of 11
Measurement of the electronic thermal conductance channels and heat capacity of
graphene at low temperature
Kin Chung Fong,
1
Emma 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
2
Department of Physics, McGill University, Montreal, Canada H3A 2T8
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
(Dated: August 13, 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 opto-electronics, plasmonics, and ultra-sensitive bolometry.
Here we
present measurements of bipolar, electron-diffusion and electron-phonon thermal conductances,
and infer the electronic specific heat, with a minimum value of 10
k
B
(10
22
JK
1
) per square
micron. We test the validity of the Wiedemann-Franz law and find the Lorenz number equals
1
.
32
×
(
π
2
/
3)(
k
B
/e
)
2
. The electron-phonon thermal conductance 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 superconducting electrodes, which
sets the stage for future graphene-based single microwave photon detection.
PACS numbers: 65.80.Ck, 68.65.-k, and 07.20.Mc
INTRODUCTION
Electrical transport in graphene has attracted much
attention due to the pseudo-chiral and relativistic nature
of the band structure[1, 2]. Since both electrons and
holes carry energy as well as charge, the thermal trans-
port of Dirac Fermions in two dimensions is expected
to be as fascinating as its electrical counterpart. Theo-
rists have suggested a number of intriguing possibilities:
the relativistic hydrodynamics of a Coulomb-interacting
electron-hole plasma may result in deviations from the
Fermi-liquid values of the Mott relation, Wiedemann-
Franz ratio[3, 4], and electronic specific heat[5] . Thermal
transport measurements may reveal the physics of a neu-
tral 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 sensitiv-
ity 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 con-
ductance over a temperature range of 300 mK to 100
K, using three different sample configurations (described
below). For temperatures below
1 K, 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 con-
ductance due to phonon emission,
G
ep
, while varying
the charge density. There has been recent theory[16]
which 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
λ
p

l
e
, where
λ
p
=
hs/
(
k
B
T
) is the dominant ther-
mal 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 constant, respec-
tively. We present measurements which both confirm the
effect of the disorder and the nature of the ep coupling
(scalar, 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 temper-
atures above the Bloch-Gr ̈uneisen temperature[24, 25], at
the charge neutrality point (CNP)[11], or without consid-
ering the effects of disorder[26]. 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.
EXPERIMENTS
We probe thermal transport of the electrons in
graphene by applying Joule heating and measuring the
electron temperature utilizing ultra-sensitive, microwave
frequency Johnson noise thermometry with a sensitiv-
ity of 2 mK/
Hz[11, 27] [see Fig. 1a and Supplemen-
tary Material (SM)]. Fig. 1b shows the expected thermal
arXiv:1308.2265v1 [cond-mat.mes-hall] 10 Aug 2013
2
model of the electron gas. With a typical coupling band-
width of 80 MHz to the graphene[11], one-dimensional
thermal transport[28] through black body radiation,
G
rad
(
'
10
15
pW/K)

(
G
wf
,
G
ep
) is expected to be neg-
ligible in this experiment. We assume both the elec-
trodes and lattice are in thermal equilibrium with the
sample stage as the ep coupling in normal metal[29] and
the boundary thermal conductance of the SiO
2
-graphene
interface[30] are large compared to the
G
wf
and
G
ep
ther-
mal channels. Three devices with different electrodes and
gating materials (see Tab. 1 and SI) are measured in two
cryostats to cover the entire sample temperature range:
0.3-1.5 K and
T >
1.5 K. For all three samples, the device
length is much longer than the inelastic scattering length,
l
e
, which avoids any issues of electron shot noise[31–33].
