N. R. Sree Harsha

Introducing a magnetic field perpendicular to the electric field in a diode turns electrons as they cross the gap and alters the electric potential’s spatial profile and the space-charge limited current density (SCLCD) compared to a nonmagnetic diode. In this paper, we apply gap capacitance to crossed-field gaps with magnetic fields below and above the Hull cutoff magnetic field BH for magnetic insulation to assess these properties. Combining capacitance and the force law gives the same relationship for the limiting current for BBH as the exact approach that extremizes the current density, while simplifying the mathematics required. Although the crossed-field geometry prohibited a direct solution for the electric field E as a function of position across the gap x for B<BH, we derived a closed form solution for x as a function of E. Using the ansatz that the electric potential ϕx∝xζ, where 4/3≤ζ≤2 accounts for 0≤B≤BH, we calculated ϕ in excellent agreement with the exact solution up to B≈0.999BH. Finally, we used this ansatz to derive a correction factor to convert the current density obtained using the charge-free capacitance to the exact solution for the SCLCD. In summary, we demonstrate the feasibility of estimating ϕx based on B/BH at the space-charge limit and the utility of charge-free capacitance for characterizing the SCLCD of a crossed-field gap.

Thermionic energy converters (TECs) offer a direct method for converting heat into electricity, providing potential applications in waste heat recovery and renewable energy generation. While extensive research has focused on developing low work function materials, the performance of vacuum-based converters remains fundamentally hindered by space-charge effects, necessitating improved theoretical characterization of the space-charge limited current (SCLC). In this study, we derive exact theories for SCLC for multidimensional diodes with nonzero initial velocity (approximating high cathode temperatures) to demonstrate how to design TECs with higher steady-state currents to enhance the conversion of heat to electricity. We show that injected currents exceeding the SCLC limit result in a sharp decrease in the collector current, defining the bifurcation condition. Simulations using the two-dimensional particle-in-cell codes XOOPIC and VSim for both the SCLC and bifurcation regimes yield SCLC within 7% of the theoretical predictions. Finally, the sharp knee characteristic of the transition from one-dimensional thermionic emission to SCLC, known as the Miram curve, is extended to multidimensional diodes characterized by a single sharpness parameter σ , and its implications are discussed.

Recent studies have applied variational calculus, conformal mapping, and point transformations to extend the one-dimensional space-charge limited current (SCLC) from planar gaps to more complicated geometries. However, introducing a magnetic field orthogonal to the diode's electric field complicates these calculations due to changes in the electron trajectory. This paper extends a recent study that applied variational calculus to determine the SCLC for a cylindrical crossed-field diode to develop a framework that is valid for any orthogonal coordinate system. We then derive equations for the SCLC for crossed-field gaps in spherical, tip-to-tip, and tip-to-plane geometries that can be solved numerically. These calculations exhibit a discontinuity at the Hull cutoff magnetic field BH corresponding to the transition to magnetic insulation as observed analytically for a planar geometry. The ratio of the crossed-field SCLC to the nonmagnetic SCLC becomes essentially independent of geometry when we fix δ=D/DM>0.6, where D is the canonical gap distance accounting for geometric effects on electric potential and DM is the effective gap distance that accounts for magnetic field and geometry. The solutions for these geometries overlap as δ→1 since the geometric corrections for electric potential and magnetic field match. This indicates the possibility of more generally accounting for the combination of geometric and magnetic effects when calculating BH and SCLC.

Recent studies have applied variational calculus, conformal mapping, and point transformations to generalize the one-dimensional (1D) space-charge limited current density (SCLCD) and electron emission mechanisms to nonplanar geometries; however, these assessments have focused on extending the Child–Langmuir law (CLL) for SCLCD in vacuum. Since the charge in the diode is independent of the coordinate system (i.e., covariant), we apply bijective point transformations to extend the Mott–Gurney law (MGL) for the SCLCD in a collisional or semiconductor gap to nonplanar 1D geometries. This yields a modified MGL that replaces the Cartesian gap distance with a canonical gap distance that may be written generally in terms of geometric scale factors that are known for multiple geometries. We tabulate results for common geometries. Such an approach may be applied to any current density, including non-space-charge limited gaps and SCLCD that may fall between the CLL and MGL.

The space-charge-limited current (SCLC) in a vacuum diode is given by the Child–Langmuir law (CLL), whose electric potential ${\phi}\text{(}\textit{x}\text{)}\propto\text{(}\textit{x} /\textit{D}\text{)}^{\text{4} /\text{3}},$ where $\textit{x}$ is the spatial coordinate across the gap and $\textit{D}$ is the gap separation distance. For a collisional diode, SCLC is given by the Mott–Gurney law (MGL) and ${\phi}\text{(} \textit{x}\text{)}\propto\text{(}\textit{x} /\textit{D}\text{)}^{\text{3} / \text{2}}$. Here, we apply a capacitance argument for SCLC and use the transit time from a recent exact solution for collisional SCLC to show that ${\phi}\text{(}\textit{x}\text{)}\propto\text{(}\textit{x}/\textit{D}\text{)}^{{\xi}}$ for a general collisional gap, where $\text{4} / \text{3}\le \xi \le\text{3}/ \text{2}$. Furthermore, ${\xi}$ is strictly a function of ${\nu \textit{T}}$, where ${\nu}$ is the collision frequency and $\textit{T}$ is the electron transit time. Using this definition of ${\xi}$, we estimate the spatial dependence of the electron velocity and use the gap capacitance to derive an analytic equation for collisional SCLC that agrees within $\sim$4.5% of the exact solution that requires solving parametrically through $\textit{T}$. This analytic equation for general ${\xi}$ asymptotically recovers the CLL as ${\nu}\to \text{0}$ and the MGL as ${\nu} \to \infty$. Matching these limits shows that ${\xi}\approx \text{1.40}$ and $\textit{V}\propto\textit{D}^{\text{2}}{\nu}^{\text{2}}$ at the transition from a vacuum to a collisional diode for any device condition.

