Journal ArticleComputer simulations show that if stress is present, steps on a vicinal surface can self-organize into a regular array of step bunches. Such self-organization can provide templates for subsequent fabrication of "quantum wire" nanostructures. The size and spacing of the bunches can be controlled independently. We analyze the requirements for optimal ordering

Liu, Feng

Tersoff, J.

English

The University of Utah: J. Willard Marriott Digital Library

VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998Self-Organization of Steps in Growth of Strained Films on Vicinal SubstratesFeng Liu,1 J. Tersoff,2 and M. G. Lagally11University of Wisconsin, Madison, Wisconsin 537062IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598(Received 23 June 1997)Computer simulations show that if stress is present, steps on a vicinal surface can self-organizeinto a regular array of step bunches. Such self-organization can provide templates for subsequentfabrication of “quantum wire” nanostructures. The size and spacing of the bunches can be controlledindependently. We analyze the requirements for optimal ordering. [S0031-9007(97)05273-3]PACS numbers: 68.35.Fx, 68.35.Bs, 81.10.AjThe continued miniaturization of electronic devices isleading us into a realm of nanostructures, which exhibitnovel electronic and optical properties, and have a widerange of potential applications such as single-electrontransistors, quantum-dot lasers, and quantum computing.Quantum well devices are already widespread. Recenteffort has been devoted to fabricating quantum wires andquantum dots, respectively, one- and zero-dimensionalanalogs to the quantum well. Just as uniform thickness isa prerequisite for a film in a quantum well device, uniformsize and spacing are prerequisites for applications of wiresand dots.Different routes to fabricating wires and dots arepossible. One can perform direct substrate patterningor lithography [1], or wirelike and dotlike structurescan be induced to “self-assemble” during the growth ofthin films. These self-assembled structures sometimesexhibit surprising uniformity [2–5], suggesting that theirapplication in actual devices is a real possibility.An approach that is, in a sense, a hybrid betweendeliberate nanopatterning and self-assembly is the useof vicinal surfaces as templates for the growth of wiresand dots. Such surfaces are cut at a small angle to theatomic planes, creating a staircase of atomic-height stepsthat can serve as preferential sites for the growth of wires[6,7] and the nucleation of clusters [8,9]. But single stepstend to meander [10], and their spacing is often irregular[7]. These drawbacks have caused difficulties in previousefforts to use steps to grow semiconductor and metallicwires [6,7] or vertical superlattices [6].Step bunches, on the other hand, tend to be muchstraighter than individual steps, due to their greater “stiff-ness” [10]. They can also have different heights depend-ing on the number of steps in the bunches, allowing thepossibility of different thicknesses for quantum wires, ordirect control of the size of quantum dots that nucleateon the step bunches [11]. Recently it was shown that thesteps on a vicinal surface can be induced to bunch byintroducing stress, through the deposition of a thin layerof a lattice mismatched film (e.g., a SiGe alloy on Si)[12,13]. The problem is to control the bunching, in orderto achieve a useful template structure.In this Letter we show, through simulations, that stress-induced step bunching can be controlled to achieveuniformly sized and uniformly spaced step bunches. Wedemonstrate that, when growing in the step-flow mode,one should in principle be able to grow step bunches ofany desired size and spacing by adjusting the growth rate,the growth temperature, the substrate step density, or thecomposition of the alloy.Step-flow growth is perhaps the simplest mode ofgrowth imaginable, with deposited atoms diffusing andattaching to existing steps. Yet it can exhibit complexbehavior. Steps are sometimes observed to bunch, mostoften due to kinetic factors [14–16]. Even highly simpli-fied models of kinetic step bunching exhibit a rich behav-ior that is not completely understood [14].For a strained layer, there exists an attractive interactionbetween steps [13,17]. This leads to a bunching instabilitythat is thermodynamically driven, and thus distinct fromkinetic bunching. We focus here on the steady-stategrowth morphology resulting from such stress-inducedstep bunching, and on the prospect of systematicallycontrolling this morphology.We use a one-dimensional (1D) model [18] for step-flow growth of a surface under stress, such as the surfaceof a heteroepitaxial layer. Steps move only by attachmentand detachment of adatoms, which diffuse across theterraces driven by gradients in the chemical potential.These