Theory and calculations of thermoelectric transport in heterostructures

TL;DR

This thesis explores theoretical frameworks and computational methods for thermoelectric transport in heterostructures, connecting fundamental formalisms with practical numerical calculations.

Contribution

It provides a comprehensive derivation of transport formalisms and introduces new computational techniques for thermoelectric properties in heterostructures.

Findings

Derived connections between transport theories

Implemented Boltzmann Monte Carlo simulations

Applied NEGF and Landauer frameworks for calculations

Abstract

This thesis is divided into two major parts. Part I provides an overview of some of the most important formalisms of transport theory, while Part II provides a presentation and discussion of my own work. The order in which these parts are read is not important, as long as the reader is prepared to accept that part II references a few topics in part I. Part 1 discusses 6 different theoretical frameworks: The Landauer B\"uttiker framework, the Keldysh field integral, linear response theory, Non-Equilibrium Green's function (NEGF) formalism, Markovian Master equations, and the semi classical Boltzmann equation. Considerable weight is put on illuminating the connections between these frameworks, and in particular an attempt is made to derive all of the latter four as a sequence of approximations starting from the Keldysh field integral, admittedly with some appeals to physical intuition. While part 1 is mostly concerned with fundamental theory, the focus of part 2 is on computational and numerical aspects. It discusses a set of practical calculations we have made, using a subset of the formalisms from part 1. In particular, we have made use of the Boltzmann Monte Carlo method to solve the Boltzmann equation, applying both direct calculations and the Green Kubo relations of linear response. We have also made some calculations in the Landauer B\"uttiker framework, utilizing B\"uttikers approximation to introduce scattering in an otherwise ballistic solver. Finally, we have performed a set of calculations within the NEGF framework, which illustrate a simple perturbative approach useful for speeding up the calculations of Gr, as well as a Monte Carlo method developed of our own.