A combined square-shaped van der Pauw (VDP) sensor is proposed for calculation of the difference of the in-plane normal-stresses in this work. In previous works, two separate measurements were required, which is so inconvenient and tricky, because the different pair of point-contacts were used for 0 and/or 90 degree oriented resistance measurement. This work analytically presented how to resolve these problems by using a combined VDP configuration and compared the in-plane normal-stresses differential sensitivities for the different silicon planes and the coordinate systems (un-primed or primed). Furthermore, we offered another approach using the voltage difference, in which the sensitivity was observed to be linearly proportional to the unstressed value of voltage $V_{ref}$ (= $I$ x $R$).

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## I. INTRODUCTION

Traditionally, van der Pauw (VDP) structures were
used for the measurement of sheet resistance, $R_{s}$.
However, as stress sensors, these are currently widely
used to measure the die stresses in electronic packages
because of their characteristics of higher sensitivities
compared to the conventional resistor sensors ^{(1)}. Van
der Pauw, et al. ^{(2-}^{4)} demonstrated how to determine the
sheet resistance for isotropic and anisotropic conductors
of constant thickness. Price ^{(5,}^{6)} developed the
resistance equations for rectangular isotropic and
anisotropic conductors. Mian, et al. ^{(1,}^{7)} showed that
van der Pauw (VDP) devices have the potential to
remove some of the limitations of resistor-based sensors
when used as stress sensors. Semiconductor piezoresistive
stress sensors are used to measure stress and
they have so many applications as sensing elements in
various transducers ^{(1,}^{7-}^{9)}. Expressions of resistance
changes for VDP stress sensors were derived for stress
measurements ^{(9)}. However, this method is difficult to
measure the normal-stress because it requires two
separate measurements for 0° and 90° resistor sensor.
However, in this work, the voltage for 0° and 90°
oriented VDP sensor pair can be measured
simultaneously and then we can easily calculate the
normal-stress. In addition, we presented another
approach using the voltage difference between 0° and 90°
oriented VDP sensor pair, in which the sensitivity was
observed to be linearly proportional to the injection
current I times unstressed resistance $R$. In this case, we
can enhance normal-stress sensitivity by increasing the
applied injection current $I$ and/or increasing the
unstressed resistance $R$ for VDP sensor pair by low
doping in the process of fabrication.

## II. BASIC THEORIES

Van der Pauw’s theorem is used to measure the
specific resistivity of an arbitrary shaped sample of
constant thickness without isolated holes [1, 2-4].
However, as a stress-sensing element, VDP sensors are known to offer 3.157 times higher
sensitivity than an
analogous two element resistor sensor rosette ^{(1)}. A
simple structure with uniform thickness is shown in Fig. 1 where A, B, C, and D are contacts on the conducting
material. Simply, we denote AB = a and BC = b as
the length of the sides of the rectangle.

Fig. 1. A simple van der Pauw test structure and its symbolic representation with respect to the primed axes.

In VDP structure, a current is injected through one pair
of the contacts (e.g., contacts A and B), and the voltage is
measured across another pair of contacts (e.g., contacts D
and C). The potential difference between the contacts D
and C divided by the current through contacts A and B
can be expressed as R_{AB,CD}=(V_{D}-V_{C})/I_{AB}. Then, the
resistance of 0° and 90° VDP can be represented as ^{(1)}

##### (1)

$$\begin{array}{l} R_{A B, \mathrm{CD}}=R_{D C, \mathrm{BA}}=\mathrm{R}_{0} \\ =-\frac{8 \sqrt{\rho^{'}_{11} \rho^{'}_{22}-\left(\rho^{'}_{12}\right)^{2}}}{\pi \mathrm{t}} \ln \prod_{\mathrm{n}=0}^{\infty}\left\{\tanh \left[\sqrt{\frac{\rho^{'}_{22}}{\rho^{'}_{11}}} \frac{\mathrm{BC}}{\mathrm{AB}}(2 \mathrm{n}+1) \frac{\pi}{2}\right]\right\} \end{array}$$

##### (2)

$$\begin{array}{l} R_{A D, \mathrm{CB}}=R_{B C, \mathrm{DA}}=\mathrm{R}_{90} \\ =-\frac{8 \sqrt{\rho^{'}_{11} \rho^{'}_{22}-\left(\rho^{'}_{12}\right)^{2}}}{\pi \mathrm{t}} \ln \prod_{\mathrm{n}=0}^{\infty}\left\{\tanh \left[\sqrt{\frac{\rho^{'}_{11}}{\rho^{'}_{22}}} \frac{\mathrm{AB}}{\mathrm{BC}}(2 \mathrm{n}+1) \frac{\pi}{2}\right]\right\} \end{array}$$
where $\rho$’_{11} and $\rho$’_{22} are the principal resistivity
components. For 0° VDP sensor measurement, current is
injected from A to B and, simultaneously, we measure
voltage between D and C. On the other hand, for 90°
VDP sensor measurement, current injection is from A to
D and voltage is measured between B and C.

