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

12

The Quantum Theory of Lattice Vibrations

In the previous chapter lattice vibrations were analyzed using the classical equations of motion. It was established that a much higher specific heat is predicted by classical statistical physics for the thermal excitation of such modes than what is observed experimentally at low temperatures. To obtain a better agreement a quantum mechanical treatment is required. In addition to providing a more precise description for the thermodynamic behavior of solids, this will also help to understand the influence of lattice vibrations on transport and optical properties (which will be discussed in Volume 2).

12.1 Quantization of Lattice Vibrations

Perhaps the most important result of the previous chapter is that the classical oscillatory motion of atoms in a lattice can be described in terms of harmonic oscillators. Starting with this, it is a fairly simple matter to derive and illustrate the quantized energy spectrum of a vibrating lattice, as the determination of the wavefunction and energy eigenvalues of a harmonic oscillator ranks among the easiest problems in quantum mechanics. Before turning to the general solution of the quantum mechanical problem, we shall present some simple models.

12.1.1 The Einstein Model

Shortly after Planck’s proposal of the quantum hypothesis,1 in 1907 Einstein extended the hypothesis to lattice vibrations, postulating that their energy is also quantized, and for a lattice vibration of frequency ν it can

1M. Planck, 1900. Max Karl Ernst Ludwig Planck (1858–1947) was awarded the Nobel Prize in 1918 “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta”.

388 12 The Quantum Theory of Lattice Vibrations

take only integral multiples of hν: εn = nhν. The mean thermal energy of a vibrational mode is then

∞∞

ε	=			=			.	(12.1.1)
		εne−εn /kBT				nhνe−nhν/kBT
		n=0				n=0
		∞				∞
		e−εn /kBT				e−nhν/kBT
		n=0				n=0
In terms of the partition function
		∞
				e−εn /kBT				(12.1.2)
		Z =
		n=0
the mean thermal energy may be written as
		ε = −		d
					ln Z ,			(12.1.3)
				dβ

where β = 1/kBT . For harmonic oscillators the partition function is the sum of an infinite geometrical series:

∞					1
	e−nhν/kBT = 1			−	e−hν/kBT			(12.1.4)
Z =							,
n=0
and so		hν
	ε =					.		(12.1.5)
		ehν/kBT − 1

For simplicity Einstein also assumed that independently of each other, atoms oscillate with the same frequency νE; in other words frequency is independent of the wave vector of lattice vibrations. The model based on the assumption that the frequency is the same for each vibrational mode is called the Einstein model. Instead of the frequency νE we shall use the angular frequency ωE. Assuming that there are N p atoms in the crystal, and each of them can oscillate in any of the three spatial directions, the internal energy of these vibrations is

ωE
E = 3N p ε = 3N p e ωE/kB T − 1 .	(12.1.6)

From this expression the specific heat due to lattice vibrations is found to be

	∂E	= 3N p kB	ωE	2	e ωE/kBT
CV =							.	(12.1.7)
	∂T		kBT		(e ωE /kBT − 1)2
On the right-hand side				x2ex
		F (x) =						(12.1.8)
				(ex − 1)2

12.1 Quantization of Lattice Vibrations

389

Fig. 12.1. The temperature dependence of specific heat in the Einstein model, fitted to experimental data for diamond [based on A. Einstein, Ann. d. Phys. 22, 180 (1907)]

is the Einstein function with the argument x = ωE/kBT . Figure 12.1 shows the temperature dependence of the specific heat, compared to the experimental data that were known at the beginning of the 20th century.

At su ciently high temperatures, where kBT ωE, the first term of the expansion of the Einstein function for small values of x,

F (x) =	x2(1 + x	+ . . . )	= 1 + O(x2)	(12.1.9)
	(x + x2/2	+ . . . )2

leads to the Dulong–Petit value of specific heat. At lower temperatures finiteenergy oscillations are more di cult to excite thermally, therefore the specific heat decreases. In the T → 0 limit asymptotically

F (x) = x2e−x,	if	x		1 ,	(12.1.10)

that is, the specific heat vanishes exponentially. Measurements performed since that time in the low-temperature region have shown a slower, power-law decay. Obviously, the excessively rapid decay of the specific heat in the Einstein model is due to the absence of long-wavelength acoustic vibrations that correspond to low-frequency – and therefore low-energy – collective modes. Such modes can be excited thermally more easily than Einstein oscillators.

