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APPLIED PHYSICS LETTERS 87, 251908 共2005兲 Self-assembly of steps and vacancy lines during the early stages of Ge/ Si„001… heteroepitaxy D. T. Tambe Division of Engineering, Brown University, Providence, Rhode Island 02912 C. V. Ciobanu Division of Engineering, Colorado School of Mines, Golden, Colorado 80401 V. B. Shenoya兲 Division of Engineering, Brown University, Providence, Rhode Island 02912 共Received 2 August 2005; accepted 21 October 2005; published online 13 December 2005兲 The wetting layer formed during the early stages of Ge/ Si共001兲 growth has been found in recent experiments to undergo a roughening process, where the SA surface steps affect the spatial organization of vacancy lines 共VLs兲 by increasing 共stretching兲 or decreasing 共squeezing兲 their average spacing. Using a combination of atomistic simulations and elastic theory of surface defects, we have computed the interaction energy of the SA steps and VLs for each of the observed defect configurations. We find that the repulsive SA-VL interactions lead to an increase in the spacing of the VLs in the “stretch” arrangement, but do not significantly affect the VL spacing in the “squeeze” configuration, providing an explanation for the observed correlations in the wetting layer roughness. © 2005 American Institute of Physics. 关DOI: 10.1063/1.2147720兴 The strained-layer Ge/ Si共001兲 system shows a remarkable variety of structural and morphological changes depending on growth conditions and Ge coverage. The system undergoes transformations from the 2 ⫻ 1 structure in the absence of Ge, to the 2 ⫻ N reconstructed wetting layer, to the formation of hut and dome quantum dots at Ge coverages larger than about three monolayers 共ML兲. While the formation of pyramid and dome clusters has been actively studied in the last two decades, in comparison, the initial roughening of the wetting layer1–3 that occurs just before the formation pyramids has remained virtually unexplored. The scanning tunneling microscopy 共STM兲 experiments of Sutter et al.3 have recently provided important insights on the initial roughening of the Ge film using microscopy techniques capable of achieving a remarkable degree of detail and image statistics over wide areas of the film surface. Their work shows that the interactions between the SA steps and the vacancy lines 共VLs兲 drive the roughening of the Ge film through the formation of ordered stripe patterns. Motivated by these elegant experiments,3 we have investigated the energetics of the observed patterns, using both atomistic simulations and elastic theory of surface defects. The aim of our calculations is to determine the interactions between the SA steps and VLs and to study the role of these interactions in the spatial ordering of these line defects. Our simulations predict that the repulsive SA-VL interactions decay with an inverse-distance law, in agreement with the behavior expected for lines of monopoles interacting with dipoles through their elastic fields. Furthermore, we find that these interactions modify the mean distance between the VLs, depending on the way the SA steps and VLs are distributed on the surface. The structural models that correspond to the stretch and the squeeze arrangements are illustrated in Fig. 1. In the a兲 Author to whom correspondence should be addressed; electronic mail: shenoyv@engin.brown.edu former case, an island of width d bounded by SA steps is sandwiched between VLs, which leads to an increase 共or stretching兲 in the mean spacing of the VLs. In contrast, the two VLs sandwiched between the similar islands are observed to be “squeezed” by the SA steps in the latter case. Using the Tersoff potential for Si–Ge,4 we have performed relaxations for 90-layer-thick computational cells with 2-ML or 3-ML Ge coverage5 and for periods D = Na 共2 ⬍ N ⬍ 40兲, where a = 3.84 Å is the lattice constant of the Si共001兲 surface. From the total energy of the slab E, we compute the surface energy ␥ = 共E − nSiSi − nGeGe兲 / A as the excess