A numerical and efficient model for thermo-field electron emission calculations
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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BibTeX
@misc{carceles2026numerical,
title = {{A numerical and efficient model for thermo-field electron emission calculations}},
author = {Salvador Barranco Cárceles and Andreas Kyritsakis and Anthony Ayari},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07013},
eprint = {2610.07013},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05795910v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A numerical and efficient model for thermo-field electron emission calculations
AU - Salvador Barranco Cárceles
AU - Andreas Kyritsakis
AU - Anthony Ayari
PY - 2026
DA - 2026/10/04
PB - HAL
DO - 10.48550/arxiv.2610.07013
UR - https://hal.science/hal-05795910v1
AB - 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.
ER -