Quantum Theory of Optical Electronic Properties of Semiconductors-Hartmut Haug

Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

QUANTUM THEORY OF THE OPTICAL AND ELECTRONIC PROPERTIES
OF SEMICONDUCTORS
Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
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For photocopying of material in this volume, please pay a copying fee through the Copyright
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photocopy is not required from the publisher.

ISBN 981-238-609-2
ISBN 981-238-756-0 (pbk)

Printed in Singapore.


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Preface

The electronic properties of semiconductors form the basis of the latest
and current technological revolution, the development of ever smaller and
more powerful computing devices, which affect not only the potential of
modern science but practically all aspects of our daily life. This dramatic
development is based on the ability to engineer the electronic properties
of semiconductors and to miniaturize devices down to the limits set by
quantum mechanics, thereby allowing a large scale integration of many
devices on a single semiconductor chip.
Parallel to the development of electronic semiconductor devices, and no
less spectacular, has been the technological use of the optical properties of
semiconductors. The fluorescent screens of television tubes are based on
the optical properties of semiconductor powders, the red light of GaAs light
emitting diodes is known to all of us from the displays of domestic appliances, and semiconductor lasers are used to read optical discs and to write
in laser printers. Furthermore, fiber-optic communications, whose light
sources, amplifiers and detectors are again semiconductor electro-optical
devices, are expanding the capacity of the communication networks dramatically.
Semiconductors are very sensitive to the addition of carriers, which can
be introduced into the system by doping the crystal with atoms from another group in the periodic system, electronic injection, or optical excitation. The electronic properties of a semiconductor are primarily determined
by transitions within one energy band, i.e., by intraband transitions, which
describe the transport of carriers in real space. Optical properties, on the
other hand, are connected with transitions between the valence and conduction bands, i.e., with interband transitions. However, a strict separation
is impossible. Electronic devices such as a p-n diode can only be under-

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Quantum Theory of the Optical and Electronic Properties of Semiconductors

stood if one considers also interband transitions, and many optical devices
cannot be understood if one does not take into account the effects of intraband scattering, carrier transport and diffusion. Hence, the optical and
electronic semiconductor properties are intimately related and should be
discussed jointly.
Modern crystal growth techniques make it possible to grow layers of
semiconductor material which are narrow enough to confine the electron
motion in one dimension. In such quantum-well structures, the electron
wave functions are quantized like the standing waves of a particle in a square
well potential. Since the electron motion perpendicular to the quantumwell layer is suppressed, the semiconductor is quasi-two-dimensional. In this
sense, it is possible to talk about low-dimensional systems such as quantum
wells, quantum wires, and quantum dots which are effectively two, one and
zero dimensional.
These few examples suffice to illustrate the need for a modern textbook
on the electronic and optical properties of semiconductors and semiconductor devices. There is a growing demand for solid-state physicists, electrical and optical engineers who understand enough of the basic microscopic
theory of semiconductors to be able to use effectively the possibilities to
engineer, design and optimize optical and electronic devices with certain
desired characteristics.
In this fourth edition, we streamlined the presentation of the material and added several new aspects. Many results in the different chapters

are developed in parallel first for bulk material, and then for quasi-twodimensional quantum wells and for quasi-one-dimensional quantum wires,
respectively. Semiconductor quantum dots are treated in a separate chapter. The semiconductor Bloch equations have been given a central position.
They have been formulated not only for free particles in various dimensions,
but have been given, e.g., also in the Landau basis for low-dimensional electrons in strong magnetic fields or in the basis of quantum dot eigenfunctions.
The Bloch equations are extended to include correlation and scattering effects at different levels of approximation. Particularly, the relaxation and
the dephasing in the Bloch equations are treated not only within the semiclassical Boltzmann kinetics, but also within quantum kinetics, which is
needed for ultrafast semiconductor spectroscopy. The applications of these
equations to time-dependent and coherent phenomena in semiconductors
have been extended considerably, e.g., by including separate chapters for
the excitonic optical Stark effect and various nonlinear wave-mixing configurations. The presentation of the nonequilibrium Green’s function theory

