We study the step meandering instability on a surface characterized by the
alternation of terraces with different properties, as in the case of Si(001).
The interplay between diffusion anisotropy and step stiffness induces a finite
wavelength instability corresponding to a meandering mode. The instability sets
in beyond a threshold value which depends on the relative magnitudes of the
destabilizing flux and the stabilizing stiffness difference. The meander
dynamics is governed by the conserved Kuramoto-Sivashinsky equation, which
display spatiotemporal coarsening.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Lett. (February 2006

A. Pimpinelli

Alberto Verga

Thomas Frisch

W. K. Burton

Y. Saito

English

arXiv

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Effect of Step Stiffness and Diffusion Anisotropy on the Meandering of a Growing Vicinal Surface

We study the step meandering instability on a surface characterized by the alternation of terraces with different properties, as in the case of Si(001). The interplay between diffusion anisotropy and step stiffness induces a finite wavelength instability corresponding to a meandering mode. The instability sets in beyond a threshold value which depends on the relative magnitudes of the destabilizing flux and the stabilizing stiffness difference. The meander dynamics is governed by the conserved Kuramoto-Sivashinsky equation, which display spatiotemporal coarsening

Frisch, Thomas

Verga, Alberto

HAL AMU

Effect of Step Stiffness and Diffusion Anisotropy on theMeandering of a Growing Vicinal SurfaceThomas Frisch, Alberto VergaTo cite this version:Thomas Frisch, Alberto Verga. Effect of Step Stiffness and Diffusion Anisotropy on the Me-andering of a Growing Vicinal Surface. Physical Review Letters, American Physical Society,2006, 96, pp.art. 166104. <10.1103/PhysRevLett.96.166104>. <hal-00082493>HAL Id: hal-00082493https://hal.archives-ouvertes.fr/hal-00082493Submitted on 28 Jun 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Effect of step stiffness and diffusion anisotropy on themeandering of a growing vicinal surfaceThomas Frisch∗ and Alberto Verga†Institut de Recherche sur les Phe´nome`nes Hors E´quilibre,UMR 6594, CNRS, Universite´ de Provence, Marseille, France(Dated: April 11, 2006)We study the step meandering instability on a surface characterized by the alternation of terraceswith different properties, as in the case of Si(001). The interplay between diffusion anisotropy andstep stiffness induces a finite wavelength instability corresponding to a meandering mode. The insta-bility sets in beyond a threshold value which depends on the relative magnitudes of the destabilizingflux and the stabilizing stiffness difference. The meander dynamics is governed by the conservedKuramoto-Sivashinsky equation, which display spatiotemporal coarsening.PACS numbers: 81.15.Hi, 68.35.Ct, 81.10.Aj, 47.20.HwMolecular beam epitaxy (MBE) is often used to grownanostructures on vicinal surfaces of semiconductor andmetallic crystals [1–5]. Under nonequilibrium growth avery rich variety of crystal surface morphologies are ex-perimentally observed resulting from the nonlinear evo-lution of step bunching and meandering instabilities [6–8]. Self-organized patterns arising from these instabilitiesmay be used for the development of technological appli-cations such as quantum dots and quantum wells [9, 10].The step meandering instability was originally predictedtheoretically by Bales and Zangwill [11] for a vicinal sur-face under growth. Its origin comes from the asymme-try between the descending and ascending currents ofadatoms. Nonlinear extensions of this work have shownthat the meander evolution can be described by ampli-tude equations showing diverse behaviors: spatiotempo-ral chaos governed