Collective formation of misfit dislocations at the critical thickness
Tobias Særkjær, Thue Christian Thann, Tomaš Stankevi?, Sergej Schuwalow, Peter Krogstrup
CCollective formation of misfit dislocations at the critical thickness
Tobias Særkjær † , ∗ ,
1, 2
Thue Christian Thann ∗ ,
1, 2
Tomaš Stankevič, Sergej Schuwalow,
1, 2
Peter Krogstrup †
1, 2 Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, 2100 Copenhagen, Denmark Microsoft Quantum Materials Lab Copenhagen, 2800 Lyngby, Denmark (Dated: February 9, 2021)The critical thickness constitutes a vital parameter in heterostructure epitaxy engineering asit determines the limit where crystal coherency is lost. By finite element modeling of the totalstrain relaxation in finite size heterostructure nanowires, we show that the equilibrium configurationchanges abruptly at the critical thickness from a fully elastically strained structure to a structurewith a network of MDs. We show how the interdependent MD relaxation changes as a function ofthe lattice mismatch. These findings suggest that a collective formation of MDs takes place whenthe growing heterostructure layer exceeds the critical thickness.
I. INTRODUCTION
The presence of misfit dislocations in epitaxial het-erostructures can have detrimental effects on the devicefunctionality as it alters the structural, mechanical,optical and electronic properties. The stress inducedfrom elastic strain originating from the mismatch be-tween the lattice parameters of a growing thin film andthe substrate, acts as a driving force for the formationof structural defects when the critical thickness isexceeded. Understanding the mechanisms that leadto misfit dislocations (MDs) at the critical thicknessis therefore of utmost importance for understandingthe limits of coherency when engineering epitaxialdevices. The transition from elastic to plastic relaxationat the critical thicknesses in thin films has for thesereasons been studied intensively in the past decades,where both equilibrium and kinetic models for the for-mation of MDs have been compared with experiments .In our framework of structural finite element method(FEM) simulation, MDs are localized ’additional’ latticeplanes in the part with the smaller lattice parameter.Unfortunately, pseudo-infinite simulations of planargrowths employ symmetric boundary conditions, whichimpede the displacement that these planes should causein the surrounding crystal. This makes FEM simulationsof MDs in large planar films particularly challenging. Weinstead turn our attention to finite size heterostructureswhere the symmetric boundary conditions are notneeded.Selective area growth (SAG) methods for synthesis ofnanowire (NW) heterostructure networks have receivedincreasing interest in the field of quantum devices andapplications . Unlike free-standing NWs grown viaunidirectional growth methods such as the Vapor-Liquid-Solid (VLS) method, the SAG method offers theopportunity to design complex networks in the plane ofthe substrate which makes it flexible and relevant forscalable and advanced quantum applications.However, as the in-plane SAG NW networks are con-strained to the requirement of underlying insulating layers, the crystal growth method is challenged by theinherent lattice parameter mismatch, much like planarfilms. Additionally, the strongly anisotropic morphologyof SAG NWs will help highlight the interplay betweenrelaxation of different strain components.We initially study purely elastic relaxation in SAGNWs to examine the strain energy as a function ofgrowth stages. The shapes studied are observed inexperiments , appearing to be approximate equilibriumshapes given lowest surface energy configuration for theNW cross section.Building on these examinations, we subsequently studyelastic and plastic relaxation in SAG NWs with disloca-tions as ’additional’ crystal planes. The density of dislo-cations is a persistent question in the study of critical di-mensions for thin film growths. Our FEM