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Simulations of the atomic structure, energetics, and cross slip of screw dislocations incopper

Rasmussen, Torben; Jacobsen, Karsten Wedel; Leffers, Torben; Pedersen, Ole Bøcker

Published in:Physical Review B

Link to article, DOI:10.1103/PhysRevB.56.2977

Publication date:1997

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Rasmussen, T., Jacobsen, K. W., Leffers, T., & Pedersen, O. B. (1997). Simulations of the atomic structure,energetics, and cross slip of screw dislocations in copper. Physical Review B, 56(6), 2977-2990.https://doi.org/10.1103/PhysRevB.56.2977

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PHYSICAL REVIEW B 1 AUGUST 1997-IIVOLUME 56, NUMBER 6

Simulations of the atomic structure, energetics, and cross slip of screw dislocations in copper

T. RasmussenCenter for Atomic-scale Materials Physics, Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denm

and Materials Research Department, Riso” National Laboratory, DK-4000 Roskilde, Denmark

K. W. JacobsenCenter for Atomic-scale Materials Physics, Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denm

T. Leffers and O. B. PedersenMaterials Research Department, Riso” National Laboratory, DK-4000 Roskilde, Denmark

~Received 12 December 1996!

Using nanoscale atomistic simulations it has been possible to address the problem of cross slip of a disso-ciated screw dislocation in an fcc metal~Cu! by a method not suffering from the limitations imposed byelasticity theory. The focus has been on different dislocation configurations relevant for cross slip via theFriedel-Escaig~FE! cross-slip mechanism. The stress free cross-slip activation energy and activation length forthis mechanism are determined. We show that the two constrictions necessary for cross slip in the FE cross-slipmechanism are not equivalent and that a dislocation configuration with just one of these constrictions isenergetically favored over two parallel Shockley partials. The effect of having the dislocation perpendicular toa free surface is investigated. The results are in qualitative agreement with transmission electron microscopyexperiments and predictions from linear elasticity theory showing recombination or repulsion of the partialsnear the free surface. Such recombination at the free surface might be important in the context of cross slipbecause it allows the creation of the above-mentioned energetically favorable constriction alone. In addition weobserve a strong preference for the partials to be in a glide plane parallel to the surface step. We haveperformed simulations of two screw dislocations of opposite signs, one simulation showing surface nucleatedcross slip leading to subsequent annihilation of the two dislocations. It was possible to monitor the annihilationprocess, thereby determining the detailed dislocation reactions during annihilation.@S0163-1829~97!06630-7#

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I. INTRODUCTION

The prominent role of dislocations and other lattice dfects in controlling the mechanical properties of ductile mterials is well established. Dislocation theory1 provides anunderstanding of the basic properties of single dislocatiand, to some extent, the interaction between dislocatioHowever, the interactions between many dislocations cansult in entangled or more ordered structures, which comcates the use of dislocation theory. In the other extreelasticity theory, which is the basis for dislocation theobreaks down close to the dislocation, meaning that the inactions between two dislocations at very close range cabe accurately described either.

One of the most famous examples of this deficiencydislocation theory is cross slip of a screw dislocation. Crslip is an important mechanism in plastic deformationductile materials. The onset of stage III in the stress-strcurves of single crystals,2 the minimum stable dipole heighof screw dislocations of opposite signs,3 and the saturation othe stress in cyclic constant strain amplitude experime4

are examples where cross slip is believed to play a crurole. Cross slip is a difficult problem to tackle, becausecontains both long-ranged elastic interactions between dcationsand atomistic effects due to recombination of dislcations. Until now cross slip has only been theoreticallyvestigated using methods derived from elasticity theory, tneglecting atomistic effects.

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Our approach to the problem of cross slip is nanoscatomistic simulations involving up to 156 060 atoms, thereincluding both purely atomistic effects and to some extlong-range elastic interactions. Due to the length scale ofproblem, cross slip has, to our knowledge, not beendressed by atomistic simulations before. Based on resfrom earlier nonatomistic simulations,5,6 the details of thecross-slip event are expected to be at the nanometer swhich implies simulations involving;100 000 atoms. Theuse of atomistic simulations will of course dispose of tshortcomings of elasticity theory. Other problems suchfinite-size effects and time-scale problems must be expeand dealt with. Furthermore, the choice of the interatompotential is very important. Simple pair potentials all obthe Cauchy relationC125C44 ~Ref. 7! and thus cannot beexpected to give a reliable description of elastic propertiesspecific materials. The use of a more sophisticated potenis therefore necessary. Such a potential can be derived fthe effective-medium theory,8 which provides a computationally efficient interatomic potential that also includemany-atom effects.

In this paper we will be concerned with cross slip ofdissociated screw dislocation from one close-packed plananother in an fcc metal~Cu!. We investigate bulk propertieas well as the effects of having the dislocation perpendicuto a free surface. More specifically we shall focus on diffeent possible dislocation configurations in connection wcross slip, and we determine the stress free activation enand activation length. We show that the two constrictio

2977 © 1997 The American Physical Society

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2978 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

necessary to produce cross slip in the Friedel-Escmodel9,10 are not equivalent, and that a dislocation configration with just one of these constrictions is energeticafavored over that of two parallel Shockley partials. We ashow that a free surface can act as a nucleation centecross slip, thereby facilitating the annihilation of two scredislocations of opposite signs. To check this we have pformed simulations of two screw dislocations of oppossigns and observed their mutual annihilation. The resmight serve as an atomic scale check of the approximatmade in the analytical theory10 and simulations of single dislocations made within the framework of elasticity theory,5,6

but also as input to the more mesoscopic nonatomisimulations11,12containing many~thousands of! dislocations,and to the modeling of experimentally characterized dislotion microstructures evolving during plastic deformation afatigue.13,14

The paper is organized as follows. In Sec. II we briepresent and discuss earlier work on cross slip. The methused in this work are introduced in Sec. III. The resultspresented and discussed in Sec. IV and finally we conclin Sec. V.

II. EARLIER WORK ON CROSS SLIP

Several possible cross-slip mechanisms have bproposed,15–17 but recently the model by Friedel9 andEscaig10 ~FE! has attracted most attention. In this model tdissociated screw dislocation must be recombined~con-stricted! over a short length, comparable to a few timesdissociation width, in the primary glide plane before the susequent redissociation in the cross-slip plane; see Fig. 1.FE model is derived on the basis of the line tension appromation following the method for calculating the constrictioenergy and shape by Stroh.18 The configuration of the partials far from the constriction, as determined in the line tesion approximation, converges towards the equilibrium p

FIG. 1. The Friedel-Escaig mechanism for cross slip.~a! Cre-ation of a Stroh-type constriction in the primary plane.~b! Disso-ciation in the cross-slip plane~shaded! creating two twisted con-strictions A and B. ~c! Two noninteracting constrictions. ThBurgers vectors are indicated with arrows. Due to the differcharacters of the Shockley partials, the two constrictions areequivalent. We denote the constrictionsA and B ‘‘edgelike’’ and‘‘screwlike,’’ respectively.

