Modeling and computation of kubo conductivity for two-dimensional incommensurate bilayers

Simon Etter, Daniel Massatt, Mitchell Luskin, Christoph Ortner

Research output: Contribution to journalArticlepeer-review

Abstract

This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. First, we derive an expression for the large-body limit of Kubo-Greenwood conductivity in terms of an integral of the conductivity function with respect to a current-current correlation measure. We then observe that the incommensurate structure can be exploited to decompose the current-current correlation measure into local contributions and deduce an approximation scheme which is exponentially convergent in terms of domain size. Second, we analyze the cost of computing local conductivities via Chebyshev approximation. Our main finding is that if the inverse temperature \beta is sufficiently small compared to the inverse relaxation time η, namely β ≤ η - 1/2, then the dominant computational cost is O (η - 3/2) inner products for a suitably truncated Chebyshev series, which significantly improves on the O (η -2) inner products required by a naive Chebyshev approximation. Third, we propose a rational approximation scheme for the low temperature regime η - 1/2 \lesssim β, where the cost of the polynomial method increases up to O (β 2), but the rational scheme scales much more mildly with respect to β.

Original languageEnglish (US)
Pages (from-to)1525-1564
Number of pages40
JournalMultiscale Modeling and Simulation
Volume18
Issue number4
DOIs
StatePublished - 2020

Bibliographical note

Funding Information:
The second and third authors were supported in part by ARO MURI Award W911NF-14-1-0247, and the third author was supported in part by NSF DMREF Award 1922165 and NSF Award DMS-1819220. The fourth author was supported by ERC Starting Grant 335120 and Leverhulme Research Project Grant RPG-2017-191. The first and fourth authors received support for visits to the Institute for Mathematics and Its Applications.

Funding Information:
\ast Received by the editors July 9, 2019; accepted for publication (in revised form) September 21, 2020; published electronically December 3, 2020. https://doi.org/10.1137/19M1273499 Funding: The second and third authors were supported in part by ARO MURI Award W911NF-14-1-0247, and the third author was supported in part by NSF DMREF Award 1922165 and NSF Award DMS-1819220. The fourth author was supported by ERC Starting Grant 335120 and Lev-erhulme Research Project Grant RPG-2017-191. The first and fourth authors received support for visits to the Institute for Mathematics and Its Applications. \dagger University of Warwick, Coventry CV4 7AL, UK (S.Etter@warwick.ac.uk). \ddagger University of Chicago, Chicago, IL 60637 USA (dmassatt@uchicago.edu). \S School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (luskin@ umn.edu). \P Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (c.ortner@warwick. ac.uk).

Keywords

  • Chebyshev
  • Kubo
  • Two-dimensional materials

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