simulations arecarried out for a wide range of mismatches and MD densi-ties, finding the equilibrium configurations at the criticalheights from comparison between the elastic and plasticconfigurations. We study in particular plastic strain re-laxation of a h i orientated NW on a (001) substrateand buffer, and find that collective onset of dislocationsis favored over singular onset - a conclusion expected tocarry over to other heterostructures subject to in-planestrain caused by a lattice mismatch. Lastly we analyzethe equilibrium MD densities and show critical heightsas functions of mismatch. II. PURELY ELASTIC STRAIN RELAXATION
Figure 1a presents a stereographic projection ofthe typical NW types available on (001) substrates.The purely elastic simulation features a translationallyinvariant segment, using three symmetry planes asillustrated in figure 1b along with an example mesh. Weassume for simplicity that the buffer (region separatingthe conducting NW channel from the substrate) isrelaxed to the underlying substrate. See SupplementalMaterial for information on strain implementation andcalculation of strain energy density (SED) in the FEM a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Figure 1:
Elastic growth of translationally invariant NWs. a)
Linear stereographic projection of SAG NWsgrown on (001) substrate. b) Sketch of a SAG NW on a substrate with indicated symmetry planes, and a meshexample in zoom-in. c) Total elastic strain energy per 100 nm section length of a h i NW as a function of InAstransport channel volume V InAs on a In . Ga . As buffer (approx. 1.8% mismatch). Insets (InAs: grey, InGaAs:black) illustrate three types of cross sectional shapes. Growth stages are described further in the main text. d) As c)except investigated as a function of transport channel thickness h InAs , where the dotted line represents a free-standingNW model at same mismatch and interfacial area. The free-standing NW is simulated as hexagonal in cross-section,protruding normal to a {111} substrate. We consider only half a stage II B) NW, as the two sides have little to nostrain field interplay.software COMSOL .Varying the geometric parameters allows for analysisof the dependence on dimensions, shape and size ofthe structure. In an actual growth environment theseparameters can be controlled by lithographic patterningand adjusting growth time, flux compositions andtemperature. See Supplemental Material for exampleresults of varying size effects.These simulations are run for an InAs NW on anIn . Ga . As buffer grown in the h i direction with { } side facets (see figure 1a). In figure 1c we showthree different variations of this morphology, wherestage I represents a transport channel grown from athin layer on the buffer to a full pyramid shape withfully formed facets. We regard this shape to be anapproximation of the lowest-energy shape as dictatedby the surface energy densities associated with differentcrystallographic orientations. The other morphologies represent overgrowth, wherestage II A) specifically represents a layer beginning toform on a fully grown stage I NW, and the transition ismarked with an arrow in figure 1c. We conclude thatfor our model, overgrowth contributes to total strainat a lower rate than stage I growth. We also see thatstage II B) can accommodate higher mismatch for lowertransport channel volumes, but also that this becomesunfavorable at larger channel volumes. All three stagesexhibit slightly sublinear increase in total strain energyfor very large transport channel volumes. We note thatthe simulated interfacial area is kept constant betweenthese morphologies for comparison.In figure 1d we investigate stage I and stage II B andthe strain energy dependence on the thickness of the InAslayer. For stage II B we consider only half of the wire (cutalong the axial direction), re-dimensionalized so that theinterfacial area is identical for both morphologies. WeFigure 2: FEM simulations of dislocations.