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allel separation as an exponentially decaying functiHence, there exist a characteristic length beyond whichtwo constrictionsA and B ~see Fig. 1! can be consideredindependent and a corresponding activation energy. Esca10

has shown that the redissociation of the screw dislocaand the motion~bow-out! in the cross-slip plane can be considered independent. It is the external stress componentsing on the opposite edge characters of the Shockley parthat control the cross slip by modifying the relative splittinwidths in the two glide planes. The driving force for thcross-slip mechanism is thus the widening of the stackifault ribbon in the cross-slip plane, and no net stress ondislocation is necessary. The process is thermally activaand the redissociation in the cross-slip plane can, in pciple, take place spontaneously without any applied stres

Experiments have been designed and carried outcopper19,20which, at least in a semiquantitative way, seemconfirm the FE model. However, other experiments onand Si,21 Cu,22 and CuAl alloys23 suggest that cross slip alsomaybe preferentially, nucleates at free surfaces.

One of the experimental problems is that it is impossito actually monitor the single cross-slip event and gaintailed information about the intrinsic characteristics of tdislocation reactions. To obtain a more detailed knowledof the structure and energetics of the cross-slipping scdislocation, computer simulations within the frameworkelasticity theory have been performed by Duesberyet al.5

and Pu¨schl and Schoeck.6 The method of Duesberyet al. isbased on numerical integration of the stresses and enealong the dislocation lines, using a relaxation method to fithe equilibrium configuration of the dislocation. The methis detailed in Refs. 5 and 24. Pu¨schl and Schoeck use thPeierls model~see, e.g., Ref. 1! to account for the atomisticstructure of the dislocation core. They approximate the dlocation shape by piecewise straight dislocation segmeand by varying the geometry of the polygon-shaped dislotion they can find a minimum energy configuration.

Although their methods are different, these authors arrat rather similar results. Both Duesberyet al. and Pu¨schl andSchoeck find that cross slip is stress aided in the senseposed by the FE model. However, the stress dependeseems to be overestimated in the original Escaig theory cpared to these works. Using the values for the relevantrameters for Cu obtained in the present work~see Table I!,the stress free cross-slip activation energy is found to be

TABLE I. Comparison between reference values of elastic cstants and intrinsic stacking-fault energy and values calculatedEMT. The values for the elastic constants are in GPa. The unitthe stacking-fault energyg is mJ/m2. The shear modulusm, calcu-lated with EMT, is that corresponding to the^110&$111% slip sys-tem.

Element C11 C44 B m g

Cu ~Ref. ! 176.2a 81.8a 137a 46, b 59 c 70 d

Cu ~org. EMT! 177 82.6 137 50 17Cu ~mod. EMT! 177 88.1 132 56 59

aRef. 38.bRef. 1, p. 462.cRef. 1, p. 430.dRef. 30.

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56 2979SIMULATIONS OF THE ATOMIC STRUCTURE . . .

eV ~Ref. 5! and 2.1 eV~Ref. 6!. In both cases as well as iEscaig theory the separation necessary for two constrictto be considered independent is of the order~50–60!b, whereb is the length of the perfect Burgers vector.

The estimated activation energies are, however, subjea considerable uncertainty for several reasons. The eladescription of a dislocation breaks down close to the discation line ~the core!. This problem is usually avoided bintroducing cutoff radii in the calculations. These cutoffs alikely to be of crucial importance to the problem of cross swhich, in some way, involves recombination of the twShockley partials. Special attention must thus be paid tocutoff radii. In considering the influence of the cutoff radand by using a refined version of the Stroh treatment oconstriction, Saada25 is led to conclude that the cross-slactivation energy can only be estimated to an order of mnitude within elasticity theory. Duesberyet al. also considerthis problem in their work. They too find a significant depedence on the cutoff radii, both quantitatively and qualitively.

Another uncertainty in the estimation of the activatienergy from elasticity theory arises from the use of the retion for the splitting of a perfect screw dislocation into twparallel Shockley partials bounding a perfect intrinsic staing fault,

d05mb2~223n!

24pg~12n!, ~1!

whered0 is the equilibrium splitting width,m is the shearmodulus,g is the intrinsic stacking-fault energy, andn isPoisson’s ratio. This relation is derived on the basis of itropic elasticity theory and can of course be expectedbreak down when applied in the context of an anisotrometal such as Cu, and it is in fact not fulfilled in our simlations. Applying relation~1! with the values forg and mobtained in this work~Table I! results in an equilibrium split-ting width of d0.4.5b. This value should be comparedthe actual equilibrium splitting width obtained from thsimulations ofd0.6b or to the experimentally determinevalue26 of d0.7b. Using the reference values forg and malso given in Table I results ind0.4b.

The calculated activation energies are usually given afunction of either the dimensionless variabled0 /b or theother dimensionless variableg/mb. However, the breakdownof the elasticity relation~1! means that for a particular system different estimates of the activation energy can betained depending on the set of variables used. To illustthis point we can consider the results by Pu¨schl and SchoeckThey calculate the activation energy as a function ofd0 /b,and if we use the value from our simulationsd0 /b56, theabove-mentioned 2.1 eV is obtained. If we instead takevalueg/mb50.0042 obtained in our simulations and uselation ~1!, we only get 1.4 eV. Similarly, the abovementioned activation energy of 2.5 eV by Duesberyet al.transforms~in this case changing fromg/mb to d0 /b) to 3.4eV. It is interesting to note that the analytical expressgiven by Escaig theory,20 when transformed into a functioof eitherd0 /b or g/mb, results in activation energies of 1.eV or 1.3 eV, respectively. Hence, there is a systematic lering of the activation energy when the dimensionless v

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able is changed fromd0 /b to g/mb. Furthermore, there isvery good agreement between the results of Escaig and tof Puschl and Schoeck, whereas the results of Duesbet al. are roughly a factor of 2 higher.

The appeal of an approach like Escaig’s lies in the fthat it provides analytical expressions for interesting propties such as activation energy and length. Even thoughproximate, such analytical expressions can provide physinsight that cannot be easily gained from numerical calcutions. Based on the analytical theory it is possible to desspecific experiments to check the theory. This has bdone,19,20 and the experimental results are in fair agreemwith the theory.

