Patterson, Bailey - Solid State Physics Introduction to theory

144 3 Electrons in Periodic Potentials

and

δExcLDA = ∫δnεxcu dτ + ∫ n							δεxcu	δn dτ
							δn
by the chain rule. So,
LDA		LDA				u			u
	= ∫	δExc	δn dτ = ∫					δεxc
δExc		δn			εxc + n			δn		δn dτ .
Thus,
		δExcLDA	u			δεxcu (n)			.
		δn	= εxc (n) + n				δn

(3.109)

(3.110)

(3.111)

The exchange correlation energy per particle can be written as a sum of exchange and correlation energies, εxc(n) = εx(n) + εc(n). The exchange part can be calculated from the equations

Ex

= 1

V

∫kM A1(k)k 2dk ,

(3.112)

π 2

2

0

and

e

2

kM

2

2

kM + k

A (k) = −

2

+

kM − k

ln

,

(3.113)

1

2π

kkM

kM − k

see (3.63), where 1/2 in Ex is inserted so as not to count interactions twice. Since

N =

V

kM3

,

π 2

3

we obtain by doing all the integrals,

E

x

3

3

N 1/ 3

= −

.

(3.114)

N

4

π

V

By applying this equation locally, we obtain the Dirac exchange energy functional

εx (n) = −cx[n(r)]1/ 3 ,					(3.115)
where
	3	3		1/ 3
cx =				.	(3.116)
	4
		π

3.1 Reduction to One-Electron Problem	145

The calculation of εc is lengthy and difficult. Defining rs so

4	πr3	= 1	,	(3.117)
3	s	n

one can derive exact expressions for εc at large and small rs. An often-used expression in atomic units (see Appendix A) is

						r
εc =		0.0252F				s		,						(3.118)
						30
where
	3			1				x		2		1
F(x) = (1+ x		) ln 1	+	x		+			− x		−		.	(3.119)
								2				3

Other expressions are often given. See, e.g., Ceperley and Alder [3.9] and Pewdew and Zunger [3.39]. More complicated expressions are necessary for the nonspin compensated case (odd number of electrons and/or spin-dependent potentials).

Reminder: Functions and Functional Derivatives A function assigns a number g(x) to a variable x, while a functional assigns a number F[g] to a function whose values are specified over a whole domain of x. If we had a function F(g1,g2,…,gn) of the function evaluated at a finite number of xi, so that g1 = g(x1), etc., the differential of the function would be

dF = ∑iN=1	∂F	dgi .	(3.120)
	∂gi

Since we are dealing with a continuous domain D of the x-values over a whole domain, we define a functional derivative in a similar way. But now, the sum becomes an integral and the functional derivative should really probably be called a functional derivative density. However, we follow current notation and determine the variation in F (δF) in the following way:

	δF = ∫		δF	δg(x)dx .	(3.121)
		x D δg(x)

This relates to more familiar ideas often encountered with, say, Lagrangians. Suppose

F[x] = ∫D L(x, x)dt ; x = dxdt , and assume δx = 0 at the boundary of D, then

δF = ∫ δδxF(t) δx(t)dt ,

146 3 Electrons in Periodic Potentials

but

δL(x, x) = ∂∂Lx δx + ∂∂Lx δx .

If

∫ ∂∂Lx δxdt = ∫ ∂∂Lx ddt δxdt = ∂∂Lx Boundary δx − ∫ ddt ∂∂Lx δxdt ,

→0

then

δF

∫D δx(t)

So

	δx(t)dt = ∫			∂L				d		∂L
						−				δx(t)dt .
					∂x			dt
			D							∂x
		δF	=	∂L	−	d ∂L				,
		δx(t)		∂x		dt ∂x

	which is the typical result of Lagrangian mechanics. For example,
	ExLDA = ∫ n(r)εxdτ ,								(3.122)
	where εx = −cxn(r)1/3, as given by the Dirac exchange. Thus,
	ExLDA = −cx ∫ n(r)4 / 3dτ
	δExLDA = −cx			4		∫ n(r)1/ 3δndτ ,			(3.123)
			3
	= ∫	δExLDA
			δn				δndτ
	so,
	δExLDA				4		1/ 3
	δn	= −					cxn(r)	.	(3.124)
					3

Further results may easily be found in the functional analysis literature (see, e.g., Parr and Yang [3.38].

