Numerical simulation of dislocation dynamics in disordered crystals with high Peierls barriers

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Numerical simulation of dislocation dynamics in disordered crystals with high Peierls barriers

I.R. Sagdeev, V.M. Vinokur

To cite this version:

I.R. Sagdeev, V.M. Vinokur. Numerical simulation of dislocation dynamics in disordered crystals with high Peierls barriers. Journal de Physique, 1987, 48 (9), pp.1395-1400.

�10.1051/jphys:019870048090139500�. �jpa-00210567�

1395

Numerical simulation of dislocation dynamics ⁱⁿ disordered crystals ^with high ^{Peierls barriers}

I.R. Sagdeev and V.M. Vinokur

Institute of Solid State Physics, ^U.S.S.R. Academy ^of Sciences, Chernogolovka, ^Moscow region,

U.S.S.R. 142 432

(Reçu ^{le 29 mai} 198 7, accept£ ^{le 8} juille t 1987)

Résumé.-Une simulation numérique ^{du mouvement} de dislocations dans des cristaux désordonnés avec de hautes barrières de Peierls est présentée. ^{Une chute} brutale de la mobilité des dislocations dans la région ^des champs

externes faibles est révélée. Quand ^la charge impulsionnelle ^est appliquée, ^{la mobilité} des dislocations décroît avec

l’accroissement de la fréquence ^des impulsions. ^{Il est montré} que la chute de mobilité des dislocations sous l’action des forces externes impulsionnelles ^et stationnaires résulte de sous-linéarités dans la propagation des marches dans un

champ ^{de force aléatoire.}

Abstract.-Numerical simulation of dislocation motion in disordered crystals ^with high Peierls relief is performed. ^A sharp drop in dislocation mobility ^{in the} region of weak external fields is revealed. When pulse loading ^is applied, ^the

dislocation mobility ^{decreases as the} pulse frequency increases. It is shown that the dislocation mobility drop ^under

the action of both stationary ^and pulse ^{external forces results from} sublinearity ^{in kink} propagation ^{in a} random force field.

J. Physique ⁴⁸ (1987) ^{1395-1400 SEPTEMBRE 1987,}

Classification

Physics ^Abstracts

61.70G

LE JOURNAL DE PHYSIQUE

The basic concepts of dislocation dynamics ⁱⁿ crystals ^with high Peierls barriers were formed by

Lothe and Hirth [1] and Kazantsev and Pokrovsky [2].

A dislocation is treated as an elastic string ^{in a} peri-

odic potential. ^{It is} assumed that initially ^{the entire}

dislocation lies in one of the valleys. Later, ^thermal

fluctuations create kink-antikink pairs ^{on the disloca-} tion, which then expand under the action of the _ap-

plied external force until they annihilate with kinks from other pairs. This results in a transverse displace-

ment of the dislocation. Experimental data show that dislocation mobility ^{is also} significantly ^effected by point defects, always present ^{in real} crystals. Specif- ically, it has been revealed [3-5] ^that doping ^of crys- tals can give ^{rise to an} increase in dislocation velocity.

This effect could be explained within the framework of the model [1,2] ^{as a} consequence of local lowering

of the Peierls barrier due to interaction between the dislocation and impurities (Petukhov [6,7,8]). ^Unfor-

tunately, ^{all the} attempts ^to compare quantitatively

theoretical and experimental data, which would make it possible to extract unambiguously ^the physical pa-

rameters, describing dislocations and defects, ^{so fair}

failed. One principal difficulty ⁱⁿ comparing ^theoreti-

cal and experimental data is that there are too many

parameters describing dislocation dynamics : ^Peierls

barrier height, kink creation _energy, concentration of

impurities ^{and the} energy of their interaction with the

dislocation, kink diffusion coeflicient along ^{the dislo-}

cation line, ^{etc. At the same time} virtually ^the only

measurable quantity ^{is the mean} (over ^an ensemble of

dislocations) dislocation displacement ^{under the ac-}

tion of an applied ^force.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090139500

