The reduction of dislocation density in GaAs by in doping: a specific interaction of in with the cores of 30° partial dislocations

HAL Id: jpa-00210804

https://hal.archives-ouvertes.fr/jpa-00210804

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The reduction of dislocation density in GaAs by in doping: a specific interaction of in with the cores of 30°

partial dislocations

F. Louchet

To cite this version:

F. Louchet. The reduction of dislocation density in GaAs by in doping: a specific interaction of in with the cores of 30° partial dislocations. Journal de Physique, 1988, 49 (7), pp.1219-1224.

�10.1051/jphys:019880049070121900�. �jpa-00210804�

The reduction of dislocation density ^{in GaAs} by ^In doping: ^a specific

interaction of In with the ^cores of 30° partial dislocations

F. Louchet

LTPCM, ENSEEG, ^{INP de} Grenoble, ^BP 75, 38402, St-Martin-d’Hères, ^France

(Requ ^{le 15} fgvrier ^1988, accepté ^{le 21 mars} 1988)

Résumé.

La réduction importante de la densité de dislocation dans GaAs monocristallin CZ par dopage ^In,

souvent mentionnée dans la littérature, ^est interprétée ^{en termes} d’intégration ^{des atomes} d’indium dans les dislocations partielles ^à 30°, ^{dont le c0153ur} peut ^{évoluer vers une structure} particulière ^de type ^shuffle interstitielle. Cette intégration ^se réalise par diffusion des ^atomes d’indium au passage des décrochements.

L’énergie d’activation correspondante ^est déduite des données expérimentales. On propose ^un critère simple

de blocage ^{des sources de} dislocations, fonction de la concentration en indium, qui permet ^{de rendre} compte des densités de dislocations observées.

Abstract.

The drastic reduction of dislocation densities in CZ grown GaAs by ^In doping reported ^{in the} literature, is interpreted ^{in terms of an} integration ^{of In atoms in 30°} partial dislocations, ^{in which a} particular

shuffle core structure can be built, through diffusion of In atoms at kinks. The corresponding activation energy is derived from experimental ^{data. A} simple criterion is given ^for locking dislocation sources as a function of In content, which agrees with observed distributions of dislocation densities.

Classification

Physics ^Abstracts

61.16D - 61.70J

61.70L - 61.70P

Introduction.

It is generally accepted ^now that In has a substantial effect on the reduction of dislocation densities obtained in CZ grown GaAs crystals. ^An interpre-

tation of this effect has been given ^{in terms of solid}

solution hardening ^{due to} the interaction with dislo- cations of dilatation centers associated with In atoms

[1]. ^{However, the} specific action of In (as compared

to Si for instance) [2], ^{and its} particular ^{influence on}

screw and a dislocations shown by velocity ^measure-

ments, suggest ^{that the} problem ^{is more} closely

related to core structures of dislocations than to their elastic strain field. After a brief review of some

experimental ^{data on this} question, ^a particular type of interaction of In atoms with a and screw dislo- cations will be proposed, ^{and its} consequences ^on dislocation densities will be discussed.

1. Analysis of available experimental ^data.

Indium addition in GaAs reduces significantly ^the density ^of dislocations obtained during ^CZ crystal growth. ^More precisely, only glide dislocations, moving ^on {111} planes under thermal stresses, ^are concerned with this effect [3], ^while growth ^dislo-

cations are not affected. Guruswamy, ^{Hirth and}

Faber [1] showed the existence of a substantial

hardening ^{due to In} doping, ^evidenced by ^hardness

measurements up ^to 500 °C and a higher plateau

stress level during creep with increasing ^{In content.}

The interpretation ^{is based on the solid solution} strengthening proposed ^earlier by Ehrenreich and Hirth [4]. ^{In this model,} the increase of about 7 % of the length of the InAs bond compared ^{to that of a}

GaAs _one, is shown to lead to a volume dilation of about 21 % around the InAs unit in the GaAs matrix. The interaction of the dislocations with these dilatation centers can either pin dislocations which bow out between them (classical ^line tension solute

hardening), ^or slow down the migration ^{of kinks} along dislocations (interaction ^of kinks with InAs

units).

