Built-In Self Test of MEMS Accelerometers
N. Deb and R. D. (Shawn) Blanton
Department of Electrical and Computer Engineering
Carnegie Mellon University, Pittsburgh, PA 15213
email: fndeb, blantong@ece.cmu.edu
Abstract
actions of multiple energy domains that can include electrical, mechanical, optical, thermal, chemical and fluidic. Ma-
A built-in self-test technique that is applicable to symmetric
jor classes of microsystems include MicroElectroMechani-
microsystems is described. A combination of existing layout
cal Systems (MEMS), MicroOptoElectroMechanical Systems
features and additional circuitry is used to make measure-
(MOEMS), and MicroElectroFluidic Systems (MEFS). Re-
ments from symmetrically-located points. In addition to the
liable manufacture of affordable microsystems naturally re-
normal sense output, self-test outputs are used to detect the
quires the use of cost-effective test methods that distinguish
presence of layout asymmetry that are caused by local, hard-
malfunctioning systems from good ones. The multi-domain
to-detect defects. Simulation results for an accelerometer re-
nature of microsystems makes them inherently complex for
veal that our self-test approach is able to distinguish misbe-
both design and test. In addition, the growing use of mi-
havior resulting from local defects and global manufacturing
crosystems in life-critical applications such as air-bags [2],
process variations. A mathematical model is developed to an-
bio-sensors [3] and satellites [4] creates a significant need for
alyze the efficacy of the differential built-in self-test method
high reliability that cannot be achieved without the use of ro-
in characterization of a wide range of local manufacturing
bust test methods.
variations affecting different regions of a device and/or wafer.
Model predictions are validated by simulation. Specifically, it
Among currently used MEMS process technologies, sur-
has been shown that by using a suitable modulation scheme,
face micromachining [5] is widely used due to its similarity
sensitivity to etch variation along a particular direction is im-
to thin-film technology used for integrated circuits. Surface
proved by nearly 30%.
micromachining is already proven to be a technology that is
commercially viable since it has supported high-volume manufacture of MEMS devices. Example applications of this tech-
1 Introduction
nology include the digital micromirror display [6] and the ac-
The increasing need for multi-functional sensor and actua-
celerometer [7, 8]. Our work in microsystem test has therefore
tor systems that are capable of real-time interaction with both
been focussed on surface-micromachined devices. However,
electrical and non-electrical environments has led to the devel-
we believe the built-in test method described here is also ap-
opment of a very broad class of “Microsystems”. Microsys-
plicable to other process technologies.
tems are heterogeneous [1] since they are based on the inter-
The objective of manufacturing test is to identify mal This
research was sponsored by the National Science Foundation under grant no. MIP-9702678 and the MARCO Gigascale Systems Research
Center.
functioning devices in a batch of fabricated devices. Device
1
malfunction is caused by various failure sources in the manu-
suitable for manufacturing test of MEMS is needed. Previous
facturing process that include but are not limited to foreign
work [13]-[19] has addressed some of these problems. For
particles [9], etch variations and stiction [10, 11], each of
example, our work [13, 14, 15] describes a BIST approach
which can lead to a variety of defects. When defects cause
that samples outputs from symmetrically-located nodes of the
the device performance to go “out-of-spec”, the device is said
MEMS microstructure. Increasing observability in this way
to have failed. However, test has limitations which can cause
allows one to identify misbehavior resulting from local de-
good devices to be rejected (yield loss) and bad devices to
fects as opposed to more benign causes such as global etch
be accepted (test escape). The cost of test escape can exceed
variation.
the cost of yield loss since a device that passes traditional,
The work in this paper is a continuation of the differential
specification-based manufacturing test may fail in the field
BIST approach [13, 14, 15] and aims to build a mathemati-
with catastrophic results [12]. One of the goals of built-in self
cal model for an integrated and comprehensive approach to
test (BIST) is the prevention of field failures due to diverse
this form of BIST, when applied to an accelerometer sensor.
failure sources (see Table 1).
We show how the BIST can be used to detect the presence
Failure source
Foreign particles
Etch variations
Stiction
Curvature
Occurrence in
manufacturing
yes
yes
yes
yes
of device asymmetry caused by different types of local de-
Occurrence in
the field
yes
no
yes
yes
fects. Simulation results reveal that local defects such as particulates, local over/under-tech, and local curl-mismatch that
are not detected by standard specification-based tests, can be
detected by our BIST. In addition to self-test, our technique
Table 1: Possibility of occurrence of various failure sources
during the device lifetime.
is shown to be useful for device and wafer characterization.
Our analyses (theoretical and simulation) validate the claim
Commercially-manufactured MEMS are typically af-
that the BIST measurements can reveal significant informa-
fected by multiple failure sources. Defects caused by many of
tion about the nature of local manufacturing variations affect-
these failure sources exhibit very similar behaviors [12] which
ing different portions of a given die.
fall within the specified tolerance ranges. Since these similarbehaving defects may have widely varying stability charac-
The remaining portions of this paper are organized as
teristics, those which are unstable over time pose a reliabil-
follows. Section 2 describes a model of the capacitor-based
ity problem. Therefore, there is a need to distinguish unsta-
sensing network used in a MEMS accelerometer. Section 3
ble defects from the stable ones. BIST can identify poten-
describes the model developed for representing local manu-
tially unstable defects [13]. In addition, simultaneously ex-
facturing variations. Section 4 discusses the extent to which
isting multiple failure sources can cause misbehavior mask-
each BIST modulation scheme is sensitive to different types
ing [12] that often leads to hard-to-detect defects because the
of local manufacturing variations. The simulation results for
misbehavior due to one defect is negated by that caused by
various types of local defects and manufacturing variations are
another defect. BIST offers a way of detecting hard-to-detect
presented in section 5. In section 6, device and wafer charac-
defects [13]. Since the complexity and range of applications
terization using BIST is described. Finally, in section 6, con-
of MEMS have grown, off-chip testing has increased in cost,
clusions are drawn based on how sensitive the various modes
which in turn has enhanced the need for on-chip self-test capa-
of BIST are to specific local manufacturing variations in the
bility. Current commercial BIST techniques are similar to the
accelerometer sensor.
one described in [17] and commonly require calibration and
therefore are not useful for manufacturing test. Hence, BIST
2
2 Capacitor Network Model
are manufacture-enhancing features used to achieve the same
In this section, we develop an electrical model that represents
level of etching for the spring beams and the combdrive fin-
the network of combdrive capacitors for an accelerometer sen-
gers. The combdrives are centers of electromechanical inter-
sor. Figure 1 illustrates the topology of a CMOS-MEMS ac-
action. The sense combdrives convert mechanical displace-
celerometer sensor [23]. The sensor mass is largely provided
ment into electrical voltage/current while the actuation comb-
11
00
00
11
00
11
00
11
D1
drives convert electrical voltage into mechanical force.
11
00
00
11
00
11
00
11
D3
1
0
1
0
B1
D2
D5
B3
0
1
1
0
0
1
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
D6
The sense operation is based on the fully differential sens-
B2
1
0
1
0
D4
ing technique [24] and the sense signals are tapped from the
D7
sense combdrives. As shown in Figure 1, the sensor has four
B4
identical, symmetrically-located sense combdrives.
D8
A combdrive is electrically represented as a pair of capacC1
itors connected in series (see magnified view of combdrive
C2
fingers in Figure 1). Note that each combdrive capacitor, in
actuation
combdrive
00
11
11
00
00
11
00
11
00
11
reality, is an array of capacitors connected in parallel. The
00
11
11
00
00
11
M6
00
11
11
00
00
11
00
11
and ‘t’ in Figures 1 and 2 denote left, right, bottom and top,
sense
combdrives
shuttle
00
11
11
00
00
11
00
11
00
11
combdrives is shown in Figure 2(a). The symbols ‘l’, ‘r’, ‘b’,
00
11
11
00
00
11
00
11
rt
lt
circuit network constituted by the capacitors in the four sense
F3
lb
respectively. The subscripts ‘1’ and ‘2’ denote the two capac-
00
11
11
00
00
11
00
11
00
11
rb
itors of a differential pair within a combdrive. For example,
C1lb is a lumped capacitor that represents the top array of capacitors of the left bottom sense combdrive. For shuttle mo-
11
00
00
11
00
11
00
11
00
11
00
11
00
11
11
00
00
11
00
11
actuation
combdrive
tion in the
+Y
direction (see Figure 1), all capacitors with a
subscript ‘1’ decrease while all capacitors with a subscript ‘2’
Y
increase. The opposite changes in the capacitance values take
beam
X
place for motion in the Y direction.
dummy finger
1
0
0
1
0
1
1
0
0
1
0
1
1
0
0
1
0
1
0
1
0
1
2.1 Modulation Schemes
1
0
0
1
0
1
0
1
0
1
1
0
0
1
The capacitors in the sensor of Figure 1 can be represented
1
0
0
1
00
11
11
00
00
11
00
11
00
11
11
00
00
11
00
11
by the equivalent network shown in Figure 2(a) and have the
anchor
following generic characteristics:
Figure 1: Topology of an accelerometer sensor.
1. Each pair of electrically-connected capacitors constitutes
by the rigid shuttle. The flexible serpentine springs that at-
a potential divider circuit. Voltage is applied to each
tach the shuttle to the anchors are made of beams (B1 -B4 in
free or modulation node of each capacitor and voltage
Figure 1). The combdrives consist of interdigitated beam-like
is sensed from their common or sense node. Thus, each
structures (called “fingers”, examples of which are M6 and F3
capacitor pair has one sense node and two modulation
in Figure 1). Fingers attached to the shuttle are movable while
nodes.
those attached to anchors are fixed. There are other beam-like
2. Each pair of capacitors must have different signals ap-
structures called dummy fingers (D1 -D8 in Figure 1) which
3
plied to their modulation nodes in order for the sense
balanced modes. For our case of n = 4, only 4 modes out of a
node to be sensitive to changes in the capacitances.
total of 16 need to be examined.
