Quantum theory of conducting matter: Newtonian equations of motion for a Bloch electron

223 Pages·2007·2.243 MB·English

by  Shigeji Fujita, Kei Ito

Quantum Theory of Conducting Matter
Shigeji Fujita and Kei Ito
Quantum Theory of
Conducting Matter
Newtonian Equations of Motion
for a Bloch Electron
ShigejiFujita
DepartmentofPhysics
UniversityatBuffalo
TheStateUniversityofNewYork
Buffalo,NY14260
[email protected]
KeiIto
ResearchDivision
TheNationalCenterforUniversityEntranceExaminations
2-19-23Komaba,Meguro
Tokyo153-8501
Japan
[email protected]
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Preface
The measurements of the Hall coefficient R and the Seebeck coefficient
H
(thermopower) S are known to give the sign of the carrier charge q. Sodium
(Na) forms a body-centered cubic (BCC) lattice, where both R and S are
H
negative, indicating that the carrier is the “electron.” Silver (Ag) forms a
face-centered cubic (FCC) lattice, where the Hall coefficient R is negative
H
but the Seebeck coefficient S is positive. This complication arises from the
Fermi surface of the metal. The “electrons” and the “holes” play important
rolesinconductingmatterphysics. The“electron”(“hole”), whichbydefini-
tion circulates counterclockwise (clockwise) around the magnetic field (flux)
vector B cannot be discussed based on the prevailing equation of motion
in the electron dynamics: (cid:2)dk/dt = q(E + v × B), where k = k-vector,
E = electric field, and v = velocity. The energy-momentum relation is not
incorporated in this equation.
In this book we shall derive Newtonian equations of motion with a sym-
metric mass tensor. We diagonalize this tensor by introducing the principal
masses and the principal axes of the inverse-mass tensor associated with the
Fermi surface. Using these equations, we demonstrate that the “electrons”
(“holes”) are generated, depending on the curvature sign of the Fermi sur-
face. The complicated Fermi surface of Ag can generate “electrons” and
“holes,” and it is responsible for the observed negative Hall coefficient R
H
and positive Seebeck coefficient S. When the Fermi surface is nonspher-
ical, the conduction electron moves anisotropically with different effective
masses (m , m , m ). The magnetic oscillations in the susceptibility χ and
1 2 3
the magnetoresistance arise from the oscillatory density of states assocated
with the Landau states upon the application of a magnetic field. The most
direct probe of the Fermi surface can be made by observing the cyclotron
resonance. A magnetic field is applied to a pure sample at liquid helium
temperatures. The sign of the charge carrier can be determined by using
the circularly polarized lasers. The data are analyzed in terms of Shockley’s
v
vi Preface
formulaoritssimplifiedversion. Mostoften,theintrinsiceffectivemassesfor
a semiconductor or metal can be determined directly after simple analyses.
Inthepresentvolume,wemainlydealwiththebehaviorsofthefermionic
conduction electrons. In the companion volume, called book2 in the text,
superconductivity and quantum Hall effect are treated. The charge carri-
ers in the supercurrent are the Cooper pairs, each composed of a pair of
electrons bound by the phonon exchange attraction. The statistics of a com-
posite particle with respect to the center-of-mass motion follows Ehrenfest–
Oppenheimer–Bethe’s rule: a composite moves as a fermion (boson) if it
contains an odd (even) number of elementary fermions. Accordingly the
Cooper pair moves as a boson since the pair contains two electrons. The
different statistics generate very different behavior. The quantum Hall effect
arises from the supercurrent generated in a two-dimensional system subject
to a magnetic field.
The text is composed of three parts: preliminaries, Bloch electron dy-
namics, and applications (fermionic systems). Part I, Chapters 1 through 6,
startswithanintroductionandthendealswiththephonons(quantaoflattice
vibrations), the free-electron model, the kinetic theory of electron transport,
the magnetic susceptibility, and the Boltzmann equation method. These
materials are normally covered in introductory solid-state physics courses;
however, they are prerequisite to the theoretical developments in Part II,
Chapters 7 through 10. The Bloch theorem, the self-consistent mean field
theory, the Fermi surface, the Bloch electron (wave packet) dynamics with
Newtonian equations of motion are discussed in Part II. In Part III, Chap-
ters 11 through 15, a selection of applications for fermionic systems (mostly
electrons) are discussed: the de Haas–van Alphen oscillations in suscepti-
bility, the Shubnikov–de Haas oscillations in magnetoresistance, the angle-
dependentcyclotron resonance, the Seebeck coefficient arising from the ther-
mal diffusion, and the infrared-laser Faraday rotation.
The present book is written for first-year graduate students in physics,
chemistry, electrical engineering, and material sciences. Dynamics, quantum
mechanics, thermodynamics, statistical mechanics, electromagnetism, and
solid-state physics at the undergraduate level are prerequisite. The authors
believe that the students should learn physics, starting from the bottom
up and following all theoretical developments with step-by-step calculations.
We have included many problems, most of them elementary excercises in
the text. The students learn key concepts more firmly by working out these
problems.