On the other hand, more accurate numerical calculatilike the ones mentioned above may provide detailed infmation about the interesting properties which cannot bepected to be accurately included in the approximate theBy suitable scaling it is even possible to achieve this infmation for a wide variety of materials. The numerical simlations can also serve as another check of the approximatmade in the analytical theory, thereby complementing th

The qualitative agreement between the analytical theand the numerical simulations indicates that the essenphysics of cross slip is well described within linear elastictheory. Quantitatively the three approaches are also inagreement, but, as we have seen, the cross-slip activaenergy is subject to rather substantial uncertainty. Ethough the nonlocal part of the interactions is well describwith elasticity theory, it must be emphasized that the overof Shockley partials and eventual recombination into a pfect screw dislocation at the constrictions is a nonelastic pnomenon, which cannot be expected to be accuratelyscribed even with the use of the most sophisticated methderived from elasticity theory.

To avoid the uncertainties resulting from the applicatiof elasticity theory there is no alternative to atomistic simlations. Such simulations may provide information aboutpartitioning of energy between elastic interactions and puratomistic effects, and possibly reveal new phenomena whcannot be dealt with in the elastic description. Atomissimulations can be seen as the last step in a successioincreasingly accurate but less general methods.

III. METHODS

A. Effective-medium theory

The effective-medium theory8 ~EMT! is an approximatetotal energy method which has been used in describinwide variety of phenomena at surfaces and for bulk meincluding dislocation generation at high strain rates,27 defor-mation of nanophase metals,28 and tribology.29 In EMT theenergy of an atomi is calculated as the energy of the atoembedded in a suitably chosen reference system withsame electronic density as in the real system and a correctaking account of the difference between the real and reence systems.

Ei5Ec~ni !1EAS~ i !. ~2!

Ec(ni) is the cohesive energy of an atom embedded inelectronic densityni from its neighboring atoms.EAS is theatomic-sphere correction which describes the change in

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2980 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

ergy when the atom is moved from the real to the referesystem. The atomic-sphere energy is represented as theference between a pair potential contribution in the realreference systems. In the present implementation of EMTreference system is chosen to be an fcc crystal. The paeters entering into the EMT potential can be found froeither first-principles calculations or from fitting to expemental values of bulk elastic and cohesive properties.8 Gen-erally EMT describes the elastic properties of metals quwell, but for Cu the energy of an intrinsic stacking faultg ismuch underestimated. In the context of dislocation reactithis low value of the intrinsic stacking-fault energy mighta serious drawback. The energy contributions from differatomic layers to the total stacking-fault energy have bcalculated30 using density functional theory for a rangetransition and noble metals. An analysis31 of these calcula-tions shows that the energetics can be understood in a sitwo-parameter model. The two parameters describe thetive contribution to the stacking-fault energy from thes andd electrons and the results of the model are in excelagreement with first-principles calculations.30 The parameterdescribing the contribution from thes electrons can be approximated by a pair potential. Furthermore, it is shown tfor the noble metals~Au, Ag, and Cu! this parameter is themain contributor to the stacking-fault energy. With thknowledge, it was decided to modify the original EMatomic-sphere potential by a small additional pair potenfitted so as to reproduce a reasonable intrinsic stacking-fenergy. The potential was chosen to be Gaussian shapedlocated around the fifth-nearest neighbor in a perfectstacking sequence. This corresponds to a distanceA(11/3)dNN wheredNN is the nearest-neighbor distance infcc crystal. As the modification is small in amplitude it wadecided to keep all the original EMT parameters unchangIn Table I we show values of relevant elastic constantsCu calculated with both the original and modified EMT ptentials. For comparison we also quote experimental vaand the value for the intrinsic stacking-fault energy obtainby the above mentioned first-principles calculation. The cculated m is the shear modulus corresponding to t^110&$111% slip system. The overall quality of the EMT potential is unaffected by the modification, and the calculavalues with the modified EMT potential are all withi;15% of the reference values.

B. Geometry of the system

In all simulations the overall geometry of the systemsbasically the same, and is shown in Fig. 2. Only the dimsions of the systems and whether the initial screw dislocais perfect or dissociated in one or two of the two possiglide planes vary. The systems contain a centered screw

location withb5 12 @110#, and may thus be seen as a stack

of ~110! planes. The screw dislocation is introduced in totherwise perfect fcc crystal by using the result for the dplacement of the atoms from linear elasticity theory,

ub5bu

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whereu is the angle between a fixed direction and the vecfrom the dislocation to the actual atom. Since the fcc sta

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ing sequence in the@110# direction is . . .ABAB. . . , this re-sults in two intertwined helix-shaped@110# planes. The twoShockley partials are introduced as screw dislocations wBurgers vectors equal to half the Burgers vector of the pfect screw dislocation, and they can be in either glide plawith variable initial separation. It is also possible to prepathe system with partials of varying separation, e.g., like aof the shapes shown in Fig. 1. In this latter case, the partare approximated with straight dislocation segments becalculating the displacements. The splittings in the two glplanes are as follows:

~111!plane:1

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In Fig. 2 we show a system consisting of 73 960 atomThe system is made up by a stacking of 80~110! planes, butmight as well be regarded as a stacking of 43 (111) planesor 43 (111) planes. The length of the dislocation~the heightof the system! is denotedl , and the width of the two$111%planesw. The two different$111% planes are not perpendicular; rather, the angle between them is 70.53°. The$111% sur-

FIG. 2. A medium-size system consisting of 73 960 atoms. Tsystem contains one centered screw dislocation parallel to the@110#direction perpendicular to the top surface. The surface step isallel to the (111) plane. The width of the two$111% planes isw59.5 nm, and the length of the dislocation~the height of thecrystal! is l 540b ~10.2 nm!. The inset shows the difference between fcc and hcp sites in the (111) plane.

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56 2981SIMULATIONS OF THE ATOMIC STRUCTURE . . .

faces are always free. The steps on the two~110! surfaces areparallel to a (111) plane, and the dislocation is located in tmiddle of the system. In Sec. IV A we always apply periodboundary conditions along the@110# direction and the posi-tion of the step is thus irrelevant. However, in Sec. IV B wremove the periodic boundary conditions and the orientaof the step becomes important. By bulk simulations~Sec.IV A ! we mean simulations with periodic boundary contions along the@110# direction, even though the$111% sur-faces are free. We will refer to simulations without any priodic boundary conditions as free surface simulations~Sec.IV B !, meaning that one or two of the~110! surfaces are nowfree.

