Supplementary Material
Sub- picoWatt/Kelvin Resistive Thermometry for Probing Nanoscale Thermal Transport
Jianlin Zheng, Matthew C. Wingert, Edward Dechaumphai, Renkun Chen#
Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093
#
Email: rkchen@ucsd.edu
1. Device Fabrication
Devices were fabricated on a <100> Si wafer using micro-fabrication techniques. A 300-nm
thick film of silicon nitride (SiNx) was deposited on Si using plasma-enhanced chemical vapor
deposition (PECVD) (Fig. S1(a)). The uniformity and low stress characteristics of the deposited
SiNx film were obtained using mixed frequency PECVD to control film stress and density. A
PMMA (poly-(methyl-methacrylate)) layer was spin-coated and patterned using electron beam
lithography (EBL). After electron beam pattern exposure, the sample was developed using
MIBK:IPA (1:3) and underwent further descum using O2 plasma to eliminate possible
organic/photoresist residuals. 10 nm thick Cr and 40 nm thick Pt films were then sputtered,
where Cr was used as an adhesion layer between SiNx and Pt. PMMA lift-off was then carried
out using acetone. EBL patterning after lift-off is shown in Fig. S1(b).
EBL was also used to pattern a window in the SiNx film in order to release the suspended
beam structure (Fig. S1(c)). The SiNx window was selectively etched by reactive ion etching
(RIE) with CHF3 and O2. PMMA was then removed using acetone, leaving a window for
selective silicon etching, shown in Fig. S1(d). The suspended structure was released by wet
etching of Si under the SiNx beams in a KOH solution at 80 oC for approximately 1 hr (etching
rate approximately ~ 1 μm/min). The sample was cleaned after KOH etching by dipping in DI
1
water and methanol. Methanol was used due to its low surface tension to ensure that the device
would not collapse during air-drying. No critical point drying was used in the fabrication
process. Fig. S1(e) shows the final fabricated device.
Figure S1. Device fabrication flow: (a) Coating of the Si wafer by low-stressed SiNx using
PECVD. (b) Patterning and lift-off to define the beams and pads made of Pt. (c) Opening of the
window for subsequent SiNx etching. (d) Etching of SiNx. (e) Release of the suspended structures
by etching the Si substrate using KOH.
2. Thermal Conductance Measurement of SiNx Beams
In this section, we showed the measurement of thermal conductance of the heating beam
using the modulated heating method, which is the same as the “3𝜔” method for suspended
structures documented in the literature1.
2
Figure S2 shows the circuit diagram for the measurement on the heating side with a
modulated heating current. In the limit of low frequency, it can be shown that the amplitude of
the temperature oscillation, averaged along the entire heating beam, can be extracted from 1,
2
̅̅̅̅̅
Δ𝑇ℎ = 3
π𝑉3𝑓ℎ
𝐼 𝑓ℎ
𝑑𝑅
−1
( 𝑑𝑇ℎ )
(S1)
where 𝐼𝑓ℎ and 𝑉3𝑓ℎ are the root-mean-squared (RMS) values of the applied current and the
measured 3rd harmonic voltage, respectively, and
𝑑𝑅ℎ
𝑑𝑇
is the temperature coefficient of resistance
(TCR) of the heating beam.
Figure S2. Circuit diagrams for the heating beam measurement with modulated heating current.
From the above measurement, we are able to extract the beam conductance. Consider the
heat conduction model in the heating beam shown in Fig. 2 (in the main manuscript) and
assuming negligible heat loss from the heating beam to the ambient (see section 3 for further
3
discussion), one could recognize the parabolic temperature distribution along the heating beam,
namely,
𝑞 ′′
Δ𝑇ℎ (𝑥) = 2𝜅
where 𝑞 ′′ =
𝐼 2 𝑅ℎ
2𝐿𝑏
𝑏 𝐴𝑏
𝑥(2𝐿𝑏 − 𝑥)
(S2)
𝑄
= 2𝐿 is the heat flux per unit length of the beam (in [𝑊𝑚−1]).
𝑏
Therefore, the average temperature of the beam, which is also the effective temperature
measured in our experiments for the AC schemes (in the low frequency limit), is,
′′ 𝐿2
𝑏
𝑞
̅̅̅̅̅
Δ𝑇ℎ = 3𝜅
𝑏 𝐴𝑏
𝑄𝐿𝑏
= 6𝜅
𝑏 𝐴𝑏
𝑄
= 3𝐺
(S3)
𝑏
or,
𝑄
𝐺b = 3Δ𝑇
̅̅̅̅̅̅
ℎ
(S4)
where 𝐺𝑏 is defined as 2𝜅𝑏 𝐴𝑏 /𝐿𝑏 , which is the effective conductance from the center of the
beam to the substrate (note: it is not the conductance of the single beam).
