Patterson, Bailey - Solid State Physics Introduction to theory

44 2 Lattice Vibrations and Thermal Properties

and the time-independent Schrödinger wave equation that we wish to solve is

H cψ(ri , Rl ) = Eψ(ri , Rl ) .

(2.9)

The time-independent Schrödinger wave equation for the electrons, if one assumes the nuclei are at fixed positions Rl, is

H φ(r , R ) = E0φ(r , R ) .		(2.10)
0 i l	i l

Born and Huang [46] have made a perturbation expansion of the solution of (2.9) in powers of K. They have shown that if the wave function is evaluated to second order in K, then a product separation of the form ψn(ri, Rl) = φn(ri)X(Rl) where n labels an electronic state, is possible. The assertion that the total wave function can be written as a product of the electronic wave function (depending only on electronic coordinates with the nuclei at fixed positions) times the nuclear wave function (depending only on nuclear coordinates with the electrons in some fixed state) is the physical content of the Born–Oppenheimer approximation (1927). In this approximation the electrons provide a potential energy for the motion of the nuclei while the moving nuclei continuously deform the wave function of the electrons (rather than causing any sudden changes). Thus this idea is also called the adiabatic approximation.

It turns out when the wave function is evaluated to second order in K that the effective potential energy of the nuclei involves nuclear displacements to fourth order and lower. Expanding the nuclear potential energy to second order in the nuclear displacements yields the harmonic approximation. Terms higher than second order are called anharmonic terms. Thus it is possible to treat anharmonic terms and still stay within the Born–Oppenheimer approximation.

If we evaluate the wave function to third order in K, it turns out that a simple product separation of the wave function is no longer possible. Thus the Born– Oppenheimer approximation breaks down. This case corresponds to an effective potential energy for the nuclei of fifth order. Thus it really does not appear to be correct to assume that there exists a nuclear potential function that includes fifth or higher power terms in the nuclear displacement, at least from the viewpoint of the perturbation expansion.

Apparently, in actual practice the adiabatic approximation does not break down quite so quickly as the above discussion suggests. To see that this might be so a somewhat simpler development of the Born–Oppenheimer approximation [46] is sometimes useful. In this development, we attempt to find a solution for ψ in (2.9) of the form

ψ(ri , Rl ) = ∑nψn (Rl )φn (ri , Rl ) .

(2.11)

The φn are eigenfunctions of (2.10). Substituting into (2.9) gives

∑n H cψnφn = E∑nψnφn ,

2.1 The Born–Oppenheimer Approximation (A) 45

or

∑n H 0ψnφn + ∑n TNψnφn = E∑nψnφn ,

or using (2.10) gives

∑n En0ψnφn + ∑n TN (ψnφn ) = E∑nψnφn .

Noting that

TN (ψnφn ) = (TNψn )φn +ψn (TNφn ) + ∑l

1

(Plφn ) (Plψn ) ,

Ml

where

TN

= ∑

l

1

Pl2 = − ∑

1

2 ,

2M l

l 2M l

Rl

we can write the above as

∑

1 φ

1 (T

N

+ E0

− E)ψ 1 + ∑ 1

ψ

1T

N

φ

1

n

n

n

n

n

n

n

.

+ ∑n1 ∑l

1

(Pl ,φ 1 ) (Pl ,ψ 1 ) = 0

Ml

n

n

Multiplying the above equation by φn* and integrating over the electronic coordinates gives

(TN + En0 − E)ψn + ∑n1 Cnn1 (Rl , Pl )ψn1 = 0 ,						(2.12)
where
C 1 = ∑li	1		li	li
		Q	1 Pli + R	1		(2.13)
nn	Ml nn			nn

(the sum over i goes from 1 to 3, labeling the x, y, and z components) and
Qli	1 =		∫φn*Pliφ 1 dτ ,	(2.14)
nn			n
Rli 1	=	1	∫φn*Pli2φ 1 dτ .	(2.15)
nn	2		n

The integration is over electronic coordinates.

For stationary states, the φs can be chosen to be real and so it is easily seen that the diagonal elements of Q vanish:

Qnnli 1 = ∫φn Pliφndτ = 2i ∂X∂li ∫φn2dτ = 0 .