The two $111% planes are not exactly equivalent with rspect to a given direction, e.g., the@110# direction, and thisresults in two different configurations of the Shockley ptials in the two possible glide planes. This difference isresult of the two kinds of hollow sites, i.e., the fcc site athe hcp site, in a$111% plane. For the (111) plane and the(111) plane the fcc sites are geometrically different wrespect to, e.g., the@110# direction. In Fig. 3 we show aclose-up of a (111) plane with two Shockley partials, and

FIG. 3. Zoom-in on a (111) plane with the two Shockley partials colored black and grey. The system has been sliced througmiddle, and the radius of the atoms has been reduced to eninspection of the stacking sequence. Perfect fcc stacking is sethe left and to the right of the partials. The stacking changes graally from fcc to bridge~at the partials! to hcp between the partialsThe inset shows the difference between fcc and hcp sites in(111) plane.

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is possible to inspect the detailed stacking sequence.perfect fcc stacking is to the left and to the right of thpartials and the perfect Burgers vector1

2@110# points up-wards in the figure. The atoms in fcc sites closest toviewer are located in hollow sites surrounded by three atoin the underlying plane comprising a triangle pointing to tright in the figure, whereas for the hcp sites betweenpartials the corresponding triangle points to the left. For(111) plane the fcc site is different and the underlying tangle points to the left; see Fig. 2. When the perfect scdislocation dissociates, this results in two different configrations of the Shockley partials. In the (111) plane the par-tial Burgers vectors point away from each other, whereasthe (111) plane the partial Burgers vectors point toward eaother. This can easily be understood in the following mannIn the (111) plane the sequence of the partials is dictatedthe formation of the stacking fault in between them. Tsymmetry operation that transforms the (111) plane into the(111) plane is just a rotation around the@110# direction fol-lowed by a 180° rotation around the plane normal. The larotation reverses the dislocation orientation, which methat the signs of the rotated Burgers vectors must be revetoo. In Fig. 4 we show the splittings of the perfect scredislocation into Shockley partials in the two different glidplanes.

C. Identification of dislocations

In a study like this it is of course essential that we cidentify the dislocations. For this purpose we use a dislo

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FIG. 4. Configuration of the Shockley partials in the two po

sible glide planes:~a! (111) plane,b512 @110#. ~b! (111) plane,

b512 @110#. ~c! (111) plane, b5

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2982 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

tion finding program developed by Schio”tz.32 The basic ideais to treat the dislocation as a topological defect as in discation theory. For each atom small ‘‘Burgers circuits’’ amade in the vicinity of the atom, thereby identifying thatoms close to the dislocation line. These atoms can thecolored by a visualization tool. Atoms near the surface ofsystem are also given a ‘‘Burgers-vector’’ by the prograwhich causes some of these atoms to be colored togewith the actual dislocation atoms. For details aboutmethod we refer to Ref. 32. The program has been usesimulations involving different defects, e.g., migrating graboundaries28 and high velocity dislocations.27

D. Simulation methods

The simulations are either molecular-dynamics~MD!simulations at a finite temperature or energy-minimizatsimulations at zero temperature. The simulations in SIV B 3 were performed using the Langevin algorithm, whea small fluctuating force and a friction term stabilize ttemperature at 580 K. All other finite-temperature simutions were performed at room temperature~RT! using theVerlet algorithm after the temperature had been stabilizedan initial application of Andersen thermalization. Finally walso use direct minimization of the energy by a moleculdynamics~MD! minimization algorithm~MDmin!. The MD-

min algorithm performs a Verlet MD time step and zerosmomentum whenever the dot product between the momtum and the force is,0, thereby bringing the system to aenergy minimum. The length of an MD time step5.4310215 s. For details about the MD algorithms and temployed MD simulation tool and visualization tool, we rfer to Ref. 33.

IV. RESULTS AND DISCUSSION

A. Bulk simulations

1. Splitting width

As a starting point the equilibrium splitting widthd0 ofthe dissociated screw dislocation was determined. Thisserve as yet another check of the reliability of the interatompotential. d0 was found by varying the widthw and thelength l of the computational cell, using both direct minmization of the energy and RT simulations. Periodic bouary conditions were applied in the@110# direction~along thedislocation line! and a perfect screw dislocation as welltwo Shockley partials of variable initial splitting width iboth $111% planes were used as initial configurations. Incasesd05761 k, wherek is the length of a1

4^112& vector.At room temperature this corresponds tod051.560.2 nm, invery good agreement with the experimental result26 ofd051.860.6 nm. Using the elastic constants from Tableand the isotropic elasticity relation~1! to determine the equilibrium splitting width results ind051.160.1 nm.

A prominent feature of the splitting of the perfect scredislocation is the in-plane smearing of the Shockley partwhich strongly influences the stacking between them. Tsmearing is also observed in computer simulations34–36 ofedge dislocations in Cu. In Fig. 3 the radius of the atomsbeen reduced to allow inspection of the stacking sequeand it is easily seen how the partials invade the ribbon

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tween them. The stacking changes from fcc sites to theand to the right of the partials to bridge sites at the partand a few columns of hcp sites between the partials. In pciple it could be possible to fit the displacements to a Peimodel of the dislocation core, and obtain a value for tdislocation ‘‘width.’’ The width of a single Shockley partiacan also be roughly estimated from Fig. 3 to be;4 k.0.9nm. The effect of the in-plane smearing is to preventfaulted ribbon between the partials from being a perfecttrinsic stacking fault, but also to increase the total widththe fault compared to the separation of the partials. Howethe in-plane smearing is only;60% of the total splittingwidth, indicating that the partials are well separated. If aisotropy can be neglected, the application of relation~1!should be valid. The equivalent relation1 for the splitting of aperfect edge dislocation into two Shockley partials predithat separation to be;2.5 nm, which should be comparedthe experimental result26 of 3.860.6 nm. Using anisotropicelasticity theory37 with a stacking-fault energy of 70 mJ/m2

results in splitting widths for a screw and an edge dislocatof 1.3 nm and 4.1 nm, respectively, in better agreement wexperiment and our result for the screw dislocation. Henthe incapability of relation~1! of reproducing the experimentally found splitting widths must be attributed to the fact ththis relation is derived from isotropic elasticity theory.

2. Cross-slip activation energy

The models based on elasticity theory estimate the crslip activation energy to approximately 2 eV. This energymuch too high to produce a rate of cross slip which cansimulated with ordinary MD simulations, where the typictime scale of a simulation is 10–100 ps. However, it is psible to simulate different optimized static configurationsthe dislocation, and thereby obtain information aboutstructures and energies. To determine the stress-free cslip activation energy in the FE model we divide the problein two. The reason for this lies again in the length scalethe problem. The separation between the two constrictionthe transition state is expected to be 55b ~14.0 nm! whichimplies the use of a computational cell at least twice tlength. We therefore decided to make simulations of isolaconstrictions in computational cells of varying sizes. As cbe seen in Fig. 1, the two constrictions necessary in themodel are not equivalent. At constrictionB the partials as-sume screwlike character, whereas at constrictionA the par-tials are more like edge dislocations. We shall distingubetween these two constrictions and refer to them as ‘‘scrlike’’ and ‘‘edgelike,’’ respectively.