12.1.2 The Debye Model

To correct the error of the Einstein model mentioned above, in 1912 P. Debye proposed that instead of individual atomic oscillations, elastic waves propagating in the solid with linear dispersion relation should be considered, and quantized according to Planck’s hypothesis. Thus Debye assumed that the energy of compressional (longitudinal) and torsional (transverse) waves of angular frequency ωL(q) (ωT(q)) can only change in steps of ωL(q) ( ωT(q)).

390 12 The Quantum Theory of Lattice Vibrations

If it is possible to excite an arbitrary number of such vibrations, the thermal energy associated with the vibration of polarization λ and wave vector q can be determined along the same lines as in the Einstein model. Making use of the result

ελ(q) =	ωλ(q)	,	(12.1.11)
	e ωλ(q)/kBT − 1

the total thermal energy of the crystal is

	ωλ(q)
		−
E =	e ωλ(q)/kBT		1	.	(12.1.12)
q,λ

By extending the classical results for elastic continua Debye assumed that the strict proportionality between ωλ(q) and the wave number holds not only in the long-wavelength region but all the way down to wavelengths on the order of atomic dimensions. For simplicity, he also assumed that longitudinal and transverse vibrations propagate with the same velocity, and that the allowed values of the vectors q fill a sphere in reciprocal space rather than the Brillouin zone determined by the specific crystal structure. As there are one longitudinal and two transverse modes for any q, the radius qD of this sphere has to be chosen such that three times the number of allowed q vectors in its interior should be equal to the number of normal modes, 3pN . Since a reciprocal-space volume (2π)3/V is associated with each vector q,

4πq3		V
D			= pN .	(12.1.13)
3		(2π)3

In crystals with a polyatomic basis excitations in the optical branches are taken into account by larger a qD than in the monatomic case.

These are the underlying assumptions of the Debye model. Before turning to the detailed discussion of its properties, we shall present a more precise formulation of quantization.

12.1.3 Quantization of the Hamiltonian

In the Einstein model the solid was regarded as a collection of atoms that vibrate at the same frequency and are independent in a sense. In contrast, in the Debye model elastic waves propagating in a continuous elastic medium are considered, and the atomic structure of the solid is ignored. Looking back at the classical description of lattice vibrations presented in the previous chapter, one may say that the Einstein model describes optical modes, while the elastic waves in the Debye model correspond to the long-wavelength limit of acoustic waves.

Depending on the basis, optical and acoustic vibrations may be simultaneously present in real crystals; the dispersion relation is then di erent from the one used in the previous approximation. Through plausible assumptions

12.1 Quantization of Lattice Vibrations

391

about the potential and the force constants the entire classical vibrational spectrum may be determined in principle, it is thus straightforward to suppose that Planck’s quantum hypothesis should be applied to these frequencies ωλ(q). The quantization condition can be formulated by stipulating that, for acoustic and optical vibrations alike, regardless of the dispersion relation, the energy of a vibration of frequency ωλ(q) can change only in quanta of ωλ(q), but the number of energy quanta can be arbitrarily large.

This also means that (12.1.11) and (12.1.12) are accepted for any dispersion relation. As we shall see, up to a temperature-independent constant term this yields the correct result for the total energy. Using (12.1.12), the thermal energy – and from it, the thermodynamic properties of the crystal – may then be determined, at least in principle, when the classical vibrational spectrum is known. To determine and physically interpret the additive constant, a more precise definition of quantization is required.