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Preface

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vii

has been modified to present both introductory material as well as applications to Coulomb carrier scattering and time-dependent screening. In
several chapters, direct comparisons of theoretical results with experiments
have been included.
This book is written for graduate-level students or researchers with general background in quantum mechanics as an introduction to the quantum

theory of semiconductors. The necessary many-particle techniques, such as
field quantization and Green’s functions are developed explicitly. Wherever
possible, we emphasize the motivation of a certain derivation and the physical meaning of the results, avoiding the discussion of formal mathematical
aspects of the theory. The book, or parts of it, can serve as textbook for
use in solid state physics courses, or for more specialized courses on electronic and optical properties of semiconductors and semiconductor devices.
Especially the later chapters establish a direct link to current research in
semicoductor physics. The material added in the fourth edition should
make the book as a whole more complete and comprehensive.
Many of our colleagues and students have helped in different ways to
complete this book and to reduce the errors and misprints. We especially
wish to thank L. Banyai, R. Binder, C. Ell, I. Galbraith, Y.Z. Hu, M. Kira,
M. Lindberg, T. Meier, and D.B. Tran-Thoai for many scientific discussions and help in several calculations. We appreciate helpful suggestions
and assistance from our present and former students S. Benner, K. ElSayed, W. Hoyer, J. Müller, M. Pereira, E. Reitsamer, D. Richardson, C.
Schlichenmaier, S. Schuster, Q.T. Vu, and T. Wicht. Last but not least we
thank R. Schmid, Marburg, for converting the manuscript to Latex and for
her excellent work on the figures.
Frankfurt and Marburg
August 2003

Hartmut Haug
Stephan W. Koch


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Quantum Theory of the Optical and Electronic Properties of Semiconductors

About the authors
Hartmut Haug obtained his Ph.D (Dr. rer. nat., 1966) in Physics at the
University of Stuttgart. From 1967 to 1969, he was a faculty member at the
Department of Electrical Engineering, University of Wisconsin in Madison.
After working as a member of the scientific staff at the Philips Research
Laboratories in Eindhoven from 1969 to 1973, he joined the Institute of
Theoretical Physics of the University of Frankfurt, where he was a full
professor from 1975 to 2001 and currently is an emeritus. He has been a
visiting scientist at many international research centers and universities.
Stephan W. Koch obtained his Ph. D. (Dr. phil. nat., 1979) in Physics
at the University of Frankfurt. Until 1993 he was a full professor both
at the Department of Physics and at the Optical Sciences Center of the
University of Arizona, Tucson (USA). In the fall of 1993, he joined the
Philipps-University of Marburg where he is a full professor of Theoretical
Physics. He is a Fellow of the Optical Society of America. He received
the Leibniz prize of the Deutsche Physikalische Gesellschaft (1997) and the
Max-Planck Research Prize of the Humboldt Foundation and the MaxPlanck Society (1999).

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Contents

Preface

v

1. Oscillator Model

1

1.1 Optical Susceptibility . . . . . . . . . . . . . . . . . . . . .
1.2 Absorption and Refraction . . . . . . . . . . . . . . . . . . .
1.3 Retarded Green’s Function . . . . . . . . . . . . . . . . . .
2. Atoms in a Classical Light Field

17

2.1 Atomic Optical Susceptibility . . . . . . . . . . . . . . . . .
2.2 Oscillator Strength . . . . . . . . . . . . . . . . . . . . . . .
2.3 Optical Stark Shift . . . . . . . . . . . . . . . . . . . . . . .
3. Periodic Lattice of Atoms
3.1
3.2
3.3
3.4

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6

12

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Reciprocal Lattice, Bloch Theorem
Tight-Binding Approximation . . .
k·p Theory . . . . . . . . . . . . .
Degenerate Valence Bands . . . . .

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4. Mesoscopic Semiconductor Structures
4.1 Envelope Function Approximation . . . . . . . . . . . . . .
4.2 Conduction Band Electrons in Quantum Wells . . . . . . .
4.3 Degenerate Hole Bands in Quantum Wells . . . . . . . . . .
5. Free Carrier Transitions

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5.1 Optical Dipole Transitions . . . . . . . . . . . . . . . . . . .
5.2 Kinetics of Optical Interband Transitions . . . . . . . . . .
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5.2.1 Quasi-D-Dimensional Semiconductors
5.2.2 Quantum Confined Semiconductors

with Subband Structure . . . . . . . .
5.3 Coherent Regime: Optical Bloch Equations .
5.4 Quasi-Equilibrium Regime:
Free Carrier Absorption . . . . . . . . . . . .