Figure S3 shows the measured ̅̅̅̅̅
Δ𝑇ℎ vs. heating power for the AC heating case. Based on
Eq. S4 and Fig. S3, the thermal conductance of the beam is determined to be ~50 nW/K. The
exact 𝐺𝑏 of a specific device depends on the beam geometry and ranges from 40-60 nW/K on the
devices fabricated and tested in this study. The conductance is about a factor of two lower than
that of previously used devices2 despite the much shorter beam length, due to the reduced
amount of beams (two vs. four to six) and narrower beam width (~ 1 µm vs. 2-3 µm) in the
present devices.
4
Average Th [K]
2.0
1.5
1.0
0.5
Gb 
Q
~ 50 nW/K
3Th
0.0
0
50
100
150
200
250
Heating Power, Q [nW]
Figure S3. Measured ̅̅̅̅̅
Δ𝑇ℎ vs. heating power for the heating beam
3. Characterization of Heat Loss from Heating Beams
3.1. Modeling of Heat Loss in Variable Length Beams
In the previous sections, we analyzed the thermal models describing the fabricated
measurement device, assuming negligible heat loss along the length of the suspending beams. In
order to verify the validity of this assumption, we measured average temperature of the selfheated microfabricated beams using the schematic shown in Fig. S4(b) and compared to results
to a theoretical model that takes the heat loss into account. In this experiment, SiNx beams of
varying lengths (100, 200, 400 µm) coated with Pt for self-heating and temperature sensing were
utilized for the comparison (Fig. S4).
5
The heat loss along the beam is modeled by a constant heat transfer coefficient, h, from
the beam to the ambient. The heat transfer equation along the beam thus follows the fin model
(Fig. S4(a)):
𝑑2 𝑇
𝑄
𝑘𝐴 𝑑𝑥 2 − ℎ𝑃(𝑇 − 𝑇𝑜 ) + 𝐿 = 0
(S5)
where A is the cross sectional area, L is the length of the beam (note that 𝐿 = 2𝐿𝑏 ), T is the
temperature along the beam, 𝑇𝑜 is the ambient temperature, h is the heat transfer coefficient, P is
the perimeter of the SiNx surfaces, and Q is the electrical power dissipated in the beam.
Figure S4. Thermal conductivity (𝜅) and heat transfer coefficient (ℎ) determination of the SiNx
beams. (a) Schematic of the thermal fin model. (b) A suspended beam of total length 𝐿 selfheated by applying a current 𝐼.
In the case when the heat loss is negligible (h=0):
𝑄𝑥
𝑥
∆𝑇 = 𝑇 − 𝑇𝑜 = 2𝑘𝐴 (1 − 𝐿 )
(S6)
̅̅̅̅ = 𝑄𝐿
∆𝑇
12𝑘𝐴
(S7)
𝑄
̅̅̅̅
∆𝑇
(S8)
=
12𝑘𝐴
𝐿
6
Whereas assuming heat loss is not negligible (h≠0)3:
𝑄
∆𝑇 = ℎ𝑃𝐿 [1 −
sinh(𝑚𝑥)+sinh⁡(𝑚(𝐿−𝑥))
sinh⁡(𝑚𝐿)
]
(S9)
where
ℎ𝑃
𝑚 = √𝑘𝐴
(S10)
and subsequently,
̅̅̅̅ = 𝑄 [1 − 2(cosh(𝑚𝐿)−1)]
∆𝑇
ℎ𝑃𝐿
𝑚𝐿⁡sinh⁡(𝑚𝐿)
𝑄
̅̅̅̅
∆𝑇
=
ℎ𝑃𝐿
2(cosh(𝑚𝐿)−1)
[1−
]
𝑚𝐿⁡sinh⁡(𝑚𝐿)
(S11)
(S12)
Figure S5 shows heating power over average beam temperature vs. beam width divided
by length. For short beam lengths (<200 μm), the negligible and non-negligible heat loss models
converge, validating the simplified thermal analysis. For longer beams, however, heat loss
becomes increasingly significant, such that neglecting heat loss can lead to an overestimated
temperature rise.
