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 ? 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 X kl and electrical resistivity r ij . 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 T c 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 X kl and an electric field E i 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 i is expressed in terms of the current density I j and applied stress X kl as [Mason and Thurston, 1957]. E i = E i (I j , X kl ) 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. ? 2000 by CRC Press LLC dE i = (?E i /?I j ) dI j + (?E i /?X kl ) dX kl + (1/2!) [(? 2 E i /?I j ?I m ) dI j dI m + (? 2 E i /?X kl ?X no ) dX kl dX no + 2 (? 2 E i /?X kl ?I j ) dX kl dI j ] + . . . H.O.T (51.2) The partial derivatives in the expansion Eq. (51.2) have the following meanings: (?E i /?I j ) = r ij (electric resistivity tensor); (?E i /?X kl ) = d ikl (converse piezoelectric tensor); (? 2 E i /?X kl ?I j ) = (?/?X kl ) (?E i /?I j ) = P ijk l (piezoresistivity tensor); (? 2 E i /?I j ?I m ) = r ijm (nonlinear resistivity tensor); (? 2 E i /?X kl ?X no ) = d iklno (nonlinear piezoelectric tensor). Replacing the differentials in Eq. (51.2) by the components themselves, we get E i = r ij I j + d ikl X kl + 1/2 r ijm I j I m + 1/2 d iklno X kl X no + ? ijkl X kl I j (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 i = S j r ij I j + S j S k S l P ijkl X kl I j (51.4) taking the partial derivatives of Eq. (51.4) with respect to the current density I j and rearranging ?E i /?I j = r ij (X) 2 r ij (0) = S k S l P ijkl X kl Thus, the specific change in resistivity with stress is given by (dr ij /r 0 ) = P ijkl X kl (51.5) the piezoresistivity tensor P ijkl 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 P ijkl The transformation law of P ijkl (a fourth rank polar tensor) is as follows: P￠ ijkl = (?x￠ i /?x m )(?x￠ j /?x n )(?x￠ k /?x o )(?x￠ l /?x p )P mnop (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 /?x m \, . . . etc. are the Jacobian of the transformation. A general fourth rank tensor has 81 independent components. The piezoresistivity tensor P ijkl has the following internal symmetry: P ijkl = P ijlk = P jilk = P jikl (51.7) which reduces the number of independent tensor components from 81 to 36 for the most general triclinic point group C 1 (1). It is convenient to use the reduced (two subscript) matrix notation P ijkl = P mn (51.8) ? 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 P mn = 2P ijk l , for m and/or n = 4, 5, 6 Thus, for example, P 1111 = P 11 , P 1122 = P 12 , 2P 2323 = P 44 , 2P 1212 = P 66 , and 2P 1112 = P 16 . Hence, Eq. (51.5) takes the form (dr i /r 0 ) = P ij X j ,(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 O h (m3m): single crystal silicon and germanium 2. Spherical (￥ ￥ mmm): polycrystalline silicon and germanium and films where P 44 = 2(P 11 2 P 12 ). Thus, three coefficients P 11 , P 12 , and P 44 are required to completely specify the piezoresistivity tensor for silicon and germanium single crystals, and only two, P 11 and P 12 , for polycrystalline films. Under hydrostatic pressure conditions, the piezoresistivity coefficient P h for the preceding two symmetry groups is a linear combination of the longitudinal P 11 and transverse P 12 components, P h = P 11 + 2P 12 . Unlike the elastic stiffness c ij (a fourth rank polar tensor), the piezoresistivity tensor P mn is not symmetric, i.e., P mn # P nm , except for the following point groups, C ￥ v (￥ ￥ mmm), O h (m3m), T d (43m), and O(432). 51.4 Geometric Corrections and Elastoresistance Tensor The experimentally derived quantity is the piezoresistance coefficient 1/R 0 (?R/?X). This must be corrected for the dimensional changes to obtain the piezoresistivity coefficient 1/r 0 (?r/?X) as follows: 1. Uniaxial tensile stress parallel to current flow 1/R 0 (?R/?X) –(s 11 – 2s 12 ) = 1/r 0 (?r/?x) = P 11 (51.10) P 11 P 12 P 12 000 P 11 P 12 000 P 11 000 P 44 00 P 44 0 P￠ 44 P 11 P 12 P 12 000 P 11 P 12 000 P 11 000 P 44 00 P 44 0 P 44 ? 