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MATHEMATICAL MODELING AND COMPUTATION OF THE
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Yuanchang Sun

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MATHEMATICAL MODELING AND COMPUTATION OF THE OPTICAL
RESPONSE FROM NANOSTRUCTURES

By

Yuanchang Sun

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Mathematics

2009

ABSTRACT

MATHEMATICAL MODELING AND COMPUTATION OF THE
OPTICAL RESPONSE FROM NANOSTRUCTURES

By

Yuanchang Sun

This dissertation studies the computational modeling for nanostructures in response
to external electromagnetic fields. Light-matter interactions on nanoscale are at the
heart of nano—optics. To fully characterize the optical interactions with nanostructures
quantum electrodynamics (QED) must be invoked, however, the required extremely
intense computation and analysis prohibit QED from applications in nano-optics.
To avoid the expensive computations and be able to seize the essential quantum
effects a semiclassical model is developed. The wellposedness of the model partial
differential equations is established. Emphasis is placed on the optical interactions
with an individual nanostructure, excitons and biexcitons effects and finite-size effects
are investigated.

The crucial step of our model is to couple the electromagnetic fields with the mo-
tion of the excited particles to yield a new dielectric constant which contains quan-
tum effects of interest. A novel feature of the dielectric constant is the wavevector-
dependence which leads to a multi-wave propagation inside the medium. Additional
boundary conditions are proposed to deal with this situation. We proceed with incor-
porating this dielectric constant to Maxwell’s equations, and by solving a scattering

problem the quantum effects can be captured in the scattered spectra.

To my parents

iii

ACKNOWLEDGMENTS

I would like to give my deepest and sincerest gratitude to my advisor, Professor
Gang Bao, for his constant encouragement and support during my graduate study at
Michigan State University. His knowledge, insights and enthusiasm are invaluable.

I would also express my thanks to Professor Andrew Christlieb, Professor Di
Liu, Professor Jianliang Qian and Professor Zhengfang Zhou for their time, efforts
and valuable suggestions. Professor Peijun Li deserves my special thanks for his
continuous help and useful discussions.

I am very grateful to my classmates Zhengfu, KiHyun, Junshan, Yuliang and
Tianshuang for the wonderful time together. Last, but not least, I want to thank

Ling, my wife, for her support during these years.

iv

TABLE OF CONTENTS

List of Figures ........................................ vi
INTRODUCTION ..................................... 1
BACKGROUND ...................................... 5
2.1 Electrodynamics in matter ........................ 5

2.1.1 Microsc0pic Maxwell’s equations ................. 8
2.2 Quantum mechanics in nano—optics ................... 10

2.2.1 Schrodinger equations ...................... 11

2.2.2 'Dirac’s notation and operators .................. 15
PHYSICAL MODELS .................................. 20
3.1 Modeling approaches ........................... 21

3.1.1 Lorentz model ........................... 22

3.1.2 Nonlocal response theory ..................... 26
3.2 Dielectric constant ............................ 35

3.2.1 Exctions and biexctions ..................... 42
3.3 Finite-size effects ............................. 47
MATHEMATICAL ANALYSIS ........................... 52
4.1 Concept of polaritons ........................... 52
4.2 Additional boundary conditions ..................... 55
4.3 Model PDEs and theoretical results ................... 61
NUMERICAL EXPERIMENTS .......................... 74
5.1 CuCl nano slab .............................. 75
5.2 Quantum dots ............................... 79
CONCLUSIONS AND FUTURE WORKS .................. 82
6.1 Plasmons in metallic nanostructures ................... 83
6.2 Quantum dots and ultra-efficient solar cells ............... 90

3.1

3.2

3.3

3.4

4.1

4.2

4.3

4.4

4.5

4.6

5.1

5.2

5.3

5.4

6.1

6.2

6.3

LIST OF FIGURES

Optical interactions with a nanoscale medium .............
Lorentz Oscillator Model. ........................
Uncoupled oscillators ...........................
An ensemble of coupled oscillators. ...................
Two waves in a semiconducting nano slab which confines excitons. . .
the wavenumbers in the vicinity of a resonance w = wex ........
The transmission and reflection spectra of a slab ............
A nano slab ................................
Two dimensional confinement ......................
Three dimensional confinement .....................
Size dependence of spectral transmission and reflection ........
Electromagnetically induced transparency ...............
A stop band for a thick slab .......................
Scattering cross sections of a quantum dot ...............
Excitation of surface plasmons ......................
Surface plasmons of a metallic nanoparticle ...............

A quantum dots solar cell ........................

vi

22

41

55

58

60

62

76

79

80

87

89

Chapter 1

INTRODUCTION

The study of the propagation of light waves in nanostructured media has gener-
ated numerous interests among scientific and industrial communities over the past
decade. Because of the tiny structural scales, some quantum effects which are neg-
ligible macroscopically dominate among factors in shaping the optical properties of
such structures. Modeling and computations of nano optical response hence become
an increasing demand in technical applications and designs of nanostructures, ex-

ploitations of quantum effects are another obvious driving force behind.

Consider an optically excited semiconductor, excitons are formed in response to
the applied field. An exciton is a quantum of electronic excitation energy traveling in
a semiconductor, lots of research works have been devoted to the study of the exciton
effects since the concept of exciton was first introduced by Frenkel in 1931 [21]. The
fundamental importance of the excitons is clear; they play an essential role in optical

effects such as luminescence, fluorescence, photographic process, photoconductivity,

 

and so on [36]. These various features enable semiconductors to be widely used in
numerous fields, including optical communication, computing, biomedical imaging,
and data storage. The optical properties of excitons in bulk semiconductors have been
extensively studied (see Chap. 13 of [33] and references therein). For the excitons in
structures of reduced dimensionality, for example, in nanostructured semiconductors
like quantum well, quantum wire and quantum dot, the states of the exciton shifts
to higher energy as the size of structure decreases. This situation is called quantum
confinement effect, which makes nanoscale semiconductor cease "to resemble bulk,
exhibits strikingly different properties instead [50, 66]. Furthermore, the interactions
between excitons should be taken into account if large density of excitons is produced
in a material. Provided the interaction is attractive, two excitons could bind to
form a biexciton. The exciton to biexciton transition confined in nanostructures has
been observed and studied by many groups, and some resonance phenomena, e.g.,
electromagnetic induced transparency, were reported in [7, 11]. While some physicists
have begun to address a few of the challenging problems in these areas[12, 32, 57, 58],
the mathematical formulation and efficient numerical algorithms remain open. The
dissertation work is an initial attempt of the mathematical modeling and numerical

computations of the quantum confinement effects of excitons.

It should be emphasized that the understanding of excitons was made possible
only after the discovery of quantum theory. Based on this understanding, the quan—

tum mechanics are invoked to characterize the excitation of the nanostructure (viewed

2

as an ensemble of charged particles), and microscopic Maxwell’s equations are em-
ployed to govern the wave propagation. The two equations are bridged via the dipole
moment operator which is a key factor for the derivation of a new dielectric constant.
The resulting equations are a coupled system which is difficult to derive and solve
mainly for two reasons; Firstly, the wavefunctions and eigenenergies on atomic scales
are usually assumed to be known, whereas the many-body Schrodinger equations are
not completely solvable. Approximations have to be made for systems containing
large numbers of particles. Secondly, a set of integro—differential equations must be
solved and usually analytical solution are not achievable. Often numerical schemes
need to be developed to attain solutions. Instead of solving a many-body problem,
we approximate the wavefunctions via a perturbation argument. The induced polar—
ization in the quantum picture then can be calculated, and it will serve as a source
term to Maxwell’s equations. By choosing suitable numerical methods, the solu-
tion of Maxwell’s equations provides an insight to investigate the exciton effects on

nanoscales.

This work is structured as follows. Chapter 2 provides a brief review of the most
important concepts of electromagnetics and quantum mechanics, the central prob-
lems in nano-optics are also introduced. Subsequently, the design and mathematical
analysis of a semiclassical model are presented in Chap. 3 and Chap. 4, respectively.
In particular, Chap. 3 is devoted to a description of modeling approaches both in

classical and quantum pictures, special emphasis is the derivation of wavevector-

3

dependent dielectric constant. In addition, the effects of finite-size are discussed as
an open issue. Chap. 4 deals with the mathematical issues in the model partial
differential equations. Boundary value problems are derived along with additional
boundary conditions. Existence and uniqueness results are obtained. The numerical
experiments in Chap. 5 are specially designed to show the validity of our model. In

Chap. 6 a summary of the thesis is presented and future directions are discussed.

Chapter 2

BACKGROUND

To describe the Optical radiation on nanoscale the field theory based on Maxwell’s
equations is required. To characterize the optical properties of the nanostructures
with which the light fields interact a quantum description is necessary. This section
summarizes the fundamentals of these principles forming a basis for the following
chapters. For more thorough treatments the reader is referred to [8, 29] on electro—
magnetism and [22, 54] on quantum mechanics. This chapter starts with Maxwell’s

equations established by James Clerk Maxwell in 18708.

2.1 Electrodynamics in matter

The spatial and temporal evolution of electromagnetic fields is described by electro—
dynamics which was founded in 19th century by Maxwell. The theory can correctly
predict the propagation of electric and magnetics fields and their interactions in the

presence of charges, currents and any type of matter, and it constitutes a set of four

5

elegant equations known as Maxwell’s equations. In the Gaussian unit, the equations

have the form

V - D(r,t) = 47rp(r, t) ,

V x E(r,t) = —%QP—é:—¢2,

(2.1.1)

 

\ V x H(r,t) = %6D( ” + 4—c’1j(r,t) ,
where E denotes the electric field, D the electric displacement, H the magnetic field,
B the magnetic induction, the constant c is known as the vacuum speed of light, j
the current density, and p the charge density. The electromagnetic properties of the

medium are most commonly discussed in terms of the polarization P and magnetiza-

tion M according to

D(r,t) = E(r, t) + 47rP(r, t) ,H(r, t) = B(r,t) — 47rM(r,t) .

Therefore, in vacuum E E D, B = H. In addition, the charge conservation is implic-
itly contained in Maxwell’s equations. Taking the divergence of the last equation in
Eq. 2.1.1, noting that V - V x H is identical zero, and substituting the first equation
for V - D one obtains the continuity equation

69(1‘, t)

at =0.

 

V-J(r,t)+

After substituting the fields D and B in the Maxwell’s curl equations and combining

the two resulting equations we obtain the following wave equations

162E 471'6 , aP
VXVXE+Z§_6t2=_;-2_BT(J+—5t_+CVXM), (2.1.2)

162H 47r aP , 62M
VXVXH+;§—a—t2———c—(VX5t—+VXJ+-5t—2—). (2.1.3)

The expression in the bracket of Eq. (2.1.2) can be associated to the total current
denszty Jt = J3 +Jc+ -— +cV x M, where J 1S split into a source current densrtyjs and

8t
. . . . 8P ,
an induced conduction current densrtyjc. The terms at— and V x M are recognized as
the polarization current density jp and the magnetization current density jm. In most
practical applications, such as scattering problems, there are no source currents or

charges present. However, to study the fields inside a medium the induced polarization

by the excited charges serves as a source term in the equations.

Since the material properties are discontinuous on the boundaries, knowledge of
the variation of fields quantity across an interface is often necessary in solving or

formulating electromagnetic problems. These boundary conditions can be given as

[8]

nx(E1-—E2)=0 onl",
nx(H1—H2)=K onf‘,

n-(Dl—D2)=0 onl",

n-(Bl—B2)=0 onF,

where n is the unit normal vector on the boundary I’. K and a are the surface
current and charges on the boundary, respectively.

Often the fields of interest vary harmonically (sinusoidally) with time. The time

dependence in the wave equations can be easily separated to obtain a harmonic dif-

ferential equation. Here, a monochromatic field can be written as

E(r, t) = Re{E(r) exp(—iwt)} .

With similar expressions for the other fields. Notice that E(r, t) is real, whereas the
spatial part E(r) is complex. The symbol E will be used for both, the real, time-
dependent field and the complex spatial part of the field. The Maxwell equations can

then be written as

V ° D0) = 4700(1) .

v x E(r) = +£‘gi13(r) ,
[ (2.1.4)

 

[ v x H<r> = —%’D(r> + igijtr).
2.1.1 Microscopic Maxwell’s equations

This set of equations together with the appropriate boundary conditions can success-
fully determine the time and spatial evolution of electromagnetic fields in the presence

of a matter. Both the above equations and Eq. (2.1.1) belong to macroscopic elec-

8

tromagnetic theory of bulk media, all the variables are averaged quantities. However,
when the size of medium reduces to nanoscale, although small compared to the wave-
length, the medium consists of many charged particles. On a macroscopic scale the
charged density p and current density j can be treated as continuous functions. But
the discrete charges are spatially separated. Thus, the microscopic structure of matter
is not considered in macroscopic Maxwell’s equations. In order to derive the optical
responses of a nanoscale system it is essential to consider the fields which are created
by atomic electric charges in motion. In this case, we must carry out a microscopic

treatment of the electromagnetic fields and set

p(7‘) = 2816(1‘ — r1) ,j(7‘) = 2‘3"er - I‘l),
l l

where v) is the velocity of the l—th charged particle.

