Op to Electronics

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Optoelectronics

Optoelectronics is a practical and self-contained graduate-level textbook and reference, whichwill be of great value to both students and practising engineers in the field. Sophisticated

concepts are introduced by the authors in a clear and coherent way, including such topics as

quantum mechanics of electronphoton interaction, quantization of the electromagnetic field,semiconductor properties, quantum theory of heterostructures, and non-linear optics. The book

builds on these concepts to describe the physics, properties, and performances of the main

optoelectronic devices: light emitting diodes, quantum well lasers, photodetectors, optical

parametric oscillators, and waveguides. Emphasis is placed on the unifying theoretical analogies

of optoelectronics, such as equivalence of quantization in heterostructure wells and waveguidemodes, entanglement of blackbody radiation and semiconductor statistics. The book concludes

by presenting the latest devices, including vertical surface emitting lasers, quantum well infrared

photodetectors, quantum cascade lasers, and optical frequency converters.


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Optoelectronics

Emmanuel RosencherResearch Director

French Aerospace Research Agency (ONERA, France)

Professor at the Ecole Polytechnique (Paris, France)

Borge VinterSenior Scientist

THALES Research and Technology

Translated by Dr Paul G. Piva


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The Pitt Building, Trumpington Street, Cambridge, United Kingdom

The Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, AustraliaRuiz de Alarcn 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

First published in printed format

ISBN 0-521-77129-3 hardbackISBN 0-521-77813-1 paperback

ISBN 0-511-03423-7 eBook

English edition Cambridge University Press 2004

Originally published in French as Optolectroniqueby Emmanuel Rosencher andBorge Vinter, Paris 1998 and Masson 1998

2002

(Adobe Reader)


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For Nadia, Anne, Julien, and Clara, for their patience

with all my love.

For Nadia who understands so many other things.


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Contents

Preface xvProperties of common semiconductors xvii

1 Quantum mechanics of the electron 1

1.1 Introduction 1

1.2 The postulates of quantum mechanics 1

1.3 The time-independent Schrodinger equation 6

1.3.1 Stationary states 6

1.3.2 Calculation of stationary states in a one-dimensional potential 7

1.4 The quantum well 81.4.1 The general case 8

1.4.2 The infinite square well 14

1.5 Time-independent perturbation theory 15

1.6 Time-dependent perturbations and transition probabilities 18

1.6.1 The general case 18

1.6.2 Sinusoidal perturbation 20

1.7 The density matrix 23

1.7.1 Pure quantum ensembles 24

1.7.2 Mixed quantum ensembles 24

1.7.3 Density matrix and relaxation time for a two-level system 26

Complement to Chapter 1 29

1.A Problems posed by continuums: the fictitious quantum box

and the density of states 29

1.B Perturbation on a degenerate state 33

1.C The quantum confined Stark effect 37

1.D The harmonic oscillator 41

1.E Transition probabilities and Rabi oscillations 50

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2 Quantum mechanics of the photon 56

2.1 Introduction 56

2.2 Maxwells equations in reciprocal space 56

2.3 Properties of the Fourier transform 58

2.4 Quantization of electromagnetic waves 61

2.5 The photon 63

2.6 The coherent state 67

2.7 Blackbody radiation 71


2.A Radiation field for an oscillating charge: the Lorentz gauge 76

2.B Thermography 84

3 Quantum mechanics of electronphoton interaction 91

3.1 Introduction 913.2 Dipolar interaction Hamiltonian for electrons and photons 91

3.3 Linear optical susceptibility obtained by the density matrix 93

3.4 Linear optical susceptibility: absorption and optical gain 96

3.5 The rate equations 100

3.5.1 Adiabatic approximation and corpuscular interpretation 100

3.5.2 Stimulated emission 101

3.5.3 Absorption saturation 102

3.6 Spontaneous emission and radiative lifetime 104

3.6.1 Spontaneous emission 104

3.6.2 The rate equations including spontaneous emission 109

3.7 Polychromatic transitions and Einsteins equations 110

3.8 Rate equations revisited 111

3.8.1 Monochromatic single-mode waves 112

3.8.2 Multimode monochromatic waves 113

3.8.3 Polychromatic waves 114


3.A Homogeneous and inhomogeneous broadening: coherence of light 115

3.A.1 Homogeneous broadening 116

3.A.2 Inhomogeneous broadening 120

3.B Second-order time-dependent perturbations 123

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3.C Einstein coefficients in two limiting cases: quasi-monochromatic

and broadband optical transitions 131

3.D Equivalence of the A p and D E Hamiltonians and the

ThomasReicheKuhn sum rule 133

4 Laser oscillations 139

4.1 Introduction 139

4.2 Population inversion and optical amplification 139

4.2.1 Population inversion 139

4.2.2 Optical amplification and gain saturation 1414.3 Three- and four-level systems 143

4.4 Optical resonators and laser threshold 146

4.5 Laser characteristics 150

4.5.1 Internal laser characteristics and gain clamping 150

4.5.2 Output power 152

4.5.3 Spectral characteristics 154

4.6 Cavity rate equations and the dynamic behaviour of lasers 156

4.6.1 Damped oscillations 1584.6.2 Laser cavity dumping by loss modulation (Q-switching) 1594.6.3 Mode locking 163


4.A The effect of spontaneous emission and photon condensation 167

4.B Saturation in laser amplifiers 171

4.C Electrodynamic laser equations: electromagnetic foundations for

mode locking 1784.D The SchawlowTownes limit and Langevin-noise force 1854.E A case study: diode pumped lasers 193

5 Semiconductor band structure 199


5.2 Crystal structures, Bloch functions, and the Brillouin zone 1995.3 Energy bands 204

5.4 Effective mass and density of states 206

5.5 Dynamic interpretation of effective mass and the concept of holes 210

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5.6 Carrier statistics in semiconductors 216

5.6.1 Fermi statistics and the Fermi level 216

5.6.2 Intrinsic semiconductors 221

5.6.3 Doped semiconductors 2225.6.4 Quasi-Fermi level in a non-equilibrium system 224


5.A The nearly free electron model 227

5.B Linear combination of atomic orbitals: the tight binding model 230

5.C Kanes k p method 2345.D Deep defects in semiconductors 242

6 Electronic properties of semiconductors 245


6.2 Boltzmanns equation 245

6.3 Scattering mechanisms 251

6.4 Hot electrons 257

6.4.1 Warm electrons 257

6.4.2 Hot electrons: saturation velocity 258

6.4.3 Hot electrons: negative differential velocity 260

6.5 Recombination 261

6.6 Transport equations in a semiconductor 266


6.A The Hall effect 2716.B Optical phonons and the Frohlich interaction 273

6.B.1 Phonons 273

6.B.2 The Frohlich interaction 280

6.C Avalanche breakdown 285

6.D Auger recombination 289

7 Optical properties of semiconductors 296


7.2 Dipolar elements in direct gap semiconductors 296

7.3 Optical susceptibility of a semiconductor 301

7.4 Absorption and spontaneous emission 306

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7.5 Bimolecular recombination coefficient 313

7.6 Conditions for optical amplification in semiconductors 316


7.A The FranzKeldysh-effect electromodulator 3217.B Optical index of semiconductors 328

7.B.1 Mid- and far-infrared regions 329

7.B.2 Near gap regime 330

7.C Free-carrier absorption 333

8 Semiconductor heterostructures and quantum wells 342


8.2 Envelope function formalism 344

8.3 The quantum well 350

8.4 Density of states and statistics in a quantum well 354

8.5 Optical interband transitions in a quantum well 358

8.5.1 Hole states in the valence bands 358

8.5.2 Optical transitions between the valence and conduction bands 359

8.6 Optical intersubband transitions in a quantum well 365

8.7 Optical absorption and angle of incidence 369

8.7.1 Summary for interband and intersubband transition rates 369

8.7.2 Influence of the angle of incidence 370


8.A Quantum wires and boxes 3778.B Excitons 380

8.B.1 Three-dimensional excitons 381

8.B.2 Two-dimensional excitons 385

8.C Quantum confined Stark effect and the SEED electromodulator 388

8.D Valence subbands 392

9 Waveguides 396


9.2 A geometrical approach to waveguides 396

9.3 An oscillatory approach to waveguides 400

9.4 Optical confinement 407

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9.5 Interaction between guided modes: coupled mode theory 410


9.A Optical coupling between guides: electro-optic switches 414

9.B Bragg waveguides 421

9.C Frequency conversion in non-linear waveguides 427

9.C.1 TE mode inTE mode out 4279.C.2 TE mode inTM mode out 432

9.D FabryPerot cavities and Bragg reflectors 4349.D.1 The FabryPerot cavity 4379.D.2 Bragg mirrors 442

