Elastic energy release rate of quantum islands in Stranski–Krastanow growthB. Yang Citation: Journal of Applied Physics 92, 3704 (2002); doi: 10.1063/1.1506386 View online: http://dx.doi.org/10.1063/1.1506386 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/92/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strain relaxation dependent island nucleation rates during the Stranski–Krastanow growth of GaN on AlN bymolecular beam epitaxy Appl. Phys. Lett. 93, 243105 (2008); 10.1063/1.3046730 Unidirectional self-assembling of SiGe Stranski-Krastanow islands on Si(113) Appl. Phys. Lett. 86, 223109 (2005); 10.1063/1.1943490 Equilibrium shape of SiGe Stranski–Krastanow islands on silicon grown by liquid phase epitaxy Appl. Phys. Lett. 84, 5228 (2004); 10.1063/1.1759070 Structure of GaN quantum dots grown under “modified Stranski–Krastanow” conditions on AlN J. Appl. Phys. 94, 2254 (2003); 10.1063/1.1592866 Investigations on the Stranski–Krastanow growth of CdSe quantum dots Appl. Phys. Lett. 76, 418 (2000); 10.1063/1.125773

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Elastic energy release rate of quantum islands in Stranski–Krastanowgrowth

B. Yanga)

Structures Technology Inc., 543 Keisler Drive, Ste. 204, Cary, North Carolina 27511

~Received 8 February 2002; accepted for publication 16 July 2002!

When constrained by a lattice-misfit substrate, a sufficiently thick epitaxial layer develops surfacequantum islands~QIs! via mass transport as a thermodynamic process for energy minimization. Bythis process, the misfit strain energy is partially released, which serves as the driving force for islandformation. In this work, we examine the elastic energy release rate~EERR, i.e., elastic relaxationenergy per unit volume! of QI growth in multilayered heteroepitaxial structures under the conditionof mass conservation. The analysis is based on a two-dimensional isotropic linear elastic continuumapproach. A parametric study first is carried out to investigate the effects of various parameters onthe variation of EERR of QI formation in an epilayer-substrate system. Furthermore, the EERR,which represents the energetics of global equilibrium, is applied to investigate the correlation ofsurface islands to buried seed islands in a multilayered heterostructure. The variation of EERRduring nucleation of new surface islands shows a tendency of vertical correlation to buried seedislands in the isotropic heterogeneous system. This relationship of correlation does not alter withchanging vertical distance between the surface and seed islands in the range examined. ©2002American Institute of Physics.@DOI: 10.1063/1.1506386#

I. INTRODUCTION

A lattice-mismatched epitaixal layer on a crystal sub-strate tends to spontaneously grow surface quantum islands~QIs! via mass transport on the nanoscales. The self-assembly process is called the Stranski–Krastanow growthof QIs,1 which has recently been utilized to fabricate semi-conductor nanoscale particles.2 The islands include quasi-zero-dimensional quantum dots and quasi-one-dimensionalquantum wires, as schematically shown in Fig. 1. The semi-conductor QIs possess certain special electronic and opticalfeatures, rendering fascinating devices, such as low-thresholdlaser and huge-capacity memory media, possible.2,3

The spontaneous formation of surface QIs can be viewedas a thermodynamic process for energy minimization. Anepitaxial layer grown on a crystal substrate for example bymolecular beam epitaxy is strained if the lattice constants ofboth materials are mismatched. The strain can be as high as7% while retaining the elastic status of the system, for ex-ample, in the semiconductor system of GaAs/InAs.4 A fieldof such high elastic strain represents a large amount of spareenergy available for dissipative actions. However, the epitax-ial layer initially grows in a flat manner on the substrate.When the layer thickness is above a critical value, surfaceQIs appear. The formation of QIs relaxes a part of the elasticstrain energy stored in the epitaxial layer. It also causes thetotal area of surface and total length of singular edges toincrease. Energetically speaking, if the released strain en-ergy, which serves as the driving force, is more than thatneeded for the surface and edge adjustments, the QI growth

on the epitaxial layer surface is favorable. Otherwise, theepitaxial layer surface remains flat.