Top row:XY-components of stress fields caused by an edge disloca-tion at the markers. a) Analytic solution by Head . b) and c) Results from FEM simulations with dislocationsmodeled as a planes indicated by markers, with strain(+1) and (-1) respectively. Panel c) additionally showsthe region near the core excluded from energy calcula-tions, size exaggerated for clarity. d) and e) FEM mod-els of h i type NW (substrate not shown) with a h i dislocations modeled as vertical planes with strain (+1)and (-1), respectively. f ) and g) Horizontal componentsof strain resulting from models d) and e) with 3% mis-match (InAs / In . Ga . As) . h) Composite image ofresults from the two models in panels f) and g).further compare to a free-standing NW with identicalinterfacial area, which is not constrained by symmetryplanes. We find this free-standing type NW to be favor-able in comparison with SAG growth at all thicknesses,which is expected since the free-standing NW is less con-strained. We note that the graphs for stage I and stage IIB) cross each other at approximately 11nm in panel 1d.This is due to stage I gaining less volume per unit layerheight, as the triangular cross section becomes thinnertowards the top. Hence, this crossing is absent in thepanel 1c displaying the energy as a function of transportchannel volume.All cases compare favorably to the planar growth of thinfilm on a planar substrate, which is shown as the dashedline in panel and 1d. The thin film is a rectangular struc-ture with symmetry planes on all four sides to emulate apseudo-infinite plane. For the thin film case, a mismatchof 1.8% (InAs / In . Ga . As) corresponds to a criticalthickness of ≈ . according to Matthews model , de-pending on the exact Burger’s vector in the strained toplayer. This highlights the morphological advantages of aSAG buffer, where the NW can relax strain by a rota-tional degree of freedom which has also been shown byother authors . III. PLASTIC STRAIN RELAXATION
As the crystal volume of a lattice mismatched het-erostructure increases during growth, the excess energyincreases until a critical thickness is reached, at whichpoint MDs are formed to lower the total energy, mostoften edge dislocations along the interfaces . Ultimatelywe are interested in understanding the limits of fullyelastically strained heterostructures as a function ofshape, volume and composition (lattice mismatch).In a simple 1-dimensional case, the spacing betweendislocations is generally given by: d = | b | / ( δ − ε ) , with | b | being the length of the Burger’s vector, δ being themismatch and ε being the average remaining elasticstrain. Therefore ’full plastic relaxation’ correspondsto ε = 0 with a corresponding density of dislocations.However, there will be a certain fraction of elastic vs.plastic relaxation that will display the minimum strainenergy, and we can not in general expect full plasticrelaxation. If we wish to engineer structurally defect freeNWs, we need to examine configurations with differentMD densities in order find the equilibrium configurationat the critical height.Figure 2d shows a model of a h i type NW withedge MDs with in-plane Burger’s vectors of type a h i ,where a is the lattice parameter. The dislocationsare modeled as planes in the buffer/substrate withthickness matching the length of the Burger’s vector andpositive unity strain (normal to the planes) simulating’additional’ crystal planes due to misfit dislocations atthe interface. This ensures the correct effect of MDs inthe transport channel, but leaves an un-physical artefactin the simulations of the buffer and substrate where the’additional’ planes will not in general be strained relativeto the surrounding material far from the interface.An alternate method simulates the same dislocations as’missing’ planes in the transport channel with negativeunity strain (figure 2e). This conversely creates thecorrect effect in the buffer and substrate, while theunwanted artefact is now simulated in the transportchannel.The dislocation planes end at the NW-buffer interfacewhere the dislocations are situated . In panels 2a-c wecompare 2-dimensional versions of these two methods tothe analytical solution of stress fields associated withdislocations found by Head . We conclude that ourmethods are entirely consistent as 3-dimensional exten-sions of the equivalence shown in 2 dimensions. Thetwo methods can be combined graphically to yield theresults shown in figure 2h. See Supplemental Materialfor further comparison between methods.As seen in figure 2a-c, a small region around thedislocation cores becomes very highly strained. As aresult, the elastic theory employed for evaluation of theSED is locally no longer valid, and an alternate methodis needed if one wishes to evaluate the strain energyFigure 3: Model and strain energy as function of film thickness. a)
