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Phonon thermal transport in N ano-transistors

Phonon thermal transport in N ano-transistors. Anes BOUCHENAK-KHELLADI Advisors : - Jér ô me Saint-Martin - Philippe DOLLFUS Institut d’Electronique Fondamentale. Contents. A - General Introduction (page 3 to 8) B - Simulation Results (page 10 to 23) 1. Fourier equation

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Phonon thermal transport in N ano-transistors

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  1. Phonon thermal transport in Nano-transistors Anes BOUCHENAK-KHELLADI Advisors : - Jérôme Saint-Martin - Philippe DOLLFUS Institutd’ElectroniqueFondamentale

  2. Contents • A - General Introduction (page 3 to 8) • B - Simulation Results(page 10 to 23) • 1. Fourier equation • 2. Boltzmann Transport equation PHONON THERMAL TRANSPORT

  3. A. General introduction What’s a phonon ? Thermal agitation In a uniform solid material and vibrate Atoms in a regular lattice : Wave propagating inside the crystal Phonon A quantum of energy of this vibration is a PHONON THERMAL TRANSPORT

  4. A. General introduction But Why are we interested in “ Phonons ” ? PHONON THERMAL TRANSPORT

  5. A. General introduction Mr. electron In metals heat But !!! In Semiconductors and insulators Lattice vibration “Mr. Phonon” PHONON THERMAL TRANSPORT

  6. A. General introduction • Some phonon characteristics : - Behave as particles (quasi-particles) and as waves. - Described by a periodic dispersion : Pulsation Energy - Particles described by a wave-packet - The group velocity of wave-packet is determined by : - Obey to Bose-Einstein statics just like photons : PHONON THERMAL TRANSPORT

  7. A. General introduction Electrons Phonons Fermi-Dirac statics Bose-Einstein statics Why not ! Would you like to come with me ? I will vibrate ! Each energy state can be occupied by any number of phonons PHONON THERMAL TRANSPORT

  8. A. General introduction • The dispersion approximation: • We have then : • 1 LA • 2 TA • 1 LO • 2 TO The slope ? ! Why this order ? ! PHONON THERMAL TRANSPORT

  9. Contents • A - General Introduction • B - Simulation Results • 1. Fourier equation • 2. Boltzmann Transport equation PHONON THERMAL TRANSPORT

  10. B. Simulation Results But what’s the “ Purpose ” ? PHONON THERMAL TRANSPORT

  11. B. Simulation Results our device : Y Thermal reservoirs at equilibrium Channel characterized by a dispersion Assumed to be ideal contacts T1 T2 Propagation axes X PHONON THERMAL TRANSPORT

  12. B. Simulation Results The goal is to get the temperature profile inside our device ! So, just solve the Heat diffusion equation (Fourier equation) ! Euhhh … ! Not exactly … ! … ? PHONON THERMAL TRANSPORT

  13. Contents • A - General Introduction • B - Simulation Results • 1. Fourier equation • 2. Boltzmann Transport equation PHONON THERMAL TRANSPORT

  14. B. Simulation results 1. Fourier equation • In a medium : - Isotropic • with constant thermo-physical parameters The heat diffusion equation is : The Fourier law : Then, at equilibrium => So, the variable is T ! but how could we resolve this equation ? PHONON THERMAL TRANSPORT

  15. B. Simulation results1. Fourier equation First step: Discretization (mesh of our silicon nano-wire) PHONON THERMAL TRANSPORT

  16. B. Simulation results1. Fourier equation Then: Simply resolve the linear system: Second step: Write the right program in MATLAB After : Check the results !! PHONON THERMAL TRANSPORT

  17. B. Simulation results1. Fourier equation Third step: Admire the results Ti = 9 nm = 150 nm TGrilles = 300 K Tsource = Tdrain = 300 K PHONON THERMAL TRANSPORT

  18. Contents • A - General Introduction • B - Simulation Results • 1. Fourier equation • 2. Boltzmann Transport equation PHONON THERMAL TRANSPORT

  19. B. Simulation results 2. Boltzmann Transport equation The general form is : What we need to resolve is this: The RTA say : Then => So, the variable is Ns ! but how could we resolve this equation ? PHONON THERMAL TRANSPORT

  20. B. Simulation results2. Boltzmann Transport equation As we work in 2D, the above equation become : First step: Discretization (mesh of our silicon nano-wire both along x and y and in the reciprocal space (the Brillouin zone)) fBrown III, Thomas W., et Edward Hensel. « Statistical phonon transport model for multiscale simulation of thermal transport in silicon: Part I – Presentation of the model ». International Journal of Heat and Mass Transfer 55, no 25‑26 (décembre 2012): 7444‑7452. PHONON THERMAL TRANSPORT

  21. B. Simulation results2. Boltzmann Transport Equation (BTE) Then: Simply resolve the linear system: with Second step: Write the right program in MATLAB After : Check the results !! Third step: Admire the results ! Euhh … ! Not yet ! • We have to find a way to compute the TemperatureusingNs (or more exactlyEs = h’.w.Ns) PHONON THERMAL TRANSPORT

  22. B. Simulation results2. Boltzmann Transport Equation (BTE) So, we compute the equilibrium local phonon energy After, we draw the E(T) graph … ! Euhh ! In fact, we drawT(E) Then, we make a polynomial fit to get T(E) PHONON THERMAL TRANSPORT

  23. B. Simulation results2. Boltzmann Transport Equation (BTE) And : The temperature profile … PHONON THERMAL TRANSPORT

  24. B. Simulation results2. Boltzmann Transport Equation (BTE) Pending Work … ! But almost done ! PHONON THERMAL TRANSPORT

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