by Ioffe for comparably doped GaAs samples. For both van der Pauw and Hall bar configurations we showed
52

16

-3

Berkun Hall bar N =6.3x10 cm
d

16

-3

Berkun VDP N =6.3x10 cm
d

5500

16

-3

17

-3

Ioffee N =10 cm
d

5000

Ioffee N =10 cm

2

Hall Mobility (cm /Vs)

d

4500

16

-3

WRS Materials N =8.80x10 cm
d

4000
3500
3000
2500
2000
1500
200

300

400

500

600

700

800

Temperature (K)

Figure 3.7 Temperature dependent mobility comparison for n-type GaAs with contact deposition

18

2.5x10

18

-3

Concentration (cm )

2x10

18

1.5x10

18

1x10

17

5x10

0
200

300

400

500

600

700

Temperature (K)

Figure 3.8 Temperature dependent concentration for n-type GaAs contact deposition

53

good agreement of the room temperature carrier concentrations reported by the sample
manufacturer (WRS Materials). We also compared temperature dependent results with
semi-empirical fit values given by Christou and Webb that we labeled as theory in our error
analysis [63]. We found less than 20% difference with the theoretical values over the 300K700K range, and less than 8% difference with the values reported for the ND = 1017 cm−3
sample reported in literature (Ioffe) [62].

3.1.3

p-Type Ge

For an additional reference material, p-type Ge was purchased from WRS Materials with
known resistivity (0.01-0.04 Ω.cm at 300K), are performed. Annealing effects at 700K to
current vs. voltage behavior is presented in Fig. 3.9.
30
300K
300K_AA

Current (mA)

20
10
0
-10
-20
-30

-1

-0.5

0

0.5

1

Voltage (V)

Figure 3.9 IV curve for p-type Ge

Measurements for resistivity show a good agreement with the given company values at
300K as shown in Fig. 3.10a. Measurements taken from 300K up to 700K and from 700K
54

back down to 300K, show the repeatability of the experiment.
0.05

0.045

0.04

Resistivity (Ohm.cm)

Resistivity (Ohm.cm)

0.045

0.05

300K-700K
700K-300K

0.035
0.03
0.025
0.02
0.015

I=20mA
I=30mA
I=40mA

0.04
0.035
0.03
0.025
0.02
0.015

0.01
300 350 400 450 500 550 600 650 700

0.01
300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) Fixed current

(b) I=20,30 and 40mA

Figure 3.10 Temperature dependent resistivity for p-type Ge

Effects of the applied current level to the resistivity measurements also have been studied
for applied current values of 20, 30 and 40mA. As expected, resistivity data is repeatable for
all three current levels as shown in Fig. 3.10b.
At room temperature, measured resistivity is around 0.22 Ω.cm which corresponds to a
concentration value around 3.5x1017 cm−3 according to semiconductor physics calculations
[11] which falls in agreement with the measured concentration shown in Fig. 3.11a and
3.11b. There is a repeatable discontinuity in the measurement between 550K - 600K due to
the transition from extrinsic region to intrinsic region.

55

18

18

5 x 10

5 x 10

18

4 x 10

4 x 10

I=20mA
I=30mA
I=40mA

-3

Concentration (cm )

-3

Concentration (cm )

300K-700K
700K-300K
18

18

3 x 10

18

2 x 10

18

3 x 10

18

2 x 10

18

1 x 10

18

1 x 10

0
300 350 400 450 500 550 600 650 700

0
300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) Fixed current

(b) I=20,30 and 40mA

Figure 3.11 Temperature dependent carrier concentration for p-type Ge

Extrinsic to intrinsic transition temperature range between 550K-600K which is repeatably observed in the temperature dependent mobility measurements as seen in Fig.3.12a and
Fig. 3.12b.
700

700
300K-700K
700K-300K

I=20mA
I=30mA
I=40mA

Mobility (cm /Vs)

600

500

500

2

2

Mobility ( cm /Vs)

600

400
300

400
300

200

200

100

100

0
300 350 400 450 500 550 600 650 700

0
300 350 400 450 500 550 600 650 700

Temperature (K)

Temperature (K)

(a) Fixed current

(b) I=20,30 and 40mA

Figure 3.12 Temperature dependent mobility for p-type Ge

56

Reference materials show good agreement between the measured transport properties,
company given values and physical equations and can be used to calibrate the system regularly.

3.2

Single Crystal Diamond

3.5 mm 3.5 mm high pressure, high temperature (HPHT) substrates from Sumitomo Electric
have been used to deposit the single crystal boron doped diamond films. Films are grown
on top of the substrates using a microwave plasma-assisted CVD reactor [64], as reported in
previous investigations [56]. Samples are acid cleaned and hydrogen plasma etched followed
by the PVD deposition of gold or titanium/gold near the four corners of the sample for van
der Pauw measurements as reported in our previous work [1].
For gold contacts, in order to achieve adhesion, a fine mesh technique has been developed using a transparency mask with four identical holes near the corners and a fine mesh
structure similar to a TEM grid. Technique includes the following steps: (I) Hydrogen termination of the sample surface (II) Au deposition with mask and mesh structure (III) Removal
of the mask and mesh structure (IV) Oxygen termination of the sample surface and (IV)
Au deposition with mask. Samples are then oxygen plasma treated to remove surface conductivity effects. Ohmic contacts with good adhesion have been achieved via this technique.
For Ti/Au contacts, annealing was sufficient to observe ohmic contacts without additional
process steps.

57

3.2.1

Influence of Ohmic contacts

The influence of Ohmic and non-Ohmic contacts to the Hall effect measurements was studied
as a part of this work. Near the origin of the current vs. voltage plots, non-Ohmic curves such
as the one shown in Fig. 3.13 as dashed line for a 500K IV measurement were observed for the
moderately doped diamond sample. In order to study the effects, temperature dependent
measurements were taken for this sample, then the contacts were cleaned off, and new
contacts deposited on the same sample. The new contacts exhibited Ohmic behavior as
shown in straight line at 500K.
40
30

non-Ohmic500K
Ohmic500K

Current (mA)

20
10
0
-10
-20
-30
-40
-15

-10

-5

0

5

10

15

Voltage (V)

Figure 3.13 Current vs. voltage comparison for Ohmic and non-Ohmic contacts for a moderately doped single crystal diamond (SND02)

The resulting Hall mobility for the non-Ohmic and Ohmic contacts shows comparable
values, but notably more scatter in the data for the non-Ohmic contacts as seen in Fig. 3.14.

58

250
non-Ohmic contacts
Ohmic contacts

2

Hall Mobility (cm /Vs)

200

150

100

50

0
300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 3.14 Hall mobility data comparison between Ohmic and non-Ohmic contacts for a
moderately doped single crystal diamond sample (SND02)

IV measurements are performed for all samples in order to monitor the effects of annealing
to the contacts. For a moderately boron doped single crystal diamond sample, IV curves
showed annealing effects of the contacts as seen in Fig. 3.15. A considerable change in
resistance and reduction in the non-linear behavior can be seen for the 300K curve measured
after annealing (AA) compared with the 300K measurement before annealing to 700K. Nonlinear behavior is observed for the lower temperature measurements at high voltages, however
linearity is maintained as the IV measurements pass through the origin.

59

Figure 3.15 IV measurements for a moderately doped sample (SND02)

Followed by these contact studies, measurements of Hall effect according to ASTM standards were used to determine carrier concentration and mobility as seen in Fig. 3.16 and
Fig. 3.17.