46 2 Lattice Vibrations and Thermal Properties

From this we see that the effect of the diagonal elements of C is a multiplication effect and not an operator effect. Therefore the diagonal elements of C can be added to E0n to give an effective potential energy Ueff.2 Equation (2.12) can be written as

(TN +Ueff − E)ψn + ∑n1(≠n) Cnn1ψn1 = 0 .

(2.16)

If the Cnn1 vanish, then we can split the discussion of the electronic and nuclear motions apart as in the adiabatic approximation. Otherwise, of course, we cannot. For metals there appears to be no reason to suppose that the effect of the C is negligible. This is because the excited states are continuous in energy with the ground state, and so the sum in (2.16) goes over into an integral. Perhaps the best way to approach this problem would be to just go ahead and make the Born– Oppenheimer approximation. Then wave functions could be evaluated so that the Cnn1 could be evaluated. One could then see if the calculations were consistent, by seeing if the C were actually negligible in (2.16).

In general, perturbation theory indicates that if there is a large energy gap between the ground and excited electronic states, then an adiabatic approximation may be valid.

Can we even speak of lattice vibrations in metals without explicitly also discussing the electrons? The above discussion might lead one to suspect that the answer is no. However, for completely free electrons (whose wave functions do not depend at all on the Rl) it is clear that all the C vanish. Thus the presence of free electrons does not make the Born–Oppenheimer approximation invalid (using the concept of completely free electrons to represent any of the electrons in a solid is, of course, unrealistic). In metals, when the electrons can be thought of as almost free, perhaps the net effect of the C is small enough to be neglected in zeroth-order approximation. We shall suppose this is so and suppose that the Born–Oppenheimer approximation can be applied to conduction electrons in metals. But we should also realize that strange effects may appear in metals due to the fact that the coupling between electrons and lattice vibrations is not negligible. In fact, as we shall see in a later chapter, the mere presence of electrical resistivity means that the Born–Oppenheimer approximation is breaking down. The phenomenon of superconductivity is also due to this coupling. At any rate, we can always write the Hamiltonian as H = H (electrons) + H (lattice vibrations) + H (coupling). It just may be that in metals, H (coupling) cannot always be regarded as a small perturbation.

Finally, it is well to note that the perturbation expansion results depend on K being fairly small. If nature had not made the mass of the proton much larger than the mass of the electron, it is not clear that there would be any valid Born– Oppenheimer approximation.3

2We have used the terms Born–Oppenheimer approximation and adiabatic approximation

interchangeably. More exactly, Born–Oppenheimer corresponds to neglecting Cnn, whereas in the adiabatic approximation Cnn is retained.

3For further details of the Born–Oppenheimer approximation, references [46], [82], [22, Vol 1, pp 611-613] and the references cited therein can be consulted.

2.2 One-Dimensional Lattices (B)	47

2.2 One-Dimensional Lattices (B)

Perhaps it would be most logical at this stage to plunge directly into the problem of solving quantum-mechanical three-dimensional lattice vibration problems either in the harmonic or in a more general adiabatic approximation. But many of the interesting features of lattice vibrations are not quantum-mechanical and do not depend on three-dimensional motion. Since our aim is to take a fairly easy path to the understanding of lattice vibrations, it is perhaps best to start with some simple classical one-dimensional problems. The classical theory of lattice vibrations is due to M. Born, and Born and Huang [2.5] contains a very complete treatment.

Even for the simple problems, we have a choice as to whether to use the harmonic approximation or the general adiabatic approximation. Since the latter involves quartic powers of the nuclear displacements while the former involves only quadratic powers, it is clear that the former will be the simplest starting place. For many purposes the harmonic approximation gives an adequate description of lattice vibrations. This chapter will be devoted almost entirely to a description of lattice vibrations in the harmonic approximation.

A very simple physical model of this approximation exists. It involves a potential with quadratic displacements of the nuclei. We could get the same potential by connecting suitable springs (which obey Hooke’s law) between appropriate atoms. This in fact is an often-used picture.

Even with the harmonic approximation there is still a problem as to what value we should assign to the “spring constants” or force constants. No one can answer this question from first principles (for a real solid). To do this we would have to know the electronic energy eigenvalues as a function of nuclear position (Rl). This is usually too complicated a many-body problem to have a solution in any useful approximation. So the “spring constants” have to be left as unknown parameters, which are determined from experiment or from a model that involves certain approximations.

