allows the possibility of a device whose output voltage is a function of a static magnetic field. If two weak links are connected in parallel, the lumped version of Faraday s law gives the voltage across the second weak link as V2=V,+(da/dt), where a is the total flux threading the loop between the links. Substituting Eq.(53. 15), integrating with respect to time, and again setting the integration constant to zero yields 2-q=(2x@)Φ (53.19) showing that the spatial change in the phase of the macroscopic wavefunction is proportional to the local magnetic flux. The structure described is known as a superconducting quantum interference device(SQuId)an can be used as a highly sensitive magnetometer by biasing it with current and measuring the resulting voltag as a function of magnetic flux. From this discussion, it is apparent that a duality exists in how fields interact with the macroscopic phase: electric fields are coupled to its rate of change in time and magnetic fields are oupled to its rate of change in space 53.4 Types of Superconductors The macroscopic quantum nature of superconductivity also affects the general electromagnetic properties previously discussed. This is most clearly illustrated by the interplay of the characteristic lengths 5, representing the scale of quantum correlations, and A, representing the scale of electromagnetic screening. Consider the scenario where a magnetic field, H, is applied parallel to the surface of a semi-infinite superconductor. The correlations of the electrons in the superconductor must lower the overall energy of the system or else the material would not be superconducting in the first place. Because the critical magnetic field H destroys all the correlations, it is convenient to define the energy density gained by the system in the superconducting state as (1/2)u,H2. The electrons in a Cooper pair are separated on a length scale of 5, however, and so the correlations cannot be fully achieved until a distance roughly S from the boundary of the superconductor. There is thus an energy per unit area, (1/2)uH25, that is lost because of the presence of the boundary. Now consider the effects of the applied magnetic field on this system. It costs the superconductor energy to maintain the Meissner effect, B=0, in its bulk; in fact the energy density required is(1/2)] H2. However, since the field can penetrate the perconductor a distance roughly A, the system need not expend an energy per unit area of(1/2)HHato screen over this volume. To summarize, more than a distance s from the boundary, the energy of the material is lowered(because it is superconducting), and more than a distance A from the boundary the energy of the material is raised(to shield the applied field) Now,ifλ<ξ,the of superconducting material greater than A from the boundary but less than s will higher in energy than that in the bulk of the material. Thus, the surface energy of the boundary is positive and so costs the total system some energy. This class of superconductors is known as type I Most elemental uperconductors, such as aluminum, tin, and lead, are type I. In addition to having A< s, type I superconductors are generally characterized by low critical temperatures(5 K)and critical fields(-005 T). Typical type I superconductors and their properties are listed in Table 53.2. TABLE 53.2 Material Parameters for Type I conductors* Material T (K) A, (nm) E(nm) A, (mev) HoH(mT) 00090 123.0 "The penetration depth A, is given at zero temperature, as are the coher- ence length Eo, the thermodynamic critical field Hee, and the energy gap a Source: R ]. Donnelly, " Cryogenics,in Physics Vade mecum, H L Ander- on, Ed, New York: American Institute of Physics, 1981. with permission. e 2000 by CRC Press LLC© 2000 by CRC Press LLC allows the possibility of a device whose output voltage is a function of a static magnetic field. If two weak links are connected in parallel, the lumped version of Faraday’s law gives the voltage across the second weak link as n2 = n1 + (dF/dt), where F is the total flux threading the loop between the links. Substituting Eq. (53.15), integrating with respect to time, and again setting the integration constant to zero yields (53.19) showing that the spatial change in the phase of the macroscopic wavefunction is proportional to the local magnetic flux. The structure described is known as a superconducting quantum interference device (SQUID) and can be used as a highly sensitive magnetometer by biasing it with current and measuring the resulting voltage as a function of magnetic flux. From this discussion, it is apparent that a duality exists in how fields interact with the macroscopic phase: electric fields are coupled to its rate of change in time and magnetic fields are coupled to its rate of change in space. 53.4 Types of Superconductors The macroscopic quantum nature of superconductivity also affects the general electromagnetic properties previously discussed. This is most clearly illustrated by the interplay of the characteristic lengths j, representing the scale of quantum correlations, and l, representing the scale of electromagnetic screening. Consider the scenario where a magnetic field, H, is applied parallel to the surface of a semi-infinite superconductor. The correlations of the electrons in the superconductor must lower the overall energy of the system or else the material would not be superconducting in the first place. Because the critical magnetic field Hc destroys all the correlations, it is convenient to define the energy density gained by the system in the superconducting state as (1/2)moHc 2 . The electrons in a Cooper pair are separated on a length scale of j, however, and so the correlations cannot be fully achieved until a distance roughly j from the boundary of the superconductor. There is thus an energy per unit area, (1/2)moHc 2 j, that is lost because of the presence of the boundary. Now consider the effects of the applied magnetic field on this system. It costs the superconductor energy to maintain the Meissner effect, B = 0, in its bulk; in fact the energy density required is (1/2)moH2 . However, since the field can penetrate the superconductor a distance roughly l, the system need not expend an energy per unit area of (1/2)moH2l to screen over this volume. To summarize, more than a distance j from the boundary, the energy of the material is lowered (because it is superconducting), and more than a distance l from the boundary the energy of the material is raised (to shield the applied field). Now, if l < j, the region of superconducting material greater than l from the boundary but less than j will be higher in energy than that in the bulk of the material. Thus, the surface energy of the boundary is positive and so costs the total system some energy. This class of superconductors is known as type I. Most elemental superconductors, such as aluminum, tin, and lead, are type I. In addition to having l < j, type I superconductors are generally characterized by low critical temperatures (;5 K) and critical fields (;0.05 T). Typical type I superconductors and their properties are listed in Table 53.2. TABLE 53.2 Material Parameters for Type I Superconductors* Material Tc (K) lo (nm) jo (nm) Do (meV) m0Hco (mT) Al 1.18 50 1600 0.18 110.5 In 3.41 65 360 0.54 123.0 Sn 3.72 50 230 0.59 130.5 Pb 7.20 40 90 1.35 180.0 Nb 9.25 85 40 1.50 198.0 *The penetration depth lo is given at zero temperature, as are the coher￾ence length jo, the thermodynamic critical field Hco, and the energy gap Do. Source: R.J. Donnelly, “Cryogenics,” in Physics Vade Mecum, H.L. Ander￾son, Ed., New York: American Institute of Physics, 1981. With permission. f f p 2 1 - = (2 F F) o