We summarize in Table 3.1 the one-electron approximations we have discussed thus far.

3.1 Reduction to One-Electron Problem 147

Table 3.1. One-electron approximations

Approximation

Equations defining

Comments

Free electrons

2

2 +V

Populate energy levels

H = −

with Fermi–Dirac

2m

statistics useful for

V = constant

simple metals.

m = effective mass

Hψk = Eψ

Ek

=

2k

2

+V

2m

ψk

= Aeik r

A = constant

Hartree

[H +V (r)]uk (r) = Ek uk (r)

V (r) = Vnucl +Vcoul

See (3.9), (3.15)

Vnucl =

e2

−

∑

+ const

4πε

r

a(nuclei)

0 ai

i(electrons)

Vcoul = ∑ j(≠k) ∫ u j (x2 )V (1,2)u j (x2 )dτ2

Vcoul arises from Coulomb interactions of

electrons

Hartree–Fock

[H +V (r) +Vexch ]uk (r) = Ek uk (r)

Ek is defined by

Vexchuk (r) =

Koopmans’ Theorem

(3.30).

− ∑ j ∫ dτ2u j (x2 )V (1,2)uk (x2 )u j (x1)

and V(r) as for Hartree (without the j ≠ k

restriction in the sum).

Hohenberg–

An external potential v(r) is uniquely

No Koopmans’ theorem.

Kohn Theorem

determined by the ground-state density of

electrons in a band system. This local

electronic

charge

density is the basic

quantity in density functional theory, rather than the wave function.

148 3 Electrons in Periodic Potentials

Table 3.1. (cont)

Approximation

Equations defining

Comments

Kohn–Sham

1

2

equations

−

+ veff (r)

− ε j

ϕ j (r) = 0

2

where n(r) = ∑Nj =1

ϕ j (r)

2

Related to Slater’s earlier

ideas (see Marder op cit

p. 219)

Local density

veff (r) = v(r) + ∫

n(r′)

dr′ + vxc (r)

See (3.90).

approximation

r − r′

ExcLDA = ∫ nεxcu [n(r)]dr,

exchange correlation energy εxc per particle

vxc (r) = δExc[n]

δn(r)

and see (3.111) and following

3.2 One-Electron Models

We now have some feeling about the approximation in which an N-electron system can be treated as N one-electron systems. The problem we are now confronted with is how to treat the motion of one electron in a three-dimensional periodic potential. Before we try to solve this problem it is useful to consider the problem of one electron in a spatially infinite one-dimensional periodic potential. This is the Kronig–Penney model.10 Since it is exactly solvable, the Kronig– Penney model is very useful for giving some feeling for electronic energy bands, Brillouin zones, and the concept of effective mass. For some further details see also Jones [58], as well as Wilson [97, p26ff].

3.2.1 The Kronig–Penney Model (B)

The potential for the Kronig–Penney model is shown schematically in Fig. 3.3. A good reference for this Section is Jones [58, Chap. 1, Sect. 6].

Rather than using a finite potential as shown in Fig. 3.3, it is mathematically convenient to let the widths a of the potential become vanishingly narrow and the

10 See Kronig and Penny [3.30].

3.2 One-Electron Models 149

V(x)

a

a1

…

u

…

0 x

Fig. 3.3. The Kronig–Penney potential

heights u become infinitely high so that their product au remains a constant. In this case, we can write the potential in terms of Dirac delta functions

V (x) = au∑nn==∞−∞δ (x − na1) ,

(3.125)

where δ(x) is Dirac’s delta function.

