Electroreflectance in a Nonuniform Field in the Small-Wave-Number Approximation and Its Measurement by Ellipsometry

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Abstract
The changes in the real and imaginary parts of the dielectric constant of a solid induced by an electric field which decreases exponentially with distance from the surface are calculated from perturbation theory for photon energies near interband transitions. This exponential model for the field is of interest because it approximates well the actual field over a fairly wide range of surface conditions and because it contains only two adjustable parameters, the surface field and the rate of decay of the exponential. The contribution from each of these parameters can be separated and identified in the results. Previous calculations of electro-reflectance in a nonuniform field have employed the one-electron Franz-Keldysh theory for a uniform field, assuming the field to vary slowly enough with distance from the surface so that a WKB approximation could be used to extract spatially averaged values of the change in dielectric constant. Our calculation is not limited by the WKB approximation, and is applicable even at very large field nonuniformities. However, when the field penetration depth is less than about three times the photon penetration depth, effective masses must be known in order to complete our calculation, but it is still valid. The theory, used to interpret a modulated-ellipsometry experiment on Ge in the 2.1-eV region, shows that illumination of the sample surface by a second light beam can decrease the field penetration depth by at least a factor of 20, and increase the surface field by at least a factor of 10, because of increased free-carrier screening.