PHYSICAL REVIEW B 67, 075316 ~2003!

Theory of strain relaxation in heteroepitaxial systems

A. C. Schindler,1 M. F. Gyure,1,2 G. D. Simms,2 D. D. Vvedensky,1 R. E. Caflisch,3,4 C. Connell,3 and Erding Luo31The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom2HRL Laboratories LLC, 3011 Malibu Canyon Road, Malibu, California 90265

3Department of Mathematics, University of California, Los Angeles, California 90095-15554California NanoSystems Institute and Department of Materials Science & Engineering,

University of California, Los Angeles, California 90095~Received 29 August 2001; revised manuscript received 20 December 2002; published 28 February 2003!

We introduce a general approach to calculating the morphological consequences of coherent strain relaxationin heteroepitaxial thin films based on lattice statics using linear elasticity. The substrate and film are describedby a simple cubic lattice of atoms with localized interactions. The boundary conditions at concave and convexcorners that appear as a result of this construction, those along straight interfacial segments, and the governingequations are obtained from a variational calculation applied to a discretized form of the total elastic energy.The continuum limit of the equations and the boundary conditions along straight boundaries reproducesstandard results of elasticity theory, but the boundary conditions at corners have no such analog. Our methodenables us to calculate quantities such as the local strain energy density for any surface morphology once thelattice misfit and the elastic constants of the constituent materials are specified. The methodology is illustratedby examining the strain, displacement, and energies of one-dimensional strained vicinal surfaces. We discussthe effects of epilayer thickness on the energy of various step configurations and suggest that coupling betweensurface and substrate steps should affect the equilibration of the surface toward the bunched state.

DOI: 10.1103/PhysRevB.67.075316 PACS number~s!: 68.55.2a, 68.35.Gy, 68.60.2p

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I. INTRODUCTION

The structural and compositional integrity of heteroetaxial films is central to the fabrication of all quantum heerostructures. The morphology of these films is determiby a number of factors, including the manner in which stris accommodated if the materials have different lattice cstants, the surface and interface energies of the materialsany effects associated with alloying and segregation. Tmodynamic arguments based on interfacial free energiesoften used to provide a classification scheme for theequilib-rium morphology of thin films.1 But while such consider-ations undoubtedly play an important role in providing toverall driving force for the morphological evolution of thifilms, they neglect a number of inherently kinetic effecThe interplay between thermodynamics and kinetics is escially germane to heteroepitaxial systems where, forample, variations in growth conditions~substrate temperature, flux, substrate misorientation! and annealing are used tmanipulate the spatial and size distributions of thrdimensional~3D! coherent islands that appear during tStranski-Krastanov growth of lattice mismatched semicductors for quantum dot applications.2

Strain relaxation in heteroepitaxial systems has beensubject of an abundance of theoretical studies, but theryet no general methodology with the versatility of thBurton-Cabrera-Frank theory,3 rate equations,4 or kineticMonte Carlo simulations5 which captures the essence of thfilm evolution in the presence of lattice misfit. There are twmain reasons for this. The rates of atomistic processesstrained surfaces are not determined solely by the localvironment of the atoms, as in the case of homoepitaxy,may depend on nonlocal features such as the heightterrace above the initial substrate, the size and shape of

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and three-dimensional islands,6,7 and their localenvironment.8 This is further complicated by the competitiobetween different strain relaxation mechanisms~e.g., alloy-ing, misfit dislocation formation, surface profile modultions!, each of which has a characteristic signature inmorphology of the substrate.9 Additionally, any general the-oretical approach must incorporate long-range elastic inactions, which are best treated within a continuum framwork, and atomistic effects such as step-adatom interactialloying, and possibly reconstruction changes during grow

The theoretical description of the formation of heterogneous interfaces falls into one of three broad categories~i!the minimization of energy functionals, including thermodnamic potentials, of various levels of sophistication to detmineequilibriumatomic positions as a function of the latticmismatch,6,10,11 ~ii ! kinetic Monte Carlo simulations, bothlattice-based7,12 and off lattice,13 where the hopping rules armodified to account for the effects of strain on diffusion aadatom attachment and detachment at step edges, and~iii !classical elasticity theory applied to the evolution of tgrowth front profile of continuous films.14,15These studies alrequire a compromise between a realistic description ofteratomic interactions and the mesoscopic effects of strelaxation due to lattice misfit.

The approach we describe in this paper is based on csical elasticity, but with the substrate and film composedan atomistic grid, as in the method of lattice statics.16–18Thelattice mismatch and the difference in elastic constantstween the film and the substrate enter explicitly into ththeory. This representation of the growing film is capableincluding both atomistic and continuum elastic aspectsmorphological energetics and kinetics. At the atomistic levthis includes the effect of strain on adatom diffusion,19 andthe kinetic6,7 and thermodynamic10,20,21 stability of islands.

