Channel E ACT) FIGURE 53.2 A lumped element model of a superconductor. are actually two sets of interpenetrating electron fluids: the uncorrelated electrons providing ohmic conduction and the correlated ones creating supercurrents. This two-fluid model is a useful way to build temperature effects into the london relations Under the two-fluid model, the electrical current density, J, is carried by both the uncorrelated(normal electrons and the superelectrons: J=J,+ J, where J, is the normal current density. The two channels are modeled in a circuit as shown in Fig. 53.2 by a parallel combination of a resistor (representing the ohmic hannel)and an inductor(representing the superconducting channel). To a good approximation, the respective temperature dependences of the conductor and inductor are forT≤T (53.10) T A(T)=A0) fort< t (53.11) (T/T) where o. is the dc conductance of the normal channel. Strictly speaking, the normal channel should also contain an inductance representing the inertia of the normal electrons, but typically such an inductor contrib utes negligibly to the overall electrical response. )Since the temperature-dependent penetration depth is defined as A(D=A(T)/u,, the effective conductance of a superconductor in the sinusoidal steady state is =+ (53.12) where the explicit temperature dependence notation has been suppressed. should be noted that the temperature dependencies given in Equations(53.10)and (53. 11)are not precisely correct for the high-T materials. It has been suggested that this is because the angular momentum of the electrons forming a Cooper pair in high-T materials is different from that in low-T ones. Nevertheless, the wo-fluid picture of transport and its associated constitutive law, Eq (53. 12), are still valid for high-T super conductors Most of the important physics associated with the classical model is embedded in Eq. (53. 12). As is cle from the lumped element model, the relative importance of the normal and superconducting channels is a e 2000 by CRC Press LLC© 2000 by CRC Press LLC are actually two sets of interpenetrating electron fluids: the uncorrelated electrons providing ohmic conduction and the correlated ones creating supercurrents. This two-fluid model is a useful way to build temperature effects into the London relations. Under the two-fluid model, the electrical current density, J, is carried by both the uncorrelated (normal) electrons and the superelectrons: J = Jn + Js where Jn is the normal current density. The two channels are modeled in a circuit as shown in Fig. 53.2 by a parallel combination of a resistor (representing the ohmic channel) and an inductor (representing the superconducting channel). To a good approximation, the respective temperature dependences of the conductor and inductor are (53.10) and (53.11) where so is the dc conductance of the normal channel. (Strictly speaking, the normal channel should also contain an inductance representing the inertia of the normal electrons, but typically such an inductor contrib￾utes negligibly to the overall electrical response.) Since the temperature-dependent penetration depth is defined as l(T) = , the effective conductance of a superconductor in the sinusoidal steady state is (53.12) where the explicit temperature dependence notation has been suppressed. It should be noted that the temperature dependencies given in Equations (53.10) and (53.11) are not precisely correct for the high-Tc materials. It has been suggested that this is because the angular momentum of the electrons forming a Cooper pair in high-Tc materials is different from that in low-Tc ones. Nevertheless, the two-fluid picture of transport and its associated constitutive law, Eq. (53.12), are still valid for high-Tc super￾conductors. Most of the important physics associated with the classical model is embedded in Eq. (53.12). As is clear from the lumped element model, the relative importance of the normal and superconducting channels is a FIGURE 53.2 A lumped element model of a superconductor. s s ˜ o o c c T T c T T ( ) = ( ) T T Ê Ë Á ˆ ¯ ˜ £ 4 for L T L T T T T c ( ) c = ( ) - ( ) Ê Ë Á Á ˆ ¯ ˜ ˜ 0 £ 1 1 4 for L(T) mo § s s w l = + m ˜ o o j 1 2