Patterson, Bailey - Solid State Physics Introduction to theory

174 3 Electrons in Periodic Potentials

free-electron approximation would have little validity. In recent years, by the method of pseudopotentials, it has been shown that the assumptions of the nearly free-electron model make more sense than one might suppose.

In this Section it will be assumed that a one-electron approximation (such as the Hartree approximation) is valid. The equation that must be solved is

		2	2
	−			+V (r) ψ	k	(r) = E ψ	(r) .	(3.216)
		2m
						k k

Let R be any direct lattice vector that connects equivalent points in two unit cells. Since V(r) = V(r + R), we know by Bloch’s theorem that we can always choose the wave functions to be of the form

ψk (r) = eik rUk (r) ,

where Uk(r) = Uk(r + R).

Since both Uk and V have the fundamental translational symmetry of the crystal, we can make a Fourier analysis [71] of them in the form

V (r) = ∑K V (K )eiK r	(3.217)
Uk (r) = ∑K U (K )eiK r .	(3.218)

In the above equations, the sum over K means to sum over all the lattice points in the reciprocal lattice. Substituting (3.217) and (3.218) into (3.216) with the Bloch condition on the wave function, we find that

2 ∑K U (K ) | k + K |2 eiK r + ∑K1,K11 V (K1)U (K11)ei(K1 + K11) r (3.219) 2m

= Ek ∑K U (K )eiK r .

By equating the coefficients of eiK·r, we find that

		2
			| k + K |2 −E	U (K ) = −∑	K	1 V (K1)U (K − K1) .	(3.220)
		2m
				k

If we had a constant potential, then all V(K) with K ≠ 0 would equal zero. Thus it makes sense to assume in the nearly free-electron approximation (in other words in the approximation that the potential is almost constant) that V(K)<<V(0). As we will see, this also implies that U(K)<<U(0).

Therefore (3.220) can be approximately written

			2	| k + K |2		−δK0 ) .
	Ek −V (0)	−			U (K ) =V (K )U (0)(1		(3.221)
			2m

3.2 One-Electron Models 175

Note that the part of the sum in (3.220) involving V(0) has already been placed in the left-hand side of (3.221). Thus (3.221) with K = 0 yields

	Ek V (0) +	2k 2	.	(3.222)
		2m

These are the free-particle eigenvalues. Using (3.222) and (3.221), we obtain for K ≠ 0 in the same approximation:

U (K )	= −	m		V (K )			.	(3.223)
U (0)		2		k K +	1	K 2
					2

Note that the above approximation obviously fails when

k K +	1	K 2	= 0 ,	(3.224)
2

if V(K) is not equal to zero.

The k that satisfy (3.224) (for each value of K) span the surface of the Brillouin zones. If we construct all Brillouin zones except those for which V(K) = 0 then we have the Jones zones.

k	k+K
θ	K
	θ
	θ
	k

Fig. 3.12. Brillouin zones and Bragg reflection

Condition (3.224) can be given an interesting interpretation in terms of Bragg reflection. This situation is illustrated in Fig. 3.12. The k in the figure satisfy (3.224). From Fig. 3.12,

k sinθ =	1	K .	(3.225)
2

But k = 2π/λ, where λ is the de Broglie wavelength of the electron, and one can find K for which K = n 2π/a, where a is the distance between a given set of parallel lattice planes (see Sect. 1.2.9 where this is discussed in more detail in connection with X-ray diffraction). Thus we conclude that (3.225) implies that

	2π	sinθ =	1	n	2π	,	(3.226)
	λ				a
		2
or that
	nπ = 2a sinθ .						(3.227)

176 3 Electrons in Periodic Potentials

Since θ can be interpreted as an angle of incidence or reflection, (3.227) will be recognized as the familiar law describing Bragg reflection. It will presently be shown that at the Jones zone, there is a gap in the E versus k energy spectrum. This happens because the electron is Bragg reflected and does not propagate, and this is what we mean by having a gap in the energy. It will also be shown that when V(K) = 0 there is no gap in the energy. This last fact is not obvious from the Bragg reflection picture. However, we now see why the Jones zones are the important physical zones. It is only at the Jones zones that the energy gaps appear. Note also that (3.225) indicates a simple way of defining the Brillouin zones by construction. We just draw reciprocal space. Starting from any point in reciprocal space, we draw straight lines connecting this point to all other points. We then bisect all these lines with planes perpendicular to the lines. Starting from the point of interest; these planes form the boundaries of the Brillouin zones. The first zone is the first enclosed volume. The second zone is the volume between the first set of planes and the second set. The idea should be clear from the two-dimensional representation in Fig. 3.13.

