Kosevich A.M. The crystal lattice

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8.4 The Long-Wave Vibration Spectrum

211

We rewrite (8.3.11), (8.3.12) as equations for the functions u(r, t) and ϕ(r, t):

(n)

∂

∂t

− λ

iklm

∇

= Aµ

∇

∂ϕ

∂t

; (8.3.17)

∂

∂t

− Aρ

(s)

∇

ϕ = µ

∂

∂t

, (8.3.18)

with µ

= ρ

−ρ

(s)

, and introduce a new elasticity modulus tensor

iklm

= λ

iklm

− Mβ

= λ

iklm

− M (δ

− β

)(δ

− β

) .

Equations (8.3.17), (8.3.18) are a complete set of equations for small mechanical

quantum crystal vibrations. They involve, apart from the tensor parameter ρ

(s)

speciﬁc

for a quantum crystal, the quantity A having the dimension of the velocity squared.

According to (8.3.15), this qua ntity is included in the linear differential relation

∂v

∂t

= A grad (η + β



) , (8.3.19)

derived by assuming the existence of the velocity potential for a superﬂuid motion

velocity. The constant A is a macroscopic quantum-mechanical characteristic of the

tunneling atomic motion in the crystal.

8.4

The Long-Wave Vibration Spectrum

Let us now discuss the dispersion laws for small vibrations of a quantum crystal.

Assuming all the variables in (8.3.17), (8.3.18) are dependent on the coordinates and

time through the multiplier exp(ikr − iωt),weﬁnd

(ω

(n)

−

iml j

+ Aµ

ωϕ = 0;

(ω

− Aρ

(s)

)ϕ + µ

ωu

= 0.

(8.4.1)

We have obtained a system of four homogeneous algebraic equations that allows us

to ﬁnd the four unknowns (u, ϕ). The solvability condition for this system determines

the d ependence ω = ω(k ), i. e., the dispersion law. But the solvability condition

for the system (8.4.1) is the zero determinant of a corresponding fourth-rank matrix.

Hence, to each value of the wave vector k there correspond four eigenfrequencies

ω, i. e., there are four branches of the quantum crystal eigenvibrations. Thus, a new

branch of mechanical vibrations generated by additional degrees of freedom appears

in a quantum crystal.

It follows from (8.4.1) that all the branches of the vibrations have the sound-type

dispersion laws: ω = s

(κ)k, κ =

, α = 1, 2, 3, 4. If the quantum properties of the

212

8 Quantum Crystals

crystal are weakly manifest (A  s

for α = 1, 2, 3 and |µ



(s)



 ρ

), then in

the main approximation the vibrations are divided into purely lattice ones and those

of quantum dilatatio n. The equation for lattice vibr ations



(ρδ

−ρ

(s)

) −

ilm j



= 0, (8.4.2)

formulated in the approximation (8.4.2), is equivalent to a set of equations



−



ilm j

(s)

nl m j





= 0.

Thus, the dynamics of quantum dilatation in the approximation linear in ρ

(s)

/ρ

can be taken into account in the renormalization of crystal elastic moduli. The ef-

fective elastic moduli λ

∗

iklm

+(1/ρ) ρ

(s)

nkl m

. The sound velocities are

renormalized to the same extent.

The fourth equation and the corresponding dispersion law can be obtained from the

last equation of (8.4.1): ω

=(1/ρ)Aρ

(s)

This dispersion law corresponds to the crystal density vibrations at ﬁxed lattice sites

(u = 0).

Part 4 Crystal Lattice Defects

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

Point Defects

9.1

Point-Defect Models in the Crystal Lattice

In Chapter 1 it was mentioned that any distortion or violation of regularity in the crys-

tal atomic arrangement can be considered as a cry stal lattice defect. The presence of

defects in a real crystal distinguishes it from an ideal crystal lattice and some proper-

ties of a real crystal are determined by its defect structure. The inﬂuence of defects

on the physical properties of the crystal depends essentially on the defect dimension-

ality. This value (dimensionality ) is the number of spatial dimensions along which the

defect has macroscopic dimensions.

A point (or zero-dimensional) defect is a lattice distortion concentrated in a volume

of the order of magnitude of the atomic volume. If a regular atomic arrangement is

broken only in the small vicinity of a certain line, the corresponding defect will be

called linear (or one-dimensional). Finally, when a regular atomic arrangement is

violated along the p art of some surface with a thickness of the order of interatomic

distances a surface (or two-dimensional) defect exists in the crystal.

