of 7
Measurement of the Electronic Thermal Conductance Channels
and Heat Capacity of Graphene at Low Temperature
Kin Chung Fong,
1
Emma E. Wollman,
1
Harish Ravi,
1
Wei Chen,
2
Aashish A. Clerk,
2
M. D. Shaw,
3
H. G. Leduc,
3
and K. C. Schwab
1,
*
1
Kavli Nanoscience Institute, California Institute of Technology, MC 128-95, Pasadena, California 91125, USA
2
Department of Physics, McGill University, Montreal H3A 2T8, Canada
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
(Received 29 June 2013; published 29 October 2013)
The ability to transport energy is a fundamental property of the two-dimensional Dirac fermions in
graphene. Electronic thermal transport in this system is relatively unexplored and is expected to show unique
fundamental properties and to play an important role in future applications of graphene, including
optoelectronics, plasmonics, and ultrasensitive bolometry. Here, we present measurements of bipolar thermal
conductances due to electron diffusion and electron-phonon coupling and infer the electronic specific heat,
with a minimum value of
10
k
B
(
10

22
J
=
K
) per square micron. We test the validity of the Wiedemann-Franz
law and find that the Lorenz number equals
1
:
32

2
=
3
Þð
k
B
=e
Þ
2
. The electron-phonon thermal conduc-
tance has a temperature power law
T
2
at high doping levels, and the coupling parameter is consistent with
recent theory, indicating its enhancement by impurity scattering. We demonstrate control of the thermal
conductance by electrical gating and by suppressing the diffusion channel using NbTiN superconducting
electrodes, which sets the stage for future graphene-based single-microwave photon detection.
DOI:
10.1103/PhysRevX.3.041008
Subject Areas: Condensed Matter Physics, Graphene
I. INTRODUCTION
Electrical transport in graphene has attracted much atten-
tion because of the pseudochiral and relativistic nature of the
band structure [
1
,
2
]. Since both electrons and holes carry
energy as well as charge, the thermal transport of Dirac
fermions in two dimensions is expected to be as fascinating
as its electrical counterpart. Theorists have suggested a num-
ber of intriguing possibilities: The relativistic hydrodynam-
ics ofaCoulomb-interacting electron-holeplasmamayresult
in deviations from the Fermi-liquid values of the Mott rela-
tion, the Wiedemann-Franz ratio [
3
,
4
], and electronic spe-
cific heat [
5
]. Thermal transport measurements may reveal
the physics of a neutral mode in the fractional quantum Hall
effect [
6
]. The thermal properties of the electron gas are also
critical to graphene-based device applications [
7
,
8
], as they
impact photodetector performance [
9
], place fundamental
limits on the mobility of charge carriers [
10
], and set the
sensitivity of terahertz and microwave-frequency bolometers
[
11
13
], which promise single-photon resolution due to the
expected minute specific heat [
11
,
14
].
We present measurements of the bipolar thermal
conductance over a temperature range of 300 mK to
100 K, using three different sample configurations
(described below). For temperatures below approximately
1K
, we identify the thermal transport due to electron
diffusion
G
WF
, test the Wiedemann-Franz (WF) law, and
infer the electronic heat capacity, with a minimum value of
10

20
J
=
K
at 300 mK, which is 9 times smaller than the
previous record [
15
]. For higher temperatures, we measure
the thermal conductance due to phonon emission
G
ep
while varying the charge density. There has been recent
theory [
16
] that explores the effects of electronic disorder
on the electron-phonon (
ep
) coupling mechanism and
predicts a substantial modification in comparison to earlier
theory in the clean limit [
17
20
]; the disordered limit is
defined by

p

l
e
, where

p
¼
hs=
ð
k
B
T
Þ
is the dominant
thermal phonon wavelength,
l
e
is the electron mean-free
path,
s
is the sound velocity of graphene acoustic phonons,
and
k
B
and
h
are Boltzmann’s and Planck’s constants,
respectively. We present measurements that confirm both
the effect of the disorder and the nature of the
ep
coupling
(scalar or vector, screened or unscreened).
Previous thermal studies of graphene have been limited
to measurements of thermoelectric power [
21
23
]or
to measurements of thermal conductance taken at tem-
peratures above the Bloch-Gru
̈
neisen (BG) temperature
[
24
,
25
], at the charge neutrality point (CNP) [
11
], or
without considering the effects of disorder [
26
]. Recent
measurements of the behavior of Josephson junctions have
provided insight into electron-phonon coupling at tempera-
tures below 1 K [
27
]. Significant discrepancies between the
theoretical [
17
20
] and measured values [
26
] of both the
ep
coupling temperature power law and the coupling
constant are found in some of these experiments.
*
To whom all correspondence should be addressed.
schwab@caltech.edu
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 3.0 License
. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
PHYSICAL REVIEW X
3
, 041008 (2013)
2160-3308
=
13
=
3(4)
=
041008(7)
041008-1
Published by the American Physical Society
II. EXPERIMENTS
We probe thermal transport of the electrons in graphene by
applying Joule heating and measuring the electron tempera-
ture utilizing ultrasensitive, microwave-frequency Johnson-
noise thermometry with a sensitivity of
2mK
=
ffiffiffiffiffiffi
Hz
p
[
11
,
28
].
[See Fig.
1(a)
and the Supplemental Material (SM) [
29
]and
Refs. [
30
33
] therein.] Figure
1(b)
shows the expected ther-
mal model of the electron gas. With a typical coupling
bandwidth of 80 MHz to the graphene [
11
], one-dimensional
thermal transport [
34
] through blackbody radiation
G
rad
ð’
10