Usually the modulation signals are applied in opposite
phases.
2.2 Modal Dependence Relations
Therefore each capacitor pair constituting a
combdrive can be viewed as a
dipole1.
The sense voltage outputs VA and VB , tapped from nodes A
and B ((see Figure 2(a)) respectively, depend on changes in
3. The number of modulation states associated with each
mP =
2
dipole is given by the permutation
m(m
the capacitors of the electrical network shown in Figure 2(a)-
1),
(c). In the fully-differential sensing topology [23], the change
where m is the number of phases of the applied mod-
in the voltage difference, VB VA , is measured by a differential
ulation signal, or in general, the number of modulation
amplifier (see Figure 3) and is of most interest. For the sake
signals. So, if one of m modulation signals is applied
to one modulation node, then there is a choice of m
of convenience, the gain of the amplifier has been assumed to
1
be unity. Here, based on electrical network theory, we derive
signals for the other modulation node.
B
C pB
Note as an aside that these characteristics are applicable to
VB − VA
−
other networks shown in Figure 2 as well.
A
C pA
For the typical choice of m = 2, each dipole can have two
Figure 3: Schematic of a differential amplifier used to measure changes in sense voltage outputs of sense combdrives.
possible states, denoted by symbols 0 and 1. Every dipole has
two capacitors, one labeled with a ‘1’ in the subscript and the
other with a ‘2’ in the subscript. If the
+
+
a relation that captures the dependence of VB
phase (Vmp) of the
VA on changes
modulation signal is applied to the free node of the capacitor
in the network capacitances. All three configurations of fully
phase (Vmn ) of the modulation
differential sensing topologies shown in Figure 2(a)-(c) have
signal, naturally, is applied to the free node of the capacitor
been known to be usable. We have arbitrarily focussed on the
with subscript ‘2’, and the dipole is said to be in state 0. By
diagonal configuration (see Figure 2(a)) since it is frequently
swapping the modulation signals, the dipole is put in state 1.
used [24, 23] and because the fundamental conclusions we
with subscript ‘1’, then the
arrive at are largely independent of the sense configuration
A capacitor network consisting of n dipoles can be in
chosen.
2n modes. However, for the purpose of our analysis, only n
Using the topology of the dipole capacitances (C ji , j
suitably-chosen modes are needed for a system of n dipoles.
f1 2g, i 2 flt rb lb rt g) illustrated in Figure 2(a) and the par;
How these n modes are selected is explained next. Since
1
;
nC
n=2
2
CAtotal
=
CBtotal
where j 2 f1; 2g; i 2 flt ; rbg
(1)
∑ C ji + Cp
where j 2 f1; 2g; i 2 flb; rt g
(2)
A;
=
B;
i; j
However, if each dipole capacitance C ji changes to C0ji , e.g.,
number of 0’s and 1’s) and the all-zero mode (or its mirror,
the all-one mode) are worth analysis because they subsume
due to shuttle displacement), then the corresponding altered
information captured by the unbalanced modes. Therefore,
1A
∑ C ji + Cp
i; j
balanced modes (i.e., modes with equal
we only need to choose n
;
capacitances at nodes A and B are found to be:
modes out of the total of 2n
are uniquely interesting. This number can be further reduced
since only the
;
asitic capacitances (C pA and C pB ) shown in Figure 3, the total
changing the state of each dipole will only change the polarity of the sense output, only 2n
2
values would be:
CA0 total
1 suitable modes from the set of
CB0 total
dipole is defined as an object with opposite polarities at its two ends.
=
∑ C0ji + Cp
where j 2 f1; 2g; i 2 flt ; rbg
(3)
∑ C0ji + Cp
where j 2 f1; 2g; i 2 flb; rt g
(4)
A;
i; j
=
B;
i; j
4
S4
S1
S4
S1
C1rt
C1lt
A
S4
S1
C1rt
C1lt
C1lt
A
B
C2rt
C2lt
C2rt
C2lt
S2
S3
S2
S5
S8
S5
C1lb
C1rb
C2lb
C2rb
S6
S7
A
B
C1lb
C1rb
C2lb
C2rb
S6
S2
S8
S5
(a)
C2rt
C2lt
S3
S2
S8
C1lb
S8
C1rb
B
C2lb
S7
C1lt
A
C2lt
S3
S1
C1rt
C1rb
C2rb
C2rb
S6
S7
(b)
S7
(c)
(d)
Figure 2: Potential divider circuit formed by a set of sense combdrive capacitors in the fully-differential sensing scheme in (a)
diagonal, (b) vertical, (c) horizontal sense configurations and (d) its simplified version, the basic differential sensing scheme.
Nodes A and B are sense nodes while S1 -S8 are modulation nodes.
Similarly, we have for ∆VB :
We define the change in dipole capacitance as
∆C ji = C0ji
C ji ; where j 2 f1; 2g; i 2 flt ; rb; lb; rt g
∆VB = ∑
(5)
i; j
If we impose the condition that the sum of the capacitances in
=
∑
i
a dipole is constant (for small displacements), then we have
=
the following:
0 + C0 ) ∆C1i = ∆C2i ; where i 2 flt ; rb; lb; rt g (6)
C1i + C2i = C1i
2i
∆C jiV ji
; where j 2 f1; 2g; i 2 flb; rt g
CBtotal
∆C1i (V1i V2i )
; where i 2 flb; rt g
CBtotal
∆C1lb (V1lb
V2lb ) + ∆C1rt (V1rt
CBtotal
V2rt )
(12)
Different modulation signals should be applied to the two
Substituting Equation 6 into Equations 1-4 leads to the con-
free nodes of each dipole. As already mentioned, the two
clusion that
phases of the modulation signal are Vmp and Vmn . Hence, if
CAtotal
0
=C
(7)
V1lt
CBtotal
0
=C
(8)
hand, if V1lt
Atotal
Btotal
= Vmp ; V2lt = Vmn ,
use Slt
Let V ji ( j 2 f1; 2g, i 2 flt ; rb; lb; rt g) be the voltage applied to
=
then dipole lt is in state 0. On the other
= Vmn ; V2lt = Vmp ,
=
1 and Slt
then dipole lt is in state 1. We
1 to represent states 0 and 1, respec-
tively, for the dipole lt. Similar reasoning applies for all other
the modulation node of dipole capacitor C ji (see Figure 2(a)).
dipoles. In general, the variable Si (i 2 flt ; rb; lb; rt g) is used
In our analysis V ji
to represent the state of dipole i such that Si
2 fVmp Vmn g but other values of V ji are
;
also possible. Let ∆VA and ∆VB be the changes in VA and VB ,
is in state 0 and Si
respectively, caused by the changes in the dipole capacitances.
Equations 11 and 12 reduce to:
Hence, from Kirchoff’s voltage law, we derive the following
relation:
∆VA = ∑
i; j
C0jiV ji
C0
Atotal
"
C jiV ji
∑ CA
i; j
total
=
∑
i; j
C0ji
CA0 total
where j 2 f1; 2g; i 2 flt ; rbg
C ji
V ji ;
CAtotal
(9)
=
∆VB
=
1, if dipole i
1, if dipole i is in state 1. Therefore,
Slt ∆C1lt + Srb ∆C1rb
(Vmp
CAtotal
Slb ∆C1lb + Srt ∆C1rt
(Vmp
CBtotal
Vmn )
(13)
Vmn )
(14)
Subtracting Equation 13 from Equation 14, we get
∆VB
Vmp
Equation 5, we have:
i; j
∆VA
#
Substituting Equation 7 into Equation 10 and then using
∆VA = ∑
=
=
∆C jiV ji
; where j 2 f1; 2g; i 2 flt ; rbg
CAtotal
∆VA
Vmn
=
Slb ∆C1lb + Srt ∆C1rt
CBtotal
Slt ∆C1lt + Srb ∆C1rb
(15)
CAtotal
Equation 15 is the unified expression for all modes for the
(10)
network configuration shown in Figure 2(a).
Substituting Equation 6 into Equation 10, we get:
Equation 11 or 12 is also applicable to basic differential
∆C1i (V1i V2i )
; where i 2 flt ; rbg
CAtotal
sensing scheme used in devices manufactured from single-
∆VA = ∑
i
=
∆C1lt (V1lt
V2lt ) + ∆C1rb (V1rb
CAtotal
V2rb )
conductor processes such as MUMPS [28]. Since the basic
differential case uses only one diagonal (sense node A for
(11)
example) of the network in Figure 2(a), it is reduced to that
5
shown in Figure 2(d). The change in voltage of the sense
crostructures forming the capacitors C ji .
node A for this simplified case is obtained by substituting
V1lt
= Vmp ,
V1rb
= Vmn ,
and V2lt
= V2rb =
Vmp +Vmn
2
The parameter λ(α; β) can be conveniently expressed as
in Equa-
λ(α; β) = k00 + X (α) + Y (β) + Z (α; β)
(18)
tion 11, which then reduces to:
∆VA
Vmp Vmn
=
∆C1lt ∆C1rb
2CAtotal
where k00 is a constant, X (α) includes only and all terms
(16)
that are functions of α alone, Y (β) includes only and all terms
It is evident from Equation 16 that ∆VA is non-zero only
when ∆C1lt
that are functions of β alone, and Z (α; β) includes only and
6= ∆C1rb , a condition that implies the presence
all cross-product terms (i.e., functions of both α and β). The
of an asymmetry/mismatch between the two combdrives con-
form of Equation 18 reveals that k00 = 1 if the nominal values
cerned.