Model along with a zoom section displayingthe mesh with increased density near the dislocations. b) Total strain energy of different plastic configurations inunits of the elastic configuration at the corresponding film thickness for a mismatch of δ = 2% . Note that the firstdislocations to cross below the unity line is not the one with just one transverse dislocation.included in regions near the dislocations. The ’Volterramethod’ or ’empirical method’ of excluding smallcylindrical cores (see figure 2c) from SED integrationsto account for singularities and the reconstructionof chemical bonds at the dislocation cores seems towork well for dislocations surrounded by a largerbody of material , but in thin films the strain fieldsnear the dislocation cores interact with the free sur-face, effectively making the radius of this small excludedcore a non-trivial function of the mismatch and geometry.Our simulations were carried out using the modelshown in figure 3a (interface width , channellength µ m ), with ’transverse’ dislocations as equidis-tantly spaced ’additional’ planes in the substrate andbuffer. This time the orientation was chosen with h i along the NW axis and {111} type side facets (see fig-ure 1a) . The material composition of the buffer wasvaried with corresponding changes in material parame-ters according to Vegard’s Law, and chosen to emulatemismatches from 1% to 4% corresponding to InAs onIn x Ga − x As with x between . and . . In all casesthe composition within each region (substrate, buffer,NW) was chosen as spatially uniform for simplicity. Theheight of the thin film was varied (akin to the methodemployed for figure 1d) to emulate different stages of ap-proximate layer-by-layer growth throughout.We are concerned with the total strain energy in thetransport channel (NW), which comprises by far thedominant energy contribution compared to the buffer andsubstrate. In order to evaluate the ’invalid regions’ men-tioned above, we modify the ’Volterra method’ by exclud-ing slightly larger cylindrical cores of radius r core = | b | / ,with b being the Burger’s vector, arguing that the dom-inant energy contribution inside this range is due to therearrangement of chemical bonds. We account for thesebonds by adding an energy per unit dislocation lengthfrom the melting approximation given as E m = Gb / π where G is the shear modulus of InAs. IV. RESULTS
Figure 3b shows the strain energy of plastic config-urations in units of the strain energy for the purelyelastic configuration as a function of film thickness fora mismatch of δ = 2% . See Supplemental Material forsimilar results for other mismatches.From closer examination it becomes obvious, that thefirst plastic configuration to become favorable is not theone with a singular dislocation. This is a general featureacross the mismatches examined, but more notable forhigher mismatches. This suggests that nucleation ofdislocations at the critical height is a collective process,which becomes stable when a certain equilibrium MDdensity is achieved.In this study we have limited ourselves to one axialdislocation running along the center of the NW, andnote that a more complete examination would have todeal with a much larger parameter space of both numberand positions of axial MDs.We also note the general feature that higher mismatchestend to favor configurations with more MDs. For the1% mismatch case the equilibrium configuration at thecritical height has only transverse dislocations (MD ⊥ ),which could be interesting for engineering of MDs inheterostructures. However, for the mismatches of 2%,3% and 4% the equilibrium configurations at the criticalheights have both the axial and transverse dislocationspresent. This could prove useful for analysis, as thelack of an axial MD from cross sectional TEM of ahigh mismatch structure could indicate that the entirestructure is purely elastically relaxed.The question of MD configuration at the criticalheight is examined further in figure 4a which shows theFigure 4: Fitting critical heights.
Panel a) shows the predicted critical heights, assuming a set number of dislo-cations for each of the four mismatches. Markers denote minima from fits. Panel b) shows the minima from a) foreach mismatch ( δ ) along with the corresponding number of dislocations found from fits. Fit types are described inthe main text.points where different configurations become favorablecompared to the purely elastic case. For a given mis-match the lowest of the critical heights is the predictedequilibrium critical height, and a specific MD density isassociated with this.The guidelines in figure 4a are fits to the form h c = an disl + b + c/ ( n disl + d ) , where n disl is the numberof dislocations. The minima from figure 4a (marked) areextracted from the fits and plotted in figure 4b alongwith the associated number of MDs and new fits ofthe simpler form h c = α/ ( δ + β ) where δ is again themismatch.The variables α ⊥ = 22 . , β ⊥ = − . , α || =12 . , and β || = − . are found from the fits forconfigurations without and with the axial dislocation, re-spectively. Notably, for mismatches below δ ∗|| = 1 . , theconfiguration at the critical height shows no axial dislo-cation.The fit forms and variables found suggest a divergence ofthe critical height at a mismatch of a quarter of a percent.However, to ensure a fully elastic growth in stage 1, wewould merely need the critical height to be larger thanthe thickness of the transport channel grown. Due tothe geometry in question, the stage 1 transport channelcan grow to a maximum height of h max = w/ √ , where w = 125nm is the width of the interface. From the model,this can be accommodated elastically at a mismatch of δ ∗ = 0 . or InAs on In . Ga . As, although a bufferthis high in In composition would probably cause issuesin terms of properly containing the wave function to theintended transport channel.Finally we note that while all the critical heights quotedare specific to the morphology, dimensions and materials examined, the method presented is directly transferablefor examination of other combinations.
