Formation, ripening, and stability of epitaxially strained island arrays

Formation, ripening, and stability of epitaxially strained island arrays

Helen R. Eisenberg* and Daniel Kandel†

Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, IsraelsReceived 20 October 2004; published 23 March 2005d

We study the formation and evolution of coherent islands on lattice-mismatched epitaxially strained filmsusing two-dimensional evolution simulations. Faceted islands form in films with anisotropic surface tension.Under annealing, these islands ripen until a stable array is formed. We show that the presence of a cusp in thesurface tension is essential for reproducing the experimentally observed characteristics of the stable islandarray. It is also shown that a single cusp in the surface energy is sufficient in order to explain the island-shapetransition and associated bimodal island-size distribution which occurs in growth experiments. In films withisotropic surface tension we observe and explain a new mode of growth in which a stable wavy surface isformed.

DOI: 10.1103/PhysRevB.71.115423 PACS numberssd: 68.55.2a, 81.15.Aa

I. INTRODUCTION

Coherent sdislocation-freed islands form to relieve thestrain associated with lattice-mismatched heteroepitaxial thinfilms. The islands can then self-organize to create periodicarrays. When such arrays are embedded in a wider band-gapmaterial, they can create quantum dot structures of electronicsignificance in semiconductor or optoelectronic devicesssee,e.g., Ref. 1d. Indeed there is currently an explosion of interestssee, e.g., Ref. 2–4d in exploring the use of self-organizedquantum dot arrays to create optoelectronic devices, in par-ticular quantum dot lasers.

Of particular interest is whether the island arrays thatform are energetically stable or metastable configurationsthat will ripen. Here we show in annealing simulations thatanisotropy in surface tension is necessary for the formationof a stablesroughly periodicd array. Moreover, we show thatthe presence of a cusp in the surface energy is essential forreproducing the experimentally observed characteristics ofthe island array, in particular the increase in island densitywith increasing film thickness.5–11 The cusped surface ten-sion produces sharply edged faceted islands, which are sta-bilized by the elastic relaxation the edges contribute. We alsoshow that when surface tension is cusped, islands form in a“chain-reaction ripple” effectsi.e., islands tend to developnear other islandsd. This mode of growth has also been ob-served in experiment.12–14

Bimodal island-size distributions have been observedexperimentally15–19 and are related to an observed change inthe shape of growing islands.15–19 Due to the relationshipbetween the shape change and the island-size distribution, anunderstanding of shape transitions is crucial in order to ob-tain the narrow uniform island size distributions necessaryfor successful device applications. Previous groupsssee, e.g.,Ref. 20 and 21d have modeled an island shape transitionwhen two cusped minima are present in the surface tensionother than at 0° and hence two distinct faceting angles arepresent in the crystal shape. We however show that the shapetransition and associated bimodal island-size distribution canoccur with only one surface tension minimum—that is, with-out an additional facet orientation at a larger angle.

We see a new phenomenon during long-term annealingafter ripening has stopped, in which islands equilibrate to

have a very uniform size distribution. We would have tocarry out our simulations on larger samples to ensure this isnot a finite-size effect. This phenomenon could be very use-ful in producing uniform quantum dot arrays.

In films with isotropic surface tension we observe andexplain a new mode of growth in which a stable wavy sur-face is formed. Stable, nonflat morphology has previouslyonly been predicted for faceting films.22–25

We study the energetic profile of the entire strained islandarray during film evolution. We consistently see free-energyplateauing during ripening events and then dropping verysharply at the end of the ripening event.

II. PROBLEM FORMULATION

The self-assembly of quantum dot arrays is not well un-derstood even on the qualitative level. Here, we try to ex-plain the general qualitative behavior observed in typical ex-perimental systems rather than quantitatively reproducesystem-specific results. We study the evolution of an elasti-cally isotropic system using continuum theory. The surfaceof the solid is aty=hsx,td and the film is in they.0 regionwith the film-substrate interface aty=0. The system is mod-eled to be invariant in thez direction and hence is essentiallytwo dimensional. This is consistent with plane strain wherethe solid extends infinitely in thez direction and hence allstrains in this direction vanish.26 All quantities are calculatedfor a section of unit width in thez direction. The islands wedescribe are equivalent to elongated island ridges or wires.