The energetics of Stranski–Krastanow growth of surfaceQIs in a heteroepitaxial system can be expressed mainly as acompetition of three energy terms:5

DE5DS EVeelasticdVD 1DS E

SesurfacedSD

1DS ELeedgedLD , ~1!

where DE is the total energy change,eelastic,esurface, andeedgeare the densities of elastic strain energy, surface energy,and edge energy, respectively, andV, S, andL represent vol-ume, surface, and line~of edge!, respectively. The total en-ergy change can be caused by a change of densities or by achange of surface area or edge length. In the Stranski–Krastanow growth of QIs, while the geometry is changed,the mass is conserved. In this case, the change of total elasticstrain energy is a matter of density change associated witheach mass point due to the mass rearrangement. It is com-monly understood that the first term of elastic strain energycan be evaluated by using a continuum-mechanicsapproach2,6 while the evaluation of the last two terms mayrequire a lattice-level simulation7 or a phenomenological ap-proach that is under development.8 It is the objective of thepresent study to elaborate, within the framework of con-tinuum mechanics, the concept of elastic energy release rate~EERR! defined as elastic relaxation energy per unit volumefor QI assembly in heteroepitaxial systems. The evaluationof the other two dissipative terms of local geometrical

a!Current address: Materials Reliability Division, National Institute of Stan-dards and Technology, Boulder, CO 80305; electronic mail:boyang@boulder.nist.gov

JOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 7 1 OCTOBER 2002

37040021-8979/2002/92(7)/3704/7/$19.00 © 2002 American Institute of Physics

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change is left to a lattice-level simulation or an experimentalmeasurement associated with a phenomenological descrip-tion.

The present work examines the variation of EERRagainst a number of parameters during growth of surface QIsin a two-dimensional~2D! heteroepitaxial structure. The ma-terials are assumed to be isotropic and linearly elastic. Thereason for us to use such a simple model is that we wouldlike to illustrate the underlying physics in a scenario as ge-neric as possible. When other parameters such as materialsanisotropy and three-dimensional~3D! effect come into play,one would be able to identify what a role they play by com-parison. Furthermore, the island shape is assumed to be tri-angular and to be self-similar during growth. When the QIsgrow, the wetting layer surface is lowered to satisfy the con-dition of mass conservation. Meanwhile, the wetting layersurface, excluding the QIs, is taken to be flat and laid in itsoriginal ~horizontal! orientation. By doing so, a kinetic lawof mass transport generally required for a QI-growth simula-tion can be bypassed.9 Based on this model, the effects ofisland size, wetting layer thickness, substrate elastic property,buried seed islands, etc., on the EERR for surface QI forma-tion are examined systematically. The analytical results sug-gest that the EERR can be used as a measure of the elasticenergy-releasing capability of a system through the forma-tion of surface QIs.

The concept of EERR is further applied to explore thevertical correlation of surface island nucleation with buriedseed islands. This issue is important due to the fact that mul-tiple sheets of QIs, also called QI superlattices, are de-manded in most of the device applications such as thosementioned earlier.2,8,10 The existence of lateral and/or verti-cal orderings of QIs plays a crucial role in determining thedevice functionalities. Also, it is necessary to understand theconfigurational stability of a QI superlattice over time forreliable device applications under the service environments.For example, a kinetically arranged QI superlattice mayevolve in time if the arrangement is not an equilibrated one.The superlattice may evolve into a different arrangement orjust destroy the current one to reach a potential well. Sincethe EERR addresses the energetics of global equilibrium of asystem, it is appropriate to use it to address the configura-tional stability of QI superlattices. Shchukinet al.11 have ap-plied the concept of system equilibrium with respect to totalenergy to investigate the interaction of two sheets of QIs in acubic material. They found that there exists a bifurcation ofcorrelation and anticorrelation in terms of distance betweenthe two sheets. In parallel, a number of other works10,12have

applied the criteria with respect to local strain or strain en-ergy density for the prediction of vertical correlation andanticorrelation of QIs. Zhanget al.13 further developed thelocal-field approach of QI formation by coupling to it a ki-netic law. The present study shows that the surface islandnucleation is correlated to buried seed islands by assumingthat the nucleation occurs at the site of maximum EERR. Thevertical anticorrelation of islands11 was not observed in therange of parameters examined in the isotropic heterostruc-ture.

