Journal of Crystal Growth 193 (1998) 257270
Equilibrium shape of epitaxially strained crystals(VolmerWeber case)
P. Muller*, R. KernCentre de Recherche sur les Me&canismes de la Croissance Cristalline1, Campus de Luminy, Case 913, F-13288 Marseille Cedex 9, France
Received 2 March 1998; accepted 5 May 1998
Three-dimensional epitaxial deposits when accomodated on a mismatched substrate only reach an equilibrium state fora given shape and a given strain distribution in the deposit A and in the substrate B. The aim of this paper is to formulate theequilibrium shape ratio r"h/l (height over lateral size ratio) of a crystal A epitaxially coherently strained on a substrateB where the natural misfit is m. Whereas for a structureless substrate WulffKaishews theorem tells that r"r
depending only on the wetting of B by A, the new theorem shows that when mO0, whatever its sign, r increases with size sothat self similarity is lost. This size dependence originates in the fact that epitaxial strain acts against wetting and thus leadsto a thickening of the equilibrium shape. The greater the parameters m, r
0and K (the substrate/deposit stiffness) the larger
the shape ratio r. For a collection of crystals, and when close enough, crystals interact by substrate deformation so that theirshapes deviate from that of isolated crystal. In spite of this elastic shape effect other behaviors are only slightly changed. (a)The nucleation barrier *G* practically is not influenced by the elastic energy provided the nucleus is small or *G*+k. (b)GibbsThomsons equation stays close to the usual one where elastic energy is omitted (m"0). As a consequencea collection of epitaxially strained crystals have an Ostwald ripening without any anomaly. All these results only hold forVolmerWeber coherent epitaxy. ( 1998 Elsevier Science B.V. All rights reserved.
1.1. Equilibrium shape
The equilibrium shape of a free crystal is thatwhich mimimizes the total surface free energy c fora given volume . It is given by the Wulff
*Corresponding author. Fax: #33 91 41 89 16; e-mail:firstname.lastname@example.org.
1Associe aux Universites AixMarseille II et III.
theorem : the equilibrium shape is the innerenvelope of the planes perpendicular to directionsn and proportional to distances c(n) measured froma point called Wulff point. There is an activationbarrier *G
0to overpass to obtain the equilibrium
shape, which is one third of the total surface energyof the crystal . Quite different is the case ofsupported crystals since, as shown independently byKaishew  then Winterbottom , the equilib-rium shape of a supported crystal is modified bythe influence of the substrate. For a structurelesssubstrate, or strictly isomorphous species, these
0022-0248/98/$ see front matter ( 1998 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 5 0 8 - 9
authors have shown that the shape of a depositedcrystal on a flat surface only differs from that ofa free crystal by a truncation. Otherwise the crystalfree faces are not modified. The truncation of thecrystal is proportional to *c"c
Bare the surface energies
of a face A and B before contact, cAB
the interfacialenergy and b
ABthe adhesion energy when the con-
tact is realized with a given plane of B. For isomet-ric crystals A and B there results an aspect ratio ofthe equilibrium shape:
where l is the lateral size, h the height of the crystaland c@
Athe lateral surface energy (c@
A. in the
case of a cubic crystal). For a free crystal bAB
is the greatest and thus, the higher the adhesionenergy, the flatter the deposit crystal A. For com-plete wetting c
Band *c"0 so r
It should be noted that the aspect ratio is volumeindependent.hatever its size the equilibrium shapeis self similar. For such heterogeneous nucleationthe activation barrier *Gb to overpass is now*Gb"*G0(*/) where */ is the truncated tonon-truncated volume ratio. This relation meansthat, since */(1, the heterogeneous nucleationon a substrate is always easier than the homogene-ous nucleation . Cahn and coworkers  triedto answer the question of the selection of mutualorientations of A over B by minimizing the activa-tion barrier *Gb in a very general phenomenologi-cal fashion by not considering however the crystalstructure.
1.2. Epitaxial deposits
Up to now a crystalline structure has not beenconsidered in spite of its presumed important role.Similar lattice planes of two structures induce strictmutual orientations of the two crystals. These ori-entations are called epitaxial orientations. Two lat-tice planes of A and B when similar come in contactand accomodate their two-dimensional misfit (see[8,9] for a review). By this means, the couple A/Bstores a certain amount of elastic energy which hasto be considered in the equilibrium shape balanceand the nucleation barrier.
Two simple tentative approaches to this problemmay be mentioned [10,11]. The Markov approach is based on the atomistic formulation ofStranski and Kaishews  equivalent to Wulffstheorem. Here the condition for equilibrium shapeis that the mean separation work of the atoms lyingin the different faces of deposit crystal A have all tobe equal. For epitaxially strained crystals, Mar-kovs treatment of the equilibrium shape considersthat (i) the crystal is homogeneously strained by fullaccomodation but nonrelaxed, (ii) that the worknecessary to disrupt a strained first neighbor bondis /e where / is the bond energy of an unstrainedbond and e the strain energy of the bond. The resultis that the shape ratio r becomes very slightly misfitdependent r"r
0/(1!Cm2) since the elastocapil-
lary number amounts typically to C+20 and theusual misfits to m+10~2. The other approach is based on the same simplification (i) but dulydistinguishes bulk and surface elasticity. Indeed,the work of surface deformation by unit area is itssurface stress s which is a surface excess quantity inthe sense of Gibbs. As a consequence surface stresssA
of the basal face contributes to the aspect ratio sothat r"r
A.2 With respect to mis-
fit m, the symmetry is broken. It should be pointedout that the same conclusions arise from these twomodels: the aspect ratio is misfit dependent butvolume independent. In other words, the equilib-rium shape is still self-similar as that of free crystals.Furthermore, both treatments imply [10,11] that3D nucleation can only occur when the thermodyn-amic supersaturation overpasses the stored bulkelastic energy density which is m2 dependent. Wewill see that all these conclusions (self-similarityand supersaturation threshold) are specific to suchsimplified models where elastic relaxation is notconsidered. When elastic relaxation is consideredthese conclusions no more stand.
2From formula (2) in Ref.  r"U/2U withU"2c
A) and after correction of a residual
error in , U@"c@A#m(s
is the Poisson ratio of the crystal A. So for usual materials(l
258 P. Muller, R. Kern / Journal of Crystal Growth 193 (1998) 257270
1.3. Elastically relaxed deposits
Elastic relaxation is a necessity for epitaxial de-posits when accommodated on a substrate. Themechanical equilibrium is only reached when freesurfaces of the deposit crystal have vanishing nor-mal stress components. Therefore, for relaxed crys-tals things are much more complicated since: (i)a 3D crystal elastically relaxes and when the inter-face is not perfectly glissile, it drags the atoms of thecontact area so that the nonhomogeneous storedelastic energy is shared between the island and itssubstrate (striction effect for coherent epitaxies),(ii) the islands and substrate strains become in-homogeneous. The nonhomogeneous lateral straine, in the contact plane, which is caused by thelattice mismatch and the striction effect, propagatesinto the deposit, from the interface to the topface of the crystal and from the interface into thedeposit. Owing to Poisson effect these non-homogeneous lateral strains also induce a verticalstrain e
Min the deposit as well as in the substrate.
Such elastic relaxation has been considered in theliterature.
Freund et al.  analyzed an elastic relaxationfor a circular cap primarily homogeneouslystrained on a foreign but elastically similar substra-te. Furthermore, putting c
one obtains according to Eq. (1), r0"0,
so that their system fullfills described the conditionof complete wetting. The relaxation strain wascomputed numerically in the deposit and the sub-strate by continuous elasticity and finite differences.The minimized elastic energy per unit volume wascomputed and found to decrease with an increasingshape ratio r"h/l, here h is the height over thebasis of the cap l; this ratio otherwise is linked tothe cap contact angle a by h/l"(sin a/2)2/sin a.his with the h/l decreasing energy density maytherefore compete with surface energy so that anequilibrium shape ratio installs. It was found that thegreater the misfit, the greater the shape ratio. Thestrain energy blows up the cap. Furthermore,the shape is no longer self-similar with volume. Atcomplete wetting and zero misfit the cap becomestotally flat. Using the same model but solved ana-lytically Ling  and Chen and Washburn came to the same conclusion showing furthermore
 that a 2D strained platlet, a"0, transformsinto a cap a0 at certain critical size, a resultwhich is significant for the 2DP3D StranskiKrastanov transition.
