I. INTRODUCTION

For many years, hydrostatic pressure sensors have been widely studied for stress measurements such as in the break system of a vehicle as sensing elements in various transducers^[1-10]. In the future, the applications of hydrostatic pressure sensors will be widened such as in the medical field for blood pressure monitoring with development of wearable devices. In addition, hydrostatic pressure sensors can be applicable to various fields of micro-electronics or optics such as wavelength tuning of laser diodes and birefringent measurements.

For hydrostatic pressure calculation, semiconductor piezo-resistive sensors are generally used for its small size. Piezo-resistive sensors are generally composed of resistor stress sensors which are conveniently fabricated into the surface of the die using current microelectronic technology^[11-15]. Expressions of resistance changes for piezo-resistive stress sensors were derived for stress measurements^[2].

So far, a single meandering-patterned resistor has been used for hydrostatic pressure measurement. However, this method required the laborious $\Delta R/R$ measurement for an individual resistor, that is, we needed to check the change in current with respect to the sweeping voltage for resistance measurement, and what is worse it exhibited a very low sensitivity. In this work, the measuring inconveniences was resolved by calibrating the mid-point voltage of the combined (p-type in the n-substrate and n-type in the p-well) resistor sensor pair, and even the sensitivity was improved drastically by controlling the applied voltage. It should be noted that we had already shown the mid-point measurement method for normal-stress and shear-stress calculation in our previous works^[15]. Now, we extended the application to the hydrostatic pressure measurement.

II. GENERAL THEORIES

Hydrostatic pressure is the force acting on a unit area for the object surrounded by gases or liquids. A schematic for the hydrostatic pressure-sensor is shown in Fig. 1.

Fig. 1. A schematic for the hydrostatic pressure-sensor system.

We analyzed the hydrostatic pressure sensitivities of silicon for the selected silicon surfaces, resistor angle with respect to the primary axis, and sensor configurations.

III. DESIGN & ANALYSIS

The test chip contains the p-type and n-type resistor. Resistors are often designed with relatively large meandering patterns to achieve acceptable resistance levels for measurement. In previous our works^[3], it was observed that the pressure coefficients for both the p- and n-type silicon have small values ($\pi _{p, p}$ = 145.9/TPa (p-type) and $\pi _{p, n}$ = 31.0/TPa (n-type) at room temperature). The layout of the test chip for hydrostatic tests is shown in Fig. 2.

Fig. 2. A simplified layout of the test chip.

The voltage for an n-substrate was set to be the same as the applied voltage V (positive value), whereas the voltage for a p-well was set to be 0 (GND). This is for electrical isolation between the doped surface resistor and the bulk of the chip by using proper reverse biasing of the resistor and substrate regions. The pad ‘X’ denotes the junction of p- and n-type resistor sensors. We applied the constant voltage between pad ‘V’ (+5V) and pad ‘GND’ (0V). Now, we proposed and analyzed a measurement-method in which the hydrostatic pressure sensitivity can be determined by the change in ($R_{p}$/$R_{n}$) with respect to hydrostatic pressure p as follows:

where x is the junction voltage.

where subscript p and n denotes p-type and n-type, respectively. Also, Rpo and Rno denotes the reference resistance value of p- and n-type during measurements, respectively.Calibration of $f ( \Delta T )$ is needed to accurately determine the pressure coefficients. Then, substitution Eq. (6) into Eq. (5) yields

Also, we can express Eq. (5) as

where $\frac{\pi_{p, p} p}{1+f(\Delta T)_{p}} << 1$ and $\frac{\pi_{p, n} p}{1+f(\Delta T)_{n}} << 1$ for any stress level. Using the approximation theory for $A << 1$ and $B << 1$, $(1+\mathrm{A}) /(1+\mathrm{B}) \approx(1+\mathrm{A})(1-\mathrm{B}) \approx 1+\mathrm{A}-\mathrm{B}$ leads to

From Eqs. (7, 9), the hydrostatic-pressure sensitivity can be expressed as the change in x with respect to p as follows:

where $\mathrm{R}_{\mathrm{po}}\left[1+\pi_{\mathrm{p}, \mathrm{p}} \mathrm{p}+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right]$ is the final value for p-type resistance during hydrostatic measurements ($\equiv \mathrm{R}_{\mathrm{p}}$) while $\mathrm{R}_{\mathrm{no}}\left[1+\pi_{\mathrm{p}, \mathrm{n}} \mathrm{p}+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right]$ is the final value for n-type resistance ($\equiv \mathrm{R}_{\mathrm{n}}$). Also, by approximation in Eq. (10), $\mathrm{R}_{\mathrm{po}}\left[1+\pi_{\mathrm{p}, \mathrm{p}} \mathrm{p}+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right] \approx \mathrm{R}_{\mathrm{po}}\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right]$ and $\mathrm{R}_{\mathrm{no}}\left[1+\pi_{\mathrm{p}, \mathrm{n}} \mathrm{p}+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right] \approx \mathrm{R}_{\mathrm{no}}\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right]$ because $\pi_{\mathrm{p}, \mathrm{p} \mathrm{p}} << \left(1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right)$ and $\pi_{\mathrm{p}, \mathrm{n}} \mathrm{p} << \left(1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right)$. Also, in Eq. (10), defining $V \cdot\left(R_{p} \cdot R_{n}\right) /\left(R_{p}+R_{n}\right)^{2} \equiv M$ leads to

where M can be increased by controlling the applied voltage V. Also, the case in which $R_{p}$ = $R_{n}$ gives the maximum M of V/4. In our cases, $R_{p}$=10.5KΩ and $R_{n}$=2.25KΩ at room temperature. Therefore, M = 0.145·V. If we set V=20, we may have approximately 3 times magnification (M=2.90) for measurement of hydrostatic pressure.

