Dislocation junctions as barriers to threading dislocation migrationSiu Sin Quek, Zhaoxuan Wu, Yong-Wei Zhang, Yang Xiang, and David J. Srolovitz Citation: Applied Physics Letters 90, 011905 (2007); doi: 10.1063/1.2426971 View online: http://dx.doi.org/10.1063/1.2426971 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Basal plane slip and formation of mixed-tilt boundaries in sublimation-grown hexagonal polytype silicon carbidesingle crystals J. Appl. Phys. 92, 778 (2002); 10.1063/1.1484229 Dislocation glide and blocking kinetics in compositionally graded SiGe/Si J. Appl. Phys. 90, 2730 (2001); 10.1063/1.1389333 Effects of doping impurity and growth orientation on dislocation generation in GaAs crystals grown from the melt:A qualitative finite-element study J. Appl. Phys. 88, 2295 (2000); 10.1063/1.1287600 Evolution of microstructure and dislocation dynamics in In x Ga 1−x P graded buffers grown on GaP bymetalorganic vapor phase epitaxy: Engineering device-quality substrate materials J. Vac. Sci. Technol. B 17, 1485 (1999); 10.1116/1.590779 Novel dislocation structure and surface morphology effects in relaxed Ge/Si-Ge(graded)/Si structures J. Appl. Phys. 81, 3108 (1997); 10.1063/1.364345

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Dislocation junctions as barriers to threading dislocation migrationSiu Sin Queka� and Zhaoxuan WuInstitute of High Performance Computing, 1 Science Park Road, 01-01, The Capricorn,Singapore Science Park II, Singapore 117528, Singapore

Yong-Wei ZhangDepartment of Materials Science and Engineering, National University of Singapore,10 Kent Ridge Crescent, Singapore 119260, Singapore

Yang XiangDepartment of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay,Kowloon, Hong Kong

David J. SrolovitzDepartment of Physics, Yeshiva College of Yeshiva University, New York, New York 10033

�Received 1 September 2006; accepted 29 November 2006; published online 3 January 2007�

Level set simulations of dislocation dynamics in biaxially strained, heteroepitaxial films revealinteresting kinetic and thermodynamic mechanisms for blocking the migration of threadingdislocations. Two dislocations on the same or on intersecting slip planes may react to form athreading dislocation segment that does not glide under the influence of the misfit strain. In thecoplanar case, a kinetic barrier exists that slows down dislocation migration. For the reactioninvolving dislocations on intersecting planes, an energetic barrier impedes other advancingdislocations. These barriers create significant and frequent impediment to threading dislocation flow,resulting in pileups and high threading dislocation densities. © 2007 American Institute of Physics.�DOI: 10.1063/1.2426971�

High threading dislocation densities in heteroepitaxialthin films are often associated with dislocation pileups.Threading dislocations have an adverse effect on many ma-terial properties, serving as easy diffusion paths for dopantsor impurities or as recombination centers that diminish car-rier mobility in electronic devices.1,2 Careful control of epi-taxial processes can be used to limit dislocation pileups inthin films. Pileups can be caused by sessile dislocations, sur-face crosshatch, and by other defects that impede the motionof threading segments. In this letter, we describe two mecha-nisms by which dislocation migration can be impeded as aresult of a dislocation reaction within the film.

Fitzgerald and co-workers showed how surface cross-hatch, formed by inhomogeneous strains in the film, can im-pede dislocation flow and results in dislocation pileups.3,4

Currie et al.5 also demonstrated that removal of the cross-hatch surface morphology by chemical mechanical polishingcan be used to produce a very low threading dislocation den-sity �TDD� film by freeing “trapped” dislocations to glide outof the semiconductor wafer or annihilate with one another.Kim et al.6 reported the presence of microstructural orbranched defects that appear to pin threading dislocationsand cause dislocation pileups in In�Ga1−�P/GaP systems.Similar dislocation pileup mechanisms have also been ob-served in the Ge�Si1−� /Si systems.5,7,8 Though McGill et al.9

characterized these branched defects, their origin and themechanisms by which they form are not well understood.Experimental images of these branched defects6,9 indicatethat branching microstructural defects often span the filmthickness. Our dislocation dynamics simulations routinely re-produce such barriers for dislocation migration that can ef-fectively impede dislocation flow within heteroepitaxial

films and resemble the branched defects observed in experi-ment.

The barriers form as a result of favorable reactions be-tween a pair of dislocations with different Burgers vectors.For the same pair of Burgers vectors involve in the reaction,there can be two different configurations, depending onwhether they occur on coplanar or intersecting slip planes.While both configurations result in different types of barri-ers, both share the common property that the biaxial misfitstrain is normal to the slip plane and therefore cannot drivetheir glide. Nonetheless, the barriers resulting from the reac-tion is glissile and can move under the influence of nonmisfitstresses, such as the stress fields of other dislocations. Hence,these barriers are fundamentally different from a Lomer-Cottrell junction or lock. In the case of intersecting slipplanes, the triple junction that results from the barrier �junc-tion� formation is thermodynamically pinned, hence provid-ing further impediment to other approaching dislocations.Such barriers can, therefore, cause dislocation pileups andhigh TDD in heteroepitaxial films.

