Equilibrium and metastable states of nanoislands in multilayered Stranski-Krastanow systems

C.-H. Chiu* and Hangyao Wang†

Department of Materials Science and Engineering, National University of Singapore, Singapore 117576, Singapore�Received 16 October 2006; published 14 March 2007�

In this paper, we analyze the total-energy change during the formation of a nanoisland on multilayeredStranski-Krastanow �SK� systems containing an embedded nanostructure. The energy analyses show that whenthe film thickness is in the vicinity of the critical value for the SK transition, the embedded nanostructure caninduce an equilibrium or metastable state in which the top nanoisland is stable against size variations. The twostates are a general feature of the SK systems that can be utilized to fabricate uniform stable arrays ofnanoislands.

DOI: 10.1103/PhysRevB.75.125416 PACS number�s�: 68.55.�a, 81.16.Dn

I. INTRODUCTION

The self-assembly of nanoislands in a multilayeredStranski-Krastanow �SK� system is a promising nanotechnol-ogy with numerous potential applications.1–10 In this self-assembly process, strained films and spacer layers are grownalternatively and epitaxially onto the system. The films, hav-ing a lattice mismatch with the substrate, generally followthe SK transition that nanoislands start to develop on a flatwetting layer after the film thickness exceeds a criticalvalue.11 The spacer layers, on the other hand, are usuallymade of the substrate material. Consequently, there is nomismatch strain in the layers and the morphology of thelayer is much smoother than that of the strained films. Thealternative growth of islanded films and smooth spacer layershas the advantage that the island size distribution is narrowerand the spatial arrangement of the nanostructures is moreordered.2,4

One crucial question in the self-assembly process is theexistence of an equilibrium state in which the nanoisland isstable against size variations. The question was extensivelystudied in the literature for systems consisting of a substrateand a single layer of film. It is now well accepted that theanswer to the question is negative12,13 unless the surfacestress is high14,15 and/or the film-substrate interaction, an im-portant driving force for the SK transition, is strong.16

In contrast to the single-layered systems, the equilibriumstate of a nanoisland on multilayered systems has beenlargely overlooked in the literature. For example, the size ofnanoislands in multilayered systems was generally calculatedby deterministic models without considering the stability ofthe island size against variations.2,17 Similarly, the size sta-bility was explored in several numerical simulations.18–21

However, the size stability in some of the studies came froma kinetic effect.18–20 Whether or not the stability corre-sponded to an equilibrium state in the systems was not fullyunderstood from the thermodynamic point of view.

In this paper, we examine the effects of embedded nano-structures on the equilibrium states of the islands on top of amultilayered SK system. The effects are determined by con-sidering the variation of the total energy of the system withthe island size. The results show that the effects of the em-bedded structures on the equilibrium states are dictated bythe thickness of the top film. The embedded structures cannot

induce an equilibrium or metastable state when the filmthickness is large. The two stable states, however, can beachieved if the thickness is in the vicinity of the critical valuefor the SK transition. The importance of the film thickness inrealizing the stable states applies to all of the multilayeredSK systems. The finding of this general feature rekindles theprospect of using embedded nanostructures to fabricate uni-form and stable island arrays on the SK systems.

II. MODEL

Figure 1 depicts the multilayered film-substrate systemexamined in this paper. The substrate in the system is a semi-infinite solid, and on top of the substrate is a strained film,which develops into a flat wetting layer and a facet pyramid.The depth of the wetting layer measured from the top surfaceof the system is H, the base width of the pyramid is De, andthe facet angle of the pyramid is �e. Above the strained filmare a spacer layer and another strained film. Similar to theembedded strained film, the top strained one also forms awetting layer and a pyramid. The thickness of the top wettinglayer is h, the base width of the island is D, and the facetangle is �. The top film surface is attached by a set of Car-tesian coordinate axes. The x and y axes are parallel to thesurface and the z axis is normal to the surface.

The materials in the multilayered system are elasticallyisotropic ones with the same shear modulus � and Poisson’sratio �. There is a mismatch strain E0 in the embedded andtop films because the lattice parameter of the two films isdifferent from that of the substrate. The spacer layer, on thecontrary, is made of the substrate material; hence, the mis-match strain in the layer is zero.

The total energy of the system is controlled by the mis-match strain E0, the surface energy density � of the film, and

FIG. 1. A schematic diagram of a nanoisland forming on a mul-tilayered system containing an embedded nanostructure.

