Fundamentals of the Physics of Solids / 12-The Quantum Theory of Lattice Vibrations

12.3 The Thermodynamics of Vibrating Lattices

417

CV /T

(J mol 1 K 2)

4	KCl
3
2
1	Cu

0					2
				T (K)
		5
0			10	15

Fig. 12.8. Temperature dependence of specific heat for KCl and Cu at low temperatures, with CV /T plotted against T 2 [based on P. H. Keesom and N. Pearlman,

Phys. Rev. 91, 1354 (1953), and H. M. Rosenberg, Low Temperature Solid State Physics (Oxford University Press, 1963)]

) K

molcal (

ecificSp eath

(I and II)

				T $D					2.5
			0.5			1	1.5	2
		6		Na	KBr
	)		Hg Cd				III
6	(III	5	Tl						6
5		4					Pb, Ag, KCl, Zn, NaCl, Cu, Al, CaF2, C		5
4		3				I			4
		II
3		Ag	Pb	Ca		Zn			3
2							C		2
							Fe
1	Al	CaF2		FeS2	Cu	KCl			1
						NaCl
			0.5			1	1.5	2	2.5

T $D

Fig. 12.9. Experimental data for the temperature dependence of the molar specific heat of some simple metals and ionic crystals, plotted against the reduced temperature T /ΘD. For better visibility measured data are partially shifted, however the solid line is always the result of the Debye interpolation formula [M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford (1954)]

In principle, the Debye temperature ΘD, which appears as a parameter in the specific heat, can be derived from the sound velocity. However, it is usually determined from specific heat data measured at low temperatures: once the electronic contribution is subtracted, (12.3.37) is fitted to the result.

418 12 The Quantum Theory of Lattice Vibrations

The experimental values of the Debye temperature are listed for some elements and compounds in Table 12.1.

Table 12.1. Debye temperature (in kelvins) for some elements and compounds, determined by fitting (12.3.37) to experimental specific heat data

Element ΘD (K)		Element ΘD (K)		Compound ΘD (K)		Compound ΘD (K)
Na	158	Mn	410	LiF	732	CaF2	510
K	91	Fe	467	NaCl	321	MgO	946
Cu	343	Ni	450	KCl	235	Fe2O3	660
Ag	225	C(d)	2230	RbCl	165	FeS2	637
Au	165	C(g)	420	CsCl	175	SiO2	470
Al	428	Ge	370	ZnS	315	Nb3Sn	228

If the assumptions of the Debye model were valid for real materials, the interpolation formula could be fitted to the measured data with a single parameter over the entire temperature range. However, the agreement is not always as good as in the cases shown in Fig. 12.9. Therefore when the Debye formula is fitted to the specific heat data over a relatively small temperature range around T , the obtained ΘD tends to be slightly di erent from the value derived from the low-temperature fit. It is in this sense that one may speak about the temperature dependence of the Debye temperature.

12.3.3 The Equation of State of the Crystal

A more complete analysis of the thermodynamic behavior of the crystal is based on the equation of state, which relates the pressure p, the volume V , and the temperature T . Starting with the free energy, pressure is given by the partial derivative

p = −	∂F	.	(12.3.38)
	∂V
		T

The free energy of the vibrating crystal is given by (12.3.20), therefore in the Debye model we have

F = E0	+ 9 p N ωD3		ωD				ω/kBT ω2dω
				ln 1 − e−
		kBT
			0
						ΘD /T	(12.3.39)
= E0 + 9 p N kBT					3		ln 1 − e−t t2 dt ,
				ΘD
				T
						0

where E0 is the ground-state energy due to zero-point vibrations.

12.3 The Thermodynamics of Vibrating Lattices	419
Volume dependence may occur through E0 and ΘD. Therefore the pressure

is

∂E

Θ

1 ∂Θ

p = −

0

+ 3 p N kBT D3

D

D

.

