Patterson, Bailey - Solid State Physics Introduction to theory

94 2 Lattice Vibrations and Thermal Properties

The density of states for mode p (which is the number of modes of type p per unit frequency) is

D p (ω) =	dn	=	ω2V	.	(2.224)
	dω		(2π 2c3p )

In (2.224), cp means the velocity of the wave in mode p.

Debye assumed (consistent with the isotropic continuum limit) that there were two transverse modes and one longitudinal mode. Thus for the total density of states, we have D(ω) = (ω2V/2π2) (1/cl3 + 2/ct3), where cl and ct are the velocities of the longitudinal and transverse modes. However, the total number of modes must be 3NK. Thus, we have

3NK = ∫0ωD D(ω)dω .

Note that when K = 2 = the number of atoms per unit cell, the assumptions we have made push the optic modes into the high-frequency part of the density of states. We thus have

3NK = ∫	V			1	+	1			dω
							ω
ωD								2		.	(2.225)
0	2π	2		3		3
			cl			ct

We have assumed only one cutoff frequency ωD. This was not necessary. We could just as well have defined a set of cutoff frequencies by the set of equations

2NK = ∫ωDt	D(ω)t dω,	(2.226)
0
NK = ∫ωDl	D(ω)l dω.
0

There are yet further alternatives. But we are already dealing with a phenomenological treatment. Such modifications may improve the agreement of our results with experiment, but they hardly increase our understanding from a fundamental point of view. Thus for simplicity let us also assume that cp = c = constant. We can regard c as some sort of average of the cp.

Equation (2.225) then gives us

	6π 2 Nc3	1/ 3
ωD =	V	K .

The Debye temperature θD is defined as

	ω	D				6π 2 NKc3	1/ 3
θD =			=				.
	k					V
				k

Combining previous results, we have for the specific heat

Cv =	3	∫ωD	( ω)2 exp( ω / kT )		V	ω 2dω ,
	kT 2				2π 2c3
		0 [exp( ω / kT ) −1]2

(2.227)

(2.228)

2.3 Three-Dimensional Lattices	95

which gives for the specific heat per unit volume (after a little manipulation)

	Cv	= 9k(NK /V )D(θD / T ) ,	(2.229)
	V

where D(θD/T) is the Debye function defined by

D(θD / T ) = (T /θD )3 ∫θD T	z 4ez dz	.	(2.230)
0	(ez −1)2

In Problem 2.13, you are asked to show that (2.230) predicts a T 3 dependence for Cv at low temperature and the classical limit of 3k(NK) at high temperature. Table 2.3 gives some typical Debye temperatures. For metals θD in K for Al is about 394, Fe about 420, and Pb about 88. See, e.g., Parker [24, p 104].

Table 2.3. Approximate Debye temperature for alkali halides at 0 K

Alkali halide	Debye temperature (K)
LiF	734
NaCl	321
KBr	173
RbI	103

Adapted with permission from Lewis JT et al. Phys

Rev 161, 877, 1967. Copyright 1967 by the American

Physical Society.

In discussing specific heats there is, as mentioned, one big difference between the one-dimensional case and the three-dimensional case. In the one-dimensional case, the dispersion relation is known exactly (for nearest-neighbor interactions) and from it the density of states can be exactly computed. In the three-dimensional case, the dispersion relation is not known, and so the dispersion relation of a classical isotropic elastic continuum is often used instead. From this dispersion relation, a density of states is derived. As already mentioned, in recent years it has been possible to determine the dispersion relation directly by the technique of neutron diffraction (which will be discussed in a later chapter). Somewhat less accurate methods are also available. From the dispersion relation we can (rather laboriously) get a fairly accurate density of states curve. Generally speaking, this density of states curve does not compare very well with the density of states used in the Debye approximation. The reason the error is not serious is that the specific heat uses only an integral over the density of states.

In Fig. 2.9 and Fig. 2.10 we have some results of dispersion curves and density of states curves that have been obtained from neutron work. Note that only in the crudest sense can we say that Debye theory fits a dispersion curve as represented by Fig. 2.10. The vibrational frequency spectrum can also be studied by other methods such as for example by X-ray scattering. See Maradudin et al [2.26, Chap. VII] and Table 2.4.

96 2 Lattice Vibrations and Thermal Properties

Table 2.4. Experimental methods of studying phonon spectra

Method

Reference

Inelastic scattering of neutrons by phonons.

Brockhouse and Stewart [2.6].

See the end of Sect. 4.3.1

Shull and Wollan [2.31]

Inelastic scattering of X-rays by phonons (in

Dorner et al [2.13]

which the diffuse background away from

Bragg peaks is measured). Synchrotron radia-

tion with high photon flux has greatly facili-

tated this technique.

Raman scattering (off optic modes) and Bril-

Vogelgesang et al [2.36].

louin scattering (off acoustic modes). See

Sect. 10.11.

