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

11.2 Vibrational Spectra of Simple Lattices

341

mentioned in Chapter 6 it is the consequence of invariance under time reversal. When interactions are not limited to nearest neighbors, the dispersion curve still starts linearly, and the group velocity still vanishes at the boundary of the Brillouin zone.

Up to now it has been assumed that atomic displacements are along the chain. Such vibrations are called longitudinal. For each q there exists one such solution. When atomic displacements are perpendicular to the chain, we speak of transverse vibrations. As there are two perpendicular directions, two transverse vibrations are associated with each value of q. In a lattice made up of N atoms 3N vibrational states are therefore possible. However, in a linear chain this is true only in principle. Displacements perpendicular to the chain modify the energy to a lesser extent than parallel ones. As mentioned in connection with (11.1.19), transverse displacements are not opposed by restoring forces in the harmonic approximation, consequently transverse vibrations cannot propagate in the linear chain. This is no longer the case in twoand three-dimensional crystals. Here transverse vibrations are of finite frequency, too, provided certain atomic bonds are not in the plane spanned by the propagation direction and the direction of vibration.

11.2.2 Vibrations of a Diatomic Chain

The vibrational spectrum is more complicated when the primitive cell of the one-dimensional chain contains two atoms. There are two limits of particular interest. In the first an atom of mass M1 is located at the lattice point and another of mass M2 at the midpoint of the cell.

We shall denote the displacement from equilibrium of the atom of mass M1 (M2) in the nth primitive cell by un (vn). Equilibrium positions and instantaneous positions at a given time are shown in Fig. 11.4.

M1 M2

(a)

a

(b)

un 1

vn 1 un

vn

un 1 vn 1

Fig. 11.4. Atomic positions in the primitive cell of a linear chain made up of two kinds of atom. (a) Equilibrium positions; (b) instantaneous displacements

Assuming that each atom feels the potential arising from its two nearest neighbors, the system can be characterized by a single force constant K. The potential energy is

Uharm = 21 K [un − vn]2 + 21 K [vn − un+1]2.

(11.2.16)

n

n

342 11 Dynamics of Crystal Lattices

Then the equations of motion for the two kinds of atom are

M1u¨n = −K [2un − vn − vn−1] ,

(11.2.17)

M2v¨n = −K [2vn − un+1 − un] .

From the results obtained for the vibrations in a monatomic linear chain the solutions are expected to be linear combinations of independently propagating waves. Choosing a component of wave number q and frequency ω, the amplitudes are assumed to be di erent on the two sublattices:

un(t) = u(q)ei(qna−ωt) ,

vn(t) = v(q)ei(qna−ωt) .

(11.2.18)

Using periodic boundary conditions means that the requirement

uN +1 = u1 ,

vN +1 = v1

(11.2.19)

applies, and the chain is closed into a ring made up of N primitive cells. The allowed values for q are the same as for the monatomic chain.

Substituting the traveling wave form into the equations of motion, elimination of the common phase factor yields

−ω2M1u(q) = −2Ku(q) + Kv(q)(1 + e−iqa) ,

(11.2.20)

−ω2M2v(q) = −2Kv(q) + Ku(q)(eiqa + 1) .

For this equation to have a nontrivial solution the determinant of the coe - cient matrix has to vanish:

ω2M + 2K

qa

2Ke−iqa/2 cos

1 qa

− 1

cos

M2 + 2

2

= 0 .

(11.2.21)

−

2

−

−

2Keiqa/2

1

ω2

K

The

vibration frequencies are then

− 4M1M2 sin2

21 qa .

ω±2 (q) = M1M2

(M1 + M2) ±

(M1 + M2)

K

"

2

(11.2.22)

Using the notation

1

1

M1M2

ω02

= 2K

M1

+

M2

, γ2

= 4

(M1 + M2)2

,

(11.2.23)

they can be written as

1 ± "

.

ω±2 (q) = 21 ω02

1 − γ2 sin2

21 qa

(11.2.24)

In contrast to the single longitudinal mode in the monatomic chain there are now two such modes with di erent frequencies. The corresponding dispersion relations are shown in Fig. 11.5.

