Signal-to-Noise Ratio improves in Critical-Point Flexure
Biosensors due to Intrinsic Low Pass Filtering
S1: Time Domain Simulation Framework for Force Noise
The time domain response of Flexure sensor in presence of white thermo-mechanical force noise is modeled
using Newton’s equation, given by𝑚
𝑑𝑣
𝜖0 𝑊𝐿𝑉𝐴2
+ 𝑏𝑣 = 𝑘(𝑦0 − 𝑦) −
+ 𝐹𝑁 (𝑡),
𝑑𝑡
2𝑦 2
(𝑆1𝑎)
𝑑𝑦
= 𝑣.
𝑑𝑡
(𝑆1𝑏)
Here, 𝑚 is the mass of movable electrode, 𝑣 is the velocity, 𝑡 is time, 𝑏 = 𝑚𝜔0 /𝑄 is the damping
coefficient. 𝐹𝑁 (𝑡) is the random noise force with autocorrelation function
⟨𝐹𝑁 (𝑡)𝐹𝑁 (𝑡 ′ )⟩ =
2𝑘𝐵 𝑇𝑏𝛿(𝑡 − 𝑡 ′ ) and one sided power spectral density 𝑆𝐹 (𝜔) = 4𝑘𝐵 𝑇𝑏. It is also important to note that, the
white Gaussian noise force is 𝐹𝑁 (𝑡) = √2𝑘𝐵 𝑇𝑏
𝑑𝑣
Using the definition 𝑑𝑡 =
(1 +
𝑣(𝑡+Δ𝑡)−𝑣(𝑡)
Δ𝑡
and
𝑑𝑦
𝑑𝑡
=
𝑑𝑊(𝑡)
,
𝑑𝑡
where 𝑊(𝑡) is the Brownian process.
𝑦(𝑡+Δ𝑡)−𝑦(𝑡)
,
Δ𝑡
Eq. S1 can be written as follows-
𝑏Δ𝑡
𝑘Δ𝑡
𝜖0 𝑊𝐿𝑉𝐴2 Δ𝑡
𝑑𝑊𝛥𝑡 (𝑡)
) 𝑣(𝑡 + Δ𝑡) = 𝑣(𝑡) +
(𝑦0 − 𝑦(𝑡)) −
+ √2𝑘𝐵 𝑇𝑏
,
2
𝑚
𝑚
2𝑚𝑦
𝑚
𝑦(𝑡 + Δ𝑡) = 𝑦(𝑡) + 𝑣(𝑡 + Δ𝑡)Δ𝑡,
(𝑆2𝑎)
(𝑆2𝑏)
where 𝑣(𝑡) and 𝑣(𝑡 + Δ𝑡) are the values of velocity, at time 𝑡 and 𝑡 + Δ𝑡, respectively. Similarly, 𝑦(𝑡) and
𝑦(𝑡 + Δ𝑡) are the values of electrode positions, at time 𝑡 and 𝑡 + Δ𝑡, respectively; here, Δ𝑡 is the time step.
Most importantly, 𝑑𝑊Δ𝑡 (𝑡) is a random variable that is normal distributed with mean zero and standard
deviation √Δ𝑡. Knowing the initial condition i.e., 𝑦(0) and 𝑣(0), Eq. S2 can be solved for any 𝑉𝐴 . At every
instant 𝑡, a new random variable 𝑑𝑊Δ𝑡 (𝑡) is generated to evaluate Eq. S2.
S2: Time Domain Simulation Framework for Stiffness Noise
The time domain response of Flexure sensor in presence of white stiffness noise is also modeled using
Newton’s equation, given by𝑚
𝑑𝑣
𝜖0 𝑊𝐿𝑉𝐴2
+ 𝑏𝑣 = (𝑘 + Δ𝑘𝑁 (𝑡))(𝑦0 − 𝑦) −
,
𝑑𝑡
2𝑦 2
𝑑𝑦
= 𝑣,
𝑑𝑡
(𝑆3𝑎)
(𝑆3𝑏)
where Δ𝑘𝑁 (𝑡) is the random noise due to stiffness fluctuations with autocorrelation ⟨Δ𝑘𝑁 (𝑡)Δ𝑘𝑁 (𝑡′)⟩ =
0.5𝑁𝑘 (𝑡 − 𝑡 ′ ) and one sided power spectral density 𝑆𝑘 (𝜔) = 𝑁𝑘 . It is also important to note that white
stiffness fluctuations are Δ𝑘𝑁 (𝑡) = √0.5𝑁𝑘
𝑑𝑊(𝑡)
.
𝑑𝑡
For numerical simulations, Eq. S3 can be written as
follows(1 +
𝑏Δ𝑡
𝑘Δ𝑡
𝜖0 𝑊𝐿𝑉𝐴2 Δ𝑡 √0.5𝑁Δ𝑘 (𝑦0 − 𝑦(𝑡))𝑑𝑊𝛥𝑡 (𝑡)
) 𝑣(𝑡 + Δ𝑡) = 𝑣(𝑡) +
+
,
(𝑦0 − 𝑦(𝑡)) −
𝑚
𝑚
2𝑚𝑦 2
𝑚
(𝑆4𝑎)
𝑦(𝑡 + Δ𝑡) = 𝑦(𝑡) + 𝑣(𝑡 + Δ𝑡)Δ𝑡.