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the film-substrate interaction. The mismatch strain E0 resultsin the strain energy in the system, which favors the formationof islands. The surface energy density is anisotropic: �=�1on the flat film and �=�2 on the island facet surface. Thefilm-substrate interaction accounts for the SK transition. Animportant mechanism of the interaction is the quantum con-finement effect, which can be modeled as a special type offilm surface energy of which density g is expressed as g�t�=g0l / �t+ l�, where t is the distance between the top film sur-face and the top film-spacer layer interface.22

III. ENERGY ANALYSES

A. Surface and interaction energies

We consider the case where the facet island forms on themultilayered system under the condition that the total massof the top film is conserved and the morphology of the em-bedded pyramid remains unchanged during the process. Theisland formation affects three types of energy in the system,namely, the surface energy Es, the interaction energy Es, andthe strain energy W. The changes of the surface and interac-tion energies during the process were derived earlier,23

�Es = �1D2G, �EI = g�h�D2I�A� , �1�

where G=−1+�2 / ��1 cos ��, A=A / �h+ l�, and

I�A� = − 1 +A

3+

2�1 + A�ln�1 + A� − 2A

A2 cos �. �2�

B. Strain energy

1. Stresses

The strain energy change �W is estimated by employingthe first-order perturbation method suggested in Ref. 24. Thefirst step is to determine the stress �� on the top islandsurface accurate to the first order of the facet slopes,

�� = ��,0 + ��,t + ��,e, �3�

where ��,0 is the biaxial stress when both the embedded andthe top films are flat, ��,t is the stress due to the top island inthe absence of the embedded nanostructure, and ��,e is thatdue to the embedded nanostructure in the absence of the topisland. The biaxial stress ��,0 is expressed as

��,0 = �T 0 0

0 T 0

0 0 0� , �4�

where T=2��1+��E0 / �1−��. The stress ��,t due to the topislands can be found in Ref. 24. The term ��,e due to theembedded island is calculated by extending the Fourier se-ries solution of embedded wires to the case of three-dimensional islands,20

��,e = − �m,n=−�

�

2TSecmn�e−kH�eik�mx+ny�

��m2 + �n2�/ �1 − ��mn/ 0

�1 − ��mn/ ��m2 + n2�/ 0

0 0 0� , �5�

where i=�−1, =m2+n2, Se=tan �e, and cmn is the coeffi-cient of the Fourier series expression for the shape of theembedded island,

fe�x,y� = �m,n=−�

�

Secmnk−1 exp�ik�mx + ny�� . �6�

The wave number k in the Fourier series is taken to be muchsmaller than 1/D to minimize the effects of the neighboringislands implied by the Fourier series expression for fe�x ,y�.

2. Strain energy density

The stress �� is substituted into the elasticity constituteequation w=�� · �S ·��� /2 to determine the strain energydensity w on the top surface, where S is the compliancetensor of isotropic materials.24 The result accurate to the firstorder of the facet slopes is found to be

w�x,y� = w03d − 2Sw0��x,y� − 2Sew0�e�x,y� , �7�

where S=tan �, w03d=2��1+��E0

2 / �1−��, w0=��1+��2E0

2 / �1−��, �e�x ,y� describes the variation of w due tothe embedded island,

�e�x,y� = �m,n=−�

�

2cmn�e−kH�eik�mx+ny�, �8�

and ��x ,y� gives that due to the top island. The expressionfor ��x ,y� is lengthy and omitted here for conciseness; thedetails can be found in Ref. 24.

3. Strain energy variation

Knowing w�x ,y�, we carry out the procedure outlined inRef. 24 to determine the strain energy change �W during themass-conserved island formation. The procedure starts withthe surface integral that relates the variation of the total en-ergy �W and that of the top island morphology �f�x ,y�,25,26

�W = w�x,y��f�x,y�d� , �9�

where f�x ,y� denotes the surface profile of the top island. Touse Eq. �9�, it is helpful to rewrite f�x ,y� as f�x ,y�=S f�x ,y�, where f�x ,y� is a shape function independent ofthe island characteristic slope S.24 The new expression forf�x ,y� implies that the top island can be thought to be gen-erated by the process in which S increases from 0 to S, while

the shape function f�x ,y� and the embedded islands remainfixed. The morphological variation �f during this process isgiven by

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�f�x,y� = f�x,y��S . �10�

The variation process is not mass conserved. Thus, as dem-onstrated later in Eq. �12�, the corresponding strain energychange is partially attributed to different film volumes andpartially to different film morphologies.

It follows from Eqs. �7�, �9�, and �10� that

dW

dS= �w0

3d − 2Sew0�e�x,y� − 2Sw0��x,y�� f�x,y�d� .

