Cooperative atomic motions and core rearrangement in dislocation cross slipSrinivas Pendurti, Sukky Jun, In-Ho Lee, and Vish Prasad Citation: Applied Physics Letters 88, 201908 (2006); doi: 10.1063/1.2204449 View online: http://dx.doi.org/10.1063/1.2204449 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/88/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Structure of screw dislocation core in Ta at high pressure J. Appl. Phys. 115, 093505 (2014); 10.1063/1.4867520 Atomistic study of temperature dependence of interaction between screw dislocation and nanosized bcc Cuprecipitate in bcc Fe J. Appl. Phys. 104, 083523 (2008); 10.1063/1.3003083 Analysis of cooperativity and localization for atomic rearrangements J. Chem. Phys. 121, 6689 (2004); 10.1063/1.1794653 Micropipes and the closure of axial screw dislocation cores in silicon carbide crystals J. Appl. Phys. 96, 348 (2004); 10.1063/1.1759082 Direct observation of the core structures of threading dislocations in GaN Appl. Phys. Lett. 72, 2680 (1998); 10.1063/1.121097

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Cooperative atomic motions and core rearrangement in dislocationcross slip

Srinivas Pendurti and Sukky Juna�

Department of Mechanical and Materials Engineering, Florida International University, Miami,Florida 33174

In-Ho LeeKorea Research Institute of Standards and Science, Daejeon 305-600, Korea

Vish PrasadDepartment of Mechanical and Materials Engineering, Florida International University, Miami,Florida 33174

�Received 28 February 2006; accepted 18 April 2006; published online 16 May 2006�

Atomistic study of cross slip of a screw dislocation in copper is presented using theaction-optimization numerical technique which seeks the most probable dynamic pathway on thepotential-energy surface of the atomic system during the cross-slip process. The observedmechanism reveals features of both competing mechanisms postulated in literature, i.e., theFleischer mechanism and the Friedel-Escaig mechanism. Due to cooperative atomic motions andcomplex core rearrangement during the process, the activation energies of the current cross-slipmechanism are around 0.5 eV less than the lowest ever reported in corresponding studies usingatomistic numerical techniques. © 2006 American Institute of Physics. �DOI: 10.1063/1.2204449�

Cross slip in fcc crystals is a process by which a screwdislocation changes its glide plane from one �111� plane toanother. It plays an important role in the plasticity of metalsand semiconductors. There have been two alternative mecha-nisms for cross slip proposed in literature: the Friedel-Escaigmechanism1,2 and the Fleischer mechanism.3 Continuumelasticity theory suffers from the limitation not being appli-cable near the core and the consequent difficulties caused bya core-cutoff radius. On the other hand, atomistic studieshave not conclusively demonstrated the predominance of ei-ther mechanism.4–7 Rasmussen et al.4 and Vegge6 have usedthe nudged elastic band method to study cross-slip process incopper. Their results tally with the Friedel-Escaig �FE� hy-pothesis, and the activation energy of the process is in therange of 3.0–3.3 eV. Recently, Mordehai et al.7 have per-formed a molecular dynamics study of cross slip in copperusing a dipole configuration. They concluded that the Fleis-cher mechanism dominates when the screw dislocationlength is short, and the FE mechanism is prevalent when thedislocation length is long. Bonneville et al.8 experimentallydetermined the activation energy of cross slip to be 1.2�±0.4� eV. Thus there is a huge discrepancy between theexperimental prediction and the theoretical values. Vegge etal.9 suggested that dislocation cross slip initiates at preexist-ing constrictions formed by jogs; hence the experimental ac-tivation energy does not involve the constriction formationand is consequently less than that observed in atomisticsimulations. A review on the dislocation cross slip can befound elsewhere.10

This letter describes the application of an action-optimization numerical technique to solve the cross-slipproblem. This methodology has lately been proposed tosearch, on the potential-energy surface, the most probabledynamic Newtonian trajectory between the given initial and

final atomic configurations of a system.11,12 Its pathway op-timization is built on minimizing the classical action alongwith an energy-conservation constraint and thus named theaction-derived molecular dynamics �ADMD�. Since the ini-tial and final configurations of the cross-slip problem areknown, ADMD is suitable to find a dynamic Newtonianpathway for the cross-slip problem. ADMD has already beenused for several studies of fullerene and nanotube syntheses,protein folding, and so on.13–15

