Engineering Quantum Mechanics

ENGINEERING QUANTUM MECHANICS

ffirs01.indd i

4/20/2011 10:51:57 AM

ENGINEERING QUANTUM MECHANICS

Doyeol Ahn Seoung-Hwan Park

IEEE PRESS

A JOHN WILEY & SONS, INC., PUBLICATION

ffirs02.indd iii

4/20/2011 10:51:58 AM

Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Ahn, Doyeol. Engineering Quantum Mechanics/Doyeol Ahn, Seoung-Hwan Park. p. cm. Includes bibliographical references and index. ISBN 978-0-470-10763-8 1. Quantum theory. 2. Stochastic processes. 3. Engineering mathematics. 4. Semiconductors–Electric properties–Mathematical models. I. Park, Seoung-Hwan. II. Title. QC174.12.A393 2011 620.001'53012–dc22 2010044304 oBook ISBN: 978-1-118-01782-1 ePDF ISBN: 978-1-118-01780-7 ePub ISBN: 978-1-118-01781-4 Printed in Singapore. 10 9 8 7 6 5 4 3 2 1

ffirs03.indd iv

4/20/2011 3:26:00 PM

This page intentionally left blank

CONTENTS Preface PART I 1

vii Fundamentals

1

Basic Quantum Mechanics

3

1.1 Measurements and Probability 1.2 Dirac Formulation 1.3 Brief Detour to Classical Mechanics 1.4 A Road to Quantum Mechanics 1.5 The Uncertainty Principle 1.6 The Harmonic Oscillator 1.7 Angular Momentum Eigenstates 1.8 Quantization of Electromagnetic Fields 1.9 Perturbation Theory Problems References

3 4 8 14 21 22 29 35 38 41 43

2

45

Basic Quantum Statistical Mechanics

2.1 Elementary Statistical Mechanics 2.2 Second Quantization 2.3 Density Operators 2.4 The Coherent State 2.5 The Squeezed State 2.6 Coherent Interactions Between Atoms and Fields 2.7 The Jaynes–Cummings Model Problems References 3

Elementary Theory of Electronic Band Structure in Semiconductors

3.1 Bloch Theorem and Effective Mass Theory 3.2 The Luttinger–Kohn Hamiltonian 3.3 The Zinc Blende Hamiltonian

45 51 54 58 62 68 69 71 72 73 73 84 105 v

ftoc.indd v

4/20/2011 10:51:59 AM

vi

CONTENTS

3.4 The Wurtzite Hamiltonian 3.5 Band Structure of Zinc Blende and Wurtzite Semiconductors 3.6 Crystal Orientation Effects on a Zinc Blende Hamiltonian 3.7 Crystal Orientation Effects on a Wurtzite Hamiltonian Problems References

135 152 168 169

PART II

171

4

Modern Applications

Quantum Information Science

4.1 4.2 4.3 4.4

ftoc.indd vi

114 123

173

Quantum Bits and Tensor Products Quantum Entanglement Quantum Teleportation Evolution of the Quantum State: Quantum Information Processing 4.5 A Measure of Information 4.6 Quantum Black Holes Appendix A: Derivation of Equation (4.82) Appendix B: Derivation of Equations (4.93) and (4.106) Problems References

180 183 184 202 203 204 205

5

207

Modern Semiconductor Laser Theory

173 175 178

5.1 Density Operator Description of Optical Interactions 5.2 The Time-Convolutionless Equation 5.3 The Theory of Non-Markovian Optical Gain in Semiconductor Lasers 5.4 Optical Gain of a Quantum Well Laser with Non-Markovian Relaxation and Many-Body Effects 5.5 Numerical Methods for Valence Band Structure in Nanostructures 5.6 Zinc Blende Bulk and Quantum Well Structures 5.7 Wurtzite Bulk and Quantum Well Structures 5.8 Quantum Wires and Quantum Dots Appendix: Fortran 77 Code for the Band Structure Problems References

209 211

235 252 258 265 274 286 287

Index

289

223 232

4/20/2011 10:51:59 AM

Preface

Quantum mechanics is becoming more important in applied science and engineering, especially with the recent developments in quantum computing, as well as the rapid progress in optoelectronic devices. This textbook is intended for graduate students and advanced undergraduate students in electrical engineering, physics, and materials science and engineering. It also provides the necessary theoretical background for researchers in optoelectronics or semiconductor devices. In the task of providing advanced instruction for both students and researchers, quantum mechanics presents special difficulties because of its hierarchical structures. The more abstract formalisms and techniques are quite meaningless until one has mastered the earlier stages in classical physics, which most engineering students are lacking. Quantum mechanics has become an essential tool for modern engineering. This book covers topics such as semiconductors and laser physics, which are traditionally quantum mechanical, as well as relatively new topics in the field, such as quantum computation and quantum information. These fields have seen an explosive growth during the past 10 years, as quantum computing or quantum information processing can have a significant impact on today’s electronics and computations. The essence of quantum computing is the direct usage of the superposition and entanglement of quantum mechanics. The most challenging research topics include the generation and manipulation of quantum entangled systems, developing the fundamental theory of entanglement, decoherence control, and the demonstration of the scalability of quantum information processing. In laser physics, there has been a growing interest in the model of semiconductor lasers with non-Markovian relaxation partially because of the dissatisfaction with the conventional model for optical gain in predicting the correct gain spectrum and the thermodynamic relations. This is mainly due to the poor convergence properties of the lineshape function, that is, the Lorentzian lineshape, used in the conventional model. In this book, a non-Markovian model for the optical gain of semiconductors is developed, taking into account the rigorous electronic band structure, many-body effects, and the non-Markovian vii

fpref.indd vii

4/20/2011 10:51:59 AM

viii

PREFACE

relaxation using the quantum statistical reduced-density operator formalism for an arbitrary driven system coupled to a stochastic reservoir. Example programs based on Fortran 77 will also be provided for band structures of zinc blende quantum wells. Many-body effects are taken into account within the time-dependent Hartree–Fock. Various semiconductor lasers including strained-layer quantum well lasers and wurtzite GaN blue-green quantum well lasers are discussed. We thank Professor Shun-Lien Chuang, Doyeol Ahn’s Ph.D. thesis adviser, for extensive enlightening and encouragement over many years. We are also grateful to many colleagues and friends, especially Frank Stern, B. D. Choe, Han Jo Lim, H. S. Min, M. S. Kim, Robert Mann, Tim Ralph, K. S. Seo, Y. S. Cho, and Chancellor Sam Bum Lee. The support of our research by the Korean Ministry of Education, Science and Technology is greatly appreciated. This book would not have been completed without the patience and continued encouragement of our editors at Wiley and above all the encouragement and understanding of Taeyeon Yim and Young-Mee An. Thanks for putting up with us. Doyeol Ahn Seoung-Hwan Park

fpref.indd viii

4/20/2011 10:51:59 AM

PART I Fundamentals

c01.indd 1

4/20/2011 10:33:28 AM

This page intentionally left blank

1 1.1

Basic Quantum Mechanics MEASUREMENTS AND PROBABILITY

In the beginning of 20th century, it was discovered that the behavior of very small particles, such as electrons, the nuclei of atoms, and molecules, cannot be described by classical mechanics, which had been quite successful in explaining the macroscopic world until then. Nonetheless, it was soon discovered that the description of these phenomena on the atomic scale is possible by the set of laws described by quantum mechanics. Both classical mechanics and quantum mechanics are based on the description of measurements of observable quantities called dynamical variables, such as position, momentum, and energy. Consider an experiment in which we can make three measurements successive in time. Let’s denote the first of observable quantities A, the second B, and the third C. We also denote a, b, and c as one of a number of possible results that could come from the measurement of A, B, and C, respectively. Let P(b | a) be the conditional probability that if the measurement of A results in a, then the measurement of B will result in b. From the elementary probability theory, the conditional probability P(b | a) can be written as follows: P(b | a) =

P(a, b) , P(a)

(1.1)

where P(a, b) is the joint probability that measurements of both A and B will give a and b, simultaneously, and P(a) is the probability that the measurement of A will give the outcome a. For three successive measurements A, B, and C, the conditional probability P(cb | a) that if the measurement of A results in a, then the measurement of B will result in b, then the measurement of C will result in c is given by: P(cb | a) = P(c | b)P(b | a).

(1.2)

Engineering Quantum Mechanics, First Edition. Doyeol Ahn, Seoung-Hwan Park. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 3

c01.indd 3

4/20/2011 10:33:28 AM

4

BASIC QUANTUM MECHANICS

Moreover, if we sum Equation (1.2) over all the mutually exclusive alternatives for b, we obtain the conditional probability P(c | a): P(c | a) =

∑ P(c | b)P(b | a).

(1.3)

b

In classical mechanics, the above relation described by Equation (1.3) is always true. However, it was found that the above relation sometimes fails on the atomic scale, and one needs to modify Equations (1.1) to (1.3) by introducing new complex quantities ϕ ba, ϕ cb, and ϕ ca, called probability amplitudes, which are related to probabilities by [1,2] P(b | a) = ϕ ba , 2

(1.4)

and P(c | a) =

∑

2

ϕ cbϕ ba .

(1.5)

b

Equations (1.4) and (1.5) describe the probability of measurement outcome in quantum mechanics. From the mathematical point of view, the probability amplitude is found to be the inner product of vectors in a special kind of vector space called the Hilbert space: ϕ ba = b a ,

(1.6)

where a is the column vector, called the “ket vector,” corresponding to the observable a, and b is the row vector, called the “bra vector,” corresponding to the observable b.

1.2

DIRAC FORMULATION

In quantum mechanics, a physical state corresponding to the observable quantity a is represented by the ket vector, a , in a complex vector space H with dimension N. For example, when N = 2, the ket vector a is a column vector given by [1,3] ⎛ a1 ⎞ a =⎜ ⎟, ⎝ a2 ⎠

c01.indd 4

(1.7)

4/20/2011 10:33:29 AM

DIRAC FORMULATION

5

where a1 and a2 are complex numbers. We can also consider the case where the dimension of the vector space is infinite. In this case, the ket vector a is represented by a column vector given by ⎛ a1 ⎞ ⎜ a2 ⎟ ⎜ ⎟ ⎜ .⎟ a =⎜ ⎟, . ⎜ ⎟ ⎜ .⎟ ⎜⎝ a ⎟⎠ ∞

(1.8)

with complex quantities a1, a2, a3, … , a∞. From now on, we will denote ket vectors simply as vectors. If a and b are vectors in H, and α and β are complex numbers, then the linear superposition of these two vectors α a + β b is also a vector in H. For each vector a in H, we can relate a row vector a called bra vector, which is given by a = ( a1*

a2* ) , for N = 2,

(1.9)

and a = ( a1*

a2*

a3* . . . . a∞* ) , for N = ∞,

(1.10)

where * is the complex conjugate. By comparing Equations (1.7) with (1.10), one can see that the ket vector a and bra vector b are related by a =(a ) , †

(1.11)

where † is the adjoint operation, which is the transposition, followed by the complex conjugation. The inner product between two vectors a and b is a complex number b a , which is given by b a = b1*a1 + b2*a2 + … + b∞* a∞ = a b *.

(1.12)

The vector space where one can define the inner product relation of Equation (1.12) is called the Hilbert space. The length of the vector a is defined by

c01.indd 5

4/20/2011 10:33:29 AM

6

BASIC QUANTUM MECHANICS

a =

(1.13)

aa,

and when the vector has the unit length, we call it a normal vector. The vectors in the Hilbert space H are mapped to different vectors ˆ which is often in the Hilbert space H ′ by the linear transformation A, called the “linear operator” in quantum mechanics. Mathematically, the ˆ on the vectors of the Hilbert space H is act of the linear operator A ˆ expressed as A : H → H ′. In most cases, the initial Hilbert space H and the final Hilbert space H ′ are the same, and unless otherwise specified explicitly, we will assume that is the case. Any two vectors a and b ˆ as in H are transformed by A ˆ (α a + β b ) = αA ˆ a + βA ˆ b, A

(1.14)

and

( Aˆ α a )

†

ˆ †, = α* a A

(1.15)

ˆ a from Equation where α and β are complex constants. If we set c = A † † ˆ (1.11), we obtain c = ( c ) = a A , and then, using Equation (1.12), we get ˆ a*= bc*= cb = aA ˆ† b . bA

(1.16)

ˆ there exists particular set of vectors For a given operator A, { a1 , a2 , … , an , …} called eigenstates, which satisfy ˆ a1 = a1 a1 , A ˆ a2 = a2 a2 , … , A ˆ an = an an , … A

(1.17)

Here, the numbers a1, a2, … , an, … are called eigenvalues. Of the many kinds of operators, one particular type of operator, called the Hermitian ˆ is a operator, plays an important role in quantum mechanics. If A Hermitian operator, then it satisfies the following properties: ˆ† = A ˆ. A Eigenvalues a1, a2, … , an, … are real. Eigenvectors corresponding to different eigenvalues are orthogonal. (IV) an an = I , where I is an identity matrix. (I) (II) (III)

∑ n

ˆ ai = ai ai The proof is as follows: From A we have

c01.indd 6

ˆ aj = aj aj , and A

4/20/2011 10:33:29 AM

DIRAC FORMULATION

(

ˆ = aj A ˆ† = A ˆ aj aj A

)

†

= a*j a j .

7

(1.18)

Then, we get ˆ ai = ai a j ai , aj A

(1.19)

ˆ ai = a*j a j ai . aj A

(1.20)

and

Subtracting Equation (1.20) from Equation (1.19), we obtain (ai − a*j ) a j ai = 0.

(1.21)

When i ≠ j , we get a j ai = 0, which proves the property (III). On the other hand, if i = j , then we have ai = ai*, thus giving the property (II). The property (III) dictates that the set of eigenvectors { ai } of a ˆ form an orthogonal basis if the state ai is propHermitian operator A erly normalized—that is, ai ai = 1 and ai a j = 0 for i ≠ j. Let the Hilbert space spanned by these basis vectors be denoted as H. Then any vector ψ that belongs to H can be expressed as ψ =

∑C

am ,

m

(1.22)

m

where C m is an expansion coefficient that can be calculated by taking an inner product between the state vectors ψ and am : C m = am ψ .

(1.23)

By substituting Equation (1.23) into Equation (1.22), we obtain

∑C =∑ a

ψ =

am

m

m

m

am ψ (1.24)

m

⎛ =⎜ ⎝

∑a

m

m

⎞ am ⎟ ψ ⎠

=I ψ , which proves the property (IV). It would be handy to memorize that given a chain of vectors or operators, we can insert the identity operator

c01.indd 7

4/20/2011 10:33:30 AM

8

BASIC QUANTUM MECHANICS

defined by (IV) in any place at our convenience. For example, the ˆ can be written as Hermitian operator A ˆ =IA ˆI A ⎛ =⎜ ⎝

∑a =∑ a

n

n

n

⎞ ˆ⎛ an ⎟ A ⎠ ⎜⎝

∑a

m

m

⎞ am ⎟ ⎠

ˆ am am an A

n, m

=

∑a

n

(1.25)

(am an am ) am

n, m

=

∑a

n

an an ,

n

ˆ which is also called the spectral representation of an operator A. ˆ ˆ ˆ ˆ ˆ ˆ. Let A and B be linear operators, and we define D as D = AB ˆ ˆ ˆ Moreover, we define b = B a and d = D a = A b . Then, from Equations (1.11) and (1.15), we obtain ˆ† d = aD ˆ† = bA

(1.26)

ˆ †, = a Bˆ † A or ˆ ˆ) ( AB

†

1.3

ˆ †. = Bˆ † A

(1.27)

BRIEF DETOUR TO CLASSICAL MECHANICS

Almost seven decades ago, Dirac made a connection between classical mechanics and quantum mechanics by assuming that the linear operators correspond to the dynamical variables at that time. By “dynamical variables,” we mean quantities such as the coordinates and the components of velocity, momentum, and angular momentum of particles, and functions of these quantities, the variables in terms of which classical mechanics is built. Even now, 70 years later, his postulates of quantum mechanics are still valid and perhaps only plausible approaches. Dirac’s postulates require that those dynamical variables shall also occur in quantum mechanics, but with the difference that they are now subject to an algebra in which the commutative axiom of multiplication does

c01.indd 8

4/20/2011 10:33:30 AM

9

BRIEF DETOUR TO CLASSICAL MECHANICS

not hold. Nonetheless, the dynamical variables of quantum mechanics still have many properties in common with their classical counterparts, and it will still be possible to build up a theory of them closely analogous to the classical theory and form a generalization of it. In this spirit, the transition from classical mechanics to quantum mechanics can be made most conveniently and easily using the Hamiltonian formulation of classical mechanics [4,5]. Classical mechanics is based on the assumption that any physically interesting variable, that is, dynamical variable, can be measured with arbitrary precision and without mutual interference from any other such measurement. On the other hand, quantum mechanics is based on the realization that the measuring process may affect the physical system. The measurement of one variable affects other variables in such a way that it prevents us from knowing what their values might have been. The mathematical formulation of the law of physics that takes this basic idea into account is very different from the mathematical formulation of classical mechanics. Hamilton’s least action principle, which is equivalent to Newton’s law, is formulated as follows. The laws of physics are such that the time integral over a certain function L(qi , q i , t ), called Lagrangian of the physical system, assumes a minimum. For mechanical systems, the variables qi on which the Lagrangian depends on are the coordinates of all independent parts of the system. A system with f degrees of freedom has f coordinates q1, q2,, q f and the time integral J, which is defined by J=

∫

t2

t1

L(qi , q i , t )dt,

(1.28)

is minimum. In other words, when δqi = 0 at t = t1, t2 , we get δJ = = =

∫

t2

∫

t2

∫

t2

t1

t1

t1

δL(qi , q i ; t )dt ∂L ⎛ ∂L ⎞ ⎜⎝ ∂q δqi + ∂q δq i ⎟⎠ dt i i

(1.29)

⎛ ∂L d ∂L ⎞ ⎜⎝ ∂q − dt ∂q ⎟⎠ δqi dt i i

= 0. Equation (1.29) dictates that the Lagrangian satisfies the following Euler equation:

c01.indd 9

4/20/2011 10:33:30 AM

10

BASIC QUANTUM MECHANICS

∂L d ⎛ ∂L ⎞ − = 0, i = 1, 2, 3, … , f . ∂qi dt ⎜⎝ ∂q i ⎟⎠

(1.30)

The behavior of the physical system is thus completely specified by Euler’s equation once the Lagrangian is known. The proper Lagrangian is the one that leads to a description of the physical system that is in agreement with experimental observations. For example, if we choose L = T − V = 21 mx 2 − V ( x) for a onedimensional particle with mass m in the potential field V(x), we obtain ∂L ∂V ( x ) ∂L =− and = mx . ∂x ∂x ∂x

(1.31)

Then the Euler equation yields mx = −

∂V ( x ) = F ( x), ∂x

(1.32)

which is the famous Newton’s first law of mechanics. For later purpose, we make the following canonical transformation: pi ≡

∂L , H≡ ∂q i

∑ p q − L, i i

(1.33)

i

where H is called the Hamiltonian of the system. Then we obtain the following, Hamilton’s equation of motion: ∂H = q , ∂pi d ⎛ ∂L ⎞ ∂L ∂ ⎛ p i = ⎜ = = ⎜ ⎟ dt ⎝ ∂q i ⎠ ∂qi ∂qi ⎝

∑ j

⎞ ∂H pj q j − H ⎟ = − , ∂qi ⎠

(1.34)

or, q i =

∂H , ∂pi

p i = −

∂H . ∂qi

(1.35)

Here, the Hamiltonian H represents the total energy of the system. For any dynamical variable F that depends on canonical variables (qi , pi ) , the time derivative of F is given by

c01.indd 10

4/20/2011 10:33:30 AM

BRIEF DETOUR TO CLASSICAL MECHANICS

dF = dt =

⎧ ∂F

∂F ⎫

11

∂F

∑ ⎨⎩ ∂q q + ∂p ⎬⎭ + ∂t i

i

i

i

⎧ ∂F ∂H ∂F ∂H ⎫ ∂F − ⎬+ ∂pi ∂qi ⎭ ∂t i ∂pi

∑ ⎨⎩ ∂q i

= {F , H } +

(1.36)

∂F , ∂t

where the Poisson bracket {A, B} for dynamical variables A,B is defined by ⎧ ∂A ∂B ∂A ∂B ⎫ − ⎬ = − {B, A} . ⎩ ∂qi ∂pi ∂pi ∂qi ⎭

{A, B} = ∑ ⎨ i

(1.37)

Equations (1.36) and (1.37) imply that dH ∂H ∂H = {H , H } + = , dt ∂t ∂t

(1.38)

which says if H does not explicitly dependents on the time t, the Hamiltonian, the total energy of the system, is conserved. Among many interesting properties of the Poisson bracket, the following relation is especially useful for the latter purpose:

{qi, pj } = δ ij .

(1.39)

For a particle moving in a one-dimensional world specified by the coordinate x, the Lagrangian is given by L = (1 / 2 m) x 2 − V ( x), as before. Then the canonical transformation yields, ∂L = mx , ∂x p2 1 H = px − L = mx 2 + V ( x) = + V ( x). 2 2m p=

(1.40)

Equation (1.40) allows us to interpret the Hamiltonian H as the total energy of the system and p as the momentum. Hamilton’s equation of motion, Equations (1.34) and (1.35), gives x =

c01.indd 11

∂H p = , ∂p m

4/20/2011 10:33:30 AM

12

BASIC QUANTUM MECHANICS

p = −

∂H ∂V ( x ) =− , ∂x ∂x

Newton’s law. If we have a charged particle in an electromagnetic field, the situa tion is abit more complex. The electric field E and  the magnetic flux density B can be expressed by a vector potential A and a scalar potential φ as    ∂A E = −∇φ − , ∂t    B = ∇ × A.

(1.41)

The divergence of a magnetic flux density is zero, so it can be written as the curl of the vector, the well-known vector potential A. In static, the curl of the electric field is zero, and it can be written as the gradient of a scalar function. In the time-varying case, the electric field and the magnetic field are related by the following Maxwell’s equations:    ∂B ∇×E=− , ∂t   ∇ ⋅ B = 0,   ∇ ⋅ D = ρ,     ∂D ∇×H = J + , ∂t

(1.42)

      with D = εE and B = μH . Here, D is the electric fulx density, H is the  magnetic field, ρ is the charge density, J is the current density, ε is the permittivity, and μ is the permeability. We have used the international system of units (SI units) to write Maxwell’s equation. From substituting Equation (1.41) into (1.42), we find that   ⎛  ∂A ⎞ ∇ ×⎜E + = 0. ⎝ ∂t ⎟⎠ Then it is obvious that we have    ∂A = −∇φ. E+ ∂t

c01.indd 12

(1.43)

4/20/2011 10:33:30 AM

BRIEF DETOUR TO CLASSICAL MECHANICS

13

Equation (1.30) implies that the generalized force Qi is defined by Qi = −

∂V d ⎛ ∂V ⎞ + . ∂qi dt ⎜⎝ ∂q i ⎟⎠

(1.44)

The force acting on the charged particle, called the Lorentz force, is     F = e {E + r × B} . (1.45) Substituting Equation (1.41) into Equation (1.45), we obtain   ⎧    dA ⎫ F = e ⎨−∇ ( φ − r ⋅ A) − ⎬, dt ⎭ ⎩

(1.46)

where we have used the following relation:          r × (∇ × A) = ∇ ( r ⋅ A) − ( r ⋅ ∇) A. Equations (1.44) and (1.46) indicate that the generalized potential in the case of a charged particle moving in an electromagnetic field is given by   V = e ( φ − r ⋅ A) , (1.47) and the corresponding Lagrangian is L=

1  2   mr + er ⋅ A − eφ. 2

(1.48)

The generalized canonical momentum is then pi =

∂L = mx i + eAi , qi = xi , ∂q i

(1.49)

and the Hamiltonian is given by H=

∑ p q − L i i

i

=

2

∑ (mx + eA ) x − 2 mr 1

i

i

i

1  2 mr + eφ 2  2  p − eA) ( = + eφ. 2m

i

  − er ⋅ A + eφ (1.50)

=

c01.indd 13

4/20/2011 10:33:31 AM

14

BASIC QUANTUM MECHANICS

1.4 A ROAD TO QUANTUM MECHANICS In classical mechanics, we dealt with functions of the coordinates qi and momentum pi such as energy; we describe these quantities collectively as “observable.” The term “observable” describes any quantity accessible to the measurement processes. We assume that every physical observable is mathematically represented by a Hermitian operator, and every measurement of the physical observable will result in one of the eigenvalues of the corresponding Hermitian operator. The eigenvector is used to characterize the state of the physical system. Quantum mechanics assumes that any arbitrary state of the physical system is characterized by a state vector that is not necessarily an eigenvector of any particular Hermitian operator. After a measurement has been performed, the state vector collapses to one of the eigenvectors with an eigenvalue En. If we describe the state of the system by a ket vector Ψ , then we can represent this state vector by Ψ = ∑ n En En Ψ , where En is an eigenvector of a particular Hermitian operator corresponding to the measurement done on the system. How does Ψ relate to possible measurements when the result of a measurement must be one of the eigenvalues {En }? For this, we need the following postulate. 1.4.1

Postulate

The quantity Ψ H Ψ represents the average value of a series of measurements on an ensemble of systems that are all described by the state vector Ψ and is given by ΨH Ψ =

∑

En Ψ En, 2

(1.51)

n

where Pn = En Ψ is the probability of obtaining the value of En as a result of a measurement. Previously, we described that in quantum mechanics the dynamical variables of classical mechanics are replaced by corresponding Hermitian operators. In classical mechanics, the dynamics of an observable are described by the Poisson bracket. We would like to extend the Poisson bracket to describe the dynamics of a quantum mechanical operator. The Poisson bracket for the classical observables A, B is defined in Equation (1.37) as 2

⎧ ∂A ∂B ∂A ∂B ⎫ − ⎬ = − {B, A} . ⎩ ∂qi ∂pi ∂pi ∂qi ⎭

{A, B} = ∑ ⎨ i

c01.indd 14

(1.37)

4/20/2011 10:33:31 AM

A ROAD TO QUANTUM MECHANICS

15

It is straightforward to show the following properties:

{A + C , B} = {A, B} + {C , B} ,

(1.52a)

{A, B + C } = {A, B} + {A, C }, {AB, C } = {A, C } B + A {B, C },

(1.52b)

{A, BC } = {A, B}C + B {A, C }.

(1.53b)

(1.53a)

Let us assume that a quantum mechanical Poisson bracket is analogous to a classical one. So we assume that a quantum mechanical Poisson bracket satisfies all the conditions of Equations (1.37), (1.52), and (1.53). We shall denote the quantum mechanical Poisson bracket ˆ and Bˆ as ⎡ A ˆ ˆ for operators A ⎣ , B ⎤⎦. We use these conditions to determine the functional form of a quantum Poisson bracket by evaluating ˆ ˆ , CD ˆ ˆ ⎤ in two different ways from Equation (1.53a,b): ⎡⎣ AB ⎦ ˆ ˆ , CD ˆ ˆ ⎤ = ⎡A ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ⎡⎣ AB ⎦ ⎣ , CD⎤⎦ B + A ⎡⎣ B, CD⎤⎦ ˆ , Cˆ ⎤ D ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = ⎡⎣ A ⎦ + C ⎡⎣ A, D⎤⎦ B + A ⎡⎣ B, C ⎤⎦ D + C ⎡⎣ B, D⎤⎦ (1.54) ˆ , Cˆ ⎤ DB ˆ ˆ + Cˆ ⎡ A ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = ⎡⎣ A ⎦ ⎣ , D⎤⎦ B + A ⎡⎣ B, C ⎤⎦ D + AC ⎡⎣ B, D⎤⎦ ,

{

}

{

}

and ˆ ˆ , CD ˆ ˆ ⎤ = ⎡ AB ˆ ˆ , Cˆ ⎤ D ˆ ˆ ˆ ˆ ˆ ⎡⎣ AB ⎦ + C ⎡⎣ AB, D⎤⎦ ⎦ ⎣ ˆ + Cˆ ⎡ A ˆ ˆ ˆ ˆ ˆ ˆ ˆ , Cˆ ⎤ Bˆ + A ˆ ⎡ Bˆ , Cˆ ⎤ D = ⎡⎣ A ⎣ ⎦ ⎦ ⎣ , D⎤⎦ B + A ⎡⎣ B, D⎤⎦ (1.55) ˆ , Cˆ ⎤ BD ˆ ˆ +A ˆ ⎡ Bˆ , Cˆ ⎤ D ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ = ⎡⎣ A ⎦ ⎣ ⎦ + C ⎡⎣ A, D⎤⎦ B + CA ⎡⎣ B, D⎤⎦ .

{

}

{

}

Equating Equations (1.54) and (1.55), we obtain

(

) (

)

ˆ , Cˆ ⎤ BD ˆ ˆ − DB ˆ ˆ + AC ˆ ˆ − CA ˆ ˆ ⎡ Bˆ , D ˆ ⎤ = 0. − ⎡⎣ A ⎦ ⎣ ⎦

(1.56)

Since the above condition holds for arbitrary Hermitian operators ˆ , Bˆ , C, ˆ and D, ˆ we must have A

(

)

(

)

ˆ , Cˆ ⎤ = AC ˆ ˆ − CA ˆ ˆ and ⎡ Bˆ , D ˆ ⎤ = BD ˆ ˆ − DB ˆ ˆ . ⎡⎣ A ⎦ ⎣ ⎦

(1.57)

From now on, we shall call a quantum Poisson bracket a commutator. We further assume that a commutator or a quantum Poisson bracket is proportional to the corresponding classical Poisson bracket:

c01.indd 15

4/20/2011 10:33:31 AM

16

BASIC QUANTUM MECHANICS

ˆ , Bˆ ⎤ = i {A, B} , ⎡⎣ A ⎦

(1.58)

where  is Planck’s constant divided by 2π. This suggests that quantum mechanics is based on the assumption that the Poisson bracket assumes the same physical meaning and the same numerical values as in classical mechanics. In particular,

[qˆ i, pˆ j ] = i {qi, pj } = iδ ij ,

(1.59)

which is the fundamental quantum condition. The time evolution of a quantum mechanical operator can be obtained from the equation of motion for the classical observable: dF ∂F = + {F , H } , dt ∂t dFˆ ∂Fˆ 1 ˆ ˆ + ⎡ F , H ⎤⎦ , = dt ∂ t i ⎣

(1.36) (1.60)

where Hˆ is the energy operator corresponding to the classical Hamiltonian. Equation (1.60) is called the equation of motion in the Heisenberg picture. This particular quantum mechanical representation assumes that the operators vary with time, while the state vector Ψ is time independent. This picture is formally analogous to classical mechanics, since the equations of motion for the operators closely resemble the corresponding classical equations.  Now, let us study a different picture, called the Schrodinger picture, where the operators are constant in time, while the time variation is expressed by the state vectors. In this picture, the explicit time dependence of an operator still remains. Different quantum mechanical pictures follow from each other by a unitary transformation. From now ˆ simply as A when we on, we denote a quantum mechanical operator A don’t need to distinguish it from the corresponding classical observable. Let us denote the state vector and operator in the new picture by Ψ ′ and A′, respectively. Then, they are related to Ψ and A of an old picture by Ψ ′ = U Ψ , A′ = UAU †

(1.61)

Ψ ′ A′ Ψ ′ = Ψ U †UAU †U Ψ = Ψ A Ψ ,

(1.62)

and

c01.indd 16

4/20/2011 10:33:31 AM

A ROAD TO QUANTUM MECHANICS

17

since U †U = UU † = I ,

(1.63)

when U is a unitary operator. Equation (1.62) indicates that the unitary transformation does not change the physical content of the theory. Also, we have dA′ dU dA † dU † AU † + U U + UA = dt dt dt dt A 1 dU † dU ∂ AU † + U U † + U [ A, H ]U † + UA . = dt ∂t i dt

(1.64)

If A is not explicitly dependent on time, ∂A / ∂t = 0, and the operator  Ain the Schrodinger picture is time independent except for an explicit time dependence. This implies that dA′ ∂A † −U U = 0. dt ∂t

(1.65)

From Equations (1.64) and (1.65), we get dU 1 dU † AU † + U [ A, H ]U † + UA dt i dt † dU dU 1 = AU † + UA + U ( AH − HA)U dt dt i † dU dU † 1 + ( A′ H ′ − H ′A′ ) = U A′ + A′ U dt dt i dU 1 ⎛ dU † 1 ⎞ =⎛ U † − H ′ ⎞ A′ + A′ ⎜ U + H ′⎟ ⎝ dt ⎝ i ⎠ dt i ⎠

(1.66)

= 0. Equation (1.66) should hold for any quantum mechanical operators A′  in the Schrodinger picture, and, as a consequence, we have i

c01.indd 17

dU = H ′U . dt

(1.67)

4/20/2011 10:33:31 AM

18

BASIC QUANTUM MECHANICS

 Since Ψ ′ = U Ψ , we obtain the following Schrodinger equation for the state vector i

d Ψ′ = H ′ Ψ′ . dt

(1.68)

 The Schrodinger picture is perhaps the most widely used because it leads directly to the formulation of wave mechanics. In this picture, eigenvectors of the position operator q are used to represent the state vector of the system. We consider the continuous set of eigenvalues q′ of the position operator q given by q q′ = q′ q .

(1.69)

We assume that corresponding eigenvectors { q′ } form a complete set. For simplicity, we consider the one-dimensional case only, but the extension to the higher dimensional case is straightforward. The position representation of the state vector is defined by Ψ(q′) = q′ Ψ .

(1.70)

From Equation (1.59), the canonical momentum operator p and the position operator q satisfy the following quantum Poisson’s bracket or the commutator relation

[q, p] = (qp − pq) = i,

(1.71)

(qp − pq) Ψ = i Ψ .

(1.72)

and

If we take an inner product of Equation (1.72) by the bra vector q′ , we obtain q′ (qp − pq) Ψ = q′ q′ p Ψ − q′ pq Ψ .

(1.73)

q′ (qp − pq) Ψ = i q′ Ψ .

(1.74)

Also,

Equations (1.73) and (1.74) are consistent only if the following condition is satisfied

c01.indd 18

4/20/2011 10:33:31 AM

A ROAD TO QUANTUM MECHANICS

q ′ p Ψ = − i

19

∂ ∂ q ′ Ψ = − i Ψ(q′), ∂q ′ ∂q ′

(1.75)

∂ (q′Ψ(q′)) . ∂q ′

(1.76)

since Equation (1.75) implies q′ pq Ψ = − i

Moreover, we have the following relations: q′ p q′′ = − i

∂ ∂ q′ q′′ = − i δ(q′ − q′′), ∂q ′ ∂q ′

(1.77)

and I=

∫

∞

−∞

dq q q .

(1.78)

Then, we get

∫ =∫

ΦΨ =

∞

−∞ ∞

−∞

Φ pΨ =

∫

∞

=

∫

∞

∫

∞

= = =

−∞

−∞

−∞

∞

dq′

∫

dq′

∫

∞

∫

∞

dq′

−∞

dq Φ q q Ψ (1.79) dq Φ*(q)Ψ(q),

dq′′ Φ q′ q ′ p q ′′ q′′ Ψ

∂ ⎛ ⎞ dq′′ Φ*(q ′) ⎜ −i δ((q′ − q′′)⎟ Ψ(q′′) ⎝ ⎠ −∞ ∂q ′ −∞

∂ ⎛ ⎞ dq′′ ⎜ i Φ*(q ′)⎟ δ(q ′ − q ′′)Ψ(q ′′) ⎝ ∂q ′ ⎠

(1.80)

∂ ⎛ ⎞ dq ′ ⎜ i Φ*(q ′)⎟ Ψ(q ′′) ⎝ ∂q ′ ⎠ −∞

∫

∞

∂ ⎞ ⎛ Ψ(q′′), dq ′ Φ*(q ′) ⎜ i ⎝ ∂q′ ⎟⎠ −∞

∫

∞

and ∂ ⎞ ⎛ q′ F (q, p) Ψ = F ⎜ q′, − i Ψ(q′). ⎝ ∂q′ ⎟⎠

c01.indd 19

(1.81)

4/20/2011 10:33:32 AM

20

BASIC QUANTUM MECHANICS

As a special case, we consider F = H , where H is the Hamiltonian of  the system. Then the Schrodinger equation (1.68) becomes i

∂ Ψ(q′, t ) = H (q′, − i∂ / ∂q′)Ψ(q′, t ), ∂t

(1.82)

of the wave mechanics. As an example, we consider a particle moving in a one-dimensional potential V ( x). If we set the position operator as q′ = x, the classical Hamiltonian is given by Equation (1.40): H=

p2 + V ( x). 2m

 Then, the corresponding Schrodinger equation is given by i

∂ ⎛ 2 ∂2 ⎞ Ψ( x, t ) = ⎜ − + V ( x)⎟ Ψ( x, t ). 2 ⎝ 2 m ∂x ⎠ ∂t

(1.83)

  For three dimensions, we simply replace q′ with r and ∂ / ∂q ′ with ∇.  For example, the Schrodinger equation for a charged particle moving in an electromagnetic field is i

   2  1 ∂    Ψ(r , t ) = − i∇ − eA(r , t )) Ψ(r , t ) + eφ(r )Ψ(r , t ). ( ∂t 2m

(1.84)

When the Hamiltonian H is time independent, the time-dependent  Schrodinger equation can be rewritten in time-independent form by trying the solution of the form   Ψ(r , t ) = exp(− iEt / )ψ E (r )

(1.85)

into Equation (1.83). Then, we obtain the following time-independent  Schrodinger equation −

2  2     ∇ ψ E (r ) + V (r )ψ E (r ) = Eψ E (r ), 2m

(1.86)

where E is the energy of the particle. One can also consider the eigenvectors of the momentum operator p , which are given by

c01.indd 20

4/20/2011 10:33:32 AM

THE UNCERTAINTY PRINCIPLE

p p′ = p′ p′ .

21

(1.87)

From q ′ p p ′ = − i

∂ q ′ p′ = p′ q ′ p′ , ∂q ′

(1.88)

we get q′ p′ = C exp (iq′ p′ /  ) ,

(1.89)

where C is a normalization constant. If we assume that p′ p′′ = δ( p′ − p′′), then it is straightforward to calculate the normalization constant, which is given by 1 2π. As a consequence, we have x p =

1 exp(ixp / ), 2π

(1.90)

and 1 ⎞   r p =⎛ ⎝ 2 π ⎠

3/ 2

  exp (ir ⋅ p /  ) .

(1.91)

1.5 THE UNCERTAINTY PRINCIPLE The most remarkable property of quantum mechanics is that the measurements of different dynamical variables are not necessarily independent. This property was first formulated by Heisenberg, and is known as the uncertainty principle. We start with a quantum Poisson bracket for the position and momentum operators q and p,

[q, p] = i. Let Ψ be a state vector, and q and p be the average values of the measurements of q and p, respectively: q= Ψ q Ψ , p= Ψ p Ψ . We now calculate the mean-square deviations of measured values,

(Δq)2 = Ψ (q − q )2 Ψ 2 = Ψ q2 Ψ − ( Ψ q Ψ ) ,

c01.indd 21

(1.92)

4/20/2011 10:33:32 AM

22

BASIC QUANTUM MECHANICS

and

(Δp)2 = Ψ ( p − p)2 Ψ 2 = Ψ p2 Ψ − ( Ψ p Ψ ) .

(1.93)

We also define α = q − q and β = p − p, and we get

[α, β] = [q, p] = i. Also,

(Δα )2 = (Δq)2 and (Δβ)2 = (Δp)2.

(1.94)

From Equation (1.94), we get

(Δq)2 (Δp)2 = (Δα )2 (Δβ)2 = Ψ α 2 Ψ Ψ β2 Ψ

(1.95)

≥ Ψ αβ Ψ , 2

from the Schwarz inequality. Since αβ = 1 / 2 (αβ − βα ) + 1 / 2 (αβ + βα ) , we also obtain Ψ αβ Ψ

2

1 Ψ [α, β ] Ψ + Ψ (αβ + βα ) Ψ 4 2 1 2 ≥ Ψ [α , β ] Ψ = . 4 4

=

2

(1.96)

Equations (1.95) and (1.96) give Heisenberg’s famous uncertainty relation:  ΔqΔp ≥ . 2

(1.97)

1.6 THE HARMONIC OSCILLATOR The one-dimensional simple harmonic oscillator is perhaps the most important and useful problem in quantum mechanics. The Lagrangian for the harmonic oscillator is L=

c01.indd 22

1 1 mx 2 − kx 2. 2 2

(1.98)

4/20/2011 10:33:32 AM

THE HARMONIC OSCILLATOR

23

The canonical momentum is p=

∂L = mx , ∂x

(1.99)

and the Hamiltonian becomes H = px − L =

p2 1 2 + kx . 2m 2

(1.100)

 equation for the harmonic oscilThe time-independent Schrodinger lator is then given by −

2 ∂2 ψ 1 2 + kx ψ = Eψ, 2 m ∂x 2 2

(1.101)

with the boundary condition ψ = 0 at x = ±∞. The solution of Equation (1.101) gives the energy eigenvalues of the harmonic oscillator and the eigenfunctions in terms of Hermite polynomials. In this section, we introduce an operator method that is based on the algebraic approach,  and is much simpler than solving the Schrodinger directly. We introduce new operators 1 (mωx + ip) , 2 mω 1 a† = (mωx + ip) , 2 mω k . ω= m a=

(1.102)

From aa† =

1 (mωx + ip)(mωx − ip), 2 mω

a† a =

1 (mωx − ip)(mωx + ip), 2 mω

we obtain i i aa† − a† a = ( px − xp) = (−i) = 1.  

c01.indd 23

(1.103)

4/20/2011 10:33:32 AM

24

BASIC QUANTUM MECHANICS

We also have  (a + a† ) , 2 mω mω † p=i ( a − a) . 2 x=

(1.104)

Substituting Equation (1.104) into Equation (1.100), and using Equation (1.103), we get p2 1 + mω 2 x 2 2m 2 1  (−1) ⎛ mω ⎞ † = (a − a) (a† − a) + mω 2 (a + a† ) (a + a† ) ⎝ ⎠ 2m 2 2 2 mω (1.105) ω † † = (aa + a a) 2 1 = ω ⎛ a† a + ⎞ . ⎝ 2⎠

H=

Let’s assume that the state is in a particular eigenstate E ′ with eigenvalue E ′, H E′ = E′ E′ .

(1.106)

Now, aH E ′ = aE ′ E ′ , 1⎞ ⎛ with aH = aω a† a + . From [a, a† ] = 1 , we get aa† a = a† aa + a, and ⎝ 2⎠ aH E ′ = Ha E ′ + ωa E ′ = E ′ E ′ .

(1.107)

Equation (1.107) can be rewritten as Ha E ′ = (E ′ − ω) E ′ .

(1.108)

In other words, if E ′ is an eigenstate of a harmonic oscillator Hamiltonian H with an eigenvalue E ′, a E′ is also an eigenstate of H but with different eigenvalue E ′ − ω.

c01.indd 24

4/20/2011 10:33:33 AM

THE HARMONIC OSCILLATOR

25

By the way, E H E = ω E a† a E +

1 ω E E 2

(1.109)

=E E E . Since

∫ = ∫ dq

E E = dq E q q E Eq

2

≥ 0,

we have

∫

E a† a E = dq q a E

2

≥ 0.

(1.110)

Combining Equations (1.109) and (1.110), the energy eigenvalue E must be a nonnegative number. On the other hand, Equation (1.108) indicates that the operation of an operator a on the energy eigenstate E lowers the energy eigenvalue by an amount ω. Nonetheless, the energy eigenvalue must be always nonnegative, so the reduction of the eigenvalue must be terminated at the ground state E0 , that is, a E0 = 0, and as a result, we obtain E0 =

1 ω. 2

(1.111)

Likewise, a† H E ′ = E ′a† E ′ , and after some mathematical manipulations, we get Ha† E ′ = ( E ′ + ω ) a† E ′ .

