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Journal of Crystal Growth 193 (1998) 257—270

Equilibrium shape of epitaxially strained crystals(Volmer—Weber 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

Abstract

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 Wulff—Kaishew’s theorem tells that r"r

0is constant

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*+k¹. (b)Gibbs—Thomson’s 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 forVolmer—Weber coherent epitaxy. ( 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

1.1. Equilibrium shape

The equilibrium shape of a free crystal is thatwhich mimimizes the total surface free energy c fora given volume [1—3]. It is given by the Wulff

*Corresponding author. Fax: #33 91 41 89 16; e-mail:[email protected].

1Associe aux Universites Aix—Marseille II et III.

theorem [4]: 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 [3]. Quite different is the case ofsupported crystals since, as shown independently byKaishew [5] then Winterbottom [6], 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

A!c

B#c

AB"

2cA!b

ABwhere c

Aand 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:

r0"

h

l"

*c2c@

A

"

2cA!b

AB2c@

A

, (1)

where l is the lateral size, h the height of the crystaland c@

Athe lateral surface energy (c@

A"c

A. in the

case of a cubic crystal). For a free crystal bAB

"0,r0

is the greatest and thus, the higher the adhesionenergy, the flatter the deposit crystal A. For com-plete wetting c

AB"0, c

A"c

Band *c"0 so r

0P0.

It should be noted that the aspect ratio is volumeindependent.¼hatever its size the equilibrium shapeis self similar. For such heterogeneous nucleationthe activation barrier *Gb to overpass is now*Gb"*G

0(»*/») where »*/» is the truncated to

non-truncated volume ratio. This relation meansthat, since »*/»(1, the heterogeneous nucleationon a substrate is always easier than the homogene-ous nucleation [5]. Cahn and coworkers [7] 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[10] is based on the atomistic formulation ofStranski and Kaishew’s [12] equivalent to Wulff’stheorem. 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-kov’s 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 [11]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

0#2(s

A!c

A)m/c@

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. [11] r"U/2U withU"2c

A!b#4m(s

A!c

A) and after correction of a residual

error in [11], U@"c@A#m(s

A!c

A)[(1!3l

A)/(1!l

A)] where

lA

is the Poisson ratio of the crystal A. So for usual materials(l

A+1/3) r"r

0#2(s

A!c

A)m/c@

A.

258 P. Muller, R. Kern / Journal of Crystal Growth 193 (1998) 257–270

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. [13] analyzed an elastic relaxationfor a circular cap primarily homogeneouslystrained on a foreign but elastically similar substra-te. Furthermore, putting c

A"c

B, c

AB"0 or

bAB

"2cA

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.¹his 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 [14] and Chen and Washburn [15]came to the same conclusion showing furthermore

[15] that a 2D strained platlet, a"0, transformsinto a cap a'0 at certain critical size, a resultwhich is significant for the 2DP3D Stranski—Krastanov 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” [16—18] which means very flatpyramids with M1 0 5N facets, being transformed ata later stage into bigger and more globular epi-taxial crystals. Some authors [19—22] 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 forStranski—Krastanov 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 [23—25] 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 [26]. 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[27] how the 2DP3D Stranski—Krastanovtransition is initiated, and to find a certain critical

P. Muller, R. Kern / Journal of Crystal Growth 193 (1998) 257–270 259

size of the 2D islands which now depends on thenumber of underlying wetting layers.

A further use of these analytical minimal elasticenergy calculations [26] will be made in this paper.The Volmer—Weber case will be inspected wherecA#c

AB!c

B"2c

A!b

AB'0 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

A!b

ABand the relative stiffness K of

the deposit and substrate. Only coherent epitaxiesare considered where the strain calculations [26]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 Volmer—Webergrowth meaning that islands are supported bya nude substrate. Equilibrium shape of Stranski—Krastanov 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

*G"!hl2*k#(2cA!b

AB)l2

#4c@Ahl#*¼

%-hl2. (2)

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

Aand c@

Atheir surface

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 [26], it writes:

*¼%-"

EA

1!lA

m2R(h, l),E.R(h, l). (3)

EA

and lA

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 *¼

%-"E

."E

Am2/(1!l

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 [26] that this approximationcan be made for misfits greater than several 10~3.

