Patterson, Bailey - Solid State Physics Introduction to theory

134 3 Electrons in Periodic Potentials

of the charge density of electrons (with the same spin state as a given electron) will be made for this model. This charge density will be found as a function of the distance from the given electron. If we have two free electrons with the same spin in states k and k′, the spatial wave function is

ψ	k,k′	(r , r	) = 1	eik r1		eik r2 .		(3.68)
		1 2	2V 2	e	ik′r1	e	ik′r2

By quantum mechanics, the probability P(r1,r2) that rl lies in the volume element drl, and r2 lies in the volume element dr2 is

P(r , r	)d3r d3r =	|ψ		k,k′		(r , r	) |2	d3r d3r
1 2	1 2					1 2		1 2				(3.69)
	=	1			{1− cos[(k			′ − k) (r	− r )]}d3r d3r .
			V2					1	2	1	2

The last term in (3.69) is obtained by using (3.68) and a little manipulation.

If we now assume that there are N electrons (half with spin 1/2 and half with spin −1/2), then there are (N/2)(N/2 − 1) N 2/4 pairs with parallel spins. Averaging over all pairs, we have for the average probability of parallel spin electron at rl and r2

(r1, r2 )d3r1d3r2 =

4

∑k,k′ ∫∫{1− cos[(k′ − k) (r1 − r2 )]}d3r1d3r2 ,

P

V2N2

and after considerable manipulation we can recast this into the form

4

1

2

4π

3

2

P (r1, r2 ) =

kM

3

N2 8π3

sin(k

r ) − k

r

cos(k

r

)

2

M

(3.70)

× 1−9

M

12

M 12

12

k3 r3

M 12

≡

1

ρ(k

r

).

V2

M 12

If there were no exchange (i.e. if we use a simple product wave function rather than a determinantal wave function), then ρ would be 1 everywhere. This means that parallel spin electrons would have no tendency to avoid each other. But as Fig. 3.1 shows, exchange tends to “correlate” the motion of parallel spin electrons in such a way that they tend to not come too close. This is, of course, just an example of the Pauli principle applied to a particular situation. This result should be compared to the Fermi hole concept introduced in a previous section. These oscillations are related to the Rudermann–Kittel oscillations of Sect. 7.2.1 and the Friedel oscillations mentioned in Sect. 9.5.3.

In later sections, the Hartree approximation on a free-electron gas with a uniform positive background charge will be used. It is surprising how many

3.1 Reduction to One-Electron Problem	135

experiments can be interpreted with this model. The main use that is made of this model is in estimating a density of states of electrons. (We will see how to do this in the Section on the specific heat of an electron gas.) Since the final results usually depend only on an integral over the density of states, we can begin to see why this model does not introduce such serious errors. More comments need to be made about the progress in understanding Coulomb correlations. These comments are made in the next section.

ρ(ε)

1.0

ε = kMr12

π/2

Fig. 3.1. Sketch of density of electrons within a distance r12 of a parallel spin electron

3.1.4 Coulomb Correlations and the Many-Electron Problem (A)

We often assume that the Coulomb interactions of electrons (and hence Coulomb correlations) can be neglected. The Coulomb force between electrons (especially at metallic densities) is not a weak force. However, many phenomena (such as Pauli paramagnetism and thermionic emission, which we will discuss later) can be fairly well explained by theories that ignore Coulomb correlations.

This apparent contradiction is explained by admitting that the electrons do interact strongly. We believe that the strongly interacting electrons in a metal form a (normal) Fermi liquid.6 The elementary energy excitations in the Fermi liquid are called Landau7 quasiparticles or quasielectrons. For every electron there is a quasielectron. The Landau theory of the Fermi liquid is discussed a little more in Sect. 4.1.

Not all quasielectrons are important. Only those that are near the Fermi level in energy are detected in most experiments. This is fortunate because it is only these quasielectrons that have fairly long lifetimes.

6A normal Fermi liquid can be thought to evolve adiabatically from a Fermi liquid in which the electrons do not interact and in which there is a 1 to 1 correspondence between noninteracting electrons and the quasiparticles. This excludes the formation of “bound” states as in superconductivity (Chap. 8).