To effectively simulate dislocation motion and interac-tions within a heteroepitaxial thin film, we use a level setapproach.10–16 The position of the dislocation lines ��t� arerepresented by the intersection of the zero contours of twolevel set functions, ��x ,y ,z , t� and ��x ,y ,z , t�, defined inthree dimensions �i.e., ��x ,y ,z , t�=��x ,y ,z , t�=0�. Weevolve the level set functions forward in time using a pair ofconvection equations driven by a common velocity field thatis obtained by extending the velocity of the dislocation lineto the entire simulation space. This description of the dislo-cation line is therefore implicit. The velocity of the disloca-tion line is the product of the Peach-Koehler �PK� force andthe mobility tensor. The elastic field caused by the disloca-tions, together with the misfit stresses in the thin film, willa�Electronic mail: quekss@ihpc.a-star.edu.sg

APPLIED PHYSICS LETTERS 90, 011905 �2007�

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determine the PK force. The elasticity equations are solvedusing fast Fourier transforms in three dimensions and cor-recting the resultant stress field to ensure zero traction freesurfaces. Details of the calculations were presented else-where �see Quek et al.13�.

We model dislocation loop�s� terminating at the free sur-face of a heteroepitaxial Si1−�Ge� alloy film �0���1 is theatomic fraction of Ge in the alloy� on a �001� Si substrate.We assume that the dislocation loops are formed as a resultof some nucleation process not explicitly considered here.The thin film is elastically isotropic and is subjected to anequal biaxial misfit strain, �xx=�yy =�o. The correspondingmisfit stress is �misfit=2��o�1+� / �1−�, where � is theelastic shear modulus and is Poisson’s ratio. Periodicboundary conditions are assumed in the x and y directionsand the simulation cell is discretized uniformly into 128128128 �500b500b500b, where b is the magnitudeof the Burgers vector� grid points. The thickness of the thinfilm used here is fixed at approximately 78b. Note that theinitial configuration of dislocation loops in our simulationswas chosen to facilitate the study of reactions and blockingmechanisms.

Figure 1 shows the dynamics of three coplanar disloca-tion loops on the �111� plane of Si0.8Ge0.2/Si. The initialBurgers vector of the concentric �blue� loops is b1= �a /2��01̄1�, while the other �magenta� loop has b2= �a /2��101̄�.Note that the motion of the dislocation lines out of the slipplane in Figs. 1�b� and 1�c� �and later in Fig. 3� correspondsto the cross slip of the screw dislocation segment. The crossslip results in interfacial dislocations running in both the

�1̄10� and �110� directions—consistent with experimentalobservations.3,8,17–19 The treatment of cross slipping is de-tailed in Ref. 13. Figure 1�b� shows that when the threadingsegments of the larger b1 �blue� loop and the b2 �magenta�loop meet, they react to form a dislocation with Burgers

vector b3=b1+b2= �a /2��11̄0�. This dislocation reaction be-gins near the free surface where the segments initially meetand continues down towards the substrate by the propagationof an inverted Y junction. The reaction satisfies the Frankcriterion �two dislocations will be attracted towards one an-other and combine to form a new dislocation provided thatthe new dislocation has a smaller energy than the disloca-tions it is replacing, b1

2+b22�b3

2�. The reacted dislocationsegment is a pure edge �b3 is perpendicular to the line direc-

tion �= �1/�6��112̄��. This reaction is commonly observed indiamond cubic heteroepitaxial systems �as discussed previ-ously by Pichaud et al.20�. Nevertheless, it is interesting tonote that the biaxial misfit stress creates a PK force on thenew segment, fmisfit= �a /�6��misfit · �111�, which is actingnormal to the �111� slip plane. This implies that there is noglide force on the reacted segment from the misfit stress andit, therefore, forms a dislocation that is stationary in the ab-sence of other forces �ignoring the slow climb process�. Notethat this dislocation junction differs from a Lomer-Cottrelljunction in that it is glissile.

Analysis of the Burgers vectors of the smaller, b1 �blue�,dislocation loop and the new dislocation segment shows thatthese two dislocation segments repel one another. Figure 2shows a schematic illustration of the forces acting on thedislocation segments. Besides the driving force from the mis-fit stresses acting on the smaller b1 loop, there is also arepulsive force from the reacted segment acting in the oppo-site direction. On the other hand, an equal and opposite re-pulsive force acts on the reacted segment pushing it towardsthe right. As observed in Figs. 1�b� and 1�c�, the reactedsegment slows down the movement of the expanding b1�blue� loop �relative to the case where there is no b2 loop13�,which is driven by the biaxial misfit stress. Figure 1�c� alsoshows that the smaller, b1 �blue� loop pushes the reactedsegment along the slip plane, indicating that it is not“locked.” As the smaller b1 dislocation approaches the re-acted segment, both the b1 loop and the reacted segment willmove with the same steady-state velocity at a fixed equilib-rium separation between them. Since the velocity of the dis-location line is the product of the PK force and the disloca-tion mobility �Quek et al.13�, the steady-state velocity of thesegment of the b1 loop near the reacted segment will be v1= fmisfitM1Mreact / �M1+Mreact�= fmisfit�M1