10 Elements of device physics 447


10.2 Surface phenomena 448

10.3 The Schottky junction 451

10.4 The pn junction 456


10.A A few variants of the diode 466

10.A.1 pn heterojunction diode 46610.A.2 The pin diode 467

10.B Diode leakage current 470

11 Semiconductor photodetectors 475


11.2 Distribution of carriers in a photoexcited semiconductor 475

11.3 Photoconductors 481

11.3.1 Photoconduction gain 481

11.3.2 Photoconductor detectivity 484

11.3.3 Time response of a photoconductor 486

11.4 Photovoltaic detectors 488

11.4.1 Photodiode detectivity 492

11.4.2 Time response of a photodiode 494

11.5 Internal emission photodetector 497

11.6 Quantum well photodetectors (QWIPs) 500

11.7 Avalanche photodetectors 509

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11.A Detector noise 513

11.A.1 Fluctuations51411.A.2 Physical origin of noise 518

11.A.3 Thermal noise 518

11.A.4 Generationrecombination noise 52111.A.5 Multiplication noise 525

11.B Detectivity limits: performance limits due to background (BLIP) 530

12 Optical frequency conversion 538


12.2 A mechanical description for second harmonic frequency generation 538

12.3 An electromagnetic description of quadratic non-linear

optical interaction 543

12.4 Optical second harmonic generation 546

12.5 ManleyRowe relations 55012.6 Parametric amplification 551

12.7 Optical parametric oscillators (OPOs) 55412.7.1 Simply resonant optical parametric oscillators (SROPOs) 554

12.7.2 Doubly resonant optical parametric oscillator (DROPO) 557

12.8 Sum frequency, difference frequency, and parametric oscillation 560


12.A A quantum model for quadratic non-linear susceptibility 565

12.B Methods for achieving phase matching in semiconductors 57212.B.1 Birefringent phase matching 573

12.B.2 Quasi-phase matching 579

12.C Pump depletion in parametric interactions 582

12.D Spectral and temporal characteristics of optical parametric

oscillators 587

12.E Parametric interactions in laser cavities 596

12.F Continuous wave optical parametric oscillator characteristics 602

12.F.1 Singly resonant OPO 60312.F.2 Doubly resonant OPO: the balanced DROPO 608

12.F.3 Doubly resonant OPO: the general case 610

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13 Light emitting diodes and laser diodes 613


13.2 Electrical injection and non-equilibrium carrier densities 613

13.3 Electroluminescent diodes 617

13.3.1 Electroluminescence 617

13.3.2 Internal and external efficiencies for LEDs 619

13.3.3 A few device issues 623

13.4 Optical amplification in heterojunction diodes 624

13.5 Double heterojunction laser diodes 629

13.5.1 Laser threshold 62913.5.2 Output power 634

13.6 Quantum well laser diodes 637

13.6.1 Optical amplification in a quantum well structure: general case 637

13.6.2 Transparency threshold 641

13.6.3 Laser threshold for a quantum well laser 647

13.6.4 Scaling rule for multi-quantum well lasers 649

13.7 Dynamic aspects of laser diodes 652

13.8 Characteristics of laser diode emission 65513.8.1 Spectral distribution 655

13.8.2 Spatial distribution 656


13.A Distributed feedback (DFB) lasers 660

13.B Strained quantum well lasers 665

13.C Vertical cavity surface emitting lasers (VCSELs) 671

13.C.1 Conditions for achieving threshold in a VCSEL 67113.C.2 VCSEL performance 675

13.D Thermal aspects of laser diodes and high power devices 676

13.E Spontaneous emission in semiconductor lasers 683

13.F Gain saturation and the K factor 69013.G Laser diode noise and linewidth 696

13.G.1 Linewidth broadening 700

13.G.2 Relative intensity noise (RIN) and optical link budget 701

13.H Unipolar quantum cascade lasers 70413.I Mode competition: cross gain modulators 708

Index 713

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Preface

The field of optoelectronics is currently in full expansion, drawing to its classrooms

and laboratories numerous science and engineering students eager to master the

discipline. From the lecturers perspective, optoelectronics is a considerable chal-

lenge to teach as it emerges from a complex interplay of separate and oftenseemingly disjointed subjects such as quantum optics, semiconductor band struc-

ture, or the physics of carrier transport in electronic devices. As a result, the

student (or lecturer) is left to navigate through a vast literature, often found to be

confusing and incoherent.

The aim of this text is to teach optoelectronics as a science in itself. To do so, a

tailored presentation of its various sub-disciplines is required, emphasizing within

each of these, those concepts which are key to the study of optoelectronics. Also,

we were determined to offer a partial description of quantum mechanics orientedtowards its application in optoelectronics. We have therefore limited ourselves to a

utilitarian treatment without elaborating on many fundamental concepts such as

electron spin or spherical harmonic solutions to the hydrogen atom. On the other

hand, we have placed emphasis on developing formalisms such as those involved

in the quantization of the electromagnetic field (well suited to a discussion of

spontaneous emission), or the density matrix formalism (of value in treating

problems in non-linear optics).

Similarly, our treatment of semiconductor physics ignores any discussion of theeffect of the crystallographic structure in these materials. Rather, a priori use is

made of the semiconductor band structures which implicitly incorporate these

effects on the electrical and optical properties of these materials. In carrying out

our rather utilitarian-minded presentation of these disciplines, we have claimed as

ours Erwin Schrodingers maxim that it mattered little whether his theory be an

exact description of reality insofar as it proved itself useful.

We have sought in this work to underline wherever possible the coherence of the

concepts touched on in each of these different areas of physics, as it is from this

vantage point that optoelectronics may be seen as a science in its own right. There

exists, for instance, a profound parallel between the behaviour of an electron in a

quantum well and that of an electromagnetic wave in an optical waveguide. As

well, one finds between the photon statistics of black bodies, the mechanics of

quantum transitions within semiconductor band structures and the statistics of

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charge carriers in these materials, an entanglement of concepts comprising the

basis for infrared detection. In the same spirit, this work does not pretend to

present an exhaustive list of all known optoelectronic devices. Such an effort could

only come at the cost of the overall coherence aimed at in this work, and add to thetype of confusion we have claimed as our enemy. The goal is rather to present

those optoelectronic concepts which will allow an overall understanding of prin-

ciples necessary in solving problems of a general or device-specific nature. Thus,

only the analysis of generic classes of optoelectronic components will be under-taken here without entering into the labyrinth offered by more particular applica-

tions.

Lastly, regarding the problem of notation (a problem inherent to any multidis-

ciplinary study), we have chosen simply to follow the lead of standard physicsnotation in any given chapter. Thus, the symbol may be used indiscriminately torepresent the permittivity, the quantum confinement energy, or the saturation

coefficient of a semiconductor laser. We could have attempted the introduction of

various notations for each of these different uses based on the Latin, Greek, and

Hebrew character sets, but we realized that even these would have soon been

exhausted. We have thus chosen merely to redefine in each chapter the correspon-

dence between the symbols and their respective notions.

The authors wish to thank all those having assisted with the preparation of thismanuscript, such as Erwan LeCochec, Andrea Fiore, Arnaud Fily, Jean-Yves

Duboz, Eric Costard, Florence Binet, Eric Herniou, Jean-Dominique Orwen,

Anna Rakovska, and Anne Rosencher among many others. This work could never

have seen the light of day without the support of ONERA and THALES (ex

THOMSON-CSF) and most particularly the encouragement of Mr Pierre Tour-

nois, formerly scientific director of THOMSON-CSF. Finally, the authors are

deeply indebted to Paul Piva, whose translation from French to English reflects

his competence, intelligence, and culture.

xvi Preface


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Prop

ertiesofcomm

onsemicondu

ctors

Si

Ge

GaAs

AlAs

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Bandgap

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Latticeconstant,a

5.43095

5.64613

5.6533

5

.6600

6.0583

5.4

505

5.8688

6.09

6

6.4794

Relativepermittivity,

/

11.9

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13.1

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Effectivemass

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0.9163

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Electrontraverse,mCR/m

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0.537

0.284

0.50

0

.79

0.40

0.6

7

0.60

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Lighthole,mJF/m

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Luttingerparameters

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Intrinsicdensity,n

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1.5;

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10

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10

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Mobility

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8000

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370

1800

400

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0

480

150

180

1500

1500

Furtherreading

Generalreferencesusefulinobtainingvaluesfor

semiconductorproperties:

K.H.Hellwege,ed.,

Landolt-BornsteinNumerica

lDataandFunctionalRelationshipsinScienceandTechnology,Springer,Berlin.

O.Madelung,ed.,Semiconductors,GroupIVElementsandIII-VCompounds,inDatainScienceandTechnolo

gy,Springer,Berlin(1996).

Recentreviewworks:

B.L.Weiss,ed.,EMISDataviewsSeries,INSPE

C,London.


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1 Quantum mechanics of the electron

1.1 Introduction

This chapter reviews the fundamental principles and techniques of quantum

mechanics that are necessary to understand the subject of optoelectronics. Often,

concepts are not presented in depth: the aim, rather, is to provide the tools and

notation required to work through this book. Thus, in spite of their immenseimportance in other areas of physics, and the severe scientific injustice resulting

from their being placed aside, we shall pass almost entirely in silence over Heisen-

bergs uncertainty principle, spherical harmonics, electron spin, etc. The reader

wishing to deepen his/her understanding of these concepts is greatly encouraged to

read or reread the remarkable work by C. Cohen-Tannoudji et al. (1992).