2000 by CRC Press LLC 2. Uniaxial tensile stress perpendicular to current flow 1/R 0 (?R/?X) + s 11 = 1/r 0 (?r/?X) = P 12 (51.11) 3. Hydrostatic pressure 1/R 0 (?R/?p) – (s 11 + 2s 12 ) = 1/r 0 (?r/?p) = P h (51.12) where s 11 and s 12 are the elastic compliances that appear in the linear elasticity equation x ij = s ijkl X kl , with x ij 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 x o as follows (dr i /r 0 ) = M io x o (51.13) the dimensionless quantity M io is the elastoresistance tensor (known as the gage factor in the sensors literature). It is related to the piezoresistivity P ik and the elastic stiffness c ko tensors by M io = P ik c ko (51.14) thus, the 3 independent elastoresistance components (gauge factors) can be expressed as follows M 11 = P 11 c 11 + 2 P 12 c 12 M 12 = P 11 c 12 + P 12 (c 11 + c 12 ) M 44 = P 44 c 44 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 O h (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 P ij and M ij for Selected Materials Resistivity (10 –11 m 2 /N) Dimensionless Material (Unstrained and RT) P 11 P 12 P 44 M 11 M 12 M 44 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 .648 Sr .35 La .002 TiO 3 ?100 (W-cm) 250 250 Thin films Si 15 Ge 30 Mn 160 (mW-cm) 3 Thick film resistors DP 1351, main constituent Bi 2 Ru 2 O 7 100 (KW /h) 13.5 ESL 2900 100 (KW /h) 13.8 11.6 ? 2000 by CRC Press LLC According to the deformation potential theory, the strain will shift the energy of all the states in a given band extremum by the same amount, i.e., the valley moves along the energy scale as a whole by an amount (the deformation potential constant) which is linearly proportional to the strain. Let’s assume that the energy of those on the y- and z-axes are lowered with respect to those on the x-axis. This effect is represented by dashed lines in Fig. 51.1. As a result, there will be electron transfer from the high to low energy valleys. The components of the mobility tensor m xy (= e t/m xy , where e is the electron charge, t is the relaxation time, and m xy is the effective mass) are illustrated by arrows in Fig. 51.1. The mobility anisotropy is due to the curvature of the conduction band near the bottom. The effective mass is inversely proportional to this curvature (1/m xy = (h/2p) –2 (? 2 E/?k x ?k y ), which is larger for a direction perpendicular to the valley. For an applied E field parallel to the stress, the conductivity will increase (i.e., the resistivity decreases) relative to the unstressed state because of the increase in the number of electrons in the four valleys (yz-plane) for which the mobility is large in the field direction. If the field is perpendicular to the stress, the conductivity will decrease (i.e., the resistivity increases) with stress. Therefore, the piezoresistivity components P 11 and P 12 have opposite signs. A shear stress about the crystallographic axes will not lift the degeneracy; hence, P 44 = 0. Similarly, a tensile stress along the <111> does not destroy the three-fold symmetry, and the degeneracy will not be lifted; thus, no piezoresistance should be there. Calculations based on the deformation potential model show that P 11 = 22 P 12 , and P 44 = 0. Information concerning the symmetry properties of the valleys can be derived from the representation surface of the longitudinal P 11 piezoresistance component. This surface can be constructed by measuring the depen- dence of P 11 on the crystallographic direction. Smith showed that P 11 is maximum in the <001> directions of n-type silicon and not quite zero in the <111> directions. Reasons for the deviation from the deformation potential model of piezoresistivity in multivalley semiconductors are discussed in Keyes [1960]. For n-type germanium P 11 is maximum in the <111> directions. This is consistent with the loci of the valleys in these two materials. Qualitatively, a piezoresistance effect is produced whenever the stress destroys the symmetry elements that are responsible for the degeneracy of the valleys. Intervalley scattering contribution to the piezoresistance of multivalley semiconductors may be comparable to that of the strain-induced electron repopulation. In this scattering process, the initial and final electron states are in different valleys. The effect of intervalley scattering can be deduced from the T –1 dependence of the elastoresistance tensor. FIGURE 51.1Two-dimensional representation of the constant energy surfaces in momentum space of a multivalley semiconductor (e.g., n-type silicon) showing only one quadrant, point group symmetry C 4v (4mm). (Source: C.S. Smith, Piezoresistance effect in germanium and silicon, Phys. Rev., vol. 94, p. 42, 1953. With permission.) ? 2000 by CRC Press LLC The influence of hydrostatic pressure on the electrical resistivity can provide additional insights on the transport properties. Some of the noted features include (1) high pressures (in the GPa range, versus MPa for tensile stresses) can be applied without destroying the crystal; (2) it does not destroy the symmetry, provided no phase transition is involved; hence, the symmetry degeneracies in the band structure are not lifted; (3) band edges which are not degenerate for symmetry reasons will be shifted; and (4) nonlinear effects could be discerned. 51.6Longitudinal Piezoresistivity P l and Maximum Sensitivity Directions Consider a long thin bar “strain gauge” cut from a piezoresistive crystal with the bar length parallel to an arbitrary direction in the crystal. Let P l , q, and j be the spherical coordinates of the longitudinal piezoresistivity tensor measured along the length of the bar. For the cubic symmetry group O h (m3m) of Si and Ge, P l is given by (Mason et al. 1957) P l = P 11 + 2(P 44 + P 12 - P 11 ) [sin 2 q cos 2 q + cos 4 q cos 2 j sin 2 j]. = P 11 + 2(P 44 + P 12 - P 11 ) F[q,j] The variation of P l with direction may be considered as a property surface. The distance from the center to any point in the surface is equal to the magnitude of P l . The function F[q,j] has a maximum for q = 54° 40` and j = 45° which is the <111> family of directions, for which P l takes the following form, P l = P 11 + 2/3 (P 44 + P 12 - P 11 ) If (P 44 + P 12 – P 11 ) and P 11 have the same sign or 2/3 * (P 44 + P 12 – P 11 ) * > P 11 then the maximum sensitivity direction occurs along <111>. If (P 44 + P 12 – P 11 ) = 0 the longitudinal effect is isotropic and equal to P 11 in all directions, otherwise it occurs along a crystal axis. The maximum sensitivity directions are shown in Fig. 51.2 for Si and Ge. 51.7 Semiconducting (PTCR) Perovskites Large hydrostatic piezoresistance P h coefficients (two orders of magnitude larger than those of silicon and germanium) have been observed in this class of polycrystalline semiconductors [Sauer et al., 1959]. PTCR compositions are synthesized by donor doping ferroelectric barium titanate BaTiO 3 , (Ba,Sr)TiO 3 , or (Ba,Pb)TiO 3 with a trivalent element (e.g., yttrium) or a pentavalent element (e.g., niobium). Below the ferroelectric transition temperature T c , Schottky barriers between the conductive ceramic grains are neutralized by the spontaneous polarization P s associated with the ferroelectric phase transition. Above T c the barrier height increases rapidly with temperature (hence the electrical resistivity) because of the disappearance of P s and the decrease of the paraelectric state dielectric constant. Analytic expressions that permit the computation of barrier heights under different elastic and thermal boundary conditions have been developed [Amin, 1989]. 51.8 Thick Film Resistors Thick film resistors consist of a conductive phase, e.g., rutile (RuO 2 ), perovskite (BaRuO 3 ), or pyrochlore (Pb 2 Ru 2 O 7-x ), and an insulating phase (e.g., lead borosilicate) dispersed in an organic vehicle. They are formed by screen printing on a substrate, usually alumina, followed by sintering at ?850 o C for 10 min. The increase of the piezoresistance properties of a commercial thick film resistor (ESL 2900 series) with sheet resistivity is illustrated in Fig. 51.3. The experimentally observed properties such as the resistance increase and decrease with tensile and compressive strains, respectively, and the increase of the elastoresistance tensor with sheet resistivity seem to support a barrier tunneling model [Canali et al., 1980]. ? 