# Low dimensional materials

Building Block approach

Ab initio framework is a powerfull tool to simulate electronic excitations of materials. Nevertheless, the size of the object is limited to a rather small number of atoms, usually much smaller than the one of systems under interest. Indeed, if we consider carbon nanotubes of large diameter, depending of their chirality, the number of atoms is rapidly out of range of ab initio calculations. One of the ideas is to calculate the dielectric properties of a much smaller system, with similar electronic properties, and to build the excitations of the full object by a building block approach. One example is given by carbon nanotubes, whose excitations can be described with a graphene layer, using a zone folding of the polarizability. Indeed for nanotubes with large diameter, the curvature prevents the overlapping of orbitals and the graphene polarizability is similar to the one of carbon nanotubes.

In such a spirit, one can also calculate the electronic excitations of heterostructures, using as basic ingredients the polarizability of the different components calculated separately in their monolayer geometry. An example has been recently achieved by simulating electron energy loss spectra of a sandwich of graphene/MoS2/graphene. This allows the prospective study of electronic properties of the stacking of layered materials with different lattice parameters.

Selected-G formalism

Most of the codes used to calculate electronic properties of crystals are based on a plane waves formalism in reciprocal space. This comes from the fact that they have been developed for 3D periodic materials. The traditional way to treat a surface in such a framework is to build a supercell, which contains a certain amount of matter to modelized the surface properties, and a certain amount of vacuum to isolate the slab of matter from the artificial replicas arising from the periodically repeated supercell.

Using the TDDFT framework, we have evidenced that the absorption spectrum calculated for an excitation perpendicular to the surface was not correct when accounting for Local Field effects. We have shown that the absorption peak was puched up to 12 eV instead of 4 eV for the bulk conterpart, and that the energy position and the amplitude were dependant of the vaccum introduced in the supercell.

We have derived a new expression for Dyson equation at the basis of TDDFT with quantities having the periodicity of the matter instead of the supercell. This new sheme called Selected-G has allows us to recover the correct absorption spectrum for the excitation perpendicular to the surface. This formalism has been succefully applied to the second harmonic generation od silicon surfaces.