©2003 The American Physical Society16-1

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A. C. SCHINDLERet al. PHYSICAL REVIEW B 67, 075316 ~2003!

Over larger length scales, there are interactions due tostrate distortion, which lead to a repulsive interaction btween islands10,22 and other surface species.23,24 While suchlong-range effects are directly amenable to a descripwithin continuum elasticity with suitably chosen materiaparameters, more localized effects can be modeled withpirical or first-principles methods, especially where a dirconnection between the atomistic and continuum formutions can be established.25 Our approach can be made cosistent with atomistic models cast in a valence force firepresentation. This allows us the flexibility to incorporaatomistic effects where needed while remaining withingeneral framework of linear continuum elasticity.

In this paper, we formulate the elasticity equations aboundary conditions for discrete substrates and films. In ctrast to the situation for continuous films, where the bouaries are smooth curves,14 the boundaries of our discrete sytems are piecewise constant, and this requires a septreatment. Accordingly, we derive the elasticity equatioand boundary conditions from a variational calculation aplied to the total elastic energy of a discretized system whrespects the local symmetries of all points. The equatiobtained for the interior region and along straight segmeof the boundary lead, in the continuum limit, to the usuequations and boundary conditions, respectively, of linelasticity. But at corners, the boundary conditions are bunderstood in quasiatomistic terms as a constraint onlocal displacement. We illustrate our method by calculatthe interactions between steps on one-dimensional stravicinal surfaces, which is relevant to the step-bunching insbility on such surfaces. We examine qualitatively and qutitatively the strain and displacement fields that arise frthe model and compare and contrast these results to knresults from continuum elasticity. We also examine the inflence of the thickness of the epilayer and the differencethe elastic properties between the film and the substratediscuss the implications of these on the evolution towardbunched state.

The outline of this paper is as follows. The basic equtions of linear elasticity are reviewed briefly in Sec. II. Thdetails of our model are then presented in Sec. III, wheredescribe our variational formulation of the discretized eqtions and boundary conditions. The complete set of equatdescribing the boundary conditions are compiled in the Apendix. In Sec. IV we describe some results of our moincluding a comparison between step-step interactionscontinuum elasticity and those calculated within our modWe also present results obtained from the application ofmodel to vicinal surfaces that are relevant to the stbunching instability on a one-dimensional strained vicinsurface. In particular, we discuss the likely influence of eilayer thickness and lattice mismatch of the substrateepilayer on the evolution of the film toward the bunchstate. Finally, in Sec. V we summarize our results and outfuture applications of our approach.

II. CLASSICAL ELASTICITY

The following discussion presumes that the system isdimensional~i.e., that the substrate is one dimensional!, but

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the extension to a 3D system~i.e., a two-dimensional substrate! is straightforward. Letuk , wherek51,2, denote theCartesian components of the displacement vector. For linelasticity in an isotropic material, the components of tstrain tensorS and stress tensorT are given in terms of thedisplacement vector by26

Skl51

2~]kul1] luk!, ~1!

Tkl5ldklSnn12mSkl , ~2!

where]1[]/]x,]2[]/]y andl,m are the Lame´ constants.Repeated indices imply summation from 1 to 2. For theteroepitaxial growth of a film with lattice constantaf on asubstrate with a lattice constantas , the normalized latticemismatch is

e5af2as

as. ~3!

The strain tensor for the film is given by

Skl8 51

2~] luk81]kul8!, ~4!

where the prime denotes quantities associated with the fiUsing Eq.~3!, S8 can be expressed in terms of displacemewith respect to the substrate lattice positions as

Skl8 5Skl1dkle. ~5!

Accordingly, the corresponding stress tensorT8 for the filmis

Tkl8 5l8dklSnn1m8Skl2~2l81m8!dkle. ~6!

In mechanical equilibrium the forces inside any volumV vanish,

¹•T50, ~7!

and the force on the boundary]V equals the external pressure~in the absence of external tractions which in the casevacuum is zero!, leading to

n•T50, ~8!

wheren5(n1 ,n2) is the vector normal to the surface. Addtionally, at the interface between the film and the substrthe normal component of the stress is continuous:

n•T5n•T8. ~9!

For a continuous film, Eqs.~7!, ~8!, and ~9! completelyspecify the distribution of stress within the system. For tpurpose of the discussion in the next section, it is usefupoint out here that the above boundary conditions canderived from a variational principle applied to the total elatic energy. The elastic energy densityE of the strained sub-strate is given by the tensor contraction1

2 S:T. In terms ofCartesian components

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

E51

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The force balance equations and boundary conditionseach point then follow from setting the variation of the elatic energy with each of the displacements equal to zero:

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whereukl5] luk .

