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.

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⁾

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,

For the un-primed axes,

Likewise, for the (111) silicon surface,

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 R₀ = (V₁₊ - V_1-)/I and R₉₀ = (V₂₊ - V_2-)/I.

Then, the following notations (V₁₊ - V_1-) $\equiv$ V₁ and (V₂₊ - V_2-) $\equiv$ 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.

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,

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

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:

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

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

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

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

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.

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

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,

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

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.

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