In Sec. II, we describe the formulation of elastic strainenergy of a body consisting of multiple subdomains of gen-erally dissimilar homogeneous media. The concept of EERRof QI formation is introduced. In Sec. III, the two-dimensional isotropic-elasticity model of QI growth is devel-oped. The EERR of QI formation is studied against variousparameters systematically. In Sec. IV, the concept of EERRis applied to explore the relationship of surface island nucle-ation to buried seed islands. Finally, conclusions based onthis study are drawn in Sec. V.

II. FORMULATION

A. Elastic strain energy

Consider a bodyD consisting of multiple subdomainsVn ~n51,...,N!, as shown in Fig. 2. The subdomains containgenerally dissimilar homogeneous and linearly elastic media.The interfaces are perfectly bonded. A global Cartesian co-ordinate system (x1 ,x2 ,x3), is attached. The equilibrium ofthe body in the absence of body force requires

s i j , j50, ~2!

wheres i j is the stress tensor,j indicates the partial differen-tiation with respect toxj , and a repeated subscript indeximplies the conventional summation over its range. The con-stitutive law for the elastic materials is given by

s i j 5Ci jkl ~ekl2ekl0 !, ~3!

where Ci jkl is the elastic stiffness tensor,ekl@[(uk,l

1ul ,k)/2# is the strain tensor, andekl0 is the eigenstrain ten-

FIG. 1. Stranski-Krastanow growth of quantum dots~left! and wires~right!in epitaxy.

FIG. 2. Schematics of a bodyD consisting of multiple subdomainsVn ofgenerally dissimilar homogeneous and linearly elastic media.

3705J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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sor, assumed to be given. In the definition of strain,ui is thedisplacement vector, and the condition of infinitesimal defor-mation of the body is assumed. Further assuming that anappropriate boundary condition that ensures a unique solu-tion is applied, we derive below the strain energy of thebody.

The elastic strain energy of the body is defined as14

W[1

2EDs i j ~e i j 2e i j

0 !dV. ~4!

Applying e i j 5(ui , j1uj ,i)/2, symmetry ofs i j , and Eq.~3!,Eq. ~4! is rewritten as

W51

2EDs i j ui , jdV2

1

2EDCi jkl e i j

0 uk,ldV

11

2EDCi jkl e i j

0 ekl0 dV. ~5!

The last term, which is a constant, represents the initial strainenergy of the eigenstrain field. The other two terms are ex-amined separately below.

Applying the Gauss divergence theorem and Eq.~2! tothe first term of Eq.~5!, we obtain

ED

s i j ui , jdV5E]D

TiuidS, ~6!

where]D is the boundary ofD, andTi([s i j nj ) is the trac-tion at a boundary point with outward normalnj . Thus, thefirst term of Eq.~5! represents the work of external boundarytraction acting on the surface]D.

In the Stranski–Krastanow growth of QIs, the eigen-strain field due to a mismatch of lattice constants is uniformin each subdomain. In this case, the second term in Eq.~5!can be reduced to a surface integral, which may lead to lesscomputational effort and shed more light on the physicalmeaning of the island misfit strain field. So, assuming thatekl

0 is uniform in each of the subdomainsVn(n51,...,N), weobtain

ED

Ci jkl e i j0 uk,ldV5 (

n51

N

Ci jkl~n! e i j

0~n!E]Vn

uknldS, ~7!

through the Gauss divergence theorem, where the superscriptn in the bracket indicates the attachment to thenth subdo-main Vn . Compared to Eq.~6!, one may considerCi jkl

(n) e i j0(n)nl as an intrinsic traction acting along the interface

between the subdomains. Thus, the second term of Eq.~5!,i.e., Eq. ~7!, represents the work of the intrinsic tractioncaused by the internal misfit-strain field during deformationof the body.

Finally, in the case of uniform eigenstrain in each sub-domainVn , the total elastic strain energy of the bodyD isgiven by

W51

2E]DTiuidS2

1

2 (n51

N E]Vn

FkukdS

11

2EDCi jkl e i j

0 ekl0 dV, ~8!

whereFk([Ci jkl e i j0 nl) is the intrinsic traction. Therefore, to

evaluate the strain energy of the body, it suffices to know thetraction and displacement along]D and the displacementalong ]Vn(n51,...,N). Only for some special cases canthese quantities be evaluated analytically, for instance, by aGreen’s function approach. In general, numerical solutionshave to be managed for the evaluation. In the present work,we use the boundary element method15 to obtain the bound-ary and interfacial fields and evaluate the total elastic strainenergy of the system of QIs. It should be remarked that theboundary element method is overwhelmingly advantageousin dealing with the elasticity problems in many aspects overconventional domain-based numerical methods such as thefinite element method and the finite difference method. Thisis especially true in the present case where knowing the in-terior fields is not necessary.

