Supplementary Information: Electronic Modulation of Near Field
Radiative Transfer in Graphene Field Effect Heterostructures
Nathan H. Thomas,
1
Michelle C. Sherrott,
2, 3
Jeremy
Broulliet,
3
Harry A. Atwater,
4, 3
and Austin J. Minnich
1,
∗
1
Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California 91125, United States
2
Current Address: Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA, 02139
3
Thomas J. Watson Laboratory of Applied Physics
4
Resnick Sustainability Institute, California Institute of Technology,
Pasadena, California 91125, United States
∗
aminnich@caltech.edu
1
S1. SAMPLE SCHEMATIC
A diagram of the sample is shown in Figure S1. The temperatures
T
1
and
T
2
are recorded
with K type thermocouples, embedded in copper heat spreaders above and below the sam-
ples. The top and bottom samples are separated by SiO
2
posts fabricated by electron beam
lithography and subsequent electron beam deposition.
FIG. S1.
Diagram of experimental setup.
The top and bottom samples are separated by posts
of SiO
2
. To measure
T
1
and
T
2
, the thermocouples are embedded in copper heat spreader plates.
2
S2. WET TRANSFER OF CVD GRAPHENE
The graphene used in these experiments is grown by chemical vapor deposition on copper
foil. The samples in these experiments are quite large, 15 mm
×
20 mm. For standard
wet transfer techniques typically, once the copper foil has been etched away, the floating
graphene must be transferred to a series of water baths. We find that the samples here were
large enough that they often tore during the transfer from bath to bath. To overcome this
problem, the transfer method shown in Figure S2 was developed. The copper foil is etched
FIG. S2.
Modified wet transfer technique.
The copper foil upon which the graphene was grown
is etched in a modified seperatory funnel. Instead of removing the graphene from the etchant to be
cleaned, the etchant is removed by opening the stopcock and then displaced with de-ionized water.
in a modified separatory funnel with additional snorkel welded above the stopcock. Once
the copper is completely etched, the stopcock is slowly opened and etchant drips out the
bottom of the funnel, while it is being displaced with de-ionized (DI) water, poured through
the snorkel. The DI water is added repeatedly until the etchant has been completely washed
away, and the graphene is sufficiently clean. Then the graphene is picked up by a sample
substrate.
The sample substrates undergo a rigorous cleaning process. First, they are sonicated in
isopropyl alchohol for 15 minutes, after which they are dried and cleaned in an O
2
plasma
3
clean for 60 minutes. Finally, to prepare the substrate surface for graphene transfer, the
substrates are soaked in a pirahna solution overnight.
4
S3. HEAT FLUX SENSOR CALIBRATION
The heat flux sensor is a critical component to the apparatus. The signal response for a
given thermal input is dependent on the sample temperature and requires careful calibration.
We calibrate the sensor at 5 temperatures from room temperature down to 90 K. First, the
heating element is pressed directly onto the copper heat spreader with thermal grease applied
to both. The heater is held in place with two actuator screws. To limit conductive loss
through the screws, low thermal conductivity ceramic washers and a glass spacer separate
the screw from the aluminum shroud that encompasses the heater (see Section S4).
The heat flux sensor signal in
μ
V was tracked in time at the given input powers into the
heater, as shown in Figure S3. After the input power and heat flux signal have equilibrated,
the average input power and sensor voltage are recorded. The regions from which the average
values are taken are represented in grey in Figure S3. A line is fit to the resulting data,
shown in Figure S4, at all five temperatures. The summary of the fits is shown in Table S1.
The Y-intercept values represent a background “DC” heat flux into the sensor that needs
to be appropriately adjusted, see Section S4.
Heat flux sensor
temperature (K)
Slope
(
μV/Wm
−
2
)
Y-intercept (
μV
)
90
1.487(6)
58(1)
132
1.817(1)
58.6(3)
180
2.087(4)
64(1)
228
2.321(1)
46.8(4)
297
2.561(1)
-14.8(2)
TABLE S1. Fitting data from all calibration plots.
5
FIG. S3.
Representative plot of real time data for heat flux sensor calibration.
The
heat flux voltage signal is measured for different input heating powers. After the signals have
equilibrated, the data are collected and averaged over an allotted time interval, shown in grey.
6
FIG. S4.
Calibration plots for heat flux sensor at different temperatures
. Each point
represents the data averaged over a single grey interval in Figure S3. The inverse of the slope from
each fitted line gives the conversion factor of measured voltage to input heat flux. The heat flux is
normalized to the area of the heat flux sensor.
7
S4. BACKGROUND HEAT FLUX CORRECTION
The non-zero y-intercept values from the calibration fits indicate that there is a small
background heat flux signal that must be addressed when converting the heat flux sensor
voltage signal to a heat flux value. As shown in the diagram in Figures S5a and b, we divide
this background “DC” heat flux into two pathways, one in which the heat flows directly into
the sensor, and the other where the heat flows through the heater. As the entire assembly
is radiation shielded, we attribute these small heat flows to heat leakage through connecting
lead and thermocouple wires.