7See Landau [3.31].

136 3 Electrons in Periodic Potentials

We may think of the quasielectrons as being weakly interacting. Thus our discussion of the N-electron problem in terms of N one-electron problems is approximately valid if we realize we are talking about quasielectrons and not electrons.

Further work on interacting electron systems has been done by Bohm, Pines, and others. Their calculations show two types of fundamental energy excitations: quasielectrons and plasmons.8 The plasmons are collective energy excitations somewhat like a wave in the electron “sea.” Since plasmons require many electron volts of energy for their creation, we may often ignore them. This leaves us with the quasielectrons that interact by shielded Coulomb forces and so interact weakly. Again we see why a free-electron picture of an interacting electron system has some validity.

Np

Np

			p						p
		pF					pF
		(a)				(b)
Fig. 3.2. The Fermi distribution at absolute zero						(a) with no interactions, and
(b) with interactions (sketched)

We should also mention that Kohn, Luttinger, and others have indicated that electron–electron interactions may change (slightly) the Fermi–Dirac distribution.8 Their results indicate that the interactions introduce a tail in the Fermi distribution as sketched in Fig. 3.2. Np is the probability per state for an electron to be in a state with momentum p. Even with interactions there is a discontinuity in the slope of Np at the Fermi momentum. However, we expect for all calculations in this book that we can use the Fermi–Dirac distribution without corrections and still achieve little error.

The study of many-electron systems is fundamental to solid-state physics. Much research remains to be done in this area. Further related comments are made in Sect. 3.2.2 and in Sect. 4.4.

8 See Pines [3.41].

3.1 Reduction to One-Electron Problem	137

3.1.5 Density Functional Approximation9 (A)

We have discussed the Hartree–Fock method in detail, but, of course, it has its difficulties. For example, a true, self-consistent Hartree–Fock approximation is very complex, and the correlations between electrons due to Coulomb repulsions are not properly treated. The density functional approximation provides another starting point for treating many-body systems, and it provides a better way of teaching electron correlations, at least for ground-state properties. One can regard the density functional method as a generalization of the much older Thomas– Fermi method discussed in Sect. 9.5.2. Sometimes density functional theory is said to be a part of The Standard Model for periodic solids [3.27].

There are really two parts to density functional theory (DFT). The first part, upon which the whole theory is based, derives from a basic theorem of P. Hohenberg and W. Kohn. This theorem reduces the solution of the many body ground state to the solution of a one-particle Schrödinger-like equation for the electron density. The electron density contains all needed information. In principle, this equation contains the Hartree potential, exchange and correlation.

In practice, an approximation is needed to make a problem treatable. This is the second part. The most common approximation is known as the local density approximation (LDA). The approximation involves treating the effective potential at a point as depending on the electron density in the same way as it would be for jellium (an electron gas neutralized by a uniform background charge). The approach can also be regarded as a generalization of the Thomas–Fermi–Dirac method.

The density functional method has met with considerable success for calculating the binding energies, lattice parameters, and bulk moduli of metals. It has been applied to a variety of other systems, including atoms, molecules, semiconductors, insulators, surfaces, and defects. It has also been used for certain properties of itinerant electron magnetism. Predicted energy gap energies in semiconductors and insulators can be too small, and the DFT has difficulty predicting excitation energies. DFT-LDA also has difficulty in predicting the ground states of open-shell, 3d, transition element atoms. In 1998, Walter Kohn was awarded a Nobel prize in chemistry for his central role in developing the density functional method [3.27].

Hohenberg–Kohn Theorem (HK Theorem) (A)

As the previous discussion indicates, the most important difficulty associated with the Hartree–Fock approximation is that electrons with opposite spin are left uncorrelated. However, it does provide a rational self-consistent calculation that is more or less practical, and it does clearly indicate the exchange effect. It is a useful starting point for improved calculations. In one sense, density functional theory can be regarded as a modern improved and generalized Hartree–Fock calculation, at least for ground-state properties. This is discussed below.

