Strained islands as step bunches: shape and growth kinetics

PERGAMON

Solid State Communications 117 (2001) 337±341

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Strained islands as step bunches: shape and growth kinetics V.M. Kaganer*, K.H. Ploog Paul-Drude-Institut fuÈr FestkoÈrperelektronik, Hausvogteiplatz 5-7, D-10117 Berlin, Germany Received 16 October 2000; accepted 16 November 2000 by H. Eschrig

Abstract We describe the growth of strained heteroepitaxial islands by the motion of atomic steps forming the island surface. The adatom attachment to a step from the down terrace and the detachment to the up terrace provide the mass transport mechanism to the island top. As the island increases in size, the bulk stress concentration at the island edges and the stress relaxation at the island top lead to steeper island sides and a ¯atter top. This effect is treated as a primary source for the shape transformation from ªpyramidsº to ªdomesº, with subsequent faceting by a slight rearrangement of steps. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; B. Epitaxy PACS: 81.10.Aj; 81.15.Hi; 68.35.Ct

The self-organized growth of heteroepitaxial strained islands (the Stranski±Krastanov growth) is a peculiarly interesting growth mode: islands are formed instead of a uniform strained ®lm, since the relaxed elastic energy exceeds the increase in surface energy. The fundamental importance of this growth mode and the potential applications of the islands in nanoscale optoelectronic devices stimulated intense experimental studies focused on island shapes [1±7], growth kinetics [6±11], as well as island size distributions, coarsening, alloying, and overgrowth. Theoretical studies [12±20] concentrate on the island energetics. The common approach is to assume that the island surface consists of planar facets of a few possible orientations, then determine the shape of the island of a given volume by minimizing the sum of elastic and surface energies, and ®nally analyze the energy gain as a function of the island size. These studies implicitly assume that an island reaches an equilibrium shape for a given volume much faster than its volume changes, and do not describe the island growth. In the present paper, we simultaneously describe the island growth kinetics and its shape transformation. We treat the island surface as a bunch of atomic steps and consider the generation and motion of steps by applying * Corresponding author. Tel.: 149-30-203-77-366; fax: 149-30203-77-201. E-mail address: [email protected] (V.M. Kaganer).

the model developed for strained vicinal surfaces [21±24]. The island energetics is hence a consequence of the appropriate kinetical growth model. The island shape continuously transforms simultaneously with (and as a result of) the island growth, without reaching an equilibrium shape at any given volume. The equilibrium distance between steps on a strained surface is due to the competition between strain-independent repulsion and strain-dependent attraction. We estimate this distance and show that it coincides with the typical terrace widths on the surface of strained islands. We take into account that the bulk stress acting on a step is in¯uenced by the elastic ®elds of all other steps. As the island increases in size, the strain relaxation at the island apex and the strain concentration at the island edges become more pronounced and give rise to a continuous transformation from equal widths of all terraces to a ¯atter top part of the island and steeper sides. This effect produces a template for the transformation of ªhutsº or ªpyramidsº to ªdomesº [1±5] by subsequent faceting (not considered in this paper), which is a crystallographic effect allowing further reduction of the total energy of the island by only a slight change in the distances between steps. An island can grow in height only by nucleating new atomic layers at the top. It is well established experimentally that the islands are primarily fed by adatoms initially deposited on the ¯at surface between the islands, while the direct atomic ¯ux from the incoming beam to the island top plays a

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(00)00484-1

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V.M. Kaganer, K.H. Ploog / Solid State Communications 117 (2001) 337±341

