We study the energetics of island formation in Stranski-Krastanow growth
within a parameter-free approach. It is shown that an optimum island size
exists for a given coverage and island density if changes in the wetting layer
morphology after the 3D transition are properly taken into account. Our
approach reproduces well the experimental island size dependence on coverage,
and indicates that the critical layer thickness depends on growth conditions.
The present study provides a new explanation for the (frequently found) rather
narrow size distribution of self-assembled coherent islands.Comment: 4 pages, 5 figures, In print, Phys. Rev. Lett. Other related
  publications can be found at http://www.fhi-berlin.mpg.de/th/paper.htm

A. Polimeni

C. Priester

C. W. Snyder

D. J. Bottomley

D. Leonard

E. Pehlke

G. S. Solomon

H. T. Dobbs

H. Yamaguchi

I. Daruka

I. N. Stranski

J. G. Amar

J. M. Moison

J. Tersoff

L. G. Wang

M. Berti

M. Bockstedte

M. Scheffler

N. Carlsson

N. Moll

P. Kratzer

Q. K. Xue

R. Heitz

S. H. Xin

S. Lee

V. A. Shchukin

Y. Chen

Y. W. Mo

English

arXiv

Wang, L.

Kratzer, P.

Scheffler, M.

Moll, N.

MPG.PuRe

VOLUME 82, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1999nyfreensityunt.icatesnewnds.4042Formation and Stability of Self-Assembled Coherent Islandsin Highly Mismatched HeteroepitaxyL. G. Wang, P. Kratzer, M. Scheffler, and N. Moll*Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germa(Received 21 December 1998)We study the energetics of island formation in Stranski-Krastanow growth within a parameter-approach. It is shown that an optimum island size exists for a given coverage and island deif changes in the wetting layer morphology after the 3D transition are properly taken into accoOur approach reproduces well the experimental island size dependence on coverage and indthat the critical layer thickness depends on growth conditions. The present study provides aexplanation for the (frequently found) rather narrow size distribution of self-assembled coherent isla[S0031-9007(99)09155-3]PACS numbers: 68.65.+g, 68.55.–a, 81.10.Ajsitymbybe-usgehecal-y-,theis-13]ur-byri-. Inw-anctsatedi-ininlyengen-wetheas.n-e,meshety.The surface morphology of overlayers in heteroepitaxigrowth has attracted intense interest because of its imptance for basic science and applications in optoelectrondevices. Experiments [1–5] showed that heteroepitain systems with a lattice constant difference$2%, suchas InAsyGaAs [2–4], GeySi [1], and InPyInGaP [5], fol-lows the so-called Stranski-Krastanow growth mode [6three-dimensional (3D) dislocation-free (so-called coheent) islands form on top of the wetting layer. These smacoherent islands are often found to have a very narrow sdistribution [2–4] and are promising to be used in quantum dot light emitting diodes and lasers.It is commonly agreed that the energetics of strarelief plays a key role in the growth process: Islandform, instead of a uniformly strained, epitaxial filmbecause the gain of elastic relaxation energy in an islaovercompensates the cost due to the increased surfenergy by islanding. It is tempting to attribute theobserved island size distribution to a minimum of the freenergy of the system. However, an equilibrium theorwith only two energetic contributions, a positive one fromthe island surface energy (E , V 2y3; V is the quantumdot volume) and a negative one from the elastic relaxatienergysE , V d, fails to predict a finite equilibrium sizeof the islands. Instead, the energy gain from strarelief always prevails for sufficiently high coveragesrendering larger islands more stable than smaller onIn order to cope with this difficulty, several additionaeffects, e.g., contributions from intrinsic surface stresor from interactions between islands [7,8], have beeinvoked. Priester and Lannoo [9] proposed a mechanisin which 2D platelets act as precursors for the formatioof 3D coherent islands, thus determining their size. Morecently, the observation of island ripening [10] has madit doubtful if the islands can be interpreted at all astructures in total equilibrium.In this Letter, we show