1.2 The postulates of quantum mechanics

We consider an electron of charge q and mass mC

subjected to a generalized

potential of the form V(r, t) varying in three-dimensional space r, and time t.Quantum mechanics tells us that the notion of a classical electron trajectory loses

its meaning when the distance over which this potential varies is of the order of the

de Broglie wavelength ("

). This length is given by:

" : 2

(2mCE: 1.23 (nm)

(V(V)(1.1)

where is Plancks constant (1.04; 10\ J s\), Vis the average potential experi-enced by the particle, and E is the energy of the particle. We will see that in acrystalline solid where electrons are subjected to spatially varying potentials of the

order of 5 eV (1 eV : 1.6; 10\ J), their de Broglie wavelength turns out to be ofthe order of 5. As this length corresponds to the interatomic distance between

atoms in a crystalline lattice, conduction electrons in this medium will be expectedto display interference effects specific to the mechanics of wave-motion. These

effects (studied in Chapter 5) are the origin of the semiconductor band gap, and

cannot readily be discussed in terms of classical theories based upon the notion of

a well-defined trajectory.

Quantum mechanics also teaches us that we must forgo the idea of a trajectory

1


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in favour of a more subtle description in terms of quantum states and wavefunc-tions. The electron is then represented by a state vector evolving in time "(t)2. Oneof the strongest postulates of quantum mechanics is that all these state vectors

span a Hilbert space. For instance, the existence of linear combinations of states(which leads to dramatic effects such as molecular stability, energy bandgaps, . . .)

is a direct consequence of this postulate. This vector space possesses a Hermitianscalar product, whose physical significance will be given later. We will use Diracnotation to represent the scalar product between two vector states "

2 and "

2 as

1"

2. Now, we recall the properties of a Hermitian scalar product:

1"2 : 1"2*

1";

2:

1"2;

1"2 (1.2)1

; "2 : *1

"2 ; *1

"2

1"2 real, positive, and zero if and only if"2 : 0

where the asterisk indicates that the complex conjugate is taken. By definition a

physical state possesses a norm of unity, which is to say that "(t)2 is a physicalstate if:

1(t)"(t)2 : 1 (1.3)

A certain number of linear operators act within this Hilbert space. A second

postulate of quantum mechanics is that classically measurable quantities such as

position, energy, etc. are represented by Hermitian operators A (i.e. operatorssuch that AR : A, where is the adjoint or Hermitian conjugate) called observables,and that the result of the measurement of such an observable can only be one of

the eigenvalues associated with the observable. If the ensemble of eigenvalues of

the observable A forms a discrete set, then the set of all possible measurements of asystem are given by the a

Lsolutions of the eigenvalue equation:

A"L2 : a

L"

L2 (1.4)

As the observable operators are Hermitian, it follows that their eigenvalues are

necessarily real (consistent with the familiar fact that the result of a physical

measurement is a real number). We also define the commutator of two operators Aand B as:

[A, B] : AB 9 BA (1.5)

It can be shown that if two operators commute (i.e. if their commutator equals

zero), then they share a complete set of simultaneous eigenvectors. A noteworthy

consequence of this is that physical states exist in which the results of measurement

of both of these observables (A and B) can be obtained simultaneously withcertainty: these are their common eigenstates.



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If the orthonormal eigenvector basis of observable A is complete, then anyphysical state "(t)2 of the electron can be described in terms of a linear combina-tion of eigenvectors:

"(t)2 : L

cL(t)"

L2 (1.6)

where the coefficients cL

are given by:

cL(t) : 1

L"(t)2 (1.7)

The probabilistic interpretation of quantum mechanics states that the square of the

norm of the coefficient "cL(t)" gives the probability of finding the electron in the"

L2 state at time t (implying that measurement of the observable A at that time

will yield the value ofaL

with equal probability "cL(t)"). A further postulate is that,

immediately after a measurement of observable A has been performed, the statefunction resides entirely in one of the eigenstates of the observable A (i.e. c

L(t) : 1

or "2 : "L2). In the event that a particular eigenvalue is degenerate, the state

function after measurement is restricted to the subspace spanned by the degenerate

eigenstates. The latter postulate, which is still the subject of intense investigation, is

necessary for the coherence of quantum mechanics.It is therefore implicit in the probabilistic interpretation that we may not, in

general, know the outcome of a measurement with certainty. We can, however,

extract the average value of an observable A taken over the course of a statisticallysignificant number of independent measurements. This value will then correspond

to an average value of all possible measurement outcomes aL

of an observable

weighted by the individual probabilities "cL(t)" of finding the system in an eigen-

state "L2 associated with this particular eigenvalue a

L:

1A2(t) : L

aL"c

L(t)" (1.8)

This average value is easily found to be:

1A2(t) : 1(t)"A"(t)2 (1.9)

Some of these A observables may be vectorial, such as the position r : (x, y, z) and

momentum p operators. For these operators, the eigenvalues belong to a continu-um of values. Therefore, the eigenvector "r2 of the position operator r is interpretedas describing the state of the system once the measurement of the position has

yielded a particular value r. We then say that the particle may be found at r withcertainty.

The decomposition of a state vector onto any particular basis set of eigenvectors

3 1.2 The postulates of quantum mechanics


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is called a representation. One important representation is the projection of thestate vector onto the eigenstates of the position operator r. Each component of thisprojection is the wavefunction (r, t) given by:

(r, t) : 1r"(t)2 (1.10)

Referring back to the probabilistic interpretation of quantum mechanics, we see

that the norm of the wavefunction "(r, t)" gives the probability of finding anelectron at r at time t. Furthermore, in the r representation, the inner product ofthe two states "

2 and "

2 may be shown to be written as:

1"2 :

*(r)(r) dr (1.11)

where the integral is evaluated over all space. Finally, evolution of the state of the

system with time is given by Schrodingers equation:

i**t

"(t)2 : H(t)"(t)2 (1.12)

Schrodingers equation

where H(t) is the Hamiltonian of the system, which yields as an observable theenergy of the system. Its general expression takes the form:

H(t) :p

2mC

; V(r, t) (1.13)

Hamiltonian for a particle with mass me

subject to a potential V

where p is the momentum operator. In the r representation (i.e. projected onto theposition eigenvectors of "r2), the correspondence principle gives the followingexpression for the p operator:

p :

i :

i

**x

**y

**z

(1.14)

and in the r representation, takes the following form when acting upon awavefunction (r, t):

The symbol is generally used when confusion may arise between a classical physical quantity (such asposition r) and its corresponding quantum observable (r).



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1r"p"(t)2 : p(r, t) :

i

**x

(r, t)

*

*y (r, t)

**z

(r, t)

(1.15)

This correspondence results from the requirement that the p operator acting uponthe de Broglie matter-waves (ekr) yields an associated momentum eigenvalue ofp : k, i.e. satisfying the de Broglie relation. The momentum operator p thereforetakes on the form /i.

The operators r and p are linked by the important commutator relation:

r p(r, t) :

i x*

*x(r, t) ; y

**y

(r, t) ; z**z

(r,t) (1.16)and

p r(r, t) :

i *

*x[x(r, t)] ;

**y

[y(r, t)] ;**z

[z(r, t)] (1.17)

from which we deduce the commutation relation:

[xG, p

H] : i

GH(1.18)

Anticommutation of position and momentum observables

leading to the first of the Heisenberg uncertainty relations

A corollary of the properties stated earlier for commuting observables, is that

non-commuting observables cannot share a common basis set of eigenvectors.

Therefore, neither of these position or momentum observables may be known

simultaneously with arbitrary precision. This is the first of Heisenbergs uncertain-

ty principles, which can be shown to lead to the following relationships between

the momentum and position uncertainties:

xpVP /2

ypWP /2 (1.19)

zpXP /2

Returning to Schrodingers equation in the position representation, we may now

write:

i**t

(r, t) : 9

2mC

(r, t) ; V(r, t)(r, t) (1.20)

where is the Laplacian operator ((*/*x) ; (*/*y) ; (*/*z)). Once given the

5 1.2 The postulates of quantum mechanics


26/743

space and time evolution of the potential, this last equation allows one, in prin-

ciple, to calculate the evolution of electron probability in the structure. We note

that this equation preserves the norm of a function, which is consistent with the

fact that every physical state evolves in time and space to some other physicalstate.