Any defect can have the following two functions affecting various crystal proper-

ties. First, a r egion of a “distorted” crystal arises near the defect and the defect looks

like a local inhomogenity in the crystal. Then, the p resence of a defect causes some

stationary deformations in the crystal lattice at a d istance from it, resulting in the dis-

placement o f atoms from their equilibrium positions in an ideal lattice. Thus, the

defect is also a displacement ﬁeld source in a crystal. The ﬁeld of atomic displace-

ments n ear the defect is dependent naturally on the character of the inﬂuence of the

defect on the surrounding lattice (matrix).

The simplest types of point defects in a crystal are as follows: interstitial atoms are

atoms occupying positions between the equilibrium positions o f id eal lattice atoms

(Fig. 9.1a); vacancies are lattice sites where atoms are absent ( Fig. 9.1b); interstitial

impurities are “strange” atoms incorporated in a crystal, i. e., those that occupy inter-

stitial positions in a lattice (Fig. 9.2a); substitutional impurities are “strange” atoms or

entire molecules that replace the host atoms in lattice sites (Fig. 9.2b).

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

216

9 Point Defects

Fig. 9.1 “Proper” defects of a crystal lattice: (

) is an interstitial atom;

(

) is a vacancy.

We consider the inﬂuence of a “proper” point defect on the surrounding matrix in

a simple cubic lattice of a metal or a nonpolar dielectric (dielectric with a covalent

bond). An interstitial atom incorporated in such a lattice locally breaks its ideality.

The sites nearest to this atom are displaced due to the interstitial atom (Fig. 9.1a). In

a simple cubic lattice this deformatio n has a cub ic symmetry.

A vacancy in a simple crystal lattice leads to the displacement of the nearest atoms

in the direction of the vacancy position (Fig. 9.1b).

Fig. 9.2 Atoms of impur ities: (

) – interstitial; (

) – substitutional.

It is seen from Fig. 9.1a,b that vacancies and interstitial atoms may be considered

as defects of opposite “sign”. In particular, a vacancy in a simple crystal lattice leads

to the displacement of the nearest atoms towards the vacancy position ( Fig. 9.1b). The

annihilation of a vacancy and an interstitial atom may b e effected, that is followed by

the disappearan ce of the vacancy and interstitial atom.

Another pro cess is possible when an interstitial atom leaves its position a nd goes

over to an interstitial site, creating a pair of defects. This scheme is effected if a crystal

is radiated by energetic particles when a particle passing through the crystal displaces

an atom from its site, transferring it to an interstice.

A vacancy and an interstitial atom po sitio ned close together are referred to as a

Frenkel pair of defects (Ya. I. Frenkel was the ﬁrst to describe these defects).

Let us note that the local environment of an interstitial atom (Fig. 9.1a) and the di-

rections of the nearest-atom displacements for crystals with a primitive Bravais lattice

(Fig. 9.3) are different from those observed in a body- or face-centered cubic lattice.

9.1 Point-Defect Models in the Crystal Lattice

217

The displacements near the interstitial atom in such lattices may have no high degrees

of symmetry, as shown in Fig. 9.3. In particular, in the vicinity of an “excess” atom

the lattice conﬁguration with two isolated atoms is possible (Fig. 9.4). Such a position

of an interstitial atom generates a dumb-bell-shaped conﬁguration of the defect.

Fig. 9.3 Atomic displacement near an interstitial in a primitive cubic

lattice.

Another possible position of an “extra” proper atom in a lattice is the crowdion con-

ﬁguration that is different in symmetry from the standard interstitial position. A crow-

dion in a 1D crystal is described in Section 5.2.

Fig. 9.4 Dumb-bell-shaped conﬁguration of an interstitial atom.

Now we turn to a line of densely packed atoms in a 3D crystal (Fig. 9.5). Of the

two possible types of defects with an extra atom of the same type (dumb-bell-shaped

and crowdion conﬁgurations) the “dumb-bell” is an energetically more advantageous

static point defect, while the crowdion is a dynamic defect capable of moving easily

along a line of densely packed atoms. The comparatively easy motion of a crowdion is

associated with the fact that with a sufﬁciently extended region of atomic condensation

along a separate line, the displacement of the crowdion center of mass is achieved by

an insigniﬁcant displacement of each atom in this line.

It is natural that in a low-symmetry lattice the vacancy may also be characterized

by a local atomic rearrangement of a dipole or “anticrowdion” type.

218

9 Point Defects

Fig. 9.5 Acrowdiononthex-axis.