that the narrow size distribution of the coherent islands can be understood as the reof the system being trapped in aconstrainedequilibrium0031-9007y99y82(20)y4042(4)$15.00alor-icxy]:r-llize-ins,ndaceeyonin,es.lsnmnstes-sultstate, where the size is determined by the island denand the nominal coverage. In the constrained equilibriutheory, the existing nuclei grow to a size determinedthe energetic balance that governs material transporttween the wetting layer and the islands. This allowsto derive an optimum islands size for a fixed coveraand island density from a parameter-free approach. Telastic energy in both the islands and the substrate isculated within continuum elasticity theory. Using densitfunctional theory within the local-density approximationaccurate surface energies are obtained with the help ofplane-wave pseudopotential method [11] for both theland facets and the wetting layer. Previous studies [12,had missed the latter contribution, assuming that the sface energy of the wetting layer would be unchangedthe 3D transition. Our theory reproduces very well expemental data for the island size dependence on coverageparticular, we improve over previous approaches by shoing that the selectivity of growth of a certain island size cbe explained without invoking delicate elastic edge effeor island interactions [7,8]. Furthermore, we demonstrhow the critical layer thickness depends on growth contions, settling this long-standing issue.We propose a view of the growth process dividedthree phases: an early nucleation phase which madetermines the island densityn, a second phase wherthe islands grow mostly on the expense of the wettilayer, and a third phase characterized by Ostwald riping. Since we are mostly interested in island sizes,concentrate on the second phase, and briefly discussthird phase later. As long as the wetting layer actsa source for material, existing nuclei will grow rapidlyHereby the island densityn remains constant [3]. Wetreat it as an input to our model noting that the island desity is determined by the growth kinetics. Furthermorwe assume the islands have identical shape and voluV . In the following, we discuss the island size in termof a constrained thermodynamic equilibrium between tislands and the wetting layer, for a fixed island densi© 1999 The American Physical SocietyVOLUME 82, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1999alez-edareesofecealiser-n-r,ernlindn-sAlthough our approach is not limited to a certain systemhere we consider, as an example, the strainedh110j pyra-midal shaped, dislocation-free InAs islands with a squabase (areaa2) on the GaAs(100) surface (with a wettinglayer). We will also discuss theh111jyh1̄1̄1̄jfaceted pyra-midal islands later. We choose this system with a lattimismatchDa ø 7%, because a large number of expeimental and theoretical studies have been done [2–4The real island shape may be more complex, but the sple island shape used here should still allow us to captuthe important features of the island formation (see beloand Ref. [13]). Figure 1 schematically illustrates the iland formation on the substrate surface.u0 andu are thenominal coverage and wetting layer thickness after islaformation, respectively. We omit the interaction betweeislands (as we will discuss below this is a very good aproximation in our case), as well as the energies of edgand corners. The entropic contribution to free energyalso neglected.The total energy gain per unit volume of a single islancan be expressed asEtotyV ­ eelis 2 eelfilm 1 fSgf 2 a2gwlsu0dgyV1 s1yn 2 a2d 3 fgwlsud 2 gwlsu0dgyV , (1)whereeelis andeelfilm are the elastic energy densities of thisland and uniformly strained film. The third term describes the change in surface energy due to the islawith gf being the surface energy of the island facets aS their area. The fourth term accounts for the thinninof the part of the wetting layer which feeds the islangwlsu0d, gwlsud are the formation energy of the wettinglayer as a function of its thicknessu, measured relative toInAs bulk kept at the GaAs lattice constant. This allowus to introduce the elastic contribution to the formatioenergy as a uniquely defined separate termelfilm, similarto earlier work [14]. From mass conservation, the volumof an islandV is given byV ­ 16 a3 tana ­ 1n su0 2 udL,where a and L are the tilt angle of island facets andthe monolayer (ML) thickness, respectively. Equation (holds true as long as the island contains a sufficiently hinumber of atoms (e.g., 5000 atoms) because elastictheory is applicable and the reconstructions on the facare completed [14].In order to obtain accurate values for the surface eergies and intrinsic surface stresses, these are compFIG. 