Design and characterization of in-plane silicon stress sensors with isotropic sensitivity

Abstract

This paper reports on the development and characterization of novel in-plane CMOS-based stress sensors featuring equal sensitivities towards the two mechanical shear stress components sigma_xy = (sigma_x'x' - sigma_y'y') / 2 and sigma_x'y'. The sensor structures are based on symmetric n-well resistors with eight contacts. A geometric parameter variation is performed using FEM simulations to adjust the stress sensitivities of different sensor layouts. For characterization, in addition to a four-point bending bridge used to exert normal stresses (sigma_x'x' - sigma_y'y'), a novel torsional bridge setup was developed to apply well-defined shear stress sigma_x'y' to the surface of silicon beams diced parallel to the <110> crystal direction of silicon and containing the stress sensor elements. The measured sensitivities are consistent with simulated values.

Full text

Design and Characterization of In-Plane Silicon
Stress Sensors with Isotropic Sensitivity
M. Herrmann1, P. Gieschke1, Z. Liu2, J. Korvink2, P. Ruther1 and O. Paul1
Department of Microsystems Engineering (IMTEK), University of Freiburg,
Georges-Koehler-Allee 103, 79110 Freiburg, Germany
1Microsystem Materials Laboratory, 2Laboratory for Simulation, Email: herrmann@imtek.de
Abstract—This paper reports on the development and
characterization of novel in-plane CMOS-based stress sensors
featuring equal sensitivities towards the two mechanical shear
stress components σxy = (σx’x’ − σy’y’) / 2 and σx’y’. The sensor
structures are based on symmetric n-well resistors with eight
contacts. A geometric parameter variation is performed using
FEM simulations to adjust the stress sensitivities of different
sensor layouts. For characterization, in addition to a four-point
bending bridge used to exert normal stresses (σx’x’ − σy’y’), a
novel torsional bridge setup was developed to apply well-
defined shear stress σx’y’ to the surface of silicon beams diced
parallel to the <110> crystal direction of silicon and containing
the stress sensor elements. The measured sensitivities are
consistent with simulated values.
I. INTRODUCTION
The analysis of packaging processes of integrated circuit
(IC) chips and microelectromechanical (MEMS) devices
gains growing attention. Typically, Wheatstone bridge
configurations of implanted piezoresistors are applied [1]
giving access to either the shear stress σx’y’ or the normal
stress difference (σx’x’ − σy’y’) at a specific chip location.
Eight-terminal CMOS-based stress sensors have been
employed to simultaneously extract both shear stress
components σxy = (σx’x’ − σy’y’) / 2 and σx’y’ from a single
sensor element [2]. However, these sensors feature highly
different voltage related stress sensitivities )(xy
V
S and )''( yx
V
S
towards these stress components due to material specific
anisotropic piezoresistive coefficients. In order to ease the
interpretation of sensor signals and reduce the required
electronic circuitry, it is desirable to find sensor
configurations featuring an identical sensitivity towards these
two in-plane shear stress components.
Four-point bending bridges (4PBB) are well-established
tools for sensor characterization to apply a single-stress state
to the surface of silicon beams [3]. However, to evaluate
both stress sensitivities )( xy
V
S and )''( yx
V
S two silicon beams cut
along the <100> and <110> crystallographic directions of
silicon are required. To reduce the cost, a novel torsional
bridge (TB) setup was developed which introduces the shear
stress σx’y’ to the silicon beam. This makes it possible to
characterize the same sensor both under normal and shear
stress, thus minimizing inaccuracy due to sample-to-sample
variations.
II. STRESS SENSOR DESIGN
Two stress sensor variants based on n-wells with eight
peripheral contacts C1 through C8 , schematically shown in
Fig. 1, are applied in this study. As illustrated in Fig. 1(a),
variant I comprises a square n-well (100×100 µm2) with a
non-conducting inner square of side length w. The size of the
inner square is varied systematically with w = 0 µm, 65 µm,
70 µm, 75 µm, 80 µm, and 85 µm. The n+-doped contacts Ci
of 9.4×3 µm2 are located at the outer rim and along the
corners of the n-well. The edges of the n-well and the non-
conducting island are aligned in parallel to the <110>
directions of the silicon substrate.
Variant II shown in Fig. 1(b) is based on narrow, 5-µm-
wide n-well resistors Ri (i = 1..8) positioned between
quadratic n+-contacts C1 through C8 (contact size 5×5 µm2).
While contacts C1 , C3 , C5 , and C7 are positioned on the
mid-points of a virtual square of 100×100 µm2 {dashed line
in Fig. 1(b)}, contacts C2 , C4 , C6 , and C8 are shifted along
the diagonals within this square. In this way, the angle α of
the n-well resistors Ri between the n+-contact diffusions is
varied and the entire stress sensor element can be considered
as an interpolation between two Wheatstone bridges aligned
with the <110> and <100> crystal directions. The angle
α = 0° corresponds to a Wheatstone bridge of 100×100 µm2
aligned parallel to the <110> direction. In contrast, α = 45°
is equivalent to a Wheatstone bridge of 70.7×70.7 µm2
aligned along the <100> directions. Stress sensor elements
with α = 0°, 15°, 30° and 45° are implemented.
Figure 1. Eight-terminal stress sensor components; (a) Variant I with non-
conducting square inner hole of side length w and (b) variant II with
eight resistors Ri (n+-doped contacts C1 through C8).

