Space-charge-limited current for nonplanar relativistic diodes with nonzero monoenergetic initial velocity using point transformations

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 gamma(0)greater than or similar to 1.1; for lower gamma(0), nonrelativistic SCLC gives a reasonable approximation. In the ultra-relativistic limit, J(r,SCLC)/J(SCLC)proportional to V-1/2, where J(r,SCLC) and J(SCLC) 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 gamma(0) 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.