phonon thermal transport in n ano-transistors
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
050
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Longueur (nm)Epaisseur (nm)
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pera
ture
(K
)
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12299.8
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Longueur (nm)Epaisseur (nm)
Tem
pera
ture
(K
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Longueur (nm)Epaisseur (nm)
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pera
ture
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390
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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Longueur (nm)
Tem
pera
ture
(K
)
LATALOTOEffectiveFourier
And : The temperature
profile …
04/21/23 24PHONON THERMAL TRANSPORT
B. Simulation results2. Boltzmann Transport Equation (BTE)
Pending Work … !
But almost done !