of 13
Ultra-sensitive and Wide Bandwidth Thermal Measurements of
Graphene at Low Temperatures
K.C. Fong
1
and K.C. Schwab
1
Applied Physics, Caltech, Pasadena, CA 91125 USA
Abstract
Graphene is a material with remarkable electronic properties[1] and exceptional thermal trans-
port properties near room temperature, which have been well examined and understood[2, 3].
However at very low temperatures the thermodynamic and thermal transport properties are much
less well explored[4, 5] and somewhat surprisingly, is expected to exhibit extreme thermal isola-
tion. Here we demonstrate an ultra-sensitive, wide-bandwidth measurement scheme to probe the
thermal transport and thermodynamic properties of the electron gas of graphene. We employ
Johnson noise thermometry at microwave frequency to sensitively measure the temperature of the
electron gas with resolution of 4
mK/
Hz
and a bandwidth of 80 MHz. We have measured the
electron-phonon coupling from 2-30 K at a charge density of 2
·
10
11
cm
2
. Utilizing bolometric
mixing, we have sensed temperature oscillations with period of 430 ps and have determined the
heat capacity of the electron gas to be 2
·
10
21
J/
(
K
·
μm
2
) at 5 K which is consistent with that
of a two dimensional, Dirac electron gas. These measurements suggest that graphene-based de-
vices together with wide bandwidth noise thermometry can generate substantial advances in the
areas of ultra-sensitive bolometry[6], calorimetry[7], microwave and terahertz photo-detection[8],
and bolometric mixing for applications in areas such as observational astronomy[9] and quantum
information and measurement[10].
1
arXiv:1202.5737v1 [cond-mat.mes-hall] 26 Feb 2012
Given the single atomic thickness and low electron density, the electron gas of graphene
is expected to be very weakly coupled to surrounding thermal baths at low temperatures.
Figure 1b shows the temperature dependence of the expected cooling channels for the two-
dimensional electron gas: cooling through electron diffusion,
G
WF
, emission of phonons,
G
ep
, and emission of photons,
G
rad
, where
G
tot
the sum of all three mechanisms. Electron
diffusion is expected to follow the usual Wiedemann-Franz form:
G
WF
=
L
0
T
e
/R
, where
L
0
,
k
B
, and
R
are the Lorentz constant, the Boltzmann constant, and the electrical resistance of
the sheet, respectively. The thermal conductance through the emission of blackbody photons
into the electrical measurement system (Johnson noise) is limited by the quantum of thermal
conductance,[11]
G
0
=
πk
2
B
T
e
/
(6
~
) and the bandwidth of the connection to the environment,
B
:
G
rad
=
G
0
(2
π
~
B/k
B
T
e
), assuming
B < k
B
T
e
/
(2
π
~
) which is the bandwidth of the black
body radiation[12–14].
The electron gas can thermalize through the emission of acoustic phonons[16–
19].
For temperatures below the Bloch-Gr ̈uneisen temperature[15, 20, 21] (
T
BG
=
2
1
/
2
~
v
F
n
1
/
2
/
(
v
F
k
B
) = 33 K for
n
= 10
11
cm
2
), heat transport between electrons
and phonons[22] is expected to follow
̇
Q
=
A
Σ(
T
4
e
T
4
p
), where
A
is the sample area,
Σ = (
π
5
/
2
k
4
B
D
2
n
1
/
2
)
/
(15
ρ
~
4
v
2
F
c
3
) is the coupling constant,
D
is the deformation potential,
v
F
= 10
6
m/s is the Fermi velocity, c = 20 km/s is the speed of sound, and
n
is the charge
carrier density. When
|
T
e
T
p
| 
T
p
,
G
ep
= 4Σ
AT
3
. Through careful engineering of the
sample geometry and the coupling to the electrical environment, it is possible to force the
heat through the phonon channel, minimizing
G
tot
(see supplementary information) and at
very low temperatures, this thermal conductance is expected to be extraordinarily weak[19]
(Fig. 1).
