Pyramid Formation and Coarsening During Homoepitaxial Growth
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ABSTRACT One of the generic scenarios in molecular beam epitaxy is the growth of large scale pyramid-like three-dimensional structures. These structures are a consequence of nonequilibrium surface diffusion currents that result from a transport of adatoms in the uphill direction in regions of the surfaces that are tilted away from a high symmetry orientation. The microscopic origin of these currents are, e.g., diffusion barriers at step edges that suppress the diffusion of adatoms to lower terraces. The temporal evolution of the resulting instabilities can be described by simple continuum equations that display slope-selection, pyramid-like structures, and coarsening. We show that similar scenarios can be found in microscopic models investigated by Monte-Carlo simulations. However, crossover phenomena can complicate the comparison of the asymptotic theory with computer simulations or experiments. We therefore discuss criteria that allow to distinguish experimentally between kinetic roughening with exponents ( c 1 and unstable pyramid-like growth. INTRODUCTION In the last few years it has become clear that the surface evolution of films grown by molecular beam epitaxy is quite often characterized by the formation pyramids, even in the case of homoepitaxial growth. In fact, theory predicts [1,2] that there are only two generic scenarios that describe the late stages of the growth process, corresponding to large film thicknesses: either unstable three-dimensional growth or kinetic surface roughening described by the Edwards-Wilkinson [3] equation leading to only logarithmically rough surfaces. Hence, kinetic surface roughening with roughness exponents larger than zero is not expected to occur as long as evaporation from the surface can be neglected. The three-dimensional growth mode was first predicted by Villain [4] and later seen in computer simulations of a simple solid-on-solid (SOS) model [5]. The temporal evolution of the instability proceeds through a coarsening mechanism that has been observed in numerical simulations [6-8,10] as well as in experiments [9,6]: the characteristic lateral dimension R of the three dimensional features grows with a power law R(t) _ tn with n somewhere in the range 0.16 to 0.26. The slope of the pyramids or mounds remains essentially constant after an initial transient. These observations can all be explained in terms of Langevin equations of the type ,9th(x, t) = -VT. j[V~h(x, t)] + F + 71(x, t)
(1)
where h(x, t) describes the surface height at substrate position x and time t, F is the deposition flux, and r is the shot noise modeling beam fluctuations. The diffusion current consists of two parts, j = jeq + jb- The first part, 251 Mat. Res. Soc. Symp. Proc. Vol. 399 a 1996 Materials Research Society
neq =
DeqVV 2h(x,t)
(2)
is also present in an equilibrium situation, where it describes surface diffusion driven by surface tension [11]. The second part is the contribution due to the nonequilibrium surface currents. In the unstable case jb is proportional to rn = Vh for ImI
1
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