For charge densities which can be reached with our ex-
periment,
n
= 10
11
10
13
cm
2
, the transition from ep
to electron-diffusion cooling is expected to occur at
1
K and should be apparent due to the difference in tem-
perature dependence of the thermal conductance:
G
ep
and
G
wf
are expected to depend on temperature as
T
δ
1
(with
δ
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
=
−∇·
(
κ
wf
T
e
) + Σ
ep
(
T
δ
e
T
δ
p
)
(1)
where ̇
q
=
E
2
is the local Joule heating (such that
̇
qd
2
r
=
̇
Q
), E is the local electric field,
ρ
is the electrical
resistivity,
κ
wf
is the thermal conductivity due to elec-
tronic diffusion, Σ
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 describes cooling
by phonon emission. In a Fermi liquid, the local Joule
heating and diffusive cooling are connected through the
wf law,
κ
wf
=
L
0
T
e
, where
L
0
is the Lorenz number
given by (
π
2
/
3)(
k
B
/e
)
2
. Since the 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 temperature, the wf relationship will be
modified to
G
wf
=
α
L
0
T
p
/R
, where
R
is the graphene
resistance and
α
= 12 (see SM for discussion.)[34]
By computing the ratio of
̇
Q
to the measured increase
in average electron temperature with ∆
T
e
/T
p

1, we
determine the thermal conductance,
G
th
. Fig. 2a shows
the results from device D1 (gold leads, top-gated) at
various charge carrier densities. There is a clear tran-
sition from a quadratic to linear temperature depen-
dence at
1K, which is expected and can be under-
stood as
G
wf
dominating at low temperatures (Fig. 2b).
We test the Wiedemann-Franz law for two-dimensional
Dirac Fermions by plotting
G
wf
versus
T/R
(Fig. 2c)
such that the slope is
α
L
0
. We also note that this
G
wf
is
not equal to zero at
T/R
= 0, which at this point is not
understood. Fig. 2d shows the measured Lorenz number
at different densities. The averaged Lorenz number val-
ues 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[35], and comparable to values ob-
tained in other materials[36]. While the electron-electron
(e-e) interaction may modify the Lorenz number in a
material[3, 4, 36], 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. 2d). At the Dirac point and at high tempera-
tures where
E
F

k
B
T
(where
E
F
is the Fermi energy),
theory predicts that the system becomes quantum crit-
ical and the interaction between massless electrons and
holes enhances the Lorenz number[3, 4]. The same the-
ory 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 screen-
ing 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. 2c.
The electronic specific heat capacity,
c
e
, in graphene
can be determined by applying a two-dimensional kinetic
model:
κ
wf
= (1
/
2)
c
e
v
F
l
e
, where
C
e
=
Ac
e
is the total
electronic heat capacity,
v
F
is the Fermi velocity, and
A
is the device area. Using
l
e
= 32 nm (Fig. 1e), we plot
C
e
on the right hand side of the y-axis in Fig. 2b. Since
G
wf
T
and
l
e
does not depend on temperature signifi-
cantly due to impurity scattering, the measured specific
heat is linear in
T
for all densities. This agrees with
theories for
|
E
F
| 
k
B
T
, as the minimum
|
E
F
|
of our
samples is limited by impurity doping. For
|
E
F
|
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[37]. The smallest specific
heat attained near the CNP is merely 10
k
B
μ
m
2
or
1000
k
B
for the whole sample, a factor of 9 smaller than
the inferred value in state-of-the-art nano-wires used for
bolometry[15]. This value is also consistent with our ear-
lier 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. 2b) as the dominant cooling
mechanism switches from
G
wf
to
G
ep
; the cross-over
temperature is given by (
α
L
0
/δRA
Σ
ep
)
1
/
(
δ
2)
. In this
regime, Eq. 1 reduces to[16, 19, 20]:
̇
Q
=
A
Σ
ep
(
T
δ
e
T
δ
p
).
(2)
Using a dc current bias, Fig. 3a plots the measured
T
e
versus the applied Joule heating power. For large
heating powers, the electron temperature converges to
3
(
̇
Q/A
Σ
ep
)
1
, independent of the initial temperature.