energy per area A = 2aD, where nGe, Ge 共nSi , Si兲 are the number of atoms and the chemical potential of Ge 共Si兲, respectively. The overall surface energy ␥ can be written as ␥ = ␥u D−d d + ␥l + , D D D 共1兲 where the surface energy ␥u 共␥l兲 of the upper 共lower兲 terrace can be obtained from separate calculations for defect-free Ge/ Si共001兲 surfaces and 共subsequently referred to as a or b for the two cases in Fig. 1兲 includes the formation energies of SA steps and VLs and their interactions. Before presenting the results for the interactions between SA steps and VLs, it is useful to separately consider the energetics of periodic arrays of each of these line defects. Since the VLs can be described as elastic-force dipoles in the plane of the surface, they interact with each other with a repulsive inverse-square distance law.6 The energy per unit length of an array of VLs with the spacing D can then be written as 0 1 + ⌳VL 2/共6D2兲, VL共D兲 = ⌳VL 共2兲 0 ⌳VL is the formation energy of a VL and the second where term is the interaction energy of the uniform array, where 1 is the strength of the dipolar interaction. The up- and ⌳VL down-SA steps on the sides of an island, on the other hand, behave like force monopoles7 due to the discontinuity of surface stress at the island edges.8,9 Following the notations 0003-6951/2005/87共25兲/251908/3/$22.50 87, 251908-1 © 2005 American Institute of Physics Downloaded 08 Feb 2006 to 138.67.11.69. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 251908-2 Appl. Phys. Lett. 87, 251908 共2005兲 Tambe, Ciobanu, and Shenoy FIG. 1. 共Color online兲 Surface unit cells for the stretch 共a兲 and squeeze 共b兲 configurations reported in Ref. 3. The Ge film is shown in black, and the Si substrate in gray. The island size is d and the distance between an SA step and a VL is p. Configuration 共b兲 requires another parameter q to denote the distance between the two VLs of the unit cell. The periods D along the 关110兴 direction are D = 2p + d and D = 2p + d + q for models 共a兲 and 共b兲, respectively. The periodic length of the cells in the 关1̄10兴 direction is 2a 共a = 3.84 Å兲. The horizontal arrows schematically show the force monopoles and dipoles at the steps and VLs, respectively. in Eq. 共2兲, the energy of a periodic array of such islands of width d and period D is given by9,10 冋 冉 冊册 SA共d,D兲 = 2⌳S0 − ⌳S1 log A A D d sin c D 共3兲 , where c = a / 2 is an atomic scale cutoff introduced to regularize the logarithmic divergence.11 The parameters ⌳0 and ⌳1 in Eqs. 共2兲 and 共3兲 obtained from atomistic simulations as described in Refs. 6 and 10 are given in Table I. We now consider the stretch configuration 关Fig. 1共a兲兴, for which the defect energy can be written as int a共d,D兲 = VL + SA + a,S A-VL 共4兲 , where VL and SA are given by Eqs. 共2兲 and 共3兲, respectively, and the third term denotes the interaction between the monopoles at the SA steps and the dipoles at the VLs. We have calculated this term from atomistic relaxations of the stretch structure 关Fig. 1共a兲兴 and plotted it in Fig. 2共a兲 as a function of the VL spacing D, for fixed island width d = 4a. In order to extract the strength of the SA-VL interaction from this data, we note that according to elasticity theory, a single SA step interacts with a single VL with a power law that decays as the inverse of the distance between them. Using ⌳S1 -VL to denote the strength of this monopole-dipole interA action, the contributions corresponding to the steps and VLs in the unit cell of the stretch configuration and its periodic images can be summed analytically to int a,S A-VL = ⌳S1 A-VL 冉 冊 2 d tan . D 2D 共5兲 the SA-VL interaction is accurately modeled using the elastic theory of surface defects. In the case of squeeze configurations 关Fig. 1共b兲兴, the defect energy can be written as int int 0 b共d,q,D兲 = 2⌳VL + SA + b,VL + b,S A-VL 共6兲 , where SA is given by Eq. 共3兲, and the second and third terms correspond to the VL-VL and SA-VL interactions, respectively. Since there are two 共one兲 VLs in the unit cell of the squeeze 共stretch兲 configuration, these terms are different for the two cases. We have computed the last two terms in Eq. 