by the Kuramoto-Sivashinsky equa-tion in the case of the Erlich-Schwoebel effect with des-orption [12]; nonlinear coarsening in the case of negligibledesorption [13–15]; and interrupted coarsening when two-dimensional anisotropy is included [16, 17]. Step mean-dering on a Si(001) vicinal surface can also be driven bya drift electromigration current in the presence of alter-nating diffusion coefficients, even for symmetric adatomattachment to the steps [18, 19].Recent experiments [20–23] on the growth of Si(001)have revealed the existence of a step bunching instabil-ity and the development of a transverse two-dimensionalcomplex structure (ripples), possibly reminiscent of a me-andering instability. We have recently shown that theobserved step bunching instability is due to the interplaybetween the elastic interactions and the alternation of thestep parameters [24]. In order to understand the rough-ening of the Si(001) surface during growth it would beimportant to know if step bunching and meandering canarise simultaneously. So far, no conclusive experimentalevidence for the presence of the Schwoebel effect leadingto meandering instability has been established.In this Letter, we show that the difference in step stiff-ness and diffusion anisotropy induce a meandering insta-yzxbn1xanxbnDaDbSbSaSbΓaΓbΞbn1y,tΞany,tFIG. 1: Sketch of the Si(001) vicinal surface showing the al-ternation of terraces and steps Sa and Sb. Lines on terracesindicate the privileged diffusion directions. Da, Db and γ˜a,γ˜b are the surface diffusion and step stiffness coefficients, re-spectively; xna , xnb and ξna (y, t), ξnb (y, t) are step positions andthe corresponding perturbations.bility which can account for the complex step structuresobserved in Si(001) epitaxial growth. We first presentthe linear two-dimensional stability analysis of a train ofsteps using a simplified extension of the model studiedin reference [24]. We show that the interplay betweendiffusion anisotropy and the step stiffness effect undergrowth conditions induces a finite wavelength instabilitywhich is maximum for the in-phase modes. We presentthe stability phase diagram in the parameters space. Ourresults are complemented by a weak nonlinear analysis ofthe step meander which reveals a coarsening dynamics.Using a simple similarity argument, we show that thecharacteristic coarsening exponent is 1/2 and that thegeneral solution of the CKS equations can be thought asa superposition of parabolas. Finally we conclude thisLetter by discussing the possible relevance of our modelto the experiment on Si growth and the consequences ofthe simultaneous existence of bunching and meanderinginstabilities.The Si(001) vicinal surface consists of a periodic se-quence of terraces where rows of 2×1 dimerized adatoms(terrace of type a) alternate with 1×2 dimerized adatoms(terrace of type b), as shown in Fig. 1 where we also in-2troduce several notations. On the reconstructed surfaceadatoms diffuse preferentially along dimer rows, givingrise to an anisotropic diffusion. Therefore, the steps sep-arating the terraces are of two kinds. The Sa steps arerather straight while the Sb ones are very corrugated [25].We shall allow each step to have a different step stiffnesscoefficient γ˜a and γ˜b [26]. For simplicity we neglect elas-tic interactions between steps and assume that the des-orption of adatoms is negligible; we also neglect Erlich-Schwoebel effects. Let us denote by xna(y, t) and xnb (y, t)the positions at time t of steps Sa and Sb respectively(cf. Fig. 1). During growth, the adatom concentrationson each terrace Cna (x, y, t) and Cnb (x, y, t), obey the fol-lowing diffusion equations [27]:Da∂2xCna +Db∂2yCna = −F , (1)Db∂2xCnb +Da∂2yCnb = −F , (2)where Da and Db are the diffusion coefficients, and F thedeposition