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(ik
₁
x) is an incident plane wave, and psi
_r
(x)=B exp(-ik
₁
x) and psi
_t
(x)= C exp(ik
₂
x) 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.