Amin, A. Piezoresistivity The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Amin, A. “Piezoresistivity” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
51 Piezoresistivity 51.1 Introduction 51.2 Equation of State 51.3 Effect of Crystal Point Group on Ili 51.4 Geometric Corrections and Elastoresistance Tensor 51.5 Multivalley Semiconductors 51.6 Longitudinal Piezoresistivity Il and Maximum Sensitivity Directions 51.7 Semiconducting(PTCR) Perovskites Ahmed Amin 51.8 Thick film resistors Texas Instruments Inc. 51.9 Design Considerations 51.1 Introduction Piezoresistivity is a linear coupling between mechanical stress X& and electrical resistivity Pi. Hence, it is represented by a fourth rank polar tensor Ilik. The piezoresistance properties of semiconducting silicon and germanium were discovered by Smith [ 1953] when he was verifying the form of their energy surfaces. Piezore- sistance measurements can provide valuable insights concerning the conduction mechanisms in solids such as strain-induced carrier repopulation and intervalley scattering in multivalley semiconductors[Herring and Vogt, 956, barrier tunneling in thick film resistors Canali et al., 1980] and barrier raising in semiconducting positive temperature coefficient of resistivity(PTCR)perovskites [Amin, 1989]. Piezoresistivity has also been investi- gated in compound semiconductors, thin metal films Rajanna et al., 1990], polycrystalline silicon and germa nium thin films [Onuma and Kamimura, 1988, heterogeneous solids [Carmona et al., 1987], and high T. uperconductors [Kennedy et al, 1989]. Several sensors that utilize this phenomenon are commercially available 51.2 Equation of State The equation of state of a crystal subjected to a stress Xu and an electric field E; is conveniently formulated in the isothermal representation. The difference between isothermal and adiabatic changes, however, is negligible [ Keyes, 1960]. Considering only infinitesimal deformations, where the linear theory of elasticity is valid, the electric field E, is expressed in terms of the current density I and applied stress Xu as [Mason and Thurston, E1=E;(,X)ikl=1,2,3 (51.1) In what follows the summation convention over repeated indices in the same term is implied, and the letter subscripts assume the values 1, 2, and 3 unless stated otherwise. Expanding in a McLaurin's series about the origin(state of zero current and stress) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 51 Piezoresistivity 51.1 Introduction 51.2 Equation of State 51.3 Effect of Crystal Point Group on ’ijkl 51.4 Geometric Corrections and Elastoresistance Tensor 51.5 Multivalley Semiconductors 51.6 Longitudinal Piezoresistivity ’l and Maximum Sensitivity Directions 51.7 Semiconducting (PTCR) Perovskites 51.8 Thick Film Resistors 51.9 Design Considerations 51.1 Introduction Piezoresistivity is a linear coupling between mechanical stress Xkl and electrical resistivity rij . Hence, it is represented by a fourth rank polar tensor ’ijkl . The piezoresistance properties of semiconducting silicon and germanium were discovered by Smith [1953] when he was verifying the form of their energy surfaces. Piezoresistance measurements can provide valuable insights concerning the conduction mechanisms in solids such as strain-induced carrier repopulation and intervalley scattering in multivalley semiconductors [Herring and Vogt, 1956], barrier tunneling in thick film resistors [Canali et al., 1980] and barrier raising in semiconducting positive temperature coefficient of resistivity (PTCR) perovskites [Amin, 1989]. Piezoresistivity has also been investigated in compound semiconductors, thin metal films [Rajanna et al., 1990], polycrystalline silicon and germanium thin films [Onuma and Kamimura, 1988], heterogeneous solids [Carmona et al., 1987], and high Tc superconductors [Kennedy et al., 1989]. Several sensors that utilize this phenomenon are commercially available. 