III. DESIGN & ANALYSIS
The test chip contains the ptype and ntype 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 ntype silicon
have small values ($\pi _{p, p}$ = 145.9/TPa (ptype) and $\pi _{p, n}$ = 31.0/TPa
(ntype) 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 nsubstrate was set to be the same as the applied voltage V (positive
value), whereas the voltage for a pwell 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 ntype resistor sensors. We applied the constant voltage between pad ‘V’
(+5V) and pad ‘GND’ (0V). Now, we proposed and analyzed a measurementmethod 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 ptype and ntype, respectively. Also, Rpo and Rno
denotes the reference resistance value of p and ntype 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 hydrostaticpressure 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 ptype 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 ntype 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 piezoresistive 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 ntype silicon over the temperature ranges. $\pi_{11}$
and $\pi_{12}$ for both p and ntype can be determined by the fourpoint bending
(4PB) technique in which only inplane 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 ntype 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 ptype and ntype 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 hydrostaticpressure sensitivity between previous works and our
newly proposed works are shown in Table 1.
Table 1. Comparison of hydrostaticpressure sensitivity between the ‘conventional’
method and ‘combined’ method
Method

Sensitivity

Expression

I. Con. (p)

$\pi_{\mathrm{p}, \mathrm{p}}$

$\Delta \mathrm{R} / \mathrm{R}=\pi_{\mathrm{p}, \mathrm{p}} \cdot \mathrm{p}+\mathrm{f}(\Delta
\mathrm{T})_{\mathrm{p}}$

II. Con. (n)

$\pi_{\mathrm{p}, \mathrm{n}}$

$\Delta \mathrm{R} / \mathrm{R}=\pi_{\mathrm{p}, \mathrm{n}} \cdot \mathrm{p}+\mathrm{f}(\Delta
\mathrm{T})_{\mathrm{n}}$

III. Combined (p & n)

$\mathrm{M} \cdot\left(\pi_{\mathrm{p}, \mathrm{p}} /\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right]\right.$
$\pi_{\mathrm{p}, \mathrm{n}} /\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right]
)$

$\mathrm{d} \mathrm{x} / \mathrm{dp}=\mathrm{M} \cdot\left(\pi_{\mathrm{p}, \mathrm{p}}
/\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right]\right.$$\pi_{\mathrm{p,n}}
/\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right] )$

In our previous works^{[5]}, we evaluated the pressure coefficient by performing the hydrostatic experiments.
By subtraction of the temperatureinduced resistance change $\mathrm{f}(\Delta \mathrm{T})$,
adjusted pressure coefficient for p and ntype 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 ptype and ntype samples. Around the room temperature with stressfree 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 temperatureinduced normalized resistance change are plotted together for p and
ntype resistor sensor (see Fig. 4). Also, adjusted resistance change for p and ntype resistor sensor is presented
in Fig. 5.
Fig. 4. An example of $\Delta R/R$ for p and ntype resistors during hydrostatic
test.
Fig. 5. Adjusted resistance change for p and ntype 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.
Table 2. Analysis in hydrostaticpressure sensitivity in our proposed method
V [v]

M

$\pi_{p, p} /\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{p}}\right]\pi_{\mathrm{p,n}}
/\left[1+\mathrm{f}(\Delta \mathrm{T})_{\mathrm{n}}\right]$ [TPa^{1}]

dp/dx [TPa^{1}]

5

0.725

114.78

83.2

10

1.450

114.78

166.4

15

2.175

114.78

249.6

20

2.900

114.78

332.8

Compared to ptype sensor, ($\pi_{p, p}$ = 145.9 TPa^{1}), this proposed method roughly enhanced the hydrostaticpressure sensitivity by 15%
for V = 10 and 130% for V=20. The sensitivity comparisons among the previous works
(ptype and ntype) and the proposed method (p and ntype combined) are shown in
Fig. 6.
Fig. 6. The sensitivity comparisons among 3 methods.