III. MODEL

In this section we describe a discrete atomistic modelstrain, designed to agree with continuum elasticity in thoregions where continuum theory has a natural discretiza~namely, in the bulk and at straight boundaries!. For our sys-tem, we have used an approach to formulating the equatof elasticity, based on lattice statics, that is especially apppriate for epitaxial systems. This methodology is explainin detail in Ref. 27. Here we outline the general structurethe method; mathematical details are included in the Appdix. The general approach is not to discretize Eq.~12! di-rectly. Instead, we construct a discrete version of the elaenergy density Eq.~10!, and then define the total energy assum of this energy density over lattice points

ET5S i , jE~ i , j !. ~13!

At each point (i , j ) of the grid, the energy contributionE( i , j )only involves terms from the nine point stencil~nearest andnext-nearest neighbors! centered at (i , j ). The energy contri-bution per siteE( i , j ) is written in its most general form inthe Appendix, as are specific cases of interest.

The principal virtue of this formulation is that it combineatomistic and continuum approaches. If the computatiogrid is the same as the underlying atomistic lattice, thendiscrete version of the energy may be considered purelymistic. This correspondence is equivalent to impositionthe Cauchy-Born hypothesis, that the atomistic lattice dplacement equals the macroscopic elastic displacement.advantage of the atomistic interpretation is that it allowsto tailor the energy density to suit specific atomistic geoetries where continuum elasticity fails to inform us aswhat discrete equations to use. In particular, we have usehere to derive numerical boundary conditions at the shcorners at the top of the film, and to derive equations atfilm/substrate interface.

Using the discrete version of Eq.~10!, the discretizedforce balance laws come from minimizing the total enerET . The resulting equations are

]ET

]uk~ i , j !50, ~14!

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which is the discrete analog of Eq.~12!. These discrete equations are formally equivalent to discretizations of the clascal elasticity equations. In particular, at interior points E~14! is a discretization of Eq.~7!, and at flat edges it is adiscretization of Eq.~8!. At corners and at material interfaces, however, these equations are new and do not admcontinuum interpretation.

The boundary conditions that arise from performingvariation on the discrete energy have the effect of regulaing the singularities that exist in continuum theory at, e.interior corner points. This regularization is not completeartificial; since our discretization is at the atomic scale, itperformed at the appropriate length scale. In a real matethe singularities present in continuum elasticity are, in faregularized by the atomic lattice. This regularization coube done in a more controlled fashion, entirely within oframework, by using an appropriately parametrized atomibond model for the energy at these singular points. Thisbeyond the scope of the present work, however, and wepect that qualitatively correct results for step equilibrium adynamics can be obtained with the present form of theergy.

Our method for treating elasticity should be placedcontext with an alternative approach,28,29 which is based onthe use of Green’s functions. This approach makes use oanalytic expression for the half-space Green’s function tdescribes the displacement at any pointx in the bulk due toa unit force acting at another pointx8. Then, assuming thathe bulk is homogeneously strained, a multipole expansiothe force distribution caused by a step can be made. Muplying each term in the expansion by the appropriate dertive of the Green’s function, the displacement at any podue to the presence of a step can be obtained as well aforce on one step due to another. This is a powerful aappealing approach, but has several serious limitations.use of the half-space Green’s function implies not only tthe epilayer is homogeneously strained, but also that theilayer thickness is much greater than the step height. Furtmore, the distance between steps must be sufficiently lathat the step interactions can be approximated by only aterms in the multipole expansion. Finally, if the material sytem is inhomogeneous, then depending on the geometryelastic parameters, the Green’s function approach mayeither difficult or impractical to implement.

These restrictions cause problems if the issue to bedressed is the interaction between steps on very thin epers on a vicinal substrate, or the interaction between islaon the first few epilayers on a singular surface. In many caof practical interest, heteroepitaxial layers are tunnel barror quantum wells whose thickness is often no more thafew layers. On a vicinal surface, the presence of steps onsubstrate causes the assumption of homogeneity of theayer strain to be violated. Even on nominally singular sfaces, the epilayers can easily be thin enough to violateassumption that the step height is much smaller thanepilayer thickness. The initial nucleation of quantum dotsof intense interest and also occurs in a regime whereGreen’s function is not valid. The epilayer is thin in thnucleation phase and island distances may be small eno

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A. C. SCHINDLERet al. PHYSICAL REVIEW B 67, 075316 ~2003!

to violate the assumptions of the multipole expansion. Oapproach suffers from none of these restrictions and, assing that linear elasticity holds down to length scales of a flattice spacings, as has been observed,30,31 our only approxi-mations are in the treatment of the atomistic effects relevvery near the step edges and at the surface. Even thesfects can be accommodated with some additional effordescribed in Sec. V.

IV. RESULTS AND DISCUSSION

As an illustration of our method, we now consider strelaxations and step-step interactions on strained vicinalfaces.