15
pW
=
K
Þð
G
WF
;G
ep
Þ
is expected to be neg-
ligible in this experiment. We assume both the electrodes and
lattice are in thermal equilibrium with the sample stage, as
the
ep
coupling in a normal metal [
35
] and the boundary
thermal conductance of the
SiO
2
-graphene interface [
36
]
are large compared to the
G
WF
and
G
ep
thermal channels.
Three devices with different electrodes and gating materials
(see Table
I
and the SM) are measured in two cryostats to
cover the entire sample temperature range: 0.3–1.5 K and
T>
1
:
5K
. For all three samples, the device length is much
longer than the inelastic scattering length
l
e
,whichavoids
any issues of electron shot noise [
38
40
]. For charge
densities that can be reached with our experiment
n
¼
10
11
10
13
cm

2
, the transition from
ep
to electron-
diffusion cooling is expected to occur at approximately
1K
and should be apparent because of the difference in tempera-
ture dependence of the thermal conductance:
G
ep
and
G
WF
are expected to depend on temperature as
T


1
(with


3
typically) and
T
,respectively.
With Joule heating
_
Q
applied to the electron gas,
the electron temperature
T
e
is expected to follow the
two-dimensional heat transfer differential equation
_
q
¼rð

WF
r
T
e
Þþ

ep
ð
T

e

T

p
Þ
;
(1)
where
_
q
¼
E
2
=
is the local Joule heating (such that
R
_
qd
2
r
¼
_
Q
),
E
is the local electric field,

is the elec-
trical resistivity,

WF
is the thermal conductivity due to
electronic diffusion,

ep
is the
ep
coupling parameter,
and
T
p
is the local phonon temperature. On the right-
hand side, the first term describes diffusive cooling
through the electron gas, while the second term de-
scribes cooling by phonon emission. In a Fermi liquid,
the local Joule heating and diffusive cooling are con-
nected through the WF law

WF
¼
L
0
T
e
=
,where
L
0
is
the Lorenz number given by
ð

2
=
3
Þð
k
B
=e
Þ
2
. Since the
(b)
T
4 K
300 K
20-dB attenuator
Directional
coupler
Circulator
LNA
Bias tee
LC matching
network
Graphene
sample
Reference
13.6 Hz
10 M
rf diode
LP
mixer
LNA
BP
~1 GHz
Lock-in 1
(a)
Reference
Lock-in 2
(e)
(d)
LP
300 mK
(c)
Device
D
2
Gold
electrode
Gold
electrode
Graphene
40
30
20
10
0
l
e
(nm
)
12
8
4
0
-4
n
(10
12
cm
-2
)
D
1
D
2
D
3
30
20
10
0
R
(k
)
12
8
4
0
-4
n
(10
12
cm
-2
)
D
1
D
2
D
3
FIG. 1. (a) Experimental setup for simultaneously measuring the thermal and electrical transport of a graphene device: LC, inductor
capacitor matching network; LP, low-pass filter; LNA, low-noise amplifier; and BP, band-pass filter. (b) Graphene thermal model. Heat
from the graphene electrons can flow out through two different channels: electronic diffusion to the electrodes
G
WF
and
ep
coupling
G
ep
. (c) Optical micrograph of device
D
2
. (d) dc graphene resistances. (e) Electrical mean-free path calculated using
l
e
¼
E
F
=ne
2
v
F
.
TABLE I. Sample information and measured quantities.

is
the electronic mobility from fitting [
37
].
l
e
and
k
F
l
e
are quoted
for data nearest to
n
¼
3
:
5