Hence the so called Microscopic Maxwell’s equations take the following form

V-E=47rp
VxE= ——1-6H

[ CW (2.1.5)
V-H=0

 

This set of equations are the most basic ones, which describe the electric field E and
the magnetic field H in vacuum, together with their sources (charge- and current-

densities). That is, it is assumed that there is no other ponderable matter in the

9

system than the charges and currents accounted for in the equations. There several
ways of deducing the macroscopic equations from their microscopic counterparts (see
[63] for example). To derive the polarization of the charge distribution we consider

the total current density

._aP

we ignored the contribution of the source current js which generates the incident
field since it is not part of the considered particle. Furthermore, we incorporate the
conduction current jc into the polarization current. The optical response is given by

the polarization P induced on the matter is

P = (Mimi!) = /\II*[1\IIdr,

where ft is the polarization operator. (\Ill [1|\II) stands for the expectation value of
operator [1 in Dirac’s notation. ‘II is the wavefunction of the matter under the applied
field. Next part will be dedicated to the Schrodinger equation with \I! being its

solution.

2.2 Quantum mechanics in nano-optics

N anostructures or nano optical materials can capture and manipulate light in many
amazing ways, and they may possess drastically different optical properties from their

bulk counterparts. To get insights of the optical responses of these structures, an ac-

10

curate characterization of excitations in them for an applied field becomes essentially
important. As we move to smaller and smaller scales the underlying physical laws
change from macroscopic to microscopic, some quantum effects start to dominate
in shaping the material properties. The quantum mechanics hence must be invoked
to fully characterize the evolution of the matter states on nanometer scales. The
following section is devoted to some fundamentals and basic concepts in quantum

mechanics. In particular the Schrédinger equations will be briefly discussed.

2.2.1 Schrodinger equations

Quantum mechanics is a set of principles underlying the most fundamental known
description of all physical systems at the submicroscopic scale. It is essential to un-
derstand the behavior of systems at atomic scales and smaller. In the development
of quantum mechanics, Planck’s quantum theory, Bohr’s postulates, and de Broglie’s
hypothesis represented very important steps. However, they are overshadowed by
the discovery of a fundamental differential equation describing the electron and ac—
counting for its wave properties, and the construction of a theory accounting for
the quantum nature of radiation. The crucial move in this connection was made by
Schrédinger in 1926 when he proposed a partial differential equation that turns out to
be generally applicable to the motion of charged particles in the nonrelativistic regime
(1) << 0). Schrodinger equation is as central to quantum mechanics as Newton’s laws

are to classical mechanics.

11

The time dependent Schrodinger equation for a general quantum system is of the
form
8

magma) = Him, t) , (2.2.1)

where i is the imaginary unit, it is the Reduced Planck’s constant (Planck’s constant
divided by 27r), the wavefunction \Il(r, t) which describes the behavior of the system
may be statistically interpreted by means of the Schrodinger theory. In particular,
the quantity \II*(r, t)\Il(r, t) = [\II|2, which plays the role of a distribution function,
represents the probability density, or probability of the system in any of possible states;
H stands for the Hamiltonian operator which is a Hermitian operator. Take a single

particle in three dimensions as an example, the Schrodinger equation takes the form

a h? 2
’ — = —— \II t
thatwr, t) 2MV (r, )+ V(r)\IJ(r,t) ,
where r = (x, y, z) is the particle’s position in three—dimensional space; \Il(r, t) is the

wavefunction, which is the amplitude for the particle to have a given position r at

any given time t; M is the mass of the particle. Here the Hamiltonian operator reads

. n2
H = _2—MV2 + V(r), V(r) is the time independent potential energy of the particle

at each position r and V2 is the Laplace operator.

Consider the optical interactions of a nanoscale medium and light which are en-
countered in various fields of research. For instance: the activity of proteins and

other macromolecules is followed by optical techniques; optically excited single mole-

12

cules are used to probe their local environment; and optical interactions with metal
nanostructures are actively investigated because of their resonant behavior important
for sensing applications. Furthermore, various nanoscale structures are encountered
in near-field optics as local light sources. All above matters or structures are small
compared to the wavelength of light, and they consists of many charged particles.
To describe the behavior of this many-particle (N particles) system in response to an
electromagnetic radiation E(r, t) (laser beam for example), the general Hamiltonian
takes the form

HM = H0+Hint ,

which is written as the sum of the Hamiltonian Hg for the matter system and an
interaction Hamiltonian, Hint: which describes the interaction of the matter with the

electromagnetic field. Usually, we take them to be the following forms [10]

H0 = Z{-1—p§ + V(rl,rj)} , (2.2.2)

2
M ml

Hint = - Z elr) - E(rl,t) , (2.2.3)
l

where 61, ml, rl and Pl are the charge, mass, coordinate and conjugate momentum of
the coordinate, respectively, for the lth particle, V(rl, rj) is the potential interaction

energy of the lth and jth particles. We assume that E(r, t) can be represented as a

13

discrete sum of (positive and negative) frequency component as

E(r, t) = ZE(r,w) exp(—iwt) .

no

So (3.2.1) can also be written as —wadrP(r) -E(r,t)exp(—z'wt), where P(r) =

Z) elrd (r — rl) is the electric dipole moment operator.

The wavefunction \Il(r, t) = \Il(r1, - -- ,rN,t) satisfies

ihgt—WUJ) = HMr(r,t) . (2.2.4)

In general, V has contributions from all four fundamental interactions so far
known, namely strong, electromagnetic, weak, and gravitational interactions. For
the behavior of electrons only the electromagnetic contribution is of importance, and
within the electromagnetic interaction the electrostatic potential is dominant. Since
the masses of nuclei are much greater than the mass of an electron, the nuclei move
much slower than the electron. This allows the electrons to practically instantaneously
follow the nuclear motion. For an electron, the nucleus appears to be at rest. This is
the essence of Born-Oppenheimer approximation which allows us to separate the nu-
clear wavefunction from the electronic one. We can restrict the index I in Eq. (2.2.2)
to run only over electron coordinates. In the case of H M = H0 we can separate the

t and r dependence as

00
\I/(r, t) = Z e"/hE'ntton(r) .

72:1

14

where cpn and En are the eigenfunction and eigenvalues of H0, i.e., Hot/271(r) =
Encpn(r) .

After the radiation field is turned on, the system experiences an external, time-
dependent perturbation represented by the interaction Hamiltonian Hint- A pertur-

bation method will be discussed to obtain the wavefunction in the next chapter.

2.2.2 Dirac’s notation and operators

In quantum mechanics it is very common to write the Schrodinger equation (2.2.1)

using the so called Dirac’s notation.
. 8 ~
malt/2(0) = Hlv) - (22-5)

In the following section I will briefly introduce the Dirac’s notation which has many
advantages, especially from the physicist’s point of view. In addition the basic math-
ematics of vector space as used in quantum mechanics will be formulated. The reader
is referred to [54] for a detailed discussion.

Ket Space. Consider a complex vector space whose dimensionality is specialized
to physical system. In quantum mechanics a physical state, for example, an atom
with definite spin orientation, is represented by a state vector in a complex vector
space. Following Dirac, we call such a vector a ket and denote it by [(1). This state
ket is postulated to contain complete information about the physical state. Two kets

can be added: Ia) + [5) = [7). The sum [7) is another ket. If la) is multiplied by

15

a complex number c, the resulting product cla) is another ket. One remark is that
la) and cla), with c # 0, represent the same physical state. In other words, only the

“direction” in vector space is of significance.

An observable, such as momentum and energy, can be represented by an oper-

ator, such as A. Generally an operator acts in a ket from the left

A - la) = Ala) , (2.2.6)

which is another ket.

In general, Ala) is not a constant times la). However, there are particular kets of

importance, known as eigenkets of operator A, denoted by la), lb), with the property

Ala) = ala),Alb) = blb) , (2.2.7)

where a, b are just numbers. The physical state corresponding to an eigenket is called

an eigenstate.

Bra Space and Inner Products. The vector space we have introduced is a ket
space. The notation of a bra space, a vector space “dual to” the ket space will be
discussed in the following. We postulate that corresponding to every ket [01) there
exists a bra, denoted by (al, in this dual, or bra, space. Roughly speaking, we can

regard as some kind of mirror image of the ket space.

The bra dual to cla) is postulated to be c* (al, not C(al, which is a very important

16

point. More generally, we have the bra dual of cala) + Cfilfi) being 3(al + cE<fil We
now define the inner product of a bra and a ket. The product is written as a bra

standing on the left and a ket standing on the right, for example

(Ala) = (WU-(10>) -

This product is, in general, a complex number. Two fundamental property of the
inner product are postulated as follows. First, (filer) = (alfi)*. Second, (ala) 2 0,
where the equality sign holds only if la) = 0, or called null ket. Two kets la) and [5)

are said to be orthogonal if (fila) = 0.

Operators. As I mentioned earlier, observables like momentum and energy are

to be represented by operators that can act on kets from the left

X - (lei) = Xia),

and the resulting product is another ket. Operators X and Y are said to be equal
if X la) = Yla) for any ket in the ket space. An operator X always acts on a bra
from the right side ((al) - X = (alX, and the resulting product is another bra. The
ket X lo) and the bra (alX are, in general, not dual to each other. We define (ale
to be the dual of X la). The operator X I is called the Hermitian adjoint of X. An
operator X is said to be Hermitian if X = X l. Operators can be added; addition

operations are commutative and associative: X + Y = Y + X, X + (Y + Z) =

17

(X + Y) + Z. Operators can be multiplied and multiplication operations are, in
general, noncommutative, that is X Y 5:5 YX. Multiplication operations are, however,

associative: X(YZ) = (XY)Z = XYZ.

We have considered the following products: (alfi), X la), (0(le and X Y. Let us

multiply [,8) and (al, in that order. The resulting product

(Ifil) - ((04) = |fi><a|

is known as the outer product of [6) and (al. In fact, lfi)(al is to be regarded
as an operator which will be emphasized in a moment. It is hence fundamentally
different from the inner product (alfi). It is clear that multiplication operations
among operators are associative. Actually the associative property is postulated to
hold to deal with multiplications among kets, bras, and operators. Dirac calls this
important postulate the associative axiom of multiplication. To illustrate the axiom

we consider an outer product acting on a ket:

(WWII) - [7),

which is the same as [(3) - ((047)), where (alry) is a complex number. So the outer
product acting on a ket is just another ket; in other words, l[)’)(a[ can be regarded

as an operator. It is easy to see that if X = lfi)(oz[, then X1r = la)(fll. In a second

18

important illustration of the associative axiom, we note that

(W!) - (Xlal) = (MIX) - (lei)-

Since the two sides are equal, we might use the more compact notation (5 [X la). With
all these concepts and properties we define the expectation value of an operator A

taken with respect to state la) as (A) = (alAla).

19

Chapter 3

PHYSICAL MODELS

The study of optical interactions with nanoscale systems (nanostructures) is at the
heart of nano-optics. Unlike their bulk counterparts, nanostructures may control and
manipulate lights in ways beyond our imaginations. The novel optical properties
have made them promising candidates in various applications. Hence the relevant
optical modeling of nanostructures has become an increasing demand. This chapter
is devoted to the modeling approaches for the optical responses of nanostructures.
Generally, it is a challenging task to build a good model for nano optical responses
for several reasons; First, it contains multiple scales for which careful analysis needs
to be made especially on the interfaces; Secondly, some quantum effects (for example,
quantum confinement) becomes more important as scales getting smaller. Hence a
rigorous characterization is a must to take the quantum effects into account; Finally,
the resulting equations are usually a coupled system which poses many issues in terms

of analysis and computations.

2O

 

To completely understand light-matter interactions quantum electrodynamics (QED)
needs to be employed. Although QED provides accurate characterizations of optical
interactions with atoms and molecules, nanostructures are usually too complex and
thus the required extremely intense computation prohibits QED from applications in
nano-optics. Reader is referred to [13, 14, 38] for a thorough introduction of QED.
Two types of modeling approaches will be introduced in this chapter: classical model
and semiclassical model. In the Lorentz model the motions of charged particles and
electromagnetic field are both treated classically. Cho’s nonlocal response theory is a
semiclassical approach which requires a quantum mechanical characterization for the
moving charges, while treats the field using classical Maxwell’s equations. In addition,
two other types of approached by Keller and Stahl will be briefly mentioned in the
chapter. Starting from the nonlocal response theory, a different semiclassical model
is developed. The crucial step is to derive a wavevector-dependent dielectric constant
which contains quantum confinement information and nonlocal effects as well. The
model problem can then be solved by combining the new dielectric constant with
the Maxwell equations. Hence by solving Maxwells equations for the fields, these

quantum effects can be captured.

3.1 Modeling approaches

Consider a nanoscale medium exposed in a laser beam as illustrated in Fig. 3.1. We

are interested in how this medium responds to the applied field, or how the optical

21

 

 

 

\

Figure 3.1: Optical interactions between a nanoscale medium and electromagnetic
radiation.

properties are determined. Physically the external field will leads to the motion
of a charged particle. The following Lorentz model offers the simplest picture of
particle-field interactions. It is purely classical; however, this model is an elegant tool
for visualizing particle—field interactions. The Lorentz Oscillator model also bears a

number of basic insights into this problem. For a basic discussion of this model see

[33, 40].