Fig. 2.9. Measured dispersion curves. The dispersion curves are for Li7F at 298 °K. The results are presented along three directions of high symmetry. Note the existence of both optic and acoustic modes. The solid lines are a best least-squares fit for a seven-parameter model. [Reprinted with permission from Dolling G, Smith HG, Nicklow RM, Vijayaraghavan PR, and Wilkinson MK, Physical Review, 168(3), 970 (1968). Copyright 1968 by the American Physical Society.] For a complete definition of all terms, reference can be made to the original paper

The Debye theory is often phenomenologically improved by letting θD = θD(T) in (2.229). Again this seems to be a curve-fitting procedure, rather than a procedure that leads to better understanding of the fundamentals. It is, however, a good

2.3 Three-Dimensional Lattices	97

Fig. 2.10. Density of states g(ν) for Li7F at 298 °K. [Reprinted with permission from Dolling G, Smith HG, Nicklow RM, Vijayaraghavan PR, and Wilkinson MK, Physical Review, 168(3), 970 (1968). Copyright 1968 by the American Physical Society.]

way of measuring the consistency of the Debye approximation. That is, the more θD varies with temperature, the less accurate the Debye density of states is in representing the true density of states.

We should mention that from a purely theoretical point we know that the Debye model must, in general, be wrong. This is because of the existence of Van Hove singularities [2.35]. A general expression for the density of states involves one over the k space gradient of the frequency (see (3.258)). Thus, Van Hove has shown that the translational symmetry of a lattice causes critical points [values of k for which kωp(k) = 0] and that these critical points (which are maxima, minima, or saddle points) in general cause singularities (e.g. a discontinuity of slope) in the density of states. See Fig. 2.10. It is interesting to note that the approximate Debye theory has no singularities except that due to the cutoff procedure.

The experimental curve for the specific heat of insulators looks very much like Fig. 2.11. The Debye expression fits this type of curve fairly well at all temperatures. Kohn has shown that there is another cause of singularities in the phonon spectrum that can occur in metals. These occur when the phonon wave vector is twice the Fermi wave vector. Related comments are made in Sects. 5.3, 6.6, and 9.5.3.

In this chapter we have set up a large mathematical apparatus for defining phonons and trying to understand what a phonon is. The only thing we have calculated that could be compared to experiment is the specific heat. Even the specific heat was not exactly evaluated. First, we made the Debye approximation. Second,

98 2 Lattice Vibrations and Thermal Properties

if we had included anharmonic terms, we would have found a small term linear in T at high T. For the experimentally minded student, this is not very satisfactory. He would want to see calculations and comparisons to experiment for a wide variety of cases. However, our plan is to defer such considerations. Phonons are one of the two most important basic energy excitations in a solid (electrons being the other) and it is important to understand, at first, just what they are.

We have reserved another chapter for the discussion of the interactions of phonons with other phonons, with other basic energy excitations of the solid, and with external probes such as light. This subject of interactions contains the real meat of solid-state physics. One topic in this area is introduced in the next section. Table 2.5 summarizes simple results for density of states and specific heat in one, two, and three dimensions.

Table 2.5. Dimensionality and frequency (ω) dependence of long-wavelength acoustic phonon density of states D(ω), and low-temperature specific heat Cv of lattice vibrations

	D(ω)		Cv
One dimension	A1		B1T
Two dimensions	A2	ω	B2	T2
Three dimensions	A3	ω2	B3	T3

Note that the Ai and Bi are constants.

cv

T

θD

Fig. 2.11. Sketch of specific heat of insulators. The curve is practically flat when the temperature is well above the Debye temperature

2kBT

2.3 Three-Dimensional Lattices	99

2.3.4Anharmonic Terms in The Potential / The Gruneisen Parameter

(A)19

We wish to address the topic of thermal expansion, which would not exist without anharmonic terms in the potential (for then the average position of the atoms would be independent of their amplitude of vibration). Other effects of the anharmonic terms are the existence of finite thermal conductivity (which we will discuss later in Sect. 4.2) and the increase of the specific heat beyond the classical Dulong and Petit value at high temperature. Here we wish to obtain an approximate expression for the coefficient of thermal expansion (which would vanish if there were no anharmonic terms).

We first derive an expression for the free energy of the lattice due to thermal vibrations. The free energy is given by

FL = −kBT ln Z ,							(2.231)
where Z is the partition function. The partition function is given by
Z = ∑	exp(−βE	) ,		β =	1	,	(2.232)
{n}	{n}				kBT
where
E{n} = ∑k, j (nk +		1	)	ω j (k)			(2.233)
		2

in the harmonic approximation and ωj(k) labels the frequency of the different modes at wave vector k. Each nk can vary from 0 to ∞. The partition function can be rewritten as

Z = ∑n ∑n	2	exp(−βE{n			k	})
1			1
= ∏ ∏ exp[−β(nk +				)		ω j (k)]
			2
k, j nk
= ∏exp[− ω j (k) / 2] ∏ exp[−βnk ω j (k)],
k, j			nk

which readily leads to

F		= k		T ∑
	L		B		k, j	ln 2sinh

ω j (k)

. (2.234)

⎠

19 [2.10, 1973, Chap. 8]

100 2 Lattice Vibrations and Thermal Properties

Equation (2.234) could have been obtained by rewriting and generalizing (2.74). We must add to this the free energy at absolute zero due to the increase in elastic energy if the crystal changes its volume by V. We call this term U0.20