1.3 The time-independent Schrodinger equation

1.3.1 Stationary states

We will interest ourselves, in this section, with the description of the physical state

of an electron subjected to a time-independent potential (i.e. a conservative system).This system could be a hydrogen atom, in which case the potential V(r) is aCoulomb field localized in space, or a crystal, where the potential V(r) is periodic(corresponding to the regular spacing of the constituent atoms). Schrodingers

equation may then be written as:

i**t

"(t)2 : H"(t)2 :p

2mC

; V(r)"(t)2 (1.21)Let us first begin by considering the eigenstates of the Hamiltonian:

H"L(t)2 : E

L"

L(t)2 (1.22)

Time-independent Schrodinger equation

For the time being we will suppose that these states are:

discrete, i.e. they can be denoted by integers;

non-degenerate, i.e. no two or more distinct quantum states may have the same

energy;

complete, i.e. any physical state may be projected in a unique fashion onto the

basis set formed by the eigenfunctions ofH of type (1.6).Substituting Eq. (1.22) into (1.21), we find the time evolution of an eigenstate "

L2

to be:

"L(t)2 : "

L(0)2e\SLR (1.23)

where

EL : L (1.24)

and L

is the Bohr oscillation frequency associated with the state "L2. Equation

(1.23) is noteworthy as it allows an important prediction to be made. Let us

suppose the system is in an eigenstate "L2 and that we seek the average value of

some observable A:



27/743

1A2(t) : 1Le\SLR"A"

Le\SLR2 : 1

L"A"

L2 (1.25)

This average value therefore does not vary over time, i.e. the eigenstates are

stationary states for all observables. These stationary states are particularly im-portant as they form states which yield unchanging values for observables. Addi-

tionally, they allow a description of the time evolution of a non-stationary state.

Let us suppose an arbitrary state "(t)2, for which we know its projection at t : 0onto the basis set of stationary states "

L2:

"(0)2 : L

cL"

L2 (1.26)

We then determine the time evolution of the coefficients cL(t). To do this we

substitute the "(t)2 stationary state decomposition into the time-independentSchrodinger equation (1.21), which gives:

L

id

dtc

L(t)"

L2 : H

L

cL(t)"

L2:

L

cL(t)E

L"

L2 (1.27)

Projecting this equation onto each eigenvector "L2 we find that:

cL(t) : c

Le\SLR (1.28)

Therefore, once we know the effect of decomposition of the state function at t : 0on the stationary states of the system, we will know the state function at any

ulterior time t.

"(t)2 : L

cLe\SLR"L2 (1.29)

This decomposition may be generalized for a basis set consisting of degenerate

eigenstates and/or forming a continuum. This generalization comes, however, at

the cost of a more cumbersome notation, and so we shall limit ourselves to its use

only in those situations in which such a treatment cannot be avoided.

1.3.2 Calculation of stationary states in a one-dimensional potential

Let us consider a one-dimensional space mapped by the co-ordinate x and let ussuppose a confinement potential V(x), such that V(x) : 0 over all space, andV(x) ; 0 as x ;


28/743

9

2mC

d

dx

L(x) ; [V(x) 9 E

L]

L(x) : 0 (1.30)

Time-independent Schrodinger equation

in xrepresentation

We recall that the unknowns are the eigenvalues EL, and the stationary state

wavefunctions L(x). For each value of E

L, Eq. (1.30) becomes a second-order

differential equation. We can show that the L(x) solutions of this equation are

continuous, as are their first derivatives dL(x)/dx over all space. These two

conditions, added to the normalization requirement of all physical states, lead to

the quantization of energy, i.e. the existence of discrete energy levels. It is thereforethe wave nature of the wavefunctions and their integrability and continuity require-ments which lead to the quantized nature of the energy levels. We will illustrate thispoint with a precise example which plays a primordial role in the remainder of this

text the quantum well.

1.4 The quantum well

1.4.1 The general case

We now consider an electron subject to a potential well as described in Fig. 1.1, i.e.

defined by:

V(x) : 0, if "x " 9a

2(1.31)

V(x) : 9V

, if"x " :a

2

The first region ("x" 9 a/2) defines the potential barrier, whereas the second region("x" : a/2) defines the well. The Schrodinger equation which governs the electronin this structure is:

2mC

d

dx(x) 9 E(x) : 0, for "x " 9

a

2(1.32)

9

2mC

d

dx(x) 9 (V

; E)(x) : 0, for "x " :a

2

We first seek solutions to this equation having energies less than the potential

barrier, i.e. E : 0. For this, we introduce three quantities, k, , and k

, having as

When the mass of a particle varies as a function of position x, in a semiconductor heterostructure forexample, it is the quantity 1/m(x)d/dx which is conserved.



29/743

500

400

300

200

100

0

Energy(meV)

20 10 0 10 20

Position (nm)

0

V0

a/2a/2

Fig. 1.1. A one-dimensional quantum well. Represented are the eigenenergies and

wavefunctions associated with the three bound states of the system. This particular quantum

well may be implemented in the GaAs/Al

Ga

As system. The difference between the first

two energy levels is 104 meV and leads to photon absorption at 11.9m.

dimension the inverse of a length, i.e. having the dimensions of a wavevector (the

number of spatial periods in 2), defined by:

E : 92m

C

V

; E :k

2mC

(1.33)

V

:k

2mCWe note that 2/k

is the de Broglie wavelength associated with the energy V

of

the confining potential.

Using this notation, the most general solutions to (1.32) are:

A(x) : A

AeIV ; B

Ae\IV, for "x " :

a

2

J(x) : AJeGV ; BJe\GV, for x : 9a

2 (1.34)

P(x) : A

PeGV ; B

Pe\GV, for x 9

a

2

where c, l, and r designate the centre, left, and right regions, respectively. We will

9 1.4 The quantum well


30/743

now illustrate the process of quantization by propagating the continuity condi-

tions of the wavefunction and its first derivative (also referred to as boundaryconditions) from 9- to ;- and, furthermore, by requiring that the results benormalized.

As the wavefunction must be normalized, its value cannot diverge as x ; 9-.Therefore, B

J: 0. Additionally, the boundary conditions at x : 9a/2 lead to:

AA

: ; ik

2ike\G>I?A

J(1.35)

BA

: 9 9 ik

2ike\G>I?A

J

AA

and BA

are related by the following useful equation:

AA

BA

: 9 ; ik 9 ik

e (1.36)

The boundary conditions at x : a/2 give:

APeG? ; B

Pe\G? : A

AeI? ; B

Ae\I?

(1.37)

APe G? 9 BPe\G?: ik

(AAeI? 9 BA

e\I?)

We propagate the boundary conditions by bringing (1.37) into (1.35) where:

AP

:[( ; ik)eI? 9 ( 9 ik)e\I?]

4ike\G?A

J(1.38)

BP

: ; k

2ksin kaA

J

As the wavefunction must remain finite as x ; ;-, this requires that AP

: 0 or

that:

9 ik ; ik

: eI? (1.39)

which may also be expressed as:

k : tan

ka

2

(1.40)

or

coska

2 :k

k

(1.41)



31/743

These solutions, of which there are two types, are expressed in the transcendental

equations that follow.

(1) Even solutions

9 ik ; ik: 9eI? (1.42)

or

coska

2 :k

k (1.43)

tan

ka2

9 0

Equation (1.36) informs us that AA

: BA, or that the solutions are even. The energy

levels, solutions of Schrodingers equation, can then be determined from Fig. 1.2,

and are represented by the intersection points where the line of slope 1/k

meets

the sinusoidal arches (dotted lines). Therefore, the energies accessible to an elec-

tron with total energy less than that of the potential barrier constitute a discrete

spectrum (implying the energy levels are quantized).

The wavefunction then takes the form:

LA

(x) : AL

cos kLx, for "x " :

a

2

LJ

(x) : BLeGV, for x : 9

a

2(1.44a)

LP(x) : B

Le\GV, for x 9

a

2

where n designates the nth even solution of the equation. The values for AL

and BL

are obtained by noting that the integral of the square of L(x) from 9- to ;-

equals 1. For the ground state (n : 1), we obtain:

A

:2

a ; 2/

(1.44b)

B :

2

a ; 2/

k

k e\G

?

where k

is the wavevector for the ground state from (1.43). Equation (1.44) shows

that the electron wavefunction penetrates into the barrier over a distance given by

1/, which signifies that the probability of finding the electron in the barrier regionis non-zero (see Fig. 1.1). This phenomenon, known as tunnelling, possesses no



32/743

1.2

1.0

0.8

0.6

0.4

0.2

0.03.02.52.01.51.00.50.0

ak k0a /

Fig. 1.2. Graphical determination of the quantized states for a symmetric quantum well

using Eqs. (1.43) and (1.46), with k

: 0.78 nm\, and a well width a : 10 nm (see the example).

classical equivalent and results from the fundamental wave nature of the electron

and recalls analogous behaviour in light. We now recall the equation relating the

energy of the eigenstate to its penetration depth into the barrier region:

:1

:

(2mC(9E)

(1.45)

(2) Odd solutionsThese correspond to an alternative solution to (1.39):

9 ik ; ik: eI?

Namely:

sin

ka

2

:

k

k (1.46)

tanka

2 : 0This time Eq. (1.36) tells us that A

A: 9B

A, i.e. that the solutions are odd. The

energy levels are now given by the intersection of the same line, with slope 1/k

,

with the other series of sinusoidal arches appearing as solid lines in Fig. 1.2.