�11�

Integrating Eq. �11� with respect to S from 0 to S and noting

that �, �e, f , and Se are independent of S gives the totalstrain energy change resulting from the variation process,

�W* = w03dV − w0SVU − 2w0SeVUe, �12�

where the asterisk in �W* highlights that the process is notmass conserved, V is the volume of the top island, and

U =1

V ��x,y�f�x,y�d� , �13�

Ue =1

V �e�x,y�f�x,y�d� . �14�

The quantity U defined in Eq. �13� is found to be 1.984 forthe top pyramid island illustrated in Fig. 1; the value is in-variant with the location, the size D, and the facet angle � ofthe island.24

The product w03dV appearing in Eq. �12� can be under-

stood as the strain energy increment when the volume of theflat film is increased by V without the formation of an island.Accordingly, the remaining two terms in Eq. �12� correspondto the strain energy change �W when the film transformsfrom a flat profile to an islanded one under the condition ofmass conservation,

�W = − w0SVU − 2w0SeVUe. �15�

The first term in Eq. �15� is the strain energy reductiondue to the top island alone,24 while the second term repre-sents the extra reduction due to the embedded nanostructure.The extra reduction is proportional to the reference strainenergy density w0, the characteristic slope Se of the embed-ded island, the volume of the top island, and the quantity Uethat is controlled by the shapes of the top and embeddedislands. The extra reduction varies with the top island loca-tion. For the current case, where the embedded island is apyramid and the elastic properties are isotropic, the reductionis found to be the largest when the top and embedded islandsalign. Based on the result, we focus on the aligning case inour subsequent energy analyses.

C. Total energy

Adding together �Es, �EI, and �W expressed in Eqs. �1�and �15� determines the total-energy change �Etot due to themass-conserved island formation,

�Etot = �Es + �EI + �W . �16�

The result is further normalized as �Etot=�Etot / ��1L2�,where L=�1 /w0 is the characteristic length of the system.

IV. EQUILIBRIUM AND METASTABLE STATES

Figures 2�a�–2�d� show four typical examples of the

variation of �Etot with the normalized top island size D=D /L. The parameters adopted in the four cases are identicalexcept the film thickness h: �=�e=11.3°, g=g0 /�1=1/256,�2 /�1=0.99, L=25 nm, H=0.3L, De=L, and /De=5. Thesevalues roughly correspond to the Si0.5Ge0.5/Si system, andany single island on the system without the embedded nano-structure would be unstable against coarsening.16 The film

thickness is normalized by h= �h+ l� /hcr, where hcr

=�g0l /SUw0 is the critical value for the SK transition with-out the embedded structure.23 According to the notation, theflat film in the absence of embedded nanoislands is the equi-

librium morphology in the thickness range h�1, and islands

develop on a flat wetting layer when h�1.

Figure 2�a� plots the curve �Etot�D� when h=1.2. Thecurve is similar to the results without the embedded island:

�Etot increases with D first, reaches a maximum, and then

decreases with D. The lack of a minimum in the curve indi-cates that the top island is still unstable against coarsening inthis case.

Figure 2�b� depicts the result of the case where h=1.05.The result is characterized by one local minimum and twomaxima, in contrast to the one maximum in Fig. 2�a�. Thelocal minimum suggests that the embedded island can inducea metastable state in which the top island is stable against

small size variations when h is slightly higher than 1.

FIG. 2. The variation of �Etot with D at different normalized

film thicknesses h. The value of h is taken to be �a� 1.2, �b� 1.05, �c�0.95, and �d� 0.6.

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Figure 2�c� illustrates the curve �Etot�D� when h is furtherdecreased to 0.95. The curve shows three features: �1� Thetotal-energy change �Etot is minimized at some finite islandsize, �2� the minimum energy change, denoted as �Eeq, isnegative, and �3� the total-energy change increases mono-tonically with D after D exceeds Deq, the size of the islandwith the minimum energy. These features demonstrate thatthe embedded nanostructure can cause the formation of topisland to be energetically favorable in the subcritical thick-ness regime of the SK transition. The result is consistent withthe experimental observations reported in Ref. 11.

The island forming in the regime h�1 has the uniqueproperty that the island is in the equilibrium state stableagainst any size variation. The equilibrium state can be acti-vated by the embedded island on the SK systems withoutsatisfying the typical equilibrium requirements suggested inthe literature, for example, a large surface stress and/or astrong film-substrate interaction.

