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arXiv · 2610.07013

A numerical and efficient model for thermo-field electron emission calculations

Salvador Barranco Cárceles·Andreas Kyritsakis·Anthony Ayari
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Abstract

As electron sources are reduced in size, fabricated in new materials, or pushed for performance, the analytical formulations for thermo-field electron emission, namely Fowler-Nordheim and Murphy-Good equations and their JWKB-based extensions, are increasingly applied out of the range of conditions for which they were derived. We present GETELEC-3, an open-source code that replaces them with a direct numerical solution of the one-dimensional time-independent Schrödinger equation (1D-TISE). Several methods to solve the 1D-TISE have been implemented to test for speed, robustness, and solution's uncertainty. The Noumerov algorithm is found to be quickest, with the solution uncertainty below 1%, and together with an optimised kernel that advances all electron energies in parallel, the cost of calculating the current density is reduced to a few milliseconds - comparable to the JWKB evaluation. To further boost the calculation speed, especially in complicated barriers, a neural network has been trained on the residual of the logarithm of the semiclassical reference and it reproduces the current density better than 0.4% at a fraction of the computational cost. The code is developed for metals and semiconductors, accepts arbitrary potentials and density of states, and comes with a user-friendly interface and compiled executable to democratise rigorous electron emission calculations and data analysis.

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Salvador Barranco Cárceles, Andreas Kyritsakis, Anthony Ayari. 2026-10-04. A numerical and efficient model for thermo-field electron emission calculations. https://arxiv.org/abs/2610.07013

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