dimensions

Stability of strained heteroepitaxial systems in„1+1… dimensions

Pierre Thibault* and Laurent J. Lewis†

Département de physique et Regroupement québécois sur les matériaux de pointe (RQMP), Université de Montréal, Case Postale 6128,Succursale centre-ville, Montréal, Québec, Canada, H3C 3J7

(Received 15 December 2003; revised manuscript received 6 April 2004; published 27 July 2004)

We present a simple analytical model for the determination of the stable phases of strained heteroepitaxialsystems ins1+1d dimensions. In order for this model to be consistent with a subsequent dynamic treatment, allexpressions are adjusted to an atomistic Lennard-Jones system. Good agreement is obtained when the totalenergy is assumed to consist of two contributions: the surface energy and the elastic energy. As a result, wedetermine the stable phases as a function of the main “control parameters”(binding energies, coverage, andlattice mismatch). We find that there exists no set of parameters leading to an array of islands as a stableconfiguration. We, however, show that a slight modification of the model can lead to the formation of stablearrays of islands.

DOI: 10.1103/PhysRevB.70.035415 PACS number(s): 68.43.Hn, 68.65.Hb, 68.35.2p

I. INTRODUCTION

It appears today that self-assembly is not only one of themost elegant avenues for the production of devices based onquantum dots(QD’s), but also one of the most promising.Basic understanding of the physics of the formation of arraysof islands should ultimately lead to the realization of suchexciting concepts as spintronics1,2 and quantum dot cellularautomata.3 Driven by these possible developments, consider-able effort has been devoted to understanding and predictingthe conditions necessary for ensuring the stability of arraysof islands. Simple arguments based on the scaling of theenergy as a function of the volume of the islands4 indicatethat any system naturally undergoes ripening and, therefore,the only relevant observation is the(very long) time scaleneeded for the system to reach equilibrium.5 It is becomingclear, however, that a realistic energy function can lead toarrays of islands as equilibrium configurations,4,6–9 as canalso be deduced from experiment(see, e.g., Ref. 10 for areview).

The dynamics of formation of arrays of islands has tosome extent been investigated using atomistic models,11–14

but is not yet fully understood. For instance, the importanceof nucleation in the early stage of array formation is stillunclear.15 The long-term goal of our work is to address suchquestions and to provide a coherent picture of island forma-tion in heteroepitaxial systems, duly taking into accountchanges in the energy landscape arising from the lattice mis-match between the two components of the system. We aim toachieve this using a kinetic Monte Carlo(KMC) model,whereby the particles evolve according to the relative prob-abilities for hopping from one site to a neighbouring site ona fixed lattice. The main difficulty of this approach, whichwe are still in the process of developing, resides in properlymodulating the energy barriers to account for elastic contri-butions generated by the lattice mismatch between the twospecies. We note that this problem could in principle be ap-proached using molecular dynamics(MD). However, thetime scales involved in the present problem are such that MDcalculations are not feasible at this time.

For lack of an accurate model for the kinetics of islandformation, there is definite interest in the investigation of the

equilibrium properties of heteroepitaxial systems, i.e., thesurface morphology as determined by the various parametersthat control the physics of the systems(lattice mismatch,coverage, binding energies, etc.). Our objective here is topresent and discuss a continuous model suited for this pur-pose. This will enable us, in particular, to identify the regionsof parameter space where the formation of stable coherentarrays of island are favored. We develop an analytical ex-pression for the zero-temperature total energy of a strainedarray of islands. To ensure consistency, an important con-straint on this model is that it should be expressed in terms ofquantities that can be “exported” to a subsequent KMCmodel which will allow the dynamics to be investigated. Wedo this by assuming Lennard-Jones interactions between thetwo types of atoms and considering a triangular“ s1+1d-dimensional” geometry. This defines a reference sys-tem (which we will call theLJ system) on which both thestatic and the dynamic models should be mapped. Note thatwe use the term “s1+1d-dimensional,” rather than “two-dimensional,” to indicate that one of the spatial dimensionsis height (the z component), not to be confused with theusual two-dimensional case where atoms move in thexyplane.

We thus obtain an expression for the difference in energyDE between a system with a flat layer of adsorbed atoms anda system where islands have formed. For a given set of con-trol parameters, this quantity is minimized with respect to thesize of the islands, the distance between the islands, and thethickness of the wetting layer, so as to determine the equi-librium state of the system. We find that good agreementbetween the LJ system and the continuous model can beachieved by considering only two contributions inDE: thesurface energy and the elastic energy, the latter arising fromboth relaxation and substrate-mediated island-island interac-tions. This general approach is in many respects similar tothat proposed by Combe, Jensen, and Barrat.4 (CJB). Apartfrom the fact that our model allows the lattice misfit and theenergy parameters to vary, the main differences lie in thedetails of the method, as discussed below.

