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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Filling the gap for comprehensive coverage of the realistic fundamentals and approaches needed to perform cutting-edge research on mesoscopic systems, this textbook allows advanced students to acquire and use the skills at a highly technical, research-qualifying level.
Starting with a brief refresher to get all readers on an equal footing, the text moves on to a broad selection of advanced topics, backed by problems with solutions for use in classrooms as well as for self-study.
Written by authors with research and teaching backgrounds from eminent institutions and based on a tried-and-tested lecture, this is a must-have for researchers, research students and instructors involved with semiconductor junctions, nanostructures and thin film systems.
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Seitenzahl: 516
Veröffentlichungsjahr: 2013
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Contents
Preface
Part One Linear Response of Low Dimensional Quantum Systems
1 Introduction
1.1 Second-Quantized Representation for Electrons
1.2 Second Quantization and Fock States
1.3 The Boson Case
1.4 The Fermion Case
1.5 The Hamiltonian of Electrons
1.6 Electron–phonon Interaction
1.7 Effective Electron–electron Interaction
1.8 Degenerate Electron Gases
1.9 Ground-State Energy in the High Density Limit
1.10 Wigner Solid
1.11 The Chemical Potential of an Ideal Bose Gas and Bose-Einstein Condensation
1.12 Problems
References
2 The Kubo–Greenwood Linear Response Theory
2.1 Fluctuations and Dissipation
2.2 Nyquist’s Relation
2.3 Linear Response Theory
2.4 The Density Matrix and Quantum Statistics
2.5 Kubo’s Theory
2.6 The Kubo Equation
2.7 Fluctuation–Dissipation Theorem
2.8 Applications
2.9 Kinetic Equation for Elastic Processes
2.10 Problems
References
3 Feynman Diagrammatic Expansion
3.1 General Formalism
3.2 Functional Derivative Techniques
3.3 Unrenormalized Expansion for G and ∑
3.4 Renormalized Expansion for Self-Energy ∑
3.5 The Schrödinger Equation in the Hartree–Fock Approximation
3.6 Screened External Potential
3.7 Retarded Polarization Function
3.8 RPA for the Polarization Function
3.9 Problems
References
4 Plasmon Excitations in Mesoscopic Structures
4.1 Linear Response Theory and Collective Excitations
4.2 A Linear Array of Nanotubes
4.3 A Linear Array of Quantum Wires
4.4 Coupled Half-Plane Superlattices
4.5 Problems
References
5 The Surface Response Function, Energy Loss and Plasma Instability
5.1 Surface Response Function
5.2 Electron Energy Loss for a Planar Surface
5.3 Plasma Instability for a Planar Surface
5.4 Energy Transfer in Nanotubes
5.5 Problems
References
6 The Rashba Spin–orbit Interaction in 2DEG
6.1 Introduction to Spin–Orbit Coupling
6.2 Spin–orbit Coupling in the Dirac Equation
6.3 Rashba Spin–orbit Coupling for a Quantum Wire
6.4 SOI Effects on Conductance and Electron-Diffusion Thermoelectric Power
6.5 Problems
References
7 Electrical Conductivity: the Kubo and Landauer–Büttiker Formulas
7.1 Quantum Mechanical Current
7.2 The Statistical Current
7.3 A Green’s Function Formalism
7.4 The Static Limit
7.5 Model and Single-Particle Eigenstates
7.6 Averaged Conductivity
7.7 Applications to One-Dimensional Density Modulated 2DEG
7.8 Scattering Theory Formalism
7.9 Quantum Hall Effect
7.10 Problems
References
8 Nonlocal Conductivity for a Spin-Split Two-Dimensional Electron Liquid
8.1 Introduction
8.2 Kubo Formula for Conductivity
8.3 The Self-Energy and Scattering Time
8.4 Drude-Type Conductivity for Spin-Split Subband Model
8.5 Vertex Corrections to the Local Conductivity
8.6 Numerical Results for Scattering Times
8.7 Related Results in 3D in the Absence of SOI
References
9 Integer Quantum Hall Effect
9.1 Basic Principles of the Integer Quantum Hall Effect
9.2 Fundamental Theories of the IQHE
9.3 Corrections to the Quantization of the Hall Conductance
References
10 Fractional Quantum Hall Effect
10.1 The Laughlin Ground State
10.2 Elementary Excitations
10.3 The Ground State: Degeneracy and Ginzburg-Landau Theory
10.4 Problems
References
11 Quantized Adiabatic Charge Transport in 2D Electron Systems and Nanotubes
11.1 Introduction
11.2 Theory for Current Quantization
11.3 Tunneling Probability and Current Quantization for Interacting Two-Electron System
11.4 Adiabatic Charge Transport in Carbon Nanotubes