Furthermore, the heat capacity of the electron gas is expected to be minute[19]:
C
e
= (2
π
3
/
2
k
2
B
n
1
/
2
T
e
)
/
(3
~
v
F
). At 100 mK and with
n
= 10
11
cm
2
, one expects
C
e
= 2
.
3
·
k
B
for a 1
μm
x 1
μm
flake. This combined with the thermal conductance, one can estimate the
thermal time constant:
τ
=
C
e
/G
tot
. Assuming
G
tot
G
ep
, one expects the maximum
thermal time constant to be
τ
= 10
ps
at 10K, 1
ns
at 1K, and 10
μs
at 10mK. Due to
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 2DEG, assuming
n
= 10
9
cm
2
.
Given the expected very weak coupling and high speed thermal response, we have imple-
2
mented microwave frequency noise thermometry to explore these delicate and high band-
width thermal properties, (Fig.1). Noise thermometry has shown itself to be an excellent
and nearly non-invasive probe of electron temperature for nanoscale devices [7, 11] with very
minimal back-action heating. Using a superconducting LC matching network, we match the
relatively high impedance of a 15
μm
×
6.8
μm
flake of graphene, 30
k
Ω at the charge neutral-
ity point (CNP), to a 50 Ω measurement circuit; the network resonates at 1.161 GHz with a
bandwidth of 80 MHz. This allows for fast, high sensitivity measurements of the electron gas
temperature which follows the Dicke Radiometer formula[23]: ∆
T
e
/
(
T
e
+
T
S
) = (
Bt
m
)
1
/
2
,
where
t
m
is the measurement time, and
T
S
= 12 K is the system noise temperature of our
cryogenic HEMT amplifier. This leads to a temperature resolution of
S
T
= 4
mK/
Hz
at
a sample temperature of 2 K. Much lower system noise temperatures are possible with the
use of high bandwidth, nearly quantum-limited SQUID amplifiers[24].
Our graphene sample is fabricated using exfoliation onto a Si wafer coated with 285 nm of
SiO; the single layer thickness is confirmed using Raman spectroscopy[25]. Ohmic contact is
made using evaporated Au/Ti leads. Figure 1e shows the measured noise spectrum in band
of the HEMT amplifier showing the Johnson noise thermometry of the sample. Given the
sample size and the electrical resistance, we expect the thermal conductance to be dominated
by
G
ep
for temperatures above 300 mK. At lower temperatures, superconducting leads can
be used to block transport through
G
WF
[11].
By applying currents through the graphene sample and producing ohmic heating,
̇
Q
, we
can measure the thermal conductance of the electron gas. We impose a small oscillating
current bias at 17.6 Hz, detect the resulting 35.2 Hz temperature oscillations of the electron
gas in the limit where ∆
T
e
/T
e
'
10
2
, and then compute the differential thermal conduc-
tance:
G
=
̇
Q/
T
e
(Fig. 2b). This data is well fitted with the expected form:
G
= 4Σ
AT
p
,
with Σ and
p
as fitting parameters. At the CNP where
n
= 2
·
10
11
cm
2
due to impu-
rities, we find Σ = 0
.
07
W/
(
m
2
K
3
) which is consistent with a deformation potential of
D
= 47 eV, and a power law of
p
= 2
.
7
±
0
.
3 The power law exponent is near the theo-
retical expectation of
p
= 3 and the deformation potential is within the range of previous
measurements[17, 20, 21, 26].
Figure 2 also shows the results of applying a wide range of DC current biases, where
T
e
can be much larger than
T
p
. We find this data to fit the expected form:
T
e
= (
̇
Q/
(
A
·
Σ) +
T
p
+1
p
)
1
/
(
p
+1)
using Σ and
p
found from the differential measurements. Finally, we applied
3
a heating signal at 1.161 GHz, at the frequency of our LC matching network where the
microwave absorption into the graphene is nearly complete (Fig. 1d), and measured the
increase in the electron gas temperature with the Johnson noise. This method also shows
the same thermal conductance and demonstrates graphene as a bolometer to microwave
frequency radiation.
