ISSN 10637761, Journal of Experimental and Theoretical Physics, 2010, Vol. 110, No. 1, pp. 41–48. © Pleiades Publishing, Inc., 2010. Original Russian Text © B.V. Petukhov, 2010, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2010, Vol. 137, No. 1, pp. 48–56.
ORDER, DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Statistical Description of the Motion of Dislocation Kinks in a Random Field of Impurities Adsorbed by a Dislocation B. V. Petukhov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiі pr. 59, Moscow, 119333 Russia email:
[email protected] Received June 10, 2009
Abstract—A model has been proposed for describing the influence of impurities adsorbed by dislocation cores on the mobility of dislocation kinks in materials with a high crystalline relief (Peierls barriers). The delay time spectrum of kinks at statistical fluctuations of the impurity density has been calculated for a sufficiently high energy of interaction between impurities and dislocations when the migration potential is not reduced to a random Gaussian potential. It has been shown that fluctuations in the impurity distribution substantially change the character of the migration of dislocation kinks due to the slow decrease in the probability of long delay times. The dependences of the position of the boundary of the dynamic phase transition to a sublinear drift of kinks x ∝ t δ (δ < 1) and the characteristics of the anomalous mobility on the physical parameters (stress, impurity concentration, experimental temperature, etc.) have been calculated. DOI: 10.1134/S1063776110010061
the delay time spectrum during motion of dislocation kink solitons (or simply kinks) in a random potential of impurities adsorbed by a dislocation.
1. INTRODUCTION The problem associated with the transport of parti cles of different nature in disordered media has long attracted the particular attention of researchers. An interesting aspect of this problem arises in materials characterized by a broad spectrum of barriers with a slowly decreasing probability of long delay times of migrating objects. A large number of these systems in physics, chemistry, biophysics, and other fields of sci ence are known to date, and a large number of works have been devoted to studying the kinetic features in these systems (see, for example, reviews [1–7] and ref erences therein). As an example, we note the disloca tion motion, dispersive charge transport in extrinsic semiconductors, diffusion of polymers and biological macromolecules in gels, transfer in fractal structures, and laser cooling of atomic gases. The existence of a broad spectrum of barriers with a slowly decreasing asymptotics of the probability of long delay times (heavy tails) leads to qualitative features in the behav ior of particles, such as the nonlinear drift in response to an external driving force, and in the absence of this force, to an anomalously slow relaxation. Thus, the determination of the delay time spectrum is a key problem in predicting anomalies in the kinetics of spe cific systems. Nonetheless, in many cases, the exist ence of slowly decreasing asymptotics of the delay time spectrum has only been postulated, even though the problem regarding their origin is noteworthy because its solution allows one to reveal the correla tion between the experimental data and the internal nature of the object under investigation. The purpose of this paper is to perform a statistical calculation of
A factor responsible (but insufficient) for the increased probability of long delay times can be one dimensionality of the system when particles have no possibilities of choosing the path of motion and getting around inconvenient obstacles. This situation naturally arises in the motion of dislocation kinks along disloca tion lines in impurity materials; in this respect, the sys tem under consideration has a longstanding history in the investigation of kinetic anomalies [8–11]. In semiconductors, metals with a bodycentered cubic structure, ceramics, and some other materials, dislocations move by overcoming a periodic potential relief of the crystal lattice (Peierls barriers) via ther mally activated nucleation of kink–antikink pairs, fol lowed by their propagation over the entire dislocation. The kink mechanism is well developed for ideal crys tals [12–14], whereas the theory for impurity and doped materials, solid solutions, and alloys is far from complete. There exist effects important for practical applications due to doping: solidsolution hardening, dislocation ageing, etc. The stable operation of semi conductor devices frequently requires immobilization or pinning of dislocations. Dislocation pinning is caused by the absorption of impurities owing to the presence of energetically favorable states in disloca tion cores, which are characterized by some binding energy Eb. Experimental data indicate that the stresses of dislocation depinning in silicon depend substan tially on the magnetic field [15], which was interpreted by the contribution of shortrange forces to the impu 41