(1.112)

This indicates that a† E ′ is also an eigenstate of H, but with a new eigenvalue E ′ + ω. We can see that the operator a† raises the eigenvalue by the amount of ω. So if En is the nth eigenstate, one can generate En from E0 by n successive applications of a†,

c01.indd 25

4/20/2011 10:33:33 AM

26

BASIC QUANTUM MECHANICS

En = An ( a† ) E0 , n

(1.113)

where An is the normalization factor. The normalization factor An is determined from the condition En En = 1 with the relations a ( a† ) = aa† ( a† ) n

n−1

= ( a† )

n−1

+ a† a (a† )

= ( a† )

n−1

+ a† ⎡⎣( a† )

= 2 ( a† )

n−1

n−1

n− 2

+ a† a ( a† )

+ ( a† ) a ( a† ) 2

n− 2

⎤ ⎦

(1.114)

n− 2

= n ( a† ) + ( a† ) a n

and an ( a† ) = nan − 1 (a† )n − 1 + an − 1 (a† )n a. n

Then we obtain En En = An2 E0 an ( a† ) E0 n

= nAn2 E0 an − 1 ( a† )

n−1

E0

= n! A E0 E0 2 n

= n! An2 = 1. The normalization factor is An =

1 n!

and the normalized eigenstate is En

c01.indd 26

n a† ) ( =

n!

E0 .

(1.115)

4/20/2011 10:33:33 AM

THE HARMONIC OSCILLATOR

27

From Equations (1.114) and (1.115), we get a En = =

a (a† )

n

n! n ( a† )

E0 n−1

n! = n En − 1

E0

(1.116)

and (a† )n + 1 E0 n! = n + 1 En + 1 .

a† En =

(1.117)

We call the operator a an annihilation operator because it destroys a single quantum of energy. The operator a† is called a creation operator since it raises (or creates) a quantum of energy. The operator a† a is called the number operator because if operating on the state, one can see that its eigenvalue is the number of energy quantum n, a† a En = a† n En −1 = n En .

(1.118)

The operator method can also be used to determine the wave function of the one-dimensional harmonic oscillator in the coordinate space. Since a E0 = 0, from Equation (1.102), we have mω 1 x a E0 = x ⎛ x+i p⎞ E0 ⎝ 2 mω 2 mω ⎠ = = 0,

mω  ∂ x x E0 + x E0 2 2 mω ∂x

(1.119)

where Equations (1.69) and (1.75) are used. If we denote ψ 0 ( x) = x E0 , then Equation (1.119) can be rewritten as d mωx ψ 0 ( x ) = 0. ψ 0 ( x) +  dx

(1.120)

Solving the first-order differential equation is straightforward, and the solution is given by

c01.indd 27

4/20/2011 10:33:33 AM

28

BASIC QUANTUM MECHANICS

mω 2 ⎞ ψ 0 ( x) = A exp ⎛ − x , ⎝ 2 ⎠

(1.121)

where A is the normalization constant given by mω ⎞ A=⎛ ⎝ π ⎠

1/ 4

.

(1.122)

In order to determine the normalization constant, we have used the condition

∫

∞

−∞

ψ 0 ( x) dx = 1, 2

(1.123)

and the Gaussian integral

∫

∞

−∞

exp(−αx 2 )dx =

π , α > 0. α

(1.124)

The excited state wave functions can be obtained by successive application of a creation operator a† on the state E0 and taking an inner product with the bra vector x . For example, the first excited state wave function is ψ 1 ( x ) = x a† E0 ⎛ mω p ⎞ x−i E0 = x⎜ ⎝ 2 2 mω ⎟⎠ mω  ∂ xψ 0 ( x ) − ψ 0 (x) 2 2 mω ∂x 1/ 4 2 mω ⎛ mω ⎞ ⎛ mωx 2 ⎞ = exp ⎜ − , x ⎝ π ⎠ ⎝ 2  ⎟⎠  =

(1.125)

and the nth state wave function is given by ψ n ( x) =

⎛ mω ⎞ ⎛ mω ⎞ 1 / 4 ⎛ mωx 2 ⎞ Hn ⎜ x⎟ exp ⎜ − . ⎝ 2  ⎟⎠ ⎝  ⎠ ⎝ π ⎠ 2 n n! 1

(1.126)

Here, H n (ζ) is the Hermite polynomials, and the first few Hermite polynomials are H 0 (ζ) = 1, H1 (ζ) = 2ζ, H 2 (ζ) = 4ζ 2 − 2, 

c01.indd 28

(1.127)

4/20/2011 10:33:33 AM

ANGULAR MOMENTUM EIGENSTATES

29

1.7 ANGULAR MOMENTUM EIGENSTATES    If the potential term V (r ) in the time-independent Schrodinger equa  tion has a spherical symmetry, then the angular momentum L = r × p is conserved. We need the following mathematical relations to prove the previous statement. 











(r × ∇) ⋅ (r × ∇) = {(r × ∇) × r} ⋅ ∇ − r ⋅ {(r × ∇) × ∇}               = {−r (∇ ⋅ r ) + ∇ (r ⋅ r )} ⋅ ∇ − r ⋅ {− r (∇ ⋅ ∇) + ∇ ( r ⋅ ∇)}        = −3r ⋅ ∇ + ∇ ( r ) ⋅ ∇ + r ∇ − ( r ⋅ ∇) ( r ⋅ ∇) 2

2

2

(1.128)

∂ ∂ ∂ + r 2∇2 − r ⎛ r ⎞ ⎝ ∂r ∂r ∂r ⎠ ∂ ∂2 = − r 2 2 − 2r + r 2 ∇ 2. ∂r ∂r

= −r

Here, we used the vector identity 





  

  

( A × B) × C = − A ( B ⋅ C ) + B (C ⋅ A)

(1.129)

∂   r ⋅∇ = r . ∂r

(1.130)

and

 If we define the orbital angular momentum operator L as    L = −ir × ∇,

(1.131)

then, from Equation (1.128), we obtain   ∂2 2 ∂ L⋅ L ∇ = 2+ − . ∂r r ∂r 2  2 r 2 2

(1.132)

 The time-independent Schrodinger equation (1.86) becomes   2 ⎛ ∂2 2 ∂ L ⋅ L⎞    − + − 2 2 ⎟ ψ E (r ) + V (r )ψ E (r ) = Eψ E (r ),, (1.133) ⎜ 2 2 2 m ⎝ ∂r r ∂r r ⎠

c01.indd 29

4/20/2011 10:33:33 AM

30

BASIC QUANTUM MECHANICS

and the Hamiltonian with spherically symmetric potential is given by   2 ⎛ ∂2 2 ∂ L ⋅ L⎞ H=− + − + V (r ) . . . (1.134) 2 m ⎜⎝ ∂r 2 r ∂r 2  2 r 2 ⎟⎠ The angular momentum operator can be written, component-wise, as ∂⎞ ⎛ ∂   Lx = ( − ir × ∇) x = − i ⎜ y − z ⎟ , ⎝ ∂z ∂y ⎠ ∂ ∂   Ly = ( − ir × ∇ ) y = −i ⎛ z − x ⎞ , ⎝ ∂x ∂z ⎠ ∂⎞ ⎛ ∂   Lz = ( − ir × ∇)z = − i ⎜ x − y ⎟ , ⎝ ∂y ∂z ⎠

(1.135)

and they satisfy the following commutation relations [Ly, Lz ] = iLx, [Lz, Lx ] = iLy,

(1.136)

[Lx , Ly ] = iLz. Now consider the infinitesimal rotation about the z-axis by an angle θ. The rotated coordinates are given by x ′ = x cos θ + y sin θ ≈ x + θy, y′ = − x sin θ + y cos θ ≈ −θx + y, z′ = z.

(1.137)

Since the Hamiltonian is invariant under the rotation, we have Hψ E ( x, y, z) = Eψ E ( x, y, z),

(1.138)

Hψ E ( x ′, y′, z′) = Eψ E ( x ′, y′, z′).

(1.139)

and

Also, we have ∂⎞ ⎛ ∂ ψ E ( x + θy, −θx + y, z) = ψ E ( x, y, z) + θ ⎜ y − x ⎟ ψ E ( x, y, z) ⎝ ∂x ∂y ⎠ iθ = ψ E ( x, y, z) − Lz ψ E ( x, y, z). 

c01.indd 30

(1.140)

4/20/2011 10:33:34 AM

ANGULAR MOMENTUM EIGENSTATES

31

From Equations (1.138) to (1.140), we obtain HLz ψ E = Lz Hψ E , or [ H , Lz ] = 0.

(1.141)

In other words, the operators H and Lz share the same set of eigenfunctions. It is straightforward to show that [Lz, L2 ] = 0,

(1.142)

which implies that we can choose a common set of eigenfunctions for the operators H , Lz, L2. Let’s assume that the wave function ψ E ( x, y, z) is given by ψ E ( x, y, z) = RE (r )Yλ (θ, φ),

(1.143)

where (r, θ, φ) is a set of spherical coordinates, and Yλ (θ, φ) is a simultaneous eigenfunction of Lz and L2 such that LzYlm (θ, φ) = mYlm (θ, φ), L2Ylm (θ, φ) = l(l + 1) 2Ylm (θ, φ), Ylm (θ, φ) = θ, φ lm .

(1.144)

 By using the definition of the differential operator ∇  ∂ ∂ ∂ ∇ = xˆ + yˆ + zˆ ∂x ∂y ∂z ∂ ∂ ∂ , = rˆ + θˆ + φˆ ∂r r∂ θ r sin θ∂φ

(1.145)

we get  ∂ = xˆ ⋅ ∇ ∂x

∂ ∂ sin φ ∂ + cos θ cos φ − , ∂r r∂θ r sin θ ∂φ

(1.146)

∂ ∂ cos φ ∂ = sin θ sin φ + cos θ sin φ + , ∂r r∂θ r sin θ ∂φ

(1.147)

= sin θ cos φ  ∂ = yˆ ⋅ ∇ ∂y

c01.indd 31

4/20/2011 10:33:34 AM

32

BASIC QUANTUM MECHANICS

and  ∂ = zˆ ⋅ ∇ ∂z ∂ ∂ = cos θ − sin θ . ∂r r∂ θ

(1.148)

Then, by combining Equations (1.135) to (1.148), we obtain ∂ ∂⎤ ⎡ Lx = i ⎢cot θ cos φ + sin φ ⎥ , ∂φ ∂θ ⎦ ⎣ ∂ ∂ ⎤ ⎡ Ly = i ⎢cot θ sin φ − cos φ ⎥ , ∂φ ∂θ ⎦ ⎣ ∂ Lz = −i . ∂φ

(1.149)

We also define the ladder operators L± as L± = Lx ± Ly ∂⎞ ⎛ ∂ =  exp(± iφ) ⎜ ± + i cot θ ⎟ . ⎝ ∂θ ∂φ ⎠

(1.150)

Then, we get the following relations among operators: [Lz, L± ] = ± L± , [L+ , L− ] = 2 Lz, L+ L− = L2 − L2z + Lz,

(1.151)

L− L+ = L − L − Lz, 2

2 z

(L± )† = L∓ . This implies that lm L+ L− lm = lm L2 − L2z + Lz lm

=  2 (l(l + 1) − m2 + m) ≥ 0,

and as a result − l ≤ m ≤ l + 1.

c01.indd 32

(1.152)

4/20/2011 10:33:34 AM

ANGULAR MOMENTUM EIGENSTATES

33

Similarly, we also have − l ≤ m ≤ l,

(1.153)

lm L− L+ lm ≥ 0.

(1.154)

lm L2 lm =  2 l(l + 1) ≥ 0.

(1.155)

from

Also, we have

Next, consider L± lm . From [Lz, L± ] = ±L±, we get Lz L± lm = ([Lz , L± ] + L± Lz ) lm = ( L± Lz ± L± ) lm

(1.156)

= (m ± 1) lm . Thus, L± lm are eigenstates of Lz with eigenvalues m ± 1: L± lm = c± lm ± 1 .

(1.157)

For this reason, we call L± the ladder operators, in particular, the angular momentum-raising and -lowering operators. The normalization constants c± are evaluated with the aid of Equation (1.151): c± = lm L∓ L± lm 2

=  2 (l(l + 1) − m(m ± 1)) .

(1.158)

If we take c± as positive, then we have the following expressions: c± =  l(l + 1) − m(m ± 1).

(1.159)

In order to find the eigenfunction of the orbital angular momentum, we start with θ, φ Lz lm = − i

∂ θ, φ lm = m θ, φ lm , ∂φ

(1.160)

which gives θ, φ lm = e imφ θ, 0 lm .

c01.indd 33

(1.161)

4/20/2011 10:33:34 AM

34

BASIC QUANTUM MECHANICS

By the way, Equations (1.157) and (1.159) imply that L+ ll = 0,

(1.162)

⎛ ∂ − l cot θ⎞ θ, φ lm = 0. ⎝ ∂θ ⎠

(1.163)

Or, from Equation (1.150),

Since the function (sinθ ) satisfies Equation (1.163), the orbital angular momentum eigenfunction has the following functional form: l

Yll (θ, φ) = θ, φ ll = αe ilφ (sin θ ) . l

The normalization constant α is evaluated from the condition 1=

∫

2π

0

dφ

= 2π α

2

= 2π α

2

1

∫ d(cos θ) α (sin θ) ∫ du (1 − u ) 2

2l

−1

1

2 l

(1.164)

−1

2 2 l + 1 (ll !)2 , (2l )!(2l + 1)

which gives Yll (θ, φ) =

(−1)l 2l l !

(2l + 1)(2l )! ilφ l e (sin θ) . 4π

(1.165)

Here, (−1)l is the conventional phase factor. In order to get Ylm (θ, φ), we apply L− to Equation (1.165) (l − m) times and from Equation (1.157) we obtain Yim (θ, φ) =

(−1)l 2l l !

(2l + 1)(l + m)! ⎛ d ⎞ ⎜ 4 π(l − m)! ⎝ d(cos θ) ⎟⎠

l−m

(sin θ)l . (1.166)

In mathematical physics, the function Ylm (θ, φ) is called as spherical harmonics.

c01.indd 34

4/20/2011 10:33:34 AM

QUANTIZATION OF ELECTROMAGNETIC FIELDS

1.8

35

QUANTIZATION OF ELECTROMAGNETIC FIELDS

There are cases where one needs to treat the electromagnetic field quantum mechanically, such as the spontaneous emission, the Lamb shift, the resonance florescence, and nonclassical light, such as squeezed states. Also, the fluctuation intensity of the laser near the threshold needs to be treated by the quantized electromagnetic fields. The classical electromagnetic fields are governed by Maxwell’s equations (1.42). In most treatment of the quantized electromagnetic fields, unbound regions are considered, and vector potentials are used. In many applications, we are primarily considering the interaction between the laser radiation with electrons in atoms or quantum structures, such as the semiconductor quantum dot within the electric-dipole approximation. In such a case, it is convenient to develop the theory in a gaugeinvariant form more appropriate for quantum electronics, emphasizing the electric and magnetic fields;    ∂D , ∇×H = ∂t    ∂B , ∇×E=− (1.167) ∂t   ∇ ⋅ B = 0,   ∇ ⋅ E = 0,     with B = μ 0 H and D = ε 0 E. We also take the electric field to have the spatial dependence appropriate for the cavity resonator with propagation axis in the z-direction such that [6] ⎛ 2Ω 2 M ⎞ Ex (z, t ) = q(t ) ⎜ ⎝ Vε 0 ⎟⎠

1/ 2

sin Kz

(1.168)

where V is the cavity volume, K = Ω / c is the wave number, M is the constant with the dimension of mass, and q(t ) is the quantum mechanical variable with the dimension of length. Then, from Maxwell’s equations (1.167),   ∂H y ∇ × H = − xˆ ∂z ⎛ 2Ω 2 M ⎞ = xˆ ε 0 q (t ) ⎜ ⎝ Vε 0 ⎟⎠

c01.indd 35

(1.169)

1/ 2

sin Kz,

4/20/2011 10:33:35 AM

36

BASIC QUANTUM MECHANICS

we obtain the magnetic field H y (z, t ) = q (t )

ε 0 ⎛ 2Ω 2 M ⎞ K ⎜⎝ Vε 0 ⎟⎠

1/ 2

cos Kz.

(1.170)

The total electromagnetic energy contained in the cavity is then 1 2 1 = 2

H=

∫

V

dv {ε 0 Ex2 + μ 0 H y2 }

2Ω 2 M 2M ⎧ ⎫ dv ⎨q2 (t ) sin 2 Kz + q 2 (t ) cos2 Kz⎬ V V V ⎩ ⎭ 2 p (t ) 1 = + MΩ 2 q2 (t ), 2M 2

∫

(1.171)

where we have defined the generalized momentum p(t ) = Mq (t ). By comparing Equation (1.105), we see that the total electromagnetic energy inside the cavity is the same as that of the simple harmonic oscillator with mass M and angular frequency Ω. We can now define the creation and annihilation operator for a single mode of electromagnetic field quanta denoted as a photon by 1 ( MΩq (t ) + ip (t )) , 2 MΩ 1 a† (t ) = ( MΩq (t ) − ip (t )) . 2 MΩ a (t ) =

(1.171)

The quantization rule for the photon is then

[q, p] = i, [a, a† ] = 1.

(1.172)

a (t ) = a (0 ) exp ( − iΩt ) , a† (t ) = a (0 ) exp ( − iΩt ) .

(1.173)

We also have

Then, the quantized electric field can be written as Ex (z, t ) = ( a + a† ) Eo sin Kz,

(1.174)

where Eo is the electric field per photon. The total electromagnetic energy contained in the cavity is

c01.indd 36

4/20/2011 10:33:35 AM

QUANTIZATION OF ELECTROMAGNETIC FIELDS

H = Ω(a a + 1 / 2),

37

(1.175)

and the n-photon state in which n photons is defined as n , where each photon has an energy Ω. We interpret the vacuum state 0 as a state which contains no photon. Photons are quanta of a single mode of the electromagnetic fields, and are not localized at any particular position within the cavity. Rather, they are spread out over the entire cavity. The quantized electromagnetic fields seem to offer amazingly satisfying accounts of a very wide range of radiative phenomena, and there is no real need to have a corpuscular theory of photons. When there is more than one mode of electromagnetic fields in the cavity, the electromagnetic fields can be expanded by multi-modes  E(z, t ) = x

∑ ∑

ε ⎛ 2Ω 2n Mn ⎞ q n (t ) 0 ⎜ ⎟ K n ⎝ Vε 0 ⎠

n

 H (z, t ) = y

1/ 2

⎛ 2Ω 2n Mn ⎞ qn (t ) ⎜ ⎝ Vε 0 ⎟⎠

n

sin K n z, (1.176)

1/ 2

cos K n z,

with Ω n = nπc / L, K n = nπ / L, and L is the length of the cavity. The total electromagnetic energy contained in the cavity is H=

1 2

⎛

∑ ⎜⎝ M Ω q n

2 2 n n

n

+

pn2 ⎞ , Mn ⎟⎠

(1.177)

and [qn, pn′ ] = iδ n,n′ , [qn, qn′ ] = [ pn, pn′ ] = 0.

(1.178)

The annihilation and creation operators for the mode n are

( Mn Ω n qn (t ) + ipn (t )) , −1 / 2 an† (t ) = ( 2 Mn Ω n ) ( Mn Ω n qn (t ) − ipn (t )) , an (t ) = ( 2 Mn Ω n )

−1 / 2

(1.179)

and a quantized electric field is given by Ex (z, t ) =

∑ (a

n

+ an† ) En sin K n z.

(1.180)

n

For a mode s, we have as† as ns = Ω s ns .

c01.indd 37

4/20/2011 10:33:35 AM

38

BASIC QUANTUM MECHANICS

The general eigenstate of H has n1 photons in the first mode, n2 in the second, ns in the sth, and so forth, and can be written as n1n2 … ns … ≡ {ns } = n1 n2 … ns …

(1.181)

The annihilation operator as destroys a photon in the ns mode alone: as n1 n2 … ns … = ns n1 n2 … ns − 1… .

(1.182)

The general state vector for the field is a superposition of these eigenstates: ψ =

∑

C n1n2 …ns … n1 n2 … ns … .

n1, n2 , …ns,…

(1.183)

This state includes the correlations between different field modes and is called an entangled state. The entangled state arises from perturbations such as the atom-field interaction coupled by an electricdipole interaction. 1.9

PERTURBATION THEORY

In this section, we deal systematically with the technique for calculating the effect of small extra potential, or Hamiltonian H ′, acting on a system governed mainly by a free Hamiltonian H 0. Suppose for simplicity that our unperturbed Hamiltonian H 0 governs the noninteracting system with H 0 Ψ 0 = E0 Ψ 0 ,

(1.184)

and the corresponding eigenstate is normalized. When the perturbation is applied, we shall need to solve the equation

(E − H0 ) Ψ = H ′ Ψ .

(1.185)

Provided we make no condition on the normalization of the new eigenstate Ψ , we may always split it into two terms [3] Ψ = Ψ0 + Φ ,

(1.186)

where the change due to the perturbation is to be orthogonal to the unperturbed state:

c01.indd 38

4/20/2011 10:33:35 AM

PERTURBATION THEORY

Ψ 0 Φ = 0.

39

(1.187)

This condition can be expressed by the formal introduction of projection operator P: P = Ψ0 Ψ0 ,

(1.188)

which acts on any quantum state Φ as P Φ = Ψ0 Ψ0 Φ . The complementary projection operator is defined as Q = 1 − P.

(1.189)

The complementary operator is idempotent, that is, Q2 = Q, as in the case of P, and it projects any state upon the manifold in Hilbert space that is orthogonal to Ψ 0 . Then one can see that, from Equations (1.186) and (1.187), one can write QΨ = Φ .

(1.190)

Applying the unperturbed Hamiltonian operator to this function, we get QH 0 Φ = H 0 Φ − Ψ 0 Ψ 0 H 0 Φ = H 0 Φ − E0 Ψ 0 Ψ 0 Φ = H0 Φ

(1.191)

= H 0Q Φ , showing that Q commutes with H 0. The one we write

(E − H 0 ) Ψ = (E − E0 ) Ψ0 + (E − H 0 ) Φ ,

(1.192)

Q ( E − H 0 ) Ψ = ( E − H 0 )Q Ψ = ( E − H 0 ) Φ ,

(1.193)

and from Equations (1.185) and (1.193), we obtain

(E − H 0 ) Φ = QH ′ Ψ .

(1.194)

Applying P to Equation (1.185), we get E = E0 + Ψ 0 H ′ Ψ .

c01.indd 39

(1.195)

4/20/2011 10:33:35 AM

40

BASIC QUANTUM MECHANICS

Equation (1.194) can also be written as Φ = ( E − H 0 ) QH ′ Ψ , −1

(1.196)

which implies that there is an operator ( E − H 0 ) which is the inverse of ( E − H 0 ) . Then, Equations (1.185), (1.186), and (1.192) can be combined into −1

Ψ = Ψ 0 + ( E − H 0 ) QH ′ Ψ . −1

(1.197)

This is an exact equation, but can only be solved approximately by iteration. Thus

(

Ψ = Ψ 0 + ( E − H 0 ) QH ′ Ψ 0 + ( E − H 0 ) QH ′ Ψ −1

−1

= Ψ 0 + ( E − H 0 ) QH ′ Ψ 0 −1

) (1.198)

+ ( E − H 0 ) QH ′ ( E − H 0 ) QH ′ Ψ 0 + . −1

−1

This series is known as the Brillouin-Wigner perturbation expansion. Let { Ψ n } be the complete eigenset of the free Hamiltonian H 0 , then we have

∑Ψ

n

Ψn = I ,

(1.199)

n

and Q Ψ n = Ψ n , n ≠ 0.

(1.200)

Thus from Equations (1.195) to (1.200), we obtain the energy equation (1.195), and represent all states and operators in eigenstates of H 0. We get E = E0 + Ψ 0 H ′ Ψ 1 ⎛ ⎞ QH ′ Ψ 0 + ⎟ = E0 + Ψ 0 H ′ ⎜ Ψ 0 + ⎝ ⎠ E − H0 = E0 + Ψ 0 H ′ Ψ 0 +

∑Ψ

0

(E − H 0 )−1 Ψ n Ψ n QH ′ Ψ0 + 

n

= E0 + Ψ 0 H ′ Ψ 0 +

∑ n≠0

Ψ0 H ′ Ψn E − En

2

+ .

(1.201)

c01.indd 40

4/20/2011 10:33:35 AM

PROBLEMS

41

The projection operator Q automatically excludes certain matrix elements from the summations in each term of the series. It is an implicit equation to be solved for E. Consider the very simple case where H ′ couples just two states of nearly equal energy. If the expectation value of H ′ is zero in each of these states, we get E = E0 +

Ψ 0 H ′ Ψ1 E − E1

2

2

= E0 +

H 01 ′ , E − E1

(1.202)

which has two roots to be obtained from the equation E 2 − ( E0 + E1 ) E + E0 E1 − H 01 ′ = 0. 2

(1.203)

Note that this is the same as the solution of the determinantal equation for the degenerate or nearly degenerate Rayleigh-Schrödinger perturbation theory, that is E − E0 H10 ′

H 01 ′ = 0. E − E1

(1.204)

PROBLEMS 1. We consider the ket vector a represented by a column vector given by ⎛ a1 ⎞ ⎜ a2 ⎟ ⎜ ⎟ ⎜ .⎟ a =⎜ ⎟, . ⎜ ⎟ ⎜ .⎟ ⎜⎝ a ⎟⎠ ∞ with complex quantities a1, a2, a3, … , a∞, and the bra vector given by b = ( b1* b2* b3* . . . . b∞* ). The outer product of vectors a and b is defined by ⎛ a1b1* ⎜ a2 b1* a b =⎜ ⎜  ⎝⎜ a∞ b1*

c01.indd 41

a1b2* a2 b2*  a∞ b2*

 a1b∞* ⎞  a2 b∞* ⎟ ⎟.  ⎟  a∞ b∞* ⎟⎠

4/20/2011 10:33:36 AM

42

BASIC QUANTUM MECHANICS

In two-dimensional Hilbert space, the basis vectors are given by ⎛ 1⎞ 0 =⎜ ⎟ ⎝ 0⎠

and

⎛ 0⎞ 1 =⎜ ⎟ . ⎝ 1⎠

The Pauli operators are defined by ⎛ 0 1⎞ ⎛ 0 −i⎞ ⎛1 0 ⎞ , σy = ⎜ , σz = ⎜ . σx = ⎜ ⎟ ⎟ ⎝ 1 0⎠ ⎝i 0⎠ ⎝ 0 −1⎟⎠ Obtain the spectral representation of Puali operators from Equation (1.25). 2. If A and B are two fixed noncommuting operators, and λ is a parameter, then show that exp(λA)B exp(− λA) = B + λ[a, b] + +

λ2 [ A,[ A, B]] 2!

λ3 [ A,[ A,[ A, B]]] + . 3!

When [ A,[ A, B]] = [ B,[ A, B]] = 0, show that 1 exp( A + B) = exp( A)exp( B)exp ⎛ − [ A, B]⎞ . ⎝ 2 ⎠ Hint: Consider the operator function f (λ ) = exp(λA)exp(λB) , and differentiate f (λ ) with respect to λ to obtain df (λ ) / dλ = ( A + exp(λA)B exp(−λA)) f (λ ), and solve the differential equation. 3. In one dimension, the Hamiltonian is given by Hˆ = pˆ 2 / 2 m + V ( xˆ ). ˆ and En be the eigenvalue. Then Let En be the eigenvector of H, show that

∑

En′′ xˆ En′

2

(En′ − En′′ ) =

n′

2 . 2m

4. In two-dimensional Hilbert space, the basis vectors are given by ⎛ 1⎞ 0 =⎜ ⎟ ⎝ 0⎠

and

⎛ 0⎞ 1 =⎜ ⎟ . ⎝ 1⎠

When the Hamiltonian of this two-dimensional system is given by

c01.indd 42

4/20/2011 10:33:36 AM

PROBLEMS

43

V(x) V0

–L/2

L/2

Figure 1.1. Finite potential well.

H = E0 0 0 + E1 1 1 + V ( 1 0 + 0 1 ) , find eigenvectors and eigenvalues of H . 5. A one-dimensional harmonic oscillator is under the electric field F, and the total Hamiltonian is given by H = p2 / 2 m + 21 ω 2 x 2 − eFx. Using the perturbation theory equation (1.201), calculate the ground state energy. 6. Find the wave functions and the energy levels of a particle in the potential well given by V ( x) = 0 if x ≤ L / 2 and V ( x) = V0 > 0 if x > L / 2 (Fig. 1.1). REFERENCES [1] J. J. Sakurai, Modern Quantum Mechanics. New York: Addison Wesley, 1994. [2] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals. New York: McGraw-Hill, 1965. [3] K. Gottfried, Quantum Mechanics. London: Benjamin Cummings, 1966. [4] H. Goldstein, Classical Mechanics. Reading, MA: Addison Wesley, 1980. [5] R. L. White, Basic Quantum Mechanics. New York: McGraw-Hill, 1966. [6] M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics. London: AddisonWesley, 1974.

c01.indd 43

4/20/2011 10:33:36 AM

This page intentionally left blank

2 2.1

Basic Quantum Statistical Mechanics

ELEMENTARY STATISTICAL MECHANICS

If a system in equilibrium can be in any of N states, then the probability of the system having energy En is given by (1 Z )exp(− En / kBT ), where the partition function Z is defined by [1] N

Z=

∑ exp(−E

n

/ kBT ).

(2.1)

n =1

Here, kB is a Boltzmann constant, and T is a temperature. For quantum mechanical system of states { n }, where n denotes the state with energy En, the expected value of the quantum mechanical operator A is defined by A =

1 Z

∑ n A n exp(−E

n

/ kBT ).

(2.2)

n

This fundamental law is the essence of statistical mechanics, with the concept of thermal equilibrium and temperature. If a system is very weakly coupled to a heat bath at a given temperature, if the coupling is indefinite or not known precisely, if the coupling has been on for a long time, and if all the “fast” processes have happened and all the “slow” processes have not, the system is said to be in thermal equilibrium. Consider two different states of a system that has the same energy, E j = Ek . The probabilities of the system being in either state j and k are then equal. Since the system remains in contact with the heat bath for a long time, one would expect states of equal energy to be equally likely. Also, states of different energies would be expected to have different probabilities.

Engineering Quantum Mechanics, First Edition. Doyeol Ahn, Seoung-Hwan Park. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 45

c02.indd 45

4/20/2011 10:35:26 AM

46

BASIC QUANTUM STATISTICAL MECHANICS

Because two states of the same energy are equally probable, the probability of a state having energy E is a function only of energy; P = P(E ). Consider a system, S, in equilibrium with a large heat bath H. It is assumed that the heat bath is extremely large and its total energy E very great. Also, the possible energy levels of the heat bath may be assumed to be quasi-continuous. Let the energy levels of S be denoted by E j . Then H i  E j for all i, j. We now consider the bath plus the system as a new system, Ti , which is also in thermal equilibrium. T has a definite energy, but as that energy is not fixed exactly (the bath is in contact with the outside world), we can assume that the energy may be anywhere in the range E0 ± Δ. If Δ is sufficiently small, we can assume that the states of the heat bath are equally likely in the range H i ± Δ. Let η ( H r ) be the number of states per unit energy range in the heat bath H around energy H r . The probability, P(Er ), that S is in a state with energy Er, is proportional to the number of ways S can have that energy. From E0 = Er + H i , the number of states around Er is the same as the number of states around H i , and is given by η(E0 − Er ) × 2 Δ. Then the ratio of the probabilities P(Er ) and P(Er′ ) is P(Er ) η(E0 − Er )2 Δ = P(Er′ ) η(E0 − Er′ )2 Δ =

exp ( log η(E0 − Er ) ) exp ( log η(E0 − Er′ ) )

(2.3)

= exp ( log η(E0 − Er ) − log η(E0 − Er′ ) ) . For Er  E0 , we have the following approximation: log η(E0 − Er ) ≈ log η(E0 ) − Er

d log η(E ) . dE E0

(2.4)

Then we have P(Er ) = exp ( −β(Er − Er′ ) ) , P(Er′ )

(2.5)

P(Er ) ∝ exp(−βEr ),

(2.6)

or

c02.indd 46

4/20/2011 10:35:26 AM

ELEMENTARY STATISTICAL MECHANICS

47

where β=

d log η(E ) . dE E0

(2.7)

From Equations (2.1), (2.6), and (2.7), we obtain P(Er ) =

exp(−βEr ) , exp(−βEr )

∑

(2.8)

r

with β = kBT .

(2.9)

We can regard Equations (2.7) and (2.9) as the definition of the temperature T. We now consider a quantum system of N identical particles and assume that there is no interaction among them. Any two configurations of the system that differ only by an interchange of two or more identical particles are regarded as one and the same state. Thus, the state of the system is determined by giving the number of particles na with energy Ea. The problem is to calculate the partition function with the condition that ∑ na = constant. The values admitted for every a

possible na may be: (a) na = 0, 1, 2,  : Bose−Einstein case, (b) na = 0, 1 : Fermi−Dirac case.

(2.10)

For Bose particles, any number of particles may occupy a given state. For Fermi particles, however, there can only be one particle at most in each state. This is due to the Pauli exclusion principle. A state of the system is described by the set of numbers na, which can take any set of values allowed both by the statistics and by the condition

∑n

a

=N

(2.11)

a

and Q=

c02.indd 47

⎛ exp ⎜ −β ⎝ n1 ,n2 ,n3…

∑

⎞

∑ n E ⎟⎠ . a

a

(2.12)

a

4/20/2011 10:35:27 AM

48

BASIC QUANTUM STATISTICAL MECHANICS

We consider the system of particles to be a box connected to a large reservoir of particles, and by assuming that it is possible for particles to pass to and from the reservoir. We further assume that the statistical mechanics of the system act as if it took energy μ to move a particle from the box to the reservoir. For a given μ, the energy levels of a particle in the box becomes Ea − μ rather than Ea. The partition function also becomes then ⎛ exp ⎜ −β ⎝ n1 , n2 , n3 …

∑

Q(μ ) = =

⎛ ⎜⎜ ⎝

∏∑ a

na

∑ n (E a

a

a

⎞ − μ) ⎟ ⎠

(2.13)

⎞ exp ( −βna ( Ea − μ ) ) ⎟⎟ ⎠

with N = ∑ na . The second line of the Equation (2.13) can be easily derived: a ⎛

⎞

∏ ⎜⎝ ∑ exp (−βn K )⎟⎠ a

a

a

na

⎛ =⎜ ⎝ =

⎞

⎞⎛

⎛

⎞

∑ exp (−βn K )⎟⎠ ⎜⎝ ∑ exp (−βn K )⎟⎠  ⎜⎝ ∑ exp (−βn K )⎟⎠ 1

1

2

n1

2

n

n2

⎛

n

nn

⎞

∑ exp ⎜⎝ −β∑ n K ⎟⎠ , a

n1 ,n2 …

a

a

where Ka ≡ Ea − μ. From simplified expression of the partition function Q(μ ) =

⎛

∑ exp ⎜⎝ −β∑ n ( E a

n1 , n2 …

a

a

⎞ − μ) ⎟ ⎠

(2.14)

≡ exp ( −βg ) , we obtain 1 g = − ln Q(μ ) . β

(2.15)

Then we get the following relations:

c02.indd 48

4/20/2011 10:35:27 AM

ELEMENTARY STATISTICAL MECHANICS

⎛ ∂ (μ ) β⎜ Q = ∂μ n1 ,n2 … ⎝

⎛

⎞

⎞ − μ) ⎟ ⎠

∑ ∑ n ⎟⎠ exp ⎜⎝ −β∑ n ( E a

a

a

a

a

⎛ βN exp ⎜ −β = ⎝ n1 ,n2 …

∑

∑ a

⎞ na ( Ea − μ ) ⎟ , ⎠

49

(2.16)

and N =

1 Q(μ )

⎛

∑ N exp ⎜⎝ −β∑ n ( E a

n1 , n2 …

a

a

⎞ − μ) ⎟ ⎠

1 ∂ (μ ) Q = βQ(μ ) ∂μ ∂ = − g. ∂μ

(2.17)

For the purpose of comparing the probability of a state, we assumed that the system acts as if it had energy E = ∑ a na ( Ea − μ ). But μ was inserted only in order to provide the weighting factor that accounts for different probabilities of different number of particles in the system. For the system of the ideal noninteracting Bose–Einstein particles, we have exp(−βg ) = Q(μ ) =

⎛

∑ exp ⎜⎝ −β∑ n ( E a

n1 ,n2 …

⎛ = ⎜⎜ ⎝

a

a

⎞ − μ) ⎟ ⎠ ⎞⎛

∑ exp ( −βn ( E − μ )) ⎟⎟⎠ ⎜⎜⎝ ∑ exp ( −βn ( E 1

1

2

n1

n2

2

⎞ − μ ) ) ⎟⎟  ⎠

for na = 0, 1, 2, =

⎛

⎞ ⎟ a − μ )) ⎠

∏ ⎜⎝ 1 − exp ( −β ( E 1

a

or g=−

c02.indd 49

1 β

∑ ln (1 − exp ( −β ( E

a

− μ ) ) ).

(2.18)

a

4/20/2011 10:35:27 AM

50

BASIC QUANTUM STATISTICAL MECHANICS

From Equations (2.17) and (2.18), the average number of particles in the system is given by N =−

∂ g ∂μ

∑ exp (β ( E =∑ n ,

1

=

a

a

(2.19)

− μ )) − 1

a

a

where na is the number of Bose–Einstein particles in the particular state with energy Ea, and is sometimes called the Bose–Einstein distribution. It is interesting to note that one can also write na =

∂ g. ∂Ea

(2.20)

For the system of the ideal noninteracting Fermi–Dirac particles, we have exp(−βg ) = Q(μ ) =

⎛

∑ exp ⎜⎝ −β∑ n ( E a

n1 ,n2 …

⎛ = ⎜⎜ ⎝

a

a

⎞ − μ) ⎟ ⎠ ⎞⎛

∑ exp ( −βn ( E − μ )) ⎟⎟⎠ ⎜⎜⎝ ∑ exp ( −βn ( E 1

1

2

n1

n2

2

⎞ − μ ) ) ⎟⎟  ⎠

but na = 0 or 1 =

∏ (1 + exp ( −β ( E

a

− μ ) ) ),

a

or g=−

1 β

∑ ln (1 + exp ( −β ( E

a

− μ ) ) ),

(2.21)

a

and we obtain na = =

c02.indd 50

∂ g ∂Ea 1 . 1 + exp ( β ( Ea − μ ) )

(2.22)

4/20/2011 10:35:27 AM

SECOND QUANTIZATION

51

Below, we summarize the distribution functions for the Bose– Einstein and the Fermi–Dirac particles: 1 ⎧ ⎪⎪ exp ( β ( Ea − μ ) ) − 1 ; Bose−Einstein particles na = ⎨ 1 ⎪ ; Fermi−Dirac particles ⎪⎩ exp ( β ( Ea − μ ) ) + 1

2.2

(2.23)

SECOND QUANTIZATION

In the previous chapter, we considered the annihilation and creation operators for simple harmonic oscillators and quantized electromagnetic fields. These operators also provide a convenient representation of the many-particle states in the system of N identical particles. The quantum mechanic of a single particle is formulated in terms of the ˆ and the convenposition operator xˆ and the momentum operator p, tional representation, the coordinate representation, is defined in terms of the eigenfunctions of the position operator. For a system of many identical particles, however, it is convenient to define operators which create or annihilate a particle in specified states. Operators of physical interest can be expressed in terms of these creation and annihilation operators. This new presentation is called the second quantization formulation. Let’s assume that the wave functions of N noninteracting identical particles ϕ1 ( x), ϕ2 ( x),  form the complete orthonormal set where x specifies the coordinates and spin variables that characterize the state of a particle. Instead of giving the complete wave functions of the system of particle, we can obviously describe the system by specifying the number of particles in a given state where the occupation numbers n1 , n2 , are new variables. In this new representation, we define the new wave function or state vector as Φ n1n2 ,  , where n1 particles are in the state 1, n2 particles in the state 2…. Consider first the case of a system of Bose–Einstein particles. In this case, the complete wave function of a system of Bose–Einstein particles is symmetric with respect to permutations of the variables corresponding to different particles. Then the wave function corresponding to the occupation numbers n1 , n2 ,  has the form [2] n ! n ! ⎞ Φ n1n2  = ⎛⎜ 1 2 ⎟ ⎝ N! ⎠

c02.indd 51

1/ 2

∑ϕ

p1

( x1 )ϕ p2 ( x2 ) ϕ pN ( xN )

(2.24)

P

4/20/2011 10:35:27 AM

52

BASIC QUANTUM STATISTICAL MECHANICS

where the pi denotes the different states, and the sum is taken over all possible permutations of different numbers pi. Now assume that F (1) is the operator that is symmetric in all particles and has the forms F (1) =

∑f

( 1) n

(2.25)

n

where fn(1) is the operator acting on functions of xn. When such an operator acts on the function Φ N1N2 it carries it either into the same function or into another function in which the state of one particle has been changed. As a result, the matrix elements of F (1) with respect to the wave function Equation (2.24) have the form F (1) =

∑f

( 1) i ii

n

for diagonal elements,

i

(F )

( 1) N i , N k − 1 N i − 1, N k

(2.26) =f

( 1) ik

ni nk

for nondiagonal elements,

where

∫

fik(1) = ϕi *( x) f (1)ϕk ( x)dx.

(2.27)

The above relation Equation (2.26) can be conveniently expressed using the annihilation and creation operators and the number states introduced in Equations (1.181) and (1.182) of Chapter 1: ai n1 n2  ni  = ni n1n2  ni − 1

(2.28)

and ai† n1n2  ni  = ni + 1 n1n2  ni + 1 where n1n2  denotes the abstract state vector where there are n1 particles in state 1, n2 particles in state 2, … and so on. Using these new notations, the operator F (1) can be written in following second quantized form F (1) =

∑f

( 1) † i k ik

aa .

(2.29)

i ,k

The matrix elements of this operator coincide with those given by Equation (2.26) when it acts on the abstract state n1n2  . The creation operator ak† and the annihilation operator ai satisfy the following commutation relations:

c02.indd 52

4/20/2011 10:35:27 AM

SECOND QUANTIZATION

[ai ,ak† ] = δik , [ai , ak ] = 0, [ai† ,ak† ] = 0.

53

(2.30)

A similar representation is possible for a symmetric operator of the form F (2) =

∑f

(2) mn

(2.31)

m, n

(2) where fmn acts on functions of xm and xn. In the second quantized representation, F ( 2 ) has the form

F (2) =

∑f

( 2) † † iklm i k m l

aaa a

(2.32)

i ,k ,l , m

where

∫

(2) fiklm = ϕi ∗ ( xa )ϕk *( xb ) fab( 2 )ϕm ( xb )ϕl ( xa )dxadxb .

(2.33)

In the case of Fermi–Dirac statistics, the complete wave function must be antisymmetric in all variables. This implies that in the case of noninteracting particles, the occupation numbers can only take the values 0 or 1, and the wave function has the form Φ n1n2  =

1 N!

∑ (−1)

P

ϕ p1 ( x1 )ϕ p 2 ( x2 ) ϕ pN ( xN )

(2.34)

P

where all the numbers p1 , p2 ,  , pN are different, and the symbol (−1)P means that the odd number of permutations appear in the sum Equation (2.34) with a minus sign. For definiteness, the term in Equation (2.34) for which p1 < p2 <  < pN

(2.35)

will be chosen with a plus sign. The operator F (1) of the type Equation (2.25) can also be written in the second quantized form F (1) =

∑f

( 1) † ik i k

cc ,

(2.36)

i ,k

∫

with fik(1) = ϕi *( x) f (1)ϕk ( x)dx. However, in Equation (2.36), the fermion annihilation operator ci and the creation operator ck† satisfy the following anticommutation relations:

c02.indd 53

4/20/2011 10:35:28 AM

54

BASIC QUANTUM STATISTICAL MECHANICS

{ci ,ck† } ≡ ci ck† + ck† ci = δik , {ci ,ck } = {ci† ,ck† } = 0.