Figure 1 illustrates a screw dislocation in a computa-tional cell �parallelepiped� representing the copper lattice.The bounding surface pairs of the parallelepiped are the

�110� and �1̄1̄0� planes, the �11̄1̄� planes, and the �1̄11̄� and

�11̄1�. The screw dislocation is at the center of the parallel-epiped and along the �110� direction. The length, width, andheight of the computational cell are denoted by l, w, and h,respectively. In all simulations presented in this letter, l=60b and w=60b, where b is the magnitude of the Burgersvector of a perfect dislocation in fcc copper. On the otherhand, the height h varies to change the size of the specimen.The Mishin embedded-atom potential for copper16 �termedEAM1 therein� was used. This interatomic potential givesgood structural and defect properties as the stacking faultenergy of 44.4 mJ/m2, which compares well with the experi-

a�Electronic mail: juns@fiu.edu FIG. 1. The configuration of computational cell.

APPLIED PHYSICS LETTERS 88, 201908 �2006�

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mental value for copper of 45 mJ/m2.17 In the first simula-tion, the height of the cell h is set as 15b. The atomic posi-tions in the parallelepiped with a perfect screw dislocation�Fig. 1� in the middle were optimized energetically. The co-ordinates of the boundary atoms on the bounding �111� sur-faces were set in accordance to the displacements for anelastic solution of a perfect screw dislocation at the center ofthe parallelepiped. These coordinates were fixed during theenergy minimization. The �110� direction facilitates the useof periodic boundary conditions, with the periodicity beingthe height h of the parallelepiped. Following minimizationwith these constraints, the perfect screw dislocation in themiddle of the computational cell splits into two Shockleypartial dislocations separated by a width of 8b, enclosing an

intrinsic stacking fault between them on the �11̄1̄� plane.This configuration served as the initial state for the ADMDcalculation. Using the same methodology and a slight pertur-bation, the perfect dislocation can be subtly manipulated to

split on the �1̄11̄� cross-slip plane during the energy minimi-zation. This served as the final configuration for the ADMDalgorithm. The ADMD procedure was then run to find a dy-namic pathway for the cross slip. The simulation time was0.798 ps, and 60 intermediate time-step indices �also calledimages� were used.

The results are depicted in Fig. 2 which is viewed fromthe top. Only the defective atoms along the core are shown,while all other atoms of regular lattice sites in the computa-tional cell are eliminated for clear visualization. Frame �a�corresponds to the initial configuration. As the simulation

proceeds, the two Shockley partials in the �11̄1̄� glide planeapproach each other, and the stacking fault shrinks. Frame�b� corresponds to the time-step index of 22, where the stack-ing fault width has shrunk to 4b. Frame �c� corresponds toindex 26 and shows an interesting phenomenon. The cross-slip process has begun, as indicated by the two rows of de-fective atoms in the cross-slip plane, but the two partials inthe original slip plane are still there and separated by a stack-ing fault of width 4b. Frame �d� corresponding to step 30 isthe transition state for the cross-slip process and showsstacking faults bound by two partials, both in the initial as

well as in the cross-slip plane. Thus there are two stackingfaults and four partials in the transition state. Frames �e�, �f�,and �g� represent the steps 34, 38, and 61, respectively.

Three important points are noted as follows. �1� As wehave four partials as well as a stair-rod-like dislocation at theintersection of the stacking faults, the four partials cannot beShockley partials. The net Burgers vector of these five dis-locations �four partials plus a stair rod� has to be equal to theBurgers vector of the perfect screw dislocation. Hence thestacking faults are bound by partials, which are similar to theones hypothesized by Duesbery18 as fractional dislocations.Diffuse constriction about core has also been reported bylattice-statics atomistic computations for Ni.5 �2� The cross-slip mechanism has elements of both the FE mechanism andthe Fleischer mechanism. The similarity to the FE mecha-nism is the tendency of the Shockley partials to move to-wards each other up to a minimum distance of 3b at thetransition state. The similarity to the Fleischer mechanism isthe presence of two stacking faults and a stair-rod-like de-fect. �3� This mechanism can be explained as the cooperativemovement of atoms to give the lowest energy cross-slip pro-cess. Subsequent frames complete the cross slip. The activa-tion energy is 1.2 eV. To verify the above result, the samestudy was repeated, this time with the tight binding potentialdue to Cleri and Rosato,19 which underestimates the intrinsicstacking fault energy as 21 mJ/m2, Ref. 20 but is otherwiseacceptable. The transition state for cross slip with this poten-tial was similar to the one in Fig. 2�d�, but the low stackingfault energy resulted in an extended structure, with the sepa-ration between the fractional dislocations on glide as well ascross-slip plane in the transition state being around 8b in-stead of 3b and higher activation energy of 2.5 eV.