Dislocation velocity measurements, by etching techniques ^or by ^X Ray transmission topography ^for instance, give ^more information than macroscopic hardening ^or creep experiments [5]. ^{In the} present

case, with temperatures ^{between 260 °C and 500} °C,

and stresses from 5 to 20 MPa, dislocations have

polygonal shapes, along ( 110 ) directions, ^and only

screw and 60° dislocations are sensitive to In addi- tions [5, 6, 2]. ^{If we focus our attention on these two}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070121900

1220

types ^of dislocations, ^{two different} regimes ^are observed, according whether the applied ^{stress is} lower ^or higher ^{than a} critical value o-c =10 MPa [5, 6]. ^In regime ^I (o- o-c), ^the longer ^{is the distance}

travelled by ^{60° and screw} dislocations, ^the _stronger

is the velocity reduction, ^{so that no} mobility ^{can be}

defined unambiguously ^{for these} dislocations in this

case. In regime ^II (a

o’c) ^{on the other hand, 60° a}

and screws are slowed down, ^but independently ^of

the distance travelled in the slip plane. ^This suggests

an interaction of 60* a and screws with fixed

impurities ⁱⁿ regime II, ^{and a} scavenging ^{of mobile} impurities ⁱⁿ regime I, ^as for instance in the case of PLC effect. This last point ^{seems to be confirmed} by

the behaviour of the critical stress crcg which in-

creases with temperature ^{or with} the duration of the temperature plateau, ⁱⁿ agreement ^{with a} thermally

actived process of incorporation ^{of In atoms in the}

dislocation cores.

The interaction mechanisms of In atoms with dislocations are then probably quite ^{different in the}

two regimes. Regime ^{II is} likely ^{to be} explained by ^a

solid solution hardening, ^as proposed by Guruswamy

et al. [1], ^{but a} specific ^{mechanism must be invoked}

for regime ^I. This mechanism operates only ^{on 60° a}

and screw dislocations, which both contain a 30° a

partial dislocation. 60° f3 dislocations are insensitive

to In, ^{and do not} contain the 30* a partial. ^This partial dislocation has then probably ^a specific

interaction with In atoms, which will be described and discussed in the following ^sections.

2. The particular ^{structure of the 30° a} partial.

There is circumstantial evidence that dislocation

cores in elemental semiconductors (Si, Ge) ^{are on}

average of glide (G) character. A similar configur-

ation can be reasonably assumed in the case of

compound semiconductors, although ^a stabilization of glide ^cores by bond reconstruction is more

questionable ^{in this case} [7]. ^{In this G} configuration,

a 60° « dislocation dissociates into a 30° a and a

90° « partial, ^{each of them} having ^{an As core} (Fig. 1), ^{while a screw} dislocation dissociates into a

30’ a partial (with ^{an As} core) ^{and a 30°} /3 partial (with ^{a Ga} core). ^{The 60° f3} dislocation dissociates into 90° f3 ^{and 30°} f3 partials (with ^Ga cores). ^All

these dislocations will be labelled G in the following.

They ^are separated by ^a stacking fault which neces-

sarily lies in the

G » plane.

Now, ^if point ^{defects are allowed to} migrate ^onto

the cores of partial dislocations, ^{several core sites}

can turn into a shuffle

S

configuration, ^labelled Si ^{if the shuffle site is} obtained from a G site by

addition of an interstitial, ^or S, if it is obtained by

addition of ^a vacancy. As noticed above, ^{the tran-}

sition to these shuffle core structures from G ones is

probably ^easier than in elemental semiconductors,

Fig. ^1.

Dissociated dislocations loop ^{in GaAs.} Burgers

vectors of partial dislocations are indicated by ^arrows.

Core atoms in the glide (G) configuration ^are schematized

by ^stars for Ga and open circles for As. The ^nature of core atoms in kinks is also mentioned.

since reconstruction of dangling ^{bonds is more}

difficult between identical As atoms (in a ^{« G »} cores) ^{or Ga atoms} (in (3

^G

cores) ^{than between}

Si or Ge ones. Different possible Si ^or Sv ^{cores for}

30° and 90° partials ^{have been} recently proposed ^and

discussed [7]. ^{The main} conclusions are that Sv ^cores

are rather hollow and difficult to reconstruct, that 90° Si ^{cores are} overcrowded, ^and probably ^too energetic ^{to be} found, ^{but that 30°} Si ^{cores have}

three possible ^and nearly equivalent low energy semi-reconstructed structures, ^{which are} shown in

Fig. ^2.

