# Piezoresistive Effect In Semiconductors

Tuesday, November 21st, 2017 -

## Piezoresistive Effect In Semiconductors

Piezoresistive effect in silicon and germanium was discovered by Charles Smith in 1954. He found that both p-type and n-type silicon and germanium exhibited much greater piezoresistive effect than metals. Piezoresistivity of silicon arises from the deformation of the energy bands as a result of applied stress.  In turn, the deformed bands affect the effective mass and the mobility of electrons and holes, hence modifying resistivity or conductivity.

To understand how the resistance change relates to the applied stress, consider an infinitesimally small cubic piezoresistive crystal element with normal stresses σxx, σyy, and σzz along the cubic crystal axes x, y, and z, respectively, and three shear stresses τyz, τzx, and τxy, as indicated in Figure 1. The piezoresistive effect in this case can be described by relating the resistance change ΔR to each of the six stress components using a matrix of 36 coefficients, πij, expressed in Pa-1, as shown in Equation 1. where the vector DR represents the change in resistance with corresponding stress components, R is the original resistance, and Π is a 6 × 6 piezoresistive coefficient matrix. If the coordinate axes coincide with the crystal axes, a cubic crystal has three independent, nonvanishing elastic components, π11, π12, and π44, that is

π11 = π22 = π33
π12 = π21 = π13 = π31 = π23 = π32
π44 = π55 = π66

Thus, the Π matrix becomes π11  and π12 are associated with the normal stress components, whereas the coefficient π44 is related to the shearing stress components. By expanding the aforementioned matrix equation (for convenience, using 1—xx, 2—yy, 3—zz, 4—yz, 5—zx, and 6—xy)

ΔR1/R = π11σ1 + π122 + σ3)
ΔR2/R = π11σ2 + π1213)
ΔR3/R = π11σ3 + π121 + σ2)
ΔR4/R = π44τ1
ΔR5/R = π44τ2
ΔR6/R = π44τ3

The actual values of these three coefficients, π11, π12, and π44, depend on the angles of the piezoresistor with respect to silicon crystal lattice. The values of these coefficients in 〈100〉 orientation at room temperature (25°C) are given in Table 1.

In practical applications, a thin strip of silicon is commonly used to make a strain gauge sensor, instead of a three-dimensional cube. In this case, change in resistance versus in-plane stresses in the longitudinal (parallel to the current) direction and transverse (perpendicular to the current) direction can be expressed as where ΔR and R are the change in resistance and the original resistance, respectively; σL and σT are the longitudinal and transverse stress, respectively; πL and πT are the piezoresistive coefficient along the longitudinal and transverse direction, respectively. Table 2. Longitudinal and Transverse Piezoresistive Coefficients for Various Combinations of Directions in Cubic Crystals

Since the values of three piezoresistive coefficients π11, π12, and π44 (defined in a coordinate system aligned to the 〈100〉 axis of the silicon crystal) are known, all the piezoresistance coefficients of silicon in an arbitrary Cartesian system can be determined using coordinate system transformation. Table 2 lists longitudinal and transverse piezoresistance coefficients for various practical directions in cubic crystals. When the piezoresistors are fabricated, their orientation with respect the silicon crystal is usually in the 〈110〉 direction. From the last row of Table 2.13, the longitudinal piezoresistive coefficient in the 〈110〉 direction is πL = 1/2(π11 + π12 + π44) and the transverse coefficient in the 〈110〉 direction is πT = 1/2(π11 + π12 − π44). Based on Table 1, π44 is more significant than π11  and π12 for p-type silicon resistors; thus Equation 3 is simplified for p-type silicon resistors as