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6. Ideal Quantum Gases
6.1 Ideal
6.1.1
6.1.2
6.2 Ideal
6.2.1
6.2.2
6.3 Ideal

89

Fermi Gas . . . . . . . . . . . . . . . .
Ideal Fermi Gas in Three Dimensions
Ideal Fermi Gas in Two Dimensions .
Bose Gas . . . . . . . . . . . . . . . .
Ideal Bose Gas in Three Dimensions
Ideal Bose Gas in Two Dimensions .
Quantum Gases in D Dimensions . . .

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. 90
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7. Interacting Electron Gas
7.1
7.2
7.3
7.4
7.5

107

The Electron Gas Hamiltonian . . .

Three-Dimensional Electron Gas . .
Two-Dimensional Electron Gas . . .
Multi-Subband Quantum Wells . . .
Quasi-One-Dimensional Electron Gas

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8. Plasmons and Plasma Screening
8.1 Plasmons and Pair Excitations . .
8.2 Plasma Screening . . . . . . . . . .
8.3 Analysis of the Lindhard Formula .
8.3.1 Three Dimensions . . . . . .
8.3.2 Two Dimensions . . . . . . .
8.3.3 One Dimension . . . . . . .

8.4 Plasmon–Pole Approximation . . .
9. Retarded Green’s Function for Electrons

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9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2 Interacting Electron Gas . . . . . . . . . . . . . . . . . . . . 152
9.3 Screened Hartree–Fock Approximation . . . . . . . . . . . . 156
10. Excitons

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10.1 The Interband Polarization . . . . . . . . .

10.2 Wannier Equation . . . . . . . . . . . . . .
10.3 Excitons . . . . . . . . . . . . . . . . . . . .
10.3.1 Three- and Two-Dimensional Cases .
10.3.2 Quasi-One-Dimensional Case . . . .
10.4 The Ionization Continuum . . . . . . . . . .
10.4.1 Three- and Two-Dimensional Cases .
10.4.2 Quasi-One-Dimensional Case . . . .
10.5 Optical Spectra . . . . . . . . . . . . . . . .
10.5.1 Three- and Two-Dimensional Cases
10.5.2 Quasi-One-Dimensional Case . . . .

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11. Polaritons
11.1 Dielectric Theory of Polaritons . . . . . . . . . . . . . .
11.1.1 Polaritons without Spatial Dispersion
and Damping . . . . . . . . . . . . . . . . . . . .
11.1.2 Polaritons with Spatial Dispersion and Damping
11.2 Hamiltonian Theory of Polaritons . . . . . . . . . . . . .
11.3 Microcavity Polaritons . . . . . . . . . . . . . . . . . . .

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12. Semiconductor Bloch Equations
12.1 Hamiltonian Equations . . . . . . . . . . . . . . . . . . . . .
12.2 Multi-Subband Microstructures . . . . . . . . . . . . . . . .
12.3 Scattering Terms . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Intraband Relaxation . . . . . . . . . . . . . . . . . .
12.3.2 Dephasing of the Interband Polarization . . . . . . .
12.3.3 Full Mean-Field Evolution of the Phonon-Assisted
Density Matrices . . . . . . . . . . . . . . . . . . . .
13. Excitonic Optical Stark Effect

164
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13.1 Quasi-Stationary Results . . . . . . . . . . . . . . . . . . . . 237
13.2 Dynamic Results . . . . . . . . . . . . . . . . . . . . . . . . 246
13.3 Correlation Effects . . . . . . . . . . . . . . . . . . . . . . . 255
14. Wave-Mixing Spectroscopy

269

14.1 Thin Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 271
14.2 Semiconductor Photon Echo . . . . . . . . . . . . . . . . . . 275
15. Optical Properties of a Quasi-Equilibrium Electron–


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Hole Plasma

283

15.1 Numerical Matrix Inversion . . . . . . .
15.2 High-Density Approximations . . . . . .
15.3 Effective Pair-Equation Approximation .
15.3.1 Bound states . . . . . . . . . . .
15.3.2 Continuum states . . . . . . . . .
15.3.3 Optical spectra . . . . . . . . . .

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16. Optical Bistability

305

16.1 The Light Field Equation . . . . . . . .
16.2 The Carrier Equation . . . . . . . . . .
16.3 Bistability in Semiconductor Resonators
16.4 Intrinsic Optical Bistability . . . . . . .