60

Berkun (E =0.29eV)
a

18

-3

18

-3

19

-3

Borst (E =0.35eV, n =1x10 cm )
a

A

Borst (E =0.34eV, n =2x10 cm )

19

10

a

A

Borst (E =0.24eV, n =2x10 cm )
a

18

A

-3

Concentration(cm )

10

17

10

16

10

15

10

3/2

p ~ T exp(-E /kT)
a

14

10

13

10
0.001

0.0015

0.002

0.0025

0.003

0.0035

-1

Reciprocal Temperature (K )

Figure 3.16 Temperature dependent concentration measurements for a moderately doped
sample (SND02) with comparison to data reported by Borst, et al[9]

Berkun

1000

18

-3

18

-3

19

-3

Borst (n =1x10 cm )
A

Borst (n =2x10 cm )
A

Borst (n =2x10 cm )
A

µ ~T -2
H

2

Mobility (cm /Vs)

µH~T

-2.4

100

µH~T -2.6
µH~T -1

10
200

400

600

800 1000

Temperature (K)

Figure 3.17 Temperature dependent mobility measurements for a moderately doped sample
(SND02) with comparison to data reported by Borst, et al.[9]

The carrier concentration is observed to be in agreement with those reported by Borst
and Weis [9], where literature data is extracted from the graphs reported and nA is the
61

fitted acceptor concentration. The resulting activation energy, obtained from the equation
in Fig. 3.16, is slightly larger in the samples fabricated at MSU. The red (square) and
blue (triangular) symbols indicate data points for measurements from 300K to 700K in
red (square), and 300K to 700K in blue (triangular) which help to give the measure of
repeatability.
The total mobility for semiconductor materials is commonly the sum of lattice-scattering
mobility (T −1.5 ) and the impurity scattering mobility (T 1.5 ). Mobility includes additional
scattering mechanisms and is observed to be proportional to T −2.6 for temperatures above
500K for the MSU diamond sample which is comparable to the T −2.4 dependence reported
by Borst and Weis [9] for similarly doped samples, however the mobility of these values are
approximately 1.7 times larger than the mobility of the MSU sample at 400K due to higher
acceptor concentration. Temperature dependence of the mobility is also comparable to the
T −2.8 value reported by Tsubota [31] for relatively smaller doping levels.

3.2.2

Effect of Applied Current to Hall Measurements

Non-linearities at higher voltages in the low temperature IV curves were the impetus for
investigating the influence of current level on the temperature dependent measurements.
The effect of applied current level for a moderately doped sample is investigated. Temperature dependent Hall effect measurements were taken between 300K and 700K using a
measurement current of 30mA. This was then repeated at measurement currents of 50mA
and 70mA. The resulting carrier concentration showed a systematic increase with increasing
current at the lower temperatures, however significantly low dependence on the current is
seen for temperatures above approximately 500K as seen in Fig. 3.18.

62

10

19

Concentration

30mA

10

18

10

17

10

16

50mA

70mA

300 350 400 450 500 550 600 650 700

Temperature(K)

Figure 3.18 Effect of applied current level to concentration measurements for a moderately
doped sample (SND02)

Similarly, as the current is increased a systematic decrease in mobility can be seen for
the low temperatures in Fig. 3.19. It is not yet clear if this is related to the non-linear IV
behavior seen at the higher current levels particularly evident in the low temperature IV
curves and/or the possibility of self heating.

63

1000
50mA

70mA

2

Hall Mobility (cm /Vs)

30mA

100

10
300 350 400 450 500 550 600 650 700

Temperature(K)

Figure 3.19 Effect of applied current to Hall mobility measurements for a moderately doped
sample (SND02)

The differences are not seen in resistivity measurements for varying current levels as
shown in Fig. 3.20. It is recognized that the measured Hall voltages are significantly smaller
than the measured electrical resistivity voltages, thus if the error source is an offset error,
this could have a stronger effect on the Hall measurements relative to the resistivity measurements. Further studies are needed to better understand such measurement dependencies.

64

10

Resistivity (OhmCm)

70mA

30mA

50mA

1

0.1
300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 3.20 Effect of applied current to resistivity measurements for a moderately doped
sample (SND02)

As a part of this work, resistivity, Hall and IV measurements of boron doped single crystal
diamond films have been performed. Effects of system temperature stability, ohmic versus
non-ohmic contacts, and measurement current amplitude have been studied.

3.3

Thermoelectric Materials

Through thermoelectric materials, heat can be directly converted to electricity using solid
state energy conversion and heating/cooling can be achieved by applying a current through
certain materials without any moving parts involved.
The efficiency of thermoelectric energy conversion depends on a materials properties,
which could be evaluated by a dimensionless figure of merit, ZT :

ZT =

S 2σ
T
K

65

(3.1)

where S, σ and K are the Seebeck coefficient, electrical conductivity and thermal conductivity of the material. Generally ZT = 1 is the benchmark for good thermoelectric materials.
In this work, M g2 (Si, Sn) compounds were synthesized using the molten salt sealing method
and were powder processed, followed by pulsed electric sintering (PECS) densification by our
group [2]. A set of M g2.08 Si0.4−X Sn0.6 SbX (0 ≤ X ≤ 0.072) were characterized by temperature dependent hall effect system.

3.3.1

Carrier Concentration and Hall Mobility

The undoped sample (x=0) shows typical intrinsic behavior which is used to estimate the
band gap, Eg , of the materials by fitting the temperature dependent carrier concentration n
to Eq. 3.2,

n=2×

2π × m∗ kB T
h2

(3.2)

where m∗ is the effective mass, kB is the Boltzmann constant, T is the temperature in Kelvin,
h is the Planck constant, EC is the energy level of the bottom of the conduction band and
EF is the Fermi level. The assumption that the Fermi level of intrinsic semiconductors are
in the middle of the bandgap was used and the temperature dependence of the band gap
was neglected, a bandgap of Eg = 0.594eV was obtained. The electric conductivity of the
compound increases with increasing concentrations of Sb for 0 ≤ x ≤ 0.048 as shown in Fig.
3.21. The Sb-doping is saturated when x ≥ 0.060 and the carrier concentrations start to
decrease as the doping level is further increased. Thus, electrical conductivities of the two
most heavily doped samples (x=0.60 and x=0.072) are lower than that of x=0.048.
Hall mobilities of un-doped and moderately doped samples (x ≤ 0.036) are shown in Fig.

66

22

10

x=0.000

x=0.036

x=0.012

x=0.048

x=0.024

x=0.072

-3

Concentration (cm )

21

10

20

10

19

10

18

10

300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 3.21 Carrier concentration versus temperature for nominal compositions of
M g2.08 Si0.4−X Sn0.6 SbX
3.22. The temperature dependence of the Hall mobilities generally follows the relation of
Eq. 3.3.

µH ∝ T j

(3.3)

where j=-0.5 corresponds to alloy scattering and j=-1.5 indicates acoustic phonon scattering. Most of the Hall mobility indicates a mixed carrier scattering mechanism, j is in
between -1.5 and -0.5. Measured hall mobility was fit to the empirical equation, Eq. 3.4 [65].

1
1
T 0.5
1
T 1.5
=
(
) +
(
)
µH
µal 300K
µph 300K

(3.4)

where µal and µph are the alloy disorder scattering mobility and acoustic phonon scattering mobility at room temperature, respectively. The alloy scattering can be related to the
formation of the M g2 Si − M g2 Sn solid solution. Disorder induced by Mg vacancies may
67

100
-0.9

~T

-2.4

2

Hall Mobility (cm /Vs)

µ=T

10
x=0
x=0.012
x=0.024
x=0.036

1
300 350 400 450 500 550 600 650 700

Temperature (K)

Figure 3.22 Hall mobilities for M g2.08 Si0.4−X Sn0.6 SbX
also be effecting the scattering. The value of µal decreases as the doping level increases,
possibly indicating filling of the Mg vacancies. The values of µph for the Sb-doped samples
are higher than for the undoped sample, except for the x=0.024 specimen. This result is in
agreement with the effect of Mg vacancies in Mg-deficient M g2 Si1−x Sbx materials reported
by Dasgupta et al. [66].
For x=0.024, µph value is lower due to internal cracks, which could also explains the lower
electrical conductivity measured using ZEM-3 system and the Hall measurement system [2].