We shall define the energy of a constriction as the diffence in minimized energy of the constricted dislocation athe minimized energy of two parallel Shockley partials of tsame length,l . The procedure of determining the constrition energy then consists of two steps: First, we minimizeenergy of two reference systems, each containing a paiparallel Shockley partials in the (111) plane and the (111)plane, respectively. Second, we extract the four top and fbottom~110! planes of the relaxed configurations. Taking ttop planes and bottom planes from different relaxed reence systems and making a sandwich with a system coning an initially constricted~and twisted! dislocation gives usone constriction. Taking the other top and bottom set w

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56 2983SIMULATIONS OF THE ATOMIC STRUCTURE . . .

another initially constricted system gives us the second cstriction. It is important to keep the four top planes andfour bottom planes static; i.e., the atoms are not allowedmove during the simulation. By using the top and bottofrom the already relaxed systems as static atoms, we areto mimic a semi-infinite bulklike dislocation on either sidethe constricted part. The choice of four static planes ensthat the dynamic atoms do not feel the vacuum abovebelow the static layers. We have performed simulations wdislocation lengths in the range~30–60!b ~7.7–15.3 nm! andwith the width perpendicular to the dislocation in the ran7.7–14.8 nm. The smallest system consisted of 36 750 atand the largest system consisted of 156 060 atoms. Calcing the two constriction energies for each system size trequires energy minimization of 4 systems, including thelaxation of the two reference systems.

It is obvious that the constraint of equilibrium separatidistance between the partials imposed by the static lamakes the energy of the constrictions depend on the leof the computational cell,l . This dependence is, howeveexpected to vanish asl is increased, thereby making thconstriction energies converge. To check this we performsimulations with a fixed width ofw511.3 nm andl varyingfrom 30b to 60b. The results of these simulations are shoin Fig. 5. The convergence of the energy is clearly seen,we can estimate a minimum length, i.e., a minimum distabetween noninteracting constrictions, of;50b. To check thedependence of the energies on the width of the systemperformed simulations with a fixed lengthl 530b andwidths in the range 7.7–14.8 nm. These simulations shono dependence of the energies on the widths for widlarger than 10 nm. For systems with widths less than 10we observed a small increase in the energies. Based onobservations we can conclude that the minimum separabetween independent constrictions is;50b, and that thecross-slip activation energy is 2.7 eV. This result for t

FIG. 5. Total energy in eV of the two constrictions as a functiof dislocation lengthl . The width of the systems is kept fixedw511.3 nm. The energy converges forl ;50b.

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activation energy agrees well with the result of Duesbet al.5 mentioned in Sec. II, and it is somewhat higher ththe estimates of Escaig20 and Pu¨schl and Schoeck.6 Also, theminimum separation between noninteracting constrictionin perfect agreement with all three estimates based on eticity theory. For clarity we only show the sum of the costriction energies in Fig. 5. Taken separately the constricenergies converge similarly to the sum, but they show a vinteresting difference: The energy of the edgelike consttion is 3.8 eV, whereas the energy of the screwlike consttion is negative, 21.1 eV. None of the mentioned workbased on elasticity theory treat the constrictions separamaking a detailed comparison impossible.

The most striking result is the negative energy of tscrewlike constriction. This indicates that the configuratiwith the dislocation in the constricted and twisted screwlstate is energetically more favorable than just two paraShockley partials in one glide plane. Intuitively it is obviouthat the screwlike constriction must be energetically pferred over the edgelike constriction. In dislocation theoryis well established that the self-energy of a screw dislocais lower than that of an edge dislocation. It is also know1

that the interaction energy between two perfect screw dications inclined at an angle of 45° is zero. The anglestween the partials involved here are not exactly 45°, but thare not very far from that. Finally the constriction reducthe area of the stacking fault, thereby lowering the energyboth constrictions.

The atomistic approach enables an even more detainvestigation of the energetics than that described above.possible to plot the energy of the systems layer by laythereby gaining insight into the ‘‘energy distribution’’ alonthe dislocation line. The decomposition of the total enerinto contributions from individual atoms is of course, in priciple, not unique. However, within the EMT there is a natral way of doing this.8 The energy of anAB ~110! plane pairis defined as the difference in energy between anAB pair inthe relaxed reference systems containing parallel partialsthe actualAB pair in the constriction system. TheAB pairsin the reference systems all have the same energy, whethe energy of theAB pairs in the constricted systems aexpected to vary along the dislocation line. In Fig. 6 wshow the results of such a projection of the energy ontoindividual AB pairs for the systems consisting of 156 06atoms havingw511.3 nm andl 560b ~15.3 nm!. We onlyshow the energy of the dynamic atoms. In this type of pthe qualitative difference between the constrictions is vapparent. Returning to the intuitive considerations about cstriction energies given above, we can try to explain thehavior. For the screwlike constriction there is a drop in eergy towards the center of the system. This mightexplained by the lowering of the self-energy of the partialsthey assume a more screwlike orientation, the reductioninteraction energy also due to acquired screw orientatand, finally, the reduction in stacking-fault area. For tedgelike constriction only the reduction in stacking-fault arwould tend to reduce the energy. The acquirement of a medgelike orientation would, according to dislocation theocause the energy to increase. Common to the two constions is, however, a peak in energy located around the ceof the system. A probable explanation is that atomis

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2984 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

effects begin to play a role. Such effects could be causedthe increasing overlap of the cores of the partials andeventual recombination into a perfect screw dislocationthe center of the system. The extension along the dislocaline of this ‘‘atomistic domain’’ can be estimated from thfigure to be approximately 6b. This rough estimate fits nicelywith the pictures~see Fig. 7! of the constrictions, where thextension of the recombined domain is also about~5–6!b byvisual inspection.

In Fig. 7 we show relaxed configurations of both constrtion types. The systems are the same as those in Fig. 6.two constriction types cannot be distinguished visuaFrom pictures similar to Fig. 7 it is possible to estimate tlength over which the partials constrict simply by countithe number of planes where the partials are nonparallel.counting was done using pictures with reduced atomic rand no coloring of the atoms, i.e., not using Fig. 7. For bconstrictions the result is~25–30!b ~6.4–7.7 nm!. Decreas-ing the lengthl of the dislocation to 40b does not changethe visual nonparallel lengths significantly. These nonparalengths are representative for all the systems we have uA nonparallel length of;30b indicates that this is the minimum reasonable length of the dislocations in simulationsthis kind.

