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 expansions^{2,3} 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.

Let us start with the solutions for the spherical and cylindrical SCLE given by DG in their equations (10), (11), (14), and (15). By inserting the expressions for the potential $\varphi $ in the Poisson equation (4), it yields

which are the generalized versions of Eqs. (11) and (15). Given a steady state, the charge conservation demands that the current density distribution *J*(*r*) satisfies the continuity equation $\u2207\xb7J=0$. Given that *J _{s}* and

*J*follow a spherical and cylindrical symmetry correspondingly, they are both radial, i.e., $J=r\u0302J(r)$. The corresponding divergences are

_{c}We see that in both cases $\u2207\xb7J\u22600$. This means that the charge is not conserved, which is clearly not physical. Therefore, DG's equations (10), (11), (14), and (15) are not a valid solution of the SCLE problem. An equivalent way to prove this would be to directly solve the continuity equation, yielding $Jc\u221dr\u22121\u2009and\u2009Js\u221dr\u22122$ and then show that (10) and (14) do not satisfy the Poisson equation (4).

However, expressions (10) and (14) are valid solutions of Eq. (7), since they are of the form $\varphi =Vg\varphi 04/3,$ where $\varphi 0$ is the solution of the Laplace equation (i.e., a harmonic function), with boundary conditions $\varphi =0$ at the cathode and $\varphi =1$ at the anode. This form is correct for the planar geometry [Eq. (8)] but fails in the general case. In a more recent paper, Harsha and Garner^{4} attempt to validate this form for the cylindrical case, by considering it as a transformation of the planar solution (8) via a conformal mapping. Although this transformation is correct, the result is not a valid solution of the SCLE problem. Such conformal mappings can be used to transform $\varphi 0$ from one geometry to another, because they preserve harmonicity.^{5} Nevertheless, the solution of the Poisson equation $\varphi $ is not a harmonic function, since $\u22072\varphi \u22600$. Thus, its conformal mapping transformation does not yield a valid solution for a different geometry.

Nevertheless, it can be shown that functions of the form $\varphi =Vg\varphi 04/3,$ satisfy Eq. (7) of Ref. 1 for any electrode geometry. We can, therefore, deduce that Eq. (7) is not valid for the spherical and cylindrical cases. Yet, it is valid for the planar geometry, as it can be derived directly from the Poisson equation (4), following the procedure outlined in Eq. (6) of Ref. 6, which is, however, specific to the planar case. Furthermore, Eq. (7) is a valid solution of the Euler–Lagrange equation (6) for the functional (5). Thus, we conclude that the error originates from expression (5), which is not a valid functional to be extremized in order to solve the SCLE problem. DG motivates this as a power minimization, but they do not justify how this results in a solution that satisfies the Poisson and continuity equations. The correct functional to be minimized here would be the integral of the electrostatic Lagrangian density, for which the Euler–Lagrange equation would lead back to Eq. (4).

Finally, DG compare their solution to the particle-in-cell (PIC) simulation results of Ref. 7, pointing out that their analytical equations reproduce them in better accuracy than the LB solutions. However, this comparison is done for only a specific case and does not hold in general. For instance, if we perform the same comparison for the rightmost point of Fig. 6 of Ref. 7, for which *R _{a}* = 10 mm and

*R*= 1 mm, we find that the LB solution deviates from the numerical one by 9.4%, while the one given by DG in Eq. (11)

_{c}^{1}deviates by 72.3%. Therefore, the conclusion that DG's solution is in better agreement with numerical PIC calculations is not supported. The small deviation between the PIC results and the classical LB solutions is due to numerical errors and the finite size of the cylindrical electrodes used in Ref. 7.

In conclusion, we have shown that the expressions derived in Refs. 1 and 4 are not valid solutions of the SCLE problem. The error is attributed to the arbitrary selection of the functional to be extremized, which does not result in a valid solution of the Poisson and continuity equations.

This work was funded by the European Union's Horizon 2020 research and innovation program, under Grant Agreement No. 856705 (ERA Chair “MATTER”).