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rity–dislocation interaction. These electronic or “chemical” forces clearly manifest themselves in the magnetoplastic effect [16]. Consequently, a more or less realistic model of dislocation motion in doped materials should be based on the inclusion of the influ ence of shortrange potentials induced by randomly arranged impurities on the kink kinetics. The systematic features of dislocation pinning have been studied using detailed experiments with prelimi nary heat treatment of samples. A change in the dura tion of heat treatment made it possible to produce a controllable excess of impurities in dislocation cores [17–19], which resulted in dislocation “ageing.” An excess content of impurities in cores can also arise in moving dislocations under conditions of socalled dynamic ageing (see, for example, [20–23]). It is nec essary to dwell in greater detail on the conditions responsible for this regime of motion, because they determine the range of applicability of the results obtained in our work. A possible model of an elementary act of disloca tion displacement by a lattice spacing under condi tions of impurity drag was particularly proposed in [24]. The main prerequisite for this regime is the pres ence of a finite energy gap between the most energeti cally favorable states of impurities in the dislocation core and their other states, which follows from the atomically discrete structure of the dislocation core and is confirmed by atomistic calculations of the impurity–dislocation binding energy. The presence of a potential well with a depth on the order of the impu rity–dislocation binding energy results in a large dif ference between the time of transition of impurities τ to the state in the dislocation core and the time of escape of impurities τ1 from this potential well to other valleys of the crystalline relief: τ1 Ⰷ τ. The condition providing the regime considered in our work can be written in the form τ ≤ τtr < τ1, where τtr is the time of dislocation displacement by the lattice spacing. The condition τ ≤ τtr ensures the impurity drag in the regime under consideration in accordance with the experimental fact of dynamic ageing of dislocations. The condition τtr < τ1 determines the upper boundary of applicability of the calculated delay time spectrum for kinks. In this work, we investigate the role of an excess content of impurities in dislocation cores, which according to our assumption is considerably higher than the impurity content average over the crystal vol ume due to the adsorption. This allows us to ignore the presence of impurities outside the dislocation cores. It is assumed that the impurity distribution over the dis location lines is completely random. In earlier works [25], the interaction of dislocation kinks with impurities was described by analogy with conventional particles with the use of localized poten tials, which at low impurity concentrations produce isolated barriers in the path of the kink motion. This interaction renormalizes the average velocity of kinks
as compared to their velocity in sufficiently pure crys tals. Another model that takes into account the spe cific features of quasiparticles of the kink type and the total contribution of impurities modifying the energy of the dislocation core was proposed in [8]. Within this model, the energy of the interaction between the kink and impurity changed in a stepwise manner by a value of the order of the impurity–dislocation binding energy Eb in the displacement of the kink through the impurity cell in the crystal lattice. Therefore, it was assumed that the kink potential in the impurity crystal contains the contribution ΔE proportional to the dif ference N1 – N2 between the numbers of impurities in two valleys connected by the kink in the crystalline relief: (1) ΔE ( x ) = E b [ N 1 ( x ) – N 2 ( x ) ]. As a result, the potential of the kink during its motion executes random walks on the energy scale with steps equal to Eb at random locations of impuri ties along the dislocation. This strongly fluctuating potential is the main factor responsible for the appear ance of high barriers and related qualitatively new fea tures of the kink kinetics. In particular, it was shown in [8] that the average time it takes for the kink to over come the barrier formed by random clusters of chaot ically distributed impurities infinitely increases at a temperature T lower than some critical temperature Tg dependent on the impurity concentration. Therefore, it was established that there exists a specific dynamic phase transition during which the average velocity of kink motion vanishes in the range T ≤ Tg. For this tem perature range, the dependence of the free path length x on the time t was described by an approximate kinetic law x ∝ P–1(t), where P(t) is the probability of the barrier meeting with an overcoming time longer than t. This type of kinetic dependence with zero aver age velocity of motion subsequently had different names: anomalous mobility, motion in the creep phase, heterogeneous dynamics, quasilocalization, motion in the field of a random force, etc. [1–11, 26–30]. The mobility in the normal phase is described by only one moment of the delay time distribution func tion, namely, the average time it takes to overcome an obstacle 〈 τ〉 , whereas the description of the motion in the anomalous phase at T ≤ Tg necessitates a more complete characteristic of the delay time spectrum. The problem regarding the determination of this spec trum was partially solved in [8–11]. In these works, it was revealed that the probability of long delay times decreases according to a power law. The present work is devoted to a more general solution for the case that corresponds to binding energies Eb higher than the thermal energy and is important for practical applica tions. The case of the same average distribution density of impurities in valleys of the crystalline relief that pro duce the statistically symmetric random potential (1) for kinks was analyzed in [8]. The asymmetry was
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Ek
43 f=0
Eki f = fb Eb 0 xi
f > fb
x
Fig. 1. Potential of interaction Ek(x) of the kink with the impurity: local peak + step (xi is the coordinate of the impurity along the dislocation line).