^{Core structure of a 30°}

partial : ^{black and}

white atoms refer to the two interpenetrating f.c.c. lattices

(Ga ^{or In in} black, ^{As in} white) (a) glide configuration ;

(b) (c) (d) : ^three possible shuffle interstitial cores. Notice

the As-Ga (or B-A) zig-zag ^chains.

figure ^{2. These 30°} Si structures are made of zig-zag shaped chains of A and B atoms, with very low distortions of tetrahedral bonding. ^{In the case of a}

30° a partial ⁱⁿ GaAs, ^{the atoms labelled A in} figure ^{2 are As} atoms, whereas B atoms can be either Ga or In ones. A mechanism of incorporation

of In atoms on B (Si) ^sites is discussed in the

following.

3. Mechanisms and kinetics of incorporation ^{of In}

atoms on 30° a Si ^sites.

Two different kinds of interaction mechanisms of In atoms with dislocation cores can be imagined, ^ac- cording ^{whether diffusion is allowed. to} operate ^or

not.

3.1 HIGH STRESS, LOW TEMPERATURES. - In this case, diffusion is not favoured. When In atoms are met by dislocation _cores, stress relaxation can pin ^for

a while the dislocation core on the associated dilatation center, giving ^a classical solid solution

hardening. ^{This is} probably ^the microscopic aspect of the solid solution hardening ^invoked by ^Gurus-

wamy et ^al. [2], ^and might correspond ^to regime ^II.

The details of this interaction are probably ^different

in the case of {3 ^{or a} partials, since in the former case

In atoms (assumed ^to occupy Ga sites in the crystal)

are in the same sublattice as the dislocation _core, whereas in the latter case the dislocation core (in ^the

As sublattice) flees ^{over the In atom. It is difficult to}

decide from core structure models in which case the

pinning ^{effect is most} effective. Nevertheless, ^the

relative insensitivity ^of {3 dislocation velocities on In

doping ⁱⁿ regime ^II [5] suggests that relaxation around the In center is larger ^{in the case of a} partials. ^{In addition, 90°} partials might ^{be more}

sensitive to In than 30° _ones, due to their larger edge

component.

3.2 LOW STRESS, HIGH TEMPERATURES. - Dif- fusion can now operate, ^{and a} progressive incorpor-

ation by diffusion of In atoms in dislocation cores can be envisaged. ^As discussed in 2, ^{the best} type ^of

core for In storage ^{is the 30° a one. As} long ^{as the In}

atom remains on the Ga sublattice, ^{it is bonded to 4}

As atoms, ^{and the} expected ^stress relaxation is small

(even smaller for a 30° a than for a 90° a partial). ^In

contrast, diffusion ^{of In onto a} Si site leads to a

reduction of In-As bonds, ^{and to an increased stress}

relaxation. This advantage ^becomes predominant

when a number of In atoms have been collected on

30° a Si ^sites, forming zig-zag ^{chains as shown in} figure 2. The shift of In atoms into shuffle sites is a

thermally activated process, in agreement ^{with the} dependence of the critical ^stress Te ^on temperature and annealing ^time (see [5] ^and 1). ^The progressive incorporation of In in the core also agrees with the continuous slowing ^{down of a and screw} dislocations in regime ^{I. The} specific ^{core structure} of In deco-

rated 30° ^a partials explains ^the specific ^{effect of In} doping ^{on a and screw} dislocations, since this type ^of

partial belongs ^{to both a and screw} dislocations, ^but

not to j3 ^ones.

3.3 ROLE OF KINKS. - In fact, ^{In atoms are not met} directly by dislocations, ^but by ^{kinks, and one has to}

look at the way by ^{which In atoms can be} incorpo-

rated into 30° a cores by ^kinks.

