Theory of Surface Excitations in Fermi Systems of Finite Thickness

Abstract

We consider a degenerate system of fermions interacting via a Yukawa potential and confined within a slab of thickness $L$. Using the semiclassical random-phase approximation, we obtain the retarded density response function ${\ensuremath{\chi}}_{\mathrm{nn}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}},\ensuremath{\omega})$, which exhibits resonances corresponding to surface, as well as surface-bulk, or mixed, collective excitations. The surface and surface-bulk density fluctuations are spatially either of a symmetric or antisymmetric character. Their dispersion relations and collisionless damping are discussed, mainly in the strong-coupling limit. A symmetric surface mode which is phononlike at long wavelengths is found, but the antisymmetric surface mode ceases to exist below a certain value of ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\ensuremath{\parallel}}$, the wave vector parallel to the surface. The mixed modes all possess an energy gap at ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\ensuremath{\parallel}}=0$. We also evaluate the dynamic structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ which describes the inelastic scattering of particles in the Born approximation. Our calculations exhibit how crucially the nature of collisionless surface modes depends on the range of interaction between quasiparticles. We argue that our model calculations should be appropriate to liquid ${\mathrm{He}}^{3}$ in situations where the restricted dimension is of order 100 \AA{} or larger.