Fundamentals of the Physics of Solids / 11-Dynamics of Crystal Lattices

11.2 Vibrational Spectra of Simple Lattices

351

Note that (11.2.55) can also be obtained by Fourier transforming (11.1.29), provided Dαβ (q) is defined as

Dαβ (q) =	1		1	− e−iq·(Rm −Rn ) Φαβ (m − n)	(11.2.57)
	M m=n
				%

when the basis consists of a single atom. The comparison of (11.1.13) and (11.2.51) immediately yields Φ%αβ (m −n), and then the force constants can be obtained directly from the previous equation.

To determine the eigenfrequencies, the eigenvalues of the 3 × 3 matrix D(q) have to be calculated. This requires the solution of a cubic equation for ω2, which gives three di erent frequencies. In special cases, in high-symmetry points of the Brillouin zone the eigenvalues may nevertheless become degenerate.

The Brillouin zone of the simple cubic crystal is shown in Fig. 7.2. For vibrations propagating along the direction [100] and characterized by the wave vector = (q, 0, 0) (that lies along the line between Γ and X) the matrix D(q) is diagonal:

D(q, 0, 0) = 0 B

0

,

A

0

0

where

0

0 B

2K1 + 4K2

qa) =

4K1 + 8K2

sin2

1

A =

2K2 M

(1 − cos4K2

2

1M

2 qa ,

B =

(1 − cos qa) =

sin

2 qa .

M

M

(11.2.58)

(11.2.59)

It is immediately recognized that the crystal has a nondegenerate and a doubly degenerate eigenfrequency:

4K1

+

8K2

4K2

ω1 = /

M

,

ω2,3

=

/ M

. (11.2.60)

sin 21 qa

sin 21 qa

In the harmonic approximation the amplitude of the vibrations can be arbitrarily large, only their directions are determined by the eigenvectors of the matrix D(q). The eigenvectors for the above eigenfrequencies are

e(1) =	0	,	e(2) =	1	,	e(3) =	0 .	(11.2.61)
	1			0			0
	0			0			1

The first solution describes atoms moving in the propagation direction of the wave; the vibration is longitudinal in this case. For the two other solutions atomic displacements are perpendicular to the propagation direction; these vibrations are therefore transverse. Note that the condition for the existence

352 11 Dynamics of Crystal Lattices

of transverse waves is that K2 be finite – that is, in a simple cubic crystal force constants should be nonzero not only along the direction of propagation or perpendicular to it but also in the diagonal direction.

The propagation velocities for the two waves are

cL = a

1

M

2

1/2

cT = a

M

1/2

(11.2.62)

K

,

.

+ 2K

K2

The matrix is diagonal for the points Z = (π/a, q, 0) along the line between X and M in Fig. 7.2, too:

A 0

0

D(π/a, q, 0) =

0 B 0

,

(11.2.63)

where

0 0 C

A =

4K1

+

2K2

(3 + cos qa) =

4K1 + 8K2

−

4K2

sin2

21 qa ,

2K1

4K1

M

4K2

4K2

2

1

M

M

M

B =

2K2

(1

−

cos qa) +

4K2

=

4K2

+

2 1

sin

2 qa ,

(11.2.64)

M

M

M

M

C =

(3 − cos qa) =

+

sin

2 qa .

M

M

M

The eigenvectors are the same as those given in (11.2.61), and the eigenfrequencies are given by the square root of A, B, and C. The vibration characterized by the eigenvector e(3) is transverse, while in the two other cases the eigenvector e is neither longitudinal nor transverse. This clearly indicates that in the general case one cannot speak of purely longitudinal and purely transverse oscillations.

The matrix is no longer diagonal for the points Σ = (q, q, 0) along the line between M and X:

A B 0

D(q, q, 0) =

B A 0

,

(11.2.65)

where

0

0 C

A =

2K1 + 2K2

(1 − cos qa) +

2K2

(1

− cos2 qa)

M

M

=

4K1 + 4K2

sin2

21 qa +

2K2

sin2 qa ,

2K2 M

2

M

B =

sin

qa ,

(11.2.66)

M

C =

4K2

(1

− cos qa) =

8K2

sin2

21 qa .