gradients arise because the steps act as sinks foradatoms, and the elastic forces cause some steps to bebetter sinks than others. One can derive an equation ofmotion for the steps by integrating the diffusion equationfor adatoms between steps, with a boundary conditiondetermined by thermodynamic equilibrium with the steps[13]. This givesdxidt­ Fµxi11 2 xi212∂2 Bµfi11 2 fixi11 2 xi2fi21 2 fixi21 2 xi∂. (1)Here xi is the position of the ith step in the directionperpendicular to the step; F is the adatom flux per site;and f is the force per unit length on the step due to elastic1268 0031-9007y98y80(6)y1268(4)$15.00 © 1998 The American Physical SocietyVOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998step-step interactions. B is closely related to the surfacediffusion constant,B ­DA2kTe2EdykT , (2)where D is the adatom diffusion coefficient, A is the areaper surface site, and Ed is the detachment energy of anadatom from a step.The force f can be written asfi ­Xjfiiµa1sxj 2 xid2a2sxj 2 xid3∂. (3)Here a1 reflects the attractive interaction arising from theelastic relaxation around each step on a strained layer, anda2 reflects the repulsion arising from the inherent stress(the “force dipole”) of each step. The elastic theory ofthese interactions has been discussed elsewhere [17].Five parameters define our model: F, B, a1, a2, andthe average step spacing Lav (which is determined bythe angle of the overall surface relative to the atomicplanes). However, with a rescaling of energy, length, andtime, the behavior depends on only two independent ratiosof characteristic lengths: L0yLav , where L0 ­pa2ya1is the equilibrium spacing of an isolated step pair, andLdyLav , where Ld ­ sBa1yFd1y3 is a length reflectingboth diffusion and strain. L0 can be adjusted through a1by changing the alloy composition, and Ld by changingthe temperature or growth rate.The parameters used here correspond to L0yLav ­0.192; the results should be qualitatively similar as longas Lav ¿ L0. For convenience, we refer to varying theflux F, with Ba1 fixed at 105, but it would be equivalent,and perhaps more practical in an experiment, to vary Bthrough its exponential dependence on temperature.We simulate the dynamics by direct integration ofEq. (1). The attractive interaction leads to step bunching[13], while the growth flux tends to break up largebunches [14]. Breakup of a bunch occurs when the largeadatom current diffusing to the leading step and drivingit forward overcomes the step-step attraction. Aftersufficient time, the system approaches a steady-state stepmorphology.Figure 1(a) shows a portion of the surface after reachingsteady-state growth for F ­ 25. It consists of rather well-ordered bunches of 3–4 steps, plus a comparable numberof “free” steps. Individual steps are emitted by one bunchand captured by the next. The step bunches themselvesare relatively static, with free steps dynamically attachingto and detaching from them.The average bunch size decreases continuously with in-creasing FyB, reflecting the competition between thermo-dynamics (strong bunching for small FyB) and kinetics(suppression of bunching [14] for large FyB). Becauselarger bunches tend to emit steps, and smaller bunches tocapture them, the tendency is for all bunches to convergeto the same size. The spatial distribution of bunches alsoFIG. 1. Sequence of step configurations in the steady-stategrowth regime (from bottom to top) for deposition total-ing about 1.2 monolayers (ML). The interval between subse-quent configurations corresponds to about 0.2 ML deposition.(a) F ­ 25. A “phase boundary” separates the left and righthalves of the figure: the left 4 bunches have 4 steps andthe right 5 bunches have 3 steps. The spacings between4-step bunches are consistently larger than those between 3-stepbunches. (b) F ­ 30. All bunches have three steps except forone with four steps, marked by the arrow.becomes rather regular, if one ignores the highly mobilefree steps, as Fig. 1(a) shows.For F ­ 25 (L0yLav ­ 0.192 and Ba1 ­ 105), mostbunches have either 3 or 4 steps. The presence of twodifferent bunch sizes limits the degree of order. Toquantify the ordering, Fig. 2(a) shows the pair correlationfunction for steps,gsrd ­1NsXijdsjxi 2 xj j 2 rd , (4)of the F ­ 25 steady state, where Ns is the total numberof steps. For a perfectly ordered step-bunch array, thiswould be a series of equally spaced sharp peaks, whosewidth corresponds to the (small) size of the individualbunches, and whose spacing reflects the spacing betweenbunches.Figure 2(a) shows a split first peak, indicating thepresence of two distinct bunch separations. These can1269VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998FIG. 2. Step-step correlation functions for steady state atdifferent deposition fluxes F. The results are averaged overten configurations well separated