The influence of hydrostatic pressure on the electrical resistivity can provide additional insights transport properties. Some of the noted features include(1)high pressures(in the GPa range, versus 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 51.6 Longitudinal Piezoresistivity Il 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 Il ,0, and op be the spherical coordinates of the longitudinal piezoresistivity tensor measured along the length of the bar. For the cubic symmetry group O, (m3m)of Si and Ge, II is given by(Mason et al. 1957) ∏1=∏1+2(∏4+∏12-I1)[sin2cos36+ cose cosp sin ∏1+2(∏4+∏12-1)F[ep) The variation of Il, 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 nl The function F[0, p] has a maximum for 0= 5440 and (p=45 which is the <111> family of directions, for which Il, takes the following form, 1=I1+2/3(∏4+∏12-∏1) If(II+In-I)and nl have the same sign or 2/3(nl +ll2-Ilu) >I,u then the maximum sensit irection occurs along <111>. If(Il4 +l2-ll)=0 the longitudinal effect is isotropic and equal to lln 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 IIn 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 oppositions are synthesized by donor doping ferroelectric barium titanate BaTiO,,(Ba, Sr)TiO3,or ith a trivalent element(e.g, yttrium) or a pentavalent element(e.g, niobium). Below the ferroelectric transition temperature T- Schottky barriers between the conductive ceramic grains are neutralized by the spontaneous polarization P, associated with the ferroelectric phase transition. Above T the barrier height increases rapidly with temperature(hence the electrical resistivity) because of the disappearance of P, 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(Ruo2), perovskite(BaRuO,), or pyrochlor (Pb,Ru, O,-), 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 =850C for 10 min. he 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 e 2000 by CRC Press LLC© 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. 51.6 Longitudinal Piezoresistivity Pl 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 Pl , q, and j be the spherical coordinates of the longitudinal piezoresistivity tensor measured along the length of the bar. For the cubic symmetry group Oh (m3m) of Si and Ge, Pl is given by (Mason et al. 1957) Pl = P11 + 2(P44 + P12 - P11) [sin2 q cos2 q + cos4 q cos2 j sin2 j]. = P11 + 2(P44 + P12 - P11) F[q,j] The variation of Pl 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 Pl . The function F[q,j] has a maximum for q = 54° 40` and j = 45° which is the <111> family of directions, for which Pl takes the following form, Pl = P11 + 2/3 (P44 + P12 - P11) If (P44 +P12 – P11) and P11 have the same sign or 2/3 * (P44 +P12 – P11) * > P11 then the maximum sensitivity direction occurs along <111>. If (P44 + P12 – P11) = 0 the longitudinal effect is isotropic and equal to P11 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 Ph 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 BaTiO3, (Ba,Sr)TiO3, or (Ba,Pb)TiO3 with a trivalent element (e.g., yttrium) or a pentavalent element (e.g., niobium). Below the ferroelectric transition temperature Tc , Schottky barriers between the conductive ceramic grains are neutralized by the spontaneous polarization Ps associated with the ferroelectric phase transition.Above Tc the barrier height increases rapidly with temperature (hence the electrical resistivity) because of the disappearance of Ps 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 (RuO2), perovskite (BaRuO3), or pyrochlore (Pb2Ru2O7-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 ª850oC 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]