11.5 Summary and Remarks
References
12 Graphene
12.1 Introduction
12.2 Electronic Properties of Graphene
12.3 Graphene Nanoribbons and Their Spectrum
12.4 Valley-Valve Effect and Perfect Transmission in GNR’s
12.5 GNR’s Electronic and Transport Properties in External Fields
12.6 Problems
12.A Energy Eigen States
12.B The Conductance
References
13 Semiclassical Theory for Linear Transport of Electrons
13.1 Roughness Scattering
13.2 Phonon Scattering
13.3 Thermoelectric Power
13.4 Electron–electron Scattering
References
Part Two Nonlinear Response of Low Dimensional Quantum Systems
14 Theory for Nonlinear Electron Transport
14.1 Semiclassical Theory
14.2 Quantum Theory
References
15 Spontaneous and Stimulated Nonlinear Wave Mixing of Multi-excitons
15.1 Spontaneous, Stimulated, Coherent and Incoherent Nonlinear Wave Mixing
15.2 n + 1 Wave Mixing in QD Fluids and Polymer QDs Molecule Solutions
15.3 Application to Two-Photon-Induced Signals
15.A Semiclassical vs. Quantum Field Derivation of Heterodyne Detected Signals
15.B Generalized Susceptibility and Its CTPL Representation
References
16 Probing Excitons and Biexcitons in Coupled QDs by Coherent Optical Spectroscopy
16.1 Model Hamiltonian for Two Coupled Quantum Dots
16.2 Single-exciton Manifold and the Absorption Spectrum
16.3 Two-exciton Manifold and the 2D Spectra
16.4 Summary
16.A Transformation of the Electron–Hole Hamiltonian Using Excitonic Variables
16.B The Nonlinear Exciton Equations
16.C The 2D Signals
References
17 Non-thermal Distribution of Hot Electrons
17.1 Introduction
17.2 Boltzmann Scattering Equation
17.3 Numerical Results for Effective Electron Temperature
17.4 Summary
References
Index
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The Authors
Dr. Godfrey Gumbs
University of New York
Department of Physics
695, Park Avenue
New York, NY 10065
USA
Dr. Danhong Huang
USAF Research Lab (AFRL/RVSS)
Adv. E/O Space Sensors Group
3550, Aberdeen Ave, SE Bldg 426
Kirtland AFB, NM 87117
USA
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Preface
There has been a considerable amount of research on “mesoscopic” structures whose sizes are intermediate, that is, between the macroscopic and the atomic scale. These include semiconductor heterojunctions, quantum dots and wires as well as carbon nanotubes and atomic layers of graphene. One is unable to explain the properties of these systems simply in terms of a single-particle Schrödinger equation since many-body effects cannot be neglected. Therefore, there is a need to combine an introduction to some typical topics of interest and the methods and techniques needed to handle them in a single volume. This book tries to achieve that goal by carefully presenting a number of topics concerned with the optical response and transport properties of low-dimensional structures. This material is supplemented by a selection of problems at the end of each chapter to give the reader a chance to apply the ideas and techniques in a challenging manner. There are several excellent textbooks which already deal with electron–electron interaction effects. However, the material we cover supplements those publications by covering more recently studied topics, that is, especially in semiconductors.
Since the aim of the book is to be self-contained, we first present some background diagrammatic methods. This is based on standard field theoretic techniques. (See, for example, the early work in [1–17].) However, here we give several examples which have useful applications to the topics covered in later chapters. We evaluate the Green’s function expansion and the polarization function which is a necessary ingredient in our investigations employing the linear response formalism which can then be used in the study of the collective plasma excitations in quantum dots and wires, electron transport, light absorption, electron energy loss spectroscopy for nanotubes, graphene containing massless Dirac fermions or layered semiconductor structures, just to name a few examples which are covered in this book. We have developed some of the methods introduced here in collaboration with co-workers. By learning the formalism and getting introduced to the novel physical properties of low-dimensional systems, the reader should be able to understand scientific papers in condensed matter physics dealing with the effects arising from many-particle interactions.
