The present paper proposes the direct calculation of the microscopic contributions to the magneto-electric coupling, using ab initio methods. The electrostrictive and the Dzyaloshinskii-Moriya contributions were evaluated individually. For this purpo
se a specific method was designed, combining DFT calculations and embedded fragments, explicitely correlated, quantum chemical calculations. This method allowed us to calculate the evolution of the magnetic couplings as a function of an applied electric field. We found that in $rm YMnO_3$ the Dzyaloshinskii-Moriya contribution to the magneto-electric effect is three orders of magnitude weaker than the electrostrictive contribution. Strictive effects are thus dominant in the magnetic exchange evolution under an applied electric field, and by extension on the magneto-electric effect. These effects remain however quite small and the modifications of the magnetic excitations under an applied electric field will be difficult to observe experimentally. Another important conclusion is that the amplitude of the magneto-electric effect is very small. Indeed, it can be shown that the linear magneto-electric tensor is null due to the inter-layer symmetry operations.
We introduce a thermodynamical model based on the two-temperature approach in order to fully understand the dynamics of the coherent A$_{1g}$ phonon in laser-excited bismuth. Using this model, we simulate the time evolution of (111) Bragg peak intens
ities measured by Fritz {it{et al}} [Science {bf 315}, 633 (2007)] in femtosecond X-ray diffraction experiments performed on a bismuth film for different laser fluences. The agreement between theoretical and experimental results is striking not only because we use fluences very close to the experimental ones but also because most of the model parameters are obtained from {it{ab-initio}} calculations performed for different electron temperatures.
