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

ME 595M: Computational Methods for
Nanoscale Thermal Transport
Lecture 10:Higher-Order BTE
Models
J. Murthy
Purdue University
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BTE Models
• Gray BTE drawbacks
 Cannot distinguish between different phonon polarizations
 Isotropic
 Relaxation time approximation does not allow direct energy transfers
between different frequencies even if “non-gray” approach were
taken
 Very simple relaxation time model
• Higher-order BTE models
 Try to resolve phonon dispersion and polarization using “bands”
 But finer granularity requires more information about scattering rates
 Various approximations in finding these rates
• Will look at
 Semi-gray models
 Full dispersion model
 Full scattering model
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Semi-Gray BTE
• This model is sometimes called the two-fluid model (Armstrong, 1981;
Ju, 1999).
• 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.
• 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.
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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
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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
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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
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Full-Dispersion 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 with no velocity.
• 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.
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Phonon Bands
Optical band
Acoustic bands
Each band characterized by
its group velocity, specific
heat and “temperature”
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Optical Mode BTE
e0

t
No ballistic term
– no transport
Nbands 1

j 1
  Toj
 
 oj   C0 dT  e0  

  Tref

 qvol
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
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Acoustic Mode BTE
Scattering to
same band
ei
   (vi sei )  (ei0  ei ) ii 
t
Tij

Nbands 

1


Ballistic

Ci dT  ei   ij 



term
 
j 1  4 Tref
 
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
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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
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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
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Silicon Bulk Thermal
Conductivity
Full-Dispersion
Model
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Full Scattering Model
f0 f
 f 




 t  scat .

Elastic Scattering
 f 



t

3 phonon
Inelastic Scattering
2
 f 



2
c
(
K
,
K
,
K
)
 (      )

3
 
3
M   
 t 3 phonon K ,K 
3
  ( f  1)( f   1) f   ff ( f   1) 
c3 (K , K , K )  
i 2 M
  
G 3v
Klemens, (1958)
Valid only for phonons satisfying conservation rules
Complicated, non-linear
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N and U Processes
• N processes do not offer resistance because there is no
change in direction or energy
k2
k1
k3
• U processes offer resistance to phonons because they turn
phonons around
k3
k1
k’3
k2
N processes
change f and
affect U
processes
indirectly
G
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General Computation Procedure for
Three-phonon Scattering Rates
• 12 unknowns
 ,  ,  ; K , K , K 
• 7 equations
    
K  K  K  G
 K
• Set 5, determine 7

One energy conservation equation
Three components of momentum
conservation equation
Three dispersion relations for
the three wave vectors
Specify K (Kx, Ky, Kz) and direction of K’ (K’x, K’y)
• Bisection algorithm developed to find all sets of 3-phonon interactions
Wang, T. and Murthy, J.Y.; Solution of Phonon Boltzmann Transport
Equation Employing Rigorous Implementation of Phonon Conservation
Rules; ASME IMECE Chicago IL, November 10-15, 2006.
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Thermal Conductivity of Bulk Silicon
4
10
•
2-10K, boundary
scattering dominant;
•
20-100K, impurity
scattering important, as
well as N and U
processes;
•
Above 300K, U
processes dominant.
Expts
with N
without N
3
K (W/mK)
10
2
10
1
10
0
10
1
2
10
10
3
10
T (K)
Experimental data from Holland (1963)
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Thermal Conductivity of Undoped Silicon
Films
Specularity Parameter p=0.4
Experimental data from Ju and Goodson (1999), and Asheghi et al. (1998, 2002)
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Conclusions
• In this lecture, we considered three 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
• Models like the semi-gray and full-dispersion models still
employ temperature-like concepts which are not
satisfactory.
• Newer models such as the full scattering model do not
employ relaxation time approximations, and temperaturelike concepts
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