		b
	b	a
		b
		b

Fig. 3.13. Construction of Brillouin zones in reciprocal space: (a) the first Brillouin zone, and (b) the second Brillouin zone. The dots are lattice points in reciprocal space. Any vector joining two dots is a K-type reciprocal vector

To finish the calculation, let us treat the case when k is near a Brillouin zone boundary so that U(K1) may be very large. Equation (3.220) then gives two equations that must be satisfied:

2

Ek −V (0)

−

| k + K

1

|2

U (K1) =V (K1)U (0),

K1 ≠ 0

,

2m

2

Ek −V (0) −

k 2 U (0)

=V (−K1)U (K1) .

2m

(3.228)

(3.229)

3.2 One-Electron Models 177

The equations have a nontrivial solution only if the following secular equation is satisfied:

Ek −V (0) −

2

(k + K1)2

−V (K1)

2m

= 0 .

(3.230)

2

1)

2

−V (−K

Ek −V (0) −

k

2m

By problem 3.7 we know that (3.230) is equivalent to

Ek =

1

(Ek0 + Ek01 ) ±

1

[4 | V (K1) |2 +(Ek0

− Ek01 )2 ]1 2 ,

(3.231)

2

2

where

Ek0 =V (0) +

2

k2 ,

(3.232)

2m

and

E0

=V (0) +

2

(k + K1)2 .

(3.233)

2m

k1

For k on the Brillouin zone surface of interest, i.e. for k2 = (k + K l)2, we see that

there is an energy gap of magnitude
Ek+ − Ek− = 2 |V (K1) | .	(3.234)
This proves our point that the gaps in energy appear whenever |V(K l)| ≠ 0.
The next question that naturally arises is: “When does V(K l)	= 0?” This

question leads to a discussion of the concept of the structure factor. The structure factor arises whenever there is more than one atom per unit cell in the Bravais lattice.

If there are m atoms located at the coordinates rb in each unit cell, if we assume each atom contributes U(r) (with the coordinate system centered at the center of the atom) to the potential, and if we assume the potential is additive, then with a fixed origin the potential in any cell can be written

V (r) = ∑m= U (r − rb ) . b 1

Since V(r) is periodic in a unit cube, we can write

V (r) = ∑K V (K )eik r ,

where

(3.235)

(3.236)

V (K ) =	1	∫ V (r)e−iK r d3r ,	(3.237)
	Ω Ω

178 3 Electrons in Periodic Potentials

and Ω is the volume of a unit cell. Combining (3.235) and (3.237), we can write the Fourier coefficient

V (K ) =	1	∑bm=1			∫ U (r − rb )e−iK r d3r
	Ω
					Ω
=	1	∑bm=1			∫ U (r′)e−iK (r′+rb )d3r′
	Ω
					Ω
=	1	∑bm=1e−iK rb ∫ U (r′)e−iK r′d3r′,
	Ω
					Ω
or
		V (K ) ≡ SK v(K ) ,				(3.238)
where
		SK ≡ ∑bm=1e−iK rb ,				(3.239)
(structure factors are also discussed in Sect. 1.2.9) and
v(K ) ≡			1		∫ U (r1)e−iK r1 d3r1 .	(3.240)
			Ω Ω

SK is the structure factor, and if it vanishes, then so does V(K). If there is only one atom per unit cell, then |SK| = 1. With the use of the structure factor, we can summarize how the first Jones zone can be constructed:

1. Determine all planes from

k K + 12 K 2 = 0 .

2.Retain those planes for which SK ≠ 0, and that enclose the smallest volume in k space.

To complete the discussion of the nearly free-electron approximation, the pseudopotential needs to be mentioned. However, the pseudopotential is also used as a practical technique for band-structure calculations, especially in semiconductors. Thus we discuss it in a later section.

The Tight Binding Approximation (B)14

This method is often called by the more descriptive name linear combination of atomic orbitals (LCAO). It was proposed by Bloch, and was one of the first types of band-structure calculation. The tight binding approximation is valid for the inner or core electrons of most solids and approximately valid for all electrons in an insulator.