The previous simulations involved short dislocationswith h=15b. Consequently, the entire dislocation crossslipped as a straight unit; no curvature effects were observed.The height of the computational cell was thus increased toh=50b, initial and final configurations prepared, and ADMDalgorithm applied to simulate cross slip. The physical timecorresponding to the simulation is 0.798 ps, and 40 interme-diate images were used. Figure 3 shows the various stages ofthe cross slip. The atomic structures are shown by perspec-tive view as in previous figures. The color scheme of theatoms in Fig. 3 corresponds to the centrosymmetryparameter,21 which is an indication of the defective environ-ment in the crystal. Dark red color corresponds to the mostdefective atoms, i.e., the centrosymmetry parameter is6.0 Å2, and blue corresponds to the least defective atoms ofwhich the centrosymmetry parameter is 0.25 Å2. Steps 0 and41 represent the initial and final configurations, which are

FIG. 2. The cross-slip mechanism for a short screw dislocation as viewedfrom the top.

FIG. 3. �Color online� Energy profileof the cross slip with selected interme-diate images of core structures alongthe approximated dynamic pathway.

201908-2 Pendurti et al. Appl. Phys. Lett. 88, 201908 �2006�

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just a stacking fault, in the glide and cross-slip planes, re-spectively. There is a gradual gradation in the defectivenessof the atoms across the stacking fault, with atoms being mostdefective near the center of the stacking fault and least de-fective near the edges. At the time-step index 9, the spacingbetween two Shockley partials has now reduced to 4b formost of the length and 5b near the edges, thus revealingcurvature. At step 11, the cross-slip process has nucleated.Two extra rows of blue colored atoms which belong to astacking fault in the cross-slip plane can be seen, and theseappear with the stacking fault in the original glide plane stillbeing present. At step 15, the cross slip which had nucleatednear the center has been completed in the central part of thescrew dislocation and is propagated by the two almost, butnot perfect, constriction points A and B in the figure movingaway from the center and from each other. The atomic ar-rangement near A and B is similar to that point in step 11where the cross slip nucleated, there are two stacking faults,and fractional dislocations are present.

To clarify this, the top view of step 15 near A has beenblown up with extraneous atoms removed. The traces of boththe glide as well as cross-slip planes are clearly indicated.This configuration is similar to that of Fig. 2�d�, indicatingthe presence of fractional dislocations, two stacking faults,and complicated core rearrangements near A and B. In thefollowing time steps, A and B move away from each other,completing the cross slip. The energy versus time index isalso plotted in Fig. 3. The plot shows that the energy risessharply till the cross slip nucleates and then stays constanttill the process is completed. The activation energy is around2.14 eV and, though there is no single transition state withmaximum energy due to successive cooperative atomic mo-tions, the cross-slip nucleation near step index 11 can betaken as a seminal event. To establish size independence theheight h of the cell has been raised to 70b and higher. Theactivation energy rose to and stabilized at 2.54 eV.

We have reported detailed processes for cross slip in-volving cooperative movement of atoms and the complexcore structures involving fractional dislocations and two si-multaneous stacking faults during cross-slip nucleation. Theactivation energies are 0.5 eV lower than other atomistic re-sults but still around 0.9 eV higher than experimental values.Part of this might be due to inherent limitation in our simu-lation in the application of boundary conditions. In fixing the�111� boundaries according to an elastic solution, with a per-

fect screw dislocation at the center, we have restricted thetotal defect to always remain at the geometrical center of thecomputational cell and also introduced unwanted additionalsymmetry in the process. This deficiency can be eliminatedby working with a quadrupole arrangement and introducingperiodicity in all three directions. We are working on suchcalculations. The present calculations were conducted on aparallel Linux cluster using 42 Opteron 1.6 GHz CPUs, andthe calculations of Fig. 3 took 15 days and involved around15�106 degrees of freedom. All images were made with thefreeware presented in Ref. 22.

This work was supported by the U.S. Air Force Office ofScientific Research �AFOSR�. One of the authors �I.-H.L.�acknowledges the support by the Ministry of Commerce, In-dustry, and Energy of Korea through “The R&D Project forKey Technology.”

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