minor role. Therefore, the island growth requires, ®rst of all, transport of adatoms to the island apex from its bottom. This transport is provided by attachment of adatoms to a step from the lower terrace with subsequent detachment to the upper terrace, to provide the equilibrium adatom concentration at the step on both sides. We assume that the adatom concentration on the ¯at surface far from the island is equal to the equilibrium adatom concentration at an isolated step. If the adatom concentration on the ¯at surface exceeds the equilibrium concentration because of the incoming ¯ux, new islands nucleate somewhere far from the island under consideration, thus reducing the concentration to the equilibrium value. In our model, the island growth starts from a single pair of steps and proceeds continuously. We ®nd that the initial growth is very fast and then slows down, which is in good agreement with the observations of the growth kinetics of the individual islands [6±9]. Our model does not require the three-dimensional (3D) nucleation which is predicted based on the competition between elastic and surface energies [12±20]. Experimentally, it would be rather dif®cult to distinguish the fast initial growth from te 3D nucleation. We also note that the island growth at low mis®ts without the 3D nucleation has recently been observed [10,11]. We postulate that a new terrace nucleates from adatoms on the top terrace of the island as soon as the size of the top terrace exceeds some critical value, independent of strain. This crude approximation is suf®cient for the purposes of the present study since nucleation is not the limiting process. If we arti®cially turn off or slow down nucleation at some stage of the growth calculation, a ¯at top facet parallel to the interface develops but the stepped side surfaces of the island are only slightly modi®ed. From the absence of such ¯at facets in the experiments we conclude that the nucleation is fast enough and does not limit the growth. We consider a two-dimensional (2D) model with straight parallel steps. Such a model describes the main qualitative features of the problem [25]. A 3D model with circular islands will be considered elsewhere. The strain ®eld of a surface step on a non-strained crystal [26,27] is that of an elastic dipole applied to the planar surface and acting on the semi-in®nite medium with a force f(x), where x is the coordinate normal to the step line along the surface. The force consists of two components The component normal to the surface fz ˆ Qz 2d…x†=2x; where d (x) is the delta-function, is the result of surface stress producing a torque. The corresponding dipole component Qz has opposite signs for up and down steps. The component parallel to the surface fx ˆ Qx 2d…x†=2x is a torque-free property of the step as a linear defect. Qx is the same for up and down steps. In addition, the bulk stress acting on the step edge produces an uncompensated force Ð a force monopole [21±24] fx …x† ˆ Pd…x†; where P ˆ ^hs xx : Here, h is the step height, s xx is the inplane bulk stress at the step location, and the two signs correspond to up and down steps. Calculating the interaction energy between two steps, we

have to take into account that the components P and Qz for up and down steps are opposite and the in-plane stress s xx is modi®ed by the interaction with the other steps. It is therefore instructive to calculate the interaction energy of two unequal steps, one of strength (Pm,Qm) and the other one of strength (Pn,Qn), located at a distance x ˆ x n 2 xm : " 1 2 n2 2 Unm …x† ˆ 2 Pn Pm 1 …Pm Qnx 2 Pn Qmx † 2x pE # 22 2 …Qnx Qmx 1 Qnz Qmz † 2 lnuxu; …1† 2x where E is the Young modulus and n the Poisson ratio. The three terms in Eq. (1) describe the monopole±monopole, monopole±dipole, and dipole±dipole interactions, respectively. Note that the monopole±dipole interaction is absent if all step parameters are equal. The length l0 ˆ Q=P; where Q2 ˆ Q2x 1 Q2z ; is the characteristic length of the problem. To estimate this length, we have to know the strengths of the dipoles Qx and Qz, which depend on the microscopic structure of the steps and can be determined from atomistic models [28±31]. We can use the data for steps on unstrained surfaces since the strain essentially in¯uences the energies of individual steps but only slightly affects the dipole±dipole interaction between the steps [32]. Both components of the dipole can be found by comparing the interaction energies of steps with the same sign (both up or both down) with that of steps with opposite signs (one up and the other down). The corresponding interaction energies in the non-strained crystal are given by the last term of Eq. (1): U…x† ˆ 22…1 2 n2 †…Q2x ^ Q2z †=…pEx2 †: Poon et al. [28,29] found for the Si(100) surface that the interaction energies for a pair of steps of the same sign and a pair of steps of opposite sign differ by not more than 12%. We conclude that Qx q Qz and use a mean value to estimate  Both components of the dipole can also be Qx < 2 eV=A: found by directly comparing the calculated atomic positions with the bulk displacement ®eld given by the elasticity  and Qz ˆ theory. In this way, the values Qx ˆ 1:46 eV=A  were obtained for the Si(111)(7 £ 7) surface [30] 0:58 eV=A  for the surfaces of Ni and Au and Qx < Qz < 0:15 eV=A [31]. Summarizing the results of atomistic simulations on different materials, we conclude that Qx is larger than Qz and Ê simple dimensional estimate the dipole strength as 1 eV/A estimate (modi®ed after Ref. [33]) is Q , Eh2 : Taking the  3 and the step Young modulus E ˆ 1011 J=m3 ˆ 0:625 eV=A  we obtain a reasonable value of Q < height h ˆ 1:4 A;  1:2 eV=A: With the values given above and taking a mis®t e0 ˆ 0:04 of the Ge-on-Si system, we ®nd a characteristic length l0 ˆ  which gives a slope of the stepped surface Q=P < 20 A;  h=l0 < 0:07: Note that double-layer steps …h ˆ 2:8 A†  instead of single-layer steps …h ˆ 1:4 A† would increase the slope by a factor of 4. These values agree with the observed slopes ( < 0.2) of Ge islands on Si.