The relaxation factor R(h, l) has been calculatedfor a ribbon [26] 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)"m#eBxx

(x, z"0). (ii) The thin filmapproximation of Hu [28] eA

xx(x ,z)"eA

xx(x, z"0),

however, is released by writing eAxx

(x, z)"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 crystal A, 0(fl(z)(1, which onlyreaches a completely relaxed state at its free surfacefor an infinite height. This is consistent with certainother models [29] but does not allow us to obtainthe sign inversion of strain in the top layers reportedby some calculations [14,15,30]. (iii) The elastic en-ergy of a 3D crystal is calculated from that of aribbon by using the superposition principle witheAxx

(x, y, z)"eAyy(x, y, z)+eA

xx(x, z) implying plane

stress conditions and neglecting border effects.Within these approximations we have found (see

[26]) that the minimized elastic energy (Eq. (3)) isdivided into two parts via its relaxation factor R:the one from deposit, A, the other from the substra-te, B.

R(h, l)"RA(h, l)#RB(h, l),

RA(h, l)"S f 2TSM2T+S f 2TSMT2 [23],

RB(h, l)"S f TSMT[1!SMT], (4)


where SMT is the mean value along the contactarea of eA

xx(x, 0)/m

SMT"1!2

P2#C1#

J2

P D2exp(!PJ2) (5)

with the parameter

P2"Kl

ph

1

S f T(6)

containing the shape ratio r"h/l, the relative sub-strate to deposit stiffness ratio

K"

EB/(1!l2

B)

EA/(1!l2

A). (7)

In Eqs. (4) and (6) S f T and S f 2 T describe thelateral relaxation of the island along z, as meanvalues of f l(z) or f 2l (z):

S f T"1

h

1!SM1Th

1!SM1T1

and S f 2 T"1

h

1!SM1T2h

1!SM1T2

,

(8)

where SM1T is similar to Eq. (5) but with P2"

P21"l/pa, a being an atomic size.Since we numerically find that R(h, l) essentially

is a function of the aspect ratio r"h/l [26], inFig. 1 we plot the energy density relaxation factorR(h, l) noted in the following R(r). Thus, in Fig. 1we can see that the softer the substrate is relative tothe deposit and the greater the aspect ratio r, thegreater the relaxation (the smaller R(r)). For a rigidsubstrate (KPR, SMTP1 according to Eqs. (4)and (5)) and in this case R(r)PSf 2T only describesthe relaxation of the deposit with a rigid basal face.That is what was calculated differently in [29]. Itcan easily be seen from Fig. 1 that substrate relax-ation is an important effect.

From Fig. 1 it can be concluded that, froma mechanical point of view, a deposited crystalwants to thicken (larger value of h or r) in order todecrease its elastic energy density by relaxation(smaller value of R(r)). Therefore, elastic relaxationis the thermodynamic driving force for thickening.Nevertheless, such a thickening can only be realizedif the corresponding surface energy cost, which in-creases with r, is overpassed. The latter will be moreclosely analysed in the following.

Fig. 1. Energy relaxation factor R(r) depending on the shaperatio r"h/l for box-shaped island for different values of relativerigidities K. Inset: RA and RB are the relaxation factor of depositA and substrate B, respectively. R"RA#RB.

2.2. Equilibrium shape

We have to minimize *G from Eqs. (2) and (3) atconstant volume, or more generally to take fromEq. (2) the first-order differential of *G at constantvolume d»"l2 dh#2hl dl"0. It is

d*G"

*G

h Kl dh#*G

l Kh

dl

"C*G

h Kl!Al

2hB*G

l KhD dh,

which gives

d*G"CE.hl2AR

hBl!1

2E.l3A

R

lBh

!(2c!b)l2

h#2c@lD dh.

Since for any arbitrary variation of dh there has tobe d*G"0, the equilibrium shape fulfills the equa-tion:

E.h%2A

R

hBl!1

2E.l%2A

R

lBh

!

(2c!b)

h%2

#2c@l%2

"0.

(10)


In order to construct the equilibrium shape, usingEq. (1) and putting h/l"r so that

R

l Kh

"!

h

l2

dR

dr,

R

h Kl"1

l

dR

dr,

Eq. (10) can be rewritten as a parametric systemin r:

G h%2"!

4c@A

3E.A1!

r0r B )C

dR

drD~1

,

l%2"h

%2/r. (11)

Giving positive values r'r0to r, calculating dR/dr

from Eqs. (4) and (8) (Fig. 1) for a given K value,then from Eq. (11), for a given r

0value and c@/E

.ratio, we plot h

%2"h

%2(l

%2/2) (see above). The result

is shown in Fig. 2 where each curve is the locus ofthe edge of the equilibrium shape. In Fig. 2a wherer0"0.1 and K"1, the curves are labelled with

the misfit m entering in the ratio c@/E.of Eq. (11).

