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

11.4 Lattice Vibrations in the Long-Wavelength Limit

371

As we have already seen, only two independent elastic constants remain in the isotropic case. In terms of these the tensor Cαβ,γδ is written as

Cαβ,γδ = λδαβ δγδ + μ (δαγ δβδ + δαδ δβγ ) .

(11.4.41)

Using this expression in the elastic energy formula, and comparing it with the corresponding expression for cubic crystals written in terms of the three independent Voigt constants, the following relations are obtained:

λ = c12 ,

μ = c44 ,

(11.4.42)

and λ + 2μ = c11. This implies the relationship

	s	≡	c11 − c12	= 1	(11.4.43)
			2c44

for isotropic materials. Therefore deviations from isotropy may be characterized by the quantity s. As the data in Table 11.2 clearly show, for the majority of cubic crystals this dimensionless quantity is significantly di erent from unity.

11.4.3 Elastic Waves in Cubic Crystals

Once the elastic properties are known, the propagation velocity of elastic waves in the crystal can be determined. We shall demonstrate this for waves propagating in directions of high symmetry in cubic crystals.

Starting with the equation

	∂		∂Uharm
ρ u¨α =				,	(11.4.44)
	∂rβ		∂εαβ
β

which is the consequence of (11.4.13) and (11.4.15), and using (11.4.35), the expression for elastic energy in cubic crystals, the equation that governs the propagation of elastic waves is written as

∂2u

∂2ux

∂2ux

∂2ux

ρ

x

= c11

+ c44

+

∂t2

∂x2

∂y2

∂z2

(11.4.45)

+ (c12 + c44)

∂2uy

+

∂2uz

;

∂x∂y

∂x∂z

similar equations apply to the y- and z-components of the displacement. Owing to the anisotropy of crystals, the propagation velocity of elastic waves depends on the crystallographic direction. For elastic waves propagating along the direction [100] strain is a function of the x-coordinate alone, consequently

ρ

∂2ux

= c11

∂2ux

∂2uy

= c44

∂2uy

∂2uz

= c44

∂2uz

(11.4.46)

, ρ

, ρ

.

∂t

2

2

∂t

2

2

∂t

2

2

∂x

∂x

∂x

372 11 Dynamics of Crystal Lattices

The propagation velocities for the longitudinal and two transverse waves of wave vector q = (q, 0, 0) are immediately read o :

cL =	ρ			1/2	cT =	ρ	1/2	(11.4.47)
				,			.
		c11				c44

Note that this is the same result as for isotropic samples, provided Lamé

constants are expressed in terms of the Voigt elastic constants c .

√ ij

For longitudinal waves of wave vector q = (q, q, q)/ 3 and propagation direction [111] the displacement is

uL(r) = √		u ei(q·r−ωt) ,
1		u
		u
	3

while for the two transverse vibrations

uT1 (r) = √		u ei(q·r−ωt),
1		u
		−0
	2
uT2 (r) = √		u	ei(q·r−ωt).
1		u
		−2u
	6

(11.4.48)

(11.4.49)

Substituting these into the equations of motion, the same velocity is found for the two transverse branches, and a di erent one for the longitudinal:

cL =

3ρ

1/2

cT =

c

11 −

3ρ

1/2

,

. (11.4.50)

c11 + 2c12

+ 4c44

c12

+ c44

Along the propagation direction [110] the longitudinal wave of wave vector

√

q = (q, q, 0)/ 2 is written as

1 uL(r) = √

2

while its transverse counterparts as

	uT1 (r) = √		u ei(q·r−ωt),
	1		u
			−

20

The propagation velocities are

u

uei(q·r−ωt),

0

0

uT2 (r) = 0 ei(q·r−ωt). u

(11.4.51)

(11.4.52)

L

2ρ

1/2

T1

2ρ

1/2

T2

ρ

1/2

c

=

c11 + c12

+ 2c44

, c

=

c11 − c12

, c

=

c44

.