M

M

It is an elementary exercise to find the eigenvectors of the matrix:

11.2 Vibrational Spectra of Simple Lattices

353

e(1) = √		1	,	e(2) = √		−1	,	e(3) =	0	. (11.2.67)
1		1		1		1			0
		0				0			1
	2				2

The corresponding eigenfrequencies are

ω1

=

√A + B = /

M

sin2

21 qa +

M

sin2 qa ,

4K1

+ 4K2

4K2

4K1 + 4K2

ω2

= √A − B = /

sin

21 qa

,

(11.2.68)

M

ω3

=

8K2

sin

1

√C = / M

2 qa

.

The highest-frequency vibration (given by the first solution) is longitudinal; the two others are transverse.

Finally, for vibrations propagating along the direction [111] and characterized by the wave vectors Λ = (q, q, q) the matrix D(q) is of the form

A B B

D(q, q, q) =

B A B ,

(11.2.69)

where

B B A

A =

2K1

(1

−

cos qa) +

4K2

(1

cos2 qa) =

4K1

sin2

21 qa

+

4K2

sin2 qa ,

2K2

2

2K2

−2

M

M

M

M

B =

(1 − cos qa) =

sin

qa .

(11.2.70)

M

M

This matrix is also straightforward to diagonalize. The eigenvectors are

e(1) = √

1 , e(2) = √

−1 ,

e(3) =

√

1

, (11.2.71)

1

1

1

1

1

1

1

0

−2

3

2

6

and the corresponding eigenfrequencies are

ω1 =

√A + 2B = /

M

sin2

21 qa +

M

sin2 qa ,

4K1

8K2

(11.2.72)

√A − B =

/

sin2

sin2 qa .

ω2,3 =

M

21 qa +

M

4K1

2K2

Once again, a longitudinal and two degenerate transverse vibrations are found. Figure 11.10 shows the dispersion relations for the lattice vibrations along the four characteristic directions of the Brillouin zone discussed above.

In this crystal structure three acoustic branches are observed; they start from zero frequency at q = 0. For wave vectors q that are along directions of

354 11 Dynamics of Crystal Lattices

Fig. 11.10. Dispersion relations for the lattice vibrations along the four characteristic directions of the Brillouin zone in a simple cubic crystal with a monatomic basis

su ciently high symmetry, atoms oscillate in the direction of wave propagation in one branch and in perpendicular directions in the two others. Starting with these excitation branches and continuing them in directions of lower symmetry, one can generally speak of longitudinal or transverse vibrations, although it is not strictly true any more that one eigenvector is parallel and the two others are perpendicular to the propagation direction. The frequency tends to be higher for longitudinal vibrations than for transverse ones. The frequencies of the two transverse vibrations are usually unequal, but degeneracy may occur in directions of high symmetry.

When only first and second neighbors are assumed to interact via central forces, the vibrational spectra for faceand body-centered cubic lattices with a monatomic basis can be calculated along the same lines. The obtained spectra are very similar to the above: they contain three acoustic branches; along directions of su ciently high symmetry one of them corresponds to longitudinal and two to transverse vibrations. This is in perfect agreement with experimental observations. Figure 11.11 shows the calculated and measured vibrational frequencies in some characteristic directions of the Brillouin zone for a face-centered cubic gold crystal. Measurements were made using neutron scattering techniques.

11.3 The General Description of Lattice Vibrations

In the foregoing we have seen that only longitudinal acoustic vibrations are possible in a linear monatomic chain. In cubic lattices with one atom per primitive cell three acoustic vibrations are possible; in directions of high symmetry one of them is longitudinal (atoms oscillate in the propagation direction) and two are transverse (atoms oscillate in a perpendicular direction). In addition

11.3 The General Description of Lattice Vibrations

355

Fig. 11.11. Room-temperature dispersion relation of lattice vibrations for gold in the principal symmetry directions. The reduced wave vector ζ = aq/2π. The solid curves were obtained from model calculations with several fitting parameters [J. W. Lynn et al., Phys. Rev. B 8, 3493 (1973)]

to acoustic vibrations, optical vibrations appear in chains with two atoms per primitive cell. These are of finite frequency even in the long-wavelength limit. Below we shall examine the most general case when the three-dimensional crystal contains p atoms per primitive cell.

