Axial Stress and StrainStress: σ=F/A (N/m2, Pascal)Strain: ε=∆L/L0 (dimensionless)Young’s modulus:(elastic modulus) εσ==strainstressEStrain Gauge and PiezoresistivityεσE=Hook’s law (1-D) Stretch: positiveContraction: negativeShear Stress and StrainLXAFG∆===γτ(rad) anglent displacemeshear stressshear Shear stress:Shear modulusof elasticityAF=τ Poisson’s RatiolttlDDLLεενεε−==∆−=∆=strain allongitudinstrain transverse,00)1(2 vEG+=Shear strain: γWhat’s the sign of v ?General CaseHooke’s law in the most generic form:Definitions of the normal stresses and shear stresses on a cubic elementThere are six independent stress componentsCorrespondingly, there are six independent strain componentsnnmnmEεσ∑==61=321321τττσσσ321321γγγεεεFor cubic-lattice crystalsWith the vector of stressOriented along the [100] axis, there are only 3 independent coefficients321321τττσσσ321321γγγεεε36 coefficientsAxial stressesShear stressesAxial strainsShear strainsStiffness coefficientsOutput voltage as a functionOf core position in a linear Differential transformer.Circuit modelPiezoresistive strain gaugePiezoresistivity- the resistivity change due to mechanical stressPiezoresistive effectStrain Gauge, GLet’s take a look at a resistor. We have the following equation for the resistance: L: lengthA: area of cross sectionρ: resistivityGeometry changeSemiconductor (Si, Ge) has much larger gauge factorjρ=ΕFor simple case (ρ is a scalar): For 3-D anisotropic crystal: Next, let’s find the the relationship between dρ/ρ and ε (strain) or σ (stress) The resistivity tensor has 6 componentsPiezoresistive coefficientsFor Si, Ge, there are only 3 independent coefficientsJoule’s law:Now we know the 6 resistive coefficients when the resistor experiences stresses.Q: how to calculate the resistance change at an arbitrary direction?g123(here E is electrical field, don’t be confused by Young’s modulus in previous slides)g123Assume l, m, n are the direction cosines associated with the unit vector g. With the help of the direction cosines, the current, electrical, stress in g direction can all be projected to the crystal axes. For exampleggggggniiimiiiliii======)cos()cos()cos(321γβααβγThe relative resistance change can be expressed byttllRRσπσπ+=∆πl: longitudinal piezoresistive coefficient πt: transverse piezoresistive coefficient σl: longitudinal stressσt: transverse stressThe general expressions for πland πt are obtained by applying coordinate transform))(())((22221222122214412111221212121212144121111nnmmllnmnlmltl++−−+=++−−−=ππππππππππWhere (l1, m1, n1) and (l2, m2, n2) are the sets of direction cosines between the longitudinal resistor direction (subscript 1) and the crystal axis, and between the transverse resistor direction (subscript 2) and the crystal axis. Example: calculate πl and πtin [110] direction. [100][010][001][110]0,2/2,2/2111=== nml0,2/2,2/2222==−= nml)(21)(21441211110,441211110,ππππππππ−+=++=tlp-type silicon in the (001) plane n-type silicon in the (001) planePiezoresistance coefficients πland πtin (100) silicon Schematic representation of the diamond lattice unit cell of silicon[110] direction:Applications of piezoresistive strain gaugePressure sensorAccelerometermaF =MicrophoneQuantum-Physical