V. CONCLUSION
The mechanisms of strain relaxation in lattice mis-matched SAG NW heterostructures are shown to bedistinctly different from planar heterostructures as wellas free-standing NW heterostructures . The additionalelastic relaxation as compared to planar thin filmheterostructures stems from the rotational degree offreedom for relaxation transverse to the NW axis whichin principle can overshoot the bulk relaxed values,giving additional room for elastic relaxation along theNW axis. We identify three different growth stages,all of which are energetically favorable compared to aplanar film growth, and all of which quickly becomeapproximately linear with different dependencies onlayer height, favoring stage II A).Our findings establish a relationship between trans-port channel layer thickness and MD density similar tothat between NW radius and misfit percentage as foundby Glas , and highlights the difference between SAG andfree-standing NWs. For comparison we quote the ex-perimentally found critical height of h c,film % = 1 . for planar thin film growth of InP on GaAs at 3.8%lattice mismatch . This confirms the ability for elasticrelaxation in SAG NWs as somewhere between thehighly constrained planar thin films and the nearlyunconstrained free-standing NWs of VLS.The simulation methods are directly transferable acrossmaterials and morphologies, and the results can becompared to physical samples by analysis of e.g. atomicresolution TEM with GPA etc.In summary we present a method for introducing plas-tic relaxation from MDs as localized FEM model fea-tures in heterostructure simulations, allowing an exam-ination covering different MD densities. This leads toour prediction of collective rather than singular onsetof MDs at the critical height. For SAG NW growth instage I, we find critical thicknesses of h c, , ⊥ = 30 . , h c, , || = 9 . , h c, , || = 4 . , and h c, , || = 3 . for 1%, 2%, 3%, and 4% mismatch, respectively, as sum-marized in figure 4. In all cases we find that collectiveformation of dislocations is favorable compared to singu-lar onset. For mismatches below δ ∗|| = 1 . mismatchwe find that the equilibrium configuration from the crit-ical height shows only transverse dislocations, while formismatches above this value both axial and transverse dislocations are expected from the critical height. ACKNOWLEDGEMENTS
This paper was supported by Microsoft Station Q, andby the European Research Council (ERC) under grantagreement No. 716655 (HEMs-DAM).The authors thank Martín Espiñeira, Anna WulffChristensen, Filip Křížek, Joachim Sestoft, Jordi Arbiol,Sara Martí-Sánchez, Kevin van Hoogdalem, Léo Bourdetand Philippe Caroff for shared data, fruitful discussionsand academic and practical inputs. ∗ : These two authors contributed equally. † : [email protected], [email protected] F. Glas
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Tobias Særkjær ∗ , † ,
1, 2
Thue Christian Thann ∗ ,
1, 2
Tomaš Stankevič, Sergej Schuwalow,
1, 2
Peter Krogstrup †
1, 2 Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Denmark Microsoft Quantum Materials Lab Copenhagen, Lyngby, Denmark (Dated: February 9, 2021)
I. COMSOL MODELS FOR FEM SIMULATION