1. Schematic illustration of the formation of coherenislands on the substrate surface.u0 and u are the nominalcoverage and wetting layer thickness, respectively.a is the tiltangle of island facets anda is the island base length.,recer-,9].im-rews-ndnp-esisde-nd,ndgd.sne1)ghityetsn-utedtTABLE I. Surface energiesgf and surface stressessx , syfor InAs surface reconstructions with the chemical potentimAs ­ mAssbulkd 2 0.2 eV.gf sx sySurface smeVyÅ2d smeVyÅ2d smeVyÅ2ds110d Cleavage 41 26 54s111d In vacancy 42 48 48s1̄1̄1̄d As trimer 49 92 92using slab models of the surfaces with the help of thpseudopotential plane-wave method [11]. After optimiing the atomic geometries using consistently calculatforces on the atoms, the total energies of the slabscomputed, and the formation energies of various surfacare obtained by subtracting the calculated total energyan appropriate amount of bulk material. We further takinto account the surface stress contribution to the surfaenergy up to the linear term [14], and a term proportionto the chemical potentialmAs of the environment in caseof nonstoichiometric surfaces. Since epitaxial growthmostly performed under As-rich conditions, all surfacenergies are evaluated close to equilibrium with bulk asenic (i.e.,mAs ­ mAssbulkd 2 0.2 eV). For each facet,we have selected the reconstruction with the lowest eergy from several candidates [14]. For the wetting layewe consider theb2s2 3 4d reconstruction which is usu-ally found on GaAs(001) and InAs(001) surfaces undmoderately As-rich conditions [15]. The results are givein Table I and Fig. 2.The elastic energy is calculated within continuumelasticity theory using the experimental elastic moduto describe the elastic properties of both the island aFIG. 2. Formation energy of the wetting layer as a functioof thicknessu, defined by gwlA ­ Etot 2 mGaAssbulkdNGa 2mstrainedInAssbulkdNIn 2 mAssNAs 2 NGa 2 NInd, whereA is the sur-face area andNisi ­ As, Ga, and Ind are the number of parti-cles of the speciesi in the supercell. Etot is the total energyof the supercell. mGaAssbulkd and mstrainedInAssbulkd are the chemicalpotential of GaAs bulk and of strained InAs bulk with the theoretical equilibrium lattice constant of GaAs bulk. From variouconfigurations withNIn ­ 8u In atoms pers2 3 4d surface unitcell, those with the lowest formation energy are presented.4043VOLUME 82, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1999-ndi-dnthx-heus-dsean-desinsitreroteege.etalthe substrate. For the island plus a 240 Å thick slarepresenting the substrate, a finite-element approachapplied to solve the nonlinear elasticity problem [14,16We calculate the elastic energy for several island shap(different tilt anglesa of island facets) with a fixed islandvolume. For each particular shape, we can extracteelisfrom the observed linear scaling relation with the islanvolume. The elastic energy densityeelfilm of the uniformlystrained film can be obtained by extrapolating the resufor the islands toa ­ 0, and we find it is in very goodagreement with the value from linear elasticity theory.In Fig. 3, we show the various energy contributionand the total energy gain per volume forn ­ 1010 cm22,u0 ­ 1.8 ML. The elastic relaxation energy [the first ansecond term in Eq. (1)] is negative due to strain reliand scales linearly with the island volume. Surface eergy [the third term in Eq. (1)] is a cost, and therefore icontribution is positive. The wetting layer energy contribution [the fourth term in Eq. (1)] is also positive anddepends complexly on the