(11.4.53)

11.4 Lattice Vibrations in the Long-Wavelength Limit

373

Along this propagation direction the velocity is di erent for each branch. The observation made for acoustic vibrations in simple cubic crystals ap-

plies to elastic waves, too: vibrations that are longitudinal and transverse with respect to the propagation direction exist only in some directions of su ciently high symmetry. In the general case u(r) is neither parallel nor perpendicular to q.

11.4.4 Optical Vibrations in Ionic Crystals

In contrast to acoustic vibrations, where atoms of the primitive cell oscillate in phase and with equal amplitudes in the long-wavelength limit, in optical vibrations atoms of the primitive cell oscillate completely out of phase and their center of mass remains stationary. Both longitudinal and transverse modes have finite frequencies, and they are di erent in general. In spite of this expectation, the longitudinal and transverse optical vibrations in silicon, shown in Fig. 11.12, are observed to be of the same frequency for q = 0. That this is not accidental is confirmed by Fig. 11.13(a), where the same is observed in the measured vibrational spectrum of diamond.

	)	2.5	LO
	rad/sec
			TO
		2.0
	14
	(10		LA
	!	1.5
	uency
	eq	1.0	TA
	Fr
		0.5

(a)

(½½½) (###) (000)

	LO	)	3.0	LO
		rad/sec
	TO
		13	2.5
	LA	(10	2.0	TO
		ncy!
			1.5
	TA	ue
		eq	LA
		Fr
			1.0
			0.5	TA
		(b)
	(00#) (001)		(½½½)	(###) (000)

LO

TO

LA

TA

(00#) (001)

Fig. 11.13. (a) Dispersion relations for the lattice vibrations of diamond in two characteristic directions of the Brillouin zone [J. L. Warren et al., Phys. Rev. 158, 805 (1967)]. (b) The same for NaI [based on A. D. B. Woods et al., Phys. Rev. 131, 1025 (1963)]

Exploiting the symmetry properties of force constants it may be shown that in a cubic crystal longitudinal and transverse optical vibrations must indeed be of the same frequency at q = 0. However, when the same measurements are performed in ionic crystals – results are shown in Fig. 11.13(b) and listed in Table 11.3 –, the two kinds of optical vibrations are found to be of unequal frequencies even in q = 0.

374 11 Dynamics of Crystal Lattices

Table 11.3. The frequency ν of optical vibrations (in units of 1012 Hz) at the center of the Brillouin zone for some covalently bonded elements and ionically bonded compounds that have two atoms per primitive cell

Element	νLO = νTO	Compound νLO νTO
C (diamond)	39.9	NaCl	7.91	4.92
Si	15.6	CsCl	4.95	2.97
Ge	9.0	GaAs	8.75	8.06

This indicates that, unlike in covalently bonded materials, the vibrational spectrum in ionic crystals cannot be represented by the mass–spring model. The underlying physical reason for the di erent behavior of ionic crystals is that the longitudinal motion of the atoms – in contrast to their transverse motion – gives rise to a periodically varying charge density, and through it to a periodically varying polarization. This generates an additional restoring force for longitudinal motions, and so such vibrations will be of higher frequency than transverse ones.

Since in the long-wavelength limit the motion of atoms is practically identical in every primitive cell, it is su cient to examine the equations of motion in a single primitive cell. However, due account must be taken of the fact that inside ionic crystals a screened local electric field Ee acts on the atoms. Denoting the displacement of the ion of charge e (−e ) and mass M+ (M−) from its equilibrium position by u+ (u−), the equations of motion are

M+

M−

d2u+	= −K (u+ − u−) + e Ee ,
dt2
d2u−											(11.4.54)
	=	−	K (u	− −	u	+	)	−	e E	e	.
dt2

Combining these into a single equation for the relative displacement u = u+ − u−:

d2u	= −Ku + e Ee ,	(11.4.55)
M dt2

where M = M+M−/(M+ + M−) is the reduced mass.