A great variety of pure materials showing typicallyan elastocapillar length of

c@/(EA/(1!l

A))+10~9 cm. (12)

is used in the following to scale the illustrations.Curves h

%2"h

%2(l

%2/2) of Fig. 2a are plotted for

different misfit values DmD"0, 1, 2, 3, and 4]10~2.The new result is that if mO0, whatever the sign ofthe misfit the equilibrium shape is in fact, not self-similar. The bigger the crystallite is, and thus thehigher the misfit, the more it “blows up”. Self-sim-ilarity only exists for m"0 since in this caseh%2"r

0l%2

/2 where r0

is the wetting ratio (Eq. (1)).For a nonstrained deposited crystal the Wulff—Kaishew theorem is valid. Note also in Fig. 2 thatall h

%2"h

%2(l

%2/2) shapes show the asymptotic be-

havior dh%2

/dl%2

Dl%2?0

"liml%2?0

(r)"r0. The proof

is easily given by rearranging Eq. (11) so as toobtain an equation of second degree of r, whosesolution is expanded. Thus, for vanishing lateralsize l dR/drP0 (see Fig. 1), there is liml?0

(r)"r0.

In Fig. 2b again with K"1, but only form"4]10~2, we look at the equilibrium shapewhen changing the wetting ratio: r

0"0.1, 0.2, 0.4,

1. The better the wetting (small r0

according toEq. (1)) the greater the strain-induced thickening.

Fig. 2. Equilibrium shape of misfit coherent and relaxed box-shaped islands of different sizes, H, height, ¸/2, half basis inatomic units. Self-similarity is lost. (a) Effect of misfit m. For thecase m"0 there is self-similarity. (b) Effect of the wetting factorr0"(2c

A!b)/c@

A. (c) Effect of the relative substrate stiffness K.

In Fig. 2c we took m"4]10~2, r0"0.1 and

changed the relative rigidity K. Thus, the greaterK the greater the strain-induced thickening. At thelimit for an infinitely rigid substrate (KPR) only


the deposited crystal A relaxes with a fixed basisface. Therefore, the R factor is greater and dR/drsmaller (see Fig. 1) so that according to Eq. (11)h%2

increases with K. For a very soft substrate(KP0) the deposited crystal is not, in fact, con-strained by its substrate and at equilibrium it hasthe same equilibrium shape r"r

0whatever its size

is. This result is similar to a crystal on a struc-tureless substrate or similar to a perfectly glissileepitaxy.

To sum up we can say that wetting flattens theequilibrium shape whereas epitaxial strain actsagainst flattening and thus “blows up” the equilib-rium shape. In the following, three points should bediscussed concerning the equilibrium shape and thelimitations.

2.2.1. Partial loss of coherenceIt is important to note that for increasingly large

crystals our description of the equilibrium shapemust become nonrealistic. Indeed, 3D islands, evenwhen partially elastically relaxed, accumulatestrain energy which may be subject to plastic relax-ation leading to a noncoherent epitaxy as alreadyknown (see experiments in [8,31]). Concerning theinfluence on the equilibrium shape there is someexperimental evidence as in the Ge/Si system. ForGe/Si(1 1 1) Voigtlander et al. [32] have measuredthe aspect ratio r"h/l of increasing crystals nearequilibrium. For small islands, in agreement withour calculations, the aspect ratio is small and in-creases with the lateral size so that there is noself-similarity. When the size is large enough, theislands partially relax by dislocation entrance, theeffective misfit drops and the aspect ratio decreasesas seen in Fig. 1a when jumping from a curve ofhigh misfit to a smaller one. For Ge/Si(1 0 0) thingsare somewhat similar but more spectacular due tothe in situ observations of [33]. These authors havereported an oscillatory behavior of slowly growingcrystals. A quick lateral size increase of crystals isobserved at each dislocation entrance followed bya slower growth. This has been interpreted by theseauthors as meaning that the shape is strain depen-dent and that at each strain release the shape cha-nges quickly by very active material exchangebetween the basal face and the lateral faces. Thegrowing crystal therefore becomes much flatter and

much larger at each dislocation entrance whereasin between two dislocation entrances it thickens.Both the critical sizes at which the crystal shapestarts to change abruptly and the successive oscilla-tions can be predicted as analyzed in our recentpaper [34]. Freund et al. [13] pertinently discussedthese observations [33] using their cap-shaped nu-merical model.