island volume, island densiand coverage. We also show the energy contributionthe edges in Fig. 3, which becomes negligible comparto the other contributions for a large island (estimateas in Ref. [14]). It is important that an energy minimum exists in the total energy gain curve. This indcates an optimum island size can be obtained under cergrowth conditions. The minimum in Fig. 3 correspondsan island with about 38 000 atoms, which compares resonably well with typical experimental values (betwee20 000 [3] and 50 000 atoms per island [17]). The quiuniform islands prior to ripening observed in Ref. [10lend further support to the existence of an optimum iland size. However, the island size strongly dependsthe island density. Figure 4(a) shows that the equilirium island volumeV is a hyperbolic function of islanddensity n. As the experimental observations [18] havshown, the island radius varies exponentially with thgrowth temperature. This was attributed to a kineticalFIG. 3. Total energy gain by islanding and various energcontributions (solid lines) forn ­ 1010 cm22, u0 ­ 1.8 ML.The dashed line is the total energy gain forn ­ 1010 cm22,u0 ­ 1.5 ML. The arrows mark the minima of the curves.4044bis].esdltssdefn-ts-ty,ofedd-i-taintoa-nte]s-onb-eelyylimited process [18]. It is consistent with the theory presented here, since the island density is known to depestrongly on the growth temperature [19,20], and thus knetics controls the growth through controlling the islannucleation density. Figure 4(b) shows that our theory careproduce very well the increase of the island radius withe amount of deposited material observed in various eperiments, by using suitable island densities as input. Tgood agreement between theory and experiment also jtifies the neglect of repulsive interactions between islanin the present study. A careful check indicates that thdistances between the islands are quite large (larger th65 and 220 nm for the highf2.3 3 1010 cm22g and lowf1.9 3 109 cm22g island densities, respectively). The island density2.3 3 1010 cm22 used to fit the experimentalresults [21] agrees well with the experimentally estimateisland density1.5 2 3 1010 cm22.The critical layer thickness is a very interesting issuand the reported values vary from 1.2 to 2 ML [21–23]. Our present theory puts us in position to discusthe critical layer thickness, because the total energy gaepends on the coverage (see Fig. 3). When we depoless material, keepingn fixed, the energy minimumrises above zero, i.e., island formation is no longefavorable. We take the critical layer thickness as thcoverage at which the minimum energy equals ze(the error 60.01 ML). Our results, in Fig. 5, indicatethat the critical layer thickness varies from 1.20 to1.79 ML when the island density varies from109 cm22to 3.5 3 1011 cm22. Our theoretical prediction matchesthe experimentally observed range (1.2–2 ML). We noFIG. 4. The dependence of the optimum island size (volumand half-base) on the island density and the nominal coveraIn (a), the total energy gains for the optimum island sizin various island densities are also shown. The experimenvalues in (b) are taken from Ref. [21] (d) and estimated fromRef. [3] (o).VOLUME 82, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1999n.rntesrydris.ettsorFIG. 5. Critical layer thickness as a function of the islandensities.that an exact determination is difficult, because in thexperiment various other factors may influence the criticlayer thickness, such as more complex island shapdetails of the growth method, and growth conditions (e.gIII yV ratio) [24,25], or a possible correlation betweenisland density and coverage [2].We have also performed a corresponding analysis fthe strained InAsh111jyh1̄1̄1̄j pyramidal islands. Theresults show similar behavior as for the InAsh110j islandsand indicate that the basic features of our model dnot depend on shape assumptions. The calculations aindicate that a larger island tends to be a steeper one, ia h111jyh1̄1̄1̄j faceted pyramidal island, due to enhanceelastic energy relief [14,26]. Our study with strainedh111jyh1̄1̄1̄j pyramidal islands