Numerical solver for the out-of-equilibrium time dependent Boltzmann collision operator: Application to 2D materials

Abstract

The Time Dependent Boltzmann equation (TDBE) is a very efficient description of a vast variety of strongly out-of-equilibrium dynamics which are becoming increasingly critical for many novel physical applications. However its applicability is greatly limited by the impractical scaling of the numerical cost with precision of its scattering integral, if addressed without any simplifying approximations (like closedness to equilibrium). In our previous works [1,2] we had proposed a numerical solver to calculate the scattering integral term in the TDBE and then improved on it [3] to include second degree momentum discretisation and adaptive time stepping. Our solver requires no close-to-equilibrium assumptions and can work with realistic band structures and scattering amplitudes. Moreover, it is numerically efficient and extremely robust against inherent numerical instabilities. While in our previous work [3] we showcased the application of our solver to 1D materials, here we showcase its applications to a simple 2D system. We show that in spite of an important increase in complexity of the scattering integrals, the 1D approach can be applied to higher dimensions. We then analyse thermalisation of a set of out-of-equilibrium initial conditions. The excitations added at higher energies were found to thermalise faster than those introduced at relatively lower energies. Also, we conclude that the thermalisation of strong out-of-equilibrium population to equilibrium values is not a simple exponential decay but rather a non-trivial function of time. Nonetheless, by fitting a double exponential function to the decay of the out-of-equilibrium population with time we were able to generate quantitative insights into the time scales involved in the thermalisations. Program summaryProgram Title: OneBandScattering-nonPerturbativeCPC Library link to program files:https://doi.org/10.17632/82bwxs5ds7.1Developer's repository link:https://github.com/MarcoBattiato/OneBandScattering-nonPerturbativeLicensing provisions: GPLv3Programming language: MATLABNature of problem: Numerical solver for the time dependent far-from-equilibrium Boltzmann scattering equation in 2D.Solution method: Ad hoc technique developed in [1,2] and here.Additional comments including restrictions and unusual features: Although extension to include a generic number of scattering legs, multiple band and multiple types of quasiparticles is straightforward, this version of the code is built for a single band electron-electron scatterings.