Patterson, Bailey - Solid State Physics Introduction to theory

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which are turned on at t = 0, we obtain for transitions between discrete states

In the text, we have loosely referred to (E.5), (E.7), or (E.9) as the Golden rule (according to which is appropriate to the physical situation).

F Derivation of The Spin-Orbit Term From Dirac’s Equation

In this appendix we will indicate how the concepts of spin and spin-orbit interaction are introduced by use of Dirac’s relativistic theory of the electron. For further details, any good quantum mechanics text such as that of Merzbacher7, or Schiff8 can be consulted. We will discuss Dirac’s equation only for fields described by a potential V. For this situation, Dirac’s equation can be written

In (F.1), c is the speed of light, α and β are 4 × 4 matrices defined below, p is the momentum operator, m0 is the rest mass of the electron, ψ is a four-component column matrix (each element of this matrix may be a function of the spatial position of the electron), and E is the total energy of the electron (including the rest mass energy that is m0c2). The α matrices are defined by

where the three components of σ are the 2 × 2 Pauli spin matrices. The definition of β is

where I is a 2 × 2 unit matrix.

For solid-state purposes we are not concerned with the fully relativistic equation (F.1), but rather we are concerned with the relativistic corrections that (F.1) predicts should be made to the nonrelativistic Schrödinger equation. That is, we want to consider the Dirac equation for the electron in the small velocity limit. More precisely, we will consider the limit of (F.1) when

7See Merzbacher [A.15 Chap. 23].

8See Schiff [A.23].

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and we want results that are valid to first order in ε, i.e. first-order corrections to the completely nonrelativistic limit. To do this, it is convenient to make the following definitions:

where both χ and φ are two-component wave functions.

If we substitute (F.5) and (F.6) into (F.1), we obtain an equation for both χ and φ. We can combine these two equations into a single equation for χ in which φ does not appear. We can then use the small velocity limit (F.4) together with several properties of the Pauli spin matrices to obtain the Schrödinger equation with relativistic corrections

This is the form that is appropriate to use in solid-state physics calculations. The term

is called the spin-orbit term. This term is often used by itself as a first-order correction to the nonrelativistic Schrödinger equation. The spin-orbit correction is often applied in band-structure calculations at certain points in the Brillouin zone where bands come together. In the case in which the potential is spherically symmetric (which is important for atomic potentials but not crystalline potentials), the spin-orbit term can be cast into the more familiar form

where L is the orbital angular momentum operator and S is the spin operator (in units of ).

It is also interesting to see how Dirac’s theory works out in the (completely) nonrelativistic limit when an external magnetic field B is present. In this case the magnetic moment of the electron is introduced by the term involving S · B. This term automatically appears from the nonrelativistic limit of Dirac’s equation. In addition, the correct ratio of magnetic moment to spin angular momentum is obtained in this way.

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GThe Second Quantization Notation for Fermions and Bosons

When the second quantization notation is used in a nonrelativistic context it is simply a notation in which we express the wave functions in occupation-number space and the operators as operators on occupation number space. It is of course of great utility in considering the many-body problem. In this formalism, the symmetry or antisymmetry of the wave functions is automatically built into the formalism. In relativistic physics, annihilation and creation operators (which are the basic operators of the second quantization notation) have physical meaning. However, we will apply the second quantization notation only in nonrelativistic situations. No derivations will be made in this section. (The appropriate results will just be concisely written down.) There are many good treatments of the second quantization or occupation number formalism. One of the most accessible is by Mattuck.9

G.1 Bose Particles

For Bose particles we deal with bi and bi† operators (or other letters where convenient): bi† creates a Bose particle in the state i; bi annihilates a Bose particle in the state f. The bi operators obey the following commutation relations:

[bi ,bj ] ≡ bib j −bjbi = 0, [bi†,b†j ] = 0,

[bi ,b†j ] =δij .

The occupation number operator whose eigenvalues are the number of particles in state i is

ni = bi†bi ,

and

ni +1 = bi bi† .

The effect of these operators acting on different occupation number kets is

where |n1,…,ni,… means the ket appropriate to the state with n1 particles in state 1, n2 particles in state 2, and so on.

9 See Mattuck [A.14].

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The occupation number operator whose eigenvalues are the number of particles in state i is

ni = ai†ai ,

and

1− ni = ai ai† .

Note that (ni)2 = ni, so that the only possible eigenvalues of ni are 0 and 1 (the Pauli principle is built in!).

The matrix elements of these operators are defined by

where ∑(1,i − 1) equals the sum of the occupation numbers of the states from 1 to i − 1.

