Kosevich A.M. The crystal lattice

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116

3 Vibrations of Polyatomic Lattices

It is evident that (3.7.7), (3.7.8) admit the solutions

ϕ = ϕ

−q|z|

ikζ−iωt

, A = A

−q|z|

ikζ−iωt

, ξ = ξ

ikζ−iωt

= k

− ω

, ζ =(x, y).

(3.7.9)

The p resence of the delta-like right-h and sides in (3.7.3) is equivalent to jumps

of the space derivative of the potentials with respect to the coordinate z at z = 0.

Therefore,

= −2πi

kξ

, A

= −2πi

ξ. (3.7.10)

We substitute (3.7.10) into (3.7.8) to obtain

mω

2πe



k(kξ

) −



. (3.7.11)

Multiplying both parts of (3.7.11) by the vector k, we write implicitly the dispersion

law for longitudinal (plasma) vibrations

2πe

q ≡

2πe



−

. (3.7.12)

Solving (3.7.12) for ω

, we renormalize the dispersion law for longitudinal vibra-

tions to include the electromagnetic wave retardatio n

= 2(ck

∗

)





1 +



∗



−1





, (3.7.13)

where k

∗

= πe

/mc

is the characteristic wave vector whose value is shared by two

regions of a different behavior of the dispersion law (3.7.13). A plot of the dispersion

law for longitudinal vibrations of a 2D electron crystal is given in Fig. 3.6.

For k  k

∗

, we get the dispersion law of electromagnetic waves in a medium

with ε = 1: ω = ck. Thus, the maximum long-wave longitudinal vibrations of an

electron crystal (k  k

∗

) prove to be consistent with ﬁeld vibrations: for ω → ck the

parameter q → 0 and the electromagnetic wave penetrates deep inside the medium.

For k  k

∗

we come back to the dispersion law (3.7.3). Thus, shorter-wave vibrations

(but under the condition ak  1) have the dispersion law that differs greatly from

that for electromagnetic waves in the medium. Therefore, the corresponding ﬁeld

vibrations cannot exist in the bulk far from a 2D electron crystal and they are localized

near it.

However, it should be noted that the region where the dispersion law (3.7.13) is

essentially renormalized is n ot important in experim e nts with a 2D electron crystal on

the helium surface. Indeed, with existing electron densities n

∼ 10

− 10

−2

we have k

∗

∼ 10

−5

−10

−3

−1

, so that the characteristic wavelength λ

∗

= 2π/k

∗

is too large for laboratory conditions.

3.8 Long-Wave Vibrations of an Ion Crystal

117

Finally, we dwell brieﬂy on the renormalization of the dispersion law for transverse

sound vibrations. Equation (3.7.11) admits no transverse vibrations (for q > 0) local-

ized near a 2D electron crystal. This is due to the fact that (3.7.11) is obtained in the

continuum approximation, completely ignoring the periodic structure of a 2D electron

crystal. Both the derivation of the dispersion law (3.7.5 ) an d its renor malization can

be obtained only on the basis of a certain discrete model.

Fig. 3.6 Transformation of limiting long-wave dispersion law for 2D

plasma vibrations.

3.8

Long-Wave Vibrations of an Ion Crystal

Let us discuss the equation s for the vibration s of a 3D lattice composed of ions. The

equations of motion of a crystal (2.8.6) in the long-wave approximation and the elas-

ticity theory equations (2.8.7) were obtained using the assumption that the interaction

of vibrating atoms decreases rapidly with increasing distance between them. Thus,

expanding the displacement ﬁelds in a series, we can use only its leading terms and

reduce the equation of crystal motion to a system of second-order differential equa-

tions. However, our experience of studying the optical vibrations of two-dimensional

dipole lattices shows that this procedure is not always possible. I f a polyatomic lattice

does not consist of neutral a toms, but of electrically or magnetically active dipoles

or ions (electrons), the interaction potential between them decreases weakly with dis-

tance and the standard method of expansion in a series over the displacement gradients

is inapp licable.

Apart from that, under lattice vibrations the moving ions excite dynamical electro-

magnetic ﬁelds in a crystal whose retardation effect is important. But, for long-wave

3D crystal vibrations the electromagnetic interaction is easily taken into account by

means of the Maxwell equations for a continuous medium.