All the results mentioned in this paper relate to vicinalsurfaces with a very small miscut angle in thez direction.Experimentally, surfaces often have such a small miscut, as itis very difficult to grow a perfect facet.

We assume that surface diffusion is the dominant masstransport mechanism, leading to the following evolutionequation27:

]h

]t=

DshV

kBT

]

]x

]m

]s, s1d

whereDs is the surface diffusion coefficient,h is the numberof atoms per unit area on the solid surface,V is the atomic

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volume,T is the temperature,kB is the Boltzmann constant,sis the arc length, andm is the chemical potential at the sur-face.

In our previous work28,29 we showed thatm can be ex-pressed as

m

V= g̃sudk +

dfels0d

dh+ US1

2Sijklsijskl −

1

2Sijklsij

s0dskls0dDU

y=hsxd,

s2d

wherek is the surface curvature,u is the angle between thenormal to the surface and they direction, andg̃sud=gsud+]2g /]u2 is the surface stiffnessfwith gsud being the surfacetensiong. Sijkl are the compliance coefficients of the material,sij is the total stress in the material,sij

s0d is the mismatchstress in the zero-strain reference state, andfel

s0dshd is thereference-state free energy per unit length in thex direction.The reference state is defined as a flat film of thicknesshconfined to have the lateral lattice constants of the substrate.

Linear stability analysis predicts that a flat film thinnerthan the linear wetting layer thicknesshc is stable at all per-turbation wavelengths and is marginally stable to perturba-tions of wavelengthlc for thicknesshc. The expressions forhc andlc are given in Ref. 28 and 29. Abovehc the flat filmis unstable to a larger and larger range of wavelengthsl−øløl+ until for infinitely thick films the film is unstable toall perturbations of wavelengths larger thanl=lc/2.

We simulated the surface evolution given by Eqs.s1d ands2d and using the numerical scheme described in our earlierwork.29 We used the cusped form of surface tension given byBonzel and Preuss,30 which shows faceting in a free crystal:gsud=g0f1+b usin spu / s2u0dd u g, where b<0.05 andu0 isthe angle of maximumg. We considered a crystal whichfacets at 0°,645° and690° with u0=p /8. This is a genericmodel of crystal surface tension and is only meant to showthe general properties of a faceting material and not to cor-respond exactly to a specific experimental system. The cuspgives rise tog̃=` at the facet angle. However, a slight mis-cut of the low-index surface along thez direction leads to arounding of the cusp, which can be described by

gsud = g0F1 + bÎsin2S p

2u0uD + G−2G , s3d

whereG is inversely proportional to the miscut angle and,for example, G=500 corresponds to a miscut angleDu<0.1° ssee Fig. 1 for surface tension plotsd. As mentionedearlier all the results mentioned in this paper relate to sur-faces with a very small miscut angle in thez direction.

dfels0dshd /dh was obtained fromab initio quantum me-

chanical calculations of Si1−xGex grown on Sis001d sfor de-tails see Ref. 33d. All our simulations start from a randomlyperturbed flat film with an initial thickness denoted byC.

III. RESULTS

When perturbations larger than a critical amplitude28,29,31

are applied to a flat film, faceted islands develop in the filmduring both annealing and growth, as illustrated in Fig. 2.

The critical perturbation amplitude was largely independentof cusp smoothnessG, unlike the linear wetting layer thick-ness which depended strongly onG. We carried out most ofour simulations forG=500 but we also ran selected simula-tions for G=5000 and perfectly cusped surface tension. Theresults were qualitatively identical and quantitatively verysimilar. This is important as in experimental systems the ex-act miscut angle varies and observable phenomena shouldnot be highly dependent on it. The film first becomes un-stable at wavelengthl,50gs0d /M«2, whereM is the plainstrain modulus and« is the lattice mismatch. The islandswhich form from this perturbation typically have a width ofabout 10% of the unstable wavelength. Both the unstablewavelength and the faceted island widths scale as«−2, asobserved in experiments32–35 in which islands develop fromlong-ripple-like structuresscorresponding to our model ofplane straind.