A. Step relaxations

We begin by examining the behavior of the displacemand strain fields given by our model for steps on a straivicinal surface. Figure 1 shows the basic geometry thatbe considered. The epilayer~shown in gray! consists of 40layers of material with isotropic elastic constantsl51 andm51 on a substrate~shown in black! of thickness 20, whichalso has elastic constantsl5m51. The buried step on theinterface between substrate and epilayer is horizontally offrom the surface step to indicate a generic nonsymmeconfiguration. Although only one step is shown, skeperiodic boundary conditions are applied so that the modescribes an infinite step train, not an isolated step.lower boundary of the lattice necessarily also has a stepbe consistent with the skew-periodic boundary conditio

FIG. 1. A vicinal surface with a single step consisting ofepilayer ofA-type atoms~shown in gray! on a substrate ofB-typeatoms~shown in black!.

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The atoms immediately on the lower boundary are heldzero displacement in all of the subsequent relaxation calations, to approximate bulk behavior.

For clarity we will examine the effects of surface stressa separate case from epilayer mismatch, with the knowlethat the effects can be combined if desired. For the casepilayer misfit with no surface stress, there is a lattice mmatch of 1% between substrate and epilayer with sign sthat the epilayer is under compressive strain. For the anate case of surface stress with no epilayer misfit, the bobetween pairs of atoms on the epilayer-vacuum interfacemodified to have a lattice constant 1% larger than the bugiving a compressive stress for the surface layer only. Inlatter case, the substrate and epilayer are indistinguishaso that the system is effectively a layer of thickness 60 wa step on the top and the bottom held at the zedisplacement bulk position.

Figure 2 shows thex and y components of displacemenproduced by the epilayer misfit model for this particular stconfiguration and Fig. 3 shows the displacements forsurface-stress only case. In Fig. 3 the coupling of the hzontal surface stress to vertical displacements is appaThe step edge breaks the planar symmetry of the surfaceallows this coupling, since the two stresses on either sidethe step produce a torque about the step edge.

Better qualitative appreciation of the results of the relaation can be gained from graphing the components ofstrain tensor. Figure 4, for the compressed epilayer, cleshows the distinct strain fields produced by the surface sand the buried interface step. The lineary displacement nolonger dominates the image as it did in Fig. 2~b!, since thestrain only contains derivatives of the displacement. Of nin the figure is the different ‘‘polarity’’ of thex andy strainsat the surface step vs. the interface step, and the strucaround each of the steps as the value of the strain compooscillates between positive and negative values as a funcof angle. Figure 5 shows the strains produced by surfstress only. In this case, the buried step experiences no stand the visible strain field is due to the surface step aloThe angular structure around the step in~a! and~c! is notice-ably different from Fig. 4, suggesting more dominant highmultipole moments. In both Figs. 2 and 3, the lower bounary of the lattice is invisible since it has the same zero vaof strain as the vacuum.

Having discussed some of the qualitative features of srelaxations produced by our model, we now turn to a mquantitative analysis. In particular, we wish to show thatdisplacements due to surface steps produced by our mare in quantitative agreement with predictions derived frcontinuum elastic theory in regimes where we expectlatter to be valid. The results from continuum elastic theowe will compare to are for displacements due to an isolastep; our method, however, most naturally treats periodicplaced steps in a step train. In order to separate the effean isolated step from that of the periodic images, we perfoa lattice sum of periodic multipole forces, which is then usfor evaluation of the multipole coefficients due to step dplacements.

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

FIG. 2. Displacement fields due to the step from Fig. 1 oncompressively strained epilayer. The color of the vacuum aabove and below the lattice indicates the zero of the scale,displacements are calculated with respect to the reference latticdescribed in the Appendix. The bottom-most layer of atoms is fiat zero displacement to approximate the bulk. Lighter shadingresents positive displacements and darker shading representstive displacements. Thex displacements are shown in~a! and theydisplacements~with a different zero of the grayscale! are shown in~b!. Because of the Poisson ratio of the material, there is an ovincreasingy displacement that visibly overwhelms the details oydisplacements due to the step itself.

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FIG. 3. Displacement fields due to the step from Fig. 1 wintrinsic surface stress in the material. The gray color in the vacuregion above and below the lattice represents the zero of the swith lighter shading indicating positive displacement and darshading indicating negative displacement with respect to the reence lattice. The shading gradient is not to the same scale as Fsince the absolute displacements from this surface stress effecsmaller. Thex displacements are shown in~a! and they displace-ments are shown in~b!.