A macroscopic approach allows one to divide the ionic interactions (for simplicity

we assume these to be point charges) into two types with an essentially different physi-

cal origin. The ﬁrst type is represented by the interactions (determined unambiguously

118

3 Vibrations of Polyatomic Lattices

by local conditions) that are described by the matrix of atomic force constants α(n).

These interactions as well as the atomic ones are responsible for the elastic properties

of nonionic crystals. The second type is generated by the action of the mean electric

ﬁeld in a crystal on the ion charge (the effect of the mean magnetic ﬁeld on moving

ions can be neglected

). This type, also involving the Coulomb ion interaction, cannot

be determined by local conditions only and is described by introducing macroscopic

ﬁelds.

We now realize the program proposed. In the ionic crystal, equations of the type

(1.8.4) should be written for each ion sublattice separately. If u

(n) is the displace-

ment of the n-th site of the sublattice with number s,wehave:

∑



+ C

ikl







+ C

iklm







} + e

, (3.8.1)

where e

is the s-th ion charge in the unit cell (by virtue of electric neutrality,

∑



= 0), E is the averaged electric ﬁeld strength in a crystal. The tensors of second,

third and fourth rank C



, are obviously connected with the elements of the matrix of

atomic force constants in a p olyatomic lattice. It follows from their deﬁnition and the

equality (3.2.4) that

∑



∑



∑



(n)=0. (3.8.2)

Furthermore, in the ionic crystal where each ion is an inversion center, C

ikl

≡ 0.

For crystals such as NaCl, we shall further use the last identity. We introduce by

(3.2.13) the displacement of the center of mass of a unit cell u,aswellastherelative

displacement of ions ξ

= u

− u. We keep on the r.h.s. of (3.8.1), the lowest-order

space derivatives of the vector u and independent displacements ξ

∑



+ C

iklm





} + e

. (3.8.3)

In this approximation, the dynamic equations for the vectors u and ξ

are separated

= c

iklm



+ e

, c

iklm

∑



iklm



;

∑



+ e

(3.8.4)

Equations (3.8.3) should be solved together with the system of macroscopic

Maxwell equations whose form is well known. We note some points that refer to

4) Even in the region of limiting vibration frequencies (ω ∼ 10

−1

)theion

motion velocities are less than the sound velocities in a crystal (v < s ∼

cm/s) and the Lorentz force generated by the magnetic ﬁeld is vanish-

ingly small compared to the electric force.

3.8 Long-Wave Vibrations of an Ion Crystal

119

writing and solving the Maxwell equations. First, ionic crystals are dielectric (in-

sulator) crystals and have no marked magnetic properties. Thus, the only quantity

that refers to th e characteristics of the medium properties and is involved in Maxwell

equations is the crystal polarization vector P. We remind ourselves that the vector P

is included in the deﬁnition of electric induction of a dielectric D: D = E + 4πP.

Then, since we consider the ions as point charges, we should relate the vector P only

with the displacement of the ions from their equilibrium positions

P =

∑

≡

∑

. (3.8.5)

We analyze the equations for crystal ion vibrations by considering NaCl-type crys-

tals that possess cubic symmetry and contain only two ions in the unit cell. Each of

the ions is a center of symmetry (Fig. 0.2). In this case (3.8.4) will be much simpliﬁed

= C

+ C

+ e

;

= C

+ C

+ e

(3.8.6)

By virtue of the cubic symmetry of a crystal and the properties (3.8.2) all the elements

of the matrix C



, are expressed through one scalar quantity:



= C



, C

= C

= −C

= −α

. (3.8.7)

To be speciﬁc, let the subscript 1 refer to a positive ion (e

= e, e

= −e). Then

(3.8.6), taking into account (3.8.7), will be

= −α

(ξ

− ξ

)eE;

= −α

(ξ

− ξ

)eE.