The solid lines in Fig. 3a are the best fit to
T
e
=
(
̇
Q/A
Σ
ep
+
T
3
p
)
1
/
3
, establishing
δ
= 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 deformation potential, consis-
tent with recent electrical transport measurements[38].
The simplified physical scenario is that impurity scat-
tering in disordered graphene prolongs the ep interac-
tion time and thus enhances the emission rates. Recent
investigations[24, 25, 39] suggest that the same power
law,
δ
= 3, may also govern the disorder-assisted cooling
rate of hot electrons for
T > T
BG
, but through a different
mechanism.
For the three devices we fabricated and measured, all
show
δ
'
3 except for device D2 (gold leads, back-gated)
at temperatures above 50 K and device D1 near the CNP
(see Fig. 4b). In both circumstances, the temperature
power law increases from
δ
= 3 to 4. This behavior
of D1 near the CNP, where transport is expected to be
dominated by disorder and charge puddles, is surpris-
ingly consistent with theoretical expectations[17–20] in
the clean limit, as reported in Ref 11. In this device, as
the charge carrier density decreases to 10
11
cm
2
near
the CNP, the screening length grows to 50 nm which is
comparable to the distance to the nearby metallic top
gate (100 nm, dielectric constant of 4). We speculate
this screening of the metallic gate may impact the ep
coupling. Moreover, near the CNP, the impurity scatter-
ing is long-range in nature and
k
F
l
e
<
1, which is outside
the regime of validity of the existing ep coupling theory
for disordered graphene[16]. More experiments and the-
ory are required to understand the nature of ep coupling
at low charge carrier density. This is of particular im-
portance to graphene-based bolometry as the ultimate
sensitivity is expected to be limited by the ep coupling
at the lowest carrier densities[11, 14].
We can further explore Eq. 2 by measuring the dif-
ferential thermal conductance at different temperatures
and carrier densities using a small ac current bias. If
T
e
T
p

T
p
, the ep thermal conductance is
δA
Σ
ep
T
δ
1
p
.
Fig. 3b shows
G
ep
as function of carrier density for de-
vice D3 (superconducting leads, back-gated). Fig. 2b
and 3c, for devices D1 and D3 respectively, show that
G
th
is limited by the ep thermal conductance at T
>
1.5
K with a power law
δ
'
3 for both devices. However,
since the electrons in a superconductor have negligible
entropy and do not conduct thermally,
G
th
of device D3
with NbTiN electrodes is not limited by
G
wf
at low tem-
peratures. The dashed line in Fig. 3c is the calculated
G
wf
, similar to the dashed line in Fig. 2b. NbTiN is
used because of its higher transition temperature (14 K)
to avoid e-e interaction that may promote hot electrons
over the superconducting bandgap[40]. The ep data at
0.4 K demonstrate the suppression of heat diffusion by
roughly 80%.
We can obtain Σ
ep
and
δ
by fitting
G
ep
as function
of lattice temperature at a constant carrier density. Re-
sults are plotted in Fig. 4. For devices D2 and D3, we
find
δ
values of approximately 3.0 and 2.8, respectively.
Near the CNP, Σ
ep
has a minimum, but is not vanishing.
Furthermore, the fitted Σ
ep
values vary by an order of
magnitude across all three devices. This variation and
also the discrete jump in measured
G
th
after thermal cy-
cling in Fig. 3c indicate that the underlying ep coupling
mechanism is strongly modified by disorder.
We compare the theory of ep coupling in disordered
graphene to both the measured electrical and ep thermal
transport data using[16]:
Σ
ep
=
2
ζ
(3)
π
2
E
F
v
3
F
ρ
M
D
2
k
3
B
~
4
l
e
s
2
(3)
where
D
is the deformation potential and
ρ
M
is the mass
density of the graphene sheet. This theory assumes a
short-range scattering impurity potential and
k
F
l
e
>
1.
The estimated values of
D
for devices D1, D2, and D3, at
a charge density of
n
'
3
.