11.3.1 The Dynamical Matrix and its Eigenvalues

The classical equations of motions were already given in (11.1.28). Just like in the simple examples, their solutions are sought in the form of an expansion in traveling waves – a Fourier series. Since the masses of the p atoms in the primitive cell are not necessarily identical, the explicit separation of a massdependent factor proves useful:

uα(m, μ, t) =		N Mμ 2π		dω	q	uμ,α(q)ei(q·Rm −ωt) .	(11.3.1)
	1		1

Substituting this form of the displacement into the equation of motion (11.1.28),

	ω2uμ,α(q) =			Dαβμν (q)uν,β (q) ,	(11.3.2)
			ν,β
where
	1
Dαβμν (q) =			Φαβμν	(m − n)e−iq·(Rm −Rn ) .	(11.3.3)
		MμMν

n

356 11 Dynamics of Crystal Lattices

Owing to the translational symmetry of the crystal, Dαβμν (q) does not depend on the particular choice of the lattice point Rn, so we shall use the equivalent expression

			1
Dμν	(q) =			Φμν	(m)e−iq·Rm .	(11.3.4)
αβ				αβ

MμMν m

This quantity – the Fourier transform of the force constants weighted by the masses – is called the dynamical matrix. The system of equations (11.3.2) has nontrivial solutions when

det Dαβμν (q) − ω2δαβ δμν = 0 .

(11.3.5)

It is important to note that the Fourier coe cients associated with different wave numbers do not mix in this general case, either: for each wave vector q only the 3p components – which correspond to the p atoms per primitive cell and the three spatial directions – are mixed. Because of this drastic simplification, the determination of vibrational frequencies is now relatively easy. By arranging the quantities Dαβμν (q) into a 3p × 3p matrix in the space spanned by the pairs (α, μ) and (β, ν), the previous equation turns out to be the eigenvalue equation of the matrix. We have already encountered a special case, when discussing simple cubic crystals. Consequently, 3p solutions are possible for each q. We shall now demonstrate that all of them lead to real and positive frequencies.

First we shall show that the dynamical matrix is Hermitian. Substituting the relation (11.1.5) for Φμναβ (m, n) into the defining equation (11.3.4), we have

	1
Dαβμν (q) =			Φβανμ (−m)e−iq·Rm = Dβανμ (−q) .	(11.3.6)
		MμMν

m

Since we started with a real potential, the complex phases in the elements of the dynamical matrix must come from the Fourier transform, so

Dαβμν (q) = Dαβμν (−q) .	(11.3.7)
Comparison of the two last equations gives
Dαβμν (q) = Dβανμ (q) .	(11.3.8)

Strictly speaking, Hermiticity only implies that the 3p eigenvalues of the dynamical matrix are all real. In reality, the eigenvalues are not only real but also nonnegative. The reason for this is that the potential is expanded around a minimum, therefore the matrix of force constants is positive definite. This feature is not lost during Fourier transformation, and so the same applies to the dynamical matrix.

Using the notation ωλ2 (q) (λ = 1, 2, . . . , 3p) for the eigenvalues, the eigenvalue equations for the dynamical matrix can also be written in terms of the components of the eigenvectors e(μ,αλ) of the λth eigenvalue:

	11.3 The General Description of Lattice Vibrations						357
	ωλ2 (q)eμ,α(λ) (q) =		Dαβμν (q)eν,β(λ)	(q) ,		(11.3.9)
			ν,β
or, alternatively,
						(11.3.10)
	ν,β Dαβμν (q) − ωλ2 (q)δαβ δμν eν,β(λ) (q) = 0 .
This formula is equivalent to (11.3.5).
When eigenvectors are normalized,
	(λ)		(λ )			(11.3.11)
	eμ,α	(q)eμ,α (q) = δλλ
	α,μ
follows from their orthogonality, and
	eμ,α(λ) (q)eν,β(λ) (q) = δαβ δμν					(11.3.12)

λ

from their completeness.