In the previous chapter we saw that using the canonically conjugate variables u(m, μ) and P (m, μ), the classical Hamiltonian (11.1.25) can be written as the sum of the kinetic energy (11.1.24) and the potential energy (11.1.21). Transition to quantum mechanics is then straightforward through canonical quantization. To obtain the quantum mechanical Hamiltonian operator from the classical Hamiltonian, the variables u and P in Tkin and Uharm are considered as operators, and the canonical commutation relation

[uα(m, μ), Pβ (n, ν)] = i δαβ δmnδμν

(12.1.14)

is imposed. This is automatically satisfied when expression

P (m, μ) =	∂	(12.1.15)
	i ∂u(m, μ)

is used.

Repeating the steps that led – through the solution of the eigenvalue problem of the dynamical matrix – from the displacements u(m, μ) to the normal coordinates, Qλ(q) also appears as an operator: with its canonically conjugate momentum Pλ(q) it satisfies the commutation relations

[Qλ(q), Pλ (q )] = i δλλ δqq ,

	[Qλ(q), Qλ (q )] = [Pλ(q), Pλ (q )] = 0 .					(12.1.16)
The Hamiltonian then reads			(q)Qλ†	(q)Qλ(q)9
Hharm = 21	q,λ	8Pλ†(q)Pλ (q) + ωλ2			,	(12.1.17)

where Q†λ(q) and Pλ†(q) are the Hermitian conjugates of Qλ(q) and Pλ (q). Note that the same result is obtained when one starts with the classical Hamiltonian (11.3.30) expressed with the normal coordinates, and changes

392 12 The Quantum Theory of Lattice Vibrations

to a quantum mechanical description by imposing the canonical commutation relations between the normal coordinates Qλ(q) and their canonically conjugate momenta Pλ(q).

The quantum mechanical Hamiltonian is then the sum of the Hamiltonians of 3pN independent quantum oscillators. Nevertheless the vibrational frequencies can still be determined classically, as they are the eigenvalues of the dynamical matrix, the second derivative of the potential.

12.1.4 The Quantum Mechanics of Harmonic Oscillators

To gain insight into the properties of solids that are due to lattice vibrations, the quantum mechanics of harmonic oscillators has to be recalled. Before absorbing the mass into the normal coordinates, the Hamiltonian reads

	p2				2		d	2
								+ 21 mω2x2.
H =		+	21 mω2x2	= −					(12.1.18)
	2m				2m		dx

Imposing the requirement of square integrability – whereby the wavefunction vanishes su ciently rapidly at infinity –, the eigenvalues of this di erential operator are expressed in terms of the Hermite polynomials (presented in Appendix C). The eigenfunctions normalized to unity are

ψn(x) = π

(2n n!)−1/2Hn

/

x e−mωx

/2 ,

(12.1.19)

mω

1/4

mω

2

and the energy eigenvalue of the state of quantum number n is

εn = ω n + 21 .

(12.1.20)

The wavefunction of the ground state is a Gaussian of finite width,

ψ0(x) =

mω

1/4

2

e−mωx

/2 .

(12.1.21)

π

According to the uncertainty principle, the oscillator performs small-amplitude vibrations, so-called zero-point vibrations even in its ground state. This is indicated by the finite value of the mean square displacement,

x2 = π		1/2	∞	x2e−mωx	/ dx =	2mω .		(12.1.22)
	mω			2
			−∞
This is why the ground-state energy is finite, ε0						=	21 ω; half of it comes

from the kinetic energy and the other half from the potential energy of the oscillator. This finite ground-state energy is called the zero-point energy.

The solutions of the eigenvalue problem were determined in terms of the Hermitian operators x and p, which satisfy canonical commutation relations.

12.1 Quantization of Lattice Vibrations

393

An easier alternative is o ered by writing the Hamiltonian in terms of the non-Hermitian operator a and its adjoint defined as

a = /		2	x + mω p		,	a† = /		2	x − mω p .		(12.1.23)
		mω		i				mω		i

Using the new operators, x and p are expressed as

x =	/				,	p = i/				a† − a .	(12.1.24)
			2mω a + a†					2
								mω

Making use of the canonical commutation relation [x, p] = i , it is immediately established that

[a, a†] =	i	(px − xp) = 1 ,	(12.1.25)
and the Hamiltonian is
H = 21 ω a†a + aa† = ω a†a + 21 .			(12.1.26)

The energy eigenstates and eigenvalues are thus related to the eigenfunctions and eigenvalues of the operator nˆ = a†a. The commutation relation

[ˆn, a] =	−	a ,	[ˆn, a†] = a†	(12.1.27)

implies that if ψn is an eigenfunction of operator nˆ with eigenvalue n, then the aψn and a†ψn are also eigenfunctions, with eigenvalues n − 1 and n + 1:

nˆ aψn = (n − 1)aψn ,

nˆ a†ψn = (n + 1)a†ψn .

