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

1.7 Soliton as a Particle in 1D Crystals 45

−∞

Equation (1.7.11) has the form of one of the Hamilton equations and W(ζ) plays the role of the soliton potential energy in an external field. If the force density f (x) is of purely elastic origin, it can be written as f (x) = ∂σe/∂x, where σe are the elastic stresses created in a 1D crystal by external loads. In this case

The first term on the r.h.s. of (1.7.12) is a constant that may not be taken into account in writing the potential energy of a soliton W(ζ).

We assume further that the field of stresses σe(x) varies smoothly in space and the characteristic distance of its variation is much larger than l (soliton width). Then we may replace (1.7.12) by

−∞

Comparing (1.7.11), (1.7.13) we conclude that the force acting on a soliton is determined by the gradient (space derivative) of elastic stresses created by external loads at the point where the soliton center of mass is located.

On the other hand, the relation (1.7.13) makes it possible to generalize the expression for the Hamilton function that, for small soliton momenta, takes the form

Using (1.7.14), a standard pair of Hamilton equations is constructed

The equations for soliton motion thus take the form of equations of the motion of a particle with mass m and effective “charge” a, reflecting its interaction with the elastic potential field, where σe(x) is the field potential.

The soliton, a collective excitation of a 1D crystal, has very different properties from the collective small harmonic vibrations of a crystal lattice. The primary difference is that the soliton is generated by the nonlinear dynamics of a 1D crystal. Thus, the usual superposition principle is inapplicable to solitons. One may expect that this

46 1 Mechanics of a One-Dimensional Crystal

restricts the possibility of using solitons to describe the excited states of a crystal. Problems may arise when we try to consider many solitons in the crystal or the interaction between soliton perturbation and small harmonic vibrations of the crystal.

The remarkable properties of (1.6.9) eliminate these problems. First, the asymptotic superposition principle is applicable to solitons. If at time t = 0 two solitons with velocities V1 and V2 (assume them moving in the opposite directions) exist in a 1D crystal, then at t → ∞ the same two solitons with velocities V1 and V2 will remain in the crystal. In other words, the asymptotic behavior of soliton solutions to the nonlinear equation (1.6.9) is analogous to the independent behavior of the eigensolutions to a linear equation.

Then, the asymptotic superposition principle is valid also for the interaction of collective excitations of different types: solitons and small harmonic vibrations. Without going into details of the nonlinear mechanics to justify the first remark, we illustrate the validity of the second one by studying the properties of small harmonic vibrations in a 1D crystal containing one crowdion (soliton).

1.8

Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink)

We use the simplest (linear) perturbation theory to describe how a harmonic vibration moves through a soliton (1.6.12). We assume a soliton to be at rest (results for a moving soliton can be obtained by means of the Lorentz transformation) and represent the soliton to (1.6.9) in the form

We substitute (1.8.1) into (1.6.9) and linearize the equation obtained in u1 . Then

We seek for the solution to this linear equation in a standard form u1 = ψ(x)e−iωt. For the function ψ(x) we have

Equation (1.8.2) is a Schrödinger stationary equation with a reflectionless potential. One localized eigenstate is among its solutions

1.8 Harmonic Vibrations in a 1D Crystal Containing a Crowdion (Kink) 47

corresponding to a zero frequency (ω = 0), and a set of harmonic vibrations of the continuum spectrum

with the dispersion law (1.6.8).

Formula (1.8.3) gives a translational mode in the linear approximation

dus(x)

ψloc(x) = dx .

It reflects the homogeneity of a 1D crystal and the possibility to choose arbitrarily the position of the center of gravity of a soliton. Indeed, the linear approximation gives

The solutions (1.8.4) describe the harmonic vibrations with the background of a crowdion at rest. Their form confirms the asymptotic superposition principle according to which the independent eigenvibrations are only slightly modulated near the soliton center and change insignificantly. Each eigenvibration is still characterized by the wave number k and the dispersion law (the vibration frequency dependence on k) does not change.

A set of functions (1.8.3), (1.8.4), as a set of eigenfunctions of the self-adjoint operator (1.8.2), forms a total basis in the space of functions of the variable x. This is the most natural basis for the representation of perturbations of a soliton solution, as it allows one to give a clear physical interpretation. A translational mode describes the motion of a soliton mass center, and the continuum spectrum modes refer to the change in its form and the resulting “radiation” of small vibrations.

It follows from (1.8.4) that on passing through a soliton, the eigenvibration reproduces its standard coordinate dependence eikx, but the vibration phase η(x, k) is shifted by

The phase shift (1.8.5) affects the vibration density in a 1D crystal. In the absence of a soliton, the expression for the density of states in the specimen of the finite dimension L follows from the requirement kL = 2πn, n = 1, 2, 3 . . ., that is a consequence of cyclicity conditions (1.1.8) for the eigenvibrations. In the presence of a soliton an additional phase shift (1.8.5) results in a change in the allowed wave-vector values: kL + η(k) = 2πn, n = 0, 1, 2, . . . . In the limit L → ∞, the spectrum of the values of k becomes continuous. The vibration density (the distribution function for the wave vector k) then equals

48 1 Mechanics of a One-Dimensional Crystal

where L/2π is the vibration density in the absence of a soliton. It is easily seen that

∞

−∞

The number of plane-wave vibrations (with a continuous frequency spectrum) scattered by the soliton is reduced by unity because of the presence of one local state with a discrete frequency ω = 0 (translational mode).

