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

12.1 Quantization of Lattice Vibrations

397

Summation over m, the label of primitive cells gives a vanishing result unless q = 0. Nevertheless q – or sometimes q itself – is called the crystal momentum of the phonon: as discussed in Chapter 6, in processes where the phonon number changes by one, a conservation law – that is valid only up to a reciprocal-lattice vector – applies to the wave vector q.

12.1.7 Acoustic Phonons as Goldstone Bosons

The Hamiltonian of a crystalline structure is invariant under arbitrary translations, while the crystal itself only under discrete ones. The crystalline state therefore breaks a continuous symmetry – or, more precisely, one along each spatial direction. According to Goldstone’s theorem (page 200) when a continuous symmetry is broken, boson-like low-energy excitations, so-called soft modes or Goldstone bosons appear. The three acoustic branches starting at the center of the Brillouin zone at zero energy have precisely these properties. In this sense acoustic phonons can be considered Goldstone bosons.

The appearance of soft Goldstone bosons may be illustrated by a simple physical picture: in the acoustic branches excitation energies vanish in the long-wavelength limit, as the uniform translation of the lattice requires no expense of energy.

It should be noted that Goldstone’s theorem does not apply for long-range interatomic forces. This can be directly demonstrated in a straightforward fashion by considering the long-wavelength limit of the frequency formula (11.2.14) obtained for monatomic chains. For small values of q the expansion yields

7

1

ω = Kp p2 qa . (12.1.44)

M

p

When Kp decreases slowly with distance, the above sum diverges, indicating that the frequency does not vanish in the q → 0 limit: it tends to a finite value.

12.1.8 Symmetries of the Vibrational Spectrum

It was shown in Chapter 6 that if a crystal is taken into itself by an element P = {α|tn} of the space group, then the same energy eigenvalue is associated with q and αq. In our case this means that the energy spectrum of phonons possesses the symmetry

ωλ(q) = ωλ(αq) .

(12.1.45)

This does not imply that the dynamical matrix itself should be invariant under the symmetry operations associated with the elements of the space group. It can nevertheless be shown that there exists a matrix RP (q) such that the dynamical matrix satisfies the congruence relation

RP (q)D(q)RP−1(q) = D(αq) .

(12.1.46)

398 12 The Quantum Theory of Lattice Vibrations

By constructing a matrix E (q) from the eigenvectors as column vectors, the eigenvalue problem of the dynamical matrix can be written as

D(q)E (q) = ω2(q)E (q) ,

(12.1.47)

where ω2(q) is a diagonal matrix, and its diagonal elements are the eigenvalues. Applying matrix RP (q) to this equation,

RP (q)D(q)E (q) = ω2(q)RP (q)E (q) .

(12.1.48)

On the other hand, from (12.1.46) we have

RP (q)D(q)E (q) = D(αq)RP (q)E (q) .

(12.1.49)

By comparing the two equations it is readily seen that the same frequencies are associated with αq and q; moreover, the relations among eigenvectors can also be established.

12.2 Density of Phonon States

When determining macroscopic quantities one frequently faces the problem of summing the contributions of individual phonons over all allowed vectors q in the Brillouin zone. If the crystal contains N1, N2, and N3 primitive cells in the directions of the three primitive translation vectors then, according to (6.2.22), the vectors q allowed by the periodic boundary conditions are separated by b1/N1, b2/N2, and b3/N3 along the directions of the three primitive vectors of the reciprocal lattice. In samples that are large compared to atomic dimensions the vectors q are spaced rather densely, therefore no significant error arises from replacing summation by integration.

Since the volume associated with each allowed point q of the reciprocal lattice is

q =

b1

·

b2

b3

=

1

[b1b2b3] =

vr

=

1 (2π)3

=

(2π)3

×

, (12.2.1)

N1

N2

N3

N

N

N v

V

where N = N1 N2 N3, and vr is the volume of the primitive cell of the reciprocal lattice, while v is that of the direct lattice, replacing the sum by an integral requires the following substitution:

		V		dq.	(12.2.2)
		→ (2π)3
	q

If the summand (integrand) is a function of the phonon energy alone, then it is usually much simpler to calculate the integral using the density of states (DOS), i.e., the number of phonons of a given energy.

12.2 Density of Phonon States

399

12.2.1 Definition of the Density of States

The total density of states g(ω) and the partial density of states gλ(ω) of a given polarization branch λ is defined through the requirement that for any function f (ω)

	V	q,λ f (ωλ(q)) =	λ			(2π)3 f (ωλ(q))
	1						dq
		=	λ		dω gλ(ω)f (ω) =				dω g(ω)f (ω) .	(12.2.3)
Then the partial and total densities of states are formally given by
		gλ(ω) =					(2π)3 δ(ω − ωλ(q)) ,			(12.2.4)
							dq
		g(ω) =			λ			(2π)3 δ(ω − ωλ(q)) .
								dq

In the general d-dimensional case (2π)3 is replaced by (2π)d.3

In the Einstein model the frequency ωE is the same for each vibration, so

gE(ω) =	(2π)3 δ(ω − ωE) =	V δ(ω − ωE) .	(12.2.5)
	dq	N

In the second step we made use of relation (5.2.18) between the volumes of the Brillouin zone and the primitive cell. This delta-like sharp peak in the density of states is often a good approximation for optical phonons, since their group velocity is low.