2.2.2. Equilibrium shape versus growth shapeA few words should be said about the growth

kinetics of supported crystals. FollowingStranski—Kaishew’s approach [10,12,35], the meandetachment energy per atom for the atoms locatedin a crystal face must have the same value for all thecrystal faces belonging to the equilibrium shape.This means that in out-of-equilibrium conditionsa net current of matter installs between two faceshaving different mean detachment energies. Thegrowth kinetics of the two faces may be quite speci-fic and different, the quickest growing faces becom-ing smaller or even disappearing on the equilibriumshape. For a free (nonsupported) cubic-box-shapedcrystal and in the absence of screw dislocations, the6 (F type) faces grow simultaneously by 2D nuclea-tion and belong to the equilibrium shape. Thiscannot be true for a box-shaped supported-crystalsince now the faces are unequivalent. Indeed,whereas the top face remains a true F face, thelateral faces, owing to contact with the substrate,can grow by attachment onto the contact line andthus lose their pure F character. For this reason,lateral faces grow faster than the top one. Note thatthis assymmetry does not exist for evaporationsince all the faces can start their evaporation fromtheir edges.The attainment of the equilibrium shapebeing the result of growth and evaporation by ma-terial exchange of contiguous faces, it can be seenthat growth shapes governed by kinetics should al-ways be flatter than the equivalent (same number ofatoms) equilibrium shape.

2.2.3. Other shapesIn Section 2.2 we only dealt with a box-shaped

crystal. This is due to the fact that our analyticalcalculation [26] of the relaxed elastic energy(Eq. (3)) cannot be used for other shapes. Clearly, ifnecessary and in certain cases, numerical methods


can be used for any shape [13], though valuable 3Dcalculations are very complicated [36,37] and donot lead to general physical insight.

Nevertheless, with our box shaped crystal we caninspect further small shape changes around theM1 0 0N faces. The energy ¼

%-.of Eq. (3) will ap-

proximately still hold for such small changes. Fora simple cubic crystal structure the counting of firstneighbour bonds leads to a gamma-plot [4] ofeight equal spheres having a common Wulff point,each diameter pointing along the longest diagonalof a cube. Due to the six inward cusps along [0 0 1],Wulff’s construction only leads to a cube withsharp corners and sharp edges for the equilibriumshape [35]. Weak second neighbor interactionsgenerate new Herring spheres and lead smallM1 1 0N facets truncating the free edges of the cube[38]. It is also known that nonzero temperaturerounds off the edges of the cube due to the thermalroughening of the step and their meandering[39—42]. In the latter case the gamma-plotalong the [1 0 0] zone axis and close to the (0 0 1)face gives a power expansion of the projectedsurface energy plot b"c(h)/cos(h)"c

A#

b1DpD#b

3DpD3. Here the vicinal inclination of the

A face is the angle h or the local slope p"!z/x.b1

is the energy of isolated steps, b3

is theirinteraction energy. Thermal meandering of steps[39] and/or dipolar elastic interactionbetween steps [42,43] give a repulsive termb3'0. b

1which decreases with temperature

gives the extent of the (0 0 1) facet. b3

leads therounding off edges near the face with a +x3@2

profile [40,41].We should ask ourselves if the former expansion

is still valid for a strained crystal. More preciselythe strain close to the free basal face (0 0 1) of thedeposit at level z"h from the substrate needs to beanalyzed.

For a crystal of high shape ratio r+1 (see Fig. 1)the top face of the crystal is not, in fact, strainedcompared to that of the interface z"0. Accordingto Eqs. (8) and (5) eA

xx(x, h)/eA

xx(x, 0)+10~2 for r+1

whatever the size is. In this case the b expansionaround this face needs no modification due to epi-taxial strain. The (0 0 1) face of the equilibriumshape is still relayed either by the M0 0 1N facets orby x3@2 profiles.

Conversely, for flat and therefore not too largedeposited crystals (r+r

0+0.1) the top surface re-

mains strained since according to Eqs. (5) and (8)there is eA

xx(x, h)/eA

xx(x, 0)+0.7 so that owing to the

local strain a new interaction between steps has tobe taken into account. Tersoff [44] introduceda new interaction for the understanding of stepbunching occurring in highly strained and miscutepitaxial layers at high temperature. The elasticdiscontinuity appearing at each step produces anattractive term [E

.a/p] ln(¸/2p) between neighbor-

ing steps separated by a distance ¸ (in atomicunits). Thus, the b expansion due to p"¸~1 iswritten with an additional term !DpD(E

.a/p) ln(2pp)

where E.+E

AmN 2/(1!l2

A) and mN the mean value of

eAxx

(x, h) along x. From Wulff’s construction it iseasy to show that there remains the (0 0 1) facetnow relayed by a different curved part cutting thefacet at a finite slope p

F"(E

.a/2pb

3)1@2. This

rounded part represents a piling-up of the steps atthe border of the facet.