The negative energy of the screwlike constriction raisome interesting questions about the behavior of a scdislocation: For example, will screw segments in a dislotion network dissociate into Shockley partials in two gliplanes connected by a screwlike constriction? And howthe mobility of a screw dislocation affected by the presenof a screwlike constriction? Another interesting aspect isapparent lack of experimental evidence of the existence

FIG. 6. Energy contribution from individual~110! plane pairs tothe total energy of the two kinds of constrictions. Upper curedgelike constriction. Lower curve: screwlike constriction. Tlength of the dislocations isl 560b ~15.3 nm! and the width of thesystems isw511.3 nm. The systems consist of 156 060 atomOnly the contributions from the dynamical atoms are shown. Nothe negativeenergy of the screwlike constriction.

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screwlike constrictions. In Sec. IV B we show how cross scan be initiated by the creation of a screwlike constrictinear a free surface. A direct consequence of such surnucleated cross slip is the annihilation of two screw dislotions of opposite signs.

:

.e

FIG. 7. Relaxed configurations of both constriction types. Tsystems are the same as those in Fig. 6. The systems have bein halves and rotated to display the dislocation in both glide plan

screwlikee planes.

56 2985SIMULATIONS OF THE ATOMIC STRUCTURE . . .

FIG. 8. Four different systems with one free~110! surface after direct energy minimization:~a! (111) plane, static atoms in the bottomof the crystal.~b! (111) plane, static atoms in the top of the crystal.~c! (111) plane, static atoms in the bottom of the crystal.~d! (111)plane, static atoms in the top of the crystal. The dislocation has cross slipped in the bottom of the crystal, thereby creating aconstriction. The differences between the two glide planes are due to the different orientation of the surface step relative to the glid

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B. Free surface simulations

1. One dislocation: One free surface

Experiments21–23have shown that cross slip of screw dilocations also, maybe preferentially, takes place at freefaces. The effect of a free surface on the configuration odissociated dislocation is very different from usual bulk bhavior. To investigate the role of a free surface, Hazzledand co-workers23 performed transmission electron microcopy ~TEM! experiments on low-stacking-fault-energy Cualloys and used elasticity theory to address the problem thretically. According to the theory the free surface actssuch a way as to modify the interactions between the parin two ways. In the vicinity of a free surface the partiaattract and they tend to rotate towards screw character. Fdissociated screw dislocation running through a slabemerging at free surfaces on both sides of the slab, theseeffects will compete on one side of the slab and reinforcethe other. The experimental results showed good agreemwith the predictions of the theory with a preference of tpartials to acquire screw character. In order to investigaterole of a free surface qualitatively, we have made simulatiwhere the dislocation is perpendicular to one or two f~110! surfaces.

The simulations with just one free surface were performin the same fashion as the simulations containing a cons

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ad

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ese

dc-

tion. By removing the periodic boundary conditions from talready relaxed configuration containing two parallel partiand making, e.g., the four top~110! layers static, it is pos-sible to mimic a semi-infinite rod with one dissociated scrdislocation ending at a free~110! surface in the bottom of thecrystal. Having the dislocation initially in either of the twpossible glide planes gives four different systems with ofree ~110! surface. The two systems with two free~110! sur-faces were generated simply by removing the perioboundary conditions from the already relaxed reference stems, thereby creating clusters with a single dislocatemerging on both free~110! surfaces. All the systems haw59.5 nm andl 540b ~10.2 nm! and consisted of 73 960atoms. We performed direct energy minimization as wellRT simulations followed by a quench byMDmin energyminimization.

Figure 8 shows the four systems with one free~110! sur-face after the direct energy minimization. For the two sytems with the partials initially parallel in the (111) plane andwith static atoms in the bottom of the crystal@Fig. 8~a!# or inthe top of the crystal@Fig. 8~b!#, the predicted effects of thefree surface are clearly seen. In the (111) plane the partiaBurgers vectors point toward each other as seen in Fig. 4~b!.For the system with static atoms in the top, one of the Sholey partials bows out away from the other thereby acquir

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2986 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

more screw character, whereas the other partial sstraight. The tendency to acquire more screw charaseems to be stronger than the tendency for the partialattract, in agreement with the experimental findings.23 How-ever, it is not easy to judge which of the effects is the strger because of the interaction with the image dislocatiparallel to the real dislocation, which would also tendattract the partials to the free (111) surfaces. With the statiatoms in the bottom of the crystal, the effect of the frsurface is to make the two partials recombine into a perscrew dislocation in the top of the crystal. There is no signredissociation of the perfect screw dislocation in the (111)plane. The recombination in the (111) plane is a result oone or both of the effects of a free surface predictedelasticity theory, and together these two effects are stronthan the interaction with the parallel image dislocatioAgain one of the partials bows out to acquire more scrcharacter, whereas the other partial stays straight.

The asymmetrical configurations of the partials nearfree surfaces in Figs. 8~a! and 8~b! must be attributed to theasymmetrical positions of the partials with respect tosurface step. Furthermore, the two$111% glide planes are noequivalent with respect to the step on the free~110! surfaces.The surface step is located in a (111) glide plane; see Fig. 2When the partials are in this plane the step is parallel topartials. However, when the partials are in a (111) glideplane the surface step is located between the partials aangle of 70.53°.

For the two systems with the Shockley partials initiaparallel in the (111) plane, the effect of the free surfacetherefore somewhat different. In this plane the partial Bgers vectors point away from each other; see Fig. 4~a!. Thesystem with static atoms in the top of the crystal@Fig. 8~d!#corresponds to Fig. 8~a!. The two partials recombine as expected into a perfect screw dislocation in the bottom ofcrystal, but in this case we observe an additional splittingthe perfect screw dislocation in the (111) plane. The redissociation in the (111) plane, not visible in Fig. 8~d!, gener-ates a screwlike constriction on the dislocation and allothe partials to be in a glide plane parallel to the surface sApparently there is no or a very-low-energy barrier for thkind of surface nucleated cross slip. For the system wstatic atoms in the bottom@Fig. 8~c!# no net effect of the freesurface is observed. The partials stay straight, even vclose to the free~110! surface. The tendency to rotate twards screw character seen in Fig. 8~b! seems to be compensated by an attraction to the surface step.

For all four systems we have also performed RT simutions followed by a quench byMDmin minimization. Thesesimulations show no qualitative differences from the abomentioned simulations. The only difference is for the systwith static atoms in the top of the crystal and the partiinitially in the (111) plane which shows surface nucleatcross slip. The redissociation in the (111) plane in the bot-tom of the crystal is much more pronounced, with the ptials in the lower part of the crystal in the (111) plane adopt-ing a configuration almost identical to that of Fig. 8~a!.