(𝑆4𝑏)
Equation S4 can be now solved for any 𝑉𝐴 to evaluate the noise response due to stiffness fluctuations.
S3: Numerical Simulations
Figures S1-S2 show the results of time domain stochastic simulations (see Table S1 for parameters used)
for thermo-mechanical noise and stiffness noise due to temperature fluctuations, respectively. For the two
cases, Eqs. S2 & S4 have been solved, respectively. We simulated the noise response at different voltages.
For the specific voltage of 𝑉𝐴 = 0.9𝑉𝑃𝐼 , the results are summarized in Figs. S1a-d and Figs. S2a-d. Figures
S1a-b show the fluctuations in the position of electrode on the potential energy landscape. Each symbol
denotes the total energy during fluctuations. As expected, electrode does random thermal vibration around
its equilibrium position, as shown in Fig. S1c. Figure S1d shows the corresponding sample average of root
mean square fluctuations i.e.,
2
1
𝑖=𝑁
𝑖
Δ𝑦𝑁 (𝑡) = √𝑁 ∑𝑖=1 𝑠(𝑦𝑖 (𝑡) − 𝑦𝑚𝑒𝑎𝑛
(𝑡)) .
𝑠
𝑖
(𝑡) =
𝑦𝑚𝑒𝑎𝑛
1
𝑁𝑠
𝑠
∑𝑖=𝑁
𝑖=1 𝑦𝑖 (𝑡).
𝑖
Here, 𝑦𝑖 (𝑡) denote the position of electrode during 𝑖 𝑡ℎ simulation at time 𝑡 and 𝑦𝑚𝑒𝑎𝑛
(𝑡) is the
corresponding mean position. 𝑁𝑠 (1000 in this article) is the number of simulations performed to calculate
the statistical average. Interestingly, Δ 𝑦𝑁 (𝑡) starts from zero and then saturates to an equilibrium value
(solid dot in Fig. S1d), which is nothing but the average noise power. Figure S1e compares the results
obtained from time domain simulations (symbols) with the ones obtained from transfer function based
analysis (solid line). In spite of the presence of the highly nonlinear electrostatic force, the results match
because the fluctuations are small (Fig. S1d and Fig. S2d), thus justifying linearization around the
equilibrium value (Eq. 2 and Eq. 3 in the main text) for transfer function based analysis. Having said that,
as we go closer to the pull-in voltage, fluctuations increase considerably, eventually leading to noise
initiated pull-in. Therefore, the linear transfer function based analysis is valid so long as we are below safe
operating voltage. Note that, similar results and similar matching between time domain and transfer
function analysis is achieved for stiffness noise as well (Fig. S2).
Parameter
Value
𝑊
1𝜇𝑚
𝐿
4𝜇𝑚
𝐻
40𝑛𝑚
𝐸
200𝐺𝑃𝑎
𝜈
0.31
𝜌
8912𝐾𝑔/𝑚3
𝑦0
100𝑛𝑚
𝑔
7.4 × 10−6 𝑊/𝐾
1 𝜕𝑘
𝑘 𝜕𝑇
10−3 /𝐾
𝜏𝑇
30𝑝𝑠
Table S1: Parameters used for calculation of SNR and LOD in the main text .
Fig: S1: Time domain stochastic numerical simulations of thermo-mechanical noise. (a)-(b) Fluctuations of movable
electrode position shown on the potential energy landscape. The region in the oval has been zoomed in Fig. S1b.
Symbols denote the total energy (kinetic + potential) of the electrode. (c) Position of electrode as a function of time.
𝑦𝑠 denote the equilibrium position. (d) Root mean square fluctuations as a function of time. (e) Equilibrium value of
root mean square fluctuations is the average noise power. Symbols denote the results from time domain numerical
simulations; whereas solid line denote the calculations from linear transfer function based analysis (Eq. 5b in the main
text).
Fig: S2: Time domain stochastic numerical simulations of stiffness noise due to temperature fluctuations. (a)-(b)
Fluctuations of movable electrode position shown on the potential energy landscape. The region in the oval has been
zoomed in Fig. S2b. Dotted black curve corresponds to the maximum stiffness; whereas magenta dotted to minimum
stiffness. Symbols denote the total energy (kinetic + potential) of the electrode. (c) Position of electrode as a function
of time. 𝑦𝑠 denote the equilibrium position. (d) Root mean square fluctuations as a function of time. (e) Equilibrium
value of root mean square fluctuations is the average noise power. Symbols denote the results from time domain
numerical simulations; whereas solid line denote the calculations from linear transfer function based analysis (Eq. 6b
in the main text).