λ(0; 0) = 1; X (0) = Y (0) = Z (0; 0) = 0 hold true.
Substituting Equation 18 in Equation 17 and assuming
that the area A is rectangular and equals (α2
3 Local Manufacturing Variations
we have:
Local manufacturing variation is the uncorrelated (mismatch)
C ji =
component of manufacturing variations and is strongly depen-
(α2
= C0
dent on device layout attributes (length, width, orientation,
C0
α1 )(β2 β1 )
k00 +
surrounding topography, etc.) [29]. In contrast, global man-
+C0
ufacturing variation is the correlated component of manufac-
1
α2
Z αZ2
β2
α 1 β1
Z
α1
α2
α1
α1 )(β2
β 1 ),
[k00 + X (α) + Y (β) + Z (α; β)] dαdβ
Z
β1 )
β2
α 2 Z β2
α1
Z
1
X (α)dα +
1
(α2
α1 )(β2
β1
β1
β2
β1
Y (β)dβ
Z (α; β)dαdβ
(19)
turing variations and is therefore independent of device layout
Using the Mean Value Theorem of calculus, it can be shown
attributes. We have focussed on local manufacturing variation
that there exists a location (α0 ; β0 ) within the rectangle area A
since our goal is to detect device asymmetry, of which mis-
such that:
X (α0 ) =
match is one of the causes.
Dipole capacitance C ji depends on physical parameters
Y (β0 ) =
such as finger width, the gap between fingers, finger thick-
Z
1
α2
α1
Z β2
β1
β1
1
β2
α2
α1
X (α)dα
(20)
Y (β)dβ
(21)
ness, finger height mismatch, etc. All of these parameters may
Equations 20-21 can be used to simplify Equation 19 to:
vary over the plane of the device, meaning each is a function
C ji = C0 k00 + X (α0 ) + Y (β0 ) + Zavg
of location. In our analyses, each point in the device layout
where
is represented by the coordinates (α, β). Here, we analyze
Zavg =
the combined effect of all local manufacturing variations on
Z
1
(α2
α1 )(β2
β1 )
α 2 Z β2
α1
β1
(22)
Z (α; β)dαdβ (23)
a dependent variable (e.g., combdrive capacitance) distributed
Equation 22 captures the concept of average manufacturing
over device area A. The dimensionless variable λ(α; β) repre-
variations. Since the combdrive capacitance is distributed
sents local manufacturing variations and its impact on comb-
over an area, the combined effect of all manufacturing vari-
drive capacitance is characterized as:
ations over the area is equated to an average multiplied by the
Z
C ji = C0
dαdβ
λ(α; β)
A
A
(17)
same area. The fact that only one sense signal comes out of the
entire combdrive/dipole imposes a practical limit on resolving
In the absence of local manufacturing variations, λ(α; β) =
local variations that are internal to the combdrive. Hence, we
1; 8α; 8β, and Equation 17 reduces to the simple relation,
will use the average value (λavg ) to represent the distributed
C ji = C0 . Hence, C0 is the nominal value of C ji .
dipole capacitance. Therefore, from Equations 17 and 22, it is
evident that
When the sensor moves, C ji changes but λ(α; β) is as-
C ji
C0
sumed to remain constant for small displacements of the mi6
= λavg = k00 + X (α0 ) + Y (β0 ) + Zavg
(24)
Equation 24 indicates that the average manufacturing varia-
show how various BIST modes are sensitive (or insensitive)
tion in a quadrant can be represented by the value of λ at a
to manufacturing variations along specific directions in the
single point in the quadrant. Therefore, the point locations
device plane, we now express λ(α; β) as a two-dimensional
(α+ ; β+ )
and (α
β
;
)
power series of α and β as follows:
are assigned to the combdrives in two
λ(α; β) =
diagonally opposite quadrants, right-top (rt) and left-bottom
quadrant:
C2rt
=
C0
= λrt = k00 + X (α ) + Y (β ) + Zrt
+
X (α+ ) =
Y (β
+
)=
Zrt
+
stant coefficients. From Equation 37 and prior definitions of
(25)
=
Z
1
α2
α1
1
α2
α1
β2
Z
X (α) =
X (α)dα
(26)
Y (β) =
Y (β)dβ
(27)
β2 β1 β1
Z Z
1 β2 α 2
Z (α; β)dαdβ
A β1 α 1
Z (α; β) =
(28)
C2lb
C0
=
= λlb = k00 + X (α ) + Y (β ) + Zlb
(
α2 )
1
(
1
A
Zlb =
Z
1
α1 )
Z
α1
α2
Z
1
Zlt
=
Z
1
A
β2 Z
β1
α1
α2
X (α)dα =
α2 α1
β1
1
Y (β)dβ =
Z
α1
X ( α)dα (30)
lowing expression for C ji :
β2
C ji = C0
Y ( β)dβ (31)
= C0
= C0
= λlt = k00 + X (α ) + Y (β ) + Zlt
+
= C0
(33)
1
A
Z
A
λ(α; β)dαdβ = C0
1 ∞;∞
kmn
A m=∑
0;n=0
(α2
A
∞;∞
∑
A m=0;n=0
1
A
Z
β1 Z
β2
α2
α1
= km0 k0n
α1 )(β2
Z
β1 ) m=∑
0;n=0
kmn
∑
β1 ) m=0;n=0
kmn
α1 )(β2
α 2 Z β2
α1
Z
∞;∞
α1 )(β2
kmn αm βn dαdβ
αm βn dαdβ; [A = (α2
1
(α2
Z
α2
α1
β1
β1 )]
αm βn dαdβ
αm dα
Z
β2
β1
1
A
Z
β2 Z α 2
βn dβ
β1
α1
∑
(41)
Z ( α; β)dαdβ (34)
where umn is a constant given by:
umn =
= λrb = k00 + X (α ) + Y (β ) + Zrb
+
umn
m=0;n=0
kmn
(m + 1)(n + 1)
(α2
+1
n+1
αm
)(β2
β1n+1 )
1
(α2 α1 )(β2 β1 )
m +1
Z (α; β)dαdβ =
(42)
(35)
From Equation 42, it is obvious that:
u00 = k00
where
Zrb =
=
(40)
∞;∞
Z (α; β)dαdβ =
C2rb
C0
Z
1
A
∞;∞
1
= C0
and right-bottom (rb):
C1rb
C0
kmn αm βn
Substituting Equation 37 into Equation 17 gives the fol-
α2
combdrives in the other two quadrants, left-top (lt):
where
∑
tion leads to the special case (already analyzed in [14]) where
(29)
It follows from Equations 26, 27, 30, and 31 that for the
C2lt
=
C0
(39)
(for all m 1; n 1), then Z (α; β) = X (α)Y (β). This condi-
β1 ) ( β2 ) β2
β2 β1 β1
Z β1Z α1
Z Z
1 β2 α 2
Z (α; β)dαdβ =
Z ( α; β)dαdβ (32)
A β1 α 1
β2
α2
C1lt
C0
∑
k0n βn
n=1
∞;∞
λ(α; β) = [1 + X (α)][1 + Y (β)] is variable separable.
where
(
(38)
m =1
∞
Note as an aside from Equations 38-40 that if kmn
quadrant:
C1lb
C0
∞
∑ km0 αm
m=1;n=1
Similarly, Equation 24 gives the following relations for the lb
)=
(37)
X (α), Y (β), and Z (α; β) in Equation 18, it follows that:
where
Y (β
kmn αm βn
where kmn (sometimes also written as km;n for clarity) are con-
C1rt
C0
)=
∑
m=0;n=0
(lb), respectively. Hence, we have from Equation 24 for the rt
X (α
∞;∞
1
A
Z
β2 Z
β1
α2
α1
(43)
km0 (αm+1 αm1 +1 ); m 1
um0 =
α2 α1 m + 1 2
1
k0n (βn+1 β1n+1 ); n 1
u0n =
β2 β1 n + 1 2
kmn
umn = um0 u0n km0 k0n
1
Z (α; β)dαdβ (36)
Equations 25, 29, 33, and 35 represent the dependence of the
combdrive capacitances on the average local manufacturing
variations in each quadrant and will be used for later analyses.
(44)
(45)
(46)
Substituting Equation 38 in Equations 26 and 30, and then
For the purpose of a more detailed analysis, which will
simplifying using Equation 44, we get
7
∞
∑ um0
X (α+ ) =
X (α
;
∞
)=
m=1
∑ um0 (
1)m
C1i have the same dependence on local variations in the comb-
(47)
drive. Hence, the dependence of ∆C1i on local variations in
m=1
Similarly, substituting Equation 39 in Equations 27 and 31,
the combdrive alone is represented by λi in Equation 54. In
and then simplifying using Equation 45, we get
addition, the combdrive displacement in each quadrant de-
Y (β
+
∞
)=
∑ u0n
Y (β
;
∞
)=
n=1
∑
u0n ( 1)n
pends on the effect of local manufacturing variations on other
(48)
n=1
structures, such as beams and plates. Hence, an additional
Also, Equation 40 is substituted into Equations 28, 32, 34 and
quantity φi (i 2 frt ; lb; lt ; rbg), that includes the indirect ef-
36, and then simplified using Equation 42 to give
fect on combdrive capacitance of local variations affecting
∞;∞
Zrt
=
∑
∞;∞
umn
;
Zlb =
m=1;n=1
∞;∞
Zlt
=
∑
umn ( 1)m
∑
umn ( 1)m+n (49)
∑
umn ( 1)n
m=1;n=1
∞;∞
;
Zrb =
m=1;n=1
other structures (such as beams and plates), appears in Equation 54. In absence of any local manufacturing variations in
the beams and the plates, φi = 1. Theoretically, in a quadrant
(50)
m=1;n=1
i, it is possible to have λi = 1 (no local variations in the combdrive) and φi
The relations described by Equations 47-50 will be used to
calculate the sensitivities of the BIST modes.