9 See Kohn [3.27] and Callaway and March [3.8].

138 3 Electrons in Periodic Potentials

We start by deriving the basic theorem for DFT for N identical spinless fermions with a nondegenerate ground state. This theorem is: The ground-state energy E0 is a unique functional of the electron density n(r), i.e. E0 = E0[n(r)]. Further, E0[n(r)] has a minimum value for n(r) having its correct value. In all variables, n is constrained, so N = ∫n(r)dr.

In deriving this theorem, the concept of an external (local) field with a local external potential plays an important role. We will basically show that the external potential v(r), and thus, all properties of the many-electron systems will be determined by the ground-state electron distribution function n(r). Let φ = φ0(r1,r2,…rN) be the normalized wave function for the nondegenerate ground state. The electron density can then be calculated from

n(r1) = N ∫ϕ0ϕ0 dr2 …drn ,

where dri = dxidyidzi. Assuming the same potential for each electron υ(r), the potential energy of all electrons in the external field is

V (r1 …rN ) = ∑iN=1υ(ri ) .

(3.71)

The proof of the theorem starts by showing that n(r) determines υ(r), (up to an additive constant, of course, changing the overall potential by a constant amount does not affect the ground state). More technically, we say that υ(r) is a unique functional of n(r). We prove this by a reductio ad absurdum argument.

We suppose υ′ determines the Hamiltonian H′ and hence the ground state φ′0, similarly, υ determines H and hence, φ0. We further assume υ′ ≠ υ but the groundstate wave functions have n′ = n. By the variational principle for nondegenerate ground states (the proof can be generalized for degenerate ground states):

E0′ < ∫ϕ0H ϕ′ 0dτ ,	(3.72)
where dτ = dr1…drN, so
E0′ < ∫ϕ0 (H −V +V ′)ϕ0dτ ,
or
E0′ < E0 + ∫ϕ0 (V ′ −V )ϕ0dτ
< E0 + ∑iN=1 ∫ϕ0 (1…N )[υ′(ri ) −υ(ri )]ϕ0 (1…N )dτ ,	(3.73)
< E0 + N ∫ϕ0 (1…N )[υ′(ri ) −υ(ri )]ϕ0 (1…N )dτ

by the symmetry of |φ0|2 under exchange of electrons. Thus, using the definitions of n(r), we can write

E0′ < E0 + N ∫[υ′(ri ) −υ(ri )](∫ϕ0 (1…N )ϕ0 (1…N ) dr2 …drN )dr1 ,

3.1 Reduction to One-Electron Problem	139
or
E0′ < E0 + ∫ n(r1)[υ′(r1) −υ(r1)]dr1 .	(3.74)

Now, n(r) is assumed to be the same for υ and υ′, so interchanging the primed and unprimed terms leads to

E0 < E0′ + ∫ n(r1)[υ(r1) −υ′(r1)]dr1 .	(3.75)
Adding the last two results, we find
E0 + E0′ < E0′ + E0 ,	(3.76)

which is, of course, a contradiction. Thus, our original assumption that n and n′ are the same must be false. Thus υ(r) is a unique functional (up to an additive constant) of n(r).

Let the Hamiltonian for all the electrons be represented by H. This Hamiltonian will include the total kinetic energy T, the total interaction energy U between electrons, and the total interaction with the external field V = ∑υ(ri). So,

H = T +U + ∑υ(ri ) .

(3.77)

We have shown n(r) determines υ(r), and hence, H, which determines the groundstate wave function φ0. Therefore, we can define the functional

F[n(r)] = ∫ϕ0 (T +U )ϕ0dτ .

(3.78)

We can also write

∫ϕ0 ∑υ(r)ϕ0dτ = ∑ ∫ϕ0 (1…N)υ(ri )ϕ0 (1…N)dτ ,

(3.79)

by the symmetry of the wave function,

∫

ϕ

0

∑υ(r)ϕ

0

dτ

= N

∫

ϕ

(1…N )υ(r )ϕ

0

(1…N )dτ

0

i

(3.80)

= ∫υ(r)n(r)dr

by definition of n(r). Thus the total energy functional can be written

E0[n] = ∫ϕ0Hϕ0dτ = F[n] + ∫ n(r)υ(r)dr .