1396

In this context one can view as a major ^advance

the series of works by Nikitenko, ^{Farber and Iunin}

[9,10], ^who investigated ^{motion of} dislocations in Peierls relief under the action of a pulse ^{force. The} technique employed ^{made it} possible ^not only ^{to track}

the displacement ^of individual dislocations, ^{but to}

trace (independently) diffusion of kinks along ^{the dis-}

locations line. A sharp drop ⁱⁿ dislocation mobility

was reported, ^{as the} frequency of the external pulse

force increases (for ^constant period-to-pulse ^duration ratio), ^{as well as} considerable slowing ^{down of dislo-}

cation motion as the spacing ^between pulses ^increases (their ^duration being unchanged). Furthermore, ^in- terpretation ^{of the} experimental data, ^{obtained in}

[9,10] corroborated the notion that dislocations in

crystals ^with high Peierls barriers move by ^successive

transitions from one valley ^to another. The principal

aim of our work was to simulate numerically ^dislo-

cation dynamics ^{in the} presence of defects. We have tested the commonly accepted ^{view on the character}

of dislocation dynamics ^{in the} presence of defects. In the work of one of the authors [111 ^{it was shown that}

moderate external forces impurities and defects can

play ^the leading part ⁱⁿ forming dislocation mobility.

Specifically, if the external force is such that the quan-

tity q ⁼ 2TujcV2 ¹ ( T is ^the temperature, ^{a is the} external applied strain, ^{V is the} energy of interaction of a defect with the dislocation, ^c is the concentration of defects), then drift of kinks along the dislocation becomes nonlinear in time (sublinear law) : ^{w - tq} (x

is the displacement ^{of a kink over time} t), ^{and, as a}

consequence, the mobility of the dislocation sharply

decreases. We have studied, among things, ^disloca-

tion dynamics ^under the action of a pulse force, ^and

found that the decrease in mobility with the increase in pulse frequency might really ^{be a} consequence of the IIX _ tqll -effect.

As a model of a dislocation for numerical simula- tion we chose that of an elastic string, ^whose equation

of motion is of the form (we believe the friction in the system ^{to be} large, ^{so that we can} neglect the inertial

term) :

It is assumed here that the dislocation initially ^lies along ^{the x} axis, u ^{is its} transverse displacement, ^{x is}

the elastic constant, Uo ^{is the} height of the Peierls

barrier, ^{r is} the kinetic coefficient, -fez) ^{is external}

force _per unit lenght ^of dislocation, U1(x, u) ^{is a ran-}

dom quantity, describing distortion of the Peierls bar- rier by defects, is the lattice parameter. ^{The last term} in the rhs of (1) ^{is a} Langevin force, ^{which is a source}

of thermal fluctuations of the dislocation line. The

angular ^brackets (...) ^denote thermodynamical ^aver- aging. The values of the coefficients in (1) ^{were chosen}

close to their actual experimental ^values.

Discretizing the dislocation into N elements (each ^of

them _may be viewed as a lattice site), ^we bring equa- tion (1) ^{to a finite} difference form :

where u(i) ^{is the} displacement of the itch element at the time instant t, u’(i) ^{is its} displacement ^{at the time}

t + At, At is the discrete time step, ci ^{= x} .r, f ⁼ F.r, u(u(i)) ⁼ r.(gU/au. The external field is either

stationary ^{or a train of} square pulses ^{with the} period

T and duration tp.

The thermal force £ ^was approximated by ^{a se-}

quence of pulses ^{of random} amplitude with normal distribution and dispersion - ^{T. The} spacing ^between

the pulses ^{had an} exponential distribution with the mean Tf .

The lattice potential ^{U was} chosen in the form U ⁼ Uo ^sin (y/27ra). ^{Defects were placed in} randomly

chosen sites with the probability ^{i = ac} (where ^{a is}

the lattice parameter, c, the volume concentration of

defects) and where assumed to be short-range (acting withing ^{one lattice} spacing).

In the course of our numerical experiment ^we

simulated motion of dislocations of length ^{N =1000}

-

4000 lattice parameters, ^{in the} absence, ^{as well}

as in _presence of defects, and both in stationary ^an pulse external fields. We computed ^mean dislocation

velocity, ^{v, kink} concentration, Nk, ^{and kink} mobility,

u.