2000 by CRC Press LLC FIGURE 51.2 Section of the longitudinal piezoresistivity surface, the maximum sensitivity directions in Si and Ge are shown [Keys, 1960]. FIGURE 51.3 Relative changes of resistance for compressive and tensile strain applied parallel to the current direction. Note the increase of gage factor with sheet resistivity. 1.0 0.8 0.6 0.4 0.2 0.2 1000 800 800 600 600 1000 400 400 200 200 0.4 0.6 0.8 1.0 1 K?/ 10 K?/ 100 K?/ GF L = 5.5 GF L = 9.8 GF L = 13.8 R R (%) Tension m Strains m Strains Compression ESL 2900 Series ? 2000 by CRC Press LLC 51.9 Design Considerations Many commercially available sensors (pressure, acceleration, vibration,… etc.) are fabricated from piezoresistive materials (see for example, Chapter 56 in this handbook.) The most commonly used geometry for pressure sensors is the edge clamped diaphragm. Four resistors are usually deposited on the diaphragm and connected to form a Wheatstone bridge. The deposition technique varies depending upon the piezoresistive material: standard IC technology and micro-machining for Si type diaphragms; sputtering for thin film metal strain gauges; bonding for wire strain gauges, and screen printing for thick film resistors. Different types of diaphragms (sapphire, metallic, ceramic,… etc.) have been reported in the literature for hybrid sensors. To design a highly accurate and sensitive sensor, it is necessary to analyze the stress–strain response of the diaphragm using plate theory and finite element techniques to take into account: (1) elastic anisotropy of the diaphragm, (2) large deflections of plate (elastic non linearities), and (3) maximum sensitivity directions of the piezoresistivity coefficient. Signal conditioning must be provided to compensate for temperature drifts of the gauge offset and sensitivity. Defining Terms r ij : Electric resistivity tensor d ik l : Converse piezoelectric tensor P ijk l : Piezoresistivity tensor r ij m : Nonlinear resistivity tensor d ikln o : Nonlinear piezoelectric tensor Related Topic 1.1 Resistors References A. Amin, “Numerical computation of the piezoresistivity matrix elements for semiconducting perovskite fer- roelectrics,” Phys. Rev. B, vol. 40, 11603, 1989. C. Canali, D. Malavasi, B. Morten, M. Prudenziati, and A.Taroni, “Piezoresistive effect in thick-film resistors,” J. Appl. Phys., vol. 51, 3282, 1980. F. Carmona, R. Canet, and P. Delhaes, “Piezoresistivity in heterogeneous solids,” J. Appl. Phys., vol. 61, 2550, 1987. C. Herring and E. Vogt, “Transport and deformation-potential theory for many valley semiconductors with anisotropic scattering,” Phys. Rev., vol. 101, 944, 1956. R. J. Kennedy, W. G. Jenks, and L. R. Testardi, “Piezoresistance measurements of YBa 2 Cu 3 O 7-x showing large magnitude temporal anomalies between 100 and 300 K,” Phys. Rev. B, vol. 40, 11313, 1989. R. W. Keyes, “The effects of elastic deformation on the electrical conductivity of semiconductors,” Solid State Phys., vol. 11, 149, 1960. W. P. Mason and R. N. Thurston, “Use of piezoresistive materials in the measurement of displacement, force, and torque,” J. Acoust. Soc. Am., vol. 10, 1096, 1957. Y. Onuma and K. K. Kamimura, “Piezoresistive elements of polycrystalline semiconductor thin films,” Sensors and Actuators, vol. 13, 71, 1988. K. Rajanna, S. Mohan, M. M. Nayak, and N. Gunasekaran, “Thin film pressure transducer with manganese film as the strain gauge,” Sensor and Actuators, vol. A 24, 35, 1990. H. A. Sauer, S. S. Flaschen, and D. C. Hoestery, “Piezoresistance and piezocapacitance effect in barium strontium titanate ceramics,” J. Am. Ceram. Soc., vol. 42, 363, 1959. C. S. Smith, “Piezoresistance effect in germanium and silicon,” Phys. Rev., vol. 94, 42, 1953. ? 2000 by CRC Press LLC Further Information M. Neuberger and S. J. Welles, Silicon, Electronic Properties Information Center, Hughes Aircraft Co., Culver City, Calif., 1969. This reference contains a useful compilation of the piezoresistance properties of silicon. Electronic databases such as Chemical Abstracts will provide an update on the current research on piezore- sistance materials and properties. ? 2000 by CRC Press LLC