An important conclusion of our work is thatno single set

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of control parametersleads to an array of islands as a stableconfiguration. This is in contradiction with the results ofCJB;4 the discrepancy arises from the arbitrariness in thechoice of the parametersz0d which represents the character-istic length for the decay of the adsorption energy above thesurface. If we relax the constraint on the choice of this pa-rameter(i.e., consistency with the LJ model is not imposed),then stable arrays of islands are found, along with a “cracks”phase, made of flat islands that touch at their base.

II. THE MODEL

A. The LJ system

As mentioned above, the reference system consists of at-oms occupying the sites of as1+1d-dimensional triangularlattice and interacting via the Lennard-Jones potential

Usrd = 4eFSs

rD12

− Ss

rD6G . s1d

The atoms are of two types: substratesSd and adsorbatesAd.Thus, there are three different types of interactions and sixdifferent Lennard-Jones parameters that define the total en-ergy: hsSS,sAA,sSAj andh«SS,«AA,«SAj. Since one of thes’sand one of the«’s fix the length scale and energy scale,respectively, there are only four free parameters. This num-ber can be reduced to three by applying the Lorentz-Berthelot combination rule

sSA=1

2ssSS+ sAAd. s2d

Hence, there is a single degree of freedom as far as lengthscales are concerned, which can be expressed in terms of thelattice mismatcha, defined as

a =sAA − sSS

sSS. s3d

The other two degrees of freedom are the binding energiesbetween adsorbate atoms«AA, and between adsorbate andsubstrate atoms«SA. These parameters are independant and«SA.«AA is the wetting condition.

In practical calculations, the LJ interaction must be cutoffat some distancerc. The choice ofrc affects slightly thephysical properties of the system, notably the equilibriuminteratomic distancereq, the cohesive energyu0, and the elas-tic constants.(See Sec. II B 2 for details on the calculation ofthe elastic constants.) Table I presents the dependence ofthese important quantities on the cutoff radius.

B. The continuous model

As discussed in the Introduction, our purpose is to evalu-ate the energy difference between a system in which theadsorbate atoms form islands and one in which they form auniform layer on top of the substrate

DE = Eisland− Elayer. s4d

This energy difference can be decomposed into surface andelastic contributions

DE = DEsurface+ DEelastic, s5d

and is a function of the following parameters:a, the latticemismatch;eSS, the binding energy between two atoms of thesubstrate;eAA, the binding energy between two atoms of theadsorbate;eSA, the binding energy between an atom of thesubstrate and an atom of the adsorbate;u, the coverage, ex-pressed in monolayers(ML ); h, the height of the islands,expressed in ML;L, the width of the islands at their base;z,the thickness of the wetting layer, expressed in ML;d, thedistance between the centers of two neighboring islands. Asmentioned earlier, one of the binding energies, sayeSS, fixesthe energy scale. The last five parameters describe the geom-etry of the system(see Fig. 1); they are integers but we willassume that they are real in order to facilitate the calcula-tions. Sinceh, u, and z are expressed in ML, the actualheight is obtained by multiplying by the thickness of onemonolayer,sÎ3/2dreq.

The conservation of atoms(or volume) between the twoconfigurations imposes a constraint on the geometric param-eters

ud = zd+ hsL − h/2d. s6d

Sincea, eSS, eSA, eAA, andu are assumed to be known fromexperiment for a given material, i.e., they can be considered

TABLE I. Computed values of some important quantities as afunction of the cutoff distancerc (and number of nearest-neighborshells): req is the equilibrium interatomic spacing in units ofs, u0 isthe cohesive energy in units of«, andm and l are the two Laméparameters(both are equal in this geometry).

rc reqssd u0sed m=lsed

1.000(1) 1.1225 −3.000 31.18

1.732(2) 1.1159 −3.222 33.48

2.000(3) 1.1132 −3.319 34.49

2.646(4) 1.1122 −3.356 34.87

3.000(5) 1.1119 −3.364 34.96

` 1.1115 −3.382 35.15

FIG. 1. Geometry of the system(a) with islands and(b) withoutislands. The shaded region is the substrate and the white region isthe adsorbate. Islands are assumed to have the shape of an isoscelestrapezoid, with contact anglep /3.

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as control parameters, the energy need only be minimisedwith respect to the three parametersL, h, and z, d beingdetermined by the constraint(6).

In the next two subsections we develop expressions forthe two energy contributions in Eq.(5) as a function of thevarious parameters of the problem. They are derived largelyfrom theoretical considerations but, in some cases,ad hocterms are introduced in order to ensure that the model isconsistent with the LJ system; in these cases, we proceed asfollows: first, we generate a set of configurations with somenumber of adsorbate atoms placed in the shape of a trap-ezoid. Then, with suitably chosen parameters(e ands), welet the whole system(including a substrate large enough tomake finite size effects negligible) relax to minimum energy.Periodic boundary conditions are used along thex direction.Figure 2 illustrates the atomic displacement of the atoms ofan island as a result of relaxation.

1. The surface energy

We first determine the adsorption energy of an island oftype S (i.e., “substrate on substrate”) whose actual height is

hsxd or hsxd when expressed in ML(see Fig. 3). This energyis the sum of bulk and surface contributions

ESisland= Vu0eSS+ sL − LdgSS, s7d

whereV is the volume of the island,u0eSS the cohesive en-

ergy per atom,sL−Ld the increase of surface due to the is-

land, and gSS the surface energy density. UsingL

=eÎdx2+dz2=eÎ1+h82sxddx, ESisland can readily be rewritten

in terms ofhsxd:

ESisland=E

0

L Fhu0eSS+ SÎ3

4h82 + 1 − 1DgSSGdx, s8d

where the factor34 comes from the substitution ofhsxd by

sÎ3/2dhsxd.The surface energy densitygSS is proportional to«SS; we

can write

gSS= CeSS, gAA = CeAA,

gSA= CseSS+ eAA − 2eSAd, s9d

whereC is a constant that depends on the number of neigh-bors taken into account in the model.

If the island is of typeA (i.e., adsorbate on substrate),now, the adsorption energy is easily obtained from Eq.(8) bysubstituting«AA and gAA for «SS and gSS and adding a termdescribing the interaction between the island and the sub-strate. If we assume nearest-neighbor interactions, this terminvolves only the atoms at the interface between the islandand the substrate. In a more general situation, the adsorptionenergy for an atom at positionz above the surface(in ML )can be written

Eadszd = Ead0 + seAA − eSAdhszd, s10d

where Ead0 is the “generic” adsorption energy for the case

«SA=«AA andhszd is a function which decreases withz, withcharacteristic lengthz0. As in similar works4,7,16 we assumethe form

hszd = Ae−z/z0; s11d

the parametersA andz0 can be determined by fitting to thetotal energy of a particular atomic model(here Lennard-Jones). Using Eq.(10), we find that the adsorption energy ofa vertical column ofh atoms is proportional to

oj=1

h

e−j /z0 =1 − e−h/z0

e1/z0 − 1. s12d

Hence, we obtain

FIG. 2. Displacements of the atoms of a relaxed islandsL=20,h=10d. (a) tensilesa=−1%d; (b) compressivesa=+1%d. Thelengths of the arrows are about 30 times the actual atomicdisplacements.

FIG. 3. Island of shape given byhsxd.

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EAisland=E

0

L Fhu0eAA + SÎ3

4h82 + 1 − 1DgAA

+ BseAA − eSAds1 − e−h/z0dGdx, s13d

whereB=A/ se1/z0−1d. It can easily be shown thatB=2C bytaking the limiting case of a very thick adsorbate. Overall,therefore, only two parameters need to be fitted to the atomicmodel, namely,B andz0. This fit was carried out on systems

with uniform adsorbed layers of thicknessu [i.e., hsxd=u⇒ h8sxd=0]. In this case, the energy difference between asystem with an adsorbate of typeA and an adsorbate of typeS is

EAlayer− ES

layer= dfuu0seAA − eSSd + BseAA − eSAds1 − e−u/z0dg.

s14d

The numerical calculations yieldB=2.53 andz0=0.39 (seeAppendix A for details).

Assuminghsxd describes a trapezoidal shape, we can fi-nally write the first term of Eq.(5) as

DEsurface= 2CseAA − eSAdfd − sd − L + z0de−z/z0 − sL − h

− z0de−sh+zd/z0 − ds1 − e−u/z0dg + CeAAh. s15d

2. Elasticity

It is an interesting fact that the theory of elasticity, whichdeals with continuous media, can accurately describe sys-tems as small as a few tens or hundreds of atoms(see Ref.17, for instance). In this work, we exploit this property toconstruct(at least in part) the analytical expressions enteringthe second term of Eq.(5). Unfortunately, we know of noanalytical solution to the elasticity differential equations for asystem with the boundary conditions illustrated in Fig. 1(a).This difficulty can be circumvented by making some as-sumptions on the force distribution caused by the island onthe substrate. As argued by Tersoff and Tromp,18 when theisland is very flatsh!Ld, one can assume that there is nostrain relaxation in thez direction. This leads to a force dis-tribution which is proportional to the derivative of the heightfunction f ~h8sxd. In more general cases, however, we foundthat a (truncated) linear force density was a better assump-tion. Since higher islands deform the substrate more effi-ciently (for a given volume), we used the following expres-sion:

fsxd = Hkx if uxu , L/2,

0 otherwise,s16d

wherek is a constant(to be determined) which depends onthe lattice mismatch, the shape of the island and the elasticmodulii of both the island and the substrate.

