Materials Science and Engineering A319–321 (2001) 115–118

Boundary conditions for dislocation core structure studies:application to the 90° partial dislocation in silicon

Karin Lin a,c, D.C. Chrzan b,c,*a Department of Physics, Uni�ersity of California, Berkeley, CA 94720, USA

b Department of Materials Science and Mineral Engineering, Uni�ersity of California, 577 E�ans Hall MS 1760, Berkeley, CA 94720, USAc Materials Sciences Di�ision, Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Abstract

The effects of boundary conditions on studies of dislocation cores are examined, using the 90° partial dislocation in Si as anillustration. The relative stability of two possible reconstructions of the core are explored using periodic supercell calculations andcylindrical boundary conditions and the results compared. It is argued that the stable core structure depends systematically on thestress field experienced by the dislocation, and that this dependence can be quantified effectively using a combination of periodicsupercells and elasticity theory. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Dislocation; Core; Silicon; Elasticity; Boundary conditions

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1. Introduction

The prediction of the mechanical properties of amaterial is of obvious importance, for applications aswell as a theoretical challenge. Central to this effort isan understanding of dislocation behavior. Rapid in-creases in computational power have allowed for thedevelopment of large-scale simulations of dislocationdynamics [1–3]. These simulations are complicated bythe long range of dislocation interactions and the fail-ure of continuum elasticity theory to describe propertiesat small distances. In order to make up for this defi-ciency, a description of the dislocation core structure atthe atomic level is needed.

Several attempts at identifying the core structure ofthe 90° partial dislocation in silicon have been made.These dislocations lie in the �110� directions in the{111} slip planes, with Burgers vectors pointing in the�112� directions. In the late 1970s, it was proposed byHirsch [4] and Jones [5] that the core of the 90° partialreconstructs by breaking the mirror symmetry along thedislocation line direction and restoring the 4-fold coor-dination of the atoms along the core (see Fig. 1a) This

reconstruction has come to be called the single-period,or SP core, in contrast with the double-period, or DPcore, recently proposed by Bennetto et al. [6]. In thisreconstruction, the 4-fold coordination of the coreatoms is retained, but the core is composed of five- andseven-membered rings and the period along the disloca-tion line is doubled (see Fig. 1b).

The relative energies of these two cores have beenstudied using a variety of atomic-scale total energycalculations [6–8], and the general consensus is that thedouble-period core is indeed more stable in most cases.However, the choice of boundary conditions can havesignificant effects on the results of the calculations. Thetwo most common approaches in the literature are touse cylindrical boundary conditions [7,9,10] or periodicsupercells [6–8,11–13]. The cylinder method has anadvantage in that, it allows the treatment of an isolateddislocation and greater control over the individualstress components. However, this method also suffersfrom a sensitivity to the exact placement of the disloca-tion, and generally requires a larger number of atoms.In this work, the cylinder method and the periodicsupercell method are carefully examined and compared.It is argued that the periodic supercell method, whileallowing for less control over the stress field experi-enced by the dislocation, nonetheless allows one to

* Corresponding author. Tel.: +1-510-6431624; fax: +1-510-6431624.

E-mail address: dcchrzan@socrates.berkeley.edu (D.C. Chrzan).

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 9 21 -5093 (01 )00940 -6

K. Lin, D.C. Chrzan / Materials Science and Engineering A319–321 (2001) 115–118116

Fig. 1. Ball and stick representation of the 90° partial in a silicon asviewed in the (111) plane, (a) the SP and (b) the DP reconstruction.The dotted lines indicate the positions of the dislocation cores.

suggests that the core energy of the relaxed configura-tion may be lower if the dislocation is displaced slightlyfrom this axis. Further, there is no reason to assumethat the optimal position for the dislocation is the samefor the SP and DP structures.