68

Chapter 4
Analysis and Discussion

4.1

Band Theory

Band hypothesis clarifies the purpose behind the contrasts between the properties of a decent
insulator and of a metal. For a material in which the base of the conduction band is an
energy Eg over the highest point of the valance band, a small number of electrons will be
energized thermally into the unfilled band. The number will be relative to a Boltzmann
factor exp(−Eg /kT ) so it shifts quickly with temperature. For insulators, energy needed to
excite an electron from the valance band to the conduction band is larger than that of visible
light and is in the order of 5 − 10eV . At 300K temperature, kT

1/40eV . Consequently,

at room temperature, the Boltzmann factor will allow a very small number of electrons
to be energized. In any case it is expected to assume that if in some material, Eg was
sufficiently small, then a considerable number of electrons could be energized at a moderate
temperature. In that material, the variety of the Boltzmann factor with temperature would
lead to an exponential variation of conductivity. At the point when this happens, electrons
in the conduction band and holes in the valance band both adds to conduction, so called
“intrinsic conduction” [10]
Sign of the Hall coefficient depends on the type of the doping which makes the semiconductor either p-type or n-type. Thus, all samples of germanium or diamond do not have the
same sign Hall coefficient. These facts drove studies to calculate the effects of conduction
69

electrons in those samples [67].
The band diagram occasionally studied assumes that the crystal is a perfectly periodic
structure containing no defects. In the case that now an additional atom is added inside
the crystal, it will impact the periodicity which will be seen in the band structure. The
quantity of levels in the conduction band will still be multiple of the number of atoms
present (counting the additional one), yet while the levels got from the ordinary atoms
will be grouped together and effect to the states of motion through the crystal. The levels
associated with the additional atom can be displaced from the fundamental group levels and
may fall into the energy gap between this band and the filled band. These levels represent
a localized state centered on the extra space. Subsequently additional atoms result to a
physical perturbation of the crystal and is reflected in the band diagram by relocation of
some of the levels which may impact the appearance of levels in the gap between the full
and empty bands as seen in Fig 4.1.

Figure 4.1 Intrinsic conduction. (a) Density of levels plotted for two separated bands separated by energy gap, g . A hole in the valance band is formed when energy not less than g
is supplied to the system. The electron from the valance band is liberated in the conduction
band. (b) Impurity levels supply electrons to the conduction band.[10]

70

An additional atom can be added in variety of ways. For instance, lead sulphide forms
an ionic crystal of rock salt structure in which P b2+ ions and S 2− ions alternate. Presume
extra Pb atoms were added into the sample, there are two possibilities, either they might
occupy an interstitial position or they might fill a typical P b2+ site. If they fill a typical
P b2+ site, a vacant S 2− site ought to be created, in a way that adding a Pb atom would be
same as extracting an S atom. In the final crystal, localized levels will be related with the
Pb atom and the system will have two more electrons in addition to those required to ionize
the entire sulphur atoms. In a defect free crystal, ideally there are just enough electrons to
fill the valance band fully. The two additional electrons currently present will take up the
localized levels associated with the additional lead atom, in the form of the lowest energy and
remain with their parent atom. In case that these levels are located in the gap between the
valance and conduction bands, then the energy needed to excite them into the conduction
band will be smaller than the energy needed to excite electrons from the valance band. As a
result this situation will supply mobile conduction electrons at a smaller temperature than
the intrinsic mechanism [10].
In the case of impurities being dominant, the quantity of conduction electrons will vary
with the concentration of impurity, therefore, different from the intrinsic mechanism, this
mechanism can clarify the large difference in conductivity of different samples of the same
type of material. Suppose an extra sulphur atom is added to lead sulphide, a similar behavior
is expected to occur, but in this specific condition the sulphur center forms levels near the
top of the valance band. Mentioned levels trap electrons and leave two holes in the band.
Conduction in that case can be obtained and the Hall coefficient will have a positive sign.
Another way of this behavior occurs when one extra sulphur atom is present and has tendency
to capture two electrons from the lead atoms. If there is one extra S atom, it will be almost
71

impossible for them to get fully ionized. At absolute zero the un-ionized atom would be
present close to the P b2+ vacant site, however at higher temperatures, electrons can add to
the conduction by differentiating their distribution on the S atoms [10].

4.2

Metal-Semiconductor Contacts

Followed by Wilson’s formula of the transport theory of semiconductors based on the band
theory of the solids in 1931 [68], as explained in Section 4.1, band theory was then applied
to the metal-semiconductor contacts. In 1938, Schottky reported that, without the presence
of a chemical layer, a potential barrier could be observed due to the stable space charges in
the semiconductor [69]. “Schottky barrier” model is coming from this observation. In the
same year, Mott came up with a more appropriate theoretical model, so called “Mott barrier”
model [70]. Bethe et al. improved upon these models in 1942, so called “thermionic-emission”
model. [71]. Further models on metal-semiconductor contacts can be found in References
[72, 73, 74, 75].
Ohmic contacts are a critical since they could drastically enhance high temperature and
the high power device capabilities, such as diamond diodes and are required in electrical
transport characterization techniques, such as Hall effect. The high operating temperatures
may convey challenge because of the diffusion processes in the contact layer and the reactions
between the contact components. These reactions could prompt to changes of the contact
properties during operation at high temperatures, and can either enhance or hinder the
performance of the devices. If the contact resistivity is not low enough, a high voltage drop
could emerge due to the high current density in the contact of the high power devices and
leading to significant power loss and heating at the contacts. This high voltage drop is not

72

a desired phenomenon due to its divergence from the linear I-V behavior of the contacts,
which also impacts Hall effect measurements.
As a result, the following properties are required in order to obtain high quality ohmic
contacts [76]:
• Work function match - The work function of the contact metal, Fmetal , needs to
be close to the work function of the semiconductor, Fsemi (thermal ionic). In some
conditions, a thin layer of narrow bandgap material between metal and semiconductor
is embedded. For low doped semiconductors, selectively highly doped regions can be
created (tunneling assisted).
• Low contact resistivity - There is a small resistance associated with an ohmic contact
between two materials. Contact resistivity, ρc , is proportional to the area of the contact,
A as seen in Eq. 4.1.

ρc = Rc ∗ A

(4.1)

• High temperature stability - This becomes a challenge for the wide band-gap semiconductors. In GaAs and Si devices, with melting temperatures of around 1238 ◦ C and
1414 ◦ C respectively, the maximum operating temperature is constrained by the metal
contacts’ stability and onset of intrinsic carriers. For materials which can perform at
the higher temperatures, such as diamond, SiC and III-V nitrides, it is required to
have thermally stable, reliable contacts which do not react with oxygen.
• Reproducibility - This criteria is essential in the case of the device production as well
as Hall effect measurements. Therefore the contact fabrication process should provide
73

reproducibility as well as high quality. Contact materials and semiconductors need to
have smooth morphology and clean surface.
The nature of the contact can be investigated further by considering the energy structure
at the metal-semiconductor interface where an energy barrier can prompt to non-ohmic,
Schottky, behavior and the lack of a potential barrier results in linear current versus voltage
behavior, or an ohmic contact.
For the condition depicted in Fig. 4.2, the Fermi level in the semiconductor is closer
to the vacuum level than the Fermi level in the metal. Thus, electrons will move from the
semiconductor to the lower energy states in the metal. This will continue until the two
Fermi levels align and the Fermi level in the semiconductor is reduced by an amount which
is identical to the difference between the two work functions.
The work function, qφm (for metal) and q(χ + φn ) (for semiconductor), is nothing but
the energy difference between the vacuum level and the Fermi level. qχ is the electron
affinity measured from the bottom of the conduction band Ec to the vacuum level, and qφn
is the energy difference between the Ec and the fermi level, EF . The contact potential,
φm − (χ + φn ), is defined as the difference between the work function of the metal and the
semiconductor. As the gap distance, δ, becomes smaller, electric field in the gap region
increases. As a result of this increased electric field, negative charge is built up at the metal
surface. In order to satisfy equilibrium, an equal amount of opposite (in this case positive)
charges must exist in the semiconductor depletion region. There is similarity here with the
n-type side of a p+ - n junction. When the metal and semiconductor come in contact, the
energy band diagram of the junction is found as shown in Fig. 4.2d. The limiting value
of the barrier height, qφBn0 , is given by the difference between the work function and the

74

electron affinity of the semiconductor in Eq. 4.2.

qφBn0 = q(φm − χ)

(4.2)

If a p-type semiconductor is used instead of the n-type semiconductor, then the barrier
height, qφBp0 , is given in Eq. 4.3.

qφBp0 = Eg − q(φm − χ)

(4.3)

For any given metal-semiconductor pair, the sum of the barrier heights for n and p-type
semiconductor is accepted to be equal to the bandgap as in Eq. 4.4.

q(φBp0 + phiBn0 ) = Eg

(4.4)

W

Figure 4.2 Energy band diagrams of metal-semiconductor contacts. Metal and semiconductor
(a) in separated systems, and (b) connected into one system. As the gap δ (c) is reduced
and (d) becomes zero. [11].