10
12
cm

2
, corresponding to a
Bloch-Gru
̈
neisen temperature
T
BG
¼
2
ð
s=v
F
Þð
E
F
=k
B
Þ’
101 K
.
The disorder temperature is given by [
16
]
T
dis
¼
hs=l
e
. At this
density,
T
dis
<T
BG
and
k
F
l
e
>
1
.
Devices
D
1
D
2
D
3
Gate
Local top Global back Global back
Electrodes
Ti/Au
Ti/Au
Ti/Au/NbTiN
Length
ð

m
Þ
15
15.4
4.6
Average width
ð

m
Þ
6.8
3.6
5.4

ð
cm
2
V

1
s

1
Þ
5800
200
5100
l
e
(nm)
21
10
22
k
F
l
e
5.5
3.3
7.5
T
dis
(K)
46
96
42
D
(eV)
19
23
51
KIN CHUNG FONG
et al.
PHYS. REV. X
3,
041008 (2013)
041008-2
temperature of the sample will not be uniform (the
middle will have a higher temperature than the leads)
and we measure the average electron-noise tem-
perature, the WF relati
onship will be modified to
G
WF
¼

L
0
T
p
=R
,where
R
is the graphene resistance
and

¼
12
(see the SM for discussion) [
41
].
By computing the ratio of
_
Q
to the measured increase in
average electron temperature with

T
e
=T
p

1
, we deter-
mine the thermal conductance
G
th
. Figure
2(a)
shows the
results from device
D
1
(gold leads, top gated) at various
charge-carrier densities. There is a clear transition from a
quadratic to a linear temperature dependence at approxi-
mately
1K
, which is expected and can be understood as
G
WF
dominating at low temperatures [Fig.
2(b)
]. We test the
Wiedemann-Franz law for two-dimensional Dirac fermions
by plotting
G
WF
versus
T=R
[Fig.
2(c)
] such that the slope is

L
0
. We also note that this
G
WF
is not equal to zero at
T=R
¼
0
, which at this point is not understood. Figure
2(d)
shows the measured Lorenz number at different densities.
The averaged Lorenz-number values for electron and hole
doping are
1
:
34
L
0
and
1
:
26
L
0
, respectively.
Our measured Lorenz number is
35%
higher than the
Fermi-liquid value,
17%
higher than the measured value in
graphite [
42
], and comparable to values obtained in other
materials [
43
]. While the electron-electron (
e
-
e
) interac-
tion may modify the Lorenz number in a material [
3
,
4
,
43
],
other effects such as contact resistance and contributions
due to graphene under the contacts could contribute to
errors in our calculation of the Lorenz number. Four-point
probe measurements of the thermal conductance may solve
this problem in the future. We also observe an increase of
the Lorenz number near the CNP [Fig.
2(d)
]. At the Dirac
point and at high temperatures where
E
F

k
B
T
(where
E
F
is the Fermi energy), theory predicts that the system
becomes quantum critical and the interaction between
massless electrons and holes enhances the Lorenz number
[
3
,
4
]. The same theory also expects that a deviation from
the Fermi-liquid value is possible in the case where
E
F

k
B
T
, as is true for our impurity-limited samples, but only if
the screening is weak [
3
]. Further experiments in cleaner
samples are needed to understand this anomalous behavior
in the Lorenz number as well as the offset in Fig.
2(c)
.
(a)
(b)
Electron-phonon
coupling regime
Wiedemann-
Franz regime
(c)
(d)
Device
D
1
Device
D
1
Device
D
1
Device
D
1
2.0
1.8
1.6
1.4
1.2
1.0
L
meas
/
L
0
-2
-1
0
1
2
n
(10
12
cm
-2
)
50
40
30
20
10
0
G
t
h
(pW
K
-1
)
100
80
60
40
20
0
T
/
R
(
μ
K/
)
Wiedemann-Franz law
Data
Fit
0.01
0.1
1
G
th
(nW
K
-
1
)
2
4
6
8
1
2
4
6
8
10
T
(K)
80
60
40
20
0
c
e
(
k
B
μ
m
-2
)
-2
-1
0
1
2
n
(10
12
cm
-2
)
60
50
40
30
20
10
0
G
th
(pW
K
-1
)
900 mK
700 mK
500 mK
311 mK
FIG. 2. Data from device
D
1
with normal metallic electrodes. (a) The bipolar
G
WF
as a function of charge-carrier density at
various sample temperatures. (b)
G
th
data as a function of temperature at
n
¼
2
:
2

10
12
cm

2
. The solid line is the power-law
fit to the
ep
thermal conductance above 1.5 K, while the dashed line is the best linear
T
fit to the WF thermal conductance. The
size of the data points represents the measurement error. (c) Wiedemann-Franz law in graphene. Each data point represents
a measurement at a different temperature and charge-carrier density. The fitted line is
G
WF
¼

L
meas
T=R
with a
y
offset;
L
meas
¼
3
:
25

0
:
02

10

8
WK

2
. (d) The measured Lorenz number
L
meas
as a function of density from fitting of
the Wiedemann-Franz law. For electrons with
n

