# A Pull-in Based Test Mechanism for Device Diagnostic and Process Characterization

Authors: L. A. Rocha; L. Mol; E. Cretu; R. F. Wolffenbuttel; J. Machado da Silva
Journal: VLSI Design
 (2008)

---

## Abstract

A test technique for capacitive MEMS accelerometers and electrostatic microactuators, based on the measurement of pull-in voltages and resonance frequency, is described. Using this combination of measurements, one can estimate process-induced variations in the device layout dimensions as well as deviations from nominal value in material properties, which can be used either for testing or device diagnostics purposes. Measurements performed on fabricated devices confirm that the 250 nm overetch observed on SEM images can be correctly estimated using the proposed technique.


---

## Body

## 1. Introduction

The use of microsystems, containing sensors and actuators,
in commercial products calls for simple and automated diagnostics and fault detection mechanisms. Despite the already wide use of these microsystems, their testing techniques tend to be complex and cost-intensive. If on one hand, the
complex nature of microsystems, where multiple energy domains interact at the microlevel,
imposes an additional level of difficulty, then on the other hand, the large number
of fabrication processes available, each of them with their own unique
characteristics, makes it difficult to have a process independent test
mechanism.Capacitive accelerometers are among
the most commercially available microelectromechanical systems (MEMS), and
several built-in self-test (BIST) schemes have been proposed to ensure high-reliability
levels [1–4]. In [1], a simple functional test is performed. An electrical test
signal is used to stimulate the device, and movement is detected. As this is a
very simple functional test, it does not fully
evaluates the device, making it unsuitable for more advanced tasks such
as diagnostic and manufacturing tests. In [2–4], differential BIST approaches are
presented. In [2, 3], the voltage that results from the self-testing operation
is used as an indication of asymmetries between capacitors caused by
fabrication defects or operational failures, while in [4] a dedicated test
signal is applied, and the capacitive readout signal holds information of
eventual asymmetries or defects occurring in the device.A more recent approach [5] proposes
the use of the current consumed during a pull-in transition to detect possible
failures. Pull-in [6] is a unique feature of capacitive MEMS devices that can
provide detailed information about their characteristics. Since the
electrostatic force due to a field is inversely proportional to the square of
the deflection, and the restoring force of the beam is, to a first
approximation, linear with deflection, an unstable system results in case of a
deflection, v, beyond a critical
value, vcrit. The pull-in
voltage, Vpi, is defined
as the voltage that is required to obtain this critical deflection and depends
mainly on dimensions, residual stress level, and design, which makes it ideal
to characterize structural materials in surface micromachining processes [7, 8].
Unlike the case of the comb drive, which is based on area-varying capacitors,
the design of most electrostatic actuators relies on gap-width varying capacitors, and the pull-in phenomenon has
to be considered [9]. Pull-in causes the displacement range due to electrostatic
forces to be limited to 1/3 of the gap between the electrodes, in case of a
motion perpendicular to the capacitor plate orientation.In this paper, the use of the pull-in
voltage as a test parameter is proposed. Furthermore, when pull-in voltage
measurements are combined with the measurement of the resonance frequency (a single
measurement is needed), the fabrication process nonidealities like overetching
and process asymmetries can be estimated. Consecutive pull-in voltage
measurements can be used to make accurate diagnostics as well as to perform
electronic calibrations.This paper is organized as follows. Section2 introduces the pull-in voltage, and in Section 3 the
details of the test mechanism are discussed. In Section 4, the test method is evaluated,
and a discussion of the experimental results is performed. Finally, in Section 5 some conclusions are presented.