To probe the thermal time constant,
τ
, 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 resistance of graphene (Fig. 1c). We first apply a
high frequency oscillating current, within the impedance matching band to heat the sample:
̇
Q
=
I
2
heat
(
t
)
·
R
(
T
e
) =
I
2
heat
R
(
T
e
)(1 +
cos
(2
ω
heat
t
))
/
2 where
ω
heat
= 2
π
·
1
.
161 GHz. Similar
to above thermal conductance measurements, the DC component of the temperature change
is observed through the Johnson noise of the sample. If the thermal time constant of the
graphene electron gas satisfies 2
ω
heat
τ <
1, then the temperature of the sample will oscillates
at 2
·
ω
heat
= 2
π
·
2
.
322 GHz. From our measurement of the
G
ep
and our expectation of
the heat capacity , we expect 2
ω
heat
τ
= 1 for
T
= 5
.
7 K. Given the weak dependence of
the sample resistance on temperature,
dR/dt
=
400Ω
/K
at 4 K, the impedance,
Z
(
ω
),
will also oscillate at 2
ω
heat
with
T
e
. By applying a second smaller modulation tone at
ω
mod
=
ω
heat
1 kHz, the impedance oscillations are transduced into a very small voltage
oscillation, typically 10-100 pV, and are mixed back into the range of our matching network:
δV
(
t
) = (
I
mod
e
mod
t
)
·
(
δZe
i
2
ω
heat
t
), where a component of
δV
(
t
) oscillates at 2
ω
heat
ω
mod
.
See supplementary information for more details of the mixing and signal processing.
For temperatures above 5 K we observe this mixed tone, resulting from bolometric mixing,
with its amplitude agreeing with our expectations, demonsrating the temperature oscillations
of the graphene sheet at 2.322 GHz. However, for temperatures below 5 K we observe a
substantial decrease in the amplitude of the mixed tone, consistent with the expected roll-off
due to the finite thermal response time of the sheet due to the heat capacity. 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[8].
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[27]. Figure 4 shows the expected sensitivity as a bolometer versus noise
4
bandwidth for various temperatures, where the noise equivalent power (NEP) is given by:
NEP
=
G
tot
·
S
T
, where
S
T
is the noise spectral density of noise thermometer. Given 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 [27, 28]:
T
2
e
=
k
B
T
2
e
/C
,
which gives
S
T
(
ω
) = 4
τk
B
T
2
e
/
(
C
(1+(
τω
)
2
)) . The maximum sensitivity versus measurement
bandwidth is result of the balance between gaining resolution in the noise thermometry by
increasing the measurement band, and increasing the thermal response by decreasing
G
rad
.
As is clear from the plots, a graphene-based bolometer may exceed the sensitivity of the
current state of the art bolometers developed for far-infrared/submillimeter wave astronomy
with a sensitivity of 6
·
10
20
W/
Hz
and a thermal time constant of
τ
= 300 ms [6], an
improvement in bandwidth of
5 orders of magnitude.
As a photon detector and calorimeter, the expected energy resolution if given by[7, 27]:
E
=
NEP
·
τ
. Given the exceptionally fast thermal time constant, one expects single
photon sensitivity to gigahertz photons(Fig. 4.) For astrophysical applications in terahertz
spectroscopy, one expects an energy resolution of one part in 1000 at 300 mK for an ab-
sorbed 1 THz photon. This satisfies the instrument resolution requirements for future NASA
missions (BLISS) at 3He-refrigerator temperatures[9]. At 10mK, the intriguing possibility
to observe single 500MHz photons appears possible.
Furthermore, for high rates of photon flux, ̇
n
, the quantization of the field produces
shot noise on the incoming power:
S
shot
= 2(
~
ω
)
2
̇
n W
2
/Hz
. For sufficiently high rates of
microwave photons, this noise will dominate the temperature fluctuations of the sample.