Eq. 4.2 and 4.3 are theoretically ideal however, experimentally it is very challenging,
75

almost impossible to achieve. The main reasons causing this deviations of experimental
barrier heights from ideal are:
• An unavoidable interface layer, δ = 0, as shown in Fig. 4.2c.
• The presence of interface states
In order to achieve the desired metal to semiconductor contacts, consideration of the
metal work functions and semiconductor electron affinities provide an initial guide. Typically
for metals, qφm is of the order of a few electron volts (2 − 6eV ), which is very sensitive to
surface contamination. Some commonly used metal contacts in this thesis are Ag, Al, Ti and
Au with qφm values of 4.26 − 4.73eV , 4.06 − 4.26eV , 4.33eV and 5.1 − 5.47eV respectively.
Values are given as a range since the numbers may differ depending on the crystal face.
The electron affinity of diamond (χ) and work function (φ) for different doping levels and
different crystal orientation is given in Fig. 4.3.
Ohmic and Schottky contacts are the two types of metal-semiconductor contacts. A
Schottky barrier is established in the Schottky metal-semiconductor contacts. The ohmic
contacts are another type of metal-semiconductor contacts, which have linear and symmetrical I-V characteristic. Ohmic contacts have a relatively small contact resistance. I-V
characteristics, such as the linearity around the origin, are used to determine the presence
of ohmic properties. A high quality ohmic contact should not hinder the performance of
the device which can also supply the current with a relatively small voltage drop. In order
the characterize an ohmic contact, specific contact resistance, so called contact resistivity, is
being used. Contact resistivity is defined as one over the derivative of the current density
with respect to the voltage across the interface, as shown in Eq. 4.5 [76].

76

(a) Low-B-doped (100) surface
T (°C)
χ (eC)
φ (eV)

H-plasma
-1.0
3.9

400
-0.8
3.9

600
0.3
4.9

1100
1.3
5.3

400
-0.5
4.4

600
0.3
4.9

1100
1.5
5.6

400
-0.6
4.2

600
-0.5
4.3

1100
1.8
5.7

400
-0.2
3.1

600
-0.3
3.3

1100
0.7
4.7

(b) Low-B-doped (111) surface
T (°C)
χ (eVI)
φ (eV)

H-plasma
-0.9
4.2

(c) High B-doped (100) surface
T (°C)
χ (eV)
φ (eV)

H-plasma
-1.1
3.7

(d) High N-doped (100) surface
T (°C)
χ (eV)
φ (eV)

H-plasma
/
/

Figure 4.3 Electron affinity of diamond (χ) and work function (φ) for different doping and
orientation [12]

∂J −1
ρc =
∂V V =0

(4.5)

Thus, the contact resistivity can be determined from the I-V characteristics of the junction and the area of the contact [77]. When forward biased, there are four main contributions to current transport across the metal semiconductor contact: 1) Emission of electrons
from the semiconductor into the metal over the top of the barrier (thermionic emission).
2) Quantum-mechanical tunnelling through the barrier (field emission). 3) Recombination
in the space-charge region. 4) Hole injection from the metal into the semiconductor [13].
Comparing the doping, E00 , to temperature, kT , decision of the current transport mechanism through the contact is realized. These mechanisms mainly include thermionic emission

77

(TE), thermionic field emission (TFE) or field emission (FE). At a low or moderate semiconductor doping level and/or moderately high temperatures, kT

E00 , standard thermionic

emission mechanism dominates with the contact resistivity exponentially dependent on the
barrier height as shown in Eq. 4.6.

ρc =

qφBn
qφBn
k
exp(
) ∝ exp(
)
A ∗ ∗T q
kT
kT

(4.6)

Since applied voltage is small, the voltage dependence of the barrier height can be neglected. In order to obtain small contact resistance, ρc , small barrier height, φBn , is required.
In the case of higher doping levels, kT ≈ E00 , the depletion layer width lowers; the barrier
becomes smaller and a part of electrons tunnel through it. In this case, current transport is
dominated by the thermionic field emission mechanism through the barrier only as shown in
Eq. 4.7.

ρc =

√
k E00 cosh(E00 /kT )coth(E00 /kT )
A ∗ ∗T q

πq(φBn − φn )

exp[

q(φBn − φn )
qφn
+
]
E00 coth(E00 /kT )
kT
qφBn
∝ exp[
]
E00 coth(E00 /kT )

(4.7)

where φn is negative for degenerate semiconductors. The factor kT /E00 (where k is
the Boltzmanns constant, and T is the absolute temperature) is the criteria for the relation between thermionic and field emission processes. The characteristic energy E00 is
related with the tunnelling probability and, as a result of the decrease of the depletion layer
width, increases with the semiconductor doping level. The thermionic emission occurs at
kT /E00 >> 1. When kT /E00 = 1 these two processes can be comparable and the current
transport mechanism is named as thermionic-field emission. The field emission is predominating at kT /E00 << 1 as shown in Eq. 4.8 [78].

78

ρc =

ksin(πc1 kT )
qφBn
qφBn
exp(
) ∝ exp(
)
A ∗ ∗T q
E00
E00

(4.8)

In order to achieve low contact resistivity, low barrier height should be used [77]. Low
barrier height is very challenging to achieve, a high quality ohmic contact is required to
operate in the tunneling regime.
Contact resistivity is a function of barrier height in all regimes, doping concentration (in
TFE and FE), and temperature (more sensitive in TE and TFE) as shown in Fig. 4.4. In
TE regime, ρc is independent of doping concentration and dependent only on the barrier
height, φB . In FE regime however,ρc is dependent on ∝ exp(N −0.5 ) in addition to φB .

Specific contact resistance,Rc (Ω-cm2)

1

φB2 ( > φB1)

10-1
φB1
10-2
10-3
10-4

TE
kT » E00

TFE
kT ≈ E00

10-5

FE
kT « E00

10-6
10-7 17
10

1018
1019
1020
Doping concentration (cm-3)

1021

Figure 4.4 Dependence of specific contact resistanceon doping concentration (and E00 ), barrier height, and temperature. TE, TFE and FE are indicated [11]

As the contact resistivity, ρc , is the most critical factor to quantify the quality of the
ohmic contacts, various methods for contact resistivity measurement have been used: two
probes method, four probes method, extrapolation method, differential method, method of
the interface probes and Transmission line model method (TLM) [13]. The commonly used
79

method is the TLM method due to relatively low measurement error with I-V characteristic
linearity determination. TLM method includes two structures, linear [79] and circular [80].
The linear TLM method needs formation of mesa structures. Circular TLM method does
not require mesa structures; however it is practical for very small sheet resistance of the
metals. The values of the contact resistivity shown here are determined using a linear TLM
method as seen in Fig. 4.6.

Figure 4.5 Sample image for transmission line measurements (TLM).