This book is inspired by a collection of lectures which were given over the years at the Graduate Center of the City University of New York. This course is usually taken by graduate students who have had some exposure to basic quantum mechanics, statistical mechanics and introductory solid state physics at the undergraduate level. What makes this book different from previously published ones on the many-body theory of solids is that it presents a variety of current topics of interest in the field of mesoscopic systems and it also provides basic formal theory which is relevant to these systems but not available in previously published books. This makes it suitable for either a “Special Topics” course in solid state physics in which a few of the chapters may be selected, or as a textbook for an advanced solid state physics course in which the methodology is taught. In any event, problem solving would be an integral part of the course. A number of problems and the references have been given at the end of each chapter for students wanting to become more familiar with the topics and their background.
We would like to express our gratitude to Dr. Oleksiy Roslyak for his generous time in helping us write the chapters on graphene, nonlinear Green’s function theory as well as on excitons and biexcitons in quantum dots. This contribution as well as his helpful comments on the manuscript are gratefully acknowledged. Our thanks also to Dr. Paula Fekete, Dr. Tibab McNeish, Dr. Oleg Berman Dr. Antonios Balassis, Andrii Lurov, Hira Ghumman, and Alisa Dearth for their helpful comments and criticisms of the manuscript.
Hunter College of the City University of New York
AFRL, New Mexico, March 2011
Godfrey Gumbs
Danhong Huang
References
1 Martin, P.C. and Schwinger, J. (1959) Theory of Many-Particle Systems. Phys. Rev., 115, 1342.
2 Abraham, M. and Becker, R. (1949) Classical Theory of Magnetism, Hafner, NY.
3 Ambegaokar, V. and Baratoff, A. (1963) Tunneling Between Superconductors. Phys. Rev. Letts., 10, 486.
4 Anderson, P.W. (1964) Lectures on theMany-Body Problem, Vol. 2 (ed. E.R. Caianello), Academic Press.
5 Anderson, P.W. (1958) Random-Phase Approximation in the Theory of Superconductivity. Phys. Rev., 112, 1900.
6 Ashcroft, N., Mermin, D., and Bardeen, J. (1956) Theory of Superconductivity, in Handbuch der Phys., Vol. 15 (ed. S. Flugge), 274.
7 Bardeen, J. (1961) Tunnelling from a Many-Particle Point of View. Phys. Rev.Lett., 6, 57.
8 Bardeen, J. and Schrieffer, J.R. (1961) in Prog. in Low Temp. Phys. (ed. C.J. Gorter), North Holland, Vol. B, p. 170.
9 Bardeen, J., Cooper, L.N., and Schrieffer, J.R. (1957) Theory of Superconductivity. Phys. Rev., 108, 1175.
10 Blatt, J.M. and Butler, S.T. (1954) Superfluidity of a Boson Gas. Phys. Rev., 96, 1149.
11 Bogoliubov, N.N. (1958) On a New Method in the Theory of Superconductivity. Nuovo Cim., 7, 794.
12 Bogoliubov, N.N. (1959) Us. Fiz. Nauk,67, 549.
13 Cohen, M.H., Falicov, L.M., and Phillips, J.C. (1962) Superconductive Tunneling. Phys. Rev. Lett., 8, 316.
14 Ferrell, R.A. and Prange, R.E. (1963) Self-Field Limiting of Josephson Tunneling of Superconducting Electron Pairs. Phys. Rev. Lett,10, 479.
15 Feynman, R.P. (1972) Statistical Mechanics, W.A Benjamin, Reading, MA.
16 Fock, V. (1932) Konfigurationsraum und zweite Quantelung. Z. Phys., 75, 622.
17 Callaway, J. (1974) Quantum Theory of the Solid State, Ch. 7, 2nd edn, Academic Press Inc., San Diego.
Part One
Linear Response of Low Dimensional Quantum Systems
1
Introduction
1.1 Second-Quantized Representation for Electrons
The use of a Schrödinger equation to describe one or more electrons already treats the electron quantum mechanically and is sometimes referred to as first quantization. As long as electrons are neither created nor destroyed, such a description is complete. However, an electron that is transferred from state n to state m is often described as the destruction of an electron in state n and creation in state m by an operator obeying an algebra of the form . It is convenient therefore to further refine the algebra of such operators analogous to the operators and bq that create and destroy phonons of wave-vector q. However, electrons are fermions rather than bosons and the state occupancy number should only be permitted to take the values zero or one. This aim is achieved by using anti-commutation rules {described by braces} or by square brackets with a + subscript, that is, […]+, instead of commutation rules described by brackets or square brackets with a – subscript, that is, […]–.