3.1.1 Lorentz model

Lorentz was a late nineteenth century physicist, and quantum mechanics had not yet
been discovered. However, he did understand the results of classical mechanics and
electromagnetic theory. Therefore, it is not surprising that he described the problem
of atom-field interactions in these terms. Lorentz thought of an atom as a mass ( the

nucleus ) connected to another smaller mass ( the electron ) by a spring, Fig. 3.2.

22

electron

 

 

nucleus

Figure 3.2: Lorentz Oscillator Model.

The spring would be set into motion by an electric field interacting with the charge of
the electron. The field would either repel or attract the electron which would result
in either compressing or stretching the spring. Lorentz was not positing the existence
of a physical spring connecting the electron to an atom; however, he did postulate
that the force binding the two could be described by Hooke’s Law, i.e., F(a:) = —fi$,

where :17 is the displacement from equilibrium.

If Lorentz’s system comes into contact with an electric field, then the electron will
simply be displaced from equilibrium. The oscillating electric field of the electromag-
netic wave will set the electron into harmonic motion. The effect of the magnetic
field can be omitted because it is miniscule compared to the electric field. Lorentz

also considered the possibility of damping in his model.

23

The result of Lorentz’s work was the Abraham—Lorentz Equation:

d2 d —iwt

This equation has one of the most well-known forms of mathematical physics. Here
we assume that the mass m of every oscillator (electron) carries a charge 6; fl = mg
is the spring constant with “’0 being the eigenfrequency. Lorentz understood the
origin of all the terms in the equation except for the damping term. He really could
not quantify 7. But this result is not surprising because the full interpretation of
7 requires at least a semiclassical treatment of this problem. The main sources are
atomic collisions, the Doppler effect, and spontaneous emission. The general solution
of Eq. (3.1.1) is a sum of the general solution of the corresponding homogeneous

equation and of a special solution of the inhomogeneous one. Correspondingly we

assume the following ansatz:
:r(t) = 2:0 expl—z'(wg — 72/4)1/2t] exp(—t7/2) + rpe—iw” . (3.1.2)

The first term on the right hand side is the solution of the homogeneous equation and
describes a transient feature. For (1)8 — 72/4 > 0, one finds a damped oscillation with
a damping-dependent eigenfrequency (w?) — 72/4)1/2. The above inequality defines

the regime of weak damping. For stronger damping one gets essentially decaying

term.

24

This transient feature disappears in any case after t > 7—1.

It is thus of im-
portance for (ultra-) fast, time—resolved spectroscopy case. For the stationary, linear
optics regime, we may safely omit this term. What is then left is a forced oscilla-
tion with amplitude 11p. Inserting the ansatz (3.1.2) into (3.1.1) we find the usual

resonance term

(3.1.3)

This oscillation is connected with a dipole moment of u = exp and a polarizability
(1(a)) = ,uEO. The cumulative effects of all individual dipole moments of all electrons
(oscillators) result in a polarization per unit volume P = N ,u, where N is the number

of electrons per unit volume. It is known that the electric displacement

D N8 2 2 —1
— E + P — l + —— _ _ ' E
50 50 (0)0 w W7)

and

N 2
5(a)) = 1+ —E—(w8 — w2
80m

— 2'1427)—1 .

Materials contain not only one type of oscillators and one resonance frequency 0J0,
but many of them - like phonons, excitons etc. In linear optics, i.e. in linear response

theory, we can just to sum over all response leading to

N82 . _
5(a)) = 1+ 2 ———(w(2)j — (1)2 — 2w7) 1.

3.1.2 Nonlocal response theory

Based on understanding the microscopic and nonlocal character of the nano optical
responses, Cho and his group has developed a microscopic nonlocal response approach
in [12]. The size— and shape-dependent response of nanoscale systems is due to the
presence of size-quantized levels which should be described microscopically (quantum
mechanically). Nonlocality appears as a straightforward consequence of quantum
mechanical calculation of susceptibilities. The applied field E(r) at a point r induce
polarization P(r), not only at the same position, but also at other positions within
the extent of the relevant wavefunctions. Therefore, P is given as a function of E(r),

namely

P(r) = [V x(r,r') - E(r,)dr' .

The framework of Cho’s theory will be presented in the following, we start with

rewriting Maxwell’s equations (2.1.5) in frequency domain

r
V - E(r) = 47rp(r)

[ v x E(r) = igmr)
(3.1.4)

 

v x H(r) = —%‘*’E(r) + 461m).

From now on ad will be dropped for simplicity purpose. Instead using the current

density j, sometimes it is more convenient to operate with polarization P since they

26

have the following relation (Fourier transformation of Eq. (21.6))
j(r) = —in(r) . (3.1.5)
From Eq. (3.1.4)a.nd (3.1.5), we eliminate the magnetic field H and obtain
V x v x E(r) — q2E(r) = 47rk2P(r) (3.1.6)

with q = w/c being the wavenumber in the free space.

For the nonlocal response as proposed in Cho’s approach the polarization is related

toEby

P(r) = fx(r,r')E(r')dr' , (3.1.7)

From the first principle the susceptibility X57) in terms of polarization operator P is

calculated as

 

 

OPrAAPr’O 013 ’AAPrO
X€n=2{< I g( )I it I n( )l) <|n(r)| X | 5( )l i}. (31.8)

,\ EA—hw—26+ + E,\+hw+z'6+

The detailed derivation will be given in next part. Here E /\ and [A) are the energy
and eigenstate of the unperturbed hamiltonian H0 and (g, 7)) are the components of
Cartesian coordinates. The two terms of (3.1.8) are the resonant and nonresonant
parts respectively. According to these two parts, the polarization P into resonant

Pr and nonresonant Pm parts. And the latter is assumed to be described by the

27

background susceptibility Xb- The equation (3.1.6) can be changed to

where Pm» = Xb9(r)E(r), and 6(r) = 1 or 0 for r inside or outside the background

medium, respectively. Then the above equation can be rewritten as
V x v x E — (125ng = 47rq2Pr (3.1.10)
with 5b = [1 + 47Tth9(r)]. The solution of (3.1.10) is given by
E(r) = E0(r) + /G(r,r’) - Pr(r')dr’ , (3.1.11)

Where E0(r) is the solution of Eq.(3.1.10) for P(r) = 0, and the Green’s function

G’ (r, r’ ) satisfies the equation
V x V x G(r,r’) — q2€bgG(r,r’) = 47r16(r,r') (3.1.12)

with I being the identity operator.

 

Denote
(Olf’ (r)IA)<A|I-°’ (r’)l0) ,,.
X3, = Z is). _ ,W _ 23+ = Z CAPA§(1‘) pintr’i . (3.1.13)
A ,\

28

where CA(w)=1/(E,\ — hw — 36+) and p/\€(r) = (0|P§(r)l)\).

Then we can write the polarization Pr(r) as the following

Pr§(r) = fdr’ Z XénEli(rl)

n=z.y.z

= ZCA(w)/dr' Z PA§(P)*P,\n(r')En(r’)
A 7)

=27)in

20A<w1pigtrr Z fdr’pintr’mntr’)
/\

”2393112

= [cutaway [dr'pA(r’)-E(r’,w)
A

Therefore

Pr(r) = ZCA(w)p/\(r)* fdr’p/(OJ) - E(r,,w) (3.1.14)
A

Plug Eq. (3.1.14) into (3.1.11), we obtain

E(r) = E0(r) + Z CA(w) fdr’ fdr”G(r, r’) - p/\(r’)*py(r”) - E(r”,w) (3.1.15)
A

The inner products of Eq. (3.1.15) with p)‘ (r) integrated over r lead to the following

linear algebraic equations as

(EA — hw —16+)F,\ + ZAMIF/v = F? (3.1.16)
2"

29

1

With FA = EA _ hw _ 16+ [MAM E(r), Fl) = fdww‘) ' E0(I‘) and

 

AAA’ 2 —/dr/dr'p)\(r) ' G(r,r') - pA/(T,)*.

As described in the linear equation system Eq. (3.1.16) in the problem of radiation-
matter interaction, we have to deal with the coexisting electromagnetic field and
charged particles. The problem is formulated as a scattering process. To compute F2
an eigenvalue problem of Schrbdinger equation must be solved, then an integral must
be carried out over the extension of the wavefunction. Once the initial condition
F9 is given Eq. (3.1.16) could be solved. Finally the field E(r) is obtained by
solving an integro—equation. This is a rather complicated process. In [12] the nonlocal
response theory is formulated in a different way from what is discussed above. In
fact, Maxwell’s equations are given in terms of vector and scalar potentials, A and

ab, respectively

v2¢+ %v - 351} = —47rp

(3.1.17)
2 162A 18 _ 4.
with
E=38—A—V¢,H=VXA.
cat

It can be verified that Maxwell’s equations in potential forms (3.1.17) are invariant

30

under the following transformations

/_ 18G
45 — ‘2???
A’=A+VG,

where G is an arbitrary and well-behaved function called the gauge function or gauge
operator. Maxwell’s equations (3.1.17) enjoy the full gauge freedom. However, a
particular gauge may be chosen with an appropriate choice of the gauge function.
For example, a gauge transformation can always be made on the potential so that

the Lorenz gauge condition is satisfied,

0=V-A’+%%=V-A+VQG+%%—%%27g. (3.1.18)
Another choice is Coulomb gauge, which leads to the equation of motion
O=V~A’=O=V-A+V2G.
With Coulomb gauge being used in [12], we can easily solve for d)
¢(r, t) = / Tr—erlpa’, t)dr' . (31.19)

In terms of this result and consider the Fourier component of . ent of the the first

31

 

equation in (3.1.17) can be rewritten as

 

4 1 I I_- I,
—(V2+q2)A(r,w) = 77rj(r,w)+E/vV ‘10. w) , (3.1.20)

lr-r’l

. . . . . w .
where a partial integration has been carried out in the integral and q = — IS the
c

wavenumber in free space. Defining the Green function Gq satisfying the equation
2 2 _
(V + q )Gq(r,w) — —47r6(r) ,
we can easily write down the solution of (3.1.20) as

A(r) = A0(r) + E/th — r’)j(r')dr’

II II.- II
+ _1_/dr’/dr”Gq(r—r’)V V J(r ,w) ,
47rc

[I'l _ r//[

 

where the argument (1} is omitted for simplicity and A0 is the free field, usually
corresponding to an incident field. Here the vector potential A is expressed as a

functional of current density j
A(r) = A0 + glj] (3.1.21)

with
1

GD] = — / Gq<r—r’>j<r’>dv~’

32

 

II I/_ - II
~l—-1—fdi"’/d7"lG'q(r—r’)V V ‘10. ,w) .
47rc

[r’ _ r/II

Next a functional j of A will be established from the first principle. The general
Hamiltonian for an assemble of charged particles in a given electromagnetic radiation
is:

1
HM = $5,7th — %A(r1.t»2 + an that) + Von} .

Where the el, m), 1"] and PI are the charge, mass, coordinate and conjugate momentum

of the coordinate, respectively, for the lth particle. And

1
H0 = Zi2—m—lpl2 + V(7"l)}
l

 

2

8 e
H- =2: 1A2——l-A .
mt l 2mlc2 mlcpl +3er V¢

are regarded as the unperturbed Hamiltonian and the interaction Hamiltonian, re—

spectively. The Schrodinger equation governs the time evolution of wavefuvntion \II
. 5‘ ~
zit—\Il(r,t) = ij‘I’(r,t) .
8t
Without giving the detailed derivation we write down the final expression

j(r,t) = (wilt) = HA) , (3.1.22)

33

 

where

1:."
II
M
g [:m

I = Z2%— [(r—r1)+6(r—r1)pl].
I

So we have the functional relationships between the induced current density j(r, t)

and the vector potential A(r, t) as

j(rit) = fIAI ,

A(r, t) = A0(r,t) + Glj] .

The former is obtained from the solution of the Schrodinger equation for a given
vector potential and the latter from the Maxwell equations for a given current den-
sity. These two equations for A and j should be solved simultaneously. Indeed, by
specifying the initial condition both j and A can be calculated at any (r, t). Reader

is referred to chapter 2 of [12] for for detailed formulation of the approach.

In addition to Cho’s nonlocal microsc0pic response theory, there are two types of
approaches by the groups of Keller [32] and Stahl [58] worth to mention. They both
take the coherence of matter—excited states into account in calculating the linear and
nonlinear optical responses of various forms of mater within the semiclassical scheme.
Keller’s method is rather similar to Cho’s approach; Two integral equations for elec-

tromagnetic field and transition current density. Stahl’s coherent wave approach uses

34

“interband transition amplitudes” to set up the constitutive equation between the

interband current density and electric field.

Inspired by Cho’s work and based on understanding the nonlocality nature of nano
optical responses, a different semiclassical model is developed. It should be pointed
out that only the transverse component of the electromagnetic fields and full Coulomb
interaction among particles are included in the matter Hamiltonian. However, a dif-
ferent Hamiltonian is used in our model. For the matter Hamiltonian (unperturbed
Hamiltonian) only the kinetic energy and Couloumb interaction of charged particles
are included, and for the matter-light interaction Hamiltonian, we take the full EM
field as the external field. By using the dipole moment approximation and introducing
the electric dipole operator, we are able to write the matter—light interaction Hamil-
tonian (perturbed part) into an integral form. After these initial steps, we derive a
formula for the calculation of the susceptibility, then a new dielectric constant which
depends on the wavevector as well as frequency. The model is applied to study various

problems related to exctions and biexcitons confined in nanoscale semiconductors.