B. Elastic energy release rate

During the growth of QIs, the geometry of a body ischanged. The geometrical change results in a change of elas-tic strain energy in the system. By applying the previousformulation, the variation of elastic strain energy of the sys-tem can be evaluated through the growth process. In order toquantify the rate of the elastic strain energy change, we in-troduce the quantity of elastic energy release rate~EERR! Gdefined as the reduction of elastic strain energy per unit vol-ume of mass transport in the system, i.e.,

G[2dW

dVmt, ~9!

whereVmt is the volume of mass transport. Equation~9! mayalso be written as

G5(n

G~n!dVmt

~n!

dVmtwith G~n![2

]W

]Vmt~n! , ~10!

where the superscriptn indicates thenth QI, andG (n) is theEERR of thenth QI. If the system is periodic with respect toeach QI,G (n)5G. In the following computation,G is evalu-ated approximately by a finite difference scheme with respectto Vmt .

III. QUANTUM ISLAND FORMATION IN AN EPILAYER-SUBSTRATE SYSTEM

Let us consider a 2D configuration of a wetting layerperfectly bonded on a substrate in the plane-strain deforma-tion condition, as shown in Fig. 3. Both the wetting layer andsubstrate are assumed to be isotropic and linearly elastic. Theelastic constantsCi jkl are thus reduced to (Ew ,vw) and(Es ,vs) for the wetting layer and the substrate, respectively,whereE is the Young’s modulus, andv is Poisson’s ratio.Note that the previous formulation of elastic strain energy ofa body is valid for 3D generally anisotropic solids. Theeigenstrain in the wetting layer is assumed to be hydrostatic,i.e., e i j

0 5e0d i j , and that in the substrate to be zero.The system geometry is assumed to consist of a series of

triangular quantum wires being nucleated and grown on thetop surface of the wetting layer, as shown in Fig. 3~a!. TheQWs are uniformly distributed in a spacing of 2L along the

3706 J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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surface. They have the same shape, and grow at the samerate, i.e., always equally sized. The bottom width of thewires is indicated bya. During the growth, the top surfaceexcluding the boundary of the QWs is taken to be flat andlaid in original ~horizontal! orientation. Also, this part of thesurface is lowered accordingly to meet the condition of massconservation. The QWs formed via mass transport are as-sumed to contain the same medium as the wetting layer. Bydoing so, a kinetic law of mass transport generally requiredfor a QI-growth simulation is bypassed. Due to the periodic-ity of the system, it suffices for us to consider a unit cell of asingle QW, as shown in Fig. 3~b!. The symmetry-boundarycondition of zero normal displacement component and zerotangential traction component is correspondingly imposedalong the unit-cell boundaries except that the top surface istraction free. The thickness of the substrate is taken to beequal to 2L. In this system, there are six parameters to vary:tw0 ~initial wetting layer thickness!, u ~tilt angle of QW!,vw , Es , vs , and e0, with L taken as the normalizationlength scale andEw as the normalization scale for quantitiesof stress dimensions. Since the behavior of the system islinearly scaled with (e0)2, only five of the six parametersremain to be examined.

First, the EERR of QW growth is evaluated at differenttw0 with the other parameters fixed atEs /Ew51, vw5vs

50.3, andu545°. Figure 4~a! shows the variation of thenormalized EERR,G/@Ew(e0)2#, with wire size ~indicatedby the bottom widtha! at three differenttw050.1, 0.3, and0.5L, respectively. In the cases oftw050.1 and 0.3L, thewetting layer material was consumed up before the growingQWs met one another, i.e., beforea52L. It can be seen thatwhile the QWs grow, the magnitude of EERR,G, which ispositive, drops. The dropping rate accelerates with increasingwire sizea. However, the initial wetting layer thickness,tw0,affects the variation ofG with wire sizea very little. Sincethe magnitude ofG is positive, the elastic strain energyserves as a driving force in the process of island formation inthe epilayer-substrate system.