introduced only by the presence of the external driving force f. The anomalously slow diffusion of particles in this potential in the absence of the external driving force (f = 0) was considered in [26]. The results obtained in [26] attracted the particular attention of both physicists and mathematicians and stimulated an extensive literature (see references in [1–7]). In order to describe the kink motion in the field of impurities adsorbed by the dislocation, it is necessary to consider an asymmetric variant of potential (1), when the main role is played by impurities located in the initial valley of the crystalline relief. In this respect, we assume that N2(x) = 0. The simplified potential of the kink–impurity interaction is schematically repre sented in Fig. 1. After the kink passes through the impurity, the dislocation is detached from the impurity and the total energy increases by the binding energy Eb. Apart from this stepwise contribution to the energy Ek(x), the local interaction between the kink and impurity results in a peak in the vicinity of the impurity cell, which additionally increases the energy by some value Eki . The superposition of stepwise contributions from all impurities chaotically distributed along the disloca tion core produces a strongly fluctuating (but increas ing, on average) energy relief for the kink motion (Fig. 2). This increase in the energy relief is equivalent to the action of the braking force fb = Eb/ l , where l is the average distance between single impurities. The force driving the kink and the external stress σ are related by the expression f = σbh, where b is the Burg ers vector of the dislocation and h is the kink height (the distance between the valleys in the crystalline relief). We take into account only the stepwise contri butions to the energy of the interaction between the kink and impurity by assuming for simplicity that Eki = 0, because the generalization to the case Eki > 0 can be easily performed through a simple modification of final relationships.
x Fig. 2. Potential relief for migration of the kink along the dislocation with adsorbed impurities for different driving forces f.
2. INFLUENCE OF IMPURITIES ON THE PROPAGATION OF KINKS The time of overcoming of a barrier τ formed by some fluctuation of a cluster of impurities is repre sented by the known formula of the theory of decay of metastable states (see, for example, [31]); that is, l U(x) τ = f exp dx. kT Dk
∫
(2)
Here, lf is the size of localization of a particle in the prebarrier state, Dk is the diffusion coefficient, and U(x) is the energy profile of the barrier. The integral is taken over the barrier size. In the case under consider ation, lf = kT/f, U(x) = EbN(x) – fx, and N(x) is the number of impurities over the length x reckoned from the position of the kink before the first impurity in the cluster. The time τ defined by formula (2) is a functional of the impurity configuration inducing the random potential. Functionals of this type in finite and infinite limits of integration have applications in various fields and have been widely studied in mathematical statis tics (see, for example, [32]). The case of the Gaussian potential popular in view of its universality is best investigated. In [8], the equation was derived for func tional (2) and the solution to this equation made it possible to find the distribution function of the integral in infinite limits for the limiting Gaussian case. This distribution function was also obtained in [33] for applications in the field of financial risks. However, in physical applications to real impurity systems, it is more adequate in many cases to consider the interac tion energy Eb higher than the thermal energy kT when the potential cannot be reduced to the Gaussian potential. It is this situation that will be considered in the present paper. For this case, the exponent of the decrease in the distribution function in the range of
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The first impurity in the cluster increases the kink energy by Eb with respect to the energy of the prebar rier state. Hence, we seek the probability Pb(τ) that, for the initial energy level Eb, the delay time at the barrier increases to τ over the length corresponding to the first intersection of the energy Ek(x) with the prebarrier level E = 0. The probability Pb(τ) is conveniently cal culated using a somewhat more general quantity, i.e., the probability that the delay time P(E, τ) for an arbi trary initial energy E increases to τ, provided the con dition Ek(x) > 0 is satisfied over the entire barrier length. This new quantity is convenient because the solution to the equation that can be easily derived for this quantity will give the desired probability Pb(τ) in the limit E Eb. The boundary condition for the function P(E, τ) is represented in the form
2' 1'
0 3
1
x Fig. 3. Composite barrier separating into two independent barriers to the kink motion. Numerals 0 and 1 indicate the prebarrier states of individual barriers, numerals 1' and 2' denote the maximum values of the barrier heights, and numeral 3 corresponds to the end of the second barrier.
