of 8
Ultrasensitive and Wide-Bandwidth Thermal Measurements of Graphene
at Low Temperatures
Kin Chung Fong
*
and K. C. Schwab
Applied Physics, California Institute of Technology, MC 128-95, Pasadena, California 91125, USA
(Received 9 April 2012; published 30 July 2012; corrected 3 August 2012)
At low temperatures, the electron gas of graphene is expected to show both very weak coupling to
thermal baths and rapid thermalization, properties which are desirable for use as a sensitive bolometer. We
demonstrate an ultrasensitive, wide-bandwidth measurement scheme based on Johnson noise to probe the
thermal-transport and thermodynamic properties of the electron gas of graphene, with a resolution of
2mK
=
ffiffiffiffiffiffi
Hz
p
and a bandwidth of 80 MHz. We have measured the electron-phonon coupling directly
through energy transport, from 2–30 K and at a charge density of
2

10
11
cm

2
. We demonstrate
bolometric mixing and utilize this effect to sense temperature oscillations with a period of 430 ps and
determine the heat capacity of the electron gas to be
2

10

21
J
=
ð
K


m
2
Þ
at 5 K, which is consistent
with that of a two-dimensional Dirac electron gas. These measurements suggest that graphene-based
devices, together with wide-bandwidth noise thermometry, can generate substantial advances in the areas
of ultrasensitive bolometry, calorimetry, microwave and terahertz photo-detection, and bolometric mixing
for applications in fields such as observational astronomy and quantum information and measurement.
DOI:
10.1103/PhysRevX.2.031006
Subject Areas: Graphene, Mesoscopics, Photonics
I. INTRODUCTION
Graphene is a material with remarkable electronic prop-
erties [
1
] and exceptional thermal-transport properties near
room temperature, which have been well examined and
understood [
2
]. In fact, recent experiments have shown that
graphene exhibits one of the highest phononic thermal
conductivities of all measured materials [
3
]. However,
at low temperatures, the thermodynamic and thermal-
transport properties are much less well explored [
4
] and
somewhat surprisingly, due to the single atomic thickness,
low electron density, linear band structure, and weak
electron-phonon coupling, the electron gas of graphene is
expected to exhibit extreme thermal isolation [
5
7
].
The very weak thermal coupling combined with excep-
tionally small electronic heat capacity of graphene leads to
projections for very high sensitivity as both a bolometer
[
8
,
9
] and as a calorimeter [
10
,
11
]. As is typical with
extremely sensitive sensors, a device readout that has
both the sensitivity and sufficiently low measurement
backaction to realize the ultimate measurement sensitivity
can also be a significant challenge. Because of the thermal
sensitivity and the relatively weak dependence of resist-
ance on temperature [
12
,
13
], the simplistic use of electrical
transport as the readout scheme cannot realize the ultimate
sensitivity of graphene, which is given by thermodynamic
fluctuations [
14
,
15
]. Here, we present a measurement
scheme based upon high-frequency Johnson noise, which
provides both wide bandwidth and sensitivity to resolve the
fundamental thermal fluctuations, with minimal distur-
bance to the thermal properties of the sample. This tech-
nique should be useful for both thermodynamic studies of
graphene [
16
19
] and for bolometric applications at very
low temperatures.
This paper is organized as follows. We present the
thermal model of the electron gas of graphene at low
temperatures in Sec.
II
. Section
III
discusses the
Johnson-noise measurement scheme and fundamental lim-
its to the sensitivity. In Sec.
IV
, we present our measure-
ments, which provide a direct and accurate measurement of
the electron-phonon coupling of graphene from 2–30 K,
and a measurement of the thermal time constant, and we
determine the heat capacity of graphene. Section
V
ex-
plores the expected sensitivity of graphene as a bolometer
and as a calorimeter in the temperature range from 10 mK
to 10 K. These estimates suggest a number of exciting
possibilities: detection of single microwave-frequency
photons, photon number resolution, and spectroscopy of
terrahertz photons [
9
,
20
22
].
II. THERMAL MODEL
Figure
1(b)
shows the temperature dependence of the
expected thermal-conductance channels for the two-
dimensional electron gas: coupling to the electrical leads
through electron diffusion,
G
WF
, coupling to the lattice,
G
ep
, and coupling to the electromagnetic environment,
G
rad
, where
G
tot
is the sum of all three mechanisms.
Thermal transport through electron diffusion in gra-
phene has not yet been measured and most theories focus
on the clean, low-density graphene [
16
,
17
]. For doped
*
kcfong@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
2
, 031006 (2012)
2160-3308
=
12
=
2(3)
=
031006(8)
031006-1
Published by the American Physical Society
samples dominated by the disorder potential of the
SiO
2
substrate [
23
], such as the sample in this report, we assume
the simple Wiedemann-Franz (WF) relationship as a start-
ing point:
G
WF
¼
12
L
0
T
e
=R
, where
T
e
is the electron
temperature, and
L
0
and
R
are the Lorenz constant and
the electrical resistance of the sheet, respectively. The
prefactor, 12, is the result of the temperature profile devel-
oped by uniform ohmic heating and by the presence of the
thermal boundary condition of the contacts in a two-
terminal device (see comment [14] in Ref. [
24
] and the
Supplemental Material [
25
]).
The thermal conductance through the emission and ab-
sorption of blackbody photons into the electromagnetic
environment formed by the electrical measurement system
is limited by the quantum of thermal conductance [
26
],
G
0
¼
k
2
B
T
e
=
ð
6
@
Þ
(where
k
B
and
@
are the Boltzmann
and Planck constants, respectively), and the bandwidth
of the connection to the environment,
B
:
G
rad
¼
G
0
ð
2