11.2 Vibrational Spectra of Simple Lattices

343

Fig. 11.5. Dispersion relations in the acoustic (ω−) and optical (ω+) branches for the vibrations in a diatomic chain

The dispersion relation of one type of vibration (ω−(q)) is very similar to that of the acoustic vibrations in a monatomic chain. It vanishes for q = 0,

and grows linearly for small q:
ω−(q) ≈ 41 ω0γ|q| a .	(11.2.25)
The other branch of the dispersion relation starts at a finite frequency:
ω+(q = 0) = ω0 .	(11.2.26)

The ratio of the amplitudes u(q) and v(q) is determined from (11.2.20).

For q = 0, i.e., at the center of the Brillouin zone
	v(0)		=	+1	in the branch ω−,	(11.2.27)
	u(0)				in the branch ω+.
				(M1/M2)

−

In the branch with frequency ω− the two atoms of the primitive cell oscillate with almost equal amplitudes and are in phase in the long-wavelength limit

– just like in a sound wave. For this reason this lower branch of vibrations is called acoustic branch in this case, too. In the branch with frequency ω+ the two atoms oscillate in opposite directions around their center of mass in the long-wavelength limit. Oscillation amplitudes on the two sublattices are inversely proportional to the masses. When the diatomic chain is an ionic crystal made up of oppositely charged ions, such vibrations may be excited by high-frequency electromagnetic fields (light); they are therefore called optical vibrations.

In the acoustic branch, at the boundary of the Brillouin zone

1

1

1

1

1/2

+

ω−(q) = √K M1

M2

− M1

− M2

(2K/M

)1/2

,

if

M

> M

2 ,

(11.2.28)

=

1

1/2

,

1

(2K/M2)

if

M2

> M1 .

344 11 Dynamics of Crystal Lattices

The situation is just the opposite in the optical branch. Here the frequency is (2K/M2)1/2 for M1 > M2 and (2K/M1)1/2 for M2 > M1 at the boundary of the Brillouin zone. Consequently there is always a finite gap between the two branches: the acoustic branch is always below the optical. Examined as functions of q, both branches of excitation are flat at the boundary of the Brillouin zone, so the group velocity vanishes there.

Note that at the boundary of the Brillouin zone the frequency of the acoustic (optical) branch depends only on the mass of the heavier (lighter) atom. This can be easily understood by taking the ratio of the two vibration amplitudes. At the boundary of the Brillouin zone it is either zero or infinity

– implying that only one type of atom participates in either vibration. Figure 11.6 shows the atomic displacements in the acoustic and optical branches for wave numbers at the center and boundary of the Brillouin zone.

(a)	q"0	acoustic
(b)	q 0	optical
(c)	q /a acoustic
(d)	q /a optical

Fig. 11.6. Atomic displacements in the acoustic and optical modes of a diatomic linear chain in the long-wavelength limit (q ≈ 0), and for a wave number at the zone boundary

Let us examine what happens when the two masses are changed continuously and become equal. Denoting the common mass by M , (11.2.22) implies

			2K
ω±2 (q) =				1 ± cos			21 qa ,				(11.2.29)
			M
that is,			1/2
					| cos
ω±(q) =		2	(K/M )1/2					41 qa	| ,		(11.2.30)
		2	(K/M )		|	sin		41 qa	|	.

The dispersion relation valid for this special case is shown in Fig. 11.7.

Both the figure and the analytical expressions show that the dispersion curves are now not perpendicular to the zone boundary at π/a. This is not surprising as we are now dealing with a chain that contains 2N equivalent atoms spaced uniformly over a length N a, i.e., separated by regular distances a/2. Therefore the chain is in fact monatomic, and the dimension of its true primitive cell is a/2. The boundary of the Brillouin zone is then at 2π/a

11.2								Vibrational Spectra of Simple Lattices									345
		!(q)									!(q)

- /a

/a

-2 /a

2 /a

Fig. 11.7. Dispersion relation for the vibrations of a diatomic linear chain in the M1 = M2 limit, shown in the Brillouin zones of chains with lattice constants a and a/2

instead of π/a. In this “large” Brillouin zone there are 2N allowed values for q, each associated with one eigenfrequency – while if the Brillouin zone that corresponds to a periodicity a is used, the number of allowed qs is N , and each of them is associated with two frequencies. In the M1 = M2 limit the optical vibrations obtained in the diatomic chain correspond to those acoustic vibrations of a monatomic chain for which the wave number is either π/a < q < −π/a or π/a < q < 2π/a. Shifting the optical branch by ±2π/a into these intervals, the dispersion curve of the monatomic chain is recovered.