Analyzing the changes in the vibration distribution, it is of interest to find the frequency spectrum function ν(ω) in a crystal with a soliton. Since the dispersion law (1.6.8) does not change in the presence of a soliton, the function ν(ω) is found by multiplying (1.8.6) by dk/dω:

If a soliton moves with velocity V, the formula for the phase shift is obtained from (1.8.5) by replacing l0 → l(V) = l0 1 − (V/s0 )2 :

Finally, if there are several solitons (crowdions with different velocities) in the system, the total phase shift of an eigenvibration is equal to the sum of the phase shifts for each soliton

where Vα is the soliton velocity with number α, N is the number of solitons. It becomes clear that a set of kinks and small harmonic vibrations can be considered as “nonlinear normal modes”. It follows from the “asymptotic independence” of these modes that the energy is the sum of soliton energies (1.7.1) and the energy of small vibrations with the density of states determined by (1.8.6), (1.8.9).

It is clear that the crowdion energy is periodically dependent on the coordinate of its center x0 that may be regarded as a quasi-particle coordinate. We set x0 = Vt in (1.6.15) using the ordinary relation between the coordinate and the velocity at V = const = 0. The part of the energy (1.9.6) that is dependent on the coordinate plays the role of the crowdion potential energy. Minima of the potential energy determine possible equilibrium states of the crowdion (x0 = a/2 ± na, where n = 0, ±1, ±2, . . .), and the crowdion can oscillate relative to these equilibrium positions with the frequency

Vibration motion with such a frequency can really be a free harmonic oscillation if the quantum energy of the ground (zero) state of the oscillator h¯ Ω0 is much smaller than U0:

h¯ 2

m a2

(1.9.7)

Under such a condition the crowdion in a discrete structure possesses an internal vibration degree of freedom with the frequency Ω0.

However, in the case l0 a, the potential energy curve (1.9.6) creates very weak potential barriers between the neighboring energy minima, and the crowdion may overcome them through quantum tunneling. Thus, the crowdion migrates really in

52 1 Mechanics of a One-Dimensional Crystal

perturbation does not change the elastic bond parameters. Let the isotope be situated at the origin (n = 0) and have a mass M different from the mass of the host atom m. With such a defect we get in the case of stationary vibrations the following set of equations

where Gε0 is the Green function for ideal lattice vibrations, ε = ω2; u(0) is a constant multiplier still to be defined.

Setting n = 0 in (1.10.5), we find that (1.10.5) is consistent only when

Equation (1.10.6) is an equation to determine the squares of frequencies ε at which the atomic displacements around an isotope have the form of (1.10.5). In the theory of crystal vibrations with a point defect, an equation such as (1.10.6) was first obtained by Lifshits, 1947.

The expression (1.1.22) for the Green function for ε > ωm2 at n = 0 is substituted into (1.10.6):

Since ε > 0, a solution to (1.10.7) exists only at U0 > 0. It is not difficult to find this solution:

ω2

ε = m . (1.10.8)

1 − (∆m/m)2

1.10 Point Defect in the 1D Crystal 53

Thus we obtain a discrete frequency corresponding to vibrations of the crystal with the single point defect. For |∆m| m this frequency is slightly shifted relative to the

Crystal vibrations with the frequencies described are called local vibrations, and the discrete frequencies are called local frequencies ωd. These names are attributed to the fact that the amplitude of the corresponding vibration is only nonzero in a small vicinity near the point defect. The local vibration amplitude is given by (1.10.6), implying its coordinate dependence is completely determined by the behavior of the ideal crystal Green function. In order to obtain the Green function using (1.1.22) for ε > ωm2 it is convenient to take the quasi-wave number in the complex form (1.1.24):

and take into account the connection of the frequency with the parameter κ (1.1.25)

Thus, the local vibration amplitude decreases if the distance na increases and this decay of the amplitude confirms the fact of the vibration localization.

Let us introduce a length of the localization region of vibrations l = 1/κ. As it results from (1.10.12), under the condition |∆m| m the length of localization is very large (l = am/∆m a). In this connection it is interesting to consider a long-wave description of problems concerning with the localization of crystal vibrations near a point defect. Returning to (1.10.4) and using (1.1.15) in the long-wave approximation the Kronecker delta δn0 in the r.h.p. of (1.10.4) can be substituted by the Dirac

54 1 Mechanics of a One-Dimensional Crystal

and the finite differences in the l.h.p. can be substituted with the partial derivatives of the function v(x) (see (1.1.16) written in the same approximation):

The presence of the delta-function in the r.h.p. of (1.10.15) is equivalent to the following boundary condition for (1.1.16)

which determines a jump of the first spatial derivative of the continuous function v(x) at the point x = 0 (on the isotopic defect).

Take a solution to (1.10.15) in the form

u(x) = u(0) exp(−κ |x|), s2 κ2 = ω2 − ωm2 ,

and find the parameter κ from (1.10.16):

In the long-wave approximation aκ 1 and ω − ωm ωm, and then the simplifi-

cation of (1.10.17) is possible:

κa = 2 |∆mm|.

The result obtained coincides with (1.10.13).

Therefore, the long-wave approximation allows us to solve problems associated with local vibrations in the frequency interval ω − ωm ωm.

1.11

Heavy Defects and 1D Superlattice

In the previous section we analyzed the local vibrations and found that only a light isotope defect could produce such a vibration with a frequency higher than all frequencies of a defectless chain. Under the conditions m − M m and ω − ωm ωm description of the problems under consideration could be performed in the long-wave approximation. It was explained why a heavy isotope defect could not produce a local vibration. Nevertheless, the heavy defect influences the continuous vibration spectrum. Obviously a heavy defect can influence low vibrational frequencies, because heavier masses are associated with lower frequencies. One can expect to find essential effects at very low frequencies ω ωm at M − m m.

Consider a periodical array of isotope defects with M > m separated by the distance d = N0 a in the linear chain. Suppose d a (N0 1) and M − m m;