0
:
18

10
12
cm

2
, averaged
L
meas
=
L
¼
1
:
34

0
:
06
, while for holes with
n
0
:
18

10
12
cm

2
, averaged
L
meas
=
L
¼
1
:
26

0
:
07
.
MEASUREMENT OF THE ELECTRONIC THERMAL
...
PHYS. REV. X
3,
041008 (2013)
041008-3
The electronic specific heat capacity
c
e
in graphene can
be determined by applying a two-dimensional kinetic
model:

WF
¼ð
1
=
2
Þ
c
e
v
F
l
e
, where
C
e
¼
Ac
e
is the total
electronic heat capacity,
v
F
is the Fermi velocity, and
A
is
the device area. Using
l
e
¼
32 nm
[Fig.
1(e)
], we plot
C
e
on the right-hand side of the
y
axis in Fig.
2(a)
. Since
G
WF
/
T
, and
l
e
does not depend on temperature signifi-
cantly because of impurity scattering, the measured specific
heat is linear in
T
for all densities. This measured linear
behavior agrees with theories for
j
E
F
j
k
B
T
, as the mini-
mum
j
E
F
j
of our samples is limited by impurity doping. For
j
E
F
j
k
B
T
, which is not accessible in this experiment, the
specific heat is expected to be proportional to
T
2
in the case
of massless Dirac fermions [
44
]. The smallest specific heat
attained near the CNP is merely
10
k
B
=
m
2
or
1000
k
B
for
the whole sample, a factor of 9 smaller than the inferred
value in state-of-the-art nanowires used for bolometry [
15
].
This value is also consistent with our earlier result at 5 K
estimated using a bolometric mixing effect [
11
].
At higher sample temperatures or for large Joule heat-
ing, the thermal conductance changes its temperature
power-law behavior [Fig.
2(b)
] as the dominant cooling
mechanism switches from
G
WF
to
G
ep
; the crossover tem-
perature is given by
ð

L
0
=RA

ep
Þ
1
=
ð


2
Þ
. In this regime,
Eq. (
1
) reduces to [
16
,
19
,
20
]
_
Q
¼
A

ep
ð
T

e

T

p
Þ
:
(2)
Using a dc current bias, Fig.
3(a)
plots the measured
T
e
versus the applied Joule heating power. For large
heating powers, the electron temperature converges to
ð
_
Q=A

ep
Þ
1
=
, independent of the initial temperature. The
solid lines in Fig.
3(a)
are the best fit to
T
e
¼ð
_
Q=A

ep
þ
T
3
p
Þ
1
=
3
, establishing

¼
3
. In light of recent theory [
16
], our
experimental data suggest that the
ep
heat transfer in the
disordered limit is primarily due to a weakly screened de-
formation potential, consistent with recent electrical trans-
port measurements [
45
]. The simplified physical scenario is
that impurity scattering in disordered graphene prolongs the
ep
interaction time and thus enhances the emission rates.
Recent investigations [
24
,
25
,
46
] suggest that the same power
law

¼
3
may also govern the disorder-assisted cooling
rate of hot electrons for
T>T
BG
, but through a different
mechanism.
(b)
(c)
D
2
D
3
(a)
2
3
4
5
6
1
2
3
4
5
6
10
2
3
4
5
6
100
2
T
e
(K
)
10
-2
10
1
10
4
10
7
Heating power (pW)
Device
D
2
8.0 K
4.0 K
3.5 K
2.0 K
Device
D
3
800 mK
420 mK
P
1/4
P
1/3
Device
D
3
250
200
150
100
50
0
G
t
h
/
A
(p
W
K
-
1
μ
m
-
2
)
-4
-2
0
2
4
n
(10
12
cm
-2
)
8 K
6 K
4 K
2 K
1
10
100
G
t
h
/
A
(pW K
-
1
μ
m
-2
)
2
4
6
8
1
2
4
6
8
10
T
(K)
Device
D
3
Data
Fit
WF limit
FIG. 3. (a) The measured electron temperature versus dc heating power applied to the graphene devices at different phonon
temperatures with
n
¼
28

10
12
cm

2
for device
D
2
and
n
¼
2
:
2

10
12
cm

2
for device
D
3
. The solid lines are fits to Eq. (
2
).
The power law
T
3
persists down to 420 mK for device
D
3
with NbTiN electrodes. (b),(c) Data from device
D
3
with NbTiN electrodes:
(b) the bipolar electron-phonon thermal conductance as a function of charge-carrier density at various temperatures and (c)
G
th
data as
a function of temperature at
n
¼
2
:
9

10
12
cm

2
. The solid line is the power-law fit to the
ep
thermal conductance above 1.5 K,
while the dashed line is the calculated
G
WF
¼

L
0
T=R
using
R
¼
1770 
. The offset between high- and low-temperature data is due
to thermal cycling of the device on two cryostats.
KIN CHUNG FONG
et al.
PHYS. REV. X
3,
041008 (2013)
041008-4