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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.00877 |
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| _version_ | 1866912333316489216 |
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| author | Garner, Allen L. Harsha, N. R. Sree |
| author_facet | Garner, Allen L. Harsha, N. R. Sree |
| contents | The space-charge limited current (SCLC) in a vacuum diode is given by the Child-Langmuir law (CLL), whose electric potential $φ(x) = (\it{x/D})^{4/3}$, 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 $φ(x) = (x/D)^{3/2}$. This Letter applies a capacitance argument for SCLC and uses the transit time from a recent exact solution for collisional SCLC to show that $φ(x) = (x/D)^ξ$ for a general collisional gap, where $4/3 \leqslant ξ\leqslant 3/2$. Furthermore, $ξ$ is strictly a function of $νT$, where $ν$ is the collision frequency and T is the electron transit time. Using this definition of $ξ$, we estimate the spatial dependence of the electron velocity and use the capacitance to derive an analytic equation for collisional SCLC that agrees within $\sim5-6\%$ of the exact solution that requires solving parametrically through T. We derive equations in the limits of $ν\longrightarrow 0$ and $ν\longrightarrow \infty$ for general $ξ$ that asymptotically recover the CLL as $ν\longrightarrow 0$ and the MGL as $ν\longrightarrow \infty$. Matching these limits shows that $ξ\simeq 1.40$ and $V \propto D^2ν^2$ at the transition from a vacuum to collisional diode for any device condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The implications of collisions on the spatial profile of electric potential and the space-charge limited current Garner, Allen L. Harsha, N. R. Sree Plasma Physics Mesoscale and Nanoscale Physics Classical Physics The space-charge limited current (SCLC) in a vacuum diode is given by the Child-Langmuir law (CLL), whose electric potential $φ(x) = (\it{x/D})^{4/3}$, 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 $φ(x) = (x/D)^{3/2}$. This Letter applies a capacitance argument for SCLC and uses the transit time from a recent exact solution for collisional SCLC to show that $φ(x) = (x/D)^ξ$ for a general collisional gap, where $4/3 \leqslant ξ\leqslant 3/2$. Furthermore, $ξ$ is strictly a function of $νT$, where $ν$ is the collision frequency and T is the electron transit time. Using this definition of $ξ$, we estimate the spatial dependence of the electron velocity and use the capacitance to derive an analytic equation for collisional SCLC that agrees within $\sim5-6\%$ of the exact solution that requires solving parametrically through T. We derive equations in the limits of $ν\longrightarrow 0$ and $ν\longrightarrow \infty$ for general $ξ$ that asymptotically recover the CLL as $ν\longrightarrow 0$ and the MGL as $ν\longrightarrow \infty$. Matching these limits shows that $ξ\simeq 1.40$ and $V \propto D^2ν^2$ at the transition from a vacuum to collisional diode for any device condition. |
| title | The implications of collisions on the spatial profile of electric potential and the space-charge limited current |
| topic | Plasma Physics Mesoscale and Nanoscale Physics Classical Physics |
| url | https://arxiv.org/abs/2410.00877 |