dE,=(dE /al)dl;+(dE /dXu)dx +(1/2!)[(dE /dI dlm)di dI, +(d E /aXxaX,o)dXdX +2(E,X0)ull+…HOT (51.2) The partial derivatives in the expansion Eq. (51.2) have the following meanings: (dE /aip)=pi (electric resistivity tensor);(aE, /aX)=dikrconverse piezoelectric tensor);(a E /aX al)=(a/aX)(aE /ar)=Ij piezoresistivity tensor);(aE /aI aIm)=Pim(nonlinear resistivity tensor);(d'E /aX aXno)=8,uno(nonlinear piezoelectric tensor) Replacing the differentials in Eq (51.2) by the components themselves, we get E;=Pi L;+ dk XH+ 1/2 Pm I Im 1/2 dino Xy Xno+ Iju Xy I (51.3) Most of the technologically important piezoresistive materials, e.g., silicon, germanium, and polycrystalline films, are centrosymmetric. The effect of center of symmetry(i.e, the inversion operator) on Eq (51.3)is to force all odd rank tensor coefficients to zero; hence, the only contribution to the resistivity change under stress will result from the piezoresistive term. Therefore, Eq. (51.3)takes the form E=2p;+222∏Xl caking the partial derivatives of Eq (51.4)with respect to the current density I; and rearranging dE /dl=P (X)2 Pi (0)=2E,Ily Xy Thus, the specific change in resistivity with stress is given by (p/po)=∏1X ne piezoresistivity tensor Ili in Eq (51.5) has the dimensions of reciprocal stress(square meters in the MKS system of units). The effects of the intrinsic symmetry of the piezoresistivity tensor and the crystal 51.3 Effect of Crystal Point Group on IlikI The transformation law of I(a fourth rank polar tensor) is as follows I=(dx /axm)(dx; /dxm)(axk/dxo)(dxi/dxp)IIm (516) where the primed and unprimed components refer to the new and old coordinate systems, respectively, and the determinants of the form dx /dxmll are the Jacobian of the transformation. A general fourth rank tensor has 81 independent components. The piezoresistivity tensor Iiu has the following internal symmetry ∏lu=∏k==k which reduces the number of independent tensor components from 81 to 36 for the most general triclinic point group C(1). It is convenient to use the reduced(two subscript) matrix notation e 2000 by CRC Press LLC© 2000 by CRC Press LLC dEi = (¶Ei/¶Ij) dIj + (¶Ei/¶Xkl) dXkl + (1/2!) [(¶2 Ei/¶Ij¶Im) dIj dIm + (¶2 Ei/¶Xk l¶Xno) dXkl dXno + 2 (¶2 Ei/¶Xk l¶Ij ) dXkl dIj ] + . . . H.O.T (51.2) The partial derivatives in the expansion Eq. (51.2) have the following meanings: (¶Ei/¶Ij ) = ri j (electric resistivity tensor); (¶Ei/¶Xkl) = dik l (converse piezoelectric tensor); (¶2 Ei/¶Xk l¶Ij ) = (¶/¶Xkl) (¶Ei/¶Ij ) = Pijk l (piezoresistivity tensor); (¶2 Ei/¶Ij¶Im) = rijm (nonlinear resistivity tensor); (¶2 Ei/¶Xk l¶Xno) = diklno (nonlinear piezoelectric tensor). Replacing the differentials in Eq. (51.2) by the components themselves, we get Ei = rijIj + dikl Xkl + 1/2 rijm IjIm + 1/2 diklno Xkl Xno + ’ijkl Xkl Ij (51.3) Most of the technologically important piezoresistive materials, e.g., silicon, germanium, and polycrystalline films, are centrosymmetric. The effect of center of symmetry (i.e., the inversion operator) on Eq. (51.3) is to force all odd rank tensor coefficients to zero; hence, the only contribution to the resistivity change under stress will result from the piezoresistive term. Therefore, Eq. (51.3) takes the form Ei = Sj rij Ij + Sj SkSl Pijkl Xkl Ij (51.4) taking the partial derivatives of Eq. (51.4) with respect to the current density Ij and rearranging ¶Ei/¶Ij = rij(X) 2 rij(0) = SkSlPijkl Xkl Thus, the specific change in resistivity with stress is given by (drij/r0) = Pijkl Xkl (51.5) the piezoresistivity tensor Pijkl in Eq. (51.5) has the dimensions of reciprocal stress (square meters per newton in the MKS system of units). The effects of the intrinsic symmetry of the piezoresistivity tensor and the crystal point group are discussed next. 51.3 Effect of Crystal Point Group on Pijkl The transformation law of Pijkl (a fourth rank polar tensor) is as follows: P¢ijkl = (¶x¢ i/¶xm)(¶x¢ j/¶xn)(¶x¢ k /¶xo)(¶x¢ l /¶xp)Pmnop (51.6) where the primed and unprimed components refer to the new and old coordinate systems, respectively, and the determinants of the form \¶x¢ i /¶xm \, . . . etc. are the Jacobian of the transformation. A general fourth rank tensor has 81 independent components. The piezoresistivity tensor Pijkl has the following internal symmetry: Pijkl = Pijlk = Pjilk = Pjikl (51.7) which reduces the number of independent tensor components from 81 to 36 for the most general triclinic point group C1(1). It is convenient to use the reduced (two subscript) matrix notation Pijkl = Pmn (51.8)