An anisotropic lattice deformation (indicated by arrows in Fig. 9.4) is also gen-

erated by an impurity molecule that has no spherical symmetry. The corresponding

static point defect is sometimes called an elastic dipole. As a rule, the same dipole-

type defect may be oriented in different ways with respect to the crystal axes and the

processes of elastic dipole reorientation are possible where its dynamic properties are

manifest.

Considering the localization of an interstitial im purity (or an interstitial atom) in a

crystal, we note th a t the unit cell of each crystal lattice has one or several equivalent

positions for interstitials that are determined by the geometrical structure of the lattice.

For instance, in a primitive cubic lattice this position is the cube center, in a FCC

lattice the positions fo r interstitials are localized either at the center of the cube or

at the middle of the cube edges. In a BCC lattice these p ositions are at the centers of

tetrahedrons constructed of two atoms at the cube vertices and two atoms at the centers

of neighboring cubes or in the middle of the cube edges (in octahedron interstices).

If the local surroundings of interstitial vacancies have no cubic symmetry (as, e. g.,

in a BCC lattice) nonisotropic deformations arise around the interstitial atoms. First, a

prerequisite for a dumb-bell or crowdion conﬁguration of an in terstitial atom appears

and, second, the impurity in such a position behaves as an elastic dipole. A classical

example of the latter is the iron crystal distortion around a carbon atom impurity.

Carbon atoms get into octahedron interstices of a BCC iron lattice and behave as

single-axis elastic dipoles oriented along the cube edges.

9.2

Defects in Quantum Crystals

In describing the point defects of a crystal lattice we proceeded from a seemingly

obvious assumption of defect localization in a certain site or interstice. However,

the existence of crystals with speciﬁc quantum properties (Chapter 8) makes such an

assumption that is purely substantiated and even unreal in some cases. This refers

9.2 Defects in Quantum Crystals

219

ﬁrst of all to the defects in quantum crystals and to the behavior of hydrogen atom

impurities in the matrix of rather heavy elements.

Since defect localization is doubtful, it is necessary to make a more rigorous analy-

sis of the physical situation arising in a crystal containing defects.

If we have a crystal with a single point defect, then its Hamiltonian function (or

Hamiltonian) is a periodic function of the coordinate of the defect with the period

of the crystal lattice. We shall study only those crystal degrees of freedom that are

described by the defect coordinates, with the temperature assumed to be zero. We

consider the localization of the defect at one of the equilibrium positions correspond-

ing to the crystal-energy minimum. The dynamical properties of a defect are then

manifest only in small oscillations near the ﬁxed equilibrium position. Thus, the crys-

tal state concept is unambiguously associated with the notion of a ﬁxed coordinate of

the defect.

Defect localization becomes impossible when quantum tunneling occurs. The de-

fect coordinate as a characteristic of the crystal state ceases to be a well-deﬁned quan-

tity and various states of a crystal with a defect should be classiﬁed by the values of

a quasi-wave vector k. The crystal energy becomes a periodic function with respect

to k.

If we subtract from this energy the energy of a crystal without a d efect, then we get

the defect energy ε

(k) . The different values of k inside the Brillouin zone determine

different energies ε

(k) . Thus, an energy band of some width ∆ε proportional to

the probability for quantum tunneling of a defect from one equilibrium position to a

neighboring one arises. This new part of a crystal energy spectrum (Fig. 9.6a) is due

to the appearance of a movable quantum defect. Thus, the defect is associated with an

additional branch of quantum single-particle excitations.

Fig. 9.6 The defecton dispersion law: (

) the appearance of a defecton

needs energy expenditure ε

;(

) the appearance of vacancies induces

the rearrangement of the crystal ground state.

It is clear that a defect in a quantum crystal is delocalized and behaves as a free

particle. It is called a defecton (Andreev and Lifshits, 1969) and the dependence of

the energy ε

(k) on k is referred to as the defecton dispersion law. An example o f a

defecton is a

He isotope atom in solid

He.

220

9 Point Defects

The defectons can collide with one another at ﬁnite concen trations, and collisions

with other crystal excitations (e. g., phonons) are also possible at ﬁnite temperatures.

An increasing number of collisions fundamentally affects the character of defecton

motion. If the frequency of collisions is small enough, we practically have a freely

moving defecton. With increasing frequency of collisio ns the defecton can approach

equilibrium with the lattice during the time when it stays within a unit cell. We then

speak about a practically localized defect.

