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

Ultrafast Thermalization Pathways of Excited Bulk and Surface States in the Ferroelectric Rashba Semiconductor GeTe

“…The were carried out using a recently developed numerical technique to solve the full, non-linearized, time-dependent Boltzmann scattering equation with no close to equilibrium approximations and simultaneous particle, momentum, and energy conservation. [66][67][68] The method uses piecewise continuous polynomial basis functions for populations, dispersions, and scattering matrix elements, and a hybrid analytical inversion and Monte Carlo integration for the scattering integrals. For the input of the calculations, the dispersions of the three electronic bands SS1, SS2, and BS were discretized by assuming cylindrical symmetry and the experimental band dispersions in Figure 1 and ref.…”

“…Finally, in order to identify which individual scattering channels are predominantly responsible for the contrasting temperature dependence of the thermalization pathways within the electronic subsystem and between the electronic subsystem and the lattice, we performed theoretical calculations within the Boltzmann approach including all types of elementary scatterings. [66][67][68] Among several existing stateof-the-art theoretical methods that have proven successful for describing nonequilibrium dynamics of excited states, such as real-time time-dependent density-functional theory (RTTD-DFT) [69] and other types of nonequilibrium DFT calculations, [70] the Boltzmann approach is a relatively inexpensive and very powerful method to identify in detail which individual scattering channels drive the different thermalization pathways. [66][67][68] To this end, we computed the temperature dependence of the scattering rates for all electron-electron and electron-phonon scatterings that can be obtained from the Boltzmann scattering equation when combining the bulk (BS) and surface (SS1, SS2) transient Rashba bands with a single effective phonon band.…”

“…[66][67][68] Among several existing stateof-the-art theoretical methods that have proven successful for describing nonequilibrium dynamics of excited states, such as real-time time-dependent density-functional theory (RTTD-DFT) [69] and other types of nonequilibrium DFT calculations, [70] the Boltzmann approach is a relatively inexpensive and very powerful method to identify in detail which individual scattering channels drive the different thermalization pathways. [66][67][68] To this end, we computed the temperature dependence of the scattering rates for all electron-electron and electron-phonon scatterings that can be obtained from the Boltzmann scattering equation when combining the bulk (BS) and surface (SS1, SS2) transient Rashba bands with a single effective phonon band. The calculations of the scattering rates account for all potential spin-dependent intraband and interband transitions under any given scattering angle in energy-momentum space.…”

“…The resulting electronelectron (electron-phonon) scattering integrals are in eight (six) dimensions and include three Dirac delta functions representing momentum and energy conservation. The integrals are performed numerically using a recently developed technique to solve the full time-dependent Boltzmann scattering integral without any close-to-equilibrium approximation and with simultaneous momentum, energy, and particle conservation [66][67][68] (see Experimental Section and Supporting Information Note S2 for more details).…”

Out-of-equilibrium dynamics in real band structures

“…[40] and ref. [41]. This material was mainly written by me, although reviewed and modified slightly by the co-authors.…”