Anes BOUCHENAK-KHELLADI Advisors : - Jérôme Saint-Martin - Philippe DOLLFUS Institut d’Electronique Fondamentale Phonon thermal transport in Nano-transistors

Contents 2PHONON THERMAL TRANSPORT 10/12/2015 A - General Introduction (page 3 to 8) B - Simulation Results (page 10 to 23) 1. Fourier equation 2. Boltzmann Transport equation

A. General introduction In a uniform solid material 10/12/2015 3PHONON THERMAL TRANSPORT What’s a phonon ? Thermal agitation Atoms in a regular lattice : Wave propagating inside the crystal A quantum of energy of this vibration is a Phonon and vibrate

A. General introduction 10/12/2015 4PHONON THERMAL TRANSPORT But Why are we interested in “ Phonons ” ?

A. General introduction 10/12/2015 5PHONON THERMAL TRANSPORT In metals But !!! In Semiconductors and insulators heat Mr. electron Lattice vibration “Mr. Phonon”

A. General introduction 10/12/2015 6PHONON THERMAL TRANSPORT  Some phonon characteristics : - Behave as particles (quasi-particles) and as waves. - Described by a periodic dispersion : - Particles described by a wave-packet - The group velocity of wave-packet is determined by : - Obey to Bose-Einstein statics just like photons : Pulsation Energy

A. General introduction 10/12/2015 7PHONON THERMAL TRANSPORT Bose-Einstein staticsFermi-Dirac statics Phonons Electrons Each energy state can be occupied by any number of phonons Would you like to come with me ? Why not ! I will vibrate !

A. General introduction 10/12/2015 8PHONON THERMAL TRANSPORT  The dispersion approximation: -We have then : 1 LA 2 TA 1 LO 2 TO ! Why this order ? ! The slope ?

Contents 9PHONON THERMAL TRANSPORT 10/12/2015 A - General Introduction B - Simulation Results 1. Fourier equation 2. Boltzmann Transport equation

B. Simulation Results 10/12/ PHONON THERMAL TRANSPORT But what’s the “ Purpose ” ?

B. Simulation Results 10/12/ PHONON THERMAL TRANSPORT our device : Y X Propagation axes T1T2 Channel characterized by a dispersion Thermal reservoirs at equilibrium Assumed to be ideal contacts

B. Simulation Results 10/12/ PHONON THERMAL TRANSPORT The goal is to get the temperature profile inside our device ! So, just solve the Heat diffusion equation (Fourier equation) ! Euhhh … ! Not exactly … ! … ?

Contents 13PHONON THERMAL TRANSPORT 10/12/2015 A - General Introduction B - Simulation Results 1. Fourier equation 2. Boltzmann Transport equation

B. Simulation results 1. Fourier equation 10/12/ PHONON THERMAL TRANSPORT Then, at equilibrium => The heat diffusion equation is : In a medium :- Isotropic - with constant thermo-physical parameters So, the variable is T ! but how could we resolve this equation ? The Fourier law :

10/12/ PHONON THERMAL TRANSPORT First step: Discretization (mesh of our silicon nano-wire) B. Simulation results 1. Fourier equation

10/12/ PHONON THERMAL TRANSPORT Second step: Write the right program in MATLAB After : Check the results !! Then: Simply resolve the linear system: B. Simulation results 1. Fourier equation

10/12/ PHONON THERMAL TRANSPORT Third step: Admire the results T source = T drain = 300 K T i = 9 nm = 150 nm T Grilles = 300 K

Contents 18PHONON THERMAL TRANSPORT 10/12/2015 A - General Introduction B - Simulation Results 1. Fourier equation 2. Boltzmann Transport equation

B. Simulation results 2. Boltzmann Transport equation 10/12/ PHONON THERMAL TRANSPORT The RTA say : The general form is : So, the variable is N s ! but how could we resolve this equation ? What we need to resolve is this: Then =>

B. Simulation results 2. Boltzmann Transport equation 10/12/ PHONON THERMAL TRANSPORT First step: Discretization (mesh of our silicon nano-wire both along x and y and in the reciprocal space (the Brillouin zone)) As we work in 2D, the above equation become : 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, n o 25 ‑ 26 (décembre 2012): 7444 ‑ 7452.

10/12/ PHONON THERMAL TRANSPORT B. Simulation results 2. 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 Temperature using Ns (or more exactly Es = h’.w.Ns)

10/12/ PHONON THERMAL TRANSPORT B. Simulation results 2. Boltzmann Transport Equation (BTE) So, we compute the equilibrium local phonon energy After, we draw the E(T) graph … ! Then, we make a polynomial fit to get T(E) Euhh ! In fact, we draw T(E)

10/12/ PHONON THERMAL TRANSPORT B. Simulation results 2. Boltzmann Transport Equation (BTE) And : The temperature profile …

10/12/ PHONON THERMAL TRANSPORT B. Simulation results 2. Boltzmann Transport Equation (BTE) Pending Work … ! But almost done !