5
×
10
12
cm
2
, are 19, 23, and 51
eV respectively (Tab. 1). The considerable scatter in the
inferred values of
D
is consistent with the range of values
obtained from electrical transport measurements[38, 41–
43]. Fig. 4b inset shows the inferred deformation po-
tential values in both thermal and electrical experiments
reported in the literature[11, 24–26, 38, 41–43]. The wide
scatter suggests that some factors may not be captured
in the ep coupling theory for disordered graphene, such
as long-range impurity scattering, substrate-induced ef-
fects, or surface-acoustic phonons[44].
CONCLUSIONS
In this report, we investigate the bipolar thermal con-
ductance of graphene in both the electron diffusion and
electron-phonon regimes. We find that the ep coupling
is strongly modified by electronic disorder and is consis-
tent with scalar coupling in the weak screening limit.[16]
ep coupling in the disordered limit is especially relevant
for ultra-sensitive device applications at low tempera-
tures, since even the cleanest samples[42, 43] yet reported
(
l
e
>
1
μ
m) would cross into the disordered limit for
temperatures below 1 K. This experiment has validated
the wf law for two-dimensional Dirac fermions. It may
be possible to study many-body physics in this system
through more precise measurements of the Lorenz ra-
tio and with cleaner samples near the CNP. The elec-
tronic specific heat inferred through the electron diffusion
measurement is merely 10
k
B
μ
m
2
. Our estimates sug-
gest that a single terahertz photon should be detectable
at 300 mK using a 1
μ
m
2
size sample and a SQUID
amplifier[11, 14, 45]. We have also demonstrated the con-
trol of heat flow in graphene by both using the field effect
and employing superconductors to suppress the thermal
4
diffusion channel. These findings point the way to future
experiments to probe both the fundamental and practi-
cal electronic thermal properties of this unique atomically
thin material.
ACKNOWLEDGMENTS
We acknowledge helpful conversations with P. Kim,
J. Hone, E. Henriksen, and D. Nandi. This work was
supported in part by (1) the FAME Center, one of six cen-
ters of STARnet, a Semiconductor Research Corporation
program sponsored by MARCO and DARPA, (2) the US
NSF (DMR-0804567), (3) the Institute for Quantum In-
formation and Matter, an NSF Physics Frontiers Center
with support of the Gordon and Betty Moore Founda-
tion, and (4) the Department of Energy Office of Sci-
ence Graduate Fellowship Program (DOE SCGF), made
possible in part by the American Recovery and Rein-
vestment Act of 2009, administered by ORISE-ORAU
under contract no. DE-AC05-06OR23100. We are grate-
ful to G. Rossman for the use of a Raman spectroscopy
setup. Device fabrication was performed at the Kavli
Nanoscience Institute (Caltech) and at the Micro Device
Laboratory (NASA/JPL), and part of the research was
carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the Na-
tional Aeronautics and Space Administration.
5
b)
T
4 K
300 K
20 dB attenuator
directional
coupler
circulator
LNA
bias tee
LC matching
network
graphene
sample
ref.
13.6Hz
10M
rf diode
LP
Mixer
LNA
BP
~1GHz
lockin 1
a)
ref.
lockin 2
e)
d)
LP
300 mK
c)
T
e
,
C
e
T
amp
T
p
T
elec
G
rad
G
ep
G
wf
̇
Q
device D2
gold
electrode
gold
electrode
graphene
μ
m
10
40
30
20
10
0
l
e
(n m)
12
8
4
0
-4
n (1 0
12
cm
-2
)
D1
D2
D3
30
20
10
0
R (k
)
12
8
4
0
-4
n (1 0
12
cm
-2
)
D1
D2
D3
FIG. 1: (a) Experimental setup for simultaneously
measuring the thermal and electrical transport of a
graphene device. (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 D2. (d) dc graphene resistances. (e) Electrical
mean-free-path calculated using
l
e
=
σE
F
/ne
2
v
F
.