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.

Figure 1. Definition of the normal stresses σi and shear stresses τi (i = 1, 2, 3)

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, 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 + π₁₂(σ₂ + σ₃)
ΔR₂/R = π₁₁σ₂ + π₁₂(σ₁ +σ₃)
ΔR₃/R = π₁₁σ₃ + π₁₂(σ₁ + 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.

Table 1. Resistivity and Piezoresistive Coefficients of Silicon at 25°C

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.

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 π_L = 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

Similarly for n-type silicon resistors, π₄₄ can be neglected:

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