S4: Safe Operating Voltage to avoid Noise Initiated Pull-in
In the main text, we argued that biasing close to pull-in improves 𝑆𝑁𝑅 and 𝐿𝑂𝐷. Here, we answer a very
important and fundamental question regarding the stability of critical-point Flexure sensors close to pull-in
point. The question is “how close to the pull-in point can one operate without making the sensor unstable?”
Note that, in Figs. 3-4 in the main text, 𝑉𝐴 was swept from 𝑉𝐴 = 0 to 𝑉𝐴 = 0.995𝑉𝑃𝐼 . We should check if
biasing at 𝑉𝐴 = 0.995𝑉𝑃𝐼 is feasible. To answer this question, we look at the behavior of movable electrode
in response to both force and stiffness noise using time domain stochastic simulations.
1
Figure S3a shows the potential energy (𝑈 = 2 𝑘(𝑦0 − 𝑦)2 −
𝜖0 𝑊𝐿 2
𝑉𝐴 )
2𝑦
landscape of Flexure sensor at
𝑉𝐴 = 0.995𝑉𝑃𝐼 . Movable electrode is stabilized at the minimum of 𝑈. In absence of noise, movable
electrode should have remained at the bottom of potential energy well in Fig. S3a (see the dotted line in
Fig. S3b also). However, thermo-mechanical noise exerts random force on the electrode making it fluctuate
(reason for noise characterized by Δ𝑦𝑁 ) around its equilibrium position as shown in Figs. S3a-b. Symbols
in Fig. S3a denote total energy (kinetic + potential) of movable electrode during fluctuations.
Due to the presence of a high energy barrier Δ𝑈𝑏 = 𝑈(𝑦𝑢 ) − 𝑈(𝑦𝑠 ) ≈ 3.75 × 103 𝑘𝐵 𝑇, (𝑦𝑠 : stable
equilibrium position and 𝑦𝑢 : unstable equilibrium position) the movable electrode only fluctuates around
the bottom of potential well in Fig. S3a, but cannot surmount the energy barrier to make the system unstable.
On the other hand, stiffness noise due to temperature fluctuations will make the potential energy landscape
fluctuate as shown in Fig. S3c. Dotted black line corresponds to the potential energy profile for maximum
stiffness; whereas dotted magenta for minimum stiffness. Due to the fluctuations in the stiffness, the
position of the electrode fluctuates around its equilibrium position as shown in Figs. S3c-d. Once again, the
stiffness fluctuations are not strong enough to make the electrode pull-in (Figs. S3c-d). Therefore, we will
classify 𝑉𝐴 = 0.995𝑉𝑃𝐼 as the safe operating voltage.
(a)
(b)
(c)
(d)
Fig. S3: Results of time domain stochastic simulations of a Flexure sensor at 𝑉𝐴 = 0.995𝑉𝑃𝐼 with Δ𝑈𝑏 ≈ 3.75 ×
103 𝑘𝐵 𝑇, due to (a)-(b) thermo-mechanical noise and (c)-(d) temperature fluctuations stiffness noise. 𝑈𝑠 denote the
potential energy at equilibrium position i.e., at the bottom of potential well. Symbols in Figs. S3a & c denote the total
energy (kinetic + potential). Dotted black line in Fig. S3c correspond to maximum stiffness; whereas magenta line to
minimum stiffness. Inset in Figs. S3a &c show the zoomed region around the bottom of potential well.
Note that, if 𝑉𝐴 is increased further, Δ𝑈𝑏 decreases and becomes Δ𝑈𝑏 ≈ 5𝑘𝐵 𝑇 at 𝑉𝐴 = 0.99994𝑉𝑃𝐼 as
shown in Fig. S4a. In this case, the movable electrode gets the sufficient energy from the surrounding to
surmount the energy barrier and gets pulled-in as shown in Figs. S4a-b. Therefore, 𝑉𝐴 = 0.99994𝑉𝑃𝐼 cannot
be classified as safe operating voltage. Interestingly, pull-in at 𝑉𝐴 = 0.99994𝑉𝑃𝐼 occurs due to thermomechanical noise, and not because of stiffness noise. However, if the voltage is increased even further, pullin can occur due to stiffness noise because of temperature fluctuations as shown in Figs. S4c-d. The bottom
line from this section is that as we go closer and closer to pull-in point, chances of noise initiated pull-in
increases. A safe operating voltage is that does not cause noise initiated pull-in or at least not in the time
duration of measurement.
(a)
(b)
(c)
(d)
maximum
minimum
Fig. S4: Noise initiated pull-in due to (a)-(b) thermo-mechanical noise at 𝑉𝐴 = 0.99994𝑉𝑃𝐼 with Δ𝑈𝑏 ≈ 5𝑘𝐵 𝑇 and
(c)-(d) stiffness noise due to temperature fluctuations at 𝑉𝐴 = 0.999995𝑉𝑃𝐼 . 𝑦𝑠 corresponds to the stable equilibrium
position; whereas 𝑦𝑢 unstable.