Introduction
The detection of electrical potential is of significant interest in surface
potential distribution characterizations and
biological and chemical analysis
. Electrometers are another
application for potential sensing devices, which can be employed for
particulate matter detection . MEMS resonant devices
have been widely used for these applications with the advantage of high
resolution and a large dynamic range.
Recently, mode-localized MEMS resonant sensors have emerged as an alternative
resonant sensing scheme , in which the mode shape of a weakly coupled resonator system
changes subject to an external stiffness perturbation caused by the
electrical potential change. Orders of magnitude improvement in sensitivity
for electrometers have already been reported. Furthermore, mode-localized
sensors exhibit better common-mode rejection capability
. Previously, mode-localized sensors
were implemented with two resonators weakly coupled electrically
or mechanically
, with the electrical
coupling element offering advantages of tunability of the sensitivity
.
In this paper, we demonstrate an alternative approach for potential sensing
applications using a 3-degree-of-freedom (3DoF) structure. The structure has already been reported
elsewhere ; however, the advantages of the 3DoF design
have not been discussed in full detail, as only improvements in sensitivity
have been shown. In this paper, we focus on the design considerations of the
3DoF structure, which we believe also have other advantages – for instance,
the alleviation of the electrical nonlinear driving force, as well as the
nonideal sensing current. Furthermore, in terms of potential sensing
applications, two sensing methods exist: (i) modulating the electrostatic
spring and (ii) directly applying an axial electrostatic force. However, the
two sensing methods were not directly compared previously. In this work, we
are able to demonstrate that, by modulating the electrostatic spring, an
improvement in sensitivity can be observed.
The paper is arranged as follows: in Sect. , the
advantages of the 3DoF structure design are discussed; in Sect. , the two potential sensing schemes by applying a DC
potential to different ports are discussed; in Sect. ,
experimental results are presented and the paper is concluded in Sect. .
(a) Schematics of the 3DoF mode-localized potential sensor, showing
three resonators coupled electrostatically with their neighbours. In
addition, electrostatic springs and forces caused by electrical potential
differences are also shown; (b) the linearized spring–mass–damper model of
the 3DoF mode-localized potential sensor.
Advantages of the 3DoF structure with electrical coupling
The schematic of the 3DoF mode-localized MEMS resonant potential sensor is
shown in Fig. . Each resonator has four suspension beams
acting as springs and a relatively large proof mass to reduce the effect of
fabrication tolerances to the mass. In addition, resonators 1 and 3 have a
tether structure that is capable of translating an axial electrostatic force
to the suspension beams. Electrical coupling was chosen due to the ability to
tune the coupling strength and thus the sensitivity of the sensor to external
perturbations. A DC voltage Vbias was applied to resonators 1
and 3, while resonator 2 was connected to ground. The voltage difference
created electrostatic springs to couple the resonators to its neighbouring
counterparts. An AC drive voltage is applied on the drive electrode,
generating the actuation force. Details of the design and its fabrication
process were reported in . However, due to the limited
scope of the previous work, the advantages of the design were not discussed
in full detail; these are presented in the following sections.
Sensitivity improvement
As reported in , the sensitivity to stiffness changes of
a 3DoF mode-localized resonant sensor can be expressed by, assuming linear
springs, K2>2K and K/Kc>10:
S3DoF=∂(Amplituderatio)∂(ΔK/K)=K(K2-K+Kc)Kc2,
where K, K2 and Kc denote the stiffness of the suspension beam of
resonator 1 (and 3), resonator 2 and the coupling spring, respectively.
Moreover, the sensitivity of a 2DoF mode-localized resonant sensor can be
expressed by :
S2DoF=∂(Amplituderatio)∂(ΔK/K)=K2Kc.
For identical K/Kc, the sensitivity of the 3DoF mode-localized resonant
sensor can be enhanced by a factor of K2-K+KcKc. Over
2
orders of magnitude improvement has already been demonstrated
.
Sensitivity improvement without sacrificing signal transduction
The electrostatic coupling Kc for a parallel plate configuration as
shown in Fig. can be expressed by
:
Kc=-ε0AV2d3,
where ε0, A and d are the dielectric constant of vacuum,
cross-sectional area and the air gap between the plates, respectively. V is
the potential difference between the two plates, which is the determining
factor of the coupling strength for a given design. Therefore, for a 2DoF
mode-localized resonant sensor, decreasing V, and thus Kc, is
beneficial for sensitivity enhancement. On the other hand, a high V is
often desirable due to the required motional current level for a reasonable
signal-to-noise ratio in the readout circuit.
This design contradiction for choosing an optimal V can be solved by
adopting a 3DoF resonant sensor configuration. An additional third parameter,
the effective spring constant of the middle resonator K2, can be altered
to maintain or even improve the sensitivity without sacrificing the readout
signal level. As shown in Eq. (), increasing K2 can
improve the sensitivity.
