In state-of-the-art MBPT neutral excitations are described through an approximate version of the Bethe-Salpeter equation (BSE) that completely neglects dynamical effects. As a consequence, in this approach finite lifetime effects are completely neglected. Moreover we are not able to describe multiple excitations that are strictly related to dynamical effects. The aim of DENE is the development of a theoretical tool for the description of dynamical effects beyond the static BSE. The project, carried out at the Laboratoire ded Solides Irradies (LSI), Ecole Polytechnique under the supervision of Dr. Lucia Reining, has been developed in three main steps: The first two (WP1 and WP2) concern the development and the implementation in a numerical code of the theory in order to include dynamical effects beyond the BSE. The last step (WP3) Is focused on the application of the theory in real materials where dynamical effects are expected to be important.
1) Dynamical effects beyond the state-of the-art: theory (WP1) and coding (WP2)
Starting from the recent results obtained from the host group in the evaluation of the single particle Green’s function through a “cumulant” approach, the aim of this part of the project was to do a further step along that direction: the generalization of the theory to the two particles Green’s function that contains all the informations about neutral excitations. To this end, I derived an integro-differential equation for the two particles Green’s function and I demonstrated that in the case of a two level system coupled with a boson its solution has a cumulant structure in analogy with what happens for the single particle Green's function where all dynamical effects are included through a frequency dependent coefficient (cumulant coefficient). In a second step I generalized the theory to real systems involving infinite crystals as well as finite systems as atoms and molecules demonstrating that also in this case it is possible to find an approximate expression for the two particle Green's function with a cumulant structure. In particular I derived an expression for the cumulant coefficient in terms of an effective exciton self-energy that describes the coupling between excitons and other neutral excitations namely other excitons and plasmons. The exciton self-energy is expressed in terms of the eigenstates and eigenvalues of the excitonic hamiltonian that is diagonalized by standard numerical codes used to solve the static BSE. This make possible the implementation of the cumulant as a post processing of the static BSE. In collaboration with other members of the LSI (Dr. Matteo Gatti and Dr. Francesco Sottile) I am implementing the evaluation of the cumulanf coefficient in the EXC code, a GPL computational code that is freely available to the whole community (
http://dp-code.org(si apre in una nuova finestra)).
2) Scientific cases (WP3)
At the beginning the method was applied to a model system characterized by two bands coupled with a boson where the excitonic effect has been treated in two limits: strong localization (Frenkel exciton) and weak localization (Wannier exciton). The study allowed inferring a general picture of the dynamical effects and how this latter are related to the properties of the exciton as the exciton Bohr radius and exciton dispersion. Moreover I identified some features that could enhance dynamical effects as the presence of localized states and charge transfer excitations.
The application of the cumulant to model systems suggests that dynamical effects are important in systems characterized by localized states. For this reason I decided to apply the new method to the study of the optical properties of 2D and layered systems.
In particular I did standard BSE calculations on single layer and bulk hBN investigating the properties of the excitons and their dispersion as a function of the momentum transfer. I demonstrated that, due to the confinement of the electronic charge on a layer, at low momentum transfer the exciton energy is characterized by a linear dispersion relation which is a feature common to all 2D systems. In addition I demonstrated that in 2D systems the exciton band structure provides a powerful way to identify the exciton degree of localization. The work represents the first step towards the investigation of exciton dynamics in 2D that is still in progress.
Finally I plan to study strongly correlated transition metal oxides (NiO and CoO) where IXS experiments revealed the presence of multiple excitations at high momentum transfer.