The contact resistance and resistivity for Ti/Au contacts after annealing were calculated
as Rc = 0.13957Ω and ρc = 3.07 ∗ 10−4 Ωcm2 respectively. The order of magnitude for the
specific contact resistance in literature were in the order of ρc = 10−4 Ωcm2 [81].
Two main techniques can be used to achieve ohmic contacts: by increasing the semiconductor doping level and/or by decreasing the barrier height as seen in Fig. 4.7.
It is challenging to achieve low resistivity ohmic contacts to p-type wide band-gap semiconductors due to the high electron affinity and large width of the band-gap. For example,
the electron affinity of SiC and GaN is 3.3eV and 1.84eV , respectively. For the commonly
used SiC materials, the band-gap width is in the range of 2.3 − 3.2eV , while for GaN it is
3.44eV for wurtzite polytype and 3.2eV of the zinc-blended structure. This results a high
Schottky barrier which is formed at the interface metal/p-type (SiC, GaN).
80

0.9

Resistance (Ohm)

0.8
0.7
0.6
0.5
0.4
0.3
50

100 150 200 250 300 350 400

Distance (μm)

Figure 4.6 Transmission line measurements (TLM).
a)

b)

qφB

qφB

EC

EC

EF

EF

EV

EV

Figure 4.7 A zone diagram of an ohmic contact with a) low barrier height and b) high doping
level [13].
A metal with a work function which is enough to yield a low barrier at the interface
for a low doped diamond sample does not generally exist. Techniques for making ohmic
contacts for such samples involves shallow diffusion, alloy regrowth, in-diffusion of a dopant
contained in the contact material. Selective doping process for a lightly boron doped single
crystal diamond sample, LB04 has been carried through as shown in Fig. 4.8.
LB04 started with a ∼ 7.5µm thick low boron doped film. Then a ∼ 0.6µm thick highly
boron film was deposited on top of the low boron doped film. Etching process was performed

81

with a ∼ 1.2µm thick Al hard mask applied through a tape mask. Using Reactive Ion Etch
(RIE), ∼ 1.6µm is etched into the sample as seen in Fig. 4.8. The low boron doped film has
shown a carrier concentration of ∼ 1.3 ∗ 1017 cm−3 from FTIR estimation.

a)

b)

c)

d)

Figure 4.8 Low boron doped diamond (LB04) selective doping process steps. a)Low boron
doped surface b)High boron doped surface c) Surface with Al mask before etching d) Surface
with Al mask after etching.

Most commonly used process for obtaining low resistivity contacts involves annealing.
Followed by the metal deposition, annealing process involves heating at the eutectic temperature for an optimal time either under vacuum environment or in an ambient of an inert
gas. Annealing results metal to be alloyed into the semiconductor lowering the barrier height
at the interface. Ohmic contacts can be obtained without annealing with molecular-beam
epitaxy (MBE), metal-organic chemical vapour deposition (MOCVD) epitaxy and ion implantation by obtaining high doping level ( 1020 cm−3 ) of the epitaxial layers during the
growth.
For this work, annealing process has been performed followed by metal contact fabrication. Properly aligned shadow mask has been positioned over the selectively doped p+
82

layers as seen in Fig. 4.9. Ti/Au (30nm and 300nm thick respectively) contacts have been
deposited by sputtering under the argon gas with a starting pressure of 2 ∗ 10−5 T orr.

D

C

A

B

Figure 4.9 Shadow mask alignment prior to Ti/Au deposition on boron doped diamond
sample (LB04)

Followed by metal contact fabrication annealing has been performed in the Hall effect
measurement system while monitoring the I-V behavior of the contacts at various temperatures for LB04. Due to selective doping, even at room temperature with no anneal, ohmic
nature of the contacts around the origin can be observed as seen in Fig. 4.10. IV behaviors for six different contact pairs at various temperature regimes have been measured and
uniformity has been observed between different contact pairs. After annealing, IV behavior
at room temperature has been observed to remain ohmic. Thus, resistivity and Hall effect
measurement have been performed on this sample.

83

80

30
AB300K
BC300K
CD300K
DA300K
AC300K
DB300K

10

AB500K
BC500K
CD500K
DA500K
AC500K
DB500K

60
40

Current (mA)

Current (mA)

20

0
-10

20
0
-20
-40

-20
-30
-150

-60
-100

-50

0

50

100

-80
-150

150

-50

0

50

Voltage (V)

Voltage (V)

(a) T=300K

(b) T=500K

100

100

150

100

150

20

AB700K
BC700K
CD700K
DA700K
AC700K
DB700K

AB300K_AA
BC300K_AA
CD300K_AA
DA300K_AA
AC300K_AA
DB300K_AA

15
10

Current (mA)

50

Current (mA)

-100

0

-50

5
0
-5
-10
-15

-100
-150

-100

-50

0

50

100

150

-20
-150

Volatage (V)

(c) T=700K

-100

-50

0

50

Voltage (V)

(d) T=300K (AA*=after anneal)

Figure 4.10 IV curves at various temperatures (300K to 700K) for boron doped diamond
sample (LB04).

4.3

Characterization of Silicon and Germanium

Silicon played an important role in the development of microwave radar during World War
II. This resulted to an in-depth study of silicon and of germanium where the impurity
content during the fabrication of the material could be precisely controlled. Intrinsic and
84

impurity conduction was studied from quantification of the variation of the Hall coefficient
and conductivity with temperature. Thus Fig. 4.11 and 4.12 shows the results of some
of the early experiments on germanium. In these figures, the logs of R and ρ are plotted
against 1/T . During the estimation of n where Boltzman term is exp( /kT ), plot yields a
straight line. Slope of this line gives the activation energy, Ea and Ed , for acceptors and
donors respectively. At low temperatures, the conductivity and Hall coefficient of different
samples may vary from one another by a factor of almost 104 however at high temperatures,
the measurements for all samples have a tendency to be on the same straight line. The line
found at high temperatures demonstrates the intrinsic region. From the slope on the intrinsic
line the energy gap, Eg between the conduction and the valance band was calculated to be
about 0.74eV [10].

Figure 4.11 Hall coefficient data for Ge [10]

Amount of impurities added to a sample effects the point at which the curve for a given
specimen leaves the intrinsic line. With decreasing temperature, the concentration of intrinsic
carriers fall down. At the point where the intrinsic carrier concentration is less than the
85

concentration of the impurities, the behavior will be dominated by the impurities.
Fig. 4.11 shows that on leaving the intrinsic line the Hall coefficient initially appears to
be independent of temperature, but at still lower temperatures the Hall coefficient starts to
rise slowly, suggesting that the carrier concentration is decreasing. This is related to the
energy required to excite carriers from the impurity levels being very small (about 0.2eV ).
Considering the room temperature is equivalent to about 0.025eV , at room temperature all
the impurity carriers are expected to be excited. In order to observe the impurity activation
energy, much lower temperatures are needed. A different behavior can be seen in the plot of
resistivity, on leaving the intrinsic range the resistivity initially decreases as the temperature
is lowered. Resistivity then passes through a minimum and then starts to increase moderately as the temperature decreases. This seems fairly conflicting but can be explained by
the relationship between the concentration of carriers and their mobility. At temperatures
just below the intrinsic range, the carrier concentration is constant, as a result, the variation
of resistivity will be determined by that mobility. It is known that, as the temperature
decreases, mobility increases. At moderate temperatures, scattering of electrons by lattice
vibrations have a dominant effect on the mobility. Significant results can be obtained by taking a closer look to the the amplitude of these vibrations which should fall as the temperature
lowers, as a result the mobility should rise and accordingly the resistivity should decrease.
In the intrinsic range, the mobility still falls as the temperature rises, but its impact to the
conductivity is dominated by the considerably fast variation of the exponential component.
At the lower temperatures, Hall coefficient starts to increase again, which shows that the
free carriers are falling into the impurity levels. The resistivity is also seen to rise at lower
temperatures due to several factors including the reduction in the carrier concentration as
well as the effects of the mobility. At the lowest temperatures, Hall coefficient does not
86

e

Figure 4.12 Resistivity data for Ge [10]
continue to increase, but either approaches to a constant value, or, often falls down. The
reason of this behavior is impurities and imperfections in the sample are dominating over
the lattice vibrations. In the past years, materials had more defects and more imperfection
than current materials, so it shows the presence of impurity scattering very clearly. As a
result, it is observed that at the lowest temperatures the resistivity increases more rapidly
than the Hall coefficient.
Common impurities for germanium include but not limited to boron, aluminum and
phosphorus. Germanium has four electrons in the outermost shell and is a member of group
IV of the periodic table. These four electrons fill the valance band. Phosphorus belongs to
Group V, and has five outer electrons, while aluminum and boron belongs to Group III, with
three outer electrons. One impurity electron is available for each phosphorus atom which
makes the sample n-type. In aluminum and boron doping case, a hole could be created by
every impurity atom which makes the sample p-type. Since a phosphorus atom has one
unit of positive charge and one additional electron on its nucleus more than the germanium
atom, a sole phosphorus atom in a germanium crystal is equivalent to a single hydrogen atom

