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Simulation of Nanoscale Thermal Transport Laurent Pilon UCLA Mechanical and Aerospace Engineering Department Copyright© 2003 Motivations • Macroscopic laws break down when: – The length of the system is comparable to the mean free path of the heat carrier – The time scale of the physical system is smaller than the relaxation time of the heat carriers • Experimentation at submicron scale poses great challenges Numerical Simulations have become critical for basic understanding and design purposes Phonon Transport Equation • Equation for Phonon Radiative Transfer – The transport of phonons is governed by the Boltzmann transport equation (BTE) – The so-called Equation for Phonon Radiative Transfer (EPRT) is an alternative formulation of the BTE T2 Dielectric thin film ℓ ℓ T1 L I I0 I v I t s advection scattering (EPRT) Method of Characteristics Along the heat carrier pathline: dx u x, y, z, dt dy vx, y, z, dt dz w x, y, z, dt the EPRT simplifies Phonon pathline DI I0 I Dt s Transform a PDE into an ODE solved along the phonon pathlines The temperature is recovered from: D T 4 I cos sin ddd 0 4 Numerical Method t+Dt (xi,yj+1,zk+1) (xi+1,yj+1,zk+1) pathline (xi,yj,zk+1) t (xi+1,yj,zk+1) (xn,yn,zn) z y (xi+1,yj+1,zk) x (xi,yj,zk) + Pre-specified grid + Backward in time + Transient and steady state (xi+1,yj,zk) + Very accurate + Multidimensional problems + Compatible with other methods Transient Ballistic Transport Dimensionless temperature, T* 0.5 Numerical solution Exact solution t*=1 0.4 I v I 0 t Black surfaces z 0.3 T2 Dielectric thin film 0.2 0 T (z ) * 0.0 0.0 0.2 0.4 0.6 0.8 Dimensionless coordinate, z*=z/L 1.0 1mm T1 t*=0.1 0.1 q * T 4 T24 T14 T24 Steady-State Ballistic Transport 20 Discontinuity Temperature (K) 19 I v I 0 t 18 17 Black surfaces 16 15 z 14 13 Analytical solution 12 Present numerical method 11 Monte Carlo (Mazumder and Majumdar, 2001) T2=10 K 0 10 T1=20 K T * 0 50 100 150 1mm Dielectric thin film 200 250 300 350 Distance along the film (nm) 400 q T 4 T24 T14 T24 Conduction Across a Diamond Thin-Film Dimensionless temperature, T+ 1.00 Fourier’s law I I0 I v I t s 0.75 Black surfaces z T2=300 K 100 nm 0.50 1 mm 1mm Diamond thin film 10 mm 0.25 0.00 0.00 0.25 0.50 0.75 Dimensionless location, z*=z/L 1.00 0 T1=301 K T T T2 T1 T2 q SiO2 Rod with Specularly Reflecting Boundaries 1.0 10 m m Dimensionless Temperature, T* 0.9 Fourier’s law I I I v I 0 t 1m m 0.8 100 nm 0.7 10 nm Reflecting Boundaries 0.6 1 mm T1 0.5 0.4 L 0.3 (2D problem) 0.2 T * 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Dimensionless Distance, y* 0.9 1.0 T 4 T24 T14 T24 T2