P ( E, τ ) = 1 for E > E M ≡ kT ln ( τ/τ ), *
long delay times was previously determined in [10]. In our work, this result will be generalized by calculating the full spectrum. The spectral asymptotics for long delay times can be determined by examining functional (2) in infinite limits, as was done in [8, 10] (perpetual functional [32]), whereas, in order to calculate the full spectrum, it is necessary to define more exactly the functional, generally speaking, with due regard for a random char acter of limits of integration. Let us refine the barrier size determining the upper limit of integration in rela tionship (2). The additional condition for the separa tion of barriers follows from the definition of impurity clusters forming a single barrier and independent bar riers. The situation is illustrated in Fig. 3. Point 2' cor responds to the global maximum. However, the height of this maximum is not the activation energy for the transition from prebarrier state 0, because the indi cated configuration is separated into individual barri ers. The height of the first barrier corresponds to the energy at point 1', and the height of the second barrier corresponds to the difference between the energies at points 2' and 1. Therefore, the condition for the end of the barrier is a decrease in the energy below the initial (prebarrier) value taken as the reference point and the integral in relationship (2) is taken to the first intersec tion of the potential with zero energy level.
E/f
P ( E, τ ) =
⎛ l⎞ ⎛
because, at a specified energy E, the minimum possi ble time τ determined by the contribution of the only impurity is given by the expression τ∗exp(E/kT), 2
where τ∗ = l f /Dk. Now, we write the relationship for the probability P(E, τ) with allowance for all positions of the impurity next in the order. The contribution of the first impu rity to the integral in relationship (2) is represented in the form τ = τ
0
*
E E'⎞ exp ⎛ ⎞ – exp ⎛ + τ', ⎝ kT⎠ ⎝ kT⎠
(4)
where E' = E – fl, l is the distance between the first and subsequent impurities, and τ' is the residual contribu tion to the delay time. The probability P(E, τ) can be written as the sum of probabilities of all variants of the position of the second impurity in the cluster that sat isfy the condition Ek(x) > 0. The probability of a par ticular variant is given as the product of the probability exp(–l/ l )dl/ l that the distance between the first and subsequent impurities is l by the probability P(E – fl + Eb, τ') that the integral of type (2) with the potential U(x), which begins with the energy E – fl + Eb reck oned from the position of the second impurity, will exceed the time τ'; as a result, we obtain
∫ exp ⎝ –l ⎠ P ⎝ E – fl + E , τ – τ* b
(3)
E – fl⎞ ⎞ dl exp ⎛ ⎞ – exp ⎛ E ⎝ kT⎠ ⎝ kT ⎠ ⎠ l (5)
E
–E E E'⎞ ⎞ = 1 exp ⎛ E' ⎞ P ( E' + E b, τ – τ * exp ⎛ ⎞ – exp ⎛ dE'. ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎠ kT kT fl fl
∫ 0
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By differentiating relationship (5) with respect to E, we derive the equation ∂P ( E, τ) = – 1 [ P ( E, τ ) – P ( E + E , τ ) ] b ∂E fl (6) τ E ∂P ( E , τ ) ⎛ ⎞ – * exp . ⎝ kT⎠ ∂τ kT We are interested in the solution to Eq. (6) for the delay times determined not by the only impurity but by the impurity cluster for the values of τ that satisfy the condition EM = kTln(τ/τ∗) > Eb. It should be noted that the quantity EM can be considered an effective barrier height and, in the limit T 0, it actually transforms into the activation energy for overcoming the obstacle. By changing over from variable τ to vari able EM, the last term in Eq. (6) takes the form E – E M⎞ ∂P ( E, E M ) exp ⎛ . ⎝ kT ⎠ ∂E M It follows from this expression that, at E Eb in the range EM – Eb Ⰷ kT, this term contains the small fac tor exp((E – EM)/kT) and can be disregarded. There fore, we arrive at the simplified equation ∂P ( E, τ) = – 1 [ P ( E, τ ) – P ( E + E , τ ) ]. (7) b ∂E fl The following conditions are imposed on the func tion P(E, τ): ⎧ 0, P ( E, τ ) = ⎨ ⎩ 1,
E≤0 E > EM .