In simple cubic crystals the local field and the macroscopic field are related

	by the Lorentz formula6	1
	Ee = E +		P .	(11.4.56)
		3 0

The polarization P comes from two sources. Denoting the polarizability of the medium due to the displacement of electrons by α, the local field induces a polarization

6 H. A. Lorentz, 1880.

	11.4 Lattice Vibrations in the Long-Wavelength Limit				375
	P el =	N	αEe	(11.4.57)
		V

in a sample of volume V made up of N primitive cells. In addition to electronic polarization, an additional contribution arises from the displacement of ions, and so

	P =		N			(e u + αEe ) .		(11.4.58)
			V
	Substituting (11.4.56) into this equation, the polarization is
	P =	N				e u + αE	.	(11.4.59)
		V 1 − (N α/3V 0)

In the high-frequency limit ions are unable to follow the rapid variations of the field, therefore they remain practically stationary, so

	P =	N			αE		(11.4.60)
						.
		V		1 − (N α/3V 0)
Comparison with the general relationship
P (ω) = (ω)E − 0E = [ r(ω) − 1] 0E							(11.4.61)
between the polarization and the dielectric constant gives
	N α/V				= [ r(∞) − 1] 0 ,		(11.4.62)
	1 − (N α/3V 0)

which leads to a relation between the polarizability α and r(∞) that is very similar to the Clausius–Mossotti relation:7

			N	α =	r(∞) − 1	0 .	(11.4.63)
					r(∞) + 2
			3V
Substituting this into (11.4.59), the expression for polarizability,
P =	N	e [ r(∞) + 2]u + [ r(∞)				− 1] 0E .	(11.4.64)
	3V

Using this term in the Lorentz formula for the local field, and introducing the

notation

1

N

ω02 =

r(∞) + 2

,

(11.4.65)

M

K − 9V 0 e 2

the equation of motion (11.4.55) takes the form

d2u

1

e r(∞) + 2 E .

M

= −M ω02u +

(11.4.66)

dt2

3

7O.-F. Mossotti, 1850, and R.√J. E. Clausius, 1879. When written in terms of the index of refraction, n = , it is known as the Lorentz–Lorenz equation

(H. A. Lorentz and L. V. Lorenz, 1880).

376 11 Dynamics of Crystal Lattices

In the static case the left-hand side of the equation vanishes. Substituting the relation between the static displacement and the electric field into the polarization formula (11.4.64), and making use of (11.4.61) at ω = 0,

r(0)

−

r( ) =

N

e 2

r(

∞

) + 2

2

,

and therefore

∞

9V M ω02

0

d2u

1/2

V M ω2

0

r(0) − r(∞)

1/2

E .

M dt2

= −M ω02u +

N 0

Changing the variable u to

s =

7

u ,

NV

M

(11.4.67)

(11.4.68)

(11.4.69)

the only remaining parameters in the equations of motion are the frequency ω0 and the dielectric constant:

d2s	= −ω02s + ω0 0( r(0) − r(∞))E .	(11.4.70)
dt2

In terms of the variable s the polarization reads

P = ω0 0( r(0) − r(∞))s + [ r(∞) − 1] 0E . (11.4.71)

The quantity s is proportional to the displacement of ions. To decompose it into components parallel (sLO) and perpendicular (sTO) to the wave vector q – which specifies the propagation direction –, the conditions

q · sTO = 0 , q × sLO = 0

(11.4.72)

are exploited. Assuming wavelike spatial propagation with time-dependent coe cients for the longitudinal and transverse components alike,

sLO = QLO(t)eLOeiq·Rm ,

sTO = QTO(t)eTOeiq·Rm .

(11.4.73)

To solve the equation of motion, it should be noted that inside ionic crystals the macroscopic field satisfies the Maxwell equations

curl E = 0 , div D = div( 0E + P ) = 0 .				(11.4.74)
Taking the curl of (11.4.70) gives
	d2QTO	2
		= −ω0	QTO	(11.4.75)
	dt2

for transverse vibrations; their frequency ωTO is seen to be equal to ω0.

