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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

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−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.