The heat that flows through the heater and into the sensor ultimately also flows through
the sample itself and should
not
be subtracted from the heat flux signal. The heat that
flows directly into the sensor, however, does not flow through the sample and should be
subtracted. We carefully measure this background “DC” signal for two cases, where the
heater is firmly pressed against the heat flux sensor and when the heater is suspended above
the sensor. In the former case, heat flow pathways are open and are plotted in orange in
Figure S6. In the latter case, heat cannot flow from the heater to the sensor (other than
through a negligible far-field radiative pathway), and is plotted in blue in Figure S6.
Both curves show a monotonic decrease in background heat flow as the sensor temperature
increases. The area beneath the blue curve represents the heat flowing directly into the
sensor, and the area between the two curves, shown in orange, represents the heat only
flowing through the heater. All heat flux modulation measurements are conducted where
the cryostat cold finger is kept at 77 K, and the heat flux sensor is at 90 K. From these
measurements, the background heat flux of 20 Wm
−
2
, area normalized to the heat flux
sensor, is subtracted. This subtraction corresponds to a reduction in the heat flux sensor
signal of 28
±
5
μ
V. The uncertainty is mainly due to the signal drift of the heat flux sensor.
As the temperature of the heat flux sensor asymptotically approaches its equilibrium value,
so too does the heat flux sensor signal. This equilibration has been found to take many
hours. Each data point in Figure S6 is collected after the sensor signal has flattened out,
but the signal was found to continue to drift by close to 5
μ
V. This signal drift is also
addressed in Section S5 for correcting the heat flux modulation measurement.
8
FIG. S5.
Diagram of thermal leakage pathways. a
, Heat flux sensor is pressed directly onto
the heat flux sensor, allowing for two paths for heat to flow into the sensor, one direct and one
indirect through the heater.
b
, The heater is elevated above the sensor, and only heat only flows
into the sensor through directly. The heat that flows through the heater ultimately also flows
through the sample, meaning that heat flow should
not
be subtracted from the end heat flux
signal. Heat that flows directly into the sensor does not flow through the sample and should be
subtracted.
FIG. S6.
Background heat flux signal versus sensor temperature
. The measurements taken
at zero input power from the resistive heater and are normalized to the area of the heat flux sensor.
9
S5. SIGNAL DRIFT
As mentioned in the previous section, the heat flux signal exhibits a linear drift (at the
end of an asymptotic equilibration). To account for this drift, we fit a line to the raw heat
flux signal at all times when zero bias is applied and then subtract off the slope
×
time, as
shown in Figure S7. This effectively rotates the heat flux values down about time
t
= 0,
and as a result reduces the absolute final heat flux values by 5 Wm
−
2
.
FIG. S7.
Signal drift correction for samples S1-S3.
The raw data are shown in
a-c
. A line
is fit to the subset shown in gray at which zero bias is applied. The drift is removed by then
subtracting off the slope
×
time from the entire data set. The result is in
d-f
.
10
S6. FITTING PROCEDURE FOR THERMAL MODEL
The radiative contribution to the heat flux is modeled in Equation S1
H
(
ω,T
1
,T
2
) = Φ(
ω
) (Θ(
ω,T
1
)
−
Θ(
ω,T
2
))
,
(S1)
where Φ(
ω
) is the transmissivity function, partitioned over propagating modes where
k
||
<
ω/c
, and evanescent modes where
k
||
> ω/c
,
Φ(
ω
) =
∑
s,p
∫
ω/c
0
dk
||
k
||
2
π
(1
−|
r
s,p
13
|
2
)(1
−|
r
s,p
23
|
2
)
|
1
−
r
s,p
13
r
s,p
23
e
i
2
k
z
0
d
|
2
+
∫
∞
ω/c
dk
||
k
||
2
π
4
=
(
r
s,p
13
)
=
(
r
s,p
23
)
|
1
−
r
s,p
13
r
s,p
23
e
−
2
|
k
z
0
|
d
|
2
,
(S2)
and the total integrated heat flux is
Q
(
T
1
,T
2
) =
∫
∞
0
dω
2
π
H
(
ω,T
1
,T
2
)
.
(S3)
The inherent dependence on Fermi levels is manifest in the Fresnel coefficients
r
13
and
r
23
.
As there is considerable conductive leakage, we also account for an added conductive thermal
pathway with an additional term proportional to ∆
T
=
T
1
−
T
2
,
Q
total
=
G
(
T
1
−
T
2
) +
Q
rad
(
μ,d,T
1
,T
2
)
.
(S4)
As the top and the bottom samples are in conductive contact, that likely means the top
and bottom graphene sheets are also shorted electrically. As a result, the Fermi levels are
assumed to be equal,
μ
1
=
μ
2
=
μ
.