It is also interesting to calculate the number of quantum levels within the well.

Inspection of Fig. 1.2 gives

N : 1 ; Int(2m

CV

a

(1.47)where Int designates the integer function.



33/743

Also, no matter how shallow the well is, there is always at least one quantized

state which lies within it. While this is a general observation which applies to all

one-dimensional wells, this may not hold in three dimensions. The quantized levels

are also referred to as: localized, as the wavefunctions have a non-negligibleamplitude only in the vicinity of the well; and bound, as the probability of findingthe electron is only significant near the well (the electrons are not mobile and

cannot participate in current flow). The energy levels residing above the barrier

(E 9 0) are called delocalized or free (consult Complement 1.A for further details).It is important to note that this line of reasoning may be generalized to any sort

of potential: i.e. that quantification of the energy levels results from propagation of

the boundary conditions from 9- to ;-, and from the requirement that the

amplitudes of the wavefunctions vanish at infinity.

ExampleWe will later see in Chapter 8, that an electron in a semiconductor heterostructure

fabricated with GaAs/Al

Ga

As is subjected to a potential well of 360 meV

depth. Furthermore, the interaction of the electron with the periodic potential of

the GaAs host crystal is taken into account by multiplication of the electron mass

by a coefficient equal to 0.067. The result of this product corresponds to the

effective mass of the electron m* : 0.067mC

. Application of Eq. (1.33) allows us to

solve for the wavevector k

:

k

:'(2;0.067; 0.9; 10\ (kg); 0.36 (eV); 1.6; 10C)/1.05; 10\ J s

or

k

: 0.78 nm\

which corresponds to a wavelength of:

8.05 nm.Let us now consider a quantum well with a width of 10 nm. As the well width is

of the order of the de Broglie wavelength

associated with V

: 360 meV, we

may expect the system to exhibit quantization. Using Eq. (1.47), we see that we can

expect three bound states in this particular system (i.e. 1 ; Int(0.78; 10/3.14)).The wavefunctions corresponding to each of these states are shown in Fig. 1.1.

The MATHEMATICA program below is very useful for solving quantum

confinement problems:

m0=0.91 10 -30 (*kg*);hbar=1.05 10 -34 (*J.s*);q=1.6 10 -19 (*C*);

meff=0.067 (* effective electron mass in GaAs*);

V0=.36 (*well depth in eV*);

a=10. (*well width in nm*);

k0=Sqrt[2*meff*m0*q*V0]*10-9/hbar (*in nm-1*)



34/743

eq1=Cos[k*a/2];

eq2=Sin[k*a/2];

eq3=k/k0;

plot1=Plot[Abs[eq1],+k,0,k0,]plot2=Plot[Abs[eq2],+k,0,k0,]plot3=Plot[Abs[eq3],+k,0,k0,]Show[plot1,plot2,plot3]

FindRoot[eq1==eq3,+k,0.2,]E1 =hbar2*(k*109)2/(2*meff*m0*q)/.%

FindRoot[eq2==eq3,+k,0.5,]E2 =hbar2*(k*109)2/(2*meff*m0*q)/.%

hnu=E2-E1 (*optical transition energy in eV*)

1.4.2 The infinite square well

A particularly important case worth investigating is that of the infinite square well

(see Fig. 1.3). In this case, the solution to Schrodingers equation is found immedi-

ately:

ka : n odd, L(x) :

2

a

cos nx

a (1.48)

ka : n even, L(x) :

2

asin n

x

a

and in both cases:

EL

: n

2mCa

: nE

(1.49)

Energy levels for the infinite square well

E

is the confinement energy. We thereby uncover an alternate interpretation of thede Broglie wavelength given in (1.1), i.e. it is the width required of an infinite squarewell to yield a confinement energy E

equal to the energy of the particle. An

important definition is the thermal de Broglie wavelength "

: this is the width of

an infinite square well necessary for a confinement energy equal to the thermal

energy kT:

" : 2

(2mCkT

(1.50)

Therefore, potential wells having widths less than "

at T: 300 K will show

quantum effects unhindered by thermal vibrations in the system. Only in these

cases may we speak of quantum wells at room temperature.



35/743

600

400

200

0

Energy(meV)

8 4 0 4 8

Position (nm)

Fig. 1.3. Infinite quantum well with a width of 10 nm. In contrast to the finite square well

solutions depicted in Fig. 1.1, the wavefunctions do not penetrate into the barriers and the

energy levels are lifted up relative to the base of the well as a result.

ExampleAssuming an infinite (although unachievable in practice) square well consisting of

GaAs, at 300 K we calculate a "

: 6.28; 1.05 ; 10\ (J s\)/'(2;0.067; 0.9; 10\ (kg) ;0.0259 (eV); 1.6; 10\ C): 12 nm.

Therefore, quantum effects will only be discernible in GaAs layers thinner thanthis value and only in these cases will we be able to speak of GaAs quantum wells

existing at room temperature.

1.5 Time-independent perturbation theory

Very few physical systems present solutions as simple as those afforded by quan-

tum wells. We find among such analytically tractable systems, the hydrogen atom(not treated in the present text) and the harmonic oscillator (treated in Comple-

ment 1.D). More general systems seldom have analytical solutions. However, by

elaborating on simpler systems possessing better known solutions, we will attempt

to approximate solutions to those that are more complex. The most popular (and

arguably the most fruitful in terms of its success in expanding our conceptual

understanding of many physical systems) is time-independent perturbation theory.

Consider an electron in a system described by a time-independent Hamiltonian

H for which the complete basis set of stationary states +"L2, consists of solutionsto Schrodingers equation:

H"

L2 : E

L"

L2 (1.51)

(Note that from this point onwards, to simplify the notation, we will drop the

used earlier to identify operators, as we assume the reader is now able to

15 1.5 Time-independent perturbation theory


36/743

distinguish between an operator and a variable.) For the present, we will suppose

that the states are discrete and non-degenerate. An important case involving the

extension of perturbation theory to degenerate systems is given in Complement

1.B. We will now submit this system to a small additional perturbation W: U,such as may be achieved by the application of an electric field to a quantum well.

By small, we mean that 1 and that the eigenvalues ofU are of the order EL

(i.e.

that U:H

or that the eigenenergies of U are roughly the same size as those ofthe unperturbed Hamiltonian H

). The eigenvalues of the new Hamiltonian

H : H

; Ware:

(H

; U)"L()2 : E

L()"

L()2 (1.52)

Then, make the important hypothesis that a sufficiently weak perturbation willallow us to consider the solutions of the modified system in terms of the original

levels of the unperturbed system (i.e. that such a small perturbation has not

distorted the original energy spectrum of the system beyond recognition). The new

eigenvalues and eigenvectors of the perturbed system are then written in terms of

the original eigenenergies and eigenvectors and the perturbation coefficient :

EL() :

;

;

;

(1.53)

" L()2 : "02 ; "12 ; "22 ;

Substitute (1.53) into (1.52) and obtain by identifying like terms in powers of:

Order 0 H"02 :

"02 (1.54a)

Order 1 (H

9

)"12 ; (U 9

)"02 : 0 (1.54b)

Order 2 (H

9

)"22 ; (U 9

)"12 9 "02 : 0 (1.54c)

0th orderAs we have assumed that the levels are non-degenerate, Eq. (1.54a) shows that "02is an eigenstate ofH

. By continuity, as ; 0, we find that "02 : "

L2. This is not

true when the levels are degenerate, as Eq. (1.54a) no longer corresponds to a single

quantum level.

1st orderProject (1.54b) on "02 : "

L2 and use the identity:

10"H 9 "12 : 0 (1.55)

to find the first-order energy correction:

: 1L"U"

L2 (1.56)

or, in terms of earlier definitions:



37/743

EL

: EL

; 1L"W"

L2 (1.57)

First-order energy perturbation

where the perturbed energy EL is expressed without reference to .To find a limited expansion for the eigenvector, we need only project (1.54b)onto the other states "

N2 with p" n:

(EN

9 EL)1

N"12 ; 1

N"U"

L2 : 0 (1.58)

We then obtain for the perturbed eigenvectors the following first-order expansion:

" L2 : "

L2 ;

N$

L

1N"W"

L2

E

L

9 E

N

"N2 (1.59)

First-order perturbation of the eigenstates

We notice that the unperturbed stationary state "L2 is contaminated by other

eigenstates "N2, and all the more so for those states "

N2 closest to "

L2 in energy.

Therefore, in describing the effect of a perturbation, we will be content to limit

ourselves to a description in terms of those levels closest in energy (see, for

example, the treatment of the Stark effect given in Complement 1.C).

2nd orderIn a certain number of cases, the first-order perturbation will be null when:

1L"W"

L2 : 0 (1.60)

This occurs as a result of symmetry considerations (as, for instance, in the case of

the perturbation of a quantum well confinement potential by an electric field). As a

result, it is often necessary to continue the perturbation expansion to higher

orders. Projecting (1.54c) onto "L2, we find:

: 1L"U"12 (1.61)

after which using (1.59) we may write for the second-order perturbation:

EL

: EL

; 1L"W"

L2 ;

N$L

"1N"W"

L2"

EL

9 EN

(1.62)

Second-order energy perturbation

where again we note that the magnitude of the contribution of any given state

increases for those closest to "L2 in energy.