@
B=k
B
T
e
Þ
, where
B
, the bandwidth of the blackbody
radiation [
27
], is assumed to be less than
k
B
T
e
=
ð
2

@
Þ
.
The electron gas can thermalize through the emission
and absorption of acoustic phonons [
28
]. For temperatures
below the Bloch-Gru
̈
neisen temperature [
13
,
29
,
30
][
T
BG
¼
2
c
1
=
2
@
v
F
n
1
=
2
=
ð
v
F
k
B
Þ¼
33 K
for
n
¼
10
11
cm

2
], heat
transport between electrons and phonons in a doped sample
in the clean limit (
k
p
l
e

1
where
k
p
and
l
e
are the phonon
wave vector and electron mean-free path, respectively) is
expected to follow
_
Q
¼
A

ð
T
4
e

T
4
p
Þ
[
5
7
,
31
], where
A
is
the sample area,

¼ð

5
=
2
k
4
B
D
2
n
1
=
2
Þ
=
ð
15

@
4
v
2
F
c
3
Þ
is the
electron-phonon coupling constant,
D
is the deformation
potential that characterizes the scattering of electrons by
phonons,
v
F
¼
10
6
m
=
s
is the Fermi velocity,
c
¼
20 km
=
s
is the speed of sound, and
n
is the charge-carrier
density. When
j
T
e

T
p
j
T
p
, the electron-phonon cou-
pling
G
ep
¼
4
AT
3
. This coupling has been inferred using
the temperature dependence of electrical transport for
T>T
BG
at a charge density of
n
¼
10
12
cm

2
[
13
] and
for both
T>T
BG
and
T<T
BG
with a very high charge
density,
n

10
13
cm

2
[
30
]. These measurements are
consistent with the expected form and magnitude of the
coupling.
Through careful engineering of the sample geometry
and the coupling to the electrical environment, it is
possible to force the graphene to thermalize primarily
through the electron-phonon channel, minimizing
G
tot
(see Supplemental Material [
25
] and Refs. [
32
38
]) and
at very low temperatures, this thermal conductance is ex-
pected to be extraordinarily weak [
7
] (Fig.
1
). The thermal
modeling of our graphene sample shows that the thermal
conductance is expected to be dominated by
G
ep
for tem-
peratures larger than 1 K, and by
G
WF
for
T<
1K
,as
shown in Fig.
1(b)
. Superconducting leads can be used to
block transport through
G
WF
[
26
] although care must be
taken to avoid processes such as multiple Andreev reflec-
tion and electron-electron scattering, which has been found
to contribute to heat transport when superconducting Al
leads are used to contact graphene [
39
].
Furthermore, for the same reasons listed above, which
result in weak thermal coupling, the heat capacity of
the electron gas is also expected to be minute. For
FIG. 1. (a) The measurement circuit: graphene (represented by a resistor), impedance matched with a 1.161 GHz resonant,
lithographic LC network (NbTiN film,
T
c
¼
13
:
5K
), and connected to a HEMT amplifier. (b) The expected thermal conductances
(
G
WF
;G
ep
;G
rad
) and heat capacity versus temperature for a bandwidth of 80 MHz,
n
¼
10
11
cm