A. Sensor realization
The test structures were fabricated using a commercial
0.6 μm CMOS process of X-FAB Semiconductor Foundries
AG, Erfurt, Germany. Fig. 2 shows optical micrographs of
both stress sensor variants comprising n-well structures with
eight terminals. Fig. 2(a) illustrates variant I with the
respective bonding pads (size 100×100 µm2) for electro-
mechanical characterization. Sensor variant II is given in
Fig. 2(b) showing the eight resistors Ri , the n+-doped
contacts and the guard ring.
B. Stress Sensitivity Calculations
The voltage related stress sensitivities V
S of both sensor
variants shown in Fig. 1 were simulated using the finite
element (FEM) package COMSOL Multiphysics in the
Conductive Media DC mode. The applied two-dimensional
(2D) conductivity tensor σ(el) of the n-well material defined
on the crystal coordinate system is given by
⎥
⎦
⎤
⎢
⎣
⎡
ρρ−
ρ−ρ
ρρ−ρρ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
σσ
σσ
xxxy
xyyy
xyxyyyxx
el
yy
el
xy
el
xy
el
xx 1
)()(
)()( , (1)
where the components ρij of the 2D resistivity tensor are
calculated using
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
σ
σ
σ
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
π
ππ
ππ
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
ρ=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
ρ
ρ
ρ
xy
yy
xx
xy
yy
xx
44
1112
1211
0
00
0
0
0
1
1, (2)
with πij and ρ0 denote the piezoresistive coefficients and the
resistivity in the stress-free state. The electrical conductivity
of the contact areas is set isotropic with σ(el ) = 6×10−7 S/m,
i.e. representing areas of constant voltage. For simulation, a
defined voltage difference Vbias is applied between opposite
contacts, i.e. C1 and C5 , and the resulting piezoresistive
offset voltage Voff due an applied mechanical shear stress σ
is extracted at the orthogonal contact pair, i.e. C3−C7. For all
simulations, mechanical shear stress of σxy = σx’y’ = 100 MPa
is applied. The voltage related stress sensitivity is given by
)(ij
V
S = Voff / (Vbias σij).
The simulation results are shown in Figs. 3(a) and (b) for
sensor variants I and II, respectively. In case of sensor
variant I without the inner non-conducting square, i.e. w = 0,
)''( yx
V
S measured either at contact pair C1−C5 or C3−C7 is
about ten times higher than )(xy
V
S measured either at contact
pair C2−C6 or C4−C8 as shown in Fig. 3(a). As the width w of
the inner non-conducting square increases, )''( yx
V
S decreases
from 0.90 mV/(V MPa) to 0.05 mV/(V MPa) while )(xy
V
S
increases slightly from 0.07 mV/(V MPa) to
0.14 mV/(V MPa). At a width of the inner square of ca.
w = 82 µm, both sensitivities are equal with )(xy
V
S =
)''( yx
V
S= 0.13 mV/(V MPa).
In case of sensor variant II, the FEM results for )(xy
V
S and
)''( yx
V
S as a function of α are shown in Fig. 3(b). In addition to
the FEM simulations, an analytical model was developed
considering the sensor geometry as eight separate resistors
Ri , each at an individual angle θi with respect to the <100>
crystal direction. The relative change ΔRi / R0 in resistance of
these individual resistors Ri upon application of mechanical
stress are given by
)2sin()2cos()(
))((5.0
44''1211
1211
0
ixyiyx
yyxx
i
R
R
θσπθσππ
σσππ
+−+
++=
Δ
(3)
with ΔRi = ΔRi
σ
– R0 where ΔRi
σ
and R0 denote the resistance
of resistor Ri under stress and in the stress-free state. The
angles θi are defined with respect to the crystal coordinate
system as θ1 = θ5 = α+45°, θ2 = θ6 = −α−45°, θ3 = θ7
= α−45°, and θ4 = θ8 = −α+45°. The resulting stress
sensitivities SV due to an applied mechanical shear stress are
calculated as
)4()(
)4()(
04321
)''(
04132
)(
RRRRRS
RRRRRS
yx
xy
V
V
σΔ−Δ−Δ+Δ=
σΔ−Δ−Δ+Δ= (4)
and shown in Fig. 3(b) as dashed lines. As illustrated,
simulated and analytical stress sensitivities are in good
agreement. The point of isotropic sensitivity of sensor
variant II is found at α ≈ 2.5°, again with a sensitivity of
)(xy
V
S= )''( yx
V
S= 0.13 mV/(V MPa).
Figure 2. Optical micrograph of eight-terminal stress sensor components
(a) with non-conducting inner square (variant I) and bonding pads for
electrical characterization and (b) variant II.
Figure 3. Simulated and calculated voltage related stress sensitivities SV
(a) for variant I as a function of width w of inner non-conducting square
and (b) for variant II as a function of the angle α.