17 1.5 Time-independent perturbation theory


38/743

1.6 Time-dependent perturbations and transition probabilities

1.6.1 The general case

Situations where exact solutions may be found to Schrodingers time-dependent

equation (1.12) are unfortunately few and far between. The time-dependent behav-

iour of an electron in a quantum well is worth citing; it may be worked out as an

exercise. Generally, we employ a perturbative approach, which will enable defini-

tion of the transition rate. Let us consider a system described by the Hamiltonian

H

which is in an initial state "G2 at time 0. At time t : 0 we turn on a perturbation

W(t) : U(t), where the conditions placed on and U(t) are same as in thepreceding section (namely that 1 and U:H

). In order to solve Schrodingers

time-dependent equation:

id

dt"(t)2 : [H

; W(t)]"(t)2 (1.63)

to describe the evolution of the system, we can expand "(t)2 in terms of the basis ofstationary states, as described in (1.6):

" (t)2 : L

cL(t)"

L2 (1.64)

Substituting (1.64) into (1.63) and identifying like terms, we obtain a system of

coupled differential equations, relating the coefficients cL(t) to one another:

id

dtc

L(t) : E

Lc

L(t) ;

N

ULN

(t)cN(t) (1.65)

where ULN are the elements in the matrix:

ULN

(t) : 1L"U(t)"

N2 (1.66)

We will suppose that, for reasons of symmetry, ULL

: 0 for any given level n. Wethen make the following change of variables:

bL(t) : c

L(t)e>#LRe (1.67)

which leads us to:

id

dtb

L(t) :

N

eSLNRULN

(t)bN

(t) (1.68)

where LN

: (EL

9 EN)/ is the Bohr oscillation frequency for the transition n ; p.

As in Section 1.5, we perform a limited expansion:



39/743

bL(t) : b

L(t) ; b

L(t) ; b

L(t) ; (1.69)

allowing us to identify like terms in after substitution of (1.69) into (1.68).

0th-order termWe find that b

L(0) is a constant which corresponds to the stationary state solutions

given by (1.29).

qth-order termWe obtain:

id

dtbO

L(t) :

N

eSLNRULN

(t)bO\N

(t) (1.70)

Therefore, once the zeroth-order solution is known, we may calculate the first-

order solution and then any other order solution by recurrence. We will interest

ourselves in the remainder of this chapter with first-order perturbations. Second-

order perturbation will be developed in Chapter 12, in the context of non-linear

optics.

At t : 0, the system is in the state "G2, with initial conditions:

bG

(t : 0) : 1(1.71)

bL

(t : 0) : 0, for i" n

To zeroth order, these values remain constant with respect to time. Inserting these

values into (1.70), we obtain the first-order time evolution equation:

id

dtb

L(t) : eSLGRU

LG(t) (1.72)

which takes the integral form:

bL

(t) :1

i

R

eSLGRYU

LG

(t)dt (1.73)

We are now in a position to calculate the probability PGD

(t) of finding the system ina final stationary state "

D2 at time t. Following the probabilistic interpretation of

quantum mechanics, this is obtained by evaluating "bD

(t)" or:

19 1.6 Time-dependent perturbations and transition probabilities


40/743

PGD

(t) :1

R

eSDGRYWDG

(t)dt

(1.74a)

Transition probability between levels iand f

under the effect of a time-varying perturbation

where

WDG

(t) : 1D"W(t)"

G2 (1.74b)

This formula is one of the most important in quantum mechanics and will be

referred to throughout this book. We will presently apply it to the particularly

interesting and useful problem of a time-varying sinusoidal perturbation.

1.6.2 Sinusoidal perturbation

This perturbation potential may be written as:

W(r, t) : W(r)sin t (1.75)

Equation (1.74) leads immediately to a time-dependent transition probability

PGD

(t) between initial and final states:

PGD

(t) :"W

DG"

4 1 9 eSDG>SR

DG

; 9

1 9 eSDG\SR

DG9

(1.76)

We therefore make what is classically referred to as the rotating phase or thequasi-resonance approximation, which ignores the contribution of the term pos-sessing the larger denominator

DG; in favour of that with

DG9 . Thus,

keeping only the second term in (1.76) we obtain:

PGD(t) : "WDG"

4 1

9e

S

DG\S

R

DG

9

: "WDG"

4 sin (DG

9)t2

(DG9

)2

(1.77)

Figure 1.4 shows the evolution of this probability as a function of time for

different frequencies (or detuning) between the perturbing field and resonanttransition frequency 9

DG. We note that as the frequency of the perturbation

field approaches that of the resonant Bohr oscillation frequency (i.e. ; DG

), the

time dependence of the transition amplitude changes from a sinusoidally varying

function to a parabola in t. In a complementary fashion, we show in Fig. 1.5 the

spectral distribution of the transition probability as a function of detuning forvarious times t. This function is a sinus cardinal multiplied by t, which tendstowards a Dirac delta function as t ;-. We therefore rewrite (1.77) as:

PGD

(t) :"W

DG"

4tsinc

(DG

9 )t2

(1.78)



41/743

10

8

6

4

2

0

Pif

(Wif

2/h2)

543210

if = 0

if= 1

if= 2

t(1/( if ))

Fig. 1.4. Time evolution of the transition probability between levels i and ffor differentdetuning values of

GD9 . In off-resonance conditions, the electrons oscillate between both

levels.

40

30

20

10

02 0 2 4

(1/arb.unit)

t= 3 a.u.

t= 5 a.u.

t= 7 a.u.

Pif(W

if2/h2)

( if )

4

Fig. 1.5. Transition probability between two levels i and fas a function of detuningfrequency for different observation times t (arb. units). At longer times, only transitions

between states satisfying the requirement of energy conservation are accessible. This behaviouris in accordance with Heisenbergs second uncertainty principle.

where sinc(x) is the sinus cardinal sin x/x. Equation (1.78), while appearing simple,is in fact rather difficult to grasp in its entirety, as it is a function of two intimately

related quantities, namely frequency and time. To investigate its behaviour better,

we will distinguish between three different cases.

Case 1: transitions induced between discrete levels by single frequency excitationIn this case, the resonant transition largely dominates the behaviour whereby:

PGD

(t) :"W

DG"

4t (1.79)

As the transition probability thus stated increases quadratically with time, this

21 1.6 Time-dependent perturbations and transition probabilities


42/743

description is clearly an approximation as its value cannot exceed unity. We will

see in Complement 1.E that this approximation holds only for very short times

over which the zeroth-order expansion employed in (1.69) may be seen as valid. We

note that the resonance condition : DG may be written, alternately, as

: ED

9 EG

: DG

(1.80)

Bohr frequency

This equation describes the conservation of energy between the energy quantum

transferred to the system and the energy difference of the system between the final

and initial states E : ED

9 EG. From a technical point of view concerning the

calculation of these quantities, we note that the ordering of the indices in these

equations ("WGD " : "WDG") plays no explicit role owing to the properties of Her-mitian products. This, however, is not the case for the Bohr oscillation frequency

GD: 9

DG!

Additionally, Fig. 1.5 shows that the transition probability becomes negligible

once:

Et 9 (1.81)

This last condition is also known as Heisenbergs second uncertainty relation and itallows the classical restriction of energy conservation to be violated by excitations

acting over short time periods.

Case 2: transitions induced between a discrete level and a continuum state by singlefrequency excitationIn this case, the final states form a continuum described by the continuous variable

DG

, and the transition probability between the discrete level and the continuum

PGA

(t) is calculated by summing the probabilities over the density of final states(

DG):

PGA

(t) :1

4t

>

\

"WGD

(DG

)"sinc(

DG9 )t2

(

DG)d

DG(1.82)

Distribution theory tells us that if a function "WGD

(DG

)"(DG

) is well behaved (i.e.

square normalizable and slowly varying), then:

lim

R;

sinc

1

2

(DG

9 )t

:

2

t

(DG

9 ) (1.83)

where is the Dirac delta function. Therefore, for long times, Eq. (1.82) takes theform:

PGA

(t) :

2"W

GD(

DG)"( :

DG)t (1.84)



43/743

The above equation tells us that when a transition occurs from a discrete state to a

continuum, the transition rate GGA

: dPGA

/dt is constant as a function of time andhas a value given by:

GGA

() :

2"W

GD(

DG)"( :

DG) (1.85a)

Fermis golden rule in terms of frequency

or

GGA

() :

2"W

GD"( : E

D9 E

G) (1.85b)

Fermis golden rule in terms of energy

This important equation is referred to as Fermis golden rule. It stipulates thatunder the influence of monochromatic excitation , only continuum levelshaving energy E

D: E

G; will be populated by the optical excitation with a

transition rate given by the above equation.