Our FEM simulations were carried out as 3-dimensional Stationary studies with the Solid Mechanicspart of the Structural Mechanics module in COMSOLMultiphysics . Materials in COMSOL can be definedwith a variety of different properties, either from scratchor from a library of predefined materials. The impor-tant properties are the Bulk Modulus, Poisson ratio andelasticity matrix. The entries of the elasticity matrix areused for calculations of strain energy density from linearelastic theory. The Linear Elastic Material sub-menu ofthe Solid Mechanics part of the interface allows for im-posing initial strain in select parts of any geometry built.Initial tensile strain is employed (in-plane with the in-terface) in the NW corresponding to a chosen lattice mis-match between NW and the buffer, from which a balanceof forces on each mesh point yields the final configurationwith forced coherence at the interface.The strain energy density (SED) is found locally fromderivatives of the displacement according to equation 1: U SED = X ijkl c ijkl ε ij ε kl (1)with c ijkl being the stiffness coefficients and ε ij components of the strain tensor. Evaluations of theresulting strain must properly account for the initialstrain imposed.The relevant bulk parameters are the lattice and elasticconstants and for In x Ga − x As, where we assume linearinterpolation between the respective parameters ofthe component materials (Vegard’s law).Some drawbacks do arise from the static and continu-ous simulations of a dynamic and atomistic physical sys-tem, and we should address those here. Drawbacks in-clude but are not limited to a lack of polarity, static elas-ticity and lattice constants, and uniform compositionswithin each region. While these drawbacks are relevantand present, plenty of results are still obtainable, andsimulations of strain relaxation and dislocations in SAGheterostructures could prove a central tool for achievingdislocation free, scalable, high mobility devices. Figure S1:
Finite length effects, 3% mismatch (InAs / In . Ga . As) . a)
Illustration of end sectionwith symmetry planes.
Box:
Example mesh of endsection. b) Distribution of strain energy density onbuffer-NW interface as well as both symmetry planesfor a fully triangular shaped NW (fully grown stage I). c) Strain energy in a central cube as a function of sizeparameters, showing length scales for decoupling of thecenter to the end effects.
II. FINITE LENGTH EFFECTS
Turning our attention to the regions near the NW ends,figure S1a shows a NW morphology for stage I growthwith only two symmetry planes imposed to examine finitelength and the corresponding gradient in elastic strainenergy density towards the end of the NW.The translationally invariant NW segments investigatedin the main text relax strain primarily by rotation of crys-tal planes in directions perpendicular to the NW axis.Near the ends of the NWs, rotation along the NW axisprovides an additional degree of freedom for relaxation. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Figure S1b shows the distribution of SED in the stage Imodel. Unsurprisingly, the general trend shows a higherSED near the interface tapering off with distance. Forthe stage I growth we see the SED decreasing as we movefrom the middle towards the NW sides.Figure S1c shows a comparison of the average strain en-ergy density of nm sections in the center and end ofthe NW. The simulations are run for a h i type NW at3% mismatch (InAs / In . Ga . As)
NW, and we inves-tigate the geometric parameters of NW length and bufferheight.We notice a clear trend of the end region converging fasterand at lower values. We also see target dimensions of abuffer in order to minimize strain energy. This methodcan be employed for examination of parameters in bothfully elastic and plastic configurations.
