The Implications of Collisions on the Spatial Profile of Electric Potential and the Space-Charge-Limited Current

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.