2. One dislocation: Two free surfaces

In order to obtain more information about the influencefree surfaces on the configuration of the partials, we a

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performed RT simulations followed by a quench byMDminminimization on systems with two free~110! surfaces; i.e.,the systems were clusters with the partials initially in eiththe (111) plane or the (111) plane. From the simulations osystems with one free~110! surface, we expect the first system to recombine into a perfect screw dislocation in theof the crystal and to split even further than the bulk splittiin the bottom of the crystal. For the latter system we expthe system to perform cross slip in the bottom of the crysbut it is not clear whether the cross slip will be completewhether the system prefers a configuration with a screwconstriction. The system with the partials initially in th(111) plane behaves as expected. The partials recombinthe top of the crystal with no sign of redissociation in t(111) plane and they repel in the bottom of the crystal. Af;11 ps~and before the quench! the dislocation is located ofcenter in the crystal, due to the attraction to the image discation; see Fig. 9~a!. The system with partials initially in the(111) plane performs a complete cross slip to the (111)plane. The partials recombine and redissociate in the botof the crystal, thereby creating a screwlike constriction. Tscrewlike constriction ascends the crystal, making the ptials in the top gradually change their glide plane from t(111) plane to the (111) plane. The cross slip is completein ;11 ps. Figure 9~b! shows an intermediate stage in thcross slip and Fig. 9~c! shows the final relaxed configuratioof the cross-slipped dislocation after the quench. Figures~a!and 9~c! are very similar to TEM pictures of dissociatescrew dislocations in thin foils in Ref. 23.

To check the significance of the orientation of the surfastep with respect to the partials we performed simulatiosimilar to the two simulations with two free surfaces, but thtime with the step parallel to a (111) plane; i.e., the surfacestep has been rotated compared to the step seen in FiThe results are equivalent to the simulations with the origistep orientation. Now the system with the partials initiallythe (111) plane cross slips to the (111) plane, while thesystem with the partials initially in the (111) plane, which isnow parallel to the step, does not. The two final configutions have the partials recombined in the bottom of the crtal and split in the top of the crystal.

These simulations show that there is a strong preferefor the dissociated screw dislocation to be in the glide plaparallel to the surface step, and that the dislocation will ada configuration with perfect screw dislocation in one end atwo nonparallel Shockley partials in the other end. It is tdesire of the partials to be in a plane parallel to the surfstep and not the possibility of making a screwlike constrtion, which controls the surface-nucleated cross slip, athere will be an energy barrier for surface nucleated crslip away from the glide plane parallel to the surface ste

Finally we performed simulations on systems with diffeent step orientations in either end of the crystal. With tpartials initially in the (111) plane or the (111) plane thisgives four possibilities. For the two systems with the stparallel to the (111) plane in the top and the step parallelthe (111) plane in the bottom we expect the dislocationadopt a configuration with a screwlike constriction. This cofiguration will allow the partials to be parallel to the stepboth ends of the crystal. However, a similar configurati

emg fault

56 2987SIMULATIONS OF THE ATOMIC STRUCTURE . . .

FIG. 9. Room-temperature simulations with two free~110! surfaces.~a! Partials initially in the (111) plane. The partials recombine in thtop of the crystal and are attracted to the free (111) surface.~b! Partials initially in the (111) plane. The dislocation has cross slipped frothe (111) plane to the (111) plane in the bottom half of the crystal, thereby creating a screwlike constriction. The trace of the stackinbetween the partials of the non-cross-slipped part of the dislocation is seen in the upper half of the crystal.~c! Partials initially on the(111) plane. The cross slip is complete, and the dislocation is entirely in the (111) plane. The configuration has been quenched byMDminminimization.

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with the partials parallel to the step in both ends, whenstep orientations have been switched, would result in cation of an edgelike constriction. The simulations with tstep in the top parallel to the (111) plane and the step in thbottom parallel to the (111) plane showed the expected bhavior. Both systems create a configuration with a screwconstriction and the partials parallel to the step in either eThe system with the partials initially parallel in the (111)plane performs cross slip in the top of the crystal whereassystem with partials initially parallel in the (111) plane per-forms cross slip in the bottom of the crystal. For the twsystems with the switched step orientation the results

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different. The partials stay in their initial glide plane, adoping a configuration with a perfect screw dislocation in oend of the crystal and two nonparallel Shockley partialsthe other end of the crystal. There are no signs of crossleading to configurations with an edgelike constriction athe partials parallel to the steps in either end.

These simulations demonstrate the influence of a f~110! surface on the configuration of a dissociated scrdislocation perpendicular to that surface. We have qualtively confirmed the predictions by elasticity theory of theffect of a free surface. In addition we have observedstrong preference for the partials to be in the glide plaparallel to the surface step.

e of the

theving the

2988 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

FIG. 10. Snapshots from the simulation showing annihilation of two screw dislocations of opposite signs.~a!–~d! (111) planes.~a! The

dislocation withb512@110# has performed a cross slip in the top of the crystal, thereby creating a screwlike constriction. The presenc

dislocation withb512@110# in a (111) plane and the non-cross-slipped part of theb5

12@110# dislocation also in a (111) plane is clearly

seen as two vertical lines of stacking fault marked SF separated by seven$111% plane spacings.~b! A new Shockley partial~vertical! has beencreated to the right in the (111) plane. The Shockley partial to the left is the originally cross-slipped1

6@121# and the small inclined partconnecting these two is the remains of the1

6@211# Shockley partial.~c! The newly created Shockley partial16@121# to the right reacts with

the 16@121# Shockley partial and they annihilate.~d! Annihilation in the top of the crystal, leaving a small stacking-fault loop comprising

16@211# and the1

6@121# Shockley partials. The stacking fault loop moves down through the crystal and disappears at the bottom, leacrystal defect free.

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3. Annihilation of two screw dislocations

An interesting parameter which is believed to be closrelated to cross slip is the minimum stable dipole heightscrew dislocations of opposite signs. Experiments3 haveshown this length to be 50–500 nm in Cu. Screw dislotions of opposite signs in different glide planes closer ththis minimum dipole height are believed to cross slip aannihilate. Due to the length and time scale, this problemnot suited for ordinary atomistic MD simulations. Howeveit might be possible to obtain insight into the cross-smechanism of a single screw dislocation, by performing Msimulations of systems containing two screw dislocationsopposite signs at very close range. In order to speed upsimulations, it was decided to use a high temperature~580 K!and rather small systems withw58.6 nm andh550b ~12.7nm!. The systems consisted of 76 050 atoms. The two sc

dislocations with Burgers vectors6 12@110# were initially in-

troduced in the system as two perfect screw dislocation

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different glide planes separated by only;2.2 nm. We per-formed a simulation with periodic boundary conditionalong the@110# direction and a simulation without periodiboundary conditions, i.e., with two free~110! surfaces.