A particular ^and interesting ^{feature of 30°} partials

is that, contrary ^{to 90°} partials, their double kinks

are made of kinks of different natures. A 30° ^a glide partial (with ^{an As} core) ^{has an As 90° kink and a Ga}

30° kink (see Fig. 1). ^{This is not the case for 90°}

partials ^{in which core atoms are of the same nature}

as those in their kinks. During ^{motion of a 30° a} partial, ^{the As 90°} kinks will probably fly ^{over In}

atoms (on ^Ga sites), ^{whereas Ga 30° kinks will} actually ^{meet In} impurities, around which bonds will be cut. In this latter _case, climb of the In atom onto a

Si site is much easier than from a 90° kink, ^since bonds are already ^{broken. A} vacancy is left ^on the

kink, ^{which can} facilitate its motion [7], leaving ^the

In atom on its Si site. Other kinks will sweep ^out the

partial, but it will be now very difficult for ^a normal kink to take the In atom out from the _core, since it would require ^a larger energy than the formation energy of ^an interstitial. If the incoming ^{kink is}

associated with a vacancy, ^as for instance a 30° kink with a Ga vacancy, it might incorporate ^{the In atom,}

and we return into the former situation, in which the

following ^30° Ga normal kink will transfer In into a

Si site in the new core position. ^{If the} incoming ^kink

is not a 30° kink with a Ga vacancy but ^a 90° kink with an As vacancy, In ^{can come} back into a glide

site only by creation of an As/Ga ^{antisite defect,}

which is rather energy consuming. ^{It seems that the}

return of In into a glide position ^is difficult, ^and

therefore that In atoms migrate ^{with the 30° a core.}

More and more atoms are then collected, leading ^to

In-As chains.

3.4 KINETICS.

The jump of In from a G onto a

Si ^{site is a} thermally activated process, characterized

by ^an activation energy Ag. ^{The effective} jump frequency ^can be written :

assuming that the vibration frequency of the In atoms in the dislocation core is the Debye frequency

vD.

If the probability ^{of reverse} jump ^is neglected, ^as

discussed in 3.3, ^{the In atom will be} actually incorporated ^{into the} Si site if the average time

1/ v necessary for jumping ^on Si is less than the time

åtk spent by ^{the kink on} the In site. otk ^{can be}

obtained from the expression of kink velocities (8)

1222

valid at relatively ^{small stresses :}

where a is the applied stress, and Wm ^the migration

energy of kinks.

The condition of incorporation ^{becomes :}

and the critical stress o-c separating regimes ^{I and II}

is obtained when Atk =1/v

i.e.:

or :

where Wm ^and Ag ^{do not} depend ^on temperature.

In current cases, o,-b3,: kT, and therefore Wm - Ag ^{0. We check} that 03C3c is ^an increasing function of T, ^{as observed} experimentally [5]. ^{If we} put Uc = 10 ^{MPa at T = 400 °C} [5], ^{we find :}

If the migration energy Wm is estimated about 0.8 eV (see [9] ^and appendix), ^this gives ^{an activation}

energy for In incorporation ^{of 0.95} eV, ^{which is not} unreasonable.

4. Role of In concentration on dislocation densities in CZ grown crystals.

The density ^of glide dislocations measured in CZ grown GaAs crystals depends drastically ^{on In}

content [3]. ^{For In} concentrations larger ^{than about}

1 %, ^the crystal ^is nearly dislocation-free, except ^{in a} thin zone close to the surface. When In content is decreased (Fig. 3), ^{the thickness of this zone in-}

creases, and for an In content less than approxi- mately ^0.5 %, ^another dislocated region appears ^at the center of the crystal.

A simple explanation ^{based on} equation (3) might

account for this kind of dislocation distribution, ^the

extension of the dislocated region depending ^{on the}

relative values of oc and of the thermal stress ath, i.e. the stress induced by temperature gradients

in the crystal, ^as sketched in figure 4a : in the regions

where _o-lh

o-c, ^we are in regime II, dislocations are

only ^{slowed down} by ^{fixed In atoms, and can} develop. ^{In the} intermediate region where Uth- 03C3c ^{we are} in regime (I), ^and dislocations are stopped

after a mean free path ^{X which} corresponds ^{to a}

Fig. ^3.

Influence of In doping ^on the dislocation distri- bution in a CZ grown GaAs crystal (after Chabli, Molva, Bunod and Bletry) (KOH etching).

saturation of 30* a cores in In. If the incorporation

occurs as soon as a Ga 30° kink meets an In atom

(see 3.3), ^{the core} will be saturated after a path :

where the lattice periodicity ^{is taken} equal ^{to the} Burgers ^{vector b for} simplicity, and where the factor 2 accounts for the fact that only ^one type ^of kink is assumed to incorporate ^{In atoms in the core.}

This estimate gives ^{X = 0.1} f.Lm for _{cIn == 1} %, which

is unfortunately much less than the extension of dislocated regions.

