Space-charge-limited (SCL) current defines the maximum current density that can flow through a vacuum diode. A first decade well-known SCL current is the classical Child–Langmuir law, where the plasma sheath is regarded as the ion collisionless plasma. In plasma sheaths, there are several plasma sheath scenarios, such as the Mott-Gurney model, which includes plasma collisional effects for the ion sheath, or Bohm's sheath that combines both collisions and electron-ion plasma. This work reviews the traditional descriptions of such models based on one-dimensional theory, including proposed precedent two-dimensional schemes and the fractional-dimensional versions.
In the first part of the thesis, the Bohm sheath is extended to two dimensions. Even though the corresponding Poisson equation is nonlinear, the analytical expressions are derived by considering regions near the anode, where approximate solutions for SCL currents are valid, for both cold and Boltzmann distributed plasmas. Both formulae have a similar mathematical structure, but different characteristic Debye lengths. Moreover, in the large gap spacing limit compared to the Debye lengths themselves, the 2D ratio currents reproduce the same one.
Furthermore, the Bohm model for one coordinate is generalized into the fractional-dimensional space using fractional-dimensional calculus, where the fractional dimension α characterizes local effects and the inhomogeneity of the medium. The model recovers the classical laws in the integer-dimensional limit (α =1). Additionally, the fractionalized Jeffe's model is proposed for collisionless ion sheaths including initial ion kinetic energy, which not only reproduces the fractional Child–Langmuir law, but also predicts the fractional SCL current in the drift-space regime.

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