V.M. Kaganer, K.H. Ploog / Solid State Communications 117 (2001) 337±341

Fig. 1. (a) Snapshots of a growing island obtained by numerical integration of the step motion equations for diffusion-limited kinetics (lK ! 0) and q ˆ 1 at the time moments t=t ˆ 5 £ 100 ; 5 £ 101 ; 5 £ 102 ; and 5 £ 103 ; (b) the same islands scaled to a common height to reveal the shape changes; (c) time evolution of the island height (measured in number of atomic layers N) and its base width w (in units of l0).

The bulk stress at the position of a step is, to zeroth order, given by the mis®t, s xx ˆ 2Ee0 =…1 2 n†: This value is corrected by the stress ds xx produced by all other steps. Calculating their contribution at the position xn of the nth step, we ®nd …n† ds xx ˆ2

2 X 0 Pm ; p m xn 2 xm

…2†

where the prime on the sum denotes that the term m ˆ n is excluded. The value of the correction thus introduced can be estimated by considering N equidistant steps separated by the characteristic length l0. Then, the additional stress (2) can be estimated as ds xx =s xx < …h=l0 † ln N and, for a slope h=l0 ˆ 0:1; the correction amounts to 25% already for a dozen steps. Restricting to the ®rst order over the small …n† parameter h/l0, one can calculate Pn ˆ h…s xx 1 ds xx † using the undisturbed value of Pm in the right-hand side of Eq. (2). The chemical potential of the step is the change of its energy when an adatom is attached to it and hence the step is advanced. Therefore, we have

mn ˆ S

X 0 2Unm …xn 2 xm † ; 2xn m

…3†

where S is the surface area per atom. We describe the step kinetics in the framework of the familiar model [34±36]. The steps move (and the island grows) by attachment of adatoms to steps. It is assumed that the movement of a step is slow compared to the equi-