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17. Semiconductor Laser

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17.1 Material Equations . . . . . . . . . . . . . .
17.2 Field Equations . . . . . . . . . . . . . . . .
17.3 Quantum Mechanical Langevin Equations .
17.4 Stochastic Laser Theory . . . . . . . . . . .
17.5 Nonlinear Dynamics with Delayed Feedback

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18. Electroabsorption
18.1 Bulk Semiconductors . . . .
18.2 Quantum Wells . . . . . . .
18.3 Exciton Electroabsorption .
18.3.1 Bulk Semiconductors
18.3.2 Quantum Wells . . .

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349

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19. Magneto-Optics

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19.1 Single Electron in a Magnetic Field . . . . . . . . . . . . . . 372
19.2 Bloch Equations for a Magneto-Plasma . . . . . . . . . . . 375
19.3 Magneto-Luminescence of Quantum Wires . . . . . . . . . . 378
20. Quantum Dots
20.1 Effective Mass Approximation
20.2 Single Particle Properties . .
20.3 Pair States . . . . . . . . . .
20.4 Dipole Transitions . . . . . .

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xiii

20.5 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . 395
20.6 Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 396
21. Coulomb Quantum Kinetics

401

21.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . 402
21.2 Second Born Approximation . . . . . . . . . . . . . . . . . . 408
21.3 Build-Up of Screening . . . . . . . . . . . . . . . . . . . . . 413
Appendix A

Field Quantization

421

A.1 Lagrange Functional . . . . . . . . . . . . . . . . . . . . . . 421
A.2 Canonical Momentum and Hamilton Function . . . . . . . . 426
A.3 Quantization of the Fields . . . . . . . . . . . . . . . . . . . 428
Appendix B

Contour-Ordered Green’s Functions

435

B.1 Interaction Representation . . . . . . . . . . . . . . . . . . . 436
B.2 Langreth Theorem . . . . . . . . . . . . . . . . . . . . . . . 439
B.3 Equilibrium Electron–Phonon Self-Energy . . . . . . . . . . 442
Index

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Chapter 1

Oscillator Model

The valence electrons, which are responsible for the binding of the atoms
in a crystal can either be tightly bound to the ions or can be free to move
through the periodic lattice. Correspondingly, we speak about insulators
and metals. Semiconductors are intermediate between these two limiting

cases. This situation makes semiconductors extremely sensitive to imperfections and impurities, but also to excitation with light. Before techniques
were developed allowing well controlled crystal growth, research in semiconductors was considered by many physicists a highly suspect enterprise.
Starting with the research on Ge and Si in the 1940’s, physicists learned
to exploit the sensitivity of semiconductors to the content of foreign atoms
in the host lattice. They learned to dope materials with specific impurities which act as donors or acceptors of electrons. Thus, they opened the
field for developing basic elements of semiconductor electronics, such as
diodes and transistors. Simultaneously, semiconductors were found to have
a rich spectrum of optical properties based on the specific properties of the
electrons in these materials.
Electrons in the ground state of a semiconductor are bound to the ions
and cannot move freely. In this state, a semiconductor is an insulator. In
the excited states, however, the electrons are free, and become similar to the
conduction electrons of a metal. The ground state and the lowest excited
state are separated by an energy gap. In the spectral range around the
energy gap, pure semiconductors exhibit interesting linear and nonlinear
optical properties. Before we discuss the quantum theory of these optical
properties, we first present a classical description of a dielectric medium
in which the electrons are assumed to be bound by harmonic forces to the
positively charged ions. If we excite such a medium with the periodic transverse electric field of a light beam, we induce an electrical polarization due

1

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Quantum Theory of the Optical and Electronic Properties of Semiconductors

to microscopic displacement of bound charges. This oscillator model for
the electric polarization was introduced in the pioneering work of Lorentz,
Planck, and Einstein. We expect the model to yield reasonably realistic
results as long as the light frequency does not exceed the frequency corresponding to the energy gap, so that the electron stays in its bound state.
We show in this chapter that the analysis of this simple model already
provides a qualitative understanding of many basic aspects of light–matter
interaction. Furthermore, it is useful to introduce such general concepts as
optical susceptibility, dielectric function, absorption and refraction, as well
as Green’s function.

