Elastic properties of porous silicon studied by acoustic transmission spectroscopy

The porosity dependence of the elastic properties of porous silicon in different crystallographic directions is studied. The velocity of longitudinal acoustic waves in porous silicon layers electrochemically etched in (100), (110), and (111) oriented wafers has been measured by acoustic spectroscopy in the gigahertz frequency range. This non-destructive method was used for porous silicon layers with porosity of 25-85% obtaining velocities in the range of about 1 to 7 km s(-1). The implication of constant Poisson's ratio of porous silicon is examined. The effect of velocity dispersion due to multiple scattering is considered. The c(11) stiffness constant can be obtained from the velocity measurement in the [100] direction of a cubic crystal. We show that, using the results for velocity in [110] or [111] directions and Keating's relation, the stiffness constants c(12) and c(44) can be obtained. The velocity dependence on porosity was fitted as v=v(0)(1 - phi)(kappa), where v(0) is the velocity in bulk silicon, phi is porosity, and kappa is a fitting parameter. It is shown that with other conditions being equal: (i) the porosity dependence of the acoustic velocity is related to the doping level of the wafer from which the porous silicon was etched (kappa depends on wafer resistivity); (ii) acoustic velocities in different crystallographic directions have the same dependence on porosity (kappa is independent of wafer orientation). This requires that all three stiffness constants c(11), c(12) and c(44) have the same dependence on porosity: cij = c(ij)(0)(1 - phi)(m); and (iii) the morphology of porous layers depends on the HF concentration in the etchant (j is used as an indicator for the disorder of the porous structure).