(12.3.40)

∂V

T

ΘD

∂V

Using (12.3.33) for the thermal energy ET due to lattice vibrations,

p = −

∂E0

+ γ

ET

(12.3.41)

,

∂V

V

where we have introduced the Grüneisen parameter 8

γ = −

V ∂ΘD

= −

∂ ln ΘD

(12.3.42)

.

ΘD

∂V

∂ ln V

In harmonic crystals phonon frequencies are independent of the volume, and therefore so is the Debye temperature. This is most easily demonstrated by noting that in crystals of orthorhombic or higher symmetry the lattice constant ai always appears in the combination qiai in the q dependence of the phonon frequencies, where qi is an integral multiple of 2π/Niai – so phonon frequencies are independent of the lattice constant. For a formal proof of general validity we shall assume that the crystal is strictly harmonic, therefore the relation

U (Rm) = U0 + 21	uα(Rm)Φαβ (Rm − Rn)uβ (Rn)	(12.3.43)

m,n,α,β

is exact. For simplicity, we have assumed that each primitive cell contains a single atom. If the lattice constant increases from a to (1+ )a, the equilibrium position changes from Rm to Rm = (1 + )Rm. If the actual ionic positions

are Rm + u(Rm) in the original system and Rm + u(Rm) with respect to the new equilibrium positions, we have

u(Rm) = Rm +

u

(Rm) .

(12.3.44)

Substituting this into the expression for the potential, and making use of the property that if Rm is the equilibrium position then all terms that are linear in the displacement u relative to it must vanish,

U (Rm) = U0 + 21 2		Rmα Φαβ (Rm − Rn)Rnβ
m,n,α,β
	(12.3.45)
+ 21	u	α(	R	m)Φαβ (Rm − Rn)	u	β (	R	n).
m,n,α,β

Since the force constants are the same as in the old system, so are the vibrational frequencies – that is to say frequencies are independent of the equilibrium volume:

8 E. Grüneisen, 1926.

420 12 The Quantum Theory of Lattice Vibrations

	∂ωλ(q)	= 0 .	(12.3.46)
	∂V

This implies that in the harmonic approximation the Grüneisen parameter must vanish, and so the equation of state is reduced to

p = −			∂E0			(12.3.47)
				.
			∂V
The pressure is then independent of temperature:
	∂p
		V = 0			,	(12.3.48)
	∂T

and therefore the specific heats at constant volume and constant pressure, and the adiabatic and isothermal compressibilities, related by the thermodynamic identities

cp = cV −	T (∂p/∂T )2		cp	=	(∂p/∂V )	(12.3.49)
	V	,			S
	V (∂p/∂V )T		cV		(∂p/∂V )T

are equal. Even more interesting is that the linear thermal expansion coe - cient of the solid, defined as

1			∂l	p	=	1		∂V	p
α =										,	(12.3.50)
	l		∂T			3V		∂T

where l is a characteristic linear dimension, vanishes in the harmonic approximation. To demonstrate this, consider the thermodynamic identity

	∂p	V		∂T	p	∂V	T	= −1	(12.3.51)
	∂T			∂V		∂p

and the bulk modulus K, the inverse of the isothermal compressibility κ defined as

K =	1	= −V	∂p	T	;	(12.3.52)
	κ		∂V

the coe cient of thermal expansion may then be written as

1			∂p	V .
α =					(12.3.53)
	3K		∂T

According to (12.3.48), pressure is independent of temperature in the harmonic approximation, therefore the coe cient of thermal expansion vanishes. To account for the observed finite thermal expansion, we have to go beyond the harmonic approximation.

12.4 Anharmonicity

421

12.4 Anharmonicity

The possibility of describing the vibrations of the lattice in terms of independent collective excitations (phonons) stems from the applicability of the harmonic approximation to the crystal potential. Keeping only the secondorder terms in the displacement, an exact diagonalization of the Hamiltonian was possible using the phonon creation and annihilation operators. When higher-order corrections become important, the harmonic approximation breaks down. Phonons will no longer be well-defined elementary excitations, as they can decay or be scattered by one another. Such processes change all the physical quantities that depend on the thermally excited states of the crystal.

