Transactions on Modelling and Simulation vol 3, 1993 WIT Press, ISSN X

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1 Morphological stability of thin films L.J. Gray," M.F. Chisholm/ T. Kaplan* " Engineering Physics and Mathematics Division, ^ Solid State Division, Oak Ridge National Laboratory, ABSTRACT The boundary element method for elastostatics is applied to a thin film stability problem arising in solid state surface science. An aim of this work is to determine the morphology of germanium deposited on a silicon substrate. Nonstandard boundary conditions at the material interface are used to model the epitaxially grown film. In addition to determining the deformed geometry it is also necessary to compute the surface stress tensor. Although the surface displacement at the junction between the interface and the silicon free surface is not differentiate, the hypersingular integral equation for surface stress can still be employed. These techniques will be described, along with results from two-dimensional calculations. INTRODUCTION Solids with technologically important physical properties can be produced by a process of growing a thin layer of one crystalline material onto a substrate of a second, e.g., germanium (Ge) deposited on silicon (Si). When the growing layer adopts a specific orientation relative to the substrate, the growth is called epitaxial. In the Ge/Si system, as in most other applications, there is a mismatch in the lattice spacing of the two crystals - in this case the atomic distance in Ge is 4% larger than Si - and thus epitaxial growth results in a strained material.

2 182 Boundary Element Technology While technologically important, strained epitaxial layers are not stable structures. As a result, the relaxation of strain in these layers has been the subject of much experimental and theoretical research. Nearly all of this work has examined the incorporation of dislocations to relieve the misfit between the substrate and the thin film. However, strain can also activate transport mechanisms to shift material in response to created differences in chemical potential. Although Gibbs demonstrated that classical macroscopic concepts can be used to predict shapes which minimize the free energy of the system/ acceptance of the fact that strained planar films can undergo morphological transitions to relieve strain is a relatively recent development.^ The transport mechanism involved in the development of morphological instabilities is driven by the reduction of the systems strain energy, and suppressed by increases in the surface area. In this paper, preliminary work on developing boundary integral tools for calculating the elastic state of the entire film/substrate system is described. These techniques can be used to examine the effect of various film shapes and feature densities on the stress state of the system. These results, when coupled to the equation for the chemical potential of the solid surface, will allow the computational simulation of the morphological evolution of a strained film during growth and service. BOUNDARY INTEGRAL MODELING For two-dimensional plane strain elastostatics, the boundary integral equation relating surface displacement and traction, u(q) and t(q), Q F, is given by^ u,-(p) + / T,,(P, Q)uk(Q) dq = / Utk(P, Q)tk(Q) dq. (1) Jr Jr The summation convention is assumed, and for isotropic materials the tensors Uik and Tik are from the Kelvin solution for a point load at P, Uik(P,Q) = [-(3-4^.<,log(r)+r,r,,] (2). (3) The notation employed is defined by r = \\Q P\\, rj denotes partial differentiation with respect to the j^ coordinate of Q, G is the shear modulus and y is Poisson's ratio. Eq. (1) is strictly valid for P inside the domain P T). However, by defining the singular integrals in terms of a limit to the boundary, ^ this equation remains valid for P F. The 'interior angle' term that usually

3 Boundary Element Technology 183 Cn Ci2 C44 O " Ge x 10" 4.83x10" 6.68 x 10*i x 10" Si x 10" 6.39x10" 7.96x10" x 10" Table 1: Elastic constants (dyn/cm^) for silicon and germanium appears in Eq. (1) for P G F is automatically calculated in the singular integral evaluation. Crystalline silicon and germanium are not isotropic materials, but for these preliminary studies, it is convenient to employ this approximation. The effects of the crystal symmetry have been somewhat taken into account by using 'Voight averaged' elastic constants. The three constants C\\, Ci2, and C^ for a cubic crystal are combined to approximate the isotropic parameters G and v by " 2(A A = C\2 H 2C<4 4- C\ (4) The values of the constants, with the exception of Poisson's ratio in units of dyn/cm\ are listed in Table 1. Boundary Conditions The domain V will consist of a large silicon substrate and a relatively thin layer of germanium. Two types of geometries will be considered, the basic configurations being shown in Figures 1 and 2. It should be emphasized that these geometries are highly simplified models of actual thin film configurations. The two models represent, respectively, growth with separate island formation, and layered*growth with island formation. With the exception of the interface between the two materials, the imposed boundary conditions are standard. For the first geometry, the bottom surface of the silicon is fixed, u = 0, while all other boundary segments, excluding the interface, are free surfaces, t = 0. In the layeded growth geometry, the only difference is that the constraints invoked on the sides of the model are designed to mimic 'periodic' boundary conditions. Thus, for the silicon, the boundary conditions on the sides are u^ = 0, ty = 0. For the germanium, ty = 0 and u^ = T, where x is the x-coordinate of

