Generalized Boltzmann relations in semiconductors including band tails

Abstract

Boltzmann relations are widely used in semiconductor physics to express the charge-carrier densities as a function of the Fermi level and temperature. However, these simple exponential relations only apply to sharp band edges of the conduction and valence bands. In this article, we present a generalization of the Boltzmann relations accounting for exponential band tails. To this end, the required Fermi–Dirac integral is first recast as a Gauss hypergeometric function followed by a suitable transformation of that special function and a zeroth-order series expansion using the hypergeometric series. This results in simple relations for the electron and hole densities that each involve two exponentials. One exponential depends on the temperature and the other one on the band-tail parameter. The proposed relations tend to the Boltzmann relations if the band-tail parameters tend to zero. This work is timely for the modeling of semiconductor devices at cryogenic temperatures for large-scale quantum computing.

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“…It is worth mentioning that the physical mechanism underlying the SS saturation limit at extremely low temperatures has been debated in the scientific community. Bohuslavskyi et al [54] and Beckers et al [53,55] proposed that the presence of an exponential band tail contributes to SS saturation at deep-cryogenic temperatures. This tail arises from various factors like crystalline disorder, surface roughness, strain, and residual impurities [54].…”

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“…Since the generated V o -related defects are mainly distributed near the bottom of the E C [29,30], the conduction band tail slope is adjusted to describe the creation of V o -related defects for a-IGZO TFTs under a light illumination process. It has been reported that the exponential band tails can be expressed by a generalization of Boltzmann relations in semiconductor materials [31,32]. Thus, in this work, the Boltzmann distribution relation is applied to describe the generated V o -related defects at the bottom of the E C under light conditions.…”

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“…This, however, imposes an upper limit on the maximum power dissipation of the CMOS circuitry since conventional DR has very low thermal budget (cooling power limited to ≈1 W at 4 K). Although this power dissipation can be scaled down by minimizing the CMOS operational voltage ( V DD ), conventional bulk‐semiconductors with their inefficient electrostatics and large band‐tails (representing exponentially decaying states within the material bandgap) limit the theoretically achievable minimum V DD , [ 264 ] and hence, become impractical for realizing highly scaled and integrated qubit systems. 2D‐M not only offers excellent electrostatics, thereby lowering voltage and power dissipation, but the layered and pristine nature of 2D‐TMDs, coupled with their low phonon DOS, result in extremely short band‐tails (Urbach energies of meV) (Figure 2e), [ 14 ] allowing for continued SS, and hence, V DD scaling till cryogenic temperatures.…”