All results discussed henceforth refer to Ge/ Sis001dthough the same trends were seen in Ge0.5Si0.5/Sis001d. Is-lands form in a “chain-reaction ripple” effectsi.e., islandstend to develop near other islandsd as is illustrated in Fig. 2.The ripple effect occurs because the growth of the islanddestabilizes the flat faceted film at its boundaries. This modeof growth has also been observed in experiments.12–14 Egg-leston and Voorhees36 observed a similar chain reaction ofisland formation in simulations of a strained film grown on asubstrate with a mesa. After initial island formation we ob-serve island ripening occurring over much longer time scalessabout 50 times longerd.

During annealing the islands are fully faceted. Their topsare faceted at 0° and their sides at 45°. This shape is pre-served as the islands grow; i.e., the islands maintain a fixedheight-diameter ratiosas seen in experiments5,15,32,37 andtheory38d. During deposition, on the other hand, an interest-ing transition is observed in the island shape. Initially, theislands are fully faceted as during annealing. However, whenthe islands reach a certain diameter, they stop growing later-ally and only vertical growth occursssee Fig. 3d. This criticaldiameter is about 40 nm for Ge islands grown on a Sis001d

FIG. 1. The figure shows the form of surface tensiong used inEq. s3d and in particular shows the effect of rounding the surfacetension cusp. In the isotropic case shown hereg=1.

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substratesduring annealing we never observed islands whichexceeded this diameterd. Thus the islands become tall andnarrow, and their sides are steeper than 45°. This shape tran-sition is observed experimentally15,17–19and is sometimes re-ferred to as the “pyramid-to-dome transition.” The drivingforce behind it is the increased elastic relaxation experiencedby tall narrow islands. Theoretical equilibrium calculationswith isotropic surface tension39 show a continuous increasein island aspect ratio with increasing island volume, as elas-tic effects dominate surface tension effects. The sharp ratherthan smooth transition in growth mode we observe is due tothe anisotropic nature of the surface tension and in particularthe presence of a surface tension cusp at 45°. In smallerislands where surface effects dominate, islands are lockedinto the desired facet angle, until a sharp transition pointwhere the elastic effects dominate and they start growingvertically. Note that contrary to existing explanations of the

island shape transitionssee, e.g., Refs. 20 and 21d, the tran-sition occurs without an additional facet orientation at alarger angle. Altering the exact facet value and/ or having asecond steeper faceting angle such as in the dome case, onlychanges specific quantitative properties of this transition,such as the island size and shape at which the transitionoccurs.

The transition in the island-shape and -growth mode isclearly reflected in the size distribution shown in Fig. 4. Nar-row island size and spacing distributions are seen duringearly depositionssee Fig. 4, 20 equivalent monolayersd.These narrow distributions are observed in manyexperiments.5,6,15–17,19,32,40,41After additional depositions30equivalent monolayersd a bimodaldistribution forms as someof the islands pass from the fully faceted to the tall narrowshape. At later timesse.g., 50 equivalent monolayersd nearlyall islands have the tall narrow shape. At this stage the dis-tribution becomes quite symmetric and evolves at a fixeddistribution widthsincreasing its meand. Similar results wereobserved in experiments.15,17–19

One of our central observations is that annealing of aperturbed flat film with anisotropic surface tension leads tothe formation of a stable array of islands. This result is con-sistent with several experimental systems7,16,42,43 and is incontrast with films of isotropic surface tension where theislands ripen indefinitely. Theoretical studies also predictstable island arrays.22–25The crucial term in determining thestability of an island array apart from anisotropic surfacetension and a film-substrate interaction is the presence of anelastic contribution due to island edges. This contribution isautomatically present in our calculations and does not needto be introduced separately. Theoretical works that ignorethis term41,44 predict continuous ripening.

Our simulations show that the density of islands in thestable array increases with increasing film thicknessssee Fig.5d. An increase in island density with film thickness has alsobeen seen in many experiments.5–11 Indeed Milleret al.7 andKamins et al.11 performed annealing experiments and Le-

FIG. 2. Evolution of a randomperturbation on a 25-monolayer-sML- d thick Ge film on a Sis001dsubstrate. In the first graph thedashed line is the initial perturba-tion and the solid line is the sur-face at t=0.028 s. The second tofifth graphs show the surface attimes t=0.044 s, 0.068 s, 0.123 sand 2.181 s. The final graph is thestable steady-state island array.Note the ripple effect in islandformation and the later island rip-ening leading to a stable island ar-ray. Periodic boundary conditionswere applied in thex direction.The same results were observed insimulations with double the periodlength.