In this book, we will be primarily concerned with low-dimensional systems such as quantum wells, dots and wires. A typical band structure of the valence and conduction bands for a heterostructure like GaAs/AlGaAs is shown in Figure 1.1. However, the formulation in this chapter and in some of the others is independent of dimensionality.
The phrase “second quantization” is descriptive of the notion that the Schrödinger wave function Ψ(r) is to be quantized, that is, treated as an operator. In terms of any complete set of states ϕk (r), we can write:
(1.1)
where the anti-commutation rules are given by
(1.2)
and
(1.3)
Figure 1.1 Valence (lower curve) and conduction (upper curve) bands of electrons in a semiconductor heterostructure.
For a single state, we can omit the subscripts and examine the consequences. Equation (1.3) implies that
(1.4)
(1.5)
Consider the two eigenstates of N:
(1.6)
Then, is c†Ψ0 also an eigenstate of N?
(1.7)
Therefore,
(1.8)
that is, c†Ψ0 is proportional to Ψ1. Evaluate the normalization:
(1.9)
Therefore, c†Ψ0 is normalized and we can simply choose
(1.10)
Similarly,
(1.11)
(1.12)
Note that
(1.13)
(1.14)
We begin by rewriting the Schrödinger equation in second quantized form. In most cases, the Hamiltonian has the form
(1.15)
(1.16)
The single-particle operator
(1.17)
in second quantized form becomes
(1.18)
(1.19)
where and are creation and annihilation operators, respectively.
1.2 Second Quantization and Fock States
For a system in which the number of particles is variable, it is essential to introduce creation and destruction operators. However, it is also possible to do so when the number of particles is conserved. In that case, of course, the perturbation operators will contain an equal number of creation and destruction operators. In that case, it is customary to describe the procedure as “second quantization”. First quantization replaces classical mechanical equations of motion, and second quantization replaces a Hamiltonian containing one-body forces, two-body forces, and so on by a Hamiltonian that is bilinear in creation and destruction operators, quadratic in creation and destruction operators, and so on. Nothing new is added, but the commutation rules of the creation and destruction operators make the bookkeeping of the states simpler than using permanents or determinants for the Schrödinger wave functions. For Bose particles, second quantization was developed by Dirac [1], and extended to Fermi particles by Wigner and Jordan [2]. A more detailed discussion is given by Fock [3] and by Landau and Lifshitz [4].
1.3 The Boson Case
(1.20)
We then introduce creation and destruction operators and Bj defined by
(1.21)
and
(1.22)
The principal simplification of second quantization is that a one-body operator
(1.23)
which can take Ψ(Ni – 1, Nk) into Ψ(Ni, Nk – 1) by “destroying” a particle in state k and creating one in state i in the Schrödinger permanent wave functions can be much more easily calculated when the operator
(1.24)
acts on the Fock states. In particular, the matrix element
(1.25)
where the first integral is over the Schrödinger space dxldx2 … dxN. The second one is thought of in terms of creation and destruction operators in a space described by the number set {Nj}, and the matrix element
(1.26)
is the usual one-body matrix element in the Schrödinger representation.
Landau and Lifshitz [4] do not derive this result. They merely state that “The calculation of these matrix elements is, in principle, very simple, it is easier to do it oneself than to follow an account of it.” It would be unfair to leave the matter there: Landau knows how to do it; let it be an exercise for the reader.
We can make the answer plausible by showing that the right-hand side of Eq. (1.25) is a product of four factors:
(1.27)
(1.28)
(1.29)
The matrix elements in Eqs. (1.21) and (1.22) are such as to insure the commutations rules
(1.30)
A natural generalization of Eq. (1.25) to two-body operators implies the replacement:
(1.31)
A compact statement of these commutation rules in Eq. (1.30) can be obtained by introducing an operator ψ(r) in the form
(1.32)
Then, the “Schrödinger operators”, ψ(r) and ψ†(r′), obey the commutation rules
(1.33)
(1.34)
where the δi j arises from Eq. (1.30) and the Dirac delta function follows from the completeness.
