Multidimentional Space Charge Limited Current Theories for Cylindrical, Tip-to-tip, and Thick Planar Electrodes

Investigates space-charge-limited current density (SCLCD) across multidimensional and nonplanar diode geometries, presenting novel analytical methodologies that expand upon traditional theoretical frameworks. Addresses critical gaps by systematically examining electron transport phenomena through variational calculus, conformal mapping, Lie-point transformations, and the general covariant formulation of SCLCD derived from Maxwell’s equations.

Comprehensive review of the classical Child-Langmuir (CL) law, outlining its historical and mathematical context, limitations, and the necessity for more versatile analytical tools to accommodate complex diode structures. Variational calculus is applied to precisely derive exact analytical solutions, avoiding approximation errors typical in conventional series expansions. Furthermore, conformal mapping is utilized to solve complex boundary-value problems by transforming intricate geometries into analytically tractable domains. Lie-point transformations provide additional analytic leverage and mathematical generality, simplifying nonlinear differential equations associated with multidimensional diode configurations.

Explicitly incorporating electrode thickness within two-dimensional (2D) diode models using Schwartz-Christoffel conformal mapping. The analytical formulations developed are validated through particle-in-cell simulations, confirming that increasing cathode thickness significantly reduces SCLC. The maximum errors presented by these theories do not exceed 12% and tend to be less than 5%.

Extends analytical modeling to multidimensional tip-to-tip and cylindrical diode geometries, employing a general covariant formulation from Maxwell’s equations. Detailed geometric scaling factors derived through canonical coordinates and orthogonal systems significantly enhance the theoretical rigor and scope of application. The resultant SCLCD expressions demonstrate substantial applicability in vacuum microelectronics, high-power microwave generation, and nanoelectronic devices, enhancing theoretical advancements and practical technological applications.