(3.8.8)

Equations (3.8.8) can be replaced by a single equation for the relative displacement

ξ = ξ

− ξ

= −α

ξ + eE, (3.8.9)

where µ is the reduced mass of a unit cell (µ = m

/M). The crystal polarization

is expressed through the displacement ξ

P =

ξ. (3.8.10)

Equations (3.8.9), (3.8.10) and also the Maxwell equations describe the combined

(coupled) optical ion crystal vibrations and the electromagnetic ﬁeld vibrations. Thus,

the above system of equations allows one to take into account the interaction of a free

120

3 Vibrations of Polyatomic Lattices

electromagnetic ﬁeld and independent optical vibrations of a crystal. The interaction

between electromagnetic waves and the optical eigenvibrations of a crystal is espe-

cially large when their frequencies and the wave vectors almost coincide (under the

resonance conditions). As a result of such an interaction, collective excitations of a

new type known as polariton vibrations appear. This is why the long-wave ion crystal

vibrations are analyzed within a macroscopic theory of polaron excitations.

It is known from electrodynamics that the displacement ﬁeld ξ is excluded from

Maxwell equations by incorporating the dielectric permeability of the medium

, ε,

relating the Fourier time components of the vectors D and E:

D(ω)=ε(ω)E(ω). (3.8.11)

To obtain the dielectric permeability, we consider the harmonic vibrations of all

ﬁelds. For stationary vibrations with frequency ω, (3.8.9) is transformed to the alge-

braic one

(ω

− ω

)ξ =

E, (3.8.12)

where ω

= α

/µ is the square of the characteristic frequency th a t is the limiting

optical frequency for a crystal wh ere there is no interaction with an electromagnetic

ﬁeld.

If ω = ω

, we obtain trivially from (3.8.10)–(3.8.12) an expression for the dielec-

tric permeability

ε(ω)=1 +

− ω

≡

+ ω

−ω

− ω

, (3.8.13)

where ω

= 4πe

/µV

is the square of the so-called plasma frequency (ω

the frequency of the eigenvibrations of an ionic 3D plasma whose ions interact only

through a macroscopic Coulomb ﬁeld), µ is the reduced mass of a pair of ions with

opposite signs, V

is the volume of this pair. If we take V

∼ 10

−23

and assume

µ ∼ 5 ×10

−23

g, it is easily seen that ω

∼ 10

−1

. Therefore, we may assume

that ω

∼ ω

With formula (3.8.1 3) for dielectric permeability, we can directly write th e dis-

persion law of transverse vibrations for a crystal–electromagnetic ﬁeld system (the

possible trivial decomposition of electromagnetic vibr ations into transverse and lon -

gitudinal ones results from the symmetry of a cubic crystal). In fact, the transverse

electromagnetic vibrations (electromagnetic waves) in a m edium with dielectric per-

meability ε(ω) have the dispersion law

ω =



ε(ω)

, (3.8.14)

5) As the NaCl-type crystal has cubic symmetry , the dielectric permeability

tensor of such a medium reduces to a single constant.

3.8 Long-Wave Vibrations of an Ion Crystal

121

where c is the light velocity in vacuum. Substitu ting (3.8.13) into (3.8.14), we write

the dispersion law for transverse vibrations as

= ω

− ω

. (3.8.15)

The longitudinal electromagnetic vibrations with nonzero frequency are possible

in the medium only when

D = 0,whenE = −4πP.ButforE = 0, it follows

from (3.8.11) that the frequencies of the corresponding vibrations are zeros of the

functions ε(ω): ε(ω)=0. Thus, longitudinal vibrations are possible only at frequen-

cies ω = ω

, ω

= ω

+ ω

. Therefore, their dispersion law reduces to a trivial

dependence; the frequency coincident with the value ω

is constant.

The dispersion laws for noninteracting electromagnetic waves and optical crystal

vibrations can be written in a simpler form by formally excludin g th e interaction in

the system concerne d, i. e., by setting e = 0. The Maxwell equations will then reduce

to the ﬁeld equations in vacuum and the dispersion law for electromagnetic waves will

be of a simple form

ω = ck. (3.8.16)

The dispersion law for optical vibrations follows from (3.8.12) with e = 0

ω = ω

. (3.8.17)

The limiting simplicity of this dispersion law follows from the condition ak  1.

We note that the plots of the dispersion laws (3.8.16), (3.8.17) (Fig. 3.7a, straight lines

1 and 2, respectively) intersect. At the intersection point of the plots the frequ e ncies

and wavelengths of vibrations of different nature coincide and the above-mentioned

cross situation arises. Therefore, when even a small interaction between vibrations oc-

curs, resonance considerably affecting the electromagnetic and mechanical processes

of the system concerned is observed.

