Patterson, Bailey - Solid State Physics Introduction to theory

204 3 Electrons in Periodic Potentials

experiment will not be made here. Later chapters give some details about the type of experimental results that need E(k) information for their interpretation. In particular, the Section on the Fermi surface gives some details on experimental results that can be obtained for the conduction electrons in metals. Further references for band-structure calculations are in Table 3.4. See also Altman [3.1].

The Spin-Orbit Interaction (B)

As shown in Appendix F, the spin-orbit effect can be correctly derived from the Dirac equation. As mentioned there, perhaps the most familiar form of the spinorbit interaction is the form that is appropriate for spherical symmetry. This form is

H ′ = f (r)L S .

(3.317)

In (3.317), H′ is the part of the Hamiltonian appropriate to the spin-orbit interaction and hence gives the energy shift for the spin-orbit interaction. In solids, spherical symmetry is not present and the contribution of the spin-orbit effect to the Hamiltonian is

H =		S ( V × p) .	(3.318)
	2m2c2
	0

There are other relativistic corrections that derive from approximating the Dirac equation but let us neglect these.

A relatively complete account of spin-orbit splitting will be found in Appendix 9 of the second volume of Slater’s book on the quantum theory of molecules and solids [89]. Here, we shall content ourselves with making a few qualitative observations. If we look at the details of the spin-orbit interaction, we find that it usually has unimportant effects for states corresponding to a general point of the Brillouin zone. At symmetry points, however, it can have important effects because degeneracies that would otherwise be present may be lifted. This lifting of degeneracy is often similar to the lifting of degeneracy in the atomic case. Let us consider, for example, an atomic case where the j = l ± ½ levels are degenerate in the absence of spin-orbit interaction. When we turn on a spin-orbit interaction, two levels arise with a splitting proportional to L · S (using J2 = L2 + S2 + 2L · S). The energy difference between the two levels is proportional to

(l + 12 )(l + 32 ) −l(l +1) − 12 ( 32 ) − (l − 12 )(l + 12 ) + l(l +1) + 12 ( 32 ) = (l + 12 )[(l + 32 ) −l + 12 ] = (l + 12 ) 2 = 2l +1.

This result is valid when l > 0. When l = 0, there is no splitting. Similar results are obtained in solids. A practical case is shown in Fig. 3.20. Note that we might have been able to guess (a) and (b) from the atomic consideration given above.

3.2 One-Electron Models 205

E E E

		k			k			k
	(a)			(b)			(c)

Fig. 3.20. Effect of spin-orbit interaction on the l = 1 level in solids: (a) no spin-orbit, six degenerate levels at k = 0 (a point of cubic symmetry), (b) spin-orbit with inversion symmetry (e.g. Ge), (c) spin-orbit without inversion symmetry (e.g. InSb). [Adapted from Ziman JM, Principles of the Theory of Solids, Cambridge University Press, New York, 1964, Fig. 54, p. 100. By permission of the publisher.]

3.2.4 Effect of Lattice Defects on Electronic States in Crystals (A)

The results that will be derived here are similar to the results that were derived for lattice vibrations with a defect (see Sect. 2.2.5). In fact, the two methods are abstractly equivalent; it is just that it is convenient to have a little different formalism for the two cases. Unified discussions of the impurity state in a crystal, including the possibility of localized spin waves, are available.24 Only the case of one-dimensional motion will be considered here; however, the method is extendible to three dimensions.

The model of defects considered here is called the Slater–Koster model.25 In the discussion below, no consideration will be given to the practical details of the calculation. The aim is to set up a general formalism that is useful in the understanding of the general features of electronic impurity states.26 The Slater– Koster model is also useful for discussing deep levels in semiconductors (see Sect. 11.3).

In order to set the notation, the Schrödinger equation for stationary states will be rewritten:

Hψn,k (x) = En (k)ψn,k (x) .

(3.319)

24See Izynmov [3.24].