## 2. Pull-in Voltage

The simplest symmetric
micromechanical system suitable for studying the pull-in voltage is composed of
three electrodes, one movable and connected to a suspension beam with a certain
spring constantk
(see Figure 1(a)),
and the other two fixed on a rigid supporting substrate. This is often the case
of capacitive accelerometers, which have separate electrodes for sensing and
actuation.Sketch of the basic device with (a) ideal conditions and (b) with overetch and asymmetries.
(a)(b)For a given applied
voltage, global stable equilibrium in the microsystem under analysis occurs, if
the second derivative of the potential energy of the system with respect to
deflection is positive:∂2Up/∂x2>0. Thus, the pull-in voltage (Vpi) results from ∂2Up/∂x2=0 and is determined by the beam material,
the beam dimensions, residual stress, and the electrodes dimensions
(electrostatic energy). The residual stress should not affect Vpi, and therefore the beam
should be suspended using folded tethers at each end [10]. This approach
ensures that the built-in strain energy component caused by longitudinal stress
is negligible.Due to the symmetry of the structure,
three pull-in voltages can be defined as shown in Figure1(a): asymmetric-right
(Vpr), asymmetric-left (Vpl), and symmetric (Vps). Assuming ideal
conditions, analytical expressions for the three pull-in voltages can be found
[6]: (1)Vpr=Vpl=827d03kε0wl,Vps=12d03kε0wl, where d0 is the capacitor initial gap, k is
the mechanical spring, ε0 = 8.8546 × 10−12
is the air permittivity, and w and l are the capacitor plate width and length, respectively.If nonideal process conditions are
now considered (see Figure1(b)) like overetching [11], capacitor gap mismatch,
and Young’s modulus (E) value deviations,
the pull-in voltage values will vary, making them suitable to estimate the nonidealities and to be used as a diagnostic mechanism.


## 3. Test Algorithm

The pull-in phenomenon is an intrinsic property of actuated microelectromechanical
systems, and therefore it can be used as a test mechanism. Fabricated devices
often exhibit smaller dimensions than the actually designed ones (e.g., due to overetching).
Overetching can be considered uniform along the microfabricated device [11],
which means that all layout dimensions will be affected by the same parameter α. This will have a uniform effect on
all three pull-in voltages. Small gap mismatches (a few nm) are also observed
in fabricated devices. In this case, the gap mismatch (β) will affect differently the three
pull-in voltages and consequently, it becomes easy to estimate β from the differences between Vpl and Vpr. The parameter α is more difficult to
estimate, because there is an extra unknown parameter: the Young’s modulus—its average bulk value is known, but it can show
large deviations. If we introduce a new measurement, the resonance frequency,
both α and E can be estimated, and a clear description of the mechanical device is achieved.A flow chart of the proposed scheme
is shown in Figure2. After this initial estimation, the parameters susceptible
of changing with aging and device operations are the Young’s modulus and the gap mismatch [12],  while the structure
dimensions remain constant. This implies that consequent pull-in measurements
are an excellent diagnostic parameter. Since the sensitivity of the device is
known, test signals can also be applied to the actuation capacitors to
calibrate the full system (device plus readout electronics).Figure 2
Flow chart to estimateα, β, and E from measurements.One disadvantage of the
proposed technique is that it relies on very accurate device models that can
predict the device electromechanical behavior. These models have to incorporate
all the nonidealities existing in the microdomain, like capacitor fringe fields
and residual stress, which makes modeling one of the critical parts of the proposed
test scheme.


## 4. Evaluation of the Method

Accelerometers fabricated within the
Bosch epipoly process [13] were used to evaluate the proposed test scheme. A
drawing of the device being used is depicted in Figure 3. Several sets of
electrodes are used for sensing while two sets of differential capacitors are
used for actuation (left and right).Figure 3
Drawing of the accelerometer.


### 4.1. Mechanical Domain

The mechanical spring of the structure is
composed of 4 folded beams. Assuming that the trusses joining the folded-beam
segments are rigid, an approximate analytical expression fork can be found [14]: (2)k(α)=46EI(L+2α)3=2Eh(b−2αL+2α)3, where I=h(b−2α)3/12 is the moment of inertia of the beams, E is
the Young’s modulus, α is the overetch parameter, and h, b, and L are the thickness, width and
length, respectively, of each beam. A finite-element model (FEM) reveals that
this expression overestimates in 4% the actual mechanical spring. This
difference is not significant and can be explained by the fact that the
analytical model does not consider the bending of the small beam joining the
two beams of the folded beam topology.