Following the guidelines presented in Ref. 19, we findthat the Green’s tensor for a semi-infinite two-dimensionalplane is

Gxx =sl + mdx2 − sl + 2mdr2 lnsrd

2pmsl + mdr2 ,

Gxz=sl + mdxz− mr2 arctans x

zd2pmsl + mdr2 ,

Gzx=sl + mdxz+ mr2 arctans x

zd2pmsl + mdr2 ,

Gzz=− sl + mdx2 − sl + 2mdr2 lnsrd

2pmsl + mdr2 , s17d

wherer2=x2+z2. Thex component of the displacement fieldon the surface of the substrate is found to be

uxsx,z= 0d =E−`

`

Gxxsx,z= 0dfsx − x8ddx8 ;L

2uS2x

LD ,

s18d

whereusjd is a scale-independant function

usjd = kF2j + s1 − j2dlogU1 + j

1 − jUG , s19d

with k=3kL/16pmS and wheremS is one of the Lamé pa-rameters of the substrate.

Because the atoms lie on a triangular lattice and interactvia a radial potential, the two Lamé parametersm andl mustbe equal, given by

l = m =Î3

8 oj

r j2U9sr jd, s20d

the sum being carried out over all lattice points within agiven cutoff radius. The Young modulus and Poisson ratiofor a two-dimensional system are24

E =4msl + md

l + 2m=

8

3m,

n =l

l + 2m=

1

3. s21d

With these assumptions, we can construct expressions forthe two principal elastic contributions to the total energyarising from the presence of a mismatched island on a sub-strate, viz., the energy due to the mutual strain between theisland and the substrate, and the island-island interaction en-ergy mediated by the substrate

DEelastic= DEstrain+ Einteraction. s22d

The elastic energy per unit volume(more precisely unitarea in the present case) of a strained uniform adsorbed layeris

dEelasticlayer

dV=

1

2EAa2, s23d

EA being the Young modulus of the adsorbate. Hence, a por-tion of width d of this sÎ3/2du-thick layer has an energy


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Eelasticlayer =

2Î3

mAa2ud. s24d

Similarly, we can write

Eelasticisland =

1

2EAa2vsL,hd, s25d

wherevsL ,hd is a function(to be determined) having units ofvolume. CJB(Ref. 4) write this as

vsL,hd = RsrdV, s26d

where r =h/L is the aspect ratio,V is the volume of theisland, andRsrd is a dimensionless function bounded be-tween 0 and 1. Evidently, for scaling reasons, anyvsL ,hd canbe written in this form. In the present work, however, wechoose to determinevsL ,hd without invoking this scalingansatz. This choice is mainly dictated by accuracy consider-ations: in order to fitRsrd to numerical data, it is necessary todivide the computed elastic energy by the volume, therebyreducing the relative importance of large islands(see Appen-dix B for more details).

We find that an excellent fit to the Lennard-Jones data isobtained with

vsL,hd =Î3

2

L2

cf1 − e−ch/sL−hdg, s27d

wherec is the only(unitless) parameter to be adjusted to thedata. Numerical calculations on the LJ system yieldc=13.5(see Appendix A for details). Ratsch and Zangwill20 havedeveloped a similar function from theoretical considerations;the only difference is the denominator of the exponent,where they useL instead ofL−h and findc=2Î3p<10.9. Insummary we have, for the elastic energy due to the strainbetween the island and the substrate

DEstrain=2Î3

mAa2FL2

cs1 − e−ch/sL−hdd + zd− udG , s28d

where we include the contribution arising from the presenceof a wetting layer of thicknessz in the system with islands.

The substrate-mediated interaction energy between twoislands a distanced apart—the second term in Eq.(22)—canbe written as a surface integral

Eisl-islsdd =E−`

`

uxsxdfsx − dddx, s29d

whereuxsxd is the displacement field of the first island andfis the surface force distribution of the second island. UsingEqs.(16) and (19), we obtain

Eisl-islsdd = −4pmSk2

pL2 F3L4 + 2d2L2 + 4dL3 lnUd − L

d + LU

+ 2sd4 − 3d2L2dlnUd2 − L2

d2 UG . s30d

Note that this expression remains finite whend→L. Sincewe are interested in the interaction energy for an array ofislands, we have to sum up all contributions coming from the

islands at position…, −2d,−d,d,2d, . . . . In order to get asimple expression for our model, we may replaceEisl-isl bythe first few terms of its asymptotic series

Eisl-isl <2p2mSL

2k2

9S 2L2

3pd2 +L4

5pd4 +3L6

35pd6 + ¯ D .

