Simulation et analyse modale du transport de chaleur dans les réseaux à dimensionnalité réduite

This thesis showcases the simulation of low-dimensional materials, specifically oscillator chains and graphene models, with regard to their phonon thermal transport properties. Understanding heat transport properties of materials has been a point of focus throughout most of history and today has repercussions in aerospace, microelectronics, power generation as well as many other topics related to energy efficiency. Current focus on low-dimensional materials stems from the explosion in the availability of these materials as well as from their exotic properties. Indeed, peculiarities related to statistical physics in one or two dimensions allows to circumvent some of the limitations coming from Fourier's law, which is almost always obeyed in three-dimensional materials, and therefore to create materials with previously unobtainable properties. Proposed contributions make use of equilibrium molecular dynamics simulations interpreted through linear response theory and normal mode analysis. Improvements to these algorithms are presented, which allow a better understanding of heat flux correlation functions. Special consideration is given to correlations effects which go beyond the phonon gas model.  The relationship between those effects and the anomalous thermal conductivity phenomenon, which refers to materials whose thermal conductivity increases with the size of the sample, is explored in one-dimensional oscillator chains toy models. Specifically, \MG{variants of the Fermi-Pasta-Ulam model, the FPU-$\beta$ and FPU-$\alpha\beta$ models, are studied. These models are anharmonic oscillator chains in one dimension, each having a slightly different anharmonic interaction term. The results are compared to those pertaining to the $\phi^4$ model,  which is also an anharmonic chain but is however a normal conductor}. Our computations show the existence of strong collective excitations in each case, disproving the hypothesis that anomalous thermal conductivity stems from these excitations. The FPU-$\beta$ is probed over a range of the anharmonicity parameter $\beta$ and clearly displays anomalous thermal conductivity in the quasi-recurrence (weak stochasticity) regime as well as in the chaotic (strong stochasticity) regime also refuting the previously held idea that chaotic dynamics is a sufficient condition for normal conductivity.  We then explore thermal transport in graphene. Our simulations, in very good agreement with experiments, show that most of thermal transport happens through the action of flexural modes. This can be explained by strong collective effects that allow these modes to have effective relaxation times which go beyond a few nanoseconds. This effect is however curbed by the interaction between the graphene layers that form a graphite crystal. Furthermore, our computations show that applying strain to graphene makes it cross from the normal conductivity regime to one where the conductivity displays a logarithmic divergence as a function of size.  Finally, we present a new algorithm that allows the use of molecular dynamics to compute relaxons eigenmodes. Relaxons are quasiparticles that embody the collective action of a phonon packet. This algorithm is put to use in the analysis of the FPU-$\beta$ system. A subset of the relaxon eigenvectors obtained in this manner show strong contributions from a small group of long wavelength phonons that act in phase while the other are made of short wavelength phonons which act out of phase. The former have fairly high group velocities while the velocities of the latter are close to zero. Relaxon lifetimes are of the same order of magnitude as phonon relaxation times although there is always at least one relaxon that has a relaxation time longer than that of any phonon.  As a whole, results presented in this thesis highlight the importance and the ubiquity of manybody effects produced by anharmonic interactions between phonon modes in low-dimensional materials. These effects were observed in every model under study and can, in some cases, yield huge contributions to the heat conductivity. Our understanding of the general properties of those collective excitations is not yet complete, which will prompt further investigation. Nevertheless, theoretical tools presented in this thesis, most importantly in chapter 4, provide an appealing basis for these future works.

Keyphrases: Phonons, Relaxons, Heat transport, Molecular dynamics, Normal mode analysis, FPU model, Graphene