12.4.1 Higher-Order Expansion of the Potential
To fourth order, the expansion of the potential is
1					∂2U
U ({u(m, μ}) = U0 +						uα(m, μ)uβ (n, ν)
		2	m,μ,α	∂uα(m, μ)∂uβ (n, ν)
			n,ν,β
1					∂3U
+
					∂uα(m, μ)∂uβ (n, ν)∂uγ (p, π)
3!			m,μ,α,n,ν,β
			p,π,γ
1			× uα(m, μ)uβ (n, ν)uγ (p, π)					(12.4.1)
						∂4U
+
					∂uα(m, μ)∂uβ (n, ν)∂uγ (p, π)∂uδ (r, ρ)
4!			m,μ,α,n,ν,β
			p,π,γ,r,ρ,δ

× uα(m, μ)uβ (n, ν)uγ (p, π)uδ (r, ρ) + . . . .

When going beyond the second order, one has to go to the fourth order at least, since the third-order potential does not have an absolute minimum, and so the crystal would become unstable to large displacements otherwise. The fourth-order term stabilizes the potential. This problem does not arise in perturbation theory, however thirdand fourth-order corrections may be of the same order because of the restrictions imposed on the third-order terms.

Since the corrections due to anharmonicity are expected to be small, we shall assume that it is su cient to take anharmonicity into account only in a relatively low order of perturbation theory. Therefore we shall pass to the quantum mechanical discussion by neglecting the anharmonic term in the first step of the usual procedure. This means that we shall determine the normal modes from the dynamical matrix made up of the harmonic terms of the potential energy, the normal coordinates from the normal mode expansion of the displacement, and the canonical momenta from the Hamiltonian (i.e., the sum of the harmonic terms of the kinetic and potential energies), and then

422 12 The Quantum Theory of Lattice Vibrations

impose canonical commutation relations on these coordinates and momenta. We then introduce phonon creation and annihilation operators, express the displacement in terms of them – as in (12.1.39) –, and use the quantized, operator-like quantity u in the thirdand fourth-order terms of the expansion of the potential.

For notational simplicity, we shall consider a crystal with a monatomic basis, and write uα(m) as uαm. The Hamiltonian of the anharmonic crystal is then

1

∂3U

qλ

αβγ

m

β γ umα unβ upγ

H =

ωλ(q) aλ†

(q)aλ(q) + 21 + 3!

α

m,n,p ∂u

∂un∂up

1

∂4U

+

umα unβ upγ urδ + . . . ,

(12.4.2)

4! m,n,p,r ∂umα ∂unβ ∂upγ ∂urδ

αβγδ

where (12.1.39) should be used for the ionic displacements. The expansion co- e cients are represented by double or triple Fourier series, exploiting the property that the potential depends only on the distance between lattice points. For example, for the third-order term

αβγ

∂3U

αβγ

iq

(R

R )

iq

(R

R )

Φ

(m, n, p) =

=

D

3

(q, q )e

·

m −

n e

·

m −

p

.

3

∂uα

∂uβ

∂uγ

m

n

p

qq

(12.4.3)

Since the displacement is the linear combination of the operators aλ(q) and a†λ(q), the Hamiltonian will contain eight terms that are cubic and sixteen that are quartic in them. To get a better idea of their structure, we write out in full detail a third-order term in which the annihilation operator is chosen in each u:

1

D3αβγ (q, q )eiq·(Rm −Rn )eiq ·(Rm −Rp )

(12.4.4)

3!

m,n,p qq

αβγ

1

1/2

1/2

1/2

q1 q2 q3

×

(2M N )3/2

ωλ1 (q1)

ωλ2 (q2)

ωλ3 (q3)

λ λ

λ1 2 3

×eα(λ1)(q1)eβ(λ2 )(q2)eγ(λ3 )(q3)aλ1 (q1)aλ2 (q2)aλ3 (q3)eiq1·Rm eiq2 ·Rn eiq3·Rp .

