We report a simple closed-form model of the total emission through a triangular potential barrier due to thermionic field emission (TFE) and field emission (FE). Such a model has not been derived previously, since the energy distribution function of emitted electrons is not analytically integrable. We overcame this difficulty using a geometrical approximation of the integration operation. Our model so derived reveals the energy location and spread of the emission, which allow estimation of the emission through any fraction of the barrier. It also yields a characteristic field parameter in terms of the barrier height, temperature, and effective mass, which can be used to identify the TFE, FE, and TE regimes of device operation.
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Karmalkar et al. (2003) studied this question.
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