87

embedded in a medium with the dielectric constant of germanium. Using the Bohr theory
of the atom, energy required to ionize this hydrogen atom can be calculated which is around
0.02eV , this is equivalent to the energy required to take the electron from the phosphorus
atom and place it in the conduction band. The ionization energy of the hydrogen atom
model is found by dividing the Rydberg energy by the square of the dielectric constant
which is around 0.04eV , for germanium. For impurities like aluminum, by considering them
as inverted hydrogen atoms with a negative nuclear charge and planetary positive hole, this
model can also hold. Ionization energy will be the identical to the energy calculated for an
electron [10].
In order to investigate and test the Hall effect system, germanium and gallium arsenide
samples have been measured and tested as seen in 4.3.1 and the following sections.

4.3.1

p-Type Ge

p-Type Ge sample purchased from WRS materials is measured in Van der Pauw (VdP)
configuration. Ohmic contacts observed without any metal deposition. Silver paste used
to connect 50µm gold wires to the sample. Sample thickness is 0.037cm, 40mA is used as
applied current, 10 readings has been taken to average. Conductivity and Hall effect have
been measured from 300K to 700K with 25K steps. To show repeatability, measurement is
also performed cooling back down from 700K to 300K.
Optimization results quantified four parameters for p-Ge as NA = 6.615∗1017 cm−3 , Ea =
0.0115eV , ND = 1 ∗ 1016 cm−3 and Ed = 1.379eV as seen in Fig. 4.13 which is comparable
with the carrier concentration of acceptors as provided by WRS materials. Future studies,
possibly measurements with varied magnetic field with the QMSA technique, are needed to
further investigate the donor concentration and activation energy for this sample.
88

Carrier Concentration (cm-3)

10 19

10 18

10 17

Calculated
Measured

10 16
250

300

350

400

450

500

550

Temperature (K)

Figure 4.13 Concentration vs temperature fit for p-type germanium

In order to study the mixed carrier conduction mechanism for p-type germanium, concentration dependence of the electron and hole mobilities have been taken into account in
Eq. 4.9 and results are shown in Fig. 4.14.

σ = µn + µp

89

(4.9)

100

Conductivity (S/cm)

80
Measured
Calculated

60

40

20

0
300

400

500

600

700

800

Temperature (K)

Figure 4.14 Conductivity vs temperature fit for p-type germanium

Both measurement and physical analysis results show a transition from extrinsic to intrinsic conduction at a temperature range of approximately around 550K.

4.4

Characterization of Diamond Materials

Boron is one of the most common acceptor impurity for diamond where Ei = 0.37eV . In
boron doped diamond samples, transition from a semiconductor to a metal and ultimately
to a superconductor can be observed as the boron doping level increases above 5x1020 cm3 .
In this thesis, the Hall mobility of single crystalline and polycrystalline boron doped diamond is investigated in a large temperature range (300K − 700K)and large doping range
for samples grown by different groups at Michigan State University. Literature studies show
that for low boron-doping level, the low- temperature mobility is controlled by ionized impurity scattering primarily because of the compensation and the high- temperature mobility
by phonon scattering (acoustical or optical with an intraband or interband process) [28].

90

For heavily boron doped diamond samples, the low-temperature mobility is extensively constrained by the neutral impurity scattering (neutral boron atoms), as a result, the maximum
room-temperature mobility is dependent of the doping level.
Boron doped single crystal diamond samples and the effects of substrate temperature
and growth rate on the doping efficiency for diamond samples, SND25, SND26, SND33 and
SND34 are reported by Demlow et al. [14]. Hall effect and resistivity has been studied and
shown in this thesis.
Ohmic contact fabrication is necessary to obtain high quality Hall effect and resistivity
measurements. An acid cleaning step is used prior to physical vapor deposition of metal
contacts consisting of Ti(30nm)/Au(150nm) on the four corners of the sample in a Van der
Pauw configuration.
IV measurements are taken between 300K and 700K for 6 different contact pairs. Linear
IV curves are measured as seen in Figure 1 and the effect of annealing is shown in Fig. 4.15.
700K 600K

10

500K

3

400K

300K AA

2

5
1

300K

300K AA
300K

0

0
-1

-5
-2

-10
-10

-5

0

5

-3
-10

10

-5

0

5

10

Voltage (V)

Voltage (V)

(a)IV curve

(b)Effect of annealing (SND26)

Figure 4.15 Temperature dependent IV curves for SND26. (a)Temperature range of 300K700K (b)Effect of annealing at 300K.

91

Followed by achieving permanent ohmic contacts, resistivity and Hall effect measurements
are performed.

Figure 4.16 Concentration vs temperature fit for SND26

Temperature dependent concentration compared with the literature is shown in Fig. 4.16.
Concentration measurements show a good agreement with the similarly doped single crystal
samples in the literature [4]. The remaining samples with relatively higher activation energy
(Ea) belongs to the low doped samples.
According to Barjon et al. [82], activation energies are different for high and low temperature regimes, which commonly shows a ratio of 2. For SND26 diamond sample, using
a similar activation energy calculation approach, a ratio of 1.9 with activation energies of
0.130eV for high temperature region and 0.248eV for low temperature region is observed as
shown in Fig. 4.17.
The comparison of single carrier model and mixed carrier model has been studied and
shown in Fig. 4.18. Hall effect measurements show a reliable repeatability between the measurements of heating up (300K to 700K) and cooling down (700K to 300K) as indicated
in red squares and blue triangles respectively. The fitting with a boron concentration of
92

-1

-1

Conductivity (Ω ·cm )

100

10
σ

-(0.130/kT)

HT

= 129.4·e

(S/cm)

1
σ

0.1
0.001

0.0015

-(0.248/kT)

LT

= 3950·e

0.002

(S/cm)

0.0025

0.003

0.0035

-1

1/Temperature (K )
Figure 4.17 Activation energy for SND26
6x1018 cm−3 where activation energy is 0.254eV with a concentration of compensating impurities of 3x1017 cm−3 where activation energy is 1.7eV for nitrogen. Single carrier model,
indicated with a dashed line, deviates from the experimental results whereas mixed carrier
model, indicated with a straight line, agrees well as shown with a straight line.