III. STRAIN FRAMEWORK
For pedagogical reasons, the main paper and this sup-plemental is phrased in terms of linear strain, that is: ε l = LL − (2)with ε l being the linear strain and L being the finallength of an object with unstrained length L . In thislinear strain framework, the strain is just the fractionalelongation of the object. The reader should note, thatseveral frameworks for strain are available, and notablythat strain in the Solid Mechanics module of COMSOL is Green-Lagrange strain: ε GL = 12 "(cid:18) LL (cid:19) − (3)Other FEM packages may employ different strainframeworks, and adequate adjustments should be takento account for this. E.g. we note that ε l = +1 cor-responds to ε GL = 3 / , while ε l = − corresponds to ε GL = − / . For the purposes of this supplemental, weshall continue to phrase strain in the linear terms, sinceit allows for a more intuitive understanding, keeping inmind that the specific implementation is recast depend-ing on the strain framework of the FEM software. IV. ANALYTIC SOLUTION AND METHODEQUIVALENCE
The default simulation including MDs modeled asplanes with +1 strain in the buffer, corresponding to the’additional’ crystal planes. As mentioned in the maintext, this ensures correct boundary conditions fromthe interface and above, while the boundary conditionsinside the lower part are obviously incorrect, since theimposed planes are not actually strained compared to Figure S2: Stress components caused by an edgedislocation near a fixed surface (left side). Top row:Analytic solution by Head . Mid row: Simulation using (+1) strain method. Bottom row: Simulation using ( − strain method. The middle column is also shownin the main text.the surrounding material, as confirmed in e.g. geometricphase analysis of atomic resolution TEM. This methodis especially viable, since we are mostly concerned withthe variations in the transport channel. Equivalentlywe can model the MDs as planes with − strain in thewire, corresponding to the ’missing’ crystal planes. Thismethod ensures correct boundary conditions from theinterface and below.The simulation results should agree with analytic so-lutions, e.g. solutions by Head . Fig. S2 shows differentcomponents of a 2D stress fields caused by an edge dis-location, as analytic solution or modeled by the methodsmentioned above. The figure shows a clear equivalencebetween all three methods, validating the simulationalmethod. For the purposes of our simulation the conceptis straight forwardly extended to three dimensions.Dislocations are associated with an energy, which isproportional to the square of the length of the Burger’svector. This makes the lowest energy MDs of zinc-blendethose with Burger’s vectors of type a h i . For a h i NW on a (001) substrate, these types of MDs will si-multaneously relax strain in two out of the three direc-tions: axial, transverse and out of plane. In the case ofa h i NW on a (001) substrate, there exist favorableMDs which relax strain in distinctly axial, transverse orout of plane directions as well as MDs which relax all di-Figure S3: Top views of a section of the NW at theinterface between NW and buffer for a {100} type NW.a) Transverse (horizontal) strain component using themethod with +1 strain in the buffer. b) Same as a,using the method with − strain in the NW. c) Out ofplane strain component, +1 strain method. d) Same asc, using − strain method.rections simultaneously. This was illustrated in the maintext with the model for formation of MDs. The methodproposed here is equally well suited for MDs with otherdirections of Burger’s vectors, as the strain associatedwith the MD can be defined independently of the MDplane. V. COMPOSITE PLOTS
Combining the two methods mentioned above (strain +1 and − ) as one composite solution relies on the two solutions producing a consistent solution at the interface.Figure S3 shows resulting strain components at the in-terface of a simulation of a h i type NW on a (001)substrate, where the initial boundary conditions are con-sidered equivalent. The h i type NW as well as otherstrain components were examined as well with similaragreement between methods. VI. ENERGY RESULTS FOR PLASTICCONFIGURATIONS
The results of the strain energy integrations for theplastic configurations with the axial MD (MD || ) areshown in figure S4. As is evident, the equilibriumconfiguration at the critical height shows no axial MD(MD || ) for the 1% mismatch (a solid line crosses belowthe unity line first), while for 2%, 3% and 4% theequilibrium configuration at the critical height shows atleast one axial MD (a dashed line crosses is the first tocross below the unity line). The results are summarizedin the main text. ∗ : These two authors contributed equally. † : [email protected], [email protected] M. E. Levinshtein, S.L. Rumyantsev
Handbook Serieson Semiconductor Parameters , vol.1 , M. Levinshtein, S.Rumyantsev and M. Shur, ed., World Scientific, London,1996, pp. 77-103. M.P. Mikhailova
Handbook Series on Semiconductor Pa-rameters , vol.1, M. Levinshtein, S. Rumyantsev and M.Shur, ed., World Scientific, London, 1996, pp. 147-168. A. K. Head
Edge Dislocations in Inhomogenous Media ,Proc. Phys. Soc. B , 793 (1953) Figure S4: