Space charge limited current in quantum regime by solving the Schröodinger-Poisson equation based on finite element method

In this work, we quantitatively investigate the space-charge-limited current in quantum regime by solving the Schrodinger-Poisson equation based on the finite element method (FEM). The paper shows the schematic of the self-consistent model of coupled Schrodinger and Poisson´s equation for a nano-diode gap, where psi_r(x)=A exp(ik₁x) is an incident plane wave, and psi_r(x)=B exp(-ik₁x) and psi_t(x)= C exp(ik₂x) are a reflected plane wave and a transmitted plane wave, respectively. The boundary conditions to Eqs. (1) and (2) are formulated as follows. Let V_g be the gap bias voltage, which may be zero, positive, or negative.