Partially coherent electron transport in terahertz quantum cascade lasers based on a Markovian master equation for the density matrix

TL;DR

This paper develops a Markovian master equation for electron density matrices in terahertz QCLs, capturing coherences and in-plane dynamics, and validates it against experiments and NEGF simulations, highlighting the importance of coherences and non-thermal distributions.

Contribution

It introduces a positivity-preserving Markovian master equation for single-electron density matrices in QCLs, including coherences and in-plane effects, and demonstrates its accuracy against experimental data and NEGF simulations.

Findings

Coherences can be a significant fraction of diagonal elements.

In-plane energy distributions deviate from Maxwellian assumptions.

Different relaxation times for current density and subband occupations.

Abstract

We derive a Markovian master equation for the single-electron density matrix, applicable to quantum cascade lasers (QCLs). The equation conserves the positivity of the density matrix, includes off-diagonal elements (coherences) as well as in-plane dynamics, and accounts for electron scattering with phonons and impurities. We use the model to simulate a terahertz-frequency QCL, and compare the results with both experiment and simulation via nonequilibrium Green's functions (NEGF). We obtain very good agreement with both experiment and NEGF when the QCL is biased for optimal lasing. For the considered device, we show that the magnitude of coherences can be a significant fraction of the diagonal matrix elements, which demonstrates their importance when describing THz QCLs. We show that the in-plane energy distribution can deviate far from a heated Maxwellian distribution, which suggests that the assumption of thermalized subbands in simplified density-matrix models is inadequate. We also show that the current density and subband occupations relax towards their steady-state values on very different time scales.