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A
Piezoresistive Sensor Chip for Measurement
of
Stress in Electronic Packaging
Richard
C.
Jaeger, Jeffrey
C.
Suhling,.
Martin
T. Carey, and R. Wayne Johnson
Electrical and Mechanical. Engineering Departments
and
Alabama Microelectronics Science and Technology Center
Auburn University. Al36849-5201
(205)844- 1871
ABSTRACT
studies, the authors and their co-workers have reviewed
and elucidated the general theory of piezoresistivity. and
have derived the primary equations needed for designing
silicon piezoresistive stress sensors for test chip
applications. The results
in
references I9.101 were for
ked temperature conditions. and in
1111.
emphasis was
placed on including the influence
of
temperature in
all
of
the theoretical developments. Expressions have been
obtained for the resistance changes experienced by in-
plane resistors which are subjected to arbitrary three-
dimensional stress states and temperature changes.
A new 22.50 "off-axis'' rosette design was
described in [9,10].and this paper presents the first
experimental results
of calibration of this off-axis
rosette. This rosette can be used
to
measure the three
individual piezoresistive coemcients of silicon,
KI
1. x12,
and
x44,
while requiring only the application of uniaxlal
stress for calibration. Of even greater potential
importance, this rosette yields inherently temperature
compensated values of the coefficients
x44
and
XD
=
(IC
11
-
~12).
The coefficients
n44
in p-type silicon and
XD
in
n-type silicon are needed for an optimized stress sensor
on
(1
00)
silicon.
A p-type implementation of the off-axls rosette has
been characterized over the
25
C
-
140 C temperature
range, and values for the temperature dependence of
x44
and
XD
are reported.
This paper presents the first experimental results
of calibration of the 22.50 "off-axis"rosette. This rosette
requires the application of only uniaxial stress for
measurement of the three individual piezoresistive
coefficients of silicon:
x11. xi2
and
~44.
Of even greater
potential import, this rosette yields inherently
temperature compensated values of the coefficients
x44
and
KD
=
(xll
-
~12).
P-type off-axis rosettes have been
characterized as a function of temperature and values
for the temperature dependencies of
x44
and
XD
are
reported. The coefficients
x44
in p-type silicon and
XD
in n-type silicon are needed for an optimized stress
sensor on (100)silicon.
INTRODUCTION
Piezoresistive stress sensors (semiconductor
strain gages) have been widely used for experimental
structural analysis of electronic packages. These
resistors are conveniently fabricated into the surface of
silicon integrated circuit die as part of the normal
processing procedure
[
1-81 with the objective of
providing non-intrusive measurements of surface stress
states on a chip within encapsulated packages. If the
piezoresistive sensors can be calibrated over a wide
temperature range, thermally-induced stresses can be
measured, and full-field stress distributions over a die's
surface may be obtained using specially designed test
chips that incorporate arrays of sensor rosettes and
multiplexing circuitry.
Successful application of these piezoresistive
sensors for stress measurement, however, requires both
properly designed sensors and accurately calibrated
values
of
the piezoresistive coefficients.
REVIEW OF THEORY FOR SILICON ROSETTES
The general expression for the resistance of a
resistor that
is
oriented at
an
angle
4
with respect to the
x'1
=
[I101
axis
on the
(100)
surface of
a
silicon wafer
and subjected to an arbitrary state of stress and
temperature (see Fig.
1)
is
19-
111
This work has been funded in-part by the National Aeronautics and Space Administration, and the Center for
the Commercial development of Space Power with funds from NASA
grant
NAGW- 1192-CCDS-AL., Auburn
University and the Center's Industrial Partners. Funding was also provided by Sandia National Laboratories
and the Alabama Microelectronics Science and Technology Center.
In recent
$3.00
O1993
IEEE
686
0569-5503/93/0000-0686
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OFF-AXIS ROSETTES
(xs
-
(
AS
+
U
COS
24))
+
R(0,T)
=
R(0,O)
[
1
+
COS
24))
+
633 x12
+
iI2
XD
sin 20
+
alT
+
U~V
+
...I
11)
where
xiJ
are
the temperature dependent piezoresistive
coefficients
Bfttle 191 proposed the off-axis rosette, Fig. 2, that
is
capable of determining the values of the three
individual piezoresistive coefficients
x
1
1,
x
12
and
xu
using only a uniaxial applied stress. In addition, it was
recently discovered that the results of calibration of both
x44
and
RD
using
this
rosette are inherently temperature
compensated if the resistor TCRS are matched.