1.1

Optical Susceptibility

The electric field, which is assumed to be polarized in the xdirection, causes a displacement x of an electron with a charge
e −1.6 10−16 C −4.8 10−10 esu from its equilibrium position. The resulting polarization, defined as dipole moment per unit volume, is
P=

P
= n0 ex = n0 d ,
L3

(1.1)

where L3 = V is the volume, d = ex is the electric dipole moment, and
n0 is the mean electron density per unit volume. Describing the electron
under the influence of the electric field E(t) (parallel to x) as a damped
driven oscillator, we can write Newton’s equation as
m0

d2 x
dx
− m0 ω02 x + eE(t) ,
= −2m0 γ
2
dt
dt

(1.2)

where γ is the damping constant, and m0 and ω0 are the mass and resonance
frequency of the oscillator, respectively. The electric field is assumed to be
monochromatic with a frequency ω, i.e., E(t) = E0 cos(ωt). Often it is
convenient to consider a complex field
E(t) = E(ω)e−iωt

(1.3)

and take the real part of it whenever a final physical result is calculated.
With the ansatz
x(t) = x(ω)e−iωt

(1.4)

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we get from Eq. (1.2)
m0 (ω 2 + i2γω − ω02 )x(ω) = −eE(ω)

(1.5)

and from Eq. (1.1)
P(ω) = −

1
n0 e 2
E(ω) .
m0 ω 2 + i2γω − ω02

(1.6)

The complex coefficient between P(ω) and E(ω) is defined as the optical

susceptibility χ(ω). For the damped driven oscillator, this optical susceptibility is

χ(ω) = −

n0 e2

1

2m0 ω0

ω − ω0 + iγ

−

1
ω + ω0 + iγ

.

(1.7)

optical susceptibility
Here,
ω0 =

ω02 − γ 2

(1.8)

is the resonance frequency that is renormalized (shifted) due to the damping. In general, the optical susceptibility is a tensor relating different vector

components of the polarization Pi and the electric field Ei . An important
feature of χ(ω) is that it becomes singular at
ω = −iγ ± ω0 .

(1.9)

This relation can only be satisfied if we formally consider complex frequencies ω = ω + iω . We see from Eq. (1.7) that χ(ω) has poles in the lower
half of the complex frequency plane, i.e. for ω < 0, but it is an analytic
function on the real frequency axis and in the whole upper half plane. This
property of the susceptibility can be related to causality, i.e., to the fact
that the polarization P(t) at time t can only be influenced by fields E(t − τ )
acting at earlier times, i.e., τ ≥ 0. Let us consider the most general linear
relation between the field and the polarization
t

P(t) =

dt χ(t, t )E(t ) .
−∞

(1.10)


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Here, we take both P(t) and E(t) as real quantities so that χ(t) is a real
quantity as well. The response function χ(t, t ) describes the memory of the
system for the action of fields at earlier times. Causality requires that fields
E(t ) which act in the future, t > t, cannot influence the polarization of the
system at time t. We now make a transformation to new time arguments
T and τ defined as
T =

t+t
and τ = t − t .
2

(1.11)

If the system is in equilibrium, the memory function χ(T, τ ) depends only
on the time difference τ and not on T , which leads to
t

P(t) =

−∞
∞

=
0

dt χ(t − t )E(t )
dτ χ(τ )E(t − τ ).

(1.12)

Next, we use a Fourier transformation to convert Eq. (1.12) into frequency
space. For this purpose, we define the Fourier transformation f (ω) of a
function f (t) through the relations
∞

f (ω) =
−∞
∞

f (t) =
−∞

dtf (t)eiωt
dω
f (ω)e−iωt .
2π

(1.13)

Using this Fourier representation for x(t) and E(t) in Eq. (1.2), we find for
x(ω) and E(ω) again the relation (1.5) and thus the resulting susceptibility
(1.7), showing that the ansatz (1.3) – (1.4) is just a shortcut for a solution
using the Fourier transformation.
Multiplying Eq. (1.12) by eiωt and integrating over t , we get

P(ω) =

∞

dτ χ(τ )eiωτ

0

+∞
−∞

dtE(t − τ )eiω(t−τ ) = χ(ω)E(ω) ,

(1.14)

where
∞

χ(ω) =
0

dτ χ(τ )eiωτ .