in time. (a) F ­ 25. Thevertical lines with and without arrows mark two separate seriesof peaks, corresponding to two distinct spacings. (b) F ­ 30.The presence of equally spaced peaks up to high order reflectsthe strong long-range order of the bunch array.be associated with bunches of 3 and 4 steps. Moreover,there are higher-order peaks for each spacing, but no low-order “mixed” spacings. Thus the system appears to besegregating into separate regions of 3-step bunches and4-step bunches. This is confirmed by direct inspectionof the steady-state configuration. Figure 1(a) shows theregion around a “domain boundary,” with 4-step buncheson the left, and 3-step bunches on the right.Our goal is not only to observe step-bunch ordering,but to understand how it may be controlled for nanofab-rication. Perfect uniformity would require all bunches tohave the same size. We therefore increased the flux toF ­ 30, to decrease the average bunch size. In the result-ing F ­ 30 steady state [Fig. 1(b)] most bunches have3 steps. The improved ordering is even more strikingin the pair correlation function, Fig. 2(b). There is asingle characteristic bunch spacing, reflected in a series ofequally spaced peaks. Moreover, the peaks remain quitedistinct up to at least 13th order, limited only by the sizeof our simulation cell.So far we have emphasized the average bunch size ascontrolling the degree of order. From a more fundamentalperspective, we must consider the dynamical stability ofa given step configuration. Consider for simplicity asystem of straight steps having a short-range attractiveinteraction. These will form bunches of n steps (includingsingle steps, n ­ 1). It is a reasonable approximationto treat each bunch as having a chemical potential msndwhich depends only on the size n, and a correspondingequilibrium density hsnd of adatoms near the bunch.The velocity of the mth bunch may then be obtained bygeneralizing the result for single steps [19],dxmdt­F2nmsxm11 2 xm21d1Dnmµhsnm11d 2 hsnmdxm11 2 xm2hsnmd 2 hsnm21dxm 2 xm21∂, (5)where nm is the size of the mth bunch.At large F, all the bunches move forward in the growthdirection, with smaller bunches moving more quicklybecause of the factor 1ynm. At very small F, however,large bunches move more quickly in the forward directionbecause of their lower chemical potential, and the smallestbunches can actually retreat. In any case, there willbe continual collisions between different-size bunches,leading to coalescence, fragmentation, and/or exchange ofsteps. Thus it seems natural to expect a complex, chaoticevolution of the surface morphology.For a stable configuration with long-range order, it mightseem necessary that all steps be in bunches of a singlesize, so that no collisions need occur. However, thereis a competition between energetics, which favors largebunches, and kinetics, which favors small bunches orsingle steps, to better incorporate the arriving atoms. Oursimulations suggest that the system automatically balancesthese factors by alternating bunches and single steps.This alternation is clearly seen in Fig. 1, especiallyin the better-ordered case in Fig. 1(b). The alternationis also reflected in Fig. 2(b). The first-order peak fallsalmost exactly on an integer spacing (in units of Lav),corresponding to four steps per period, i.e., one 3-stepbunch and one free step per period. There is alsoa small shoulder on the right side of the first-orderpeak, reflecting the fact that there are a few “defect”bunches containing four steps. These defect bunches arepresumably responsible for much of the damping andbroadening of the high-order peaks in Fig. 2(b). Thisalternation of bunches and free steps is consistent withprevious 2D simulations of Kandel and Weeks [14], if1270VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998one notices that for a given lateral position there is onlyone “crossing step” (the 2D analog of our single steps)between bunches.This arrangement of alternating single steps andbunches has a special behavior that may account for itsoccurrence. Consider a periodic structure with one bunchand one single step per period. In this case, from Eq. (5)the velocities of all the bunches are the same, as are thoseof all the single steps, independent of the positions ofthe single steps relative to the bunches. So one may ineffect have two coexisting periodic arrays, one of singlesteps and one of bunches, moving independently of eachother.There is one further ingredient needed to make thearrangement of alternating bunches and steps dynamicallystable: the bunches must remain in a dynamic equilibriumwith the free steps. Whenever a single step collides witha bunch, the bunch must emit a single step from itsother side, so that the distribution of sizes is preserved.Thus a bunch of some size M must be stable for thegiven conditions, while a