Download Simulation of submicron thermal transport

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
 vx, 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 ddd
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