Indeed, we come back to the dispersion law of transverse vibrations (3.8.15) where

we have taken into account the interaction required. If ω  ω

, (3.8.15) gives the

dispersion law for electromagnetic waves in the medium with a certain static dielectric

permeability

ω =



ε(0)

, ε(0)=1 +





Under such low-frequency v ibrations, the lattice manages to adapt to the electro-

magnetic ﬁeld, causing a decrease in the electromagnetic wave velocity only.

If ω  ω

the dispersion law for electromagnetic waves in vacuo (3.8.16) follows

from (3.8.15). The electromagnetic waves with such frequencies that arise in a crystal

6) The condition D = 0 for ω = 0 is a result of Maxwell’s equation for the

dielectric

curl H =

∂D

∂t

= −i

and the longitudinal character of vibrations (curl H = 0).

122

3 Vibrations of Polyatomic Lattices

Fig. 3.7 Dispersion law of polariton vibrations: (

) dispersion curves

of independent ﬁeld and crystal vibrations when there is no interaction;

(

) removal of degeneracy; (

) vibration frequencies when the retarda-

tion (c = ∞) is neglected.

do not force the lattice to move because of its inertia, so that the crystal does not react

to the wave transmission.

In the frequency range ω ∼ ω

, the dispersion law ceases to be linear (Fig. 3.7b).

A radical rearrangement of the dispersion law at frequencies ω ∼ ω

testiﬁes to

a resonance character of the interaction between the electromagnetic ﬁeld and the

optical vibrations at ω = ω

It follows from (3.8.13) that at frequencies that do not correspond to the crystal vi-

brations concerned, the d ielectric p ermeability becomes negative (total wave reﬂection

from a crystal).

Finally, we consider an extreme form of the dispersion law for transverse vibrations

in a special case when the retardation of the electromagnetic waves are entirely ne-

glected (c → ∞). In this case for k = 0, (3.8.15) yields the relation ω = ω

that is

the same as the dispersion law of optical vibrations (3.8.17) for the lattice noninter-

acting with the ﬁeld. For ω = ω

, (3.8.15) is consistent only for k = 0.

The longitudinal vibrations are retardation independent (they are quasi-static), so

that the plots of the dispersion laws at c = ∞ have the form shown in Fig. 3.7c.

The difference in the plots in Fig. 3.7b is manifest at wavelengths for which

<a(ω

/c) ∼ s/c,wheres is the sound velocity in a crystal. As the ratio

s/c ∼ 10

−5

, the retardation effects the crystal vibration spectrum in a small part of

the allowed interval of k. Thus, in all problems where the vibrations with wavelengths

satisfying the condition s/c  ak  1 are dominant, the retardation is insigniﬁcant

and the optical vibrations in an ionic crystal should be associated with the frequency

for transverse mechanical vibrations, and with the frequency ω

for longitudinal

vibrations, caused by a purely static electric ﬁeld.

The ratio between transverse and longitudinal vibration frequencies is related to

macroscopic characteristics of the ionic crystal, namely, the limiting values of its d i-

3.8 Long-Wave Vibrations of an Ion Crystal

123

electric permeability. In fact, it directly follows from (3.8.13) th at





ε(0)

ε(∞)

. (3.8.18)

Although, using (3.8.13) it is possible to conclude that ε(∞)=1. But this property

of the model concerned is not used in (3.8.18). If we reject the point charges model

and take into account the electric structure of ions that allows one to consider them

being polarizable in an external ﬁeld, we c ome to the conclusion that ε( ∞) = 1,but

in this case too (3.8.18) remains valid

3.8.1

Problems

1. Find the dispersion relation for longitudinal vibrations of a 1D chain with equidis-

tant oscillators. Discuss the possible existence of similar 1D systems with an inhomo-

geneous ground state.

Hint. Assume each atom possesses the energy U

=(1/2)mω

and take into

account the interaction of nearest neighbors only.

Solution. The dispersion law is

= ω

2α(0)

sin

if α(0) < 0 and 2 |α(0)| > mω

then for a certain k = k

in the r ange 0 < k <

π/a the vibration frequency ω vanishes. Thus, such a system has an equilibrium

superlattice with period b = π/k

2. Find the long-wave dispersion relation for librational vibrations of a 2D dipole

lattice, taking into account the electromagnetic wave retardation.