In order to explore these issues more carefully, thefollowing procedure is performed. The trial cylinder istaken to have a radius of 20 A� and is periodic along thedislocation line direction. Atoms within 5 A� of thesurface are held fixed. For each core reconstruction, thedislocation position is varied in increments of 0.1 A�along each direction within a circle of radius b, where bis the magnitude of the Burgers vector (about 2.2 A� ).The resulting configurations are relaxed using Tersoffpotentials [14] and the energies recorded. It can beshown with a simple calculation that small displace-ments of the dislocation from the cylinder axis result innegligible changes in the elastic energy. Therefore, it isappropriate to take the position corresponding to theminimum relaxed core energy as the optimal placementof the dislocation. For the SP core, this position isfound to be x= −1.4, y= −0.5, where the x-axis is inthe [11� 2� ] direction and the y-axis is in the [11� 1] direc-tion. For the DP core, the optimal position is x=0.3,y= −0.7. The energy differences associated with theplacement of the core can be significant, resulting invariations of E(DP)−E(SP) of up to 5 meV A� −1,which is on the order of E(DP)−E(SP) itself, calcu-lated using the same potentials.

Using these optimal positions for the placements ofthe SP and DP cores, the difference in energiesE(DP)−E(SP) is calculated as a function of cylinderradius. The thickness of the outer cylinder of fixedatoms is kept constant at 5 A� ; supplemental trialssuggest that variations in this thickness are insignifi-cant, especially as the cylinder radius is increased. Fig.2 shows the predicted results for E(DP)−E(SP), calcu-lated using Tersoff potentials. In agreement with previ-ous workers’ findings, the isolated DP core is indeedmore stable than the isolated SP core. For cylinders ofradii 60 and 70 A� containing 4350 and 5922 atoms,respectively, the energy difference is about 6.8 meVA� −1.

3. Periodic boundary conditions

The method of periodic boundary conditions, orperiodic supercells, has an advantage in that the dislo-cations are now embedded in an infinite medium. Thus,the surface effects which required careful treatment inthe cylindrical method are no longer present. However,the periodic supercells require some considerations oftheir own. Most notably, each supercell must have a netzero Burgers vector to avoid a divergence in the elasticstrain energy. The simplest cell, then, contains twodislocations with opposite Burgers vector.

Fig. 2. Energy difference E(DP)−E(SP) for silicon, calculated usingTersoff potentials and cylindrical boundary conditions, as a functionof the cylinder radius.

explore a wide range of parameter space using feweratoms, as well as eliminating some of the ambiguitiesinherent in the cylinder method.

2. Cylindrical boundary conditions

A typical practice for imposing cylindrical boundaryconditions is to generate the initial positions of theatoms within anisotropic elasticity theory, fix the posi-tions of the atoms on the surface of the cylinder, andallow the remaining atoms to relax. Unlike the case ofperiodic supercells, the initial placement of the disloca-tion is important because it determines for all time thepositions of the outer atoms. However, the attempt totreat a discrete atomic lattice with continuum elasticitytheory introduces an ambiguity; a dislocation’s positioncan only be defined down to the interatomic spacing.While one approach is simply to place the dislocationon the center axis of the cylinder, the lack of perfectradial symmetry about this axis at the atomic level

K. Lin, D.C. Chrzan / Materials Science and Engineering A319–321 (2001) 115–118 117

Following the notation of Bennetto et al. [6], thedimensions of unit cells viewed in the (110) plane aredefined by the unit vectors a=a/2[11� 2� ], b=a/2[110],c=a [11� 1]. The cells have height D, width L, dislocationseparation w, and are offset by a distance T (See Fig.

3). L, w, and T are in units of �a�, and D is in units of�c�.

As first noted by Bigger et al. [11], the value of theoffset T requires some consideration in order to avoid alattice mismatch at the cell boundaries. Previously, itwas thought that this problem could be solved only byusing a quadrupolar lattice, i.e. T=L/2. However,Lehto and O� berg [7] pointed out that this restriction isunnecessary. When two oppositely signed dislocationsare introduced into a perfect solid and displaced fromeach other by a distance w, the elastic strain introducedis b(w/L), where L is the width of the solid along theslip direction and b is the magnitude of the Burgersvector [15]. Provided that the offset T is adjusted bythis amount, there are no restrictions (other than thoseimposed by the lattice periodicity itself) on L, D, w, andT.

