Space-charge-limited current density for nonplanar diodes with monoenergetic emission using Lie-point symmetries

A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

“…After demonstrating that our alternate solution recovered the SCLC solution from Jaffé [51], we calculated the SCLC for cylindrical and spherical geometries. The results matched those derived analytically using Lie point symmetries [59], obeying the relation defined in (37) where the zero initial velocity solutions are written in terms of those found by Darr and Garner [41]. We also observed the relationship between J and v 0 for V → 0, demonstrating J ∝ v 3 0 .…”

“…Recently, the method of Lie point symmetries has been utilized to extend analytic 1D planar nonzero initial velocity SCLC solutions to cylindrical and spherical geometries [59]. Analogous to (18) for the bifurcation solution, applying Lie point symmetries gives SCLC as (37) for any orthogonal coordinate system, in agreement with Jaffé's correction term for planar geometry.…”

“…Comparison of the bifurcation solution and the SCLC for nonzero initial velocity found both exactly and numerically using NDSolve in Wolfram Mathematica for discrete v 0 , normalized to the zero initial velocity SCLC J SCLC /J v 0 =0,SCLC for a one-dimensional diode of any geometry. For each geometry, the bifurcation solution was found using (18), the numerical SCLC using ( 10) and ( 29) with the appropriate Laplacian and virtual cathode boundary condition, and the exact SCLC using (37) from [59]. The numerical results matched the expected results from (37) for each geometry, making the above figure universal for any orthogonal geometry due to the consequences of ( 18) and ( 37).…”

“…For spherical coordinates, χ = 1/r [59]. Thus, h 2 = h θ = r sin φ, h 3 = h φ = r (we assume φ = π/2), and…”

“…Numerical analysis from the current paper confirms these findings, with the solution of (10) after substituting the numerical integration of (29) using ( 27) and ( 31)- (33) as the virtual cathode boundary condition for each geometry matching the analytic result from (37) to a similar negligible relative error (∼10 −6 ) from the solver. Consequently, based on (37) from reference [59] and (18) from this manuscript, figure 6 is universal for any geometry after replacing the appropriate zero initial velocity solution, as reflected in our choice of vertical axis label.…”