Comment on 'A coordinate system invariant formulation for space-charge limited current in vacuum' [Appl. Phys. Lett. 115, 054101 (2019)]

摘要

In a recent letter, Darr and Garner (DG)1 used variational calculus (VC) to derive exact closed-form analytical formulas for the space charge limited electron emission (SCLE) current from the coaxial cylindrical and concentric spherical emitting diodes. The solutions derived by the authors are not in agreement with the classical Langmuir-Blodgett (LB) series expansions2,3 for the same problem. The authors claim that this disagreement can be attributed to the fact that the LB solutions "do not account for all relevant physics."In this Comment, we show that the solutions presented by DG do not satisfy the Poisson and the continuity equations and, therefore, are not valid solutions of the SCLE problem. Thus, we restore the validity of the classical LB equations,2,3 under the assumptions they were derived, i.e., a continuous charge flow described by classical mechanics, negligible injection velocity, and negligible relativistic effects. Finally, we attribute the invalidation of DG's results to the usage of invalid minimization functional in their VC derivation.