It should be mentioned that our approach (which we could call the unrestricted force constants approach) to discussing lattice vibration is probably as straightforward as any and it also is probably as good a way to begin discussing the lattice vibration problem as any. However, there has been a considerable amount of progress in discussing lattice vibration problems beyond that of our approach. In large part this progress has to do with the way the interaction between atoms is viewed. In particular, the shell model4 has been applied with good results to ionic and covalent crystals.5 The shell model consists in regarding each atom as consisting of a core (the nucleus and inner electrons) plus a shell. The core and shell are coupled together on each atom. The shells of nearest-neighbor atoms are coupled. Since the cores can move relative to the shells, it is possible to polarize the atoms. Electric dipole interactions can then be included in neighbor interactions.

4See Dick and Overhauser [2.12].

5See, for example, Cochran [2.9].

48 2 Lattice Vibrations and Thermal Properties

Lattice vibrations in metals can be particularly difficult to treat by starting from the standpoint of force constants as we do. A special way of looking at lattice vibrations in metals has been given.6 Some metals can apparently be described by a model in which the restoring forces between ions are either of the bondstretching or axially symmetric bond-bending variety.7

We have listed some other methods for looking at the vibrational problems in Table 2.1. Methods, besides the Debye approximation (Sect. 2.3.3), for approximating the frequency distribution include root sampling and others [2.26, Chap. 3]. Montroll8 has given an elegant way for estimating the frequency distribution, at least away from singularities. This method involves taking a trace of the Dynamical Matrix (2.3.2) and is called the moment-trace method. Some later references for lattice dynamics calculations are summarized in Table 2.1.

Table 2.1. References for Lattice vibration calculations

Lattice vibrational	Reference
calculations
Einstein	Kittel [23, Chap. 5]
Debye	Chap. 2, this book
Rigid Ion Models	Bilz and Kress [2.3]
Shell Model	Jones and March [2.20, Chap. 3]. Also footnotes 4 and 5.
ab initio models	Kunc et al [2.22].
	Strauch et al [2.33]. Density Functional Techniques are used
	(see Chap. 3).
General reference	Maradudin et al [2.26]. See also Born and Huang [46]

2.2.1Classical Two-Atom Lattice with Periodic Boundary Conditions

(B)

We start our discussion of lattice vibrations by considering the simplest problem that has any connection with real lattice vibrations. Periodic boundary conditions will be used on the two-atom lattice because these are the boundary conditions that are used on large lattices where the effects of the surface are relatively unimportant. Periodic boundary conditions mean that when we come to the end of the lattice we assume that the lattice (including its motion) identically repeats itself. It will be assumed that adjacent atoms are coupled with springs of spring constant γ. Only nearest-neighbor coupling will be assumed (for a two-atom lattice, you couldn’t assume anything else).

6See Toya [2.34].

7See Lehman et al [2.23]. For a more general discussion, see Srivastava [2.32].

8See Montroll [2.28].

2.2 One-Dimensional Lattices (B)	49

As should already be clear from the Born–Oppenheimer approximation, in a lattice all motions of sufficiently small amplitude are describable by Hooke’s law forces. This is true no matter what the physical origin (ionic, van der Waals, etc.) of the forces. This follows directly from a Taylor series expansion of the potential energy using the fact that the first derivative of the potential evaluated at the equilibrium position must vanish.

The two-atom lattice is shown in Fig. 2.1, where a is the equilibrium separation of atoms, x1 and x2 are coordinates measuring the displacement of atoms 1 and 2 from equilibrium, and m is the mass of atom 1 or 2. The idea of periodic boundary conditions is shown by repeating the structure outside the vertical dashed lines.

With periodic boundary conditions, Newton’s second law for each of the two atoms is

mx1 = γ (x2 − x1 ) −γ (x1 − x2 ) ,	(2.17)
mx2 = γ (x1 − x2 ) −γ (x2 − x1 ) .

In (2.17), each dot means a derivative with respect to time.

Solutions of (2.17) will be sought in which both atoms vibrate with the same frequency. Such solutions are called normal mode solutions (see Appendix B). Substituting

xn = un exp(iωt)

(2.18)

in (2.17) gives

−ω 2 mu1 = γ (u2 −u1 ) −γ (u1 −u2 ) ,

−ω 2 mu2 = γ (u1 −u2 ) −γ (u2 −u1 ) .