In this notation, any one-particle operator can be written in the form

f0(1) = ∑i, j i f (1) j ai†a j ,

where the |j are any complete set of one-particle eigenstates. In a similar fashion, any two-particle operator can be written in the form

fop(2) = ∑i, j,k,l i(1) j(2) f (2) k(1)l(2) a†j ai†ak al .

Operators that create or destroy Fermi particles at a given point in space (rather than in a given state) are given by

ψ(r) = ∑α uα (r)aα ,

where uα(r) is the single-particle wave function corresponding to state α, and ψ† (r) = ∑α uα (r)aα† .

These operators obey the anticommutation relations

{ψ(r),ψ† (r)} = δ (r − r′) .

The operators also allow a convenient way of writing Slater determinants, e.g.,

|0 is known as the vacuum ket.

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as a generalization of the Thomas–Fermi approximation. Then came the famous Kohn–Sham equations that reduced the Hohenberg–Kohn formulation to the problem of noninteracting electrons in an effective potential (somewhat analogous to the Hartree equations, for example). However, part of the potential, the exchange correlation part could only be approximately evaluated, e.g. in the local density approximation (LDA) – which assumed a locally homogeneous electron gas. A problem with DFT-LDA is that it is not necessarily clear what the size of the errors are, however, the DFT is certainly a good way to calculate, ab initio, certain ground-state properties of finite electronic systems, such as the ionization energies of atoms. It is also very useful for computing the electronic ground-state properties of periodic solids, such as cohesion and stability. Excited states, as well as approximations for the exchange correlation term in N-electron systems continue to give problems. For a nice brief summary of DFT see Mattsson [A.13].

For quantum electrodynamics, a brief and useful graphical summary can be found at: http://www2slac.standford.edu/vvc/theory/feynman.html. We now present a brief summary of the use of diagrams in many-body physics.

In some ways, trying to do solid-state physics without Feynman diagrams is a little like doing electricity and magnetism (EM) without resorting to drawing Faraday’s lines of electric and magnetic fields. However, just as field lines have limitations in describing EM interactions, so do diagrams for discussing the many-body problem [A.1]. The use of diagrams can certainly augment one’s understanding.

The distinction between quasior dressed particles and collective excitations is important and perhaps is made clearer from a diagrammatic point of view. Both are ‘particles’ and are also elementary energy excitations. But after all a polaron (a quasi-particle) is not the same kind of beast as a magnon (a collective excitation). Not everybody makes this distinction. Some call all ‘particles’ quasiparticles. Bogolons are particles of another type, as are excitons (see below for definitions of both). All are elementary excitations and particles, but not really collective excitations or dressed particles in the usual sense.

H.1 Propagators

These are the basic quantities. Their representation is given in the next section. The single-particle propagator is a sum of probability amplitudes for all the ways of going from r1, t1 to r2, t2 (adding a particle at 1 and taking out at 2).

The two-particle propagator is the sum of the probability amplitudes for all the ways two particles can enter a system, undergo interactions and emerge again.

H.2 Green Functions

Propagators are represented by Green functions. There are both advanced and retarded propagators. Advanced propagators can describe particles traveling backward in time, i.e. holes. The use of Fourier transforms of time-dependent propaga-

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Phonon – Quanta of normal mode vibration of a lattice of ions. Also often represented by wavy line.

Magnon – Quanta of low-energy collective excitations in the spins, or quanta of waves in the spins.

Plasmon – Quanta of energy excitation in the density of electrons in an interacting electron gas (viewing, e.g., the positive ions as a uniform background of charge).

Other Elementary Energy Excitations – Excited energy levels of many-particle systems.

Bogolon – Linear combinations of electrons in a state +k with ‘up’ spin and −k with ‘down’ spin. Elementary excitations in a superconductor.

Exciton – Bound electron–hole pairs.

Some examples of interactions represented by vertices (time going to the right):

An electron emitting a phonon

A hole emitting a phonon.

Diagrams are built out of vertices with conservation of momentum satisfied at the vertices. For example

represents a coulomb interaction with time going up.

H.5 Diagrams and the Hartree and Hartree–Fock Approximations

In order to make these concepts clearer it is perhaps better to discuss an example that we have already worked out without diagrams. Here, starting from the Hamiltonian we will discuss briefly how to construct diagrams, then explain how to associate single-particle Green functions with the diagrams and how to do the partial sums representing these approximations. For details, the references must be consulted.

In the second quantization notation, a Hamiltonian for interacting electrons