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

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Abstract
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.
Funding Information
  • Air Force Office of Scientific Research (FA9550-19-1-0101)
  • Air Force Office of Scientific Research (FA9550-22-1-0499)
  • Purdue Doctoral Fellowship
  • Purdue Nuclear Engineering Undergraduate Research Scholarship

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