International Workshop on Energy Conversion and Information Processing Devices, Nice, France
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
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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 x
and x+dx during t
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
RW and diffusion equation
Density at time t+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
Earlier work : Peterson (1994), Mazumder and Majumdar (2001)
System divided in cells
Phonon energy and number in cells
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International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Initialization
Weight
Too many phonons
Spectral discretization
Nb spectral bins
Distribution function
Phonons drawn in cell until
Polarization
Direction
Two numbers drawn to choose de phonon direction
9/16
International Workshop on Energy Conversion and Information Processing Devices, Nice, France
Drift and scattering
Drift
Phonon scattering
Relaxation time t due to anharmonic processes and impurities
Modified distribution function
10/16
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
11/16
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
Diffusion balistic regime transtion
Phys. Rev. B, 72, 064305 (2005)
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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
t 1   2
•Impurities
• Anharmonic interactions => new estimation of t
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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.
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