To evaluate the density of states numerically from the energy spectrum, it should be remembered that gλ(ω) dω is the number of states with energies between ω and (ω + dω) divided by V . It is sometimes more convenient to determine the density of states from Nλ(ω), which is the number of vibrational states with polarization λ and frequency less than ω in the total volume of the sample. The two quantities are related by

gλ(ω) =	1		dNλ(ω)	,	(12.2.6)
	V dω

which is why Nλ(ω) is sometimes called the integrated density of states (IDS). To determine the density of states, consider the surfaces of constant energy

Sλ(ω) and Sλ(ω + dω) in reciprocal space, shown in Fig. 12.2.

3As the eigenvalues of the dynamical matrix are given as functions of ω2, the density of states is often written in terms of D(ω2) = g(ω)/(2ω) rather than g(ω). D(ω2)dω2 is the number of those vibrational states whose squared frequency is between ω2 and ω2 + dω2.

400 12 The Quantum Theory of Lattice Vibrations

Fig. 12.2. The surfaces of constant energy ω and (ω + dω). The density of states is given by the number of allowed wave vectors between the two surfaces

Apart from a factor 1/V , gλ(ω)dω is, by definition, the number of allowed vectors q in the region between the two surfaces. This is obtained by dividing the volume of this region by the volume (2π)3/V associated with the vector q. Denoting the perpendicular distance of the two surfaces by dq (q) and the surface element by dS for any vector q that satisfies the condition ωλ(q) = ω, the volume of the region between the two surfaces is

		dSdq (q) ,				(12.2.7)
	Sλ (ω)
	and so the number of states inside the region is
	gλ(ω) dω =			(2π)3 dq (q) .		(12.2.8)
				dS
		Sλ(ω)
	Using a linear expansion for the dispersion relation,
	ω + dω = ω + | q ωλ(q)|dq (q) ,					(12.2.9)
	from which			dω
	dq (q) =				.	(12.2.10)
			| q ωλ(q)|

Substituting this into (12.2.8) leads to the following formula for the density of states:

gλ(ω) = (2π)3			| q ωλ(q)| .		(12.2.11)
1			dS
		Sλ (ω)

If the dispersion relation is approximated by that of elastic waves,

ωλ(q) = cλ|q|

(12.2.12)

12.2 Density of Phonon States

401

as in the Debye model, but with polarization-dependent propagation velocity, then the size of the constant-energy surface of energy ω is

Sλ(ω) = 4π			2	(12.2.13)
	cλ .
		ω

The partial density of states for phonons of polarization λ is then

gλ(ω) =

1

4π

ω

2 1

=

1 ω2

(12.2.14)

.

(2π)3

cλ

cλ

2π2

cλ3

Assuming that the propagation velocity is the same for both transverse vibrations, the total density of states is

g(ω) = 2π2 ω2		cL3	+ cT3	.	(12.2.15)
1		1	2

As it has been mentioned, the same velocity cD is associated with each of the three vibrational branches in the Debye model. Naturally, this value has to be chosen in such a way that the correct density of states is recovered for long-wavelength phonons. It follows directly from the above formula that

3	=	1	+	2	.	(12.2.16)
cD3		cL3
				cT3

However, the ensuing formula for the density of states,

gD(ω) =	3 ω2			(12.2.17)
	2π2		c3
			D

cannot be correct for very large energies: there is a maximum wave number qD in the Debye model, determined by (12.1.13), and therefore vibrational frequencies cannot exceed the Debye frequency

ωD = cD qD .

(12.2.18)

Expressed in terms of this quantity, the density of states is

gD(ω) =	9p	N ω2				if	ω ≤ ωD ,	(12.2.19)
		V ωD3 ,
	0 ,					if	ω > ωD .

This form of the density of states is the consequence of the linearity of the dispersion relation and the dimensionality (3) of the direct and reciprocal space. The quadratic increase of the density of states is true quite generally, even beyond the Debye model – but only for small values of ω, where the dispersion relation of acoustic phonons is nearly linear. In the general case of a d-dimensional crystal and in the same limit (i.e., for small values of ω) the density of states is proportional to the d − 1st power of frequency,

g(ω) ωd−1.

