Transcript Aucun titre de diapositive - University of California

Prediction of the thermal conductivity of a multilayer nanowire
Patrice Chantrenne, Séverine Gomés
CETHIL UMR 5008 INSA/UCBL1/CNRS
Arnaud Brioude, David Cornu
LMI UMR 5615 UCBL1/CNRS
Jean-Louis Barrat
LPMCN UMR 5586 UCBL1/CNRS
Thanks to
Laurent David, CETHIL Lyon
Florian Lagrange, LCTS Bordeaux
LAYOUT
Motivations
Nanowire description
Models
Motivations : applications
Microelectronic components
- length scale lower than 30 nm
- film thickness less than10 nm
Require temperature measurements in
order to ensure the reliability of the
microsystem
Motivations : applications
Nanostructured materials (nanoporous, nanosequences, nanolayered)
Nanostructures (nanoparticles, nanotubes, nanowires, nanofilms…)
SiO2/SiC
nanowire
SiC/SiO2/BN
nanowire
Require experimental caracterisations
limitation until now : almost one experimental device has
been developped for each nanostructure
SiC/graphitelike C
nanosequence matérial
Motivations : development of a new sensor
Temperature measurement
Thermophysical properties
measurement
High spatial resolution
below 100 nm
Quantitative measurement
lower the uncertainty
and higher sensitivity
The most popular commercial sensor actually used with an AFM
Diameter : 5 µm
Length : 200 µm
Curvature radius : 15-20 µm
Motivations : development of a new sensor
Temperature measurement
- qualitative values only
- quantitative measurement
require a calibration
- spatial resolution limited by the
tip geometry and surface
roughness
Modèle de
Lefèvre
Modèle de David
L. David Ph D, CETHIL
S. Gomès & Dj. Ziane, 2003, Solid State Electronics
47 pp 919-922
S. Gomès et al., IEEE Transactions on
Components and Packaging Technologies, 2006
Thermal conductivity
- low sensitivity at high thermal
conductivity values
- uncertainty of about 20 % at low
thermal conductivity values
Motivations : development of a new sensor
The new sensor : a functionalised multilayer nanowire
Interfaces
nanowire
nanolayers
Core : BN, SiC
crystalline / periodic defect (mâcle)
layers :
metallic
dielectric crystal (SiC)/amorphous (SiO2)
10-50 nm
eventually sharpened
The sensor should exhibit a low thermal conductivity
in order to a good temperature and thermal conductivity
sensitivity
The prediction of the thermal conductivity is
essential to optimize the design of the sensor.
Model : macroscopic approach
Heat transfer across the nanowire depends on heat transfer
- in the core (dielectric crystal) c rc , l 
Prediction for nanowire
- in metallic nanolayer
m em , l 
Use the bulk value
- in amorphous nanolayer a ea , l 
- in dielectric nanolayer d ed , l 
Prediction for nanofilm
- across the interfaces
forecoming studies
Rcm Rma Rd m
Radius of the core rc
Thermal conductivity versus
thermal conductance/thermal
resistance ?
R ,   

  ,  
Length l
Tip end
Thicknesses e1 e2 e3 ...
Model for dielectric crystals
In dielectric crystaline material, heat carriers are
PHONON
=
Atomic collective vibration modes of energy

These vibration modes may be characterised by
Wave vector K, polarization p, dispersion curves
number of phonon per vibration mode
Phonon liftime
 K, p
e
 K, p
1
/ k b T
1
Model for dielectric crystals
The kinetic theory of gaz allow to write
x K , p   CK , p v K , p  K , p cos ² K , x 
2
The total thermal conductivity = sum of individual thermal
conductivity of each vibration modes (K,p)
x   C K , p v²K , p  K , p  cos ² K , x 
K
with
p
ex
C K , p   k b x ² x
V e 1 ²
Spécific heat
d  K , p 
v
dK
Group velocity
 