The off-axis rosette configuration consists of a
three-element 00-450-900 rosette rotated 22.50 from the
[loo]
axis.
(2)
T
2
+...
qj =qj(T)=Zi+di)T+hj
(2)
For notational compactness,
xs
=
(x
11
+
x12)
and
m
=
(xi
1
-
1~12)
have been introduced, and T
=
Tm-Tref
represents the difference between the measurement
temperature Tm and the chosen reference temperature
Tref. The stresses
0'11, 0'12,
and
0'22
are evaluated
in
i
X;
[iio]
X,[O101
I
@
22.5'
x2
General
(
100) Silicon Wafer
Figure 1
:
the
xi
=
[l
lo]
and
x2
=
Ill01 coordinate system in
the
(100)
plane where the resistor
is
fabricated, and
0'33
is
in the
x3
=
I0011 direction normal to the (100)
surface of the wafer.
%e
normalized change
in
resistance is given by
22s0
Figure 2: 22.50 Off-Axis Rosette
The general equations for the 22.50 off-axis rosette can
be derived using Eqn.
(3)
with the standard equations
for transforming the in-plane stress components from
one coordinate system to another:
AB
=qxs
+&$4COS24)]+
yxs
-i€l.4cos2$]
x12
R2
(3)
2
+
o12
ZD
sin
241
+
alT
+
a2Tz
+...I
I
where
sln2e
-2sineco~e
Standard uniaxial calibration of rosettes based
upon Eqn.
3
yields values for only
x44
and.
xQ.
Application of hydrostatic pressure has been used to
obtain enough information to separately determine
xi1
and
x
12.
However, this technique significantly
complicates
an
already difficult calibration procedure.
The three-element off-axis rosette offers
a
solution to
problem of determining
the
individual values of
ql.
1~12
and
~44.
012
In these equations,
f3
represents the angle of clockwise
rotation of the
x"-
y" coordinate system with respect to
the
x'
-
y' coordinate system. Using Eqns.
3
and
4.
the
results for the off-axis rosette (for
ai
=
0)
are [9.101:
l1
;
"
-K
+IC
2+K44
3Kll+K12+K4 +0i2X11+3X12-K44
4
4
+U12
R1
2=
(5)
~11+3~12+~44
"
3~11+"12-~44
~11-~12+w
a;l
+a22
2
+U12
l1
;
&=u;lA]1+31C12-lt
"
3K11+K12+%44
-X
2-%44
'*
K
44+a22
R3
687
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in which the stress components are now measured in
the new double-primed coordinate system.
Calibration of this off-axis rosette uses uniaxial
stress applied along the x"1 or x"2 axis. For the case of
an
applied uniaxial stress
o"22
=
o.
[;1
EL
R1
0.2500 0.7500 -0.2500
[
0.7500 0.2500 0.2500
=
0.7500 0.2500 -0.2500
The piezoresistance coefficients are determined by first
measuring the variation of ARi/Ri versus stress for each
resistor.
A
least squares fit to the data
is
then used to
find the best fit values of the slopes of ARi/Ri versus
0:
Figure
3:
Layout of Off-Axis Chip with
0°-450-900
Rosettes, Heater and Diode Temperature
Sensors
apparent in the figure was not used in this series of
calibration experiments.
SIMS
analysis indicates that
the peak doping concentration in the resistors
is
approximately 1.5 x lO18/cm3, and the sheet resistance
of
the resistors
is
280
Q/O.
Each wafer was designed to have seven strips with
a single test chip per strip, and the layout for each strip
is
shown in Fig.
4.
Aluminum traces on the wafer run
from the
1/0
pads on the chip to thirty-six 1.27 mm x
1.27 mm aluminum pads. Large 0.25 mm diameter
aluminum wires were ultrasonically bonded to these
pads to provide reliable electrical contact and to
eliminate the need for probes.