7
No Heat Loss
Q/Tavg [nW/K]
140
Heat Loss (w=1.30 m)
Heat Loss (w=1.05 m)
120
Heat Loss (w=1.25 m)
L=100 m (w=1.30 m)
L=200 m (w=1.05 m)
L=400 m (w=1.25 m)
100
80
60
40
20
0
0
2
4
6
8
10
12
14
16
w/L [10-3]
Figure S5. Heating power over average temperature rise vs. width over length for suspended
beams with and without heat loss, solid and dashed lines, respectively. Three microfabricated
beams of different lengths (100, 200, 400 μm) were measured (solid dots) and compared to
theoretical curves with fitted thermal conductivity and heat transfer coefficient.
The effective thermal conductivity of the Pt coated SiNx can be found from fitting the
short length beams to the negligible or non-negligible heat loss models. With knowledge of the
effective thermal conductivity of the beams, we can calculate the heat transfer coefficient
describing the heat loss for longer beams. After calculating the effective thermal conductivity of
the beams from 300 to 450 K using the 100-µm-long device, we calculated the heat transfer
coefficient at each ambient temperature using the 400-µm-long beam, as shown in Fig. S6. In the
limit of small temperature rise due to the self-heating (< 10 K), one can also estimate the heat
transfer coefficient due to radiation heat transfer from the SiNx beam to the ambient from:
8
ℎ𝑟 = 4𝜀𝜎𝑇 3
(S13)
where ε and σ are the emissivity (0.88 for SiNx4) and Stefan-Boltzmann constant, respectively.
The calculated ℎ𝑟 is also plotted as the dash line in Fig. S6. The measured and calculated heat
transfer coefficient values are in good agreement with each other, leading us to believe that that
heat loss in the suspended structures is primarily due to radiative thermal exchange with the
sample surroundings.
Figure S6. Measured (dots) and calculated (dash line, Eq. S13) heat transfer coefficient for the
400-μm-long beam from 300 to 450 K.
3.2. Verification of Negligible Heat Loss in Short-Beam Devices
In section S3.1, it was shown that the heat loss is negligible in devices with 𝐿 = 100⁡𝜇𝑚 (or
𝐿𝑏 = 50⁡𝜇𝑚). Therefore, these short-beam suspended devices were chosen in this study for
thermal measurements in order to ensure accurate thermal analysis (in addition to the fact that
9
the short-beam devices have smaller thermal time constant and higher roll-off frequency). To
directly verify negligible heat loss from the beams in the short-beam devices, we fabricated
suspended devices with the same beam length (𝐿𝑏 = 50⁡𝜇𝑚) but containing pads with serpentine
Pt lines in order to measure temperature of both the beams and the pads (Fig. S7).
Figure S7. SEM image of microfabricated short-beam (𝐿𝑏 = 50⁡𝜇𝑚) suspended device with
pads containing serpentine Pt lines.
The average temperature rise of the self-heated beams, non-self-heated beams, and the
suspended pads were measured and are plotted in Fig. S8. It can be shown that, for beams with
negligible heat loss, the average temperature rise for the self-heated and non-self-heated beams is
2/3 and 1/2 of that for the pad, respectively, as we observed experimentally (Fig. S8). Therefore,
we have verified experimentally that the heat loss from devices with 𝐿𝑏 = 50⁡𝜇𝑚 is negligible at
room temperature.
10
Tavg/Tpad,max
1.0
Pad
Beam (No Self Heating)
Beam (Self Heating)
1/2 Pad T
2/3 Pad T
0.8
0.6
0.4
0.2
0.0
0
400
800
1200
1600
Square of Heating Current [(A)2]
Figure S8. Average measured temperature rise of suspended pad and suspending beams for two
cases, self-heating (green square) and no self-heating (red triangle). Dashed and solid lines
represent 2/3 and 1/2 of the pad temperature rise (black circle), respectively.
4. Effect of Sensing Current Amplitude
From Eq. 6 in the main manuscript, it is clear that NETs is inversely correlated with the
bridge sensing current, 𝐼𝑠 , and a lower NETs can be achieved with a higher 𝐼𝑠 , as we have
previously demonstrated with the DC-heating bridge5. Fig. S9 shows the results of the
modulated-heating bridge scheme for another device (𝑅𝑠 = 1250⁡Ω⁡) measured using various
values of 𝐼𝑠 : 12.5, 25, and 50 𝜇𝐴. As shown in Fig. S9, the corresponding noise floor values for
each sensing current is 217, 102, and 44 𝜇𝐾, respectively, which correlate well with the values of
𝐼𝑠 according to Eq. 6. Despite the lower resistance (1250⁡Ω⁡vs. 3100 Ω), the resolution of 44 𝜇𝐾
on this device is similar to that obtained on the device shown in the main manuscript owing to
the higher 𝐼𝑠 used here.