(1.15)


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The convolution integral in time, Eq. (1.12), becomes a product in Fourier
space, Eq. (1.14). The time-dependent response function χ(t) relates two
real quantities, E(t) and P(t), and therefore has to be a real function itself.
Hence, Eq. (1.15) implies directly that χ∗ (ω) = χ(−ω) or χ (ω) = χ (−ω)
and χ (ω) = −χ (−ω). Moreover, it also follows that χ(ω) is analytic for
ω ≥ 0, because the factor e−ω τ forces the integrand to zero at the upper
boundary, where τ → ∞.
Since χ(ω) is an analytic function for real frequencies we can use the
Cauchy relation to write
+∞

χ(ν)
dν
,
2πi ν − ω − iδ

χ(ω) =
−∞

(1.16)

where δ is a positive infinitesimal number. The integral can be evaluated

using the Dirac identity (see problem (1.1))
lim

δ→0

1
1
= P + iπδ(ω) ,
ω − iδ
ω

(1.17)

where P denotes the principal value of an integral under which this relation
is used. We find
+∞

+∞

1
dν χ(ν)
+
2πi ν − ω 2

χ(ω) = P
−∞

−∞

dνχ(ν)δ(ν − ω) .

(1.18)

For the real and imaginary parts of the susceptibility, we obtain separately
+∞

χ (ω) = P
χ (ω) = −P

dν χ (ν)
π ν−ω

−∞
+∞
−∞

(1.19)

dν χ (ν)
.
π ν −ω

(1.20)

Splitting the integral into two parts
0

χ (ω) = P
−∞

dν χ (ν)
+P
π ν −ω

+∞
0

dν χ (ν)
π ν −ω

(1.21)

and using the relation χ (ω) = −χ (−ω), we find
+∞

χ (ω) = P
0

dν
χ (ν)
π

1
1
+
ν+ω
ν−ω

Combining the two terms yields

.

(1.22)


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+∞

χ (ω) = P
0

dν
π

χ (ν)

2ν
ν2

− ω2

.

(1.23)

Kramers–Kronig relation
This is the Kramers–Kronig relation, which allows us to calculate the real
part of χ(ω) if the imaginary part is known for all positive frequencies. In
realistic situations, one has to be careful with the use of Eq. (1.23), because
χ (ω) is often known only in a finite frequency range. A relation similar
to Eq. (1.23) can be derived for χ using (1.20) and χ (ω) = χ (−ω), see
problem (1.3).

1.2

Absorption and Refraction

Before we give any physical interpretation of the susceptibility obtained
with the oscillator model we will establish some relations to other important
optical coefficients. The displacement field D(ω) can be expressed in terms
of the polarization P(ω) and the electric field1
D(ω) = E(ω) + 4πP(ω) = [1 + 4πχ(ω)]E(ω) = (ω)E(ω) ,

(1.24)

where the optical (or transverse) dielectric function (ω) is obtained from
the optical susceptibility (1.7) as

(ω) = 1 + 4πχ(ω) = 1−

2
ωpl

2ω0

1
1
−
ω −ω0 +iγ
ω +ω0 +iγ

.
(1.25)

optical dielectric function
Here, ωpl denotes the plasma frequency of an electron plasma with the mean
density n0 :
1 We

use cgs units in most parts of this book.


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ωpl =

4πn0 e2
m0

.

7

(1.26)

plasma frequency
The plasma frequency is the eigenfrequency of the electron plasma density
oscillations around the position of the ions. To illustrate this fact, let us
consider an electron plasma of density n(r, t) close to equilibrium. The
equation of continuity is
e

∂n
+ div j = 0
∂t

(1.27)

with the current density
j(r, t) = en(r, t)v(r, t) .

(1.28)

The source equation for the electric field is
divE = 4πe(n − n0 )

(1.29)

and Newton’s equation for free carriers can be written as
m0

∂v
= eE .
∂t

(1.30)

We now linearize Eqs. (1.27) – (1.29) around the equilibrium state where
the velocity is zero and no fields exist. Inserting

n = n0 + δn1 + O(δ 2 )
v = δv1 + O(δ 2 )
E = δE1 + O(δ 2 )

(1.31)

into Eqs. (1.27) – (1.30) and keeping only terms linear in δ, we obtain
∂n1
+ n0 divv1 = 0 ,
∂t

(1.32)

divE1 = 4πen1 ,

(1.33)


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and
m0

∂v1
= eE1 .
∂t

(1.34)