2.3. Activation barrier for nucleation

When mapping the *G(h, l) of Eq. (2) for differ-ent supersaturation levels *k among the differenthills and valleys given by the zero of *G/hDl or*G/lD

h, there is for the smallest values of h and

l a couple h*, l* where the partial derivativessimultaneously are zero and the second deriva-tives negative. For a given *k this gives theactivation barrier *G(h*, l*) for nucleation.Putting these partial derivatives zero, two indepen-dent relations between h and l are obtained.Combining them, and solving them, h* and l* areobtained:

h*"2(2c

A!b)

*k!E.(R(r)!2r dR(r)/dr)

,

l*"4c@

A*k!E

.(R(r)#r dR(r)/dr)

. (13)

The shape ratio r*"h*/l* only depends on *kand is implicitly given from Eq. (13) by

r*"r0

*k/E.![R#r dR/dr]*

*k/E.![R!2r dR/dr]*

. (14)


Fig. 3. The activation free energy of nucleation splits into three parts (Eq. (15)). Here the elastic part F depending on the reducedsupersaturation *k/E

.is plotted. In the inset the ratio r*/r

0as a function of *k/E

.necessary for the calculation of F is plotted.

The amplitude of the activation barrier is thus isobtained by substituting Eq. (13) into Eq. (2) andcan be written after some rearrangement as

*G(r*, *k)"(4c@

A)3

*k2r0F A

*kE.

, r*B. (15)

The different physical aspects are nicely uncoupledin Eq. (15). The first factor of Eq. (15) is the usualactivation energy of the so-called homogeneousnucleation. When multiplied by r

0it is reduced due

to wetting of the substrate. The last factor thereforecharacterises the elastic behavior due to epitaxialstrain mO0, which can be written in detail as

FA*kE0

, r*B"A*k/E

.*k/E

.!(R#r* dR/dr)B

2

]A*k/E

.!R

*k/E.!(R!2r*dR/dr)B. (16)

In Fig. 3 we plot for K"1 and r0"0.1 the re-

duced activation barrier *G/((4c@A)3r

0/*k2) as a

function of the reduced supersaturation *k/E0. It is

calculated from Eq. (16) solving Eq. (14) for each

*k/E0. In the inset of Fig. 3 we plot the ratio r*/r

0solution of Eq. (14) for K"1. We see that thereduced activation barrier decreases with increas-ing *k/E

0. For *k/E

0+6, F and r*/r

0go close to

unity. Nucleation rate Jexp(!*G*/k¹) becomesreally effective only when *G*+k¹. Experimentsshow [35] that nucleation needs a degree ofsupersaturation a'1, say a"e so that*k+k¹/a3 ln a"k¹/a3

.Since for usual misfit

k¹/a3AE.it results from Fig. 3 that in the nuclea-

tion regime r+r0

and F+1. So when nucleationeffectively takes place, elasticity is negligible. Inother words, at the *k where nucleation effectivelyproceeds, the nucleus has a small size. Then, ac-cording to Fig. 2 the equilibrium shape does notdeviate from straight line m"0 (see also the com-ment in text among Fig. 2). This should also be truewhen atomistic versions of nucleation are con-sidered [35].

In theories where elastic relaxation is not con-sidered, see end of Section 1.2, *G becomes infinitefor *k/E

0"1 and thus nucleation is impossible for

*k(E0. Clearly, we have seen that this is not the

case when elastic relaxation is duly considered.There is no supersaturation threshold.


3. Equilibrium shape of interacting crystals

Up to now we only described an isolated epi-taxial crystal. Generally, one has to do with a col-lection of such crystals so that when their meandistance DM approaches their mean size lM , thesecrystals may interact. We will distinguish two typesof interactions.

Even when the crystals are far from one another,lM @DM , they may exchange atoms by surface diffu-sion on the substrate provided the surface diffus-ivity is high enough. At the temperature where theequilibrium shape installs by surface diffusion onthe crystals (see Section 2.2.2), diffusion on the sub-strate may also become efficient for matter ex-change between the crystals. So we have to observehow the so-called Oswald ripening eventually ismodified by the strain. Furthermore, At smallerdistances lM +DM , we have to consider the elasticinteraction of the crystals via the substrate [26].Such interactions change the strain of the indi-vidual crystals, therefore the equilibrium shapeshould be modified. We shall now look more close-ly at each of these effects.