11
We also note that the temperature rise on the sensing beam due to Joule heating from 𝐼𝑠
does not affect the thermal conductance measurement, as long as the temperature rise is small
compared to the global ambient temperature (such that the measurement is still within the linear
regime). This is because 1) the applied DC (𝐼𝑠 ) would cause a constant temperature rise on the
sensing sides, while the temperature measurements on the sensing and heating sides are based on
the 2𝜔 signals, which are not affected by the constant temperature changes. 2) the thermal
conductance of the sample is determined from the slope of heat current vs. the temperature
difference between the heating and sensing sides, which is unchanged when (𝑇ℎ − 𝑇𝑠 ) or the heat
current is slightly shifted..
Fig. S10 shows the measured modulated Δ𝑇𝑠 (at 2𝑓ℎ ) and Δ𝑇ℎ (at 2𝑓ℎ ) as a function of the
heating power for the three applied 𝐼𝑠 values mentioned above. The figure unambiguously shows
that the slopes of the Δ𝑇𝑠 (2𝑓ℎ ) and Δ𝑇ℎ (at 2𝑓ℎ ) vs. power curves, and hence the thermal
conductance, are identical within the measurement uncertainty, even when the (un-modulated)
temperature rise on the sensing side is increased by 22.9 K with 𝐼𝑠 = 50⁡𝜇𝐴. It is also worth
noting that the Johnson (thermal) noise only increases slightly (<3.3 %) when the sensing side
temperature rise is lower than 20 K (note: Δ𝑉𝐽𝑜ℎ𝑛𝑠𝑜𝑛 = √4𝑘𝐵 𝑅𝑠 𝑇). Practically, it is preferable to
limit the temperature rise on the sensing beam to less than 5% of the global ambient temperature
such that the heat transfer measurement is performed in the linear regime, which for room
temperature is around 15 K. This constraint ultimately limits the 𝐼𝑠 that can be applied to the
sensing side.
12
Figure S9. Measured noise floor of the sensing side temperature rise vs. applied heating power
for different amplitudes of sensing current: (a) 12.5 µA, (b) 25 µA, and (c) 50 µA.
13
Figure S10. Measured modulated Δ𝑇𝑠 (at 2𝑓ℎ ) (a) and Δ𝑇ℎ (at 2𝑓ℎ ) (b) as a function of heating
power for three amplitudes of 𝐼𝑠 .
14
5. Background Conductance and Cancellation
The enhanced resolution of bridge-based thermal measurements also captures the background
conductance transferred between the suspended beams. While this is always present, we can use
the bridge system’s inherent symmetry to subtract this signal. As shown in Fig. S11(a), one can
construct a ‘cancelling’ bridge circuit to measure the conductance difference between a device
with a nanowire sample (called device 1 hereafter) and a blank pair device without a nanowire
(called device 2 hereafter). In this ‘cancelling’ scheme, an identical heating current (𝐼ℎ ) is
applied to the heating sides (𝑅ℎ and 𝑅ℎ,𝑝 ) of both devices, and the difference in the temperature
rises (or resistance changes on 𝑅𝑠 and 𝑅𝑠,𝑝 ) on the sensing sides of the two devices is directly
measured using a Wheatstone bridge. Since the devices are almost identical and their background
conductance values are about the same (as we will show later), we can directly obtain the
nanowire conductance from a single measurement based on this scheme (i.e., 𝐺𝑁𝑊 =
𝐺𝑁𝑊+𝐵𝐺,1 − 𝐺𝐵𝐺,2 ).
Figure S11(b) shows the measurement results for devices with 400-𝜇𝑚 beam length we used
previously (Ref. 5). The long-beam devices were used here for demonstration purposes because
these devices possess 𝐺𝐵𝐺 values of the same order of magnitude as a nanowire sample (~100
pW/K). First, the total conductance of device 1 containing a nanowire sample (𝐺𝑁𝑊+𝐵𝐺,1 ) was
measured without the canceling scheme (shown by the red triangles in Fig. S11(b)), followed by
the previously-described canceling scheme, which essentially directly measured 𝐺𝑁𝑊
(𝐺𝑁𝑊+𝐵𝐺,1 − 𝐺𝐵𝐺,2, shown as black circles). The difference in these two measurements (green
squares) yields the background conductance of device 2 (𝐺𝐵𝐺,2 ). Subsequently, the nanowire in
device 1 was cut using a FIB, and its background conductance (𝐺𝐵𝐺,1 ) was measured (blue
15
triangles). The difference in 𝐺𝑁𝑊+𝐵𝐺,1 and (𝐺𝐵𝐺,1 ) yields the intrinsic conductance of the NW
(blue hexagons), which essentially is the conductance determined by the method reported in Ref.