Bijective point transformations were recently used to derive the classical space-charge-limited current (SCLC) in one-dimensional (1D) nonplanar devices for electrons emitted into vacuum with nonzero monoenergetic initial velocity. Using these transformations, we first derive a canonical form of SCLC for a relativistic diode with zero initial velocity that holds for any orthogonal 1D geometry and recovers the previously derived planar result. We extend this result to derive a canonical form of SCLC that accounts for nonzero monoenergetic initial velocity and relativistic effects, while recovering SCLC for nonrelativistic diodes with zero and nonzero initial velocity and the relativistic diode with zero initial velocity in appropriate limits. We then use appropriate bijective point transformations to convert from the canonical solution to concentric cylindrical and spherical coordinates. This equation has no closed form solution and must be numerically integrated. The relativistic effects of initial velocity do not become significant until the Lorentz factor γ0≳1.1; for lower γ0, nonrelativistic SCLC gives a reasonable approximation. In the ultra-relativistic limit, Jr,SCLC/JSCLC∝V1/2, where Jr,SCLC and JSCLC are the SCLC for the relativistic diode with general initial velocity and nonrelativistic diode with zero initial velocity, respectively. These asymptotic equations match the exact solutions for sufficiently large γ0 and V. This analysis provides an exact, numerical solution for SCLC for nonzero monoenergetic initial velocity that incorporates relativistic effects for any 1D orthogonal geometry.

Theoretically and computationally describing the operation of nanodiodes requires characterizing the transitions between multiple electron emission mechanisms for nanodiodes with complicated geometries. This motivates our development of techniques to determine when simplified theories for individual mechanisms suffice compared to more complete, but more computationally expensive, models. Leveraging recent theories that define a canonical gap distance to translate planar theory to nonplanar diodes, we derive the conditions for the transitions among thermal emission, field emission, and space-charge-limited current density (SCLCD) in vacuum and with collisions for non-Cartesian coordinate systems, including spherical, cylindrical, and prolate spheroidal coordinate systems. Particle-in-cell (PIC) simulations of the current density as a function of applied voltage for a tip-to-plate geometry in vacuum agreed qualitatively with the asymptotes for thermal emission at low voltage and SCLCD at higher voltage using the canonical gap distance. This demonstrates the utility of this approach for guiding system design and suggests future extensions to save simulation time for more realistic geometries that are more computationally expensive.

Prior studies have developed theories for the maximum permissible current, or critical current, for one-dimensional planar and cylindrical crossed-field diodes where the magnetic field is below the Hull cutoff, meaning that an electron emitted from the cathode reaches the anode. Here, we develop semi-empirical and analytical models to predict the critical current for a two-dimensional (2D) planar diode with nonzero monoenergetic initial velocity. The semi-empirical method considers the geometry, nonzero initial velocity, and magnetic field as multiplicative corrections to the Child–Langmuir law for space-charge limited current in a one-dimensional planar diode with an initial velocity of zero. These results agree well with 2D particle-in-cell (PIC) simulations using the over-injection method to assess virtual cathode formation for different emission widths, magnetic field strengths, and initial velocities. The analytical solution agrees better with PIC results because it accounts for the coupling of the magnetic field, geometry, and initial velocity that the semi-empirical approach does not.

The critical current in a one-dimensional (1D) crossed-field gap is defined by the transition from a cycloidal flow to a near-Brillouin (nB) state characterized by electron flow orthogonal to both the electric and magnetic fields and uniform virtual cathode formation. Motivated by recent studies on space-charge-limited current in non-magnetic diodes, we assess the meaning of critical current in a magentically insulated two-dimensional (2D) planar crossed-field geometry. Particle-in-cell (PIC) simulations demonstrate that binary behavior between a laminar and turbulent state does not occur in 2D because the virtual cathode is nonuniform. Rather than a distinct nB state above the critical current as in 1D, there is an increase in Brillouin contribution with the presence of cycloidal components and noise even at low currents. To evaluate the electron flows in a 2D crossed-field gap in the absence of a binary transition, we developed two metrics to assess the Brillouin and cycloidal components in a 2D planar crossed-field gap for various emission widths and injection current densities by comparing the phase space plots from PIC simulations to analytical solutions for cycloidal and Brillouin flow. For a smaller emission width, less Brillouin contribution occurs for a given injection current, while maximizing the cycloidal noise requires a larger injection current. Once the virtual cathode starts to form and expand with increasing injection current, the cycloidal noise reaches its peak and then decreases while the Brillouin components become significant and increase.