Anes BOUCHENAK-KHELLADI

Advisors : - Jérôme Saint-Martin - Philippe DOLLFUS

Institut d’Electronique Fondamentale

Phonon thermal transport

in Nano-transistors

Contents

2PHONON THERMAL TRANSPORT04/21/23

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

04/21/23 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

04/21/23 4PHONON THERMAL TRANSPORT

But Why are we interested in “ Phonons ” ?

A. General introduction

04/21/23 5PHONON THERMAL TRANSPORT

In metals

But !!!

In Semiconductors and insulators

heat

Mr. electron

Lattice vibration “Mr.

Phonon”

A. General introduction

04/21/23 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 :

PulsationEnergy

A. General introduction

04/21/23 7PHONON THERMAL TRANSPORT

Bose-Einstein statics

Fermi-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

04/21/23 8PHONON THERMAL TRANSPORT

The dispersion approximation:

-We have then :• 1 LA• 2 TA• 1 LO• 2 TO

0 2 4 6 8 10

x 109

0

10

20

30

40

50

60

70

Vecteur d'onde (m-1)

Ener

gie (

meV

)

LATALOTO

! Why this order ? !

The slope ?

Contents

9PHONON THERMAL TRANSPORT04/21/23

A - General Introduction

B - Simulation Results1. Fourier equation

2. Boltzmann Transport equation

B. Simulation Results

04/21/23 10PHONON THERMAL TRANSPORT

But what’s the “ Purpose

” ?

B. Simulation Results

04/21/23 11PHONON THERMAL TRANSPORT

our device :

Y

XPropagation

axes

T1 T2Channel

characterized by a dispersion

Thermal reservoirs at equilibrium

Assumed to be ideal contacts

B. Simulation Results

04/21/23 12PHONON 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 TRANSPORT04/21/23

A - General Introduction

B - Simulation Results1. Fourier equation

2. Boltzmann Transport equation

B. Simulation results

1. Fourier equation

04/21/23 14PHONON THERMAL TRANSPORT

Then, at equilibrium =>

The heat diffusion equation is :

So, the variable is T !

but how could we resolve this equation ?

The Fourier law :

04/21/23 15PHONON THERMAL TRANSPORT

First step: Discretization (mesh of our silicon nano-wire)

B. Simulation results1. Fourier equation

04/21/23 16PHONON THERMAL TRANSPORT

Second step: Write the right program in MATLAB After : Check the results !!

Then: Simply resolve the linear system:

B. Simulation results1. Fourier equation

B. Simulation results1. Fourier equation

04/21/23 17PHONON THERMAL TRANSPORT

Third step: Admire the results

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Tsource = Tdrain = 300 K

Ti = 9 nm

= 150 nm

TGrilles = 300 K

Contents

18PHONON THERMAL TRANSPORT04/21/23

A - General Introduction

B - Simulation Results1. Fourier equation

2. Boltzmann Transport equation

B. Simulation results

2. Boltzmann Transport equation

04/21/23 19PHONON THERMAL TRANSPORT

The RTA say :

The general form is :

So, the variable is Ns !

but how could we resolve this equation ?

What we need to resolve is this:

Then =>

B. Simulation results2. Boltzmann Transport equation

04/21/23 20PHONON 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, no 25‑26 (décembre 2012): 7444‑7452.

04/21/23 21PHONON THERMAL TRANSPORT

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 Temperature using Ns (or more exactly Es = h’.w.Ns)

04/21/23 22PHONON THERMAL TRANSPORT

B. Simulation results2. 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)

04/21/23 23PHONON THERMAL TRANSPORT

B. Simulation results2. Boltzmann Transport Equation (BTE)

0 20 40 60 80 100 120 140 160 180 200 220300

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pera

ture

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)

LATALOTOEffectiveFourier

And : The temperature

profile …

04/21/23 24PHONON THERMAL TRANSPORT

B. Simulation results2. Boltzmann Transport Equation (BTE)

Pending Work … !

But almost done !