### 4.2. Electrical Domain

The device under study has 12
actuation capacitors. The total electrostatic energy can be written as
(neglecting fringe fields):(3)Uelect(α)=1212C(α)V2=1212ε0w(l−2α)d+2αV2, where d is the capacitor gap distance, w (thickness of the mechanical layer) is the capacitor width, and l is the capacitor length. The analytical (3) does not take into
account the effect of fringe fields, and since in electrostatically actuated MEMS
these are very difficult to quantify, numerical methods (FEM) have been used
instead. Capacitive FEM simulations for changing α were computed and compared with (∂C/∂d)(α).
Comparative results are shown in Figure 4, and a small deviation between models
can be noticed. The analytical model underestimates the effect of the fringe
fields (about 14%). These results prove that fringe-fields contribute to the
total electrostatic force and therefore cannot be neglected. The electrostatic
force used for the computation of the pull-in voltages is (4)Felect=1.14∂Uelect(α)∂d.Figure 4
Comparison between FEM model and analytical models.


### 4.3. Fabricated Devices

The fabricated accelerometers (see Figure5) are composed of four folded springs, 340 μm long and 3 μm wide (layout dimensions), connected
to two rigid central bars of about 1 mm long. Parallel-plate capacitors with a 2-μm gap are used for actuation. The displacement measurement
involves sensing the changes of various sets of differential capacitors. The main device layout parameters and bulk material properties are shown in Table 1.Table 1
Main nominal
parameters of the device (layout dimensions and bulk material mean values).
ParameterValueSpring length (l)340μmSpring width (b)3μmMechanical layer thickness (h)10.6μmCapacitor length (l)282μmCapacitor width (w)10.6μmCapacitor gap (d)2μmYoung’s modulus (E)163 GPa (Poly-Si)Density (ρ)2.5 g cm−3Figure 5
Fabricated
device.


### 4.4. Experimental Results

A batch of 16 devices, fabricated on the same run, was used in the
experiments. The devices are encapsulated in vacuum which facilitates the measurement of the resonance frequency. The resonance frequency is determined by acquiring the devices free oscillations followed by a FFT (see Figure6). The
pull-in voltages are retrieved from displacement measurements (see Figure 7) performed while increasing the actuation voltage from zero until an abrupt
change is detected. The voltage at which this abrupt change is detected corresponds
to the pull-in voltage.Resonance frequency measurements.Figure 7
Pull-in measurements.In case of nonfunctional devices, no displacement is observed when
actuation voltages are applied. Stiction is detected in devices showing full
displacement with no voltage applied.The measured pull-in voltages showed large deviations (both for
asymmetric and symmetric actuation) from device to device, while the deviations
between resonance frequencies were small (ranging from 2700 till 2760 Hz). The
experimentally measured values for three of the devices are shown in Table2.
Figure 8 shows the histograms of several measurements that have been performed.Table 2
Pull-in and resonance frequency measured values.
ParameterValueDevice 1Asymmetric right-Vpr3.665 VAsymmetric left-Vpl3.935 VSymmetric-Vs4.495 VResonance frequency-f02700 HzDevice 2Asymmetric right-Vpr3.781 VAsymmetric left-Vpl3.955 VSymmetric-Vs4.70 VResonance frequency-f02730 HzDevice 3Asymmetric right-Vpr3.751 VAsymmetric left-Vpl3.911 VSymmetric-Vs4.679 VResonance frequency-f02720 HzHistogram of the pull-in voltages and resonance frequency measurements.After applying the algorithm of Figure2 to the set of values retrieved from the measurements, the parameters α, β, and E are estimated.
The analytical expressions introduced in the previous subsections are used for the computation of the pull-in voltages. Table 3 presents the estimated values of α, β, Young’s modulus, resonance frequency, and pull-in voltages regarding the devices whose measurements are shown in Table 2. Figure
9 shows the histograms of the estimated parameters from all the tested devices.
The symmetric pull-in does not contribute with extra information but can be
used to confirm the estimated values for α, β, and E.Table 3
Estimatedα, β, and
E values and simulated pull-in voltages and resonance frequency.
DeviceαβE1253 nm60 nm142.8 GPa2257 nm38 nm148 GPa3255 nm35 nm145.2 GPaDeviceVprVplVsf013.665 V3.934 V4.482 V2700 Hz23.782 V3.956 V4.693 V2735 Hz33.750 V3.910 V4.665 V2716 HzHistograms of estimated overetch, mismatch, and Young’s modulus obtained from the tested devices.


