Theory of potential distributions in abrupt heterojunction crystalline semiconductor devices: Treatment of Schottky barriers and rectifiers

Abstract

A theoretical method for potential distribution in abrupt heterojunctions (HJs) made of uniformly doped degenerate semiconductors has been developed. The method reduces automatically to that in HJs from nondegenerate semiconductors in the limits of low carrier concentrations. For the development of the method the rigid band approximation of degenerate semiconductors has been considered to be valid. The transport equations of Marshak and Van Vliet [Solid-State Electron. 21, 417 (1978)] and an analytical approximation for the Fermi–Dirac integral of order half by the present author [Solid-State Electron. 30, 713 (1987)] have been employed for the formulation. The average of the scattered experimental data for band-gap narrowing of n-Si, n-Ge, p-GaAs, and n-InP have been fitted to the same form as that for the Fermi–Dirac integral of order 1/2 to ease this formulation. Local electrostatic field and local electrostatic potentials obtained from the formulation reduce to those of Chatterjee and Marshak [Solid-State Electron. 24, 1111 (1981)], Cserveny [Int. J. Electron. 25, 65 (1968)], and Kroemer [J. Appl. Phys. 52, 873 (1981)] under special conditions. It is noted that band-gap narrowing and consideration of Fermi–Dirac statistics represent opposite effects for effective intrinsic carrier concentration and local electrostatic field. At some critical concentration belonging to the degenerate limit of a semiconductor, these two effects cancel the influence of each other on effective intrinsic carrier concentration of the semiconductor and on transition region properties of an HJ. Below this critical concentration, band-gap narrowing rather than a consideration of Fermi–Dirac statistics dominantly influences the device properties. However, above this critical concentration, consideration of Fermi–Dirac statistics dominates over the other. Applications of electrostatic field and electrostatic potential to isotype and anisotype HJs have been discussed. On the basis of present formulas a general form for potential distribution in Schottky barriers has been derived. The relation reduces to that of Gummel and Scarfetter [J. Appl. Phys. 38, 2148 (1967)] under special conditions. Theoretical reasons underlying the lack of rectification in various n-N HJs have been analyzed. In light of this analysis, a theoretical model in terms of many-body electron-electron and electron-donor interactions, and in terms of lowering of band edge in the vicinity of transition region, has been proposed. Numerical results obtained from this model for n-GaAs agree remarkably with observations from experimental measurements.