3. Find the long-wave dispersion relation for a 2D electron crystal in a magnetic ﬁeld

H perpendicular to the crystal plane.

Solution.





∗

(k)

7) By the frequency ω = ∞ we mean, in the last case, the frequenc y satisfying

the condition ω  ω

, but still remaining small compared to the character -

istic frequencies of the interatomic electron motion.

Frequency Spectrum and Its Connection with the Green

Function

4.1

Constant-Frequency Surface

A set of the polarization vectors e(k, α) of crystal vibrations and the dispersion law

ω = ω

(k) (4.1.1)

provide full information on the character of vibrational states of the lattice. However,

to understand some phenomena generated by lattice vibrations, the geometric (more

exactly, topographic) representation of the dispersion law seems to be more instru-

mental than its analytical notation (4.1.1). This can be done by the introduction and

analysis of so-called isofrequency surfaces.

Isofrequency surfaces are constructed independently for each branch of vibrations,

so that in writing the dispersion law (4.1.1), we omit the index α, implying one of the

branches.

An isofrequency surface or a constant-frequency surface is the surface in k-space

described by

ω(k)=ω, ω = constant. (4.1.2)

Let us elucidate how the form of isofrequency surfaces changes when the frequency

ranges from ω = 0 up to the maximum possible one. For small frequencies the

dispersion law of a crystal coincides with the dispersion law of sound waves, so that

the isofrequency surface is determined by

k =

s(κ)

, κ =

, (4.1.3)

where s(κ) is the sound velocity. Expression (4.1.3) is a reason to introduce another

name of the isofrequency surface for low-frequency sound vibrations, namely, a slow-

ness surface.

As the sound velocity in a crystal is a ﬁnite quantity for all directions of κ, it follows

from (4.1.3) that the corresponding slowness surfaces are closed (and similar to each

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich

 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

126

4 Frequency Spectr um and Its Connection with the Green Function

other). If we look at the ﬁgure where these surfaces are crossed by the plane that goes

through a point k = 0 (Fig. 4.1a), we see that the freq uencies ω

and ω

satisfy the

condition ω

< ω

(an isofrequency surface for a smaller frequency is inside that for

larger frequency). The slowness surfaces for ω → 0 are not necessarily convex, they

can be “pillow”-shaped (Fig . 4.1b).

The fact that the constant-frequency surface even of long-wave crystal vibrations

may be other than convex leads to some interesting physical results. The parts where

the cross section of an isofrequency surface is convex are isolated from those where

it is concave by the points with zero cross-sectional curvature (Fig. 4.1b). In a 3D

k-space, the convex parts of an isofrequency surface are isolated from the concave

ones by the lines along which the Gaussian (total) surface curvature vanishes. As an

example, we consider a constant-frequency surface for one of the transverse vibration

modes in a germanium crystal, where the larger part of the mode is convex. Since the

Ge crystal has cubic symmetry, only one octant is shown in Fig. 4.2. The thick lines

illustrate the geometrical position of the points with zero Gaussian cu rvature. The

characteristic directions, for which the curvature is concave, are also shown.

Fig. 4.1 Cross sections: (

) of convex isofrequency surfaces; (

)of

nonconvex isofrequency surfaces.

The isofrequency surfaces for frequencies close to the maximum frequency of

acoustic vibrations ω

have a much simpler form. Indeed, it follows from (1.4.4)

that constant frequency surfaces are as ellipsoids with centers at the point k = k

corresponding to the maximum frequency ω

. Counting the vector k from this point,

we get the following equation for an ellipsoid

= ω

− ω, ω

− ω  ω

. (4.1.4)

A set of ellipsoids (4.1.4) crossed by the plane that goes through their common

center may be compared with those in Fig. 4.1a, but now ω

> ω

(an isofrequency

surface for larger frequency is placed inside that for smaller frequency).

On clearing up the topology of isofrequency surfaces at the boundaries of the eigen-

frequency band of acoustic vibrations, we note that the points where ω = 0 and

ω = ω

repeat periodically in a reciprocal space due to the periodicity properties