The stress field experienced by a dislocation in thiselastic medium depends, naturally, on the cell parame-ters L, D, w, and T. Determining this stress fieldinvolves performing an infinite Madelung-like sum, andhence must be treated with caution [12]. However, arecently developed method [16] (K. Lin and D.C. Chr-zan, unpublished research) allows this sum to be calcu-lated rapidly and in a well-defined manner. The infinitearray of dislocations can be considered not only as aperiodic arrangement of dipolar unit cells, but as a 1-Dstack of tilt boundaries (Fig. 4). The stress field from atilt boundary can be calculated analytically fromisotropic elasticity theory [17], and is found to decayexponentially with distance from the boundary. Thus, a2-D sum of long-ranged stress fields can be transformedinto a much more tractable 1-D sum of short-rangedstress fields. This method has been shown to be effec-tive in predicting shear moduli and core radii for 90°partial dislocations in diamond [16] and silicon [17].

The periodic supercell method, in conjunction withthis elasticity calculation, can now be used to study thedependence of core energies on the actual stress state ofthe dislocation. While the cylinder method requires oneto apply a stress by imposing the atomic displacementsexplicitly, the periodic supercell method allows one tovary the stress simply by the choice of unit cell. Lehtoand O� berg have already noted that the relative stabilityof the two core reconstructions is dependent on theenvironment in which the dislocation is placed [7]. Thelink between the choice of unit cell and the stress stateof the dislocation, provided by elasticity theory, nowallows this dependence to be quantified.

Fig. 5 shows E(DP)−E(SP) as a function of theshear stress �xy again calculated from Tersoff potentialsbut using periodic supercells. Each curve correspondsto a set of parameters L, D, and T, and the variouspoints on a given curve correspond to different choicesof w. As expected, the DP core is found to be stable inthe vast majority of cases, but by an amount that

Fig. 3. Unit cells for periodic boundary conditions.

Fig. 4. Periodic array of dislocations, considered as a 1-D stack of tiltboundaries.

Fig. 5. Energy difference E(DP)−E(SP) for silicon, calculated usingTersoff potentials and periodic boundary conditions, as a function ofthe shear stress.

K. Lin, D.C. Chrzan / Materials Science and Engineering A319–321 (2001) 115–118118

depends on the stress state of the dislocation. Thoughnot shown in the plot, the most Qutlying curves, corre-sponding to the smallest unit cells (L=6, D=2), havethe largest values of the diagonal stress components �xx

and �yy. The cells for which T=0 and L/2 producenearly coincident curves (L=6, D=8 and L=8, D=12) have very small diagonal stress components. Fur-thermore, the w=L/2 case results in a zero shear stress,but the predicted energy difference for �xy=0 is clearlynot the same for all choices of unit cell. This suggeststhat the relative stability of the cores is dependent onnot only the stress fields experienced by the dislocation,but also their gradients.

In order to compare the results of the periodic super-cell calculations with those using the cylindricalmethod, it seems most reasonable to consider the pointsfor zero shear stress in the largest unit cells, for it is inthose cells that the dislocations are most isolated. Asseen in the inset to Fig. 5, E(DP)−E(SP) ranges fromabout −6.5 to −8 meV A� −1. This is in excellentagreement with the result of −6.8 meV A� −1 from thecylindrical boundary conditions. Thus, one may con-clude that the periodic supercell calculations can repro-duce a result obtained by a more complex cylindercalculation, but also allow one to observe simply thesystematic dependence of relative core energies on thestress field environment of the dislocation.

Two final points regarding the periodic supercellcalculation are worth mentioning. First, the results ofFig. 5 suggest that the small cells used by previousworkers [6,8] to determine the relative stability of theSP and DP cores may subject the dislocations to stressfield conditions that drastically affect the observedvalue of the energy difference E(DP)−E(SP). Second,it should be emphasized that the trends observed hereshould be evaluated with more accurate electronicstructure techniques, since the Tersoff potentials areprobably not accurate enough to resolve the smallenergy differences.

4. Conclusions

Using the 90° partial dislocation as an illustration,two approaches to choosing boundary conditions for

atomic-scale calculations of dislocation core energy areexamined. Issues relating to cylindrical boundary condi-tions are explored and the results compared with calcu-lations from periodic supercells. It is shown that theperiodic supercell method, applied in conjunction withelasticity theory, gives results similar to those fromcylinder calculations, but without the ambiguities asso-ciated with surface effects. The relative stability of twopossible reconstructions of the core is found to dependsystematically on the stress field experienced by thedislocation.

Acknowledgements

The support of the Director, Office of Energy Re-search, Office of Basic Energy Sciences, US Departmentof Energy under contract No. DE-AC03-76SF00098 isgratefully acknowledged.

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