Electro-Thermal Transport in Metallic
Single-Wall Carbon Nanotubes
Eric Pop, David Mann, John Reifenberg
Profs. Kenneth Goodson and Hongjie Dai
Stanford University
E. Pop
IEDM 2005
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Carbon Nanotubes for Electronics
• Great electrical & thermal conductors
– Semiconducting transistors
– Metallic interconnects
• (Some) open questions:
– Thermal conductivity of singlewalled carbon nanotubes (SWNTs)
– Great thermal conductivity, poor
thermal conductance (small d)
Carbon nanotube
transistor
HfO2
E. Pop
2000
CNT
S (Pd)
D (Pd)
SiO2
back gate
(p++ Si)
– Transport issues
1995
top gate (Al)
2005
IEDM 2005
2010
2015
?
2
Properties of Metallic SWNTs
(what we know so far)
• Electrical properties (on insulating substrates)
– Current saturation near 20 µA for tubes longer than 1 µm
I (mA)
– Yao (PRL’00), Javey (PRL’04), Park (NL’04)
Park (NL’05)
– Larger currents (up to 100 µA) for short ~10 nm tubes
V (V)
• Thermal properties
– Yu (NL’05), Mingo (NL’05), Pop (PRL ’05)
– Thermal conductivity dominated by phonon transport
– 3500 Wm-1K-1 around 300 K, decreases as 1/T above
Thermal Cond. (W/cm/K)
Grujicic (JMS’05)
Temperature (K)
E. Pop
IEDM 2005
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Summary of This Talk
• Metallic single-wall nanotubes
– On insulating substrate (silicon nitride or oxide)
• Coupled electrical and thermal transport
– Essential for predicting breakdown voltage of SWNTs on
substrates exposed to air (oxygen)
– Use breakdown voltage scaling to find heat loss coefficient
• Unified electro-thermal model
– Must include optical phonon absorption (usually neglected) to
explain “upkick” in low-bias resistance vs. T
E. Pop
IEDM 2005
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Back-of-Envelope Estimate (Suspended)
E. Pop et al., Phys. Rev. Lett. 95, 155505 (2005)
• Thermal conductivity k ~ 3000 W/m/K
• Typical L ~ 1 µm, d ~ 1 nm suspended nanotube:
– Thermal conductance ~ 25 nW/K
– Thermal resistance ~ 40 K/µW
• Moderate power ~ 10 µW (10 µA @ 1 V) ∆T = 400 K!
∆T
k
Pt
SiO2
E. Pop
IEDM 2005
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Back-of-Envelope Estimate (On-Substrate)
• SWNT on insulating solid substrate
• Heat dissipated into substrate rather than along tube length
• What is the heat loss coefficient g?
• [A: need some gauge of the tube temperature]
∆T
Pt
g
SiO2
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IEDM 2005
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Nanotube Temperature Gauge
Pt
g
SiO2
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Nanotube Temperature Gauge
• Doesn’t exist
• But… oxidation (burning) temperature is known
O2
Pt
g
SiO2
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IEDM 2005
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Breakdown of SWNTs in Air (Oxygen)
25
Model
Data
VBD (V)
Weight (%)
20
15
10
5
0
T
0
1
2
3
4
5
L (µm)
(oC)
K. Hata, Science 306, 1362 (2004)
I. Chiang, JPCB 105, 8297 (2001)
E. Pop, Proc. IEDM (2005)
A. Javey, PRL 92, 106804 (2004)
•
Thermogravimetric (TGA) data shows SWNTs exposed to air break
down by oxidation at 500 < TBD < 700 oC (800–1000 K)
•
Joule breakdown voltage data shows VBD scales with L in air
•
Supports cooling mechanism along the length, into the substrate
E. Pop
IEDM 2005
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Breakdown of SWNTs: Analysis
25
A∇(k∇T ) + p '− g (T − T0 ) = 0
p' = I BDVBD / L
VBD = gL(TBD − T0 ) / I BD
20
VBD (V)
At breakdown:
Model
Data
15
10
5
0
0
1
2
3
4
5
L (µm)
•
Empirically note that:
– VBD vs. L in air scales approx. 5 V/µm
– Breakdown currents for L > 1 µm always about IBD ≈ 20 µA
•
Analytic solution of heat conduction equation
– Heat loss per unit length: g ≈ 0.14 – 0.2 WK-1m-1 given range of TBD
•
E. Pop
No assumption was made about electrical transport model
IEDM 2005
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Back-of-Envelope Estimate (Revisit)
• Using g ≈ 0.15 WK-1m-1 1/g ≈ 6 Kµm/µW
• Consistent with typical solid-solid thermal interface resistance
values, given the SWNT-substrate contact area (~Lxd)
• At moderate power per length p’ ≈ 10 µW/µm ∆T ≈ 60 K
