Resistive component of piezoresistivity

• Resistivity in an anisotropic material defined as a relation the electric field, E to the current flow, J
• E and J in 3-dimensions
• Resistivity, r defined as a second-rank tensor that has nine elements expressed in a 3׳ matrix
• These elements reduce to six independent values from symmetry considerations

• Case of silicon crystal aligned to the (100) axes
  Cubic lattice structure
  Entries ρ₁, ρ₂ and ρ₃ will be equal, as they all represent resistance along the (100) axes (denoted by ρ)
  Remaining components represent cross-axis resistivities, will be zero because unstressed silicon is electrically isotropic
• When stress is applied to silicon, the components in the resistivity matrix change:

• Matrix of relative resistivity d:
        where  Dr ...  resistivity variation

                       r    ...  unstrained resistivity

• Electric field - current relation in more general form:

• Relative resistivity
  • Related to the stress by the piezoresistive effect
  • Stress and resistivity are a second-rank tensors
  • Piezoresistive effect requires a four-rank tensor of piezoresistive coefficients for its full description
  • Due to symmetry conditions, this tensor is populated by only three non-zero components, as shown:

• Piezoresistive tensor coefficients, πij have units of Pa-1
• They may be either positive or negative and are sensitive to doping type, doping level and operating temperature
• π₁₁ relates the resistivity in any direction to stress in the same direction
• π₁₂ and π₄₄ are cross-terms
• Typical room-temperature values of piezoresistance coefficients and resistivity for p-type and n-type silicon:

• Relative resistance change
  Preferred representation expressing the fractional change in an arbitrarily oriented diffused resistor

Where

π_L and T_L are the piezoresistive coefficient and stress parallel to the direction of current flow in the resistor
                           (i.e., parallel to its length)

π_T and T_T are the values in the transverse direction

• Orientations of piezoresistor relatively to the applied stress
  correspond to the most common situations in piezoresistive sensor devices
  
  • Longitudinal orientation
  • Transversal orientation

• Piezoresistive coefficients for two possible gauge orientations in the case of (100) silicon wafers