A Mathematical Model for Epitaxial Semiconductor Crystal Growth from the Vapor Phase on a Masked Substrate

Certain materials used in lasers are made by a process called epitaxial semiconductor crystal growth. In this report a mathematical model is developed for this growth process which occurs on a substrate at the junction between a masked region and exposed substrate in a vapor. This new model consists of two partial differential equations; one for the surface dynamics and one for the crystal growth on the exposed substrate. An analysis of the steady state solutions is furnished. Approximate solutions for timedependent cases are found using two numerical methods. An asymptotic analysis is also carried out to determine transient solution behavior. The undesireable "bump" structure at the mask/substrate junction which has been observed experimentally is present in the solutions found by each method. Report Contributors: Kirk Brattkus (SMU), Pam Cook (University of Delaware), Seongjai Kim (University of Kentucky), and Andrew Roosen (NIST). Final Oral Presentation: Andrew Roosen (NIST). Industry Representative: Michael Mauk (AstroPower, Inc.). Other Project Participants: Gamze Berensel (University of Delaware), Alistair Fitt (University of Southampton), John King (University of Nottingham), Andrew Lacey (HeriotWatt University), Danny Petrasek (CaltechjUCLA), and Colin Please (University of Southampton). Crystals grown on patterned, masked substrates are used in lasers and detectors. The masked substrates may also be used as a diagnostic tool; the crystals grown in artificial geometries may indicate what would be successful or unsuccessful conditions for producing a desired product. The masked substrates may also be used simply for fundamental understanding of the crystal growth process. Figure 1: A schematic of expitaxial growth from a masked substrate. Atoms deposited from the vapor accumulate in or near the unmasked region and result in a growing crystal under appropriate conditions. The outward growth is indicated by the arrows. The growth process that we wish to study is epitaxial crystal growth; epitaxy involves a substrate material such as siliconor other suitable material and the crystal structure and orientation of the substrate is preserved in the growing crystal. The crystal may be grown from particles in a vapor which resides above the surface of the masked substrate. The crystal may also be grown from a very high vacuum process with a beam supplying the material for the crystal; this is molecular beam epitaxy or MBE. The crystal can be grown from the liquid phase as well (liquid phase epitaxy or LPE). We will primarily consider growth from the vapor here. A two-dimensional schematic is shown in Figure 1. A considerable amount ofwork has been done on the problem of an unmasked substrate; some recent reviews include [PV, BS, N]. Typically, for electronic applications, one would like to grow a relatively smooth crystal; this will be the case in this work as well. Relatively little work has been done in the case of a partially masked substrate. The thin mask is, in this case, a silicon dioxide film. A large number of experimental results and associated questions on epitaxial semiconductor crystal growth were brought to the Mathematical Problems in Industry workshop by Michael Mauk of AstroPower, Inc. The working group focused on the specific problem of undesireable crystal growth the formation of a "bump" at junctions between the. mask and exposed substrate, which are at the perimeter of the crystal. The bumps may appear in growth from the liquid or vapor phases, depending on the material system. In LPE, one would certainly expect bulk diffusion in the liquid to be important, and it may contribute in growth from the vapor. Geometric growth models based on the kinetic anisotropy have also been developed for highly anisotropic crystals grown from selectively masked substrates [JSLH]. In these models, the growth rate must be known as a function of orientation; from that empirical data, the Wulff shape may be developed that appears to have some success in predicting facetted crystal shapes. Some impressive computations have been carried out using level set methods for geometric growth models of various situations in microchip fabrication ([Se], both with and without anistropy). Some previous work in this area suggested that bulk diffusion may be a cause of bumps in some situations [BBK]; the diffusion model for vapor phase epitaxy studied there predicted elevated crystal surfaces near the edge of the mask and a thinner region in the middle of the crystal. That behavior is characteristic of the measured profiles found at AstroPower [M]. The cause of the bump appears to be the "point effect" in the diffusion model of [BBK]. In growth from the vapor, there may be some experimental conditions and/or material systems for which surface diffusion or other surface effects may be important [PV,BS]. It was the intuition of Mauk that the competition of bulk and surface diffusion may be relevant in bump formation. The working group focussed on the role of surface diffusion, sublimation and growth. Continuum models of the surface concentration on the mask and of the crystal surface shape were developed for this purpose. Subsequent models offer the possibility of combining these effects with bulk diffusion to study their competition. One must understand the mathematical problem with surface d:iffusionalone before letting them both compete. In this report, a mathematical model consisting of two partial differential equations is formulated for this situation. Both differential equations govern the transport of atoms that will eventually form the growing crystal. One of the equations governs the surface concentration of the atoms on the mask while the other involves the height of the growing crystal on the exposed substrate. Surface diffusion is the main physical phenomenon modeled in the first equation as well as the amount of crystal particles leaving and arriving from the vapor above. The other equation models the motion of the interface between the vapor and the crystal originating above the substrate. The equations and added physical boundary conditions and initial conditions are nondimensionalized using physical constants obtained from the industry representative's experiments. A parameter € which is the quotient of the diffusion on the crystal over the diffusion on the mask is small. This fact is exploited in an asymptotic analysis of a linearized version of the original problem. A transient "bump" is found in the asymptotic solutions. A finite difference method is used to find approximate solutions to the nondimensionalized equations. The numerical solutions from both the linearized and nonlinear equations can also exhibit the transient "bump". The steady state situation is considered and an explicit solution is found. A result of this analysis is the following: a steady state solution exists only if a parameter ~, which is proportional to the surface energy on the crystal divided by the diffusion on the mask, is sufficiently large. An outline of this report is as follows. In Section 2 we formulate the mathematical model,