25See [3.49, 3.50]

26Wannier [95, p181ff]

206 3 Electrons in Periodic Potentials

In (3.319), H is the Hamiltonian without defects, n labels the different bands, and k labels the states within each band. The solutions of (3.319) are assumed known.

We shall now suppose that there is a localized perturbation (described by V) on one of the lattice sites of the crystal. For the perturbed crystal, the equation that must be solved is

(H +V )ψ = Eψ .

(3.320)

(This equation is true by definition; H + V is by definition the total Hamiltonian of the crystal with defect.)

Green’s function for the problem is defined by

HGE (x, x0 ) − EGE (x, x0 ) = −4πδ(x − x0 ) .

(3.321)

Green’s function is required to satisfy the same boundary conditions as ψnk(x). Writing ψnk = ψm, and using the fact that the ψm form a complete set, we can write

GE (x, x0 ) = ∑m Amψm (x) .

Substituting (3.322) into the equation defining Green’s function, we obtain ∑m Am (Em − E)ψm (x) = −4πδ(x − x0 ) .

Multiplying both sides of (3.323) by ψn*(x) and integrating, we find

A	= −4π	ψ	(x	0		)	.
			n
n		En − E
Combining (3.324) with (3.322) gives
			ψ		(x		0	)ψ	n	(x)
GE (x, x0 ) = −4π∑m			m								.
					Em			− E

(3.322)

(3.323)

(3.324)

(3.325)

Green’s function has the property that it can be used to convert a differential equation into an integral equation. This property can be demonstrated. Multiply (3.320) by GE* and integrate:

∫ GE Hψdx − E∫ GEψdx = −∫ GEVψdx .	(3.326)
Multiply the complex conjugate of (3.321) by ψ and integrate:
∫ψHGEdx − E∫ GEψdx = −4πψ(x0 ) .	(3.327)
Since H is Hermitian,
∫ GE Hψdx = ∫ψHGE dx .	(3.328)

3.2 One-Electron Models 207

Thus subtracting (3.326) from (3.327), we obtain

	ψ(x0 ) =	1	∫ GE (x, x0 )V (x)ψ(x)dx .	(3.329)
		4π

Therefore the equation governing the impurity problem can be formally written as

	ψ(x0 ) = −∑n,k	ψn,k (x0 )	∫ψn,k (x)V (x)ψ(x)dx .	(3.330)
		En (k) − E

Since the ψn,k(x) form a complete orthonormal set of wave functions, we can define another complete orthonormal set of wave functions through the use of a unitary transformation. The unitary transformation most convenient to use in the present problem is

ψn,k (x) =	1	∑ j eik ( ja) An (x − ja) .	(3.331)
	N

Equation (3.331) should be compared to (3.244), which was used in the tight binding approximation. We see the φ0(r − Ri) are analogous to the An(x − ja). The φ0(r − Ri) are localized atomic wave functions, so that it is not hard to believe that the An(x − ja) are localized. The An(x − ja) are called Wannier functions.27

In (3.331), a is the spacing between atoms in a one-dimensional crystal (with N unit cells) and so the ja (for j an integer) labels the coordinates of the various atoms. The inverse of (3.331) is given by

An (x − ja) =	1	∑k (a Brillouin zone) e−ik( ja)ψn,k (x) .	(3.332)
	N

If we write the ψn,k as functions satisfying the Bloch condition, it is possible to give a somewhat simpler form for (3.332). However, for our purposes (3.332) is sufficient.

Since (3.332) form a complete set, we can expand the impurity-state wave function ψ in terms of them:

					ψ(x) = ∑l,iUl (ia)Al (x − ia) .			(3.333)
Substituting (3.331) and (3.333) into (3.330) gives
∑l,i′Ul (i′a)Al (x −i′a)
n,k 1 eikja						−ikj′a
= −∑l,i′					An (x0 − ja)∫ e		An (x − j′a)VUl (i′a)Al (x −i′a)dx.
				(k)
j, j′ N E − En
								(3.334)

27 See Wannier [3.56].

208 3 Electrons in Periodic Potentials

Multiplying the above equation by Am*(x0 – pa), integrating over all space, using the orthonormality of the Am, and defining

Vn,l ( j′,i) = ∫ An (x − j′a)VAl (x −ia)dx ,

(3.335)

we find

′

m

p

1

eik( pa− j′a)

′ ′

U

(i a) δ

δ

+

V

( j

,i )

= 0 .

