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 $\phi (x)=(x/D{)}^{4/3}$, 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 $\phi (x)=(x/D{)}^{3/2}$. This Letter applies a capacitance argument for SCLC and uses the transit time from a recent exact solution for collisional SCLC to show that $\phi (x)=(x/D{)}^{\xi }$ for a general collisional gap, where $4/3⩽\xi ⩽3/2$. 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 capacitance to derive an analytic equation for collisional SCLC that agrees within $\sim 5-6\%$ of the exact solution that requires solving parametrically through T. We derive equations in the limits of $\nu ⟶0$ and $\nu ⟶\infty$ for general $\xi$ that asymptotically recover the CLL as $\nu ⟶0$ and the MGL as $\nu ⟶\infty$. Matching these limits shows that $\xi \simeq 1.40$ and $V\propto D^2{\nu }^2$ at the transition from a vacuum to collisional diode for any device condition.

23 pages, 4 figures

References in corpus (2)

• Space-charge-limited current density for nonplanar diodes with monoenergetic emission using Lie-point symmetries
• Physical factors governing the shape of the Miram curve knee in thermionic emission

Cited by in corpus (2)

• Extending the Mott-Gurney law to one-dimensional nonplanar diodes using point transformations
• A generalized framework for the critical current for a one-dimensional crossed-field gap in an orthogonal coordinate system