In order to test the numerical model and the program, ^we studied dislocation dynamics ^{in a} per- fect crystal ^and compared ^our numerical results at the temperature ^{T = 0.15} Uo ^{with the} analytical ^re- sults, ^obtained by Buttiker and Landauer [12], ^who

had studied the model (1) without defects. Figure

1 presents ^the dislocation velocity ^{as a} function of external force according to [12] (dashed curve) ^and

our numerical result. We also obtained the disloca- tion velocity ^v and kink concentration Nk ^{as func-}

tions of a stationary external force (the ^{solid curves}

in Fig. 2). ^The commonly accepted view is that at

large forces the kink concentration and dislocation ve-

locity ^should depend ^{on force} exponentially [1]. ^For

force F Uo kink concentration is almost force-

independent ^{and the} dislocation velocity ^{should be} simply proportional ^{to F} (linear law). ^{Our numer-}

ical results are in fairly good agreement ^{with these} predictions, the transition from exponential ^{to linear}

law taking place ^at FIUO c-- 0.12, confirming ^{the the-}

Fig. 1.-Comparison ^{of our numerical data and} analytical ^results [12] (dashed line). Dislocation velocity ^{as a} function of applied

force.

Fig.2.-Kink concentration Nk (per thousand lattice _parame-

ters), dislocation velocity ^{v and kink} mobility p ⁼ v / Nk / F ^as

functions of exernal force F in perfect crystals (solid curves).

a ⁼ 0.5, Ud ^{= 0.5} Uo. ^The force is in units of Peierls barrier.

Units of mobility ^and velocity ^{are their values at F = 0.1} Uo.

T ⁼ 0.5 Uo.

oretical estimates of [1]. Furthermore, ^we studied the

dependence ^{of v and} Nk ^{on the} frequency ^{of a} pulse

external force (in the nonlinear region), ^{for a constant} period-to-duration ration 2:1 (Fig.2). ^{It is clear that}

the dislocation velocity ^{under the action of a} high- frequency force should be approximately equal ^{to that}

under the action of a half-amplitude low-frequency

force. Our numerical results (Fig.2) ^{agree well with}

this idea. Note that in the case under consideration

(nonlinear ^{region, no} defects) ^the frequency depen-

dence of dislocation velocity ^is completely ^determined by that of kink concentration.

Kink concentration, dislocation velocity ^{and kink} mobility ^as functions of external force in a crystal

with defects ^are depicted ⁱⁿ figure ² (dashed lines).

In the _range of external forces 0.15 Uo ^0.4 Uo ^the

dislocation velocity in disordered crystals ^is larger,

while outside of this _range, smaller, ^{than in a} crys- tal without defects (Fig.2b). ^{Such a} behaviour of the

velocity ^{could be} easily explained within the frame- work of the model [111. ^{At forces 0.4 Uo F Uo}

the kink creation _energy is sufficiently small, ^{and the}

contribution of defects to generation of kinks is in-

significant. ^{In this case} the defects only slightly ^slow

down the motion of kink along the dislocation line. At F 0.4Uo ^a majority ^{of kinks are created} by ^clusters

of defects, locally lowering the Peierls barrier. Thus,

at F 0.4 Uo ^defects promote generation ^{of kinks}

(Fig.2a), while their braking of kink motion is still

insignificant and the dislocation velocity ^is greated

than in a perfect crystal. As the external force fur- ther decrease, ^an abrupt change in the character of kink motion along the dislocation takes place ^{in disor-}

dered crystals, according ^to [11] (at ^a threshold force F ⁼ Fo, ^when q ⁼ 2 FoT/cUd =1) : the law of kink motion becomes sublinear. The sharp drop ^{in dis-}

location mobility (and, therefore, velocity) ^observed

at F 0.15 Uo ⁱⁿ disordered crystals (Fig.2c), ^{is a}

consequence of the above-mentioned slowing ^{down of}

kink spreading ^{at F 0t15} Uo. ^The experimental

value of the parameter q =1.2 agrees well with that

predicted ⁱⁿ [111 (q ⁼ 1). ^{It is easy to see that the} mobility drop ^takes place within the force _range, in which the kink concentration Nk depends ^{no more on}

the applied ^force i.e., within the linear _range.