3.2 Dielectric constant

In this part a new dielectric constant will derived from the first principle. Starting
from the microscopic Maxwell’s equation (2.1.5) and nonlocal polarization (3.1.7), a

new Hamiltonian for an assemble of charged particles in an applied electromagnetic

35

radiation is introduced to read
HM : H0 + Hint t

which is written as the sum of the Hamiltonian H0 for the matter system and an
interaction Hamiltonian, Hinta which describes the interaction of the matter with the

electromagnetic field. Usually, we take them to be the following forms [10]

H0 = ;{271nl Pl + V00]

Hint" — — ZelrrE (rlit) 7 (3.2.1)

where el, ml, r) and Pl are the charge, mass, coordinate and conjugate momentum
of the coordinate, respectively, for the lth particle, V(rl) is the Couloumb interaction
of charged particles. We assume that E(r, t) can be represented as a discrete sum of

(positive and negative) frequency component as

)=E(Z (r w) exp (—z'wt).

w

~

So (3.2.1) can also be written as — Z/drP(r) - E(r,t)exp(—iwt), where P(r) =
w

21 elrd (r — rl) is the electric dipole moment operator. The time evolution of matter

36

state is described by the Schrodinger equation
. 6
zha—tiw» = (H0 +H...t)iw>. (3.2.2)

The matter state prepared at time t = to as an eigenstate of H0 will experience time

evolution after switching on the radiation-matter interaction.

The susceptibility tensor with respect to the induced polarization can be derived

from the conventional time—dependent perturbation theory.

By introducing [7303» = exp(z'H0t/ fi)lt/1(t)), Equation (3.2.2) can be rewritten as
. 3 ~ - ~
malt/4t» = Hit/1) (3-2-3)

with

A

H = exp(z'H0t/h)H,-m exp(—z'H0t/h) .

Solving Equation (3.2.3) up to the first-order of H, we have

By assuming that the matter is in the ground state lg) at t = —oo, the first-order

perturbed states are given by

. 7: t i ——w—i 7' /" / ~ /
Wt» -—- lg) 7289/ drew” 7/”) (ail-[drptri-Etrilim
w 5 0°

37

 

_ ({l—fdr’P(r’)-E(r )’Ig>ez' (E /n— 44— wet
— I9) {383 E€_,,,,_,, 5

)

where a factor exp(%r), 7 —> 0+ is introduced to indicate that at time to -—> —00, the
perturbation Hint is turned on adiabatically. This factor serves mainly the purpose
of keeping the derivation of all mathematical quantities properly behaved, i.e., non-
singular. Here, Eg indicates the eigenenergy of excited state [5) measured from that

of the ground state lg).

The expectation value of the polarization can be calculated as

P(nt) = (43(4)IeiHOt/hf)(rie—iHot/hlzitt»

 

_ . <ai—fdr'Ptr'>-E<r'>|g> -1... it
— —;Z<gIP<r>Is> E€_,w_,, e W)

 

_ [A I . ~ I .
_ ggmpmg, <91 [3:11-51 :3 marten/mt .

Then the Fourier component P(r) may be written as

 

(€|(fdr'P(1") E(l‘ g)(9[ fdr'P( r') E(I‘ )IE)
MW; glP(r ”’6 Eg— — 2'7 M; (€[P(r) E5 + fiw + i7

(3.2.4)

By ignoring the second term of Equation (3.2.4) that corresponds to the antires-

onant part, the induced polarization takes the form

= [dr'x(r,r’)-E(r’,w)

38

 

with x(r, r’; w) being the susceptibility tensor

"(1) = (glf’(r)|€)(€ll3(r’)lg>
X(riri) ; E£_fiw_i7 .

 

In the case of bulk materials, the center of mass-motion of the exciton is approx-

imately represented by a plane wave [4]

(4113mm = 7",; epr‘k - r) . (3.2.5)

where g represents the exciton state with the wave vector k, V is the volume of the

medium, and u is the intensity of induced polarization obtained from the bulk limit,

2 _ CbgALT
_ 47r

 

, and A LT is the splitting energy of longitudinal and transverse mode

of exciton, Ebg is the background dielectric constant.

From (3.2.5), it follows that

 

/_H_ e _ _ r
x(r,r)— VgEk—hw—h—X“ r . (3.2.6)

When the valence and conduction bands are both parabolic in the k—region, the

energy of the exciton may be further given by

where E0 = hwo, and M is the mass of the exciton.

39

 

Next, define the Fourier component in the k—space of the susceptibility

X(r-r')=WZeik(r “’> X(k),

then we have

X(k)— - — ”2

\/1_E0+E2271§W--fiw-i7

 

Since the polarization induced by the exciton is P(r) = / dr'x(r, r’,w) - E(r,,w),

we can rewrite this relation in the k—space as

at

_ zk'-r’ I 1 ik(r
M21; )Wge E()k .72“

k
= Zetertk) - Ewan.
k

1
where we used the fact I7 / dr exp(—z'k’ - r) exp(z’k - r) = 6k kl. Hence, X(k,w) may

be defined as follows:

2

X(k,w) = Wx(k;w)= ,2 2;; .
E0 + W — ha) - ’i’)’

 

The new dielectric constant takes the form

 

A
€(k,w) = 5b + 47r)2(k,w) = Eb + :bz LT , (3.2.7)
%+%fi-W—

which depends explicitly on the wavevector k and the frequency w. It should be

40

 

 

 

 

(a) (b)

Figure 3.3: Uncoupled oscillators (electrons attached to nuclei via springs ) in their
equilibrium position (a); elongated with a incident light (b).

noted that this dielectric constant applies to a case in which the exciton polarization
and the electric field are parallel, and both perpendicular to k [28]. In general, the
wavevector dependence of the dielectric constant is referred to as spatial dispersion
for which Hopfield and Thmoas did a systematical study in [28]. Indeed the nonlocal
effect of the optical response is implicitly contained in the dielectric constant (3.2.7),
and there is an intuitive physical interpretation. Take the Lorentz model as an ex-
ample, we have assumed zero coupling between neighboring electrons (oscillators) as
shown in Fig. (3.3). The spatial nonlocal effects from the neighboring oscillators are
not considered in Lorentz model. However, a more realistic coupling between the
neighboring oscillators, or nonlocal response effect, is contained in the wavevector-
dependence dielectric constant (3.2.7). As illustrated in Fig. (3.4), the neighboring
oscillators are coupled via weak springs. The most important consequence is that the
eigenfrequency is now function of k, namely w6(k) = wo + 211:7; For incident light,
the coupling springs are elongated and increase the “effective” spring constant. As a

41

 

Figure 3.4: An ensemble of coupled oscillators.

consequence, the eigenfrequency increases with increasing It. For our model system,
this is also true for excitons whose effects will be discussed later on. The fact that the
eigenfrequency w6 of some excitation of a solid depends on k is often called “spatial
dispersion”. The development of the spatial dispersion in optical spectra of excitons

has in great part been due to Pekar [49], Hopfield and Thomas [28].

3.2.1 Exctions and biexctions

In this part the dielectric constant (3.2.7) will be employed to model the excitons
and biexcitons effects in nanocrystals. Some definitions and concepts will be briefly
introduced at first. [23, 33] are very good references for a thorough study on excitons
and biexcitons. A paper [36] by Liang also provides some fundamentals on excitons.

An exciton is a bound state of an electron-hole pair in semiconductors. To visual-
ize the formation of excitons consider the simplest two-band model [23], the electrons

are accumulated at lower band (valence band) and the higher band (conduction band)

42

is empty. Electrons can be moved from valence to conduction band by an electromag-
netic field having the energy comparable to the band gap, and meanwhile positively
charged holes are left in the valence band. The electron and the hole form a pair
by attracting each other via Coulomb force, this electron—hole pair is called exciton.
Furthermore, the interactions between excitons should be taken into account if large
density of excitons is created in a material. Provided that the interaction is attrac-
tive, two excitons could bind to form a biexciton. A great deal of research has been
devoted to the study of the exciton effects since the concept of exciton was first in-
troduced by Frenkel in 1931 [21]. The fundamental importance of the excitons in
semiconductors is clear. They are found to be playing an essential role in optical
effects such as luminescence, fluorescence, photographic process, photoconductivity,
and so on [36]. These various features enable semiconductors to be widely used in
numerous fields, including optical communication, computing, biomedical imaging,
and data storage. The electric and optical properties of excitons in bulk semicon-
ductors have been extensively studied (see Chap. 13 of [33] and references therein).
For the excitons in structures of reduced dimensionality, for example, in nanoscale
structures such as quantum well, quantum wire and quantum dot, the states of the
exciton shift to higher energy as the size of structure decreases. This situation is
called quantum confinement effect, which makes nanoscale semiconductor cease to
resemble bulk, exhibits strikingly different properties instead [50, 66]. For example,

Hanamura showed in [24] theoretically that the nonlinear optical polarisability can

43

be enhanced greatly in semiconductor microcrystallites where the exciton becomes
quantized due to the confinement. His calculation suggested that in case of CuCl
microcrystal of size about 6.4 nanometer, an enhancement of the order of 104 for
X(3) can be expected; Recently the study ultra—efficient solar cells has been attracted
a great deal of research efforts, Quantum dots are offering the possibilities for improv-
ing the efficiency of solar cells in at least two respects, by extending the band gap of
solar cells for harvesting more of the light in the solar spectrum,and by generating
more charges from a single photon [17]. Therefore, as the size of a semiconductor
is getting smaller, excitons and exciton-biexciton coupling inside are no more minor
perturbation as in the comparable bulk system, but actually play a very important

role in defining the optical properties.

For simple parabolic bands and direct-gap semiconductors one can separate the
relative motion of electron and hole and the motion of the center of mass. Usually,
the motion of an exciton is quite different in two limiting situations characterized by
the ratio of the system’s size L to the effective Bohr radius a B of the exciton in bulk
material [4]. Therefore, we have two cases, a) L >> a B (weak confinement regime).
In this limit the size quantization of the exciton is brought about, and the e-h relative
motion stays almost as in the bulk material and only the center-of—mass motion is
affected by the confinement; b) L << a B (strong confinement regime). This is opposite
to case a), the size quantization effect of the electron and hole is much larger than the

exciton effect, the energy of an exciton is mainly determined by the individual size

44

quantization with a small correction due to the Coulomb interaction. The dissertation
concerns the first case in which the internal structure of the exciton remains nearly
the same as in the bulk, but its motion is quantized due to the confinement. Then
the energy of the exciton Eex in nanocrystals include two parts: E9 = fiwo which is

2k2
the energy for overcoming the bandgap; —— is the kinetic energy possessed by the

2M
exciton. Hence the nonlocal exciton effects on the optical properties is contained in
the k—dependent dielectric constant (3.2.7). In comparison, the classical dielectric
theory of optical properties studies the dielectric constant depending only on the
frequency. The dielectric behavior is a sum over resonances, each resonance occurring
at a particular frequency. Moreover, the transport of energy is neglected by any
mechanism other than electromagnetic waves. However, we are investigating the

optical response of confined excitons in a nano medium, so the energy transports by

excitons and the electromagnetic field are equally important.

Exciton is an elementary excitation, or quasiparticle of a semiconductor. In cur-
rent research, the bound electron and hole pairs (excitons) provide a means to trans-
port energy without transporting net charge [36]. Since an exciton is a bound state
of an electron and a hole, the overall charge for this quasiparticle is zero. Hence it
carries no electric current. Under certain conditions two excitons could bind to form
a new quasiparticle, the so called biexciton or exciton molecule [35]. The exciton to
biexciton transition in nanostructures has been observed and studied by many groups,

and some resonance phenomena, e.g., electromagnetic induced transparency, were re-

45

ported in [7, 9, 11]. The nonlinear optical response of biexciton in semiconductors
has been studied recently by several groups in relation to its potential applications in
optical bistability. Three different approaches have been proposed to solve this prob-
lem: Green functions [39], Bloch equations [26] and unitary transformation [1—3]. In
[39] the author derived a non-perturbative expression of nonlinear dielectric constant

of the exciton-biexciton transition

 

 

5(a)) = Eb + 47ru§xGx(w) , (3.2.8)
1 . . . .
where Gx(w) = _ IS the renormalized Green function With the dy-
QJex - w + 7/7 + 2
#2blEl2
namical self-energy E = X , which represents the contribution to the ex-
“(bi — 2w + 23

citon’s energy due to the interactions between the excitons and the system it is part
of. Here 5b is the background dielectric constant of the medium and assumed to be
homogeneous, #gx and be are the transition probabilities from ground state to the
exciton state (frequency wex) and from the exciton to the biexciton state (frequency

wbi), respectively. 7 and B are damping constants.