Considering only in-plane stress ($\sigma$_{11}, $\sigma$_{22}, and $\sigma$_{12}),
yields the calculated equations for VDP sensors in (001)
silicon surface. For the primed axes,

##### (3)

$$\frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157 \pi_{44}\left(\sigma_{11}^{\prime}-\sigma_{22}^{\prime}\right)$$

Table 1. Pi-coefficients for silicon [TPa^{-1}] ^{(10)}

Pi-coefficients |
p-type |
n-type |

$\pi$ |
66 |
-1022 |

$\pi$ |
-11 |
534 |

$\pi$ |
77 |
-1556 |

$\pi$ |
1380 |
-136 |

B |
718 |
-312 |

B |
-228 |
297 |

B |
946 |
-609 |

For the un-primed axes,

##### (4)

$$\frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(\pi_{11}-\pi_{22}\right)\left(\sigma_{11}-\sigma_{22}\right)$$Likewise, for the (111) silicon surface,

##### (5)

$$\frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(B_{1}-B_{2}\right)\left(\sigma'_{11}-\sigma'_{22}\right)$$
Table 1 presents the summary in sensitivity for the
difference of the in-plane normal-stresses for p- and ntype
silicon. For (001) silicon plane, p44 is much larger
than ($\pi$_{11}-$\pi$_{12}) for p-type silicon whereas ($\pi$_{11}-$\pi$_{12}) is much
larger than $\pi$_{44} for n-type silicon. Thus, for in-plane
differential stress sensors, picking the primed coordinate
system is recommended for p-type silicon while the
unprimed coordinate system is better for n-type silicon.

## III. CHIP DESIGN & ANALYSIS

As described above, for in-plane normal-stress
differential calculation, measuring the VDP sensors
versus the applied stress is so inconvenient and tricky
because we have the different pair of point-contacts for 0
and 90-degree oriented resistance measurement. In order
to resolve these problems, we proposed the combined
VDP configuration in this work as presented in Fig. 2 where R_{0} = (V_{1+} - V_{1-})/I and R_{90} = (V_{2+} - V_{2-})/I.

Then, the following notations (V_{1+} - V_{1-}) $\equiv$ V_{1} and (V_{2+}
- V_{2-}) $\equiv$ V_{2} lead to R_{0} = V_{1}/I and R_{90} = V_{2}/I. It is to be
noted that V_{1} $\cong$ V_{2} for the unstressed conditions. Also,
injection current I is maintained through 0° and 90°
combined VDP sensors. Hence, inverse current injection
(from 90° to 0°) does not make any difference as long as
the injection current I is maintained in this structure.

Fig. 2. A 0° and 90° combined VDP sensors for calculation of the difference of the in-plane normal-stresses.

Our Ideas start from the mathematical theory. If we let $\mathrm{A} \equiv \frac{\mathrm{V}_{1}}{\mathrm{V}_{2}},$ then $\frac{\Delta A}{A} \cong \frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)} \cdot$ Also, $\frac{\Delta A}{A} \cong \frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}$ for $\Delta \mathrm{V}_{1} \ll \mathrm{V}_{1}$ and $\Delta \mathrm{V}_{2} \ll \mathrm{V}_{2} .$ Hence, $\frac{\Delta A}{A}=\frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)} \cong$ $\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}$.

Note that pi-coefficients for silicon generally have the
unit of (tens~hundreds)/TPa (= 10^{-11} ~ 10^{-10} order) for
any doping-level. Also, all the stress components are
restricted to less than 100 MPa due to the stiff
characteristic of silicon. Hence, ΔV_{1} and ΔV_{2} arising
from the applied normal stresses are very small
compared to V_{1} and V_{2}, respectively.

