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 (σxx) and out-of-plane (σzz) 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≪ EF where the Fermi surface is an open, corrugated cylinder. Within the relaxation-time approximation with a power-law energy-dependent scattering time τ(E)∝ Ep, we study the seven models p=0,±1,±3/2,±2 as functions of the 2D electron density n per layer and temperature T. For p=−2 (in plane) and p≤ −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 σzz(T) at T∼ EF. 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→0 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.
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BibTeX
@misc{sarma2026anisotropic,
title = {{Anisotropic Boltzmann transport in a tight-binding superlattice: Exact results for power-law energy-dependent scattering}},
author = {Sankar Das Sarma and Yuqing Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41101},
eprint = {2610.41101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41101},
}
RIS
TY - UNPB
TI - Anisotropic Boltzmann transport in a tight-binding superlattice: Exact results for power-law energy-dependent scattering
AU - Sankar Das Sarma
AU - Yuqing Lin
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41101
UR - https://arxiv.org/abs/2610.41101
AB - 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≪ 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)∝ E^p, we study the seven models p=0,±1,±3/2,±2 as functions of the 2D electron density n per layer and temperature T. For p=−2 (in plane) and p≤ −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∼ 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→0 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.
ER -