Transcript ME 597F: Micro- and Nano-Scale Energy

ME 595M: Computational Methods for
Nanoscale Thermal Transport
Lecture 11:Extensions and
Modifications
J. Murthy
Purdue University
ME 595M J.Murthy
1
Drawbacks
• Gray BTE




Cannot distinguish between different phonon polarizations
Isotropic
Relaxation time approximation does not allow energy transfers
between different frequencies even if “non-gray” approach were
taken
Very simple relaxation time model
• Numerical Method


“Ray” effect and “false scattering”
Sequential procedure fails at high acoustic thicknesses
• We will consider remedies for each of these problems in the
next two lectures
ME 595M J.Murthy
2
Semi-Gray BTE
• This model is sometimes called the two-fluid model (Armstrong, 1981).
• Idea is to divide phonons into two groups
 “Reservoir mode” phonons do not move; capture capacitative effects
 “Propagation mode” phonons have non-zero group velocity and
capture transport effects. Are primarily responsible for thermal
conductivity.
• Ju (1999) used this idea to devise a model for nano-scale thermal
transport
• Model involves a single equation for reservoir mode “temperature” with
no angular dependence
• Propogation mode involves a set of BTEs for the different directions, like
gray BTE
• Reservoir and propagation modes coupled through energy exchange
terms
Armstrong, B.H., 1981, "Two-Fluid Theory of Thermal Conductivity of Dielectric
Crystals", Physical Review B, 23(2), pp. 883-899.
Ju, Y.S., 1999, "Microscale Heat Conduction in Integrated Circuits and Their
Constituent Films", Ph.D. thesis, Department of Mechanical Engineering,
Stanford University.
ME 595M J.Murthy
3
Propagating Mode Equations
1
''
''
C
(
T

T
)

e
P
L
ref
p
e p
''
4

   (v p se p ) 
t

CP (TP  Tref ) 
''
e
 pd

Propagating model
scatters to a bath at
lattice temperature
TL with relaxation
time 

“Temperature” of
propagating mode,
TP, is a measure of
propagating mode
energy in all
directions together

CP is specific heat
of propagating
mode phonons
4
ME 595M J.Murthy
4
Reservoir Mode Equation
CR
(TR  Tref )
t

CR (TL  Tref )  CR (TR  Tref )

 qvol
• Note absence of velocity term
• No angular dependence – equation is for total energy of
reservoir mode
• TR, the reservoir mode “temperature” is a measure of
reservoir mode energy
• CR is the specific heat of reservoir mode phonons
• Reservoir mode also scatters to a bath at TLwith
relaxation time 
• The term qvol is an energy source per unit volume – can be
used to model electron-phonon scattering
ME 595M J.Murthy
5
Lattice Temperature
By definition, lattice temperature is a measure of the total
(reservoir +propagating mode) energy:
(CR  CP ) TL  Tref   CR TR  Tref   CP TP  Tref

Therefore:
CRTR  CPTP
TL 
(CR  CP )
ME 595M J.Murthy
6
Discussion
• Model contains two unknown constants: vg and 
• Can show that in the thick limit, the model satisfies:
 CP  CR 
TL
1
   k TL  qvol with k  CP vg2
t
3
• Choose vg as before; find  to satisfy bulk k.
• Which modes constitute reservoir and propagating modes?



Perhaps put longitudinal acoustic phonons in propagating mode ?
Transverse acoustic and optical phonons put in reservoir mode ?
Choice determines how big  comes out
• Main flaw is that  comes out very large to satisfy bulk k

Can be an order-of-magnitude larger than optical-to-acoustic relaxation
times
• In FET simulation, optical-to acoustic relaxation time determines hot
spot temperature
• Need more detailed description of scattering rates
ME 595M J.Murthy
7
Non-Gray BTE
• Details in Narumanchi et al (2004,2005).
• Objective is to include more granularity in phonon representation.
• Divide phonon spectrum and polarizations into “bands”. Each band has
a set of BTE’s in all directions
• Put all optical modes into a single “reservoir” mode.
• Model scattering terms to allow interactions between frequencies.
Ensure Fourier limit is recovered by proper modeling
• Model relaxation times for all these scattering interactions based on
perturbation theory (Han and Klemens,1983)
• Model assumes isotropy, using [100] direction dispersion curves in all
directions
Narumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Sub-Micron Heat Transport Model in Silicon
Accounting for Phonon Dispersion and Polarization; ASME Journal of Heat Transfer, Vol. 126,
pp. 946—955, 2004.
Narumanchi, S.V.J., Murthy, J.Y., and Amon, C.H.; Comparison of Different Phonon Transport
Models in Predicting Heat Conduction in Sub-Micron Silicon-On-Insulator Transistors; ASME
Journal of Heat Transfer, 2005 (in press).
Han, Y.-J. and P.G. Klemens, Anharmonic Thermal Resistivity of Dielectric Crystals at Low
Temperatures. Physical Review B, 1983. 48: p. 6033-6042.
ME 595M J.Murthy
8
Phonon Bands
Optical band
Acoustic bands
Each band characterized by
its group velocity, specific
heat and “temperature”
ME 595M J.Murthy
9
Optical Mode BTE
e0