Figure 4.18 Fitting vs experimental results using single carrier (dashed line) and mixed
carrier (straight line) model for SND26

Assuming the concentration of the compensating donors is zero, the simulation results of
93

varied carrier concentration with the activation energy of 0.25eV is shown in Fig. 4.19.
1019

10 × 10 19 (cm -3 )
8
6
4
2
Na = 1 × 10 19 (cm -3 )

1018

1017
(Ea - E v ) = 0.25 (eV)
Nd = 0

1016
200

300

400

500

600

700

800

Temperature (K)

Figure 4.19 Fitting with varied Na for SND26

Simulation results suggest the presence of the compensating impurities in the sample.
Activation energy has a big impact to the carrier concentration as seen in Fig. 4.20 where
the activation energy is ranging between 0.10eV to 0.30eV . There assumed to be no compensation doping involved.
1019

1018

Ea = 0.10 eV
Ea = 0.15 eV
Ea = 0.20 eV

1017

16

10

Ea = 0.25 eV

Ea = 0.30 eV

200

300

Na = 1 × 10 19 (cm -3 )
(Ea - E v ) = 0.10 - 0.30 (eV)
Nd = 0

Ea = 0.35 eV

400

500

600

700

800

Temperature (K)

Figure 4.20 Fitting with varied Ea for SND26
94

Effects of compensating donor concentration was studied and presented in Fig. 4.21
where Nd varies from 1015 cm−3 to 1019 cm−3 . Fittings results with N d values of 1015 cm−3 ,
1016 cm−3 and 1017 cm−3 did not show a significant difference whereas N d = 1018 cm−3 and
N d = 1019 cm−3 shows significant difference in carrier concentration.
1019

1018

1017

1016

Nd
1015
1016
Na = 6 × 10 19 (cm -3 )
(Ea - E v ) = 0.25 (eV)

1017

mp /m0 = 0.7
1018 (cm -3 )

1015
200

Nd = 1 × 10 19 (cm -3 )

300

400

500

600

700

800

Temperature (K)

Figure 4.21 Fitting with varied Nd for SND26

1020
1 × 10 21 (cm -3 )

1019

1 × 10 20 (cm -3 )
6 × 10 19 (cm -3 )

1018

Na = various
(Ea - E v ) = 0.25 (eV)

1017

mp /m0 = 0.7
16

10

1015
200

Nd = 1 × 10 19 (cm -3 )
(Ec - E d ) = 1.7 (eV)

300

400

500

600

700

800

Temperature (K)

Figure 4.22 Fitting with varied Na for SND26
The effect of the effective mass to carrier concentration is shown in Fig. 4.23 where
95

mp/mo is ranging between 0.7 to 1. There is a minimal change in the carrier concentration
with varied effective mass.
1019

1.0
0.9
0.8
mp /m0 = 0.7

1018

1017

Na = 6 × 10 19 (cm -3 )
(Ea - E v ) = 0.25 (eV)
Nd = 0

16

10

200

300

400

500

600

700

800

Temperature (K)

Figure 4.23 Fitting with varied mp/mo for SND26

Mobility analysis were also performed for sample SND26 using the empirical model parameters implemented in Silvaco finite element based software, reported by Marechal et al.
[26]. Analysis shown in Fig. 4.24 takes the doping concentration and temperature dependency of the mobility into account.
Our activation energy calculation studies using the same approach shows 2.2 with activation energies of 0.115eV for high temperature region and 0.254eV for low temperature
region for SND25 as shown in Fig. 4.25.
A ratio of 1.150 with activation energies of 0.130eV for high temperature region and
0.224eV for low temperature region is observed as shown in Fig. 4.26.
A ratio of 1.38 with activation energies of 0.140eV for high temperature region and
0.193eV for low temperature region is observed as shown in Fig. 4.27.
Temperature dependency of the mobility ranges from around T −1.78 to T 2 as seen in
96

300
Calculated
Experimental

200

2

Mobility (cm /Vs)

250

150
100
50
0
200

300

400

500

600

700

Temperature (K)
Figure 4.24 Mobility analysis, simulation and experimental values for SND25

Figure 4.25 Activation energy for SND25
Fig. 4.28. Mobility is impurity dependent and includes a combination of different scattering
mechanisms. Volpe et al. reported an extensive study of temperature dependency of the
mobility for boron doped single crystal diamond and revealed the temperature dependency

97

-1

-1

Conductivity (Ω ·cm )

100

10
σ

-(0.150/kT)

HT

= 205·e

(S/cm)

1
σ

0.1
0.001

0.0015

-(0.224/kT)

LT

= 1737·e

0.002

(S/cm)

0.0025

0.003

0.0035

-1

1/Temperature (K )
Figure 4.26 Activation energy for SND33

-1

-1

Conductivity (Ω ·cm )

100

10
σ

-(0.140/kT)

HT

= 265·e

(S/cm)

1
σ

0.1
0.001

0.0015

-(0.193/kT)

LT

= 1384·e

0.002

(S/cm)

0.0025

0.003

0.0035

-1

1/Temperature (K )
Figure 4.27 Activation energy for SND34
as T −β where βmax = 3.11 for pure material and βmin = 0 highly doped material [5].
Concentration measurements for diamond films with varied doping levels can be seen in
Fig. 4.29.
Reliable high temperature measurements of resistivity and Hall effect for boron doped
single crystal diamond can be achieved in the presence of high quality ohmic contacts. Phys98

2

Mobility (cm /Vs)

1000

100

~T
10

~T

-1.78
2

1

0.1

200

300

400
500
Temperature (K)

600

700

Figure 4.28 Temperature dependent mobility
21

10

20

-3

Concentration (cm )

10

19

10

18

10

17

10

16

10

15

10

200

300

400

500

600

700

Temperature (K)

Figure 4.29 Temperature dependent concentration
ical analysis for the identification of unintentional doping can be a very essential tool for
providing feedback to MPCVD fabrication of single crystal diamond.
In our studies, we have reported the effect of annealing to the Ohmic characteristics of
the contacts. Simulation results have shown the effect of each variable, N a, N d, Ea, Ed,
mp/mo to the carrier concentration. Nitrogen was assumed to be one of the unintentional
99

impurities and mixed carrier concentration simulation results showed a good agreement with
the experimental results.
Carrier concentration of all the samples are consistent with the FTIR and SIMS studies
as shown in Demlow et al.s work [14].

4.5

Physical Analysis Results

Least squares method, using f mincon algorithm has been implemented for optimization
analysis. Details can be found in Sec. 4.6. First analysis carried through with equal weight
on each temperature point in the whole temperature range.
Optimization results quantified four parameters for SND25 as NA = 9.11 ∗ 1018 cm−3 ,

Carrier Concentration (cm-3)

Ea = 0.277eV , ND = 1 ∗ 1016 cm−3 and Ed = 1.489eV as seen in Fig. 4.30.
1020
1019
1018
1017
1016
250

Calculated
Measured

300

350

400

450

500

550

600

650

700

750

Temperature (K)

Figure 4.30 Concentration vs temperature fit for SND25

Optimization results quantified four parameters for SND34 as NA = 3.64 ∗ 1020 cm−3 ,
Ea = 0.13eV , ND = 1 ∗ 1016 cm−3 and Ed = 1.06eV as seen in Fig. 4.31.
Deviation of Ed from the ionization energy of nitrogen, 1.7eV , may be related to the
broadening of the energy levels based on the Schrodinger equation and/or possible existence
100

Carrier Concentration (cm-3)

of unidentified defects contributed as donors.
1020

10

19

Calculated
Measured

1018

10

17

250

300

350

400

450

500

550

600

650

700

750

Temperature (K)

Figure 4.31 Concentration vs temperature fit for SND34

Deposition temperature for SND25 is 850 ◦ C, whereas, for SND34, deposition temperature
is 950 ◦ C as reported by Demlow et al.[14]. Depth profile of SIMS measurements showing
uniformity of doping profile through the sample depth. SIMS results for samples SND25,
SND26, SND33 and SND34 are in good agreement with Hall effect measurments as seen in
Fig. 4.32 even after the possible error reported in the SIMS calibration.
The heavily boron doped sample SND34 grown at 950 ◦ C, was measured by SIMS to
have 39% more incorporated boron than SND33, which was grown under the same reactor
conditions but at 850 ◦ C.
Heavily boron-doped diamond samples grown with a substrate temperature of 950 ◦ C
show fewer defects and higher doping efficiency than those grown at 850 ◦ C.
Finally, for the selectively doped diamond sample, LB04, where IV measurements were
presented in Fig. 4.10, carrier concentration fit using NA = 5 ∗ 1017 cm−3 , Ea = 0.35eV ,
ND = 2.25 ∗ 1015 cm−3 and Ed = 1.7eV is compared with the measurement result in Fig.
4.33.