(2.37)

Similarly, the two-particle operator F ( 2 ) has the following second quantized representation: F (2) =

∑f

( 2) † † iklm i k m l

cc c c,

(2.38)

i ,k ,l , m

where

∫

(2) fiklm = ϕi *( xa )ϕk *( xb ) fab( 2 )ϕm ( xb )ϕl ( xa )dxadxb .

2.3

DENSITY OPERATORS

When we solve complicated quantum mechanical problems, we usually divide the universe into two parts—the system in which we are interested and the rest of the universe. Then, normally, we assume that the system in which we are interested comprises the entire universe. Let us see what happens if we include something outside of the system. In this case, we assume ϕi to be a complete set of state vectors in the vector space describing the quantum system, and Ri to be a complete set for the rest of the universe or the reservoir. Then the most general wave function can be written as Ψ =

∑C

ij

ϕi ⊗ R j .

(2.39)

i, j

ˆ be an operator that acts only on the system and does not Now let A ˆ acts on the tensor product states, act on the Rj . In other words, when A we mean that   A ϕ ⊗ ψ = (A ϕ )⊗ ψ . (2.40) Then

ˆ Ψ A = ΨA =

∑C

ij

* Ckl Rj

i , j ,k ,l

=

∑C

ij

(ϕ

i

ˆ ϕk * Ckj ϕi A

ˆ ϕk A

)R

l

(2.41)

i , j ,k

=

∑ϕ

i

ˆ ϕk ρki A

i ,k

c02.indd 54

4/20/2011 10:35:28 AM

DENSITY OPERATORS

with ρki =

∑C

ij

* Ckj ,

55

(2.42)

j

where ρki is the matrix element of the density operator ρ, and is called the density matrix. The density operator is defined as ρki = ϕk ρ ϕi .

(2.43)

Then, the expectation value of the operator is expressed as

∑ϕ =∑ ϕ

ˆ = A

i

ˆ ϕk ϕk ρ ϕi A

i

ˆ ρ ϕi A

i ,k

i

(2.44)

( )

ˆ . = Tr ρA From Equation (2.42), it is obvious that the density operator ρ is Hermitian. Therefore, it can be diagonalized with a complete orthonormal set of eigenvectors ψ i and real eigenvalues wi [1] ρ=

∑w

i

ψi ψi ,

(2.45)

∑ w = 1.

(2.46)

i

ˆ = Iˆ , we obtain and for A Iˆ = Trρ =

i

i

ˆ = ψ k ψ k , we have When A wk = Ψ ( ψ k ψ k ) Ψ =

∑ Ψ(ψ

k

Rj Rj ψ k ) Ψ

j

=

∑

2

Rj ⊗ ψ k Ψ

(2.47)

j

≥ 0. Therefore, we have

∑ w = 1 and w ≥ 0, i

i

(2.48)

i

c02.indd 55

4/20/2011 10:35:28 AM

56

BASIC QUANTUM STATISTICAL MECHANICS

which means wi can be interpreted as a probability of finding the system in the state ψ i . Above results suggest that we can reformulate quantum mechanics using density operator ρ, where ρ is of the form ˆ the ρ = ∑ i wi ψ i ψ i , with ∑ i wi = 1 and wi ≥ 0. Given an operator A, ˆ is given by A ˆ = TrρA ˆ . If all but one of the wi expectation value of A are zero, we say that the system is in a pure state; otherwise, it is in a mixed state. We now consider the operator Ψ Ψ and consider the partial trace of Ψ Ψ with respect to the environment state Rj , then, from Equations (2.39), (2.42), and (2.43), we obtain

∑ R (Ψ = ∑ ∑C C

TrE ( Ψ Ψ ) =

Ψ ) Rm

m

m

kj

i ,k

ij

* ϕk ϕi

(2.49)

j

= ρ. When there is no prospect of confusion, we will also denote the partial trace of the operator Ψ Ψ as Ψ Ψ . Then the density operator, more exactly the reduced density operator ρ , can be defined simply as

ρ =( Ψ Ψ )

(2.50)

Since the quantum state Ψ satisfies the Schrodinger equation i

∂ Ψ =H Ψ , ∂t

the density operator satisfies the following equation of motion ∂ρ 1 1 = ( Hρ − ρH ) = [ H , ρ]. ∂t i i

(2.51)

Now we would like to express the density operator in second quantization. Consider the system of electrons in solid with two energy bands. Details of the band structure will be discussed in the following chapter, and here we consider a model system. It is often convenient to introduce the new operator, called field operator, which is defined as

c02.indd 56

4/20/2011 10:35:28 AM

DENSITY OPERATORS

Ψ( x) =

∑a

ck

ϕck ( x) +

k

∑a

vk

ϕvk ( x),

57

(2.52)

k

where we assume the electrons to be a two-band semiconductor. Details of the semiconductor band structure are described in the next chapter, and we can treat it as a model in this section. Here c stands for the conduction band, v the valence band, and k the wave vector. We introduce the following simplified notations: c ≡ 1, c1k = ack v ≡ 2, c2 k = avk , and the state vector Ψ =

∑(c

1k + c2 k 2k ),

1k

Ψ =

k

∑ ( 1k c

†

1k

†

)

+ 2 k c2 k .

k

(2.53)

The density operator is then defined as ρ= Ψ Ψ =

∑ (c

† 1k 1k ′

c

1k 1k ′ + c1k c2†k ′ 1k 2k ′ + c2 k c1†k ′ 2k

k .k ′

(2.54)

)

1k ′ + c2 k c2†k ′ 2k 2k ′ . Using random phase approximation, we obtain ρ=

∑ρ

αβ

(k ) αk βk , α, β = 1, 2,

(2.55)

α ,β,k

where ραβ ( k ) = cβ†k cαk .

(2.56)

We assume that the Hamiltonian of the system is given by

∑E

H=

αβ

( k ) cα† k cβk .

(2.57)

α ,β,k

Then, in the Heisenberg picture, we have i

∂ † cβk cαk = [ cβ†k cαk , H ] ∂t =

∑E

αβ′

β′

c02.indd 57

(k )cβ†k cβ′k −

∑E

α′β

(k )cα† ′k cαk .

(2.58)

α′

4/20/2011 10:35:28 AM

58

BASIC QUANTUM STATISTICAL MECHANICS

From Equations (2.55) to (2.58), we obtain i

∂ ρ= ∂t

∑E

αβ′

β′

( k ) ρβ′β − ∑ ραα′ Eα′β ( k ) α′

(2.59)

= ⎡⎣ Eˆ ( k ) , ρ ⎤⎦ where Eˆ ( k ) is the matrix representation of the Hamiltonian Equation (2.57). When two-particle interaction such as Coulomb interaction becomes important, the Hamiltonian Equation (2.57) need to be modified into H=

∑E

αβ

( k ) cα† k cβk

α ,β,k

+

∑

1 † † † V ( q ) [ c1†k +qc1†k′−qc1k′c2 k + c2†k +qc2k k′−q c2 k′c2 k + 2c1k + q c2 k′−q c2 k′c1k ] . 2 k ,k ′ ,q ≠ 0 (2.60)

2.4 THE COHERENT STATE In Chapter 1, we considered the quantum mechanical description of the radiation field through the use of a particular set of quantum states for the radiation field. These are the stationary states of the noninteracting field, which correspond to the presence of a precisely defined number of photons. At this stage, it is interesting to ask what wave function most closely describe the classical electromagnetic field given by E0 cos ( Ωt + φ ) and H 0 sin ( Ωt + φ ). It is known that the wave function, which corresponds most closely to the classical field, turns out to be the “coherent” wave function, a displaced ground state of the simple harmonic oscillator. The corresponding state vector is the “coherent state,” which is the eigenstate of the positive frequency part of the electric field operator. The coherent state α for the single mode oscillator is defined as an eigenstate of the annihilation operator [3]: a α =α α .

(2.61)

We can expand the coherent state α in terms of the number state n developed in Chapter 1 as

c02.indd 58

4/20/2011 10:35:28 AM

THE COHERENT STATE

∑n nα = ∑ n 0 (a

59

α =

n

n

)

/ n! α

n

= 0α

∑ n

(2.62)

αn n, n!

where we have used Equation (1.115) of Chapter 1. Using the 2 normalization condition α α = 1, we obtain that 0 α = e −(1 2 ) α . Then, we have 1 2 α = exp ⎛⎜ − α ⎞⎟ ⎝ 2 ⎠

∑ n

αn n. n!

(2.63)

The photon number probability distribution P(n) for the coherent state α is then given by P(n) = n α

2

= exp ( − α

2

)α

2n

/ n!

(2.64)

which turns out to be the Poisson distribution. One can show that the coherent state α is generated by the action on the vacuum state 0 of the Glauber displacement operator D ( α ) = exp ( αa† − α * a ) ,

(2.65)

with the Baker–Hausdorff relation for operators A and B exp ( A + B ) = exp( A)exp( B)exp ( − [ A, B] / 2 ) ,

(2.66)

α = D(α) 0 .

(2.67)

such that

The Glauber displacement operator satisfies the following properties: D(−α)aD(α) = a + α,

(2.68)

D ( −α ) a† D ( α ) = a† + α.

(2.69)

and

c02.indd 59

4/20/2011 10:35:29 AM

60

BASIC QUANTUM STATISTICAL MECHANICS

One property of the coherent state α is that, in contrast to the number states, the coherent states are not mutually orthogonal. The overlap between two coherent states α and β is α β = exp ( − α / 2 − β / 2 ) 2

2

(α*)n (β)m

∑ (n! m!)

1/ 2

nm

n. m

= exp ( α * β − α / 2 − β / 2 ) , 2

(2.70)

2

and hence αβ

2

= exp ( − α − β

2

).

(2.71)

Even though the coherent states lack mutual orthogonality, they form an overcomplete set such that the identity relation can be resolved in terms of the coherent states as 1 2 1 d αα α = d(Re α)d(Im α) α α = I . π π

∫

∫

(2.72)

The overcomplete relation Equation (2.72) can be derived using

∫ d α(α*) α exp(− α ) exp [i(m − n)θ ] exp(− α =∫ α 2

n

m

2

n+ m

2

) α d α dθ

(2.73)

= π(n!)δ n. m ,

and α = α e iθ .

(2.74)

The use of coherent states leads to considerable simplification in the calculation of statistical averages related to the radiation field. The radiation field can be described by the density operator, which is defined in the previous section. The density operator is the outer product ρ= ψ ψ

(2.75)

when the complete state ψ is known. However, when one has only the statistical knowledge that the system has probability wi of being the state ψ i , the density operator is given by Equation (2.45)

c02.indd 60

4/20/2011 10:35:29 AM

THE COHERENT STATE

ρ=

∑w

i

61

ψi ψi

i

The density operator is often more easily applicable to the real problem than the state vector, and it can be expanded in terms of the coherent states. The values of the expansion coefficients express the degree to which the radiation field is actually coherent in the correct quantum mechanical sense. Any density operator ρ may be represented in a unique way by means of a function of two complex variables R(α*, β), which is analytic throughout the finite α * and β planes, and is defined by R(α*, β) = α ρ β exp ( α / 2 + β / 2 ) . 2

2

(2.76)

We note that the coherent states form an overcomplete set, and then we can expand ρ as 1 d 2 αd 2 β α α ρ β β π2 1 2 2 = 2 d 2 αd 2βR(α*, β)exp ( − α / 2 − β / 2 ) α β . π

ρ=

∫ ∫

(2.77)

By expanding the coherent states in terms of number states, we also get R(α*, β) =

(α*)n βm

∑ (n! m!)

1/ 2

nρm.

(2.78)

n, m

From the trace-preserving condition of the density operator, that is, Trρ = 1, we obtain Tr(ρ) =

1 2 2 d αR(α *.α)exp(− α ) = 1. π

∫

(2.79)

Although this expansion of the density operator is straightforward, it is quite complicated. We can often obtain a simpler expansion with real coefficient P(α):

∫

ρ = d 2 α P (α ) α α .

(2.80)

The expectation value of the operator A is given by Tr(ρA).

c02.indd 61

4/20/2011 10:35:29 AM

62

BASIC QUANTUM STATISTICAL MECHANICS

Then A = Tr(ρA) =

∑ n ∫ d α P (α ) α 2

α An

n

∑ α An

∫

= d 2 α P (α )

nα

(2.81)

n

∫ = ∫ d αP(α) A(α),

= d 2 α P (α ) α A α 2

where A(α) = α A α .

(2.82)

This leads to a simple calculation involving only c numbers, provided that the operator A is expressed in normal order, that is, so that creation operators stand to the left of the annihilation operators in the product appearing in A. 2.5 THE SQUEEZED STATE Squeezed states arise in simple quantum models of a number of nonlinear optical processes including optical parametric oscillation and four-wave mixing [4–6]. The Heisenberg uncertainty principle ΔAΔB ≥ 21 [ A, B] between the standard deviation of two arbitrary observables, ΔA =

( A − A )2 , and similarly for ΔB , has a built-in

degree of freedom: One can squeeze the standard deviation of one observable provided that one stretches that for the conjugate observable. All quantum mechanics requires is that the product be bounded from below. Given a field described by the annihilation operator a, we form two Hermitian conjugate operators giving its two quadratures as xλ =

1 ( aeiλ + a†e − iλ ) , 2

pλ =

1 ( aeiλ − a†e − iλ ) , 2i

(2.83)

with [ xλ , pλ ] = i / 2 so Δxλ Δpλ ≥ 1 / 4. These two operators correspond to position and momentum for the case of a mechanical oscillator. The variance Δxλ2 is given by Δxλ2 = xλ2 − xλ . 2

c02.indd 62

(2.84)

4/20/2011 10:35:29 AM

THE SQUEEZED STATE

63

We now consider a quantum state such that the expectation values of the two Hermitian conjugate operators are zero, xλ = pλ = a = a† = 0. Then we have 1 1 † 1 + a a + ( aa e 2 iλ + a†a† e −2 iλ ) 4 2 4 1 1 † 1 = + a a + Re ( aa e 2 iλ ) . 4 2 2

Δxλ 2 =

(2.85)

The minimum variance is Δxλ2 =

1 1 † 1 + a a − aa . 4 2 2

(2.86)

Likewise, the variance of the conjugate operator is Δpλ2 =

1 1 † 1 + a a − Re ( aa e 2 iλ ) , 4 2 2

(2.87)

so the minimum variance is given by Δpλ2 =

1 1 † 1 + a a + aa , 4 2 2

(2.88)

which is certainly greater than Equation (2.86). Equations (2.86) and (2.88) give the uncertainty principle Δxλ Δpλ =

1 4

(1 + 2

a†a − 2 aa

) (1 + 2

a†a + 2 aa

)≥

1 . 4

(2.89)

Squeezing occurs for xλ if Δxλ2 becomes smaller than 41 , that is, squeezed below the minimum uncertainty product value. This, however, does not violate the uncertainty principle, since pλ has a correspondingly increased variance. In this case, it is the aa term that leads to squeezing. The simplest single-mode squeezed states are generated by the action on the vacuum state 0 of the squeezing operator [4,5] ζ ζ* ⎞ S ( ζ ) = exp ⎛⎜ − a†a† + aa ⎟ 2 ⎝ 2 ⎠

(2.90)

where ζ = r exp(iϕ) is the squeezing parameter with modulus r and argument ϕ. It is easy to show that the squeezing operator is

c02.indd 63

4/20/2011 10:35:29 AM

64

BASIC QUANTUM STATISTICAL MECHANICS

unitary, that is, S † ( ζ ) = S ( −ζ ). The single mode squeezed state ζ is defined by ζ = S(ζ) 0 ,

(2.91)

and is given by ζ =

1 cosh r

∞

∑ n =0

n

(2 n)! ⎡ 1 − exp(iϕ)tanh r ⎤⎥ 2 n . n! ⎢⎣ 2 ⎦

(2.92)

Here, we have used the definition of the exponential function of an operator A as exp( A) ≡

∑ n! A 1

n

(2.93)

n

with the fact that ( a†a + aa† ) 0 = 0 , and the ordered form of S(ζ) [4]: 1 1 2 S ( ζ ) = exp ⎡⎢ − ( a† ) exp ( iϕ ) tanh r ⎤⎥ exp ⎡⎢ − ( aa†a†a ) ln ( cosh r ) ⎤⎥ ⎦ ⎦ ⎣ 2 ⎣ 2 1 2 × exp ⎡⎢ ( a ) exp ( −iϕ ) tanh r ⎤⎥ . ⎦ ⎣2

(2.94)

The squeezed state ζ is a right eigenstate of a ζ -dependent linear combination of a and a† , with eigenvalue 0:

[a cosh r + a

exp(iϕ)sinh r ] ζ = 0.

(2.95)

For this reason, ζ is termed the “squeezed vacuum.” In the following, we derive the operator ordering theorem equation (2.98). For two operators A and B, which commute with their commutator [ A, B], we have 1 exp [ θ ( A + B )] = exp(θA)exp(θB)exp ⎛⎜ − θ2 [ A, B] ⎞⎟ . ⎝ 2 ⎠

(2.96)

Generalized angular momentum operators J 3 (= J z ), J + and J − satisfy the commutation relations [ J 3 , J ± ] = ± J ± and [ J + , J − ] = 2 J 3. We assume that F (θ) = exp [ iθ ( J + + J − )] can be written as F (θ) = exp [ip(θ) J + ] exp [iq(θ) J 3 ] exp [ir(θ) J − ].

c02.indd 64

(2.97)

4/20/2011 10:35:29 AM

65

THE SQUEEZED STATE

Then, d F (θ) dθ = i ( J + + J − ) F (θ) = ip′(θ) J + exp [ ip(θ) J + ] exp [ iq(θ) J 3 ] exp [ ir(θ) J − ] + exp [ ip(θ) J + ]{iq′(θ) J 3 } exp [ iq(θ) J 3 ] exp [ ir(θ) J − ] + exp [ ip(θ) J + ] exp [ iq(θ) J 3 ]{ir′(θ) J − } exp [ ir(θ) J − ]

(2.98)

= i [( p′(θ) − iq′(θ) p(θ) + r′(θ)exp(−iq(θ) p2(θ) ) J + + r′(θ)exp(−iq(θ) J − ] F(θ) + [( q′(θ) + 2ir′(θ) p(θ)exp(−iq(θ) ) J 3 ] F (θ),

where we have used exp( A)B exp(− A) = B + [ A, B] +

1 [ A.[ A, B]] +  2!

(2.99)

in the derivation of Equation (2.98), which in turn gives the following coupled differential equations: p′ − ipq′ + r ′ exp(− iq) p2 = 1, r ′ exp(− iq) = 1, q′ + 2ir ′p exp(− iq) = 0.

(2.100)

The initial conditions are: p(0) = q(0) = r(0) = 0 . If we set q(θ) = i log ( cos2 θ )

(2.101)

r(θ) = tan θ,

(2.102)

p(θ) = tan θ.

(2.103)

then we also get

and

As a result, we obtain exp [ iθ ( J + + J − )] = exp ( iJ + tan θ ) exp ( − J 3 log [ cos2 θ]) exp ( iJ − tan θ ) . (2.104)

c02.indd 65

4/20/2011 10:35:30 AM

66

BASIC QUANTUM STATISTICAL MECHANICS

Likewise, for operators K + , K − and K 3 which satisfy the commutation relations

[K3 , K± ] = ±K±

(2.105)

[ K + , K − ] = −2K 3 ,

(2.106)

and

we obtain exp [ iθ( K + + K − )] = exp [ iK + tanh θ ] exp [ − K 3 log ( cosh 2 θ )] exp [ iK − tanh θ],

(2.107)

which is generalized to exp [ γ + K + + γ − K − + γ 3 K 3 ]

= exp (Γ + K + ) exp ( K 3 log Γ 3 ) exp (Γ − K − )

(2.108)

= exp (Γ − K − ) exp ( − K 3 log Γ 3 ) exp (Γ − K − ) , where Γ± =

2 γ ± sinh β , 2β cosh β − γ 3 sinh β

(2.109)

2

γ ⎞ ⎛ Γ 3 = ⎜ cosh β − 3 sinh β ⎟ , 2β ⎝ ⎠

(2.110)

and 1 β = ⎛⎜ γ 23 − γ + γ − ⎞⎟ ⎝4 ⎠

1/ 2

.

(2.111)

If we define 1 † ( a a + aa† ) , 4

(2.112)

1 † 2 ( a ) = ( K− )† , 2

(2.113)

K3 = and K+ =

c02.indd 66

4/20/2011 10:35:30 AM

THE SQUEEZED STATE

67

one can easily show that they satisfy the commutation relations equations (2.105) and (2.106). By setting γ + = −ζ, γ − = ζ * and γ 3 = 0, and using equations (2.108)–(2.111), we obtain the operator ordering relation equation (2.94). For two modes A and B, with annihilation operators a and b, the two-mode squeezed state ζ AB is generated by the action on the twomode vacuum state 0 A ; 0 B of the two-mode squeezing operator [4,5] SAB ( ζ ) = exp ( −ζa†b† + ζ * ba ) ,

(2.114)

where ζ = r exp(iϕ) is any complex number. Both single-mode and two-mode squeezing arise in models of two-photon nonlinear optics being associated with degenerate and nondegenerate processes, respectively. It is also shown that the quantum black hole first suggested by Hawking can also be represented by a two-mode squeezed state. By setting K + = a†b† = ( K − ) ,

(2.115)

1 † ( a a + b†b ) , 2

(2.116)

†

K3 =

γ + = −ζ, γ − = ζ *, and γ 3 = 0, we obtain SAB ( ζ ) = exp [ −a†b†exp ( iϕ ) tanhr ]

× exp [ − ( a†a + b†b ) log ( coshr )] exp [ baexp ( −iϕ ) tanhr ],

(2.117)

from Equations (2.108) to (2.111). The two-mode squeezed state ζ AB is given by ζ AB = SAB (ζ) 0 A ; 0 B ,

(2.118)

and it then follows from Equation (2.117) that ζ AB =

1 cosh r

∞

∑[− exp(iϕ)tanh r ]

n

nA ; nB .

(2.119)

n=0

The state ζ AB is a superposition only of states in which the two modes contain the same number of photons and are normalized. Note that ζ AB is an entangled state in that it is not the product of a state for mode A and one for mode B. This leads to the correlations between the properties of the modes.

c02.indd 67

4/20/2011 10:35:30 AM

68

BASIC QUANTUM STATISTICAL MECHANICS

2.6 COHERENT INTERACTIONS BETWEEN ATOMS AND FIELDS The simplest model of an atom includes only two discrete, orthonormal states 1 and 2 , which, in the absence of coupling, are energy eigenstates with energies ω1 and ω2 > ω1, respectively. A general pure state ψ is given by the superposition ψ = c1 1 + c2 2 , where c1 2 + c2 2 = 1. For this system, we define the four Pauli operators: I= 2 2 + 1 2,

(2.120)

σ3 = 2 2 − 1 1 ,

(2.121)

σ+ = 2 1 ,

(2.122)

σ− = 1 2 = ( σ+ ) . †

(2.123)

We also define the Hermitian operators σ1 = σ + + σ −

(2.124)

σ2 = i ( σ− − σ+ ) .

(2.125)

and

The Hamiltonian describing the two-state atom interacting with a classical electric field E(t ) within the electric dipole approximation is [4–6]     H = ( ω1 + ω2 ) I + ( ω2 − ω1 ) σ 3 − d ⋅ E(t ) ( σ + + σ − ) , 2 2

(2.126)

  H o = ( ω2 + ω1 ) I + ( ω2 − ω1 ) σ 3 2 2 ω  0 ⎛ 2 ⎞ =⎜ ⎟. 0  ω ⎝ 2⎠

(2.127)

 where d is the atomic dipole operator. The first two terms in this Hamiltonian are free Hamiltonian H o for the atom having eigenstates 1 and 2 :

In the quantum theory of light, the electric and magnetic fields are Hermitian operators, and the interaction between two-state atoms and the quantized electric field is described by the electric dipole  Hamiltonian Equation (2.126), with E(t ) replaced by the quantized  electric field operator. For an atom placed at the origin r = 0

c02.indd 68

4/20/2011 10:35:30 AM

THE JAYNES–CUMMINGS MODEL

69

interacting with a single plane wave mode of frequency ω, the Hamiltonian becomes   H1 = ( ω2 + ω1 ) I + ( ω2 − ω1 ) σ 3 2 2 − iλ ( σ + + σ − ) [ aexp ( −iωt ) + a†exp ( iωt )]

(2.128)

1   H 2 = ( ω2 + ω1 ) I + ( ω2 − ω1 ) σ 3 + ω ⎛⎜ a† a + ⎞⎟ 2 2 2⎠ ⎝ † − iλ ( σ + + σ − ) ( a + a ) .

(2.129)

 1/ 2 where λ = d ⋅ εˆ ( ω / 2 ε oV ) . An equivalent and time-independent form of Equation (2.128) is

2.7 THE JAYNES–CUMMINGS MODEL In quantum optics, we are often interested in the dynamics of atoms, molecules, or even nanostructures coupled to an electromagnetic field. The field may be described either classically or fully quantum mechanically, and may be monochromatic, have time-dependent amplitude or phase, or be broadband. In this section, we consider the Jaynes–Cummings model of the interaction between a two-state atom and a single-mode quantized electromagnetic field. The time-independent Hamiltonian is   1 H = ( ω2 + ω1 ) I + ( ω2 − ω1 ) σ 3 + ω ⎛⎜ a† a + ⎞⎟ 2 2 2⎠ ⎝ † − iλ ( σ + + σ − ) ( a + a ) ,

(2.130)

as given in the previous section. Quantum mechanics provides a number of equivalent descriptions or pictures of the dynamics of the system as discussed in the previous chapter. These are related by unitary transformations of both the state and the operators acting on it. In the Schrödinger picture, the operators are time independent, and all the dynamics are contained within the state. On the other hand, in the Heisenberg picture, the operators evolve while the state is time-independent. Intermediate between them are the interaction pictures in which part of the dynamics, usually associated with a free, uncoupled Hamiltonian, is contained within the operators, while that arising from the coupling appears in the state. Alternatively, the free evolution may be contained within the state, with the operators containing the effect of the coupling. These pictures are

c02.indd 69

4/20/2011 10:35:30 AM

70

BASIC QUANTUM STATISTICAL MECHANICS

related to each other by unitary transformations. A measurable quantity is the expectation of the operator A in a state ψ given by ψ A ψ , and must be independent of the picture in which it is evaluated. The interaction picture Hamiltonian H I can be derived from the Schrödinger equation [5,6] i

∂ ψ =H ψ , ∂t

(2.131)

together with the interaction picture state ψ I = U ψ , described by the unitary operator U, and satisfies the transformed Schrödinger equation i

∂ ψI = HI ψI . ∂t

(2.132)

It is now straightforward to show that the interaction picture Hamiltonian is given by  † + UHU † . H I = iUU

(2.133)

We now consider the unitary transformation associated with the unitary operator i U = exp ⎛⎜ [( ω2 + ω1 ) I + ( ω) σ 3 ] t ⎞⎟ 2 ⎝ ⎠ = exp ( i [ ω2 − ω − Δ / 2 ] t ) 1 1 + exp ( i [ ω2 − Δ / 2 ]) 2 2 ,

(2.134)

where Δ = ω2 − ω1 − ω is the detuning between the atomic transition energy and that of the driving field. The Hamiltonian in the interaction picture is then obtained by HI =

 1 Δσ 3 + ω ⎛⎜ a† a + ⎞⎟ 2 2⎠ ⎝ − iλ [ σ+ exp(iωt ) + σ− exp(−iωt )] ( a + a† ) .

We apply the second unitary transformation exp ⎡i ∑ ( n + 21 ) ωt n n ⎤ on Equation (2.135), and get ⎣ n ⎦ HI =

c02.indd 70

(2.135)

U2 =

 Δσ 3 − iλ [ σ + exp ( iωt ) + σ − exp ( −iωt )][ aexp ( −iωt ) + a†exp ( iωt )]. 2 (2.136)

4/20/2011 10:35:31 AM

PROBLEMS

71

If we ignore terms with exp(± 2iωt ) dependences, known as rotating wave approximation, we obtain the Jaynes–Cummings Hamiltonian H JC =

 Δσ 3 − iλ [ σ + a + σ − a† ], 2

(2.137)

in which the explicit time dependence has been removed. Sometimes, the Jaynes–Cummings Hamiltonian is written as 1 ω0 + ωa† a + η ( a† σ + + aσ − ) 2 1 = ω0 + ωa† a + V , 2

H JC =

(2.138)

where V = η ( a†σ + + aσ − ). The unitary operator U is given by U = exp ( iVt /  ), then the matrix representation of U is ⎛ ⎜ cos ⎡ηt ( aa† )1/ 2 ⎤ ⎣ ⎦ ⎜ U =⎜ ⎜ sin ⎡ηt ( aa† )1/ 2 ⎤ ⎣ ⎦ a† ⎜ −i † 1/ 2 ⎜ ( aa ) ⎝

sin ⎡⎣ηt ( aa† )

⎤ ⎞ ⎦ a⎟ † 1/ 2 ⎟ aa ( ) ⎟. ⎟ 1 / 2 cos ⎡⎣ηt ( a†a ) ⎤⎦ ⎟ ⎟ ⎠

−i

1/ 2

(2.139)

PROBLEMS 1. Assuming that a particle has discrete energy spectrum E0 , E1 , , and the probability of a particle with energy Er is given by Equation (2.8). We define the partition function Z by Z = ∑ n exp(−βEn ). Then show that the average energy of a particle is given by E = −(∂ ∂β)log Z. 2. Evaluate the following vacuum expectation value of 0 ai a†j ak al† 0 , where the fermion annihilation operator am satisfies the relation am 0 = 0. 3. For bosonic annihilation operator b, which satisfies the commutation n n−1 2 relation [ b,b† ] = 1, show that ⎡⎣b, ( b† ) ⎤⎦ = 2b and ⎡⎣b, ( b† ) ⎤⎦ = n ( b† ) .

4. Consider a two-level system with bases given by { 0 , 1 }. The Hamiltonian of the system H is given by the sum of an unperturbed

c02.indd 71

4/20/2011 10:35:31 AM

72

BASIC QUANTUM STATISTICAL MECHANICS

term H 0 and an interaction energy V, that is H = H 0 + V with H 0 0 = E0 0 and H 0 1 = E1 1 . The density operator of this two dimensional system is given by ⎛ ρ00 ρ=⎜ ⎝ ρ10

ρ01 ⎞ . ρ11 ⎟⎠

Using Equation (2.51), show that ∂ρ00 1 = ( V01ρ10 − ρ01V10 ) ∂t i and ∂ρ01 1 1 = − ( E1 − E0 ) − V01 (ρ00 − ρ11 ) . ∂t i i Here, V01 = 0 V 1 and diagonal elements of V are assumed to be zero. 5. Using Equations (2.62) to (2.66), derive Equation (2.67). 6. From Equations (2.96), (2.97) and (2.99), derive Equations (2.98) and (2.104).

REFERENCES [1] R. P. Feynman, Statistical Mechanics. Reading, MA: W. A. Benjamins, 1972. [2] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics. New York: Dover, 1963. [3] M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics. London: AddisonWesley, 1974. [4] S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics. New York: Oxford University Press, 1997. [5] W. H. Louisell, Quantum Statistical Properties of Radiation. New York: Wiley Interscience, 1990. [6] P. Meystre and M. Sargent III, Elements of Quantum Optics. Berlin, Germany: Springer-Verlag, 1993.

c02.indd 72

4/20/2011 10:35:31 AM

3

Elementary Theory of Electronic Band Structure in Semiconductors

In this chapter, we derive the 8 × 8 Hamiltonian using the multiband effective theory. To understand the optical properties of semiconductors, we need to know the electronic band structure. In particular, the band structure calculations of the nanostructures, such as the quantum well, quantum wire, and quantum dot, are of great interest, because these structures have existing and potential applications for many important optoelectronic devices. First, we review several basic theories for band structure calculation, in particular, the effective mass theory. Next, we introduce the band structure calculation for nanostructures. Specially, we focus on the band structure of the quantum well and the crystal orientation effect on electronic properties. 3.1 3.1.1

BLOCH THEOREM AND EFFECTIVE MASS THEORY Bloch Theorem

A fundamental theorem concerning electrons in a crystal was proved by Bloch in 1928. It states that the wave functions of the electrons in a crystal have the Bloch form Ψ k (r) = e ik⋅r uk (r),

(3.1)

where uk (r) is a function that has the periodicity of the lattice, that is, uk (r) = uk (r + Tm ) where Tm is any primitive lattice translation. The Bloch function, in general, depends on the wave vector k, and is just a free electron wave function, exp (ik ⋅ r), modulated by a function that has the periodicity of the lattice. That is, it is a modulated plane wave. We present a proof of Bloch’s theorem in the one-dimensional case. Consider a crystal of length L = Na, where N is the number of primitive unit cells, and a is the length of each cell. The wave equation is Engineering Quantum Mechanics, First Edition. Doyeol Ahn, Seoung-Hwan Park. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 73

c03.indd 73

4/20/2011 10:43:32 AM

74

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

−

2 d 2Ψ + V ( x)Ψ = EΨ 2m dx 2

(3.2)

with V ( x) = V ( x + a). The periodic boundary condition requires that Ψ( x) = Ψ( x + L), and the translational symmetry requires that the charge density is the same at x as at x + ma, where m is any integer, 2 2 that is, Ψ( x + ma) = Ψ( x) . Therefore Ψ( x + a) = AΨ( x),

(3.3)

Ψ( x + ma) = AmΨ( x),

(3.4)

and

where A is a complex number such that A* A = 1 (the charge density is proportional to Ψ*Ψ). By applying the translation N times, we have Ψ( x + Na) = AN Ψ( x) = Ψ( x)

(3.5)

AN = 1

(3.6)

A = exp(2 πim/N ), m = 0, ±1, ±2,…

(3.7)

Equation (3.5) shows that A is just one of the roots of unity, which is written explicitly in Equation (3.7). Thus, the wave functions must be of the form Ψ( x + a) = exp(2 πim/N )Ψ( x)

(3.8)

which is the Bloch condition on the wave function. A function that satisfies this condition has the requirements demanded by translational symmetry. This result suggests that in seeking an electron wave function (i.e., a function that satisfies Eq. (3.2)), we should try a modulated free electron function, Ψ( x) = exp(ikx)u( x), and determine the conditions that allow this function to satisfy the Bloch condition, Equation (3.8). Substituting this modulated plane wave in both sides of Equation (3.8), we see that the equality is kept if

c03.indd 74

k = 2 πm/Na

(3.9)

u( x + a) = u( x).

(3.10)

4/20/2011 10:43:32 AM

75

BLOCH THEOREM AND EFFECTIVE MASS THEORY

If u( x) is a constant, then the free electron wave function with the same condition on k is recovered. Thus, a solution of the wave equation with the lattice periodicity can be written in one dimension in the form Ψ( x) = exp(ikx)u( x)

(3.11)

with k = 2 πm/Na and u( x + a) = u( x). This proves Bloch’s theorem in one dimension. The extension to three dimensions, Equation (3.1), follows straightforwardly. 3.1.2 The Schrödinger Equation and the Effective Mass Equation The effective mass equations have been widely used to treat problems involving valence bands, such as quantum wells and superlattices. In this section, we will give a brief heuristic derivation of the main results. The reader should consult Reference [1] for additional details. The one-electron Schrödinger equation describing the dynamics of an individual electron is given by i

∂ 2 2 Ψ 0 ( r, t ) = − ∇ Ψ 0 (r, t ) + UT (r, t )Ψ 0 (r, t ), 2 m0 ∂t

(3.12)

where UT (r, t ) is the total potential energy. Here, we used the subscript “0” for Ψ to distinguish it from the corresponding symbol in the effective mass equation, as discussed below. The potential energy UT (r, t ) can be separated into macroscopic part and a microscopic part. The macroscopic potential U E (r, t ) arises from macroscopic electric fields due to any externally applied voltage or macroscopic space charge. Also, an electron feels the microscopic electric fields due to the nearly periodic lattice of ions and the other valence electrons. We can separate it into two parts: A perfectly periodic potential U L (r, t ) due to a perfect static lattice, plus a random fluctuating potential U S (r, t ) representing the deviations. That is, UT (r, t ) = U L (r, t ) + U S (r, t ) + U E (r, t ).

(3.13)

The randomly fluctuating potential U S (r, t ) is called the scattering potential, which can arise from impurity scattering, phonon scattering, or electron–electron scattering. Then, Equation (3.12) is written as i

c03.indd 75

∂ Ψ 0 (r, t ) = H 0 Ψ 0 (r, t ) + U (r, t )Ψ 0 (r, t ), ∂t

(3.14)

4/20/2011 10:43:33 AM

76

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

where U (r, t ) = U E (r, t ) + U S (r, t ), H0 = −

2 2 ∇ + U L (r, t ). 2 m0

(3.15)

If we consider only the periodic potential U L (r, t ), the solutions to the Schrödinger equation can be written in the form of Bloch waves, as discussed in Section 3.1.1. The wave function of an electron in a band n with wave vector k is written as Ψ 0 (r, t ) = un,k (r) exp(ik ⋅ r) exp(− iε n (k)t/),

(3.16)

where un,k is a periodic function like U L (r) that is different for each n and each k. The function ε n (k) is known as energy band structures. The calculated band structures can be used to write an effective mass equation in which U L (r) does not appear explicitly. That is, if we are interested in electrons in a band n , then the effective mass equation is written as i

∂ Ψ(r, t ) = ε n (− i∇)Ψ(r, t ) + U (r, t )Ψ(r, t ). ∂t

(3.17)

The periodic lattice potential U L does not appear directly in the effective mass equation. This makes Equation (3.17) much easier to solve than the original Schrödinger equation, Equation (3.14). Here, ε n (− i∇) is obtained by replacing k with − i∇ in the function ε n (k). That is, kx with − i∂/∂x , k y with − i∂/∂y, and kz with − i∂/∂z. Equation (3.17) is known as the single-band effective mass equation, and can be used if the potential U (r, t ) introduces only negligible coupling to the other bands. As a result, it is inapplicable to the valence band, where multiple bands (“heavy” and “light” holes) overlap in energy, and even a weak static potential can induce interband transitions. In that case, we need multiband effective mass equation. We will consider the relation between the envelope function Ψ(k, t ) and the real wave function Ψ 0 (k, t ) in Equation (3.12) for two cases, single and multiband effective mass equations, in the following two subsections, to understand the approximations involved in using Equation (3.17). 3.1.3 The Single-Band Effective Mass Equation The eigenfunctions of the operator H 0 in Equation (3.15) can be written using two indices—the band index n and the wave vector k.

c03.indd 76

4/20/2011 10:43:33 AM

BLOCH THEOREM AND EFFECTIVE MASS THEORY

H 0 ψ nk = ε n ( k ) ψ nk ,

77

(3.18)

where ψ nk = e ik⋅r unk (r) =

e ik⋅r unk (r). V

(3.19)

The functions ε n (k) are readily available for most semiconductors, though the periodic functions unk are not available. The functions ψ nk in Equation (3.19), being eigenfunctions of the Hermitian operator H 0, are orthogonal, and are normalized to some arbitrary volume V ( ψ nk ψ n′k′ = δ nn′ δ kk′ ). That is,

∫

(3.20)

= δ nn′.

(3.21)

ψ nk ψ n′k = δ nn′ = d 3run*k unk. Hence,

∫ d ru 3

* nk n ′ k

u

Since the functions unk are periodic with the same period as the lattice, the integral in Equation (3.21) yields the same value in each unit cell. If the volume V contains N unit cells, we can write

∫

unit cell

d 3run*k un′k = 1/Nδ nn′

(3.22)

As the periodic potential U L (r) goes to zero, the functions unk in Equation (3.19) become constants equal to one. That is, Equation (3.19) then becomes just the plane wave states. 3.1.3.1 Actual Wave Function Y0 (r , t) The wave functions Ψ 0 (r, t ) in Equation (3.14) are expanded by using eigenfunctions ψ nk as a basis set. Ψ 0 ( r, t ) =

∑ϕ

nk

(t ) ψ nk .

(3.23)

nk

Then, Equation (3.14) can be written in matrix form as follows: i

d ϕ nk ( t ) = dt

∑ψ

nk

H 0 + U ψ nk′ ϕ n′k′ (t ).

(3.24)

n′ k ′

Since the function ψ nk are eigenfunctions of H 0 with eigenvalues ε n (k), we have

c03.indd 77

4/20/2011 10:43:34 AM

78

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

ψ nk H 0 ψ n′k′ = ε n′ (k ′) ψ nk H 0 ψ n′k′ = ε n (k)δ nn′ δ kk′.

(3.25)

Hence, we can simplify Equation (3.24) to i

d ϕ nk (t ) = ε n (k)ϕ nk (t ) + dt

∑U

nk ; n ′ k ′

ϕ n′k′ (t ),

(3.26)

n′ k ′

where

∫

U nk ;n′k′ = ψ nk U ψ n′k′ ≡ d 3run*k (r)un′k′ (r)e ik⋅r e i ( k′− k )⋅rU (r, t ).

(3.27)

3.1.3.2 Envelope Function Y(t , t) Next, we use the plane waves ψ k = e ik⋅r / V to expand the envelope function Ψ(k, t ). Ψ(r, t ) =

∑ ϕ (t )ψ . k

(3.28)

k

k

The effective mass equation can then be written in matrix form as follows: i

d ϕ k (t ) = dt

∑ψ

k

En (− i∇) + U ψ k′ ϕ k′ (t ).

(3.29)

k′

Using the relation q

⎛ − i ∂ ⎞ ⎛ − i ∂ ⎞ ⎛ − i ∂ ⎞ ψ = ( k ) p ( k )q ( k ) r ψ , x′ y′ z′ k′ ⎝ ∂x ⎠ ⎜⎝ ∂y ⎟⎠ ⎝ ∂z ⎠ k′ p

r

(3.30)

we obtain ε n ( − i ∇ )ψ k ′ = ε n ( k )ψ k ′ .

(3.31)

Thus, using the orthogonality relation ψ k ψ k′ = δ kk′, we obtain ψ k ε n (− i∇) ψ k′ = ε n (k)δ kk′.

(3.32)

Using Equation (3.32), we can write Equation (3.29) as i

c03.indd 78

d ϕ k (t ) = ε n (k )ϕ n,k (t ) + dt

∑U

k ;k ′

ϕ k′ (t ).

(3.33)

k′

4/20/2011 10:43:34 AM

BLOCH THEOREM AND EFFECTIVE MASS THEORY

79

where U kk′ = ψ k U ψ k′ ≡

∫

d 3r i ( k −k )⋅r e U (r, t ). V

(3.34)

3.1.3.3 The Relation betwen Y0 (r , t) and Y(t , t) Now let us compare Equations (3.26) and (3.33). Equation (3.26) is the matrix representation of the Schrödinger equation (Eq. (3.15)) using the Bloch function ψ nk as the basis, while Equation (3.33) is the matrix representation of the effective mass equation (Eq. (3.17)) for a particular band using the plane waves ψ k as the basis. The two equations would be identical if the following were true: ψ nk U ψ n′k′ = ψ k U ψ k′ δ nn′.