6= 1 (local variations in the beams and plates).
Under such conditions, the undisplaced combdrive capacitors
have the nominal value (C1i = C2i = C0 from Equation 51) but
during displacement, ∆C1i = δφi 6=
4 BIST Mode Sensitivity Analyses
δ (from Equation 54),
which is expected since the combdrive displacement is not
Here, we analyze how the sense combdrive capacitors, un-
nominal. Now, for the sake of convenience, the term φi can be
der different modulation schemes, produce a sense signal that
absorbed into λi so that from now on it is understood that λi
may or may not depend on first-order2 local manufacturing
includes the effect of local variations on all structures (comb-
variations, device rotation, DC offset, and combdrive config-
drives, beams, plates, etc.) in quadrant i. Therefore, Equa-
uration.
tion 54 reduces to:
∆C1i =
∆C2i =
δλi ; i 2 flt ; rb; lb; rt g:
(55)
4.1 Impact of Mismatch
In the presence of local manufacturing variations, the dipole
capacitors have these values for zero shuttle displacement:
C ji = C0 λi ; where i 2 flt ; rb; lb; rt g; j 2 f1; 2g:
From Equation 55, we have ∆C1lt
(51)
δλrb , ∆C1lb = δλlb , and ∆C1rt
=
C pA and CBtotal
drive capacitors change to:
=
δλlt , ∆C1rb
= 2C0 (λlt + λrb ) +
2C0 (λlb + λrt ) + C pB . Substitution of these
values in Equation 15 gives
0 = C0 λi + ∆C1i ; i 2 flt ; rb; lb; rt g
C1i
(52)
0 = C0 λi + ∆C2i ; i 2 flt ; rb; lb; rt g
C2i
(53)
∆VB
Vmp
=
δλrt . Combining Equa-
tion 51 and Equations 1- 2, we have CAtotal
When the shuttle is displaced, the expressions for the comb-
=
∆VA
Slt λlt + Srb λrb
=δ
Vmn
2C0 (λlt + λrb ) + CpA
=δ F (Slt ; Srb ; Slb ; Srt )
Slb λlb + Srt λrt
(56)
2C0 (λlb + λrt ) + CpB
(57)
where
∆C1i =
∆C2i =
δ λi φi ; i 2 flt ; rb; lb; rt g
where F is given by
(54)
F (x1 ; x2 ; x3 ; x4 )=
The quantity ∆C1i (in Equation 54) depends on both the lo-
x1 λlt + x2 λrb
2C0 (λlt + λrb ) + CpA
x3 λlb + x4 λrt
(58)
2C0 (λlb + λrt ) + CpB
cal variations in the combdrive as well as the displacement
Equation 56 allows us to represent the modes of the capacitor
of the movable part of the combdrive. We assume that λi
network (shown in Figure 2(a)) in the form of a truth table
(i 2 frt ; lb; lt ; rbg) remains constant for small displacements
(see Table 2). Every row in Table 2 corresponds to a mode of
(j∆C1i =C0 j 1) of the microstructure. Therefore, ∆C1i and
2 The
h
first-order dependence of variable y on variable x is
the capacitor network.
i
Using Equations 47-48, we define the quantities:
dy
dx x=0 .
8
Mode
(lt :rb:lb:rt)
C1lt : C2lt
Dipole modulation voltages
C1rb : C2rb C1lb : C2lb C1rt : C2rt
Normalized sense output
∆VA )=(Vmp Vmn )
= (∆VB
Mode name
0011
Vmp : Vmn
Vmp : Vmn
Vmn : Vmp
Vmn : Vmp
δ 2C (λλlt++λλrb)+C + δ 2C (λλlb++λλrt)+C
0 lt
rb
pA
0 lb
rt
pB
0110
Vmp : Vmn
Vmn : Vmp
Vmn : Vmp
Vmp : Vmn
δ 2C
0101
Vmp : Vmn
Vmn : Vmp
Vmp : Vmn
Vmn : Vmp
δ 2C
0000
Vmp : Vmn
Vmp : Vmn
Vmp : Vmn
Vmp : Vmn
δ 2C
λlt λrb
0 (λlt +λrb )+C pA
+δ
Normal sense mode
λlb λrt
2C0 (λlb +λrt )+C pB
Self-test X-mode
λlb λrt
0 (λlb +λrt )+C pB
Self-test Y-mode
λlb +λrt
0 (λlb +λrt )+C pB
Self-test XY-mode
λlt λrb
0 (λlt +λrb )+C pA
δ 2C
λlt +λrb
0 (λlt +λrb )+C pA
δ 2C
Table 2: Expressions for sense output voltage of various modes of the capacitor network of Figure 2(a) in presence of local
manufacturing variations.
Xd =X (α+ )
Yd =Y (β
+
X (α
Xs =X (α
+
m=1
∞
Y (β
)
)=
) + X (α
Ys =Y (β+ ) + Y (β
∞
∑ um0 [1
)=
∑ u0n [1
1)m ] = 2
(
n
( 1) ] = 2
∞
∑ u2m+1 0 (59)
λlb + λrt =
∑ u0 2n+1
∑
J2m+1;2n+1 ]
umn [( 1)m+n + 1] =
∑
(62)
(69)
(λlt + λrb )
(70)
(λlb + λrt ) =
right (top) halves of the device and Xs (Ys ) represents the average local manufacturing variations along the X (Y ) direction.
In addition, using Equations 59-62, the following four
variables are defined for the sake of convenience:
∞;∞
=
1+
Xs
Ys
1+
2
2
∞;∞
∑
u2m;2n
+
m=1;n=1
C pA
1
∑
k0;2n k2m;0
(63)
k2m;2n
Xd Yd
4
=
∞;∞
+
u2m+1;2n+1 1
m=0;n=0
k0;2n+1 k2m+1;0
k2m+1;2n+1
∑
∑
u2m+1;2n
m=0;n=0
Xd
Ys
=
1+
2
2
∑
+
u2m+1;2n
m=0;n=1
Yd
Xs
=
1+
2
2
(64)
1
∞;∞
∑
+
u2m;2n+1
m=1;n=0
we get these sums and differences:
umn [( 1)m + ( 1)n ] =
m=0;n=0
(λlt
λrb ) + (λlb
(72)
λrt ) = 4J2m;2n+1
(73)
λrt ) =
(74)
4J2m+1;2n
and define the following quantities for the
(75)
(76)
and then simplifying with the expressions of Equations 63-76
k0;2n k2m+1;0
(65)
k2m+1;2n
gives the differential sense output for various modes as tabu-
Several special cases can be identified:
1
k0;2n+1 k2m;0
(66)
k2m;2n+1
1. For odd3 variation along X direction, X ( α) =
2. For odd variation along Y direction, Y ( β) =
∞;∞
∑
2[u2m;2n
X (α).
Imposing this condition in Equation 61 leads to Xs = 0.
Substituting Equations 63-66 in Equations 25,29, 33, and 35,
∑
(λlb
2[J2m;2n+1 + J2m+1;2n ]
lated in Table 3.
u2m;2n+1
∞;∞
λrb )
=
(71)
Substituting Equations 25, 29, 33, and 35 into Equation 56
∞;∞
∑
m=0;n=0
λlt + λrb =
λrt
Cp
4C0
1
Hmn = [λlb + λrt + 2µ] [λlt + λrb + 2µ]
4
2
2
= µ + J2m;2n
J2m
+1;2n+1
∞;∞
J2m;2n+1 =
λlb
J2m+1;2n ]
µ=
∞;∞
J2m+1;2n =
λrb = 2[J2m;2n+1
sake of developing compact expressions.
u2m+1;2n+1
m=0;n=0
λlt
(λlt
= C pB = C p ,
∞;∞
J2m+1;2n+1 =
4J2m+1;2n+1
For simplicity, we assume equal parasitic capacitance, i.e.,
u2m;2n
m=0;n=0
(68)
(λlt + λrb ) + (λlb + λrt ) = 4J2m;2n
local manufacturing variations between the left (bottom) and
∑
∑
2[u2m;2n + u2m+1;2n+1 ]
m=0;n=0
(61)
where Xd (Yd ) is a measure of the difference in the level of
J2m;2n =
(67)
∞;∞
= 2[J2m;2n + J2m+1;2n+1 ]
∑
∑
∞;∞
m=0;n=0
(60)
;
n=1
n=0
∞
∞
)=
um0 [1 + ( 1)m ] = 2
u2m;0
m=1
m=1
∞
∞
)=
u0n [1 + ( 1)n ] = 2
u0;2n
n=1
n=1
∑
= 2[J2m;2n
;
m=0
∞
u2m+1;2n+1 ]
m=0;n=0
3 f (x)
9
is said to be an odd function of x if f ( x) =
f (x).
Y (β).
Mode
(lt :rb:lb:rt)
0011
δ 1
C0 Hmn
0110
δ 1
C0 Hmn
0101
δ 1
C0 Hmn
0000
Normalized sense output
∆VA )=(Vmp Vmn )
= (∆VB
h
J2m;2n µ + J2m;2n
2
J2m
+1;2n+1
J2m+1;2n µ + J2m;2n
J2m;2n+1 µ + J2m;2n
δ µ
C0 Hmn
J2m+1;2n+1
i
+ J2m;2n+1
J2m
+1;2n+1
J2m+1;2n J2m+1;2n+1
Table 3: Changes in sense voltage outputs for various modes of accelerometer sensor operation, after simplifications based on
expressions in Table 2.