The sum over the lattice points vanishes unless

q1 + q + q = G1 ,

q2 − q = G2 ,

q3 − q = G3 ,

(12.4.5)

where Gi is a reciprocal-lattice vector, that is

q1 + q2 + q3 = G .

(12.4.6)

12.4 Anharmonicity

423

This term of the Hamiltonian corresponds to a process in which three phonons are absorbed simultaneously. The sum of the three wave vectors is then required to be equal to a reciprocal-lattice vector. Normal processes occur for G = 0, while G = 0 is associated with umklapp processes.

Terms that contain one creation and two annihilation operators, or two creation and one annihilation operator, or three creation operators also appear.

12.4.2 Interaction Among the Phonons

The possible processes that correspond to the various terms of the third-order expansion are illustrated diagrammatically in Fig. 12.10. Each wavy line represents a phonon. Moving from left to right, a line leaving a vertex corresponds to the emission of a phonon, while a line entering a vertex corresponds to the absorption of a phonon. The wave vectors cannot be chosen arbitrarily. As the consequence of discrete translational symmetry, the conservation of crystal momentum must hold in each process.

Fig. 12.10. Anharmonic processes with three phonons

The simultaneous absorption or emission of three phonons is important when calculating the total energy of the crystal but they are irrelevant to the physical properties that are of interest here. Such processes may exist because the intermediate (virtual) states live for a short time only, and, on account of the uncertainty principle, their energy may be arbitrary; energy conservation is required only when transition probabilities are calculated between an initial and a final state. Much more important are those terms in the Hamiltonian that contain one creation and two annihilation operators, or two creation and one annihilation operator.

These correspond to real physical processes in which two phonons merge into one or one phonon is split into two. When the conservation of crystal momentum and energy are imposed simultaneously, severe restrictions apply to the phonons that participate in the process. Suppose that in addition to

q1 + q2 = q3 + G	(12.4.7)
the condition
ωλ1 (q1) + ωλ2 (q2) = ωλ3 (q3)	(12.4.8)

is also fulfilled. Using the Debye model it is immediately seen that only a triplet of collinear wave vectors can satisfy both conditions simultaneously.

424 12 The Quantum Theory of Lattice Vibrations

For more realistic dispersion relations an even stricter restriction is obtained. As the dispersion curve of acoustic phonons is concave downward, energy and momentum cannot be simultaneously conserved in a process where the three phonons belong to the same acoustic branch.

Figure 12.11 shows the dispersion relation of acoustic phonons along a selected direction, assuming that transverse vibrations are degenerate. Plotting the dispersion curves once again, this time from the point that corresponds to the wave vector q1, the intersection points of the two families of curves give the possible values of q2 and q3. As longitudinal phonons are usually more energetic than transverse ones, a longitudinal phonon can easily decay into two transverse phonons or into a longitudinal and a transverse phonon, while the decay of transverse phonons may occur only via umklapp processes.

	!		!
	!3		!3
	!2
				!2
	!1				!1
a	q1 q2 a	a	0		a	a
(a)	q3	(b)	q3	q1	q2

Fig. 12.11. Illustration of energy and momentum conservation for three-phonon processes: (a) normal process; (b) umklapp process

As phonons may decay as well as merge, they can no longer be considered as infinitely long-lived elementary excitations: their lifetime is necessarily fi- nite. Because of the same processes, the energy of phonons is modified – renormalized – as a result of its interactions with other phonons. The contributions of such processes can be determined applying the methods used in many-body problems.

The role and contributions of fourth-order processes can be analyzed along the same lines. Simultaneous absorption and emission of four phonons are of little interest. The physically relevant processes are shown in Fig. 12.12.