s31d

The sum of all possible contributions can now be carried outexactly:

Einteraction= oj=−`

jÞ0

`

Eisl-isls jdd <4p3mSL

2k2

81SL2

d2 +p2L4

50d4

+p4L6

1225d6D . s32d

We still have to determine the dependence ofk [see Eq.(19)] on the parameters of our model. Let us first recall thatthis quantity is proportional to the amplitude of the forcedistribution exerted on the substrate by the island and, there-fore, it also determines the amplitude of the displacementfield in the substrate. We assumek to have the followingform:

k = akmsmA,mSdkgeosh,Ld, s33d

wherekm depends only on the elastic coefficients of the sub-strate and the adsorbate andkgeo depends only on the geom-etry of the island.k is linear ina because Eq.(19) has itselfbeen derived within the framework of the linear theory ofelasticity.

For km we propose thead hocexpression

km =mA

mA + mSs34d

which satisfies the three important limiting cases

mS@ mA ⇒ km → 0,

mS! mA ⇒ km → 1,

mS= mA ⇒ km =1

2. s35d

The first relation corresponds to the case of a substrate whichis much more rigid than the adsorbate, the second is theopposite case, and the third is the case of equally rigid sub-strate and adsorbate. The geometry factorkgeo can be deter-mined by fitting to the LJ data. We find the following expres-sion to yield good results:

kgeo=1

4s1 − e−a1L+a2ds1 − e−b1h/sL−hd+b2d. s36d

The first factor14 is such that]xux, the x component of the

substrate’s strain tensor, is always between 0 anda. Thesecond factor ensures thatkgeo goes rapidly to 1 whenL islarger than a few atoms. The last factor is a scale invariantterm which is maximum when the aspect ratioh/L is 1. After


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fitting the displacement field to the LJ data(see AppendixA), we find the following set of values for the parametersentering the above expression:

a1 = 12.6/req, a2 = 0.028,

b1 = 0.033, b2 = − 1.35, s37d

wherereq is the equilibrium interatomic distance(see previ-ous section).

Putting everything together, we obtain the following longexpression for the interaction energy in Eq.(22):

Einteraction=4p3mSL

2

81Fa

4

mA

mA + mSs1 − e−a1L+a2d

3s1 − e−b1h/sL−hd+b2dG2SL2

d2 +p2L4

50d4 +p4L6

1225d6D .

s38d

We are now in position to construct the phase diagram of ourmodel as a function of the various control parameters.

III. RESULTS AND DISCUSSION

A. Phase diagrams

The energy differenceDE between a system in which theadsorbate atoms form islands and a system in which theyform a uniform layer on top of the substrate, Eq.(4), can beexpressed as

DE = DEsurface+ DEstrain+ Einteraction, s39d

where the three terms are given by Eqs.(15), (28), and(38),respectively. We now proceed to determine the equilibriumconfigurations of the system as a function of the control pa-rameters, viz. binding energies, mismatch, and coverage. Theparameter space is sampled as follows:

eAA = 0.7,0.8, . . . ,1.2,1.3,

eSA= 0.7,0.8, . . . ,1.2,1.3,

a = 0%,1%, . . . ,9%,10%,

u = 1, . . . ,14,15,

andeSS=1 sets the energy scale. We took positive values ofaonly sinceDE depends quadratically on this parameter. Forevery possible combination of the above parameters, we de-termineh, L, d, andz such thatDE is minimal. If the mini-mumDE is larger than zero, the equilibrium configuration isa uniform adsorbate layer on the substrate; this phase isknown as the “Frank-Van der Merwe” phase.21 If it is lessthan zero, the equilibrium phase depends on the values ofhmin, Lmin, dmin, andzmin which minimizeDE. A few differentsituations can arise(we omit the “min” subscript to facilitatereading).

(1) If d→`, the islands undergo ripening, either directlyon the substratesz=0d or on a wetting layerszÞ0d;

(2) If d is finite andL,d, the equilibrium configuration isan array of islands either on the substrate(“Volmer-Weber”

phase22) or on a wetting layer (“Stranski-Krastanov”phase23).

(3) If d is finite andL=d, the system is “cracked”, that is,islands touch at their base.

Following the example of Darukaet al.,7 we produce a setof phase diagrams in thea-u plane. Figure 4 shows a typicalphase diagram for specific values of the binding energies(eAA=1 andeSA=1.1). Such plots are collected in Fig. 5 forthe whole array of values ofeAA andeSA investigated. A wordof caution about the range of misfit values investigated: Ourmethod is strictly valid only in the limit of linear elasticity.For values ofa greater than,5% or so, nonlinear effects arecertainly not negligible. In particular, one may expect islandsto undergo dislocations for lattice mismatches of the orderof, say, 10% or more. The behavior of our generic system forlarger values of the mismatch should therefore be viewed as

FIG. 4. (Color online) A typical phase diagram; hereeAA=1 andeSA=1.1. The three phases are as follows: light gray(yellow) Frank-van der Merwe, medium gray(green) ripening islands with wettinglayer, dark gray(red) ripening islands without wetting layer.