### 4.5. Method Verification

The values estimated from the measurements of the 16 devices
are in very good agreement (α and E estimated values
present a very low standard deviation), which was expected since they all were
fabricated in the same run.  However, it is interesting to notice that while the
estimated values for α and E have a very low standard deviation,
the estimated mismatch (β) values exhibit a large standard deviation. Since the
deviations on the pull-in voltages are caused by gap mismatches (β) and these have influence on device
performance, the correct mismatch estimation is one of the strong points of the
proposed test approach. The observed mismatches have to do with lateral
gradient stresses that often are neglected, or with overetch asymmetries that
originate small deviations in the gaps.In order to verify the estimated
average values,  some devices were observed using a scanning electron microscope
(SEM). Two illustrative SEM images are shown in Figure10. These SEM images
reveal that the devices present an overetching very close to the average one
obtained after the test method being proposed. This very good agreement proves
that pull-in voltage measurements can be used to accurately estimate process
deviations and device performance.SEM photographs of (a) a folded beam and (b) capacitor gap at the stopper.
(a)(b)


### 4.6. Discussion

Like several other MEMS test techniques, the proposed test
scheme uses a variable electrical signal to stimulate the device, and the device response can be used to obtain very important device information and most of all to obtain confidence on device functionality (a damaged device will present no pull-in behavior). So far, it has been shown that with this technique the device performance can be characterized but that is not enough to
achieve a full BIST technique. Another important parameter in production is testing time. For the measurements
presented in this paper, time was not an issue and the pull-in measurements
ranged from several seconds to a few minutes (depending on the resolution of
the increasing voltage steps used). The minimum achievable test time is
directly connected to the dynamics of the device, that is, as the pull-in
voltage assumes a quasistatic behavior, the time between steps must guarantee
that the device gets to a stable state. Therefore, if a window of 200 mV is
considered around the pull-in voltage (the actuation voltages does not start at
zero but at a value close to the pull-in expected voltage), with a step
resolution of  1 mV and a settling time below 10 milliseconds, a pull-in voltage measurement would require around 2 seconds.In order to achieve a full BIST technique, on-chip test
circuits must be integrated for generating the necessary electrical signals for
actuating the device and analyzing the responses. A very simple diagnostic
mechanism can be the storage of the initial pull-in voltages on-chip, so that
they can be compared with consecutive pull-in voltage measurements. Pull-in
voltage deviations from the original values can be used for diagnostic purposes.
Before proposing an on-chip test method, long-term measurements must be
performed to check and correlate known failure mechanisms with long-term
pull-in deviations. First experimental results from the proposed test mechanism
prove that the differences between expected pull-in voltages (using layout
dimensions) and measured pull-in voltages can be explained from fabrication
(overetch, gap mismatches, and Young’s modulus deviations). We expect that
long-term measurements will be able to give us information on device failures
through shifts in the pull-in voltages, and it is likely that different failure modes will present different pull-in deviations in time.Another advantage of this technique is the fact that it can be used to electrically calibrate an accelerometer. Normal accelerometer calibration (as an example we consider a ±1 g accelerometer) is done by applying a 1 g acceleration followed by a −1 g acceleration (putting the sensitivity axis along the earth gravity field) while checking the response. Usually, this is done manually and it is not a good solution for remotely placed sensors or sensors with difficult access. Since a very accurate model is obtained with the proposed scheme, the actuation voltages that give the same response as a ±1 g can be computed and used to electrically calibrate
the sensor.


## 4.1. Mechanical Domain

The mechanical spring of the structure is
composed of 4 folded beams. Assuming that the trusses joining the folded-beam
segments are rigid, an approximate analytical expression fork can be found [14]: (2)k(α)=46EI(L+2α)3=2Eh(b−2αL+2α)3, where I=h(b−2α)3/12 is the moment of inertia of the beams, E is
the Young’s modulus, α is the overetch parameter, and h, b, and L are the thickness, width and
length, respectively, of each beam. A finite-element model (FEM) reveals that
this expression overestimates in 4% the actual mechanical spring. This
difference is not significant and can be explained by the fact that the
analytical model does not consider the bending of the small beam joining the
two beams of the folded beam topology.


