Materials Science and Engineering A 400–401 (2005) 84–88

Metastablity of the undissociated state of dissociated dislocations

Shin Takeuchi∗

Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan

Received 13 September 2004; received in revised form 3 December 2004; accepted 28 March 2005

Abstract

Undissociated, metastable dislocations have been observed in various crystals in addition to stable dissociated dislocations by high-resolutiontransmission electron microscopy. The origin of the metastablity of the undissociated state has been discussed specifically for the dissociationinto Shockley partial dislocations in fcc or hcp lattice. It is shown that the metastability is due either to a high Peierls–Nabarro stress largerthan a few percent of the shear modulus of the partial dislocations and/or to the increase of the total core energy by an increase of the danglingbonds. The metastablity of undissociated dislocations in zincblende III–V compounds is concluded to be due to a contribution of the lattereffect.© 2005 Elsevier B.V. All rights reserved.

K

1

dfidtoEhaLwaeIasIt

d by

the-

they is

the-

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de-d in a

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0d

eywords: Dislocation dissociation; Shockley partial dislocation; Metastability; Peierls–Nabarro stress; Dangling bond

. Introduction

It is well established that dislocations in crystals areissociated into partial dislocations bounding a stacking

ault between them when there exists a stable stacking faultn the crystal. The separation between the partials or theissociation widthw is determined so as to minimize the

otal energy or by the balance between the repulsive forcef the partials and the surface tension of the stacking fault.lectron microscopy observation of dislocations has shown,owever, that in many occasions undissociated dislocationsre also present in addition to the dissociated dislocations.ong constricted segments have been observed in Si byeak-beam technique[1]. Both dissociated and undissoci-ted core structures have been observed by high-resolutionlectron microscopy for III–V compounds of GaAs[2,3],

nP[4] and GaN[5], for II–VI compounds of CdS, CdSe[6]nd ZnO[7]. These observations indicate that undissociatedtate can be metastable for the stable dissociated dislocation.n this paper, we discuss the origin of the metastability ofhe undissociated state.

2. Strain energy consideration

The energy of a perfect dislocation can be representethe sum of the elastic energy outside the core radiusrc wherethe elastic strain can be described by the linear elasticityory, and the core energy inside the radiusrc. The value ofrcis (2–3)b, whereb is the strength of the Burgers vector ofdislocation. Assuming elastic isotropy, the elastic energexpressed as:

Eel = KGb2

4πln

R

rc, (1)

whereR is the outer-cutoff radius which is of the order ofaverage dislocation spacing in the crystal,G the shear modulus of the crystal and the factorK depends on the dislocaticharacter:K = 1 for screw dislocation andK = (1− ν)−1 (ν:Poisson’s ratio) for edge dislocation. The core energypends on the type of the crystal and cannot be expresseuniversal way; so that the core energy contribution is usincorporated in the total energy by using an effective cuparameterr0 (<rc) and thus the total energyEu (subscript u

∗ Tel.: +81 4 7124 1501x4305; fax: +81 4 7123 9362.E-mail address:takeuchi@rs.noda.tus.ac.jp.

stands for undissociated dislocation) is written as:

Eu = KGb2

4πln

R

r0. (2)

d.

921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserveoi:10.1016/j.msea.2005.03.054

S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 84–88 85

In metallic crystalsr0 is aboutb/3 [8–11]. In real situations,the outer-cutoff radius is variable and so we may regardR/r0in Eq.(2) a variable parameter. For a typical dislocation den-sity of 106 cm−2, R/r0 ≈ 105.

The total energy of a dissociated dislocation is the sum ofthe self-energies of the two partial dislocations, the interac-tion energy between the partial dislocations and the energyof the stacking fault between the partials. Letb1 andb2 bethe strength of the partial Burgers vector,θ1 andθ2 be theangle between the dislocation line and the Burgers vector ofpartial 1 and partial 2, respectively, andΓ be the stackingfault energy per unit area andx the distance between the par-tial 1 and partial 2. Then, the total energy of the dissociateddislocationEd (subscript d stands for dissociated dislocation)is given as:

Ed(x) = K1Gb12

4πln

R

r′0

+ K2Gb22

4πln

R

r′0

+ βGb1b2

2πln

R

x+ Γx with

K1,2 = 1 − ν cos2 θ1,2

1 − ν

β = cosθ1 cosθ2 + 1

1 − νsinθ1 sinθ2. (3)

isd

w

W .i ,w

tionp oca-t tions n fcca rtiald tionf

I thatP iusr tor.W toffr diusb ion ast

unitl ckleyp1 usd or

Fig. 1. Change of the strain energy of an edge dislocation with dissociationinto Shockley partial dislocations for three different stacking fault energiesgiving equilibrium dissociation widths ofw = 5b, 10b and 20b.

