The microscopic response method: Theory of transport for systems with both topological and thermal disorder

Abstract

In this paper, we review and substantially develop the recently proposed ''Microscopic Response Method,'' which has been devised to compute transport coefficients and especially associated temperature dependence in complex materials. The conductivity and Hall mobility of amorphous semiconductors (ASs) and semiconducting polymers are systematically derived, and shown to be more practical than the Kubo formalism. The effect of a quantized lattice (phonons) on transport coefficients is fully included and then integrated out, providing the primary temperature dependence for the transport coefficients. For higher-order processes, using a diagrammatic expansion, one can consistently include all important contributions to a given order and directly write out the expressions of transport coefficients for various processes.

Theory of temperature coefficient of resistivity: Application to amorphous Si and Ge

“…In addition, the MRM categorizes transport processes with diagrams computed to any given order of residual interactions [6]. We have seen that even to zero order in the residual interactions, LE and EL transitions contribute to conductivity [6]. Indeed, if one calculates the electrical conductivity of an AS from the full density matrix rather than its diagonal elements (master equation), one sees that LE and EL transitions contribute directly to electrical conduction.…”

“…We first calculate the conductivity from the LE transitions (line 2b of table 4 in ref. [6]). When k B T ≥ ω (the first peak of vibrational spectrum, ω = 232 K for a-Si [23]), the two time integrals I B1A± can be approximated by an asymptotic expansion [22], and the vibrational degrees of freedom are integrated out.…”

“…Next we consider the conductivity from EL transitions driven by a field (line 6a of table 5 in ref. [6]). Because the field-matter coupling is Hermitian, the conductivity for EL transition driven by a field may be obtained from eq.…”

“…If there are several processes contributing to conductivity in a material, according to MRM, the total conductivity σ of the material is σ = j σ j ,w h e r eσ j is the conductivity from the j-th process (LL, LE and EL etc.) [6]. The overall TCR κ relates to the TCR for each process by κ = σ −1 j σ j κ j ,whereκ j = −σ −1 j dσ j /dT .…”

“…In conclusion, the microscopic response method expresses transport coefficients with transition amplitudes [5,6] rather than with transition probability per unit time, which enables the method to be used with amorphous semiconductors for which the Landau-Peierls condition is violated [3]. The conductivities from the three simplest transitions, LL, LE and EL transitions driven by a field, are expressed by several material parameters.…”