Imposing this condition in Equation 62 leads to Ys = 0.
Combining Equations 77 and 78, we have:
∆VB
Vmp
Therefore, when X (α) and Y (β) are odd functions, Equations 63-66 and Equation 76 reduce to:
∞;∞
J2m;2n =1 +
∑
u2m;2n 1
k0;2n k2m;0
k2m;2n
m=1;n=1
∞;∞
X Y
u2m+1;2n+1 1
J2m+1;2n+1 = d d +
4 m=0;n=0
∞;∞
X
J2m+1;2n = d +
u2m+1;2n 1
2 m=0;n=1
∞;∞
Y
u2m;2n+1 1
J2m;2n+1 = d +
2 m=1;n=0
∑
Hmn [1 + µ]2
=(δ + δrot )
1
=δ
k0;2n+1 k2m+1;0
k2m+1;2n+1
∑
k0;2n k2m+1;0
k2m+1;2n
∑
k0;2n+1 k2m;0
k2m;2n+1
Xd Yd
4
2
∆VA
Vmn
+δrot
X2d
(δ
δrot )
Slb λlb + Srt λrt
2C0 (λlb + λrt ) + CpB
Slb λlb + Srt λrt
2C0 (λlb + λrt ) + CpB
Slt λlt + Srb λrb
2C0 (λlt + λrb ) + CpA
+
Slb λlb + Srt λrt
2C0 (λlb + λrt ) + CpB
(79)
Using Equation 58, Equation 79 can be represented as
∆VB ∆VA
=δ F (Slt ; Srb ; Slb ; Srt ) + δrot F (Slt ; Srb ; Slb ; Srt ) (80)
Vmp Vmn
Yd
2
Equation 80 reveals that this a mixed-mode case, which is
[1 + µ]2
equivalent to two different modes being active simultaneously.
Basically, if the externally applied mode is M = abcd, an in-
The cross-product terms have been neglected. In addition,
ternally generated mode due to rotation will be Mrot
Xd and Yd (defined in Equation 59-60) are much smaller than
h
unity for a normal manufacturing process so that
Slt λlt + Srb λrb
2C0 (λlt + λrb ) + CpA
Xd4Yd
Slt λlt + Srb λrb
2C0 (λlt + λrb ) + CpA
Xd Yd
4
i2
1.
or Mrot
= āb̄cd.
=
abc̄d¯
Hence, the combined effect is a linear super-
Under such conditions, the expressions of mode sensitivities
position of modes, M + Mrot . The two modes may reinforce
in Table 3 reduce to the ones listed in Table 4. Table 4 reveals
each other or oppose each other, depending on their relative
how local manufacturing variations affect various mode sensi-
magnitudes and signs. Thus, the effect of in-plane rotation
tivities and how modes can be used to estimate manufacturing
can be captured by using the principle of linear superposition.
Out-of-plane rotation/curvature usually causes all comb
variations along specific directions in the device layout.
capacitances involved to decrease. Hence, it can be modeled
4.2 Rotation
as any other manufacturing variation.
In-plane rotation of the device causes the capacitors in
diagonally-located combs to either increase or decrease. For
4.3 DC Offset
example, if Clb and Crt increase, Clt and Crb will decrease. If
DC offset can originate from the sensor microstructure or the
the change in capacitance of one combdrive due to rotation
pre-amplifier (see Figure 3). A device can be affected by ei-
alone is δrot , from Equations 13 and 14, we have these sense
ther or both offsets. Using Equation 56, we have shown in
voltages in presence of both translation and rotation:
∆VA =
∆VB =
Slt λlt + Srb λrb
(Vmp
2C0 (λlt + λrb ) + CpA
Slb λlb + Srt λrt
δrot )
(Vmp
2C0 (λlb + λrt ) + CpB
(δ + δrot )
(δ
Vmn )
[30] that both offsets can be eliminated and (optionally) esti-
(77)
mated using 2n number of modes , where n is the number of
Vmn )
(78)
dipoles in the sensor.
10
Mode
(lt :rb:lb:rt)
Normalized sense output
∆VA )=(Vmp Vmn )
= (∆VB
δ
C0
0011
1
Independent of X and Y variations
C
1+ 4Cp
0
C
1+ 4Cp
1 2
4 Yd
Cp 2
(1+ 4C )
0
0110
δ
2C0 (
0101
1+ 4Cp 14 Xd2
δ
0
(+
Y
)
Cp
d
2C0
(1+ 4C )2
Xd )
δ
4C0 (
0
0000
Dependence on manufacturing variations
C
Xd Yd )
0
Cp
4C0
Cp
(1+ 4C )2
0
X variations only (to 1st order), i.e, variations between left and right halves
Y variations only (to 1st order), i.e, variations between top and bottom halves
Diagonal variations, i.e, variations between the diagonal formed by quadrants [lt, rb] and
the diagonal formed by quadrants [lb, rt]
Table 4: Changes in sense voltage outputs, as obtained by simplifying the expressions in Table 3, for various modes of accelerometer operation (in diagonal sense configuration) and their relationship to specific types of manufacturing variations.
Normalized sense output = (∆VB
Horizontal
Mode
(lt :rb:lb:rt)
λlb λrb
0 (λlb +λrb )+C p
λ λ
+ δ 2C (λ lt+λ rt)+C
0 lt
rt
p
δ 2C
λlb +λrb
0 (λlb +λrb )+C p
λ +λ
+ δ 2C (λ lt+λ rt)+C
0 lt
rt
p
δ 2C
λlb +λrb
0 (λlb +λrb )+C p
+δ
λlt λrt
2C0 (λlt +λrt )+C p
δ 2C
0011
δ 2C
0110
δ 2C
0101
δ 2C
0000
λlb +λrb
0 (λlb +λrb )+C p
δ 2C
+δ
λlt +λrt
2C0 (λlt +λrt )+C p
∆VA )=(Vmp Vmn )
Vertical
λlb +λlt
0 (λlb +λlt )+C p
λrb λrt
0 (λrb +λrt )+C p
δ 2C
λlb +λlt
0 (λlb +λlt )+C p
λ λ
+ δ 2C (λ rb+λ rt)+C
0 rb
rt
p
λlb +λlt
0 (λlb +λlt )+C p
+δ
λlb +λlt
0 (λlb +λlt )+C p
δ 2C
λrb +λrt
2C0 (λrb +λrt )+C p
λrb +λrt
0 (λrb +λrt )+C p
δ 2C
Table 5: Mode sensitivities for horizontal and vertical sense configurations.
4.4 Different Network Configurations
along the X and Y axes. Moreover, ‘0.1’ has been used to imply a weak dependence on the corresponding variable while
Our previous analyses are applicable to a specific type of con-
‘1’ implies a strong dependence. Finally, the entries in Table 6
figuration, where combdrive pairs are electrically connected
only indicate a certain level of dependence on local manufac-
diagonally ([lb,rt] and [lt,rb], as in Figure 2(a)). Alternative
turing variations and not the exact magnitude. Even though
configurations of the capacitor network are also possible, such
Table 6 lists only odd and even type of variations, its scope is
as vertical ([lb,lt] and [rb,rt], as shown in Figure 2(b)) and hor-
general since any function can be expressed4 as a sum of odd
izontal ([lb,rb] and [lt,rt], as in Figure 2(c)). Table 5 lists the
and even functions. Below we give additional details about
sensitivities for selected modes for these alternative configu-
the more interesting configurations.
rations. Based on the expressions in Tables 2 and 5, Table 6
lists how specific modes are sensitive or immune to certain
1. For horizontal configuration, in the presence of odd vari-
types of local manufacturing variations for a particular con-
ations along both the X and Y axes, mode 0011 is sensi-
figuration. Column 1 lists the type of capacitor network con-
tive to variation along the X axis only.
figuration. Columns 2 and 3 list the type of variation (odd
2. For diagonal configuration, in the presence of odd vari-
or even) along the X and Y directions, respectively. The re-
ations along the X axis, mode 0110 is sensitive to varia-
maining four columns list the sensitivity of four modes to the
tions along the X axis only.
particular type of variation. Note that ‘1X’ (‘1Y’) is used to
3. For horizontal configuration, in the presence of odd vari-
indicate a strong first order dependence on local manufacturing variation along the X (Y ) axis. Similarly, ‘1XY’ implies
4 f (x) = f
odd (x) + f even (x), where f odd (x)
feven (x) = [ f (x) + f ( x)]=2.
dependence on the product of local manufacturing variations
11
= [ f (x)
f ( x)]=2 and
Combdrive
configuration
Diagonal
Horizontal
Vertical
Variation type
X axis Y axis
Odd
Odd
Odd
Even
Even
Odd
Even
Even
Odd
Odd
Odd
Even
Even
Odd
Even
Even
Odd
Odd
Odd
Even
Even
Odd
Even
Even
0011
0
0.1Y
0.1X
0.1(X+Y)
1X
1X
0
0
1Y
0
1Y
0
Modes (lt :rb:lb:rt)
0110
0101
1X
1Y
1X
0
0
1Y
0
0
0
1XY
0.1Y
0
0.1X
0
0.1(X+Y)
0
1XY
0
0
0.1Y
0
0.1X
0
0.1(X+Y)
0000
1XY
0
0
0
1Y
0
1Y
0
1X
1X
0
0
Table 6: Mode sensitivities under different types of local manufacturing variations and for different sense configurations.