The equation of motion for n1 can be derived by taking the time derivative
of Eq. (1.32) and using Eqs. (1.33) and (1.34) to get

n0 e
∂v1
∂ 2 n1
2
=−
= −n0 div
divE1 = −ωpl
n1 .
∂t2
∂t
m0

(1.35)

This simple harmonic oscillator equation is the classical equation for charge
density oscillations with the eigenfrequency ωpl around the equilibrium density n0 .
Returning to the discussion of the optical dielectric function (1.25), we
note that (ω) has poles at ω = ±ω0 − iγ, describing the resonant and the
nonresonant part, respectively. If we are interested in the optical response in
the spectral region around ω0 and if ω0 is sufficiently large, the nonresonant
part gives only a small contribution and it is often a good approximation
to neglect it completely.
In order to simplify the resulting expressions, we now consider only
the resonant part of the dielectric function and assume ω0 >> γ, so that
ω0 ω0 and
(ω) = 1 −

2
ωpl
1

.
2ω0 ω − ω0 + iγ

(1.36)

For the real part of the dielectric function, we thus get the relation
(ω) − 1 = −

2
ωpl
ω − ω0
,
2ω0 (ω − ω0 )2 + γ 2

(1.37)

while the imaginary part has the following resonance structure
(ω) =

2
ωpl
2γ
.
4ω0 (ω − ω0 )2 + γ 2

(1.38)

Examples of the spectral variations described by Eqs. (1.37) and (1.38) are
shown in Fig. 1.1. The spectral shape of the imaginary part is determined
by the Lorentzian line-shape function 2γ/[(ω − ω0 )2 + γ 2 ]. It decreases

asymptomatically like 1/(ω − ω0 )2 , while the real part of (ω) decreases like
1/(ω − ω0 ) far away from the resonance.


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e’’(w)

e’(w)

(w-w0)/w0
Fig. 1.1 Dispersion of the real and imaginary part of the dielectric function, Eq. (1.37)
2 /2γω .
and (1.38), respectively. The broadening is taken as γ/ω0 = 0.1 and max = ωpl
o

In order to understand the physical information contained in (ω) and
(ω), we consider how a light beam propagates in the dielectric medium.
From Maxwell’s equations
1 ∂

D(r, t)
c ∂t
1 ∂
B(r, t)
curl E(r, t) = −
c ∂t

(1.39)

curl H(r, t) =

(1.40)

we find with B(r, t) = H(r, t), which holds at optical frequencies,
curl curl E(r, t) = −

1 ∂2
1 ∂
curl H(r, t) = − 2 2 D(r, t) .
c ∂t
c ∂t

(1.41)

Using curl curl = grad div − ∆, we get for a transverse electric field with
divE(r, t) = 0, the wave equation
∆E(r, t) −

1 ∂2
D(r, t) = 0 .

c2 ∂t2

(1.42)

Here, ∆ ≡ ∇2 is the Laplace operator.
Eq. (1.42) with respect to time yields
∆E(r, ω) +

ω2
ω2
(ω)E(r, ω) + i 2
2
c
c

A Fourier transformation of

(ω)E(r, ω) = 0 .

(1.43)


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For a plane wave propagating with wave number k(ω) and extinction coefficient κ(ω) in the z direction,

E(r, ω) = E0 (ω)ei[k(ω)+iκ(ω)]z ,

(1.44)

we get from Eq. (1.43)

[k(ω) + iκ(ω)]2 =

ω2
[ (ω) + i (ω)] .
c2

(1.45)

Separating real and imaginary part of this equation yields

k 2 (ω) − κ2 (ω) =

2κ(ω)k(ω) =

ω2
c2

ω2

(ω) ,
c2

(ω) .

(1.46)

(1.47)

Next, we introduce the index of refraction n(ω) as the ratio between the
wave number k(ω) in the medium and the vacuum wave number k0 = ω/c

k(ω) = n(ω)

ω
c

(1.48)

and the absorption coefficient α(ω) as

α(ω) = 2κ(ω) .

(1.49)

The absorption coefficient determines the decay of the intensity I ∝ |E|2 in
real space. 1/α is the length, over which the intensity decreases by a factor


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1/e. From Eqs. (1.46) – (1.49) we obtain the relations
1

n(ω) =

2

(ω) +

2 (ω)

+

2 (ω)

(1.50)

index of refraction
and

α(ω) =

ω
n(ω)c

(ω) .