The second quantized Hamiltonian of a boson system with one and two-body forces can be written in the form
(1.35)
In addition to the correspondence
(1.36)
for one-body forces, we have a similar correspondence for two-body forces:
(1.37)
where
(1.38)
We note that the commutation rules, Eqs. (1.30), (1.33), and (1.34) for bosons are the same as the ones we are familiar with for harmonic oscillators and phonons, which are of course bosons.
1.4 The Fermion Case
In the fermion case, the Pauli principle requires that the wave function be antisymmetric. The simplest example of a set of independent fermions is then described by a determinant
(1.39)
in terms of the set of functions ϕi(r). The latter are usually taken as members of a complete set of eigen-functions of the one-body Hamiltonian. Here, N is the total number of eigen-functions appearing in the determinant, that is, the total number of occupied states. The set of numbers p1, p2,…, pN are some chosen ordering of the set {i}. To make the sign of the determinant unique, a fixed order must be chosen. It is conventional to choose the ordering
(1.40)
This is not necessary, but a fixed choice must be maintained in the ensuing discussion.
The result of second quantization for fermions will look similar to that for bosons in the sense that Eq. (1.32) is replaced by
(1.41)
(1.42)
rather than the commutation rules used in the boson case. In particular, Eqs. (1.33) and (1.34) are replaced by
(1.43)
(1.44)
which follows directly from Eq. (1.42).
To see that the anti-commutation rules, Eq. (1.42), accomplish the desired objectives, we first consider a single state ϕi(r) with operators Fi and , and omit the index i.
(1.45)
(1.46)
so that
(1.47)
Consider the eigenstates of N:
(1.48)
Then, is F†Ψ0 also an eigenstate of N?
(1.49)
Therefore,
(1.50)
that is, F†Ψ0 is proportional to Ψ1. Evaluate the normalization:
(1.51)
Therefore, F†Ψ0 is normalized and we can choose
(1.52)
Similarly,
(1.53)
(1.54)
Note that
(1.55)
(1.56)
The above discussion has established that the anti-commutation rules generate a set of states with occupancies zero and one. The full correspondence between first and second quantization requires that we establish the analogue of Eq. (1.36):
(1.57)
This involves the matrix element of the one-body operator V between two determinantal states. In effect, a transition in which a fermion in state k is destroyed and one in state i is created was found to have the matrix element
(1.58)
between determinantal states.
[4, Eq. (61.3)] allege (without proof) that the result in Eq. (1.58) should instead be
(1.59)
where the symbol
(1.60)
This discrepancy can be resolved as follows. In our calculation, we obtained the final wave function (before anti-symmetrization) from the initial wave function simply by replacing ϕk(ra) with ϕi(rb). However, this procedure does not preserve the chosen ordering, Eq. (1.40). To restore the chosen ordering, one must interchange row i and k in the final determinant. These gain a factor (−1)Σ where Σ is the number of occupied states between i and k.
(1.61)
and
(1.62)
(1.63)
then acquires just the extra factor (–1)Σ(i+1,k–1) demanded by Eq. (1.59). Moreover, it is easy to see why
(1.64)
because one of m, n (say m) is higher in the sequence of states. Then, the matrix for Fm in Fm Fn is reversed in sign because Fn has acted and eliminated the state n below m. In the reverse order, Fn Fm, Fn is unaffected by the elimination of state m above it. Hence, the two orders differ by a factor –1 to yield the desired anticommutation rule. These remarks are stated clearly in [5].
1.5 The Hamiltonian of Electrons
We first consider the case of a single electron, or of a set of non-interacting electrons. The Hamiltonian can be written in the form:
(1.65)
Here, ψ(r) and ψ†(r) are regarded as operators that can be expanded in an arbitrary orthonormal set ϕn(r):
(1.66)
(1.67)
The Hamiltonian then takes the form
(1.68)
where represents the matrix for destruction of electrons in n and its creation in m. If the original Schrödinger equation is
(1.69)
where m* represents the mass of an electron, then U is the operator defined by
(1.70)
whose matrix element is
(1.71)
To best understand the eigenstates of the operator
(1.72)
we can choose the ϕn to be the eigenstates of U with eigenvalues En. Then,
(1.73)
(1.74)
(1.75)
1.6 Electron–Phonon Interaction
Following Callaway [6] the Hamiltonian is written as an electron energy, plus a phonon energy, plus an electron–phonon interaction:
(1.76)
Quasi-momentum conservation is built into the above expression and σ is the index for electron spin. The original form for phonon absorption was
(1.77)
(1.78)
(1.79)
which is a representation of two-body interactions in second quantized form.