2
, and
A
¼
10

10
m
2
. The inset
shows an optical micrograph of the graphene sample. The scale bar is
15

m
long. (c) The two-terminal resistance of the graphene
versus gate voltage, taken from 1.65–12 K. (d) The reflected microwave response versus gate voltage. The absorption dip at 1.161 GHz
shows that the graphene is well matched to
50 
in an 80-MHz band. (e) Spectra of the measured noise power taken at various sample
temperatures, demonstrating the Johnson-noise signal of the graphene in the impedance-matching band.
KIN CHUNG FONG AND KEITH C. SCHWAB
PHYS. REV. X
2,
031006 (2012)
031006-2
doped graphene, the heat capacity is expected to be
C
e
¼ð
2

3
=
2
k
2
B
n
1
=
2
T
e
Þ
=
ð
3
@
v
F
Þ
[
7
,
18
]. For perfectly pris-
tine graphene, the heat capacity is expected to follow a
T
2
ln
T
temperature dependence [
19
]; this is not the situ-
ation for a sample with disorder on a
SiO
2
substrate. At
100 mK and with
n
¼
10
11
cm

2
, one expects
C
e
¼
2
:
3
k
B
for a
1

1
-

m
flake. With this, combined with
the thermal conductance, one can estimate the thermal time
constant:

¼
C
e
=G
tot
. Assuming
G
tot

G
ep
, one expects
the maximum thermal time constant to be

¼
10 ps
at
10 K, 1 ns at 1 K, and
10

s
at 10 mK. Because of the
linear bands of graphene, and the correspondingly high
Fermi temperature, the heat capacity of graphene can be 50
times lower than that of a heterostructure two-dimensional
electron gas, for an assumed
n
¼
10
9
cm

2
.
III. NOISE THERMOMETRY AND
SAMPLE FABRICATION
Given the expected very weak coupling and high-speed
thermal response, we have implemented microwave-
frequency noise thermometry to explore these delicate
and wide-bandwidth thermal properties (Fig.
1
). Noise
thermometry has shown itself to be an excellent and nearly
noninvasive probe of electron temperature for nanoscale
devices with very minimal backaction heating [
11
,
26
,
40
].
Curiously, the dominant effect of this measurement
scheme on the graphene sample is to provide a thermal
conductance channel for cooling through the emission of
photons into the measurement channel,
G
rad
. This radia-
tive channel will be present for any electrical readout
scheme; however, in bolometers, which are sensed using
electrical transport, heating due to Ohmic loss is the
dominant perturbation for an optimized measurement
[
8
,
10
].
The analysis of hot-electron bolometers with a resistive
readout and operated with electrothermal feedback shows
the optimized energy resolution to be

E
2
=
ffiffiffiffi

p
Þ

E
th
[
10
,
15
,
22
,
41
], where

¼ð
T=R
Þ
dR=dT
and

E
th
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
C
e
k
B
T
p
is the energy resolution limited by thermody-
namic fluctuations of the bolometer. At 2 K, we measure

0
:
03
, which will limit the sensitivity of a resistively
read-out graphene bolometer to a sensitivity
12
E
th
. This
situation becomes much worse at lower temperatures as
both
T
and
dR=dT
decrease. Our analysis shows that an
approach based upon noise thermometry is capable of
approaching the thermodynamic limit, which leads us to
believe that noise thermometry is preferable for experi-
ments at very low temperatures.
Our graphene sample is fabricated using exfoliation onto
a Si wafer coated with 285 nm of SiO; the single atomic
layer thickness is confirmed using Raman spectroscopy
[
42
]. The high-resistivity wafer (
1
10 
-
cm
at 300 K) is
insulating at the cryogenic temperature, and the electrical
insulation minimizes the stray capacitance between the
electrodes and the ground, which would otherwise capaci-
tively load our impedance-matching network.
We match the relatively high impedance of a
15