III. MEASUREMENT SETUP
In order to measure both stress sensitivities )( xy
V
S and
)''( yx
V
S, two strategies are applicable. Either two silicon strips
cut along the <100> and <110> directions containing the
same stress sensor components are applied in a four-point-
bending bridge (4PBB) setup or a single silicon strip cut
along the <110> direction is sequentially exposed to the
shear stress components σxy and σx’y’ in a 4PBB and TB
setup, respectively. As the first strategy requires two silicon
strips cut from two CMOS wafers, we opted for the second
solution. Aside from an existing 4PBB [4] that applies
σxy = σx’x’ / 2, the shear stress component σx’y’ is applied to
the silicon beam through torque along its longitudinal axis. A
schematic of such a TB setup is shown in Fig. 4. The silicon
beam containing the sensor structures is mounted between
two clamps. One clamp is fixed, while a moment Mx’ is
applied to the second clamp along the longitudinal axis of the
beam. Electrical connection to a printed circuit board (PCB)
is established using conventional wire bonding and can be
applied in both the 4PBB and the TB setup. During
measurement, the silicon beam is completely released from
the PCB and only the thin bond wires introduce a minimal
mechanical impact.
A. Calculation of Shear Stress
The resulting shear stress σx’y’ at the surface of the beam in
the TB is given analytically by [5]
2
'' bt
Mx
yx
η
σ
′
= (5)
where b and t, and η denote the beam width and thickness,
respectively, and a geometrical correction factor. As an
example, for b/t = 15, η is about 0.32.
In order to extract the spatial stress distribution on the
silicon beams, FEM simulations were performed using
COMSOL Multiphysics. Silicon beams with length, width
and height of 100 mm, 9 mm, and 680 μm, respectively, are
exposed to a moment of Mx’ = 0.01 Nm. Fig. 5 shows the
resulting σx’y’ distribution on top of the silicon beam
{Fig. 5(a)} and in the beam cross section {Fig. 5(b)}. The
sensor position is in the origin of the coordinate system.
Fig. 5(c) and (d) illustrate the respective dependencies of the
applied shear stress σx’y’ on the beam surface along the y’-
axis and z’-axis, respectively. As indicated in Fig. 5(c), a
homogeneous stress distribution is obtained in the inner
6 mm of the silicon beam. Within 1.5 mm close to the edges
of the silicon strip, the shear stress decreases from 7.6 MPa
to 1.4 MPa. Further, the shear stress decreases linearly over
the thickness of the silicon beam and vanishes in the beam
center.
B. Setup
The TB setup developed for this study is shown in Fig. 6.
The silicon beam is fixed between an upper and a lower
clamp. The upper clamp is mounted on a profile stand by an
xyz linear stage. The lower clamp is mounted on a torque
sensor (TD70, ME-Systeme, Germany, measurement range
0.15 Nm) attached to a motor driven rotation stage
(DMT100, OWIS, Germany, resolution 0.001°). The sensor
structure on the silicon strip is connected to a PCB through
thin bond wires minimizing any mechanical influence on the
silicon structure. This PCB can also be applied in the 4PBB
setup.
During the experiments, the torque sensor and respective
clamp are rotated with respect to the fixed clamp. The
resulting moment Mx’ allows to directly extract the applied
shear stress in the surface of the silicon strip using Eq. (5).
The xyz stage is used to adjust the position of the upper
clamp with respect to the axis of the rotary stage.
IV. EXPERIMENTAL RESULTS
The silicon beams used in this study were 100 mm long,
9 mm wide and 680 μm thick and cut along the <110>
direction of 6-inch silicon wafers. Rotations applied to the
torque sensor are ±5° leading to an applied torque of
Mx’ = ±40×10−3 Nm and resulting surface shear stresses
σx’y’ = ±30 MPa.
Figure 5. Simulated stress distribution σx’y’ in a beam under torsional load.
(a) Top view and (b) cross section (z’-direction not to scale).
Figure 4. Schematic of the torsional bridge setup with rotated clamp,
fixed clamp, and silicon beam with teststructures connected to a PCB.