In order to study ^{the influence} of defects on dislo- cation dynamics under the action of a nonstationary

force in the nonlinear region, ^we applied ^{to the dis-}

location a train of _square pulses ^{with the} amplitude

F ⁼ 0.5 Uo. ^The period-to-pulse ^{duration ratio was}

chosen to be 2:1. This was exactly ^{the same kink of}

pulse ^{force as} that used in [9,10]. ^The temperature

T was chosen to be T = 3 ^Uo, the concentration of

defects and their potential ^{c =0.3 and Ud± O.lx} Uo,

respectively (the ^{sign is} random). It turned out that

the frequency dependence ^of dislocation velocity ⁱⁿ

1398

the _presence, as well as in the absence of defects, ^ac- tually copies that of kink concentration Nk. ^What’s

more, in the _presence of defects the kink concentra-

tion, and, therefore, dislocation velocity ^{are 1.5 times} greater ^{than in a} perfect crystal. ^Our results show that the role of defects in the nonlinear region ^{is re-}

stricted to increasing ^the nucleation rate : it is ^eas- ier for kink-antikink pairs ^{to be born at} clusters of

barrier-lowering ^defects. However, ^the defects do not

change ^the general character of the frequency depen-

dence of dislocation velocity.

Fig.3.-Kink concentration (triangles) and dislocation velocity (circles) ^{as functions of the} period ^of pulse ^{external force in} perfect crystal ^{at T = 0.5} Uo, F ^{= 0.5} Uo (all ^{units same as}

in Fig.2, except ^{for the} pulse period, normalized by ^{time of} displacement ^of dislocation by ^{1 lattice} parameter ^{under sta-} tionary ^external force).

In the linear region, ^{as noted} above, ^{a more} sig-

nificant role is played by braking ^{of kink motion} by

defects. We expect this effect to be particularly ^im- portant ^{in the case of a} dislocation, moving ^under

the action of pulse ^force in the linear region, ^{as in}

the experiments [9,10]. ^{Depicted in figure 4 are the}

kink concentration and dislocation velocity ^{as func-}

tions of the period ^of constant-amplitude ^{and con-}

stant period-to duration ratio pulses ^{in a} crystal ^with

defects. The amplitude ^{of the} pulse ^{force was chosen} slightly higher ^{than its threshold value for a station-}

ary force (F ⁼ O.lC/o)’ ^{One can see that at low pulse} frequencies the kink concentration and dislocation ^ve-

locity ^{are the same as in the case of} stationary ^force (when normalized to active time). ^{However, as the}

Fig.4.-Kink concentration (triangles) ^and dislocation velocity (circles) ^{as functions of} external force pulse period ⁱⁿ crystal

with defects (E ⁼ 0.5, Ud ⁼ 0.5 Uo).

pulse ^force frequency increases, ^a sharp drop ^{of dislo-}

cation velocity ^{is observed. A} peculiarity ^{of this} drop

is that it is not accompanied (as ^{was the case in the}

linear region) ^{by a} decrease in the kink concentra- tion. On the contrary, ^{the latter even} grows ^as the

frequency increases. Thus, the above data show that the decrease in dislocation velocity ^{is due to} braking

of kink motion by defects, rather than to suppression

of kink generation ^at high frequencies.

In addition to the frequency dependences ^{of dis-}

location velocity ^{and mean kink} concentration, ^we

also obtained the dependences ^{of these} quantities ^on

the period-to pulse ^{duration of the} force, ^{for constant}

duration of the pulses. ^These dependences (not ^nor-

malized to active time) ^{are shown in} figure ^{5 and are}

in qualitative agreement ^{with the} experimental ^re-

sults of [9,10].