6
:
8

m
flake of graphene,
30 k
at the charge-neutrality
point (CNP), to a
50 
measurement circuit using a litho-
graphic, superconducting NbTiN LC network, which is
placed a few millimeters from the graphene sample. The
FIG. 2. (a) The integrated noise power versus refrigerator temperature, demonstrating the expected temperature dependence of the
Johnson-noise signal. The deviation at temperatures above 8 K are due to the temperature dependence of the NbTiN inductor. The inset
shows the precision of the noise thermometry taken with two measurement bandwidths, 4 and 80 MHz, versus integration time. A
resolution of 100 ppm is achieved, in agreement with the Dicke radiometer formula (shown as lines). (b) The results of the differential
thermal-conductance measurements. The inset shows a time trace, taken at 4 K, of the small heating current at 17.6 Hz, and the
resulting 12-mK temperature oscillations at 35.2 Hz detected with the noise thermometer. The red curve is a power-law fit with
exponent
2
:
7
0
:
3
. (c) The results of applying large dc heating currents at various sample temperatures,
T
p
(points). The expected
form is shown as the blue line. Also shown is the heating of the electron gas versus applied microwave power at 1.161 GHz, also
showing a similar heating curve and demonstrating the microwave bolometric effect with graphene.
ULTRASENSITIVE AND WIDE-BANDWIDTH THERMAL
...
PHYS. REV. X
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031006 (2012)
031006-3
LC network (
L
125 nH
,
C
115 aF
) resonates at
1.161 GHz with a bandwidth of 80 MHz. As shown in
the scattering parameter,
S
11
, in Fig.
1(d)
, absorption of
microwave power is as high as 97% within the matched
bandwidth.
In the same frequency band where we have engineered
high absorption, the graphene is able to efficiently radiate
Johnson noise; this is shown as the peak in the noise spectra
at 1.2 GHz in Fig.
1(e)
. Analyzing these thermal noise
spectrums below 8K in the matched bandwidth shows the
expected linear dependence of the Johnson noise with
temperature [Fig.
2(a)
], i.e.,
P
int
¼
G
ð
k
B
T
þ
k
B
T
S
Þ
where
P
int
is the integrated Johnson noise power,
G
is the cali-
bration gain for the noise thermometry, and
T
S
is the noise
temperature of our system, due to the added noise of the
amplifier. For
T>
8
K
, the LC matching network begins to
substantially shift its frequency as the temperature ap-
proaches the superconducting transition temperature and
kinetic inductance effects become significant [
43
].
The temperature-axis intercept of Fig.
2(a)
determines
T
S
12 K
.
The sensitivity of our noise thermometry,
T
e
, follows
the Dicke radiometer formula [
44
]:
T
e
=
ð
T
e
þ
T
S
Þ¼
ð
Bt
m
Þ

1
=
2
, where
t
m
is the measurement time, and
B
is
the measurement bandwidth. For
B
¼
80 MH
z
, this leads
to an electron temperature noise power density of
ffiffiffiffiffiffiffi
S
T
e
p
¼
2
mK=
ffiffiffiffiffiffiffi
Hz
p
at 2 K. The inset of Fig.
2(a)
shows the mea-
sured normalized standard deviation of the noise tempera-
ture versus the measurement time, plotted for two
measurement bandwidths (4 and 80 MHz); the data follows
the Dicke radiometer formula. We have recently realized
much lower noise temperatures for our system of
T
s
<
1K
with the implementation of a wide bandwidth, nearly
quantum-limited SQUID amplifier [
45
].
Ohmic contact is made using evaporated
Au
=
Ti
leads.
However, an advantage of very high-frequency measure-
ments is that high-transparency contacts are not required
since capacitive coupling to a metal film deposited on the
graphene can also be utilized. A 100-nm-thick, Au electro-
static gate is insulated from the graphene using a 100-nm-
thick, overexposed electron beam resist (polymethyl
methacrylate) as a dielectric [
46
] (see the blue area in the
inset of Fig.
1(b)
). Figure
1(c)
shows the measured resist-
ance of the graphene device versus the gate voltage,
V
gate
,
with mobility of approximately
3500 cm
2
=
Vs
at a low
temperature, corresponding to a mean-free path of about
20 nm. The charge-carrier density at the CNP is approxi-
mately
2