(12.2.20)

402 12 The Quantum Theory of Lattice Vibrations

12.2.2 The Density of States in Oneand Two-Dimensional

Systems

Before turning to the discussion of the singularities in the density of states for three-dimensional crystals it is instructive to study two simpler cases: those of one-dimensional chains and of two-dimensional lattices.

In the one-dimensional case the allowed values of q are spaced 2π/L apart. Since the same frequency is associated with q and −q, the number of states in dω is

L dq
π dω dω .	(12.2.21)

The density of states, that is, the number of states with frequency ω per unit length of the chain is then

g1d(ω) =	1	1		.	(12.2.22)
	π		dω/dq

When writing the dispersion relation (11.2.9) for monatomic chains in the form ω(q) = ωmax| sin(qa/2)|, the following relation emerges:

g1d(ω) =		2	ωmax2	− ω2 −	1/2	,	if	ω ≤ ωmax ,	(12.2.23)
		πa
		0 ,					if	ω > ωmax .

The density of states is seen to be singular for phonon energies at the Brillouin zone boundary. This inverse-square-root singularity is characteristic of the one-dimensional case. Similar singularities are observed in the density of states of the optical vibrations in diatomic chains. However, in the latter case the inverse-square-root singularities occur not only for frequency maxima but for frequency minima as well. In the acoustic branch the density of states does not show any singularity around ω = 0, as for small values of q the dispersion relation is not quadratic but linear. This apparently nonanalytic behavior – ω |q| – is the consequence of the fact that the eigenvalues of the dynamical matrix lead to analytical expressions in q for ω2 (rather than ω).

In two-dimensional crystals, too, each phonon branch contains at least one point q0 where the dispersion relation attains a maximum. In the vicinity of this point the energy spectrum is described by a quadratic form of the components of the vector ξ = q −q0. The principal axis transformation of the quadratic expression gives

ωλ(q) = ω0 − α1ξ12 − α2ξ22 .

(12.2.24)

Using the variables xi = α1i /2ξi, the density of states can be determined with the help of the integral

gλ(ω) = (2π)2		(α1	α2)1/2		dx1 dx2 δ(ω − ω0 + x12 + x22) .	(12.2.25)
1			1

									12.2 Density of Phonon States 403
By changing to polar coordinates,
gλ(ω) =	1			1					r dr dϕ δ(ω − ω0 + r2)
		(2π)2			(α1α2)1/2
=	1			1				if	ω ≤ ω0 ,	(12.2.26)
	4π (α1α2)1/2 ,
			0 ,					if	ω > ω0 .

This means that the density of states drops to zero from a finite value at the maximum frequency. A similar method can be applied to the case when the dispersion curve has a minimum around q0; the density of states then jumps from zero to a finite value at the corresponding energy. In line with the general considerations presented above, the dispersion relation starts linearly at q = 0 in the acoustic branches, and therefore the density of states starts linearly at the bottom of the spectrum.

In each phonon branch there is at least one so-called saddle point where the dispersion relation attains a minimum in one direction and a maximum in the other. This is easily demonstrated by allowing equivalent wave vectors and considering the q over the entire reciprocal space. Then, without loss of generality, the Brillouin zone can be centered at the point q0 where the dispersion curve attains a minimum. At any equivalent point q0 + G, where G is a primitive vector of the reciprocal lattice, the dispersion relation attains another minimum. When points q0 and q0 + G are connected through various paths in q-space, there exists a local maximum along each path. These paths and the local maxima along them are shown in Fig. 12.3. Next the curve connecting these maxima is considered: its points at the boundaries of the Brillouin zone are equivalent, as they di er by a reciprocal-lattice vector. Consequently there must be a local minimum along this curve – which is a saddle point.

q0

P

q0+G

Fig. 12.3. Determination of the locus of the saddle point in the extended reciprocal space. Crosses (×) mark the maxima of the dispersion curve along the paths connecting q0, the center of the Brillouin zone, with an equivalent point q0 + G. The minimum along the curve connecting these maxima is the saddle point P

In the vicinity of the saddle point the dispersion relation takes the form

ωλ(q) = ω0 + α1ξ12 − α2ξ22

(12.2.27)

404 12 The Quantum Theory of Lattice Vibrations

in the system of principal axes, with positive coe cients αi. As shown in Fig. 12.4, the lines of constant energy are hyperbolas.