FIG. 5. (Color online) Phase diagrams for all values of the bind-ing energies considered; hereeSS=1 andz0=0.39. Each small imageis a phase diagram in thea-u plane similar to that of Fig. 4. Thecolors correspond to the different phases: light gray(yellow) Frank-van der Merwe, medium gray(green) ripening islands with wettinglayer, dark gray(red) ripening islands without wetting layer.


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indicative of a trend rather than a prediction of the actualbehavior. In any case, the linear term is expected to dominatein the whole range of values examined.

We observe, first, that only one phase is possible wheneAA.eSA (above the main diagonal of the plot), namely, rip-ening islands without wetting layer. This is not surprisingsince it corresponds to the nonwetting case in which allterms of DE but Einteraction are negative. A more importantconclusion that can be drawn from Fig. 5 is thatno set ofcontrol parameters leads to an array of islands as a stableconfiguration. This is in disagreement with the results ofCJB.4 The discrepancy can be traced back to the choice ofz0,the characteristic length for the decay of the adsorption en-ergy at the surface, discussed in Sec. II B 1. The value of thisparameter(0.39) was obtained, in the present model, by fit-ting to the LJ potential, while CJB chose it in anad hocmanner.

B. Extended model

We may relax the constraint on the value ofz0 for a mo-ment and setz0=3, close to the value used by CJB, resultingin an increase of the influence of the substrate on the ada-toms higher above the interface. The introduction of this newlength scale in the problem allows some new features toappear. The corresponding phase diagrams are displayed inFig. 6. Consistent with CJB, we now observe the existence oftwo different stable phases. In addition to the stable array ofislands, we now find an equilibrium state consisting of is-lands touching at their base. Since this morphology is some-what better described by a flat layer with very narrowtroughs, we will refer to it as the “cracks” phase(shown inred in Fig. 6). The Stranski-Krastanov phase never appearedin the parameter space we considered.

It is not clear that potentials withz0=3 do in fact exist,but it is certainly conceivable, in the case for instance ofsemiconductors, that the decay length of the surface energyis significantly larger than that of the LJ potential and, as a

consequence, stable phases would exist. Consideration ofthis extended system is consistent with our long term objec-tives sincez0 can be included as an adjustable parameter inthe KMC calculations.

In order to get more detailed information about the stablephases, we selected one of the phase diagrams of Fig. 6seSA=1.3, eAA=1d and repeated the calculations on a finergrid s2003200d. The resulting diagram is shown in Fig. 7.Note that, in this graph, the layout of the phase boundarieshave been drawn as a “guide to the eye” using the originaldata(shown in the inset); we do not know the analytical form(if it exists) of these curves.(The slightly “zigzagging” be-havior of the boundary is a consequence of allowingz to takeonly integer values in the minimization process.) We findthat the transition is not sharp between the Volmer-Weber(VW) and “cracks”(C) phases(hence the dashed line). Fig-ure 7 has some features similar to the phase diagram com-puted by Darukaet al.,7 in particular the shape of the FM,R1, and R2 phases.

The Frank-van der Merwe(FM) and the two ripening(R1and R2) phases have no interesting intrinsic features; the firstis flat, and the two others correspond to the minimizing pa-rameters going to infinity. We are therefore more concernedabout the two stable phases(C and VW). A characteristicquantity often measured is the island densityn on the sur-face. In our model, since the center-to-center distance be-tween islands isd, we simply haven=1/d. Figure 8 showshow the density varies as a function of coverage and latticemisfit for the binding energies selected. In the C phase, thisdensity is simply the cracks density, that is, the number ofcracks per unit length.

We have also computed the aspect ratio of the islands as afunction of coverage and lattice mismatch; this is shown inFig. 9. The global behavior is as expected: asa increases, theelastic relaxation process becomes more and more efficientand the islands can afford an increase of their surface; this

FIG. 6. (Color online) Same as Fig. 5, but forz0=3.0. Thevarious phases are as follows: light gray(yellow) Frank-van derMerwe, light/medium gray(green) ripening islands with wettinglayer, medium/dark gray(red) ripening islands without wettinglayer, dark gray(purple) cracks; black(blue) Volmer-Weber.

FIG. 7. (Color online) Phase diagram for the system withz0

=3.0, eAA=1, andeSA=1.3. The various phases are Frank–van derMerwe (FM), ripening islands with wetting layer(R1), ripeningislands without wetting layer(R2), cracks(C), and Volmer-Weber(VW). The dashed line indicates the smooth transition between theVW and the C phases. Inset: original phase diagram used to con-struct the main graph.