At 100 mK, and with 10 GHz photons, for fluxes greater than 10
6
photons/s the noise of
the bolometer should be dominated by the shot noise of the microwave field. In this way,
graphene would act as a photodetector for microwaves: square law response, absorptive,
and sensitive to the shot noise of the incoming field. We know of no other microwave
detector which has these characteristics and would open the door to novel quantum optics
experiments with microwave photons.
Note: During the writing of this work, we have become aware to two other experimental
works which touch on some these concepts [29, 30].
K.C.F developed these concepts, measurement and sample designs, fabricated devices,
and performed data collection and analysis. K.C.S instigated the work, developed these
concepts, and over saw the work.
5
We acknowledge help with microfabricated LC resonators from M. Shaw, and helpful
conversations with P. Kim, J. Hone, E. Hendrickson, J. P. Eisentien, 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 the FCRP Center on Functional
Engineering Nano Architectonics (FENA) and US NSF (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).
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to K.C.S (email:
schwab@caltech.edu).
6
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9
FIG. 1:
A
shows the measurement circuit: graphene, impedance matched with 1.161 GHz resonant,
lithographic LC network (NbTiN film,
T
c
= 13
.
5 K), and connected to HEMT amplifier.
B
shows
the expected thermal conductances (
G
WF
,
G
ep
,
G
rad
) and heat capacity versus temperature for
assuming a bandwidth of 80 MHz,
n
= 10
11
cm
2
, and
A
= 10
10
m
2
. The inset shows an optical
micrograph of the graphene sample. Scale bar is 15
μ
m long.
C
shows the two-terminal resistance
of the graphene vs gate voltage, taken from 1.65-12 K.
D
shows the reflected microwave response
versus gate voltage. The absorption dip at 1.161 GHz shows that the graphene is well matched to
50 Ω in a 80 MHz band.
E
shows spectra of the measured noise power taken at various sample
temperatures, demonstrating the Johnson noise signal of the graphene in the impedance matching
band.
10
FIG. 2:
A
shows the integrated noise power vs refrigerator temperature, demonstrating the ex-
pected temperature dependence of the Johnson noise signal. The deviation at temperatures above
8K 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, vs integration
time. A resolution of 100 ppm is achieved, in agreement with the Dicke Radiometer formula (shown
as lines).
B
shows the results of the differential thermal conductance measurements. The inset
show 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
shows 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 vs applied microwave power at 1.161 GHz, also showing
a similar heating curve and demonstrating the microwave bolometric effect with graphene.
11
FIG. 3:
A
shows the spectrum, refered to the input of the HEMT, due to bolometric mixing.
The central, 600 pV red tone is due to a heating signal at
ω
heat
/
2
π
= 1
.
161 GHz, which produces
a temperature response at 2.322 GHz, which is then mixed down to
ω
bolo
=
ω
heat
+ 2
π
·
1 kHz
using a small modulation tone at
ω
mod
=
ω
heat
2
π
·
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:
ω
bolo
= 2
ω
heat
ω
mod
.
B
shows a plot of the measured bolometric mixing tone normalized by our
expected signal assuming the graphene thermal time constant
τ
= 0. The red points are measured
with a heating tone of
ω
heat
/
2
π
= 1
.
161 GHz, and blue points with
ω
heat
/
2
π
= 17
.
6 Hz, both with
a modulation tone of 1.161 GHz-1 kHz. The dashed line shows he expected rolloff of the bolometric
signal when 2
ω
heat
/
2
π
= 2
.
32 GHz and
τ
=
C
e
/G
ep
.
12
FIG. 4:
A
shows the expected sensitivity as a bolometer assuming
n
= 10
9
cm
2
and
A
= 10
11
m
2
versus coupling bandwidth, for various cryogenic operating temperatures, with
B
showing the
optimal value versus temperature.
C
shows the expected energy sensitivity to single photons.
D
shows the threshold for detection of shot noise of an incident microwave field of various frequencies.
13