(3.336)

∑l,i′

l

l

N ∑k

, j′ Em (k) − E

i′

m,l

For a nontrivial solution, we must have

1

e

ik ( p− j′)a

det δ mδ p

+

V

( j′,i′)

= 0 .

(3.337)

N

l

i′

∑k, j′ Em (k) − E m,l

This appears to be a very difficult equation to solve, but if Vml(j′, i) = 0 for all but a finite number of terms, then the determinant would be drastically simplified.

Once the energy of a state has been found, the expansion coefficients may be found by going back to (3.334).

To show the type of information that can be obtained from the Slater–Koster model, the potential will be assumed to be short range (centered on j = 0), and it will be assumed that only one band is involved. Explicitly, it will be assumed that

Vm,l ( j′,i) = δlbδmb δ 0j′δi0′V0 .

(3.338)

Note that the local character of the functions defined by (3.332) is needed to make such an approximation.

From (3.337) and (3.338) we find that the condition on the energy is

f (E) ≡	N	+ ∑k			1	= 0 .	(3.339)
	V		E	b	(k) − E
0

Equation (3.339) has N real roots. If V0 = 0, the solutions are just the unperturbed energies Eb(k). If V0 ≠ 0, then we can use graphical methods to find E such that f(E) is zero. See Fig. 3.21. In the figure, V0 is assumed to be negative.

The crosses in Fig. 3.21 are the perturbed energies; these are the roots of f(E). The poles of f(E) are the unperturbed levels. The roots are all smaller than the unperturbed roots if V0 is negative and larger if V0 is positive. The size of the shift in E due to V0 is small (negligible for large N) for all roots but one. This is characterized by saying that all but one level is “pinned” in between two unperturbed levels. As expected, these results are similar to the lattice defect vibration problem. It should be intuitive, if not obvious, that the state that splits off from the band for V0 negative is a localized state. We would get one such state for each band.

Problems 209

f(E)

f = 0 f = N/V0

E

Eb(k1) Eb(k2) …

Fig. 3.21. A qualitative plot of f(E) versus E for the Slater–Koster model. The crosses determine the energies that are solutions of (3.339)

This Section has discussed the effects of isolated impurities on electronic states. We have found, except for the formation of isolated localized states, that the Bloch view of a solid is basically unchanged. A related question is what happens to the concept of Bloch states and energy bands in a disordered alloy. Since we do not have periodicity here, we might expect these concepts to be meaningless. In fact, the destruction of periodicity may have much less effect on Bloch states than one might imagine. The changes caused by going from a periodic potential to a potential for a disordered lattice may tend to cancel one another out.28 However, the entire subject is complex and incompletely understood. For example, sufficiently large disorder can cause localization of electron states.29

Problems

3.1Use the variational principle to find the approximate ground-state energy of the helium atom (two electrons). Assume a trial wave function of the form exp[−η(r1+r2)], where rl and r2 are the radial coordinates of the electron.

3.2By use of (3.17) and (3.18) show that ∫ |ψ|2dτ = N! |M|2.

3.3Derive (3.31) and explain physically why ∑1N εk ≠ E .

28For a discussion of these and related questions, see Stern [3.53], and references cited therein.

29See Cusack [3.15].

210 3 Electrons in Periodic Potentials

3.4For singly charged ion cores whose charge is smeared out uniformly and for

plane-wave solutions so that |ψj| = 1, show that the second and third terms on the left-hand side of (3.50) cancel.

3.5Show that

	kM2 − k 2				kM + k				= 2 ,
lim			ln
		kkM			kM − k
k →0
and
		kM2 − k 2				kM + k				= 0 ,
lim				ln
						kM − k
k →kM		kkM
relate to (3.64) and (3.65).