共6兲 analytically by summing the series of dipole-dipole and monopole-dipole contributions: int 1 = ⌳VL b,VL int b,S A-VL 冋 冉 冊册 冋 冉 冊 冉 2 2 q 2 1 + 3 csc 3D D = ⌳S1 A-VL 共7兲 , 2 共p + d兲 p cot − cot D D D 冊册 . 共8兲 Since the strengths of all defect interactions have already been determined, each term in Eq. 共6兲 can be evaluated without the need for atomistic relaxations of the squeeze structures. We have, however, performed these calculations to further confirm that the parameters in Table I accurately describe the SA-VL interactions, as shown in Fig. 2共b兲. With the information on the defect formation and interaction energies, we can now discuss their spatial ordering in the wetting layer. In the absence of SA-bounded islands, the It can be seen from Fig. 2共a兲 that this functional form provides an excellent fit to the results of the simulations. Furthermore, the coefficient ⌳S1 -VL = 254.7 meV obtained from A the fitting procedure is very close to the geometric average 1 ⌳VL ⌳S1 = 247.4 meV that is expected12 for the interaction A of a monopole with a dipole. These observations confirm that 冑 TABLE I. Defect formation energies ⌳0 and interaction strength parameters ⌳1 computed using the Tersoff interatomic potential 共see Ref. 4兲. FIG. 2. 共Color online兲 Interaction energy of SA steps and VLs obtained from atomistic simulations in the stretch configuration 关共a兲, triangles up兴 and in the squeeze pattern 关共b兲, triangles down兴, along with the fitting curves given VL −133.934± 0.201 13434± 89 meV Å by Eq. 共5兲 and Eq. 共8兲, respectively. The dashed curves show the interaction −1.56± 0.11 4.556± 0.076 meV/ Å SA energy between VLs 关refer to Eqs. 共2兲 and 共7兲兴 in the two configurations. The island size is d = 4a in both cases, and the SA-VL distance in case 共b兲 is SA-VL 254.677± 1.077 meV p = 2.5a. Downloaded 08 Feb 2006 to 138.67.11.69. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp Defects ⌳0 共meV/Å兲 ⌳1 251908-3 Appl. Phys. Lett. 87, 251908 共2005兲 Tambe, Ciobanu, and Shenoy FIG. 3. 共Color online兲 Surface energy contributions a / D 共at d = 4a兲 and b / D 共at d = 4a, p = 2.5a兲 corresponding to the arrays of line defects shown in Figs. 1共a兲 and 1共b兲, respectively. The case of free VLs 共i.e., without steps兲 is also plotted for comparison. In the stretch and free configurations, the VL-VL distance is simply the period along 关110兴, D. For the squeeze case, the VL-VL separation is q, as shown in Fig. 1共b兲. The curves correspond to the analytical expressions given in Eqs. 共2兲, 共4兲, and 共6兲, while the data points are calculated from atomistic simulations. The optimum VL separations determined from STM experiments 共expt.兲 共see Ref. 3兲 are shown along with error estimates. spacing between VLs is determined by minimizing the defect energy per period of the defect array, which is VL / D 关refer to Eq. 共2兲兴. As shown in Ref. 6, a competition between the negative formation energy of VLs and their repulsive interactions leads to a mean spacing D = 6a, which compares favorably with the experimental value of 共8 ± 0.9兲a.3 To see how this spacing is altered in the two structures in Fig. 1, we have plotted a,b / D 关from Eq. 共4兲 and Eq. 共6兲兴 as functions of the spacing between the VLs in Fig. 3, choosing a typical island width of d = 4a. While the island width d and the VL-VL separation D determine the SA-VL spacing 共共D − d兲 / 2兲 in the stretch configuration, this spacing 共p兲 is an independent parameter for the squeeze structures. We have found, however, that the defect energy per period b / D is minimum at the SA-VL separation of p = 2.5a for island widths in the range 2a 艋 d 艋 8a. This result is in agreement with the observations of Sutter et al., who find that SA-VL spacings of 1.5a and 2.5a occur much more frequently than any other SA-VL separation. In what follows, we discuss the variation of the defect energy / D as a function of VL-VL distance. A comparative analysis of the VL-VL spacing between stretch, free and squeeze