(3.35)

That is, the matrix elements of the external potential between states from different bands should be negligible. This is the basic approximation on which the accuracy of the effective mass equation rests. It is fairly accurate for potentials U that are slowly varying on an atomic scale. It is easy to see that for a constant potential U (r, t ) = U o , Equation (3.35) is exact because of the orthogonality of the eigenfunctions. ψ nk U ψ n′k′ = U o ψ nk ψ n′k′ = δ nn′U oδ kk′.

(3.36)

ψ k U o ψ k′ = U oδ kk′.

(3.37)

In the case of a spatially varying potential U, we start from Equation (3.27) to prove Equation (3.35). The right-hand side requires us to integrate over the entire sample volume V . But first, let us consider the effect of integrating over a single unit cell, say the nth one.

∫

nth unit cell

d 3run*k (r)un′k′ (r)e i ( k′− k )⋅rU (r, t )

∫

 e i ( k′− k )⋅rnU (rn, t )

nth unit cell

d 3run*k (r)un′k′ (r)

(3.38)

 e i ( k′− k )⋅rnU (rn, t )δ nn′ (1/N ), where rn is the coordinate of the center of the nth unit cell. Here we have made two assumptions: (1) The factors exp[i(k ′ − k) ⋅ r] and U (r, t ) vary slowly enough that they can be considered approximately constant * (r) do not change very much with over a unit cell. (2) The functions unk k. That is, unk′ (r)  unk (r) over the range of values of k ′ − k , for which

c03.indd 79

4/20/2011 10:43:34 AM

80

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

U kk′ is significant. These approximations are inherent in the use of the effective mass equation, since its validity rests on the accuracy of Equation (3.35). Then,

∫

d 3run*k (r)un′k′ (r) 

unit cell

∫

d 3run*k (r)un′k (r) = δ nn′ (1/N ).

(3.39)

unit cell

We can now use Equation (3.38) to write Equation (3.27) as

∫ = ∑ ∫ d ru

ψ nk U ψ n′k ′ = d 3 run*k (r)un′k ′ (r)e ik⋅r e i ( k ′− k )⋅rU (r, t ) N

3

* nk

(rn )un′k ′ (rn )e ik⋅rn e i ( k ′− k )⋅rnU (rn, t )

n=1

 δ nn′  δ nn′ = δ nn′

1 N

N

∑e

i ( k ′− k )⋅rn

U (rn, t ) [from Eq. (3.38)]

(3.40)

n=1

d 3 r i ( k ′− k )⋅r e U ( r, t ) V ψ k ′ U ψ k [from Eq. (3.34)].

∫

This proves Equation (3.35). We can derive a very simple relation between the envelope function Ψ(r, t ) and the real wave function Ψ o (r, t ) . We can rewrite Equation (3.23) as Ψ 0 ( r, t ) =

∑ϕ

nk

(t )e ik⋅r unk (r).

n,k

∑ϕ

 unk (r)

(3.41) nk

(t )e

ik⋅r

.

n,k

Using Equation (3.28), Ψ 0 (r, t )  unk (r)Ψ(r, t ).

(3.42)

The true wave function is thus approximately equal to the product of the envelope function and the periodic part of the Bloch wave function. 3.1.4 The Multiband Effective Mass Equation The multiband effective mass equation uses a slightly different basis set χ nk to the wave function Ψ 0 (r, t ).

c03.indd 80

4/20/2011 10:43:35 AM

BLOCH THEOREM AND EFFECTIVE MASS THEORY

Ψ 0 ( r , t ) = ϕ nk ( t ) ψ nk ,

81

(3.43)

where ψ nk =

e ik⋅r un 0 (r) = e ik⋅r un 0 (r). V

(3.44)

Comparing Equation (3.44) with Equation (3.19), we can see that the difference is that we are now using the functions unk (r) at k = 0. Consequently, in our derivation, it will not be necessary to assume that the function unk (r) does not change with k. However, we encounter the complexity of having to solve multiple coupled differential equations instead of the single differential equation, Equation (3.17). Using this new basis set, the Schrödinger equation is written as before. i

∑

d ϕ n , k (t ) = ψ nk H 0 + U ψ n′k′ ϕ n′ ,k′ (t ). dt n′ , k ′

(3.45)

Consider first the matrix elements of U ,

∫

ψ nk U ψ n′k′ = d 3re i ( k′− k )⋅r un* 0 (r)un′ 0 (r)U (r, t ).

(3.46)

Using Equation (3.40), ψ nk U ψ n′k′  δ nn′

1 N

N

∑e

i ( k ′− k )⋅rn

U (rn, t )

n=1

d 3r i ( k′− k )⋅r e U ( r, t ) V = δ nn′U kk′,  δ nn′

∫

(3.47)

where U kk′ =

∫

d 3r i ( k′− k )⋅r e U (r, t ). V

(3.48)

In the last step, we have used Equation (3.34); ψ k corresponds to the plane wave states as in Equation (3.28). Note that we do not need the additional assumption that unk (r) does not vary much with k. However, the matrix representation of H 0 is no longer diagonal, since the basis functions in Equation (3.44) are not eigenfunctions of H 0 , unlike the basis functions in Equation (3.19). It can be shown that

c03.indd 81

4/20/2011 10:43:35 AM

82

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

ψ nk H 0 ψ n′k′ = H 0nn′ (k)δ k′k

(3.49)

⎧ 2k 2 ⎪⎪ 2 m + ε n 0, n′ = n o nn′ H 0 (k) = ⎨ ⎪ Pnn′ ⋅ k , n′ ≠ n ⎪⎩ mo

(3.50)

where

Pnn′ is called the momentum matrix element between bands n and n′, and is defined by Pnn′ = (−i) un*′ 0 ∇un 0

(3.51)

ε n0 is the energy at the edge (k = 0, Γ-point) of band n. The derivation is as follows: We have H0 ψ n0 = ε n0 ψ n0

(3.52)

since ε n0 is the energy eigenvalue of band n at k = 0. From Equation (3.44), ψ nk = e ik⋅r ψ n 0.

(3.53)

− i∇ψ nk = e ik⋅r (k − i∇)ψ n 0

(3.54)

−  2∇ 2 ψ nk = e ik⋅r ( 2 k 2 − 2i 2 k ⋅∇ −  2∇ 2 )ψ n 0.

(3.55)

Hence,

Using H 0 from Equation (3.15), we have i 2 k  2∇ 2 ⎞ ⎛ ⋅∇ + H 0 ψ nk = e ik⋅r ⎜ H 0 − ψ n0 ⎝ 2 mo ⎟⎠ mo

(3.56)

2k 2 ⎞ e ik⋅r k ⎛ = ⎜ ε n0 + ψ nk + ⋅ (− ∇ψ n 0 ). ⎟ ⎝ 2 mo ⎠ mo

(3.57)

Hence, 2k 2 ⎞ k ⎛ ψ nk H 0 ψ n ′ k ′ = ⎜ ε n 0 + δ n′nδ k′k + ⋅ ψ n′ 0 e i ( k − k′ )⋅r (−i∇) ψ n 0 ⎟ ⎝ 2 mo ⎠ mo (3.58)

c03.indd 82

4/20/2011 10:43:35 AM

BLOCH THEOREM AND EFFECTIVE MASS THEORY

2k 2 ⎞ k ⎛  ⎜ ε n0 + δ n′nδ k′k + ⋅ ψ n′ 0 e i ( k − k′ )⋅r (−i∇) ψ n 0 . ⎝ 2 mo ⎟⎠ mo

83

(3.59)

Here, ψ n′ 0 e i ( k − k ′ )⋅r (− i∇) ψ n 0

∫ ⎡ = ⎢∑ e ⎣

= d 3 run* ′ ,0 e i ( k − k ′ )⋅r (−i∇)un 0 ⎤ 3 * ⎥ (− i) nth unit cell d run′ 0 ∇un 0 from Equattion (3.22) ⎦ n=1 (3.60) N ⎤ 1⎡ i ( k − k ′ )⋅rn 3 * = ⎢ e ⎥ N (− i) nth unit cell d run′ 0 ∇un 0 N ⎣ n=1 ⎦ 1 i ( k − k ′ )⋅r 3 ⎤ e d r ⎥ (− i) d 3 run* ′ 0 ∇un 0 = ⎡⎢ ⎦ ⎣V = δ kk ′ Pnn′ (from Eq. (3.51)) N

∫

i ( k − k ′ )⋅rn

∑

∫

∫

∫

2k 2 ⎞ Pnn′ ⋅ k ⎛ ψ n ′ k ′ H 0 ψ nk = ⎜ ε n 0 + δ k′k. ⎟ δ n′nδ k′k + ⎝ 2 mo ⎠ mo

(3.61)

Noting that Pnn′ is usually zero. Using Equations (3.46) and (3.49), we can write Equation (3.45) as i

d ϕ nk ( t ) = dt

∑H

nn′ 0

ϕ n′ k +

n′

∑U

kk ′

ϕ nk ′ .

(3.62)

k′

The corresponding multiband effective mass equation is i

d Ψ n ( r, t ) = dt

∑H

nn′ 0

(− i∇)Ψ n (r, t ) + U (r, t )Ψ n (r, t ).

(3.63)

n′

To show that Equation (3.62) is equivalent to Equation (3.63), we just have to write down the matrix representation of Equation (3.63) using a plane wave basis set to expand Ψ n (r, t ). Ψ n ( r, t ) =

∑ϕ

nk

(t )ψ k .

(3.64)

k

The matrix form of Equation (3.63) then follows from our usual prescription.

c03.indd 83

4/20/2011 10:43:36 AM

84

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

i

∂ ϕ nk = ∂t

∑∑ ψ k

k

H 0nn′ (− i∇) ψ k′ ϕ n′k′ +

n′

∑ψ

k

U ψ k′ ϕ nk′. (3.65)

k′

Equation (3.62) follows from Equation (3.65) if we note that ψ k H 0nn′ (− i∇) ψ k′ = H 0nn′ (k )δ kk′.

(3.66)

This relation is proved in exactly the same way we proved Equation (3.49). The relation between the actual wave function and the multiband envelope function follows readily from Equation (3.43).

∑ = ∑u

Ψ 0 ( r, t ) =

ϕ nk ( t )

nk

n0

e ik⋅r un,0 (r). V

∑ϕ

(r)

nk

(3.67)

(t )ψ k .

nk

n

Using Equation (3.64), Ψ 0 ( r, t ) =

∑u

n0

(r)Ψ n (r, t ).

(3.68)

n

Note that this result is exact, and we do not need to assume that unk (r) does not change much with k. 3.2 THE LUTTINGER–KOHN HAMILTONIAN The difficulty with the multiband effective mass equation of Equation (3.63) is that the differential equations for different bands n are all coupled together, unlike the single-band effective mass equation of Equation (3.17). For practical use, we need to limit ourselves to a few bands. However, if we simply truncate the multiband effective mass equation to the desired number of bands, we will run into some problems. One method is to renormalize the elements of the truncated matrix to account for the bands we are neglecting using perturbation theory. On the other hand, this method is difficult because the functions un,0 (r) are often not known accurately for remote bands. Another method is to write each of the matrix elements H n,n′ (k) as a secondorder polynomial in k , and treat the polynomial coefficients as fitting parameters so that eigenvalues of the matrix (H(k)) reproduce the correct energy dispersion relation for the bands. Here, we will review the derivation of the 8 × 8 Hamiltonian in the valence band following the approach of Luttinger and Kohn [2].

c03.indd 84

4/20/2011 10:43:36 AM

85

THE LUTTINGER–KOHN HAMILTONIAN

3.2.1

Nondegenerate Bands Without Spin–Orbit Coupling

Let H 0 be the Hamiltonian of the electron in the periodic potential U L (r) . The eigenfunctions of H 0 will be denoted by ψ nk and the corresponding eivenvalues by ε n (k). Here, n labels the band, and k is the wave vector in the first Brillouin zone. Then, the one-electron Schrödinger equation is given by ( H 0 + U )ψ = εψ ,

(3.69)

where U (r) = U E (r) + U S (r), 2 2 ∇ + U L (r), 2 m0 = ε n ( k )ψ nk .

H0 = − H 0 ψ nk

(3.70)

If we consider only the periodic potential U L (r), the solutions to the Schrödinger equation can be written in the form of Bloch waves, as discussed in Section 3.1.1. The wave function of an electron in a band n with wave vector k in the first Brillouin zone of the crystal is written as ψ nk = e ik⋅r unk (r)

(3.71)

where unk is a periodic function like U L (r) that is different for each n and each k. Now, it is necessary to choose a complete set of functions in which to expand ψ. The ψ nk form, of course, is a complete set of functions, in which any wave function can be expanded. However, we consider another set of functions as defined in Equation (3.44) χ nk = e ik⋅r un 0.

(3.72)

These form a complete set if ψ nk is a complete set, as proved below. Assume that any function f (r) is expanded in the ψ nk: f (r) =

∑ ∫ d k g ( k )ψ n

n

nk

=

∑ ∫ dkg (k)e n

ik⋅r

unk .

(3.73)

n

Also, the periodic function unk can be expressed in terms of the Bloch functions at the bottom of the band. That is,

c03.indd 85

4/20/2011 10:43:36 AM

86

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

unk =

∑b

nn′

(k)un′ 0.

(3.74)

n

Substituting this into Equation (3.73), we obtain the result f (r) =

∑ ∫ dkg

n′

(k)e ik⋅r

n′

=

n′ n

(k)un 0

n

⎛

∑ ∫ dk ⎜⎝ ∑ g

n′

n′

n

≡

∑b

∑ ∫ dkg (k)χ n

⎞ (k)bn′n (k)⎟ ( e ik⋅r un 0 ) ⎠

(3.75)

nk

n

with gn (k) = ∑ n′ gn′ (k)bn′n (k). The χ nk satisfy the orthonomality if the ψ nk do. That is, the Bloch waves satisfy the following condition ψ nk ψ n ′ k ′ =

∫

entire crystal

ψ *nk ψ n′k′ dr = δ nn′ δ k − k′.

(3.76)

Thus,

∫

entire crystal

e i ( k′− k )⋅r unk (r)un′k (r)dr = δ nn′

1 e i ( k′− k )⋅r dr. (2 π)3

∫

(3.77)

For k ′ = 0 and k = 0,

∫

entire crystal

V . (2 π)3

(3.78)

e i ( k′− k )⋅r un* 0un′ 0 dr.

(3.79)

un 0 (r)un′ 0 (r)dr = δ nn′

For the χ nk, χ nk χ n ′ k ′ =

∫

entire crystal

Since un* 0un′ 0 has the lattice periodicity, we can expand it in a Fourier series as follows: un* 0un′ 0 =

∑B

nn′ − iK m ⋅r m

e

,

(3.80)

m

where the Bmnn′ are just numerical coefficients, and K m are the reciprocal lattice vectors. Inserting Equation (3.80) in Equation (3.79), we obtain

c03.indd 86

4/20/2011 10:43:36 AM

THE LUTTINGER–KOHN HAMILTONIAN

χ nk χ n′k′ = (2 π)3

∑B

nn′ m

δ(k ′ − k − K m ),

87

(3.81)

m

from the definition of the delta function δ( k ′ − k − K m ) =

1 e i ( k′− k −K m )⋅r dr. (2 π)3

∫

(3.82)

However, since k ′ and k are both in the first Brillouin zone, k ′ − k = K m is only possible if m = 0. χ nk χ n′k′ = (2 π)3 B0nn′ δ(k ′ − k).

(3.83)

Using Fourier’s theorem with Ω as the volume of the unit cell, the Bmnn′ are given by 1 NΩ

Bmnn′ =

∫

entire crystal

e i ( K m )⋅r un* 0un′ 0 dr = 1 Ω

B0nn′ =

∫

cell

1 Ω

un* 0un′ 0 dr =

∫

cell

e i ( K m )⋅r un* 0un′ 0 dr,

1 δ nn′, (2 π)3

(3.84) (3.85)

using Equation (3.78). Thus, from Equation (3.83), χ nk χ n′k ′ = δ nn′ δ(k ′ − k)

(3.86)

which is the required orthonormality. In Equation (3.69), we now make the Ansatz Ψ=

∑ ∫ d k ′A

′n

( k ′ )χ n ′ k ′ ,

(3.87)

n′

which gives the equation

∑ ∫ dk′A (k′)(H n′

0

+ U )χ n ′ k ′ = ε

n′

∑ ∫ dk′A (k′)∫ χ n′

nk

( H 0 + U )χ n ′ k ′ d r = ε

n′

nk

∑ ∫ d k ′A ( k ′ ) χ

nk

n′

n′

n′

n′ k ′

,

n′

∑ ∫ dk′A (k′)∫ χ n′

nk

χ n ′ k ′ d r,

n′

∑ ∫ d k ′A ( k ′ ) χ n′

∑ ∫ d k ′A ( k ′ )χ

( H 0 + U ) χ n′ k ′ = ε

∑ ∫ d k ′A

′n

( k ′ ) χ n′ k ′ χ n′ k ′ ,

n′

( H 0 + U ) χ n′k′ = εAn (k).

(3.88)

n′

c03.indd 87

4/20/2011 10:43:37 AM

88

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

The matrix elements can be evaluated in the following manner. For H 0, we have ⎡ 2 2 ⎤ H 0 χ nk = ⎢ − ∇ + U L (r)⎥ e ik⋅r un 0 2 m ⎣ ⎦ 0 2 2  ∇  2 2 ⎡ k ⎤ = e ik⋅r ⎢ + k⋅ − ∇ + U L (r)⎥ un 0 2 m m i 2 m ⎣ ⎦ 0 0 0

(3.89)

 2k 2 ⎤ ⎡ = e ik⋅r ⎢ H 0 + k⋅p+ un 0. m0 2 m0 ⎥⎦ ⎣ Thus, χ nk H 0 χ n ′ k ′

∫ = ∫e

= e − ik⋅r un* 0 H 0 e ik′⋅r un′ 0 dr  2k 2 ⎤ ⎡ k⋅p + u ⎢ H0 + un 0 dr m0 2 m0 ⎥⎦ ⎣  2k 2 ⎤ ⎡ = e i ( k′− k )⋅r un* 0 ⎢ ε n′ (0) + k⋅p + un 0 dr, m0 2 m0 ⎥⎦ ⎣ i ( k ′− k )⋅r * n0

(3.90)

∫

where ε n′ (0) is the energy at the bottom of the nth band, and p is the momentum operator (/i)∇. Since the entire factor multiplying e i( k′− k )⋅r is periodic, the same argument that leads from Equation (3.79) to (3.86) yields  2k 2 ⎤ ⎡ χ nk H 0 χ n′k′ = e i ( k′− k )⋅r un* 0 ⎢ ε n′ (0) + k⋅p+ un 0 dr, m0 2 m0 ⎥⎦ ⎣

∫

(3.91)

and  2k 2 ⎤ ⎡ k⋅p+ un* 0 ⎢ ε n′ (0) + un 0 = m0 2 m0 ⎥⎦ ⎣

∑C

nn′ − iK m ⋅r m

e

.

(3.92)

m

Thus, χ nk H 0 χ n′k′ = (2 π)3

∑C

nn′ m

δ(k ′ − k − K m ).

(3.93)

m

c03.indd 88

4/20/2011 10:43:37 AM

THE LUTTINGER–KOHN HAMILTONIAN

89

Similarly, since k ′ − k = K m is only possible if m = 0, χ nk H 0 χ n′k′ = (2 π)3 C0nn′ δ(k ′ − k).

(3.94)

Using Fourier’s theorem with Ω, the volume of the unit cell, the C mnn′ are given by  2k 2 ⎤ ⎡ e i ( K m )⋅r un* 0 ⎢ ε n′ (0) + k⋅p + un 0 dr cell m0 2 m0 ⎥⎦ ⎣ 1  2k 2 ⎤ ⎡ = un* 0 ⎢ ε n′ (0) + un 0 dr k⋅p+ m0 2 m0 ⎥⎦ Ω cell ⎣ 1 Ω

C mnn′ = C0nn′

∫

(3.95)

∫

Thus,  2k 2 ⎤ ⎡ un* 0 ⎢ ε n′ (0) + k⋅p+ un′ 0 dr cell m0 2 m0 ⎥⎦ ⎣ (3.96) α 2k 2 ⎞ kα pnn ⎡⎛ ⎤ ′ = δ(k ′ − k) ⎢⎜ ε n (0) + δ nn′ + 2 m0 ⎟⎠ m0 ⎥⎦ ⎣⎝

χ nk H 0 χ n ′ k ′ =

(2 π)3 δ( k ′ − k ) Ω

∫

Here, dummy indices mean a summation over α = x, y, z, and the α momentum matrix element pnn′ at the bottom of the band is defined by α pnn ′ =

(2 π)3 Ω

 un* 0 ⎛ ∇α ⎞ un′ 0 dr. ⎝i ⎠ cell

∫

(3.97)

α (k) of the momentum is given by Also, the expectation value pnn α pnn (k) = m∂ε n (k)/∂kα. Thus,

α pnn =

(2 π)3 Ω

 (2 π) un* 0 ⎛ ∇α ⎞ un 0 dr = ⎝i ⎠ cell Ω

∫

3

∫

cell

 ψ *n 0 ⎛ ∇α ⎞ ψ n 0 dr ⎝i ⎠

(3.98)

is the expectation value at the bottom of the band and becomes zero at the minimum. In addition, they have the following properties: α α α * pnn ′ = pn′ n = ( pn′ n )

(3.99)

which follows if a center of symmetry exists in the crystal. For the matrix elements of U , we proceed similarly:

c03.indd 89

4/20/2011 10:43:37 AM

90

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

χ nk U χ n ′ k ′

∫ = ∫e = ∑∫e

= e − ik⋅r un* 0Ue ik′⋅r un′ 0 dr i ( k ′− k )⋅r * n0 n0

u u Udr

i ( k ′− k )⋅r

(3.100)

nn′ − iK m ⋅r m

B e

Udr

m

= (2 π)3

∑∫ B

nn′ m

U (k − k ′ + K m ),

m

where U (k − k ′ + K m ) =

1 dre − i ( k − k′+ K m )⋅rU (r). 3 (2 π)

∫

(3.101)

The right-hand side requires us to integrate over the entire sample volume V . But first, let us consider the effect of integrating over a single unit cell, say the nth one. 1 (2 π)3

∫

nth unit cell

1 U (rn ) dre − i ( k − k′+ K m )⋅r nth unit cell (2 π)3 1 = U (rn )δ(k − k ′ + K m ) (3.102) N

∫

dre − i ( k − k′+ K m )⋅rU (r) 

where rn is the coordinate of the center of the nth unit cell. Here we assumed that U (r, t ) varies slowly enough that it can be considered approximately constant over a unit cell. Similarly, k ′ − k = K m is only possible if m = 0. χ nk U χ n′k′ = (2 π)3 B0nn′U (k − k ′) = δ nn′U (k − k ′)

(3.103)

from Equation (3.85). From Equations (3.91) and (3.103), Equation (3.90) becomes α 2k 2 ⎞ kα pnn ⎛ ′ An′ (k) + dk ′U (k − k ′) An (k) ⎜⎝ ε n (0) + ⎟⎠ An (k) + 2 m0 m0 n′≠ n

∑

= εAn (k).

∫

(3.104)

α Equation (3.104) still contains terms involving pnn′ , which are proportional to k, and represent coupling between bands. Since the inter-

c03.indd 90

4/20/2011 10:43:38 AM

THE LUTTINGER–KOHN HAMILTONIAN

91

α band matrix elements pnn′ are causing the trouble, we try to find a transformation to yield equations with no interband elements to the first order in k. To do so, we remove them to the first order by a canonical transformation T. That is, we put

An (k) =

∑ ∫ dk χ

nk

T χ n′k′ Bn′ (k).

(3.105)

n′

It is convenient to write this as HA = εA,

(3.106)

A = TA ≡ e S B.

(3.107)

and

Substituting Equation (3.107) into Equation (3.106), we obtain HA = He S B = εA = εe S B e − S He S B = εB ≡ HB.

(3.108)

Clearly, 1 H = e − S He S = H + [ H , S] + [[ H , S], S] + … 2

(3.109)

Let us write H = H (0) + H (1) + U , where 2k 2 ⎞ ⎛ nk H (0) n′ k ′ = ⎜ ε n (0) + δ nn′ δ(k ′ − k) ⎝ 2 m0 ⎟⎠ nk H (1) n′ k ′ =

kα pnαn′ δ( k ′ − k ) m0

(3.110)

nk U n′ k ′ = U (k ′ − k)δ nn′. Then, H = H (0) + H (1) + U + [ H (0), S] + [ H (1) , S] + [U , S] 1 1 + [[ H (0), S], S] + [[U , S], S] +…, 2 2

(3.111)

the omitted terms being of order S 3 or more. By choosing S such that

c03.indd 91

4/20/2011 10:43:38 AM

92

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

H (1) + [ H (0), S] = 0,

(3.112)

we succeeded in eliminating the interband transitions to the first order in k. Going back to the χ nk representation, we see at once that χ nk H (1) χ n′k ′ + χ nk H (0)S − SH (0) χ n′k ′ = 0 α kα pnn ′ δ(k ′ − k) + (ε n − ε n′ ) χ nk S χ n′k ′ = 0 m0 α ⎧ kα pnn ′ δ(k ′ − k), n ≠ n′ ⎪− = ⎨ m0ω nn′ , ⎪⎩ 0, n = n′

χ nk S χ n′k ′

(3.113)

(3.114)

where ω n,n′ = ε n − ε n′. Consequently, the second-order terms from H (0) and H (1) are 1 1 [ H (1), S] + [[ H (0), S], S] = [ H (1), S], 2 2

(3.115)

which becomes 1 1 χ nk [ H (1), S] χ n′k ′ = 2 2 − 1 = 2

∑χ

nk

H (1) χ n′′k ′′ χ n′′k ′′ S χ n′k ′

n′′

1 2

∑χ

S χ n′′k ′′ χ n′′k ′′ H (1) χ n′k ′

n′′

∑ n′′

nk

α kα pnn ⎡ kβ′′pnβ′′n′ ⎤ ′′ δ(k − k ′′) ⎢ − δ(k ′′ − k ′)⎥ m0 ⎣ m0ω n′′n′ ⎦

β ⎡ kβ pnn ⎤1 ′′ − ⎢− δ(k − k ′′)⎥ ⎣ m0ω nn′′ ⎦2

∑ n′′

kα′′pnα′′n′ δ(k ′′ − k ′) m0 (3.116)

Using δ function, 1 χ nk [ H (1), S] χ n′k ′ 2 kα kβ ⎡ 1 ⎞⎤ ⎛ 1 β α = + δ(k − k ′), p p ⎢ nn ′′ n n ⎜ ′′ ′ ⎝ ω nn′′ ω n′n′′ ⎟⎠ ⎥⎥ 2 m02 ⎢⎣ n′′≠ n,n′ ⎦

(3.117)

∑

c03.indd 92

4/20/2011 10:43:38 AM

THE LUTTINGER–KOHN HAMILTONIAN

93

in the χ nk representation. The correction terms to U are also easily found. Consider first χ nk [U , S] χ n′k ′ . Using Equations (3.110) and (3.114), we obtain χ nk [U , S] χ n′k ′ =

∑χ

nk

U χ n′′k ′′ χ n′′k ′′ S χ n′k ′ −

n′′

=

∑ n′′

−

∑χ

nk

S χ n′′k ′′ χ n′′k ′′ U χ n′k ′

n′′

⎡ k ′′pα ⎤ δ nn′′U (k − k ′′) ⎢ − α n′′n′ δ(k ′′ − k ′)⎥ ω m ⎣ ⎦ 0 n′′ n′

∑ n′′

(3.118)

α ⎡ kα pnn ⎤ ′′ − ⎢⎣ m ω δ(k − k ′′)⎥⎦ δ n′n′′U (k ′ − k ′′). 0 nn′′

Thus, α n ≠ n′ ⎧(kα − kα′ )U (k − k ′′) pnn ′ /( m0ω nn′ ), . χ nk [U , S] χ n′k ′ = ⎨ 0, n = n′ ⎩

(3.119)

Here, k ~ 1/ai, where ai is the extent of the impurity state when the U is the potential due to the impurities. pnn′ ~ 1/a and ω nn′ ~ 1/(moa2 ), where a is the lattice spacing. Equation (3.119) is of the order of the square of the ratio of a/ai , and can be neglected. Similarly, the next term, 1 2 [[U , S ], S ], is also omitted because it is reduced over [U , S ] by a factor S, which is of the squared order of a/ai. In the limit of very extended impurity states, Equation (3.104) becomes ⎛ ⎤⎞  2 k 2  2 kα kβ ⎡ β ⎛ 1 ⎞ α ε (0) + + pnn B (k) ⎢ n ′′ pn′′ n ⎜ ⎟ ⎜ 2 ⎝ ω nn′′ ⎠ ⎥⎥⎟⎠ n 2 m0 m0 ⎢⎣ n′′ ,≠ n ⎝ ⎦

∑

(3.120)

∫

+ dk ′U (k − k ′)Bn (k ′) = εBn (k) where we have neglected terms of the order k 3 and interband elements of order k 2. Since the coefficient of Bn (k) is exactly the expansion of ε n (k) to second order, we can write

∫

ε n (k)Bn (k) + dk ′U (k − k ′)Bn (k ′) = εBn (k).

(3.121)

Equation (3.121) is the well-known effective mass equation written in momentum space. To get the more usual formulation, we introduce a function

c03.indd 93

4/20/2011 10:43:38 AM

94

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

∫

Fn (r) = e ik⋅r Bn (k)dk,

(3.122)

where the integration is over the first Brillouin zone. Inverting Equation (3.122), we obtain Bn (k) =

Ω (2 π)3

∑e

− ik⋅R m

(3.123)

Fn (R m ),

m

since Bn (k) is periodic in k, R m being the vector to the lattice point m. Then, Equation (3.121) becomes simply

∫

ε n (− i∇)Fn (r) + dr ′Δ(r − r ′)U (r ′)Fn (r ′) = εFn (r),

(3.124)

where Δ(r − r ′) =

1 dke ik⋅( r − r′ ) . (2 π)3

∫

(3.125)

Since Δr is a δ-like function of extension ~ a, we have [ε n (− i∇) + U (r)]Fn (r) = εFn (r).

(3.126)

From Equation (3.87), the leading term in the wave function becomes Ψ=

∑ ∫ dkB (k)e n

n

3.2.2

ik⋅r

un 0 (r) =

∑ F (r)u n

n0

(r) =

n

∑ F (r)Ψ n

n0

(r).

(3.127)

n

Degenerate Bands Without Spin–Orbit Coupling

We now consider the case of a degenerate band neglecting spin–orbit coupling. For simplicity, we shall limit ourselves to the case where the degeneracy occurs at the point k = 0. Let us call the degenerate functions at this point φ1,φ 2,…,φr, a typical one being denoted by φ j. The degeneracy implies that they all have the same energy: H 0 φ j = ε 0 φ j.

(3.128)

Let us further make the convention that the wave functions at k = 0 for the other bands be denoted by φi (i ≠ j ). We now introduce a complete set of functions φ nk defined by φ nk = e ik⋅r φ n,

c03.indd 94

(3.129)

4/20/2011 10:43:39 AM

THE LUTTINGER–KOHN HAMILTONIAN

95

the index n running over both j and i. We then make the Ansatz Ψ=

∑ ∫ dkA (k)φ n

nk

(3.130)

.

n

For the impurity scattering problem,

∑ ∫ dk ′ nk H

0

+ U n′ k ′ An′ (k ′) = εAn (k ′).

(3.131)

n′

The analysis leading from Equation (3.90) to Equation (3.104) is again valid due to the orthonormality of the φ nk. On specializing n to j in Equation (3.131), we obtain 2k 2 ⎞ ⎛ ε + ⎜⎝ 0 2 ⎟⎠ Aj (k) + m0

∑ i

kα pαji Ai (k) + dk ′U (k − k ′) Aj (k ′) = εAj (k) m0 (3.132)

∫

This is the equation that replaces Equation (3.104) for the degenerate case. We again transform the interband elements pαji by a transformation analogous to Equations (3.107) and (3.114). Here, the matrix elements of p vanish between states of the same parity, so that we can write pαjj′ = 0.

(3.133)

Then,

χ nk S χ n′k ′

α ⎧ kα pnn ′ δ(k ′ − k), n = j, n′ ≠ j ⎪− . = ⎨ m0ω nn′ ⎪⎩ 0, n = j , n′ = j

(3.134)

Since we are interested in degenerate bands, putting n = j and n′ = j ′ in Equation (3.105) leads to the result, to the second-order terms in k:

∑ j′

⎛  2 kα kβ 2k 2 ε δ + δ + 0 jj jj ′ ′ ⎜⎝ 2m m2 0

0

∑ i

pαji pijβ′ ⎞ B j ′ ( k ) + d k ′U ( k − k ′ ) B j ( k ′ ) ε 0 − ε i ⎟⎠

∫

= εBj (k).

(3.135)

If we set the energy zero at ε 0 , then we can write r

∑∑D

αβ jj ′ α β

αβ

c03.indd 95

j ′=1

∫

k k Bj ′ (k) + dk ′U (k − k ′)Bj (k ′) = εBj (k),

(3.136)

4/20/2011 10:43:39 AM

96

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

where Dαβ jj ′ ≡

2 2 δ jj ′ δ αβ + 2 m0 2 m0

∑ i

pαji pijβ′ . ε0 − εi

(3.137)

This set of numbers Dαβ jj′ plays the same role in the theory of degenerate bands that the effective masses do for a simple band. We again introduce a function Fj (r) as Equation (3.122),

∫

Fj (r) = e ik⋅r Bj (r)dk.

(3.138)

Then, Equation (3.136) becomes r

∑ ∑ ⎡⎢⎣D

αβ jj ′

αβ

j ′=1

⎛ 1 ∇ ⎞ ⎛ 1 ∇ ⎞ + U (r)δ ⎤ F (r) = εF (r)), jj ′ j ⎥⎦ j ′ ⎝ i α⎠ ⎝ i β⎠

(3.139)

and the leading term in the wave function is r

Ψ=

∑ F (r)φ (r). j

(3.140)

j

j =1

The Hamitonian can be written as the sum of a band-edge contribution and a k-dependent contribution H (k) jj ′ = E j (k = 0) +

∑D

αβ jj ′ α β

k k,

(3.141)

αβ

where E j (k = 0) = ε 0δ jj ′.

3.2.3

Spin–Orbit Coupling

In this section, we explain the spin–orbit coupling effect and introduce the concept of the spinor for the wave function. So far, we have treated the electronic wave function Ψ as a complex scalar. However, Ψ is neither a scalar nor a vector; it is a spinor. We know that we need one component to specify a scalar, while we need three components along the three coordinate axes to specify a vector. On the other hand, to specify a spinor, we need two components along the “up” and “down” directions. The two spin components are often uncoupled, and we can

c03.indd 96

4/20/2011 10:43:39 AM

THE LUTTINGER–KOHN HAMILTONIAN

97

treat Ψ as a scalar and simply multiply the results by 2 at the end to account for the two spin polarizations. However, this is not always true. In a magnetic field, for example, the energies of two polarizations are different. Moreover, an electron moving in a strong electric field sees an effective magnetic field due to relativistic effects even if there are no external magnetic fields. This is known as the “spin–orbit interaction.” We include effects of spin–orbit coupling as follows: H 0 = H 0 + H so,

(3.142)

where H so =

 (σ × ∇V ) ⋅ p, 4 m02 c 2

(3.143)

and σ x, σ y, σ z are the Pauli spin matrices defined as ⎛ 0 1⎞ ⎛ 0 −i⎞ ⎛1 0 ⎞ ,σy = ⎜ ,σy = ⎜ . σx = ⎜ ⎟ ⎟ ⎝ 1 0⎠ ⎝i 0⎠ ⎝ 0 −1⎟⎠

(3.144)

Since V is periodic and p invariant under translations, the total unperturbed Hamiltonian H 0 will still have Bloch waves as solutions. Call these ψ nk = e ik⋅r u nk, with the energy ε n (k). We may again introduce a set of functions analogous to the χ nk, say χ nk, defined by χ nk = e ik⋅r u n 0.

(3.145)

Then, the Ansatz ψ=

∑ ∫ dkA (k)χ n

nk

(3.146)

n

leads to the same Schrödinger equation for An (k) as the one we had α without spin–orbit coupling, except that the matrix elements pnn′ are α replaced by π nn′, defined by π αnn′ =

(2 π)3 Ω

∫

cell

 ⎛ ⎞ u n* 0 ⎜ ∇α + (σ × ∇V )α ⎟ u n′ 0 dr 2 2 ⎝i ⎠ 4 m0 c

(3.147)

This extra term arises from the fact that the spin–orbit coupling contains the differential operation p.

c03.indd 97

4/20/2011 10:43:40 AM

98

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

Degenerate bands are split, in general, into several bands by the spin–orbit coupling. If the spin–orbit interaction were absent, the six valence band wave functions would simply be ⎛X ⎞ ⎛ 0 ⎜⎝ 0 ⎟⎠ , ⎜⎝ X

⎞ ⎛Y ⎟⎠ , ⎜⎝ 0

⎞ ⎛ 0 ⎟⎠ , ⎜⎝ Y

⎞ ⎛Z⎞ ⎛ 0 ⎟⎠ , ⎜⎝ 0 ⎟⎠ , ⎜⎝ Z

⎞ ⎟⎠ ,

(3.148)

and we could continue to treat Ψ as a scalar and multiply at the end by 2. In many common semiconductors like gallium arsenide (GaAs), due to crystal symmetry, the functions un0 (r) for the conduction and valence bands can be expressed in terms of these functions S , X , Y , and Z . Electrons in the conduction band have completely symmetric wave functions S . Here, S is completely symmetric in x, y, and z [like ( x 2 + y2 + z2 )−1/2]; X is antisymmetric in x but symmetric in y and z [like x( x 2 + y2 + z2 )−1/2]: Y is antisymmetric in y but symmetric in z and x; and Z is antisymmetric in z but symmetric in x and y. In the absence of spin–orbit coupling, we shall denote three degenerate space functions at k = 0 by X , Y , and Z . Each has an additional double degeneracy due to spin. We will construct a matrix D defined by Djj ′ =

∑k k D α β

αβ jj ′

.

(3.149)

α ,β

using these functions. Then, as can be seen from Equation (3.137), the symmetry of the diamond lattice requires that D have the form Ckx k y Ckx kz ⎛ Akx2 + B(k y2 + kz2 ) ⎞ 2 2 2 ⎜ ⎟ D= Ckx k y Ak y + B(kx + kz ) Ck ykz ⎜ ⎟ Ckx kz Ck ykz Akz2 + B(k y2 + kx2 )⎠ ⎝

(3.150)

for each spin, where A, B, C are three real constants defined by A=

2 2 + 2 2 m0 m0

B=

  + 2 2 m0 m0

C=

c03.indd 98

2

2 m02

2

∑ i

∑ i

x x pXi piX ε0 − εi

∑

y y pXi piX ε0 − εi

i

(3.151)

y y x x pXi piY piY + pXi . ε0 − εi

4/20/2011 3:25:34 PM

THE LUTTINGER–KOHN HAMILTONIAN

99

As an example of the calculation, consider DXX , DXX =

∑ αβ

⎛ 2 2 δ δ + XX αβ ⎜⎝ 2 m m2 0

0

∑ i

β α ⎞ pXi piX kα kβ, ε 0 − ε i ⎟⎠

(3.152)

where  ∇α i i

β α pXi piX = X

i

 ∇β X . i

(3.153)

Equation (3.153) is zero except for α = β = X , Y , Z , because of its antisymmetry. That is, for α = X and β = Y,  ∇x i i

X Y pXi piX = X

i

 ∇y X , i

(3.154)

which is antisymmetric in x or y, and becomes zero. Thus, ⎛ 2 2 DXX = ⎜ + 2 ⎝ 2 m0 m0

∑ i

⎛   +⎜ + 2 ⎝ 2 m0 m0 2

X X ⎛ 2 pXi piX ⎞ 2 k k + + x x ⎜⎝ 2 m m2 ε 0 − ε i ⎟⎠ 0 0

2

i

p p ⎞ ⎟ kzkz. 0 − εi ⎠

∑ε i

∑

pYXi piYX ⎞ k yk y ε 0 − ε i ⎟⎠

Z Xi

Z iX

(3.155)

y y z z Using pXi , piX = pXi piX

⎛ 2 2 DXX = ⎜ + 2 ⎝ 2 m0 m0

∑ i

X X pXi piX ⎞ 2 ⎛  2 2 k + + x ⎜⎝ 2 m m2 ε 0 − ε i ⎟⎠ 0 0

∑ i

Y ⎞ 2 pYXi piX (k y + kz2 ) ε 0 − ε i ⎟⎠

≡ Ak + B(k + k ) 2 x

2 y

2 z

(3.156)

For DXY , only terms with α = X, β = Y or α = Z, β = X are nonzero. Similarly, we can obtain the following relations for nonzero terms: α α α α α α pXi piX , pYi piY , pZi piZ Y Xi

X iY

X Yi

Y iX

Z Xi

X iZ

for α = X , Y , Z X Zi

Z Z Y Y Z p p , p p , p p , p piX , pYi piZ, pZi piY ,

(3.157)

where X X Y Y Z Z pXi piX = pYi piY = pZi piZ Y Z Z Z Z pYXi piX = pYi piY = pXi piX = … etc.

c03.indd 99

(3.158)

4/20/2011 10:43:41 AM

100

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

3.2.4 Wave Functions Modified by Spin–Orbit Coupling The introduction of spin–orbit coupling splits some of the degeneracy of the band at k = 0. Treating H so as a perturbation, we can obtain the correct wave functions. The wave function Ψ(r, t ) for electrons is actually not a scalar but is a two-component spinor ⎛ α(r, t )⎞ ⎜⎝ β(r, t ) ⎟⎠ .

(3.159)

Thus, the Schrödinger equation is not a single differential equation, but two coupled differential equations as follows. ∂ α = H11α + H12β ∂t ∂ i β = H 21α + H 22β. ∂t i

(3.160) (3.161)

Or, in matrix form, i

∂ ⎛ α ⎞ ⎡ H11 = ∂t ⎜⎝ β ⎟⎠ ⎢⎣ H 21

H12 ⎤ ⎛ α ⎞ . H 22 ⎥⎦ ⎜⎝ β ⎟⎠

(3.162)

In general, we assume that H12 = H 21 = 0 2 2 H11 = H12 = − ∇ + U L ≡ H o. 2 mo

(3.163) (3.164)

Equation (3.162) then decouples into two identical Schrödinger equations—one for α and one for β. Consequently, we can just solve one Schrödinger equation for a scalar and multiply the results at the end by 2 to account for the other spin component. However, this simplicity is lost if H12 and H 21 are nonzero. Let us now consider how the wave functions and energy levels in a semiconductor are modified by the spin–orbit interaction. The Schrödinger equation reads i

∂Ψ = ( H + H so )Ψ, ∂t

(3.165)

where H is the 2 × 2 matrix operator defined in Equations (3.163) and (3.164), Ψ is a 2 × 1 spinor, and H so is the 2 × 2 spin–orbit operator

c03.indd 100

4/20/2011 10:43:41 AM

THE LUTTINGER–KOHN HAMILTONIAN

101

defined below. The spin–orbit coupling in Equation (3.143) can be written as H so = − i

2 σ ⋅ [∇U × ∇]. 4 m02 c 2

(3.166)

H so is a 2 × 2 matrix operator whose individual elements can be written out with differential operators using the definition of the Pauli matrices σ x , y,z from Equation (3.144). ⎛ ∂U ∂ ∂U ∂ ⎞ ( H so )11 = − iC ⎜ − ⎝ ∂x ∂y ∂y ∂x ⎟⎠ ⎧⎛ ∂U ∂ ∂U ∂ ⎞ ⎛ ∂U ∂ ∂U ( H so )12 = −iC ⎨⎜ − − ⎟ −i ⎩⎝ ∂y ∂z ∂z ∂y ⎠ ⎝ ∂z ∂x ∂x ⎧⎛ ∂U ∂ ∂U ∂ ⎞ ⎛ ∂U ∂ ∂U ( H so )21 = − iC ⎨⎜ − − ⎟ +i ⎩⎝ ∂y ∂z ∂z ∂y ⎠ ⎝ ∂z ∂x ∂x ⎛ ∂U ∂ ∂U ∂ ⎞ ( H so )22 = iC ⎜ − , ⎝ ∂x ∂y ∂y ∂x ⎟⎠ where C =

(3.167) ∂ ⎞⎫ ⎬ ∂z ⎠ ⎭ ∂ ⎞⎫ ⎬ ∂z ⎠ ⎭

(3.168) (3.169) (3.170)

2 . 4 m02 c 2

We are interested in the energy eigenfunctions and eigenvalues at k = 0. Writing Ψ(r, t ) = ξ(r) exp(− iEt/) [since k = 0, exp(ik ⋅ r) = 1], we have the eigenvalue equation. Eξ = ( H + H so )ξ

(3.171)

This is actually two coupled differential equations that can be written out using Equations (3.163) and (3.164) as follows, assuming ⎛ α(r)⎞ ξ(r) = ⎜ . ⎝ β(r) ⎟⎠ Eα = [ H 0 + ( H so )11 ]α + ( H so )12 β (3.172) Eβ = [( H so )21 ]α + ( H 0 + H so )22 β. The functions S , X , Y , and Z That is,

(3.173)

are eigenfunctions of H 0.