The chosen asymmetries are guided by our interaction
ations along the Y axis, mode 0000 is sensitive to varia-
with industry as well as our own experience. For example,
tions along the Y axis only.
particles can originate from the clean room but also from the
4. For diagonal configuration, in the presence of even vari-
removal of the sacrificial layer during the release step. Parti-
ations along the Y axis and odd variations along the X
cles formed out of the sacrificial layer can be as large as a few
axis, mode 0011 is weakly sensitive to variations along
µm and are therefore large enough to act as bridges between
the Y axis only.
structures.
5 Simulation Results
Parameter
design 1 design 2
Resonant frequency (kHz)
12.5
5.6
Sensor sensitivity (mV =G)
0.88
5.2
Modulation voltage amplitude (V )
5
3
Actuation voltage amplitude
(V
)
1.5
1.5
p
Input referred noise (µG= Hz)
100
100
Bandwidth of baseband sense signal (Hz)
500
500
Nominal value of normal sense mode (mV )
10.5
34.4
Resonant frequency range (kHz)
9.4-15.6 4.2-7.0
Normal sense output range (mV )
8.4-12.6 27.5-41.2
Noise voltage floor (µV )
2
10
We present simulation results for two different accelerometer sensor designs, whose features are listed in Table 7. Both
designs are derived by modifying existing CMOS-MEMS accelerometer designs to include the necessary features required
to implement our BIST capability. Specifically, the BIST includes analog multiplexers for switching modulation signals
via simple digital control logic [13]. For both designs, the
Table 7: Nominal values for design and measured parameters
of the two accelerometer designs used for simulation.
sense combdrives have the diagonal configuration. Simulation experiments are performed to examine the capability of
Simulation experiments were conducted using NODAS
our BIST to detect asymmetry caused by:
[27], a library of mixed-domain, behavioral models written
1. A single dielectric particle acting as a bridge between a
in AHDL (Analog Hardware Descriptive Language) that has
pair of structures where at least one is movable;
been integrated into the flow of a commercial circuit simulation tool. NODAS has been shown to closely match exper-
2. A variation in vertical misalignment between fixed and
imental results [26]. The efficacy of NODAS as a reliable
movable fingers caused by curl mismatch [25];
and much faster simulator than finite element analysis has
3. A variation in local etch [26];
also been demonstrated [22]. We simulate both the electro-
4. Unequal parasitics in the interconnects from the self-test
mechanical microstructure and electronic circuitry of the ac-
sense points to the differential sense amplifier.
celerometer. The electronic subcircuit is based on a design
12
Defect
location
(%)
None
10
20
30
Resonant
frequency
(kHz)
12.53
13.12
14.00
15.32
Finger sense output for a mode
(V)
0011
0110
10.52m
-3µ
9.58m
-3µ
8.40m
-2µ
6.98m
-2µ
Beam sense output for a mode
(V)
00
-2µ
-27µ
-73µ
-133µ
Table 8: Design 1 outputs for a single bridge defect located at different points between spring beam B1 and dummy finger D1
of Figure 1.
Defect
location
None
right end of beam B2 and dummy finger D4
left end of beam B4 and dummy finger D8
Resonant
frequency
(kHz)
5.58
5.63
5.91
Finger sense output for a mode
(V)
0011
0110 0101 0000
34.37m
0
0
0
33.81m
25µ
15p
5.8n
30.72m 110µ 293p 23.3n
Beam sense output for a mode
(V)
00
0
384µ
482µ
Table 9: Design 2 outputs for a single bridge defect located at different points of the spring, between beam B2 (B4 ) and dummy
finger D4 (D8 ), in the right upper quadrant of Figure 1.
[23] that has been fabricated and validated. A simplified ver-
sor is based on the well-known principle that an electrostatic
sion of the routing of the actuation, modulation, and sense
force of attraction is generated between the positive and neg-
signals is illustrated in Figure 4. The details of the dry etch
ative plates of an electric capacitor and is described in [17].
process, design and representative die-shots are presented in
One of the parameters used to decide pass/fail for an ac-
[23]. Self-test by electrostatic actuation of accelerometer sen-
celerometer is its resonant frequency for translation in the
Y direction (see Figure 1). The acceptable range for reso-
actuation lines
nant frequency includes a maximum deviation of 25% from
the nominal value. Another pass/fail test, called the sensitivity test, uses the normal sense signal (output of mode 0011)
which is allowed to deviate at most
20% from its nominal
value. Note that for a given design, any sense signal is considered significant if and only if the signal magnitude exceeds
the noise floor for that design. All of these design-specific
parameters are listed in Table 7.
modulation lines
In the application of our BIST approach to the accelerom-
modulation lines
eter, self-test outputs are created from normal sense fingers
and spring beams [13]. Depending on the particular nature
of an asymmetry, one output may be more sensitive than the
other at observing the effects of a defect. Also, the asymmetries detected at one output need not be a subset of those
sense lines
detected at the other. Hence, the use of self-test outputs from
actuation lines
both beams and fingers, and possibly other sites, may be nec-
Figure 4: Routing of actuation, modulation, and sense signals
through an accelerometer sensor.
essary to minimize defect escapes.
13
Defect
location
(%)
None
0
10
20
Resonant
frequency
(kHz)
12.53
13.74
14.09
14.57
Finger sense output for a mode
(V)
0011
0110
10.52m
-3µ
8.30m
783µ
7.85m
848µ
7.27m
906µ
Beam sense output for a mode
(V)
00
-2µ
-2µ
-1µ
-1µ
Table 10: Design 1 outputs for a single bridge defect located at different points between movable finger M6 and fixed finger S3
of Figure 1.
δHright
(µm)
0
+0.5
+1.0
+1.5
+2.0
Resonant
frequency
(kHz)
12.53
12.51
12.51
12.51
12.51
Finger sense output for a mode
(V)
0011
0110
10.52m
-3µ
10.44m
55µ
10.23m
223µ
9.87m
482µ
9.42m
806µ
Beam sense output for a mode
(V)
00
-2µ
5µ
21µ
48µ
80µ
Table 11: Design 1 outputs for height mismatch between all fixed and movable fingers on the right side of the sensor.
In the following discussion, for the output from the sense
of Table 9 indicates that the defect is located between beam
fingers, only four BIST modes (0011, 0110, 0101, and 0000)
B2 (B4 ) and dummy finger D4 (D8 ) of the upper right quad-
have been considered for design 2 while a smaller subset
rant of Figure 1. Columns 2 and 3 list the values of resonant
(modes 0011 and 0110) have been analyzed for design 1. The
frequency and normal sense output, respectively, for bridge
sense output from beams [13] involves only two dipoles and
defects located at various locations along the beam(s).
can have four possible BIST modes, out of which only two are
As the defect location moves from the anchor end of
unique (e.g., 00 and 01). For the purpose of detecting asym-
the spring to the end where it is attached to the shuttle, the
metry, the beam sense mode 00 is sufficient and therefore is
beam stiffness increases. Consequently, shuttle displacement
the only mode considered. It is also assumed that both the
decreases. Also the layout asymmetry becomes more pro-
finger sense and beam sense amplifiers are identical.
nounced resulting in an increased output from finger sense
mode 0110 (design 2 only) and beam sense mode 00.
5.1 Beam Bridges
The listed resonant frequency values shown in Tables 8-
A bridge defect can be caused by particulate matter that at-
9 are all within the acceptable range, indicating these defects
taches a movable beam to an adjacent structure (e.g, a dummy
will pass a resonant frequency test. For design 1, the normal
finger) thereby hindering its motion. In this analysis the ma-
sense output is barely outside its acceptable range only for
terial of the bridging defect is assumed to be dielectric since
one defect location (at the 30% point). Hence, in a major-
such defects are harder to detect. Due to the four-fold sym-
ity of cases, resonant frequency and sensitivity tests will be
metry of the accelerometer, simulation of a bridge defect has
ineffective. The finger sense mode 0110 is insensitive for de-
been limited to one quadrant of the layout. Specifically, beam
sign 1 and weakly sensitive for design 2. Other finger sense
B1 and dummy finger D1 of the upper left quadrant of Fig-
modes (0101 and 0000 for design 2 only) are insensitive too.
ure 1 are used for design 1. Column 1 of Table 8 indicates the
This is largely because the stiff shuttle leads to virtually equal
defect location expressed as a percentage of beam length. The
displacements on both sides. However, for both designs, the
0% point is the anchored end of the beam and the 100% point
beam self-test mode 00 is very sensitive and strongly indicates
is where the beam meets the shuttle. For design 2, column 1
14
δHright
(µm)
0
+0.1
+0.2
+0.5
+0.7
+0.8
+1.0
Resonant
frequency
(kHz)
5.58
5.58
5.58
5.59
5.59
5.60
5.60
Finger sense output for a mode
(V)
0011
0110 0101 0000
34.37m
0
0
0
34.34m
5.6µ
0
1.2p
34.31m 22.4µ
0
4.9p
34.15m 139µ
0
30p
33.93m 269µ
0
58p
33.79m 349µ
0
75p
33.48m 537µ
1.9p 115p
Beam sense output for a mode
(V)
00
0
1.3µ
2.1µ
7.7µ
14µ
18µ
27µ
Table 12: Design 2 outputs for height mismatch between all fixed and movable fingers on the right side of the sensor.
Etch
variation
(µm)
+0.025
+0.020
+0.010
0
-0.010
-0.020
-0.025
Resonant
frequency
(kHz)
12.33
12.37
12.45
12.53
12.61
12.69
12.73
Finger sense output for a mode
(V)
0011
0110
10.30m
507µ
10.34m
408µ
10.43m
205µ
10.52m
-3µ
10.61m
-217µ
10.71m
-438µ
10.76m
-551µ
Beam sense output for a mode
(V)
00
48µ
39µ
19µ
-2µ
-23µ
-45µ
-56µ
Table 13: Design 1 outputs for etch variations between the left and right sides of the sensor.
the presence of an asymmetry.
too far from the beams to affect the output of the beam selftest mode. However, the finger self-test mode 0110 clearly
5.2 Finger Bridges
indicates the presence of asymmetry.