R= 104b but shifted by 0.275Gb2 downwards or upwards,respectively.Fig. 2shows the results for a mixed dislocationwhere the dislocation line and the Burgers vector make 60◦,andFig. 3the results for a screw dislocation again forw = 5b,10b and 20b. In these figures, we have assumed that the in-teraction term in Eq.(3) is valid approximately forx≥ 2b. Inthe case of 60◦dislocation and screw dislocation, the change

F ni rgiesg

The equilibrium widthw of the dissociated dislocationetermined by the condition dEd(x)/dx= 0, which gives

= βGb1b2

2πΓ. (4)

e should note that the interaction energy term in Eq(3)s valid only forx larger than the core radiusr′

c for partialshere the linear elasticity theory can apply.Now, we consider the energy change in the dissocia

rocess of an undissociated dislocation into partial dislions. In the following, we treat the most popular dissociacheme of Heidenreich–Shockley extended dislocation ind hcp lattices, i.e., the dissociation into Shockley paislocations, which is represented by the following equa

or the case of fcc crystals.

1

2[1 1 0] = 1

6[2 1 1] + stacking fault+ 1

6[1 21̄] (5)

n the numerical evaluation of the energy, we assumeoisson’s ratio is 1/3 and the effective inner-cutoff rad

0 or r′0 is one-third of the corresponding Burgers vec

e should note that the ambiguity of taking the inner-cuadius is equivalent to the ambiguity on the outer-cutoff raecause these two values appear in the energy equat

heir ratio.Fig. 1 shows the energy of an edge dislocation for

ength as a function of the separation distance of the Shoartials for three equilibrium dissociation widths ofw = 5b,0band 20b, and forR= 104b. We note that the energy versistance curves forR= 103band 105bare the same as that f

ig. 2. Change of the strain energy of a 60◦-dislocation with dissociationto Shockley partial dislocations for three different stacking fault eneiving equilibrium dissociation widths ofw = 5b, 10b and 20b.

86 S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 84–88

Fig. 3. Change of the strain energy of a screw dislocation with dissociationinto Shockley partial dislocations for three different stacking fault energiesgiving equilibrium dissociation widths ofw = 5b, 10b and 20b.

of theR-value by one order of magnitude shifts the curve by0.206Gb2 and 0.183Gb2, respectively.

Based on the above elastic energy consideration, the undis-sociated state of the edge dislocation and the 60◦-dislocationare definitely unstable, so that the undissociated dislocationwill spontaneously dissociate into Shockley partial disloca-tions. For the screw dislocation, on the other hand, when theequilibrium separation of the partials is rather narrow, lessthan 10b, the undissociated dislocation can be in a metastablestate although the energy barrier is low and when the equi-librium width is wide the metastablity of the undissociateddislocation is marginal. The origin of the different stabilityof undissociated dislocations for edge and screw dislocationsconsists in the special nature of the Heidenreich–Shockleytype dissociation, i.e., dissociation into non-parallel Burg-ers vectors producing extra-Burgers vector components. Theelastic energy of an edge dislocation is larger by a factor1/(1− ν) than that of a screw dislocation with the same Burg-ers vector. When an edge dislocation with initially large elas-tic energy for the Burgers vector is dissociated into partials,they produce screw components, whereas when a screw dislo-cation with the minimum elastic energy for the Burgers vectoris dissociated, the partial dislocations produce edge compo-nents. Hence, the energy gain accompanying the dissociationis much smaller for the screw dislocation than for the edge dis-location as can be seen by comparingFigs. 1 and 3. This factr sso-c ation.T ateds ctiono

rews ven ifa to thed even-t ugh

propagation of kink parts of Shockley partials. Then, a ques-tion arises what is the origin of stability of the experimentallyobserved undissociated dislocation for otherwise stable dis-sociated dislocation. There exist two factors which have notbeen taken into consideration in the above elastic energy con-sideration.

