AMaSiS 2018 Workshop: Abstracts

On a Bloch-type model with electron–phonon interactions: modeling and numerical simulations

Brigitte Bidégaray-Fesquet${}^{\mathbf{\text{(1)}}}$, Clément Jourdana${}^{\mathbf{\text{(1)}}}$, and Kole Keita${}^{\mathbf{\text{(2)}}}$

(1) Université Grenoble Alpes, CNRS, Grenoble INP${}^{†}$, Laboratoire Jean Kuntzmann

${}^{†}$ Institute of Engineering Univ. Grenoble Alpes

(2) Université Jean Lorougnon Guédé (UJLoG)

In this work, we discuss how to take into account electron-phonon (e-ph) interactions in a Bloch type model for the description of quantum dots.

Quantum dots are usually described using electrons and holes. As already detailed in [2], we prefer here a conduction and valence electron description, where valence electrons can be seen as an absence of holes in a valence band. Due to the 3D confinement, energy levels are quantized for each species of electrons. To describe these energy level occupations, we define a density matrix $\rho$ whose diagonal terms, called populations, represent the energy level occupation probabilities and off-diagonal terms, called coherences, describe the intra-band and inter-band transitions. The time evolution of $\rho$, described by a Liouville equation, is driven by a free electron Hamiltonian associated to the electron level energies and the interaction with an electromagnetic field which is solution of Maxwell equations (see e.g. [2] for details).

Similarly to the approach proposed in [1] where the addition of Coulomb interactions is discussed, the starting point to take into account e-ph interactions in such a model is to use field quantification to write an e-ph Hamiltonian. In this work, only polar coupling to optical phonons is considered. It is described by a Frölich interaction Hamiltonian (see e.g. [3, 4]). After making explicit commutators involving this e-ph Hamiltonian, the final model that we obtain consists in coupling the Liouville equation on the density matrix $\rho$ with a set of equations on quantities $S_{𝐪}$ called phonon-assisted densities, one for each phonon mode $𝐪$.

After a description of the model derivation, we discuss how to discretize efficiently this non-linear coupling in view of numerical simulations. In particular, equations on $\rho$ and $S_{𝐪}$ are discretized working on a staggered grid in time and each equation is solved using a Strang splitting procedure. An advantage of this splitting is that it numerically allows to preserve positiveness for each quantity. Finally, we present numerical simulations performed for a collection of quantum dots which are scattered in a one dimensional space and interact not directly but through the interaction with the electromagnetic field.