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Monte Carlo phonon transport at nanoscales Karl Joulain, Damian Terris, Denis Lemonnier Laboratoire d’études thermiques, ENSMA, Futuroscope France David Lacroix LEMTA, Univ Henri Poincaré, Nancy, France
Einstein 1905 Density of particle at x and t. Probability to travel on a distance between x and x+dx during t RW and diffusion equation
Density at time t+t Density expansion RW and diffusion equation
Diffusion equation 100000 particles at the origin at t=0. After 40 jumps: RW and diffusion equation
Nanoscale conductive heat transfer Distribution function Boltzmann Equation Relaxation time approximation
Boltzmann equation resolution methods • Kinetic theory • Radiative transfer equation methods • P1 • Discrete ordinate • Monte Carlo methods Advantages • Geometry • Separation of relaxation times
System divided in cells Phonon energy and number in cells Monte Carlo simulation Earlier work : Peterson (1994), Mazumder and Majumdar (2001)
Too many phonons Weight Nb spectral bins Spectral discretization Distribution function Phonons drawn in cell until Direction Two numbers drawn to choose de phonon direction Initialization Polarization
Drift Phonon scattering Relaxation time t due to anharmonic processes and impurities Modified distribution function Drift and scattering
Temperature imposed at both end of the system Extrem cells are phonon blackbodies Boundary scattering Diffuse or specular reflexion at boundaries Crystal dispersion Boundary conditions
Transient results in bulk Bulk simulation : specular reflection at boundaries Diffusion regime Phys. Rev. B, 72, 064305 (2005)
Results in bulk Diffusion balistic regime transtion Ballistic regime Phys. Rev. B, 72, 064305 (2005)
Nanowires Boundary collisions : purely diffuse Appl. Phys. Lett, 89, 103104 (2006)
Mode resolution for nanowires Relaxation times • No collision at lateral boundaries • Impurities • Anharmonic interactions => new estimation of t Perspectives
Perspectives • 1D kinetic theory. • 1D direct integration of Boltzmann equation. • 1D Monte Carlo simulations. • 3D integration of Boltzmann equation by discrete ordinate method.