Moreover, ^this explanation cannot account for the

dependence of dislocation densities on In content. It

can be improved by taking ^{into account} dislocation

multiplication. ^{The basic idea is} that, ⁱⁿ regime ^{I, a}

dislocation source can operate only ^{if the mean free} path X ^before saturation is larger than the distance necessary ^to activate a Frank Read _source, i.e.

roughly Gblo-th, where crth is the thermal stress.

If X > Gb the source can produce ^a complete

ath

loop, containing ^{both screw, a and} f3 dislocations.

The former ones will stop ^further on, while the latter

can propagate freely ^until they ^{activate another}

source.

If X Gb ^on the other hand, dislocations will

03C3th

Fig. ^4.

(a) ^and (b) : ^a comparison ^{of the} (schematic)

thermal stress radial profile in the CZ crystal ^{with the}

critical stress shows the regions where dislocation motion

obeys regimes ^{I or II.} The absence of dislocations at the center of the crystal ^for high ^{In contents} (Fig. 3) suggests that the real situation is that of (b) () crm! ! I U c

I UM I); (c) : ^within regime ^I region, ^a comparison ^of Gb / Uth (full lines) with the different values of the dislocation path X, ^which depends ^{on In content} (dashed lines), shows the existence of a ring-shaped dislocation free region (DFR) ^for Gb / Uth :> X. The width of this

region increases when X decreases (i.e. ^{when In concen-}

tration c increases).

stop ^{before the source} has reached its critical _pos- ition, ^and multiplication ^{is hindered.}

This is sketched in figure ^{4 : in} figure 4a, crj I o, m I - I cr m I, and a bundle of dislocations is always present ^{at the center of the} crystal, ^whatever

cIn might be. This is obviously ^{not the} experimental

situation. In figure 4b, I U c I ^{is larger, i.e. tempera-}

ture is also larger ^{than in} figure 4a, according ^to equation (3). Some dislocations in this case remain close to the surface (regime II), ^{but would} disappear

at higher temperatures. ^{The main} part ^{of the} speci-

men is in regime ^I (oth - (7c). Multiplication ^will

occur or will be hindered depending ^{on the} respec- tive values of X and Gb / U th, which ^are sketched in

figure ^{4c. At} high ^In concentrations, ^{X =} X(cl) ^is

less than Gb / U th everywhere except ^{close to the} surface. The crystal ^{will be} nearly dislocation free. If cIn is increased _up to C2, X

X(C2):::. ^Gb I Uth in the surface region, but also in the center, ^{where some} dislocations appear. At low In concentrations

(X = X(C3) large), ^the dislocation-free region ^is

reduced to a narrow ring ^{in the} region ^where

°’tn is minimum.

A rough ^{estimate of} the order of magnitude ^of G6/o’th ^{can be derived from} computed ^stress profiles [10], ^which give ^{maximum stresses} of several tens of MPa. With G

60 GPa, ^this gives minimum values of Gb / U th slightly less than 1 _f.Lm, which is not inconsistent with estimates of X in actual exper- imental conditions (between ^{0,1 and 1 03BCm when}

cIn varies from 1 % to 0.1 %).

Conclusion.

A simple model has been proposed ^{for the effect of}

In addition on dislocation mobilities in GaAs. It is based on a specific ^{shuffle core structure of 30° a} partials, ^{made of} zig-zag shaped ^As-In chains, ^which

can be built by ^a progressive incorporation ^{of In}

atoms in the 30° a core through ^{30° Ga kinks. This}

mechanism is thought ^to operate ^at relatively high temperatures and moderate stresses, in order to allow diffusion of In atoms onto interstitial shuffle sites of the 30° a partial. ^Another type of interaction between dislocations and In obstacles probably ^oc-

curs at higher ^stresses, leading ^{to classical solute} hardening by ^fixed impurities, which has been al-

ready predicted in the literature. A relation between the critical stress 03C3c (which separates ^{these two}

regimes) ^{and the} temperature ^{has been} derived, ^and

is in agreement with the few experimental ^available

results. From the existence of these two regimes ⁱⁿ

which dislocations interact respectively ^{with mobile}

or fixed impurities, ^{one can} expect ^a Portevin le Chatelier behaviour of In doped ^GaAs, if strained with a suitable strain rate. Experimental ^{values of}

cc and T taken from [5] ^{have been} introduced in the

U c (T) ^relation, giving ^an activation energy for In

integration ^{in 30° a cores of about 0.95} eV.

