Nonlocal dielectric response of an N‐quantum‐well superlattice embedded in bulk and semi‐infinite plasmas in magnetic field

Abstract

We derive explicit analytic results for the dynamic, nonlocal and inhomogeneous screening functions of a finite superlattice array of N equally spaced 2D quantum well plasmas embedded in infinite and semiinfinite host bulk-plasma-like media. The closed-form results presented here for the screening functions, which are space-time matrix inverses of the spatially inhomogeneous dielectric functions, are valid for any number of wells, and can accomodate nonlocality in both the 2D and host plasmas, as well as an ambient normal magnetic field.1 Introduction The most useful description of the dielectric response of a nonlocal, dynamic and inhomogeneous solid state plasma is provided by the screening function K( r, t; r , t ), which relates an impressed potential U (2) = U ( r 2 , t 2 ) to the effective potential V (1) = V ( r 1 , t 1 ) as V (1) = d2K(1, 2)U (2). K is the space-time matrix inverse of the direct dielectric function ε(1, 2) in the sense d3ε(1, 3)K(3, 2) = δ(1 − 2). Here, we solve for K(1, 2) for a N -quantum well system embedded in infinite and semi-infinite host plasmas. Since the system is translationally invariant in the x − y plane, we Fourier transform in the plane to a 2D wavevectorp, and in time to frequency ω. With this, the inversion relation can be rewritten as a RPA-type integral equation in z-variables as (suppress q, ω)