With delta function singularities in the potential, the boundary conditions on the wave functions must be discussed rather carefully. In the vicinity of the origin, the wave function must satisfy

−

2 d 2ψ

+ auδ (x)ψ = Eψ .

2m dx2

Integrating across the origin, we find

2

∫ε d2ψ dx − au∫ε

δ (x)ψ(x)dx = −E∫ε

ψdx ,

2m −ε dx2

−ε

−ε

or

2

dψ

ε

− auψ(0) = −E∫

ε

ψdx .

2m dx

−ε

−ε

Taking the limit as ε → 0, we find

dψ

dψ

2m(au)

−

=

ψ(0) .

2

dx +

dx −

(3.126)

(3.127)

Equation (3.127) is the appropriate boundary condition to apply across the Dirac delta function potential.

Our problem now is to solve the Schrödinger equation with periodic Dirac delta function potentials with the aid of the boundary condition given by (3.127). The

150 3 Electrons in Periodic Potentials

periodic nature of the potential greatly aids our solution. By Appendix C we know that Bloch’s theorem can be applied. This theorem states, for our case, that the wave equation has stationary-state solutions that can always be chosen to be of the form

ψk (x) = eikxuk (x) ,	(3.128)
where
uk (x + a1) = uk (x) .	(3.129)

Knowing the boundary conditions to apply at a singular potential, and knowing the consequences of the periodicity of the potential, we can make short work of the Kronig–Penney model. We have already chosen the origin so that the potential is symmetric in x, i.e. V(x) = V(−x). This implies that H(x) = H(−x). Thus if ψ(x) is a stationary-state wave function,

H (x)ψ(x) = Eψ(x) .

By a dummy variable change

H (−x)ψ(−x) = Eψ(−x) ,

so that

H (x)ψ(−x) = Eψ(−x) .

This little argument says that if ψ(x) is a solution, then so is ψ(−x). In fact, any linear combination of ψ(x) and ψ(−x) is then a solution. In particular, we can always choose the stationary-state solutions to be even zs(x) or odd za(x):

z	s	(x) =	1			[ψ(x) +ψ(−x)] ,	(3.130)
		2
z	s	(x) =		1		[ψ(x) −ψ(−x)].	(3.131)
		2

To avoid confusion, it should be pointed out that this result does not necessarily imply that there is always a two-fold degeneracy in the solutions; zs(x) or za(x) could vanish. In this problem, however, there always is a two-fold degeneracy.

It is always possible to write a solution as

ψ(x) = Azs (x) + Bza (x) .	(3.132)
From Bloch’s theorem
ψ(a1 / 2) = eika1ψ(−a1 / 2) ,	(3.133)
and
ψ′(a1 / 2) = eika1ψ′(−a1 / 2) ,	(3.134)
where the prime means the derivative of the wave function.

3.2 One-Electron Models 151

Combining (3.132), (3.133), and (3.134), we find that

A[zs (a1 / 2) − eika1 zs (−a1 / 2)] = B[eika1 za (−a1 / 2) − za (a1 / 2)] , (3.135)

and

A[z′s (a1 / 2) − eika1 z′s (−a1 / 2)] = B[eika1 z′a (−a1 / 2) − za′ (a1 / 2)] . (3.136)

Recalling that zs, za′ are even, and za, zs′ are odd, we can combine (3.135) and (3.136) to find that

1

− e

ika1

2

1

1

=

z′s (a / 2)za

(a / 2)

.

(3.137)

ika

1

z

1

1

1

+ e

s

(a / 2)z′

(a / 2)

a

Using the fact that the left-hand side is

− tan2 ka1

= −tan2 θ

=1−

1

,

cos2 (θ / 2)

2

2

and cos2 (θ/2) = (1 + cos θ)/2, we can write (3.137) as

cos(ka1) = −1+

2z

s

(a1 / 2)z′ (a1

/ 2)

a

,

(3.138)

W

where

W =

zs

za

.