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FIG. 4. Components of thestrain field due to a step on a compressively strained epilayer, withthe same step geometry as prevous figures. In each sub-figure, thshading of the vacuum regionabove the lattice represents thzero of the scale, with white representing positive values anblack representing negative vaues. The components shown arefollows: ~a! xx component,~b! yycomponent, ~c! xy component,and~d! hydrostatic strain~trace ofstrain tensor!.

die

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The effect of a single isolated step in an~horizontally!infinite domain is addressed by superposition in perioboundary conditions. The computational domain, whskew-periodic boundaries are applied, is effectively infinand is populated by an infinite number of equally spacsteps. The distortion fields produced by each of these slinearly superposes, however, and may be summed. Forample, a one-dimensional ‘‘dipole potential’’ term32 ofstrengthm1 for an isolated step atx0,

V115

m1

x2x0~15!

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V1`5 (

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L~x2x0! D , ~16!

whereL is the periodicity of the system. The summed dipoforce is found as the negative derivative of the potential:

f 1`52

d

dxV1

`5m1S p

L D 2

cosec2S p

L~x2x0! D . ~17!

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

FIG. 5. Components of the strain field due to a step with intrinsic surface stress in the material. In each subfigure, the shadivacuum region above the lattice represents the zero of the scale, with lighter shades representing positive values and darrepresenting negative values. The components shown are as follows:~a! xx component,~b! yy component,~c! xy component, and~d!hydrostatic strain~trace of strain tensor!.

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A. C. SCHINDLERet al. PHYSICAL REVIEW B 67, 075316 ~2003!

All higher order multipole terms may be summed anagously. The single step monopole potential term

V015m0 ln~x2x0! ~18!

cannot be directly summed and instead the corresponforce

f 0152

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dxV0

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x2x0~19!

is summed, and then integrated to give the lattice-summmonopole potential

V0`5m0 lnFsinS p

L~x2x0! D G . ~20!

These periodic functions are best examined on a singleriod extending over theterrace between two steps~Fig. 6!,

FIG. 6. Displacements at the surface layer~from Fig. 2! for astrained epilayer with no surface stress. Thex displacements~a! andy displacements~b! are shown as a function of position from onstep to the next periodic image step. The dots are the displacemproduced by the model and the solid lines are fits to the funcdescribed in the text, up to dipole order.~The first and last pointsare left out of the fit.!

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and not on a domain containing a pole, which would requfitting with an infinite discontinuity between left- and righhand sides.

The coefficientsm0 , m1 , m2, etc., of these lattice-summed terms are identical to the coefficients of the isolastep continuum multipole expansion. This allows the coecients from a fit to our model on a terrace to be compadirectly to the coefficients derived from the continuutheory for a single step.

Figure 6 shows thex andy displacements along a terracfor the case of a uniformly compressed epilayer. Includedfits to the functional form described above to dipole ordWe also fit the data to a 12 term series and extract theresponding multipole coefficients of the isolated step; thare shown in Fig. 7. We first note that the logarithmic monpole is dominant, and the falloff with increasing multipoorder is rapid and uniform. The dipole coefficient, for eample, is more than an order of magnitude smaller thanmonopole coefficient. Dominance of the logarithmic monpole is also evidenced by the fact that using only the mopole and dipole terms results in the excellent fit shownFig. 6. The simplest continuum elastic theory predictsexistence of a monopole term only and should be valid inlimit of the step height small compared to the distancetween steps in our model. That condition is reasonably wsatified for the case shown here, hence the relative donance of the monopole term. The existence of other mupole terms, though small, nevertheless shows that our mis capturing some of the atomistic effects that are presenttherefore goes beyond a purely continuum description.

Figure 8 shows the monopole coefficientm0x for the xcomponent of the displacement along the surface as a ftion of lattice mismatche. For eache, m0x is calculatedfrom the simulation data for 16 different values of the elasparametersl and m. These data are compared to the theretical predictionm0x52(l12m)/e, which can be derivedeither from continuum theory33 or from a discrete model.27,34

The agreement between the simulation data and the anaexpression is quite good, but not exact. This is again whatwould expect since we are in a regime in which the atomis

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FIG. 7. Magnitude ofx and y displacement multipole coeffi-cients for first 12 multipolesm0•••m11 for a strained epilayer withno surface stress.

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

effects captured by our discrete model are small, butcompletely negligible. Thus, we have shown from this athe preceding data that our model reproduces the expecontinuum results in the approriate limit, but with correctiodue to the discrete nature of the model that may be imporin other regimes.

B. Step-step interaction energy

We now turn our attention to the interaction energy btween steps produced by our model. For a general linelastic material with a stepped surface, there are forcestween steps even in the absence of misfit strain. InGreen’s function approach discussed above, these foseparate~conceptually! into two types: a repulsive ‘‘dipole’’interaction,33 which is due to the intrinsic surface stressthe steps, and a logarithmic repulsion between inequivasteps in the form of a force ‘‘monopole,’’35,36which is due tothe elastic distortion of the surface.

