Thermionic field (T-F) emission in uniformly doped metal-semiconductor (Schottky) barriers is analyzed to yield a normalized solution in closed form for the forward and reverse current (I)-voltage (V) relationship. A quasi one-dimensional approach and Maxwell-Boltzmann statistics are used. The formulation is expressed in terms of the 'flat-band' current density Im, the band bending Eb in the semiconductor depletion region, the materials constant E0 0 (ln[ I Im] = - Eb E0 0 at 0°K in the WKB approximation), and kT. The kinetic energy in units of Eb at which the maximum injection of carriers occurs in the semiconductor is shown to be cosh -2(kT/E0 0). Current flow in the temperature range between pure thermionic emission (kT/E0 0 ≫ 1) and pure field emission (kT/E0 0 ≪ 1) is analyzed and criteria for the transition of T-F emission to thermionic and to field emission are given. Computer solutions for the energy distribution of the injected carriers and for the normalized I-V characteristic are presented in graphical form. The results permit a straightforward calculation of the barrier height and the impurity concentration in the semiconductor from measurements of current density and differential resistance at a single applied bias. Application of these results explains a reported discrepancy between barrier heights deduced from photothreshold, C-V and I-V characteristics of WGaAs and AuGaAs Schottky barriers. A relatively constant excess temperature T0 (i.e., ln I ∝ (T + T0) when V ≫ kT/q) is predicted in the case of large Eb/E0 0 in the higher kT/E0 0 range where thermionic emission is nearly predominant. I ∝ [exp(qV/kT) - 1] is shown to be a general expectation for all Schottky barriers near zero bias when the I-V characteristic is dominated by either thermionic or thermionic-field emission. The assumption of a Gaussian energy distribution of carriers leads to values for the slope of ln I vs. V in reasonable agreement with the results of the computer analysis, but the prediction of the absolute value of the current density deviates rapidly from the computed value when kT/E0 0 departs appreciably from unity. The Gaussian distribution also does not provide the smooth transition from T-F to thermionic emission characteristic of the computer solution. © 1969.