Second, the case of varyingEs /Ew at fixed vw5vs

50.3, u545°, andtw050.5L is studied. The variation of thenormalized EERR,G/@Ew(e0)2#, with wire sizea at threedifferent Es /Ew50.5, 1, and 1.5, respectively, is plotted inFig. 4~b!. This figure shows that a lower modulus of thesubstrate enhances the EERR of QW formation on the top ofthe epitaxial layer, whose deformation is constrained by thesubstrate. It is understood that the lower the modulus of the

substrate, the weaker its holding capability on the epitaxiallayer to relax misfit strain through the creation of inclinedfacets. However, this effect is only seen when the wire size issufficiently large as compared to the wires spacing. Note thatthe magnitude of this effect should also be affected by theinitial wetting layer thickness,tw0. The thinner the initialwetting layer thickness, the larger magnitude of this effect ofmodulus ratio on the EERR, at a fixed island sizea/L.

Third, the case of varyingvs at fixed Es /Ew51, vw

50.3, u545°, andtw050.5L is studied. The results of set-ting vs50.2, 0.3, and 0.4, respectively, are plotted in Fig.4~c!. It is shown that Poisson’s ratio of the substrate plays asecondary role in determining the EERR of QW growth untila/L is sufficiently large. It is observed that the larger is thePoisson’s ratio of the substrate, the lower the EERR of QWgrowth.

Fourth, the case of varyingvw at fixed Es /Ew51, vs

50.3, u545°, andtw050.5L is studied. The results of set-ting vw50.2, 0.3, and 0.4, respectively, are plotted in Fig.4~d!. This figure shows that the Poisson’s ratio of wettinglayer plays a significant role in determining the EERR ofQW growth, not only at large values of wire size but also atits small values. Forvw varying from 0.2 to 0.4, the magni-tude of EERR is nearly doubled for the entire range ofa/L,in contrast to the secondary effect of varying the Young’smodulus and Poisson’s ratio of substrate. It is also observedthat the larger the Poisson’s ratio of wetting layer, the higherthe EERR of QW growth. This dependence is also in contrastto that of Poisson’s ratio of substrate, as shown in Fig. 4~c!.

The last parameter, tilt angleu, characterizes the shapeof a triangular QW while the bottom widtha characterizes itssize ~Fig. 3!. With Es /Ew51, vw5vs50.3, and tw0

50.5L,, the variation ofG with a is evaluated atu530° and60°, respectively. The results together with that ofu545° areplotted in Fig. 4~e!. It is shown that the magnitude ofG/@Ew(e0)2# is significantly influenced byu, in the entirerange of wire sizea/ l that was examined. The larger the tiltangle, the higher the EERR for the QW growth.

Based on the above parametric study, we can concludethat the EERR of QW formation in the isotropic epilayer-substrate system is strongly dependent on QW shape, elasticproperty of wetting layer, and misfit strain between wettinglayer and substrate. This strong dependence exists regardlessof the size of QWs. In contrast, the other parameters all playa secondary role in determining the EERR of QW growth,including the initial wetting layer thicknesstw0, which seemsto have no influence at all. These conclusions lead to thefollowing expression of EERR for QW formation in the sys-tem:

G

Ew~e0!2 5G0~vw ,u!1a

LG1S Es ,vs

a

L D , ~11!

where G0(vw ,u) and G1(Es ,vs ,a/L) are functions. Whena/L approaches zero, the normalizedG/@Ew(e0)2# is equalto G0, and henceG0 represents the normalized EERR fornucleation of QWs in the system. It is only dependent on thePoisson’s ratio of the wetting layer and the shape of islands.For the range ofvw from 0.2 to 0.4, the variation ofG0 with

FIG. 3. ~a! A two-dimensional epilayer-substrate system growing surfacequantum wires;~b! a unit-cell of the system in~a!. The dashed line in~a!indicates the initial thickness of the epilayer before growing the quantumwires.

