Tunneling across an inhomogeneous delta-barrier

The authors deal with the tunneling of electrons across an inhomogeneous de lta-barrier defined by the potential energy V(r) = [eta + mu (x(2) + y(2))] delta (z) (where eta > 0 and mu > 0 are two constants). In particular, the perpendicular incidence of an electron with a given value k(0) of the wave vector k(0) = (0, 0, k(0)) is considered. The electron is forward-scattered into the region behind the barrier (region 2: z > 0), i.e. the wave functi on psi (2)(r) is composed of plane waves with all wave vectors k(2) such th at \k(2)\ = k(0) and k(2z) = rootk(0)(2)-q(2) > 0) (where q = (k(2x), k(2y) , 0), q = \q \). Therefore, if z > 0, the wave function of the electron is represented as psi (2)(r) = integral d(2)q U-2(q) exp[i(q.u + rootk(0)(2)-q (2))z], where u = (x, y, 0). An approximate formula is derived for the ampl itude U2(q). The authors pay a special attention to the flow density J(2) ( r) = (h/m) Im psi (*)(2)(r)del psi (2) (r) and calculate this function in t wo cases: 1. for the plane z = 0 and 2. for high values of R = \r \ (z = Rc os theta, i.e. theta is an element of (0, pi /2) is the diffraction angle), The authors discuss the relevance of their diffraction problem in a prospe ctive quantum-mechanical theory of the tunneling of electrons across a rand omly inhomogeneous Schottky barrier.