FIG. 3. Island growth during material deposition. In this particu-lar case the island grew in the fixed height-width ratio mode up tot=0.7 s and then switched to the vertical growth mode. In this fig-ure Ge was grown on top of a Sis001d substrate. The initial flat filmis 10 ML thick and Ge is deposited at a rate of 5.2 nm/s.

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onardet al.8 performed experiments with very small deposi-tion rates. These three experiments clearly show the increasein island density as film thickness increases. This result waspredicted by Daruka and Barabási23 in minimal, energy equi-librium calculations. Here we show that the increase in is-land density also results from evolution simulations. Thisobservation is particularly important, since other evolutionstudies22 predicted a decrease in island density with increas-ing film thickness. We believe this is due to the smooth formof surface tension used in Ref. 22. Indeed, when we carriedout simulations with a smooth form of surface tension simi-lar to that used in Ref. 22, we also observed a decrease inisland density. This clearly demonstrates the importance ofusing a cusped form of surface tension to accurately modelevolution of faceting surfaces. As we stated earlier the elasticrelaxation due to the sharp island edges of faceted islandsappears to be critical in order to correctly predict the prop-erties of a faceted island array.

The increase in island density which we observe is anenergetic rather than kinetic effect, as the data is obtainedfrom long annealing simulations. Many simulations with dif-ferent initial surface morphologies but the same flat filmthickness have the same final island density, further indicat-ing that this is an equilibrium result.

As can be seen in Fig. 5, the island size also shows aslight increase with increasing film thickness, with islandsincreasing in width from 25 nm to 40 nm and cross-sectionalarea from 100 nm2 to 500 nm2 srecall that islands are infi-nitely long in thez directiond. The island size at 7 ML islarger than expected due to finite-size effects. Note that eventhe smallest islands have a finite nonzero size. Experimentsindeed see islands forming only above a certain size whichincreases with increasing film thickness.5,6,11,16,41However,as the annealing experiments which showed stable island ar-rays tended not to vary the film thickness, it is difficult tocompare our results with experimental observations. Our re-sult is in accordance with that predicted in equilibrium cal-culations by Daruka and Barabási.23

The islands in the final island array, during long-term an-nealing after ripening has stopped, tend to equilibrate so thatmost islands have exactly the same size apart from occasion-ally one or two larger or smaller onesssee Figs. 2 and 6d.This is unlike the island distribution during evolution ordeposition where the size distribution is of finite widthsseeFigs. 6 and 7d. When deposition is halted and the sampleannealed, the islands equilibrate so that most have exactlythe same size. This is a new effect not seen in other studies.

FIG. 4. Distribution of island cross-sectional areaA srecall thatislands are infinitely long in thez directiond, during directed depo-sition of Ge on a Sis001d substrate. The rate of deposition is 5.2nm/s, and the initial film height is 10 monolayers. The dashed ver-tical line shows the separation between the early growth mode inwhich the island height-width ratio is preserved and the later verti-cal growth mode.

FIG. 5. Island densitysbottom graphd and average island cross-sectional areaA stop graphd srecall that islands are infinitely long inthez directiond, in stable arrays of Ge islands on a Sis001d substrateafter ripening has ended.C is the initial flat film thickness, andhml

is the thickness of one monolayer. The error bars refer to the fol-lowing: Bottom graph: the span of island densities observed withdifferent initial surface morphologies of the same film thickness.Top graph: the standard deviation in island sizes observed through-out all samples of the same film thickness.


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We need to check that this effect also occurs in samples oflarger size before we can claim that it is an experimentallyobservable effect. There do not appear to be any experimentswhich compare island-size distributions after ripening hasstopped during long-time annealing in the absence of dislo-cated huge islandssKamins et al.11 who see stable islandarrays forming during annealing do compare size distribu-tions but not in the absence of dislocated “superdomes”d.This effect could be very useful in producing uniform quan-tum dot arrays.