51.2 Equation of State The equation of state of a crystal subjected to a stress Xkl and an electric field Ei is conveniently formulated in the isothermal representation. The difference between isothermal and adiabatic changes, however, is negligible [Keyes, 1960]. Considering only infinitesimal deformations, where the linear theory of elasticity is valid, the electric field Ei is expressed in terms of the current density Ij and applied stress Xkl as [Mason and Thurston, 1957]. Ei = Ei (Ij , Xkl) i,j,k,l = 1,2,3 (51.1) In what follows the summation convention over repeated indices in the same term is implied, and the letter subscripts assume the values 1, 2, and 3 unless stated otherwise. Expanding in a McLaurin’s series about the origin (state of zero current and stress) Ahmed Amin Texas Instruments, Inc
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)
where m, n=1, 2,3,...6. The relation between the subscripts in both notations is Tensor:11223323,3213,3112,2 Matrix: 1 2 3 4 IImn= 2Ilik, for m and/or n=4,5,6 Thus, for example, Ill Il, I112=I1, 2[2323=I4, 2Il212= 166, and 21l 12=1116. Hence, Eq(51.5) takes the form (卻/P)=∏X,(i,j=1,2,…,6) Further reduction of the remaining 36 piezoresistivity tensor components is obtained by applying the generating elements of the point group to the piezoresistivity tensor transformation law Eq.(51.6)and demanding invariance. The following are two commonly encountered piezoresistivity matrices: 1 Cubic O,(m3m): single crystal silicon and germanium ∏, I1200 0000 2. Spherical(oo oo mmm): polycrystalline silicon and germanium and films I lI 000 000 00000L where Iu= 2(I 2 1). Thus, three coefficients I Ily, and Ilu are required to completely specify the piezoresistivity tensor for silicon and germanium single crystals, and only two, Iln and Il, for polycrystalline films Under hydrostatic pressure conditions, the piezoresistivity coefficient II, for the preceding two symmetry groups is a linear combination of the longitudinally and transverse 2 components, In=Il+ 2nl12. Unlike the elastic stiffness c; (a fourth rank polar tensor), the piezoresistivity tensor II mm is not symmetric, i. e, II mm# IIm, except for the following point groups, Co (oooo mmm),O, (m3m), T,(43m), and O(432) 51.4 Geometric Corrections and Elastoresistance Tensor The experimentally derived quantity is the piezoresistance coefficient 1/Ro(aR/aX). This must be corrected for the dimensional changes to obtain the piezoresistivity coefficient 1/p(dp/aX) as follows: Uniaxial tensile stress parallel to current flow 1/Ro(OR/dX)-(su-25,2)=1/po(dp/dx)=IM (51.10) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where m,n = 1, 2, 3,…, 6. The relation between the subscripts in both notations is Tensor: 11 22 33 23, 32 13, 31 12, 21 Matrix: 1 2 3 4 5 6 and Pmn = 2Pijk l , for m and/or n = 4, 5, 6 Thus, for example, P1111 = P11 , P1122 = P12 , 2P2323 = P44 , 2P1212 = P66 , and 2P1112 = P16 . Hence, Eq. (51.5) takes the form (dri/r0) = Pij Xj , (i, j = 1, 2,…,6) (51.9) Further reduction of the remaining 36 piezoresistivity tensor components is obtained by applying the generating elements of the point group to the piezoresistivity tensor transformation law Eq. (51.6) and demanding invariance. The following are two commonly encountered piezoresistivity matrices: 1. Cubic Oh(m3m): single crystal silicon and germanium 2. Spherical (• • mmm): polycrystalline silicon and germanium and films where P44 = 2(P11 2 P12). Thus, three coefficients P11, P12 , and P44 are required to completely specify the piezoresistivity tensor for silicon and germanium single crystals, and only two, P11 and P12 , for polycrystalline films. Under hydrostatic pressure conditions, the piezoresistivity coefficient Ph for the preceding two symmetry groups is a linear combination of the longitudinal P11 and transverse P12 components, Ph = P11 + 2P12 . Unlike the elastic stiffness cij (a fourth rank polar tensor), the piezoresistivity tensor Pmn is not symmetric, i.e., Pmn # Pnm, except for the following point groups, C• v(• • mmm), Oh(m3m), Td (43m), and O(432). 51.4 Geometric Corrections and Elastoresistance Tensor The experimentally derived quantity is the piezoresistance coefficient 1/R0(¶R/¶X). This must be corrected for the dimensional changes to obtain the piezoresistivity coefficient 1/r0(¶r/¶X) as follows: 1. Uniaxial tensile stress parallel to current flow 1/R0(¶R/¶X) –(s11 – 2s12) = 1/r0(¶r/¶x) = P11 (51.10) P11 P12 P12 000 P11 P12 000 P11 000 P44 0 0 P44 0 P¢44 P11 P12 P12 000 P11 P12 000 P11 000 P44 0 0 P44 0 P44