For the unstressed case, V_{1} $\cong$ V_{2},

##### (6)

$$\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right) \cong \frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}$$Then, dividing both terms of Eq. (1) by Δ$\sigma$ yields

##### (7)

$$\frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\Delta \sigma}=\frac{\left(\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}\right)-\left(\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}\right)}{\Delta \sigma}$$Also, Eq. (7) can be expressed as the differential form, in which both terms in Eq. (6) are differentiated with respect to $\sigma$, as follows:

##### (8)

$$\frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\Delta \sigma} \equiv \frac{d}{d \sigma}\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)=\frac{d}{d \sigma}\left(\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}\right)-\frac{d}{d \sigma}\left(\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}\right)$$In Eqs. (3)-(5), we can rewrite ΔR_{0}/R_{0}-ΔR_{90}/R_{90} as

##### (9)

$$\begin{aligned} \frac{\Delta \mathrm{R}_{0}}{\mathrm{R}_{0}}-\frac{\Delta \mathrm{R}_{90}}{\mathrm{R}_{90}} &=\frac{\left(\mathrm{V}_{1}+\Delta \mathrm{V}_{1}\right) / \mathrm{I}-\mathrm{V}_{1} / \mathrm{I}}{\mathrm{V}_{1} / \mathrm{I}}-\frac{\left(\mathrm{V}_{2}+\Delta \mathrm{V}_{2}\right) / \mathrm{I}-\mathrm{V}_{2} / \mathrm{I}}{\mathrm{V}_{2} / \mathrm{I}} \\ &=\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}} \end{aligned}$$Then, combining Eq. (6) into Eq. (9) leads to

##### (10)

$$\frac{\Delta \mathrm{R}_{0}}{\mathrm{R}_{0}}-\frac{\Delta \mathrm{R}_{90}}{\mathrm{R}_{90}}=\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}=\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)$$Substitution Eq. (10) into Eqs. (3)-(5) yields

##### (11)

$$\begin{aligned} \Delta\left(\frac{V_{1}}{V_{2}}\right) &=3.157 \pi_{44}\left(\sigma_{11}^{\prime}-\sigma_{22}^{\prime}\right) \\ &=3.157\left(\pi_{11}-\pi_{22}\right)\left(\sigma_{11}-\sigma_{22}\right) \\ &=3.157\left(B_{1}-B_{2}\right)\left(\sigma_{11}-\sigma_{22}\right) \end{aligned}$$Then, we let the difference of the in-plane normalstresses,
$\sigma$_{ND} $\equiv$ ($\sigma$'_{11} - $\sigma$'_{22}) = ($\sigma$_{11} - $\sigma$_{22}). Differentiating Eq. (11) with respect to $\sigma$_{ND} leads to

##### (12)

$$\begin{aligned} \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) &=3.157 \pi_{44} \\ &=3.157\left(\pi_{11}-\pi_{22}\right) \\ &=3.157\left(B_{1}-B_{2}\right) \end{aligned}$$Compared to Eqs. (3)-(5), we can more easily
calculate the sensitivity with respect to $\sigma$_{ND} by the ratio
of V_{1}/V_{2} using a 0°/90° combined VDP configuration. It
is also to be noted that we can calculate the sensitivity by
two simultaneous measurements unlike the previous
works where two separate measurements were required.

Now, we present a new approach to sensitivity using
the difference between V_{1} and V_{2}.

##### (13)

$$\begin{aligned} \frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}} &=\frac{d\left(V_{1}\right)}{d \sigma_{N D}}-\frac{d\left(V_{2}\right)}{d \sigma_{N D}}=I \times \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-I \times \frac{d\left(R_{90}\right)}{d \sigma_{N D}} \\ &=I \times\left[\frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}\right] \end{aligned}$$By re-using the notation, $\sigma$_{ND} $\equiv$ ($\sigma$'_{11} - $\sigma$'_{22}) = ($\sigma$_{11} - $\sigma$_{22}),
we can rewrite Eqs. (3)-(5) as follows:

##### (14)

$$\begin{array}{l} \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157 \pi_{44} \times \sigma_{N D} \\ \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(\pi_{11}-\pi_{22}\right) \times \sigma_{N D} \\ \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(B_{1}-B_{2}\right) \times \sigma_{N D} \end{array}$$where R_{0} and R_{90} are the unstressed reference resistance.
Under unstressed conditions, it is obvious that R_{0} = R_{90}.
Adopting the notation R $\equiv$ R_{0} = R_{90} and differentiating
Eq. (14) with respect to $\sigma$_{ND} leads to,

##### (15)

$$\begin{array}{l} \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157 \pi_{44} \times R \\ \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157\left(\pi_{11}-\pi_{22}\right) \times R \\ \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157\left(B_{1}-B_{2}\right) \times R \end{array}$$Then, substitution Eq. (15) into Eq. (12) yields

