Tunneling across an inhomogeneous delta-barrier

Abstract

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy \(V\left( r \right) = \left[ {\eta + \mu \left( {x^2 + y^2 } \right)} \right]\delta \left( z \right)\) (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 \(k_0 = \left( {0,0,k_0 } \right)\)is considered. The electron is forward-scattered into the region behind the barrier (region 2: \(z >0\)), i. e. the wave function \(\psi _2 \left( r \right)\) is composed of plane waves with all wave vectors \(k_2 \) such that \(\left| {k_2 } \right| = k_0 \) and \(k_{2z} = \sqrt {k_{_0 }^2 - q^2 >\left. 0 \right)} \)) (where \(q = \left( {k_{2x} ,k_{2y} ,0} \right),q = \left| q \right|\)). Therefore, if \(z >0\), the wave function of the electron is represented as \(\psi _2 \left( r \right) = \int {d^2 qU_2 \left( q \right)\exp \left[ {{\text{i}}\left( {q.u + \sqrt {k_{_0 }^2 - q^2 } } \right)z} \right]} \), where \(u = \left( {x,y,0} \right)\). An approximate formula is derived for the amplitude \(U_2 \left( q \right)\). The authors pay a special attention to the flow density \(J_2 \left( r \right) = \left( {\hbar /m} \right)\operatorname{Im} \psi _{_2 }^* \left( r \right)\nabla \psi _2 \left( r \right)\) and calculate this function in two cases: 1. for the plane \(z = 0\) and 2. for high values of \(R = \left| r \right|\left( {z = R{cos}\vartheta ,{i}{.e}{.}\vartheta \in \left( {0,{\pi}/2} \right.} \right)\) is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.