In a film under an externally imposed strain, such as tderived from coherent epitaxy to a lattice-mismatched sstrate material, there is an additional, attractive interacbetween steps due to a force ‘‘monopole’’36 which is loga-rithmic. The unstrained monopole force of the previous pagraph is fundamentally different from this strained monpole, in that the presence or absence of the former depon relative step orientations, while the latter is present forsteps. In the calculations described below, all steps facesame direction, obviating any repulsive monopole.

The force f m on a single step at positionxm is approxi-mated, to dipole order, by26,28,33

f m5 (n

nÞm

S a1

~xn2xm!2

a2

~xn2xm!3D , ~21!

wherea1 is determined by the elastic constants of the marials and the lattice mismatch, anda2 is determined by theelastic constants of the epilayer material and the intrinsurface stresses.

FIG. 8. Monopole coefficientm0x for the x component of thedisplacement along the surface. Data is plotted as a functiolattice misfit e for 128 values of elastic coefficientsl, m, ande.Values from the simulation (h) and from theory (s) are plotted.

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In light of the above discussion, we now compare telastic energy due to step-step interaction obtained inmodel to the predicted interaction. To calculate the total etic energy as a function of step distance, we use the geomshown in Fig. 9, with skew periodic boundary conditionsappropriate for modeling an infinite step train. The two steare moved by equal amounts in opposite directions to cserve the mass of the system, thereby preventing the untrolled introduction of ‘‘background’’ bulk elastic energy ithe process. With the steps at each lattice position betwconfigurationsA andB, we record the total elastic energy.

We expect that the relevant quantities determiningqualitative behavior of the system arem i /l i and ms /me ,where the subscriptss and e are for substrate and epilayeproperties, andi P$s,e%. Every property that depends on latice mismatche does so linearly, so that scales out as weFor simplicity we calculate withl5m51, ande51%. ForSi, m/l'1.1; for Ge,m/l'1.2, andmSi /mGe'1.2, so thequalitative results should hold for that physical system aothers with similar scaled elastic properties. In this comptation, the grid spacing was chosen such that each atolattice spacing is one numerical grid point across. The syshas a lateral size of 40 lattice spacings, which wouldequivalent to a physical size of approximately 22 nm foSi/Ge system.

The first case we consider has an epilayer thickness oML, large enough that we expect this case to behave as ifepilayer were uniformly strained. The results for the toenergy as a function of step separation are shown bysquare data points in Fig. 10. The agreement betweendata from our model and the solid line is excellent, indicatithat our model reproduces the expected logarithmic intetion between steps on a homogeneously strained epilayedistances larger than one atom.

of

FIG. 9. Geometry used for calculating two-step interactionergy. Left and right boundaries are skew periodic. Steps are mofrom configurationA to configurationB in such a way that the totaarea remains constant~mass is conserved!.

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A. C. SCHINDLERet al. PHYSICAL REVIEW B 67, 075316 ~2003!

The circle data points in Fig. 10 show the total elasenergy for an epilayer thickness of only 5 ML. Note that tbehavior is qualitatively different than the thick epilaycase. There is a ‘‘dip’’ in the energy as the surface steps pover the terrace midway between configurationsA and B.This local minimum in energy becomes more pronouncedepilayers thinner than 5 monolayers. Figure 11 showshydrostatic strains for two of the configurations whose engies are plotted in the preceding figure. Note that theresignificant elastic interaction between the epilayer stepssubsurface features at the epilayer-substrate interface.interaction is what leads to the local minimum in enerwhen the epilayer steps are moved with respect to the busteps, and is an effect not present in models assuminhomogeneously strained epilayer. In fact, this effect wobe very difficult to capture with any Green’s function-basapproach.

We expect this to have considerable implications fordynamics of step bunching. Although only metastable, sconfigurations that experience significant coupling tosubstrate may be expected to slow or even completelypress the step bunching phenomenon normally expectevicinal surfaces even under annealing conditions. A fullvestigation of step dynamics using this elastic model coupto an equilibrium model for step dynamics28 is in progresswill be reported on elsewhere.

V. SUMMARY AND FUTURE APPLICATIONS

We have presented a new method to study the influencelastic interactions in strained one-dimensional systemsing an approach based on lattice statics using linear elasttheory. An application is the problem of step relaxations astep-step interactions on vicinal surfaces. We have examin detail the behavior of our model for this particular prolem and found the step relaxations to be consistent withpredictions of continuum elastic theory in the limit where t

FIG. 10. Total system energy for a step moving from configration A (s50) to configurationB (s520), for both a 30 atomthick epilayer ~squares, left axis! and a 5 atom thick epilaye~circles, right axis!. Both calculations are on a 30-layer substraThe solid line is a logarithmic fit to the thick epilayer data, and tinset is the central portion of the thin epilayer data with 203 ver-tical exaggeration.