Preface vii
The prevalent equations of motion for the electron in a crystal is chal-
lengedinthisbook,butallargumentsleadingtothenewequationsofmotion
are based on the principles of quantum statistical mechanics (Heisenberg
uncertainty and Pauli exclusion principles). Condensed matter physicists,
chemists, and material scientists, theoretical and experimental, are invited
to examine this text.
Theauthorsthankthefollowingindivisualsforvaluablecriticisms,discus-
sions, and readings: Professor M. de Llano, Universidad Nacional Auto´noma
deMexico; ProfessorT.Obata, GunmaNationalCollegeofTechnology, Pro-
fessor Robert Kohler, Buffalo State University; and Dr. Hung-Chuk Ho,
Notre Dame University. They also thank Sachiko, Michio, Isao, Yoshiko,
Eriko, George Redden, Karen Roth, and Kurt Borchardt for their encour-
agement and reading of the drafts.
Shigeji Fujita, Buffalo, New York, USA
and
Kei Ito, Tokyo, Japan
August 2007
Contents
Preface v
Constants, Signs, Symbols, and General Remarks xiii
I Preliminaries 1
1 Introduction 3
1.1 Crystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Metals and Conduction Electrons . . . . . . . . . . . . 5
1.2.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . 6
1.2.3 Heisenberg Uncertainty Principle . . . . . . . . . . . . 6
1.2.4 Bosons and Fermions . . . . . . . . . . . . . . . . . . . 7
1.2.5 Fermi and Bose Distribution Functions . . . . . . . . . 7
1.2.6 Composite Particles. . . . . . . . . . . . . . . . . . . . 7
1.2.7 Quasifree Electron Model . . . . . . . . . . . . . . . . 8
1.2.8 “Electrons” and “Holes” . . . . . . . . . . . . . . . . . 8
2 Lattice Vibrations and Heat Capacity 11
2.1 Einstein’s Theory of Heat Capacity . . . . . . . . . . . . . . . 11
2.2 Debye’s Theory of Heat Capacity . . . . . . . . . . . . . . . . 15
3 Free Electrons and Heat Capacity 25
3.1 Free Electrons and the Fermi Energy . . . . . . . . . . . . . . 25
3.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Qualitative Discussions . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Quantitative Calculations . . . . . . . . . . . . . . . . . . . . 38
ix
x Contents
4 Electric Conduction and the Hall Effect 43
4.1 Ohm’s Law and Matthiessen’s Rule . . . . . . . . . . . . . . . 43
4.2 Motion of a Charged Particle in
Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . 46
4.3 The Landau States and Levels . . . . . . . . . . . . . . . . . . 48
4.4 The Degeneracy of the Landau Levels . . . . . . . . . . . . . . 51
4.5 The Hall Effect: “Electrons” and “Holes” . . . . . . . . . . . . 56
5 Magnetic Susceptibility 61
5.1 The Magnetogyric Ratio . . . . . . . . . . . . . . . . . . . . . 61
5.2 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . . . 67
6 Boltzmann Equation Method 75
6.1 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . 75
6.2 The Current Relaxation Rate . . . . . . . . . . . . . . . . . . 78
II Bloch Electron Dynamics 83
7 Bloch Theorem 85
7.1 The Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 The Kronig–Penney Model . . . . . . . . . . . . . . . . . . . . 91
8 The Fermi Liquid Model 97
8.1 The Self-consistent Field Approximation . . . . . . . . . . . . 97
8.2 Fermi Liquid Model . . . . . . . . . . . . . . . . . . . . . . . . 99
9 The Fermi Surface 103
9.1 Monovalent Metals (Na, Cu) . . . . . . . . . . . . . . . . . . . 103
9.2 Multivalent Metals . . . . . . . . . . . . . . . . . . . . . . . . 107
9.3 Electronic Heat Capacity and Density of States . . . . . . . . 111
10 Bloch Electron Dynamics 115
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.2 Newtonian Equations of Motion . . . . . . . . . . . . . . . . . 117
10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Contents xi
III Applications. Fermionic Systems (Electrons) 131
11 De Haas–Van Alphen Oscillations 133
11.1 Onsager’s Formula . . . . . . . . . . . . . . . . . . . . . . . . 133
11.2 Statistical Mechanical Calculations: 3D . . . . . . . . . . . . . 139
11.3 Statistical Mechanical Calculations: 2D . . . . . . . . . . . . . 142
11.4 Two-Dimensional Conductors . . . . . . . . . . . . . . . . . . 147
12 Magnetoresistance 151
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.2 Anisotropic Magnetoresistance in Cu . . . . . . . . . . . . . . 153
12.3 Shubnikov–De Haas Oscillations . . . . . . . . . . . . . . . . . 155
12.4 Heterojunction GaAs/AlGaAs . . . . . . . . . . . . . . . . . . 161
13 Cyclotron Resonance 171
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.2 Cyclotron Resonance in Ge and Si . . . . . . . . . . . . . . . . 172
13.3 Cyclotron Resonance in Al . . . . . . . . . . . . . . . . . . . . 184
13.4 Cyclotron Resonance in Pb. . . . . . . . . . . . . . . . . . . . 188
13.5 Cyclotron Resonance in Zn and Cd (HCP) . . . . . . . . . . . 192
14 Seebeck Coefficient (Thermopower) 195
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14.2 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
15 Infrared Hall Effect 205
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
15.2 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 209
15.2.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 209
15.2.2 Hall Coefficient . . . . . . . . . . . . . . . . . . . . . . 209
15.2.3 Hall Angle . . . . . . . . . . . . . . . . . . . . . . . . . 210
15.2.4 Dynamic Coefficients . . . . . . . . . . . . . . . . . . . 210
15.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Appendix A: Electromagnetic Potentials 217