(3.139)

z

′

z′

s

a

The solutions of the Schrödinger equation for this problem will have to be sinusoidal solutions. The odd solutions will be of the form

za (x) = sin(rx), − a1 / 2 ≤ x ≤ a1 / 2 ,		(3.140)
and the even solution can be chosen to be of the form [58]
zs (x) = cos r(x + K ),	0 ≤ x ≤ a1 / 2,	(3.141)
zs (x) = cos r(−x + K ),	− a1 / 2 ≤ x ≤ 0.	(3.142)

At first glance, we might be tempted to chose the even solution to be of the form cos(rx). However, we would quickly find that it is impossible to satisfy the boundary condition (3.127). Applying the boundary condition to the odd solution, we simply find the identity 0 = 0. Applying the boundary condition to the even solution, we find

− 2r sin rK = (cosrK ) 2mau /	2 ,
or in other words, K is determined from
tan rK = − m(au) .	(3.143)
r 2

152 3 Electrons in Periodic Potentials

Putting (3.140) and (3.141) into (3.139), we find
	W = r cos rK .					(3.144)
Combining (3.138), (3.140), (3.141), and (3.144), we find
cos ka1 = −1+	2r cos[r(a1 / 2 + K )]cos(ra1 / 2)				.	(3.145)
		r cos(rK )
Using (3.143), this last result can be written
1	1	m(au)	1 sin ra1			(3.146)
cos ka = cos ra +		2	a	1 .
				ra

Note the fundamental 2π periodicity of kal. This is the usual Brillouin zone periodicity.

Equation (3.146) is the basic equation describing the energy eigenvalues of the Kronig–Penney model. The reason that (3.146) gives the energy eigenvalue relation is that r is proportional to the square root of the energy. If we substitute

(3.141) into the Schrödinger equation, we find that
r = 2mE .	(3.147)

Thus (3.146) and (3.147) explicitly determine the energy eigenvalue relation (E versus k; this is also called the dispersion relationship) for electrons propagating in a periodic crystal.

The easiest thing to get out of this dispersion relation is that there are allowed and disallowed energy bands. If we plot the right-hand side of (3.146) versus ra, the results are somewhat as sketched in Fig. 3.4.

Right-hand side of (3.152)

+1
0	ra1
–1

Fig. 3.4. Sketch showing how to get energy bands from the Kronig–Penney model

3.2 One-Electron Models 153

From (3.146), however, we see we have a solution only when the right-hand side is between +1 and −1 (because these are the bounds of cos kal, with real k). Hence the only allowed values of ral are those values in the shaded regions of Fig. 3.4. But by (3.147) this leads to the concept of energy bands.

Detailed numerical analysis of (3.146) and (3.147) will yield a plot similar to Fig. 3.5 for the first band of energies as plotted in the first Brillouin zone. Other bands could be similarly obtained.

E

k

Fig. 3.5. Sketch of the first band of energies in the Kronig–Penney model (an arbitrary k = 0 energy is added in)

Figure 3.5 looks somewhat like the plot of the dispersion relation for a onedimensional lattice vibration. This is no accident. In both cases we have waves propagating through periodic media. There are significant differences that distinguish the dispersion relation for electrons from the dispersion relation for lattice vibrations. For electrons in the lowest band as k → 0, E k2, whereas for phonons we found E |k|. Also, for lattice vibrations there is only a finite number of energy bands (equal to the number of atoms per unit cell times 3). For electrons, there are infinitely many bands of allowed electronic energies (however, for realistic models the bands eventually overlap and so form a continuum).

We can easily check the results of the Kronig–Penney model in two limiting cases. To do this, the equation will be rewritten slightly:

cos(ka1) = cos(ra1) + μ sin ra1 ≡ P(ra1) , ra1

where

μ ≡ ma1(au) .

2

(3.148)

(3.149)