When surface tension is isotropic, corresponding to filmsabove the roughening transition temperature, flat film evolu-tion during annealing is very different from that described

above. Perturbations in films thinner thanhc decay, and flatfilms with thicknesshc,h,hc+D, whereD<1 monolayer,develop a stable smooth wavy morphology atlc. That is,perturbations of other wavelengths decay and perturbationsof wavelengthlc grow to a finite amplitude. This is a modeof growth neither seen nor predicted before. Stable, nonflatmorphology has previously only been predicted for facetingfilms.22–25 In fact, other groups maintain that isotropic filmsshould be unstable to ripening.22,25,39,45While films are lin-early unstable to perturbations of wavelengthsl−øløl+,our simulations show that the nonlinearity stabilizes thegrowth of wavelengths close tol− and l+. As a result the

FIG. 7. Surface morphology when Ge is deposited at a rate of5.2 nm/s onto a 20-ML Ge film on a Sis001d substrate. The dashedline is the surface morphology after 0.14 s of depositions,5 ML ofGe deposited on top of the 20-ML filmd. The solid line is the surfacemorphology after deposition is interrupted and the surface is al-lowed to equilibrate. During deposition the island-size distributionis narrow but finite. In the equilibrium array islands equilibrate, somost have exactly the same size.

FIG. 8. Evolution of a randomperturbation on an isotropic filmof height C=7.2 ML=hc

+1.9 ML. «=2%. The dotted linerepresents the linear wetting layerthickness. In the first graph, thedashed line represents the initialrandomly perturbed film and thesolid line the surface att=4s. Thesecond graph shows the surface att=40 s, the third att=96s and thelast graph the surface att=255 s.hml is the thickness of onemonolayer.

FIG. 6. Surface morphology when a 25-ML film of Ge on aSis001d substrate is annealed. Periodic boundary conditions wereapplied in thex direction. The solid line shows the steady-state filmafter annealingst=7.2 sd the dashed line shows the film duringisland evolution and ripeningst=1.5 sd. During evolution theisland-size distribution is narrow but finite. In the equilibrium arrayislands equilibrate so most have exactly the same size.


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growth of the perturbation saturates and stops at a finite am-plitude, as seen by Spencer and Meiron46 for infinitely thickfilms. When films are sufficiently close to the linear wettinglayer thickness, the range of nonlinear saturation extendsover the entire range of linearly unstable wavelengths and soa stable wavy morphology is observed. For films thicker thanhc+D, initially a wavy structure at the most unstable wave-length forms. The hills of these waves then ripen on largerand larger length scales, until isolated islands are left thatcontinue ripeningssee Fig. 8d.

In all our simulations we saw several common aspects tothe ripened morphology which we also observed in filmswith faceted islands, suggesting that these features arise dueto the interplay between relieving the linear elastic strainenergy and the nonlinear elastic energysthe wetting layertermd and are independent of the form of the surface tension.These features were also observed in many experiments. Allof them can be observed in the well separated island mor-phologies during the later evolution in Fig. 8. The linearwetting layer disappears once islands begin to form and anew thinner flat wetting layer of 2–3 ML remains betweenthe islandssobserved in experiments in Refs. 32, 33, 37, and42d. Note the inter island wetting layer thickness was thesame independent of lattice mismatch unlike other morpho-logical features such as island size. This is because it is the

decay length ofdfels0d /dh which governs behavior very near to

the interface and this is independent of lattice mismatch. Adepression around the base of the islands is observedsexperi-mental evidence in Ref. 15 and 32, also seen insimulations22d.

IV. ENERGY CALCULATIONS

We calculated the total free energyfEq. s4dg of the film-substrate system for particular surface profiles, in order tomeasure how the free energy changes during ripening events.

The free energy of the whole substrate-film system isgiven by

F = Fel +E dxgÎ1 + s]h/]xd2, s4d

whereFel is the elastic free energy including any elastic con-tributions to the surface tension:

Fel =E dxfels0d +E dxE

−`

hsxd

dyS1

2Sijklsijskl −

1

2Sijklsij

s0dskls0dD

s5d

In order to calculate the stresses at the surface we solveda boundary integral equation in terms of the complex Gour-sat functions, the details of which can be found in the paperof Spencer and Meiron.46. The stress throughout the solidwas then determined by utilizing the fact that the Goursatfunctions are analytical in the solid region and using theCauchy integral formula.