11.5 Localized Lattice Vibrations

377

To write up the equation for the longitudinal component, an equation derived from div D = 0 and (11.4.71) is used:

	r( )	7	0				(11.4.76)
div E +	ω0			r(0) − r(∞)	div s = 0 .
	∞

The normal coordinates of longitudinal optical vibrations must therefore satisfy

d2QLO	2	r(0)
	= −ω0		QLO ,	(11.4.77)
dt2		r(∞)

while the frequencies of LO vibrations are given by

	ωLO =	r(0)		1/2
				ωTO .	(11.4.78)
		r(∞)

This is the Lyddane–Sachs–Teller relation.8 As we have seen, the di erence between longitudinal and transverse optical frequencies is easily observed in ionic crystals, and ωLO > ωTO is generally valid. The ratios calculated from relative dielectric constants measured at low and high frequencies are usually in good agreement with the ratios of the frequencies measured directly.

11.5 Localized Lattice Vibrations

It was mentioned in Chapter 9 in connection with the structure of real crystals that even in samples of the highest purity there are always foreign atoms, impurities. These obviously change the lattice dynamics, since both their mass and interactions with the neighbors are di erent from those of other atoms. Lattice vibrations are distorted around the impurity, hence their spectrum is also modified. To understand how an impurity may influence the spectrum of lattice vibrations, we shall discuss the simple case when a single atom of an ideal crystal is replaced by an impurity.

The mathematical description becomes rather complicated as the impurity breaks the translational symmetry of the crystal. Consequently, the states of the system cannot be characterized by the wave vector, as it is no longer a good quantum number. Nevertheless the frequency distribution of vibrational states remains a meaningful quantity.

11.5.1 Vibrations in a Chain with an Impurity

Once again, we shall use the example of a one-dimensional chain to demonstrate the method to account for the e ects of an impurity. Consider a chain

8 R. H. Lyddane, R. G. Sachs, and E. P. Teller, 1941.

378 11 Dynamics of Crystal Lattices

made up of atoms of mass M , with the atom at site n = 0 replaced by one of di erent mass (M0). For simplicity we shall assume that all force constants remain the same around the impurity. Following the steps of the discussion of monatomic linear chains, classical equations of motion are derived for the atomic displacements un. If only nearest neighbors interact, (11.2.3)

M u¨n = −K [2un − un−1 − un+1] ,

(11.5.1)

continues to be valid everywhere except for n = 0, where

M0u¨0 = −K [2u0 − u1 − u−1]

(11.5.2)

because of the di erent mass of the atom. The two equations can be written jointly as

M u¨n + K [2un − un−1 − un+1] = δn,0 (M − M0)¨u0 .

(11.5.3)

In spite of the impurity, vibrations are expected to propagate in the lattice; un(t) is therefore Fourier transformed with respect to the time variable. The obtained Fourier coe cient, un(ω) then satisfies the equation

−ω2M un + K [2un − un+1 − un−1] = δn,0 M ω2u0 ,	(11.5.4)
where M = M0 − M . Introducing the notation ω02 = 4K/M ,
−ω2un + 41 ω02 [2un − un+1 − un−1] = δn,0 (ΔM/M )ω2u0 .	(11.5.5)

Apart from the immediate vicinity of the impurity, atomic displacements of the form un = Aλn or un = Aλ−n are sought. It is easily shown that λ is complex when ω < ω0, and so the vibrations of the chain are wavelike. On the other hand, λ is real when ω > ω0; vibrational amplitudes then decrease exponentially with the distance from the impurity, therefore only the impurity and its vicinity participate in the vibration.