339

libration in the adatom distribution. Hence, the adatom concentration c(x) satis®es the diffusion equation Ds 22 c=2x2 ˆ 0; where Ds is the surface diffusion coef®cient. The direct atomic ¯ux to the island is neglected, as noted above. We make the usual assumption that the attachment kinetics of adatoms to the step are described by the kinetic coef®cients K^, where the two signs correspond to attachment from the up and down terraces: K^ ‰c…xn ^ 0† 2 cn Š ˆ Ds 2c=2xuxn ^0 : The equilibrium adatom concentration at the step cn is related to the chemical potential of the step m n by cn ˆ c0 exp…mn =kT† < c0 …1 1 mn =kT†; where c0 is the equilibrium adatom concentration at an isolated step. The ¯uxes of adatoms to the nth step from the left and the right terraces contribute to the step velocity according to the mass conservation condition vn^ ˆ ^SDs 2c=2xuxˆxn ^0 : Solving the diffusion equation, we obtain the step velocity dxn =dt ˆ vn1 1 vn2 equal to   dxn c SDs mn11 2 mn mn 2 mn21 ; …4† 2 ˆ 0 dt kT xn11 2 xn 1 lK xn 2 xn21 1 lK 21 21 where the length lK ˆ Ds …K1 1 K2 † describes the attachment kinetics. Two limits are the diffusion-limited kinetics, lK ! 0, and the attachment-limited kinetics, lK ! 1. The incoming atomic ¯ux is included in our model implicitly, as a source of adatoms on the ¯at surface between the islands. The ¯ux is assumed to be suf®ciently weak and the Ehrlich±Schwoebel effect is turned off, i.e. the difference between the two kinetic coef®cients K1 and K2 does not affect the step motion. The effects of the elastic step±adatom interaction, which are proportional to the ¯ux [37±39], are also neglected. The equations can be reduced to a dimensionless form by using l0 as a length unit, m0 ˆ 2…1 2 n2 †…hs xx †2 =…pEl0 † as a unit of chemical potential, and t ˆ kTl20 =…c0 SDs m0 † as a time unit. There are three dimensionless parameters in the model: the height of the individual steps h/l0 entering Eq. (2), the attachment kinetics length lK/l0, and the ratio of the dipole components q ˆ Qx =…Q2x 1 Q2z †1=2 : In addition to Eqs. (1)±(4) describing the motion of the existing steps, we need rules describing the nucleation of new steps on the top terrace of the island. The theories of nucleation (see Ref. [40] and references therein) consider adatoms delivered by the incoming ¯ux and do not take into account the equilibrium concentration of adatoms near the steps. Hence, they cannot be applied in our analysis which neglects the ¯ux. We assume that a new atomic island is nucleated on the top terrace as soon as the terrace size exceeds some critical size. In the numerical calculations below, a new terrace 0.2l0 wide is nucleated as soon as the width of the top terrace exceeds 2l0. The results presented below are insensitive to the nucleus size (we varied it from 0.05l0 to l0). Variation of the critical terrace width (from 0.5l0 to 5l0) does not change the results qualitatively but in¯uences the exponents (described below) which increase with an increased critical terrace width. We tried to turn off the nucleation at some stage of the calculations, or

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Fig. 2. (a) Shapes of islands of the same height …N ˆ 55 layers) for different heights of the individual steps h/l0, and time dependencies of the island height (b) and the base width (c).

arti®cially increase the critical terrace width. In these cases, the sides of the island do not change their shape (the distance between steps remains close to l0) but run in opposite directions, leaving in between a ¯at top surface parallel to the substrate. Such behavior is not observed in the experiments. We conclude that the nucleation is fast enough and hence not a limiting process for the problem under consideration. The top terrace of the island grows if the up and down steps forming the top terrace repel each other at short distances. This condition is ful®lled if Qx . Qz : The values of the dipole strengths derived from the atomic models meet this condition. Fig. 1 presents the results of numerical solution of Eqs. (1)±(4). Calculations start with a single pair of steps separated by 0.2l0. As soon as the distance between the steps exceeds 2l0, a new pair separated by 0.2l0 is added. Fig. 1(a) shows the increasing island size during growth. In Fig. 1(b), the curves from Fig. 1(a) are scaled to a common height. As the island grows, it continuously proceeds from a constant slope of the surface to steeper edges and a ¯atter top. This effect is caused by the strain relaxation at the top of the island and strain concentration at the edges. The monopole strength P ˆ h…s xx 1 ds xx † decreases at the island top, which results in increased step distances. Similarly, the strain concentration at the island edges results in smaller step separations. This behavior can be considered as a driving force for the observed transition from ªhutsº or ªpyramidsº with a constant slope of the facets to larger ªdomesº with ¯atter top facets in the Ge/Si system [1±5]. The observed faceting of the islands may be a further decrease of the island energy caused by crystallography which can be achieved by only slight changes in the distances between steps. Both the island height (measured in layers) and the base