3. Effect of Peierls potential

In real crystals, due to the lattice discreteness the self-energy of a dislocation always oscillates with the periodicityof the lattice in the glide plane: the periodic potential energyis called the Peierls–Nabarro potential (abbreviated as P–Npotential henceforth), the magnitude of which varies largelydepending on type of the crystal. Thus, in real crystals a peri-odic P–N potential is superimposed on the curves inFigs. 1–3.In this section, we consider the effect of the P–N potentialfor the stability of undissociated edge dislocation.

The maximum negative slopes of the curves inFig. 1areapproximately 0.014Gb, 0.019Gband 0.021Gb for the caseof w = 5b, 10band 20b, respectively. Thus, ifτPb1,2value (τP:the P–N stress of the partial dislocation) is larger than thesevalues, the undissociated dislocation should be trapped at theP–N potential valley without dissociation. The critical P–Nstressτc necessary to prevent the core from the dissociationw fτ

1

4

hasb f theB dis-l hend ergy.I dia-m tendta islo-ct th ane onds;i ingb 30p d in9 hus,w lingb rac twod tion,a -gt of

esults in the difference in the metastability of the undiiated state between edge dislocation and screw disloche initial slight increase of the self-energy of undissocicrew dislocation upon dissociation is due to the produf edge components in the partials.

In the multiplication process of a dislocation loop, scegment is always linked to edge segment, and hence en undissociated screw dislocation is metastable, dueissociated edge part, the undissociated screw part will

ually be dissociated to the equilibrium stable state thro

Pithout thermal activation is larger for the larger value ow;

cP�2.4× 10–2G, 3.2× 10−2G and 3.6× 10−2G for w = 5b,0b and 20b, respectively.

. Effect of dangling bonds

In the above calculations, the inner-cutoff radiuseen taken to be equal to one-third of the strength ourgers vector for both undissociated and dissociated

ocations. This is not necessarily valid particularly wangling bonds at the core contribute to the core en

n tetrahedrally coordinated crystals with either theond, zincblende or wurtzite structures, dislocations

o lie along 〈1 1 0〉 direction or 〈1 12̄ 0〉 direction due todeep P–N valley along these directions. Thus, the d

ations are either screw dislocation or 60◦-dislocation. Inhese tetrahedrally coordinated crystals, dislocations widge component should be accompanied by dangling b

n undissociated 60◦-dislocation, there exists one danglond per atomic distance along the dislocation line, in◦-artial also one dangling bond per atomic distance an0◦-partial two dangling bonds per atomic distance. Then an undissociated screw dislocation with no dangond dissociates into two 30◦-partial dislocations, an extore energy should be produced due to formation ofangling bonds per atomic distance along the dislocand when an undissociated 60◦dislocation with one danling bond per atomic distance dissociates into 30◦ par-

ial and 90◦-partial dislocations, again an extra energy

S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 84–88 87

two dangling bonds per atomic distance should be pro-duced.

Dangling bonds may be reconstructed to form bondingstates between nearest neighbor dangling bonds but the sit-uation of an increase of the extra core energy is unchangedthough the amount of the energy increase becomes less. Dis-locations in semiconducting crystals may be charged elec-tronically by capturing carriers, yielding an electrostatic en-ergy. Dislocation core in crystals having an ionic characterwill have an extra electrostatic energy contribution. All theseextra energy contribution to the core energy may be termed“dangling bond effects” in a broad sense.

Assuming that the extra energy due to the dangling bondsemerges for the dissociation wider than the core radius ofabout 3b, for the undissociated 60◦-dislocation to be sta-ble the extra energy should be larger than about 0.039Gb2

and 0.045Gb2 for w = 10b andw = 20b, respectively. Tak-ing the core radius as 3b, the strain energy of the coreis ∼(KGb2/4π)ln 9≈ 0.2Gb2. Hence, if the extra danglingbond energy is larger than about one-fourth of the core strainenergy, the undissociated state is to be stable.

If we apply this result to Si, the extra energy due to extradangling bond formation per dislocation lengthb necessaryto stabilize the 60◦-dislocation is calculated to be 0.6 eV. Thisvalue is of the same order as the energy of a dangling bondin Si theoretically estimated[12].

5

dis-ls uslyd pro-c tables esultsb in them om-p e or-d tedd r of1 oci-a lutionep isq

hcpl ordi-n stiona iates highP y.

rallyc atedt ft

Thus, there is a possibility that the stability is due to the highPeierls potential of partials. However, the following argumentleads us to the conclusion that the dangling bond energy mustcontribute to the stability.