3.1. Generalized Gibbs—¹homson equation

The equilibrium shape described in Section 2.2 is*k independant. However, for a given size and thena given equilibrium shape there is a unique value of*k. This is another way of giving the Wulff’s the-orem known as the generalized Gibbs—Thomsonequation [45]

Let us take a dilute, lM @DM , collection of crystalsof different shape and size. They have differentchemical potentials, which means different vaporpressure or solubility, surface concentration etc.When such intensive properties are equalizedaround each crystal, the equilibrium shape of eachcrystal is established and relation (10) or (11) isfullfilled. Now, the collection of crystals is onlycharacterized by the size distribution, volume » ornumber of atoms N in each crystal. Owing to thedifferent sizes, thermodynamic forces exist betweenthe crystals. Changing the volume of a crystal eitherat h"const. or l"const. gives from Eq. (2), re-spectively, at equilibrium:

*G

» Kh

"!*k#2c@l#

2c!bh

#E.CR(h, l)#

l

2

R(h, l)

l KhD"0, (17)

*G

» Kl"!*k#4c@l

#E.CR(h, l)#

R(h, l)

l KlD"0. (18)

When the equilibrium shape is realized accordingto Eq. (10), h

%2"h

%2(l

%2), relations (17) and (18) are

not independent so that from Eqs. (10) and (17) orEqs. (10) and (18) the generalized Gibbs—Thomsonequation follows:

*k"4c@/l%2#E

.CR(r%2

)#r%2

R

r Kr%2D. (19)

Relation (19) says that an equilibrium crystal offinite size l

%2has an excess chemical potential *k

with respect to a bulk and nonstrained crystal.Owing to surface free energy c@, there is an hyper-bolical decrease of *k with size l

%2as in the classi-

cal Gibbs—Thomson equation. The second term isspecific to coherent epitaxy when mO0 and thusE.'0 according to Eq. (3) and it represents the

contribution to chemical potential of the strainedbut relaxed crystal at its equilibrium shape ratio r

%2.

The second is composed of the relaxation straindensity and its change with shape. From Fig. 1 andnumerically this contribution is found positive atany K, r

0'0. Furthermore, it is negligible with

respect to the first term and it decreases faster as[l

%2]~1. For the usual elastocapillar length

(Eq. (12)) it amounts to less than 5% for misfits ashigh as m"8]10~2.

Let us make three remarks.

1. Even if in Eq. (19) the second term is negligible,it does not mean that the chemical overpotential*k is not altered by elasticity since in the firstterm of Eq. (19) l

%2(and h

%2) is elasticity depen-

dent (see relations (11)).2. It is tempting to write down directly “by hands”

the local chemical potential in the case wherec is continuous and isotropic by an extendedHerring [4] equation. In paper [46] we find*k"a3[c

AB(i

1#i

2)#¼

4] where i

1, i

2are the


principal curvatures and w4said to be the strain

energy density at the surface. In fact, comparingthe latter expression with Eq. (19) shows that theprincipal curvatures must depend on elasticityand that w

sis not simply a surface contribution.

3. One of the special aims of Gibbs—Thomson’sequation is to govern Ostwald ripening [35].This means that inside a collection of crystals ofa given size distribution, the smaller crystalsdisappears in aid of bigger crystals. The sizeevolution of such epitaxial crystals is governedby a kinetic theory based on Gibbs—Thomson’sequation as treated by [47,48]. However, theimplication of strain energy has not been con-sidered but different mass transport mechanismshave been studied.

In the case of Volmer—Weber growth our analysishas stressed that the thermodynamical leadingequation is Eq. (19) where the second term is most-ly negligible. By this way the smaller the equilib-rium shape l

%2(and then from Eq. (11) h

%2) the

higher the chemical overpotential *k around thecrystal. So the usual Ostwald ripening still holds:small crystals lose molecules in aid of the biggerone. Of course, there is a limit to the upper size aswe discussed in Section 2.2.1 where the strain dropsplastically. Then for such a crystal becoming flatterat constant volume, l

%2is increasing so that its *k

decreases and its ripening proceeds further.The only behavior which may be influenced in

a special fashion is the kinetic behavior. When theleading kinetic is volume diffusion in the gaseous orliquid surrounding medium then epitaxial strainwould not have any predictable effect on it. How-ever, when surface diffusion is prevailing the adsor-bed molecules as elastic dipoles may be pre-ferentially attracted to the smaller islands becauseof the smaller strain field they induce in the sub-strate.3 Such a kinetic is, however, not able to invertthe thermodynamic tendency of Gibbs—Thomson’sequation but only slows down its effect. Therefore,the usual Ostwald ripening is the normal behaviorfor Volmer—Weber growth. In the case of

3For stepped surfaces (steps of same sign) under epitaxialstrain a similar mechanism has been described in Refs. [21,22].