Consistently for both isotropic and anisotropic surfacetension the free energy plateaued during the ripening eventand only dropped sharply at the very end of the ripeningevent. For example, see Figs. 9–11, which show how thetotal free energy and film morphology change when an islanddisappears during a ripening event in a Ge film on a Sis001dsubstrate. For anisotropic surface tension there was a verysharp drop in energy during initial faceting and island forma-tion.

FIG. 9. Top graph: evolution of island peaks when 10 ML of Geis annealed on a Sis001d surface. Four long-living islands have de-veloped by 0.1 s. Peak 1 is represented by a thick black line peak 2by a thin black line, peak 3 by a dotted line, and peak 4 by a dashedline. As can be seen a ripening event occurs in which peak 3 dis-appears. Bottom graph: the free energy as the system evolves. Notethat the free energy plateaued during the ripening event and onlydropped sharply at the very end of the ripening event. The energy isgiven in relation to the free energy of a flat film of 10 ML.

FIG. 10. Free energy over the entire simulation time range. Notethe sharp energy drop during faceting and the ripening energyplateau.


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V. SUMMARY

Like other groups before us22,25we show that surface ten-sion anisotropy is crucial to the formation of stable islandarrays and the prevention of continuous ripening. However,the type of anisotropy used in these previous works22,25 wassmooth and not cusped and hence did not lead to faceting. Aswe show, having cusped surface tension is crucial in cor-rectly determining the characteristics of the island array, inparticular in showing an increasing rather than decreasingisland density with increasing film thickness. The cusped sur-face tension is essential in producing sharply edged facetedislands and therefore the stabilizing elastic relaxation theseisland edges contribute. We also show that when surface ten-sion is cusped islands form in a “chain-reaction ripple” effectsi.e., islands tend to develop near other islandsd. This modeof growth has also been observed in experiment.12–14

We see a new phenomenon during long-term annealingafter ripening has stopped, in which islands equilibrate tohave a very uniform size distribution. We would have tocarry out our simulations over larger samples to ensure this isa nonlocal effect. There do not appear to be any experimentswhich compare island-size distributions after ripening hasstopped during long-time annealing in the absence of dislo-cated huge islands. This could be very useful in producinguniform quantum dot arrays.

We show that the island-shape transition can occur withonly one surface tension minimum—that is, without an ad-ditional facet orientation at a larger angle.

In films with isotropic surface tension we observe andexplain a new mode of growth in which a stable wavy sur-face is formed. Stable, nonflat morphology has previouslyonly been predicted for faceting films.22–25 Experimentsabove the roughening temperature would need to be carriedout to verify our prediction.

We studied the energetic profile of the entire strained is-land array during film evolution. We consistently see free-energy plateauing during ripening events and then droppingvery sharply at the end of the ripening event.

APPENDIX

In this appendix we calculate expressions for the strainenergy of a perturbed uniaxially and biaxially strained film.As these calculations show, the two expressions differ onlyby a constant dependent only on the mismatch strain andindependent of the surface morphology. This leads only to achange in energy and length scalessdue to the differing mis-match stressd but not in the qualitative features of the ob-served phenomena.

We use the following definition of strain:

FIG. 11. Evolution of island peaks when 10 ML of Ge is annealed on a Sis001d surface.sad t=0.15 s. Four long-living faceted islandshave now fully developed. The energy has plummeted from the energy of the initial perturbed flat film corresponding to the highlyenergetically favorable process of faceted island formation. Energy now plateaus while small changes in relative island height takes place.sbd t=5.98 s The middle of the energy plateau and the beginning of the ripening process in which peak 3 will disappear. From now on theother peaks will grow at peak 3’s expense.scd t=9.06s. The start of the drop in the energy plateau, corresponding to the first vertical line inFig. 9. The morphological ripening is, however, very advanced and peak 3 is much smaller than the other peaks.sdd t=9.20 s. The end ofthe ripening event, corresponding to the second vertical line in Fig. 9. Note that the big energy drop takes place only at the end of theripening event.