width w show a power-law time dependence, N / ta and w / tb : The exponents are not universal, but weakly depend on the parameters used, which are the attachment kinetics length lK, the dipole component ratio q, and the step height h. By varying these parameters, we ®nd that a varies between 0.34 and 0.47 and b between 0.25 and 0.32. This spread of the exponents is illustrated in Fig. 2(b) and (c) for different step heights h. The power law time dependence of the island size is in qualitative agreement with experimental observations [6±9]. The experiments yield smaller values of the exponents, which can be explained by the difference between the 2D model considered here and the 3D islands in the experiments. Fig. 2(a) compares island shapes of the same number of layers for different step heights. The larger the ratio h/l0, the stronger is the correction (2) for the bulk stress at the position of a given step due to all other steps. This means that the pyramid±dome transition takes place at smaller island heights. The faceting of the islands can be considered as a subsequent stage in the reduction of the island energy by a slight rearrangement of the steps with an energy gain by developing crystallographic facets whose orientations are close to these given by the distances between steps. Hence, the orientations of the facets depend on the value of the mis®t, the step height and the strength of the dipole±dipole interaction between steps, as well as orientation of the substrate. The prominent (105) facets of the Ge/Si(001) islands [1] are thus non-universal but related to the dipole strength of steps on Ge as well as to the mis®t of Ge with respect to Si. It also follows from our model that the separate calculation of surface energies of different facets of an unstrained crystal and the elastic energy of the strained island with a

V.M. Kaganer, K.H. Ploog / Solid State Communications 117 (2001) 337±341

subsequent search for a minimum of the total energy [19,20] is not suf®cient to determine the island shapes. In summary, the proposed model gives the following qualitative picture of island growth. The growth proceeds by nucleation and subsequent expansion of the monolayer height islands at the top of the island. The mass transport to the island top is provided by attachment of adatoms to steps on the island surface from the lower terrace with subsequent detachment to the upper terrace, yielding an equilibrium adatom concentration on both sides of the step. The island continuously grows because of the elastic step±step repulsion. The time dependencies of both the island height and its base width follow power laws. The bulk elastic strain produced by the steps gives rise to stress concentration at the island edges and stress relaxation at the island top. The mean distance between steps decreases at the edges and increases at the top, which results in steeper island edges and more gentle slopes towards the top. Subsequent faceting may further reduce the island energy by locking into one or a few crystallographic orientations close to the orientations given by the distances between steps. References [1] Y.-W. Mo, D.E. Savage, B.S. Swartzentruber, M.G. Lagally, Phys. Rev. Lett. 65 (1990) 1020. [2] G. Medeiros-Ribeiro, T.I. Kamins, D.A.A. Ohlberg, R.S. Williams, Phys. Rev. B 58 (1998) 3533. [3] T.I. Kamins, G. Medeiros-Ribeiro, D.A.A. Ohlberg, R.S. Williams, Appl. Phys. A 67 (1998) 727. [4] G. Medeiros-Ribeiro, A.M. Bratkovski, T.I. Kamins, D.A.A. Ohlberg, R.S. Williams, Science 279 (1998) 353. [5] S.A. Chaparro, Y. Zhang, J. Drucker, D. Chandrasekhar, D.J. Smith, J. Appl. Phys. 87 (2000) 2245. [6] F.M. Ross, J. Tersoff, R.M. Tromp, Phys. Rev. Lett. 80 (1998) 984. [7] F.M. Ross, R.M. Tromp, M.C. Reuter, Science 286 (1999) 1931. [8] I. Goldfarb, P.T. Hayden, J.H.G. Owen, G.A.D. Briggs, Phys. Rev. B 56 (1997) 10459. [9] M. KaÈstner, B. VoigtlaÈnder, Phys. Rev. Lett. 82 (1999) 2745. [10] P. Sutter, M.G. Lagally, Phys. Rev. Lett. 84 (2000) 4637. [11] R.M. Tromp, F.M. Ross, M.C. Reuter, Phys. Rev. Lett. 84 (2000) 4641.

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