# Piezoresistive Effect In Semiconductors

**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 σ_{xx}, σ_{yy}, and σ_{zz} along the cubic crystal axes *x, y,* and *z,* respectively, and three shear stresses τ_{yz}, τ_{zx}, and τ_{xy}, 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, π_{ij}, 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, π_{11}, π_{1}2, and π_{44}, that is

π_{11} = π_{22} = π_{33}

π_{12} = π_{21} = π_{13} = π_{31} = π_{23} = π_{32}

π_{44 }= π_{55} = π_{66}

Thus, the Π matrix becomes

π_{11} and π_{12} are associated with the normal stress components, whereas the coefficient π_{44} 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*)

ΔR_{1}/R = π_{11}σ_{1} + π_{12}(σ_{2} + σ_{3})

ΔR_{2}/R = π_{11}σ_{2} + π_{12}(σ_{1} +σ_{3})

ΔR_{3}/R = π_{11}σ_{3} + π_{12}(σ_{1} + σ_{2})

ΔR_{4}/R = π_{44}τ_{1}

ΔR_{5}/R = π_{44}τ_{2}

ΔR_{6}/R = π_{44}τ_{3}

The actual values of these three coefficients, π_{11}, π_{12}, and π_{44}, 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; σ_{L} and σ_{T} are the longitudinal and transverse stress, respectively; π_{L} and π_{T} are the piezoresistive coefficient along the longitudinal and transverse direction, respectively.

Since the values of three piezoresistive coefficients π_{11}, π_{12}, and π_{44} (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 π_{L} = 1/2(π_{11} + π_{12} + π_{44}) and the transverse coefficient in the 〈110〉 direction is πT = 1/2(π_{11} + π_{12} − π_{44}). Based on Table 1, π_{44} is more significant than π_{11} and π_{12} for p-type silicon resistors; thus Equation 3 is simplified for p-type silicon resistors as

Similarly for n-type silicon resistors, π_{44} can be neglected: