Real- and reciprocal-space attributes of band tail states

John F Wager

doi:10.1063/1.5008521

Abstract

Band tail states are localized electronic states existing near conduction and valence band edges. Band tail states are invariably found to exhibit an exponential distribution defined by a characteristic (Urbach) energy. To a large extent, the band tail state density of states determines the electronic performance of an amorphous semiconductor (or insulator) in terms of its mobility. Real-space assessment of a suitable density of states model for extended (delocalized) conduction or valence band states and nearby localized band tail states leads to an expression for the peak density of band tail states at the mobility edge and for the total band tail state density. Assuming a continuous density of states and its derivative with respect to energy across the mobility edge, these densities are found to depend on only two parameters – the Urbach energy and an effective mass characterizing the extended state density above the mobility edge. Reciprocal-space assessment is then employed to deduce a probability density function associated with band tail states. The full width at half maximum of the resulting Gaussian probability density function is found to be equal to the average real-space distance of separation between band tail states, as estimated from the total band tail state density. This real- and reciprocal-space insight may be useful for developing new high-performance amorphous semiconductors and for modeling their electronic properties.

Highlights

Band tail states are localized electronic states existing just below the conduction band or right above the valence band
If it is assumed that the density of states and its derivative with respect to energy are continuous across the mobility edge, it is possible to derive simple expressions for the peak density of band tail states (NTA) and for the total band tail state density, both of which depend on only two parameters, i.e., the Urbach energy (WTA) and an effective mass for the extended states beyond the mobility edge
Assessment of the physics of conduction band tail states reveals the existence of a reciprocal space in which wave vectors are imaginary since they correspond to decaying, evanescent, localized states, rather than the propagating states of ‘normal’ k-space

Summary

INTRODUCTION

Band tail states are localized electronic states existing just below the conduction band or right above the valence band. We model a semiconductor (or insulator) as possessing both localized band tail states (with an exponential distribution) and extended (or delocalized) conduction or valence band states. Localized and extended states are distinguished by their energy with respect to a mobility edge This near-band-edge density of states picture is our launching pad. If it is assumed that the density of states and its derivative with respect to energy are continuous across the mobility edge, it is possible to derive simple expressions for the (conduction band) peak density of band tail states (NTA) and for the total band tail state density (nTOTAL), both of which depend on only two parameters, i.e., the Urbach energy (WTA) and an effective mass for the extended states beyond the mobility edge (me*). The real- and reciprocal-space topics investigated in this contribution are found to be linked since, it turns out, the average real space distance between band tail states () is approximately equal to the full width at half maximum of the Gaussian pdf (FWHM) for the four semiconductors considered (i.e., ≈ FWHM)

TOTAL BAND TAIL STATE DENSITY

BAND TAIL STATES PROBABILITY DENSITY FUNCTION

ASSESSMENT OF FOUR SEMICONDUCTORS

CONCLUSIONS