also shows that the criticalayer thickness somewhat depends on the island sha(see Fig. 5). However, we can still predict trends, e.gfor high growth temperatures (having a small islannucleation density) the critical layer is thinner. Our studalso indicates that in case of a thinner critical layer, thisland embryo should be larger than that for a thicker onThis can be understood in terms of a larger energy barrwhich must be overcome by the embryo when nucleatinon a thinner wetting layer.Finally, we briefly comment on the ripening of theislands. When no more material is supplied by the wettinlayer, the island density is no longer constant, becausmaller islands will dissolve again. Allowing the islanddensity to vary, we find that larger islands at a lowedensity are energetically preferred [see Fig. 4(a)]. Thuour theory is in accord with the observed Ostwald ripenin[10]. However, since noticeable changes in the island siand density resulting from ripening typically take mandays, ripening is not important for device applicationswhere the islands are covered by a capping layer aftedeales,.,orolso.e.,dlpe.,dyee.ierggsers,gzey,r avery short growth interruption, and was not observed iprevious experiments performed on a shorter time scaleIn conclusion, we presented a novel explanation fothe selection of particular sizes of self-assembled cohereislands in highly mismatched heteroepitaxy. It is possiblto select the island size by changing the growth conditionand the nominal coverage. Our theory reproduces vewell the experimental trends observed in the islangrowth. We have also shown how the critical layethickness depends on growth conditions and settled thlong-standing issue.The authors thank E. Pehlke for fruitful discussionsThis work was supported by Sfb 296 of the DeutschForschungsgemeinschaft.*Present address: Department of Physics, MassachuseInstitute of Technology, Cambridge, MA 02139[1] Y. W. Mo et al., Phys. Rev. Lett.65, 1020 (1990).[2] D. Leonardet al., Phys. Rev. B50, 11 687 (1994).[3] J. M. Moisonet al., Appl. Phys. Lett.64, 196 (1994).[4] R. Heitz et al., Phys. Rev. Lett.78, 4071 (1997).[5] N. Carlssonet al., Appl. Phys. Lett.65, 3093 (1994).[6] I. N. Stranski and L. von Krastanow, Akad. Wiss. Lit.Mainz Math.-Natur. Kl. IIb146, 797 (1939).[7] V. A. Shchukinet al., Phys. Rev. Lett.75, 2968 (1995).[8] I. Daruka and A.-L. Barabási, Phys. Rev. Lett.79, 3708(1997).[9] C. Priester and M. Lannoo, Phys. Rev. Lett.75, 93 (1995),and references therein.[10] S. Leeet al., Phys. Rev. Lett.81, 3479 (1998); S. H. Xinet al., Appl. Phys. Lett.69, 3884 (1996).[11] M. Bockstedteet al., Comput. Phys. Commun.107, 187(1997); http://www.fhi-berlin.mpg.de/th/fhimd/code.html[12] Y. Chen and J. Washburn, Phys. Rev. Lett.77, 4046(1996).[13] J. Tersoff and R. M. Tromp, Phys. Rev. Lett.70, 2782(1993).[14] N. Moll et al., Phys. Rev. B58, 4566 (1998).[15] H. Yamaguchi and Y. Horikoshi, Phys. Rev. B51, 9836(1995).[16] E. Pehlkeet al., Appl. Phys. A65, 525 (1997).[17] D. Leonardet al., Appl. Phys. Lett.63, 3203 (1993).[18] G. S. Solomonet al., Appl. Phys. Lett.66, 3203 (1995).[19] J. G. Amar and F. Family, Phys. Rev. Lett.74, 2066(1995).[20] H. T. Dobbset al., Phys. Rev. Lett.79, 897 (1997).[21] A. Polimeni et al., Phys. Rev. B53, R4213 (1996), andreferences therein. We take the experimental values fthe coverage thicker than 1.5 ML.[22] M. Berti et al., J. Vac. Sci. Technol. B15, 1794 (1997).[23] D. J. Bottomley, Appl. Phys. Lett.72, 783 (1998).[24] C. W. Snyderet al., Appl. Phys. Lett.62, 46 (1993).[25] Q. K. Xue et al., J. Vac. Sci. Technol. B15, 1270 (1997).[26] V. A. Shchukinet al., Surf. Sci.352–354, 117 (1996).4045

Formation and stability of self-assembled coherent islands in highly mismatched heteroepitaxy

Crossref

Formation and Stability of Self-Assembled Coherent Islands in Highly Mismatched Heteroepitaxy

Formation and stability of self-assembled coherent islands in highly
  mismatched heteroepitaxy

Formation and stability of self-assembled coherent islands in highly mismatched heteroepitaxy

Abstract

Similar works

Full text

Available Versions

MPG.PuRe

MPG.PuRe

Crossref