1.7 Effective Electron–Electron Interaction
In an electromagnetic field, the charge-1 acts on the field and the field acts on charge-2. If we can eliminate the field, we obtain a direct interaction between charge-1 and charge-2. Here, the field is the phonon field. After we eliminate the electron–phonon interaction, we should obtain an effective electron–electron interaction.
Let
(1.80)
where H1 is the interaction Hamiltonian. The transformed Hamiltonian
(1.81)
to second order of S. To dispose of H1 to lowest order, we set
(1.82)
Then,
(1.83)
Let | m〉 and | n〉 be energy eigenstates of the complete system of electrons and phonons. Then, we get
(1.84)
However,
(1.85)
Writing S as operators cs (electronic part) and as a matrix in the vibrational part, we obtain
(1.86)
Here, a phonon of wave vector q is absorbed and an electron is scattered from k to k + q at the same time, that is,
(1.87)
In Eq. (1.87), a phonon of wave vector q is created and an electron is scattered from k to k − q at the same time.
We are concerned with the effective second-order interaction
and by this we mean the part diagonal in the phonon numbers. (The off-diagonal elements can be transformed away to give still higher-order interactions). We can write
(1.88)
where the intermediate states |nq ± 1〉 are summed over. We write out one term explicitly:
(1.89)
(1.90)
With the replacement k → k + q, we get
(1.91)
Note that the effective Hamiltonian is independent of temperature. For E(k)−E(k − q) < ħωq, the interaction becomes attractive.
An alternate derivation of the above H1 is obtained in a semiclassical way by Rickayzen [7, p. 117–121], by considering an electron fluid and an ion fluid and retarded interactions between the two components.
We will be dealing with systems of many interacting particles and, as a result, we need to include the inter-particle potential in the Schrödinger equation. This problem is the basis of the present book. The N-particle wave function in configuration space contains all the possible information. However, a direct solution of the Schrödinger equation is not practical. We therefore need other techniques which involve (a) second quantization, (b) quantum field theory, and (c) Green’s functions.
Second quantization describes the creation and annihilation of particles and quantum statistics as well as simplifying the problem of many interacting particles. This approach reformulates the Schrödinger equation. The advantage it has is that we avoid the awkward use of symmetrized and anti-symmetrized product of single-particle wave functions. With the method of quantum field theory, we avoid dealing with the wave functions and thus the coordinates of all the particles – bosons and fermions.
Green’s functions can be used to calculate many physical quantities such as (1) the ground state energy, (2) thermodynamic functions, (3) the energy and lifetime of excited states, and (4) linear response to external perturbations. The exact Green’s functions are also difficult to calculate which means we must use perturbation theory. This is presented with the use of Feynman diagrams. This approach allows us to calculate physical quantities to any order of perturbation theory. We use functional derivative techniques in the perturbation expansion of the Green’s function determined by the Dyson equation and show that only linked diagrams contribute. Wick’s theorem which forms all possible pairs of the field operators is not used in this approach.
1.8 Degenerate Electron Gases
We now illustrate the usefulness of the second quantization representation by applying it to obtain some qualitative results for a metal. The simple model we use is that of an interacting electron gas with a uniform positive background so that the total system is neutral. We ignore the motion of the ions/positive charge. We do not consider any surface effects by restricting our attention to the bulk medium. We insert the system into a large box of side L and apply periodic boundary conditions; this ensures invariance under spatial translations of all physical quantities. The single-particle states are plane waves
(1.92)
(1.93)
where
(1.94)
(1.95)
(1.96)
(1.97)
(1.98)
Therefore, the total Hamiltonian is
(1.99)
Forming a linear combination of the creation and destruction operators as
(1.100)
we rewrite He in second quantized form and the total Hamiltonian is
(1.101)
(1.102)
(1.103)
where is the number operator. The ground state expectation value of Eq. (1.103) is
(1.104)
where the first term in Eq. (1.104) cancels the first term of the Hamiltonian in Eq. (1.99) and the second term in Eq. (1.104) gives −(e2/2)(4π/VK2) as an energy per particle. This second term vanishes when the thermodynamic limit is taken first. Therefore, the Hamiltonian for a bulk electron gas in a uniform positive background is
(1.105)
1.9 Ground-State Energy in the High Density Limit
(1.106)
Therefore,
(1.107)
where a, b and c are numerical constants. As a matter of fact, the “a” term corresponds to the ground state energy E(0) of a free Fermi gas, the “b” term gives the first-order energy shift E(1).