As a result, instead of Eq. (7), we obtain the equation dQ ( ε) = – γQ ( ε – 1 ), (8) dε where γ = (fb/f)exp(–fb/f) and we introduce the dimensionless variable ε = (EM – E)/Eb. The following conditions are imposed on the function Q(ε): ε ≥ E M /E b ε –1 is obvious from the defini tion of the quantity γ. The second solution s0 < –1 can be found by solving the transcendental equation (15). In what follows (and for comparison with the results of the previous works), it will be convenient to use not the quantity s0 but the quantity ϕ = –s0 – fb/f, which satis fies the equation following from Eq. (15): ϕ
e –1 f (16) = . ϕ fb Here, ϕ > 0 at f > fb, ϕ < 0 at f < fb, and ϕ ~ 2(f/fb – 1) at f fb. Substitution of expression (14) into Eq. (12) gives 1 – exp ( ϕE/E b ) δ , (17) P ( E, τ ) ≈ 1 + [ ( 1 – f b /f – ϕ )/ ( 1 – f b /f ) ] ( τ/τ * ) where δ = T/Tg and Tg = Eb/kϕ. The delay time distri bution is written in the form ϕ
1–e (18) . P b ( τ ) ≈ δ 1 + [ ( 1 – f b /f – ϕ )/ ( 1 – f b /f ) ] ( τ/τ * ) This relationship is valid on both sides of the threshold fb and, for large values of the delay times, transforms into the following expressions: at f > fb (δ > 0), ϕ ( f/f b – 1 ) ⎛ τ ⎞ δ P b ( τ ) ≈ (19) * ; ϕ + f b /f – 1 ⎝ τ ⎠ and at f < fb (δ < 0), P b ( τ ) ≈ – ϕf/f b . (20) In the range f > fb, the probability that a kink encoun ters a barrier to its motion with a delay time exceeding τ decreases in a power manner to zero with an increase in the time τ. The exponent is in agreement with that previously obtained in [10]. In the range f < fb, this probability tends to a finite value, as can be seen from Fig. 4. This means that the kink can propagate only over a finite distance. Therefore, the driving force f = fb corresponds to the percolation threshold. 3. DYNAMIC PHASE TRANSITION Now, we turn to the application of the calculated delay time spectrum Pb(τ) to the description of kink
migration over long distances. The simplest character istic of the kink kinetics in the random potential is the average delay time 〈 τ〉 at obstacles, which allows us to determine the kink velocity vk = lav/ 〈 τ〉 for the known average distance lav between obstacles: ∞
〈 τ〉 =
∫ 0
dP b ( τ ) τ dτ. dτ
(21)
Analysis of Eq. (19) shows that, with an increase in τ, the function dPb(τ)dτ decreases as τ–1 – δ; hence, the integral in relationship (21) has a finite value only at δ > 1 or T > Tg and infinitely increases at T Tg: τ ϕ ( f/f b – 1 ) * (22) 〈 τ〉 ≈ . ( ϕ + f b /f – 1 ) ( T/T g – 1 ) At T ≤ Tg, the integral in relationship (21) diverges and the average delay time at random obstacles becomes infinite and the velocity of motion vk van ishes. This is a manifestation of the dynamic phase transition previously predicted in [8] with a qualitative change in the character of the kink mobility. The n higher moments of the distribution function 〈 τ 〉 diverge over a wider range of temperatures T ≤ nTg. In particular, a change in the character of diffusion spreading of the front of motion around the average drift displacement (determined by the second moment of the distribution function) in the temperature range Tg < T < 2Tg can be of physical interest [3]. 4. KINETICS OF MOTION OF KINKS AT T < Tg Let us describe the specific features of the kink motion in the anomalous phase at temperatures T < Tg. When the kink moves in response to the external force over large distances, it sequentially overcomes a number of obstacles and the delay times at each obsta cle are added. The fact that the average delay time at one obstacle becomes infinite means that the main contribution to the total time of the motion is deter mined not by typical obstacles in the material but by the largest term. The qualitative pattern of the path length migration can be represented as follows [8]. The free path length for the time t is predominantly determined by the average length l between obstacles with the delay time equal to or exceeding t. With the use of the asymptotics of the delay time distribution function (19), this average distance in the above threshold range f > fb is estimated as δ
l ≈ l av /P b ( t ) = Δl ( t/τ ) , *
(23)
where l av ( ϕ + f b /f – 1 ) Δl = ϕ ( f/f b – 1 ) and lav is the average distance between obstacles. A more complete description can be achieved using the statistical theory of distribution of sums of random
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terms [34, 35]. The distance between obstacles is equal to the sum of the barrier width and the free gap to the subsequent impurity. It can be shown that the average size of the random barrier at f > fb behaves as l /(1 – fb/f). Therefore, the quantity lav = l [1 + 1/(1 – fb/f)] is finite, in contrast to the average delay time. As fol lows from the law of large numbers, approximately N ≈ x/lav obstacles occurs over a sufficiently large free path length x. According to [34] (see also [3, 35]), the dis tribution density of the normalized sum of N random positive terms t = τ1 + τ2 + … + τN, so that the distribu tion of each of them has an asymptotics B/τδ (τ ∞), is described by the Lévy function t ⎞ L δ, 1 ⎛ = 1 ⎝ N 1/δ⎠ 2πi
47
The characteristic dislocation length for which it is necessary to take into account the multiple nucleation of kink pairs is given by the formula δ 1/ ( 1 + δ )
Δl L tr ≈ x ( t tr ) ≈ ⎛ ⎞ ⎝ Γτ 1⎠
.