−1+Mreact−1 �−1 , where

M1 and Mreact are the glide mobilities of the b1 loop andreacted segment, respectively. If the mobility of the reactedsegment approaches infinity, as in the hypothetical case ofzero lattice resistance, the reacted segment will not act as abarrier at all. Nevertheless, if the two dislocations havenearly the same glide mobilities �the common case�, the ve-locity of the b1 loop will be approximately halved �i.e., M=M1=Mreact and v1=0.5fmisfit�. The reacted segment, there-fore, acts as a kinetic barrier to the threading dislocation,slowing its motion through the heteroepitaxial film.

The results in Fig. 3 show the formation of a junction viathe reaction of two dislocation segments initially gliding onintersecting planes. Similar to Fig. 1, the blue dislocations

have b1= �a /2��01̄1� and the magenta dislocation has b2

= �a /2��101̄�. The reaction of the two nearest segments isfavorable according to Frank’s criterion, and the force actingon the reacted segment caused by the biaxial misfit stress isagain normal to its slip plane and does not drive reactedsegment glide. Analysis of the forces, as in the coplanar case,shows that the reacted segment is a kinetic barrier to the

FIG. 1. �Color online� Three views of the expansion of three initially co-

planar dislocation loops on the �111� plane with �blue� b1= �a /2��01̄1� and

�magenta� b2= �a /2��101̄� in a Si0.8Ge0.2 film deposited on Si substrate. �a�t*=0.0, �b� t*=0.024, and �c� t*=0.08. The inclined plane is the �111� planeand the blue horizontal plane is the film-substrate interface. The substrate inthe simulation is approximately five times thicker than the film.

FIG. 2. �Color online� Schematic of forces acting on the reacted segmentand the expanding b1 loop.

011905-2 Quek et al. Appl. Phys. Lett. 90, 011905 �2007�

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approaching b1 loop. More important, however, is the factthat motion of the reacted segment necessarily trails an in-terfacial, screw segment that does not relieve the misfitstress, as shown schematically in Fig. 4. Therefore, motionof the reacted segment comes at a substantial energetic costand, hence, represents a strong barrier to the motion of thethreading dislocation—evident in Figs. 3�b� and 3�c�. It isinteresting to note the similarities in the pinning of thethreading dislocations in Fig. 3 by this reacted segment withthat by the branched defects.6,9 As an aside, Figs. 1 and 3also show the interfacial misfit segment �blue� being pushedinto the substrate by the repulsive force from the smaller, b1�blue� loop.

There are a total of four possible reactions �one in eachslip plane� to form such barriers in a diamond cubic filmunder an equal biaxial misfit stress �only one such reaction isshown here�. Each reaction can involve coplanar dislocationsor dislocations on intersecting slip planes. This suggests thatthe formation of such barriers is not rare. The present resultsrepresent heretofore unknown barriers to threading disloca-tion motion in heteroepitaxial semiconductor films.

Using dislocation dynamics simulations of thin, het-eroepitaxial films, we have described mechanisms by whichdislocation junction formation results in barriers against slipof additional dislocations that is fundamentally different

from the well-known Lomer-Cottrell junction. These barriersare formed from the reaction of dislocation segments withdifferent Burgers vectors, resulting in a dislocation segment,which does not glide under the influence of a biaxial misfitstress. Unlike the case of the Lomer-Cottrell junction, thisreacted segment is not locked but rather can move under theinfluence of the stress fields associated with other disloca-tions. For the case of a threading segment formed from twointersecting slip planes, the barrier formed is energetic andstrong, while that involving coplanar dislocations is purelykinetic and much weaker. As many reactions similar to thatshown here exist, the formation of such junctions/barriersshould be frequent and their effect on misfit relaxation sub-stantial.

The authors acknowledge the support of Singapore’sAgency for Science Technology and Research through theVisiting Investigator Program. One of the authors �Y.X.� alsoacknowledges the support of the Hong Kong RGC Competi-tive Earmarked Research Grant No. 604604.

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FIG. 3. �Color online� Expansion of two initially coplanar dislocation loops

�blue� on the �111� plane with b1= �a /2��01̄1� and one loop �magenta� on

the �11̄1� plane with b2= �a /2��101̄� in a Si0.8Ge0.2 film deposited on Sisubstrate. �a� t*=0.0, �b� t*=0.012, and �c� t*=0.06.

FIG. 4. �Color online� Schematic of formation of new screw segment that isenergetically unfavorable if the barrier is to glide on the �111� plane.

011905-3 Quek et al. Appl. Phys. Lett. 90, 011905 �2007�

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