11.3.2 Normal Coordinates and Normal Modes

Because of the linearity of the equations of motion, the amplitude of the vibration can be chosen at will, and displacements associated with vibrations of di erent wave numbers can be freely superposed. By expanding atomic displacements into Fourier series in the spatial variables only, the Fourier component uμ,α(q, t) can now be written as the linear combination of the appropriate components of the unit vectors specifying polarization – i.e., the direction of the displacement of the μth atom relative to the propagation direction determined by q. (Hence the name polarization vector for e(μλ)(q).) The time dependence is absorbed into the amplitude Qλ(q, t):

uμ,α(q, t) =	eμ,α(λ) (q)Qλ(q, t) .	(11.3.13)

λ

The quantity Qλ(q, t) is the normal coordinate of the vibrational mode. The instantaneous displacement of the atoms is then written as

	(m, μ, t) =		1		(q)Q		(q, t)eiq·Rm .
uα				e(λ)		λ		(11.3.14)

N Mμ q,λ

Since normal coordinates are complex, the number of possible values for q being N and that of the label λ being 3p apparently implies that the motion of atoms is specified by 6pN free parameters. However, as displacements are real, the normal coordinates associated with the wave vectors q and −q are

358 11 Dynamics of Crystal Lattices

each others’ complex conjugate. Therefore the number of independent normal coordinates is only 3pN , as it should be.

To prove this, a particularly important consequence of the Hermiticity of the dynamical matrix and the relation (11.3.7) is exploited, namely that the same eigenvalues belong to q and −q:

ωλ2 (q) = ωλ2 (−q) .

(11.3.15)

As demonstrated in Chapter 6, this can be regarded as the consequence of the invariance of the equations of motion under time reversal. Since atomic displacements are real, the relation

		uμ,α(λ) (−q, t) = uμ,α(λ) (q, t)							(11.3.16)
must be satisfied, that is,
e	(λ)	( q)Q	( q, t) = e		(λ)			(q, t) .
	μ,α				μ,α	(q)Q			(11.3.17)
		−	λ −				λ
Without loss of generality the relation
		eμ,α(λ) (q) = eμ,α(λ) (−q)							(11.3.18)
may be imposed, in which case
			Q (q) = Q		(	q)			(11.3.19)
			λ	λ −

is indeed satisfied. Note that for crystals with a monatomic basis the vectors e(λ)(q) can be chosen real.

For future convenience it is useful to represent the decomposition into independent vibrations in yet another way. The classical form (11.1.21) of the potential energy in the harmonic approximation,

Uharm = 12 Φμναβ (m − n)uα(m, μ)uβ (n, ν) (11.3.20)

m,μ,α

n,ν,β

is first written in terms of the normal coordinates. Substituting uα(m, μ) from (11.3.14) into this formula gives

μν

1

)

iq

R

Uharm = 1

Φ

(m

−

n)

eμ,α(λ (q )Qλ (q )e

·

m

2

αβ

N Mμ q ,λ

m,μ,α

n,ν,β

(11.3.21)

1

(λ)

×

√

q,λ eν,β (q)Qλ(q)eiq·Rn .

N Mν

To perform the summation over the lattice points, the expressions obtained in (11.3.4) for the dynamical matrix have to be exploited. As Φμναβ depends only

11.3 The General Description of Lattice Vibrations

359

on the di erence of the lattice vectors, only the terms q = −q contribute, and so

Uharm = 1	Dμν		(q)e(λ )(	q)Qλ	(		q)e(λ) (q)Qλ(q) .	(11.3.22)
			μ,α	−	−		ν,β
2	αβ
q,λ,λ	μ,α
	ν,β
Making use of the eigenvalue equation (11.3.9),
Uharm = 1	ω2	(q)e(λ )( q)Qλ (				q)eμ,α(λ) (q)Qλ(q) .		(11.3.23)
q			μ,α −		−
2	λ
,λ,λ	μ,α