Stranski—Krastanov growth (*c(0), as experi-mentally observed by several authors, a size selec-tion is not excluded. However, no proper andconvincing theoretical treatment has been given upto now in this case. In a further paper [49] we willtreat this case of the Stranski—Krastanov equilib-rium shape crystals and closely discuss their ripen-ing behavior.

3.2. Elastic interactions via the substrate

When a collection of crystals becomes moredense lM )DM , elastic interaction becomes effective.In our model the strain distribution inside thecrystal !l/2(x(l/2 is given by eA

xx(x, z)"

mM(x)fl(z) [26], where the functions M(x) and fl(z)have mean values giving the strain energy (Eqs. (3)and (4)). Just beneath the crystal A, the strain in thesubstrate is given by eB,*/

xx(x, 0)"eA

xx(x, 0)!m. Out-

side the contact area of the crystal A, DxD'l/2,there is a deformation field whose xx componentalong the interface z"0 approximately is [26]

eB,065xx

(x, 0)"mA2

pPB2

CAx

l/2B2!1D

~1, (20)

where P is given by Eq. (6).For a set of islands the corresponding stresses

pB,*/xx

(x, 0) and pB,065xx

(x, 0) can overlap and add. InFig. 4a we draw two islands 1 and 2 far enough(4l"D) so that their stresses can hardly overlap.We draw schematically in between the islands thestresses pB,065

xx, 1(x, 0) and pB,065

xx, 2(x, 0) due to islands

1 and 2, respectively, and in the crystals 1 and 2 theinside stresses pA

xx,1(x, 0) and pA

xx,2(x, 0). In Fig. 4b

the two islands are closer (2l+D) so thatpB,065xx,1

(x, 0) and pAxx,2

(x, 0) and then pB,065xx,2

(x, 0) andpAxx,1

(x, 0) overlap. Such an overlapping increasesthe stress inside each island so that the new equilib-rium state of stress inside the interacting islands1 and 2 now can be written as pA,*/5

xx,1(x, 0)"

pAxx,1

(x, 0)#pB,065xx,2

(x, 0) and pA,*/5xx,2

(x, 0)"pAxx,2

(x, 0)#pB,065xx,1

(x, 0). This means that when close enough, theislands 1 and 2 communicate via the substrate bystressing each other and increasing their elasticenergy.

Owing to the asymmetry of the overlappingsthe local strain energy at the inner border of the


Fig. 4. Two elastically interacting crystals 1 and 2 of size l anddistance D so that their pB

xx,1(x, 0) and pB

xx,2(x, 0) overlap. In

Fig. 4a their distance D"4l is too great to have significantoverlapping. In Fig. 4b the overlapping becomes important andthe stresses inside and in between the “islands” increase.

crystals are higher than that at the outer borders sothat a material transport from each inner to outerborder may become effective. Therefore, the zerostrain center of each crystal is displaced and thedistance between the crystals increases indefinitely.¹here is an effective thermodynamic elastic force ofelastic origin between the crystals leading to bringeach crystal far apart.

In an infinite periodic set of crystals of equidis-tance D and same size l the overlapping of thestress fields are symmetric and the effective elasticforce on each crystal is zero. For a one-dimensionalset, there is the strain inside an “island” of the setdue to Eqs. (20) and (6):

eA, */5xx

(x, 0)"eAxx

(x, 0)

#

2h

l

1

pK

=+n/1C CA

x!nD

l/2 B2!1D

~1

#CAx#nD

l/2 B2!1D

~1

D, (21)

which means it is the strain of a free island (firstterm in Eq. (21)) wherein the effect of the com-pression field due to the neighboring islands isadded.

