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. 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, 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 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 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