10
11
cm

2
, which is estimated by the width of
the resistance maximum [
12
,
47
].
IV. MEASUREMENTS
We measure the thermal conductance of the electron gas
by simultaneously applying currents through the graphene
to produce Ohmic heating
_
Q
while measuring the elec-
tronic noise temperature. We have investigated the heat
transfer in several limits: first, by measuring the differential
thermal conductance with small quasistatic
_
Q
at various
temperatures; second, by using a larger dc-current bias
which produces temperature changes comparable to or
larger than the starting sample temperature; and finally,
by using microwave-frequency heating currents. All of
these measurements show similar results and are consistent
with the existing theory of the electron-phonon coupling
[
5
7
,
31
].
First, we impose a small oscillating current bias (
I
heat
)at
17.6 Hz, detect the resulting 35.2-Hz temperature oscilla-
tions of the electron gas in the limit where

T
e
=T
e
10

2
,
and then compute the differential thermal conductance:
G
th
¼
_
Q=

T
e
[Fig.
2(b)
]. This data is well fitted with
the expected form:
G
th
¼ð
p
þ
1
Þ

AT
p
, with

and
p
as
fitting parameters, and with values
0
:
07 W
=
ð
m
2
K
3
Þ
and
2
:
7
0
:
3
, respectively. The power-law exponent is near
the theoretical expectation of
p
¼
3
. Figure
2(c)
also shows
the results of applying a wide range of dc current biases such
that
T
e
canbemuchlargerthan
T
p
. We find this data to fit
the expected form:
T
e
¼½
_
Q=
ð
A

Þþ
T
p
þ
1
p
1
=
ð
p
þ
1
Þ
using

and
p
, which are found from the differential measurements.
We also apply a heating signal at 1.161 GHz, at the
center frequency of our impedance-matching (LC) net-
work where the absorption of the microwave power into
the graphene is nearly complete [Fig.
1(d)
], and measure
the increase in the electron-gas temperature with the
Johnson noise. This method also shows the same thermal
conductance that we found with quasistatic heating and
demonstrates graphene as a bolometer to microwave-
frequency radiation. Using the measured sensitivity of
the Johnson-noise thermometry and thermal conductance,
the noise-equivalent power (NEP) of our graphene bolome-
ter in this work is about
0
:
4pW
=
ffiffiffiffiffiffi
Hz
p
at 2 K. Below, we
show that substantial improvements should be possible at
lower temperatures and electron densities.
To probe the thermal time constant

and reveal the heat
capacity of the graphene, we utilize the microwave-
frequency impedance-matching network together with
the small temperature dependence of the electrical resist-
ance of graphene [Fig.
1(c)
]. This is a modified
3
-
!
method and is essentially a bolometric mixer [
24
].
We first apply a high-frequency oscillating current
within the impedance-matching band to heat the sample;
_
Q
¼
I
2
heat
ð
t
Þ
R
ð
T
e
Þ¼
I
2
heat
R
ð
T
e
Þ½
1
þ
cos
ð
2
!
heat
t
Þ
=
2
, where
!
heat
¼
2


1
:
161 GHz
. The DC component of the tem-
perature change is observed through the Johnson noise of
the sample, which is similar to the above thermal-
conductance measurements. If the thermal time constant
of the graphene electron gas satisfies
2
!
heat
<
1
, then the
temperature of the sample will oscillate at
2

!
heat
¼
2


2
:
322 GHz
. From our measurement of the
G
ep
and our
expectation of the heat capacity, we expect
2
!
heat

¼
1
for
T
5
:
7K
. Given the weak dependence of the sample
resistance on temperature,
dR=dT

400 
=
K
at 2 K, the
KIN CHUNG FONG AND KEITH C. SCHWAB
PHYS. REV. X
2,
031006 (2012)
031006-4
impedance,
Z
ð
!
Þ
, will also oscillate at
2
!
heat
with
T
e
.By
applying a second smaller modulation tone across the drain
source of the graphene sample at
!
mod
¼
!
heat

