1 MODELING MATTER AT NANOSCALES 1. Introduction and overview.
Monte Carlo phonon transport at nanoscales
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Monte Carlo phonon transport at nanoscales
Karl Joulain, Damian Terris,
Denis LemonnierLaboratoire d’études thermiques, ENSMA, Futuroscope France
David Lacroix
LEMTA, Univ Henri Poincaré, Nancy, France
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Random walk and diffusion equation
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
RW and diffusion equation
Einstein 1905
Density of particle at x and t.
Probability to travel on a distance between xand x+dx during
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
RW and diffusion equation
Density at time t+
Density expansion
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
RW and diffusion equation
Diffusion equation
100000 particles at the origin at t=0.
After 40 jumps:
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Nanoscale conductive heat transfer
Distribution function
Boltzmann Equation
Relaxation time approximation
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Boltzmann equation resolution methods
• Kinetic theory• Radiative transfer equation methods
– P1– Discrete ordinate
• Monte Carlo methods Advantages
– Geometry– Separation of relaxation times
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Monte Carlo simulation
System divided in cells
Earlier work : Peterson (1994), Mazumder and Majumdar (2001)
Phonon energy and number in cells
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Initialization
Polarization
Weight Too many phonons
Spectral discretization Nb spectral bins
Direction
Two numbers drawn to choose de phonon direction
Phonons drawn in cell until
Distribution function
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Drift and scattering
Drift
Phonon scattering
Relaxation time due to anharmonic processes and impurities
Modified distribution function
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
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
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Transient results in bulk
Bulk simulation : specular reflection at boundaries
Diffusion regime
Phys. Rev. B, 72, 064305 (2005)
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Results in bulk
Ballistic regime
Phys. Rev. B, 72, 064305 (2005)
Diffusion balistic regime transtion
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Nanowires
Boundary collisions : purely diffuse
Appl. Phys. Lett, 89, 103104 (2006)
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Perspectives
Mode resolution for nanowires
Relaxation times
•No collision at lateral boundaries•Impurities• Anharmonic interactions => new estimation of
€
−1 ∝ω2
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Perspectives
• 1D kinetic theory.
• 1D direct integration of Boltzmann equation.
• 1D Monte Carlo simulations.
• 3D integration of Boltzmann equation by discrete ordinate method.