(12.1.28)

For this reason a and a† are called ladder operators or shift operators. To obtain, as expected on physical grounds, a spectrum that is bounded from below, the eigenvalues of nˆ have to be integers, and the lowest of them zero. The eigenvalues of nˆ are then nonnegative integers,

nψˆ n = nψn

n N0 .

(12.1.29)

Energy eigenvalues are indeed given by (12.1.20). The restriction that the quantum number n can take only nonnegative integers justifies the picture that in state ψn there are n quanta of energy ω present, and a† is their creation operator while a is their annihilation (destruction) operator.

Starting with the ground-state wavefunction ψ0, every eigenstate can be constructed by means of the creation operator a†. The state of quantum number n that is normalized to unity is

1					a†	n
		ψn =	√			ψ0 .			(12.1.30)
				n!
Adding a further quantum to this state or taking away one from it,
√						√
a†ψn = n + 1 ψn+1,					a ψn =		n	ψn−1 .	(12.1.31)

394 12 The Quantum Theory of Lattice Vibrations

12.1.5 Creation and Annihilation Operators of Vibrational Modes

By generalizing the results of the quantum mechanical treatment of the harmonic oscillator, we shall introduce the shift operators for the oscillator that corresponds to a lattice vibration of polarization λ and wave vector q:

aλ(q) =

/

2

Qλ

(q) + ωλ(q) Pλ†(q)

i

ωλ(q)

=

/

2

Qλ

(q) + ωλ(q) Pλ

(−q) ,

(12.1.32-a)

i

ωλ(q)

aλ† (q) =

/

2

Qλ† (q) − ωλ(q) Pλ

(q)

i

ωλ(q)

=

/

2

Qλ

(−q) − ωλ(q) Pλ (q) .

(12.1.32-b)

i

ωλ(q)

It follows immediately from the commutation relations between normal coordinates and their conjugate momenta that these operators satisfy the bosonic commutation relations

aλ(q), a†λ (q ) = δλ,λ δq,q ,

aλ(q), aλ (q ) = a†λ(q), a†λ (q ) = 0 .

The inverse formulas are then

Qλ(q) = 7		2ωλ(q)				aλ(q) + aλ† (−q) ,
Pλ(q) = −i/					2				aλ(−q) − aλ† (q) .
				ωλ(q)

In terms of these operators, the Hamiltonian is

(12.1.33)

(12.1.34)

H = 21	q,λ ωλ(q) aλ† (q)aλ(q) + aλ(−q)aλ†		(−q) .			(12.1.35)
Alternatively, by exploiting that ωλ(q) is an even function of q,
						(12.1.36)
		H = q,λ ωλ(q) aλ† (q)aλ(q) +	21	.

As the eigenvalues of operator nˆλ(q) = a†λ(q)aλ(q) are nonnegative integers, the vibrations of the crystal lattice are quantized in such a way that their energy can change by integral multiples of ωλ(q). Operator a†λ(q) takes an initial state into one whose energy is higher by ωλ(q) – while operator aλ(q)

12.1 Quantization of Lattice Vibrations

395

into another whose energy is lower by the same amount. These operators can be regarded as the creation and annihilation operators of a vibrational quantum.

Making use of the above form of the Hamiltonian, the time dependence of the creation and annihilation operators is

aλ(q, t) =eiHt/ aλ(q)e−iHt/ = aλ(q)e−iωλ(q)t ,

(12.1.37)

a†λ(q, t) =eiHt/ a†λ(q)e−iHt/ = a†λ(q)eiωλ (q)t.

