Steady-State Structures in Composition-Modulated Alloys: Kinetic Phase Transition Between 1d and 2D Patterns
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327 Mat. Res. Soc. Symp. Proc. Vol. 583 ©2000 Materials Research Society
(a) T Tc"
(b) T
inhomogeneous
2D
T
2 2D+1D
T2 --homogeneous 0
Vo
homogeqeous
0v
v
V2
1
V3v
Figure 1: Phase diagram of the alloy growth. (a): Linear stability phase diagram. (b): Steady-state phase diagram, which contains regions of homogeneous growth, a region of the growth of a 1D modulated structure, a region of the growth of a 2D modulated structure, and a region where the stable growth of both ID and 2D structures is possible. For a set of material parameters typical for semiconductor alloys, calculated values of vL, v2 , and v 3 are as follows: vi = 1.2Ak-1, v 2 = 4.0A s-1, and v3 = 96A s-1. Thus, a kinetic phase transition between the growth of a 1D structure and the growth of a 2D structure can indeed
occur at v
< V2,
i. e. at growth velocities typical for MBE.
fluctuations O(r) at the growing surface and of the surface profile fluctuations ((x, y), Ot
= a D'--Vivj
i9€ -
=
D_ f-
JF
[v
9(]¢
ViVJ--v +-I a" ICBT ' [ O6j a(1
X
Here F is the total Helmholtz free energy, D$s is the diffusion coefficient tensor related to the evolution of the surface, Di- is the one related to the substitutional diffusion of alloy components on the surface, and a is the lattice parameter. The main axes of the diffusion coefficient tensors are symmetry axes of the (001) surface of a zinc-blend semiconductor, namely [110] and [110]. The free energy equals F = Frhem + Fgrad ± Fsurf + Fetast, where Fch,, is the chemical free energy of the alloy, Fgr,,d is the gradient energy, _ f(VO) 2dV, F,.r. is the surface energy, and Feast is the elastic energy induced by composition fluctuations. Contrary to Refs. [3, 5], we take into account, in FeIast, the cubic elastic anisotropy of zincblend semiconductors. The last term in the first equation of (1) is related to a non-linear coupling between substitutional migration of atoms A and B on the surface of a given profile and their joint migration which changes the surface profile. LINEAR STABILITY ANALYSIS To perform the linear stability analysis of Eqs. (1), we expand the free energy up to quadratic terms in fluctuations of composition 0 and in fluctuations of the surface profile ( and substitute O(t; r,,) exp(-yt)exp(ik11 r1 1), and ((t;r 1l) - exp(yt)exp(ikjjrll) into linear equations. This yields the following characteristic equation for the amplification rate y, -
Df9(T) kliiklllj kk=
O'h,
[
c
T;c
+K
2
+
8=1
R, 3p)k1v
V(2
(2)
11 vI+a()k
Here fchm(T; c) is the density of the chemical free energy, K is the coefficient entering the gradient energy, kB is the Boltzmann constant. The summation over s includes contributions of three static analogues of acoustic waves into elastic interaction, R, (W)is the elastic interaction energy, and a. (y) is the attenuation coefficient of the static acoustic wave. Due to elastic anisotropy both R1(ýo) and a,(so) depend on the orientation of the wave vector ki, in the surface plane, i. e., on the angle cp.
328
(a) 11
Iii
rji [21
(b)C1IC2 IC
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