Analytical derivation of the effective temperature for field-dependent hopping conductivity

The electric-field enhancement of hopping conductivity in amorphous solids can be described in terms of an effective temperature T-eff given by T(eff)(2)approximate toT(2)+[C(F)gamma(-1) eF/k](2) for hopping, at a temperature T and field F, within a uniform distribution of electronic states with localization length gamma(-1). We have derived this expression analytically and find at low fields a value for C of 1/2(1/2), which is consistent with the F-independent value C=0.67 previously obtained numerically for band-tail hopping. With increasing electric field, the decrease in C(F) matches the hopping transport characteristics in amorphous semiconductors (hydrogenated amorphous silicon and hydrogenated amorphous carbon nitride) better than previous numerical simulations.