Dependence of the crystallization rate of vacuum-deposited amorphous antimony films on the film thickness

The crystallization rate of vacuum-deposited amorphous antimony (a-Sb) films was investigated as a function of the film thickness d. Experimental plots of the observed growth rate v against d^-1 for systems such as Sb/glass and Sb/Ge show that the relationship between v and d^-1 are convex towards the origin. Such a feature is well interpreted by a model in which

$\text{v=}\text{1}\text{d}\text{∫}\text{d}\text{0}\text{u}\text{d}\text{z}$

where u is the actual growth rate of the crystallite in the a-Sb film and z the distance from the film surface adjacent to the substrate. The quantity u is assumed to vary as follows: u = αzⁿ + u_s when 0 ? z ? d_s0, u = u_i when d_s0 ? z ? d ? d_v0 and u = β(d ? z)ⁿ + u_v, when d ? d_v0 ? z ? d where $\text{α =}\text{(u}{}_{\mathrm{i}}\text{? u}{}_{\mathrm{<mtext>s</mtext>}}\text{)}\text{d}{}_{\mathrm{<mtext>s</mtext><mtext>0</mtext>}}{}^{\mathrm{n}}$, $\text{β =}\text{(u}{}_{\mathrm{i}}\text{? u}{}_{\mathrm{<mtext>v</mtext>}}\text{)}\text{d}{}_{\mathrm{<mtext>v</mtext><mtext>0</mtext>}}{}^{\mathrm{n}}$, d_s0 and d_v0 are thickness of surface regions near the substrate and the vacuum respectively, u_i is the growth rate inside the film, u_s and u_v are the rates at surfaces adjacent to the substrate and the vacuum respectively and n an adjustable numerical parameter. As a typical example, for the Sb/Ge system at 30°C, u_i, u_s and u_v are estimated to be 139 μm s^-1, 25.4 μm s^-1 and 0.2 μm s^-1 respectively and d_s0 and d_v0 to be 133 Å and 143 Å respectively with n = 3.