Prediction of the thermal conductivity of a multilayer nanowire
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Prediction of the thermal conductivity of a multilayer nanowire
Patrice Chantrenne, Séverine GomésCETHIL UMR 5008 INSA/UCBL1/CNRS
Arnaud Brioude, David CornuLMI UMR 5615 UCBL1/CNRS
MotivationsNanowire descriptionModels
Thanks to Laurent David, CETHIL LyonFlorian Lagrange, LCTS Bordeaux
LAYOUT
Jean-Louis BarratLPMCN UMR 5586 UCBL1/CNRS
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Microelectronic components- length scale lower than 30 nm- film thickness less than10 nm
Motivations : applications
Require temperature measurements in order to ensure the reliability of the microsystem
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Nanostructured materials (nanoporous, nanosequences, nanolayered)
Nanostructures (nanoparticles, nanotubes, nanowires, nanofilms…)
Require experimental caracterisationslimitation until now : almost one experimental device has been developped for each nanostructure
Motivations : applications
SiC/graphitelike Cnanosequence matérial
SiO2/SiCnanowire
SiC/SiO2/BNnanowire
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Temperature measurement
Thermophysical properties measurement
High spatial resolutionbelow 100 nm
Quantitative measurementlower the uncertainty and higher sensitivity
Motivations : development of a new sensor
The most popular commercial sensor actually used with an AFM
Diameter : 5 µm
Length : 200 µm
Curvature radius : 15-20 µm
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Motivations : development of a new sensor
Modèle de LefèvreModèle de David
Thermal conductivity- low sensitivity at high thermal conductivity values- uncertainty of about 20 % at low thermal conductivity values
S. Gomès & Dj. Ziane, 2003, Solid State Electronics 47 pp 919-922
L. David Ph D, CETHIL
S. Gomès et al., IEEE Transactions on Components and Packaging Technologies, 2006
Temperature measurement- qualitative values only- quantitative measurement require a calibration - spatial resolution limited by the tip geometry and surface roughness
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Interfaces
nanowire
nanolayers
Core : BN, SiCcrystalline / periodic defect (mâcle)
layers :metallicdielectric crystal (SiC)/amorphous (SiO2)
The new sensor : a functionalised multilayer nanowire
The sensor should exhibit a low thermal conductivityin order to a good temperature and thermal conductivity sensitivity
The prediction of the thermal conductivity is essential to optimize the design of the sensor.
Motivations : development of a new sensor
10-50 nmeventually sharpened
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Thermal conductivity versus thermal conductance/thermal resistance ?
Length l
Heat transfer across the nanowire depends on heat transfer
- in the core (dielectric crystal)- in metallic nanolayer- in amorphous nanolayer- in dielectric nanolayer- across the interfaces forecoming studies
Thicknesses e1 e2 e3 ...
Radius of the core rc
Tip end
lrcc , lemm ,
leaa , ledd ,
mcR amR mdR
,, R
Model : macroscopic approach
Use the bulk value
Prediction for nanowire
Prediction for nanofilm
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Atomic collective vibration modes of energy
Model for dielectric crystals
In dielectric crystaline material, heat carriers are
Wave vector K, polarization p, dispersion curves
number of phonon per vibration mode
K p,1
1e k Tb /
PHONON=
Phonon liftime
These vibration modes may be characterised by
pK,
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The total thermal conductivity = sum of individual thermal conductivity of each vibration modes
(K,p)
²1²,
x
x
b eV
exkpKC
Tk
pKx
b
,
dK
pKdv
,
xKK p
x pKpKvpKC ,²cos,,², xKK p
x pKpKvpKC ,²cos,,²,
The kinetic theory of gaz allow to write
xKx pKpKvpKCpK ,2 ²cos,,,,
with Spécific heat
Group velocity
Model for dielectric crystals
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Thermal conductivity calculation
require the knowledge of - vibration modes- dispersion curves- relaxation time parameters
main assumption of the modelvibrational properties of a cristalline nanostructure
= vibrational properties of the bulk crystal
Validation of the model for Silicon...
1 1 1 1
K p K p K p K pph ph CL D, , , ,
1
u K p
A
T
B
T,exp
1
CL K p
v K p
F d K,
,
. ( )
1 4
D K pD
,
Model for dielectric crystals
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Silicon structurein the real space
diamond structure
the elementary cell contains two atoms
a0
a2
a1a3
a0 = 0.543 nm
x
yz
Model for dielectric crystals
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x
y
z
4 0 / a
ba
i j k10
ba
i j k30
ba
i j k20
Vibration modesIn the reciprocal space
- K = linear combination of de b1, b2, b3
- K belong to the first Brillouin ’s zone
- nomber of wave vectors K : number of elementary cells
- Number of polarisations p = 6
i
jk
Model for dielectric crystals
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Dispersion curves
B.N. Brockhouse, P.R.L. 2, 256 (1959)
Linear fit of the experimental dispersion curvesin the [1,0,0] direction
S. Wei et M.Y. Chou, PRB, 50, 2221 (1994)
The optical mode contribution to the thermal conductivity is negligible if T < 1000 K
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0 50 100 150 200 250 300 350
Temperature (K)
Cv/
(3R
)
P. Flubacher et al., Philos. Mag, 4,273 (1959)
0,00E+00
2,50E+12
5,00E+12
7,50E+12
1,00E+13
1,25E+13
1,50E+13
0 0,2 0,4 0,6 0,8 1
k/kmax
f (H
z)
LA
TA
Model for dielectric crystals
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Relaxation time parameters determination
M.G. Holland, PR, 132, 2461 (1963)
Fit of the thermal conductivity of a Si crystal (L = 7,16 mm) function of the temperature
Transverse modeA = 7 10-13
B = 0= 1= 4
Longitudinal modeA = 3 10-21
B = 0= 2= 1.5
F = 0.55D = 1.32 10-45 s-3
1
u K p
A
T
B
T,exp
1 4
D K pD
,
LF
pKv
pKCL .
,
,
1
10
100
1000
10000
1 10 100 1000
Temperature (K)
(W
m-1
K-1
)
Model for dielectric crystals
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0
5
10
15
20
25
30
35
40
45
50
0 50 100 150 200 250 300 350
Temperature (K)
(W
m-1
K-1
)
22 nm
37 nm
56 nm
115 nm
D. Li, et al., A.P.L, 83, 2934 (2003)
Excellent agreement except for the 22 nm wide
nanowire
Thermal conductivity of Si nanowires
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1
10
100
1000
20 60 100 140 180 220 260 300
Temperature (K)
(W
m-1
K-1
)
20 nm
100 nm
0,42 µm
0,83 µm1,6 µm3 µm
M. Asheghi et al., ASME JHT, 120, 30 (1998)M.Z. Bazant, PRB, 56, 8542 (1997)
Excellent agreement with the experimental
resutls
Thermal conductivity of Si nanofilms
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0
20
40
60
80
100
120
0 500 1000 1500 2000 2500 3000
film thickness (nm)
(W
/(m
K))
in plane
cross plane
Prediction of the thermal conductivity function of the heat transfer direction
T= 300K
Thermal conductivity of Si nanofilms
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CONCLUSION
Thermal conductivity of dielectric nanofilms and nanowires
Thermal conductivity of metallics and amorphous nanofilms
Thermal conctact resistance
Confident to get a accurate value
The bulk value overestimate the real value
Still a Problem, several models may be used However, one need to evaluate the maximun value of the thermal conductivity