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e

continuum theory would be expected to hold~the thick epil-ayer limit!. We have also demonstrated, however, thatdiscrete nature of the model allows us to capture manyfects that are essentially atomistic in nature. When consiing step-step interactions, we found that there is a significinfluence of perturbations of the elastic field on the surfadue to buried substrate morphologies. These may well lealong relaxation times for very thin epilayers, where the atual critical thickness depends on the misfite as well as onthe elastic constants of the two involved materials. Upincrease of the epilayer thickness, we observe a decay osubstrate influence until at epilayer heights of roughly 30–monolayers the effects of the initial substrate configuratvanish.

We expect this general methodology to have wide apcability to problems involving strain in epitaxial growthHere, the method was formulated for a simple cubic lattice2D, but it is easily extendible to noncubic lattices and 3D.addition, the atomistic nature of our formulation makesapplicable to including surface and step edge effects, pvided that a suitable description of the local energy at thsites is available. At the atomistic level, our model isexample of a valence force field model. Valence force fimodels, such as the Keating model, have been validatedbulk elastic properties of many real materials.37–39 Valenceforce field models accurately describing the energy of atoconfigurations at surfaces and step edges are not geneavailable, but there are no fundamental obstacles to deveing and validating such models.

One of the main motivations for this work is our intentioof incorporating elastic effects into the level-set method40 fordescribing the morphological evolution of epitaxial films.41

This technique is based on the representation of the mogrowth front ~the step edges! in terms of an auxiliary func-tion ~the level set function! which permits a straightforwardsolution of the associated Stefan problem and handlesnatural way the topological changes associated withnucleation of islands and their coalescence. By treatingx-y variables as continuous, but thez direction as discretethis method is ideally suited both to coupling continuofields to island motion and to describing abrupt atomiseffects associated with the initial stages of heteroepitagrowth, such as the 2D-3D transition during StransKrastanov growth.42 The coupling of the adatom diffusionfield to island-boundary motion has already beaccomplished,43 so we now turn to the effect of elasticity othe motion of island boundaries.

The new method opens up a vast field of possible apcations. Beginning with the most fundamental aspectsgrowth, we can examine the effect of strain on adathopping,44 nucleation, and island stability,45 as well as inves-tigate the issue of kinetic versus equilibrium effects in tisland statistics.46,47 These factors, in turn, impact upon thextent of lateral2 and vertical ordering48,49 of 2D and 3Dislands. Growth phenomena on patterned substrates iseasily within the scope of this method.50

ACKNOWLEDGMENTS

We gratefully acknowledge discussions with FraGrosse, David Srolovitz, and Jerry Tersoff. This work w

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

FIG. 11. Hydrostatic strain for sample configurations from the Fig. 10 calculation. The top two subfigures~a!, ~b! are a 30-monolayer filmand the bottom subfigures~c!, ~d! are a 5-monolayer film. The subfigures on the left~a!, ~c! are fors50 and those on the right~b!, ~d! arefor s5L/2520.

ivtee

ro

ti-

-

supported by the NSF and DARPA through cooperatagreement No. DMS-9615854 as part of the Virtual Ingrated Prototyping Initiative, the NSF through Focused Rsearch Grant~FRG! No. DMS-0074152, and by the~UK!EPSRC. A.C.S. was supported by ‘‘Hochschulsonderpgramm III von Bund und La¨ndern’’ ~Germany!.

APPENDIX: ELASTIC ENERGY

For a two-dimensional cubic lattice with lattice constanhand lattice coordinatesi5( i 1 ,i 2), denote the reference postion as x5(x1 ,x2), the elastically deformed position asX5(X1 ,X2), and the displacement asu5(u1 ,u2)5X2x. De-

07531

e--

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fine the translation operatorsTk6 and finite-difference opera

tors Dk6 ,Dk

0 as follows:

Tk6 f ~ i!5 f ~ i6ek!, ~A1!

Dk1 f ~ i!5h21~Tk

121! f ~ i!,

Dk2 f ~ i!5h21~12Tk

2! f ~ i!, ~A2!

Dk0f ~ i!5~2h!21~Tk

12Tk2! f ~ i!,

in which ek is the unit vector in thekth direction for k51,2. Define the bond displacementdk6 at the pointi as

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A. C. SCHINDLERet al. PHYSICAL REVIEW B 67, 075316 ~2003!

dk6~ i!5~d1k6 ,d2

k6!5Dk6u~ i!2eek . ~A3!

As discussed in Sec. II, the lattice mismatch parametere inthe epilayer is the relative difference between the equilibrilattice constant and the lattice constant imposed on the eayer by the substrate.

The discrete strain components at a pointi are defined as

Skk6 5dk

k6 ,

Sklpq5~dk

lq1dlkp!/2 ~A4!

in which the values ofk and l are 1 or 2 and the values ofpandq are1 or 2. The strain componentSkk

6 corresponds toa bond in the6ek direction from the pointi; the componentSkl

pq corresponds to two interacting orthogonal bonds inpek andqel directions from the pointi.