H 0 S = Ec 0 S , H 0 X = Ev′0 X , H 0 Y = Ev′0 Y , H 0 Z = Ev′0 Z . (3.174)

c03.indd 101

4/20/2011 10:43:41 AM

102

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

If the spin–orbit interaction H so were absent, the energy levels and the wave functions at the zone center (k = 0) will be written as shown below. v1 : α = X , β = 0 v2 : α = Y , β = 0 v3 : α = Z , β = 0

(3.175)

v4 : α = 0 , β = X v5 : α = 0 , β = Y v6 : α = 0 , β = Z . Now, let us use these functions to expand α(r) and β(r) . α( r ) = α s S + α x X + α y Y + α z Z ≡

∑

β(r) =

∑

αi i

i = S , X ,Y ,Z

βi i .

(3.176) (3.177)

i = S , X ,Y ,Z

We can write the differential equations in Equations (3.172) and (3.173) as a matrix equation.

∑ i H + (H ) j α + ∑ i (H Eβ = ∑ i ( H ) j α + ∑ i H + ( H

Eα i =

o

so 11

j

j

i

so 21

j

so 12

)

j β j.

(3.178)

)

j β j.

(3.179)

j

j

o

so 22

j

Since the functions i , i = S, X , Y , Z are eigenfunctions of H o, it is easy to write down the matrix elements of H o. if i = S ⎧ Ec 0δ ij, i H0 j = ⎨ . ⎩Ev′0δ ij, if i = X , Y , Z

(3.180)

Using the fact that the atomic potential U has cubic symmetry, we can show that i ( H SO )11 j or i ( H SO )22 j is nonzero only if i = X , j = Y or i = Y , j = X . Then, X ( H SO )11 Y = − X ( H SO )22 Y ≡ −iΔ/ 3

(3.181)

⎛ ∂U ∂ ∂U ∂ ⎞ Δ = 3C X ⎜ − Y . ⎝ ∂y ∂x ∂x ∂y ⎟⎠

(3.182)

where

c03.indd 102

4/20/2011 10:43:42 AM

THE LUTTINGER–KOHN HAMILTONIAN

103

Similarly, it can be shown that Y ( H SO )12 Z = Y ( H SO )21 Z = − iΔ/ 3 X ( H SO )12 Z = − X ( H SO )21 Z = Δ/ 3.

(3.183)

The other matrix elements are all zero. Also, in general, i ( H SO )nm j = − j ( H SO )mn i .

(3.184)

Using the matrix elements in Equations (3.181), (3.183), and (3.184), we can write out Equations (3.178) and (3.179) as follows: ⎛ α s ⎞ ⎛ Ec 0 ⎜ αx ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜αy⎟ ⎜ 0 ⎜α ⎟ ⎜ 0 z E⎜ ⎟ = ⎜ ⎜ βs ⎟ ⎜ 0 ⎜β ⎟ ⎜ 0 ⎜ x⎟ ⎜ ⎜ βy ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ βz ⎠ ⎝ 0

0 0 0 0 Ev′0 − iΔ/ 3 0 0 iΔ/ 3 Ev′0 0 0 Ev′0 0 0 0 0 0 0 Ec 0 0 0 0 − Δ/ 3 0 0 − iΔ/ 3 0 0 0 Δ/ 3 iΔ/ 3

0 0 0 ⎞ ⎛ αs ⎞ 0 0 Δ/ 3 ⎟ ⎜ α x ⎟ ⎟⎜ ⎟ 0 0 −iΔ/ 3⎟ ⎜ α y ⎟ − Δ/ 3 iΔ/ 3 0 ⎟ ⎜ αz ⎟ ⎟⎜ ⎟. 0 0 0 ⎟ ⎜ βs ⎟ 0 ⎟⎟ ⎜⎜ β x ⎟⎟ Ev′0 iΔ/3 − iΔ/ 3 Ev′0 0 ⎟ ⎜ βy ⎟ ⎟⎜ ⎟ 0 0 Ev′0 ⎠ ⎝ βz ⎠ (3.185)

Here, we neglected the coupling between the conduction and the valence bands, and will consider it in next section. By rearranging the rows and columns in Equation (3.185), we can get the matrix in blockdiagonalized form. ⎛ α s ⎞ ⎛ Ec 0 ⎜ βs ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ αx ⎟ ⎜ 0 ⎜α ⎟ ⎜ 0 y E⎜ ⎟ = ⎜ ⎜ βz ⎟ ⎜ 0 ⎜β ⎟ ⎜ 0 ⎜ x⎟ ⎜ ⎜ βy ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ αz ⎠ ⎝ 0

c03.indd 103

0 Ec 0 0 0 0 0 0 0

0 0 0 0 0 0 ⎞ ⎛ αs ⎞ 0 0 0 0 0 0 ⎟ ⎜ βs ⎟ ⎟⎜ ⎟ Ev′0 −iΔ/ 3 Δ/ 3 0 0 0 ⎟ ⎜αx ⎟ 0 0 0 ⎟ ⎜αy⎟ iΔ/ 3 Ev′0 −iΔ/ 3 ⎟⎜ ⎟. Δ/ 3 iΔ/ 3 Ev′0 0 0 0 ⎟ ⎜ βz ⎟ 0 0 0 Ev′0 iΔ/ 3 − Δ/ 3 ⎟⎟ ⎜⎜ β x ⎟⎟ 0 0 0 − iΔ/ 3 Ev′0 − iΔ/ 3⎟ ⎜ β y ⎟ ⎟⎜ ⎟ 0 0 0 − Δ/ 3 iΔ/ 3 Ev′0 ⎠ ⎝ α z ⎠ (3.186)

4/20/2011 10:43:42 AM

104

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

The eigenvalues and eigenfunctions for each block can be calculated individually. Note that the conduction band levels are unaffected by the spin–orbit interaction. The eigenvalues and eigenvetors of the upper 3 × 3 block are given by Δ = Ev 0 (heavy hole, h1) 3 α x = −1/ 2, α y = − i/ 2, βz = 0

Eh = Ev′0 +

Δ = Ev 0 (light hole, l 2) 3 α x = 1/ 6, α y = − i/ 6, βz = 2/3

(3.187)

El = Ev′0 +

2Δ = Ev 0 − Δ (split-off , s 2) 3 α x = 1/ 3, α y = − i/ 3, βz = −1/ 3 .

(3.188)

Es = Ev′0 −

(3.189)

Note that, at k = 0, the first two eigenvalues are equal, so that any linear combination of the heavy- and light-hole eigenvectors is also an eigenvector. In fact, there are numerous other combinations possible for Equations (3.187) and (3.188). The reason for using this choice is that if we consider electrons in motion along the z-direction (kx = k y = 0, kz ≠ 0), the heavy- and light-hole eigenvectors are decoupled. The lower 3 × 3 block in Equation (3.186) gives the same eigenvalues, but slightly different eigenvectors. Δ = Ev 0 (heavy hole, h2) 3 β x = 1/ 2, β y = − i/ 2, α z = 0

Eh = Ev′0 +

Δ = Ev 0 (light hole, l 1) 3 β x = −1/ 6, β y = − i/ 6, α z = 2/3

(3.190)

El = Ev′0 +

2Δ = Ev 0 − Δ (split-off , s1) 3 β x = 1/ 3, β y = i/ 3, α z = 1/ 3 .

(3.191)

Es = Ev′0 −

(3.192)

The introduction of spin–orbit coupling splits some of the degeneracy of the band at k = 0. The wave functions fall into a group of four and a group of two:

c03.indd 104

4/20/2011 10:43:42 AM

THE ZINC BLENDE HAMILTONIAN

3 3 , 2 2 3 1 l1 : , 2 2 3 1 l2 : , − 2 2 3 3 h2 : , − 2 2 1 1 s1 : , 2 2 1 1 s2 : , − 2 2 h1 :

1 [ X +i Y ]↑ 2 1 [( X + i Y ) ↓ −2 Z ↑] =− 6 1 = [( X − i Y ) ↑ +2 Z ↓] 6 1 = [ X −i Y ]↓ 2 1 = [( X + i Y ) ↓ + Z ↑] 3 1 = [( X − i Y ) ↑ − Z ↓]. 3

105

=−

(3.193)

where ↑ and ↓ are the spin functions corresponding to spin “up” and “down,” respectively. 3.3 THE ZINC BLENDE HAMILTONIAN 3.3.1

Matrix Elements for the 6 ¥ 6 Hamiltonian

We shall consider the situation in which the spin–orbit coupling is sufficiently small so that the zeroth-order functions of Equation (3.193) may be used. Considering both the J = 3/2 and 1/2 bands together, and using Equation (3.137) having p instead π , and using Equation (3.141), we find ⎛ h1 : 3 , 3 ⎜ 2 2 ⎜ ⎜ l1 : 3 , 1 ⎜ 2 2 ⎜ ⎜ l2 : 3 , − 1 ⎜ 2 2 H = −⎜ ⎜ h2 : 3 , − 3 ⎜ 2 2 ⎜ 1 1 ⎜ s1 : , 2 2 ⎜ ⎜ 1 1 ⎜ s2 : , − ⎝ 2 2

⎞⎛ ⎟ ⎜ P +Q ⎟⎜ ⎟⎜ † ⎟⎜ −S ⎟⎜ ⎟⎜ † ⎟⎜ R ⎟⎜ ⎟⎜ 0 ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ − S† / 2 ⎟⎜ ⎟⎜ ⎠ ⎝ 2 R†

−S

R

0

− S/ 2

P −Q

0

R

− 2Q

0

P −Q

S

3/2S †

R†

S†

P +Q

− 2 R†

− 2Q†

3/2S

− 2R

P+Δ

3/2S †

2Q†

− S/ 2

0

⎞ ⎟ ⎟ ⎟ 3/2S ⎟ ⎟ ⎟ 2Q ⎟ ⎟, ⎟ −S† / 2 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ P+Δ ⎠ 2R

(3.194)

c03.indd 105

4/20/2011 10:43:43 AM

106

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

where P=

2 γ 1 (kx2 + k y2 + kz2 ), 2 m0

Q=

2 γ 1 (kx2 + k y2 − 2kz2 ), 2 m0

R=

2 [− 3γ 2 (kx2 − k y2 ) + i 2 3γ 3kx k y ], 2 m0

(3.195)

2 2 3γ 3 (kx − ik y )kz, 2 m0 Δ = spin-orbit splitting at k = 0, S=

where we put Ev0 = 0 as follows: ⎧ E +Δ =E =0 for l = h1, l 1, h2, l 2 v′ 0 v0 ⎪⎪ 3 E j (k = 0) = ⎨ . ⎪Ev′0 − 2 Δ = Ev 0 − Δ = − Δ forr l = s1, s 2 ⎪⎩ 3

(3.196)

As an example of the calculation, we consider H h1,h1. H h1,h1 =

∑ αβ

⎛ 2  ⎜⎝ 2 m δ h1h1δ αβ + m2 0

0

∑ i

phα1i pihβ 1 ⎞ kα kβ, ε 0 − ε i ⎟⎠

(3.197)

where   phα1i pihβ 1 = h1 ∇α i i ∇β h1 i i   1 = X − iY ∇α i i ∇β X + iY i i 2 (3.198) 1 α β β α β α α β = ( pXi piX − ipYi piX + pXi piY + pYi piY ) 2 1 α β α β = ( pXi piX + pYi piY ) from Equ uations (3.157) and (3.158). 2 This becomes zero except for α = β .

c03.indd 106

4/20/2011 10:43:43 AM

THE ZINC BLENDE HAMILTONIAN

⎛ 2  H h1,h1 = kx2 ⎜ + ⎝ 2 m0 2 m02

⎛   + + k y2 ⎜ ⎝ 2 m0 2 m02 ⎛   + kz2 ⎜ + ⎝ 2 m0 2 m02 2

X X X ⎞ pXi piX 1 + pYiX piY ⎟⎠ ε0 − εi

∑

2

107

i

∑ i

Y Y Y pYXi piX piY ⎞ + pYi ⎟⎠ ε0 − εi

∑

p p +p p ⎞ ⎟⎠ ε0 − εi

i

Z Xi

Z iX

Z Yi

(3.199)

Z iY

1 1 = kx2 ( A + B) + k y2 ( A + B) + kz2 B from m Equation (3.151). 2 2 Now, it is convenient to introduce the dimensionless Luttinger effective mass parameters γ 1 , γ 2 , and γ 3 by A=

2 (γ 1 + 4 γ 2 ) 2 m0

B=

2 (γ 1 − 2 γ 2 ) 2 m0

C=−

(3.200)

2 (6 γ 3 ). 2 m0

Then, Equation (3.199) becomes H h1,h1 =

2 2 2 ( γ 1 + γ 2 )kx2 + ( γ 1 + γ 2 )k y2 + ( γ 1 − 2 γ 2 )kz2 2 m0 2 m0 2 m0

2 2 = γ 1 (kx2 + k y2 + kz2 ) + γ 2 (kx2 + k y2 − 2kz2 ) ≡ P + Q 2 m0 2 m0

(3.201)

We always have to ensure that whatever H(k) we use, its eigenvalues should give us the correct dispersion laws for the bands. Thus, note that the Luttinger parameters γ 1 , γ 2 , and γ 3 will be slightly different depending on whether we use the 4 × 4 matrix, the 6 × 6 matrix, or the 8 × 8 matrix, including the spin–orbit band. 3.3.2 Matrix Elements of the Hamiltonian between Conduction and Valence Bands Matrix elements between conduction and valence bands can be derived by Equation (3.50) when we consider the coupling between conduction

c03.indd 107

4/20/2011 10:43:43 AM

108

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

and valence bands. That is, we first calculate momentum matrix eleα ments pnn′ between conduction and valence bands, then substitute them into Equation (3.50). Since the functions un0 (r) are spinors of the form ⎛ α n,0 ⎞ ⎜⎝ β ⎟⎠ , n,0

(3.202)

pαnn′ = − i α n,0 ∇α n′ ,0 − i β n,0 ∇β n′ ,0 .

(3.203)

Equation (3.50) becomes

In many common semiconductors like GaAs, due to crystal symmetry, the functions un,0 (r) for the conduction and valence bands can be expressed in terms of these functions S , X , Y , and Z . Electrons in the conduction band have completely symmetric wave functions S . The up-spin and down-spin wave functions corresponding to c1 and c2 are written as ⎛S⎞ uc1,0 (r) = ⎜ ⎟ ⎝ 0⎠

(3.204)

⎛ 0⎞ uc2,0 (r) = ⎜ ⎟ . ⎝S⎠

(3.205)

α To calculate momentum matrix elements pnn′ between conduction and valence bands, we substitute the functions α and β from Equations (3.193) and (3.205). Due to the symmetry of the functions S , X , Y , and Z , only the following integrals are nonzero.

S − i

∂ ∂ ∂ X = S − i Y = S − i Z ≡ P0. ∂x ∂y ∂z

(3.206)

All other integrals, such as S − i

c03.indd 108

∂ Y , ∂x

(3.207)

4/20/2011 10:43:43 AM

109

THE ZINC BLENDE HAMILTONIAN

are zero. Using Equation (3.206), it is straightforward to evaluate pnn′. For example, suppose n = c1 and n′ = h1. Using Equation (3.193), we have from Equation (3.203), p c 1h 1 = −

i ( S ∇ X + i S ∇ Y ). 2

(3.208)

Using Equation (3.206), p c 1h 1 =

P0 (eˆ x + ieˆ y ), 2

(3.209)

where eˆ x and eˆ y are unit vectors along the x, y, and z directions. Then, using Equation (3.50), matrix elements of Hamiltonian between conduction and valence bands are given by ⎛ h1 ⎜ ⎜ ⎜ c1 − π (k + ik ) x y ⎜ 2 ⎜ ⎜ 0 ⎜⎝ c 2 s1 π kz 3 π (kx + ik y ) 3

h2

l1

0

2 πkz 3

π (kx − ik y ) 2 ⎞ s2 ⎟ ⎟ π (kx − ik y )⎟ , ⎟ 3 ⎟ π kz ⎟ − ⎠ 3

π (kx + ik y ) 6

l2 π (kx − ik y ) 6 2 πkz 3

(3.210)

where π = P0 /m0 = ( 2 / 2 m0 )E p , and E p is a parameter that controls mixing of the valence and conduction bands.

3.3.3 The 8 ¥ 8 Hamiltonian Combining Equations (3.194) and (3.210), we can obtain the 8 × 8 Hamiltonian given by

c03.indd 109

4/20/2011 10:43:43 AM

110

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ Ac ⎜ ⎜ 0 ⎜ ⎜ − 3V ⎜ ⎜ 2U † ⎜ H = ⎜ V† ⎜ ⎜ ⎜ 0 ⎜ ⎜ U† ⎜ ⎜ ⎜ 2V † ⎝

0 Ac 0 −V 2U † 3V † 2V −U †

2V ⎞ c1 ⎟ 0 2U 3V 2V −V −U ⎟ c 2 ⎟ 1 S 0 S −( P + Q) −R − 2R ⎟ h1 2 ⎟ −( P − Q) 0 −R 2Q − 3/2S ⎟⎟ l 1 S† − R† 0 −( P − Q) −S − 3/2S † − 2Q ⎟ l 2 , ⎟ 1 † ⎟ 2 R† S ⎟ h2 0 − R† −S† −( P + Q) 2 ⎟ 1 † ⎟ s1 † 2Q 2R 0 S − 3/2S −( P + Δ) ⎟ 2 ⎟ 1 − 2 R† − 3/2S † − 2Q† S 0 −( P + Δ)⎟ s 2 ⎠ 2 − 3V †

2U

V

0

†

U

†

(3.211) with Ac = Eg +

2 (kx2 + k y2 + kz2 ), 2 m0

1 π(kx − ik y ), 6 1 U= πkz . 3 V=

(3.212)

Here, the basis set is summarized as follows: c1 : S ↑ c2 : S ↓ 3 3 , 2 2 3 1 l1 : , 2 2 3 1 l2 : , − 2 2 3 3 h2 : , − 2 2 1 1 s1 : , 2 2 1 1 s2 : , − 2 2 h1 :

c03.indd 110

1 [ X +i Y ]↑ 2 1 =− [( X + i Y ) ↓ −2 Z ↑] 6 1 [( X − i Y ) ↑ +2 Z ↓] = 6 1 [ X −i Y ]↓ = 2 1 = [( X + i Y ) ↓ + Z ↑] 3 1 = [( X − i Y ) ↑ − Z ↓]. 3 =−

(3.213)

4/20/2011 10:43:44 AM

THE ZINC BLENDE HAMILTONIAN

111

3.3.4 The Strained Hamiltonian In this section, we will introduce the strained 8 × 8 Hamiltonian. We ˆ and zˆ in the unstrained system. Under the ˆ y, defined the unit vectors x, influence of a uniform deformation, these axes are distorted to x, y, and z. The new axes are related to the old one by x = (1 + ε xx ) xˆ + ε xy yˆ + ε xzzˆ , y = ε yx xˆ + (1 + ε yy ) yˆ + ε yzzˆ ,

(3.214)

z = ε zx xˆ + ε zy yˆ + (1 + ε zz )zˆ . The coefficients are strain tensors and define the deformation in the system. The new axes are not orthogonal in general. To label a position A (or atom A) in the unstrained system, we have r = xxˆ + yyˆ + zzˆ .

(3.215)

After the distortion, the same atom can be labeled as r ′ = xx ′ + yy ′ + zz ′.

(3.216)

Note that the coefficients x, y, z of the vectors are unchanged. The displacement of the deformation is given by R = r′ − r′ = x(x ′ − xˆ ) + y(y ′ − yˆ ) + z(z′ − zˆ ) = ( xε xx + yε yx + zε zx ) xˆ + ( xε xy + yε yy + zε zy ) yˆ + ( xε xz + yε yz + zε zz )zˆ = u(r) xˆ + v(r) yˆ + w(r)zˆ .

(3.217)

For a nonuniform distortion, one must define a position-dependent strain. Taking the origin of r close to the point and assuming a small strain, we have ε xx =

∂u ∂u ∂u ; ε yx = ; ε zx = ,etc. ∂x ∂y ∂z

(3.218)

We assume a homogeneous strain and ε ij = ε ji . We define six strain components as

c03.indd 111

4/20/2011 10:43:44 AM

112

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

e1 = ε xx , e2 = ε yy , e3 = ε zz e4 = ε xy + ε yx

(3.219)

e5 = ε yz + ε zy e6 = ε zx + ε xz ,

keeping only the linear terms in strain. It is useful to define the net fractional change in the volume produced by the distortion. This quantity is called dilation. The initially cubic volume of unity after distortion has a volume 1 + ε xx V ′ = x ′ ⋅ (y ′ × z ′) = ε yx ε zx

ε xy 1 + ε yy ε zy

ε xz ε yz  1 + ε xx + ε yy + ε zz. 1 + ε xz

(3.220)

The dilation is then δ=

V′ −V  ε xx + ε yy + ε zz. V

(3.221)

The force acting on a unit area in the solid is called stress, and Hooke’s law states that the strain is linearly proportional to the stress, or, conversely, that the stress is linearly proportional to the strain. In general, Tij = cijkl Skl, i, j, k, l = x, y, z,

(3.222)

with summation over the repeated subscript k and l. The first subscript letter indicates the direction of the force, and the second subscript letter is the direction of the normal to the plane on which the stress is acting. The number of stress components decreases to six if we impose the condition on a cubic system, since there is no torque in the system. Then, Txy = Tyx; Tyz = Tzx;Txz = Tzx. We then have the six independent components: Txx,Tyy,Tzz,Txy,Tyz,Tzx. The cijkl are called elastic stiffness constants. If the coordinate axes x, y, z are chosen to coincide with the cubic crystal axes, the nonzero elastic stiffness constants are cxxxx = cyyyy = czzzz cxxyy = cyyxx = cxxzz = cyyzz = czzyy cyxyx = cyxxy = czyyz = czyzy

(3.223)

= cxzxz = cxzzx = czxxz = czxzx = cxyxy = cxyyx = cyxxy = cyxyx.

c03.indd 112

4/20/2011 10:43:44 AM

THE ZINC BLENDE HAMILTONIAN

113

We use abbreviated subscript notation: 1 = xx ; 2 = yy; 3 = zz; 4 = yz; 5 = zx; 6 = xy . Then, the stiffness Equation (3.222) reduce to TI = cIJ SJ ,

(3.224)

where I , J = 1, 2, 3, 4, 5, 6. In abbreviated subscript notation, the nonzero elastic stiffness constants are c11 = c22 = c33 c12 = c21 = c13 = c31 = c23 = c32 c44 = c55 = c66,

(3.225)

and the stiffness matrix is ⎛ c11 ⎜ c12 ⎜ ⎜ c12 [c] = ⎜ 0 ⎜ ⎜ 0 ⎜⎝ 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0⎞ 0⎟ ⎟ 0⎟ . 0⎟ ⎟ 0⎟ c44 ⎟⎠

(3.226)

The strain energy of the system is a quadratic function of the strain, and can be written as U=

1 2

6

6

∑∑c e e . ij i j

(3.227)

i =1 j =1

For cubic systems, U is given by U=

1 c11 (e12 + e22 e32 ) + C 44 (e42 + e52 + e62 ) + C12 (e1e2 + e2 e3 + e3e2 ). 2

(3.228)

Using the relations between the deformed coordinates and the undeformed coordinates, the Hamiltonian for a strained semiconductor can be derived. The strained Hamiltonian introduces extra terms that can be obtained by using the following correspondences for the 8 × 8 Luttinger–Kohn Hamiltonian given by Equation (3.211) kα kβ ↔ ε αβ 2 ↔ ac 2 me* 2 γ 1 ↔ − av 2 m0

c03.indd 113

4/20/2011 10:43:44 AM

114

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

b 2 γ 2 ↔− 2 m0 2 d 2 γ 3 ↔− 2 m0 2 3 kα ↔ −

∑ε

k

αβ β

β

⎛ Acε 0 − 3Vε† 2U ε Vε 0 Uε 2Vε ⎞ ⎜ ⎟ † † −Vε −U ε ⎟ Acε 0 2U ε 3Vε 2Vε ⎜ 0 ⎜ ⎟ c1 1 − Rε Sε 0 Sε − 2 Rε ⎟ c 2 ⎜ − 3Vε 0 −( Pε + Qε ) 2 ⎜ ⎟ ⎜ 2U † −V Sε† −( Pε − Qε ) 0 − Rε 2Qε − 3/2Sε ⎟ h1 ε ε ⎜ ⎟ l1 2U ε† − Rε† 0 −( Pε − Qε ) −Sε − 3/2Sε† − 2Qε ⎟ , H ε = ⎜ Vε† ⎜ ⎟ l2 1 † ⎟ ⎜ 3Vε† 0 − Rε† − Sε† −( Pε + Qε ) 2 Rε† Sε ⎟ h2 ⎜ 0 2 ⎜ ⎟ s1 1 † ⎜ U† ⎟ † 2 V S 2 Q − 3/2 S 2 R − ( P 0 ε ε ε ε ε ε ε) ⎜ ⎟ s2 2 ⎜ ⎟ 1 ⎜ 2V † −U † − 2 Rε† − 3/2Sε† − 2Qε† Sε 0 − Pε ⎟ ε ε ⎝ ⎠ 2

(3.229) with Acε = ac (ε xx + ε yy + ε zz ), Vε = − Uε = −

1 π 6

∑ (ε

xj

1 π 3

∑ε

k j,

− iε yj )k j,

j

zj

j

Pε = − av (ε xx + ε yy + ε zz ),

(3.230)

b Qε = − (ε xx + ε yy − 2 ε zz ), 2 Sε = −d(ε xz − ε yz ), Rε =

3 b(ε xx − ε yy ) − idε xy. 2

3.4 THE WURTZITE HAMILTONIAN In the section, we consider the k ⋅ p method for strained wurtzite semiconductors. Similar to Section 3.2.4, we can write the matrix element of H o:

c03.indd 114

4/20/2011 10:43:45 AM

THE WURTZITE HAMILTONIAN

115

S H 0 S = Ec 0, X H 0 X = Y H 0 Y = Ev 0 + Δ 1, Z H 0 Z = Ev 0, X ( H SO )11 Y = − X ( H SO )22 Y ≡ −iΔ 2,

(3.231)

Y ( H SO )12 Z = Y ( H SO )21 Z = − iΔ 3, X ( H SO )12 Z = − X ( H SO )21 Z = Δ 3, where ⎛ ∂U ∂ ∂U ∂ ⎞ Δ2 = C X ⎜ − Y ⎝ ∂y ∂x ∂x ∂y ⎟⎠

(3.232)

and ∂U ∂ ∂U ∂ ⎞ ⎛ ∂U ∂ ∂U ∂ ⎞ Δ3 = C Y ⎜ − − Z =C X ⎛ Z . ⎝ ∂x ∂z ∂z ∂ x ⎠ ⎝ ∂y ∂z ∂z ∂y ⎟⎠

(3.233)

In the above, we used the sixfold symmetry on the x − y plane. For example, ( x, y) mapping to π/3 rotation around the z-axis leaves the Hamiltonian invariant, and the wave functions X and Y transform to 1 3 X + Y , 2 2 3 1 Y′ = − X + Y . 2 2

X′ =

(3.234)

Here,

Y H0 Y = Y ′ H0 Y ′ =

3 1 X H0 X + Y H0 Y . 4 4

(3.235)

Hence, Y H 0 Y = X H 0 X . Similarly, we obtain X H 0 Y = 0 . Using the matrix elements in Equation (3.231) for wurtzite semiconductors, we can write Equations (3.178) and (3.179) as follows:

c03.indd 115

4/20/2011 10:43:45 AM

116

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ α s ⎞ ⎛ Ec 0 ⎜ αx ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜αy⎟ ⎜ 0 ⎜α ⎟ ⎜ 0 z E⎜ ⎟ = ⎜ ⎜ βs ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ βx ⎟ ⎜ 0 ⎜ βy ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ βz ⎠ ⎝ 0

0 Ev 0 + Δ 1 iΔ 2 0 0 0 0 Δ3

0 −iΔ 2 Ev 0 + Δ 1 0 0 0 0 iΔ 3

0 0 0 Ev 0 0 −Δ 3 − iΔ 3 0

0 0 0 0 Ec 0 0 0 0

0 0 0 −Δ 3 0 Ev 0 + Δ 1 − iΔ 2 0

0 ⎞ ⎛ αs ⎞ Δ3 ⎟ ⎜ αx ⎟ ⎟⎜ ⎟ − iΔ 3 ⎟ ⎜ α y ⎟ 0 ⎟ ⎜ αz ⎟ ⎟⎜ ⎟. 0 ⎟ ⎜ βs ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ βx ⎟ 0 ⎟ ⎜ βy ⎟ ⎟⎜ ⎟ Ev 0 ⎠ ⎝ βz ⎠

0 0 0 iΔ 3 0 iΔ 2 Ev 0 + Δ 1 0

(3.236) By using a method similar to that used for zinc blende structure, we can get the matrix in block-diagonalized form ⎛ α s ⎞ ⎛ Ec 0 ⎜ βs ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ αx ⎟ ⎜ 0 ⎜α ⎟ ⎜ 0 y E⎜ ⎟ = ⎜ ⎜ βz ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ βx ⎟ ⎜ 0 ⎜ βy ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ αz ⎠ ⎝ 0

0 Ec 0 0 0 0 0 0 0

0 0 Ev 0 + Δ 1 iΔ 2 Δ3 0 0 0

0 0 − iΔ 2 Ev 0 + Δ 1 iΔ 3 0 0 0

0 0 Δ3 − iΔ 3 Ev 0 0 0 0

0 0 0 0 0 Ev 0 + Δ 1 − iΔ 2 −Δ 3

0 0 0 0 0 iΔ 2 Ev 0 + Δ 1 iΔ 3

0 ⎞ ⎛ αs ⎞ 0 ⎟ ⎜ βs ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ αx ⎟ 0 ⎟ ⎜αy⎟ ⎟⎜ ⎟. 0 ⎟ ⎜ βz ⎟ ⎟⎜ ⎟ −Δ 3 ⎟ ⎜ βx ⎟ − iΔ 3 ⎟ ⎜ β y ⎟ ⎟⎜ ⎟ Ev 0 ⎠ ⎝ α z ⎠

(3.237) The eigenvalues and eigenvectors of the upper 3 × 3 block are given by Eh = Ev 0 + Δ 1 + Δ 2

(heavy hole, h1)

(3.238)

α x = −1/ 2, α y = − i/ 2, βz = 0 Δ − Δ2 Δ − Δ2 ⎞ + 2 Δ 23 El = Ev 0 + 1 + ⎛ 1 ⎝ ⎠ 2 2 2

(light hole, l 2)

α x = Δ 3 / E + 2 Δ , α y = − iΔ 3 / E + 2 Δ , βz = − Ec / E + 2 Δ 2 c

2 3

2 c

Δ1 − Δ 2 Δ − Δ2 ⎞ + 2 Δ 23 − ⎛ 1 ⎝ 2 ⎠ 2

2 3

2 c

(3.239) 2 3

2

Ec = Ev 0 +

(crystal-field split-off holee, c 2)

α x = Δ 3 / El2 + 2 Δ 23 , α y = −iΔ 3 / El2 + 2 Δ 23 , βz = − El / El2 + 2 Δ 23

(3.240)

The lower 3 × 3 block in Equation (3.237) gives the same eigenvalues and eigenvectors.

c03.indd 116

4/20/2011 10:43:45 AM

THE WURTZITE HAMILTONIAN

117

If the spin–orbit interaction H so were absent, Δ 2 = Δ 3 = 0 . Then, the energy levels and the wave functions at the zone center ( k = 0 ) are written as shown below. Eh = Ev 0 + Δ 1 (heavy hole, h1) α x = −1/ 2, α y = − i/ 2, βz = 0 El = Ev 0 + Δ 1 (light hole, l 2) α x = 1/ 2, α y = − i/ 2, βz = 0

(3.241)

Ec = Ev 0 (crystal-field split-off hole, c 2) α x = 0, α y = 0, βz = 1 Those for the lower 3 × 3 block can be obtained by using a similar process. Here, the first two eigenvalues are equal so that any linear combination of the heavy- and light-hole eigenvectors is also an eigenvector. In the wurtzite Hamitonian, we will use the following basis functions: c1 : iS ↑ c 2 : iS ↓ h1 : u1 = −

1 [ X +i Y ]↑ 2

1 [( X − i Y ] ↑ 2 c1 : u3 = Z ↑ l 1 : u2 =

(3.242)

1 [( X − i Y ] ↓ 2 1 h2 : u5 = − [ X + i Y ]↓ 2 c 2 : u6 = Z ↓ l 2 : u4 =

Similarly to a zinc blende case, we will construct a matrix D defined by Djj ′ =

∑k k D α β

αβ

αβ jj ′

,

(3.243)

where j, j ′ = X , Y , Z . Then, the 3 × 3 matrix has the form

c03.indd 117

4/20/2011 10:43:45 AM

118

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ L1kx2 + M1k y2 + M2 kz2 D=⎜ N 1k x k y ⎜ N 2 kx kz ⎝

N 2 kx kz ⎞ ⎟, N 2 k y kz ⎟ M3 kx2 + M3 k y2 + L2 kz2 ⎠

N 1k x k y M k + L1k y2 + M2 kz2 N 2 k y kz 2 1 x

(3.244) for each spin, where the band-structure parameters L, M , N are three real constants, which are similar to the Luttinger–Kohn parameters γ 1, γ 2 , and γ 3 for zinc blende structures,

L1 =

2 2 + 2 2 m0 m0

L2 =

  + 2 2 m0 m0

M1 =

  + 2 m0 m02

2

2

2

2

  M2 = + 2 2 m0 m0 2

M3 =

2

  + 2 2 m0 m0 2

2 N1 = 2 m0

2

∑ i

∑ i

x x pXi piX 2 2 = + 2 ε 0 − ε i 2 m0 m0

∑ε i

z Zi

∑ i

y pYiy piY ε0 − εi

∑ i

x pYix piY ε0 − εi z z pYi piY ε0 − εi

z iZ

p p 0 − εi

∑ i

y y pXi piX 2 2 = + 2 ε 0 − ε i 2 m0 m0

∑

p p   = + 2 ε 0 − ε i 2 m0 m0

∑

p p   = + 2 2 m0 m0 0 − εi

∑ε

i

z Xi

∑ε i

x Zi

z iX

x iZ

2

2

2

2

i

i

(3.245)

y y pZi piZ 0 − εi

y y x x pXi piY piY + pXi ε0 − εi

as can be seen from Equation (3.137). Using the results in Equation (3.244), the 6 × 6 Hamiltonian in the bases {u1, u2, … , u6 } can be easily derived using

D6×6

⎛ D11 ⎜ D21 ⎜ ⎜ − D23 =⎜ ⎜ ⎜ ⎜ ⎝

* D21 D11 * D23

0

* − D23 D23 D33

D11 * D21 * D23

0 0 D21 D11 − D23

⎞ ⎟ ⎟ ⎟ , D23 ⎟ ⎟ − D2*3 ⎟ ⎟ D33 ⎠

(3.246)

where

c03.indd 118

4/20/2011 10:43:45 AM

THE WURTZITE HAMILTONIAN

119

L1 + M1 ⎞ 2 (kx + k y2 ) + M2 kz2 2 ⎠ 1 1 D21 = − [(L1 − M1 )(kx2 − k y2 ) + 2iN 1kx k y ] = − N 1 (kx + ik y )2, 2 2 (3.247) 1 D23 = N 2 (kx + ik y )kz , 2 D33 = M2 (kx2 + k y2 ) + L2 kz2. D11 = ⎛ ⎝

Also, we obtain consideration:

the

following

relation

from

L1 − M1 = N 1.

symmetry

(3.248)

The 6 × 6 Hamiltonian at the zone center in the bases {u1, u2, … , u6 } can be obtained from Equation (3.236):

⎛ Ev 0 + Δ 1 + Δ 2 ⎜ 0 ⎜ ⎜ 0 H 6×6 (k = 0) = ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜⎝ 0

0

0

0

0

Ev 0 + Δ 1Δ 2

0

0

0

0 0

Ev 0 0

0 Ev 0 + Δ 1 + Δ 2

0

2Δ 3

0

Ev 0 + Δ 1 − Δ 2

2Δ 3

0

0

0

2Δ 3 0

0 ⎞ u ⎛ 1⎞ 2 Δ 3 ⎟⎟ ⎜ u2 ⎟ ⎜ ⎟ 0 ⎟ ⎜ u3 ⎟ ⎟⎜ ⎟. 0 ⎟ ⎜ u4 ⎟ ⎟ Ev 0 ⎟ ⎜ u5 ⎟ ⎜ ⎟ ⎟ 0 ⎠ ⎝ u6 ⎠

(3.249) Hence, the full Hamiltonian, H = H 6×6 (k = 0) + D6×6, can be written as ⎛ F ⎜ −K ⎜ ⎜ −H H 6×6 (k) = ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0

−K * G H* 0 0 Δ

−H * H λ 0 Δ 0

0 0 0 F −K * H*

0 0 Δ −K G −H

0 ⎞ ⎛ u1 ⎞ Δ ⎟ ⎜ u2 ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ u3 ⎟ , H ⎟ ⎜ u4 ⎟ ⎟⎜ ⎟ − H * ⎟ ⎜ u5 ⎟ ⎟⎜ ⎟ λ ⎠ ⎝ u6 ⎠

(3.250)

where

c03.indd 119

4/20/2011 10:43:46 AM

120

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

F = Δ 1 + Δ 2 + λ + θ, G = Δ 1 − Δ 2 + λ + θ, λ=

2 [ A1kz2 + A2 (kx2 + k y2 )], 2 mo

θ=

2 [ A3kz2 + A4 (kx2 + k y2 )], 2 mo

K=

 A5 (kx + ik y )2, 2 mo

H=

2 A6 (kx + ik y )kz, 2 mo

(3.251)

2

Δ = 2 Δ 3. The relations between the band-structure parameters L, M , N derived from the k ⋅ p method and Ai parameters used in the Bir–Pikusmodel are given by 2 2 2 A1 = L2 , A2 = M3 , A3 = M 2 − L2, 2 mo 2 mo 2 mo

(3.252) 2 2 N1 2 L1 + M1 N2 . − M3, A5 = A4 = , A6 = 2 mo 2 2 mo 2 2 mo 2 The Hamiltonian for a strained semiconductor can be easily derived by replacing Ai with the deformation potentials Di and a straightforward addition of corresponding terms: kα kβ → ε αβ. F = Δ 1 + Δ 2 + λ + θ, G = Δ 1 − Δ 2 + λ + θ, λ=

2 [ A1kz2 + A2 (kx2 + k y2 )] + λ ε, 2 mo

θ=

2 [ A3kz2 + A4 (kx2 + k y2 )] + θ ε, 2 mo

K=

2 A5 (kx + ik y )2 + D5 ε +, 2 mo

(3.253)

(3.254)

2 A6 (kx + ik y )kz + D6 ε z+, 2 mo λ ε = D1ε zz + D2 (ε xx + ε yy ), H=

θ ε = D3 ε zz + D4 (ε xx + ε yy ), Δ = 2 Δ 3,

c03.indd 120

4/20/2011 10:43:46 AM

THE WURTZITE HAMILTONIAN

121

where ε ± = ε xx ± 2iε xy − ε yy,

(3.255)

ε z± = ε zx ± iε yz.

Matrix elements between conduction and valence bands can be derived by using a process similar to the one we have used for a zinc blende case in Section 3.3.2: Here, we use two Kane’s parameters, P1 and P2 , which are defined as ∂ m Z = 0 P1, ∂z  ∂ m ∂ iS − i X = iS − i Y = 0 P2. ∂x ∂y  iS −i

(3.256)

Then, from Equation (3.203), matrix elements of Hamiltonian between conduction and valence bands under bases functions of Equation (3.242) are given by ⎛ h1 h2 ⎜ ⎜ P ⎜ c1 − 2 (kx + ik y ) 0 ⎜ 2 ⎜ P ⎜ c2 0 − 2 (kx + ik y ) ⎝ 2

l1

l2

s1

P2 (kx − ik y ) 2

0

P1kz

0

P2 (kx − ik y ) 2

0

⎞ s2 ⎟ ⎟ 0 ⎟, ⎟ ⎟ P1kz ⎟ ⎠

(3.257) For example, suppose n = c1 and n′ = h1 . Then, we have p c 1h 1 =

i ( iS ∇ X + i iS ∇ Y ). 2

(3.258)

Using Equation (3.256), H c 1h 1 = −

P2 (kx + ik y ). 2

(3.259)

Combining Equations (3.250) and (3.257), we can obtain the 8 × 8 Hamiltonian given by

c03.indd 121

4/20/2011 10:43:46 AM

122

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ Ac ⎜ 0 ⎜ * ⎜V ⎜ 0 H =⎜ ⎜ −V ⎜ ⎜ 0 ⎜U* ⎜ ⎝ 0

0 Ac 0 V* 0 −V 0 U

*

V 0 F −K −H 0 0 0

−V * 0 −H * H λ 0

0 V −K * G H* 0 0

Δ 0

Δ

0 −V * 0 0 0 F −K * H*

U 0 0 0 Δ −K G −H

⎞ c1 ⎟ c2 ⎟ ⎟ h1 ⎟ l1 ⎟ , ⎟ c1 ⎟ H ⎟ l2 − H * ⎟ h2 ⎟ λ ⎠ c2 0 U 0 Δ 0

(3.260)

with 2 (kx2 + k y2 + kz2 ) + ac (ε xx + ε yy + ε zz ), 2 m0 P P V = − 2 (kx + ik y ) + 2 (ε xj + iε yj )k j, 2 j 2

Ac = Eg +

∑

U = P1kz − P1

∑ε

zj

(3.261)

k j,

j

where P1 and P2 are the conduction band transverse (longitudinal) deformation potentials. Here, the basis set for a wurtzite crystal is summarized as follows: c1 : iS ↑ c 2 : iS ↓ h1 : u1 = −

1 [ X +i Y ]↑ 2

1 [( X − i Y ] ↑ 2 c1 : u3 = Z ↑ l 1 : u2 =

(3.262)

1 [( X − i Y ] ↓ 2 1 h2 : u5 = − [ X +i Y ]↓ 2 c 2 : u6 = Z ↓ . l 2 : u4 =

c03.indd 122

4/20/2011 10:43:46 AM

BAND STRUCTURE OF SEMICONDUCTORS

123

3.5 BAND STRUCTURE OF ZINC BLENDE AND WURTZITE SEMICONDUCTORS Usually, the coupling between the conduction and the valence bands can be neglected for a semiconductor with a large bandgap, and, in that case, we can use the 6 × 6 Hamiltonian for the valence band. In this section, we show the block-diagonalized 3 × 3 Hamiltonian for zinc blende and wurtzite bulk crystals [3–8]. Also, the Hamiltonian for zinc blende bulk crystal is written using the basis functions of wurtzite crystals for a consistent block diagonalization into two 3 × 3 Hamiltonians using almost the same transformation.