Table 2. Normal-stress sensitivity comparisons [TPa$^{-1}$]

Sensitivity |
(111) |
(001) primed |
(001) unprimed |

Expression |
3.157(B |
3.157$\pi$ |
3.157($\pi$ |

p-type |
+2987 |
+4357 |
+243.1 |

n-type |
-1923 |
-429.4 |
-4912 |

##### (16)

$$\begin{aligned} &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157 \pi_{44} \times I \cdot R\\ &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157\left(\pi_{11}-\pi_{22}\right) \times I \cdot R \\ &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157\left(B_{1}-B_{2}\right) \times I \cdot R \end{aligned}$$Thus, as presented in Eq. (17), it is concluded that
sensitivity using the voltage difference (=V_{1}-V_{2})
measurement shows ($I$ x $R$) times that of the (V_{1}/V_{2})
measurement.

##### (17)

$$\begin{aligned} \frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}} &=(I \times R) \cdot \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) \\ &=V_{r e f} \cdot \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) \end{aligned}$$where we defined V_{ref} as the unstressed reference voltage.
And, it is obvious that V_{ref} $\equiv$ V_{1}-V_{2} if the sensor is
unstressed.

## IV. CONCLUSIONS

In this paper, using a combined van der Pauw (VDP)
configuration, the difference of the in-plane normalstresses,
$\sigma$_{ND} can be extracted by measurements of (V_{1}/V_{2}).
And, the normal stresses can be generated by 4PB (fourpoint-
bending) apparatus. Although a combined VDP
configuration exhibited the same sensitivity compared to
the traditional single VDP sensor, it offered more easier
and simpler measurement method with the simultaneous
measurements, which are not possible with the traditional
single VDP. We analytically and mathematically derived
and validated the equations of sensitivity for different
silicon planes and the coordinate systems. In addition, we
offered another approach using the voltage difference, in
which the sensitivity can be increased by the unstressed
value of voltage V_{ref} (= $I$ x $R$). In Table 2, the sensitivity
for silicon plane (with its coordinate systems) are
summarized.

For the approach of voltage-difference measurement,
the sensitivity can be obtained by multiplying the
sensitivity expressions in Table 2 by the unstressed value
of voltage V_{ref} (= $I$ x $R$). Generally, the unstressed value
of voltage in a VDP sensor is less than 1. Therefore, a
much lower doping in the VDP sensor-fabrication
processes is required for a higher sensitivity because the
resistivity is high for the low doping concentrations.

### ACKNOWLEDGMENTS

This work was supported by 2020 Hongik University Research Fund, and this work was supported by Basic Science Research Program through the Ministry of Education of the Republic of Korea and National Research Foundation of Korea (2016R1D1A1B0393 5561). Also, this work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1F1A10 59031).

### REFERENCES

## Author

Chun-Hyung Cho received the B.S. degree in Electrical Engineering from the Seoul National University, Seoul, South Korea, in 1997, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from Auburn University, Auburn, AL, in 2001 and 2007, respectively.

In 2009, he joined Hongik University, Sejong where he is currently an Associated professor in the Department of Electronic & Electrical engineering.

His research interests include the application of analytical and experimental methods of piezoresistive sensors to problems in electronic packaging.

Jonghoek Kim is an assistant professor at Hongik University. From 2011 to 2017, he worked as a senior researcher at Agency for Defense Development in South Korea.

His current research is on target tracking, control theory, robotics, and optimal estimation. In 2011, he earned a Ph.D. degree co-advised by Dr. Fumin Zhang and Dr. Magnus Egerstedt.

His Ph.D. research focuses on developing motion control law and motion planning algorithms for mobile robots, robotic sensor networks, and multi-agent system. Jonghoek Kim received his M.S. in Electrical and Computer Engineering from Georgia Institute of Technology in 2008 and his B.S. in Electrical and Computer Engineering from Yonsei university, South Korea in 2006.

Hyuk-Kee Sung received the B.S. and M.S. degrees in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 1999 and 2001, respectively, and Ph.D. degree in electrical engineering and computer sciences from the University of California, Berkeley, in 2006.

He was a Postdoctoral Researcher with the University of California, Berkeley.

He is now with the School of Electronic and Electrical Engineering, Hongik University, Seoul, Korea.

His research interests are in the area of optoelectronic devices, optical injection locking of semiconductor lasers, and optoelectronic oscillators.