Fig.5.-Kink concentration (triangles) and dislocation velocity (circles) ^as functions of spacing ^between pulses, normalized to constant pulse duration in crystal with defects (E ⁼ 0.5, Ud ⁼

0.5 Uo).

We have studied, by ^means of numerical simula-

tion, ^{the influence} of defects on dislocation dynam-

ics in Peierls relief under both stationary ^and pulse loading. ^{It was} demonstrated that in the case of sta-

tionary ^and pulse loading. ^{It was} demonstrated that

in the case of stationary external force defects lower

dislocation mobility in the linear region (F ^0.2 Uo),

by braking ^kink propagation, ^{while at} greater ^forces

(F > 0.2 !7o) clusters of barrier-lowering defects rise nucleation rate, ^there by increasing dislocation mobil-

ity ^and plasticity ^{of the} crystal. The threshold force

value, ^{at which a} sharp drop in dislocation mobility

is observed was found to be close to the critical force

Fo, ^{at which} the transition from linear x N t to sub- linear x N t7, q 1 law, predicted ⁱⁿ [111, ^{takes place.}

A study ^{of the} dependence ^{of the} velocity ^{of an in-}

dividual kink on a stationary external force showed that in crossing ^{the threshold f = Fo} the charac- ter of the kink motion undergoes ^{a radical} change,

and in the case of low forces a kink spends ^{most of}

the time in traps (formed by ^defect clusters). ^{It was}

demonstrated that defects do not alter the character of dislocation dynamics ^{in the case of} pulse loading

with large amplitudes, corresponding ^to the nonlin-

ear region. However, ^{at small} amplitudes of external

force, ^{close to} Fo (the sublinear kink motion thresh-

old), ^{a sharp drop} in dislocation velocity ^as frequency

increases is observed. The results obtained enable us

to come to the conclusion that the sharp drop ^{in dis-}

location mobility, ^observed experimentally [9,10] ^at

high frequencies ^{of the} pulse loading, might ^{be a con-}

sequence of braking ^{of kink} propagation by defects,

rather than of suppression ^{of kink} generation ^at high frequencies. The latter effects is altogether ^{absent in}

the linear region. Indeed, dislocation velocity ^{in a} perfect crystal ^is [1,2]

where u is the kink propagation velocity, W(F) ^{is the}

energy of creation of a kink-antikink pair, depending

on the stationary external force F. If the frequency

of the external force is much lower than the mean

nucleation frequency, corresponding ^{to a} stationary

force of the same amplitude, ^then

where the angular brackets denote averaging ^{over a} period. ^{In the} opposite limiting ^{case of} high (i.e.,

much higher that the nucleation frequency) frequen-

cies

The transition from vlif to vl f should be sufficiently gradual, ^as illustrated by figure ^3. In the linear regime

the nucleation _energy W is virtually independent ^of

the applied force, ^{so that vi.f m Vh. f. In other} words,

in a perfect crystal ^{in linear} region ^{one should not ex-} pect ^a significant change in dislocation velocity during

the transition from high (in ^{the above} sense) ^{to low}

frequencies ^{of the external} pulse ^force.

Note that in the linear region, as one can see

from (4) ^and (5), ^the following ^{relation must hold} (after normalization to the active time) :

where v(F) is the dislocation velocity ^{in a} stationary

external field F, equal ^{to the} amplitude ^{of the} applied pulse ^force. Comparing ^the results, obtained for the

case of stationary loading ^{in the absence of defects}

with those shown in figure 3, one can see that relation

(6) ^{holds to a} good accuracy.

All the values of parameters, ^{used in our numer-} ical simulation were close to actual experimental ^val-

ues for semiconductor crystals ^with high ^{Peierls bar-} riers, ^{with the} exception ^of temperature, ^{which was} by ^far higher ^{than in} laboratory experiment. ^Such

a choice of T was dictated by the limitations of the computers ^available.

Summing up, ^we would like to point ^{out that our}

results show that point defects in crystal lattice have

an extremely profound ^{effect on} dislocation dynamics.

One might expect ^{that at lower} temperatures, ^closer

to actual experimental values, the effects we have ob- served (dislocation mobility threshold, frequency ^de- pendence of dislocation velocity in nonlinear region, nonlinearity ^{in kink} motion) would be much sharper

and more pronounced.

Acknowledgements

We are grateful ^{to V.I.} Nikitenko, B.Ya. Farber and V.Ya. Kravchenko for useful discussions of the model and results.

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