Equation (2.19) can be written in matrix form as

	2γ −ω	2	m	− 2γ	u	1		= 0 .
	− 2γ			2γ −ω
					2 m u	2

		x1		x2
m	γ	m	γ	m	γ	m
			a

(2.19)

(2.20)

Fig. 2.1. The two-atom lattice (with periodic boundary conditions schematically indicated)

50 2 Lattice Vibrations and Thermal Properties

For nontrivial solutions (u1 and u2 not both equal to zero) of (2.20) the determinant (written det below) of the matrix of coefficients must be zero or

2γ −ω		2m − 2γ		= 0 .	(2.21)
det	− 2γ	2γ −ω
			2m

Equation (2.21) is known as the secular equation, and the two frequencies that satisfy (2.21) are known as eigenfrequencies.

These two eigenfrequencies are

ω12 = 0 ,	(2.22)
and
ω22 = 4γ / m .	(2.23)
For (2.22), u1 = u2 and for (2.23),
(2γ − 4γ )u1 = 2γu2 or u1 = −u2 .

Thus, according to Appendix B, the normalized eigenvectors corresponding to the frequencies ω1 and ω2 are

E	= (1,1) ,	(2.24)
1	2
and
E2	= (1,−1) .	(2.25)
	2

The first term in the row matrix of (2.24) or (2.25) gives the relative amplitude of u1 and the second term gives the relative amplitude of u2. Equation (2.25) says that in mode 2, u2/u1 = −1, which checks our previous results. Equation (2.24) describes a pure translation of the crystal. If we are interested in a fixed crystal, this solution is of no interest. Equation (2.25) corresponds to a motion in which the center of mass of the crystal remains fixed.

Since the quantum-mechanical energies of a harmonic oscillator are En = (n + 1/2)ћω, where ωis the classical frequency of the harmonic oscillator, it follows that the quantum-mechanical energies of the fixed two-atom crystal are given by

	1		4γ	.	(2.26)
En = n +	2		m

This is our first encounter with normal modes, and since we shall encounter them continually throughout this chapter, it is perhaps worthwhile to make a few

2.2 One-Dimensional Lattices (B)	51

more comments. The sets E1 and E2 determine the normal coordinates of the normal mode. They do this by defining a transformation. In this simple example, the theory of small oscillations tells us that the normal coordinates are

X1	= u1	+ u2	and X 2	= u1	− u2 .
	2	2		2	2

Note that X1, X2 are given by

X		1			E	u		=	1	1	1 u
				=	1	1					1	.
	X								2
		2		E2		u2				1	−1 u2

X1 and X2 are the amplitudes of the normal modes. If we want the time-dependent normal coordinates, we would multiply the first set by exp(iω1t) and the second set by exp(iω2t). In most applications when we say normal coordinates it should be obvious which set (time-dependent or otherwise) we are talking about.

The following comments are also relevant:

1.In an n-dimensional problem with m atoms, there are (n m) normal coordinates corresponding to nm different independent motions.

2.In the harmonic approximation, each normal coordinate describes an independent mode of vibration with a single frequency.

3.In a normal mode, all atoms vibrate with the same frequency.

4.Any vibration in the crystal is a superposition of normal modes.

2.2.2Classical, Large, Perfect Monatomic Lattice, and Introduction to Brillouin Zones (B)

Our calculation will still be classical and one-dimensional but we shall assume that our chain of atoms is long. Further, we shall give brief consideration to the possibility that the forces are not harmonic or nearest-neighbor. By a long crystal will be meant a crystal in which it is not very important what happens at the boundaries. However, since the crystal is finite, some choice of boundary conditions must be made. Periodic boundary conditions (sometimes called Born–von Kárman or cyclic boundary conditions) will be used. These boundary conditions can be viewed as the large line of atoms being bent around to form a ring (although it is not topologically possible analogously to represent periodic boundary conditions in three dimensions). A perfect crystal will mean here that the forces between any two atoms depend only on the separation of the atoms and that there are no defect atoms. Perfect monatomic further implies that all atoms are identical.