11.2.3 Vibrations of a Dimerized Chain

In the other limit the two atoms in the primitive cell are of equal mass but are not uniformly spaced along the chain: the separation between nearest neighbors alternates regularly between a smaller and a larger value. Such a configuration – illustrated in Fig. 11.8 – is called a dimerized chain.

K1 K2

(a) d

a

(b)

un 1

vn 1

un vn

un 1 vn 1

Fig. 11.8. Equilibrium positions and instantaneous displacements of the atoms in a dimerized chain

Let the equilibrium atomic positions in the nth cell be denoted by na and na + d, and the displacements from them by un and vn. Depending on whether neighboring atoms are separated by d or a−d, the pair potential takes di erent values. For this reason even when only nearest-neighbor interactions are taken into account, two force constants have to be introduced. In the

346 11 Dynamics of Crystal Lattices

harmonic approximation the expression for the potential reads

	K1		2	K2		2
Uharm =			[un − vn] +			[vn − un+1] .	(11.2.31)
	2	n		2	n

When d ≤ a/2, it is plausible to assume that K1 ≥ K2.

Determining the force on the atoms from the energy expression, the equations of motion are

M u¨n = −

∂Uharm

= −K1

[un − vn] − K2 [un − vn−1] ,

∂un

(11.2.32)

∂Uharm

M v¨n = −

= −K1

[vn − un] − K2 [vn − un+1] .

∂vn

Traveling-wave solutions are sought in the form

un(t) = u(q)ei(qna−ωt),

vn(t) = v(q)ei(qna−ωt) ;

(11.2.33)

substitution into the equations of motion then gives

2M u(q) =

(K

+ K

)u(q) + K

+ K e−iqa

v(q) ,

−ω2

−

1

2

1

2 iqa

(11.2.34)

−ω

M v(q) = −(K1 + K2)v(q) + K1 + K2e

u(q) .

Nontrivial solutions exist if the determinant of the coe cients of u(q) and v(q) vanishes, i.e.,

−ω2M + (K1

+ K2) 2

=

K1 + K2e−iqa K1 + K2eiqa

(11.2.35)

=

1

+ K2 + 2K1K2 cos qa .

K2

2

From this equation the allowed frequencies are

ω±2 (q) = M (K1 + K2) ± "K12 + K22 + 2K1K2 cos qa ,

(11.2.36)

1

and the amplitude ratio is

u(q)

= 7

K1 + K2e−iqa

.

(11.2.37)

v(q)

K1 + K2eiqa

Vibrational frequencies can again be written in the form of (11.2.24) with

ω2	= (K		+ K	)	2	,	γ2 = 4	K1K2	.	(11.2.38)
								(K1 + K2)2
0		1	2		M

Similarly to diatomic chains, the spectrum has two branches. The lower

branch starts linearly,
ω−(q) ≈ 41 ω0γ|q| a ,	(11.2.39)

11.2 Vibrational Spectra of Simple Lattices

347

and since for q small u(q) ≈ v(q), atoms oscillate almost in phase in the long-wavelength limit. This is the acoustic branch.

In contrast, the upper branch starts at a nonzero frequency ω0 at the center of the Brillouin zone,

ω+(q) = ω0 − O(qa)2 .

(11.2.40)

Since for q small v(q) ≈ −u(q) now, the two atoms oscillate in opposite phases. This branch is called the optical branch here, too.

At the boundary of the Brillouin zone, for q = π/a

ω±2 (π/a) =	1	[(K1 + K2) ± |K1 − K2|] .	(11.2.41)
	M
For K1 > K2 this gives
ω+(π/a) = (2K1/M )1/2 ,		ω−(π/a) = (2K2/M )1/2 .	(11.2.42)

The ratio of the amplitudes of the displacements is found to be 1. This means that atomic displacements are such that atoms separated by d in one branch and by a − d in the other branch oscillate in phase – and so only one of two springs is stretched in either branch.