3.5.1 The Hamiltonian of a Zinc Blende Bulk Semiconductor As shown in Section 3.2, the valence-band structure of a strained (001)-oriented zinc blende semiconductor can be described by the following 6 × 6 Hamiltonian in the envelope-function space: (001) H zinc blende

⎛ P +Q ⎜ ⎜ ⎜ * ⎜ R ⎜ ⎜ * ⎜ 2R = −⎜ ⎜ * ⎜ −S / 2 ⎜ ⎜ ⎜ −S* ⎜ ⎜ ⎝ 0

R

2R

− S/ 2

−S

P −Q

2Q

3/2S *

2Q

P+Δ

0

3/2S *

3/2S

0

P+Δ

− 2Q

0

0

3/2S

− 2Q

P −Q

S*

−S* / 2

− 2 R*

R*

⎞ ⎟ ⎟ ⎟ S ⎟ ⎟ ⎟ − S/ 2 ⎟ ⎟ ⎟ − 2R ⎟ ⎟ ⎟ R ⎟ ⎟ ⎟ P +Q ⎠ 0

3 3 , 2 2 3 1 ,− 2 2 1 1 ,− 2 2 1 1 , 2 2 3 1 , 2 2 3 3 ,− , 2 2 (3.263)

where P = Pk + Pε Q = Qk + Qε R = Rk + Rε

c03.indd 123

4/20/2011 10:43:46 AM

124

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

S = Sk + Sε Pk =

2 γ 1 (kx2 + k y2 + kz2 ) 2 mo

Qk =

2 γ 2 (kx2 + k y2 − 2kz2 ) 2 mo

Rk =

2 3[− γ 2 (kx2 − k y2 ) + 2iγ 3kx k y ] 2 mo

(3.264)

2 2 3γ 3 (kx − ik y )kz 2 mo Pε = − av (ε xx + ε yy + ε zz )

Sk =

Qε = − b/2(ε xx + ε yy − 2 ε zz ) 3 b(ε xx − ε yy ) − idε xy 2 Sε = − d(ε zx − iε yz ),

Rε =

where ki (i = x, y, z) is the wave vector component in the unprimed ( x, y, z) coordinate system. The zinc blende basis set is given in Equation (3.213). The biaxial strain components for the (001)-oriented zinc blende crystal are given by ε xx = ε yy = εc =

asc − aec , aec

c C12 εc, c C11 ε xy = ε yz = ε zx = 0,

(3.265)

ε zz = −2

where asc and aec are the lattice constants of the substrate and epilayer c c materials, and C11 and C12 are the stiffness constants in the strained epilayers. The superscript c means those for zinc blende (cubic) crystals. The 6 × 6 Hamiltonian matrix Equation (3.263) can be block diagonalized using the unitary transformation: ⎛ HU H = UHU † = ⎜ ⎝ 0

c03.indd 124

0 ⎞ , H L ,⎟⎠

(3.266)

4/20/2011 10:43:46 AM

BAND STRUCTURE OF SEMICONDUCTORS

125

where the upper and lower blocks, H U and H L, are given by ⎛ Pk + Qk ⎜ ⎜ U H = −⎜ Rk + i Sk ⎜ ⎜ ⎜ 2 R −i 1 S k k ⎜⎝ 2

Rk − i Sk

1 Sk ⎞ ⎟ 2 ⎟ 3 2Qk + i Sk ⎟ , ⎟ 2 ⎟ ⎟ Pk + Δ ⎟⎠

2 Rk + i

Pk − Qk 2Qk − i

3 Sk 2

(3.267)

and ⎛ Pk + Δ ⎜ ⎜ 3 ⎜ L H = − ⎜ 2Qk − i Sk 2 ⎜ ⎜ 2 R −i 1 S k k ⎜ 2 ⎝

2Qk + i

⎞ 1 Sk ⎟ 2 ⎟ ⎟ Rk − i Sk ⎟, ⎟ ⎟ Pk + Qk ⎟ ⎠

3 Sk 2

2 Rk + i

Pk − Qk Rk − i Sk

(3.268) where Rk =

2 γ + γ3⎞ 2 kt 3⎛ 2 ⎝ 2 mo 2 ⎠

Sk =

2 2 3γ 3kt kz 2 mo

(3.269)

kt2 = kx2 + k y2. The unitary transformation U is defined by ⎛ α* 0 ⎜ 0 β* ⎜ ⎜0 0 T =⎜ 0 0 ⎜ ⎜ 0 β* ⎜ * ⎝α 0

0 0 γ* γ* 0 0

0 0 γ −γ 0 0

0 −α ⎞ −β 0 ⎟ ⎟ 0 0⎟ , 0 0⎟ ⎟ 0⎟ β ⎟ 0 α⎠

(3.270)

where α=

c03.indd 125

1 i⎛⎝⎜ e 2

3 π 3φ ⎞ − ⎟ 4 2⎠

, β=γ =

π φ

1 i⎛⎝⎜ − 4 + 2 ⎞⎠⎟ , e 2

(3.271)

4/20/2011 10:43:47 AM

126

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

and φ is the angle defined by φ = tan −1 (k y /kx ). For the strain potential, H εσ is given by ⎛ Pε + Qε H = −⎜ 0 ⎜ ⎜⎝ 0

0 Pε + Qε

U ε

2Qε

0 ⎞ 2Qε ⎟ , ⎟ Pε ⎟⎠

(3.272)

⎞ ⎟ 0 ⎟. Pε + Qε ⎟⎠

(3.273)

and ⎛ Pε ⎜ H εL = − ⎜ 2Qε ⎜ 0 ⎝

2Qε Pε − Qε 0

0

Under the unitary transformation, the upper- and lower-block Hamiltonians, H U and H L, are decoupled. Let us denote the upper- and lower-block envelope functions by Ψ σmk (r) =

∑g

ν m

(k , z)e ik⋅r ν ,

ν

(3.274)

where gmν (k , z) are the envelope functions, { ν } denotes the transformed Bloch bases at the zone center, m is the quantum well subband index, k  = kx xˆ + k y yˆ , r = xxˆ + yyˆ , and σ = U (or L ) refers to the upper (or lower) blocks, respectively. We note that the summation is for ν = 1, 2, 3 for σ = U, and for ν = 4, 5,6 for σ = L. The transformed Bloch bases are given by 1 = α* 2 = β* 3 = γ* 4 =γ

*

5 = β* 6 = α*

c03.indd 126

3 3 3 3 , − α ,− 2 2 2 2 3 1 3 1 ,− − β , 2 2 2 2 1 1 1 1 ,− + γ , 2 2 2 2 1 1 1 1 ,− − γ , 2 2 2 2 3 1 3 1 ,− + β , 2 2 2 2 3 3 3 3 , + α ,− . 2 2 2 2

(3.275)

4/20/2011 10:43:47 AM

BAND STRUCTURE OF SEMICONDUCTORS

3.5.2

127

Zinc Blende Hamiltonians in the Wurtzite Basis Functions

Sometimes, it is convenient to work on a wurtzite basis for zinc blende Hamiltonians. We define a matrix Mzb , which relates the zinc blende bases given in Equation (3.213) to ( X ↑, Y ↑, Z ↑, X ↓, Y ↓, Z ↓)t . (zinc blende bases) = Mzb ( X ↑, Y ↑, Z ↑, X ↓, Y ↓, Z ↓)t,

(3.276)

where ⎛ −1/ 2 ⎜ ⎜ 0 ⎜ 1/ 6 Mzb = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 1/ 3

− i/ 2 0

0 2/ 3

0

0

0

−1/ 6

− i/ 6

0

− i/ 6

0

0

0

2/

0

0

1/ 2

− i/ 2

0

0

1/ 3

1/ 3

i/ 3

0

− i/ 3

0

0

0

−1/

⎞ ⎟ ⎟ 3 ⎟⎟ . ⎟ ⎟ ⎟ ⎟ 3⎠

(3.277)

Similarly, we define another matrix Mwz , which relates the wurtzite bases to ( X ↑, Y ↑, Z ↑, X ↓, Y ↓, Z ↓)t . (U 1,U 2,U 3,U 4,U 5,U 6 )t = Mwz ( X ↑, Y ↑, Z ↑, X ↓, Y ↓, Z ↓)t,

(3.278)

where the wurtzite bases are given by U1 = −

1 ( X + iY ) ↑ 2

1 ( X − iY ) ↑ 2 U3 = Z ↑ ,

U2 =

1 U4 = ( X − iY ) ↓ 2 1 U5 = − ( X + iY ) ↓ 2 U6 = Z ↓ ,

(3.279)

and the matrix is

c03.indd 127

4/20/2011 10:43:47 AM

128

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ −1/ 2 − i/ 2 ⎜ ⎜ 1/ 2 − i/ 2 ⎜ 0 0 Mwz = ⎜ ⎜ 0 0 ⎜ 0 ⎜ 0 ⎜⎝ 0 0 The zinc blende Hamiltonian in by the relation

0⎞ ⎟ 0 0 0 0⎟ 1 0 0 0⎟ ⎟. (3.280) 0 1/ 2 − i/ 2 0⎟ ⎟ 0 −1/ 2 − i/ 2 0⎟ 0 0 0 1⎟⎠ the wurtzite bases can be obtained 0

0

0

(001) *  H (001) = Mbases H zinc blende M bases,

(3.281)

−1 Mbases = Mwz Mzb .

(3.282)

where

We can obtain (001)-oriented zinc blende Hamiltonian with wurtzite bases:

H (001)

(1) ⎛ H11 ⎜ − H (1) 21 ⎜ (1) ⎜ − H 23 =⎜ ⎜ 0 ⎜ 0 ⎜ ⎝ 0

(1)* − H 21 (1) H 22 (1)* H 23 0 0 Δ (1)

(1)* − H 23 (1) H 23 (1) H 33 0 Δ (1) 0

0 0 0 (1) H11 (1)* − H 21 (1)* H 23

0 0 Δ (1) (1) − H 21 (1) H 22 (1) − H 23

0 ⎞ Δ (1) ⎟ ⎟ 0 ⎟ (1) ⎟ H 23 ⎟ (1)* ⎟ − H 23 (1) ⎟ H 33 ⎠

U1 U2 U3 , (3.283) U4 U5 U6

where the superscript (1) means (001)-orientated zinc blende crystals in the wurtzite bases, and the matrix elements are: 2 (1) H11 =− [(γ 1 + γ 2 )(kx2 + ky2 ) + (γ 1 − 2 γ 2 )kz2 ] − Pε − Qε, 2 mo 2 (1) H 22 = H1(1) Δ, 1 − 3 Δ 2 (1) H 33 =− − ( γ 1 − 2 γ 2 )(kx2 + k y2 ) + ( γ 1 + 4 γ 2 )kz2 ] − Pε + 2Qε, [ 3 2 mo (3.284) 2 3 (1) 2 2 (1) (1) H 21 = −3 γ ( k − k ) + 2 i γ k k − b ε − ε ), ( [ 2 x y 3 x y] xx yy 2 mo 2 (1) H 23 =−

Δ (1) =

c03.indd 128

2 3 2 γ 3 (kx + ik y )kz, 2 mo

2 Δ. 3

4/20/2011 10:43:47 AM

BAND STRUCTURE OF SEMICONDUCTORS

129

3.5.3 The Hamiltonian of Wurtzite Bulk Semiconductors From Section 3.4, we have the full Hamiltonian for (0001)-orientated wurtzite crystal:

H (0001)

(2) ⎛ H11 ⎜ − H (2) 21 ⎜ (2) ⎜ − H 23 =⎜ ⎜ 0 ⎜ 0 ⎜ ⎝ 0

(2)* − H 21 (2) H 22 (2)* H 23 0 0 Δ (2)

(2)* − H 23 (2) H 23 (2) H 33 0 Δ (2) 0

0 0 0 (2) H11 * − H 2(2) 1 (2)* H 23

0 0 Δ (2) (2) − H 21 (2) H 22 (2) − H 23

0 ⎞ Δ (2) ⎟ ⎟ 0 ⎟ (2) ⎟ H 23 ⎟ (2)* ⎟ − H 23 (2) ⎟ H 33 ⎠

U1 U2 U3 , U4 U5 U6 (3.285)

where the superscript (2) stands for (0001)-oriented wurtzite crystals. The matrix elements are: (2) = Δ 1 + Δ 2 + λ + θ, H11 (2) = Δ 1 − Δ 2 + λ + θ, H 22 ( 2) = λ + λ ε, H 33

λ=

2 [ A1kz2 + A2 (kx2 + k y2 )] + λ ε, 2 mo

θ=

2 [ A3kz2 + A4 (kx2 + k y2 )] + θ ε, 2 mo

( 2) H 21 =

2 A5 (kx + ik y )2 + D5 ε(+2 ), 2 mo

(2) H 23 =

2 A6 (kx + ik y )kz + D6 ε(z2+), 2 mo

(3.286)

λ ε = D1ε(zz2 ) + D2 (ε(xx2 ) + ε(yy2 ) ), θ ε = D3 ε(zz2 ) + D4 (ε(xx2 ) + ε(yy2 ) ), ε(+2 ) = ε(xx2 ) − ε(yy2 ) + 2iε(xy2 ), ε(z2+) = ε(xz2 ) + iε(yz2 ), Δ ( 2 ) = 2 Δ 3. Here, the Ai s are the valence-band effective mass parameters, which are similar to the Luttinger parameters in a zinc blende crystal, the Di s are the deformation potentials for wurtzite crystals, Δ 1 is the crystal

c03.indd 129

4/20/2011 10:43:47 AM

130

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

field split energy, and Δ 2 and Δ 3 account for spin–orbit interactions in wurtzite crystals. For the biaxial strain, the strain tensor has only the following nonvanishing diagonal elements: ε(xx2 ) = ε(yy2 ) = ε

(2) zz

asw − aew , aew

2C w = − w13 ε(xx2 ), C 33

(3.287)

w w where C13 and C 33 are the stiffness constants for wurtzite structure.

3.5.4 Block Diagonalization of Zinc Blende and Wurtzite Hamiltonians The full 6 × 6 Hamiltonian matrix can be block diagonalized using the ( n) ( n) basis transformation. The off-diagonal terms such as H 21 and H 23 contain the φ( n) dependence, and we can write ( n)

( n) ( n) i 2 φ H 21 = K 21 e , ( n)

( n) ( n ) iφ H 23 = K 23 e ,

(3.288)

( n) ( n) where kx + ik y = kt exp(iφ( n) ) for n = 1, 2. Here, the K 21 and K 23 matrix elements for two Hamiltonians are given by

(1) K 21 =

K

(1) 23

2 2 mo

⎡ − 3 ( γ + γ ) ⎤ k 2, 3 ⎢⎣ 2 2 ⎥⎦ t

2 ⎡ −3 2 γ 3 ⎤⎦ kt kz, = 2 mo ⎣

(3.289)

for the first Hamiltonian H (001) , and (2) K 21 =

K

(2) 23

2 A5kt2, 2 mo

2 A6 kt kz. = 2 mo

(3.290)

for the second Hamiltonian H (0001). We define the transformation matrix T ( n) as

c03.indd 130

4/20/2011 10:43:47 AM

BAND STRUCTURE OF SEMICONDUCTORS

T ( n)

⎛ α ( n)* 0 ⎜ 0 β( n ) ⎜ 0 ⎜ 0 = ⎜ ( n)* α 0 ⎜ ⎜ 0 β( n ) ⎜ ⎝ 0 0

0 0 β

( n )*

0 0 −β( n)*

α ( n) 0 0 −α ( n ) 0 0

0 β

( n )*

0 0 −β( n)* 0

0 ⎞ 0 ⎟ ⎟ β( n ) ⎟ , 0 ⎟ ⎟ 0 ⎟ ⎟ β( n ) ⎠

131

(3.291)

where α ( n)

⎛ 3 π 3 φ( n ) ⎞ + 4 2 ⎟⎠

1 i⎜⎝ = e 2

,

β( n )

⎛ π φ( n ) ⎞ 2 ⎟⎠

1 i⎜⎝ 4 + = e 2

.

(3.292)

The block-diagonalized Hamiltonian, H B( n) = T ( n)* HT ( n)t , is given by ⎛ H ( n)U H B( n) = ⎜ ⎝ 0

0 ⎞ , H ( n) L ⎟⎠

(3.293)

where the upper 3 × 3 Hamiltonian is

H ( n)U

( n) ⎛ H11 ( n) = ⎜ K 21 ⎜ ( n) ⎜⎝ iK 23

( n) K 21 ( n) H 22 ( n) Δ ( n) + iK 23

( n) −iK 23 ⎞ ( n) ⎟ ( n) Δ − iK 23 , ⎟ ( n) ⎟⎠ H 33

(3.294)

and the lower 3 × 3 Hamiltonian is

H ( n)L

( n) ⎛ H11 ( n) = ⎜ K 21 ⎜ ⎜⎝ − iK ( n) 23

( n) K 21 ( n) H 22 ( n) Δ ( n) − iK 23

( n) iK 23 ⎞ ( n) ⎟ ( n) Δ + iK 23 , ⎟ ( n) ⎟⎠ H 33

(3.295)

where n = 1 and 2. The lower Hamiltonian is the complex conjugate of the upper Hamiltonian. Therefore, they have exactly the same eigenvalues, since the eigenenergies are real. For the wurtzite quantum well structure, there will be a lifting of Kramers degeneracy in the in-plane dispersion relations because of the internal field due to piezoelectric and spontaneous polarizations. The wave functions of the lower Hamiltonian are the complex conjugates of the corresponding wave functions of the upper Hamiltonian.

c03.indd 131

4/20/2011 10:43:48 AM

132

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

3.5.5 Analytical Solutions for Valence Band Energies and Wave Functions in Bulk Semiconductors The eigenequation for the valence band energy E ( n) at the band edge is given by ( n) − E ( n) ⎛ H11 ⎜ K ( n) 21 ⎜ ⎜⎝ iK ( n) 23

( n) K 21 ( n) H 22 − E ( n) Δ ( n) + iK 23

( n) − iK 23 ⎞ ⎛ g1(n) ⎞ ( n) ⎟ ⎜ ( n) ⎟ Δ ( n) − iK 23 g = 0. ⎟⎜ 2 ⎟ ( n) ( n) ⎟ ⎜ ( n) ⎟ H 33 − E ⎠ ⎝ g3 ⎠

(3.296)

The eigenequation is a third-order polynomial equation that can always be solved analytically. The complete Bloch wave functions for the valence bands are given by Ψ(vn) (r, k ) =

1 ik⋅r e V

3

∑ g (k) ν ν

( n)

,

(3.297)

ν= 1

where 1

( n)

= α ( n)* U 1 + α ( n) U 4 ,

2

( n)

= β( n) U 2 + β( n)* U 5 ,

3

( n)

= β( n)* U 3 + β( n) U 6 .

(3.298)

Similar expressions hold for the lower Hamiltonian (Eq. (3.295)) ( n) ( n) with bases 4 , 5 , and 6 ( n) . The conduction bands can be characterized by a parabolic band model, and the effective mass Hamiltonians for the three crystal structures are (1) H cond (k) =

(2) H cond (k ′) =

2 (1) (1) (kx2 + ky2 + kz2 ) + Egc + ac (ε(1) xx + ε yy + ε zz ), c 2 me  2 ⎛ kx′ 2 + k y′ 2 kz′ 2 ⎞ w (2) (2) + w ⎟ + Δ 1 + Δ 2 + Egw + acz ε zz + actw (ε(2) xx + ε yy ), 2 ⎜⎝ metw mez ⎠ (3.299)

where mec is the effective mass in the conduction band for zinc blende w structure, mez and metw are the longitudinal and transverse effective masses for wurtzite structure, ac is the conduction band deformation w potential for zinc blende structure, and acz and actw are the conduction band deformation potentials along the c-axis and perpendicular to the c-axis, respectively. The conduction band wave function is of the form

c03.indd 132

4/20/2011 10:43:48 AM

BAND STRUCTURE OF SEMICONDUCTORS

Ψ(cnη) (r, k) =

1 ik⋅r e Sη , V

133

(3.300)

where η is the electron spin ↑ and ↓, and S is spherically symmetric wave function.

3.5.6 Interband Optical Momentum Matrix Elements in Bulk Semiconductors The optical momentum matrix elements are given by 2

2

eˆ ⋅ M η( n) = Ψ(cnη) eˆ ⋅ p Ψ(vn) .

(3.301)

Using the expressions given in Equations (3.297) and (3.300), and taking the φ( n) integration of the momentum matrix elements, we obtain the following momentum matrix elements: TE-polarization (eˆ = xˆ or yˆ ) :

M (pxn) ( n)2 ( g1 + g2(n)2 ) 4 M (pzn) ( n)2 2 Mzη( n) = g3 2 2

M xη( n) =

TM-polarization (eˆ = zˆ ) :

(3.302) (3.303)

where (1) M (1) px = M pz = S px X

(1) 2

= S pz Z

(1) 2

=

mo ⎛ mo ⎞ (Egc + Δ )Egc − 1⎟ c , ⎜ ⎠ (Eg + 2 Δ/3) 2 ⎝ mec (3.304)

and M (2) px = S px X M (2) pz = S pz Z

(2) 2

(3) 2

= =

2 mo ⎛ mo ⎞ Eg [(Eg + Δ 1 + Δ 2 )(Eg + 2 Δ 2 ) − 2 Δ 3 ] − 1 , ⎜ ⎟⎠ 2 ⎝ metw (Eg + Δ 1 + Δ 2 )(Eg + Δ 2 ) − Δ 32

mo ⎛ mo ⎞ (Eg + Δ 1 + Δ 2 )(Eg + 2 Δ 2 ) − 2 Δ 32 . ⎜⎝ w − 1⎟⎠ Eg + 2 Δ 2 2 mez (3.305)

Substituting E ( n) = E1( n), E2(n), and E3(n) into the wave function components, gi, i = 1, 2, 3, we can the obtain moment matrix element for the interband optical transitions between the conduction (C) band and the HH, LH, and SO or CH bands, respectively.

c03.indd 133

4/20/2011 10:43:48 AM

134

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

3.5.7 Interband Optical Momentum Matrix Elements in a Quantum Well Structure For a quantum well structure, the band-edge discontinuities have to be added properly in the effective mass equation. The valence subband structure for the upper Hamiltonian (Eq. (3.294)) can be determined by solving 3

∑ ⎛⎝ H

( n )U ij

j =1

⎛ k = − i ∂ ⎞ + δ V ( n)0 (z)⎞ g ( n)( j ) (z; k ) = E ( n)U g ( n)( i ) (z; k ), ij v t m m t ⎝ z ⎠ m ∂z ⎠ (3.306)

where gm( n)( i ), i = 1, 2,…,6 are the envelope functions of the mth valence subbands, and Vv( n)0 (z) is the QW potential profile of the unstrained valence band reference energy. If we choose the reference energy for the well to be zero, the reference energy for the barrier is given by the band-edge discontinuity ΔEv0 = Qv ΔEg, where ΔEg is the bandgap difference between the unstrained barrier and the well, and Qv is the partition ratio. The hole wave function in a quantum well can be written as Ψ(mn)U (z; kt ) =

Ψ(mn) L (z; kt ) =

e ikt ⋅rt ( n)(1) ( n) ( gm (z; kt ) 1 + gm( n)(2) (z; kt ) 2 A e

ik t ⋅rt

A

( gm( n)(4) (z; kt ) 4

( n)

+ gm( n)(5) (z; kt ) 5

( n)

+ gm( n)(3) (z; kt ) 3

( n)

)

(3.307) ( n)

+ gm( n)(6) (z; kt ) 6

( n)

)

(3.308)

where gm( n)( i ) follow the normalization rules 3

∑ ∫ dz g

(i ) m

2

(z; kt ) = 1,

i =1

(3.309)

6

∑ ∫ dz g

(i ) m

2

(z; kt ) = 1.

i=4

Similarly, the valence band structure for the lower Hamiltonian, Em( n) L, can be obtained using H ij( n) L + δ ijVv( n)0 (z).

c03.indd 134

4/20/2011 10:43:48 AM

135

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

3.6 CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN 3.6.1 Zinc Blende Hamiltonian with General Crystal Orientation under Transformation to Wurtzite Bases The Hamiltonian for an arbitrary crystal orientation [9–11] can be obtained using a rotation matrix ⎛ cos θ cos φ cos θ sin φ − sin θ⎞ U = ⎜ − sin φ cos φ 0 ⎟. ⎜ ⎟ ⎝ sin θ cos φ sin θ sin φ cos θ ⎠

(3.310)

The rotation with the Euler angles θ and φ transforms physical quantities from the ( x, y, z) coordinates to the ( x ′, y′, z′) coordinates. Figure 3.1 shows configuration of coordinate systems in (hkl ) -oriented QW structures. The relation between the coordinate systems for vectors and tensors is expressed as ki′ = U iα kα ε ′ij = U iαU jβ ε αβ Cijkl ′ = U iαU jβU iγU jδCαβγδ

or kα = U iα ki′, or ε αβ = U iαU jβ ε ′ij,

(3.311)

or Cαβγδ = U iαU jβU iγU jδCijkl ′ ,

where i, j, α, β, γ , δ = x, y, z . The rotation matrix for the two spins is given by

[001] z

[hkl] z'

x' y x

y' QW

Figure 3.1. Configuration of coordinate systems in (hkl )-oriented QW structures.

c03.indd 135

4/20/2011 10:43:49 AM

136

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

θ ⎛ − i φ2 ⎜ e cos 2 Ms = ⎜ φ ⎜ − e − i 2 sin θ ⎝ 2

φ i θ⎞ e 2 sin ⎟ 2 ⎟. φ i θ e 2 cos ⎟⎠ 2

(3.312)

Combining the matrices U and M s, we can obtain the matrix Mc that relates ( X ′ ↑′, Y ′ ↑′, Z ′ ↑′, X ′ ↓′, Y ′ ↓′, Z ′ ↓′)t and ( X ↑, Y ↑, Z ↑, X ↓, Y ↓, Z ↓)t: θ − i φ2 θ − i φ2 θ − i φ2 ⎛ cos φ cos θ cos sin φ cos θ cos − sin θ cos e e e ⎜ 2 2 2 ⎜ φ θ −i φ ⎜ − sin φ cos θ e − i 2 cos φ cos e 2 0 ⎜ 2 2 ⎜ φ θ −i φ θ −i φ ⎜ cos φ sin θ cos θ e − i 2 sin φ sin θ cos e 2 cos θ cos e 2 ⎜ 2 2 2 Mc = ⎜ φ φ − i − i θ −i φ ⎜ − cos φ cos θ sin θ e 2 − sin φ cos θ sin θ e 2 sin θ sin e 2 ⎜ 2 2 2 ⎜ φ φ −i −i θ θ ⎜ sin φ sin e 2 − cos φ sin e 2 0 ⎜ 2 2 ⎜ φ φ φ ⎜ − cos φ sin θ sin θ e − i 2 − sin φ sin θ sin θ e − i 2 − cos θ sin θ e − i 2 ⎝ 2 2 2 φ φ φ θ i θ i θ i ⎞ cos φ cos θ sin e 2 sin φ cos θ sin e 2 − sin θ sin e 2 ⎟ 2 2 2 ⎟ φ φ θ i θ i ⎟ − sin φ sin e 2 cos φ sin e 2 0 ⎟ 2 2 φ φ φ ⎟ θ i θ i θ i cos φ sin θ sin e 2 sin φ sin θ sin e 2 cos θ sin e 2 ⎟ ⎟ 2 2 2 φ φ φ⎟ θ i θ i θ i cos φ cos θ cos e 2 sin φ cos θ cos e 2 − sin θ cos e 2 ⎟ 2 2 2 ⎟ ⎟ φ φ θ i2 θ i2 ⎟ − sin φ cos e cos φ cos e 0 ⎟ 2 2 φ φ φ ⎟ θ i θ i θ i cos φ sin θ cos e 2 − sin φ sin θ cos e 2 cos θ cos e 2 , ⎟⎠ 2 2 2 (3.313) where the superscript t means taking the transpose of the vector or matrix.

c03.indd 136

4/20/2011 10:43:49 AM

137

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

The zinc blende Hamiltonian in the wurtzite bases for an arbitrary orientation can be obtained by the relation (001) *  H ( lmn) = Mtotal H zinc blende M total,

(3.314)

−1 Mtotal = Mwz Mc Mzb ,

(3.315)

where

using matrices Mwz and Mzb defined in Equations (3.277) and (3.280). If we want to get the Hamiltonian with the zinc blende bases given −1 by Equation (3.263), Mtotal = Mwz Mc Mzb should be replaced with −1 Mtotal = Mzb Mc Mzb . The above method to obtain the Hamiltonian for general crystal orientation is the most general and straightforward. However, the expression of Hamiltonian elements is not usually simplified. Thus, an invariant method to give the same result as the above is often used for general crystal orientation. Below, we first treat 4 × 4 Hamiltonian with a strain effect, and then extend the result to the 6 × 6 Hamiltonian. 3.6.2 Invariant Method to Obtain Zinc Blende Hamiltonians for General Crystal Orientation The 4 × 4 Hamiltonian first proposed by Luttinger to describe the interaction between the heavy-hole and light-hole bands along with the Bir–Pikus Hamiltonian for strain can be formally written as H tv = H 0v (k) + H sv,

(3.316)

with H 0v (k) representing the valence band Hamiltonian, and H sv the valence band strain or Bir–Pikus Hamiltonian. The Luttinger formulation of the most general Hamiltonian for the (001) crystal orientation is given by 2 ( γ 1k 2 I 4 − 2 γ 2 [( J x2 − 1/3 J 2 )kx2 + ( J y2 − 1/3 J 2 )k y2 + ( J z2 − 1/3 J 2 )kz2 ] 2 mo − 4 γ 3 [{ J x J y }{kx k y } + { J y J z }{k y kz } + { J z J x }{kz kx }]), (3.317) v H s = − av (ε xx + ε yy + ε zz )I 4 + b[( J x2 − 1/3 J 2 )ε xx + ( J y2 − 1/3 J 2 )ε yy H ov =

+ ( J z2 − 1/3 J 2 )ε zz ] + 2d/ 3[{ J x J y }{ε xy } + { J y J z }{ε yz } + { J z J x }{ε zx }]), where the J i s are the angular momentum matrices for a state with spin 3/2, I 4 is a 4 × 4 identity matrix, and {kx k y } = (kx k y + k ykx )/2. Here,

c03.indd 137

4/20/2011 10:43:49 AM

138

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

ki (i = x, y, z) is the wave vector component in the unprimed ( x, y, z) coordinate system, γ i (i = 1, 2, 3) are Luttinger parameters, and av, b , and d are deformation potentials in the valence band. Note that the valence-band strain Hamiltonian H sv was obtained by replacing ki k j in H ov by ε ij, with γ 1, γ 2 , and γ 3 being replaced by −av, − b/2, and − d/(2 3), respectively. The angular momentum matrices are any three matrices, ( J x J y, J z ), which satisfy the rules of commutation for angular momentum. These matrices given by Luttinger are as follows:

⎛ ⎜ Jx = ⎜ ⎜ ⎜ ⎜⎝

0

0

3/2

0

0

1

3/2

1

0

⎛ 0 0 ⎞ ⎜ ⎟ 3/2⎟ 0 , Jy = ⎜ ⎜ ⎟ 0 ⎟ ⎜ i 3/2 ⎜⎝ 0 i 0 ⎟⎠

0 3/2 0 0 0 0 ⎞ ⎛ 3/2 ⎛1 0 ⎜ 0 −1/2 0 ⎜0 1 0 ⎟ ⎟ , I4 = ⎜ Jz = ⎜ 0 1/2 0 ⎟ ⎜ 0 ⎜0 0 ⎜⎝ 0 ⎟ ⎜⎝ 0 0 0 0 −3/2⎠

0 0 1 0

0

− i 3/2

0

i

−i

0

3/2

0

0⎞ 0⎟ ⎟. 0⎟ 1⎟⎠

⎞ ⎟ − i 3/2⎟ , 0 ⎟⎟ 0 ⎟⎠ 0

(3.318)

We can obtain the (001)-oriented Hamiltonian substituting Equation (3.318) into Equation (3.317): R −S 0 ⎞ ⎛P +Q * ⎜ R 0 P −Q S* ⎟ ⎟. H 0v = ⎜ * 0 P −Q R ⎟ ⎜ −S ⎜⎝ 0 S* R* P + Q⎟⎠

(3.319)

This is exactly the same form as the 4 × 4 Hamiltonian part given in Equation (3.263). Note that the ordering of the basis functions for the choice of J i s is 23 , 23 , 23 , − 21 , 23 , 21 , and 23 , − 23 . The z′-axis, or, equivalently, the (lmn)-axis, is the direction normal to the growth surface. Using the relation kα = U iα ki′ in Equation (3.311), the most general valence band Hamiltonian obtained from these considerations can be written as

c03.indd 138

4/20/2011 10:43:49 AM

139

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

P=−

Q=−

2 2 mo

2 γ 1 [ kx2′ + k y2′ + kz2′ ] 2 mo

(3.320)

⎧ ⎡ ⎛ η6 − 4 η6 n2 − 6k 4 n2 + η2 (6k 2 n2 + n4 ) ⎞ ⎨⎢ γ 2 ⎜ ⎟⎠ η2 ( η2 + n2 )2 ⎩⎣ ⎝

⎛ ( η4 − η2 k 2 + k 4 )n2 ⎞ ⎤ 2 + 6γ 3 ⎜ kx ′ ⎝ η2 ( η2 + n2 )2 ⎟⎠ ⎥⎦ ⎡ ⎛ η4 + 6k 4 + η2 (−6k 2 + n2 ) ⎞ ⎛ ( η2 − k 2 )k 2 ⎞ ⎤ 2 + ⎢γ 1 + γ 2 ⎜ + 6γ 3 ⎜ 2 2 k y′ ⎟ 2 2 2 2 ⎝ ⎠ ⎝ η ( η + n2 )2 ⎟⎠ ⎥⎦ η (η + n ) ⎣ ⎛ η4 + 3k 4 + n4 − η2 (3k 2 + n2 ) ⎞ ⎡ ⎛ k 4 − η2 (k 2 + n2 ) ⎞ ⎤ 2 6 γ + ⎢γ 1 − 2γ 2 ⎜ + 3 ⎟⎠ ⎜⎝ ( η2 + n2 )2 ⎟⎠ ⎥ kz′ ⎝ ( η2 + n2 )2 ⎣ ⎦ ⎡⎛ 6kn( η2 − 2k 2 ) η2 − k 2 ⎞ ⎤ ( γ − γ ) + ⎢⎜ 2 3 ⎥ kx ′ k y′ ⎟ η2 ( η2 + n 2 )3/2 ⎠ ⎣⎝ ⎦ 2 2 2 2 ⎡ ⎛ 6 ηk ( η − 2 k ) η − k ⎞ ⎤ ( γ 2 − γ 3 )⎥ k y′ kz′ + ⎢⎜ ⎟ 2 2 2 3/2 η (η + n ) ⎠ ⎣⎝ ⎦ 4 4 2 2 2 ⎡⎛ 6 ηn( η + 2k − η (2k + n )) ⎞ ⎤ ⎫ + ⎢⎜ ( γ 3 − γ 2 )⎥ kx ′ kz′ ⎬ ⎟ 2 2 2 2 ⎠ η (η + n ) ⎣⎝ ⎦ ⎭ (3.321) ⎡ ⎛ η6 + 4k 4 n4 + 2k 2 n2 (k 2 − 2 m2 ) + η2 (−2k 2 n 2 + n 4 ) R = ⎢ − 3γ 2 ⎜ ⎝ η2 ( η2 + n2 )2 ⎣ 2k( η2 − 2k 2 )n3 η2 − k 2 η2 + n2 ⎞ +i ⎟ η4 ( η2 + n2 )2 ⎠ ⎛ η6 + η4 k 2 − η2 (k 4 − 2k 2 n2 ) − 2k 4 n2 − 2 3n 2 γ 3 ⎜ ⎝ η4 ( η2 + n2 )2 −i

km η2 − k 2 η2 + n2 ( η2 − 2k 2 ) ⎞ ⎤ 2 ⎟ ⎥ kx ′ η4 ( η2 + n 2 ) 2 ⎠⎦

⎡ ⎛ η6 + 2 η2 ( η4 − 3η2 k 2 + 3k 4 ) + 3n( η2 − 2k 2 )( η2 n − 2k 2 n) + ⎢γ 2 ⎜ 3η4 ( η2 + n2 ) ⎣ ⎝ 6kn( η2 − 2k 2 ) η2 − k 2 η2 + n 2 ⎞ +i ⎟ 3η4 ( η2 + n 2 ) ⎠

c03.indd 139

4/20/2011 10:43:50 AM

140

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ η2 k η2 − k 2 + 2k η2 − k 2 n2 + 2 3k η2 − k 2 γ 3 ⎜ η4 ( η2 + n2 ) ⎝ −i

η2 n η2 + n2 − 2k 2 n η2 + n2 ⎞ ⎤ 2 ⎟ ⎥ k y′ η4 ( η2 + n2 ) ⎠⎦

⎡ ⎛ 2 k 4 n 2 + η4 ( − k 2 + n 2 ) + η 2 ( k 4 − 2 k 2 n 2 ) + ⎢ −2 3( γ 2 − γ 3 ) ⎜ ⎝ η2 ( η2 + n2 )2 ⎣ kn( η2 − 2k 2 ) η2 − k 2 η2 + n2 ⎞ ⎤ 2 +i ⎟ ⎥ kz′ η2 ( η2 + n2 )2 ⎠⎦ ⎡ 2 3kn η2 − k 2 +⎢ 4 2 γ 2 ( η2 − 2k 2 ) η2 + n2 ( η2 + 2 n2 ) 2 2 ⎣ η (η + n )

(

+ i(4 nk( η2 − 2k 2 ) η2 − k 2 ) +

)

2 3 γ ( kn η2 + n2 η2 − k 2 ( η2 + 2 n2 )(− η2 + 2k 2 ) 2 2 2 3 η (η + n ) 4

⎤ + i( η8 + 2 η6 n2 + η4 n2 (n2 − 4k 2 ) + 4 η2 n 2 k 2 (k 2 − n2 ) + 4k 4 n4 ))⎥ kx ′ k y′ ⎦ ⎡ 2 3kη +⎢ 4 2 γ ( η 2 − 2 k 2 ) η 2 + n 2 ( η2 + 2 n 2 ) 2 2 2 η η + n ( ) ⎣

(

+ i(4 nk( η2 + n2 ) η2 − k 2 ) +

)

2 3kη ⎛ γ − η2 + n 2 η2 − k 2 ( η2 − 2k 2 )( η2 + 2 n 2 ) 2 2 2 3⎜ ⎝ η (η + n ) 4

⎞⎤ + i 4k(− η2 + k 2 )( η2 + n2 )⎟ ⎥ k y′ kz′ ⎠⎦ ⎡ 2 nη +⎢ γ (3η2 ( η4 + 2 η2 k 2 − 2k 2 ) − 3( η2 − 2k 2 )2 n2 ) 4 2 2 2 2 3 η ( η + n ) ⎣

(

− i(6 nk( η2 − 2k 2 ) η2 − k 2 η2 + n2 ) +

)

2 3n η ⎛ γ η6 − 4k 4 n2 + η4 (2k 2 − n2 ) − 2 η2 (k 4 − 2k 2 n2 ) 2 2 2 3⎜ ⎝ η (η + n ) 4

⎞⎤ + i 2kn(− η2 + 2k 2 ) η2 − k 2 η2 + n2 ⎟ ⎥ kx ′ kz′ ⎠⎦ (3.322)

c03.indd 140

4/20/2011 10:43:50 AM

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

141

⎡ 2 3n ( γ 2 − γ 3 ) ( η6 − η4 n2 + 2 η2 k 2 n2 − 2k 4 n2 −S = ⎢ 3 2 2 2 ⎣ η (η + n ) ⎤ − ikn( η2 − 2k 2 ) η2 − k 2 η2 + n2 ⎥ kx2′ ⎦ ⎡ 2 3k +⎢ 3 2 ( γ 2 − γ 3 ) ( 2 k ( − η2 + k 2 ) 2 η ( η + n ) ⎣

)

⎤ + i( η2 − 2k 2 ) η2 − k 2 η2 + n2 )⎥ k y2′ ⎦ ⎡ 2 3 ( γ 2 − γ 3 ) ( n(− η4 − 2k 2 + η2 (2k 2 + n2 )) +⎢ 2 2 2 η ( η + n ) ⎣ ⎤ − ik( η2 − 2k 2 ) η2 − k 2 η2 + n2 )⎥ kz2′ ⎦ ⎡ 4 3km +⎢ 3 2 ( γ 2 − γ 3 ) n( η2 − 2k 2 ) η2 − k 2 η2 + n2 2 2 η η + ( n ) ⎣

(

⎤ + i 2k( η2 − k 2 )( η2 + n2 ) ⎥ kx ′ k y′ ⎦ ⎡ 2 3km −⎢ 2 2 +2( γ 2 − γ 3 )k( η2 − 2k 2 ) η2 − k 2 η2 + n2 2 2 ( ) η η + n ⎣

)

(

⎤ + i( η2 + n 2 )[ η4 γ 3 − 4( γ 2 − γ 3 )k 4 + 4 η2 ( γ 2 − γ 3 ) + η2 γ 3 n2 ])⎥ k y′ kz′ ⎦ ⎡ 2 3 +⎢ 2 2 ((γ 2 − γ 3 )(2 η4 n2 + 4k 4 n2 − 4 η2 k 2 n2 ) + γ 3 (η6 + η2 n4 ) 2 2 η ( η + n ) ⎣ ⎤ + γ 2 2 η4 n2 + i 2( γ 2 − γ 3 )kn( η2 − 2k 2 ) η2 − k 2 η2 + n2 )⎥ k y′ kz′ , ⎦ (3.323) where η2 = h2 + n2 . Here, we can write the crystalline surface in terms of the azimuthal angle φ and the polar angle θ as follows: tan φ =

k h

and tan θ =

h2 + k 2 . h

(3.324)

Next, we consider the specific case of the (1ln) surface where the angle φ is fixed to be π/4.

c03.indd 141

4/20/2011 10:43:50 AM

142

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

3.6.3

(11n)-Oriented Zinc Blende Hamiltonian

The relation between the coordinate systems for vectors and tensors is given by Equation (3.311). Our object is to obtain P ′, Q′, R′, and S ′ in ( x ′, y′, z′) coordinates from the Hamiltonian in Equation (3.319), which can be done by substituting Equation (3.311) into Equation (3.317). Here, we consider the case of quantum wells with ( 11n ) orientations with θ = arctan( 2/n) and φ = π/4 . Then, from Equation (3.320) to (3.323),