(1.108)
(1.109)
where |g〉 is the ground state for non-interacting electrons. The states (k, λ1) and (p, λ2) must be occupied, and the states (k + q, λ1) and (p − q, λ2) must also be occupied. Therefore, we must have either
(1.110)
The choice given as (a) is forbidden since q ≠ 0 and the matrix element in Eq. (1.109) is
(1.111)
so that
(1.112)
Thus, by combining the results for E(0) and E(1), we obtain the energy per particle in the limit as rs → 0 to be given by
(1.113)
The exchange term is not the total that arises from the electron–electron interaction. All that is left out is called the correlation energy. The leading contribution to the correlation energy of the degenerate electron gas will be obtained using Feynman graph techniques. However, we note that EGS/N has a minimum at a negative value of the energy, that is, the system is bound, as shown in Figure 1.2. The Rayleigh–Ritz variational principle tells us that the exact ground state energy of a quantum mechanical system always has a lower energy than that evaluated using a normalized state for the expectation value of the Hamiltonian. The exact solution must also be that for a bound system with energy below our approximate solution and the binding energy is that of vaporization for metals.
1.10 Wigner Solid
The energy of the Fermi gas can be lowered if the electrons crystallize into a Wigner solid. The range of values of rs for metals is 1.8 rs 6.0. At low densities, Wigner suggested that the electrons will become localized and form a regular lattice. This lattice could be a closed packed structure such as bcc, fcc or hcp. The electrons would vibrate around their equilibrium positions and the positive charge is still spread out in the system. The vibrational modes of the electrons would be at the plasmon frequency. For large rs, the potential energy is much larger than the kinetic energy and there could be localization. In our discussion, the unit cell is taken as a sphere of radius rsa0 with the electron at the center. The total charge within the sphere is zero. Outside each sphere, the electric field is zero and consequently the spheres do not exert any electric fields on each other.
Figure 1.2 The energy per particle as a function of a dimensionless density parameter rs, where rs → 0 corresponds to the high density limit, while rs → ∞ corresponds to the low density limit.
The potential energy between the electron and the uniform positive background is
(1.114)
The potential energy due to the interaction of the positive charge with itself is obtained as follows. Let V(r) be the potential energy from the positive charge at distance r from the center. The electric field is E(r) where
(1.115)
Integrating to obtain V(r) gives a constant of integration. This is obtained by observing that the total potential from the electron and positive charge must vanish on the surface of the sphere and we obtain
(1.116)
The interaction of the positive charge with itself is found by using
(1.117)
Therefore, the total potential energy for the Wigner lattice in the Wigner–Seitz approximation is
(1.118)
This is larger than the exchange contribution for the free particle system. This system has gained energy by the localization of the electrons. Stroll has calculated the actual energy for several lattices. His results, expressed as −A/rs, in unit of (e2/2a0) are given as follows:
Lattice
A
sc
1.76
fcc
1.79175
bcc
1.79186
hcp
1.79168
1.11 The Chemical Potential of an Ideal Bose Gas and Bose–Einstein Condensation
(1.119)
(1.120)
then we get
(1.121)
which is the result of the Boltzmann distribution, where is used. Solving this equation for μ we obtain
(1.122)
A plot of this classical result is shown in Figure 1.3.
(1.123)
(1.124)
1.12 Problems
References
1 Dirac, P.A.M. (1927) The Quantum Theory of the Emission and Absorption of Radiation, Proc. R. Soc. Lond. A, 114, 243.
2 Jordan, P. and Wigner, E. (1928) Über Paulisches Äquivalenzverbot, Z. Phys., 47, 631.
3 Fock, V. (1932) Konfigurationsraum und zweite Quantelung, Z. Phys., 75, 622.
4 Landau, L.D. and Lifshitz, E.M. (1958) Statistical Physics, Pergamon Press, London.
5 Wentzel, G. (1949) Quantum Theory of Fields, Interscience, New York, Chap. 1.
6 Callaway, J. (1974) Quantum Theory of the Solid State, 2nd edn, Academic Press Inc., San Diego, Chap. 7.
7 Rickayzen, G. (1965) Theory of Superconductivity, 1st edn, John Wiley & Sons, Inc., New York.
2
The Kubo–Greenwood Linear Response Theory
In this chapter, we present the linear response theory of Kubo and Greenwood. It lays the foundations for calculating the electrical transport coefficients of conductivity and mobility as well as the density–density response function and its relationship to fluctuations in the system. This formalism is adopted from the pioneering papers listed in the references which is not specific to any dimensionality [1–14].