The piezoresistive coefficients can then be found from
-0.5000
1.oooO
0.5000
1.5000
-l.m
0.5000
-2.m 2.m0.m
Note that the piezoresistive coefficient calculations for
1144
and
AD
involve only the differences of variations in
two resistor values. Thus,
if
the temperature
dependencies (al.
a2.
etc.) of the individual resistors are
matched, then they will cancel when the difference in
the two AR/R terms
is
calculated, and the results will
exhibit first order temperature compensation.
Temperature compensation has been found to be
an
extremely important property with respect to
measurement of accurate coefficient values
[
12-
141.
Figure
4:
Layout of calibration test strip showing
bonding pads for large diameter aluminum
wire bonds.
OFF-AXIS ROSETTE CALIBRATION
RESULTS
OFF-AXIS TEST CHIP DESIGN
The test chip of Fig. 3 has been fabricated to test
the off-axis rosette theory and to provide preliminary
values of the piezoresistive coefficients and their
temperature dependencies.
The stress sensors were calibrated over a range of
stresses and temperatures 1161. For calibration, the
wafer was sawed into strips. and a four point bending
fixture [171 was used to apply a uniaxial stress
o"22
=
o.
Prior to application of stress, the die were calibrated in a
temperature chamber to determine the temperature
coefficients of resistance (TCR
=
al)
with a typical result
displayed in Fig.
5.
It can be seen that the first order
temperature coefficients are tightly matched, and the
second order coefficients also match to within
approximately flp!.
The rosettes were next calibrated at room
temperature to determine the values of the piezoresistive
coemcients
~11.
For the initial rosette evaluation, p-type off-axis rosettes
were fabricated in (100) n-type silicon wafers.
An
anisotropic etching process delineates the exact position
of the crystallographic axes I15.161. and a special two-
step process was used to align the circuit pattern
at
the
desired angle of
22.5'
with
respect to the
[I
IO]
axis
of
the crystal.
The test chips feature seven rosettes, each with
three diffused resistors. and three p-n junction diodes.
The large serpentine metal heater element that
is
1112,
7144
and
KD.
The results of the each
688
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From
Fig.
6.
a 2.5% change in the value of
R1
can
be observed for
a
total applied uniaxial stress of
100
MPa.
TNs
same change in resistance corresponds to
a
temperature change of less than
15
C. This behavior
underscores the facts that diffused resistors tend to be
more sensitive to temperature variations than stress
changes and great care must be exercised in the
Calibration and application of stress sensing test chips.
Cancellation of the
TCR
effects during rosette calibration
and application
is
required in order to obtain accurate
stress state data
[12-141.
During data analysis,
it
was found that an error
had been made in the design of the mask set, and the
rosette
is
actually positioned at an angle of
18.40
with
respect to the
I1001
axis.
This causes the equations
relating the piezoresistive coefficients
to
the three
normalized resistance changes to become
cr:
.
a
0
50
100
150
T
("C)
1
[
ARl/R1
I
-
3.101903
+
1.8064C3T
+
1.%737C-6TA2
W/R2
r:
-
2.633b-3
+
1.8017&ST
+
2.070&-6TA2
AR3/R3
=
-
2.7-3
-0.3749 0.7497 0.6251
1.3749 -0.7497 0.3749
0.3338 -2.6676 2.3338
+
1.80(]3e3T
+
1.9793oSTc.2
(7)
Figure
5:
Change in Resistance versus Temperature
for Rosette
441
with
(T
=
0.
test were plotted, and a
typical
graph for the normalized
change
in
resistance with respect to stress
is
shown in
Fig.
6. The piezoresistive coefficients are calculated
directly from the slopes of the normalized change in
resistance data versus stress. For example, the set of
measurements
in
Fig. 6 yields (using Eqn. 7 below)
~11=
12
x
-1.7500
1.5000
0.2500
The equations for
~44
and
KD
now involve all three
resistors, but the calculations remain temperature
compensated since the coefficients in the third and
fourth rows of the matrix still sum to zero. It can be
shown
[14]
that this is a property of this rosette at all
angles for which the original calibration matrix does not
become singular.
Accurate knowledge of the off-axis angle
is
critical
because the extracted values of the piezoresistive
coefficients are highly dependent upon this angle.
Figure 7 depicts the values of the three piezoresistive
Pa-l,
x12
=
15
x
10-l2
Pa-1
and
~44
=
890
x
10-
12
Pa-1.