Electrostatic nonlinearity reduction
For an ideal 3DoF mode-localized resonant sensor with identical resonators 1
and 3 and negligible damping, there are three fundamental modes of vibration
: in the first mode, all three resonators vibrate
in phase with each other; in the second mode, resonators 1 and 3 vibrate
out of phase, whereas resonator 2 is statutory; in the third mode, each
resonator vibrates out of phase with its neighbours, and resonators 1 and 3
are in phase. The second mode, which is referred to as out-of-phase mode, is
the focus of this study. The balanced and perturbed mode shapes of the
out-of-phase mode of a 3DoF resonant structure are illustrated in Fig. .
Illustration of the mode shapes of the out-of-phase mode of a
representative 3DoF resonant structure, simulated using CoventorWare FEM
tool: (top) without perturbations and (bottom) with stiffness perturbations. The
perturbation in the bottom panel is a positive stiffness perturbation applied to resonator 1. It should be pointed out that the same mode shape is generated given an
identical negative stiffness perturbation applied to resonator 3.
When a stiffness perturbation is introduced, resonator 2 starts to vibrate
due to mode localization. However, in the case of weak coupling, the
amplitude of resonator 2 is orders of magnitude lower than the resonator with
highest amplitude (e.g. resonator 1) . This can
also be seen qualitatively from Fig. .
Consider an abstract model of the drive electrode, resonator 1 and resonator 2 as shown in Fig. . Only resonator 1 is
considered because, under normal operating conditions, resonator 1 has a
higher amplitude than resonator 3, meaning that it is more susceptible to
nonlinear effects, as will be shown in Sect. .
Abstract model of the drive electrode, resonator 1 and resonator 2
illustrated as parallel plates. X1 and X2 are the displacement of resonator
1 and resonator 2, respectively. The cross-sectional area A and gap d are
supposed to be identical for all electrodes considered.
Assuming vac≪Vbias and neglecting nonlinear terms with
orders higher than 3, the total electrostatic force exerted on resonator 1
can be approximated by
Ftotal,elec≈ηA,Pvacsinωt+ε0Vbias2Ad3X1+ε0Vbias2Ad3(X1-X2)-3ε0Vbias2A2d4[X12-(X1-X2)2]+2ε0Vbias2Ad5[X13+(X1-X2)3].
For a 3DoF mode-localized sensor with X2≪X1, or
quasi-static motion of resonator 2, the second-order nonlinear term of
the electrostatic actuation (i.e. between drive electrode and resonator 1)
cancels out that of the electrostatic coupling (i.e. between resonators 1
and 2), therefore rendering the total second-order electrostatic nonlinearity
negligible. Thus the overall nonlinearity is reduced. This is often desirable
for resonator design, leaving only the third-order electrostatic nonlinearity
which, in turn, can be used to eliminate the third-order mechanical
nonlinearity intrinsic to the vibrating beams .
As for a 2DoF mode-localized sensor with comparable X1 and X2, the
second-order nonlinear terms remains; thus the total nonlinearity is higher
than that of a 3DoF mode-localized resonant sensor.
Reduction of nonideal sense current components
An SEM image of one pair of comb fingers attached to the proof mass
of resonator 1 for vibration motion sensing. (As shown in Fig. c,
another pair of comb fingers with identical configuration is also attached to the proof mass of resonator 3.)
The top and bottom sets of the comb fingers on the proof mass are reversely configured relative to the
stationary sense electrodes 1 and 2; this enables differential sensing. Therefore, a reduction in
nonideal sense current components is achieved by common-mode rejection.
Due to the amplitude detecting method used in a mode-localized sensor, it is
important to obtain a measure of the linear motion of the resonators with
high accuracy. From the structure design perspective, it is desirable to use
differential sensing to cancel out the common-mode second-order nonlinearity
while doubling the magnitude of the first-order term. This is
shown in Fig. . In addition to the second-order
nonlinear term of the motional current, another common-mode nonideal current
component, the feed-through current , can also be
cancelled out.
Potential sensing methods
Using two different approaches, we investigated the application of the device
to detect a DC electrical potential applied to either “port 1” or “port 2”,
as illustrated in Fig. a. When a DC potential is
applied to port 1, an axial electrostatic force change is created, modulating
the stress in the suspension beams and thus leading to a stiffness
perturbation of resonator 1. Alternatively, when a DC potential is applied to
port 2, a change in the electrostatic spring, instead of an electrostatic
force, modulates the spring softening effect compared to the case in which no
potential is applied; this is equivalent to introducing a stiffness
perturbation to resonator 3.