All solids with periodic potentials have allowed and forbidden regions of energy. Thus it is no great surprise that the tight binding approximation predicts

14 For further details see Mott and Jones [71].

3.2 One-Electron Models 179

a band structure in the energy. In order to keep things simple, the tight binding approximation will be done only for the s-band (the band of energy formed by s- electron states).

To find the energy bands one must solve the Schrödinger equation

Hψ0 = E0ψ0 ,

(3.241)

where the subscript zero refers to s-state wave functions. In the spirit of the tight binding approximation, we attempt to construct the crystalline wave functions by using a superposition of atomic wave functions

ψ0 (r) = ∑iN=1diφ0 (r − Ri ) .

(3.242)

In (3.242), N is the number of the lattice ions, φ0 is an atomic s-state wave function, and the Ri are the vectors labeling the location of the atoms.

If the di are chosen to be of the form

di = eik Ri ,

(3.243)

then ψ0(r) satisfies the Bloch condition. This is easily proved:

ψ(r + Rk ) = ∑i eik Ri φ0 (r + Rk − Ri )

= eik Rk ∑i eik (Ri − Rk )φ0[r − (Ri − Rk )] = eik Rkψ(r).

Note that this argument assumes only one atom per unit cell. Actually a much more rigorous argument for

ψ0 (r) = ∑iN=1eik Ri φ0 (r − Ri )

(3.244)

can be given by the use of projection operators.15 Equation (3.244) is only an approximate equation for ψ0(r).

Using (3.244), the energy eigenvalues are given approximately by

	E0		∫ψ0Hψ0dτ	,	(3.245)
			∫ψ0ψ0dτ

where H is the crystal Hamiltonian. We define an atomic Hamiltonian

H	i	= −( 2 / 2m) 2 +V (r − R ) ,		(3.246)
		0	i
where V0(r − Ri) is the atomic potential. Then
H iφ0 (r − Ri ) = E00φ0 (r − Ri ) ,				(3.247)

15 See Löwdin [3.33].

180 3 Electrons in Periodic Potentials

and

H − H i =V (r) −V0 (r − Ri ) ,

(3.248)

where E00 and φ0 are atomic eigenvalues and eigenfunctions, and V is the crystal

potential energy.

Using (3.244), we can now write

Hψ0 = ∑iN=1eik Ri [H i + (H − H i )]φ0 (r − Ri ) ,

or

Hψ0 = E00ψ0 + ∑iN=1eik Ri [V (r) −V0 (r − Ri )]φ0 (r − Ri ) . Combining (3.245) and (3.249), we readily find

E0 − E00	∑N	eik Ri ∫ψ0[V (r) −V0 (r − Ri )]φ0	(r − Ri )dτ	.
	i=1	∫ψ0ψ0dτ

(3.249)

(3.250)

Using (3.244) once more, this last equation becomes

E0 − E00		∑i, j eik (Ri − R j ) ∫φ0 (r − R j )[V (r) −V0 (r − Ri )]φ0 (r − Ri )dτ
							.	(3.251)
				∑i, j eik (Ri − R j ) ∫φ0 (r − R j )φ0 (r − Ri )dτ
Neglecting overlap, we have approximately
					∫φ0 (r − R j )φ0 (r − Ri )dτ δi, j .
Combining (3.250) and (3.251) and using the periodicity of V(r), we have
E0	− E00		1	∑i, j eik (Ri − R j ) ∫φ0 [r − (R j − Ri )][V (r) −V0 (r)]φ0 (r)dτ ,
			N
or
		E0 − E00			1	∑l e−ik Rl ∫φ0 (r − Rl )[V (r) −V0 (r)]φ0 (r)dτ .		(3.252)
					N

Assuming that the terms in the sum of (3.252) are very small beyond nearest neighbors, and realizing that only s-wave functions (which are isotropic) are involved, then it is useful to define two parameters:

∫φ0 (r)[V (r) −V0 (r)]φ0 (r)dτ = −α ,	(3.253)
∫φ0 (r + Rl′)[V (r) −V0 (r)]φ0 (r)dτ = −γ ,	(3.254)

where Rl′ is a vector of the form Rl for nearest neighbors.

3.2 One-Electron Models 181

Thus the tight binding approximation reduces to a two-parameter (α, γ) theory with the dispersion relationship (i.e. the E versus k relationship) for the s-band given by

E0 − (E00 −α) = −γ ∑ j(n.n.) eik R′j .

