Supplementary Materials
Sub-picowatt Resolution Calorimetry with Niobium Nitride Thin-film Thermometer
Edward Dechaumphai1, Renkun Chen1†
1
Department of Mechanical and Aerospace Engineering, University of California, San
Diego, La Jolla, California 92093, USA
†Corresponding Author: rkchen@ucsd.edu
1.
Analysis of the Wheatstone bridge
We consider the Wheatstone bridge shown in Fig. 1(a) to measure the temperature
change of the sensing side (𝑅𝑠 ) by measuring its resistance change (𝛿𝑅𝑠 ). Assuming the
bridge is initially balanced and all the four resistors have the same resistance of 𝑅𝑠 . When
there is a heat flux modulated at 2𝜔 transferred to the sensing side, the corresponding
change in the resistance is 𝛿𝑅𝑠 , as shown in Fig. S1.
Figure S1. A Wheatstone bridge (taken from Fig. 1(a)) with four identical resistors to
measure a small change in the resistance of the sensing beam (𝛿𝑅𝑠 ).
The 2nd harmonic voltage across the nodes A and B is:
1
𝑉𝐴𝐵 = 𝑉𝐴 − 𝑉𝐵 = (2𝑅
𝑅𝑠
(2𝑅𝑠 +𝛿𝑅𝑠 )2𝑅𝑠
1
𝑠 +𝛿𝑅𝑠
− 2) 2𝐼𝑠 (
(4𝑅𝑠 +𝛿𝑅𝑠 )
)
(S1)
Because 𝛿𝑅𝑠 ≪ 𝑅𝑠 , Eq.(S1) can be simplified to:
1
𝑉𝐴𝐵 = − 2 𝐼𝑠 δR s
(2)
Note the factor of 1/2 on the right hand side of Eq.(2) which is a result of this particular
bridge configuration.
Based on Eq. (S2), we can then relate the NEV (measured from A to B) to the
noise equivalent resistance (NER) of the sensing beam via:
𝑁𝐸𝑅 = 2
𝑁𝐸𝑉
(S3)
𝐼𝑠
, which is corresponding to a NET on the middle pad of the beam via:
𝑁𝐸𝑅
𝑁𝐸𝑇 = 2√2 𝑅
(S4)
𝑠 𝛼𝑠
where the factor of 2 accounts for the conversion from rms values of the voltage to
amplitude values of the temperature and the factor of 2 accounts for the fact that the
temperature of the middle pad of the beam is twice of the average temperature across
the entire suspended sensing beam (𝑅𝑠 ).
Substituting Eq. (S3) into Eq. (S4), one yields:
𝑁𝐸𝑉
𝑁𝐸𝑇 = 4√2 𝐼 𝑅
𝑠 𝑠 𝛼𝑠
(S5)
Note the extra factor of 2 in Eq. (S5) compared to Eq. (2) in the main manuscript, due
to the particular bridge configuration and different definitions of NEV. In Eq. (S5),
NEV is defined as the voltage noise across the nodes A and B in the bridge, whereas
in Eq. (2), NEV is defined as the voltage noise across Rs.
2
Now consider the same bridge shown in Fig. (S1), and assume the noise on each
resistor is Johnson noise, it can be shown that the overall noise measured between the
nodes A and B is the same as the noise of a single resistor, namely,
𝑁𝐸𝑉𝐴𝐵 = 𝑁𝐸𝑉𝑅𝑠 = √4𝑘𝐵 𝑇𝑅𝑠 Δ𝑓
(S6)
Eq. (S6) can be understood from the fact that the effective resistance between A and B is
exactly 𝑅𝑠 , and has also been experimentally verified from the power spectral density
measurement of the noise between nodes A and B (as shown in Fig. 2 in the main
manuscript).
Substituting Eq. (S6) into (S5), one yields:
NET 
8 2k BTRs f
I s Rs s
which is the Eq. (5) shown in the main manuscript.
3
(S7)
2.
Signal attenuation at high frequencies
500
2
1.5
Ts [mK]
D
D
Th [mK]
450
400
350
1
0.5
300
(a)
250
2
(b)
0
2
4
6 8 10
20
Heating current frequency [Hz]
4
6 8 10
20
Heating current frequency [Hz]
Figure S2. (a) Measured heating side temperature as a function of frequency (blue
circles). (b) Measured sensing side temperature as a function of frequency (red circles).
Signal attenuation depends on the thermal penetration depth, where 𝐿𝑝 ~1/√𝑓. Hence,
signals is attenuated at high frequency. For our device, at f = 6 Hz, the signal will
attenuate roughly 10% and drop off quickly at higher frequencies. Therefore, we selected
4 Hz in our NET measurements to obtain the power resolution of the device with
minimum signal attenuation.
4
3.
Effect of heating current frequency on the NET
(a) 500
(b) 500
1.5 Hz
4 Hz
T [mK]
400
300
300
s
s
T [mK]
400
200
100
100
75 mK
0
0
200
50 mK
0
0
0.5
1
Heating Power [nW]
(c) 500
(d) 500
8 Hz
18 Hz
400
T [mK]
400
300
300
s
s
T [mK]
0.5
1
Heating Power [nW]
200
100
100
45 mK
0
0
200
40 mK
0
0
0.5
1
Heating Power [nW]
0.5
1
Heating Power [nW]
Figure S3. Sensing side temperature with different joule heating power to determine the
NET at different frequencies: (a) 1.5 Hz, (b) 4 Hz, (c) 8 Hz, and (d) 18 Hz. The NET is
higher when the heating power is modulated at a lower frequency, which is resulted from
larger 1/f thermal drift at lower frequency. As shown in these plots, for heating
frequency of 1.5 Hz, the NET is approximately 75 μK, higher than the NET of ~50 μK at
4 Hz. Likewise, NETs at higher frequencies (8 Hz and 18 Hz) are lower, getting closer to
the Johnson noise limit, but the signal attenuation is significant at these frequencies (as
shown in Fig. (S2)).
5
4.
Effect of sensing current amplitude on the NET
Figure S4. Measured sensing side temperature rise vs. heating power at different applied
sensing current of (a) 712 nA and (b) 356 nA. Since NET  1 / I s , a lower NET is
expected with a higher sensing current. Our measurements indeed showed that when the
sensing current was doubled, the NET was reduced by approximately half (from 90 μK to
45 μK).
6