The formation of an island in the equilibrium state is lim-ited to the locations aligning with the embedded nanostruc-tures. On the remaining top film surface, the development ofthe flat film into islands is still energetically unfavorable. Thecontrast between the aligning locations and the remainingsurface demonstrates that the embedded structures have the

capability to control the formation of nanoislands when h isless than 1.

Similar to the equilibrium state, the metastable state illus-trated in Fig. 2�b� is also general on the multilayered SKsystems. The difference between the two stable states is thestability of the remaining film surface against island forma-tion. In the equilibrium case, the island formation on theremaining top surface is completely prohibited. In the meta-stable case, on the other hand, the island formation is onlyimpeded kinetically.27

Figure 2�d� depicts the result of the case where h=0.6.

The total energy �Etot increases with D in this case, whichmeans that the flat film is the equilibrium morphology whenthe film thickness is sufficiently small.

The emergence of the metastable and the equilibriumstates due to the embedded nanoisland can be understood asfollows. The embedded nanoisland induces a nonuniformdistribution of stress on the top film surface and accordinglyan extra strain energy reduction for island formation. Theextra energy reduction is particularly significant for a rangeof island size, resulting in the tendency that the metastableand the equilibrium states can exhibit in the multilayeredsystem. The occurrence of the two stable states of islands,however, is hindered by other energies in the system, whichfavor either a flat film or coarsening islands. Therefore, thetwo stable states are suppressed if those opposing energiesare stronger. This happens in thick films, which are domi-nated by energies leading to coarsening islands, and in thinfilms, which are dictated by energies yielding flat film mor-phology. On the other hand, the two stable states are acti-vated if the extra energy reduction becomes significant. Thisoccurs when the film thickness is in the vicinity of the criti-cal value for the SK transition, where the energies favoring aflat film balance those causing coarsening islands.

V. EQUILIBRIUM ISLAND SIZE

Figure 3 plots the contours of the equilibrium island sizeratio Deq /De as a function of H / and De /L for the case

where h=0.9. The other parameters are identical to those inFig. 2. The contours are found to be bounded by one line,corresponding to the condition that the minimum total-energy change �Eeq is zero. In the regime above the line,�Eeq�0; thus, the flat film is the equilibrium morphology. Inthe regime below the line, in contrast, �Eeq�0, and theequilibrium state exists.

The contours in Fig. 3 indicate that the equilibrium islandsize ratio Deq /De increases with the size of the embeddednanostructure, and the ratio decreases when the embeddedstructure is at a deeper location of the system. The values ofthe contours of Deq /De range from 0.9 to 1.3 in the figure,demonstrating the feasibility of adjusting the spacer layerthickness to control the size of the top equilibrium island.

That the equilibrium size ratio Deq /De can be less than 1is useful for the self-assembly of a three-dimensional arrayof stable and uniform islands. In particular, the self-assemblyprocess can start with a two-dimensional array of uniformislands but with much larger island size. Since the island sizedoes not need to be extremely small, it is possible to producethe first island array by the conventional schemes such as thelithography method or by the new schemes proposed in theliterature.28 In any case, the first array is overgrown by aspacer layer and a strained SK film to induce an equilibriumisland array with smaller size. The reduction of the islandsize is repeated until the size reaches the intended value.Afterwards, the uniform and stable island array can be rep-licated by similar overgrowing processes to self-assemble athree-dimensional array of nanoislands.

As a remark, the equilibrium states of the subsequentovergrowing processes can be determined by adopting theformulas presented in this paper.29 The details of the resultsare different from those shown in Fig. 3, while the essencewould be the same: The embedded nanostructures can inducean equilibrium state when the film thickness is below thecritical value for the SK transition, and the size of the top

FIG. 3. The contours of the equilibrium island size ratio Deq /De

as a function of the normalized depth H / and size De /L of the

embedded island for the case where h=0.9.

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equilibrium islands can be less than that of the embeddedones.

VI. CONCLUSION

In conclusion, an embedded nanostructure can activate anequilibrium or metastable state for the nanoisland on the topsurface of a multilayered SK system. The equilibriums andthe metastable states, respectively, are induced when the film

thickness is slightly lower and higher than the critical valuefor the SK transition. The two states are a general feature ofthe multilayered SK systems that can be exploited for fabri-cating uniform and stable arrays of nanoislands.

ACKNOWLEDGMENT

The project is supported by the National University ofSingapore �Contract No. R-152-000-062-112�.

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29 For multiple layers of islands, �e�x�� in Eq. �7� is given by thesum of the contributions from each layer of islands, and thecontribution of each layer can be calculated by Eq. �8� with Hbeing the depth of the layer and cmn being the Fourier coefficientof the corresponding surface profile.

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