Tomasch Effect as a Probe of the Dispersion Relation of Electrons in "Gapless" Superconductors

The recent theory of McMillan and Anderson is used to discuss the effect of nonmagnetic impurities and depairing mechanisms on the period and damping of the Tomasch oscillations. These oscillations in the density of states seem to provide a unique way of directly measuring the dispersion relation of electronic excitations in superconductors [$\omega$ versus $\mathrm{Re}{({\tilde{\omega }}^2-{\tilde{\Delta }}^2)}^{\frac{1}{2}}\propto$ pseudomomentum $k^{*}$, where $\tilde{\omega }$ and $\Delta$ are the usual renormalized diagonal and off-diagonal self-energies]. We shall discuss the theoretical dispersion curves for films with a small concentration of magnetic impurities and for dirty films ($l\ll {\xi }_0$) in the presence of a parallel magnetic field. The two situations lead to quite different dispersion curves at small values of $k^{*}$, for the same value of the depairing parameter. Using the Tomasch effect to plot out the dispersion curves becomes more difficult as $\omega$ decreases, since the damping of the oscillations [$\propto \mathrm{Im}{({\tilde{\omega }}^2-{\tilde{\Delta }}^2)}^{\frac{1}{2}}$] increases.