Based on understanding the k-dependence and inspired by the expression (3.2.8),
a modified dielectric function of exciton-biexciton coupling can be obtained by adding
the wavevector

47T/t2
E(k, w) = 5b + gx

 

, (3.2.9)
ex w + m + 2’7 + k
“341) E

.2
wbi—2w+fij[‘7r

2

 

 

where a k—dependent self-energy 2k = is obtained. The effec-

 

+25

46

tive mass of the biexciton is assumed to be just twice that of the exciton. The energy
h2k2
of the biexciton is given by Ebi = hwbi + 7, and the wavevector of the biexciton

is assumed to be 2k which is twice that of the exciton.

The formation of a biexciton is essentially due to the two-photon absorption res—
onance. Physically the biexciton can be produced under several different situations.
For example, two excitons interact with each other to form a biexciton in a high dense
excitonic system in which the dielectric function takes the form (3.2.9); Consider a
system which is probed (by probe light to) being disturbed by a pump light (wp). If
the probe and pump light frequencies are both chosen in resonance with some of the

transition energies of the system, then the optical property is modified to take a form
HiblEplz

which is similar to (3.2.9) with a different self-energy 2k = 2 .
cubi—-uq)——uJ+-€£f-+-fl3

 

Here [Epl2 is the intensity of the applied pump light.

3.3 Finite-size effects

The nonlocal effects of optical responses of two-band nanocrystals is reflected in the
wavevector-dependent dielectric constants. More realistically, finite-size effects have
to be taken into account when studying the optical properties of nanocrystals. Finite-
size effects, or sometimes called quantum confinement effects, have been observed for
small colloidal CdS nanocrystal [51, 53] and semiconducting nanoparticles embedded
in a glass matrix [16, 64]. In the early 19803 experiments were done with colloidal

solutions of quantum dots with applications towards solar energy conversion and pho-

47

tocatalysis. It was found that colloidal solutions of the same semiconductor showed
strikingly different colors when the size of the quantum dots was varied. This ob-
servation (called coloration) is attributed to the finite-size effect. Once the size of
a nanocrystal approaches the limit of the Bohr radius of an exciton, the states of
the exciton shift to higher energy as the confinement energy increases [46]. The con-
finement energy arises from the fact that according to the Heisenberg uncertainty
principle the momentum of a particle increases if its position becomes well defined.
In [24] Hannamura has made a theoretical study of the finite-size effects on the os-
cillator strength and third-order Optical polarisability X(B) of nanocrystals. From an
engineering point of view, material with a large optical nonlinearity is required for op—
tical shutters or optical information processors. He showed that the extremely small
scales of nanocrystals actually result in two major eflects on their optical properties.
One is the size quantization of excitons. The other is deviation of the electronic ex-
citation from an ideal harmonic oscillator. In weak confinement regime the former

effect is significant while the latter effect is less important. Hence the discussion will

be centered on the size quantization of excitons.

In the limit of weak confinement the size quantization of the exciton is brought
about. As in [24] consider first a cubic box containing N 3 unit cells. For the parabolic
conduction and valence bands with effective masses me and mh, respectively, the size

quantization is governed by the relationship among the medium size L = rm, and the

2

effective Bohr radius of electron ae = 5251) /mee and hole ah = h2sb/mhez, where

48

u is the length of the unit cell and 5b is the background dielectric constant. The size

quantization energies can be expressed for the electron and hole as

The Coulomb interaction between the electron and the hole is

C2 82
Vexc = — = —-—-——— ,
Eba eb(ae + ah)

where a is the Bohr radius of the exciton. The weak confinement condition L >>
(ae + ah) = a. is equivalent to Vexc >> ABC + AEU. In this case, the energy of the

exciton in the box is

h27r2n2

EFEg‘ngcJFW’

(3.3.1)

where ngc = ue4/2h25§ is the exciton binding energy, it = memh/(me + mh), M =
me + mh, and n = (712;, nymz). For the case of a quantum sphere with an infinite
wall at R 2 R0, the radius of the sphere, the energy of the optically allowed exciton

is
527r2n2

b
E7100 = Eg _ Eexc + W.
O

The size-quantization effect on the absorption spectrum was observed in [51, 53] and
theoretically studied in [16, 64]. In [24] the effect of very large oscillator strength

was brought about for weak confinement case. Since the sum of the whole oscillator

49

strength is constant as long as the concentration of the medium is kept constant.
Hence the oscillator strength is concentrated in the lower excitation states. Without

giving details the expression for the oscillator strength per quantum box is given in

[24] by

 

And the result for a quantum sphere with R0 = uN is

38N3

2m
anO = YWanchslswllzu 7m

Motivated by Hannamura’s work on the finite-size effects several refinements to the
wavevector-dependent dielectric constants need to be made. Most importantly, the

exciton’s energy (for the lowest excitation state) has to be modified E1 = hwo -

h2k2 r1222

b
E —— —.
+ 2M +2ML2

The eigenfrequency of the exciton in the lowest excite state is

iik2 .
w = w — Eb h + —. Hence the dielectric constant due to the excitons effect 1S
1 0 2M

changed to

+ 47m2
hwl — fin) — Z") '

 

e(k,w) = 5b

It is valid to use the bulk limit for u in the weak confinement regime. However,
the finite-size effects on the transition dipole moment have to be taken into account
in order to obtain a more accurate model. For the onedimensional confinement
(quantum slab) the squared transition dipole moment [H[2 is increased by the factor

E18N/E07r2, and it should be multiplied by the factor E18N3/E07r for a quantum

50

dot. In order to fully understand the structure of the electronic states in a semicon-
ductor nanocrystal other effects, like the band—filling effect and the screening effects
as well as exchange interaction between electrons and holes also need to be taken into

account [44].

51

Chapter 4

MATHEMATICAL ANALYSIS

As we have discussed in previous chapters, some physicists have begun to address
the challenging problems in modeling the optical interaction with nanostructures, the
underlying mathematical analysis and numerical computation remain open. Initial
attempts have been made recently in [5, 6, 59] during my doctoral study to deal with
some mathematical issues arising from these problems which is also the central topic

of this chapter.

4.1 Concept of polaritons

To better understand what is actually propagating when “light” travels through mat-
ter the concept of polariton has to be introduced. Simply speaking, polaritons are
quasiparticles resulting from strong coupling of electromagnetic waves with an electric
or magnetic dipole—carrying excitation. More details are going to be discussed [33].

In vacuum the light is a transverse electromagnetic wave, the quanta of which

52

are known as photons. To describe the interaction of light with matter one way is
to treat the electromagnetic field and the excitations of the matter as independent
quantities, this is called perturbative treatment or weak coupling. In this case a
photon is absorbed and the matter goes from the ground state to the excited state,
and that is it. It is sufficient to use this approach for many purposes, but, if we look
closer, we find that this is not the whole story. The optically excited state of the
matter is necessarily connected with some polarization P. Otherwise the transition
would be optically forbidden, i.e., it would not couple to the electromagnetic field
e.g. via the dipole operator. On the other hand, every oscillating polarization emits
an electromagnetic wave which may act back to onto the incident electromagnetic
field. This interplay leads us to the strong coupling between light and matter and to
the concept of polaritons which was first introduced for crystalline solids [28]. For
our interest the discussion is centered on exciton-polariton, resulting from coupling

of visible light with an exciton.

As mentioned earlier the term “dispersion relation” or simply “dispersion” means
the relation w(k) for all wave-like excitations independent of the functional depen-
dence. It can be simply a horizontal line, a linear or parabolic relation, or something
more complicated. Every excitation which has a wave-like character has a dispersion
relation. For the exciton-polariton (from now on we use polariton without causing
confusion) to determine the its dispersion relation, we must combine the polariton

(:32
equation 2 = 5 with the dielectric function E(k, w). For the new spatially disper-
w

 

53

sive dielectric constant (3.2.7) due to the exciton effects, the polariton’s dispersion

relation is

 

 

c2k2 47rp2
2,2 = 6b + 221,2 . '
MD + W — fiw — 2h”)!

4 2M . 2 2 22M .

k + [—h—(w0 -— w — 27) —— ebgq k — ebgq ——h—(w0 —w — 27 + ALT/h) = 0 , (4.1.1)
where ebA LT = 47ru2. It can be easily seen that Eq. (4.1.1) leads to four complex
solutions and two of them k1, kg with positive imaginary parts corresponding to phys-
ically meaningful hence acceptable modes. It means that in a medium characterized
by (3.2.7) there are two propagating waves (polariton modes), which is a striking

property of such medium.

Similarly, for an exciton-biexciton coupling medium described by (3.2.9) the dis-
persion relation

26 + 21(2))24 + b(w)k:2 + C(21) = 0 (4.1.2)

 

with

M , , w 2
(1(0)) 2 X(wbi — Cdpu + 2wa — 3w — 2,6 — 227) — (z) Eb,

M ' , w 2 2M , 87W! 2 w 2
(9(0)) - 7.;(Wbi — “Jpn — w - 15) [(8) 5b + ‘h—(Wex — w — 27)] — 7”gx(z)

2M w 2 . 2 22M2
‘1‘“(3) 5b(wex — w ’ W) + #xblEO[ F,
2M2 #2 [EU]2 . . w 2

54

medium 1 medium 2 medium 1
(vacuum) (excitons) (vacuum)

hi 9 k1 kt

kT

k7" / kt

A
l

 

 

ki

Figure 4.1: Two waves in a semiconducting nano slab which confines excitons.

It can be verified that the above equation has six complex roots, and three of them
k1, kg and k3 with positive imaginary parts corresponding to physically meaningful
hence acceptable modes.

It can be seen when the exciton / biexciton medium characterized by (3.2.7)/(3.2.9)
is irradiated with light of frequency w, more than one waves can arise inside the
medium with the same frequency but with different wavenumbers. Therefore, the
need of additional boundary conditions (abc) arise naturally to handle the multi-

mode waves.

4.2 Additional boundary conditions

To make the situation clear, we show in Fig. 4.1 the wavevectors for such cases for
normal and oblique incidence. The incident and reflected beams obeys the usual law

of reflection, their composition parallel to the surface are equal. The same is true

55

for the transmitted beams. As mentioned the abc is required to handle the multiple
waves inside the medium, although the abc can not be deduced from Maxwell’s equa-
tions. Their capacity is exhausted with the continuity of the tangential components
of fields on the interface. Since the complex index of refraction around the resonance
is different for the k1 and k2 branch, which therefore contribute differently to the
reflection and transmission spectra, the abc should contain information about the
“branching ratio”, i.e., which fractions of the incident beam couple in the medium to

the k1 branch and to the k2 branch as a function of frequency.

The abc are based mainly on arguments of physical plausibility. On the vacuum
side of the interface in Fig. 4.1 the polarization induced by excitons is zero since the
excitons are not allowed to escape from inside the medium. To avoid an unphysical
discontinuity in the polarization, we propose that the tangential component of the

polarization should vanish at the interface

n x P = 0. (4.2.1)

Another argument says that the polarization should vary smoothly across the inter-
face, implying that the derivation with respect to the normal direction has to be zero,

resulting in

E?— = 0 . (4.2.2)
dn

56

Sometimes a linear combination of (4.2.1) and (4.2.2) is another possible abc.
nxP+a:—II:-=O with —1§a§1. (4.2.3)

The abc was first studied by Pekar in [49] where the condition (4.2.1) was introduced.
It turns out that experimentally observed spectra , e.g., of the exciton resonance ,

can be fitted with all above mentioned abc.

For a semiconductor with exciton-biexciton coupling confined inside, three branches
of polariton waves exist. In order to deal with these three waves two abc are needed,
which makes the analysis and computation very complicated and expansive. For-
tunately, for finite but very small damping constants an asymptotic analysis shows
that among the three modes only two dominate, and the third one can be safely
eliminated without affecting the essential physics. Thus the complexity of the model
and its computational cost, especially for the high dimensional case, can be greatly

reduced.

For the analysis, a resonance case of w = wax in Eq. (4.1.2) is of interest and
will be considered. We start with 7 and [3 being zero, a(wex) < 0, b(wex) < 0
and C(wex) > O can be seen for the medium, in addition C(wex) is found to be an
extremely small number. It can be shown via a perturbation analysis that y3 +
a(wex)y2 + b(wex)y + C(wex) = 0 has three real roots (let y = k2); k2, k? and kg.
Furthermore the relations k? - kg - k2 = —c(wex) < 0 and k? + kg +19% = —a(wex) > 0

guarantee that one root is negative (e.g., kg) and the other two are positive (kg, kg).

57

l complex plane

 

 

\ - —:_‘___‘————-'—:—:-:;:':— kBA
——————— \\
kg \\

 

Figure 4.2: A sketch of the wavenumbers in the vicinity of a resonance w = wex with
very small damping constants is shown in the plot.

Next consider very small 7 and 5, then the coefficients a, b and c are going to be
perturbed by very small complex numbers, so are the k%, kg and kg. Clearly, both k1
and k3 have very small imaginary parts, whereas the imaginary part of k2 is relatively
large, which are shown in Fig. 4.2. It should be noted that modes k1 and k3 will
make considerable contribution to the light scattering. However, the contribution
from mode k2 is negligible. To calculate the reflection and transmission spectra of
the slab, an additional boundary condition is still needed for these two waves, one
possible argument is that the polarization induced by the exciton-biexciton transition

must be zero at the interface.

