Causal Green’s Function Method for Simulating Nanoscale Heat Transport

Heat management in semiconductor devices at nanoscale is a subject of strong topical interest. It requires materials of high as well low thermal conductivities and involves phonons of a wide frequency range. Robust mathematical tools are needed for simulation at the nano length scales, that are comparable to or even less than the phonon mean free path. At these scales, the continuum theory is not valid, and a fully discrete lattice or multiscale simulation must be carried out.<br/>In addition to the length scales, simulation at multiple time scales is also a challenging problem. The conventional molecular dynamics (MD) is usually limited to a very small temporal range for reasonable convergence. In practice, the integration of the temporal part of the MD equations needs the integration interval to be less than a few femto seconds, whereas the physical processes of interest require simulation up to nanoseconds or even microseconds. This requires at least a few million temporal iterations, may be closer to 100 million iterations. That is computationally too expensive. Various schemes for accelerating the MD are available in the literature but much more is needed for the proper simulation of nanoscale heat transport.<br/>We show that the use of causal Green’s function (GF) in MD can accelerate its temporal convergence by several orders of magnitude. To see this, we recall that in MD, at every time step, we expand the potential energy of the system to first order in atomic displacements and then use the linear Newtonian solution to calculate the atomic displacements at the end of that time interval. The process is continued iteratively until the displacements converge. The process of iteration effectively accounts for the nonlinear terms in the potential. Attempts to improve upon the temporal convergence consist of better solution of the temporal part of the MD.<br/>We approach the problem by a totally different technique. We expand the potential energy up to quadratic terms in displacements and not just linear terms as in MD. It would appear at first sight that the quadratic terms would make the solution of the temporal problem more difficult. We show that the temporal equations can still be solved exactly by using the GF. To ensure causality, we use the retarded GF, for which we use the Laplace transform. We show that the Laplace as well as inverse Laplace transforms are obtained analytically.<br/>Cubic and higher order terms in the potential are accounted through MD type iterations, but the advantage is that much fewer iterations are needed and the time interval can be much larger. We show in certain idealized test cases that the temporal convergence can be improved by 7-8 orders of magnitude.<br/>Physically, the improvement is significant for phonons and elastic properties materials because the phonons’ characteristics of a material are determined by the quandratic terms in the potential. Our GF technique includes the phonon terms exactly, whereas the MD accounts for these term only through computaionally expensive iterations.<br/>We will illustrate our causal GF technique by applying it to real 2D solids of contemporary interest such as graphene or silicene. This will bring out the important characteristics of the nanoscale heat transport such as the size effect, non-Fourier heat conduction and show the power of the causal GF technique for simulating nanoscale heat transport.