flux. Cna and Cnb are the difference of adatomsconcentrations with respect to the uniform equilibriumconcentration C0 (taken to be the same on both terracetypes). We assume for Cna and Cnb , the following bound-ary conditions:Cna (xn−1b ) = Cn−1eq,b , Cna (xna) = Cneq,a , (3)Cnb (xna) = Cneq,a , Cnb (xnb ) = Cneq,b , (4)which correspond to instantaneous attachment kinetics,and can be considered as the simplest ones capturing themain physical effects (no diffusion along steps and negli-gible transparency). The adatom equilibrium concentra-tions Cneq,a and Cneq,b depend on the step curvatures κnaand κnb [12]:Cneq,a = C0Γaκna , Cneq,b = C0Γbκnb , (5)with Γb = Ωγ˜b/kBT and Γa = Ωγ˜a/kBT (Ω is the unitatomic surface, T the temperature, and kB the Boltz-mann constant). The normal velocities of each step arevna = x˙na/(1+ (∂yxna)2)1/2 and vnb = x˙nb /(1+ (∂yxnb )2)1/2,wherex˙na = Ω [(Db∂xCnb −Da(∂yxna)∂yCnb )−(Da∂xCna −Db(∂yxna)∂yCna )]x=xna , (6)x˙nb = Ω[(Da∂xCn+1a −Db(∂yxnb )∂yCn+1a )−(Db ∂xCnb −Da(∂yxnb )∂yCnb )]x=xnb . (7)In order to get a nondimensional version of equations(1)-(7), we set the unit of length to be the initial sizeof the terrace l0 and the unit of time l30/(C0ΓaΩDa).The system is controlled by three independent positivenondimensional parameters:f0 =Fl30C0ΓaDa, α0 =DbDa, δ0 =Γa − ΓbΓa, (8)1.5 1 0.5 0 0.5 1 1.5 2Δ0051015200StableUnstable1 2 3 4q642024Σ0.5 1 1.5 2q64202ΣFIG. 2: Stability diagram in the plane (f0, δ0), for α0 > 1.The thick gray line f0 = f0c given by Eq. (9), separates theunstable region (right side), from the stable one (left side).The dispersion relation σ(q) with its two branches is shownin each region.which respectively relate to the flux, diffusion anisotropyand step stiffness difference.We investigate now the linear stability of a train ofequidistant steps traveling at a constant velocity f0 whenperturbed transversally. The shape of the steps can bedecomposed in Fourier modes of the form xna(y, t) =f0t+ 2n+ ξna (y, t) and xnb (y, t) = f0t+ 2n+ 1 + ξnb (y, t)with ξna and ξnb the perturbation amplitudes varying asexp(σ(q, φ)t + iqy + inφ), where q is the wavenumberand φ the phase (see Fig. 1). Inserting these expres-sions into (1)-(7), we obtain the general dispersion re-lation σ = σ(q, φ). The dispersion relation possesses,for each φ, two branches corresponding to a stable σsand an unstable σu mode. The maximum growth rate isreached for the in-phase perturbation φ = 0 (Fig. 2). Inthe following we consider only this in-phase mode, thusneglecting the n dependence ξna = ξa, and ξnb = ξb (thesystem is periodic in the x-direction with period 2l0). Wefind that the instability appears above a flux thresholdf0c given by,f0 > f0c = −12α0δ0/(α0 − 1) , (9)where f0 > 0. The stability domain in the plan (f0, δ0)is shown in Fig. 2. Typically the instability is related toa large diffusion Db on terrace b (α0  1) accompaniedby a small stiffness γ˜b of step Sb (δ0 > 0). Althoughthe full expression of the dispersion relation is cumber-some, near the instability threshold we can introduce asmall parameter  measuring the distance to the thresh-old. This parameter arises naturally when consideringthe long wavelength limit, in which q → q. In thislimit the relevant scaled parameters are chosen to be:f0 = f, δ0 = δ, α0 = 1+ 2α, together with a rescalingof the stiffness Γa → 2Γa. This scaling will lead to aconsistent weak nonlinear expansion, as it will be shownbelow. To lowest order the two branches of σ = σ(q, φ)3are,σs = −44q2 , (10)σu =648fα(fα+ 12δ)q2 − 6q4 . (11)The growth rate σu is maximum for the wavenumberqm = [fα(fα + 12δ)]1/2/4√6. Even for vanishing δ (nodifference between step stiffness) an instability is stillpresent driven by diffusion anisotropy and growth effect,σu(qm) ∼ f4α4. The physical mechanism of the insta-bility