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eij =1

2S ]ui

]xj+

]uj

]xiD ,

where u are material displacements. The shear modulus ism=E/2s1+nd, whereE is Young’s modulus andn is Pois-son’s ratio.

The stress-strain relations are then

eij =1

Efs1 + ndssij − sij

s0dd − nsskk − skks0dddijg, sA1d

wheres is the total stress in the material andss0d is the stressin the zero-strain reference state.

From the above expression we obtain the compliance co-efficients Sijklfeij =Sijklsskl−skl

s0ddg. The elastic free-energydensity can be written as

fv = fvs0d +

1

2Sijklsijskl −

1

2Sijklsij

s0dskls0d, sA2d

where fvs0d is the free-energy density in the zero-strain refer-

ence state.

Uniaxial strained film

The plane strain conditions areezz=exz=eyz=0.Using the stress-strain relationssA1d these conditions can

be rewritten in terms of stresses:szz−szzs0d=nfssxx−sxx

s0dd+ssyy−syy

s0ddg, sxz=sxzs0d, and syz=syz

s0d.Substituting forszz, sxz, andsyz, in Eq. sA2d and explic-

itly writing the compliance coefficients, we obtain

fv = fvs0d +

1 + n

2Ehs1 − ndssxx

2 + syy2 d − 2nsxxsyy + 2sxy

2 − s1 − nd

3fssxxs0dd2 + ssyy

s0dd2g + 2nsxxs0dsyy

s0d − 2ssxys0dd2j. sA3d

We are interested in the free-energy densityfv at the sur-face and hence can use the zero-force surface boundary con-ditions, which yield

nxsxx + nysxy = 0,

nxsxy + nysyy = 0,

wheren is the normal to the surface. The same conditionsalso apply separately to the reference state stresses.

Combining the two boundary condition equations aboveleads to the following relation between the stress compo-nents at the surface:

sxxsyy = sxy2 .

Inserting this relation into the expression for the elastic free-energy density, Eq.sA3d, gives a much simplified expressionfor the free-energy density at the surface:

fv = fvs0d + ssxx + syyd2/s2Md − ssxx

s0d + syys0dd2/s2Md,

whereM =E/ s1−n2d is the plane strain modulus. The mis-match stress in a uniaxially strained film is given bysxx

s0d

=M«.

Biaxially strained film

In the case of biaxial strain,exz=eyz=0, ezz=«, and szz

−szzs0d=nssxx−sxx

s0d+syy−syys0dd+E«. Hence,

fv = fvsplane straind + E«2/2 + fszzs0d − nssxx

s0d + syys0ddg«.

The mismatch stress in a biaxially strained film is givenby sxx

s0d=szzs0d=fE/ s1−ndg«=Ms1+nd«. Hence the free energy

under biaxial stress differs from that in plane strain by aconstant which has no effect on the morphological evolutionof the surface.

The increase in mismatch stress in the biaxial case justhas the effect of changing energy and length scales. Theevolution equationsfEq. s1d together with Eq.s2dg can bemade spatially dimensionless by scaling all lengths byl0=g0/S0 whereg0 is the surface tension of the flat film andS0is the elastic energy density of the flat film, and all times byt=g0

3kBT/DshV2S04. The strain energy density is scaled by

S0. Note thatdfels0d /dh can be written asS0f8sh/hmld where

f8sh/hmld→1 ash→`. Hence

]h

]t=

DshV2

kBT

]

]x

]fg̃k + S+ S0f8sh/hmldg]s

, sA4d

becomes

]h̃

] t̃=

]

] x̃

]fg̃k̃ + S̃+ f8sh̃/h̃mldg] s̃

, sA5d

where all variables are written in terms of the scaled vari-ables and are represented by a tilda above the original sym-bols.

Hence having a finite rather than zero strain in thez di-rection does not change any qualitative features of our re-sults.

*Electronic address: [email protected]†Present address: KLA-Tencor CorporationsIsraeld, 4 Hatikshoret

St., Migdal-Haemek, Israel.1D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G.

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Documents

Formation, ripening, and stability of epitaxially strained island arrays