(3.81)

The ground-state energy E0 is a unique functional of the ground-state electron density. We now need to show that E0 is a minimum when n(r) assumes the correct electron density. Let n be the correct density function, and let us vary n → n′, so υ → υ′ and φ → φ′ (the ground-state wave function). All variations are subject to N = ∫n(r)dr = ∫n′(r)dr being constant. We have

E0[n′] = ∫ϕ0′Hϕ0′dτ
= ∫ϕ0′ (T +U )ϕ0′dτ + ∫ϕ0′ ∑υ(ri )ϕ0′dτ	(3.82)
= F[n′] + ∫υn′dr.

140 3 Electrons in Periodic Potentials

By the principle ∫φ′0Hφ′0dτ > ∫φ0Hφ0dτ, we have
E0[n′] > E0[n] ,	(3.83)

as desired. Thus, the HK Theorem is proved.

The HK Theorem can be extended to the more realistic case of electrons with spin and also to finite temperature. To include spin, one must consider both a spin density s(r), as well as a particle density n(r). The HK Theorem then states that the ground state is a unique functional of both these densities.

Variational Procedure (A)

Just as the single particle Hartree–Fock equations can be derived from a variational procedure, analogous single-particle equations can be derived from the density functional expressions. In DFT, the energy functional is the sum of ∫υndτ and F[n]. In turn, F[n] can be split into a kinetic energy term, an exchangecorrelation term and an electrostatic energy term. We may formally write (using Gaussian units so 1/4πε0 can be left out)

	F[n] = FKE[n] + Exc[n] +	e2	∫	n(r)n(r′)dτdτ′	.	(3.84)
		2		| r − r′|

Equation (3.84), in fact, serves as the definition of Exc[n]. The variational principle then states that

	δE0[n] = 0 ,					(3.85)
	subject to δ∫n(r)dτ = δN = 0, where
	E0[n] = FKE[n] + Exc[n] +	e2	∫	n(r)n(r′)dτdτ′	+ ∫υ(r)n(r)dτ .	(3.86)
		2		| r − r′|

Using a Lagrange multiplier μ to build in the constraint of a constant number of particles, and making

	e2		∫	n(r)n(r′)dτdτ′		= e2 ∫δn(r)∫	n(r′)dτ′dτ
	δ							,	(3.87)
		2		| r − r′|			| r − r′|

we can write
	δF [n]	+υ(r) + e2 ∫	n(r′)dτ′		δE	xc	[n]
∫δn(r)	KE			+				dτ − μ∫δndτ = 0	. (3.88)
			| r − r′|
	δn(r)				δn(r)

3.1 Reduction to One-Electron Problem		141
Defining
υxc (r) =	δExc[n]	(3.89)
	δn(r)

(an exchange correlation potential which, in general may be nonlocal), we can then define an effective potential as

veff (r) =υ(r) +υxc (r) + e		2	∫	n(r′)dτ′	.	(3.90)
				| r − r′|
The Euler–Lagrange equations can now be written as
δFKE[n] + v (r) = μ .						(3.91)
δn(r)	eff

Kohn–Sham Equations (A)

We need to find usable expressions for the kinetic energy and the exchange correlation potential. Kohn and Sham assumed that there existed some N singleparticle wave functions ui(r), which could be used to determine the electron density. They assumed that if this made an error in calculating the kinetic energy, then this error could be lumped into the exchange correlation potential. Thus,

n(r) = ∑iN=1| ui (r) |2 ,									(3.92)
and assume the kinetic energy can be written as
FKE (n) =	1	∑iN=1 ∫ ui ui dτ
2				1				,	(3.93)
		N				2
= ∑i=1			∫ ui −				ui dτ
				2

where units are used so ћ2/m = 1. Notice this is a kinetic energy for non interacting particles In order for FKE to represent the kinetic energy, the ui must be orthogonal. Now, without loss in generality, we can write

δn = ∑iN=1(δui )ui ,

(3.94)

with the ui constrained to be orthogonal so ∫ui*ui = δij. The energy functional E0[n] is now given by