bunch of M 1 1 is unstableand immediately loses a step. In that way, single stepsappear to pass through the bunch, although the actualevent is capture followed by emission of a different step.If this special configuration acts as a sort of dynamical“attractor,” it may account for the two-phase behaviorobserved for F ­ 25.We emphasize that we do not yet understand thedynamics in detail, especially the “phase separation.”Nevertheless, we have demonstrated that considerablecontrol of the ordering is possible. In actual growth,such control would probably require in situ diffractionmeasurements, so that flux or temperature could be variedto optimize the degree of order. The two-phase behavior,in particular, could simplify this procedure. In that casetwo distinct periods would coexist on the sample, andone could simply vary the growth conditions until thediffraction spot for one period disappears.In conclusion, in step-flow growth of strained lay-ers there is a competition between thermodynamic stepbunching and kinetic debunching. This leads to a dy-namic steady state, with finite-size step bunches exhibit-ing considerable uniformity and long-range order. Mostimportant, the bunch size and spacing can be directly con-trolled and the ordering can be systematically improvedby tuning the temperature and/or the flux. Through con-trol of growth parameters, degree of stress, and choiceof substrate miscut, it should be possible to create anydesired size and spacing of step bunches. Such self-assembled and self-organized step-bunch arrays hold con-siderable promise as templates for subsequent fabricationof quantum wires and quantum dots.Supported in part by NSF Grants No. DMR 93-04912and No. DMR 96-32527.[1] See, for example, E. Kapon, D. M. Hwang, andR. Bhat, Phys. Rev. Lett. 63, 430 (1989); R. Nötzel, N. N.Ledentsov, L. Däweritz, M. Hohenstein, and K. Ploog,Phys. Rev. Lett. 67, 3812 (1991).[2] S. P. Ahrenkiel, S. H. Xin, P. M. Reimer, J. J. Berry,H. Luo, S. Short, M. Bode, M. Al-Jassim, J. R. Buschert,and J. K. Furdyna, Phys. Rev. Lett. 75, 1586 (1995).[3] R. Nötzel, Semicond. Sci. Technol. 11, 1365 (1996).[4] Phys. Today 49, No. 5, 22 (1996).[5] J. Tersoff, C. Teichert, and M. G. Lagally, Phys. Rev. Lett.76, 1675 (1996).[6] K. Pond, A. C. Gossard, A. Lorke, and P. M. Petroff,Mater. Sci. Eng. B 30, 121 (1995); M. S. Miller,H. Weman, C. E. Pryor, M. Krishnamurthy, and P. M.Petroff, Phys. Rev. Lett. 68, 3464 (1992).[7] T. Jung, R. Schlittler, J. K. Gimzewski, and F. J. Himpsel,Appl. Phys. A 61, 467 (1995).[8] V. Bressler-Hill, A. Lorke, S. Varma, P. M. Petroff,K. Pond, and W. H. Weinberg, Phys. Rev. B 50, 8479(1994).[9] C. Teichert, M. G. Lagally, L. J. Peticolas, J. C. Bean, andJ. Tersoff, Phys. Rev. B 53, 16 334 (1996); C. Teichert,Y. H. Phang, L. J. Peticolas, J. C. Bean, and M. G. Lagally,in Surface Diffusion: Atomistic and Collective Processes,edited by M. C. Tringides, NATO-ASI Series (PlenumPress, New York, to be published).[10] Individual steps meander because this increases theirentropy. In contrast, a bunch of N steps fluctuatingtogether has (roughly speaking) the same available phasespace but a stiffness N times larger than a single step,and so thermal fluctuations are greatly suppressed. (Thesteps cannot fluctuate independently without breaking upthe bunch, which costs substantial energy.)[11] Yasuhiko Arakawa (unpublished).[12] Y. H. Phang, C. Teichert, M. G. Lagally, L. J. Peticolas,J. C. Bean, and E. Kasper, Phys. Rev. B 50, 14 435 (1994).[13] J. Tersoff, Y. H. Phang, Z. Zhang, and M. G. Lagally,Phys. Rev. Lett. 75, 2730 (1995).[14] D. Kandel and J. D. Weeks, Phys. Rev. Lett. 69, 3758(1992); 72, 1678 (1994); Phys. Rev. B 49, 5554 (1994);Phys. Rev. Lett. 74, 3632 (1995).[15] W. W. Mullins and J. P. Hirth, J. Phys. Chem. Solids 24,1391 (1963); R. L. Schwoebel, J. Appl. Phys. 40, 614(1969).[16] M. Sato and M. Uwaha, Phys. Rev. B 51, 11 172 (1995).[17] V. I. Marchenko and A. Ya. Parshin, Sov. Phys. JETP 52,129 (1980); V. I. Marchenko, JETP Lett. 33, 381 (1981);O. L. Alerhand, D. Vanderbilt, R. D. Meade, and J. D.Joannopoulos, Phys. Rev. Lett. 61, 1973 (1988); E. Pehlkeand J. Tersoff, Phys. Rev. Lett. 67, 465 (1991); J. Tersoff,Phys. Rev. Lett. 74, 4962 (1995).[18] A 1D model is realistic provided that in the real 2Dsurface, no step deviates from the average orientationby more than a small angle. Barring an unfavorablestep-energy anisotropy, this condition will be satisfied ifthe average step spacing is sufficiently small or the stepstiffness is sufficiently large.[19] J. Tersoff, Phys. Rev. Lett. 77, 2017 (1996).1271

Self-organization of steps in growth of strained films on vicinal substrates

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Self-organization of steps in growth of strained films on vicinal substrates

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