3707J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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wire tilt angle u is plotted in Fig. 5. Note that the abovediscussion is based on the configuration of a series of equallysized QWs as shown in Fig. 3~a!. If only the process of wirenucleation is of interest, the result shown in Fig. 5 can beapplied to every individual wire on a smooth wetting layersurface. Therefore, the EERR can be used as a measure ofthe elastic energy-releasing capability of a system throughthe formation of a QI of a given shape at a surface point.

IV. VERTICAL CORRELATION OF QUANTUMISLANDS

Suppose that QWs are grown to a certain size in theprevious epilayer-substrate system. The QWs are then cov-ered coherently by a spacer of the same medium as the sub-

FIG. 4. Variation of the EERR during surface quantum wire growth against various parameters:~a! initial wetting layer thickness,~b! modulus of substrate,~c! Poisson’s ratio of substrate,~d! Poisson’s ratio of wetting layer, and~e! tilt angle of wires.

FIG. 5. Variation of the EERR for surface wire nucleation with tilt angle atdifferent values of the Poisson’s ratio of wetting layer.

3708 J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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strate. A new wetting layer is then deposited coherently onthe top of the spacer layer. Because the interfaces are coher-ent, and because the lattice constants between adjacent mediaare mismatched, a misfit-strain field is built up in the system.Driven by the misfit-strain field, a second round of QW for-mation would occur on the top surface provided that thefresh wetting layer is sufficiently thick. In the following, weexamine the EERR for formation of a surface QW under theinfluence of buried seed QWs in such a multilayered system.Intuitively, we assume that a surface QI would nucleate pref-erably at the site of maximum EERR while considering thatthe dissipative terms in Eq.~1! are not changed by the dif-ferent stress state at the possible sites.

Let us consider the case of dilute QWs in an infinitehalf-plane, as shown in Fig. 6. The half-plane consists ofalternating media of substrate, wetting layer, spacer, andfresh wetting layer. On the top of the sandwiched wettinglayer, a buried QW is assumed, which is covered by thespacer matrix. Its tilt angle,useed, is equal to 45°. Its bottomwidth a is taken to be the normalization length scale below.Further assumed is a surface QW of bottom width 0.2a andtilt angle usurf(545°) on the top of fresh wetting layer. Thewetting layers and QWs contain the same isotropic and lin-early elastic material with constants (Ew ,vw). The substrateand spacer contain a different isotropic and linearly elasticmaterial with constants (Es ,vs). The eigenstrain in the wet-ting layers and QWs is assumed to be hydrostatic, i.e.,e i j

0

5e0d i j , and that in the substrate and spacer to be zero. Thethickness of both wetting layers is taken to 0.2a. Since herewe are interested in the correlation of a new QW with aburied seed QW, only the critical EERR for island nucle-ation, i.e.,G0 in Eq. ~11!, is analyzed. Note that the~rela-tively! small surface island can be regarded as a perturbationto the freshly deposited~flat! wetting layer surface. By intro-ducing the surface perturbation at different locations relativeto the buried seed QW, the variation of the EERR for nucle-ation of a new island can be evaluated along the surface.Based on the criterion of nucleation at the site of maximumEERR, the effect of buried seed island on the nucleation of asurface island can be examined.

At fixed Es /Ew51 and vw5vs50.3 the variation ofEERR with locationx1 ~Fig. 6! is evaluated at differentspacer thicknessts50.6a, 0.9a, 1.2a, and 1.8a, respec-tively. The results are shown in Fig. 7. In the first two cases,the variation of EERR over a distance of about 3a from theseed island is flat. The magnitude is equal to the criticalvalue of EERR for island nucleation in the absence of seedisland in the previous epilayer-substrate system~Fig. 4!.When the perturbative surface island approaches the seed

island, the EERR first decreases and then increases. Itreaches a maximum value at the location right above the tipof the seed island. In the other two cases, the EERR mono-tonically increases from the far-field value when the surfaceisland approaches the seed island. When the spacer thicknessincreases, the amplitude of the EERR variation decreases,indicating a decay of the effect of seed island on surfaceisland formation. According to the present criterion of QInucleation at the site of maximum EERR, a new surface QIis expected to form right above the tip of a seed island in thedilute island system. When the spacer thickness increases,the characteristic of vertical correlation between surface andburied seed islands is not altered. However, since the ampli-tude of the EERR variation diminishes, the correlation losesits certainty as if any environmental uncertainty exists. Sincethe quantities of strain and strain energy density were used todetermine the QI nucleation site in the literature, we havealso examined the variation of these quantities along the un-perturbed surface above the seed island, which is not shown.The two quantities exhibit, respectively, a maximum and aminimum value right above the tip of the seed island. Thecharacteristics are not altered with changing spacer thick-ness. These variations all suggest that a surface island shouldnucleate on the top of the seed island according to the pre-viously proposed criteria.10,12