(3.255)

Explicit expressions for (3.255) are easily obtained in three cases 1. The simple cubic lattice. Here

R′j = (±a,0,0), (0,±a,0), (0,0,±a) ,

and

E0 − (E00 −α) = −2γ (cos kxa + cos kya + cos kz a) . The bandwidth in this case is given by 12γ.

2. The body-centered cubic lattice. Here there are eight nearest neighbors at

R′j = 12 (±a,±a,±a) .

Equation (3.255) and a little algebra gives

E0 − (E00		k	x	a		k ya		k	z	a
	−α) = −8γ cos				cos		cos				.
			2			2			2

The bandwidth in this case is 16γ.

3. The face-centered cubic lattice. Here the 12 nearest neighbors are at

R′j = 12 (0,±a,±a), 12 (±a,0,±a), 12 (±a,±a,0) .

A little algebra gives

− (E00

k ya

k

z

a

k

z

a

k

x

a

E0

−α) = −4γ cos

cos

+ cos

cos

2

2

2

2

k ya

k

x

a

+ cos

2

cos

2

.

The bandwidth for this case is 16γ. The tight binding approximation is valid when γ is small, i.e., when the bands are narrow.

182 3 Electrons in Periodic Potentials

As must be fairly obvious by now, one of the most important results that we get out of an electronic energy calculation is the density of states. It was fairly easy to get the density of states in the free-electron approximation (or more generally when E is a quadratic function |k|). The question that now arises is how we can get a density of states from a general dispersion relation similar to (3.255).

Since the k in reciprocal space are uniformly distributed, the number of states in a small volume dk of phase space (per unit volume of real space) is

2 d3k . (2π)3

Now look at Fig. 3.14 that shows a small volume between two constant electronic energy surfaces in k-space.

ds

ε + dε

dk

ε

Fig. 3.14. Infinitesimal volume between constant energy surfaces in k-space

From the figure we can write

d3k = dsdk .

But

dε = | kε(k) | dk ,

so that if D(ε) is the number of states between ε and ε + dε, we have

D(ε) =	2	∫s	ds
				.	(3.256)
	(2π)3		| kε(k) |
Equation (3.256) can always be	used	to	calculate a density of		states when

a dispersion relation is known. As must be obvious from the derivation, (3.256) applies also to lattice vibrations when we take into account that phonons have different polarizations (rather than the different spin directions that we must consider for the case of electrons).

3.2 One-Electron Models 183

Table 3.3. Simple models of electronic bands

Model				Energies
Nearly free electron				Ek = 12 (Ek0 + Ek0′)
near Brillouin zone boundary on
surface where				± 1	(Ek0				− Ek0′)2 + 4V (K ) 2
k K +	1	K 2	= 0	2
2				Ek0 =V (0) +					2k 2
									2m
				Ek0′ =V (0) +					2	(k + K )2
								2m
					1
				V (K ) =		∫			V (r)e−iK r dV
					Ω Ω

Ω = unit cell volume

Tight binding simple cube

body-centered cubic

face-centered cubic

A, B appropriately chosen parameters. a = cell side.

Ek = A − B(cos kxa + cos k ya + cos kza)

Ek

= A −

4B cos

kxa

k ya

kza

cos

cos

2

2

2

k

x

a

k ya

E

k

= A −

2B cos

cos

2

2

a

k ya

k

z

a

k

z

a

k

x

+ cos

cos

+ cos

cos

2

2

2

2

Kronig–Penny

r =	2mE	P = mub a
	2	2

a – distance between barriers u – height of barriers

b – width of barrier

cos ka = cos ra + P sin ka ra

determines energies in b → 0, ua → constant limit

Tight binding approximation calculations are more complicated for p, d., etc., bands, and also when there is an overlapping of bands. When things get too complicated, it may be easier to use another method such as one of those that will be discussed in the next section.

The tight binding method and its generalizations are often subsumed under the name linear combination of atomic orbital (LCAO) methods. The tight binding method here gave the energy of an s-band as a function of k. This energy depended on the interpolation parameters α and γ. The method can be generalized to include other interpolation parameters. For example, the overlap integrals that were neglected could be treated as interpolation parameters. Similarly, the integrals for the energy involved only nearest neighbors in the sum. If we summed