Potential detection using port 1
The stiffness perturbation of resonator 1, ΔK1, as a function of
an applied potential V1 to port 1, can be expressed as
ΔK1=1.2ε0A1(-2VbiasV1+V12)d12L,
where ε0 is the dielectric constant of free space. Typically,
to initially avoid mode aliasing, a negative stiffness perturbation,
Kp<0, created by a constant voltage Vp on port 2 is introduced
. With an applied Kp, the sensing mechanism in
response to a stiffness perturbation caused by the potential change is
explained below. Assuming |ΔK1|≪|Kp|, based on a transfer
function model of the 3DoF weakly coupled resonators device described in
, the mode frequencies of interest can be calculated as
ω≈K′+Kc+12ΔK′-2K′γ±ΔK′2+2K′γ2M,
where K′=K+ΔK1, ΔK′=ΔK1-Kp,
γ=K(K2-K+Kc)Kc2, M is the effective
mass of each of all three resonators, K is the stiffness of resonators 1 and 3,
K2 is the stiffness of resonator 2 and Kc=-ε0AVbias2d3
is the electrostatic coupling stiffness between neighbouring resonators. The
positive and negative sign is for the out-of-phase and in-phase mode, respectively.
From Eq. (), it can be derived
that the out-of-phase mode has a more significant response subject to a
stiffness perturbation. Also, assuming V1≪Vbias and weak
coupling, Kc≪K, we can find an expression of the sensitivity for
frequency shift as an output signal, Sf,1, with respect to
V1:
Sf,1=∂Δff∂(ΔK)∂(ΔK)∂(V1)≈-1.2ε0A1Vbiasd12LK.
It can be seen that Eq. () is
similar to a conventional single DoF resonant sensor with frequency shift as
an output signal , allowing a direct comparison to
using amplitude ratio as an output signal.
If a mode-localized sensing approach is used, the linearized sensitivity of
amplitude ratio as an output with respect to the potential,
SAR, can be calculated based on the assumption of
weak coupling as elaborated in , and V1≪Vbias:
SAR,1=∂X1(jωop)X3(jωop)∂(ΔK)∂(ΔK)∂(V1)≈-2.4A1(K2-K+Kc)d4ε0LA2Vbias3,
where L is the length of the suspension beams, and Ai and di are the
overlapping cross-sectional area and the gap of the parallel plate for the
ith potential port (i=1 and 2), respectively; A and d are the cross-sectional area and the gap of the electrodes for the electrostatic coupling,
respectively.
It can be seen that the improvement in sensitivity is 2γ. For weak
coupling Kc<K/10<K2/20, the improvement is at least 2 orders of
magnitude .
Potential detection using port 2
For a potential applied to port 2, V2≪Vbias, cancelling
the common term to both resonators 1 and 3 proportional to
Vbias2, the stiffness perturbation of resonator 3 can be
approximated as a linear function of V2:
ΔK3≈2ε0A2VbiasV2d23.
For |ΔK3|≪|Kp|, the in-phase mode frequency has a stronger
response . The sensitivity for the in-phase mode
frequency shift, as well as the amplitude ratio as an output signal, can be
approximated by
Sf,2=∂Δff∂(ΔK)∂(ΔK)∂(V2)≈ε0A2Vbiasd22LKSAR,2=∂X1(jωop)X3(jωop)∂(ΔK)∂(ΔK)∂(V2)≈-2A2(K2-K+Kc)d3ε0A2Vbias3.
It should be noted that the length of the suspension beams is 350 µm,
while the capacitive gap is 4.5 µm. Consequently, for the dimensions of
this device, ΔK1 is around 2 orders of magnitude lower than
ΔK3. Therefore, applying the potential to port 2 should induce a
more significant stiffness perturbation, hence a higher output signal.
Conclusions
In this paper, we have demonstrated the design of a 3DoF weakly coupled
resonant sensor for potential sensing applications, which could also be
extended to an electrometer. We have presented the design advantages for the
3DoF structure in theory, including sensitivity improvement, electrical
nonlinearity and nonideal current reduction. We have shown that, by using the
device as a mode-localized sensor, the sensitivity can be improved by over
4 orders of magnitude, compared to conventional frequency shift as an
output. In addition, we have also compared the sensitivity of the
mode-localized sensor for different bias voltages. We demonstrated that the
lower the bias voltage, the higher the sensitivity. Finally, we have shown
two viable methods for sensing an electrical potential. The more sensitive
approach is by applying the potential to port 2, where a change in
electrostatic spring is used to perturb the stiffness. The best sensitivity
improvement compared to the state-of-the-art mode-localized sensor is 123
times. If employed as an electrometer, the best resolution can also be
improved by 2.5 times compared to the state of the art.