4 184 Boundary Element Technology the boundary point. This last condition is a consequence of the 4% in the lattice constants. mismatch - Ge - Si Figure 1: Skematic thin film geometry for isolated island growth. In elastostatic calculations involving two or more different materials, the boundary conditions are generally equal displacements and equal and opposite tractions at coincident nodes along a material interface. At the Si/Ge interface however, the boundary conditions must try to mimic the epitaxial growth of the germanium layer, taking into account that the lattice constant for germanium is 4% larger than silicon. This has been accomplished by building the difference in the lattice spacing into the discretization. Thus, if P^ = (^,0) is a germanium interface node, the corresponding silicon node is initially located at P (0.96a;6,0) (note the extra length of the germanium layer in Fig. 2). The first of the two equations required at the interface then states that the tractions at corresponding nodes must balance, t(p^) = t(p ). The second models the epitaxial growth, requiring that P% and P% end up at the same location, (5) Surface Stress After determining the deformed geometry, it is also necessary to calculate the surface stress tensor. At present, this quantity is used to determine which

5 Boundary Element Technology 185 Si Figure 2: Skematic thin film model for layered growth with island formation. geometries are energetically more favorable, but eventually it will be employed in a dynamical simulation of the surface growth process. The boundary integral form for the (interior and surface) stress tensor is obtained by differentiating Eq. (1), resulting in + I Sijk(P,Q)uk(Q) dq= Wijk(P,Q)tk(Q) J i J I (6) (formulas for the kernel functions can be found in the thesis by Lutz^). Evaluating the stress for an interior point P (not too close to the boundary) presents no difficulties, but for P 6 F, the kernel Sijk is hypersingular. Although indirect methods for computing surface stress which successfully bypass the hypersingular integral have been proposed,^ a direct evaluation of the hypersingular integral is possible. The definition to be adopted is the 'limit to the boundary'/3'^ wherein Eq. (6) is first evaluated for an interior point and then the limiting value as this point approaches the boundary is computed. This method for computing surface stress is computationally simpler than an indirect evaluation and, moreover, is readily adaptable to the special circumstances arising in this thin film problem. For the geometry shown in Fig. 1, note that the stress is not continuous (equivalently, the displacement is not differentiate) at the juncture between the free Si surface and the Ge/Si interface. The traction is zero approaching this point along the free surface, while this is not necessarily the case approaching

6 186 Boundary Element Technology along the interface. The question then arises as to whether Eq. (6) can be used to calculate the two values of the stress tensor at this point, especially considering that for a continuous stress, the differentiability of the displacement is relied upon to cancel potential singular terms in the hypersingular integral.^'^ Ge/Si Interface Island model Initial Deformed Figure 3: Deformed Ge/Si interface for thin rectangular film. The discontinuous stress can be handled within the limit technique by suitably defining the approach to the boundary. The idea is similar to that first employed for treating a boundary corner,^ and it will therefore only be briefly indicated here; further details will be published elsewhere. Let PQ denote the point of discontinuity of the stress, <r'(po) and cr^(fo) the values of the stress approaching PQ from the left and right along the boundary. To compute a\po), evaluate the stress at the boundary point PS, where /b PS\\ = $, and PS is to the left of PQ. For small 8, PS will be interior to an element, and thus the approximation to the displacement will be differentiable, and all potentially singular terms (as the interior point approaches P$) vanish. Next, the limit 6» 0 is considered. Singular terms of the form log(<5) will occur, but it can be shown that these terms necessarily cancel. The second 'one-sided limit' ^(PQ) can be computed in a similar fashion.

