Density matrix to quantum master equation (QME) model for arrays of Coulomb coupled quantum dots in the sequential tunneling regime

Abstract

Coulomb coupled quantum dot arrays with staircase ground state configuration have been proposed in literature for enhancing heat-harvesting and refrigeration performance (Erdman et al 2018 Phys. Rev. B 98, 045433; Walldorf et al 2017 Phys. Rev. B 96, 115415; Daré 2019 Phys. Rev. B 100 195427; Zhang and Chen 2019 Physica E 114, 113635; Daré and Lombardo 2017 Phys. Rev. B 96, 115414; Zhang et al 2016 Energy 95, 593; Sánchez and Büttiker 2011 Phys. Rev. B 83 085428; Singha 2018 Phys. Lett. A 382, 3026). Due to their mutual Coulomb interaction, a performance analysis of such systems remains complicated and necessitates consideration of microscopic physics using density matrix formulation. However the path of transport analysis starting from the system Hamiltonian to density matrix formulation is complicated and lacks the simplicity and intuitive aspect of sequential electron transport conveyed by the quantum master equation (QME) approach. In this paper, starting from the system Hamiltonian and employing the density matrix formulation, I derive the QME of a system of three quantum dots, two of which are electro-statically coupled. The framework elaborated in this paper can be further extended to derive QME of systems with higher number of Coulomb coupled quantum dots. Hence, the formulation developed in this paper can pave the way towards an intuitive analysis of transport physics for an array of Coulomb coupled quantum dots in the sequential tunneling regime.