is the point effect, increasing the diffusion aroundthe maxima, combined with the anisotropy in diffusionbetween two consecutive terraces, locally impeding thecompensation of the currents leaving and reaching thesteps. If the anisotropy was absent, a simple differencein diffusion coefficients would not lead to an instabilityas a consequence of mass conservation.Taking for example typical values for Si(001) at twodifferent temperatures T = 1000K and T = 800K (withl0 = 210−8m and F = 0.1ML s−1), f0 = 0.3, and 4.3,α0 = 45, and 119, and δ0 = 0.9, we obtain that the typi-cal wavelength of the meanders is of the order of 250 nmand 60 nm, respectively. These sizes are in the range ofthe transverse modulations (ripples) of the step bunchesobserved in the experiments [20–23].We study at present the nonlinear evolution of the me-andering instability, in the limit of weak amplitudes andlong wavelengths. An inspection of Eqs. (10)-(11) forthe damping and growth rates, suggests that we shouldconsider different time scales and amplitudes for the sta-ble and unstable branches. We introduce the functionss(y, t) and u(y, t) corresponding to the amplitudes of thestable and the unstable branches respectively. These am-plitudes are related to the step shape by,(ξaξb)= M0()( su), (12)whereM0 is the matrix formed with the eigenvectors as-sociated to the linear dispersion relation and depends onthe physical parameters (f0, α0, δ0). In order to obtainthe relevant nonlinear dynamics we use a standard multi-scale method. Adatoms concentrations and step shapesare expanded in powers of . The amplitudes of thisexpansion depend on the slowly varying space y andtwo time variables 4t and 6t. In particular the stableand unstable shape functions are given by s = s(4t, y)and u = u(6t, y). Solving diffusion equations (1-2) andboundary conditions (3-4) up to order 7, and insertingthe results into the step velocity equations (6-7) we findthe equation for the unstable mode:∂tu = −∂2y[fα48(fα+ 12δ)u+ ∂2yu+f12(∂yu)2], (13)where we renamed the slow variables 6t → t and y →y. After rescaling we can write Eq. (13) in the form100 200 300 400 500 600 700 800 900 100001002003004005006007008009001000yu(y,t)FIG. 3: Spacetime plot of u(y, t) with nondimensional y andt axes, given by the numerical solution of the CKS equation.The coarse-graining of structures leads to a superposition ofparabolas, with a size 〈u2〉1/2 ∼ t. In the long time state allthe parabolas tend to have unity curvature at their maximum,and width increasing as√t.∂tu = −∂2y [u+ ∂2yu+ (∂yu)2/2]. This is a conserved ver-sion of the Kuramoto-Sivashinsky equation (CKS). Thenumerical simulations [28] of a related equation describ-ing bunches created by an electromigration current, thushaving an extra symmetry breaking term in ∂3yu, haverevealed a non interrupted coarsening dynamics with acharacteristic size of coalesced step bunches increasing ast1/2. In our context this ∂3yu term, leading to a dispersivedrift, must be absent. Our direct simulations confirmeda t1/2 scaling and also demonstrate a linear time growthof the characteristic meander amplitude 〈u2〉1/2 ∼ t (spa-tial average is denoted 〈· · · 〉). A typical spatiotemporalevolution from a random initial condition is shown inFig. 