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  1. (Department of Electronic & Electrical Engineering, College of Science and Technology, Hongik University, Sejong, 30016, Korea)
  2. (School of Electronic & Electrical Engineering, College of Engineering, Hongik University, Wowsan-Ro 94, Seoul, 04066, Korea)

Terms—Semiconductor sensor, van der Pauw, stress-sensor, pressure-senor, normal-stress


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.


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 RAB,CD=(VD-VC)/IAB. Then, the resistance of 0° and 90° VDP can be represented as (1)

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

$$\begin{array}{l} R_{A D, \mathrm{CB}}=R_{B C, \mathrm{DA}}=\mathrm{R}_{90} \\ =-\frac{8 \sqrt{\rho^{'}_{11} \rho^{'}_{22}-\left(\rho^{'}_{12}\right)^{2}}}{\pi \mathrm{t}} \ln \prod_{\mathrm{n}=0}^{\infty}\left\{\tanh \left[\sqrt{\frac{\rho^{'}_{11}}{\rho^{'}_{22}}} \frac{\mathrm{AB}}{\mathrm{BC}}(2 \mathrm{n}+1) \frac{\pi}{2}\right]\right\} \end{array}$$

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,

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

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

























For the un-primed axes,

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

Likewise, for the (111) silicon surface,

$$\frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(B_{1}-B_{2}\right)\left(\sigma'_{11}-\sigma'_{22}\right)$$

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.


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 R0 = (V1+ - V1-)/I and R90 = (V2+ - V2-)/I.

Then, the following notations (V1+ - V1-) $\equiv$ V1 and (V2+ - V2-) $\equiv$ V2 lead to R0 = V1/I and R90 = V2/I. It is to be noted that V1 $\cong$ V2 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, ΔV1 and ΔV2 arising from the applied normal stresses are very small compared to V1 and V2, respectively.

For the unstressed case, V1 $\cong$ V2,

$$\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right) \cong \frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}$$

Then, dividing both terms of Eq. (1) by Δ$\sigma$ yields

$$\frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\Delta \sigma}=\frac{\left(\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}\right)-\left(\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}\right)}{\Delta \sigma}$$

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:

$$\frac{\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)}{\Delta \sigma} \equiv \frac{d}{d \sigma}\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)=\frac{d}{d \sigma}\left(\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}\right)-\frac{d}{d \sigma}\left(\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}\right)$$

In Eqs. (3)-(5), we can rewrite ΔR0/R0-ΔR90/R90 as

$$\begin{aligned} \frac{\Delta \mathrm{R}_{0}}{\mathrm{R}_{0}}-\frac{\Delta \mathrm{R}_{90}}{\mathrm{R}_{90}} &=\frac{\left(\mathrm{V}_{1}+\Delta \mathrm{V}_{1}\right) / \mathrm{I}-\mathrm{V}_{1} / \mathrm{I}}{\mathrm{V}_{1} / \mathrm{I}}-\frac{\left(\mathrm{V}_{2}+\Delta \mathrm{V}_{2}\right) / \mathrm{I}-\mathrm{V}_{2} / \mathrm{I}}{\mathrm{V}_{2} / \mathrm{I}} \\ &=\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}} \end{aligned}$$

Then, combining Eq. (6) into Eq. (9) leads to

$$\frac{\Delta \mathrm{R}_{0}}{\mathrm{R}_{0}}-\frac{\Delta \mathrm{R}_{90}}{\mathrm{R}_{90}}=\frac{\Delta \mathrm{V}_{1}}{\mathrm{V}_{1}}-\frac{\Delta \mathrm{V}_{2}}{\mathrm{V}_{2}}=\Delta\left(\frac{\mathrm{V}_{1}}{\mathrm{V}_{2}}\right)$$

Substitution Eq. (10) into Eqs. (3)-(5) yields

$$\begin{aligned} \Delta\left(\frac{V_{1}}{V_{2}}\right) &=3.157 \pi_{44}\left(\sigma_{11}^{\prime}-\sigma_{22}^{\prime}\right) \\ &=3.157\left(\pi_{11}-\pi_{22}\right)\left(\sigma_{11}-\sigma_{22}\right) \\ &=3.157\left(B_{1}-B_{2}\right)\left(\sigma_{11}-\sigma_{22}\right) \end{aligned}$$

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

$$\begin{aligned} \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) &=3.157 \pi_{44} \\ &=3.157\left(\pi_{11}-\pi_{22}\right) \\ &=3.157\left(B_{1}-B_{2}\right) \end{aligned}$$

Compared to Eqs. (3)-(5), we can more easily calculate the sensitivity with respect to $\sigma$ND by the ratio of V1/V2 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 V1 and V2.

$$\begin{aligned} \frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}} &=\frac{d\left(V_{1}\right)}{d \sigma_{N D}}-\frac{d\left(V_{2}\right)}{d \sigma_{N D}}=I \times \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-I \times \frac{d\left(R_{90}\right)}{d \sigma_{N D}} \\ &=I \times\left[\frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}\right] \end{aligned}$$

By re-using the notation, $\sigma$ND $\equiv$ ($\sigma$'11 - $\sigma$'22) = ($\sigma$11 - $\sigma$22), we can rewrite Eqs. (3)-(5) as follows:

$$\begin{array}{l} \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157 \pi_{44} \times \sigma_{N D} \\ \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(\pi_{11}-\pi_{22}\right) \times \sigma_{N D} \\ \frac{\Delta R_{0}}{R_{0}}-\frac{\Delta R_{90}}{R_{90}}=3.157\left(B_{1}-B_{2}\right) \times \sigma_{N D} \end{array}$$

where R0 and R90 are the unstressed reference resistance. Under unstressed conditions, it is obvious that R0 = R90. Adopting the notation R $\equiv$ R0 = R90 and differentiating Eq. (14) with respect to $\sigma$ND leads to,

$$\begin{array}{l} \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157 \pi_{44} \times R \\ \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157\left(\pi_{11}-\pi_{22}\right) \times R \\ \frac{d\left(R_{0}\right)}{d \sigma_{N D}}-\frac{d\left(R_{90}\right)}{d \sigma_{N D}}=3.157\left(B_{1}-B_{2}\right) \times R \end{array}$$

Then, substitution Eq. (15) into Eq. (12) yields

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



(001) primed

(001) unprimed













$$\begin{aligned} &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157 \pi_{44} \times I \cdot R\\ &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157\left(\pi_{11}-\pi_{22}\right) \times I \cdot R \\ &\frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}}=3.157\left(B_{1}-B_{2}\right) \times I \cdot R \end{aligned}$$

Thus, as presented in Eq. (17), it is concluded that sensitivity using the voltage difference (=V1-V2) measurement shows ($I$ x $R$) times that of the (V1/V2) measurement.

$$\begin{aligned} \frac{d\left(V_{1}-V_{2}\right)}{d \sigma_{N D}} &=(I \times R) \cdot \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) \\ &=V_{r e f} \cdot \frac{d}{d \sigma_{N D}}\left(\frac{V_{1}}{V_{2}}\right) \end{aligned}$$

where we defined Vref as the unstressed reference voltage. And, it is obvious that Vref $\equiv$ V1-V2 if the sensor is unstressed.


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 (V1/V2). 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 Vref (= $I$ 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 Vref (= $I$ 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.


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).


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Chun-Hyung Cho

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

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

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.