The Implications of Collisions on the Spatial Profile of Electric Potential and the Space-Charge-Limited Current
Year: 2025
Type: article
Abstract: The space-charge-limited current (SCLC) in a vacuum diode is given by the Child-Langmuir law (CLL), whose electric potential${\phi }\text {(}{x}\text {)}\propto \text {(}{x} /{D}\text {)}^{{4} /{3}}$, where x is the spatial coordinate across the gap and D is the gap separation distance. For a collisional diode, SCLC is given by the Mott-Gurney law (MGL) and${\phi }\text {(} {x}\text {)}\propto \text {(}{x} /{D}\text {)}^{{3} / {2}}$. Here, we apply a capacitance argument for SCLC and use the transit time from a recent exact solution for collisional SCLC to show that${\phi }\text {(}{x}\text {)}\propto \text {(}{x}/{D}\text {)}^{\xi }$for a general collisional gap, where${4} / {3}\le \xi \le {3}/ {2}$. Furthermore,$\xi $is strictly a function of$\nu {T}$, where$\nu $is the collision frequency and T is the electron transit time. Using this definition of$\xi $, we estimate the spatial dependence of the electron velocity and use the gap capacitance to derive an analytic equation for collisional SCLC that agrees within ~4.5% of the exact solution that requires solving parametrically through T. This analytic equation for general$\xi $asymptotically recovers the CLL as${\nu }\to {0}$and the MGL as${\nu } \to \infty $. Matching these limits shows that${\xi }\approx {1.40}$and${V}\propto {D}^{{2}}{\nu }^{{2}}$at the transition from a vacuum to a collisional diode for any device condition.
Authors: Allen L. Garner, N. R. Sree Harsha
Institution: Purdue University West Lafayette
Language: English
Subfield: Electrical and Electronic Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407129368