In most cases the tunneling probability is relatively small. Therefore, to calculate

the defecton dispersion law, we may use the strong coupling approximation known in

electron theory. The function ε

(k) is found in this case explicitly at all values of k.

For instance, for a simple cubic lattice we h ave

(k)=ε

−ε

(cos ka

+ cos ka

), (9.2.1)

where ε

, ε

are constant values, |ε

| is proportional to the quantum-tunneling proba-

bility; a

are the fundamental lattice translation vectors, α = 1, 2, 3. The width of the

defecton energy band is ∆ε = 6 |ε

In an isotropic approximation, the expansion (9.2.1) near the minimum value of ε

(the energy band bottom) has the form

= ε

¯h

∗

, ε

= ε

−3ε

, (9.2.2)

where m

∗

is the defecton effective mass (of order of magnitude ¯h

∗

∼ a

∆ε).

The presence of a defecton in a quantum crystal allows one to explain the physical

nature of quantum dilatatio n (Chapter 8). We assume that in a crystal free of impurities

the “defectiveness” arises only due to the excitation of vacancies. The possibility of

tunneling transforms a vacancy into a defecton, or a vacancy wave with the dispersion

relation (9.2.2).

Vacancy wave generation with k = 0 does not break the ideal periodicity of a

crystal. However, the number of crystal lattice sites becomes unequal to the number

of atoms. The defecton energy with k = 0,i.e.,ε

, is dependent on the state of

a crystal, in particular on its volume V changing under the inﬂuence of an external

pressure. It may turn out that at a certain volume V

the parameter ε

vanishes. We

assume that near this point

= λ

−V

. (9.2.3)

We shall set, for deﬁniteness, λ > 0 and consider the defectons obeying Bose

statistics (such as vacancies in solid

He).

It then follows from (9.2.3) that for V < V

, ε

> 0 and the energy spectrum of

defectons is separated from the ground-state energy (without defectons) by a gap. A

ﬁnite activation energy is needed to form a vacancy and thus at T = 0 the number of

defectons equals zero.

9.2 Defects in Quantum Crystals

221

For V > V

, ε

< 0 (Fig. 9.6b) and defecton generation becomes advantageous

even at T = 0. Vacancies tend to assemble into a state with the last energy (k = 0)

that is promoted by Bose statistics. The defects are accumulated (condensed) in this

state until defect repulsion effects start to manifest themselves. It is clear that with any

vacancy repulsion law the energy minimum at T = 0 corresponds to a nonzero equi-

librium co ncentration of defectons with k = 0. Since with a ﬁxed number of atoms

each vacancy increases the number of crystal sites by unity, a ﬁnite defect concentra-

tion actually generates a certain crystal dilatation. This is just the quantum dilatation.

A narrow energy band is typical for the defecton dispersion law (9.2.1). According

to experiments, its width ∼ 10

−5

−10

−4

K ∼ 10

−9

−10

−8

eV ∼ 10

−21

−10

−20

erg

for the

He atom playing the role of an “impuriton” in solid

He. Thus, the energy

band width of the quasi-particle motion of this well-studied defecton is very small

compared to any energies on an atomic scale. This makes the defecton dynamics in

external inhomogeneous ﬁelds that arise in a crystal, essentially different from the

dynamics of ordinary free particles.

Let the defecton be in an external ﬁeld providing the potential energy U(x) but not

inﬂuencing its kinetic energy (9.2.1). The total energy of the defecton E( k, x) can

then be written as E(k, x)=ε

(k)+U(x).

If, for the characteristic distances l  a, a the potential energy changes by δU 

∆ε the defecton energy will ﬁll a narrow strongly distorted band (Fig. 9.7). The ﬁxed

energy of a defecton E(k, x)=E = const corresponds to its motion localized in a

space region with dimensions δx ∼ l∆ε/δU  l. Such a localization is independent

of the sign of grad U (Fig. 9.7).

Fig. 9.7 Localization of defecton motion in an external potential ﬁeld.

Let us assume that the external ﬁeld is changed at a distance l ∼ 100a ∼ 10

−6

in a certain direction by δU eV. In this case, for ∆ε ∼ 10

−8

the defecton is localized

along the direction of grad U at a distance ∆x ∼ 10

−8

cm and it moves in fact along

the surface of constant potential energy U(x)=const (to be precise, in a thin layer

with a thickness that is comparable with the interatomic distance).

We note that in a strong magnetic ﬁeld the

He impuritons with different orientation

of nuclear spin are in different energy bands. Indeed, the nuclear magnetic moment of