A bridge defect affecting fingers is similar to a beam bridge
For design 2, a finger bridge defect causes failure for both
defect except that it is located between a movable finger and a
resonant frequency and sensitivity tests. Hence, the various
fixed finger. Naturally, it acts as a hindrance to shuttle motion.
BIST modes do not reveal any additional information.
Similar to beam bridge defects, the material of the defect is
assumed to be dielectric. Like before, the symmetry of the
5.3 Finger Height Mismatch
accelerometer is used to limit simulations to the upper right
Ideally, the fingers should all be at the same height above the
quadrant of the layout. A defect that bridges fingers M6 and S3
die surface. But variations in parameters such as tempera-
in the upper right quadrant of Figure 1 is considered without
ture and residual stress can lead to finger height mismatch
loss of generality. A finger bridge defect is modeled using the
[25]. Height mismatch between the fixed and movable fin-
approach described in [22]. Defect location is expressed as
gers reduces inter-finger overlap and hence inter-finger capac-
a percentage of movable finger length. The 0% point is the
itance. We show how left-right asymmetry caused by height
movable finger tip, and the 100% point is the movable finger
mismatch can be detected by our BIST approach. Without
base where it is attached to the shuttle.
loss of generality, the finger height mismatch is assumed to
exist on the right side of the accelerometer only (δHle f t
The results in Table 10 for design 1 indicate that a finger
= 0).
bridge defect may pass a resonant frequency test but will fail
Tables 11-12 give the simulation results for the two designs.
a sensitivity test. However, the normal sense output by itself
Column 1 lists the relative height mismatch, expressed as
does not indicate an asymmetry. The beam self-test mode 00
δHright
does not detect the finger defect because the defect location is
match have not been separately simulated since we believe
15
δHle f t
= δHright .
Results for negative values of mis-
Etch
variation
(µm)
25
20
10
0
-10
-20
-25
Resonant
frequency
(kHz)
5.49
5.51
5.55
5.58
5.60
5.63
5.65
Finger sense output for a mode
(V)
0011
0110
0101
0000
33.68m
1.08m
-5.7p
179p
33.84m
0.87m
-3.7p
143p
34.10m
0.44m
-0.9p
74p
34.37m
0
0
0
34.68m -0.46m
1.0p
-78p
35.02m -0.93m
4.1p -155p
35.18m -1.17m
6.5p -196p
Beam sense output for a mode
(V)
00
3.7µ
3.1µ
1.9µ
0
-2.3µ
-3.5µ
-4.1µ
Table 14: Design 2 outputs for etch variations between the left and right sides of the sensor.
Parasitic
mismatch
(%)
0
1.25
2.5
3.75
5.0
Resonant
frequency
(kHz)
12.53
12.51
12.51
12.51
12.51
Finger sense output for a mode
(V)
0011
0110
10.52m
-3µ
10.50m
-26µ
10.49m
-49µ
10.48m
-71µ
10.47m
-94µ
Beam sense output for a mode
(V)
00
-2µ
-7µ
-11µ
-17µ
-22µ
Table 15: Design 1 outputs for variations in the parasitic interconnect capacitance of the differential amplifier.
they would yield similar results. CMOS-MEMS exhibits this
be subjected to more than the intended etch by a length δ, re-
nearly symmetric behavior since the gap of 20µm between the
sulting in dimensions [l
substrate and the sensor fingers is large.
that each side-wall of the rectangular structure shifts inwards
2δ; w
2δ]. This is due to the fact
With increasing height mismatch, the difference in capac-
by δ so that each dimension reduces by 2δ. This type of etch
itance between the left and right increases, as evident from
variation is called over-etch. In a similar fashion, under-etch
finger sense mode 0110 and beam sense mode 00 (see Ta-
causes an oversized structure of size [l + 2δ; w + 2δ].
bles 11-12). The resonant frequency however remains virtu-
Etch variation can also be local in nature. Consider two
ally unchanged. The normal sense output reveals a reduced
rectangular structures that are designed to be identical but dur-
but acceptable voltage. Hence, tests based on resonant fre-
ing fabrication they are subjected to different etch variations,
quency and normal sense output will be ineffective in detect-
δ1 and δ2 . As a result, the two structures will have different
ing the asymmetry. However, the output of mode 0110 clearly
dimensions, causing a mismatch.
indicates the presence of asymmetry. Although not as sensi-
Without loss of generality, we assumed in simulation that
tive, the beam self-test mode 00 also varies with the amount
the accelerometer’s left side has nominal etch while the right
of height mismatch and therefore indicates an asymmetry as
side has either over- or under-etch. Tables 13-14 give the sim-
well.
ulation results for various levels of etch variations. Column 1
lists the etch mismatch between the two sides. The mismatch
5.4 Local Etch Variation
in etch variation is positive when the right side is more etched
An etching process is used in fabrication to remove sacrificial
than the left side.
material to free the micromechanical sensor. Material removal
As the relative over-etch increases, the capacitance of the
through an etching process varies with time and space even
right side reduces because of the increase in the inter-finger
though such variation is not desirable. For example, a rect-
gap. Consequently, the outputs of finger sense mode 0110 and
angular structure designed to have length l and width w may
16
Parasitic
mismatch
(%)
0
+2.5
+5.0
Resonant
frequency
(kHz)
5.58
5.58
5.58
Finger sense output for a mode
(V)
0011
0110 0101 0000
34.37m
0
0
0
34.32m
3n
5p
47.3µ
34.28m
5n
10p
94.5µ
Beam sense output for a mode
(V)
00
0
4.3µ
7.7µ
Table 16: Design 2 outputs for variations in the parasitic interconnect capacitance of the differential amplifier in Figure 3.
beam sense mode 00 increase. For increasing levels of relative
inal interconnect capacitance is assumed to be 40 f F. A max-
under-etch, two counteracting effects become significant. The
imum mismatch of 5% between the two interconnects is con-
increased beam thickness on the right side causes increased
sidered, assuming that a good layout design and a stable pro-
stiffness which in turn reduces displacement. However, the
cess can restrict such variations to the presumed limit. With-
reduced inter-finger gap causes higher inter-finger capacitance
out loss of generality, we assumed that C p1 is at its nominal
which more than offsets the reduced displacement. In any
value while C p2 is higher (for both finger and beam outputs).
case, increasing levels of local etch variation lead to greater
The simulation results are listed in Tables 15-16.
outputs from finger sense mode 0110 and beam sense mode
For both designs, the parasitic mismatch has a more pro-
00.
nounced effect on the beam self-test mode 00 as compared
Neither a resonant frequency test nor a sensitivity test will
to the finger self-test modes since the beam sense signals are
detect the presence of this type of asymmetry because both
much weaker. For design 1, the finger mode 0110 is much less
parameters are within their respective tolerance ranges. How-
sensitive to variations in the interconnect when compared to
ever, the outputs of finger self-test mode 0110 and beam self-
local etch variations and finger height mismatch. The same is
test mode 00 are indicators of asymmetry. As in the case of
largely true for the beam self-test mode even though it is more
finger height mismatch, the finger self-test mode 0110 is a
sensitive compared to the finger self-test mode. For design 2,
stronger indicator of this type of asymmetry as compared to
all finger sense modes except 0000 are immune to the parasitic
the beam self-test mode 00.
mismatch. The beam self-test mode is marginally sensitive to
variations in the interconnect when compared to variations in
5.5 Parasitic Variation
local etch and finger height.
Ideally, the sensing circuitry for self-test should only be sen-
Interconnect capacitance mismatch does indeed cause
sitive to asymmetries in the micromechanical sensor and not
self-test outputs to exceed the noise floor in some cases. But,
the external electronics (including interconnects). In reality, a
in a majority of the cases considered, the output magnitude
difference signal may be due to variation external to the sen-
does not rival that produced by the other defects. For design 1,
sor area, such as in the interconnects which carry the self-test
only a beam bridge defect that is located close to its anchor
sense signals to the inputs of the differential amplifier (see
point will produce a beam self-test sense signal that is com-
Figure 3). With reference to Figure 3, assume the intercon-
parable to that produced by interconnect mismatch. In case of
nect capacitances (C p1 and C p2 ) are unequal. The objective
design 2, only the finger self-test mode 0000 will produce a
is to determine the extent to which the parasitic capacitance
weak signal. This implies that variations of up to 5% in the
mismatch due to such interconnect asymmetry will produce a
interconnect capacitance are unlikely to falsely indicate asym-
significant sense amplifier output. The maximum value of the
metry in the micromechanical structure of the accelerometer.
parasitic mismatch can be used to decide a suitable threshold
Even though our simulation has considered the effect
for detection of sensor asymmetry during self-test. The nom17
of single defects, two or more of the afore-mentioned de-
This is confirmed by row 4 in Table 17. As expected,
fects/perturbations may occur simultaneously [12]. It is not
the output for mode 0011 is practically unchanged (see
always possible to determine the individual contributions of
rows 1 and 4 in Table 17).
the perturbations and in many cases it is even difficult to determine all the contributing perturbations themselves. This is
3. For a diagonal configuration, mode 0000 is expected to
largely due to the phenomena of (mis)behavior overlap [12].
give significant output only when there are odd variations
The many-to-one nature of the defect-to-(mis)behavior map-
along both X and Y axes. This is evident from Table 17
ping greatly raises the difficulty in the identification of con-
where the output of mode 0000 is negligible in rows 2-3
tributing as well as non-contributing perturbations. More-
but significant in row 4.
over, the individual perturbations and their relative contributions are largely dependent on various factors including
4. It can be shown that mode 0011 in the vertical configu-
the manufacturing process and the accelerometer design.
ration is more sensitive to a odd etch variation along the
In addition, the effective determination of the perturbations
Y axis than mode 0000 in the horizontal configuration.
and/or their relative contributions is highly dependent on the
This is confirmed by rows 5-6 of Table 17 which show
test/measurement methodology. Therefore, in a limited num-
the output of mode 0011 in the vertical configuration to
ber of cases, it is possible to predict if a certain misbehavior
be about 29% more that the output of mode 0000 in the
is very likely due to a particular defect type [30].
horizontal configuration.