						ω j (k)
F = k		T ∑		ln 2sinh				+U		.
						2kBT
	B		k, j						0

We calculate the volume coefficient of thermal expansion α

		1		∂V
	α =			.
		V
				∂T P

But,

	∂V		∂P			∂T
							= −1.
	∂T P		∂V T			∂P V

(2.235)

(2.236)

The isothermal compressibility is defined as

		1		∂V
	κ = −						,
		V
				∂P			T
	then we have
				∂P
	α = κ					.
				∂T V
	But
				∂F
	P = −						,
				∂V T

so

	∂U	0						ω j (k)
P = −			− k	B	T ∑	k, j	coth
	∂V							2kBT

			∂ω j (k)
				.
	2kBT		∂V

(2.237)

(2.238)

(2.239)

The anharmonic terms come into play by assuming the ωj(k) depend on volume. Since the average number of phonons in the mode k, j is

1

1

ω j (k)

n j (k) =

−1 .

ω j (k)

=

2

coth

2kBT

(2.240)

exp

−1

kBT

20U0 is included for completeness, but we end up only using a vanishing temperature derivative so it could be left out.

2.3 Three-Dimensional Lattices	101

Thus

P = −

∂U

0

− ∑

k, j

(n

j

(k) +

1

)

∂ω j (k)

.

(2.241)

∂V

∂V

2

We define the Gruneisen parameter for the mode k, j as

γ

j (k) = −

V

∂ω j (k)

= −

∂lnω j (k)

.

ω j (q)

∂V

∂lnV

Thus

P = −

∂

[U

0

+ ∑

1

ω

j

(k)] + ∑

n

j

(k)

ω j (k)γ j

.

∂V

k, j 2

k, j

V

However, the lattice internal energy is (in the harmonic approximation) U = ∑k, j (n j (k) + 12 ) ω j (k) .

So

∂U

= ∑k, j

ω j (k)

∂n j (k)

,

∂T

∂T

cv =

1

∂U =

1

ωj (k)

∂

j (k)

= ∑cv j (k) ,

∑

n

k , j

V ∂T V

∂T

(2.242)

(2.243)

(2.244)

(2.245)

(2.246)

which defines a specific heat for each mode. Since the first term of P in (2.243) is independent of T at constant V, and using

α= κ ∂P

,

∂T V

we have

	α =κ		1	∑k, j	ω j (k)γ j (k)	∂n j (k)	.	(2.247)
		V				∂T
	Thus
		α = κ∑k, j γ j (k) cv j (k) .						(2.248)

Let us define the overall Gruneisen parameter γT as the average Gruneisen parameter for mode k, j weighted by the specific heat for that mode. Then by (2.242) and (2.246) we have

cvγT = ∑k, j γ j (k) cv j (k) .

(2.249)

102 2 Lattice Vibrations and Thermal Properties

We then find

α = κγT cv .

(2.250)

If γT (the Gruneisen parameter) were actually a constant α would tend to follow the changes of cV, which happens for some materials.

From thermodynamics

c	P	= c	+	α2T	,	(2.251)
		V		κ

so cp = cv(1 + γαT) and γ is often between 1 and 2.

Table 2.6. Gruneisen constants

Temperature		LiF	NaCl	KBr	KI
0	K	1.7 ± 0.05	0.9 ± 0.03	0.29 ± 0.03	0.28 ± 0.02
283	K	1.58	1.57	1.49	1.47

Adaptation of Table 3 from White GK, Proc Roy Soc London A286, 204, 1965. By permission of The Royal Society.

2.3.5Wave Propagation in an Elastic Crystalline Continuum21 (MET, MS)

In the limit of long waves, classical mechanics can be used for the discussion of elastic waves in a crystal. The relevant wave equations can be derived from Newton’s second law and a form of Hooke’s law. The appropriate generalized form of Hooke’s law says the stress and strain are linearly related. Thus we start by defining the stress and strain tensors.

The Stress Tensor (σij) (MET, MS)

We define the stress tensor σij in such a way that

	σ yx =	Fy		(2.252)
		y	z

for an infinitesimal cube. See Fig. 2.12. Thus i labels the force (positive for tension) per unit area in the i direction and j indicates which face the force acts on (the face is normal to the j direction). The stress tensor is symmetric in the absence of body torques, and it transforms as the products of vectors so it truly is a tensor.

21 See, e.g., Ghatak and Kothari [2.16, Chap. 4] or Brown [2.7, Chap. 5].

2.3 Three-Dimensional Lattices	103

By considering Fig. 2.13, we derive a useful expression for the stress that we will use later. The normal to dS is n and σindS is the force on dS in the ith direction. Thus for equilibrium

σindS = σix nxdS + σiy n y dS + σiz nz dS ,

so that

σin = ∑ j σij n j .						(2.253)
y
			Fy
					y	z
	y					x
			z
	x

z

Fig. 2.12. Schematic definition of stress tensor σij

y

n ( to plane of dS)

σiynydS dS

x

z

Fig. 2.13. Useful pictorial of stress tensor σij