t
No ballistic term
– no transport
Nbands 1

j 1
  Toj
 
 oj   C0 dT  e0    qvol

  Tref

Energy exchange due to
scattering with jth acoustic
mode
Electronphonon
energy
source
oj is the inverse relaxation time for energy exchange between
the optical band and the jth acoustic band
Toj is a “bath” temperature shared by the optical and j bands.
In the absence of other terms, this is the common
temperature achieved by both bands at equilibrium
ME 595M J.Murthy
10
Acoustic Mode BTE
Scattering to
same band
ei
   (vi sei )  (ei0  ei ) ii 
t
Tij

Nbands 

1



Ci dT  ei   ij 



Ballistic
 
j 1  4 Tref
 
term
j  i 
Energy exchange with
other bands
ij is the inverse relaxation time for energy exchange between
bands i and j
Tij is a “bath” temperature shared by the i and j bands. In the
absence of other terms, this is the common temperature
achieved by both bands at equilibrium
ME 595M J.Murthy
11
Lattice Temperature
etotal 
TL
C
total
Tref
dT 
To
Nbands 1
 C dT  
o
Tref
i 1
 Ti

  Ci dT 


 Tref

• Lattice “temperature” is a measure of the energy in all
acoustic and optical modes combined
ME 595M J.Murthy
12
Model Attributes
• Satisfies energy conservation
• In the acoustically thick limit, the model can be shown to
satisfy
Ctotal
TL
   ( K TL )  qvol
t
K
Nbands 1

i 1
1 vi 2Ci (TL )
3 Nbands
(   ij )
Fourier heat
diffusion
equation
Thermal
conductivity
j 1
ME 595M J.Murthy
13
Properties of Full-Dispersion Model
In acousticallythick limit, full
dispersion model
• Recovers Fourier
conduction in
steady state
• Parabolic heat
conduction in
unsteady state
1-D transient diffusion, with
3X3X1 spectral bands
ME 595M J.Murthy
14
Silicon Bulk Thermal
Conductivity
Non-Gray Model
ME 595M J.Murthy
15
Thermal Conductivity of Doped Silicon
Thin Films • 3.0 micron boron-doped
•
•
silicon thin films.
Experimental data is from
Asheghi et. al (2002)
p=0.4 is used for
numerical predictions
Boron dopings of 1.0e+24
and 1.0e+25 atoms/m3
considered
Full-Dispersion
Model
ME 595M J.Murthy
16
Thin Layer Si Thermal Conductivity
Specularity
Factor p=0.6
ME 595M J.Murthy
17
Thermal Modeling of SOI FET
Heat generation
region
(100nmx10nm) • Heat source assumed
known at 6x1017 W/m3 in
heat generation region
72 nm
Si
• Lower boundaries at 300K
SiO2
SiO2
1633 nm
315 nm
•Top boundary diffuse
reflector
•BTE in Si layer
•Fourier in SiO2 region
•Interface energy balance
ME 595M J.Murthy
18
Temperature Prediction -- Full
Dispersion Model
Tmax =393.1 K
= 7.2 ps for optical to
acoustic modes
ME 595M J.Murthy
19
Mode Temperatures
Optical mode
Low frequency LA
mode
ME 595M J.Murthy
20
Maximum Temperature Comparison
Model
Heat Gen.
In O-Mode
Heat Gen.
In A-Modes
Fourier
320.7 K
Gray
326.4 K
Semi-Gray
504.9 K
365.5 K
FullDispersion
393.1 K
364.4 K
ME 595M J.Murthy
21
Conclusions
• In this lecture, we considered two extensions to the gray
BTE which account for more granularity in the
representation of phonons
• More granularity means more scattering rates to be
determined – need to invoke scattering theory
• Current models still employ temperature-like concepts not in
keeping with non-equilibrium transport
• Newer models are being developed which do not employ
relaxation time approximations, and admit direct
computation of the full scattering term
ME 595M J.Murthy
22