Case 3: transitions induced between two discrete levels by multi-frequency

excitationIn this case, the perturbation consists of a continuum of excitation frequencies:

WDG

(t) :

g()WDG

()sin(t)d (1.86)

where g() is the excitation spectrum and WDG

is the matrix element of the

interaction Hamiltonian at each particular wavelength. A development strictly

equivalent to the one given above leads to a transition rate:

GGD

() :

2"W

GD"g( : E

D9 E

G) (1.87)

Transition rate for broad frequency excitation

1.7 The density matrix

Two kinds of uncertainty coexist within the description of a physical system. There

is a purely quantum uncertainty related to the probabilistic interpretation of the

results of the operator algebra applied to the system. There is also uncertainty

resulting from the thermal agitation of the systems constituent parts, which is

described by statistical mechanics. Density matrix formalism presents itself as a

23 1.7 The density matrix


44/743

very powerful and elegant framework which integrates these two notions into a

single mathematical description.

1.7.1 Pure quantum ensembles

Let us consider a quantum system in a state "(t)2 described by (1.64). We wish toknow the average value of an operator A. Following Eq. (1.8), this average value1A2 may be written in a particular basis "

I2 as:

1A2 : LK

AKL

c*K

(t)cL(t) (1.88)

where AKL

is an element of the matrix:

AKL

: 1K

"A"L2 (1.89)

Equation (1.88) is somewhat deceiving as it seems to suggest that a privileged basis

set exists in which to carry out the decomposition. If we change this basis, however,

the AKL

, cK

(t), and cL(t) entries will change in such a manner as to leave (1.88)

invariant. This inconvenience is eliminated by introducing the density matrix,whose elements are:

LK

(t) : c*K

(t)cL(t) (1.90)

In fact, the matrix (t) may be written as:

(t) : " (t)21(t)" (1.91)

With this definition, Eq. (1.88) then becomes:

1A2 : Tr(A) (1.92)

which is independent of the decomposition basis : " (t)21(t)" as its trace is alinear operator whose value is independent of the basis in which it is evaluated.

Furthermore, using (1.91), we immediately see that the evolution of (t) as afunction of time is given by:

id

dt(t) : [H(t), (t)] : H(t)(t) 9 (t)H(t) (1.93)

Schrodingers equation in density matrix formalism

1.7.2 Mixed quantum ensembles

We now consider a system consisting of a statistically distributed mixture of states

+"G2,. This system has a thermodynamic probability p

Gof being in a state "

G2.



45/743

Neglecting quantum interferences between thermodynamically blurred states, it

seems natural to define the average value of observable A as:

1A2 : G

pG1A2

G(1.94)

where 1A2Gis the average value of observable A when the system is in the state

"G2. Following Eq. (1.92), this may be written as:

1A2 : G

pGTr(

GA) : Tr(A) (1.95)

where

is the density matrix of the mixed ensemble:

: G

pG

G:

G

pG"

G21

G" (1.96)

In Eq. (1.96) we see the advantage obtained by introducing the density matrix. It is

the linear dependence of1A2 on the density matrix which allows the introduc-tion of the density or averaging operator .

As each matrix allows the same time-evolution equation (1.93), the density

matrix for the mixed ensemble may be written:

id

dt(t) : [H, (t)] (1.97)

Schrodingers equation for a mixed ensemble

The fundamental equations of the density matrix are (1.95)(1.97). Within thedensity matrix itself, we may differentiate between two conceptually distinct

constituents.

(a) Diagonal elementsFrom equations (1.90) to (1.96), the diagonal terms may be expressed in the

stationary state basis as:

II

: G

pG"cG

I"

where cGI

is the "G2 component in the "I2 basis. An immediate physical interpreta-

tion of the diagonal terms in

II is that they represent the probability of finding thesystem, upon measurement, in a stationary state "I2 given both the quantum and

statistical uncertainties. Therefore, II

represents the population of the state "I2.

As these elements result from the summation of positive terms, they may not be

zero unless the value of each of these terms is zero (i.e. that the occupation of each

state is null).



46/743

(b) Off-diagonal elementsThe significance of these terms is a little more difficult to understand. They are

sometimes referred to as the coherence elements: they describe the quantum

behaviour of the system. When thermal fluctuations completely smear out thequantum interference effects, these are the terms that become zero.

In Section 1.7.3, we will give an example of a two-level system which will allow

us to grasp the usefulness of this powerful and elegant formalism better. We note in

closing that the distinction between population and coherence depends on the

decomposition basis +"I2,.

1.7.3 Density matrix and relaxation time for a two-level system

We consider a two-level system with a Hamiltonian H

, possessing eigenenergies

E

and E

, and stationary states "12 and "22 (i.e. H"i2 : E

G"i2). In the stationary

state basis, the Hamiltonian H

may be written:

H

:E

0

0 E (1.98)

We subject this system at time t : t to a sinusoidal perturbation W(t) which maybe expressed in the basis of "12 and "22 as:

W:m

m

m

m

cos t (1.99)

where mGH

: 1i"W"j2. We may assume by symmetry, that the elements m

and m

are null, and that the terms m

and m

are real and thus equal. The general case

may be determined as an exercise. Equation (1.97) then may be written as:

d

dt: 9i

m

(

9

)cost

d

dt(

;

) : 0 (1.100)

d

dt: 9i

; i

m

(

9

)cost

The second equation of (1.100) states that the total population is conserved (i.e.

;

: 1). Solutions to this very important set of coupled differential equa-

tions are given in Complement 1.E. Nonetheless, we may investigate the transitory

behaviour of the system at this point. For instance, it is clear that the terms in

cost will act to drive the system into oscillation. If the excitation ceases (i.e.



47/743

supposing we set m

to zero), the diagonal terms will remain constant and the

off-diagonal terms will continue to oscillate with frequency

.

Intuitively, we may expect that once the excitation stops, the populations GG

will

tend toward their thermodynamic equilibrium levels GG with a certain timeconstant resulting from stochastic interactions. This time constant is often referred

to as the diagonal relaxation time or the population lifetime, to name but a few. It iswritten as T

when its value is independent of i in

GG, i.e. level independent. In

converse situations, the custom is to make use of the relaxation rate GG

. In the

same way, we expect the off-diagonal elements to lose coherence with a time

constant of\GH

or T

if this time is independent of ij. Introducing these differentrelaxation times, the equations in the density matrix become:

d

dt: 9i

m

(

9

)cost 9

9

T

d

dt: i

m

(

9

)cost 9

9

T

(1.101a)

ddt : 9i ; i

m ( 9 )cost 9

T

Time-evolution of elements in a density matrix

for a two-level system

or

d

dt: 9i

m

(

9

)cost 9

(

9

)

d

dt: i

m

(

9

)cost 9

(

9

) (1.101b)

d

dt: 9i(

9 i

)

; i

m

(

9

)cost

Time-evolution of elements in a density matrix

for a two-level system

These last two expressions are one of the major conclusions from this first chapter

and they will be used intensively throughout this book. Complement 1.E gives as

Interestingly, the introduction of a relaxation time reintroduces the set of stationary states as a privilegedbasis. This observation is of theoretical interest, however, and will not receive further consideration by us.



48/743

an example an application whose treatment using this theory leads to the optical

Bloch equations.

FUR TH ER R EA DING

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, New York (1992).C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, AtomPhoton Interactions: Basic

Processes and Applications, Wiley, New York (1998).R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol III:

Quantum Mechanics, Addison-Wesley, Reading, Mass. (1966).R. Loudon, The Quantum Theory ofLight, Clarendon Press, Oxford (1973).

E. Merzbacher, Quantum Mechanics, Wiley, New York (1970).A. Messiah, Quantum Mechanics, vols 1 and 2, Wiley, New York (1966).L. I. Schiff, Quantum Mechanics, 2nd edn, McGraw-Hill, New York (1955).



49/743

Complement to Chapter 1

1.A Problems posed by continuums: the fictitious quantum box and the

density of states

The quantum description of delocalized states, which therefore belong to a con-

tinuum, makes reference to the theory of distributions. It attempts to deal with

difficult problems such as the normalization of wavefunctions in a null potential

from 9- to ;-. In this book, systematic use is made of the theoretical trickafforded by introduction of a fictitious infinite square well of width L in which themotion of the continuum electrons are later shown to be pseudo-quantified. What

we mean by pseudo-quantification is that when we take L tending towards infinityin the expressions obtained, the dependence ofL will conveniently disappear from

physical predictions. There is no moral in this; only the tutelar protection of

distribution theory! We now proceed to an illustrative example: photoemission

from a one-dimensional well.