1 kHz
,
the impedance oscillations are then transduced into a very
small voltage oscillation, typically 10–100 pV, and are
mixed back into the range of our matching network;
V
ð
t
Þ¼ð
I
mod
e

i!
mod
t
Þð
Ze

i
2
!
heat
t
Þ
, where a component
of
V
ð
t
Þ
oscillates at
2
!
heat

!
mod
. The power spectral
density of the voltage across the graphene,
S
V
, in Fig.
3(a)
shows that the mixing tone depends on the input frequen-
cies as expected. See the Supplemental Material for more
details of the mixing and signal processing [
25
].
For temperatures above 5 K, the observed amplitude and
frequency of resulting mix tone agrees with our expecta-
tion due to a bolometric mixing effect. This demonstrates
the detection of temperature oscillation of graphene sheet
at 2.322 GHz. However, for temperatures below 5 K, we
observe a substantial decrease in the amplitude of the
mixed tone, which is consistent with the expected roll-off
due to the finite thermal-response time of the sheet due to
the heat capacity [Fig.
3(b)
]. Figure
3(b)
shows the mea-
sured bolometric mixing tone in graphene,
V
bolo
, normal-
ized to the theoretical mixing tone value,
V
calc
,if

¼
0
.
(See the Supplemental Material for mathematical details
[
31
].) As a control of the experiment, we have perform the
same measurement procedures with
!
heat
set to
2


17
:
6Hz
, and
!
mod
¼
2


1
:
160 999 GHz
. No roll-off of
the bolometric mixed tone at
!
mod
þ
2

2


17
:
6Hz
ver-
sus temperature was observed, which is as expected since
2
!
heat

th

1
in this case. At 5.25 K, we calculate a heat
capacity of
12 000
k
B
, which is comparable to the smallest
heat capacity measured to date [
9
]. We also believe this
to be the first measurement of the heat capacity of a two-
dimensional electron gas at zero field.
V. DISCUSSION
We compare our thermal-conductance data to the exist-
ing theoretical predications of the electron-phonon cou-
pling [
5
7
]. Using 30 eVas the deformation potential from
the electrical-transport measurements in single-layer gra-
phene [
12
,
13
,
30
], we find that our data is consistent with an
effective charge carrier density of
4
:
9

10
11
cm

2
. This is
within a factor of 2.5 of the charge density estimated from
the measurements of the resistance as a function of the gate
voltage (
2

10
11
cm

2
). The agreement of our data to the
theory, both in magnitude and in temperature dependence,
is somewhat surprising for two reasons. First, we make our
measurements at the CNP where the electron density is a
result of a large disorder potential that is typical with
SiO
2
substrates [
23
]. Second, our sample is deep in the diffusive
limit, i.e.,
k
p
l
e

1
. It is known from work on conven-
tional two-dimensional electron-gas structures that this
diffusive limit and screening can substantially alter the
electron-phonon coupling [
48
]. To date, we are aware of
no published literature on the electron-phonon coupling in
graphene in the diffusive limit and near the CNP.
Theoretical work on this limit will be very useful.
The Wiedemann-Franz thermal conductance estimated
here is based on that of a two-dimensional electron gas [
16
].
At a low charge-carrier density, theories have suggested that
deviations from the WF relationship are due to the relativ-
istic band structure [
17
]. This physics can be probed by
performing this experiment at a lower temperature.
The data we have gathered on the thermal-transport and
thermodynamic properties of graphene from 2–30 K and the
sensitivity of our wide-bandwidth thermometry system
motivates an estimation of the sensitivity of graphene as
a bolometer and photon detector at lower temperatures
[
8
,
10
]. Figure
4
shows the expected sensitivity as a bolome-
ter versus noise bandwidth for various temperatures, where
the NEP is given by
NEP
¼
G
tot

ffiffiffiffiffiffi
S
T
p
, where
S
T
is the
noise spectral density of the noise thermometer. Given
FIG. 3. Panel(a)showsthepowerspectraldensity,attheinputof
the HEMT, due to bolometric mixing. The central, 600-pV red
tone is due to a heating signal at
!
heat
=
2

¼
1
:
161 GHz
, which
produces a temperature response at 2.322 GHz, which is then
mixed down to
!
bolo
¼
!
heat
þ
2


1 kHz
using a small modu-
lation tone at
!
mod
¼
!
heat

2


1 kHz
. The blue spike, shifted
down by 100 Hz, is the result of increasing the modulation tone by
100 Hz. The green spike, shifted up by 200 Hz, is due to increasing
the heating tone by 100 Hz, validating the expected relationship:
!
bolo
¼
2
!
heat