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explanation holds for the cracks phase as well.In both Figs. 8 and 9, the quantities shown by the color

scale are conspicuously continuous at the VW-C boundary.This was to be expected since the very definition of thesephases implies no jump in any quantity(VW phase forL,d,C phase forL=d). This boundary actually is the only secondorder phase transition; all others are first order.

The cracks phase is well-known in cases wherea,0(tensile strain). Under compressionsa.0d, the system mayexhibit a buckling phase. Our model, however, is insensitiveto the sign of the strain. These observations suggest that,whereas cracking and buckling are two different mecha-nisms, they arise from the same relaxation principles. In ourmodel, the occurence of the cracks phase is obviously relatedto the interaction energy term. Have we had assumed thatEinteractionis infinite when the islands touch[as did CJB(Ref.4)], the C phase would have been entirely precluded. A stron-ger island-island repulsion might also lead to the appearanceof a Stranski-Krastanov(SK) phase for intermediate cover-age. While it remains to be verified, we expect this cracks

phase to be very close, at finite temperature for instance, to aSK phase; this is, however, beyond the scope of the presentwork.

IV. CONCLUSION

We have presented a simple analytical model for the de-termination of the stable phases of strained heteroepitaxialsystems ins1+1d dimensions. The model was developedwith a view of carrying out KMC simulations of the dynam-ics of the formation of islands. This is a very difficult task,but already we have made progress in this direction, onwhich we will report in a subsequent publication. In order forthe present model to be exportable to a KMC code, all ex-pressions were adjusted to an atomistic Lennard-Jones sys-tem. The present calculations reveal that, for parameterswhich are consistent with the Lennard-Jones model, the arrayof islands is not a stable configuration of the system. If fullconsistency of the parameters is not imposed, and in particu-lar if we relax the value of the decay length for the adsorp-tion energysz0d, then a stable array of islands arises. Ourcalculations also reveal, in these conditions, the formation ofa cracks phase—an array of islands touching at their base.

Evidently, the present model should be viewed as genericand is not intended to provide an accurate description of

FIG. 10. Comparison between the numerical values ofC1 andC2 [in Eq. (A1)] and the theoretical curves(see text). (a) The curvehas no free parameter.(b) The curve is the best fit to the data.

FIG. 8. (Color online) Island density 1/d in the two stablephases as a function of coverageu and lattice mismatcha. Here,z0=3.0, eAA=1, andeSA=1.3.

FIG. 9. (Color online) Aspect ratioh/L in the two stable phasesas a function of coverageu and lattice mismatcha. Herez0=3.0,eAA=1, andeSA=1.3.


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“real materials”—in particular semiconductors since themodel is two-dimensional and the interatomic potential ispurely radial. Extending the model to three dimensions for apotential that would be appropriate to semiconductors(e.g.,Stillinger-Weber) is a long-term objective, beyond the scopeof this work. In this perspective, it is justified, as notedabove, to treatz0 as an adjustable parameter. We note in factthat the value ofz0 is mainly determined by the two-bodyterm of the potential, and should thus be qualitatively,but notquantitatively, the same in various types of materials. Hence,we expect our conclusions to be rather general, i.e., nonma-terial specific. While much work remains to be done, thepresent model is a first step in our aim to better understandthe formation of dislocation-free arrays of islands.

ACKNOWLEDGMENTS

One of us(P.T.) is grateful to P. Jensen and J.-L. Barrat,from the “Laboratoire de physique de la matière condenséeet nanostructures, Université Claude-Bernard Lyon-1,” forhospitality and fruitful discussions. This work was supportedby grants from the Natural Sciences and Engineering Re-search Council(NSERC) of Canada and the “Fonds québé-

cois de la recherche sur la nature et les technologies”(FQRNT) of the Province of Québec. We are indebted to the“Réseau québécois de calcul de haute performance”(RQCHP) for generous allocations of computer resources.

APPENDIX A: COMPUTATIONAL DETAILS

We present here some details of the method used to fit thefree parameters arising in the derivation of the continuousmodel presented in Sec. II. In what follows, the cutoff radiusof the potential has been fixed torc=3.2, i.e., midway be-tween the fifth and sixth neighbor shells. The equilibriumdistance between substrate atoms is set to 1 so thatsSS=1/1.1119=0.8993(see Table I). In all calculations, the sub-strate has thickness between 50 and 100 layers, with thelower three maintained fixed to mimic the presence of thebulk. For every configuration, the energy was determined byrelaxing the positions of the atoms using a conjugate-gradient algorithm.

1. 1. Surface energy

The parametersB and z0 in Eq. (13) have been fitted tosystems composed of 50 substrate layers and coverage

FIG. 11. Numerical data and theoretical curves forkgeogiven byEq. (36).