TABLE 51.1 Numerical values of n- and M for Selected materials Resistivity (10-m2/N) Dimensionless Material Unstrained and RT) 11.7(9-cm) 102.253.4-13.6-72.6864-10.8 78(-cm) 6.6-1.1138.1 0.527110 LaoozTiO 100(9cm) Thick film resistors DP 1351, main constituent Bi,Ru, O, 100(K④ 13.5 ESL 2900 100(Kg/ 13.8116 2. Uniaxial tensile stress perpendicular to current flow 1/R(OR/aX)+Sm= 1/p(dp/aX)=Il12 3. Hydrost ressure 1/Ro(OR/Op)-(su+ 2s12)=1/po(dp/ap)=lln (51.12) where su and su2 are the elastic compliances that appear in the linear elasticity equation x,=Sin Xu, with xi, the infinitesimal strain components. Details on the different geometries and methods of measuring the piezore sistance effect can be found in the References. Equation(51.9)could be written in terms of the strain conjugate (δp/P)=Max (51.13) the dimensionless quantity M is the elastoresistance tensor(known as the gage factor in the sensors literature) It is related to the piezoresistivity Il k and the elastic stiffness ck tensors by thus, the 3 independent elastoresistance components(gauge factors) can be expressed as follows ∏I +2 12C12 ∏1c12+∏12(c1 51.5 Multivalley semiconductors For a multivalley semiconductor, e.g., n-type silicon, the energy minima(ellipsoids of revolutions) of unstrained state in momentum space are along the six <100> cubic symmetry directions; they possess the symmetry group O,(m3m). A tensile stress in the x-direction, for example, will strain the lattice in the xy-plane and destroy the three-fold symmetry, thereby lifting the degeneracy of the energy minima. However, the four-fold symmetry along the x-direction will be preserved. Thus, the two valleys along the direction of stress will be shifted relative to the four valleys in the perpendicular directions e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 2. Uniaxial tensile stress perpendicular to current flow 1/R0(¶R/¶X) + s11 = 1/r0(¶r/¶X) = P12 (51.11) 3. Hydrostatic pressure 1/R0(¶R/¶p) – (s11+ 2s12) = 1/r0(¶r/¶p) = Ph (51.12) where s11 and s12 are the elastic compliances that appear in the linear elasticity equation xij = sijkl Xkl , with xij the infinitesimal strain components. Details on the different geometries and methods of measuring the piezoresistance effect can be found in the References. Equation (51.9) could be written in terms of the strain conjugate xo as follows (dri /r0) = Mio xo (51.13) the dimensionless quantity Mio is the elastoresistance tensor (known as the gage factor in the sensors literature). It is related to the piezoresistivity Pik and the elastic stiffness cko tensors by Mio = Pik cko (51.14) thus, the 3 independent elastoresistance components (gauge factors) can be expressed as follows M11 = P11 c11 + 2 P12 c12 M12 = P11 c12 + P12 (c11 + c12) M44 = P44 c44 51.5 Multivalley Semiconductors For a multivalley semiconductor, e.g., n-type silicon, the energy minima (ellipsoids of revolutions) of the unstrained state in momentum space are along the six <100> cubic symmetry directions; they possess the symmetry group Oh(m3m). A tensile stress in the x-direction, for example, will strain the lattice in the xy-plane and destroy the three-fold symmetry, thereby lifting the degeneracy of the energy minima. However, the four-fold symmetry along the x-direction will be preserved. Thus, the two valleys along the direction of stress will be shifted relative to the four valleys in the perpendicular directions. TABLE 51.1 Numerical Values of Pij and Mij for Selected Materials Resistivity (10–11 m2 /N) Dimensionless Material (Unstrained and RT) P11 P12 P44 M11 M12 M44 Silicon n-type 11.7 (W-cm) –102.2 53.4 –13.6 –72.6 86.4 –10.8 p-type 7.8 (W-cm) 6.6 –1.1 138.1 10.5 2.7 110 Ba.648Sr.35 La.002TiO3 ª100 (W-cm) 250 250 Thin films Si 15 Ge 30 Mn 160 (mW-cm) 3 Thick film resistors DP 1351, main constituent Bi2Ru2O7 100 (KW/h) 13.5 ESL 2900 100 (KW/h) 13.8 11.6