configurations reveals a remarkable consistency with experimental findings. As seen in Fig. 3, the optimum VL-VL distance in the stretch case is D = 10a, which is larger than the spacing of free VLs 共Ref. 6兲 by four Si共001兲 lattice constants. A similar increase in the VL spacing has been observed by Sutter et al., who report separations of 共8.0± 0.9兲a and 共11.9± 1.7兲a for the free and stretch configurations, respectively. For squeeze structures with d = 4a and p = 2.5a, we find nearly the same optimum VL-VL distance 共q ⯝ 6a兲 as in the case of free VLs. This finding is also consistent with the STM observations,3 which show only small change in the VL-VL distance from 共8.0± 0.9兲a 共free兲 to 共7.1± 0.9兲a 共squeeze兲. The close agreement of the computational and analytical results with experiments suggests that the empirical potential4 used in our work is able to capture the key features of the interactions of line defects on Ge/ Si共001兲. The pronounced difference between the statistics of VL separations in the stretch and squeeze configurations3 can be understood by considering the relative contributions of the monopole-dipole and dipole-dipole interactions to the surface energy of each configuration. In the stretch case, the SA-VL 共monopole-dipole兲 interactions are stronger than the VL-VL interactions, as seen in Fig. 2共a兲. Therefore, the optimum spacing of 6a determined for a uniform array of VLs 共see Ref. 6兲 will be increased significantly by the presence of islands between the VLs 关Fig. 1共a兲兴. In the squeeze case 关Fig. 2共b兲兴, since the islands are located between pairs of VLs, the VL-VL repulsion dominates the SA-VL interaction. Indeed, the monopole-dipole terms are found to have little influence on the VL-VL distance in the squeeze configuration. In summary, we have studied the energetics of SA-VL stretch and squeeze patterns using a linear elasticity based approach combined with atomistic simulations. Our study demonstrates the existence of an effective repulsion between steps and VLs, which has been identified as a key factor in the initial roughening of the Ge/ Si共001兲 system. Our analysis also predicts optimal separations for the VLs, in agreement with recent microscopy experiments.3 While the experimental distributions of VL spacings contain information on thermal fluctuations, these effects are not included in the present work, but will be addressed in the future. The research support of the NSF through Grant Nos. CMS-0093714 and CMS-0210095 and the Brown University MRSEC program 共DMR-0079964兲 is gratefully acknowledged. P. Sutter and M. G. Lagally, Phys. Rev. Lett. 84, 4637 共2000兲. R. M. Tromp, F. M. Ross, and M. C. Reuter, Phys. Rev. Lett. 84, 4640 共2000兲. 3 P. Sutter, I. Schick, W. Ernst, and E. Sutter, Phys. Rev. Lett. 91, 176102 共2003兲. 4 J. Tersoff, Phys. Rev. B 39, 5566 共1989兲. 5 The two coverages used in the study 共2 ML and 3 ML兲 lead to very similar conclusions; we therefore report only the results obtained for the 3-ML case. 6 C. V. Ciobanu, D. T. Tambe, and V. B. Shenoy, Surf. Sci. 556, 171 共2004兲. 7 In general, surface steps also have dipole moments. For the SA steps, the dipole strength is negligible due to the absence of rebonding at the stepedge 共see Ref. 10兲. 8 V. I. Marchenko, Sov. Phys. JETP 54, 605 共1981兲. 9 O. L. Alerhand, D. Vanderbilt, R. D. Meade, and J. D. Joannopoulos, Phys. Rev. Lett. 61, 1973 共1988兲. 10 T. W. Poon, S. Yip, P. S. Ho, and F. F. Abraham, Phys. Rev. B 45, 3521 共1992兲. 11 The cutoff radius c is of the order of a, but otherwise arbitrary. Equation 共3兲 shows that the choice of c affects the formation energy ⌳S0 but does A not change the interaction coefficient ⌳S1 . A 12 Denoting the monopole 共dipole兲 strength by Ax 共Axx兲, the interaction co2 1 = ␣Axx , ⌳S1 = ␣Ax2, ⌳S1 -VL = ␣AxAxx, efficients ⌳1 can be written as ⌳VL A A where ␣ is related to the elastic moduli of the substrate 共see Ref. 8兲. It then 1 1 1 follows that ⌳S -VL = ⌳S ⌳VL. 1 2 A 冑 A Downloaded 08 Feb 2006 to 138.67.11.69. 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