Although the Hamiltonian of the system cannot be written as a sum of independent normal modes, un(ω) may be Fourier transformed with respect to the spatial variable as well, leading to the new variable

u(q, ω) =	√1
		un(ω)e−iqna .	(11.5.6)
	N	n

By Fourier transforming the quantity Fn = δn,0 (ΔM/M )ω2u0(ω) on the right-hand side of (11.5.5), the equation of motion becomes

−ω2u(q, ω) + 41 ω02 2 − e−iqa − eiqa u(q, ω) = F (q) ,		(11.5.7)
which can be simplified to
−ω2u(q, ω) + ω02 sin2	21 qa u(q, ω) = F (q) ,	(11.5.8)

11.5

Localized Lattice Vibrations 379

where

1

M

1

F (q) =

√

Fne−iqna =

√

ω2u0(ω) .

(11.5.9)

M

N

n

N

Neglecting the expression on the right-hand side, the solutions of the equation are the previously determined eigenfrequencies ω(q) = ω0 |sin(qa/2)| of the vibrations in a regular lattice. In terms of these the equation governing the vibrations of the lattice with an impurity reads

ω2(q) − ω2 u(q, ω) = F (q) .

(11.5.10)

Assuming that, owing to the impurity, each vibrational frequency is slightly shifted compared to the pure crystal, the formal solution of the equation for the vibrational amplitudes is

u(q, ω) =	F (q)	(11.5.11)
	ω2(q) − ω2 .

Fourier transforming this expression back into real space, and making use of the formula derived for F (q),

	M 1				ω2u0(ω)eiqna
un(ω) = M N					ω2(q)	−	ω2 .		(11.5.12)
				q

The solution is self-consistent if the left-hand side gives the same u0 for the displacement of the impurity atom at n = 0 as what appears on the right-hand side, in which case

M ω2					1
				ω2	(q)	−	ω2 = 1 .		(11.5.13)
M N
			q

The solutions to this equation give the eigenfrequencies of the allowed vibrations of the chain with the impurity. To illustrate the equations graphically, it is first rearranged as

f (ω2) ≡	ω2			1			=	M		−1	(11.5.14)
										.
	N	q ω2		(q)	−	ω2		M

The function f (ω2) of the left-hand side is plotted against ω2 in Fig. 11.14. The vibrational frequencies in the chain with an impurity are the horizontal coordinates (ω) of the intersection points of these curves with the horizontal line at M/ M .

Since the left-hand side is discontinuous at each value ω2(q) that corresponds to an eigenfrequency of the ideal chain, between any two adjacent eigenfrequencies there will be an intersection point, i.e., an allowed vibrational frequency of the chain with an impurity. If the impurity atom is heavier than the other atoms in the chain ( M > 0) then each vibrational frequency is shifted slightly downward compared to the frequencies of the ideal chain, but

380 11 Dynamics of Crystal Lattices

Fig. 11.14. Graphical determination of the eigenfrequencies of a chain with an impurity from the intersection points of f (ω2) with the horizontal line at M/ M

they cannot be lower than the next eigenfrequency. Although not visible in this equation, the original one nevertheless implies that ω = 0 is also a solution, and it corresponds to a uniform translation for a chain with an impurity, too. Therefore vibrations fill – practically continuously – the same range as for a pure crystal. If the impurity atom is lighter than the others ( M < 0) then the frequency of each vibrational mode is shifted slightly upward, but they cannot exceed that of the next eigenfrequency. The highest-frequency vibration is an exception, which appears as a separate mode above the continuum. Since ω > ω0 for this vibration, the vibrational amplitude exponentially decreases with the distance from the impurity, as mentioned above. The vibration is localized, only the light impurity atom and its small vicinity participates in it with significant amplitudes. In diatomic chains such localized excitations may appear between the acoustic and optical branches or above the optical branch.

It is worth taking a closer look at the case when M0 > M and M is comparable to the mass of the atoms. As we have seen, no localized vibration is possible in this case. Compared to the pure sample, vibrational frequencies are shifted downward, however, this shift is not uniform. Going back to the graphical solution of (11.5.14): if the right-hand side is of order unity, the frequencies determined by the intersection points are shifted only slightly with respect to the original ones at the low end of the spectrum, while at its high end