3. In the context of Bales-Zangwill meandering in-stability, it was shown that the dynamics of steps is fullynonlinear, excluding CKS equation, although it would becompatible with the basic symmetries of the system [15].Simple similarity and matching arguments lead to acomplete picture for the long time behavior of (13). Itis worth noting that the CKS equation admits an exactparticular solution in the form of a stationary parabolau(y, t) = −y2/2. We also note that, for rapidly decreas-ing or bounded functions, the moment of order one ofu(y, t) is conserved while the second order moment sat-isfiesddt12∫u2dy =∫[(∂yu)2 − (∂2yu)2]dy , (14)showing that the amplitude of u tends to increase in re-gions where the gradient term ∂yu inside the integral4dominates the curvature term ∂2yu. This suggests thatthe dynamics of large amplitude smooth regions of u(y, t)is almost independent of the four derivative (stabilizing)term in (13). Trying a similarity solutionu = taϕ(y/tb) , Y = y/tb (15)of ∂tu = −∂2y [u + (∂yu)2/2], we find immediately theexponents b = 1/2 and a = 1, which agree with ournumerical results. The fourth order derivative term be-haves as ∂4yu ∼ ∂4Y ϕ/t, which is consistent with the aboveassumption that it should be negligible in the consid-ered regime. Moreover, the similarity equation for ϕ:ϕ− (Y/2)∂Y ϕ+ ∂2Y [ϕ+ (∂Y ϕ)2/2] = 0 has a solution inthe form of a bounded parabola:u(y, t) = −y2/2, |y| < y0(t) (16)and zero elsewhere. The parabola edge y0(t) is deter-mined by the conditionddt14∫ y0(t)0y4dy = 2∫ y0(t)0y2dy , (17)a consequence of (14), which gives y0(t) = (16t/3)1/2.The general, asymptotic solution of (13) can be thoughtas a superposition of parabolas satisfying (16) (seeFig. 3). The joining region between the parabolaspossesses a high curvature and can be described bythe reduced inner equation ∂2yu(y) + (1/2)(∂yu(y))2 =2k2 = const., whose solution is of the form u(y) =2 log[cosh k(y − y0(t))]. Matching with the outer solu-tion (16) we find that k = y0(t)/2 ∼√t. Therefore, thecurvature of the joining line increases as κ ∼ t.In this Letter, we have shown that the effect of stepstiffness difference and diffusion anisotropy induces a me-andering instability during surface growth. We have firstpresented a linear stability analysis of a train of steps us-ing a simplified two-dimensional extension of the modelstudied in Ref. [24]. We have shown that the inter-play between diffusion anisotropy and the step stiffnesseffect under growth conditions leads to a finite wave-length instability which is maximum for the in-phasemode. Our results are complemented by a weak non-linear analysis of the step dynamics which reveals thatthe amplitude of the meanders is governed by the con-served Kuramoto-Sivashinsky equation (CKS) which dis-plays non-interrupted coarsening. We believe that themorphology observed in experiments of molecular beamepitaxy on Si(001) slightly disoriented towards the [110]direction, reported in Refs. [20–23], can be explained bythe nonlinear evolution of the step bunching and stepmeandering instabilities arising simultaneously. Indeed,we will present elsewhere an investigation of the two-dimensional dynamics originated by the nonlinear cou-pling between these kinetic effects, and we will discusstheir role in the formation of the ripples shown for in-stance in Fig. 1 of Ref. [21].We would like to thank J.-N. Aqua, I. Berbe´zier, M.Dufay, P. Politi, and C. Misbah for stimulating discus-sions.∗ Electronic address: frisch@irphe.univ-mrs.fr† Electronic address: Alberto.Verga@irphe.univ-mrs.fr[1] J. Stangl, V. Holy, and G. Bauer, Rev. Mod. Phys. 76,725 (2004).[2] A. Pimpinelli and J. Villain, Physics of Crystal Growth(Cambdrige University Press, 1998).[3] Y. Saito, Statistical Physics of Crystal Growth (WorldScientific, 1998).[4] N. Ne´el, T. Maroutian, L. Douillard, and H.-J. Ernst,Phys. Rev. Lett. 91, 226103 (2003).[5] F. Nita and A. Pimpinelli, Phy. Rev. Lett. 95, 106104(2005).[6] H.-C. Jeong and E. D. Williams, Surf. Sci. Rep. 34, 171(1999).[7] P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain,Phys. 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Effect of step stiffness and diffusion anisotropy on the meandering of a
  growing vicinal surface

Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface

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