It should be noted that ($\pi_{p, p}-\pi_{p, n}$) in Eqs. (10, 11) must be modified along with the reference temperature. Because the piezo-resistive coefficients decrease as temperature goes up, the magnitude in $\pi_{p, p}$, $\pi_{p, n}$, and therefore the magnitude in ($\pi_{p, p}-\pi_{p, n}$) also decrease with rising temperature. In order to cover the wide temperature ranges, we must have the data of $\pi_{\mathrm{p}}\left(=-\left(\pi_{11}+2 \pi_{12}\right)\right)$ for p- and n-type silicon over the temperature ranges. $\pi_{11}$ and $\pi_{12}$ for both p- and n-type can be determined by the four-point bending (4PB) technique in which only in-plane normal stress $\sigma_{11}$ (and/or $\sigma _{11}^{\prime}$) is induced. During hydrostatic tests, the die is put into the pressure vessel, whose set up is shown in Fig. 3. A pump connected to the vessel is used to generate pressure.

Fig. 3. Hydrostatic test chamber and setup^[5].

During the application of pressure, a change in T is inevitable. Also, for determining $\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}$ and $\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}$ in Eq. (10), TCR (temperature coefficient of resistance) measurements are required. $\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}$ and $\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}$ may be extracted by measuring the normalized resistance change with respect to a temperature change. To this purpose, extensive calibrations of resistance with varying temperatures for p- and n-type are performed in a temperature controllable chamber. During the TCR measurements, no stress is applied. For high temperatures, we used a resistance heater inside the pressure vessel to increase the temperature of fluid. To lower the temperature of fluid, we used liquid nitrogen, which is injected into a specially designed box surrounding the pressure vessel. The recording of the actual temperature is made by a thermistor inside the vessel. At a given reference temperature, a quadratic equation is enough to fit $\mathrm{f}(\Delta \mathrm{T})$ over any small range of temperature, especially for the hydrostatic tests, for both p-type and n-type samples. However, if the temperature is assumed to be maintained constant at the reference temperature ($\mathrm{f}(\Delta \mathrm{T})$=0) during the measurement, Eq. (10) reduces to

The comparisons of hydrostatic-pressure sensitivity between previous works and our newly proposed works are shown in Table 1.

In our previous works^[5], we evaluated the pressure coefficient by performing the hydrostatic experiments. By subtraction of the temperature-induced resistance change $\mathrm{f}(\Delta \mathrm{T})$, adjusted pressure coefficient for p- and n-type sensors were obtained ($\pi _{p, p}$ = 145.9 TPa^-1, $\pi _{p, n}$ = 31.0 TPa^-1). At room temperature, a linear (or quadratic) equation is enough to fit $\mathrm{f}(\Delta \mathrm{T})$ over any small range of temperature, especially for the hydrostatic tests, for both p-type and n-type samples. Around the room temperature with stress-free conditions, $\mathrm{f}(\Delta \mathrm{T})_{p}$ = 1.56·10^-3 $\Delta T$/℃ and $\mathrm{f}(\Delta \mathrm{T})_{n}$ = 1.70·10^-3 $\Delta T$/℃. During the application of hydrostatic pressure, p up to 13.3 MPa, the temperature change of the hydraulic fluid was about 0.67℃. An example of measured and temperature-induced normalized resistance change are plotted together for p- and n-type resistor sensor (see Fig. 4). Also, adjusted resistance change for p- and n-type resistor sensor is presented in Fig. 5.

Fig. 4. An example of $\Delta R/R$ for p- and n-type resistors during hydrostatic test.

Fig. 5. Adjusted resistance change for p- and n-type resistors during hydrostatic test.

If we assume the same conditions as above (hydrostatic pressure, p = 13.3 MPa, $\Delta T$ = 0.67 ℃), the hydrostatic pressure sensitivity ‘dx/dp’ in Eq. (11) is calculated in Table 2.

Compared to p-type sensor, ($\pi_{p, p}$ = 145.9 TPa^-1), this proposed method roughly enhanced the hydrostatic-pressure sensitivity by 15% for V = 10 and 130% for V=20. The sensitivity comparisons among the previous works (p-type and n-type) and the proposed method (p- and n-type combined) are shown in Fig. 6.

Fig. 6. The sensitivity comparisons among 3 methods.

IV. CONCLUSIONS

For the accurate hydrostatic pressure calculation, we derived the mathematical and analytical expressions in which a temperature change during measurements was considered. In this work, we could achieve the high hydrostatic-pressure sensitivity, compared to the previous results of our works, by controlling the applied voltage V. Here, we exploited the sensitivity difference between p- and n-type sensors. And then, we amplified its difference by increasing the applied voltage V. We just needed to measure the junctional mid-point voltage ‘x’ in p- and n-type combined semiconductor resistor sensors. So far, for the measurement of the hydrostatic-pressure, we have used one type single resistor sensor whose resistance change is very difficult to measure due to its low sensitivity. Also, we analyzed the sensitivity in hydrostatic-pressure between conventional method and our newly proposed method. The magnitude in sensitivity of the mid-point voltage with respect to hydrostatic-pressure, p, was approximately increased by 14% for V = 10 (128% for V =20).