Suzuki et al.[18] showed that the plastic deformation ofIII–V compounds below a certain temperatureTc is governedby perfect dislocation glide while above this temperatureby dissociated dislocation glide, as schematically shown inFig. 4(a). This transition occurs because the Peierls stress forthe undissociated dislocation is lower than those for partialdislocations as a result of the special character of the dia-mond type lattice, whereas the kink energy is larger for theundissociated dislocation than for partial dislocations.

The fact that below the critical temperatureTc plasticity iscarried by undissociated dislocations means that the undis-sociated state of the dislocations in the III–V compounds isstable up toTc without undergoing core transition to the sta-ble dissociated state. On the other hand, atTc the dissociateddislocations are glissile at a rather low stress, meaning thatthe P–N potential of the partials can be overcome by thermalactivation. If the stability of the undissociated state is duesolely to the P–N potential of the partials, then at tempera-ture well belowTc, the metastable undissociated dislocationshould undergo thermally activated transition to the stabledissociated state because in the dissociation process the par-tials are repelled from each other by a quite high repulsives ,g soci-a ationsa et e liket ofF ustb te ofd

enti n fort as-s ma-t lizedP re

F yields -p eierlsp

. Discussion

Computer simulation studies for atomistic models ofocations in fcc metals[8,9,11,13]and for hcp metals[14–16]howed that initially undissociated dislocation spontaneoissociates into Shockley partials during the relaxationess, indicating the undissociated state is not in metastate. These results are consistent with the present recause the equilibrium separations between partialsodels are around 5b and the P–N stresses, though not cuted in these papers, are probably much lower than ther of 10−2G. Experimentally, the P–N stress of dissociaislocations in fcc metals and hcp metals is of the orde0−4 to−5G [17], and hence the fact that no long undissted dislocations have ever been observed by high-resolectron microscopy for the glide dislocations on{1 1 1}lane in fcc metals and on (0 0 0 1)plane in hcp metalsuite reasonable.

Experimentally, undissociated dislocations in fcc orattice have been observed mostly in tetrahedrally coated crystals as mentioned in the introduction. The querises whether the origin of the stability of the undissoctate in tetrahedrally coordinated crystals is due to the–N stress of the partials or to the dangling bond energThe P–N stresses of partial dislocations in tetrahed

oordinated crystals are difficult to evaluate but are estimo be in the range of 10−1 to 10−2G from extrapolation ohe critical resolved shear stress vs. temperature curves[18].

tress from each other of the order of 10−2G. As a resultlide dislocations cannot stay in the metastable, undisted state at the temperature where dissociated dislocre mobile at the stress of the order of 10−2G, and henc

he temperature dependence of the yield stress should bhat given inFig. 4(b), contradictory to the observationig. 4(a). Thus, a dangling bond energy contribution me concerned with the stability of the undissociated staislocations in III–V compounds.

Finally, it should be mentioned that the present treatms based on the simplest model of the Voltera dislocatiohe estimation of the dislocation energy with simplifiedumptions, thus yielding the results of only a first approxiion. As a better approximation, we may use the generaeierls–Nabarro model[19], which can provide much mo

ig. 4. (a) Schematic illustration of temperature dependence of thetress in III–V compounds experimentally observed[18]. (b) Expected temerature dependence of the yield stress if the metastability is due to Potential of partials.

88 S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 84–88

reliable results especially for narrow dissociations. As shownby Schocek[20], equilibrium separation of partials dependson the maximum of theΓ surface[21]. Furthermore, thismodel automatically incorporates the P–N ptotential. Thus,the next step to approach the present problem is to use the gen-eralized P–N model by systematically varying theΓ surface.It is to be noted, however, that no dangling bond is involvedin theΓ surface and hence the dangling bond contribution tothe core energy must be treated separately in the generalizedP–N model.

6. Conclusion

The undissociated state of dissociated dislocations canbe stable either due to high Peierls–Nabarro (P–N) stress ofthe partial dislocations or to dangling bond effects. For theHedenreich–Shockley dissociation, the undissosiated statebecomes stable when the P–N stress is as high as a fewpercents of the shear modulus or when the extra danglingbond energy due to dissociation is∼0.04Gb2 per unit dis-location length. From the experimentally observed crossoverof the deformation mode with temperature in III–V com-pounds, the stability of undissociated dislocation is con-cluded to be due to the contribution of a dangling bondeffect.

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