Pk ′ = Qk ′ =

2 γ 1 [ kx′ 2 + k y′ 2 + kz′ 2 ], 2 mo 2 9 n2 ⎧ ⎡ (n2 − 4)(n2 − 1) ⎤ γ 1 ⎨⎢ γ + γ 3 kx′ 2 2 2 mo ⎩ ⎣ (n2 + 2)2 (n2 + 2)2 ⎥⎦ 3 ⎡ (n2 + 2)(n2 − 1) ⎤ γ 3 ⎥ k y′ 2 +⎢ γ2 + 2 2 2 ( n + 2) ⎦ ⎣ (n + 2)

6 + 12 n2 ⎤ 2 ⎫ ⎡ (n2 − 1)2 + ⎢ −2 2 γ − γ 3 kz′ ⎬ , 2 2 (n2 + 2)2 ⎥⎦ ⎣ (n + 2) ⎭ 2 2 2 4  ⎧ ⎡ (n − 4) 5n + n ⎤ Rk ′ = γ − 2 γ 3 kx′ 2 ⎨ 3⎢ 2 2 2 2 mo ⎩ ⎣ (n + 2) (n + 2)2 ⎥⎦ 1 + n2 ⎡ 1 ⎤ γ2 + 2 γ k ′2 + 3⎢ 2 2 3⎥ y (n + 2) ⎣ (n + 2) ⎦ 2 n2 γ 2 + 2 γ 3 ⎡ (n − 1) ⎤ 2 + 2 3 ⎢− 2 ( γ − γ ) k + i 2 3 kx′ k y′ ′ 3 z 2 ⎥ 2 (n2 + 2) ⎣ (n + 2) ⎦

(3.325)

n n ⎡ ⎤ ⎡ ⎤ ⎫ + 2 6 ⎢i 2 ( γ 2 − γ 3 )⎥ k y′ kz′ + 6 6 ⎢ 2 ( γ 2 − γ 3 )⎥ kx′ kz′ ⎬ , 2 ( n + 2) ( n + 2) ⎣ ⎦ ⎣ ⎦ ⎭ 2 2  ⎧ ⎡ n(4 − n ) n ⎤ ⎡ ⎤ − Sk ′ = ( γ 2 − γ 3 )⎥ kx′ 2 + ⎢ 6 2 ( γ 3 − γ 2 )⎥ k y′ 2 ⎨⎢ 6 2 2 2 mo ⎩ ⎣ (n + 2) ⎦ ⎣ (n + 2) ⎦ n(n 2 − 1) n ⎡ ⎤ + ⎢2 6 2 ( γ 2 − γ 3 )⎥ kz′ 2 + i 2 6 2 ( γ 2 − γ 3 )kx′ k y′ (n + 2) (n + 2) ⎣ ⎦ ⎡ 2 γ + n2 γ 3 ⎤ + i2 3 ⎢ 2 2 ⎥ k y′ kz′ ⎣ (n + 2) ⎦ 12 n2 (4 − 2 n2 + n4 ) ⎤ ⎡ ⎫ + 3 ⎢− 2 γ − 2 γ 3 ⎥ kx′ kz′ ⎬ , 2 2 2 2 (n + 2) ⎣ (n + 2) ⎦ ⎭

c03.indd 142

4/20/2011 10:43:50 AM

143

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

Pε′ = − av (ε ′xx + ε ′yy + ε ′zz ), b(n2 − 1)(− Γ + 3 2 ε ′xz n + n2 Γ) Qε′ = −

+ d 3[Γ + 2 ε ′xz n + n2 (2Γ − 2 ε ′xz n)] , (2 + n2 )2

Rε′ = −

b 3(− Γ + 3 2 ε ′xz n + n2 Γ) + d[Γ − 3 2 ε ′xz n − n2 Γ) , (2 + n2 )2

(3.326)

bn 3( 2Γ − 6 nε ′xz − 2 n2 Γ) − Sε′ = −

+ d[ 2 n(n2 − 1)Γ + ε ′xz (−4 + 2 n2 − n4 ) , (2 + n2 )2

where Γ = ε − ε ′zz, ε = (asc − aec )/aec . The biaxial strain components for a general crystal orientation are determined from the condition that the layer is grown pseudomorphically and these strain coefficients should minimize the strain energy of the layer simultaneously, as discussed in Section 3.6.5. It is straightforward to obtain the 6 × 6 Hamiltonian of the (11n)oriented zinc blende crystal by using components P ′, Q′, R′, and S ′ in ( x ′, y′, z′) coordinates. That is, the 6 × 6 Hamiltonian for the valenceband structure in ( x ′, y′, z′) coordinates can be written as ⎛ P ′ + Q′ ⎜ * ⎜ −S ′ ⎜ R′ * H′ = −⎜ ⎜ 0 ⎜ ⎜ −S ′* / 2 ⎜ ⎝ 2 R′ *

−S ′

R′

0

− S ′/ 2

P ′ − Q′

0

R′

− 2Q′

0

P ′ − Q′

S′

3/2S ′*

R′ *

S ′*

P ′ + Q′

− 2 R′ *

− 2Q′

3/2S ′

− 2 R′

P′ + Δ

3/2S ′*

2Q′

− S ′/ 2

0

2 R′ ⎞ u ⎟ 1 3/2S ′ ⎟ u2 2Q′ ⎟⎟ u3 − S ′* / 2 ⎟ u4 ⎟ 0 ⎟ u5 ⎟ u P′ + Δ ⎠ 6

′ ′ ′ , ′ ′ ′,

(3.327) where P ′ = Pk ′ + Pε′, Q′ = Qk ′ + Qε′, R′ = Rk ′ + Rε′, S ′ = Sk ′ + Sε′. Here, the basis set used in ( x ′, y′, z′) coordinates consists of the following basis functions:

c03.indd 143

4/20/2011 10:43:50 AM

144

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

u1 ′ =

3 3 ′ 1 , =− ( X ′ + iY ′) ↑′ 2 2 2

u2 ′ =

3 1 ′ 1 , = −( X ′ + iY ′) ↓′ +2Z ′ ↑′ 2 2 6

u3 ′ =

3 1 ′ 1 = ,− ( X ′ − iY ′) ↑′ +2Z ′ ↓′ 2 2 6

3 3 ′ 1 = u4 ′ = , − ( X ′ − iY ′) ↓′ 2 2 2 u5 ′ =

1 1 1 ′ ( X ′ + iY ′) ↓′ + Z ′ ↑′ = , 2 2 3

u6 ′ =

1 1 ′ 1 ,− ( X ′ − iY ′) ↑′ − Z ′ ↓′ . = 2 2 3

(3.328)

3.6.4 Interband Optical-Matrix Elements in a (11n)-Oriented Zinc Blende Quantum Well The hole wave function in a quantum well can be written as e ikt ⋅rt Ψ m (z; kt ) = A

6

∑ g′

( ν) m

(z; kt ) uν ′,

(3.329)

ν= 1

where gm′( ν) ( ν = 1, 2, 3, 4, 5, and 6) is the wave function for the mth subband in ( x ′, y′, z′) coordinates, uν ′ is given by Equation (3.328), and gm′( i ) follows the normalization rules 6

∑ ∫ dz g ′

( ν) m

2

(z; kt ) = 1.

(3.330)

ν= 1

Since we solve one Schrödinger equation for electrons and a 6 × 6 Hamiltonian for holes, we need only the optical matrix elements for two cases: spin up and spin down. The polarization-dependent interband momentum matrix elements are written as TE-polarization (eˆ ′ = cos φ′xˆ ′ + sin φ′yˆ ′):

c03.indd 144

4/20/2011 10:43:51 AM

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

{

145

}

1 1 1 Px gm′(1) φl + Px gm′(3) φl + Px gm′(6) φl 2 6 3 1 1 (3.331) + sin φ′ − i Px gm′(1) φl − i Px gm′(3) φl 2 6 2 1 −i Px gm′(6) φl 3

2

eˆ ′ ⋅ M ′ ↑ = cos φ′ −

2

{

{

}

}

1 1 1 Px gm′(2) φl + Px gm′(4) φl + Px gm′(5) φl 6 2 3 1 1 + sin φ′ − i Px gm′ (2) φl − i Px gm′(4) φl 6 2 (3.332) 2 1 (5) +i Px gm′ φl 3

eˆ ′ ⋅ M ′ ↓ = cos φ′ −

{

}

TM-polarization ( eˆ ′ = zˆ ′ ): 2

2

2 1 Pz gm′(2) φl + Pz gm′(5) φl 6 3 2 2 1 = Px gm′(3) φl − Px gm′(6) φl . 6 3

eˆ ′ ⋅ M ′ ↑ = − eˆ ′ ⋅ M ′ ↓

2

(3.333) (3.334)

3.6.5 Strain Tensors on Zinc Blende Semiconductors for a General Crystal Orientation 3.6.5.1 Derivation from Hooke’s Law In general, the in-plane strain tensor components are known to be ε ′11 = ε ′22 = ε′ for biaxial tensile or compressive strain, and ε 12 ′ = ε ′21 = 0 due to no in-plane shear. ′ = ε ′23 = ε ′33 ) The other three independent stain tensor elements (ε 13 remain to be determined. The substrate applies a uniform in-plane stress to the film. This is the only external stress applied to the film. Thus, T33′ = T23′ = T13′ = 0. On the other hand, T11′ , T22′ , and T12′ should be determined. Also, the stress and strain tensor components are related to Hooke’s law within the linear elastic domain: 3

Tαβ ′ +

∑C′

ε′ = 0

αβij ij

(3.335)

i , j =1

′ = 0 , as given above, for α, β = 1, 2, 3 . For (αβ) = (33) , (23), and (13), Tαβ and Hooke’s law gives

c03.indd 145

4/20/2011 10:43:51 AM

146

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE 3

∑C′

ε ′ = 0,(αβ) = (33),(23),(13).

αβij ij

(3.336)

i , j =1

′ ij and ε ′ij , Using the symmetry of Cαβ Cαβ ′ 11ε 11 ′ + Cαβ ′ 22 ε ′22 + Cαβ ′ 33 ε ′33 + 2Cαβ ′ 12 ε ′12 + 2Cαβ ′ 31ε ′31 + 2Cαβ ′ 23 ε ′23 = 0, Cαβ ′ 33 ε ′33 + 2Cαβ ′ 31ε ′31 + 2Cαβ ′ 23 ε ′23 = −(Cαβ ′ 11 + Cαβ ′ 22 )ε′ , (αβ) = (33),(23),(13).

(3.337)

This can be written as a matrix equation: C 3323 C3331 ′ ′ ′ ⎞ ⎛ ε ′33 /2⎞ ′ + C 3322 ′ ⎞ ⎛ C 3333 ⎛ C 3311 ε ′  ⎜ C 2333 C 2323 C2331 ′ ′ ′ ⎟ ⎜ ε ′23 ⎟ = − ⎜ C 2311 ′ + C 232 ′ 2⎟ . ⎟ ⎜ ⎟⎜ ⎟ 2⎜ ⎝ C3133 ⎝ C3111 C3123 C3131 ′ ′ ′ ⎠ ⎝ ε ′31 ⎠ ′ + C3122 ′ ⎠

(3.338)

The elastic stiffness tensor of a material in the cubic crystal class Oh (m3m) has only three distinct components, which are c11, c12, and c44, in the standard contracted index notation. That is, the first two and the last two subscripts in cijkl are replaced by a single subscript running from 1 to 6 as follows: tensor notation 11 22 33 23, 32 31,13 12, 21 . (3.339) matrix notation 1 2 3 4 5 6 Then, using the relation 3

C γδ′ kl =

∑U

αγ

U βδU ikU jlCαβij,

(3.340)

α ,β ,i , j =1

with c11 = c22 = c33, c12 = c23 = c13 , and c44 = c55 = c66 for a cubic crystal, we obtain C γδ′ kl = c11U 1γU 1δU 1kU 1l + c22U 2 γU 2 δU 2 kU 2 l + c33U 3 γU 3δU 3kU 3l + c12 (U 1γU 1δU 2 kU 2 l + U 2 γU 2 δU 1kU 1l ) + c23 (U 2 γU 2 δU 3kU 3l + U 3 γU 3δU 2 kU 2 l ) + c13 (U 1γU 1δU 3kU 3l + U 3 γU 3δU 1kU 1l ) + c44 (U 2 γU 3δU 2 kU 3l + U 2 γU 3δU 3kU 2 l (3.341) + U 3 γU 2 δU 2 kU 3l + U 3 γU 2 δU 3kU 2 l ) + c55 (U 3 γU 1δU 3kU 1l + U 3 γU 1δU 1kU 3l + U 1γU 3δU 3kU 1l + U 1γU 3δU 1kU 3l ) + c66 (U 1γU 2 δU 1kU 2 l + U 1γU 2 δU 2 kU 1l + U 2 γU 1δU 1kU 2 l + U 2 γU 1δU 2 kU 1l )

c03.indd 146

4/20/2011 10:43:51 AM

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

147

3

C γδ′ kl = c11

∑U

αγ

U αδU αkU αl

α =1

3

+ c12

β −1

∑ ∑ (U

αγ

U αδU βkU βl + U βγU βδU αkU αl )

(3.342)

β = 2 α =1 3

+ c44

β −1

∑ ∑ (U

αγ

U βδ + U βγU αδ )(U αkU βl + U βkU αl )

β = 2 α =1

Having calculated the various elements C γδ′ kl , Equation (3.338) can be inverted numerically to determine ε ′33, ε ′23, and ε ′31. Up to this point, the discussion is perfectly general and applicable to cases of substrates of any arbitrary orientation. We can obtain ε ij from Equation (3.338) and, in turn, obtain ε ′ij from the relation Equation (3.311). For example, the components of the strain tensor for (11n) surfaces are given by ε ′xx = ε ′yy = ε, ε ′xy = ε ′yz = 0, K1 ε, 2K2 K ε ′zz = 3 ε, K2

ε ′xz =

(3.343)

where K1 = (C11 + 2C12 )(−C11 + C12 + 2C44 )n(n2 − 1), 2 2 K 2 = 2C11C44 + 2C12C44 + 4C 424 + (C11 + C11C12 − 2C12 + 2C11C 44 2 2 − 4C12C 44 + C11C 44 n )n , (3.344) 2 2 2 K 3 = −2[C11C44 + 3C12C44 − 2C44 + (C11 + C11C12 − 2C12 − 2C11C44 + C12C 44 n2 )n2 ]. 3.6.5.2 Derivation from the Condition Grown Pseudomorphically The strain coefficients in the ( x, y, z) coordinates for a general crystal orientation can be also determined from the condition that the layer is grown pseudomorphically, and these strain coefficients should minimize the strain energy of the layer simultaneously. We define the unit vectors xˆ ′, yˆ ′, and zˆ ′ along the x ′-, y ′- , and z′-axes , and they are related to unit vectors xˆ , yˆ , and zˆ along the x-, y-, and z-axes through the rotation matrix Equation (3.310).

c03.indd 147

4/20/2011 10:43:52 AM

148

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

The zinc blende primitive translational vectors are ai (xˆ + yˆ ), 2 a b i = i (yˆ + zˆ ), 2 ai g i = (xˆ + zˆ ), 2

ai =

(3.345)

where a is the lattice constant and i labels the epilayer (e) and the substrate (s). When the crystal is translated, the primitive translational vectors become ai (x i + y i ), 2 a bi′ = i (y i + z i ), 2 a g i′ = i (x i + z i ), 2

a i′ =

(3.346)

where x i = (1 + ε ixx )xˆ + ε ixy yˆ + ε ixz zˆ , y i = ε iyx xˆ + (1 + ε iyy )yˆ + ε iyz zˆ ,

(3.347)

z i = ε izx xˆ + ε izy yˆ + (1 + ε izz )zˆ . ˆ l } so that N ˆ 1 and N ˆ2 We defined the set of orthonormal vectors {N ˆ 3 is orthogonal to this plane as lie in the superlattice plane, and N follows: ˆ 1 = cos θ cos φiˆ + cos θ sin φjˆ − sin θkˆ , N ˆ 1 = − sin θiˆ + cos φjˆ , N

(3.348)

ˆ 1 = sin θ cos φiˆ + sin θ sin φjˆ + cos θkˆ . N When the first atomic layers are deposited on the substrate, these layers will be strained to match the substrate, and a pseudomorphic (or coherent) interface will be formed. Thus, the condition for a pseudomorphic interface means that the projections of the strain-distorted primitive translational vectors of the epilayer and the substrate on the growth plane should be equal:

c03.indd 148

4/20/2011 10:43:52 AM

149

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

ˆ 1 = a ′s ⋅ N ˆ 1, a ′e ⋅ N ˆ 2 = a ′s ⋅ N ˆ 2, a ′e ⋅ N ˆ 1 = b ′s ⋅ N ˆ 1, b ′e ⋅ N ˆ 2, ˆ 2 = b ′s ⋅ N b ′e ⋅ N

(3.349)

ˆ 1 = g ′s ⋅ N ˆ 1, g ′e ⋅ N ˆ 2 = g ′s ⋅ N ˆ 2. g ′e ⋅ N The expression of the strain component for general crystal orientation is complicated, and thus we will explain the case of the quantum well with the (111) orientation as an example. Then, angles θ and φ are given as θ = arctan( 2) and φ = π/4. Equation (3.349) is easily calculated with the relations of ε xx = ε yy = ε zz, ε xy = ε yz = ε zx. Then, condiˆ 1 = α s′ ⋅ N ˆ 1 and α e′ ⋅ N ˆ 2 give the following ˆ 2 = α s′ ⋅ N tions of α e′ ⋅ N relation ae (1 + ε exx + ε exy )/ 6 − ae ε exy 2/3 = as (1 + ε sxx + ε sxy )/ 6 − as ε sxy 2/3 − ae (1 + ε exx + ε exy )/(2 2) + ae ε exy / 2 = −as (1 + ε sxx + ε sxy )/(2 2) + as ε sxy / 2 , (3.350) with analogous conditions for β′i and γ ′i . These constraint equations are linearized in the lattice constant difference ae − as. Only five of the constraint equations are linearly independent because the fourth and the fifth equations in Equation (3.349) give the same form. Four unknowns ε exx, ε exy, ε sxx, and ε sxy are determined by the condition of the minimization of the strain energy density, in addition to constraint equations (Eq. (3.350)). The strain energy density of a zinc blende structure material is 1 i U i = C11 [(ε ixx )2 + (ε iyy )2 + (ε izz )2 ] 2 i + 2C44 [(ε ixy )2 + (ε iyz )2 + (ε ixz )2 ] i + C12 [(ε iyy )(ε izz ) + (ε ixx )(ε izz ) + (ε ixx )(ε iyy )],

(3.351)

where Cij s are the elastic constants. For a stained-layer superlattice made up of two zinc blende materials, the strain energy density is U es =

c03.indd 149

U e he + U s hs , he + hs

(3.352)

4/20/2011 10:43:52 AM

150

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

where he (hs ) is the layer thickness of material e(s) in the superlattice. In a case of the single quantum well, we put hs to be infinite in a final result. For (111) crystal orientation, for example, U i (i = e, s) is given by i i U i = ( 3C11 /2 + 3C12 ) ε ixx 2 + 6C44i ε ixy2.

(3.353)

To obtain ε exy in terms of ε exx, first, we replace the ε exy in U es with the expression obtained from Equation (3.350). Then, we differentiate U es with respect to ε exx and obtain ∂U es ⎛ ∂U e ∂U s ⎞ + =⎜ ⎟ (he + hs ) = 0. ∂ε exx ⎝ ∂ε exx ∂ε sxx ⎠

(3.354)

e ∂U e e e e e e ∂ε xy = 3 C /2 + 3 C 2 ε + 6 C ε = 0, ( ) 11 12 xx 44 xy ∂ε exx ∂ε exx

(3.355)

Thus,

with ∂ε exy /∂ε exx = 1/2 from Equation (3.350). Finally, we obtain the expression ε exy = −

(C11e + 2C12e ) (ε e e 4C 44

xx

) ≡ − Ae ε exx,

(3.356)

e e + 2C12 where Ae = (C11 ) /(4C44e ) . Similarly, we can obtain

ε sxy = − As ε sxx,

(3.357)

s s + 2C12 where As = (C11 ) /(4C44s ) . Next, we replace the ε sxy in U es with the expression obtained from Equation (3.350), and differentiate U es with respect to ε exx . Then,

∂U es ⎛ ∂U e ∂U s ⎞ =⎜ + ⎟ (he + hs ) = 0, ∂ε exx ⎝ ∂ε exx ∂ε sxx ⎠

(3.358)

where s ∂U e ⎛ 3 l ⎛ 3 C s + 3C s ⎞ 2(ε s ) ∂ε xy h s, e ⎞ e e C C x h + = + ε 3 2( ) 11 12 x 11 12 xx ⎝2 ⎠ ⎠ ∂ε exx ∂ε exx ⎝ 2

(3.359)

and ∂ε sxy /∂ε exx = ae /a s from Equation (3.350). Thus,

c03.indd 150

4/20/2011 10:43:53 AM

CRYSTAL ORIENTATION EFFECTS ON A ZINC BLENDE HAMILTONIAN

ε sxx = −

(C11e + 2C12e ) (ε e ) he as ≡ − B he as ε e , xx h s ae (C11s + 2C12s ) xx hs ae

151

(3.360)

e e + 2C12 where B = (C11 ) / (C11s + 2C12s ) . The ε exx is obtained by substituting Equations (3.356), (3.357), and (3.360) into Equation (3.350) as follows:

ε exx = −

1 ⎛ as ⎞ − 1⎟ . e s s e 2 ⎜ e ⎝ ⎠ (1 + A ) + (1 + A )B(he /hs )(a /a ) a

(3.361)

In the case of a single quantum well, taking hs to be infinite, we obtain ε exx = −

e 4C 44 e e e 4C 44 + C11 + 2C12

(3.362)

The strain components for ε ′ij can be obtained using the rotational matrix and the obtained ε ij. Similarly, we can obtain the expressions ε ij for a general crystal orientation (hkl ).

3.6.6 Piezoelectric Fields in the (11n)-Oriented Zinc Blende Hamiltonian Off-diagonal strains in zinc blende structure semiconductors induce a polarization given by PiS = 2e14 ε jk,

(3.363)

where P S is the induced polarization, e14 is the piezoelectric constant, and i, j, k = x, y, z . Here, i, j, k are in cyclic order. A strained layer with a (001) growth direction has only diagonal strains. Thus, the (001)oriented layer will not have strain-induced piezoelectric polarization fields. In addition, in the cubic gallium nitride (GaN) crystal structure (c-GaN), spontaneous polarization does not exist due to the higher crystal symmetry. However, strained layers with any other growth direction will have piezoelectric polarization fields. The piezoelectric polarization along the growth direction is important, because the electric field in the quantum well originates from polarization charges at heterojunction interfaces. The electric field induced perpendicular to the growth direction due to the strain can be calculated as

c03.indd 151

4/20/2011 10:43:53 AM

152

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

E⊥ = −

PS ⋅ uˆ , ε

(3.364)

where ε is the dielectric constant, and the unit vector along the growth direction is given by uˆ = 1/(n2 + 2)( ˆi + ˆj + nkˆ ) . For a (11n) growth direction, E⊥ = −

2e14 n2 + 2

(n2 ε xy + ε yz + ε zx ),

(3.365)

where 1 [− ε ′xx + 2 nε ′xz + 2 nε ′zz ], (n + 2) 1 ε yz = [−2 nε ′xx − 2(2 − n 2 )ε ′xz + 2 nε ′zz ], 2 2(n + 2) ε zx = ε yz. ε xy =

2

(3.366)

3.7 CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN 3.7.1

Strain Tensors on Wurtzite Semiconductors

We consider only the θ dependence of physical quantities in the following for simplicity [12]. However, we can easily extend the results to a general case with φ ≠ 0 for special cases, such as nonpolar planes. Figure 3.2 shows (a) a configuration of the coordinate systems (x ′, y′, z′) in (hkil )-oriented crystals, and (b) a coordinate system in a wurtzite primitive cell and nonpolar a- and m-planes with the growth direction parallel to the c-axis. The z-axis corresponds to the c-axis (0001), and the growth axis (defined as the z′-axis) is normal to the quantum well plane (hkil ). The coordinate system (x, y, z) denotes the primary crystallographic axes. The relation between the coordinate systems for vectors and tensors is given by Equation (3.311). The strain coefficients in the (x, y, z) coordinates for a general crystal orientation are determined from the condition that the layer is grown pseudomorphically, and these strain coefficients should minimize the strain energy of the layer simultaneously. We define the unit vectors xˆ ′, yˆ ′, and zˆ ′ along the x ′-, y ′-, and z′-axes, and they are related to unit vectors xˆ , yˆ , and zˆ along the x-, y-, and z-axes through the rotation matrix Equation (3.310). The hexagonal primitive translational vectors are

c03.indd 152

4/20/2011 10:43:53 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN z(c-axis) [0001]

z(c-axis) [0001] z' [hkil]

q

A F

x'

y

j x

153

y' (a)

(ABCD) a-plane (ϕ = 0) D

E

B

(BCEF) m-plane C (ϕ = π/6)

X ϕ = π/6 (b)

Figure 3.2. (a) Configuration of the coordinate systems ( x ′, y′, z′) in (hkil )-oriented crystals. The growth axis or the z′-axis is normal to the substrate surface (hkil ), and the coordinate system ( x, y, z) denotes the primary crystallographic axes. (b) The coordinate system in a wurtzite primitive cell and nonpolar a- and m-planes with the growth direction parallel to the c-axis. θ = π/2 with φ = 0 (a-plane) corresponds to the z′ = [1120] growth direction, and θ = π/2 with φ = π/6 (m-plane) corresponds to the z′ = [1010] growth direction.

a i = ai xˆ , 3ai ai xˆ + yˆ , 2 2 g i = ci zˆ ,

bi = −

(3.367)

where a is the lattice constant, and i labels the epilayer (e) and the substrate (s). When the crystal is translated, the primitive translational vectors become a i′′= ai xˆ ′′, 3ai ai xˆ ′′ + yˆ ′′, 2 2 g i′′= ci zˆ ′′,

bi′′= −

(3.368)

where xˆ ′′ = (1 + ε xx )xˆ + ε xy yˆ + ε xz zˆ , yˆ ′′ = ε yx xˆ + (1 + ε yy )yˆ + ε yz zˆ ,

(3.369)

zˆ ′′ = ε zx xˆ + ε zy yˆ + (1 + ε zz )zˆ . When the first atomic layers are deposited on the substrate, these layers will be strained to match the substrate, and a pseudomorphic (or coherent) interface will be formed. Thus, the condition for a

c03.indd 153

4/20/2011 10:43:54 AM

154

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

pseudomorphic interface means that the projections of the straindistorted primitive translational vectors of the epilayer and the substrate on the growth plane should be equal: ˆ ′ = a ′′⋅ ˆ ′, a ′′⋅ e x s x ˆ ′ = a ′′⋅ ˆ ′, a ′′⋅ e y s y

(3.370)

with analogous conditions on β′′ and γ ′′. Here, we assume that the substrate is unstrained. Then, the constraints (Eq. (3.370)) yield the following relations for the strain tensors: ε xx = ε(xx0 ) + ε xz

sin θ , cos θ

ε yy = ε(xx0 ),

(3.371)

cos θ ( 0 ) ε zz = ε xz + ε zz , sin θ ε xy = ε yz = 0,

(0) where ε(0) xx = (as − ae )/ae and ε zz = (cs − ce )/ce. Under the engineering notation, the strain energy density is given by

W=

1 [C11ε 2xx + C11ε 2yy + C33 ε zz2 + 2C12 ε xx ε yy 2 + 2C13 (ε xx ε zz + ε yy ε zz ) + 4C 44 ε 2xz ].

(3.372)

Using the above relations, the strain energy can be expressed through only one strain component, ε xz , which can be found by minimizing the strain energy with respect to the variable ε xz . This procedure gives the following expression for ε xz

ε xz = −

(0) 2 [(c11 + c12 + c13 ε(0) zz /ε xx ) sin θ (0) 2 (0) + (2c13 + c33 ε(0) zz /ε xx ) cos θ ] ε xx cos θ sin θ

c11 sin θ + 2(c13 + 2c44 ) sin θ cos θ + c33 cos θ 4

2

2

4

.

(3.373)

Hence, we obtain a 6 × 6 Hamiltonian in the ( x ′, y′, z′) coordinates by substituting the transformation relation for the vector k in Equation (3.311), and the strain coefficients for a general crystal orientation into the Hamiltonian for the (0001) orientation. 3.7.2 Valence Band Edges of Bulk Wurtzite Semiconductors At the band edges, kx′ = k y′ = kz′ = 0 , the Hamiltonian in the ( x ′, y′, z′) coordinates can be simplified

c03.indd 154

4/20/2011 10:43:54 AM

155

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

⎛ F ′ − K ′* ⎜ − K ′ G′ ⎜ ⎜ − H ′ H ′* H ′(k ′ = 0, ε) = ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎝ 0 Δ

− H ′* H′ λ′ 0 Δ 0

0 0 0 F′ − K ′* H ′*

0 ⎞ Δ ⎟ ⎟ 0 ⎟ H′ ⎟ ⎟ G′ − H ′* ⎟ ⎟ −H ′ λ′ ⎠ 0 0 Δ −K ′

U1 U2 U3 U4 U5 U6

′ ′ ′ , ′ ′ ′ (3.374)

where F ′ = Δ 1 + Δ 2 + λ ′ε + θ′ε, G′ = Δ 1 − Δ 2 + λ ′ε + θ′ε, K ′ = D5 (ε xx − ε yy ), H ′ = D6 ε xz, λ ′ε = D1ε zz + D2 (ε xx + ε yy ),

(3.375)

θ′ε = D3 ε zz + D4 (ε xx + ε yy ), Δ = 2 Δ 3, and the strain coefficients ε ij for a general crystal orientation θ are given by Equations (3.371) and (3.373). Note that K ′ and H ′ are not zero because ε xx ≠ ε yy, and ε xz ≠ 0 in the general coordinates. Also, all elements are constant. This Hamiltonian can be block diagonalized. The transformation matrix is defined as ⎛ α* 0 0 ⎜0 β 0 ⎜ ⎜ 0 0 β* T =⎜ * α 0 0 ⎜ ⎜0 β 0 ⎜ ⎝ 0 0 −β*

0 0⎞ α 0 β* 0 ⎟ ⎟ 0 0 β⎟ , −α 0 0⎟ ⎟ 0 −β* 0⎟ ⎟ 0 0 β⎠

(3.376)

where α=

1 i (3 π/4) 1 i ( π/4) e ,β = e . 2 2

(3.377)

The block-diagonalized Hamiltonian, H ′′ = T * H ′T t , is given by

c03.indd 155

4/20/2011 10:43:54 AM

156

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ H ′U H ′′ = ⎜ ⎝ 0

0 ⎞ , H ′ L ⎟⎠

(3.378)

where the upper 3 × 3 Hamiltonian is K′ − iH ′ ⎞ ⎛ F′ ⎜ H′ = K′ G′ Δ − iH ′⎟ ⎜ ⎟ ⎝ iH ′ Δ + iH ′ λ′ ⎠ U

(3.379)

and the lower 3 × 3 Hamiltonian is K′ iH ′ ⎞ ⎛ F′ ⎜ H′ = K′ G′ Δ + iH ′⎟ . ⎜ ⎟ ⎝ − iH ′ Δ − iH ′ λ′ ⎠ L

(3.380)

The lower Hamiltonian is the complex conjugate of the upper Hamiltonian. Therefore, they have exactly the same eigenvalues, since the eigenenergies are real. The piezoelectric field in bulk crystal does not change the in-plane dispersion curves. The wave functions of the lower Hamiltonian are the complex conjugates of the corresponding wave functions of the upper Hamiltonian. The eigenequation for the valence band energy E ′ at the band edge is given by − iH ′ ⎞ ⎛ g1′ ⎞ K′ ⎛ F ′ − E′ ⎜ K′ G′ − E ′ Δ − iH ′⎟ ⎜ g2′ ⎟ = 0. ⎜ ⎟⎜ ⎟ ⎝ iH ′ Δ + iH ′ λ ′ − E ′ ⎠ ⎝ g3′ ⎠

(3.381)

The eigenequation is a third-order polynomial equation, which can always be solved analytically. These three analytical solutions can be expressed as C2 , 3 1 3 C (S1 − S2 ), E2′ = − (S1 + S2 ) − 2 + i 2 3 2 1 3 C (S1 − S2 ), E3′ = − (S1 + S2 ) − 2 − i 2 3 2 E1′ = 2(S1 + S2 ) −

(3.382)

where

c03.indd 156

4/20/2011 10:43:55 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

1 1 q = C1 − C 22, 3 9 C3 1 r = (C1C 2 − 3C0 ) − 2 , 6 27

157

(3.383)

S1 = [r + q3 + r 2 ]1/3, S2 = [r − q3 + r 2 ]1/3, and C0 = −(F ′G′λ ′ + 2 K ′H ′ 2 − H ′ 2G′ − K ′ 2 λ ′ − F ′Δ 2 − F ′H ′ 2 ), C1 = G′λ ′ + F ′ λ ′ + F ′G′ − Δ 2 − 2 H ′ 2 − K ′ 2, C 2 = −(G′ + F ′ + λ).

(3.384)

The envelope functions corresponding to the three eigenvalues are determined by ⎛ g1′ ⎞ ⎛ D1 ⎞ 1 ⎜ g2′ ⎟ = ⎜ D2 ⎟ ⎜ ⎟ D⎜ ⎟ ⎝ g3′ ⎠ ⎝ D3 ⎠

(3.385)

where D1 = iH ′(G′ − E ′) + (Δ − iH ′)K ′, D2 = iH ′K ′(F ′ − E ′)(Δ − iH ′),

(3.386)

D3 = (F ′ − E ′)(G′ − E ′) − K ′ , 2

and D=

D1 + D2 + D3 . 2

2

2

(3.387)

The complete Bloch wave functions for the valence band are given by 1 ik′⋅r′ Ψ ′ ( r ′, k ′ = 0 ) = e V v

3

∑ g′(k′) i ′, i

(3.388)

i =1

where 1 ′ = α * U 1 ′ + α U 4 ′, 2 ′ = β U 2 ′ + β * U 5 ′,

(3.389)

3 ′ = β U 3 ′ + β U 6 ′, *

c03.indd 157

4/20/2011 10:43:55 AM

158

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

and we can obtain similar expressions for the lower Hamiltonian (Eq. (3.380)). 3.7.3 Polarization-Dependent Interband Optical-Matrix Elements of Bulk Wurtzite Semiconductors The optical momentum matrix elements in the ( x ′, y′, z′) coordinates are given by 2

2

eˆ ′ ⋅ M′ η = Ψ′ cη eˆ ′ ⋅ p′ Ψ′ v .

(3.390)

Using the expressions for the band-edge wave functions of the conduction and valence bands, we obtain the polarization-dependent interband momentum matrix elements: TE-polarization (eˆ ′ = xˆ ′ or yˆ ′): 1 g1′α * cos θ S px X 2 1 + g2′β cos θ S px X − g3′β* sin θ S pz Z 2 for σ = U

2

M x′ ′η = −

1 g4′ α * cos θ S px X 2 1 + g5′β cos θ S px X + g6′β* sin θ S pz Z 2 for σ = L

2

(3.391)

= −

2

TM-polarization (eˆ ′ = zˆ ′) : 2

1 g1′α * sin θ S px X 2 1 + g2′β sin θ S px X + g3′β* cos θ S pz Z 2 for σ = U

Mz′′η = −

1 = − g4′ α * sin θ S px X 2 1 + g5′β sin θ S px X − g6′β* cos θ S pz Z 2 for σ = L.

c03.indd 158

2

(3.392)

2

4/20/2011 10:43:55 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

159

The above analytical results include the crystal orientation dependence, and the band-edge matrix elements are as follows: mo m P2, Pz = S pz Z = o P1,   2 2  ⎛ mo ⎞ (Eg + Δ 1 + Δ 2 )(Eg + 2 Δ 2 ) − 2 Δ 3 , P12 = ⎜⎝ z − 1⎟⎠ (3.393) Eg + 2 Δ 2 2 mo me Px = Py = S px X = S py Y =

P22 =

2  2 ⎛ mo ⎞ Eg [(Eg + Δ 1 + Δ 2 )(Eg + 2 Δ 2 ) − 2 Δ 3 ] − 1 . ⎜ ⎟⎠ 2 mo ⎝ met (Eg + Δ 1 + Δ 2 )(Eg + Δ 2 ) − Δ 32

3.7.4 Piezoelectric Fields and Spontaneous Polarizations in a Quantum Well Structure If a stress τ jk is applied to GaN crystals, there is an induced piezoelectric polarization with a magnitude proportional to the applied stress τ jk: Pi = dijk τ jk,

(3.394)

where dijk are the piezoelectric moduli or piezoelectric tensor elements. The stress τ ij is related to the strain by τ ij = cijkl ε kl,

(3.395)

where cijkl are the stiffness constants of the wurtzite GaN crystal. Here, it is convenient to replace the tensor notation with the engineering notation for dijk and cijkl using their symmetry properties. That is, the second and third subscripts in dijk, and the first two and the last two subscripts in cijkl , are replaced by a single subscript running from 1 to 6 as follows: ⎛ tensor notation 11 22 33 23, 32 31,13 12, 21⎞ . ⎜⎝ matrix notation 1 2 3 4 5 6 ⎟⎠

(3.396)

Then, the piezoelectric polarization in the ( x, y, z) coordinates for a general crystal orientation is given by

⎛ Px ⎞ ⎛ 0 ⎜ Py ⎟ = ⎜ 0 ⎜ ⎟ ⎜ ⎝ Pz ⎠ ⎝ d31

c03.indd 159

0 0 d31

0 0 d33

0 d15 0

d15 0 0

⎛ c11 ⎜ c12 0⎞ ⎜ ⎜ c13 0⎟ ⎜ ⎟ 0 0⎠ ⎜ ⎜ 0 ⎜⎝ 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 ⎞ ⎛ ε xx ⎞ 0 ⎟ ⎜ ε yy ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ ε zz ⎟ . 0 ⎟⎜ 0 ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ 2 ε xz ⎟ c66 ⎟⎠ ⎜⎝ 0 ⎟⎠ (3.397)

4/20/2011 10:43:55 AM

160

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

We then obtain the polarization components along the x-, y-, and z-axes: Px = 2d15 c44 ε xz, Py = 0,

(3.398)

Pz = [d31 (c11 + c12 ) + d33 c13 ](ε xx + ε yy ) + [2d31c13 + d33 c33 ]ε zz. The normal polarization with respect to the growth plane (along the growth direction) can be obtained by taking the scalar product between the polarization vector P in the ( x, y, z) coordinates and the unit vector zˆ ′: Pz′ = P ⋅ zˆ ′ = Px sin θ + Pz cos θ.

(3.399)

Similarly, we can obtain the parallel polarization on the growth plane: Px′ = P ⋅ xˆ ′ = Px cos θ − Pz sin θ, Py′ = P ⋅ yˆ ′ = 0.

(3.400)

The strain-induced polarization Pi will lead to an electric field Ei, given by Di′ = ε ijp′ E j ′ + Pi′,

(3.401)

where Di′ is the electric displacement, and ε ijp′ is the permittivity tensor for the epilayer given by ε ′ = Uε pU −1

(3.402)

and ⎛ εt ε =⎜0 ⎜ ⎝0 p

0 εt 0

0⎞ 0⎟. ⎟ εz ⎠

(3.403)

Here, ε t and ε z are permittivities perpendicular and along the c-axes for the (0001) crystal orientation, respectively. Then, the electric displacement Di′ is given by Dx′ = (ε t cos2 θ + ε z sin 2 θ)Ex′ + (ε t − ε z ) sin θ cos θEz′ + Px′, Dy′ = ε t Ey′ + Py′,

(3.404)

Dz′ = (ε t − ε z ) sin θ cos θEx′ + (ε t sin θ + ε z cos θ)Ez′ + Pz′. 2

c03.indd 160

2

4/20/2011 10:43:55 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

161

If there are no external charges, and ε t  ε z, Dz′ vanishes, and the electric field reduces to Ez′  −

Pz′ . εt

(3.405)

The electric fields in the well and barrier due to the spontaneous (SP) and piezoelectric (PZ) polarization along the (0001)-direction for (0001)-oriented wurtzite structure can be estimated from the periodic boundary condition for a superlattice structure as follows: b w w (FSP + F b − FSP − FPZ )Lb , Lb ε w + Lw ε b L Fzb′ = − w Fzw′ , Lb

Fz′ =

(3.406)

where superscripts w and b mean the well and barrier, respectively, and L and ε are the layer thickness and the static dielectric constant, respeci tively. The strain-induced piezoelectric field FPZ (i = w, b) is given by i FPZ =

i i 2 2d31 2C13 ⎛ i ⎞ i ( 3) i C + C − 11 12 ⎟ ε ′xx , i ⎜ i ε ⎝ C33 ⎠

(3.407)

where d31 is the piezoelectric constant. The spontaneous polarization PSP along the growth direction in the ( 0001) cos θ, arbitrary crystal orientation is estimated by the relation PSP = PSP since the spontaneous polarization in wurtzite (0001) GaN-based quantum well structures exists along the (0001) axis. The strain-induced piezoelectric polarization normal with respect to the growth plane (along the growth direction) in an arbitrary crystal orientation can be expressed as PPZ = Px sin θ + Pz cos θ.

(3.408)

3.7.5 Valence Band Structure of a Quantum Well and SelfConsistent Calculations with the Screening Effect The constant energy contour of the (0001) orientation is isotropic. On the other hand, the valence band structure of the quantum well structures with general crystal orientations shows anisotropy in the quantum well plane unlike the (0001)-oriented structure. Thus, the valence band structure and the momentum matrix element should be calculated as

c03.indd 161

4/20/2011 10:43:56 AM

162

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

a function of the angle Φ in the quantum well plane. That is, the angle is related to k by kx′ = k cosΦ and k y′ = k sin Φ. Here, kx′ and k y′ are the in-plane wave vectors in the ( x ′, y′, z′) coordinate system. Please note that, for simplicity, we omit the angle Φ dependence in the notation of physical functions such as the wave function and the momentum matrix element. For a quantum well structure of a width Lw, the valence band structure is calculated by using 6 × 6 Hamiltonian for an arbitrary crystal orientation, which is obtained by using Equations (3.285) and (3.311). The envelope functions {gm( ν) } satisfy

∑ ⎡⎢⎣ H ν′

νν′

⎛ k , k , − i ∂ ⎞ + (V (z ) + qE z )δ ⎤ g ( ν) (k , k , z ) ′ ′ ′ ν z′ ′ νν′ ⎥⎦ m x′ y′ ′ ⎝ x y ∂z ′ ⎠ (3.409)

= Em′ (kx′, k y′ ) gm( ν) (kx′, k y′, z′),

where ν, ν′ = 1, 2, 3, 4, 5, 6, and Ez′ is given in Equation (3.405). These coupled differential equations can be solved using a finite-difference method. The self-consistent band structures and wave functions are obtained by iteratively solving the Schrödinger equation for electrons, the blockdiagonalized Hamiltonian for holes, and Poisson’s equation. The total potential profiles for the electrons and the holes are Vc (z) = Vcw (z) − e φ(z),

(3.410)

Vv (z) = Vvw (z) − e φ(z),

where Vcw (z) and Vvw (z) are the square-well potential plus the potential due to the internal field without the screening effect for the conduction band and the valence band, respectively, and φ(z) is the screening potential induced by the charged carriers and satisfies the Poisson equation d ⎛ d ε(z) ⎞ φ(z) = − e [ p(z) − n(z)], dz ⎝ dz ⎠

(3.411)

where ε(z) is the dielectric constant. The electron and the hole concentrations, p(z) and n(z), are related to the wave functions of the nth conduction subband and the mth valence subband by n(z) =

c03.indd 162

kTme π 2

∑ f (z) ln (1 + e 2

n

[ E fc − Ecn (0)]/kT

)

(3.412)

n

4/20/2011 10:43:56 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

163

and p(z) =

∑ ∫ dk 2π ∑ g 

k

( ν) mk

ν

m

2 1 ⎞, (z) ⎛ [ E − Evm ( k )]/kT ⎠ fv ⎝ 1+ e

(3.413)

where me is the effective mass of electrons,  is Planck’s constant divided by 2π, n and m are the quantized subband indices for the conduction and the valence bands, E fc and E fv are the quasi-Fermi levels of the electrons and the holes, respectively, Ecn (0) is the quantized energy level of the electrons, Evm (k ) is the energy for the mth subband in the valence band, k is the in-plane wave vector, ν refers to the new ( ν) bases for the Hamiltonian, and fn (z) and gmk (z) are the envelope functions in the conduction and the valence bands, respectively. The potential φ(z) is obtained by integration: φ(z) = −

∫

z

− L/2

E(z′)dz′,

(3.414)

where E(z) =

1 ρ(z′)dz′. − L/2 ε(z)

∫

z

(3.415)

Note that the origin of the coordinate is chosen at the center of the well, and 2L is the total length containing the well and the barrier. The procedures for the self-consistent calculations consist of the following steps: (1) Start with the potential profiles Vc and Vv with φ(0) (z) = 0 in Equation (3.410); (2) solve the Schrödinger equation (for electrons) and the 6 × 6 Hamiltonian (for holes) with the potential profiles φ( n−1) (z) in step (1) to obtain band structures and wave functions; (3) for a given carrier density, obtain the Fermi-energies by using the band structures and the charge distribution by using the wave functions; (4) solve Poisson’s equation to find φ( n) (z); (5) check if φ( n) (z) converges to φ( n−1) (z). If not, set φ( n) (z) = wφ( n) (z) + (1 − w)φ( n−1) (z), n = n + 1; then, return to step (2). If yes, the band structures and the wave functions obtained with φ( n−1) (z) are solutions. An adjustable parameter w (0 < w < 1) is typically set to 0.5 at low carrier densities. With increasing carrier densities, a smaller value of w is needed for rapid convergence. The procedure for the self-consistent calculation is shown in Figure 3.3.

c03.indd 163

4/20/2011 10:43:56 AM

164

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

Solve Poisson’s equation to find (n) . φ(n)(z), i.e., Vc,υ Start with the potential profile Vc and Vυ with the square-well potential plus the potential due to the internal field without the screening effect. That is, with φ(0)(z) = 0.