!
mod
. (b) A plot of the measured bolometric
mixing tone normalized by our expected signal under the assump-
tion that the graphene thermal time constant

¼
0
. The red points
are measured with a heating tone of
!
heat
=
2

¼
1
:
161 GHz
, the
blue points are measured with
!
heat
=
2

¼
17
:
6Hz
, and both are
measured with a modulation tone of 1.161 GHz–1 kHz. The
dashed line shows the expected roll-off of the bolometric signal
when
2
!
heat
=
2

¼
2
:
32 GHz
and

¼
C
e
=G
ep
.
ULTRASENSITIVE AND WIDE-BANDWIDTH THERMAL
...
PHYS. REV. X
2,
031006 (2012)
031006-5
the minute heat capacity for
T<
1K
, the temperature
resolution is expected to be limited by the thermodynamic
fluctuations of the energy of the electron gas [
8
,
49
];
h

T
2
e
k
B
T
2
e
=C
, which gives
S
T
ð
!
Þ¼
4
k
B
T
2
e
=
f
C
½
1
þð
!
Þ
2
g
. The maximum sensitivity versus measure-
ment bandwidth is a result of the balance between gaining
resolution in the noise thermometry by increasing the mea-
surement band, and increasing the thermal response by
decreasing
G
rad
. As is clear from Figs.
4(a)
and
4(b)
,a
graphene-based bolometer may exceed the sensitivity
of the current state-of-the-art bolometers developed for
far-infrared or submillimeter-wave astronomy with a
sensitivity of
6

10

20
W
=
ffiffiffiffiffiffi
Hz
p
and a thermal time con-
stant of

¼
300 ms
[
20
], an improvement in bandwidth of
approximately 5 orders of magnitude.
As a photon detector and calorimeter, the expected
energy resolution is given by [
10
,
11
]

E
¼
NEP

ffiffiffi

p
.
Given the exceptionally fast thermal time constant, one
expects single-photon sensitivity to gigahertz photons
[Fig.
4(c)
]. For astrophysical applications in terahertz spec-
troscopy, one expects an energy resolution of four parts in
100 at 300 mK for an absorbed 1 THz photon. This satisfies
the instrument resolution requirements for future NASA
missions (BLISS) at
3
He
-refrigerator temperatures [
50
].
Compared to the recently proposed superconducting hot-
electron photon counter for THz applications at 300 mK
[
21
], the NEP of graphene-based bolometers is about 10
times less sensitive. However, the energy resolution for
graphene is expected to be about 7 times better. In this way,
graphene bolometers are a possible solution for low-flux
photon counting in the THz regime [
21
]. At 10 mK, the
intriguing possibility to observe single 800 MHz photons
appears possible.
Furthermore, for high photon flux,
_
n
, the quantization of
the field produces shot noise on the incoming power:
S
shot
¼
2
ð
@
!
Þ
2
_
n
W
2
=
Hz
. For sufficiently high microwave
photon flux, this noise will dominate the temperature
fluctuations of the sample [Fig.
4(d)
]. At 100 mK, and
with 10-GHz photons, for fluxes greater than
10
6
photons
=
s
, the noise of the bolometer should be domi-
nated by the shot noise of the microwave field. In this way,
graphene would act as a photodetector for microwaves,
having a square-law response and being absorptive and
sensitive to the shot noise of the incoming field. We
know of no other microwave detector that has these char-
acteristics and this would open the door to novel quantum
optics experiments with microwave photons [
51
].
ACKNOWLEDGMENTS
We acknowledge help with the microfabricated LC
resonators from M. Shaw, and helpful conversations with
P. Kim, J. Hone, D. Prober, E. Henriksen, J. P. Eisenstein,
A. Clerk, P. Hung, E. Wollman, A. Weinstein, B.-I. Wu, D.
Nandi, J. Zmuidzinas, J. Stern, W. H. Holmes, and P.
Echternach. This work has been supported by the FCRP
Center on Functional Engineering Nano Architectonics
(FENA) and U.S. NSF Contract No. (DMR-0804567).
We are grateful 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).
Note added.—
During the writing of this work, we be-
came aware of three other experimental works that touch
on some of these concepts [
52
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