For calculating the elastic energy according toEqs. (3) and (4) the mean value of eA

xx(x, 0), SMT,

has to be replaced by that of Eq. (21)

SM*/5

T"SMT!2

p

1

K

h

lSfT ln C

sin(pl/D)

pl/D D. (22)

Then substituting the new elastic energy*¼

*/5"E

0R

*/5(r, l/D) but with now E

A/

(1!l2A)m2,2E

.in the free energy (Eq. (2)), the

equilibrium shape of a given island in a periodic setfollows similarly to Section 2.2 from the parametricsystem analogous to Eq. (11) but now containingpartial derivatives of R

*/5(r, l/D)

Gh*/5%2"!C

4c@A

3E.A1!

r0r B#

1

3

l2

D

R*/5

(r, l/D)

(l/D) KrD

]CR

*/5(r, l/D)

r Kl@DD~1

,

l*/5%2"h*/5

%2/r . (23)

Note that for DPR, SM*/5

TPSMT, so thatR

*/5(r, D)PR(r) and then Eq. (23) becomes equiva-

lent to Eq. (11), which means the equilibrium shapeof far enough crystals is that of isolated ones.

The equilibrium shape h*/5%2"h*/5

%2(l*/5

%2) can again

be plotted by resolving numerically the first equa-tion of Eq. (23) in h for given l and D. In Fig. 5 weplot the equilibrium height H*/5 as a function of theequilibrium lateral size ¸*/5 (in atomic units) butsuccessively for D"R (noninteracting crystals),D"25, 50 and 100. We took K"1, r

0"0.1 and

m"4]10~2.Interacting crystals have a shape ratio r which

deviates from that of isolated crystals. More pre-cisely, the near equilibrium growing crystals preferto thicken rather than moving closer to the bordersand finally coalescing. For example, in Fig. 5a crystal having a lateral size ¸*/5"45 has anequilibrium height H*/5"20, 15 or 10 when thecrystal belongs to a set of periodicity D"50, 100 orinfinitely far away respectively.

Furthermore such a thickening induces a stressincrease in the substrate in between the islands thatamplifies the tendency of the loss of coherence(see Section 2.2.1). The more rigid the substrate (Kincreases), the weaker is the substrate deformation(see Eqs. (20) and (6)) and then the weaker theelastic interaction. So the harder the substrate, the


Fig. 5. Equilibrium shape of a box-shaped crystal of height H*/5,basis ¸*/5 when inside a 1D periodic set of equidistance D'¸ (inatomic units). r

0"0.1; K"1, m"4]10~2, D"25, 50, 100,

R, are the different equidistances.

weaker the deviation from the equilibrium shape ofisolated crystals.

Let us note that the height divergences that ap-pear in Fig. 5 for ¸*/5PD are due to the logarith-mic term in Eq. (22) and disappears if, as usually,a cut off is introduced. This originates in the factthat elasticity theory cannot be applied at veryclose distances (typically four atomic distances) sothat the macroscopic description fails when theisland edges are closer than four atomic distancesfrom each other. Lastly, let us note that for interac-ting crystals Eq. (19) is modified, but Ostwaldripening still holds except close to coalescence forD/¸*/5K1 (but dislocations enter before as de-scribed in Section 2.2.1).

4. Conclusions

We have shown that for coherent epitaxial de-posits with incomplete wetting (Volmer—Webercase *c'0) strain energy opposes wetting. Whennatural misfit m is zero the smaller the *c/c ratio,the flatter the equilibrium shape (since r

0"

h/l"*c/c). When mO0, the shape ratio r in-creases with the size of the deposit. The physicalorigin of this phenomenon is the so-called accumu-lated strain energy after relaxation. This elasticrelaxation is described by the R(r) factor (Fig. 1)which enters by its derivative dR/dr in the extended

Wulff—Kaishew’s theorem [11,26] or in its alter-native form (Eq. (19)) called generalized Gibbs—Thomson equation.

We also considered a collection of epitaxial crys-tals interacting elastically via the substrate. ¼henthe borders become closer the crystals mutually in-crease their strain (back stress effect) and the shaperatio shows an extra increase. During continuousdeposition, crystals prefer to avoid each other byincreasing their height. However, there is somelimit for accumulated elastic strain. We only gavea short discussion in Section 2.2.1 how plastic re-laxation may relay the elastic one when the sizeincreases. This point will be considered in detail ina future paper [50]. Finally, the case *c(0 whereStranski—Krastanov growth occurs should be con-sidered, i.e. when layer by layer growth proceeds.Then how these layers become metastable andtransform in 3D crystals as experimentally knownhas to be considered. In this case there are severalpeculiarities to consider with respect to wetting andstrain energy so that the equilibrium shape of suchcrystals behave quite differently [49].

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