The simulation with periodic boundary conditions showno sign of cross slip. The two dislocations dissociate in tparallel (111) planes as in Figs. 4~a! and 4~c! separated byseven$111% plane spacings, with splitting widths fluctuatinaround an average value of;1 nm. The small value of thesplitting width must be attributed to the presence of the otdislocation, because no significant temperature effectobserved in the simulations of just one dislocation in SIV A 1. Occasionally the partials in the same glide plawere so close that the dislocation might be thought ofrecombined. However, as mentioned for the partials in SIV A 1, there is an in-plane smearing of the dislocatiowhich confines it to a particular glide plane and inhibits croslip. The simulation used 4600 time steps corresponding

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56 2989SIMULATIONS OF THE ATOMIC STRUCTURE . . .

;25 ps. In the context of cross slip this is a very shsimulation, and it is in no way possible to rule out the posibility of some kind of bulk cross slip on the basis of thsimulation.

To investigate the role of free surfaces in this kindsystem, we removed the periodic boundary conditions althe @110# direction from the above-mentioned system. Thproduced cross slip of one of the dislocations and ledannihilation of the two screw dislocations through successdislocation reactions. It was possible to monitor the detadislocation reactions, thereby enabling exact specificatiothe different stages leading to the annihilation. Whenperiodic boundary conditions were removed, the two scrdislocations were split into Shockley partials in two paral(111) planes separated by seven$111% plane spacings. Theannihilation of the two screw dislocations is initiated at ttop of the crystal, and proceeds downwards through the ctal in the following manner. The dislocation wit

b5 12@110# performs cross slip to the (111) plane in the top

of the crystal, thereby creating a screwlike constrictionthe dislocation; see Fig. 10~a!. The Shockley partial16@211#, glissile in the (111) plane, is attracted to the Shocley partial 1

6@121# glissile in the (111) plane, and the twopartials react to produce a sessile stair-rod dislocation:1

1

6@211#~111!1

1

6@121#~111!→

1

6@110#. ~5!

The stair-rod dislocation, not shown in Fig. 10, is locatedthe intersection of the (111) plane containing the

b5 12@110# dislocation and the (111) plane containing the

newly cross-slipped part of theb5 12@110# dislocation.

Hence, the stair-rod dislocation is in the same (111) plane asthe 1

6@211# Shockley partial@see, e.g., Fig. 4~a!# and attractsthis:

1

6@110#1

1

6@211#~111!→

1

6@121#~111! . ~6!

The resultant Shockley partial16@121# is glissile in the same

(111) plane as the cross-slipped part of theb5 12@110# dis-

location; see Figs. 10~b! and 10~c!. The newly generatedShockley partial reacts with the remaining1

6@121# Shockleypartial in the (111) plane, and the two partials annihilate:

1

6@121#~111!1

1

6@121#~111!→0 . ~7!

The reactions create a stacking fault loop in the (111) planewhich moves down through the crystal while the annihilatitakes place; see Fig. 10~d!. Eventually the loop reaches thbottom of the crystal and disappears, leaving the crystalfect free. Hence, once one of the dislocations has perforcross slip creating a screwlike constriction, there is no overy low-energy barrier for the annihilation.

V. CONCLUSION

In this paper we have addressed the problem of crossin Cu with a purely atomistic method. The result for th

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minimum separation between noninteracting constrictionin perfect agreement with results obtained from elastictheory. Our result for the stress-free activation energy inFriedel-Escaig cross-slip mechanism is in very good agrment with an earlier nonatomistic approach,5 but it is some-what higher than other estimates10,6 derived from elasticitytheory.

Our atomistic simulations also show that the two constrtions necessary in the Friedel-Escaig cross-slip mechanare not equivalent, and that one of them, the screwlike cstriction, is energetically favored compared to two paraShockley partials. The nonequivalence of the two consttions is a fact not usually appreciated in the literature. Tdifferences have been investigated, qualitatively and quatatively.

As we have pointed out, the activation energies obtainfrom elasticity theory are subject to substantial uncertaidue to the break down of the isotropic elasticity relatidescribing the splitting of a perfect screw dislocatioWhether the overall quantitative agreement between themistic and the elastic approaches is an indication of a deeconcordance or merely fortuitous can only be resolveddetailed elastic work on isolated constrictions.

The effect of having the dislocation perpendicular tofree surface has been investigated, and surface-nuclecross slip observed. The important feature in surfanucleated cross slip is the possibility of creating a screwlconstriction without the accompanying edgelike constrictneeded for bulk cross slip. On the other hand, we haveobserved a strong preference for the partials to be in a gplane parallel to the surface step, meaning that there isenergy barrier for cross slip into the glide plane not parato the surface step. Hence, more quantitative work onenergetics of different dislocation configurations close tofree surface needs to be done before we draw conclusabout the role of surface-nucleated cross slip.

In a simulation of two screw dislocations of opposisigns, surface-nucleated cross slip initiated the annihilawhich proceeded via successive energetically favorable rtions. The atomistic approach allowed monitoring of the dtailed dislocation reactions and thereby exact specificatiothe intermediate stages leading to the annihilation.

The role of cross slip in different macroscopic phenomesuch as fatigue and plastic deformation is today well estlished. However, the present understanding of the intrinproperties of cross slip is still rather nebulous. The resuobtained in the present work may help to establish a beunderstanding. Altogether we may conclude that atomisimulations are well suited for problems involving disloction interactions at the nanoscale, where the use of methbased on elasticity theory is questionable.

ACKNOWLEDGMENTS

We are grateful to J. Bilde-So”rensen for drawing our attention to the experiments involving free surfaces, andthank Jakob Schio”tz for providing the code for the dislocation finding program. We also thank Jim Sethna for helpdiscussions. Center for Atomic-scale Materials Phys~CAMP! is sponsored by the Danish National Resea

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2990 56RASMUSSEN, JACOBSEN, LEFFERS, AND PEDERSEN

Foundation. The present work was in part financed by TDanish Research Councils through Grant No. 9501775was done as a collaboration between CAMP and the Eneering Science Center for Structural Characterization

tal

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Modelling of Materials at the Materials Research Depament, Riso”. K.W.J. also acknowledges support from the UDepartment of Energy through Grant No. DEFG088ER45364.

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