To validate the approximation an Optically excited CuCl nano slab is considered,
the model Maxwell equations in the slab are

(12‘s.,-

d 2 +kz-2Ei ,7;=1,3 (4.2.4)
(I:

 

58

and the total field E = E1 + E3. Besides the additional boundary condition, the
tangential components of the field and its derivative need to be continuous across the

interface.

To illustrate the validation of our approximation, we present numerical computa-
tions of the reflection and transmission spectra of the slab. The thickness of the slab
is 340 nm, and all other parameters are from CuCl semiconductor: 5b = 5.59,wex =
3.20228V,wbi = 6.372ev, hwpu = 3.1698ev, M = 2.371104%X = 2.513mev, “3b =
2 x 10—3mev,7 = lmev,fi = 0.015mev, where m0 is the mass of an electron. If
all the three polariton modes are considered, another boundary condition will be
required. One physically plausible argument is that the polarization should vary
smoothly across the interface, i.e., d5; = 0. The numerical simulations are plotted in
Figures 4.3 (a) and (b), the peaks in the plots actually imply that exciton-biexciton
transmission happens in the vicinity of w = wex. In the resonance region, our ap-
proximation show good agreements with the three wave results. As we can see three
polariton modes with same frequency but different wavenumbers are generated due
to the exciton-biexciton transition in a thin semiconducting slab. We have shown
that one of the modes can only penetrate a very small depth in the medium, hence it
contributes little to the light scattering even if multiple reflections and transmissions
are considered. Our criterion is to eliminate this mode and keep the other two. The

approximate model is established via coupling Maxwell’s equations with additional

boundary conditions, which agrees well with the three waves model in the resonance

59

 

0.9 - .4

0.6 *

Transmission
O
'01

 

 

 

3.202 3.2025
Incident light (ev)

 

3.201 3.2015 3.203 3.2035

 

Reflection

 

 

 

 

1 _
3.203 3.2035

n - I
3.201 3.2015

3.202 3.2025
Incident light (ev)

Figure 4.3: The transmission (a) and reflection (b) spectra of a CuCl slab of 340 nm
thick. The solid curves are obtained from the two modes approximation, the dashed
curves are resulted from the three modes computations.

60

region of interest.
In the following section we will establish the model partial differential equations
and present several mathematical results concerning the existence and uniqueness of

the solutions.

4.3 Model PDEs and theoretical results

Based on the criterion we proposed for exciton-biexciton coupling model, we are
dealing with two waves for both exciton and biexciton cases. Mathematical models
for the exciton and the coupling of exciton-biexciton are identical in terms of partial
differential equations and additional boundary conditions. Eq. (3.1.10) for the electric

field inside the medium is changed to

v x v x E — (3)25(k,w)E = 0, (4.3.1)

where E = Ea + Eb, and Ea, Eb satisfy following two equations

vxVan—k§E=0,VxVbe—k§E=o. (4.3.2)

On the boundary I‘, we impose additional boundary conditions as well as Maxwell’s
boundary conditions. Next we will specify the mathematical models for three special
nanoscale structures, namely, slab, wire, and sphere. The well-posedness of the model

partial differential equations is also presented.

61

Eoeikoz

 

TEoeikoz

<—

E'T REoe—ikoz

 

0M1;

 

Figure 4.4: A nano slab. Left: the full view; Right: the one dimensional confinement
View.

1. Nano slab. Consider the light scattering by a nano slab depicted in Fig.
4.4 for an incident plane wave. The exciton (biexciton) is confined in z axis due
to the nanometer scale of the slab in that direction. As illustrated in Fig. 4.4, the
incident field is Eoeikox, the reflected field is RE0e_ik0$, and the transmitted field
is TEoeikO‘T. To calculate the reflection and transmission coefficients R, T, we set up
the equations inside the slab along with the boundary conditions. Here, E = Ea + Eb

is the total field inside the slab from two waves ka, kb.

Define the domain 9 = (0, L), we have derived the following boundary value

62

problem in [59]

2 .
#Ea+kgea=0 mo,
d2 E k2E — 0 ' a
£2 b + b b — 1n ,
193(0) + 133(0) = —ik0(Ea(0) + Eb(0)) + 2ik0E0 ,
< (4.3.3)

EQ(L) + EISUJ) = ik0(Ea(L) + Eb(L)) ,

XEa(0) + Eb(0) = 0 a

 

XEa(L) + Eb(L) = 0 a

 

 

where x— — xii? By introducing u(:r)- — an(x ) + Eb(:z:), v(:1:) = Ea(:c) + Eb(:c),
the system (4.3. ) may be rewritten as

 
 

r 52” + My, +———1—b—X(k§:k2)v = 0 in Q,

5212+ ng _k2v+fb2——l;3u=0in§l,

< v'(0) = —ik0u(0) + 24140190 , (4.3.4)
u'(L) = ik0v(L) ,

u(0) = u(L) = 0 .

 

Our well-posedness result of the model problem is stated as follows

Theorem 4.3.1. For all but a possibly discrete set of the frequencies w, the model
problem (4.3.4) attains a unique solution (u,v) E H6(Q) x H1(Q).

63

Proof. Multiplying the first and second equations in (4.3.4) by the test functions
6 6 H662) and 77 E H1(Q), we get after some simple integration by parts and making

use of boundary conditions
/VU~VV+/ AU-V+z'k0/ BU-V=22‘kOBV, (4.3.5)
Q Q 30

where U = [u,u], V = [£377], VU = [u’,v’], the overline denotes the complex conju-

 

gation, and
2143—43 x<k2—k§) 0 0
Jr —1
A = A(w) = — X X B = . (4.3.6)
32—193 ku—kQ

J’xTr—xa-Tll 0'1

Defining W(Sl) = H662) >< H1(Q), we introduce the bilinear form a : W(Q) x W(Q) —>
C,

a(U, V) = (VU, W) + ik0(BU, V)aQ + (AU, V) , (4.3.7)

and the linear functional on W(Q),
b(V) = 2ikOBV .
Next, we decompose a into a1 + a2, where

a1(U, V) = (VU,VV) + ik0(BU, V>3fl ,a2(U, V) = (AU, V) .

64

Using the fact that k0 is positive and Poicaré’s inequality [19], we see that

e W + W + k0|v(0)|2
S2 52

cliuil§,1(,,,+ c<L.ko)HvH§,1(m

[01(U, U)I

IV

IV

IV

manna/(Q, ,

where “Imam = Hunglm, + ”in21“,).

Then the coercivity of a1 is obtained, thus the Lax-Milgram lemma implies that
the operator A1 defined by (A1 U, V) = a1(U, V) has a bounded inverse. Note that
the operator A2 defined by (AgU, V) = a2( U, V) is compact. To emphasize that both
A1 and A2 depend on the frequency w, we write A1(w), .Ag(w).

Holding “’1 fixed, consider the operator A(w1,w) = A1(w1) + A2(w). Since .41 is

—1

bounded invertible and A2 is compact, we see that A(w1,w) exists by Fredholm

theory for all w ¢ E(wl), where E(wl) is some discrete set. It is clear that
IIA1(w) - A1(w1)I| -+ 0 38w -* wi-

Thus since ||.A(w,w) — A(w1,w)[[ = [[A1(w) — A1(w1)|[ is small for [w — w1| sufli-
ciently small, it follows from the stability of bounded invertibility (see, e,g., [31]) that
A(w,w)‘1 exists and is bounded for [w — col] sufficiently small, to a: 8 ((421). Since
wl > 0 can be an arbitrary real number, we have shown that A(w, w)_1 exists for all

but a discrete set of points. El

 

{I}

Figure 4.5: A nano wire structure confines the exciton and biexciton in :r, y directions
but allows them to move along z-axis.

2. Nano wire. An infinite cylindrical nano wire with cross—section Q is depicted
in Fig. 4.5. For normally incident field (0,0, Ei (1:,y)), there exist two fields Ea, Eb
inside the wire with wavenumbers ka, kb, the governing equations are the Helomholtz
equations,

(v5? + k2)Ea = 0 ,(vi + k§)Eb = 0,

where V2 = 6% + 62, and the total field inside is E = Ea + Eb- The total field
outside the of the wire consists of the incident field E1. and the scattered field Es.
The scattered field is outgoing and this is imposed by requiring the scattered field to
satisfy the following Sommerfeld radiation condition:

lim (53%? + ikOEs) = 0,

12—00

where p = \/ at? + y2. In practice, it is convenient to reduce the problem to a bounded

66

domain by introducing an artificial boundary. Assume that R > 0 is a constant such
that Q is contained in the disk D = {:r E R2 : |z2+y2] < R2}. Let S be the boundary
of the disk, denote by u the outward unit normal to S. A suitable boundary conditions
then has to be imposed on S. For instance, we use the first order absorbing boundary
condition [18] as

(TEES + ikOES = 0 on S. (4.3.8)

Then we can derive the following system

V%.u+a1u+b1u=0 inn,

V%~v+a2u+b2u=0 inQ,
( V§m+kgv=0 in D/n, (4.3.9)
u=0 on 89,

%u + ikov = 5%Ez + ikOEI on S,

 

k2 _ k2 k2 _ k2
where em) = 2% + Eb. v<ey> = Ea + Eb, a1 = 5%.?“ bl = 3417193,

kZ-kZ k2-k2
wflmdg: b a_
x—l

 

(12:

Notice that the field u(:r, y) is compactly supported by domain 9, we extend u by
zero outside of fl. To state our variational problem, we first introduce the Sobolev
space:

Htl(D) = {w E H1(D) : w E 0 for (:r,y) E closure(D)/Q} ,

67

Then we obtain the following weak form

v av V—/ W av V+k2f BUoV+ik [311.17,
[0 T T we T T 0 12/9 0 s

_ . a .
. = - i _ i
“I'LAU V A(ikoE + 811E )7],

where the same notations as in the slab case are used, and A, B are defined in (4.3.6).

Define

a(U, V) = (VTU, VTV)Q — (BVTU, VTV)D/Q + 33(BU, V) D )9 .

+ik0(BU, V)S + (AU, V)Q .
We write a(U, V) = a1(U, V) + a2(U, V) with

a1(U, V) = (VTUwVTl/lfl-(BVTUwVTle/Q‘I'ikMBUel/lsv

a2(U,V) = kg(BU,V)D/Q+(AU,V)Q.

By writing (1:, y) into the polar coordinates system (r, 6) with :1: = rcos 6, y = r sin 0,

we get

R
323,9). = 332W)- / (pegoenpdp

Ir
2 R 2 R
= Rv (R.9)-/ '0 (p.9)dp-2/ vvp(p.9)dp
T 7'

R

?

3 12112019) — 2/

7‘

WV, . Md,
p

68

R 27r 2 2
f / [u (r,6)]rdrd0 S c1] Iv] +c2/ |v[|Vu|,
0 0 S D

2 2 2
v c Vi) +k/u).
[II (fl I OSII

The coercivity of a1 can be obtained by

I/\

a U,U 2 c / Vu2+/ Vu2+k/u2)
|1()| (9| I Dll osll
2 cllullH1(D) + C(R.w)liv|l§{1(0)
2
> eIIUHWw,
where W(D) = Ht1(D)xH1(D),||U|]%/V(D) =||u||§11<D>+I|vH21<Dyandrydepends

on the size of D and w.

Then by applying the similar argument as in the slab case, we can prove that
the operator A defined by (AU, V) = a(U, V) has a bounded inverse for all but a
discrete set of points. Hence the model system (4.3.9) obtains a unique solution
(u, u) E Ht1(D) x H1(D) for all but a countable set of the frequencies to.

A parallel waveguide case has recently been considered in [59], where the exciton
is confined in a nano square imbedded between the plates of the waveguide. The
well-posedness of that model follows immediately from a similar proof as the above.

3. Nano sphere. Suppose a semiconducting nano sphere with radius R is

exposed to a plane wave (Ei, Hi), the Maxwell equations in three dimensions are

v x E =. z‘gH ,V x H = —z'e“—(:E, (4.3.10)

69

 

A Z
\\\ T.
: S, ——» i. o
\\ I!
\ I
\\ 60 \ [I I .

 

SJ
{Sir

Figure 4.6: Left: A nano sphere can give the confinement in all three directions
Right: Spherical polar coordinate system centered at the sphere.

where

50 outside
5' :
e(k,w) inside
The divergence of (4.3.10) gives V - E = V - H = 0 because the medium is homoge-

neous. Then the governing equations become

v2E+k§E=0,v2H+k3H=0,
v2E+k§E=o,v2H+k§-H=0.

r,0, ((5 in Fig. 4.6 defined by

It is more appropriate in this case to present problem in spherical polar coordinates,

a: =rsin0cos¢ ,y =rsin08inqb ,z = rcosB.

70

Consider an x-polarized incident plane wave, written in spherical coordinates E2 =

Eoeikax, where
x = sin 0 cos (be,- + cos 0 cos (beg — sin ¢e¢ .