There are 5 parameters in the final model Equation S4. As shown in Section S1, the
temperatures of the copper heat spreaders above and below the sample are tracked with
thermocouples. To calculate the temperatures of each graphene surface, we employ a thermal
resistor model, using the measured heat flux. As thermal grease is applied to each heat
spreader, the interfacial resistance is negligible compared to the thermal resistance between
the two graphene sheets [2]. As the thermal conductivity of the bottom silicon wafer is
comparatively high, the bottom graphene temperature
T
2
is approximately equal to the
measured temperature of the heat flux sensor. However, as the thermal conductivity of the
silica optical flat is comparatively low, there is a temperature drop from the resistive heater
to the top graphene surface
T
1
, proportional to the thermal conductivity of silica.
11
The heat transfer coefficient
G
, the Fermi levels, and the gap spacing can be determined
in the following manner. The Fermi levels can be determined by finding the charge neutral
point at maximum surface resistance, and then using a capacitor model to solve for Fermi
level as a function of applied voltage, shown in Figure S9d-f for samples S1, S2, and S3,
respectively. Once the Fermi levels are established, the absolute change in heat flux ∆
Q
rad
is only a function of vacuum gap spacing
d
. As a result, from the measured heat flux change,
we uniquely determine the value for
d
as shown in Figure S8.
G
is then found to account
for the heat flux discrepency between
Q
total
and
Q
rad
. This discrepancy was found to be
731
±
9, 1590
±
30, and 2011
±
20 Wm
−
2
for samples S1-S3, respectively, indicating that
parasitic conduction is responsible for 85
±
1, 81
±
1, and 84.1
±
0.6 %, respectively, of the
total heat flux. The final parameter values for are tabulated in Table S3. Uncertainties are
determined in SI Sec. S7.
As the top sample consists of an optical flat and graphene, it is transparent to visible
light and we can set upper limits on the gap spacing using interferometry, shown for samples
S1 through S3 in Figure S9 and tabulated for different points on each sample in Table S2.
The estimated distance is quite variable for S1, indicating that there is likely a piece of dust
beneath the optical flat, causing it to be cantilevered. For S3 the spacings are uniform and
close to the distance that gives the appropriate level of heat flux modulation. However, for S2
the measure gap is about 1
μm
larger than expected. During the heat flux measurement, the
spring-loaded heater presses the optical flat to the underlying substrate, and in an analogous
experiment by simply pressing on the sample with tweezers, applying pressure was found
to reduce the spacing by upwards of 50%, as shown for Sample S4 in Figure S10. As a
result, the gap spacing measurements only provide an upper limit on the gap spacing at a
particular point on the sample since the spacing is highly dependent on applied pressure.
The fitting procedure is modified for the sample S4, since the charge neutral point could
not be found under forward bias as dielectric breakdown occurred at smaller biases than for
the previous samples. Therefore the zero-bias Fermi level is also a fitting parameter where
the previous three samples provide an approximate range for the charge neutral point. The
three parameters
G
,
d
, and
μ
are found in the zero-bias case, and then a zero-parameter fit
is applied to the biased case with the corrected Fermi level.
12
FIG. S8.
Heat flux change due to Fermi level change versus vacuum gap distance
. The
Fermi level sweeps between 0.05 eV to 0.30 eV (blue line), where the shaded region outlines the
heat flux change accounting for the uncertainty in the Fermi level of
±
0
.
05 eV. The vertical error
bars for each sample indicate the uncertainty in heat flux measurement, and the horizontal error
bars indicate the uncertainty in the solved distance
d
. The inset shows the heat flux change at
distances from 400 nm to 3
μ
m.
Measurement
Location
S1
S2
S3
S4
#1
5.63
μm
3.20
μm
2.46
μm
1.43
μm
#2
1.28
μm
4.08
μm
2.70
μm
0.65
μm
*
#3
3.74
μm
3.45
μm
2.31
μm
TABLE S2. Vacuum gap distances for each sample. The sample S1-3, the interferometric mea-
surements were taken at different locations around the optical flat. For sample S4, the samples
were taken when pressure was applied (indicated by the *), and when no pressure was applied.
13
FIG. S9.
Measurements for model parameters. a-c
, Measurements for sample gap spacing
for S1-S3, respectively. A transfer matrix model with a variable gap spacing and amplitude is fit to
the measured signal.
d-f
, Graphene surface resistance versus applied bias for S1-S3, respectively.
The peak in the surface resistance measurement is the charge neutral point where the graphene
Fermi level is near zero.
S1
S2
S3
S4
G (Wm
−
2
K
−
1
)
6.59
8.88
11.20
5.10
d (
μ
m)
2.5
2.3
2.3
0.56
T
1
(K)
197
270
269
-
T
2
(K)
86
91
90
-
CNP (V)
5
15
7
-
TABLE S3. Parameters for the model in Equation S4. The values
G
and
d
are fit to heat flux
data. The temperatures are taken from measurement, accounting for the thermal resistance of the
optical flat. The charge neutral point (CNP) is the voltage at which the graphene surface resistance
is maximized and the Fermi level is near zero.
14