 K , p 
x
kbT
Model for dielectric crystals
Thermal conductivity calculation
require the knowledge of
- vibration modes
- dispersion curves
- relaxation time parameters
1
1
1
1



 K, p  ph  ph  K, p  CL  K, p  D  K, p
1
A 
 B

exp  
 T
 u  K, p
T
v K, p
1

 CL  K, p F. d ( K)
1
 D 4
 D  K, p
main assumption of the model
vibrational properties of a cristalline nanostructure
=
vibrational properties of the bulk crystal
Validation of the model for Silicon...
Model for dielectric crystals
Silicon structure
in the real space
diamond structure
the elementary cell contains two
atoms
a3
z
a1
y
x
a2
a0
a0 = 0.543 nm
Model for dielectric crystals
Vibration modes
z
In the reciprocal space
- K = linear combination of de b1, b2,
k
b3
- K belong to the first Brillouin ’s
zone
- nomber of wave vectors K :
number of elementary cells
- Number of polarisations p = 6
y
j
i
4 / a 0
x

b1  i  j  k
a0

b2 
  i  j  k
a0

b3  i  j  k
a0
Model for dielectric crystals
Dispersion curves
Linear fit of the experimental dispersion curves
in the [1,0,0] direction
1,50E+13
0,9
0,8
1,25E+13
0,7
LA
0,6
Cv/(3R)
f (Hz)
1,00E+13
7,50E+12
TA
5,00E+12
0,5
0,4
0,3
0,2
2,50E+12
0,1
0,00E+00
0
0
0,2
0,4
0,6
0,8
k/kmax
B.N. Brockhouse, P.R.L. 2, 256 (1959)
S. Wei et M.Y. Chou, PRB, 50, 2221 (1994)
1
0
50
100
150
200
250
300
350
Temperature (K)
P. Flubacher et al., Philos. Mag, 4,273 (1959)
The optical mode contribution to the thermal conductivity is negligible if T < 1000 K
Model for dielectric crystals
Relaxation time parameters determination
Fit of the thermal conductivity of a Si crystal (L = 7,16
mm) function of the temperature
1
v K , p 
1
1
A 
 B
 D 4


exp  
 D  K, p
 T
 CL K , p 
F .L
 u  K, p
T
10000
F = 0.55
D = 1.32 10-45 s-3
1000
-1 -1
Longitudinal mode
A = 3 10-21
B=0
=2
 = 1.5
 (Wm K )
Transverse mode
A = 7 10-13
B=0
=1
=4
100
10
1
10
100
1000
Temperature (K)
M.G. Holland, PR, 132, 2461 (1963)
Thermal conductivity of Si nanowires
50
45
115 nm
40
-1 -1
 (Wm K )
35
56 nm
30
Excellent agreement
except for the 22 nm wide
nanowire
25
37 nm
20
15
10
22 nm
5
0
0
50
100
150
200
250
Temperature (K)
D. Li, et al., A.P.L, 83, 2934 (2003)
300
350
Thermal conductivity of Si nanofilms
1000
1,6 µm
3 µm
0,83 µm
0,42 µm
100
Excellent agreement
with the experimental
resutls
-1 -1
 (Wm K )
100 nm
20 nm
10
1
20
60
100
140
180
Temperature (K)
220
260
300
M. Asheghi et al., ASME JHT, 120, 30 (1998)
M.Z. Bazant, PRB, 56, 8542 (1997)
Thermal conductivity of Si nanofilms
Prediction of the thermal conductivity function of the heat
transfer direction
120
in plane
 (W/(mK))
100
80
cross plane
60
40
20
0
0
500
1000
1500
2000
film thickness (nm )
T= 300K
2500
3000
CONCLUSION
Thermal conductivity of dielectric nanofilms and nanowires
Confident to get a accurate value
Thermal conductivity of metallics and amorphous nanofilms
The bulk value overestimate the real value
Thermal conctact resistance
Still a Problem, several models may be used
However, one need to evaluate the maximun value of
the thermal conductivity