3.6 Show that (3.230) is equivalent to

Ek = 12 (Ek0 + Ek0′) ± 14 [4 | V (K ′) |2 +(Ek0 − Ek0′)2 ]12 ,

where

	Ek0 =V (0) +	2k2	and	Ek0′ =V (0) +	2	(k + K ′)2 .
		2m			2m

3.7Construct the first Jones zone for the simple cubic lattice, face-centered cubic lattice, and body-centered cubic lattice. Describe the fcc and bcc with a sc lattice with basis. Assume identical atoms at each lattice point.

3.8Use (3.255) to derive E0 for the simple cubic lattice, the body-centered cubic lattice, and the face-centered cubic lattice.

3.9Use (3.256) to derive the density of states for free electrons. Show that your results check (3.164).

3.10For the one-dimensional potential well shown in Fig. 3.22 discuss either mathematically or physically the behavior of the low-lying energy levels as a function of V0, b, and a. Do you see any analogies to band structure?

3.11How does soft X-ray emission differ from the more ordinary type of X-ray emission?

3.12Suppose the first Brillouin zone of a two-dimensional crystal is as shown in Fig. 3.23 (the shaded portion). Suppose that the surfaces of constant energy

are either circles or pieces of circles as shown. Suppose also that where k is on a sphere or a spherical piece that E = (ћ2/2m)k2. With all of these assumptions, compute the density of states.

3.13Use Fermi–Dirac statistics to evaluate approximately the low-temperature specific heat of quasi free electrons in a two-dimensional crystal.

3.14For a free-electron gas at absolute zero in one dimension, show the average energy per electron is one third of the Fermi energy.

Problems 211

V+∞

V+∞

V

a b/2 b/2 a

V0

x

Fig. 3.22. A one-dimensional potential well

ky
π/a
	Surfaces of
	constant energy
π/a	kx

Fig. 3.23. First Brillouin zone and surfaces of constant energy in a simple two-dimensional reciprocal lattice

4The Interaction of Electrons and Lattice Vibrations

4.1 Particles and Interactions of Solid-state Physics (B)

There are, in fact, two classes of types of interactions that are of interest. One type involves interactions of the solid with external probes (such as electrons, positrons, neutrons, and photons). Perhaps the prime example of this is the study of the structure of a solid by the use of X-rays as discussed in Chap. 1. In this chapter, however, we are more concerned with the other class of interactions; those that involve interactions of the elementary energy excitations among themselves.

So far the only energy excitations that we have discussed are phonons (Chap. 2) and electrons (Chap. 3). Thus the kinds of internal interactions that we consider at present are electron–phonon, phonon–phonon, and electron–electron. There are of course several othe kinds of elementary energy excitations in solids and thus there are many other examples of interaction. Several of these will be treated in later parts of this book. A summary of most kinds of possible pair wise interactions is given in Table 4.1.

The concept of the “particle” as an entity by itself makes sense only if its life time in a given state is fairly long even with the interactions. In fact interactions between particles may be of such character as to form new “particles.” Only a limited number of these interactions will be important in discussing any given experiment. Most of them may be important in discussing all possible experiments. Some of them may not become important until entirely new types of solids have been formed. In view of the fact that only a few of these interactions have actually been treated in detail, it is easy to believe that the field of solid-state physics still has a considerable amount of growing to do.

We have not yet defined all of the fundamental energy excitations.1 Several of the excitations given in Table 4.1 are defined in Table 4.2. Neutrons, positrons, and photons, while not solid-state particles, can be used as external probes. For some purposes, it may be useful to make the distinctions in terminology that are noted in Table 4.3. However, in this book, we hope the meaning of our terms will be clear from the context in which they are used.

1 A simplified approach to these ideas is in Patterson [4.33]. See also Mattuck [17, Chap. 1].

214 4 The Interaction of Electrons and Lattice Vibrations

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