Journal ArticleA model is presented for growth by so-called composition pulling, wherein an epitaxial deposit grows coherently but with a composition different from that which would be in bulk equilibrium with the liquid phase from which growth occurs. The breakdown of coherent growth occurs when a dislocation nucleates at a ledge at the growing solid-liquid interface. An expression for the critical condition for breakdown is presented

Hirth, J.P.

Stringfellow, Gerald B.

English

The University of Utah: J. Willard Marriott Digital Library

Downloaded 10 Oct 2007 to 155.97.12.90. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspA mechanism for liquid-phase epitaxial growth of nonequilibrium compositions producing a coherent interface J. P. Hirth Metallurgical Engineering Department, Ohio State University, Columbus, Ohio 43210 G. B. Stringfellow Solid State Laboratory, Hewlett-Packard Company, Palo Alto, California 94304 (Received 4 October 1976; accepted for publication 7 January 1977) A model is presented for growth by so-called composition pulling, wherein an epitaxial deposit grows coherently but with a composition different from that which would be in bulk equilibrium with the liquid phase from which growth occurs. The breakdown of coherent growth occurs when a dislocation nucleates at a ledge at the growing solid-liquid interface. An expression for the critical condition for breakdown is presented. PACS numbers: 68.SS.+b, 64.70.Dv, 61.50.Cj In a study of growth of Gax1n1_xP by LPE on a GaAs substrate, Stringfellow1 observed that growth from liquids with a range of compositions produced the same solid composition. Liquids within about ± 10% of the zero-Iattice-parameter composition yielded solid with x = 0.51, the zero-Iattice-parameter mismatch com-position. Other reports have confirmed this effect in the Ga-In-P system2,3 and in the Ga-As-Sb and In-As-Sb systems, 4 as well as in such dissimilar systems as magnetic garnet materials grown from a flux. 5 How-ever, in the quaternary system Ga-In-As-P, the effect is not observed. 0,7 Stringfellow1 attempted to calculate the magnitude of the effect by computing the free energy of the nearly lattice- matched overgrowth, determining the minimum free-energy composition including strain-energy terms and chemical terms. Reduction in the strain-energy contribution was found to offset the increase in the chemical contribution in this model. The observed vari-ation of nearly lattice-matching composition with liquid composition was found to agree very well with the model. The ultimate breakdown to a dislocation interface was also rationalized by a free-energy argument. For large enough mismatch, the free energy of a dislocated over-growth of equilibrium composition was computed to be less than the lattice-matching overgrowth. Again, the computed transition composition agreed well with the computations. However, the dislocation energy was probably overestimated because of neglect of image stresses, so the latter part of the correlation is in question. Here, we suggest a kinetic model for the ultimate breakdown, which nevertheless retains the thermody-namic portion of the correlation when lattice matching occurs. In the depOSition of a one-component epitaxial over-growth on a substrate with near lattice matching, the initial nuclei are coherent because the attendant reduc-tion in surface energy offsets the strain energy in the nucleus. 8-10 As the nuclei grow they can become semi-coherent in several ways. For lateral growth, nuclea-tion of an interface dislocation at the periphery of the 1813 Journal of Applied Physics, Vol. 48, No.5, May 1977 nucleus becomes favorable for large particles, and dis-locations are injected, reducing the strain energy and creating lattice mismatch. 8,9 For thick overgrowth, dis-locations can also be injected from the free surface. 11 However, image forces prevent this injection below a critical thickness of the order of 1-2 nm. 12 For growth from a multicomponent system, such as Ga-In-P, an alternative model is possible. Again the nucleus should be coherent. Now as growth occurs, how-ever, rather than accumulating elastic strain energy, the particle can initially grow with nearly lattice-match-ing composition as suggested by Stringfellow. 