N atoms of mass M will be assumed. The equilibrium spacing of the atoms will be a. xn will be the displacement of the nth atom from equilibrium. V will be the

52 2 Lattice Vibrations and Thermal Properties

potential energy of the interacting atoms, so that V = V(x1,…,xn). By the Born– Oppenheimer approximation it makes sense to expand the potential energy to fourth order in displacements:

V (x1,…, xN ) =

2

(0,…,0)+

1

∂

V

V

∑

n,n′

xn xn′

2

∂x

∂x

n

n′

,…,x

)=0

(x

N

3

1

1

∂

+

∑

V

x

n

x

n

′x

′′

(2.27)

6

∂x

∂x

n,n′,n′′ ∂x

n

n′

n′′

,…,x

)=

0

n

(x

N

1

1

∂

4

V

+

∑

n,n′,n′′,n′′′

xn xn′xn′′xn′′′.

24

∂x

∂x

∂x

∂x

n

n′

n′′

,…,x

)=0

n′′′ (x

N

1

In (2.27), V(0,…,0) is just a constant and the zero of the potential energy can be

chosen so that this constant is zero. The first-order term (∂V/∂x)(x1,…,xN)=0 is the negative of the force acting on atom n in equilibrium; hence it is zero and was left

out of (2.27). The second-order terms are the terms that one would use in the harmonic approximation. The last two terms are the anharmonic terms.

Note in the summations that there is no restriction that says that n′ and n must refer to adjacent atoms. Hence (2.27), as it stands, includes the possibility of forces between all pairs of atoms.

The dynamical problem that (2.27) gives rise to is only exactly solvable in closed form if the anharmonic terms are neglected. For small oscillations, their effect is presumably much smaller than the harmonic terms. The cubic and higherorder terms are responsible for certain effects that completely vanish if they are left out. Whether or not one can neglect them depends on what one wants to describe. We need anharmonic terms to explain thermal expansion, a small correction (linear in temperature) to the specific heat of an insulator at high temperatures, and the thermal resistivity of insulators at high temperatures. The effect of the anharmonic terms is to introduce interactions between the various normal modes of the lattice vibrations. A separate chapter is devoted to interactions and so they will be neglected here. This still leaves us with the possibility of forces of greater range than nearest-neighbors.

It is convenient to define

			∂		2
V	n,n′	=				V				.	(2.28)
			∂x			∂x
				n						)=0
							n′ (x ,…,x		N
							1

Vn,n′ has several properties. The order of taking partial derivatives doesn’t matter, so that

Vn,n′ =Vn′,n .

(2.29)

2.2 One-Dimensional Lattices (B)	53

Two further restrictions on the V may be obtained from the equations of motion. These equations are simply obtained by Lagrangian mechanics [2]. From our model, the Lagrangian is

L = (M / 2)∑	n	x2	−	1	∑	n,n′	V	′x	n	x ′ .	(2.30)
		n	2					n,n		n

The sums extend over the one-dimensional crystal. The Lagrange equations are

	d		∂L	−	∂L	= 0 .	(2.31)
	dt ∂xn				∂xn
The equation of motion is easily found by combining (2.30) and (2.31):
M xn = −∑n′Vn,n′xn′ .							(2.32)

If all atoms are displaced a constant amount, this corresponds to a translation of the crystal, and in this case the resulting force on each atom must be zero. Therefore

∑n′Vn,n′ = 0 .

(2.33)

If all atoms except the kth are at their equilibrium position, then the force on the nth atom is the force acting between the kth and nth atoms,

F = M xn = −Vnk xk .

But because of periodic boundary conditions and translational symmetry, this force can depend only on the relative positions of n and k, and hence on their difference, so that

Vn,k	=V (n − k) .	(2.34)
With these restrictions on the V in mind, the next step is to solve (2.32).
Normal mode solutions of the form
xn	= uneiωt	(2.35)

will be sought. The un are assumed to be time independent. Substituting (2.35) into (2.32) gives

pu	n	≡ Mω2u	n	− ∑	n′	V (n′ − n)u	′ = 0 .	(2.36)
						n
Equation (2.36) is a difference equation						with constant coefficients. Note that
a new operator p is defined by (2.36).

This difference equation has a nice property due to its translational symmetry.

Let n go to n + 1 in (2.36). We obtain
Mω2u	n+1	− ∑	n′	V (n′ − n −1)u	′ = 0 .	(2.37)
				n