Surface plasmon dispersion of semiconductors with depletion or accumulation layers

We have calculated the effect on the surface-plasmon-polariton dispersion of a space-charge layer at a semiconductor surface, representing it by a dielectric function $\epsilon \left(\omega \right)$ that has an exponentially varying part with a $\frac{1}{e}$ decay depth of the order of the layer thickness. Since we are interested in frequencies below those where interband transitions are important, we have used a free-electron model for $\epsilon \left(\omega \right)$, with the plasma frequency varying continuously from ${\omega }_{\mathrm{ps}}$ at the surface to ${\omega }_{\mathrm{pb}}$ in the bulk. Particular attention has been paid to the frequency range for which the real part of $\epsilon$ vanishes within the sample. Use of a local relation between dielectric displacement and electric field is justified, even in this range, by the fact that we use complex $\epsilon$ and the imaginary part varies little with depth. Evaluating the dispersion numerically, with damping included, we obtain for the dispersion of a depletion layer (${\omega }_{\mathrm{ps}}<{\omega }_{\mathrm{pb}}$) a single branch that starts at the light line, is reentrant at ${\omega }_{\mathrm{ps}}$, and goes asymptotically to the frequency for which $\epsilon$ at the surface ${\epsilon }_s$ equals $-\epsilon$ of the medium above. For samples with thick enough depletion layers, additional branches, corresponding to guided modes, are found both above and below ${\omega }_{\mathrm{pb}}$. For an accumulation layer (${\omega }_{\mathrm{ps}}>{\omega }_{\mathrm{pb}}$) there is always one branch which starts at the light line and goes asymptotically to the frequency for which ${\epsilon }_s=-\epsilon$ of the medium above. For large enough values of $d$, a second branch appears, lying between ${\omega }_{\mathrm{pb}}$ and ${\omega }_{\mathrm{ps}}$, curving upward in contradiction to results obtained earlier. Comparison of this theory with experimental data for InSb, some for samples with disturbed surfaces, leads to reasonable estimates for the thickness of the surface depletion layers.