	N				1		2
E0[n] = ∑i		=1		∫ ui −	2			uidτ + Exc[n]
	e2									(3.95)
				n(r)n(r′)dτdτ′
+			∫						+ ∫υ(r)n(r)dτ.
	2			| r − r′|

142 3 Electrons in Periodic Potentials

Using Lagrange multipliers εij to put in the orthogonality constraints, the variational principle becomes

	δE0[n] − ∑iN=1εij ∫δui uidτ = 0 .							(3.96)
This leads to
N		−	1		2			(3.97)
∑i=1	∫δui		2			+ veff (r) ui − ∑ j εijui dτ = 0 .

Since the ui* can be treated as independent, the terms in the bracket can be set equal to zero. Further, since εij is Hermitian, it can be diagonalized without affecting the Hamiltonian or the density. We finally obtain one form of the Kohn– Sham equations

−

1

2

+ v

= ε

u

,

(3.98)

2

(r) u

eff

i

i

i

where veff(r) has already been defined. There is no Koopmans’ Theorem in DFT and care is necessary in the interpretation of εi. In general, for DFT results for excited states, the literature should be consulted. We can further derive an expression for the ground state energy. Just as for the Hartree–Fock case, the ground-state energy does not equal ∑εi. However, using the definition of n,

1

2

2

n(r′)dτ′

∑i εi = ∑i ∫ ui −

+υ(r) + e

∫

+υxc (r) uidτ

2

| r − r′|

(3.99)

2

n(r′)n(r)dτdτ′

= FKE[n] + ∫ nυdτ + ∫ nυxcdτ

+ e

∫

.

| r − r′|

Equations (3.90), (3.92), and (3.98) are the Kohn–Sham equations. If υxc were zero these would just be the Hartree equations. Substituting the expression into the equation for the ground-state energy, we find

	E0[n] = ∑εi −	e2	∫	n(r)n(r′)dτdτ′	− ∫υxc (r)n(r)dτ + Exc[n] . (3.100)
		2		| r − r′|

We now want to look at what happens when we include spin. We must define both spin-up and spin-down densities, n↑ and n↓. The total density n would then be a sum of these two, and the exchange correlation energy would be a functional of both. This is shown as follows:

Exc = Exc[n↑,n↓] .		(3.101)
We also assume single-particle states exist, so
N↑	(r) |2 ,
n↑(r) = ∑| ui↑		(3.102)
i=1

3.1 Reduction to One-Electron Problem	143

and

N↓
n↓(r) = ∑| ui↓(r) |2 .	(3.103)
i=1

Similarly, there would be both spin-up and spin-down exchange correlation energy as follows:

	υ	xc↑	=	δExc[n↑, n↓]	,	(3.104)
				δn↑
	and
	υ	xc↓	=	δExc[n↑, n↓]	.	(3.105)
				δn↓

Using σ to represent either ↑ or ↓, we can find both the single-particle equations and the expression for the ground-state energy

1

2

2

∫

n(r′)dτ′

−

+υ(r) + e

+υxcσ (r) uiσ

= εiσ uiσ ,

(3.106)

2

| r − r′|

E0[n] = ∑i,σ εiσ

−

e2

∫

n(r)n(r′)dτdτ′

(3.107)

2

| r − r′|

− ∑σ ∫υxcσ (r)nσ (r)dτ + Exc[n],

over N lowest εiσ.

Local Density Approximation (LDA) to υxc (A)

The equations are still not in a tractable form because we have no expression for υxc. We assume the local density approximation of Kohn and Sham, in which we assume that locally Exc can be calculated as if it were a uniform electron gas. That is, we assume for the spinless case

ExcLDA = ∫ nεxcuniform[n(r)]dτ ,

and for the spin 1/2 case,

ExcLDA = ∫nεxcu [n↑(r),n↓(r)]dτ ,

where εxc represents the energy per electron. For the spinless case, the exchangecorrelation potential can be written

LDA	(r) =	δExcLDA	,	(3.108)
υxc		δn(r)