(a)	q"0	acoustic
(b)	q 0	optical
(c)	q /a acoustic
(d)	q /a optical

Fig. 11.9. Atomic displacements in the acoustic and optical modes of a dimerized linear chain in the long-wavelength limit for q = π/a

In the limit K1 K2,
ω−(q) = (2K2/M )1/2 |sin(qa/2)|[1 + O(K2/K1)] ,	v(q) ≈ u(q) ,
	(11.2.43)
ω+(q) = (2K1/M )1/2[1 + O(K2/K1)] ,	v(q) ≈ −u(q) .

Comparison with the vibrational frequencies (11.2.9) shows that the acoustic branch is just like for a monatomic linear chain with atoms of mass 2M and springs of force constant K2. This can be interpreted as follows: each strong spring of force constant K1 binds two atoms into an almost perfectly rigid “molecule” of mass 2M , and the chain made up of such “molecules”

348 11 Dynamics of Crystal Lattices

produces acoustic vibrations. On the other hand, each vibration in the optical branch has practically the same frequency, regardless of the wavelength. This vibrational frequency is the frequency of internal oscillations of a “diatomic molecule”, made up of two atoms of mass M and held together by a spring of force constant K1.

In the more general case, when the dimerized chain is built up of two atoms of unequal mass, the equations of motion for the two atoms are

M1u¨n = −K1 [un − vn] − K2 [un − vn−1] ,

(11.2.44)

M2v¨n = −K1 [vn − un] − K2 [vn − un+1] .

Seeking traveling-wave solutions, the amplitudes need to satisfy the homogeneous linear system of equations

(−ω2M1 + K1 + K2)u(q) − (K1 + K2e−iqa)v(q) = 0 ,

(11.2.45)

−(K1 + K2eiqa)u(q) + (−ω2M2 + K1 + K2)v(q) = 0 .

Nontrivial solutions exist when the determinant of coe cients is zero; the frequencies of the vibrations are then written as

1 ± "

,

ω±2 (q) = 21 ω02

1 − γ2 sin2

21 qa

(11.2.46)

where

M1

+ M2

(11.2.47)

ω02 = (K1 + K2)

1

1

and

K1K2

M1M2

γ2

= 16

.

(11.2.48)

(K1 + K2)2 (M1

+ M2)2

In the K1 = K2 (M1 = M2) limit the results derived for diatomic (dimerized) chains are recovered.

In the general case an acoustic branch starting o at zero frequency and an optical branch starting o at some finite frequency are found. The vibrational frequencies at the center and boundary of the Brillouin zone are

ω (q) =			41 0				| |
−				1 ω			γ q a		1		γ2
						ω0		1
			√				"	−		−
					2
in the acoustic		branch and
ω (q) =				1
−				ω0		ω0		1 +	1		γ2
							"
			√							−
					2
in the optical	branch.

at q → 0,

(11.2.49)

at q = ±π/a,

at q = 0,

(11.2.50)

at q = ±π/a.

11.2 Vibrational Spectra of Simple Lattices

349

11.2.4 Vibrations of a Simple Cubic Lattice

Having examined some one-dimensional examples, let us now turn to the vibrational spectrum of simple cubic crystals with a monatomic basis, employing the approximation that the only firstand second-neighbor interactions contribute to the potential.

Lattice points will be specified by their coordinate indices, and for each point (i, j, k) the six first neighbors (i ± 1, j, k), (i, j ± 1, k), (i, j, k ± 1) and the twelve second neighbors (i ±1, j ±1, k), (i ±1, j, k ±1), (i, j ±1, k ±1) will be taken into account. Calculations are highly simplified by the remark made in connection with (11.1.19): in the harmonic approximation restoring forces arise only for displacements along the line joining the atoms. For displacements in the perpendicular direction the change in the length of the spring is of second order, and can therefore be neglected. This way only two force constants remain: one characterizes the change in energy due to the relative displacement of nearest-neighbor atoms along the line joining them, and the second is related to the change in energy due to the relative displacement of second-neighbor atoms along the face diagonal:

Uharm =	21 K1	ijk	8[ux(i, j, k) − ux(i + 1, j, k)]2
	+ [uy (i, j, k) − uy (i, j + 1, k)]2 + [uz (i, j, k) − uz(i, j, k + 1)]29
+	41 K2		8	[ux(i, j, k) − ux(i + 1, j + 1, k)	(11.2.51)

ijk

+uy (i, j, k) − uy (i + 1, j + 1, k)]2

+[ux(i, j, k) − ux(i + 1, j − 1, k) − uy (i, j, k) + uy (i + 1, j − 1, k)]2

+[ux(i, j, k) − ux(i + 1, j, k + 1) + uz(i, j, k) − uz (i + 1, j, k + 1)]2

+[ux(i, j, k) − ux(i + 1, j, k − 1) − uz(i, j, k) + uz (i + 1, j, k − 1)]2

+[uy(i, j, k) − uy(i, j + 1, k + 1) + uz (i, j, k) − uz (i, j + 1, k + 1)]2

9

+ [uy(i, j, k) − uy(i, j + 1, k − 1) − uz (i, j, k) + uz (i, j + 1, k − 1)]2 .

The force on the atom sitting at the lattice point labeled (i, j, k) is the derivative of the potential:

350	11 Dynamics of Crystal Lattices
	Fx(i, j, k) = −	∂Uharm
		∂ux(i, j, k)

=K1 [2ux(i, j, k) − ux(i + 1, j, k) − ux(i − 1, j, k)]

+21 K2 2ux(i, j, k) − ux(i + 1, j + 1, k) − ux(i − 1, j − 1, k)

+2ux(i, j, k) − ux(i + 1, j − 1, k) − ux(i − 1, j + 1, k)

+2ux(i, j, k) − ux(i + 1, j, k + 1) − ux(i − 1, j, k − 1)

+2ux(i, j, k) − ux(i + 1, j, k − 1) − ux(i − 1, j, k + 1) − uy(i + 1, j + 1, k) − uy (i − 1, j − 1, k)

+ uy(i + 1, j − 1, k) + uy (i − 1, j + 1, k)

(11.2.52)

− uz(i + 1, j, k + 1) − uz (i − 1, j, k − 1)

+ uz(i + 1, j, k − 1) + uz (i − 1, j, k + 1) .

The y- and z-components of the force are given by similar expressions. Substituting them into the equation of motion, and seeking traveling-wave solutions,

uα(R, t) = uα(q) exp{i(qxRx + qy Ry + qz Rz − ωt)} .

(11.2.53)

Using the Born–von Kármán boundary conditions, the allowed values of q are once again expressed in terms of the primitive vectors of the reciprocal

lattice as
q =	n1	b1	+	n2	b2	+	n3	b3 ,	(11.2.54)
	N1			N2			N3

where n1, n2, and n3 are integers. Two vectors q that di er by a reciprocallattice vector describe the same atomic displacement, consequently there are only N = N1 N2 N3 physically di erent vectors q. Here, too, it is useful to choose them inside the Brillouin zone.

The coe cients uα(q) are determined by the homogeneous linear system of equations

					Dαβ (q)uβ (q) = ω2uα(q) ,			(11.2.55)
				β
	where
	Dxx(q) =	2K1	(1 − cos qxa) +			2K2	(2 − cos qxa cos qy a − cos qxa cos qz a) ,
		M				M
	Dyy(q) =	2K1	(1 − cos qy a) +			2K2	(2 − cos qy a cos qxa − cos qy a cos qz a) ,
		M				M
	Dzz (q) =	2K1	(1 − cos qz a) +			2K2	(2 − cos qz a cos qxa − cos qz a cos qy a) ,
		M				M
	Dxy(q) = Dyx(q) =			2K2	sin qxa sin qy a ,
				M
	Dxz(q) = Dzx(q) =			2K2	sin qxa sin qz a ,			(11.2.56)
				M
	Dyz(q) = Dzy(q) =			2K2	sin qy a sin qz a .
				M