There are three kinds of fields in our system: the incident field (Ei, Hi), the scattered
field (E3,Hs), and the field (Ew,Hw) within the sphere. Following Lorenz-Mie

scattering theory, we expand all those fields in terms of vector spherical harmonics,
(1) .— °° <1) (1)
:2: E "(M01121 ZNeln) Hi 2 _ 60 Z E"(Meln + iNoln) ’
n=1

OO 00
2 . 2 2 2
E3 = Z En<a3M£l — 21232133.) H = we Z M b32413. + wiNél.)

n=1 n=1

and

:2 En(a agiMolln_ ibj "Nell)n)’ Hw = -\/— Z En(bjnM(()11n _ zagleallii) ’
n: 1

where j = 1,2 and 51 = E(ka,w), 52 = E(kb,w), En = in(2n +1)/n(n +1), and we
append the superscript (1) and (2) to vector spherical harmonics for which the radial
dependence of the generating functions are specified by Bessel function of first kind
jn(r) and Hankel function of first kind hn(r) = jn(r) + iyn(r), where

_.m .
Memn = m Sln m¢P,T(cos 6)zn(p)e9 — cos m¢(P,T,n(cos 6))9zn()o)e¢> ,

71

m .
Momn = 555 cos m¢P7T(cos 9)zn(p)e9 — sm m¢(P,7,n(cos 6))92n(p)e¢ ,

1
Nemn = Z_n’(_p_) cos mqbn(n + 1)P,T,n(cos 9)e1~ + cos m<b(P,7[n(cos 9))9;(pzn(p))peg

Pf,"(cos 0) 1

—m sin "NA—m— ; (Pzn (Pllpe¢ ,

Nomn = 2722p) sin mgbn(n + 1)P.,T,n(cos 6)er + sin m¢(P,7,n(cos 0))9%(pzn(p))pe9

 

m
+m cos m¢P"—8i(:(;—s@ 115 (mu (p))pe¢

with zn is any of the four spherical Bessel functions jn, yn, kg) = jn(r) + iyn(r)

and hi?) = jn(r) — iyn(r). The derivation of Memn, Momn, Nemn, and Nam”, is a

standard tOpic in Mie theory and will not be repeated here, see for instance [8, 62].

The unknown coefficients a?“ bfz, a?) bit are determined by the boundary condi-

tions at r = R,

Eg+Eg=Efg+Egg,
E23+E3 =E$¢+E§U¢,
H§+H§=H¥9+H§Um
H;+H3¢=H[U¢+H§U¢,
X(ka,w)Ei"g + X(k37w)E§"g = 0,

X(kaalequ) + X(kbalegip = 0-

72

With these boundary conditions, we obtain:

an:

32,:

jnRoarxxeaRjnman’ — (koRjnRoi’(./ezjn<ea)- -x¢e—gjn(Rb)9§§‘b§,§’;%;]] )

 

hnRomxkaRjnmaw - (koRhnRowe—ajnma) — warnaflfgfififik') ’

jnR0(\/Ea(kaRjn(Ra))’ — x,/e—5(kijn(Rb))’§"(R“ ) — (kORjnRolljn(Ra)(Xr)

 

hnRo(./e—a(kaRjn(Ra)) —xr<kijn<Rb))”—'l[&7)— (koRhnRo)'jn(Ra)(x.~)

where the derivatives of these functions are denoted by primes, Xr = 1— x, R0 = kOR,

Ra = kaR and Rb = kbR. The scattering cross section is given by the standard Mie

formula

Qsca‘ — R027, 21(2n + 1)(abs(an)2 + abs(bs )).

73

Chapter 5

NUMERICAL EXPERIMENTS

In this part the dielectric constant model will be applied to CuCl nanocrystals of
different dimensions. There are several reasons why CuCl is a rather well studied
model materials [30]. CuCl is a typical material for exciton /biexciton confinements
[60], and its excitons have a relative small Bohr radius (about 0.7nm), and a large
binding energy (about 2ev). Very recently, CuCl has attracted research interests
once more, due to the ability of the excitons in CuCl to form biexcitons, which
are potential candidates to undergo a Bose-Einstein condensation. In fact, evidence
has been obtained in [25, 42] that excitonic excitations in semiconductors like CuCl
exhibit features expected in a weakly Bose gas. In the following we will show the
optical response results for excitons and biexcitons confined in one, two and three
dimensional CuCl nanocrystals. The parameters used in the numerical simulations are
Eb = 5.59, 60 = 1.0, liwo = 3.2022ev, M = 2.3m0,u5x = 2.513mev,7 = 1mev, limp =

3.168ev,fiwb = 6.372ev, Mb = 2M,fi = 5 x 10—7ev2,7b = 0.015mev , where m0 is

74

the mass of an electron.

5.1 CuCl nano slab

The Optical responses by exciton/exciton-biexciton coupling in a CuCl slab are cal-
culated. The reflection and transmission spectra are plotted. For the numerical
simulations a plane wave with wavelength /\ in the range of [380, 400] nm are used.
First the exciton effects are considered, hence the dielectric constant (3.2.7) is used.
Fig. 5.1 shows the spectra of the transmitted and reflected lights by CuCl slab of
different sizes. The resonant structures showed in the figures are due to the exciton
mode, interfering with the size resonance of the slab. The excitons in a semiconduc-
tor slab is a well studied case for various thicknesses including a semi-infinite limit
[12, 15, 28]. In theory, if all the multiple reflection effects are neglected, two kinds of
oscillatory effects should be observed. One is periodic modulation of the transmission
at fixed energy as a function of the thickness. The other is a periodic modulation
of the transmission at fixed thickness as a function of the energy. Both these two
effects are clearly verified in our numerical experiments. Compared with the normal
incidence reflection experiments in CdS [15], the experimental line shape, especially
the resonance peaks are well produced. Also from Figure 5.1, the peaks of spectral
transmission and reflection clearly indicate enhanced optical absorptions, and those
peaks move to the right as the CuCl slab thickness decreases, which is the well known

blue shift phenomenon.

75

d

 

9.0.0
\IGO

ission spectra
.0
O)

Tanm
.0 s: .o .0 .o
.1 N (A) k 01

 

 

 

 

 

 

 

 

0 l l l l l l l
3.18 3.185 3.19 3.195 3.2 3.205 3.21 3.215 3.22
Incident energy (ev)

 

 

—20 nm
0.9- --‘--10nm"

------ 5nm
0.8

 

 

 

0.7
0.6

spectra

5 0.5

0.4

Reflect

0.3
0.2
0.1

 

 

 

 

0 l l l I l I l
3.18 3.185 3.19 3.195 3.2 3.205 3.21 3.215 3.22
Incident energy (ev)

Figure 5.1: Size dependence of spectral transmission and reflection of the CuCl slab.
The horizontal axis is the incident energy (ev). Top: transmission spectrum. Bottom:
reflection spectrum.

76

Then for the exciton-biexciton coupling in the CuCl slab, numerical results are
obtained based on the two dominant modes approximation. If we view the two
boundary points of the slab (:1: = 0 ,L) as two dipole emitters, then a separation

Ar between the dipoles

defines the axial resolution (or the depth resolution) of the system. n is the refractive

index of the medium.

The axial resolution sets a limit to the ability of an optical system in the optical
sectioning and three dimensional imaging. In order to achieve a better image, the
axial resolution has to be improved. Many approaches have been proposed to enhance
the axial resolution in nano-optics, for example, one important technique is to use
the nonlinear excitation. we refer the reader to the book [46] for more details and

references.

When the thickness L of the CuCl slab is bigger than the axial resolution Ar z
340nm, we receive almost no contribution from the left dipole (a: = 0) in the transmis-
sion spectrum, or we can get a stop band within which the light is nearly completely
reflected in Figure 5.3. The second plot in Figure 5.2 indicates a nonlinear response
of the medium, the exciton-biexciton coupling opens a narrow window in the trans-
mission band, which confirms the enhancement of the axial resolution which benefits

from the nonlinear excitation.

77

L=340nm with no pumplight
1 I I l I

 

 

 

0-9 - —transmission spectrum -
0 a _ - - -reflection spectrum _
0.77 a ‘—l— — - - ~ ~ ~ ‘

v T ‘

0.6- ,-——'

0.4 -
0.3 -
0.2

0.1 -

 

 

 

 

n - l I ~
3.193 3.2 3.202 3.204 3.206 3.203 3.21
Incrdent energy (ev)

L = 340nm with pump light
1 I I I I I

 

 

0‘9_ —transmission spectrum

\I\l

 

08 ---reflection spectrum ‘_-—-"”"

 

 

0.7; _—"'"~

,’ i
0.6 - \ I
0.5 -
0.4 ~ ‘
I
I
0.3 - I
I
I
0.2 - ‘

0.1-

    

n

 

 

 

3.201 3.202 3.203 3.204 3.205 3.206 3.207 3.203
Incrdent energy (ev)

Figure 5.2: Tom: A stop band for the excitonic transmission without pumping light.
Bottom: Due to the exciton-biexciton coupling in a pumping light limp = 3.168ev, B =

2 x 6ev2, the nonlinear enhancement happens at fin} = 3.204ev which is the binding
energy to form a biexciton.

78

 

 

 

 

 

      

 

 

 

 

—340 nm l
0.9 r
- - -80 nm ‘ .......
0.8- ------- 20 nm ~
9
*5 0.7 - , A
m C: ’I
9)- 0.5 r I, a
C "- , . ------ " I
o
'5 0.5- ,’ 3
.9 N ~ " -‘ I
I
g 0.4 - ~ ‘ ‘ ’ g
c L ‘ s ,’
E 0.3 s I
l- ‘ '. .- I
\ H" ‘.‘ ’
0.2 I
I
I
0.1 , ’
’ I
0 ~ " - -
3.198 3.2 3.202 3.204 3.206 3.208 3.21

Incident energy (ev)

Figure 5.3: A stop band in transmission spectra is observed for L = 340nm. More
light is transmitted through the slab as the thicknesses decrease.

5.2 Quantum dots

From the simulation results in the three dimensional case, we plot the scattering cross
sections in Fig. 5.4. Clearly blue shift phenomena are observed when the radii of the
sphere decrease. T be blue shift is the shortening of a transmitted signal’s wavelength,
and an increase in its frequency. Furthermore, in the exciton case the peaks of the
cross sections are around fiw = 3.2022ev, in fact this is the exact amount of energy
required to form an exciton. For the biexciton case, a pump light is applied with
energy 3.168ev in addition to a probe light with the energy varying. To supply enough
energy to bind two excitons forming a biexction, we need 6.372ev —3.168€V = 3.204ev,
and this amount of energy is provided from the probe light. Therefore, the peaks in

the biexciton case happen at around fiw = 3.204ev.

79

 

1.2 I I f fir l I

 

- - -90nm
------- 60nm

 

 

 

 

Scattering cross section

 

 

 

0 l l J l l l
3.2 3.2005 3.201 3.2015 3.202 3.2025 3.203 3.2035

The engergy of the probe light (ev)

 

 

 

 

 

 

 

1.2 l l T l ‘1 I l l
- - -60nm
1_ ------ 40nm
c —30nm
.9
.0—0
8 0.8~
U)
(D
8
5 0.5-
O)
C
.5 I
I
g 0.4- ,
o I
(I) I
I
o.2~ ’I .
” “.*“‘ ‘\ ’I’ "'..'H':.V;‘a "
fl' ‘‘‘‘‘ ~- ’

 

 

 

0 l l l l l I l 1
3.2039 3.204 3.204 3.2041 3.2041 3.2042 3.2043 3.2043 3.2043
The engergy of the probe light (ev)

Figure 5.4: Top: Scattering cross sections of the exciton response. Bottom: Scatter-
ing cross sections of the exciton-biexciton coupling response. The cross sections are
normalized so that their maxima will be unity.

80

Based on the results presented above, we can conclude that, due to the excitation
of exciton polariton, the optical properties of a nano sphere (quantum dot) vary
strongly around the ground-state exciton resonance frequency 020. For example, if
we consider a single CuCl quantum dot with a radius of 60 nm, the corresponding
resonant wavelength is about 380 nm. An electromagnetic wave with a wavelength
of about 380 nm (in optical regime) will then be significantly manipulated by such a
QD whose thickness is as small as 60 nm. These facts have a bearing on the use of

exciton polaritons for beating the diffraction limit of light.

81

Chapter 6

CONCLUSIONS AND FUTURE

WORKS

This thesis has presented a modeling and computational study in optical responses
of nanostructured media. Particular interests have been placed on semiconducting
materials due to their extensive applications in optical sciences. Under the condition
of weak confinement a nonlocal dispersive dielectric constant (depends on k and w)
has been derived via coupling Maxwell’s equations and Schrodinger equation, a semi-
classical model hence is established by incorporating this new dielectric constant to
the Maxwell’s equation. The wellposedness of the model partial differential equations
is studied for the first time, and the exciton and biexction effects on the nanometer
scale have been extensively investigated in the thesis. It has been shown that the
blue shift phenomena and nonlinear electromagnetically induced transparency effect

can be well predicted from the semiclassical model. In addition to nonlocal effects

82

of the optical response, the finite size effects on the optical properties are also in-
vestigated in terms of eigenfrequencies and oscillator strengths of the excitations in
nanocrystals. Although semiconducting nanostructures have been the primary focus
of the thesis, the semiclassical framework of the present approach should be able to
generalized to other materials, for example, metallic nanostructures which have been
recently gaining extensive research efforts. While physicists and engineers have de-
signed and fabricated many novel metallic nanostructures being used in laboratories,
little is known about the rigorous modeling and efficient computation of the opti-
cal responses. A future work is to investigate the optical interactions with metallic
nanostructures. The study of optical phenomena related to electromagnetic response
of metals is termed plasmonics or nanoplasmonics which has been growing really
rapid over the past few years. It mostly deals with the control of optical radiation on
the nanometer scales. Many innovative concepts and applications have been devel-
oped in recent years and the following section will introduce a few examples. Most

importantly some open and challenging problems will be discussed.