∆T
Pt
g
SiO2
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IEDM 2005
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Combined Electro-Thermal Model
Rcontact
Rtube
Pt
h
R (V , T ) = RC + 2
4q
d
A∇ (k∇T ) + p '− g (T − T0 ) = 0
g ~ 0.15 Wm-1K-1
Lcontact
Ltube
 L + λeff (V , T ) 


(
,
)
λ
V
T
 eff

where
SiO2
p ' = I 2 ( R − RC ) / L
k (T ) = 3500(300 / T )
• Landauer-type electrical resistance
• Include Joule heating, couple with heat conduction equation
• Self-consistent solution
• Need temperature-dependent mean free paths (MFPs)
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IEDM 2005
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Combined Electro-Thermal Model
L = 3 µm
Rcontact
Rtube
Pt
20
d
T = 100, 200, 293 K
g ~ 0.15 Wm-1K-1
Lcontact
Ltube
I (µA)
15
10
Data
5
Isothermal model
SiO2
T−dependent model
0
0
0.5
1
V (V)
1.5
2
• Landauer-type electrical resistance
• Include Joule heating, couple with heat conduction equation
• Self-consistent solution
• Need temperature-dependent mean free paths (MFPs)
E. Pop
IEDM 2005
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Temperature Dependence of MFPs
MFPs at low bias (~ 50 mV)
−1
−1
λ eff = (λ −AC1 + λOP
, ems + λ OP , abs )
OP−abs
OP−ems
AC
Total MFP
−1
λOP ,abs (T ) = λOP ,300
…
N OP (300) + 1
N OP (T )
MFPs (µm)
λ AC = λ AC ,300 (300 / T )
100
10
1
100
150
200
250 300
T (K)
350
400
450
• Include scattering by optical phonon (OP) absorption
• Note OP emission may occur after:
– (1) A carrier gains enough energy (ħωOP ~ 0.18 eV) from E-field
– (2) An OP absorption event
• OP absorption is a strong function of T via N OP = [exp(hωOP / k BT ) − 1]
−1
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IEDM 2005
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Role of Optical Phonon Absorption
55
120
Predictions
L~3 µm
100
Including OP abs.
L=15 µm
50
45
R (kΩ)
R (kΩ)
80
Data
L=10 µm
60
L=5 µm
40
40
L=3 µm
Excluding OP abs.
20
L=1 µm
35
100
150
200
250
T (K)
300
0
100
350
150
200
250 300
T (K)
350
400
450
•
Low-bias resistance of metallic SWNTs
•
Subtle effect in the “up-kick” of SWNT low-bias resistance near room T
•
Important for temperature coefficient of resistivity (TCR) of SWNT
interconnects: ~0.0026 K-1, comparable to 40 nm Cu vias
E. Pop
IEDM 2005
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Electro-Thermal Modeling of SWNTs
600
25
L=2 µm
V=1,2,3,4,5 V
550
L=5 µm
15
450
10
L=15 µm
400
5
Isothermal
350
With self−heating
0
0
1
L=2 µm
500
T (K)
I (µA)
20
2
3
4
5
300
−8
V (V)
L=5 µm
L=15 µm
−6
−4
−2
0
2
X (µm)
4
6
8
•
Long tubes heat up less: better heat-sinking, lower power density
•
Model suggests current saturates near 20 µA due to self-heating
•
Self-heating may be neglected when p’ < 5 µW/µm (design goal?)
•
Current enhancement (> 20 µA) in shortest (< 1 µm) SWNTs very
likely aided by Joule heating shifting towards the contacts
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IEDM 2005
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Metallic SWNT Resistance Components
k ≈ 3500 Wm-1K-1
Ltube ≈ 2 µm, d ≈ 2 nm
Lcontact ≈ 2 µm
Contacts
3x
108
Pt
d
SiO2
Ltube
Lcontact
Thermal Res.
Phonons
Tube
Rtube
Rcontact
K/W
6 x 106 K/W
Thermal Res.
Electrons
2x
109
K/W
2 x 109 K/W
Electrical Res.
Electrons
WFL
~ 15 kΩ
Ω
~ 15 kΩ
Ω
• Relec. ~ 10 x Rphon. phonons dominate heat conduction
• Rcontact < 0.1 x Rtube as long as Lcontact > 0.8/Ltube (µm)
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IEDM 2005
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Conclusions
• High-bias breakdown in air (oxidation)
– Extract thermal conductance to substrate along SWNT length
– g ≈ 0.15 Wm-1K-1, limited by nanotube-substrate interface
• Electro-thermal model for transport in metallic SWNTs
– Estimate MFPs(T) and include optical phonon absorption
– Relevant in long (> 1 µm) SWNTs even at low bias (T > 300 K)
– Can we break the 20 µA current saturation “limit” for long metallic
SWNTs by engineering (increasing) g ?
• Contact:
– Email: [email protected]
– Web: http://www.stanford.edu/~epop/research.html
E. Pop
IEDM 2005
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