2.1 Fluctuations and Dissipation
Let us consider a gas or liquid which contains a density ni (z) of impurities, for example, ions. There will be a net flow current of ions in the condensed matter if either (a) a density gradient exists or (b) there is an external force, such as, an electric field, acting on the impurities. The current due to a density gradient in terms of the diffusion coefficient D is
(2.1)
while the external force F will produce a net drift velocity μF, where μ is the mobility and consequently a net current
(2.2)
Suppose now that both a density gradient and an external field exist, and act against each other so that there is no net flow of the impurities. In this case,
(2.3)
or
(2.4)
(2.5)
From this, we obtain
(2.6)
This relation was derived by Nernst [1] and independently by Einstein [2]. It is the earliest form of the fluctuation–dissipation theorem, giving a link between the diffusive and the drift motion of the impurities. The impurities diffuse because of the fluctuations in the forces exerted on them by an external force. This drift is dissipative, generating Joule heat by doing work against the internal forces. The diffusive motion is not.
2.2 Nyquist’s Relation
In 1928, Johnson [3] discovered the existence of a noise voltage in conductors, and attributed it to thermal motion of the electrons. The spectrum of the fluctuating voltage was related by Nyquist [4–6] to the frequency-dependent impedance of the conductor as well as the temperature. The argument presented by Nyquist is as follows.
Let us consider two conductors with resistance R in series and at the same temperature T. The voltage Va due to charge fluctuations in Ra makes a current V/2R flow. This current transfers power from “a” to “b”. In the same way, the charge fluctuations in Rb transfer power from “b” to “a”. Since Rb and Ra are at the same temperature, the average power flowing in one direction must be exactly equal to the average power flowing in the other direction, according to the Second Law of thermodynamics. This equality must hold not only for total power exchange, but also for the powers exchanged in any given frequency range. For, if there were a frequency range in which Ra delivered more power than it received, then by connecting a non-dissipative network (shown below), the Second Law could be violated. It follows that the voltage developed due to current fluctuations is a universal function of the temperature, resistance and frequency, and only of these variables. Since it is a universal function, it may be determined by solving any particular model. Nyquist took the following: Two resistors R connected by a long loss-less line of length l, with inductance and capacitance per unit length, chosen so that the characteristic impedance of the line is equal to R. The line is then matched at each end, so that all energy traveling down the line will be absorbed without reflection. The number of waves in wave number range dk is ldk/2π, each with energy εk. The average rate of flow of energy in each direction in the wave number range dk is therefore
(2.7)
However, this must be equal to
(2.8)
Therefore, we have
(2.9)
The mean energy of the mode of frequency ω at temperature T is
(2.10)
This has the high-temperature limit kBT, and Nyquist’s relation becomes
(2.11)
Equations (2.9) and (2.11) are another form of the fluctuation–dissipation theorem, relating the voltage fluctuations at frequency ω to the dissipative properties given by Rf at the same frequency.
2.3 Linear Response Theory
2.3.1 Generalized Susceptibility
There is an important class of physical problems where the effect of an external force F(t) is to change the Hamiltonian of the system by an operator of the form
(2.12)
and where, furthermore, the steady state of the system after the force is switched on is close enough to equilibrium that it is sufficient to consider terms linear in F in calculating the expectation value of the generalized displacement operator X. The most general way of describing the linear response of the system, consistent with the principle of causality, is to write
(2.13)
where the response function φ depends on the temperature, and the properties of the system such as spatial variables. We note that Eq. (2.13) is written in such a way that x(t) depends on values of F at times less than t, that is, the response is causal.
It is useful to resolve both the force and the displacement into frequency components
(2.14)
and, we obtain
(2.15)
Since the response is linear, different frequency components will not mix and we obtain
(2.16)
where
(2.17)
(2.18)
Thus, the real and imaginary parts of χ(ω) are respectively even and odd functions of frequency. These relations express the fact that X must be real for any real F(t). For example, take the real force
(2.19)
Then,
(2.20)
2.3.2 Kronig–Kramers Relations
Now, consider the integral
(2.21)
where C is the contour shown in Figure 2.1.
Since χ(ω) has no singularities in the upper half-plane, and the point Ω