Using them in the time derivative of equations (12.1.34) for Qλ(q) and Pλ(q), it can be directly shown that (11.3.28), (11.3.29), and (11.3.31) are indeed satisfied.

Substituting (12.1.34) in the formula for atomic displacements, the displacement operator (11.3.14) may also be expressed in terms of these creation and annihilation operators:

uα(m, μ) =	7								(q) aλ(q) + aλ†	(−q) eiq·Rm	(12.1.38)
			2N Mμωλ(q) eμ,α(λ)
q,λ
=	7		2N Mμωλ(q) eμ,α(λ) (q)aλ(q)eiq·Rm eμ,α(λ) (q)aλ† (q)e−iq·Rm .
q,λ
Writing out the time dependence explicitly,
uα(m, μ, t) =			7						eμ,α(λ) (q)aλ(q)ei(q·Rm −ωλ (q)t)
					2N Mμωλ(q)
	q,λ										(12.1.39)

+e(μ,αλ) (q)a†λ(q)e−i(q·Rm −ωλ (q)t) .

12.1.6Phonons as Elementary Excitations

Apart from the term due to zero-point vibrations, the above form of the Hamiltonian of a thermally vibrating lattice is the same as the Hamiltonian of a free gas of bosonic particles with energy ωλ(q). The same formula is obtained when the Hamiltonian of the radiation field is expressed in terms of the creation and annihilation operators of photons, the quanta of the electromagnetic field. Since acoustic elastic waves are related to sound propagation, the quanta of lattice vibrations are called phonons by this analogy, regardless of whether they are obtained through the quantization of acoustic or optical classical lattice vibrations. Consequently one may distinguish acoustic and optical phonons, and for both types longitudinal and transverse modes. The dispersion relation of acoustic phonons always starts linearly at q = 0, while

396 12 The Quantum Theory of Lattice Vibrations

that of optical phonons starts at a nonvanishing value, and generally varies only slightly.

The thermally excited collective vibrational states of a lattice can be regarded as states of a gas of fictitious particles – phonons – propagating in the lattice, independently of each other. States of the system can be characterized by the number of excited phonons. Since phonon creation and annihilation operators satisfy bosonic commutation relations, the Bose–Einstein statistics2 applies to these fictitious particles. The thermal population of phonon states, i.e., the expectation value of the particle number operator is given by

nλ(q) ≡ aλ† (q)aλ(q) =	1	.	(12.1.40)
	e ωλ(q)/kBT − 1

As the number of phonons is not conserved, the chemical potential is zero. Starting with the ground state that has only zero-point vibrations, each

subsequent action of the operators a†λ(q) increases the number of phonons by one and the energy by ωλ(q). Just like in the classical model, frequencies are determined by the eigenvalues of the dynamical matrix. Consequently, a†λ(q) is the creation operator of a phonon of wave vector q and polarization λ. Similarly, the annihilation operator aλ(q) reduces the number of phonons by one and the energy of the state by ωλ(q).

The energy contribution of lattice vibrations is calculated as the expectation value of the Hamiltonian (12.1.36),

E = H = q,λ ωλ(q)		aλ†	(q)aλ(q) +				21	.		(12.1.41)
Using Bose–Einstein statistics,	e ωλ(q)/kBT								.
E = q,λ ωλ(q)					1		+ 21			(12.1.42)
			1
				−

This essentially confirms the result obtained via naive quantization. An important di erence with those formulas is the presence of the term 1/2, which corresponds to zero-point vibrations. We shall discuss its significance later.

Due care must be exercised in the calculation of the momentum of a vibrating lattice. Starting with expression (12.1.39), and considering the contribution of a phonon of specific wave vector and polarization, the momentum associated with the vibration is

m,μ Mμu˙ α(m, μ) = −i m,μ /			2N	eμ,α(λ) (q)aλ(q)ei(q·Rm −ωλ (q)t)
			Mμωλ(q)

− e(μ,αλ) (q)a†λ(q)e−i(q·Rm −ωλ(q)t) . (12.1.43)

2 S. N. Bose, 1924; A. Einstein, 1924.