Physics and Control of Si/Ge Heterointerfaces
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lI.Results and discussion 111-1. Vertical interface effects IlI-l-1. Revisiting the Ge surface segregation during MBE As is well documented in the literature, one observes the Ge surface segregation when growing Si/Ge heterointerfaces by solid-source MBE. The Ge segregation has turned out to be describable reasonably well by the two-state exchange model or surface bilayer model analogous to those established for dopant segregation [1-121. The problem of these simple-minded schemes is the assumption of ballistic or instantaneous deposition of the Si cap. Recently, further sophistication of calculation has met success to some extent[9,161. However, the essence is almost fully contained in the surface bilayer model and hence a more advanced approach will be left over as the subject of future study. Instead we will focus on the surface bilayer model and draw important results in the perspective of thermodynamics which are directly derived from that model. 111-1-2. Thermodynamic interpretation of "self-limiting" site exchange A cut-away view along of the surface and subsurface sites is illustrated in Fig. 1. As was already pointed out by the authors, the key feature of the Ge surface segregation within the surface bilayer model is the "self-limiting" site exchange 15,71 that distinguishes the segregation of Ge from those of dopants. The point is that only a Si-Ge exchange contributes to the Ge segregation. Conversely, a Ge-Ge exchange event has only the effect equivalent to an enhanced kinetic limitation on the Ge segregation. Standard energy diagram is assumed for simulation and the relevant energies are the kinetic barrier El and the segregation energy Eb. Let ni and n2 denote the population of surface and subsurface Ge, respectively, and the rate equation for the kinetics controlling the Ge site exchange becomes of the form, an1
-O-n= -2pnl (l-n2) + 2qn2 (1-nl) ,
(1)
with nl+n2=const. (mass balance) unless the loss due to Ge thermal desorption is pronounced. Note that p=foexp(-EJkT) and q=p exp(-Eb/kT) where fo is an exchange frequency. As the temperature is raised, the thermal equilibrium is reached early. Then the Ge segregation becomes insensitive to the kinetic barrier, Ea. This is true for T,>450'C under normal MBE growth conditions. The steady-state solution rather than that at thermal equilibrium is obtained by putting l.h.s. of (1) to zero, or M/t =0. One finds after a simple calculus, n2 nl1 Eb 1-n2 = 1-nj exp( kT
(
,
i.e., the formula for the equilibrium segregation of a binary alloy. On the other hand, this is readily obtained from thermodynamic arguments of free energy F containing the entropy of mixing, F = n1 Eb - kT {-n2 In n2 -(1-n2) ln(l-n2) - n1 In n, -(1-nl) ln(1-nl)}.
(3)
The bracket contains the configurational (mixing) entropy terms. Differentiating Eq.(3) with respect to n1 and putting OF/Onl --0 with On2 /Onl =-l (normalization) yields Eq.(2). Therefore, it is seen that the "self-limitation" drawn from the purely kinetic argument is of physical significance and the
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Surface
n2 Su
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