The Bethe condition for thermionic emission near an absorbing boundary

The Bethe condition for thermionic emission in Schottky diodes is analysed in terms of the transport of carriers near the top of an absorbing barrier, under forward bias. It is shown that the usually accepted values of the drift velocity v_s and of the carrier density n(0) at the barrier maximum are not correct, v_s being underestimated by a factor of 2, and n(0) being overestimated by about the same factor. Fortunately, these discrepancies compensate in the product n(0)v_s, which yields the correct forward current. It is shown also by means of examples that the Bethe condition is satisfied only over a relatively limited range.