6.1 Plasmons in metallic nanostructures

The interaction of metals with electromagnetic radiation is largely dictated by the free
conduction electrons in the metal. Collective oscillations of the conduction electrons,
termed plasmons, strongly influence the optical properties of metal nanostructures

and are of great interest for future photonic devices. Many (but not all) properties of

83

real metals, including their optical properties as described by the frequency dependent
dielectric function 5(a)), are surprisingly well predicted from the so called Drude-

Sommerfeld model [46]. The resulting equation is:

with tap the so—called plasma frequency and '70 the electron relaxation rate. 500 in-
cludes the contribution of the bound electrons to the polarizability and should have
the value of 1 if only the conduction band electrons contribute to the dielectric func-
tion. The plasma frequency is given by cup = ‘/Ne2/€0m* with N and m* being
the density and effective mass of the conduction electrons, respectively. The electron
relaxation time can be calculated from the DC conductivity 0 by 7' = 0771* /N 62.
Although the Drude-Sommerfeld theory give quite accurate predictions for bulk met-
als, some quantum effects such as surface scattering and quantum confinement effects
become more important in nanosized metallic particles. For small particles with high
surface to volume ratio surface scattering leads to additional collisions of the conduc-
tion electrons with a rate 7-1, proportional to the Fermi-velocity v F (about 1.4 nm/

fs in Au and Ag). The additional damping surface is given empirically by

'73 1 = ADP/7‘,

84

where r is the particle radius and an empirical parameter A describing the loss of co-
herence by the scattering event. A is found to be dependent on surface chemistry, i.e.
the type and strength of chemical interaction of adsorbates to the surface. Typical
values in silver are between 0.1 and 0.7. Confinement of free electrons leads to no-
table quantum confinement effects as the dimension of the particle reaches the level of
the Fermi wavelength of the free electron (about 0.7 nm), discrete, quantum-confined
electronic transitions appears [65]. It should be emphasized that the understanding
of quantum confinement is made possible only when a full quantum mechanical treat-
ment is invoked. However, little work has been done in this regard and it deserves

more research efforts.

The eigenmodes of collective oscillations of the quasi-free electrons in metals are
called plasmons. The existence of plasmons is a characteristic of the interaction of
metallic nanostructures with light. Similar behavior cannot be simply reproduced
in other spectral ranges using the scale invariance of Maxwell’s equations since the
material parameters change considerably with frequency. Because electrons carry
a charge, these oscillations are inherently associated with an electromagnetic field.
Therefore a theoretical description has to include this interplay of charges and fields.
The boundary conditions for electromagnetic fields lead to different conditions for
the occurrence of plasmons for the cases of planar metal-dielectric interfaces and
metal particles. The surface charge density of oscillation associated with surface

plasmons at the interface between a metal and dielectric can give rise to strongly

85

enhanced optical near-fields which is confined near the metal surface. Similarly, if
the electron gas is confined in two dimensions or three dimensions, as in the cases of
a nano wire and nanoparticle, the overall displacement of the electrons with respect
to the positively charged lattice leads to a restoring force, which in turn gives rise to
specific particle-plasmon resonance. For the understanding of surface plasmons, it is
very useful to study the results of simple electrodynamical theory applied to an ideal
model interface. Obviously this approach neglects many real effects due to the exact
surface properties, screening effects, quantum mechanical corrections, etc.. However,
the main features of surface plasmons can be understood in this simple model. The
main result obtained for plasmons at a flat metal-dielectric interface is a relation
between the lateral momentum k” and the frequency w, the dispersion relation

2 _ 2 Emed
k” — kphoton—Em + 5d (6.1.1)

with the dielectric functions of metal 5m and dielectric 5d. kphoton = w/c. The
derivation of Eq. (6.1.1) can be found in reference [46]. Because the surface plasmon
dispersion is below the photon dispersion for all energies, it is clear that surface
plasmons cannot be be excited by plane waves incident on the interface from the
dielectric medium because regardless of the angle of incidence, the surface plasmon
modes of the same frequency have a larger momentum. In order to excite surface
plasmons, additional momentum has to be somehow provided. In practice, this is

usually done either by placing a regular grating structure at the interface (which also

86

surface palsmons

  

air ggp §ur ace palsmons

 

prism

 

Figure 6.1: Excitation of surface plasmons. Left: Otto configuration. Right:
Kretschmann configuration.

disturbs the plasmon mode) or by letting the excitation light pass through a medium
with a high refractive index (e.g. a prism). In the latter case, the excitation light
can either come from the side of the dielectricum (so—called Otto-configuration, Otto
(1968)) or from the metal side (Kretschmann configuration, Kretschmann (1972)), cf.
Fig. 6.1. In the Otto configuration, there has to be a small gap between the dielectric
and the metal surface, because otherwise the surface plasmon dispersion would also
be altered. In the Kretschmann configuration the metal film has to be very thin
in order to allow the light field to reach through the film. The distinct resonance
condition associated with excitation of surface plasmons has found applications in
various sensors ranging from biological binding arrays to environmental sensing. For
reviews see e.g. [27, 37].

For surface plasmons discussed above propagating on plane interface we observed

that the electromagnetic field is strongly localized in one dimension, i.e., normal to

87

the interface. In the context of nano-optics, we are also interested in establishing
field confinement in two or even three dimensions. Because the characteristic scale of
the structure is much smaller than the wavelength of light, it is valid to assume that
all points of the structure respond simultaneously to an incoming field. In the long
wavelength approximation the Helmholtz equation reduces to the Laplace equation.

The electric field E = -V<I> and the potential has to satisfy

v2<1> = 0 (6.1.2)

along with the boundary conditions. Let us consider a metallic nanowire with radius
a centered at the origin and extending along the z—axis to infinity. The wire is exposed

to an x-polarized plane radiation as depicted in Fig. 6.2. The solution of (6.1.2) turns

out to be
25d
E = ————n ,
1 05m “l" Ed (I:
Em —' 8d 0.2 , 2
E = E n +E (1 — 23m n
2 0 CC 0"—— 5m + E dp2( 90) a:
_ 502102
+2E —— sin cos n
0:772. +€d p—'2 90 9'9 y
From the solutions the fields diverge when Re(€m) = —€d, which is the resonance

condition for a collective electron oscillation in a wire that is excited by an electric
field polarized perpendicular to the wire axis. Notice that no resonances exist if the

electric field is polarized along wire axis. The field distribution is computed for a thin

88

 

 

Figure 6.2: Cut through a thin wire that is illuminated by an x—polarized plane wave.

silver wire with radius 25 nm and it can be seen that the field is distributed around
the wire from the second plot in Fig. 6.2. Similarly a 3D nanoparticle also support

the plasmons, the solution of Eq. (6.1.2) can be obtained

35d
E = E ——
1 0£m+25dnz
3
. 5171, "Ed 0. _
E = E 0 — 9 ——-—-—-E 2 0 0 .
2 0(cos nr sm n0)+€m+2€d p3 0( cos nr+sm me)

A strongly localized field will be obtained as Re(€m) = —2Ed.

The near—fields around metallic nanostructures could be greatly enhanced due to
the plasmons. This novel feature has enabled metallic nanostructures widely used
in near-field optics, optical imaging and solar cell technology. Recently the study of
surface enhanced Raman scattering (SERS) in molecule detection due to plasmons

has been attracted a great deal of scientists. It was reported in 1974 that the Raman

89

scattering cross-section can be considerable increased if the molecules are adsorbed
on rough metal surface [20], which initiated extensive researches on SERS, see e.g.
[34, 43, 45, 48]. Despite all the activities in clarifying the underlying physics of
SERS current theories of explaining the fundamental origin of the effect are far from
satisfactory. It is accepted the largest contribution to the great signal enhancement
stems from the enhanced localized electric fields in the vicinity of rough metal surfaces,
a rigorous and clear physical model is still missing. Hence mathematical modeling

and investigation of SERS could be a promising future direction.

6.2 Quantum dots and ultra-efficient solar cells

A solar cell or photovoltaic cell is a device that converts sunlight into electricity by
photovoltaic effect. The efficiency of solar cells is the electrical power it puts out as
percentage of the power in incident sunlight. One is the most fundamental limita-
tions on the efficiency of a solar cell is the ‘band gap’ of the semiconducting material
used on conventional solar cells: the energy required to boost an electron from the
bound valence band into the mobile conduction band. When an electron is knocked
loose from the valence band, it goes into the conduction band as a negative charge,
leaving behind a hole of positive charge. Both electron and hole can migrate through
the material. The maximum single band gap solar cell conversion efficiency is calcu-
lated to be 31 percent, termed the Shockley-Queisser limit [56]. In practice, the best

achievable is about 25 percent. Stacking semiconductors with different band gaps to-

90

gether is one possible way to improve the efficiency. Theoretically the efficiency can
be increased to higher than 70 percent by stacking dozens of different layers together
in multi—junction cells. However, the crystal layers may suffer strain damages if too
many layers are stacked. The most efficient multi-junction solar cell is one that has
three layers: GaInP / GaAs/ Ge made by the national center for photovoltaics in the
US, which achieved an efficiency of 34 percent in 2001. Recent studies have shown
that quantum dots offer a new possibility for improving the efficiency of solar cells. In
contrast to their bulks, quantum dots possess tunable bandgap due to the quantum
confinement effects. A mixture of quantum dots of different sizes used in solar cell
could help harvest the maximum portion of the incident sunlight. Another advan-
tage of quantum dots is that they can be molded into a variety of different form, in
sheets or 3D arrays. A strong electronic coupling can occur between st if they are
very close, and photongenerated excitons will have longer life, facilitating the collec-
tion and transport of electrons to produce electricity at high voltage. In addition,
quantum dots array provides a possibility of multiple exciton generation (MEG) or
carrier multiplication from one single photon. Researchers led by Arthur Nozik at
the National Renewable Energy Laboratory Golden, Colorado in the United Sates
demonstrated that the absorption of a single photon by their quantum dots yielded—
not one exciton as usual the case—but three of them [17]. Also very efficient MEG was
observed recently in PbSe nanocrystals by Schaller and Klimov [55]. The formation

of multiple excitons per absorbed photon happens when the energy of the photon

91

absorbed is far greater than the semiconductor bandgap. This phenomenon does not
readily occur in bulk semiconductors where the excess energy simply dissipates away
as heat (interaction with phonon) before it can cause other electron-hole pairs to
form. But in semiconducting quantum dots, the rate of energy dissipation is signifi-
cantly reduced, and the charge carriers are confined within a minute volume, thereby
increasing their interactions and enhancing the probability for multiple excitons to
form. The microscopic origin of MEG is still under debate and several possibilities
have been suggested in [61]: (1) Impact ionization; (2) Coherent superposition of
single and multiexciton states; (3) Multiexciton formation through a virtual state.
Clearly it requires more theoretical researches to clarify the underlying mechanism
of MEG. To achieve a clearer picture quantum electrodynamics has to be invoked.
On the other hand, optimization of structural parameters is a very important part of

practical solar cell design and much more work could be done in this direction [52].

One of the ways for enhancing the conversion efficiency (increased photovoltage
or increased photocurrent ) can be accessed, in principle, QD solar cell configuration
shown in Fig 6.3. As an attempt to a compute the its efficiency Maxwell-Garnett
mixing formula (see [41] for details) is combined with the dielectric constant (3.2.7)
to yield an effective dielectric constant for the colloidal quantum dot film

€i—Ee
5i + 25,; — f(5i — ere)

 

Eeff = ee + 3fse (6.2.1)

This mixing rule predicts the effective dielectric Eeff of a heterogeneous medium

92

 

 

 

 

 

 

s . ---f= 0.75

0 [- I I T - T — T 1
3.198 3.2 3.202 3.204 3.206 3.208 3.21
incident energy (ev)

 

 

 

 

Figure 6.3: Top: A colloidal QD array used as a photoelectrode for a photochemical or
as the i—region of a p—i—n solar cell; Bottom: The transmission as a function of incident
energy and volume fraction of QD. The peaks indicate absorbtion enhancements.

93

where homogeneous quantum dots with 5,- are dilutely mixed into istropic environ-
ment 56. The inclusions occupy a volume fraction f. It should be noted that the
formula above assumes a uniformly distribution and well separations of quantum dots.
The numerical results Show that resonant peaks depend on several parameters, the
thickness of the film L, the volume fraction of quantum dots, the incident energy.
Those peaks indicate that absorptions of incident light achieve maxima and hence

conversion efficiencies are increased around these peaks.

94

Bibliography

[1] I. Abram, Nonlinear-optical properties of biexcitons: Single-beam propagation,
Phys. Rev. B, 28 (1983) pp. 4433—4443.

[2] I. Abram and A. Maruani, Calculation of the nonlinear dielectric function in
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