0.03
I
1
O.OOe+O
5.00e+7
1.00e+8
1.50e+8
Stress [Pal
ARl/RI
=
-
1.1698e-4
-
1.452Oe-1Ox
=/U2
=
+
2.376oe-4
-
1.9983e-1Ox
AFZ3/R3
=
+
4.4538e-5
+
1.7148e-10x
1
10
-500
Figure 6: Typical Changes in Resistance versus
Uniaxial Stress
20
30
40
Off-Axls
Angle
Figure
7:
Piezoresistive Coemcient Values versus Off-
axis
Angle
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ked uniaxial stresses and varying the temperature at
each stress level. Data for
x44
and
ZD
versus
temperature were collected from several rosettes. A
typical graph of the results is given in Fig. 8, and
and
111
present the results from three similar well-
behaved rosettes.
coefficients
n11,
~12
extracted from a fEed set of
resistor variations as a function of the assumed value of
the
off-axis
angle. In this case, the extracted value of
x44
is
650
x
10-l2
Pa-l at an angle of
250
but is
890
x
10-12
Pa-' for
an
assumed angle of
17.50.
The results of calibration of
15
rosettes are given
in Table
I
including the averages and standard
deviations for the room temperature values of
xii.
x12.
X44
and
AD.
and
~44
I
Table
I1
-
XAA
TemDeratUre DeDendence
PL
strip
#
44
1
444
447
flu
(00C)
915 x 10-l2
Pa-'
-3.6
x
Pa-'C-'
886
x
10-'2 Pa-'
-2.6
x
Pa-lC-'
Table
I
-
Calibration Results From
15
p-type Rosettes
916x 10-12
Pa-l
-4.9x
Pa-lC-l
Standard
Deviation
Piezoresistive
Average
Coefficient
x5
(00
c)
0.84
x
10-l2
Pa-'
-1.2
x
10-13
Pa-'C-1
4.23
x
10-12 Pa-'
-4.9x
strip
##
441
444
447
PA
Pa-lC-l
21.0
x
10-12
Pa-1
-1.4
x
Pa-lC-l
The extracted value of
1244
is
consistent with other
measurements
[18-21).
The values of
~11
From
and
111.
it can be seen
that
the
temperature dependence of the values
of
both
x44
and
XD
is
relatively small.
A
negative temperature coemcient
of
300-500 ppm/C was measured for the value of
~44.
The value of
XD
was found to be a weak function of
temperature.
The value of
x44
varies by less than
6%
over a temperature range extending between
25
"C and
140
"C,
and the error associated with assuming a
constant value for
x44
over this temperature range is
within the experimental error for the calibration process
itself.
and
n12
are
small in lightly doped p-type silicon, and heavy doping
further reduces their values
(211.
In addition. because
of the wide disparity between the values of
~11.
x12.
and
x44
in p-type silicon, the extracted values of
IF^
1
and x12
are greatly affected by small measurement and thermal
errors. These errors lead to the negative extracted value
for
KD
and its relatively large standard deviation relative
to its mean.
The final step in this calibration effort was to
measure piezoresistive coefficient behavior over the
temperature range of
25
-
140C
by applying a set of
A
NEW
PREFERRED ROSETTE CONFIGURATION
FOR
STRESS
MEASUREMENT
1000
Based upon earlier work
19-1
11.
it has been shown
that high quality stress measurements with silicon
piezoresistive stress sensors require the use of
temperature compensated sensors
[
12-141.
An
optimized rosette configuration on
(100)
silicon consists
of the four-element dual polarity rosette shown in Fig.
9
with
two
p-type resistors oriented at
00
and
900
and two
n-type resistors oriented at
f450.
Using
Eqn.
3
for resistors
R1
and
&
oriented
at
00
and
900
with respect to the
[l
101 axis and with a
possible rotational alignment error of
A@,
y
=
885.64
-
0.26333X
U=
-
=
m
m
-
800
-
a
t
v
-
-
Pi-44
Pi-44
t
"O01
v
600
-
600
-
c
m
0
c
m
0
.
.
8
400-
a
.-
8
400-
a
.-
E
E
-
-
.f
200
-
.f
200
-
y
=
4.2288
-
4.9395e-2x
,
-200
,
.I.1.
1.
1.
1.
I
.
In p-type silicon,
x44
is large and
XD
is small. and these
rosette elements yield a temperature compensated
690
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