1 This form of growth greatly suppresses the probability of dislocation nucleation and permits growth of macro-scopic coherent la1tice-matching overgrowths. The ultimate breakdown in such coherent overgrowth is then supposed to be dislocation nucleation controlled in a ledge-type growth modeL 13 The observations re-ported by Strauss7 for growth of Ga-In-As-P, indicating more rapid growth of (100) surfaces than for (111) sur-faces having the same temperature dependence of growth, strongly suggest such a ledge-type model. As atoms attach at kink sites on ledges, the strain-chemi-cal free-energy model applies locally and atoms are selected to reduce local strain buildup of the type shown in Fig. 1. With sufficient chemical driving force, the nucleation of a dislocation loop, as shown in Fig. 2, will eventually lead to breakdown of the coherent interface. Conventional nucleation theory10 indicates that the free energy of such a configuration is (1) FIG. 1. Atom being added at a ledge. A: unstrained configura-tion; B: locally strained configuration. Copyright © 1977 American Institute of Physics 1813 Downloaded 10 Oct 2007 to 155.97.12.90. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspbl I ~', ! ~Dislocotion I I I I I I J I I I I j Dislocation Kink FIG. 2. Dislocation being created at a ledge: (a) cross-sec-tional view, (b) view normal to solid-liquid interface. where t.G is the negative chemical free-energy differ-ence per atom between solid of equilibrium bulk com-position and solid of lattice-matching composition, 1 is defined in Fig" 2, b is the lattice spacing, h /lb 2 is the roughly core-type energy per unit length of such a dis-location, and /lb 4ll is the interaction energy between the dislocation kinks at the ends of the length 10 Minimiza-tion of the free energy with respect to l gives a critical length ( /lbs ) 1/2 l* = h /lb3 + t.G and a critical free energy t.G * = 2(Jlb3)1/2(fu JlIJ3 + t.G)1/2. Theory predicts nucleation when t.G* = J? T In(nv I J), (2) (3) (4) where n = 2.5 X 109 m-1 is the number of sites per meter of ledge, v = 1013 S-l is the atomic vibration frequency (the attempt frequency for nucleation of a loop), and J -102 m-1 S-l is the critical observable nucleation rate required for nucleation. Substituting Eq. (4) into Eq. (3) gives a prediction of t.G required for nucleation as a function of known parameters. For Ga-As-P grown on Ga-As, with the use of the value Jl = 3. 39x 1010 Pa from the work of Hakki et al., 14 b = O. 4 nm, and T = 1060 K, this procedure yields a value of t.G of - 97.9 kJ/mole. In terms of the computations of Stringfellow1 for this system, this corresponds to the free-energy difference between solids with x = 0.31 or 0.71 and the 1814 J. Appl. Phys., Vol. 48, No.5, May 1977 lattice- matching solid with x c"-' 0.51. These predicted values of t.x = 0.20 at which the epitaxial layer growing on a Ga-As substrate should become noncoherent com-pare with an experimentally observed value of t. \ 0.11. Thus, there is only fair agreement between expec-tation and experience, but it is reasonable in view of the approximations in the above estimates. The domi--nant factor in the predicted values is the core energy: a value of f.r JlIJ3 (much smaller because of the nonlinear relation between t.G and t.x) would be required for this parameter to give perfect agreement. Core energies this small are possible, but to resolve the parameter more accurately than above, atomic calculations, in-cluding surface-image effects, would be rquired for the nonlinear elastic core-type configuration shown in Fig. 2. ACKNOWLEDGMENT This research was supported by the Advanced Re-search Projects Agency under contract DAHC 15-71-C-0253 with the University of Michigan. lG.B. Stringfellow, J. Appl. Phys. 43, :3455 (1972). 2W.R. Hitchens, N. Holonyak, Jr., M.H. Lee, and J. C. Campbell, J. Cryst. Growth 27, 1H (1974). :lII. Beneking, N. Grote, P. Mischel, and G. Schul, in GaAs and Related Compounds (Institute of Physics and the Physical Society, London, 1975), p. 11:1. lJ. S. HarriS, Report of the ARPA Materials Research Council (University of Michigan, Ann Arbor, Mich., 1977). 'D.C. Miller andR. Caruso, J. Cryst. Growth 27,274 (1974). GG. Antypas (private communication). lA. Strauss, in Ref. 4. 8N. Cabrera, Surf. Sci. 2, 230 (1964): Mem. Sci. Rev. Metall. 62, 205 (1965). 9yV.A. Jesser and D.Kuhlmann-Wilsdorf, Phys. Status Solidi 19, 95 (1967). IOJ.P. Hirth and G.M. Pound, Condensation and Evaporatioll (Pergamon, New York, 196:1). l1J. W. Matthews and J. L. Crawford, Thin Solid Films 5, 187 (1970). 12K. Shinohara and J. P. Hirth, Philos. Mag. 18, 88:1 (197:1). 13W.K. Burton, N • Cabrera, and F.C. Frank, Philos. Trans. R. Soc. London A 243, 299 (1951). 14B.L. Hakki, A. Jayaraman, and C.K. Kim, J. Appl. Phys. 39, 5291 (1970). J.P. Hirth and G.B. Stringfellow 1814 

Mechanism for liquid phase epitaxial growth of nonequilibrium compositions producing a coherent interface

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Mechanism for liquid phase epitaxial growth of nonequilibrium compositions producing a coherent interface

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