(26)
Relationships (25) and (26) generalize the wellknown results of the Hirth–Lothe theory [12] for pure mate rials to the case of the strong influence of impurities and transform into the corresponding expressions at δ 1. In the subthreshold range f < fb, the free path lengths of the kinks are finite and the dislocation motion occurs through the formation of multistage configurations with an exponentially low probability. 6. CONCLUSIONS
d + i∞
×
δ st – πB exp s ds 1/δ sin ( πδ )Γ ( δ ) N d – i∞
∫
1/δ ∞
N = – πt
πBN
⎞ ∑ ⎛⎝ – Γ ( δ ) sin ( πδ )t ⎠
(24)
k
δ
k=1
Γ ( 1 + kδ ) sin ( πδk ) × , k! where Γ(x) is the gamma function. Function (24) rather accurately describes the propagation of the front of the kink motion in a field of randomly distrib uted impurities. It can be seen from relationship (24) that the characteristic scales of the time of motion t and the free path length x are actually related by expression (23) following from the qualitative pattern. 5. DISLOCATION VELOCITY
T/T
The generalization to the case of the presence of the local peak Eki in the interaction of the kink with the impurity is carried out using the replacement of τ∗ by τ1 ~ τ∗exp(Eki/kT) in the relationship for the free path length of the kink. The time of dislocation motion ttr to the subsequent valley of the crystalline relief can be selfconsistently evaluated by equating the average free path length of the kink x(ttr) ~ Δl(ttr/τ1)δ to the average distance 1/Γttr between kink pairs nucleated by the time ttr (where Γ is the average frequency of nucle ation of kink pairs per unit length of the dislocation). Then, the dislocation velocity V is calculated as V = h/ttr and represented by the expression 1/ ( 1 + δ ) h V ≈ ( Γτ 1 Δl ) . τ1
Thus, in this paper, we developed the theory of motion of dislocation kinks in a random potential induced by impurities adsorbed by a dislocation. The most important results can be summarized as follows. The delay time spectrum of kinks at random clus ters of impurities was calculated for a sufficiently high energy of interaction between impurities and disloca tions Eb Ⰷ kT, when the potential of kink migration is not reduced to a random Gaussian potential. The explicit form of this spectrum is represented by rela tionship (13) and can be useful, for example, in describing the asymmetric broadening of internal fric tion peaks for extrinsic semiconductors, bodycen tered cubic metals, and other materials. A slow power decrease in the probability of long delay times leads to a dynamic phase transition at the temperature T = Tg, below which the motion of kinks acquires the character of a nonlinear drift with the dependence of the free path length x on the time t in
(25)
the form x ∝ t g . The dependences of the position of the transition boundary and the characteristics of the anomalous mobility on physical parameters, such as the driving force, experimental temperature, concen tration of impurities, and energy of their interaction with a dislocation, were calculated. The spreading of the front of the kink motion was explicitly described using the Lévy function. As a result, it was demonstrated that the influence of impu rities qualitatively changes the kinetics of kink migra tion and its dependences on the stress and tempera ture. It should also be noted that, although the disloca tion kink was used in our work for illustration pur poses, the migration potential under investigation (see relationship (1)) has a general form and a considerably wider field of applicability, which makes it possible to extend the results to objects of different nature.
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JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
Translated by N. Wadhwa
Vol. 110
No. 1
2010