The profile of dislocation density ^{in a} CZ grown

crystal ^{as a} function of In content has also been

derived, assuming ^that dislocation sources can oper- ate if the mean free path X ^{of a and screw}

dislocations (before being stopped by ^In integration

in their cores) ^is larger ^than Gb / U th ^{where the}

thermal stress profile U th has been taken from [10].

This approach ^{accounts for} the different types ^of radial dislocation distributions found when In con-

centration varies from 0.1 to 1 %.

1224

Appendix.

DETERMINATION OF THE MIGRATION ENERGY OF KINKS FROM DISLOCATION VELOCITY MEASURE- MENTS.

The migration energy of kinks Wm ^{can be}

derived from dislocation velocity measurements, ^as

soon as the dependence ^of velocities on dislocation

lengths ^{is known.}

If L is the dislocation length, ^{and X the mean free} path ^{of kinks} along ^the dislocation, the dislocation

velocity ^can be written [8] :

where F(o-) is the nucleation energy of double

kinks, Wm ^the migration energy of kinks, ^{and a the} applied ^stress.

The mean free path ^{of kinks X} (i.e. the average

separation ^{between kinks on the} dislocation) ^{can be}

written [11] :

where Voo ^{is the} velocity ^{of a} dislocation of infinite

length (given by (A.2)), ^and Vo ^the slope V /L ^{of the} V(L) ^curve for very short dislocations (given by (A.1)). ^{This mean free} path ^{can also be related to}

A measurement of V /L ^{can therefore allow an}

estimation of F (a) ⁺ W. through (A.1), ^{while an}

estimate of Voo ^{and of} Vo

V /L gives the value of

F(o-) through (A.3) ^and (A.4). ^The difference of these two values give Wm.

The velocity measurements performed by ^Caillard

et al. [9] give, ^{for screw} dislocations at T

350 °C,

a

50 MPa :

Taking v D =1 O 13 S-1, ^{and b}

^{3.98 A :}

and therefore

which is an average between the two types ^{of kinks} found on a 30° (a ^or (3) partial.

Acknowledgments.

The author thanks Drs A. Chabli and F. Molva for

helpful discussions.

References

[1] GURUSWAMY, S., HIRTH, J. P. and FARBER, K. T., J. Appl. Phys. ⁶⁰ (1986) ^4136.

[2] YONENAGA, I., TAKEBE, ^{M. and} SUMINO, K., ^Int.

Symposium ^{on structure and} properties ^{of dislo-}

cations in semi-conductors (Moscow) ^March

1986.

[3] CHABLI, A., MOLVA, E., GEORGE, A., BERTIN, F., BUNOD, ^{P. and} BLETRY, J., Proc. Conf. Adv.

Materials for Telecommunications, ^{Ed. P. A.}

Glasow (Strasbourg) 1986, p. 27.

[4] EHRENREICH, ^{H. and} HIRTH, ^J. P., Appl. Phys. ^Lett.

46 (1985) ^668.

[5] BURLE-DURBEC, N., PICHAUD, ^{B. and} MINARI, F., Philos. Mag. ^{Lett. 56} (1987) ^173.

[6] YONENAGA, I., SUMINO, ^{K. and} YAMADA, K., Appl.

Phys. ^{Lett. 48} (1986) ^326.

[7] ^{LOUCHET, F. and} THIBAULT-DESSEAUX, J., Revue

Phys. Appl. ²² (1987) ^207.

[8] ^HIRTH, J. P. and LOTHE, J., Theory of Dislocations

(McGraw Hill) ^1968.

[9] CAILLARD, D., CLÉMENT, N., COURET, A., ^AN- DROUSSI, Y., LEFEBVRE, ^{A. and VAN-} DERSCHAEVE, G., Microscopy ^of Semi-Conduct-

ing Materials V (Oxford) ^1987.

[10] ^{JORDAN, A.} S., CARUSO, R. and VON NEIDA, ^A. R., Bell Syst. Tech. J. 39 (1980) ^593.

[11] LOUCHET, F., COCHET-MUCHY, D., BRECHET, ^Y.

and PELISSIER, J., ^Philos. Mag. A ⁵⁷ (1988) ^327.