Finally, we examine the case of dense QWs in a multi-layered system, as shown in Fig. 8. A periodic array of bur-ied seed QWs with spacing 2L (L5a) is assumed along thesandwiched wetting layer surface. The model is otherwisethe same as the previous one~Fig. 6!. Figure 9 shows thevariation of EERR for a small surface QW evaluated at dif-ferent spacer thicknessts50.6a, 0.9a, 1.2a, and 1.8a, re-

FIG. 6. A model of surface island formation under the influence of diluteburied seed islands in a multilayered heterostructure.

FIG. 7. Variation of the EERR for surface island nucleation along the topsurface at different values of spacer thickness.

FIG. 8. A model of surface island formation under the influence of a peri-odic array of dense seed islands in a multilayered heterostructure.

3709J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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spectively, and at fixedEs /Ew51 andvw5vs50.3. In thecalculation, a system of nine buried seed islands was used,with the symmetry-boundary condition applied along theboundary except that the top surface is traction-free. Theresults shown in Fig. 9 were evaluated with the perturbativesurface island around the central seed island, whose tip islocated atx150. It can be seen that the EERR for surfaceisland nucleation reaches a maximum value right above thetip of the seed island and a minimum value right above themiddle point between two adjacent seed islands. By increas-ing the spacer thickness, i.e., increasing the distance of inter-action between the surface and seed islands, the amplitude ofthe EERR variation decreases, consistent with the observa-tion made in the previous system of dilute seed islands. Inthe case ofts51.8a, the variation of EERR is negligible ascompared to the mean value. However, the locations ofmaximum and minimum EERRs are held unaltered duringthe change of spacer thickness. In addition, the fields ofstrain and strain energy density along the unperturbed topsurface were examined. These variations again suggest a re-lationship of correlation between the surface and seed is-lands.

V. CONCLUSIONS

In this work, we have examined the elastic energy-releasing capability of a heteroepitaxial nanostructurethrough the formation of surface QIs. This capability is char-acterized by the parameter EERR, defined as the elastic re-laxation energy per unit volume of island growth. The analy-sis is based on a 2D isotropic linear elastic continuumapproach. In the epilayer-substrate system, we have system-atically studied the dependence of EERR@normalized byEw(e0)2] during the formation of triangular QWs on variousparameters including the initial wetting layer thickness, the

modulus and Poisson’s ratio of substrate, the Poisson’s ratioof wetting layer, and the wire tilt angle. It is found that thefirst three parameters play a secondary role in determiningthe EERR of surface QW growth. In the other words, they donot affect the EERR during the nucleation of surface QWs.Meanwhile, the EERR of QW nucleation is a function of thelast two parameters. The parametric study suggests that theEERR can be used as a measure of the elastic energy-releasing capability of a heteroepitaxial system through theformation of island at a~smooth! surface point.

Furthermore, the variation of EERR during QW forma-tion under the influence of buried seed islands is examined.By assuming a surface island to nucleate at the site of maxi-mum EERR, the vertical relationship between surface andseed islands is determined. We found that the vertical rela-tionship established through the criterion of maximum EERRis of the correlation type. The correlation relationship doesnot alter with changing distance between the surface andseed islands. However, since the amplitude of EERR varia-tion decays quickly with this distance, the correlation rela-tionship may quickly lose its certainty as if any environmen-tal uncertainty exists.

ACKNOWLEDGMENTS

The author would like to thank Dr. E. Pan of STI, andDr. V. K. Tewary and Dr. R. E. Kumon of NIST, Boulder,Colorado, for discussions.

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FIG. 9. Variation of the EERR for surface island nucleation along the topsurface at different values of spacer thickness.

3710 J. Appl. Phys., Vol. 92, No. 7, 1 October 2002 B. Yang

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