Surface electromagnetic modes in layered conductors in a magnetic field

Abstract

A transfer-matrix approach is developed for studies of the collective electromagnetic modes in a semi-infinite layered conductor subjected to a quantizing external magnetic field perpendicular to the layers. The dispersion relations for the surface and bulk modes are derived. It is shown that the surface mode has a gap in the long-wavelength limit and exists only if the absolute value of the in-plane wave vector q exceeds the threshold value q*=−1/(a ln|Δ|). Depending on the sign of the parameter Δ=(ε−ε₀)/(ε₀+ε), the frequency of the surface mode ωs(q,Δ) goes either above (for Δ>0) or below (for Δ<0) the bulk-mode frequency ω(q,k)=ω(q,k+2π/a) for any value of k. At nonzero magnetic field H the bulk mode has a singular point q₀(H) at which the bulk band twists in such a way that its top and bottom bounds swap. Small variations of q near this point change dramatically the shape of the dispersion function ω(q,k) in the variable k. The surface mode has no dispersion across the layers, since its amplitude decays exponentially into the bulk of the sample. Both bulk and surface modes have in the region qa≫1 a similar asymptotic behavior ω∝q¹/², but ωs(q,Δ) lies above or below ω(q,k), respectively, for Δ>0 and Δ<0 (a is the interlayer separation; ε0 and ε stand for the dielectric constants of the media outside the sample and between the layers; q and k are the components of the wave vector in the plane and perpendicular to the layers, respectively).