Open System Study of Conductance Quantization in 2D Chern Insulators

Abstract

This thesis aims to provide a microscopic understanding of the quantized conductance observed in experimental setups of 2D topological insulators (TIs), such as HgTe quantum wells, and in numerical simulations of corresponding microscopic models. These materials exhibit dissipationless edge transport in specific parameter regimes, with the quantization of two-terminal conductance serving as a key experimental signature of their topological nature. Conductance measurements are typically performed in a Hall bar geometry, where the system is connected to metallic leads. The conventional explanation for the quantization of two-terminal conductance observed in 2D TI setups relies on the Landauer-Buttiker formalism, which equates it to the Hall conductance under two key assumptions: (i) there is effectively a single channel between the metallic leads and the system, and (ii) electron transmission at the metal-TI junction is perfect. However, it remains unclear why electron transmission should be perfect through this single channel regardless of the junction details. To the best of our knowledge, no analytical proof of these assumptions exists in the literature. In this work, we study a microscopic model of a Chern insulator coupled to metallic leads that closely mimics experimental transport setups. Using the non-equilibrium Green’s function (NEGF) formalism, we analyze the roles of coupling and finite-size effects on the two-terminal conductance. Furthermore, by adopting a cylindrical geometry with idealized reservoirs, we derive an analytic expression for the Hall conductance in an integral form. The obtained expression is analytically shown to be quantized in the weak coupling limit. This integral turns out to be independent of coupling (even though the integrand depends on coupling) which suggests it’s possible connection to a topological invariant similar to the TKNN expression (Thouless et. al. 1982) for closed periodic system where Hall conductivity is related to Chern number of filled bands. Our results provide a theoretical foundation for understanding the quantization and robustness of conductance in an open system setup using NEGF. This study is a stepping stone towards bridging the gap between the experimentally measured two-terminal conductance and the theoretical concept of topological invariants, offering insights that could possibly guide the design of improved experimental setups for realizing 2D topological insulators.