Anisotropic Boltzmann transport in a tight-binding superlattice: Exact results for power-law energy-dependent scattering

Abstract

We develop a complete Boltzmann transport theory for the in-plane ($σ_{xx}$) and out-of-plane ($σ_{zz}$) conductivities of an electronic superlattice consisting of a periodic stack of two-dimensional (2D) parabolic-band layers coupled along the superlattice axis by nearest-neighbor tunneling $t$, in the physically relevant regime $t\ll E_F$ where the Fermi surface is an open, corrugated cylinder. Within the relaxation-time approximation with a power-law energy-dependent scattering time $τ(E)\propto E^{p}$, we study the seven models $p=0,\pm1,\pm3/2,\pm2$ as functions of the 2D electron density $n$ per layer and temperature $T$. For $p=-2$ (in plane) and $p\le -1$ (out of plane), long-lived carriers at the miniband bottom can dominate transport at finite temperature. For $p=-2$, these carriers produce a pronounced peak in $σ_{zz}(T)$ at $T\sim E_F$. We find that the temperature dependence of the anisotropic metallic dc conductivity can be highly counterintuitive depending on $t$, $T$, and $p$, with no generic increase or decrease with temperature. The Lorenz ratio is determined by the energy variance of the current-carrying electrons; the Wiedemann-Franz (WF) law is exact as $T\to0$ for all models at fixed $t$, with anisotropic corrections at finite temperature. Band-edge carriers first enhance the interlayer Lorenz ratio up to several times its low-temperature value through thermal occupation of separated energy ranges within the same electron band and then suppress it at high temperature by energy filtering. Strong violations of the Wiedemann-Franz law can therefore arise from miniband structure and energy-dependent elastic scattering alone.