A Lagrangian Formulation of Collisional Space-Charge-Limited Current

Recent studies have used point transformations to extend the Child-Langmuir (CL) and Mott-Gurney (MG) laws for the space-charge limited current density (SCLCD) to nonplanar geometries for vacuum1 and solids2, respectively. While the vacuum solution matched the result obtained using the Euler-Lagrange (EL) equation to extremize the current density,1,2 applying this result to a collisional gap requires modifying the EL equation. Here, we introduce Rayleigh dissipation (RD) to the EL equation to account for collisions and recover the result obtained using point transformations to extend the MG law.2 We further generalize RD to derive the SCLCD for electron mobility (or collision frequency) from vacuum to solid. This extends prior derivations of planar SCLCD for general collisionality3 and demonstrates the covariance of SCLCD across all values of electron mobility, which is relevant for semiconductors and plasmas.

1. N. R. S. Harsha, J. M. Halpern, A. M. Darr, and A. L. Garner, Phys. Rev. E 106, L063201 (2022).

2. A. L. Garner and N. R. S. Harsha, IEEE Trans. Electron Devices 72, 1419-1425 (2025).

3. A. L. Garner, N. R. S. Harsha, and A. M. Loveless, J. Appl. Phys. 137, 234501 (2025).

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