(1.51)

absorption coefficient
Hence, Eqs. (1.38) and (1.51) yield a Lorentzian absorption line, and
Eqs. (1.37) and (1.50) describe the corresponding frequency-dependent index of refraction. Note that for (ω) << (ω), which is often true in
semiconductors, Eq. (1.50) simplifies to
n(ω)

(1.52)

(ω) .

Furthermore, if the refractive index n(ω) is only weakly frequencydependent for the ω-values of interest, one may approximate Eq. (1.51)
as
α(ω)

ω
nb c

(ω) =

4πω
χ (ω) ,

nb c

(1.53)

where nb is the background refractive index.
For the case γ → 0, i.e., vanishing absorption line width, the line-shape
function approaches a delta function (see problem 1.3)
lim

γ→0

2γ
2

(ω − ω0 ) + γ 2

= 2πδ (ω − ω0 ) .

(1.54)

In this case, we get

(ω) = π

2
ωpl
δ(ω − ω0 )
2ω0

(1.55)

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and the real part becomes
(ω) = 1 −

1.3

2
ωpl
1
.
2ω0 ω − ω0

(1.56)

Retarded Green’s Function

An alternative way of solving the inhomogeneous differential equation

m0

∂2
∂
+ 2γ + ω02 x(t) = eE(t)
2
∂t
∂t

(1.57)

is obtained by using the Green’s function of Eq. (1.57). The so-called
retarded Green’s function G(t − t ) is defined as the solution of Eq. (1.57),
where the inhomogeneous term eE(t) is replaced by a delta function
m0

∂2
∂
+ 2γ + ω02 G(t − t ) = δ(t − t ) .
∂t2
∂t

(1.58)

Fourier transformation yields

1
1
2
m0 ω + i2γω − ω02

1
1
1
−
=−
2m0 ω0 ω − ω0 + iγ
ω + ω0 + iγ

G(ω) = −

,

(1.59)

retarded Green’s function of an oscillator
where ω0 is defined in Eq. (1.8). In terms of G(t − t ), the solution of
Eq. (1.57) is then
+∞

x(t) =
−∞

dt G(t − t )eE(t ) ,

(1.60)

as can be verified by inserting (1.60) into (1.57). Note, that the general solution of an inhomogeneous linear differential equation is obtained by adding
the solution (1.60) of the inhomogeneous equation to the general solution
of the homogeneous equation. However, since we are only interested in the
induced polarization, we just keep the solution (1.60).

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In general, the retarded Green’s function G(t − t ) has the properties
G(t − t ) =

f inite
0

t≥t
t
for

(1.61)

or
G(τ ) ∝ θ(τ ) ,
where θ(τ ) is the unit-step or Heavyside function

θ(τ ) =

1
0

τ ≥0
τ <0

for

(1.62)

.

For τ < 0 we can close in (1.60) the integral by a circle with an infinite
radius in the upper half of the complex frequency plane since
lim

|ω|→∞

ei(ω +iω

)|τ |

= lim

|ω|→∞

eiω τ e−ω

|τ |

=0 .

(1.63)

As can be seen from (1.59), G(ω) has no poles in the upper half plane
making the integral zero for τ < 0. For τ ≥ 0 we have to close the contour
integral in the lower half plane, denoted by C , and get
1
1
1
dω −iωτ
e
−
θ(τ )
2m0 ω0
ω − ω0 + iγ
ω + ω0 + iγ
C 2π
1
[e−(iω0 +γ)τ − e(iω0 −γ)τ ] .
= iθ(τ )
2m0 ω0

G(τ ) = −

(1.64)

The property that G(τ ) = 0 for τ < 0 is the reason for the name retarded

Green’s function which is often indicated by a superscript r, i.e.,
Gr (τ ) = 0 for τ < 0 ←→ Gr (ω) = analytic for ω ≥ 0 .

(1.65)

The Fourier transform of Eq. (1.60) is
+∞

x(ω) =

+∞

dt
−∞

−∞

dt eiω(t−t ) G(t − t )eiωt eE(t )

= e G(ω)E(ω) .

(1.66)

With P(ω) = en0 x(ω) = χ(ω)E(ω) we obtain
χ(ω) = n0 e2 G(ω)

(1.67)