For every variation type listed in Table 6, it is possible to de-
6 Process Characterization via BIST
termine the mode and the network configuration which is most
In the following, we give examples of how sensitive various
sensitive. Thus, it is possible to select a minimal set of modes
modes are to particular instances of local process variations,
(and network configurations) that can capture the same infor-
namely, etch variation. In the following, over-etch and under-
mation that all four modes under each of the three network
etch are arbitrarily assigned positive and negative polarities,
configurations together can capture.
respectively.
In order to characterize the device, the parameters λi (i 2
flt rb lb rt g) need to be computed from measurements from
;
1. For a diagonal configuration and an odd etch variation
;
;
no more than four suitably-chosen BIST modes. Let T1 -T4 be
along the X axis, the 0011 mode is much less sensitive
the measured sense outputs (∆VB
than the 0110 mode. Simulation results (see rows 1-3 in
listed in Table 4 for the diagonal configuration. Using Equation 56, we can compute the values of λi (i 2 flt ; rb; lb; rt g) as
Table 17) show that the absolute change in voltage output of mode 0110 is about 1:5 times that of mode 0011.
follows:
In fact, the voltage output of mode 0011 mode changes
by only
∆VA ) for the four modes
2 4% while the voltage output of mode 0110
λlt
:
changes by several orders of magnitude. Incidentally,
=
λrb =
the sense output for mode 0101 is practically zero as exλlb =
pected (see rows 2-3 in Table 17).
λrt
2. For a diagonal configuration, if there are identical odd
=
Cp
4C0
Cp
4C0
Cp
4C0
Cp
4C0
T00 + (T2 + T3 )
T00 (T1 + T4 )
T00 (T2 + T3 )
T00 (T1 + T4 )
T00 + (T2 T3 )
T00 (T1 T4 )
T00 (T2 T3 )
T00 (T1 T4 )
where
etch variations along the X and Y axes, modes 0110 and
T00 = (Vmp
0101 should produce outputs of the same magnitude.
18
Vmn )
δ
C0
1
(81)
1
(82)
1
(83)
1
(84)
Average etch variation Resonant
Sense output for a mode = ∆VB ∆VA (V)
Calculated local variation
Combdrive
in a quadrant (nm)
frequency
T1
T2
T3
T4
parameters (%)
configuration lt
rb
lb
rt
(kHz)
(0011)
(0110)
(0101)
(0000)
λlt
λrb
λlb
λrt
Diagonal
0
0
0
0
5.58
34.37m ( 0%)
0
0
0
99.98 99.98 99.98 99.98
Diagonal
0
+25
0
+25
5.49
33.68m (-2.0%)
1.08m
-5.7p
179p 99.73 93.53 99.73 93.53
Diagonal
0
-25
0
-25
5.65
35.18m (+2.4%) -1.17m
6.5p
-196p 100.60 107.52 100.60 107.52
Diagonal
0
0
+25 -25
5.57
34.47m (+0.3%) -1.13m
1.13m
-46µ 100.25 100.25 94.11 107.30
Horizontal
+25
0
0
+25
5.49
0
33.70m
0
0.84m
Vertical
+25
0
0
+25
5.49
1.08m
0
33.71m
0
Table 17: Design 2 test outputs of various BIST modes for various forms of etch variation across the sensor.
In a typical example case, where quadrants rt and rb are
erence. However, for the X (0110), Y (0101) and XY (0000)
over-etched by 25nm (see row 2 in Table 17), we have
modes, signature verification has a whole new meaning since
from NODAS [27] simulation: Vmp = +1:5V , Vmn =
1:5V,
in a good device, these modes will have practically zero
1:08mV,
output. Therefore, the issue of signature verification boils
T00
=
83:39mV,
Cp
4C0 =
1:426, T1
=
33:68mV, T2
=
T4 0. This gives the calculated values of λi (i 2
flt rb lb rt g) in row 2 of Table 17 and these values are exT3
;
;
down to arbitrarily choosing a threshold for each of these
three modes where the choice of such thresholds will depend
;
pected since the combdrive capacitances on the right half are
on process, design, performance and reliability parameters.
lower.
Without prior knowledge of all of these parameters it is impractical to choose such thresholds since that may lead to false
Wafer characterization is an extension of device charac-
positives and/or false negatives.
terization, since collection of data from multiple dice (one die
may have one or more devices) will give an estimate of how
the λ parameter varies across the wafer. Basically, a discrete
8 Conclusions
two-dimensional plot of λ over the entire wafer can be used as
Our differential self-test method described in this work is fo-
a measure of manufacturing variations across the wafer.
cused on enhancing observation and therefore complements
the existing built-in stimulus generation techniques used in
7 Stimuli Generation and Signature
industry and proposed in the literature. We have demonstrated the ability to detect the presence of three defect types
Analysis
that cause local asymmetry which are not detectable by typi-
The actuation voltage amplitude listed in Table 7 refers to the
cal specification-based tests that measure either resonant fre-
electrostatic actuation-based stimuli used in our simulation
quency or sensitivity. It has been established, in principle and
experiments. In general, our D-BIST technique is compatible
by simulation, that a test method using pairwise comparison of
with all industry-standard stimuli generation methods that are
multiple outputs from corresponding sub-parts of symmetric
also used with other BIST techniques. For example, in case of
devices can substantially enhance detection of hard-to-detect
the accelerometer, both electrostatic actuation and mechanical
defects. While our D-BIST technique can detect some types
vibration can be used to generate stimuli for our BIST.
of misbehavior masking that are not detected by traditional
Signature verification for our BIST is achievable in differ-
BIST, we do not claim that it is 100% immune to all types of
ent ways. In case of an accelerometer sensor using diagonal
misbehavior masking. In fact, since our method relies on the
sense configuration, signature verification for the “normal”
extraction of signals from device regions of finite and non-
(0011) mode is no different from current industry-standard
zero area, it is clearly based on the average behavior exhib-
BIST techniques which include comparison to a golden ref-
ited by each of the device regions. By definition, an average
19
can include contributions that are both less and more than it-
[4] S. Cass, “MEMS in Space,” IEEE Spectrum, Vol. 38, No. 7, pp.
56-61, July 2001.
self which implies that averaging inherently leads to quantities canceling each other partially or totally. Therefore, our
[5] T. A. Lober and R. T. Howe, “Surface-micromachining Pro-
method may not detect some types of misbehavior masking
cesses for Electrostatic Microactuator Fabrication,” IEEE
Solid-State Sensor and Actuator Workshop, Technical Digest,
where all the contributing perturbations are localized in the
pp. 59-62, June 1988.
same device region (same as ”quadrant” in our specific exam-
[6] D. C. Hutchison, K. Ohara and A. Takeda, “Application of
ple) from which an output signal is measured.
Second Generation Advanced Multimedia Display Processor
Our model, when applied to a symmetric microstructure
(AMDP2) in a Digital Micro-mirror Array based HDTV,” In-
(e.g., the accelerometer sensor), shows that the differential
ternational Conference on Consumer Electronics (ICCE), pp.
self-test method [13, 14, 15] is a very broad and versatile tech-
294-295, June 2001.
nique that can also be used for characterization, if appropriate
[7] R. S. Payne, S. Sherman, S. Lewis and R. T. Howe, “Surface
modulation schemes are implemented. Die and wafer char-
Micromachining: From Vision to Reality to Vision (accelerom-
acterization can be accomplished if the dependence of sense
eter),” Proc. of International Solid State Circuits Conference,
capacitances on local manufacturing variations is known. The
pp. 164-165, Feb. 1995.
electrical sensitivities measured from the various BIST modes
[8] R. Oboe, “Use of MEMS based Accelerometers in Hard Disk
can then be mapped to values of equivalent local manufactur-
Drives,” Proc. of International Conference on Advanced Intel-
ing variations. The current mathematical model also explores
ligent Mechatronics, Vol. 2, pp. 1142 -1147, 2001.
a large space and enables customization for characterization
[9] T. Jiang and R. D. Blanton, “Particulate Failures for Surface-
of specific types of manufacturing variations. The model is
Micromachined MEMS,” Proc. of International Test Confer-
generic and can be easily applied to a device with any num-
ence, pp. 329-337, Sept. 1999.
ber of sub-parts that are identical by design, as described in
[10] A. Hartzell and D. Woodilla, “Reliability Methodology for Pre-
[15]. The trade-off for our BIST is a marginal increase in
diction of Micromachined Accelerometer Stiction,” Proc. of
design complexity and device area for additional electronics.
Reliability Physics Symposium, pp. 202-205, March 1999.
However, there is no degradation in performance since the ad-
[11] A. Kolpekwar, R. D. Blanton and D. Woodilla, “Failure Modes
ditional parasitics introduced by the switching circuitry are
for Stiction in Surface-Micromachined MEMS,” Proc. of Inter-
adequately driven by the strong modulation signals while the
national Test Conference, pp. 551-556, Oct. 1998.
weak sense signals are not over-loaded.
[12] N. Deb and R. D. Blanton, “Analysis of Failure Sources in
Surface-Micromachined MEMS,” Proc. of International Test
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