## 4.2. Electrical Domain

The device under study has 12
actuation capacitors. The total electrostatic energy can be written as
(neglecting fringe fields):(3)Uelect(α)=1212C(α)V2=1212ε0w(l−2α)d+2αV2, where d is the capacitor gap distance, w (thickness of the mechanical layer) is the capacitor width, and l is the capacitor length. The analytical (3) does not take into
account the effect of fringe fields, and since in electrostatically actuated MEMS
these are very difficult to quantify, numerical methods (FEM) have been used
instead. Capacitive FEM simulations for changing α were computed and compared with (∂C/∂d)(α).
Comparative results are shown in Figure 4, and a small deviation between models
can be noticed. The analytical model underestimates the effect of the fringe
fields (about 14%). These results prove that fringe-fields contribute to the
total electrostatic force and therefore cannot be neglected. The electrostatic
force used for the computation of the pull-in voltages is (4)Felect=1.14∂Uelect(α)∂d.Figure 4
Comparison between FEM model and analytical models.


## 4.3. Fabricated Devices

The fabricated accelerometers (see Figure5) are composed of four folded springs, 340 μm long and 3 μm wide (layout dimensions), connected
to two rigid central bars of about 1 mm long. Parallel-plate capacitors with a 2-μm gap are used for actuation. The displacement measurement
involves sensing the changes of various sets of differential capacitors. The main device layout parameters and bulk material properties are shown in Table 1.Table 1
Main nominal
parameters of the device (layout dimensions and bulk material mean values).
ParameterValueSpring length (l)340μmSpring width (b)3μmMechanical layer thickness (h)10.6μmCapacitor length (l)282μmCapacitor width (w)10.6μmCapacitor gap (d)2μmYoung’s modulus (E)163 GPa (Poly-Si)Density (ρ)2.5 g cm−3Figure 5
Fabricated
device.


## 4.4. Experimental Results

A batch of 16 devices, fabricated on the same run, was used in the
experiments. The devices are encapsulated in vacuum which facilitates the measurement of the resonance frequency. The resonance frequency is determined by acquiring the devices free oscillations followed by a FFT (see Figure6). The
pull-in voltages are retrieved from displacement measurements (see Figure 7) performed while increasing the actuation voltage from zero until an abrupt
change is detected. The voltage at which this abrupt change is detected corresponds
to the pull-in voltage.Resonance frequency measurements.Figure 7
Pull-in measurements.In case of nonfunctional devices, no displacement is observed when
actuation voltages are applied. Stiction is detected in devices showing full
displacement with no voltage applied.The measured pull-in voltages showed large deviations (both for
asymmetric and symmetric actuation) from device to device, while the deviations
between resonance frequencies were small (ranging from 2700 till 2760 Hz). The
experimentally measured values for three of the devices are shown in Table2.
Figure 8 shows the histograms of several measurements that have been performed.Table 2
Pull-in and resonance frequency measured values.
ParameterValueDevice 1Asymmetric right-Vpr3.665 VAsymmetric left-Vpl3.935 VSymmetric-Vs4.495 VResonance frequency-f02700 HzDevice 2Asymmetric right-Vpr3.781 VAsymmetric left-Vpl3.955 VSymmetric-Vs4.70 VResonance frequency-f02730 HzDevice 3Asymmetric right-Vpr3.751 VAsymmetric left-Vpl3.911 VSymmetric-Vs4.679 VResonance frequency-f02720 HzHistogram of the pull-in voltages and resonance frequency measurements.After applying the algorithm of Figure2 to the set of values retrieved from the measurements, the parameters α, β, and E are estimated.
The analytical expressions introduced in the previous subsections are used for the computation of the pull-in voltages. Table 3 presents the estimated values of α, β, Young’s modulus, resonance frequency, and pull-in voltages regarding the devices whose measurements are shown in Table 2. Figure
9 shows the histograms of the estimated parameters from all the tested devices.
The symmetric pull-in does not contribute with extra information but can be
used to confirm the estimated values for α, β, and E.Table 3
Estimatedα, β, and
E values and simulated pull-in voltages and resonance frequency.
DeviceαβE1253 nm60 nm142.8 GPa2257 nm38 nm148 GPa3255 nm35 nm145.2 GPaDeviceVprVplVsf013.665 V3.934 V4.482 V2700 Hz23.782 V3.956 V4.693 V2735 Hz33.750 V3.910 V4.665 V2716 HzHistograms of estimated overetch, mismatch, and Young’s modulus obtained from the tested devices.