1. Micromechanical model

The elastic energy used here has a micromechanical inpretation as consisting of nearest neighbor springs, diagsprings, and bond bending terms. For nearest neighbor lisprings and linearized bond bending springs, in whichspring constants area and b, respectively, the energy atpoint is

Ennbb51

2a (

p56,k51,2~Skk

p !211

2b (

p56,q56~S12

pq!2.

~A5!

In order to retain maximal locality, we use ‘‘virtual’’ diagonaspring with spring constantc, defined for example betweenpoint (0,0) and the average of its nearest neighbors (1,0)(0,1), for which the energy is

E1151

2c~e11e2!• H @u~1,0!1u~0,1!#Y22u~0,0!J 2

5c~S111 1S22

1 12S1211!2/8. ~A6!

More generally forp56,q56 define

Epq5c~S11p 1S22

q 12pqS12pq!2/8. ~A7!

The energy densityE is a combination of these four virtuadiagonal springs; i.e.,

E5 (p56,q56

Epq. ~A8!

Add and rearrange Eqs.~A5! and ~A8!, to obtain the result-ing energy

E5a (p56,k51,2

~Skkp !21 (

p56,q56$2b~S12

pq!21gS11p S22

q %

~A9!

in which

a5~a1c!/25C11/4,

b5~b1c!/25C44/4,

07531

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er-alare

nd

g5c/85C12/4, ~A10!

andCi j are the Voigt constants. The energy density~A9! isthe discrete analogue of the continuum energy for elastiwith cubic symmetry. For isotropic elasticity, as in Sec.the coefficients should be chosen as

a5~l12m!/4,

b5m/4,

g5l/4, ~A11!

i.e., a5m2l, b5m/222l, c52l.The discrete energy density~A9! has been chosen to b

maximally localized; so that the energy densityE at a pointiis a quadratic function of displacementu at the five pointstencil consisting of the pointi and its nearest neighbors, anthe corresponding force balance equations involve onlynine point stencil consisting of the pointi and its nearest andnext-nearest neighbors.

2. Interfaces

For problems in which the underlying lattice has cubsymmetry but the material geometry includes interfaces,generalize the energy in Eq.~A9! by only keeping bond in-teractions that are consistent with cubic symmetry butimposing a symmetry constraint on the strength of the inactions. The resulting energy has the form

E5 (p56,k51,2

akp~Skk

p !21 (p56,q56

bpq~S12pq!21gpqS11

p S22q .

~A12!

Each coefficientakp , as well as the lattice mismatch pa

rametere, corresponds to a bond between two atoms; eof the coefficientsbpq andgpq corresponds to the interactioof two bonds in orthogonal directions from a central poiwhich defines a square ‘‘cell.’’ We assume that the valuesak

6 ande (bpq andgpq) depend only on the material type othe two ~four! atoms at the endpoints of the correspondibond ~cell!.

Consider a system consisting of two materials with elasparametersam,bm,gm,em for m51,2. Denote a cell or bondto be ‘‘pure’’ if all of its vertices are of a single material typand ‘‘mixed’’ otherwise. For maximal simplicity, we makthe following assumptions, which could easily be generized.

~1! For pure bonds~pure cells! in materialm, akp5am and

e5em, (bpq5bm andgpq5gm).~2! For mixed bonds~mixed cells! in a two-material sys-

tem, akp5 1

2 (a11a2) and e5 12 (e11e2) @bpq5 1

2 (b11b2)andgpq5 1

2 (g11g2)].~3! For a bond~cell! in which one of the vertices is in the

vacuumakp50 (bpq5gpq50).

3. Force balance equations

Assumptions~1!–~3! provide an algorithm by which theelastic coefficients and elastic energy can be determined

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THEORY OF STRAIN RELAXATION IN . . . PHYSICAL REVIEW B67, 075316 ~2003!

any material configuration involving two materials~i.e., asubstrate and an epilayer! and a vacuum. Once the energyEis determined, the force balance equations at each pointi arethe minimization conditions

]E/]u~ i!50. ~A13!

For the quadratic energy described above, the derivativ

rs

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lid

.

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r.

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lc

Leisd

07531

in

Eq. ~A13! is a linear function ofu( i8) over values ofi8 thatare equal to, nearest neighbors of, or next-nearest neighof i. The coefficients can be exactly determined as a findifference ofE with respect tou( i) andu( i8). Then a linearequation solver is used to findu by solving Eq.~A13!. Thisprocedure does not require analytic determination offorce balance equation, which is an advantage becauseanalysis has many different cases.27

p.

s

er,

B

s.

ev.

-

et top

po-

, J.

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