1. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 1/16 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

2. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 2/16 Random walk and diffusion equation

3. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 3/16 RW and diffusion equation Einstein 1905 Density of particle at x and t. Probability to travel on a distance between x and x+dx during 

4. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 4/16 RW and diffusion equation Density at time t+  Density expansion

5. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 5/16 RW and diffusion equation Diffusion equation 100000 particles at the origin at t=0. After 40 jumps:

6. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 6/16 Nanoscale conductive heat transfer Distribution function Boltzmann Equation Relaxation time approximation

7. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 7/16 Boltzmann equation resolution methods Kinetic theory Radiative transfer equation methods –P1 –Discrete ordinate Monte Carlo methods Advantages –Geometry –Separation of relaxation times

8. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 8/16 Monte Carlo simulation System divided in cells Earlier work : Peterson (1994), Mazumder and Majumdar (2001) Phonon energy and number in cells

9. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 9/16 Initialization Polarization Weight Too many phonons Spectral discretization N b spectral bins Direction Two numbers drawn to choose de phonon direction Phonons drawn in cell until Distribution function

10. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 10/16 Drift and scattering Drift Phonon scattering Relaxation time  due to anharmonic processes and impurities Modified distribution function

11. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 11/16 Boundary conditions Temperature imposed at both end of the system Extrem cells are phonon blackbodies Boundary scattering Diffuse or specular reflexion at boundaries Crystal dispersion

12. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 12/16 Transient results in bulk Bulk simulation : specular reflection at boundaries Diffusion regime Phys. Rev. B, 72, 064305 (2005)

13. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 13/16 Results in bulk Ballistic regime Phys. Rev. B, 72, 064305 (2005) Diffusion balistic regime transtion

14. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 14/16 Nanowires Boundary collisions : purely diffuse Appl. Phys. Lett, 89, 103104 (2006)

15. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 15/16 Perspectives Mode resolution for nanowires Relaxation times No collision at lateral boundaries Impurities Anharmonic interactions => new estimation of 

16. International Workshop on Energy Conversion and Information Processing Devices, Nice, France 16/16 Perspectives 1D kinetic theory. 1D direct integration of Boltzmann equation. 1D Monte Carlo simulations. 3D integration of Boltzmann equation by discrete ordinate method.