To extract the voltage related stress sensitivities, a bias
voltage Vbias is applied between opposite contacts. The stress
related offset voltage Voff is measured at the corresponding
orthogonal contact pair. This results in eight independent
input voltage switching directions at angles ψ = 0°, 45°, ..,
315° to the [110] direction applying contact pairs C3−C7 ,
C2−C6 , .., C8−C4 , respectively. However, as a rotation by
180° does not change the behavior of the sensor, only the
first four switching modes ψ = 0°, 45°, 90°, and 135 ° are
considered here.
As an example, Fig. 7 shows offset voltages Voff under
different angles ψ at Vbias = 1 V measured with sensor
variant I with an inner square of w = 65 µm as a function of
the applied moment Mx’. The slight stress sensitivity under
ψ = 45° and ψ = 135° is either due to a possible
misalignment of the sensor element with respect to the
crystallographic directions resulting in a cross-sensitivity
towards the shear stress σx’y’ or a misalignment of the silicon
beam in the TB setup resulting in the unintended application
of shear stress σxy. The stress sensitivities SV are extracted
from linear fits to such data and complementary data
obtained from 4PBB measurements. For both sensor
variants, )''( yx
V
S is obtained from measurements under ψ = 0°
or 90° on the TB setup while )( xy
V
S is extracted under ψ = 45°
or 135° on the 4PBB. The respective results are shown in
Fig. 8 in comparison to the simulated sensitivities indicating
an excellent agreement.
V. CONCLUSIONS
We reported on the development and characterization of
eight-contact in-plane CMOS-based stress sensors with
isotropic sensitivities towards the distinct mechanical stress
components σxy and σx’y’. Isotropic sensor configurations are
found by geometric parameter variation of two sensor
configurations. In combination with a four-point bending
bridge, a novel torsional bridge setup allows for the first time
to apply both σxy and σx’y’ to the same silicon beam. The
shear stress distribution of the torsional bridge is simulated.
Excellent agreement between theory, model, and reality is
obtained for both sensor geometries.
ACKNOWLEDGMENT
The authors gratefully acknowledge the technical support
by Josef Joos in the development of the TB setup.
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[3] J. Richter, M. Arnoldus, O. Hansen, and E. V. Thomsen, “Four point
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Figure 6. Torsional bridge setup with mounted silicon beam.
Figure 7. Offset voltages Voff of variant I with w = 65 μm as a function of
applied moment Mx’ for different voltage angles ψ.
Figure 8. Comparison between calculated and measured stress
sensitivities )( xy
V
S and )''( yx
V
S for sensor structures shown in Fig. 1.