Other authors tackled more complex situationsto rationalize the appearence of very flat crystals orsmall flat pyramidal deposits on strained layers.For Ge/Si, InAs/GaAs(1 0 0) systems such islandsare called huts  which means very flatpyramids with M1 0 5N facets, being transformed ata later stage into bigger and more globular epi-taxial crystals. Some authors  modelled theelastic behavior of the deposit by the superpositionof elastic monopoles or dipoles on the substrate,thereby calculating, in fact, only the energy storedin the substrate. Such an elastic energy when takenpositively represents the elastic energy induced bystriction in a nonstrained substrate. When takennegative it represents the release of elastic energyin a prestrained substrate as it is the case forStranskiKrastanov wetting layers. In the lattercase the authors call it relaxation energy. It isopposed to surface energy and an activation barriercan be calculated. At constant volume it changesthe shape, e.g. a truncated pyramid eliminates itstop facet. However, the selection of the pyramidangle is difficult to foresee. It should be noted thatthe M1 0 5N facets do not appear on non-supportedequilibrium shape crystals of tetrahedral-typestructures  so that the selection of theM1 0 5N pyramids has still to be explained by elastic-ity considerations and more likely by consideringspecific surface strain effects inducing surface re-construction.
Recently, we have analytically calculated theequilibrium strain components for a rectangularshaped model crystal in continuous isotropic elas-ticity and carried out an approximation . Thestrain components decrease exponentially from thecontact area inside the substrate and the depositwith an extinction distance depending on the shaperatio r
0"h/l. The strain now also depends on
the relative substrate to deposit stiffness ratio. Thenonhomogeneous minimal elastic energy is cal-culated for the deposit A and the substrate B. Oneof the first uses of these calculations was to show how the 2DP3D StranskiKrastanovtransition is initiated, and to find a certain critical
P. Muller, R. Kern / Journal of Crystal Growth 193 (1998) 257270 259
size of the 2D islands which now depends on thenumber of underlying wetting layers.
A further use of these analytical minimal elasticenergy calculations  will be made in this paper.The VolmerWeber case will be inspected wherecA#c
AB0 and the wetting
layers are unstable, in order to see the influence ofepitaxial strain energy on the equilibrium shape.More specifically, we will look at the equilibriumshape ratio r"h/l of cubical (c"c@) or tetragonal(cOc@) shaped crystals as a function of misfit m,wetting 2c
ABand the relative stiffness K of
the deposit and substrate. Only coherent epitaxiesare considered where the strain calculations are valid, glissile epitaxies store such a smallamount of strain energy that their shape ratio is notaltered.
2. Equilibrium shape of an isolated strained crystal
In this paper, we only deal with VolmerWebergrowth meaning that islands are supported bya nude substrate. Equilibrium shape of StranskiKrastanov growth is not considered here and willbe published later on.
2.1. Free energy change during condensation
The equilibrium shape is calculated by minimiz-ing the total free energy *G needed to forma three-dimensional (3D) crystal A onto a flat lat-tice mismatched substrate B. For crystal A (squarebox shaped with height h and basis area l]l) *G is
The first term is the chemical work to form crystalA, *k being the chemical potential difference(supersaturation). The second and third terms arethe capillarity energy spent to create the basal andlateral faces of A with c
energies, respectively, and bAB
the adhesion energybetween A and B (It is supposed that the depositand substrate do not mix.) The last term is theminimal elastic energy spent to accomodate A ontoB after elastic relaxation. For a box-shaped crystal
this term has been calculated in , it writes:
m2R(h, l),E.R(h, l). (3)
are the Young modulus and Poissonratio of the deposit in isotropic elasticity,m"(b!a)/a the natural misfit and where the func-tion 0(R(h, l)(1 describes the elastic relaxation.In absence of any elastic relaxation R(h, l)"1 sothat *
A) is the usual elastic
energy density change for accommodating a crystalperfectly. In the process (Eqs. (2) and (3)) we havenot given specific elastic properties to the surfaces.We have shown in  that this approximationcan be made for misfits greater than several 10~3.
The relaxation factor R(h, l) has been calculatedfor a ribbon  with the following assumptions: (i)The interface between A and B is coherent, whichmeans that there is continuity of the displacementfield so that in the interface (z"0) the strains in A,eAxx
(x, z"0) and B, eBxx
(x, z"0), are connected byeAxx
(x, z"0). (ii) The thin filmapproximation of Hu  eA
however, is released by writing eAxx
(x, z"0) fl (z ) where fl(z) is evaluated by consid-ering that each layer of A feels the relaxed strain ofthe underlying layer as a new substrate. Note thatthis approximation leads to decreasing stress andstrain inside the crysta...