7 CALCULATIONS Boundary Element Technology 187 The boundary integral equation has been reduced to a finite system of linear equations by employing the isoparametric Overhauser spline interpolation. *~ The advantage of this approximation is that it produces a C* displacement function, and is therefore suitable for evaluating the surface stress. 1.0 Deformed Ge/Si Interface Layered model Initial \ Deformed \ X Figure 4: Deformed Ge/Si interface. The preliminary test calculations employed the two geometries shown in Figs. 1 and 2. The deformed top surface of the silicon substrate for the 'isolated island' model is shown in Fig. 3 (the units are Angstroms A). First note, as discussed above, the non-differentiable displacement at the junction of the free surface and the interface (x ±50/1). However, of primary interest in this result is that the maximum deflection of the silicon substrate is approximately 0.3A. As the interlayer spacing in silicon is 1.36/1, these values represent a significant distortion of the substrate, and would indicate that it is essential to include the substrate in the modeling of these systems/ In the 'layered' growth model, the germanium film consisted of a 20/1 thick flat base, followed by a sinewave with amplitude 20/i. and frequency 2?r/300A. As indicated in Fig. 2, the length of the germanium layer is 4% larger than the silicon substrate, and the depth of the substrate was 2000/1. Fig. 4 displays

8 188 Boundary Element Technology the deformation of the Ge/Si interface. In this case, the displacement into the substrate is an even more significant fraction of the interlayer spacing. Fig. 5 plots the initial and deformed top surface of the germanium layer. CONCLUSIONS Preliminary efforts to develop a boundary integral simulation of thin film growth have been described. Appropriate boundary conditions for modeling epitaxial growth have been developed, allowing the substrate to be included in a natural fashion. The initial calculations indicate that the substrate strain is in fact a significant component in the system, and should therefore be included in simulations of surface growth. so.o Germanium Surface Layered model > Initial Deformed -2OO.O -1OO.O O.O 1OO.O 2OO. X Figure 5: Initial sinewave and deformed Ge top surface. Future efforts will be directed towards modeling the growth of the film, taking into account the state of strain at the surface. Boundary integral methods will be highly advantageous for this work, as the surface stress can be accurately computed. Moreover, the required iterative remeshing of the evolving geometry is clearly a much simpler task when dealing only with the boundary.

9 ACKNOWLEDGMENTS Boundary Element Technology 189 This work was supported by the Applied Mathematical Sciences Subprogram, and the Division of Materials Science, Office of Basic Energy Sciences, U.S. Department of Energy, under contract DE-AC05-840R21400 with Martin Marietta Energy Systems Inc. REFERENCES 1. Gibbs, J. W 'On the Equilibrium of Heterogeneous Substances' In Collected Works, pp , Longmans, Green & Co., New York, Bruinsma, R. and Zangwill, A. 'Morphological Transitions in Solid Epitaxial Overlayers' Europhysics Letters, Vol. 4, pp , Gilmer, G.H. and Grabow M.H. 'Equilibrium and Metastable Limits to the Thickness of Coherent Epitaxial Films' J. Metals, Vol. 39, pp. 19, Chisholm, M.F. and Smith, D.A. 'Low Energy Interfaces in Thin Film Structures' Cryst. Prop. Prep., Vol. 16, pp , Srolovitz, DJ. 'On the Stability of Surfaces of Stressed Solids' Ada Metall., Vol. 37, pp , Snyder, C.W., Orr, B.C., Kessler, D. and Sander, L.M. 'Effect of Strain on Surface Morphology in Highly Strained InGaAs Films' Phys. Rev. Lett., Vol. 66, pp , Spencer, B.J., Voorhees, P.W. and Davis, S.H. 'Morphological Instability in Epitaxially Dislocation-free Solid Films' Phys. Rev. Lett., Vol. 67, pp , Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C. Boundary Element Techniques. Springer-Verlag, Berlin and New York, Lutz, E.D. and Gray, L.J. 'Exact Evaluation of Singular Boundary Integrals without CPV Communications in Applied Numerical Methods, (in press). 10. Eaglesham, D.J. and Cerullo, M. 'Dislocation-free Stranski-Krastanow Growth of Ge on Si(lOO)' Phys. Rev. Lett., Vol. 64, pp , Lutz, E.D. Singular and Nearly Singular Integrals. PhD thesis, Cornell University, Okada, H., Rajiyah, H. and Atluri, S.N. 'A Novel Displacement Gradi-

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