FIG. 12. Numerical data and theoretical curves for the strainenergy given by(28).

FIG. 13. Elastic energy for an island of given volumeV as afunction of its aspect ratioh/L.

FIG. 14. Elastic energy for an island of given widthL as afunction of its aspect heighth.


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u[ h1,2,3,4,5,10j. These configurations have been relaxedfor a set of combinations of energy parametersseAA,eSAd[ h0.8,0.9,1,1.1,1.2j2. Altogether, the minimumtotal energies of 150 different systems have been computed.For every system, the difference between the total energyand the total energy of the equivalent system witheSA=eAA=eSS has been calculated. The numerical values forEA

layer

−ESlayer in Eq. (13) have then been fitted to an equation of the

form

EAlayer− ES

layer= C1sudseAA − eSSd + C2sudseAA − eSAd.

sA1d

Figure 10 shows the dependence ofC1 andC2 on coverage.C1 is completely determined(i.e., there are no free param-eters), given by C1=du0u, whereu0=3.364 is the cohesiveenergy for a cutoff radius of 3(see Table I). C2 was fitted toa curve of the formBs1−e−u/z0d; the fit yieldsB=2.53 andz0=0.39.

2. 2. Elastic energy and island-island interaction energy

The parametersa1, a2, b1, b2, andc of Eqs.(27) and(36)have been fitted to the same configurations as above. 30 dif-ferent islands have been generated, of widthL in the range20 to 80(six values) and heighth in the range 1 toL (fivevalues). Each of these configurations was relaxed with mis-fits of +1 and −1%. While the relaxed positions of the atomsdepend on the sign ofa, the energy does not, as could beexpected.

The numerical value ofkgeo has been found using thedisplacements of the atoms of the substrate; for every sys-tem, the amplitude of the theoretical displacement field[which is close to, but not equal to Eq.(19) because of pe-riodic boundary conditions] has been adjusted to the dis-placements of the substrate atoms. Figure 11 shows theagreement between the curves and the data.

Figure 12 shows the good agreement between the strainenergy obtained from the simulations and the analytical ex-

pression(28). In B we elaborate on the choice of this equa-tion.

APPENDIX B: FUNCTIONAL FORM OF Eq. (28)

We mentioned in Sec. II B 2 that a better fit to the numeri-cal data is obtained with a function of the formEsL ,hd=CvsL ,hd [wherevsL ,hd has units of volume], rather thanEsL ,hd=CVRsh/Ld. We show here a comparison betweenour expression forvsL ,Hd, Eq. (27), and that used by CJB;4

the latter is obtained by writing, first, the functionRsh/Ld interms of the quantities used in the present paper:

RCJBsrd = A + Be−Cr/s1−r/2d, sB1d

with A=0.13, B=0.87, andC=−4.811 andr =h/L. SincevsL ,hd=Rsh/LdV, it is a simple matter to connect the twofunctional forms. We get

vCJBsL,hd =Î3

2hsL − h/2dsA + Be−Ch/sL−h/2dd, sB2d

which must be compared with Eq.(27), and

Rsrd =s1 − e−cr/s1−rdd

crs1 − r/2d. sB3d

Figure 13 shows the the elastic energy of an island of fixedvolumeV as a function of its aspect ratio, according to Eqs.(B3) and(B1). The two curves are different, but their generalbehavior is very similar. Note in particular that the startingpoints coincide and that the end points are less than 2%apart.

The situation is however very different for the elastic en-ergy of an island of fixed width as a function of height(Fig.14). CJB’s expression for the energy yields a peak aroundh/L=0.2 and has a minimum aroundh/L=0.6; in contrast,our expression increases monotonically with height. Ourpreference for the form of Eq.(28) is mainly based on thisobservation and the quality of the fit to the LJ data(see Fig.12).

*Present address: Laboratory of Atomic and Solid State Physics,Cornell University, Ithaca, NY 14825, USA. Email address:[email protected]

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Phys. Rev. Lett.75, 2968(1995).7I. Daruka and A.L. Barabási, Phys. Rev. Lett.79, 3708(1997).8Y. W. Zhang, Phys. Rev. B61, 10388(2000).9J. E. Prieto and I. Markov, inQuantum dots: Fundamentals, Ap-

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11A. L. Barabási, Appl. Phys. Lett.70, 2565(1997).12H.M. Koduvely and A. Zangwill, Phys. Rev. B60, R2204(1999).13K. E. Khor and S. Das Sarma, Phys. Rev. B62, 16657(2000).14M. Meixner, E. Schöll, V.A. Shchukin, and D. Bimberg, Phys.

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19L. D. Landau and E. M. Lifshitz, Theory of Elasticity(Butterworth-Heinemann, London, 1995).

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