Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

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) ∂2 uk									∂ ϕ
ρik				− λiklm k l um = Aµik k						;	(8.3.17)
		∂t2							∂t
			∂2 ϕ		(s)	i k ϕ = µik	∂ ik
	ρ0				− Aρik			,			(8.3.18)
			∂t2				∂t
with µik = ρ0 βik − ρik(s), and introduce a new elasticity modulus tensor
˜						0	− βik) (δlm − βlm) .
λiklm = λiklm	− Mβik βlm = λiklm − M (δik

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

(s)

quantum crystal vibrations. They involve, apart from the tensor parameter ρik specific for a quantum crystal, the quantity A having the dimension of the velocity squared. According to (8.3.15), this quantity is included in the linear differential relation

∂vs
	= A grad (η + βik ik) ,	(8.3.19)
∂t

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), we find

2	(n)	˜	= 0;
(ω	ρij	− λiml jkl km )uj + Aµil kl ω ϕ
			(8.4.1)
	(ω2 ρ0 − Aρil(s)ki kl )ϕ + µil ki ωul		= 0.

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 dependence 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

k

dispersion laws: ω = sα (κ)k, κ = k , α = 1, 2, 3, 4. If the quantum properties of the

(s) il

/ρ0,

212 8 Quantum Crystals

crystal are weakly manifest (A s2α for α = 1, 2, 3 and |µik |, ρ(iks) ρ0), then in the main approximation the vibrations are divided into purely lattice ones and those

of quantum dilatation. The equation for lattice vibrations

			ω	2		(s)			˜	uj = 0,	(8.4.2)
					(ρδij − ρij		) − λilmjkl km
formulated in the approximation (8.4.2), is equivalent to a set of equations
	2					˜	1		(s)
ω		ρ0	δij	−		λilmj +		ρ0	ρin λnlmj	kl km uj = 0.

Thus, the dynamics of quantum dilatation in the approximation linear in ρ(s)

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

fective elastic moduli λ	= λ˜ iklm + (1/ρ)ρ(s)	λ0	. The sound velocities are
iklm	in	nklm

renormalized to the same extent.

The fourth equation and the corresponding dispersion law can be obtained from the last equation of (8.4.1): ω2 = (1/ρ)Aρ ki kl .

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

ISBN: 3-527-40508-9

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 crystal atomic arrangement can be considered as a crystal lattice defect. The presence of defects in a real crystal distinguishes it from an ideal crystal lattice and some properties 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 dimensionality. 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 part 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 properties. First, a region of a “distorted” crystal arises near the defect and the defect looks like a local inhomogenity in the crystal. Then, the presence of a defect causes some stationary deformations in the crystal lattice at a distance from it, resulting in the displacement of 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 displacements near 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 of ideal 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 interstitial 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 Copyright c 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: (a) is an interstitial atom; (b) 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 deformation has a cubic 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 impurities: (a) – interstitial; (b) – 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 be effected, that is followed by the disappearance of the vacancy and interstitial atom.

Another process is possible when an interstitial atom leaves its position and 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 positioned 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 directions of the nearest-atom displacements for crystals with a primitive Bravais lattice (Fig. 9.3) are different from those observed in a bodyor 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 configuration that is different in symmetry from the standard interstitial position. A crowdion 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 A crowdion on the x-axis.

An anisotropic lattice deformation (indicated by arrows in Fig. 9.4) is also generated 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 dipoletype 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 impurity (or an interstitial atom) in a crystal, we note that 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 for interstitials are localized either at the center of the cube or at the middle of the cube edges. In a BCC lattice these positions 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 interstitial 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 analysis 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 corresponding 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 crystal state concept is unambiguously associated with the notion of a fixed coordinate of the defect.

Defect localization becomes impossible when quantum tunneling occurs. The defect coordinate as a characteristic of the crystal state ceases to be a well-defined quantity 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 defect, then we get the defect energy ε D (k). The different values of k inside the Brillouin zone determine different energies ε D(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: (a) the appearance of a defecton needs energy expenditure ε 0 ; (b) 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 D (k) on k is referred to as the defecton dispersion law. An example of a defecton is a 3 He isotope atom in solid 4 He.

220 9 Point Defects

The defectons can collide with one another at finite concentrations, 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 collisions 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 D (k) is found in this case explicitly at all values of k. For instance, for a simple cubic lattice we have

ε D(k) = ε1 − ε2 (cos ka1 + cos ka2 + cos ka3 ),

(9.2.1)

where ε1 , ε2 are constant values, |ε2 | is proportional to the quantum-tunneling probability; aα are the fundamental lattice translation vectors, α = 1, 2, 3. The width of the defecton energy band is ∆ε = 6 |ε2 |.

In an isotropic approximation, the expansion (9.2.1) near the minimum value of ε0

(the energy band bottom) has the form
	h¯ 2 k2
ε D = ε0 +	2m , ε0 = ε1 − 3ε2,	(9.2.2)

where m is the defecton effective mass (of order of magnitude h¯ 2 /m a2 ∆ε). The presence of a defecton in a quantum crystal allows one to explain the physical

nature of quantum dilatation (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., ε0 , 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 Vk the parameter ε0 vanishes. We

assume that near this point		Vk − V
ε0	= λ		.	(9.2.3)
		Vk

We shall set, for definiteness, λ > 0 and consider the defectons obeying Bose statistics (such as vacancies in solid 4He).

It then follows from (9.2.3) that for V < Vk, ε0 > 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 > Vk, ε0 < 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 equilibrium concentration 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 concentration 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 3 He atom playing the role of an “impuriton” in solid 4 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 D (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 cm 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 3 He impuritons with different orientation of nuclear spin are in different energy bands. Indeed, the nuclear magnetic moment of