Fig. 12.4. Lines of constant energy around the saddle point for ω < ω0 and ω > ω0. Phonon energies are lower in the shaded regions than on the curves bounding them

Once again, we introduce the variables xi = α1i /2ξi; the density of states is then given by

gλ(ω) = (2π)2		(α1	α2)1/2		dx1 dx2 δ(ω − ω0 − x12 + x22) .	(12.2.28)
1			1

To evaluate the integral, we change to coordinates that are adapted to the hyperbola. A good choice is

		x1		= r sinh θ ,		x2 = ±r cosh θ	(12.2.29)
when ω < ω0	, and
		x1		= ±r cosh θ ,		x2 = r sinh θ	(12.2.30)
when ω > ω0	. Using the Jacobian of the new variables,
gλ(ω) = 4π2			(α1α2)1/2		r dr dθ δ(ω − ω0 ± r2) .		(12.2.31)
	1			1

The sign (±) in front of r2 corresponds to the cases ω < ω0 and ω > ω0. In either case, the contribution vanishes unless r = |ω−ω0|1/2. However, the value of the integral depends on the choice of the range of integration for the variable θ. Assuming that the quadratic approximation for the dispersion curve is valid in a finite neighborhood of the saddle point, integration is performed in the region where

x12 + x22 ≤ R2 .

(12.2.32)

12.2 Density of Phonon States

405

Even though the cuto R is introduced arbitrarily, the physically meaningful singularity does not depend on its particular choice.

Since the two branches of the hyperbola give equal contributions, and so do the two arms of each branch, integration needs to be performed only in one quadrant, e.g., x1 > 0, x2 > 0. To include only points within the circle of radius R, the following restriction must be imposed on the variable θ:

										R2			1/2
			0 ≤ θ ≤ arsinh				2								(12.2.33)
									r2 − 1 .
							1
Integration then gives
1				1								R2		1/2
								2							(12.2.34)
gλ(ω) = 2π2 (α1				α2)1/2 arsinh					|ω0 − ω| − 1 .
								1
Using the logarithmic expression
				arsinh x = ln[x + (x2 + 1)1/2]											(12.2.35)

for the inverse hyperbolic function, arsinh x ≈ ln 2x in the vicinity of the saddle point, where the argument is large. Therefore a logarithmic singularity appears in the density of states:

gλ(ω) ≈ −4π2			(α1	α2)1/2 ln 1 −		ω0	.	(12.2.36)
1				1		ω

12.2.3 Van Hove Singularities

In the one-dimensional case the density of states has inverse-square-root singularities at frequencies where the spectrum has maxima or minima. In two dimensions, in addition to maxima and minima, the spectrum may also have saddle points; the density of states then features finite jumps and logarithmic singularities, respectively. In the Debye model of three-dimensional crystals the spectrum is assumed to be isotropic, therefore the density of states is smooth everywhere except for an abrupt cuto at the maximum energy. This is not the case for real crystals. Energy extrema may occur at several points of the phonon spectrum, and then for the corresponding q0 value(s) | q ωλ(q)| = 0 along every direction. In other cases the gradient of the dispersion relation vanishes in certain directions only. As we saw in Chapter 6, it always vanishes at the boundaries of the Brillouin zone in the perpendicular direction if the boundary is related to the opposite one through reflection symmetry. At the corners of the Brillouin zone and at the centers of the boundary faces the gradients in other directions may vanish as well. Those points where the gradient vanishes in every direction are called analytical critical points. At such points the integrand is singular in expression (12.2.11) for the density of states. Nevertheless the integral – and so the density of states – is finite

406 12 The Quantum Theory of Lattice Vibrations

for any value of the frequency. The points where its derivative is singular are called Van Hove singularities.4 Figure 12.5 shows the density of states derived from the theoretically determined phonon spectrum of aluminum. Singularities come from transverse branches at low energies and from longitudinal phonons at the top of the spectrum.

) –1

g (ν) (THz

0.4
		Al
0.3	T = 80 K
0.2
0.1
0 0	2	4	6	8	10
		ν (THz)

Fig. 12.5. Calculated phonon density of states in aluminum, with the characteristic Van Hove singularities [C. B. Walker, Phys. Rev. 103, 547 (1956)]

When the phonon dispersion relation is expanded in a series of the components of ξ = q−q0 around an analytical critical point q0, the linear terms must be absent. The principal axis transformation of the second-order expression gives

ωλ(q) = ω0	+ α1	ξ2	+ α2	ξ2	+ α3	ξ2	,	(12.2.37)
		1		2		3

where ω0 = ωλ(q0). In contrast to the minima, maxima, and saddle points of the two-dimensional case, four types of critical points are now distinguished.

In P0-type points each coe cient αi is positive, and the spectrum has a minimum at q0. We shall now show that when ω is close to the corresponding frequency, the density of states is

gλ(ω) =	1		(ω − ω0)1/2	.	(12.2.38)
	4π2 (α1α2α3)1/2

We shall first determine the number of states on the constant-energy surface associated with the frequency ω. This surface is an ellipsoid that intersects the ξi-axis in ±[(ω − ω0)/αi]1/2. The volume of the ellipsoid is

4π		(ω − ω0)3/2	.	(12.2.39)
3 (α1α2α3)1/2

The number of states within gives the integrated density of states:

4 L. Van Hove, 1953.