## 4.5. Method Verification

The values estimated from the measurements of the 16 devices
are in very good agreement (α and E estimated values
present a very low standard deviation), which was expected since they all were
fabricated in the same run.  However, it is interesting to notice that while the
estimated values for α and E have a very low standard deviation,
the estimated mismatch (β) values exhibit a large standard deviation. Since the
deviations on the pull-in voltages are caused by gap mismatches (β) and these have influence on device
performance, the correct mismatch estimation is one of the strong points of the
proposed test approach. The observed mismatches have to do with lateral
gradient stresses that often are neglected, or with overetch asymmetries that
originate small deviations in the gaps.In order to verify the estimated
average values,  some devices were observed using a scanning electron microscope
(SEM). Two illustrative SEM images are shown in Figure10. These SEM images
reveal that the devices present an overetching very close to the average one
obtained after the test method being proposed. This very good agreement proves
that pull-in voltage measurements can be used to accurately estimate process
deviations and device performance.SEM photographs of (a) a folded beam and (b) capacitor gap at the stopper.
(a)(b)


## 4.6. Discussion

Like several other MEMS test techniques, the proposed test
scheme uses a variable electrical signal to stimulate the device, and the device response can be used to obtain very important device information and most of all to obtain confidence on device functionality (a damaged device will present no pull-in behavior). So far, it has been shown that with this technique the device performance can be characterized but that is not enough to
achieve a full BIST technique. Another important parameter in production is testing time. For the measurements
presented in this paper, time was not an issue and the pull-in measurements
ranged from several seconds to a few minutes (depending on the resolution of
the increasing voltage steps used). The minimum achievable test time is
directly connected to the dynamics of the device, that is, as the pull-in
voltage assumes a quasistatic behavior, the time between steps must guarantee
that the device gets to a stable state. Therefore, if a window of 200 mV is
considered around the pull-in voltage (the actuation voltages does not start at
zero but at a value close to the pull-in expected voltage), with a step
resolution of  1 mV and a settling time below 10 milliseconds, a pull-in voltage measurement would require around 2 seconds.In order to achieve a full BIST technique, on-chip test
circuits must be integrated for generating the necessary electrical signals for
actuating the device and analyzing the responses. A very simple diagnostic
mechanism can be the storage of the initial pull-in voltages on-chip, so that
they can be compared with consecutive pull-in voltage measurements. Pull-in
voltage deviations from the original values can be used for diagnostic purposes.
Before proposing an on-chip test method, long-term measurements must be
performed to check and correlate known failure mechanisms with long-term
pull-in deviations. First experimental results from the proposed test mechanism
prove that the differences between expected pull-in voltages (using layout
dimensions) and measured pull-in voltages can be explained from fabrication
(overetch, gap mismatches, and Young’s modulus deviations). We expect that
long-term measurements will be able to give us information on device failures
through shifts in the pull-in voltages, and it is likely that different failure modes will present different pull-in deviations in time.Another advantage of this technique is the fact that it can be used to electrically calibrate an accelerometer. Normal accelerometer calibration (as an example we consider a ±1 g accelerometer) is done by applying a 1 g acceleration followed by a −1 g acceleration (putting the sensitivity axis along the earth gravity field) while checking the response. Usually, this is done manually and it is not a good solution for remotely placed sensors or sensors with difficult access. Since a very accurate model is obtained with the proposed scheme, the actuation voltages that give the same response as a ±1 g can be computed and used to electrically calibrate
the sensor.


## 5. Conclusions and Future Work

A novel technique is presented which allows estimating
overetching, mismatch, and Young’s modulus parameters in capacitive MEMS
accelerometers or electrostatic microactuators, relying on the measurement of
pull-in voltages and resonance frequency. The underlining theoretical
justification is described, and preliminary results obtained with a set of
fabricated devices show that results in very good agreement with the expected
ones can be obtained, confirming thus the validity of the used MEMS models and the
feasibility of the method as a testing technique. The method was tested within
a surface micromachining process, but it can be extended to other surface
micromachining processes or to more recent technologies like SOI-based
processes. The important features are the correctness of the device models and
a readout mechanism for pull-in voltage measurements.In the future, long-term pull-in
measurements are needed to check if pull-in deviations from the initial
measured values will give information on device’s performance deterioration
with aging. Future work includes also the identification of failure modes and the
estimation of errors due to uncertainty on the measurements and how those can
influence the correct estimation of the device parameters.



---

*Source: 283451-2008-04-06.xml*