5. Finally, another cancelling measurement on the two blank devices (devices 1 & 2) showed
essentially negligible conductance (yellow diamonds) within the measurement uncertainty,
which proved that the background conductance of the two devices are the same (i.e., 𝐺𝐵𝐺,1 =
𝐺𝐵𝐺,2 ). In effect, the cancelling scheme shown in Fig. 11 (a) is capable of directly measuring the
nanowire conductance (i.e., 𝐺𝑁𝑊+𝐵𝐺,1 − 𝐺𝐵𝐺,2 = 𝐺𝑁𝑊 ).
16
Figure S11. Characterization of background conductance. (a) Schematic of the canceling
scheme for directly measuring the conductance of a nanowire. (b) Measurement results based on
400-𝜇𝑚-long devices used in Ref. 5. See the text for details.
Further examination of the background signal of a blank device was conducted through
measurements at various chamber pressures (Fig. S12). At high-vacuum pressures (~10-4 torr)
17
with the turbo pump switched on, the background signal was found to be stable around 215±7
pW/K. This was verified over multiple measurement runs and with multiple devices with the
same gap distance between the heating and sending pads (all with gaps of ~4 µm wide and with
400-µm long suspending beams). The background conductance was slightly increased at higher
Background Conductance [pW/K]
pressures when only the mechanical pump was turned on.
245
240
235
Mechanical Pump
Turbo Pump 1st Run
Turbo Pump 2nd Run
230
225
220
215
210
205
0.01
0.1
1
10
100
Pressure [mtorr]
Figure S12. Measured background conductance vs. chamber pressure. At high vacuum, the
background conductance was measured consecutively over a 4 hour period on separate days after
removing and replacing samples (1st and 2nd runs). The measured conductance has little variation
(±3%) as long as the pressure is below 10 mtorr.
18
6. Calculation of Background Conductance due to Blackbody Radiation
The expected background conductance due to blackbody radiation in the new short-beam
devices can be calculated based on the measured radiative heat transfer coefficient, h, and the
view factor between the suspended beams. The radiative heat transfer coefficient was previously
found to be ≈4.2 W/m-K in section S3.1, meanwhile, the view factor, f, can be calculated by3:
2
(1+𝑋 2 )(1+𝑌 2 )
𝑓 = 𝜋𝑋𝑌 {[ln √
1+𝑋 2 +𝑌 2
𝑋
𝑌
] + 𝑋√1 + 𝑌 2 tan−1 (√1+𝑌 2) + 𝑌√1 + 𝑋 2 tan−1 (√1+𝑋 2) −
𝑋 tan−1 𝑋 − 𝑌 tan−1 𝑌}
(S14)
where 𝑋 = 𝑡/𝑑 and 𝑌 = 𝐿/𝑑 are the dimensionless beam thickness (𝑡) and length (𝐿), where 𝑑 is
the gap distance between the beams. The calculated view factors for the 100 µm long, 300 nm
thick beams were 0.0205, 0.0098, and 0.0019 for gap sizes of 7, 14, and 54 µm, respectively.
The corresponding conductance can be calculated as 𝐺𝐵𝐺 = 𝑓ℎ𝑡𝐿 and are 2.6, 1.2, and 0.24
pW/K for the 7, 14, and 54 µm gaps, respectively. The calculated values are about one order of
magnitude lower than the measured conductance for the corresponding devices (measured at
29.82, 13.70, and 2.45 pW/K at 300 K for gap distance of 7, 14, and 54 µm,). This discrepancy
could be caused by near field radiation effects and/or other experimental factors, which warrants
further investigation.
19
References:
1
2
3
4
5
L. Lu, W. Yi, and D. L. Zhang, Rev Sci Instrum 72 (7), 2996-3003 (2001).
L. Shi, D. Y. Li, C. H. Yu, W. Y. Jang, D. Kim, Z. Yao, P. Kim, and A. Majumdar, J Heat Trans-Trans
ASME 125 (5), 881-888 (2003).
Frank P. Incropera, Fundamentals of heat and mass transfer, 6th ed. (John Wiley, Hoboken, NJ,
2007), pp.xxv, 997 p.
N. M. Ravindra, S. Abedrabbo, W. Chen, F. M. Tong, A. K. Nanda, and A. C. Speranza, IEEE Trans.
Semicond. Manuf. 11 (1), 30-39 (1998).
M. C. Wingert, Z. C. Y. Chen, S. Kwon, J. Xiang, and R. K. Chen, Rev Sci Instrum 83 (2) (2012).
20