Fig. 12.12. Anharmonic processes with four phonons

12.4 Anharmonicity

425

12.4.3 Thermal Expansion and Thermal Conductivity in Crystals

In (12.3.53), using some general thermodynamic considerations, the coe cient of thermal expansion was written as

1			∂p	V .
α =					(12.4.9)
	3K		∂T

It was also demonstrated in the same section that α vanishes in the harmonic approximation. To examine the role of anharmonicity, we shall start with formula (12.3.41) that establishes the relationship between pressure and internal energy. This implies

	∂p	= γ	CV	= γ cV ,	(12.4.10)
	∂T		V
		V

from which the Grüneisen relation9 between the coe cient of thermal expansion and specific heat

	α = γ	cV	(12.4.11)
		3K

is easily derived. Assuming as a first approximation a temperature-independent bulk modulus K, the coe cient of thermal expansion is found to be proportional to specific heat, that is, they exhibit the same temperature dependence.

Although the Grüneisen parameter γ vanishes in the harmonic approximation, in real crystals it does not because of the anharmonic terms. Substituting the experimental data for the coe cient of thermal expansion and specific heat into (12.4.11), a nonzero value is obtained for γ, which in general depends on the temperature. The Grüneisen relation nevertheless makes sense, since at low temperatures the coe cient of thermal expansion shows the same T 3 dependence as the specific heat, therefore the Grüneisen relation can be satisfied by a temperature-independent γ over a fairly broad temperature range. The situation is similar at room temperatures or above (i.e., at temperatures comparable to or higher than the Debye temperature), where the specific heat and the coe cient of thermal expansion are both constant. Table 12.2 shows the room-temperature Grüneisen parameter for a few elements.

When the thermal expansion coe cient is calculated for metals, the contribution of the electrons should not be ignored. We shall not revisit this question in Volume 2, where we focus on electrons, therefore it should be mentioned here that for the gas of free electrons

	pel =	2 Eel			,	(12.4.12)
		3 V

as we shall prove it in Chapter 16. This implies

∂pel	=	2	cel .	(12.4.13)
∂T
		3 V

9 E. Grüneisen, 1926.

426 12 The Quantum Theory of Lattice Vibrations

Table 12.2. The Grüneisen parameter for a few elements, measured at room temperature

Element	γ	Element	γ
Ag	2.40	Fe	1.60
Al	2.17	Co	1.87
Au	3.03	Si	0.45
Cu	1.96	Ge	0.73

The total thermal expansion coe cient is then
α =	1	γ cVion +	2	cVel	.	(12.4.14)
	3K		3

As we shall see, the specific heat of the electron gas is proportional to the temperature. At low temperatures, where the electronic contribution to the specific heat exceeds the phonon contribution, the coe cient of thermal expansion is proportional to T .

The previous considerations were based on the equation of state in the Debye model. For a more general interpretation of the Grüneisen parameter, the thermodynamic relations have to be applied to a system with a general dispersion relation. Starting from the free energy F = E − T S, the temperature dependence of pressure can be determined using (12.3.38). When the contribution of lattice vibrations is studied, (12.3.20) can be used for the free energy of independent phonons, with the assumption that phonon frequencies

may be volume dependent:

1

∂

∂

1

p = −

ωλ(q) + q,λ

−

ωλ(q)

.

(12.4.15)

2

∂V q,λ

∂V

eβ ωλ(q)

−

1

The first term gives the contribution of the ground-state energy, and the second term accounts for the temperature dependence. Substituting this into (12.3.53) for the coe cient of thermal expansion,

α =	1		−	∂	ωλ(q)	∂		(12.4.16)
	3K qλ			∂V		∂T nλ(q) .

To relate this formula to the form

cV = 1

ωλ(q) ∂ nλ(q) =

cV λ(q)

V

q,λ

∂T

q,λ

of the specific heat, we introduce a new quantity

γλ(q) = −

V ∂ωλ(q)

= −

∂ ln ωλ(q)

ωλ(q)

∂V

∂ ln V

(12.4.17)

(12.4.18)