Since displacements are real, and eigenvectors are orthogonal, as expressed in (11.3.17) and (11.3.11), we have

				(q)Q ( q)Q		(q) = 1		(q)Q	(q)Q	(q) .
U	harm	= 1	ω2				ω2				(11.3.24)
		2	λ	λ −	λ	2	λ	λ	λ
		q,λ					q,λ

The same argument can be used for the expression of the kinetic energy, whereby the formula

Tkin =	21 Mμu˙ α2 (m, μ)	(11.3.25)

m,μ,α

can be rewritten as

			˙
Tkin =	1	˙
	2	Qλ(−q)Qλ(q) =

q,λ

1

2

Q˙ λ(q)Q˙ λ(q) .	(11.3.26)

q,λ

Once the Lagrangian L = Tkin − Uharm is known, the equations of motion for the normal coordinates are readily established from Lagrange’s equation2

	∂		∂L		−	∂L		= 0 .	(11.3.27)
	∂t					∂Q (
			˙	−			q)
			∂Qλ(			λ −
				q)
Then			¨			2			(11.3.28)
			Qλ(q) = −ωλ(q)Qλ(q)

for each normal coordinate – that is, using normal coordinates, the vibrations of the system are automatically separated into those of independent harmonic oscillators. These are called the normal modes of vibration. Since λ can take 3p di erent values in (11.3.28), and the number of possible values for q is N , the three-dimensional coupled vibrations of the crystal (made up of pN atoms) can be described with 3pN independent oscillators.

To prepare the quantum mechanical discussion of the next chapter, it is useful to write down the classical Hamiltonian using normal coordinates.

2More precisely: Lagrange’s equation of the second kind. Since they are of the same form as Euler’s equation in variational calculus, the name Euler–Lagrange equation is also used.

360 11 Dynamics of Crystal Lattices

Following the steps of the Hamiltonian formulation in classical mechanics, we first introduce the canonical conjugate to the normal coordinate Qλ(q), which

					˙
is the derivative of the Lagrangian with respect to Qλ(q):
	P (q) =		∂L	= Q˙ (q) ,				(11.3.29)
			˙
	λ			λ
			∂Qλ(q)
and then the Hamiltonian
		˙
Hharm =	Pλ (q)Qλ(q) − L
q,λ							!
= 1				(q)Q (q)Q
	P (q)P (q) + ω2					λ	(q)	(11.3.30)
2	λ	λ	λ	λ
	q,λ				!
= 21
	|Pλ(q)|2 + ωλ2 (q)|Qλ(q)|2				.

q,λ

The full Hamiltonian is the sum of the Hamiltonians for individual normal modes. The equations of motion obtained in the Hamiltonian formulation,

Q˙ λ(q) =				∂H	= Pλ (q) ,
			∂Pλ(q)						(11.3.31)
P˙				∂H
	(q) =	−			=	−	ω2	(q)Q	(q)
λ				∂Qλ(q)			λ	λ

are equivalent to (11.3.28).

11.3.3 Acoustic and Optical Vibrations

In the previous subsections we saw that if the basis contains p atoms, there exist 3p vibrational frequencies for each vector q of the Brillouin zone. In the one-dimensional case we also saw that some of them correspond to acoustic vibrations, while others to optical ones. As a generalization we shall demonstrate that among the branches formed by the 3p vibrational frequencies three can always be called acoustic, since they start from ω = 0 at q = 0, and the others optical, as their frequency does not vanish at q = 0.

Acoustic Vibrations

A branch of vibrations is called acoustic if it has a vanishing frequency at q = 0, the center of the Brillouin zone. We shall now prove that there always exist three such branches. When the eigenvalue equation (11.3.9) is examined at q = 0, and the value of the dynamical matrix at q = 0 is taken from (11.3.4), we have

ω2 (0)e(λ) (0) =	Dμν	(0)e(λ) (0) =		1		Φμν	(m)e(λ) (0) .
λ μ,α		ν,β					ν,β
	αβ					αβ
	ν,β		m,ν,β		MμMν

(11.3.32)