Solve the Schrödinger equation (for electrons) and the block-diagonalized Hamiltonian (for holes) with the potential profiles φ(n – 1)(z) to obtain band structures and wave functions.

Check if φ(n)(z) converges to φ(n – 1)(z).

Set φ(n)(z) = wφ(n)(z) + (1 – w)φ(n – 1)(z), n = n + 1.

Solutions For a given carrier density, obtain the Fermi-energies by using the band structures and the charge distribution by using the wave functions.

Figure 3.3. Procedure for the self-consistent calculation.

3.7.6 Polarization-Dependent Interband Optical-Matrix Elements of Wurtzite Quantum Wells The optical momentum matrix elements in the ( x ′, y′, z′) coordinates for a general crystal orientation are given by Equation (3.390). Since we solve one Schrödinger equation for electrons and a 6 × 6 Hamiltonian for holes, we need only the optical matrix elements for two cases: spin up and spin down. On the other hand, for a block-diagonalized 3 × 3 Hamiltonian, we needed the optical matrix elements for four cases: spin up and spin down for upper block, spin up and spin down for lower Hamiltonian. The polarization-dependent interband momentum matrix elements for a wurtizte quantum well structure can be written as TEpolarization (eˆ ′ = cos φ ′xˆ ′ + sin φ ′yˆ ′):

{

2

eˆ ′ ⋅ M ′ ↑ = cos φ ′ −

1 cos θ Px gm′(1) φl 2

}

1 cos θ Px gm′( 2 ) φl − sin θ Pz gm′( 3) φl 2 1 1 + sin φ ′ − i Px gm′(1) φl − i Px gm′( 2 ) φl 2 2 +

{

c03.indd 164

(3.416)

}

2

4/20/2011 10:43:57 AM

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

2

eˆ ′ ⋅ M ′ ↓ = cos φ ′

{

1 cos θ Px gm′( 4 ) φl 2

}

1 cos θ Px gm′( 5) φl − sin θ Pz gm′( 6 ) φl 2 1 1 + sin φ ′ − i Px gm′( 4 ) φl − i Px gm′( 5) φl 2 2 −

{

165

(3.417)

}

2

TM-polarization (eˆ ′ = zˆ ′): 2

eˆ ′ ⋅ M ′ ↑ = −

1 1 sin θ Px gm′(1) φl + sin θ Px gm′( 2 ) φl 2 2 2

+ cos θ Pz g ′

( 3) m

φl

1 1 sin θ Px gm′( 4 ) φl − sin θ Px gm′( 5) φl 2 2

2

eˆ ′ ⋅ M ′ ↓ =

2

+ cos θ Pz g ′

(6) m

(3.418)

(3.419)

φl ,

where gm′( ν) ( ν = 1, 2, 3, 4, 5, and 6) is the wave function for the mth subband in ( x ′, y′, z′) coordinates. Also, we can obtain the optical matrix element for the (0001) crystal orientation by substituting θ = 0 in Equation (3.419): TE-polarization (eˆ = xˆ or yˆ ): 2

1 1 Px gm(1) φl + Px gm(2) φl 2 2 1 1 = Px gm′(4) φl − Px gm′(5) φl 2 2

2

eˆ ⋅ M ↑ = − eˆ ⋅ M ↓

2

(3.420) 2

(3.421)

TM-polarization (eˆ = zˆ ) : 2

2

(3.422)

2

2

(3.423)

eˆ ⋅ M ↑ = Pz gm′(3) φl

eˆ ⋅ M ↓ = Pz gm′(6) φl . 3.7.7

Band Structure of Nonpolar a and m-Planes

3.7.7.1 {1010} m-Plane Hamiltonian The m-plane Hamiltonian with a set of faces {1010} can be obtained by substituting φ = π /6 and θ = π /2 into Equation (3.311) [13].

c03.indd 165

4/20/2011 10:43:57 AM

166

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

⎛ F ′ − K ′* ⎜ − K ′ G′ ⎜ ⎜ − H ′ H ′* H ( k ′, ε ′ ) = ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎝ 0 Δ

− H ′* H′ λ′ 0 Δ 0

0 0 0 F′ − K ′* H ′*

0 0 ⎞ Δ ⎟ 0 ⎟ Δ 0 ⎟ −K ′ H′ ⎟ ⎟ G′ − H ′ * ⎟ ⎟ −H ′ λ′ ⎠

U1 U2 U3 U4 U5 U6

′ ′ ′ ′ ′ ′,

(3.424)

where F ′ = Δ 1 + Δ 2 + λ ′ + θ ′, G ′ = Δ 1 − Δ 2 + λ ′ + θ ′, 2 2 ⎞ (0) S , [ A1kx′ 2 + A2 (k y′ 2 + kz′2 )] + D1ε zz + D2 ⎛ 2ε (0) + ⎝ xx 2 mo 3 ⎠ 2 2 ⎞ (0) + D4 ⎛ 2ε (0) + S , θ′ = [ A3kx′ 2 + A4 (k y′ 2 + kz′2 )] + D3ε zz ⎝ xx 2 mo 3 ⎠

λ′ =

K′ =

2 ⎡ 1+ i 3 2 − A5 kx′ + A5 (− 3 + i)k y′ kz′ ⎢ 2 mo ⎣ 2 ⎤ 1 + i 3 2 D5 + A5 kz′ + (1 + 3i)S ⎥ , 2 3 ⎦

(3.425)

⎡ 1− i 3 ⎤ 3−i ⎢ A6 2 kx′ k y′ − A6 2 kx′ kz′ ⎥ , ⎣ ⎦ (0) (0) (0) (C + 3C12 )(ε xx + 1) + C12 (−3 + ε xx ) + C11 (−1 + 3ε (0) xx ) + 4C13ε zz S = − 3 11 . 5C11 + 3C12

H′ =

2 2 mo

Note that the old bases are used in Equation (3.424), which were defined in Equation (3.279). This use of old bases is convenient since Px, Py, and Pz in the old bases are known quantities, as defined in Equation (3.374). The optical momentum matrix elements for the mplane are given by 2

2

eˆ ′ ⋅ M ′η = Ψ′l cη eˆ ′ ⋅ p′ Ψ′mv ,

(3.426)

where Ψ′ c and Ψ′ v are wave functions for the conduction and the valence bands, respectively, and η = ↑ and ↓ for both electron spins. The polarization-dependent interband momentum matrix elements can be written as

c03.indd 166

4/20/2011 10:43:57 AM

167

CRYSTAL ORIENTATION EFFECTS ON A WURTZITE HAMILTONIAN

TE-polarization eˆ ′ = cos φ ′xˆ ′ + sin φ ′yˆ ′: 2

eˆ ′ ⋅ M′ ↑ = − cos φ ′Pz gm′(3) φl + sin φ ′ − (1 + 3i) g ′

(2) m

φl

}

2

− (1 − i 3) g ′

φl

}

(3.427)

2

eˆ ′ ⋅ M ′ ↓ = − cos φ ′Pz gm′(6) φl − sin φ ′ (5) m

{

Px (1 − 3i) gm′(1) φl 2 2

{

Px (1 + i 3) gm′(4) φl 2 2

(3.428)

2

TM-polarization eˆ ′ = zˆ ′: 2

eˆ ′ ⋅ M ′ ↑ = − 2

eˆ ′ ⋅ M ′ ↓ =

{

Px ( 3 + i) gm′(1) φl − ( 3 − i) gm′(2) φl 2 2

{

Px ( 3 − i) gm′(4) φl − ( 3 + i) gm′(5) φl 2 2

}

2

(3.429)

2

}.

(3.430)

3.7.7.2 {1120} a-Plane Hamiltonian The a-plane Hamiltonian with a set of faces {1120} can be obtained by substituting φ = 0 and θ = π /2 into Equation (3.311). The matrix elements in Equation (3.424) are given by F ′ = Δ 1 + Δ 2 + λ ′ + θ ′, G ′ = Δ 1 − Δ 2 + λ ′ + θ ′,

λ′ =

2 C C ⎛ ⎞ (0) [ A1kx′ 2 + A2 (k y′ 2 + kz′2 )] + D1ε zz + D2 ⎜ 2ε (0) − 12 ε (0) − 13 ε (0) , ⎝ xx C11 xx C11 zz ⎟⎠ 2 mo

θ′ =

C 2 C ⎛ ⎞ (0) + D4 ⎜ 2ε (0) − 12 ε (0) − 13 ε (0) , [ A3kx′ 2 + A4 (k y′ 2 + kz′2 )] + D3ε zz ⎝ xx C11 xx C11 zz ⎟⎠ 2 mo

K′ =

2 C C13 (0) (0) ⎞ ε zz − ε xx ⎟ , [ − A5ky′ 2 + 2iA5ky′ kz′ + A5kz′ 2 ] + D5 ⎛⎜⎝ − C12 ε (0) xx − ⎠ 2 mo C11 11

H′ =

2 [ −iA6kx′ ky′ − A6kx′ kz′ ]. 2 mo

(3.431)

The polarization-dependent interband momentum matrix elements for an a-plane are given by

c03.indd 167

4/20/2011 10:43:58 AM

168

ELEMENTARY THEORY OF ELECTRONIC BAND STRUCTURE

TE-polarization eˆ ′ = cos φ ′xˆ ′ + sin φ ′yˆ ′: 2

eˆ ′ ⋅ M ′ ↑ = − Pz cos φ ′ gm′(3) φl + sin φ ′

Px {−i gm′(1) φl − i gm′(2) φl 2

}

2

(3.432) 2

eˆ ′ ⋅ M ′ ↓ = − Pz cos φ ′ gm′(6) φl + sin φ ′

Px {−i gm′(4) φl − i gm′(5) φl 2

}

2

(3.433) TM-polarization eˆ ′ = zˆ ′: 2

eˆ ′ ⋅ M ′ ↑ = − 2

eˆ ′ ⋅ M′ ↓ =

1 1 Px gm′(1) φl + Px gm′(2) φl 2 2

2

(3.434)

2

1 1 Px gm′(4)φl − Px gm′(5 φl . 2 2

(3.435)

PROBLEMS 1. Derive DYY , DZZ, DXY , DYZ, and DZX in Section 3.2.3. 2. Derive H 0c 1l 1 and H 0c 1s 1 in Hamiltonian Equation (3.210) of Section 3.3.2. 3. Show that a (001)-oriented Hamiltonian with wurtzite bases is given by Equation (3.283) in Section 3.5.2. 4. Derive Equation (3.134) for a degenerate band without spin–orbit coupling. 5. Derive the expression Equation (3.373) for ε xz by minimizing the strain energy Equation (3.372) with respect to the variable ε xz using Equation (3.371) in Section 3.7.1. 6. Derive three analytical solutions Equation (3.382) for 3 × 3 Hamiltonian Equation (3.381). 7. Derive the block-diagonalized Hamiltonian expression Equation (3.293) using the transformation matrix Equation (3.291). 8. Derive matrix form Equation (3.185) for Equations (3.178) and (3.179) using the matrix elements in Equations (3.181), (3.183), and (3.184).

c03.indd 168

4/20/2011 10:43:58 AM

REFERENCES

169

REFERENCES [1] S. Datta, Quantum Phenomena. Modular Series on Solid State Devices, Vol. 8, New York: Addison-Wesley, 1989. [2] J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phy. Rev., 97, p. 869, 1955. [3] S. L. Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995. [4] J. F. Nye, Physical Properties of Crystals. Oxford: Clarendon, 1989. [5] D. Ann, S. J. Yoon, S. L. Chuang, and C.-S. Chang, “Theory of optical gain in strained-layer quantum wells within the 6 × 6 Luttinger-Kohn model,” J. Appl. Phys., 78, p. 15, 1995. [6] G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors. New York: Wiley, 1974. [7] S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B, 54, p. 2491, 1996. [8] S. H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” J. Appl. Phys., 87, p. 353, 2000. [9] S. H. Park, “Optical anisotropy in (110)-oriented zinc-blende GaN/AlGaN quantum wells,” Semicond. Sci. Technol., 23, p. 1, 2008. [10] Y. Kajikawa, “Optical anisotropy of (111)-oriented strained quantumwells calculated with the effect of the spin-orbit split-off band,” J. Appl. Phys., 86, p. 5663, 1999. [11] D. L. Smith and C. Mailhiot, “Theory of semiconductor superlattice electronic structure,” Rev. Mod. Phys., 62, p. 173, 1990. [12] S. H. Park arid S. L. Chuang, “Crystal-orientation effects on the piezoelectric field and electronic properties of strained wurtzite semiconductors,” Phys. Rev. B, 59, p. 4725, 1999. [13] S. H. Park, D. Ahn, and S. L. Chuang, “Electronic and optical properties of a- and m-plane wurtzite InGaN/GaN quantum wells,” IEEE J. Quantum Electron., 43, p. 1175, 2007.

c03.indd 169

4/20/2011 10:43:58 AM

This page intentionally left blank

PART II Modern Applications

c04.indd 171

4/20/2011 10:46:42 AM

This page intentionally left blank

4 4.1

Quantum Information Science QUANTUM BITS AND TENSOR PRODUCTS

Quantum information science is an interdisciplinary subject that includes physics, mathematics, chemistry, electrical engineering, and computer science, and has the potential to revolutionize many areas of science and technology. It exploits fundamentally new modes of computation and information processing, because it is based on the laws of quantum mechanics instead of those of classical physics. It holds the promise of immense computing power beyond the capabilities of any classical computer, and it is directly linked to emerging quantum technologies, such as quantum teleportation. The key element of quantum information science is a quantum system with two states that can represent “0” and “1,” the basic building blocks of a digital logic circuit called “qubit.” A qubit is a unit vector in a two-dimensional Hilbert space, whose basis vectors (computational basis) are defined as [1] ⎛0⎞ ⎛ 1⎞ 0 = ⎜ ⎟ and 1 = ⎜ ⎟ . ⎝0⎠ ⎝ 1⎠

(4.1)

In order to implement any useful quantum information processing unit, we need to deal with many qubits at a time, and the appropriate model is a tensor product of qubits. Specifically, if we have n qubits, each with a given computational basis in a two dimensional Hilbert space H i, then the tensor product is a 2 n-dimensional Hilbert space with a computational basis consisting of 2 n vectors i1i2  i n = i1 ⊗ i2 ⊗  ⊗ in ,

(4.2)

where the indices ik take on the values 0 and 1. In the following, we give a more detailed description of a tensor product. Let A be an m × n matrix, and B be a p × q matrix. Engineering Quantum Mechanics, First Edition. Doyeol Ahn, Seoung-Hwan Park. © 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 173

c04.indd 173

4/20/2011 10:46:42 AM

174

QUANTUM INFORMATION SCIENCE

Then ⎛ a11B a12 B  a1n B ⎞ ⎜ a B a B  a B⎟ 21 22 2n ⎟ A⊗ B = ⎜ ⎜ ⎟  ⎜ ⎟ ⎝ am1B am 2 B  amn B ⎠

(4.3)

is an mp × nq matrix called the tensor product of A and B. The tensor ⎛a⎞ ⎛c⎞ product of vectors ψ = ⎜ ⎟ and ϕ = ⎜ ⎟ is then given by ⎝ b⎠ ⎝d⎠ ⎛ ac ⎞ ⎜ ⎟ ⎛ a ϕ ⎞ ⎜ ad ⎟ ψ ⊗ ϕ =⎜ . ⎟= ⎝ b ϕ ⎠ ⎜ bc ⎟ ⎜ ⎟ ⎝ bd ⎠

(4.4)

The tensor product ψ ⊗ ϕ is often denoted as ψ ϕ or ψϕ when there is no cause for confusion. If C is an n × r matrix, and D is a q × s matrix, then one can show that

( A ⊗ B ) (C ⊗ D) = ( AC ) ⊗ ( BD) .

(4.5)

Moreover, we obtain

( A ⊗ B )† = A† ⊗ B†

(4.6)

( A ⊗ B )−1 = A−1 ⊗ B−1 .

(4.7)

and

When Ak and ψ k are the linear operator and the vector in the Hilbert space H k, respectively, then Equation (4.5) implies that

( A1 ⊗  ⊗ An ) ( ψ 1 ⊗  ⊗ ψ n ) = ( A1 ψ 1 ) ⊗  ⊗ ( An ψ n ) . (4.8) Let A = A1 ⊗  ⊗ An and ψ = ψ 1 ⊗  ⊗ ψ n , then ψ A ψ = ψ 1 A1 ψ 1  ψ n An ψ n .

c04.indd 174

(4.9)

4/20/2011 10:46:42 AM

QUANTUM ENTANGLEMENT

4.2

175

QUANTUM ENTANGLEMENT

Suppose we have n qubit states. An entangled state is defined as the superposition of tensor product states, which is not necessarily decomposable into a tensor product form. A general entangled state is represented as [1,2] ψ =

∑a

i1i2in

i1 ⊗ i2 ⊗  ⊗ in .

(4.10)

ik =0 ,1

Entangled states play very important roles in such aspects of quantum information science as quantum computing and quantum teleportation. Consider a system of two qubits. The basis for the two-qubit system is { 00 , 01 , 10 , 11 }. More generally, the basis for a system of n qubits could be written as { cn−1cn−2  c0 }, where cn−1 , cn−2 ,, c0 ∈ {0, 1}. We can also represent this basis in terms of the decimal system. In decimal system, we denote x = cn−1cn−2  c0 ,

(4.11)

x = cn−1 2 n−1 + cn−2 2 n−2 +  + c0

(4.12)

where

Using this decimal notation, the basis for the two-qubit system can be written as { 0 , 1 , 2 , 3 }. The basic principles of quantum computation can be described easily using the entangled state with decimal notation. Assuming that we have the system of n qubits from which we form following 2n qubit state: Ψx = x ⊗ 0 .

(4.13)

Let U f be a unitary operator, which leaves the first n qubit, but changes the second n qubit as follows: U f ( x ⊗ 0 ) = x ⊗ f ( x) .

(4.14)

Now, we have the new state formed by the superposition of the first n qubits given by Φ =

∑

ax x ⊗ 0 .

(4.15)

x =0 ,1,2 n −1

c04.indd 175

4/20/2011 10:46:43 AM

176

QUANTUM INFORMATION SCIENCE

If we apply the unitary operator U f on Φ , we obtain

∑a U ( x ⊗ 0 ) = ∑ a x ⊗ f ( x) .

Uf Φ =

x

f

x

(4.16)

x

x

The above result suggests that 2 n different computations can be done at once as the result of unitary operation. Imagine that each computation takes 1 second. When n is equal to 32, one needs to do four billion computations, which takes approximately 32 years using the conventional computer. On the other hand, if one has a quantum processor, the same computation can be done in one second. That’s a billionfold increase in computation speed! In quantum information science, the unitary operators are often called quantum gates. The most elementary gates are single-qubit gates and two-qubit gates. General single qubit gate can be built from the following Pauli operators X, Y, Z, and the identity operator I: ⎛1 I =⎜ ⎝0 ⎛0 X = σx = ⎜ ⎝1 ⎛0 Y = σy = ⎜ ⎝i ⎛1 Z = σz = ⎜ ⎝0

0⎞ , 1 ⎟⎠ 1⎞ , 0 ⎟⎠ −i ⎞ , 0 ⎟⎠ 0⎞ . −1 ⎟⎠

(4.17)

The operation I is the identity transformation, while X is the negation (NOT), Z the phase shift, and Y = iXZ the combination of both X and Z. The CNOT (controlled-NOT) gate is the most important two-qubit gate in quantum computation. The gate flips the second qubit (the target qubit) when the first qubit (the control qubit) is 1 , while leaving the second qubit unchanged when the first qubit is 0 . Let { 00 , 01 , 10 , 11 } be the basis for the two-qubit system. The action of CNOT gate, whose matrix expression will be written as U CNOT, is U CNOT : 00  00 , 01  01 , 10  11 , 11  10 . (4.18) The matrix expression of U CNOT is given by

c04.indd 176

4/20/2011 10:46:43 AM

QUANTUM ENTANGLEMENT

U CNOT

⎛1 ⎜0 =⎜ ⎜0 ⎜ ⎝0 = 0

0 0 1 0 0 0 0 1 0 ⊗I +

0⎞ 0⎟ ⎟ 1⎟ ⎟ 0⎠ 1 1 ⊗ X.

177

(4.19)

Here, the outer product u v between two vectors ⎛ u1 ⎞ ⎜u ⎟ 2 u =⎜ ⎟ ⎜  ⎟ ⎜ ⎟ ⎝ un ⎠

and

⎛ v1 ⎞ ⎜v ⎟ 2 v =⎜ ⎟ ⎜  ⎟ ⎜ ⎟ ⎝ vm ⎠

is defined as ⎛ u1v1 u1v2  u1vm ⎞ ⎜u v u v  u v ⎟ 2 1 2 2 2 m ⎟. u v =⎜ ⎜ ⎟  ⎜ ⎟ ⎝ unv1 unv2  unvm ⎠

(4.20)

Another important two-qubit gate is the SWAP gate, which swaps the first and second qubits: U SWAP ψ ⊗ φ = φ ⊗ ψ ,

(4.21)

U SWAP = Φ + Φ + + Φ − Φ − + Ψ + Ψ + + e iπ Ψ − Ψ − ,

(4.22)

or

in terms of the Bell states Φ± =

1 ( 00 ± 11 ) and 2

Ψ± =

1 ( 01 ± 10 ) . 2

(4.23)

Recently, it was shown that an arbitrary two-qubit operation can be α implemented using only three ( SWAP ) gates and six single-qubit α operations. The ( SWAP ) gate is defined as follows

c04.indd 177

4/20/2011 10:46:43 AM

178

QUANTUM INFORMATION SCIENCE

U (SWAP )α = Φ + Φ + + Φ − Φ − + Ψ + Ψ + + e iαπ Ψ − Ψ − 0 ⎛1 ⎜ iαπ ⎜0 1+ e ⎜ 2 =⎜ 1 + e iα π ⎜0 2 ⎜ ⎜0 0 ⎝

0⎞ ⎟ 0⎟ ⎟ ⎟, 0⎟ ⎟ 1 ⎟⎠

0 iαπ

1+ e 2 1 + e iαπ 2 0

(4.24)

where the exponent α is a control parameter. The general two-qubit operation can be implemented when the Hamiltonian for the twoqubit system is of the form given below: H = hx σ x ⊗ σ x + hyσ y ⊗ σ y + hzσz ⊗ σz ,

(4.25)

where π / 4 ≥ hx ≥ hy ≥ hz ≥ 0. In this case, H can be written as H = λ 00 Φ + Φ + + λ 01 Ψ + Ψ + + λ 10 Φ − Φ − + λ 11 Ψ − Ψ − , λ 00 = hx − hy + hz , λ 01 = hx + hy − hz ,

(4.26)

λ 10 = −hx + hy + hz , λ 11 = −hx − hy − hz . 4.3

QUANTUM TELEPORTATION

Quantum teleportation is perhaps the most unique feature of quantum information science, in which an unknown quantum state of a qubit is transmitted to the recipient who can reproduce exactly the same state as the original qubit state. It is important to note that the qubit itself is not transmitted, but the information required to reproduce the quantum state is sent to the recipient. The original quantum state is destroyed in the process. Assume that Alice and Bob share an entangled state, for example, Φ+

AB

=

1 ( 00 2

AB

+ 11

AB

)

(4.27)

and Alice wants to teleport an unknown quantum state belonging to Charlie φ

c04.indd 178

C

=a 0

C

+b 1 C

(4.28)

4/20/2011 10:46:43 AM

QUANTUM TELEPORTATION

179

to Bob. Then the total state is given by Ψ = φ

C

⊗ Φ+

AB

1 ⎛ ⎞ ⎛ ⎞ =⎜ a 0 C +b 1 C ⎟⊗ ⎜ 00 AB + 11 AB ⎟ (4.29) 2⎝ ⎠ ⎝ ⎠ 1 = ( a 000 CAB + a 011 CAB + b 100 CAB + b 111 CAB ) . 2 Now Alice has the first two qubits while Bob has the third qubit. First, Alice does the Bell state measurement of her two qubits. In quantum mechanics, by taking a measurement of a system, we project the state vector to one of the basis vectors that the measurement defines. In order to do this, we define the measurement operator Mm such that the probability p(m) of obtaining the outcome m in the state ψ is [1] p(m) = ψ Mm Mm ψ ,

(4.30)

and the state immediately after the measurement is m =

Mm ψ . p(m)

(4.31)

The measurement operators satisfy the completeness relation,

∑M

† m

Mm = I .

(4.32)

m

Before Alice performs her measurement, it is convenient to rewrite the Equation (4.29) into [1,2] 1 ( a 000 CAB + a 011 CAB + b 100 CAB + b 111 CAB ) 2 1 ⎛ a a = Φ + CA + Φ − CA } ⊗ 0 B + { { Ψ + CA + Ψ − CA } ⊗ 1 B ⎜ ⎝ 2 2 2 b b + { Ψ + CA − Ψ − CA } ⊗ 0 B + { Φ + CA − Φ − CA } ⊗ 1 B ⎞⎟⎠ 2 2 1 = ( Φ + CA ⊗ {a 0 B + b 1 B } + Φ − CA ⊗ {a 0 B − b 1 B } 2 (4.33) + Ψ + CA ⊗ {a 1 B + b 0 B } + Ψ − CA ⊗ {a 1 B − b 0 B }) .

Ψ =

c04.indd 179

4/20/2011 10:46:43 AM

180

QUANTUM INFORMATION SCIENCE

If Alice measures the two qubits in her hand, she will obtain one of the states Φ + , Φ − , Ψ + , or Ψ − with equal probability ¼. Bob’s qubit immediately collapses to a 0 + b 1 , a 0 − b 1 , a 1 + b 0 , or a 1 − b 0 , respectively, depending on the result of Alice’s measurement. Alice then sends her results to Bob using two classical bits through a classical communication channel, such as a telephone. Once Bob learns the result of Alice’s measurement, he knows which qubit state is in his hand. Bob reconstructs the initial state φ by applying unitary operators I, Z, X, or Y, respectively, depending on Alice’s result. As an example, suppose Alice lets Bob know that her measurement result was Φ − . Then Bob applies an operator Z on his state to reconstruct φ as follows: Z (a 0 − b 1 ) = a 0 + b 1 = φ .

(4.34)

4.4 EVOLUTION OF THE QUANTUM STATE: QUANTUM INFORMATION PROCESSING In previous sections, we have seen that quantum entanglement plays a very important role in aspects of quantum information science such as quantum computing and quantum teleportation. Considerable progress has been made in developing a general quantum theory of information processing, analogous to classical information theory founded by Shannon 60 years ago. In this section, we would like to understand how the quantum state evolves through the transmission process [1,2]. Suppose R and Q are two quantum systems, and Q is described by a Hilbert space HQ of finite dimension d. Initially, the joint system RQ is prepared in an entangled state Ψ RQ , whose vector is uniquely specified. Such a state is called a pure state. On the other hand, it may happen that a quantum system under consideration is in the state Ψ i with a probability pi so that we cannot say definitely which state the system is in. Such a system is said to be in a mixed state. The system R is dynamically isolated and zero internal Hamiltonian, while the system Q undergoes evolution that possibly involves interaction with the environment E. The evolution of Q may represent a coding, transmission, and decoding process via some quantum mechanical channel or gate for the quantum information in Q. The final state of RQ is possibly mixed, and is described by the density operator ρRQ′. In order to characterize the process, we first define the fidelity of the quantum mechanical process by Fe = Ψ RQ ρRQ′ Ψ RQ ,

c04.indd 180

(4.35)

4/20/2011 10:46:43 AM

EVOLUTION OF THE QUANTUM STATE

181

which is the probability that the final state ρRQ′ would pass a test checking whether it agreed with the initial state Ψ RQ . Fe measures how successfully the quantum process preserves the entanglement of Q with the reference system R. Imagine that the system Q is prepared in an initial state ρQ, and then subjected to some dynamical process, after which the state becomes ρQ′. The dynamical process is described by a map εQ, so that the evolution is ρQ  ρQ′ = εQ (ρQ ).

(4.36)

In the most general case, the map εQ must be a trace-preserving, completely positive linear map. In other words, we have the following [1,2]: (1) εQ must be linear in the density operators. That is, if ρQ = p1ρQ1 + p2ρQ2 , then εQ (ρQ ) = εQ ( p1ρQ1 + p2ρQ2 ) = p1εQ (ρQ1 ) + p2 εQ (ρQ2 ).

(4.37)

A probabilistic mixture of inputs to εQ leads to a probabilistic mixture of outputs. Here, εQ is a superoperator that acts on the space of linear operators, such as density operators on HQ . (2) εQ must be trace preserving, so that Tr(ρQ ) = Tr(ρQ′ ) = 1.

(4.38)

(3) εQ must be positive. We use the term “positive” to refer to operators that are Hermitian and have no negative eigenvalues. This means that if ρQ is positive, then ρQ′ = εQ (ρQ ) must be positive. These properties mean that the superoperator εQ takes normalized density operators to normalized density operators in a reasonable way. (4) εQ must be completely positive. This means that if we extend the evolution superoperator εQ to an evolution superoperator for the compound system RQ, yielding I R ⊗ εQ, where I R is the identity superoperator on R states. This means adjoining a system R that has trivial dynamics (no state of R is changed), and which does not interact with Q. If, for all such trivial extensions, the resulting superoperator I R ⊗ εQ is positive, the superoperator εQ is said to be “completely positive.”

c04.indd 181

4/20/2011 10:46:44 AM

182

QUANTUM INFORMATION SCIENCE

A completely positive map is not only a reasonable map from density operators to density operators for Q, but also, it is extensible in a trivial way to a reasonable map from density operators to density operators on any larger system RQ. Since we cannot exclude a priori that our system Q is in fact initially entangled with some distant isolated system R, any acceptable εQ must satisfy this condition. Completely positive, trace-preserving linear maps include all unitary evolutions of the state ρQ′ = U QρQU Q † .

(4.39)

They also include unitary evolutions involving interactions with an external system. Suppose we consider an environment system that is initially in the pure state 0 E . Then we could have εQ ( ρQ ) = TrE (U QE ( ρQ ⊗ 0 E 0 E )U QE † ) ,

(4.40)

where U QE is the unitary evolution on the joint system QE. This map is also trace preserving and completely positive. If we can write a superoperator εQ as a unitary evolution on an extended system QE, followed by a partial trace over E, we say that we have a “unitary representation” of the superoperator. Such a representation is not unique, since many different unitary operators U QE will lead to the same εQ. There is another useful representation for completely positive maps that employs only operators on HQ. Let AμQ be a collection of such operators indexed by μ . Then the map εQ given by εQ (ρQ ) =

∑A ρ

Q Q μ

AμQ

(4.41)

μ

is a completely positive map. If, in addition, AμQ operators satisfy

∑A

Q† μ

AμQ = I Q ,

(4.42)

μ

then the map is also trace preserving. Such a representation for εQ in terms of operators AμQ is called an “operator representation” for εQ. A single εQ will admit many different operator–sum representations. In fact, the “unitary representation” and the “operator–sum representation” are related to each other. Suppose that ρQ = φQ φQ , and let { μ E } be a complete orthonormal set of states of E.

c04.indd 182

4/20/2011 10:46:44 AM

A MEASURE OF INFORMATION

183

Then εQ ( ρQ ) = TrE {U QE ( φQ φQ ⊗ 0 E 0 E )U QE † } =

∑μ

E

U QE ( φQ φQ ⊗ 0 E 0 E )U QE † μ E

E

U QE ( φQ ⊗ 0 E

μ

=

∑μ μ

=

∑A

Q μ

)(

φQ ⊗ 0 E )U QE † μ E

(4.43)

φQ φQ AμQ † ,

μ

if we define the operator AμQ by AμQ φQ = μ E U QE ( φQ ⊗ 0 E ) .

(4.44)

It can be shown that every trace-preserving, completely positive linear map εQ has a unitary representation and an operator–sum representation. These are the class of allowed evolutions of a quantum system. 4.5 A MEASURE OF INFORMATION In classical communication systems involving transmission and processing of data, the information or data and the channel are appropriately represented by probabilistic models. Suppose the nonnegative numbers p1 , p2 , sum to unity and thus form a probability distribution. The  Shannon entropy H ( p) of this probability distribution is [1,2]  H ( p) = −



∑ p log p , p = ( p , p ,). k

k

1

2

(4.45)

k

 We specify the base of our logarithms to be 2. If p forms the probability for some random variable X, so that p( xk ) = pk for various values xk of a random variable X, then we often write this entropy as H ( X ). The Shannon entropy H(X) is the fundamental measure in classical information theory, and it represents the average number of binary digits (bits) required to represent the value of X. It can be thought of as a measure of the uncertainty in the value of X expressed by the probability distribution. We can use it to define various information theoretic quantities, such as the conditional entropy H(X | Y ) =

∑ p( y )H (X | y ) = −∑ p(x , y k

k

c04.indd 183

k

j

k)

log p( x j | yk )

(4.46)

j ,k

4/20/2011 10:46:44 AM

184

QUANTUM INFORMATION SCIENCE

for a joint distribution p( x j , yk ) and the conditional probability p( x j | yk ) over values of two variables X and Y. H ( X | Y ) is interpreted as the average information (over x j and yk required to specify x j after yk is known. A particularly important quantity is the mutual information I ( X : Y ) between two random variables X and Y; I ( X : Y ) = H ( X ) − H ( X | Y ),

(4.47)

which is the average amount that the uncertainty about X decreases when the value of Y is known. If X represents the input of a communication channel, and Y represents the output, then I ( X : Y ) represents the amount of information conveyed by the channel. It turns out that I ( X : Y ) = I (Y : X ).

(4.48)

The quantum mechanical definition of entropy was first given by von Neumann. Suppose ρQ is the density operator representing a mixed state of Q. Then the entropy is S(ρQ ) = −Tr ( ρQ log ρQ ) .

(4.49)

If λ 1 , λ 2 , are the eigenvalues of ρQ, then   S(ρQ ) = H (λ), λ = (λ 1 , λ 2 ,).

(4.50)

The von Neumann entropy is the average number of two-level quantum systems (or qubits) needed to faithfully represent one of the pure states of an ensemble described by ρQ. 4.6

QUANTUM BLACK HOLES

Quantum entanglement is the most valuable resource in quantum information science. It is found that quantum entanglement is universally found not only in microscopic quantum systems, but also in cosmological objects such as black holes. A black hole is a singularity formed when a celestial body collapses beyond a certain radius called the event horizon. More than 30 years ago, Hawking discovered that a black hole can be regarded a quantum mechanical object that emits quantum particles called Hawking radiation. This remarkable phenomenon has been called the Hawking effect ever since the discovery. Before the Hawking effect was discovered, it was generally believed

c04.indd 184

4/20/2011 10:46:44 AM

QUANTUM BLACK HOLES

185

that a black hole absorbs everything surround it and nothing comes out. Hawking effects implies that a black hole also emits radiation and eventually evaporates. The question of whether the information consumed by a black hole is destroyed and lost forever, or might be recovered from the Hawking radiation that is emitted as the black hole evaporates, is one of the fundamental problems in modern physics. It requires a clear understanding of phenomena ranging from gravity to information theory. Hawking’s semiclassical argument predicts that the process of black hole formation and evaporation is not unitary [3,4]. The vacuum state under the strong gravity of the black hole is composed of infalling and outgoing particle states inside and outside the event horizon. To an observer outside the black hole, the unitarity is lost because he would not be able to make any measurement in the interior of the event horizon, and as a result he would be forced to make an average over the states in H in that corresponds to the Hilbert space inside of the horizon to obtain the density operator in H out, the Hilbert space outside the black hole. On the other hand, there is some evidence in string theory that the formation and evaporation of black hole is a unitary process [4]. Nonetheless, the Hawking effect, discovered nearly 30 years ago, is generally accepted as very credible, and considered as would be an essential ingredient of the yet unknown correct theory of quantum gravity. 4.6.1

Microcanonical Model of a Black Hole Evaporation

Horowitz and Maldacena (HM) proposed a final-state boundary condition (FBC) in microcanonical form to reconcile the unitarity of the black hole evaporation with Hawking’s semiclassical reasoning [5]. The essence of the HM proposal is to impose a unique final boundary condition at the black hole singularity, such that no information is absorbed by the singularity. The final boundary state is a maximally entangled state of collapsing matter and infalling Hawking radiation. When a black hole evaporates, particles are created in entangled pairs, with one falling into the black hole, and the other radiated to infinity. The projection of the final boundary state at the black hole singularity collapses the state into one associated with the collapsing matter, and transfers the information to the outgoing Hawking radiation. The HM model is further refined by including the unitary interactions between the collapsing matter and infalling Hawking radiation, and a random purification of the final boundary state. The black hole evaporation process as seen by the observer outside the black hole is a unitary process, which looks like a quantum teleportation process

c04.indd 185

4/20/2011 10:46:44 AM

186

QUANTUM INFORMATION SCIENCE

without the classical information transmitted. This indicates that nonlocal physics would be required to transmit the information outside the black hole, and inside and outside the Hilbert spaces do not have independent existence. Then one could raise the following question: “In quantum theory, which is more fundamental, unitarity or locality?” What locality requires is that there be no influence on an object due to any action taken in a region that is at spacelike separation with respect to the object. If one forces the quantum gravity be unitary to observers outside the event horizon, then nonlocality should be an essential feature of the theory. In the HM model, the boundary state outside the event horizon is assumed to be the Unruh vacuum state [4]. As a matter of fact, Hawking’s original discovery can be regarded as imposing a boundary condition at the event horizon. We would like to denote it as Hawking boundary condition (HBC), in contrast with the FBC proposed by HM (Fig. 4.1). HBC dictates that the quantum states inside and outside r=0 (Origin) M=0 Complete evaporation

+

M = M0 (Initial mass)

r=0 (Singularity) Hin HM FBC

r=0 (Origin)

HBC Hout

–

Figure 4.1. Penrose diagram of for the black hole formation and evaporation processes. A semiclassical theory that predicts the unitary evolution of black hole formation and evaporation has two boundary conditions: (1) Hawking boundary conditions (HBC) at the event horizon with quantum states in H in and H out maximally entangled, and (2) the final-state boundary condition (FBC) inside the black hole for the quantum states of collapsing matter in H M and the infalling Hawking radiation in H in maximally entangled. J + and J − are future and past null infinity respectively. (This figure is reproduced from D. Ahn, Phys. Rev. D, 74, 084010 [2006].)

c04.indd 186

4/20/2011 10:46:45 AM

QUANTUM BLACK HOLES

187

the event horizon of the black hole are maximally entangled. The significance of the HM proposal is that the black hole formation and evaporation process can be put into a unified picture by combining HBC together with FBC. Moreover, the process can be unitary as predicted by string theory. It is shown [6] that the black hole evaporation process will be affected by the boundary condition outside the event horizon. Boundary states outside the event horizon affect the final state projection because the quantum states inside and outside the event horizon are entangled by HBC. One of the critical assumptions in the HM proposal is that the internal quantum state of the black hole can be represented by maximally entangled states of collapsing matter and infalling Hawking radiation. The HM model is also based on the simplified microcanonical form, and it would be an interesting question how the FBC will look in a more general case, such as a Schwarzschild black hole. Here, it is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The outgoing Hawking radiation is obtained by the final state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer outside the black hole is a unitary process, but nonlocality is required to transmit the information outside the black hole. It is also shown that the final state projection by the evaporation process is affected by HBC at the event horizon, which clearly violates the locality principle. The purpose of this section is to demonstrate that the FBC of Horowitz and Maldacena necessarily implies a breakdown of locality in whatever quantum theory of gravity one might construct that incorporates this proposal. We proceed by (1) reviewing the result that the original vacuum outside of a black hole evolves into a maximally entangled two-mode squeezed state on H in and H M , and then (2) showing that the interior state of the black hole is a maximally entangled two-mode squeezed state on H in and H M , where H M is the Hilbert space for the collapsing matter. The FBC is then applied to the latter state, and the outgoing radiation is obtained by projection of this onto the former state. However, if one took an excited state outside of the black hole as the HBC, then it too could be written as a maximally entangled two-mode squeezed state on H in and H out —but the state of outgoing radiation that one would obtain via the FBC is orthogonal to what one obtains from the vacuum state (Eq. (4.30)). Hence, one obtains

c04.indd 187

4/20/2011 10:46:45 AM

188

QUANTUM INFORMATION SCIENCE

the result that the final outgoing particle state for black hole evaporation is dependent on the HBC. Since the interior and the exterior regions of the event horizon are causally disconnected, this is contrary to expectations based on locality. We interpret this as indicating that nonlocality is required to transmit information outside a black hole. We first review the original HM proposal briefly. We assume that the quantum state of the collapsing matter belongs to a Hilbert space H M with dimension N and ψ M be the initial quantum state of the collapsing matter. The Hilbert space of fluctuations on the background spacetime for black hole formation and evaporation is separated into H in and H out, which contain quantum states localized inside and outside the event horizon, respectively. In the HM proposal, HBC is assumed to be the Unruh vacuum state Φ 0 in⊗out belonging to H in ⊗ H out in microcanonical form: Φ0

in⊗out

1 N

=

∑i

in

⊗i

out

,

(4.51)

i

where { i in } and { i out } are orthonormal bases for H in and H out, respectively. The FBC imposed at the singularity requires a maximally entangled quantum state in ψ M, which is called the final boundary state, and is given by M ⊗in

Ψ =

1 N

∑

M

l ⊗in l (S ⊗ I ),

(4.52)

l

where S is a random unitary transformation. The initial matter state ψ M evolves into a state in H M ⊗ H in ⊗ H out under HBC,