We consider a quantum well of width d as represented in Fig. 1.A.1. Thisquantum well admits a quantized level "i2 described by a square integrablewavefunction

G(z) and a quantized energy level 9E

'(where the index I stands for

ionization, for reasons which shall soon be clear). We further presume that the wellis sufficiently deep for G(z) to be considered the wavefunction for the ground state

of the infinite well such that:

G(z) :

2

dcos

d

z (1.A.1)

This well also admits delocalized states, where the electrons may take on any value

of positive energy. We will neglect the influence of the well on the free electrons, i.e.we will suppose that the free electrons are subject to a null potential once they are

in the continuum. To avoid problems involving the normalization of these

wavefunctions, we introduce a fictitious square well of width L within which the

continuum electrons are trapped. The corresponding eigenenergies and eigenfunc-

tions of the unbound states are:

29


50/743

Energy

Position

L

0

EI

Fig. 1.A.1. Procedure for pseudo-quantification of the potential barrier states. The width Lof the infinite quantum well is arbitrary.

eL

: ne

L(z) :

2

Lsin nk

*z, ifn is even (1.A.2)

L(z) :

2

L cos nk*z, ifn is odd

where e

is the confinement energy of the fictitious well:

e

:

2m*k

*(1.A.3)

with wavevector k*

k* : L (1.A.4)

IfL takes on dimensions of centimetres, then e

is of the order of 10\ eV. In this

sense, such a box would be fictitious as the energy level spacings would be

infinitesimal in comparison with typical interaction or thermal energies (of the

order of meV). The energy levels given in (1.A.2) are so close that rather than

attempting to take each one into account individually, we group them together by

means of infinitesimal batches of the density of states.

Let us consider a certain wavevector interval dk. In this interval the individualstates of the fictitious box are separated in the wavevector by /L. Without takinginto account electron spin, the number of states in this interval is clearly:

dn :L

dk (1.A.5)

30 Complement to Chapter 1


51/743

The density of states dn/dk is then given by:

(k) :dn

dk

:L

(1.A.6)

In this same interval, as obtained by differentiation of (1.A.3), the corresponding

change in energy relates to dk as:

dk :1

2

(2m*

dE

E(1.A.7)

The energy density of states dn/dE is then finally:

(E) : L2

(2m*

1

(E(1.A.8)

One-dimensional density of states without spin

We note that as L tends towards infinity, (E) increases without bound. This is tobe expected as more and more states become available over the same energy range

as the energy separation between levels decreases.

We will now calculate the transition probability between an initial quantized

state "i2 and the continuum under the effect of a sinusoidally varying dipoleperturbation:

W(z, t) : 9qFz cos(t) (1.A.9)

From Fermis golden rule (1.85b), the transition probability may be written as:

GGA

() :qF

2"z

GD"( : E

D9 E

G) (1.A.10)

The transition element zGD is non-zero only for odd parity final states and is givenby:

"zGD

" : "1G"z"

D2" :

2

(Ld

B

\B

cosd

z sin(kDz)zdz

(1.A.11)

or

"zGD" : 4d

L f(ED) (1.A.12)

where f (E) is the dimensionless integral in equation (1.A.11) and is found to be:

f (ED

) :

9 dkDsin

kDd

29

4kD

d

9 dkD

cosk

Dd

2 (1.A.13)

31 1.A Problems posed by continuums


52/743

and kD

is the wavevector of the associated state with energy ED

:

E

D

: 9 E'

:k

D

2m*

(1.A.14)

We then substitute the expression for "zGD

" into (1.A.10). Notice that the width of thefictitious box L, which appears in the denominator of the transition element and in thenumerator of the density of states, cancels out as advertised. In physical terms, thewider the width of the pseudo-well, the greater the final density of states; however,

this effect on the transition probability is cancelled out as the increased width also

serves to dilute the probability density of the electron states above the quantum

well (of width d) by a similar amount.Taking into account the fact that the density of final states is only one-half of the

expression we derived in (1.A.8) (because of the spin of the electrons, which is

conserved in the transition) and since only the odd parity wavefunctions partici-

pate in the transitions, we therefore find for the transition probability from an

initial quantized state to the continuum:

GGD

() : qFd(m*

f( 9 E')

(2( 9 E'

)(1.A.15)

The behaviour of the system is therefore found to be independent of the size of the

fictitious box we introduced at the onset. This technique is referred to as pseudo-quantification, and is in fact a very powerful tool in spite of its simplistic appear-ance. Figure 1.A.2 shows the variation of the transition rate as a function of the

excitation frequency .We notice the presence of an ionization threshold for the transition probability.

The cut-off energy for detected photons corresponds to the ionization energy E'.

Furthermore, the absorption near the detection threshold, i.e. for photons with:E

'is given by:

GGD.( 9 E

', for :E

'(1.A.16)

A second characteristic is the quasi-resonant nature of the transition probability

near the energy threshold for photoionization. This quasi-resonance results from

decreases in both the density of states (in k\D

) and in the dipole moment (in k\D

)

which leads to:

GGD.

1

( 9 E')

, for E'

(1.A.17)

These expressions give a reasonable description of the spectral response of quan-

tum well based detectors.



53/743

12 10+6

10

8

6

4

2

0

Transitionrate

(s1)

201510505

Photon energy normalized to EI

Fig. 1.A.2. Ionization transition rate (s\) of a quantum well as a function of incident photonenergy in multiples of the ionization energy, E

'.

1.B Perturbation on a degenerate state

Consider a system described by a Hamiltonian H

which possesses a degenerate

state EL with degeneracy gL (i.e. that gL independent eigenvectors +"GL2, i :1 , . . . , g

L,, forming an eigenvector subspace, share the same eigenenergy E

L). We

seek the perturbation induced in this eigenvector subspace by a perturbing field

W: U. Equation (1.54a) stipulates that the perturbed state always belongs to theeigenvector subspace but does not allow one to find the perturbed state "02 sinceall linear combinations of the eigenvectors "G

L2 form possible solutions of this

equation. It is therefore necessary to use (1.54b) to obtain the perturbed state "02.Projecting (1.54b) onto the vectors "G

L2, we obtain:

1GL"U"02 :

1G

L"02 (1.B.1)

We recall that the unknowns are the new perturbed state "02 and the perturbationE :

. Equation (1.B.1) is nothing other than the eigenvalue and eigenvector

equation of the perturbation operator W in the eigenvector subspace +"GL2, i :

1 , . . . , gL,. More convincingly, for the time being let us designate c

Gas the compo-

nent of "02 on the basis "GL2 (i.e. c

G: 1G

L"02) and w

GHas the element of the

perturbation matrix wGH

: 1GL

"W"HL

2. Written in matrix form (1.B.1) then be-comes:

. . .

. wGH

.

. . ..

cG

. E.

cG

. (1.B.2)

33 1.B Perturbation on a degenerate state


54/743

We recognize the above secular equation as corresponding to the diagonalization

of the perturbation operator Win the subspace spanned by the eigenvectors "GL2.

In order to refine our ideas, we now turn to an example illustrating the use of

this formalism. We consider two identical physical systems far removed from oneanother (see Fig. 1.B.1). The system on the left is centred at r

J, while the system on

the right is centred at rP. Their respective Hamiltonians are:

HJ

:p

2m; V(r

J)

(1.B.3)

HP

:p

2m; V(r

P)

These two systems share similar Hamiltonians and therefore have identical

eigenenergies. For example, both "l2 and "r2 share the same ground state energy:

HJ"l2

HP"r2

: E"l2: E"r2

(1.B.4)

Let us now bring these two systems into closer proximity with one another in such

a manner as to cause each system to engender a perturbing influence on the other.

We may therefore represent this effect by a perturbation Hamilton W. Equation(1.B.2) then takes the form:

w

PPw

PJw

JPw

JJ

cP

cJ: E

cP

cJ (1.B.5)

Without loss of generality, we may suppose wPP

: wJJ

: 0; and wPJ

to be real and

equal to A. Equation (1.B.5) may then be rewritten as:

0 AA 0

cPc

J: E

cPc

J(1.B.6)

From which the system admits the following eigenenergies and eigenvectors:

E> : E ; A ";2 :1

(2("r2 ; "l2)

(1.B.7)

E\: E 9 A "92 :1

(2("r2 9 "l2)

The ground state energy E is then no longer an eigenenergy of the states "l2 and "r2.We say that the interaction between the systems has lifted the degeneracy and thatthe states "l2 and "r2 have hybridized. This mechanism is represented schematicallyin Fig. 1.B.2. This mechanism forms the basis for chemical bonding. Let us



55/743

l|

r

r

l rr

V V

E E

l r

L

l

r|

Fig. 1.B.1. A degenerate system consisting of two identical atoms separated spatially by L.

| + >= ( |r>+ |l ) / 2

2A|l |r

| + >= ( |r>+ |l ) / 2

|

|

Fig. 1.B.2. When the two states "l2 and "r2 are brought into close proximity, they hybridize,thereby lifting the degeneracy of the system.

consider the simple example of two hydrogen atoms. Far away from one another,

the energy of their two electrons is 2E (where E represents each electrons groundstate energy). As they are brought together, the strength of their interaction W

increases, lifting the degeneracy and the two electrons on the "2 level have nowonly a total energy of 2(E 9 A) (thanks to spin degeneracy). We continue below

with a particularly illuminating example.

Example: coupled quantum wellsAdvances in the growth of high quality, cr

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