Supplementary Information:
Deflection measurement resolution:
(i) Actual measurement condition: The Johnson noise and thermo-mechanical noise for a
cantilever are given by1-3
Johnson noise (V): vnoise (rms) = inoise (rms) × R DS = √4k B TR DS ∆𝑓
(1)
Thermo-mechanical noise at resonance (nm): TMres (rms) = √4k B TQ∆𝑓⁄(kω0 )
(2)
Thermo-mechanical noise off resonance (nm): TMoff,res (rms) = √4k B TQ∆𝑓⁄(kω0 )
(3)
We have not considered 1/f noise in our calculations since it is significant only at lower resonant
frequencies.1-3 For our calculations we have assumed kBT = 0.026 eV at 300 K and RDS = 22 KΩ,
quality factor, Q = 200, spring constant, k = 1.5 N/m, and a bandwidth f = 1 Hz. The Johnson
noise is then calculated using (1) 19.14 nV. Since
𝐺𝐹𝑑𝑦𝑛(𝑝𝑒𝑎𝑘) =
𝑅𝑑𝑠 ⁄𝑅0
𝜖𝑝𝑒𝑎𝑘
=
𝑉𝑑𝑠 ⁄𝑉0
𝜖𝑝𝑒𝑎𝑘
,
(4)
where the Rds and Vds are the peak changes in RDS and Vds due to oscillations, respectively, V0
is the drain-source voltage off resonance, and peak is the peak stress. Assuming the amplitude of
cantilever oscillations to be the same as the mechanical base (23 nm) for off resonance
conditions, the maximum amplitude at resonance is 4.6 m due to quality factor enhancement (Q
= 200), which gives rise to a peak strain peak of 4.6  1.865×10-5 = 8.579  10-5. With an initial
voltage V0 of 0.22 V, and ΔVds of 0.822 = 1.16 mV. Since ΔVds = 1.16 mV at resonance
corresponds to an oscillation amplitude of 4.6 µm, the equivalent noise amplitude is found to be
19.14 nV  4.6 m/1.16 mV = 75.9 pm. The thermo-mechanical noise is calculated for the same
frequency,
using
(2)
as
2.8 pm.
The total
RMS
noise is then calculated as
√(𝑇𝑀𝑛𝑜𝑖𝑠𝑒 )2 + (𝐽𝑜ℎ𝑛𝑠𝑜𝑛 𝑛𝑜𝑖𝑠𝑒)2 = 75.95 𝑝𝑚 at 1 Hz bandwidth.
(ii)
Optimized Vds: The Johnson noise is much higher than the TM noise due to low voltage
sensitivity of deflection, 1.16 mV for 4.6 µm oscillation amplitude, which is due to low Ids. We
used a constant current of Ids = 10 µA which results in a bias voltage of 0.22 V (= 10 µA × 22
KΩ), which is very low compared to Si piezoresistive cantilever (5-10 V) resulting in a drain
current in the mA range.1-3 If a more generic Vds of 10 V is used, then ΔVds would increase by a
1
factor of 45.45 (=10/0.22), which yields an equivalent Johnson noise amplitude of 75.9/45.45 =
1.67 pm. Thus the overall noise will be 3.26 pm at 1 Hz bandwidth or 103.1 pm at 1 KHz
bandwidth.
(iii)
Off resonance static and low frequency bending: Since Si cantilever noise are calculated
for off-resonance conditions,1-3 for accurate comparison with our GaN microcantilever the TM
noise needs to be calculated off resonance, which comes out to be 2.8/200 pm = 0.014 pm, thus
the overall noise becomes basically only the Johnson noise. In off resonance conditions, we had
a voltage change ΔVds = 0.22 V  1.17% = 2.574 mV for 1 m deflection (results in manuscript).
If a generic voltage of Vds = 10 V is used, ΔVds = 117 mV, thus the equivalent deflection noise
becomes 19.14 nV  1 m /117 mV = 0.1635 pm at 1 Hz bandwidth or 5.173 pm at 1 KHz
bandwidth. Therefore, for an optimized Vds = 10 V, at static or low frequency ac conditions, the
noise limited deflection sensitivity is much better than the best estimate for optimized Si
cantilevers reported so far of 20 pm at 1 KHz bandwidth,
1-3
and even better than the MOSFET
embedded Si microcantilever with a deflection noise of 10 pm (sensitivity of 0.1 – 0.2 mA/nm,
with the current noise of 40 – 60 nA for 1 KHz bandwidth, which gives a detection limit of ~10
pm).4
Pressure measurement resolution:
From the acoustic pressure calculations in page 8 in the manuscript it is found that pressure
change of 0.476 Pa resulted in a resonance amplitude change of 792 nm. Thus, the noise limited
pressure resolution of the cantilever at resonance is calculated as 1.96 µPa (= 0.4763.26 pm/792
nm) for a measurement bandwidth of 1 Hz, and 62 µPa for a bandwidth of 1 KHz.
References:
[1] J. A. Harley and T. W. Kenny, Journal of Microelectromechanical Systems 9, 226 (2000).
[2] X. M. Yu, J. Thaysen, O. Hansen, and A. Boisen, J. Appl. Phys. 92, 6296 (2002).
[3] J. C. Doll, S. Park, and B. L. Pruitt, J. Appl. Phys. 106, 064310 (2009).
[4] G. Shekhawat, S. Tark, and V. P. Dravid, Science 311, 1592 (2006)
2