101

Figure 4.32 SIMS measurements for diamond samples [14]
18

-3

Carrier Concentration (cm )

1x10

17

1x10

16

1x10

15

-3

Nd = 2.25 × 10 (cm )
1

(Ec-Ed ) = 1.7 (eV)
1

15

1x10

17

-3

Na = 5 × 10 (cm )
1

(Ea -Ev) = 0.35 (eV)
1

14

1x10

200

300

400

500

600

700

800

Temperature (K)

Figure 4.33 Concentration vs temperature fit for LB04

4.6

Software Tools for Analysis

In semiconductor physics, the neutrality equation can be approximated as shown in Eq. 4.10
[11].

102

n + NA− = p + ND+

(4.10)

The total negative charges are shown on the left hand side of the equation with n and
NA , electrons and ionized acceptors, respectively. Positive charges are shown on the right
hand side of the equation with p and ND , holes and ionized donors, respectively.
Electron concentration has been calculated as shown in Eq. 4.11 where NC is the effective
density of states in the conduction band and F1/2 is the Fermi-Dirac integral.

2
EF − EC
n = NC √ ∗ F1/2 ∗ (
)
kT
π

(4.11)

In most cases, there is both intentional doping and unintentional doping during the
fabrication of single cyrtsal diamond samples. Thus, in this work, we have implemented two
acceptor concentrations with two activation energies in order to calculate ionized acceptors
as shown in Eq. 4.12a where NA1 , NA2 and Ea1 , Ea2 are the user input values, corresponding
to the acceptor concentrations and their activation energies, respectively.

−
−
NA− = NA1
+ NA2

(4.12a)

−
NA1
= (NA1 /(1 + 4 ∗ exp(

Ea1
)))
kB ∗ T

(4.12b)

−
NA2
= (NA2 /(1 + 4 ∗ exp(

Ea2
)))
kB ∗ T

(4.12c)

Hole concentration has been calculated as shown in Eq. 4.13 where NV is the effective
density of states in the valance band and F1/2 is the Fermi-Dirac integral.

103

2
EV − EF
p = NV √ ∗ F1/2 ∗ (
)
kT
π

(4.13)

Similar to the acceptor concentration approach shown above, we have implemented two
donor concentrations with two activation energies in order to calculate ionized donors as
shown in Eq. 4.14a where ND1 , ND2 and Ed1 , Ed2 are the user input values, corresponding
to the donor concentrations and their activation energies, respectively.

+
+
ND+ = ND1
+ ND2

(4.14a)

+
ND1
= (ND1 /(1 + 2 ∗ exp(

Ed1
)))
kB ∗ T

(4.14b)

+
ND2
= (ND2 /(1 + 2 ∗ exp(

Ed2
)))
kB ∗ T

(4.14c)

In order to the satisfy the neutrality equation, Eq. 4.10, using the physical components
described above, Fermi-Dirac integral has been calculated with changing variables, η =
(E − EC )/kT and ηF = (EF − EC )/kT as shown in Eq. 4.15a - 4.15c.

F1/2 (

EF − EC
) ≡ F1/2 (ηF )
kT
∞

F1/2 (ηF ) =
EC

[(E − EC )/kT ]1/2 dE
1 + exp[(E − EF )/kT ] kT
∞

F1/2 (ηF ) =
0

η 1/2
dη
1 + exp(η − ηF )

(4.15a)
(4.15b)

(4.15c)

Approximation of Boltzmann statistics have not been applied in this work. At the value
of the chosen Fermi level, where neutrality equation (Eq. 4.10) is satisfied and emphasized
with a yellow circle on the left and shown in more detail on the right in Fig. 4.34.

104

n + NA-

n + NA-

n

p + ND+

p + ND+

p

p

(EF – Ev) (eV)

(EF – Ev) (eV)

Figure 4.34 Left: All the postivite and negative charges are shown. Right: Zoomed in image
where the neutrality equation is satisfied.
Using the LabVIEW based software for physical analysis simulations, expected intentional and unintentional concentrations of the impurities can be calculated. In order to
make the analysis computationally inexpensive, MATLAB has been used for physical analysis. A similar approach has been implemented in order to calculate the acceptor and donor
concentrations. However, instead of a graphical technique, minimization technique has been
used.
To solve the optimization problem, we use the Matlab built-in function fmincon. f mincon
function aims to find a constrained minimum of a scalar function of numerous variables
starting at an initial estimate. Number and duration of iterations depend on the initial guess
provided to the program. This is generally referred to as constrained nonlinear optimization
or nonlinear programming.
To apply f mincon, the gradient of the objective function must be given. In addition,
the Hessian matrix of the objective function can greatly speed up the convergence of the
optimization problem. But the new problem is that the cost of computing the Hessian matrix
is too much. In order to solve this problem, we can use a good approximation to the Hessian
105

and the cost of calculating the approximation can be accepted relatively.
For any given starting point, we can find a local minimizer through optimization process,
but it is hard to determine whether it is a global minimizer or it is sufficiently close to
the global minimizer. Therefore, a good choice of the starting point acepptor concentration
and activation energy,Na , Ea and donor concentration and activation energy Nd and Ed ,
respectively can increase the probability of finding the global minimizer and speed up the
convergence rate of the method. In addition, the computational cost of the optimization
algorithm can be greatly reduced with a good starting point.

106

Chapter 5
Conclusions and Future Work

5.1

Conclusion

A temperature dependent Hall Effect measurement system with software based data acquisition and control was built and tested in hardware and software automation level as a part
of this PhD work [1].
The system differs from the previously reported systems by adding several control features including a software based PID temperature controller, temperature dependent I-V
measurements for monitoring the annealing effects on contact resistances, and the ability to
accommodate samples for both van der Pauw and inline configurations.
Transport measurements are primarily shown for boron-doped single crystal diamond
(SCD) films deposited in a microwave plasma-assisted chemical vapor deposition (MPCVD)
reactor, as well as phosphorus doped single crystal diamond films, poly crystal diamond
films, thermoelectric bulk samples and reference samples including germanium and gallium
arsenide. The influence of the ohmic contacts and stability of the temperature control to the
measurements are studied. For a temperature range of 300K-700K IV curves, Hall mobilities
and carrier concentrations are presented. Software analysis and optimization algorithms
based on the neutrality equation are developed and results are compared with the measured
data.
There are some systematic errors related to temperature gradients present in the speci107

mens subjected to Hall measurements. If a longitudinal temperature gradient

Tx is present

in a specimen then a transverse voltage VN appears in it by virtue of the Ettingshausen-Nerst
effect and also a transverse temperature gradient

Ty can generate a thermoelectric error

voltage between the Hall electrodes if the specimen and its Hall contacts have a significant
thermoelectric power between them and if the thermoelectric power is itself a significant
function of the magnetic field [83].
In addition to the Ettingshausen-Nerst effect, there is also Righi-Leduc effect which
basically occurs when a magnetic field is applied at proper angles to the direction of a temperature gradient in a conductor, a new temperature gradient is produced perpendicular
to both the direction of the original temperature gradient and to the magnetic field [84].
The Ettingshausen-Nerst and Righi-Leduc effects can be eliminated by preventing longitudinal heat flow [85]. Since the Ettingshausen-Nerst effect is a function of electron flow, in a
magnetic field, the transverse temperature gradient ∂T /∂y is always present under usual experimental conditions and the Hall voltage is than adiabatic. To avoid those errors, choosing
an appropriate specimen symmetry is helpful but not enough. Permutation of the current
polarity and magnetic field orientation are needed to get rid of the errors introduced by such
thermomagnetic effects on Hall measurements. Thus, the Hall effect measurement system in
this work includes these permutations and eliminates the non-Hall contributions.

5.2

Future Work

Future work includes the following :
• An improved fitting routine will be investigated upon the existing analysis for further
investigation of the transport properties of the semiconductor materials. New fitting
108

routine will be computationally less expensive.
• Thermal-power measurement will be coupled with hall effect to detect effective mass.
Followed by the addition of Nernst effect, effective mass can be detected. From thermalpower measurements, fermi level location can be also determined. Diamond materials
may require a significant power flow, as a result, effective mass measurements can be
more challenging compared to other semiconductor materials.
• Analysis and software implementation of resulting measurements including mixed carrier conduction modeling (QMSA).

109

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110

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