Delin, K.A., Orlando, T.P. Superconductivity The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Delin, K.A., Orlando, T.P. “Superconductivity” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
53 Superconductivity Kevin A. Delin 53.1 Introduction Terry P. Orlando 53.2 General Electromagnetic Properties Massachusetts Institute of 53.3 Superconducting Electronics 53.4 Types of Superconductors 53.1 Introduction The fundamental idea behind all of a superconductor's unique properties is that superconductivity is a quantum mechanical phenomenon on a macroscopic scale created when the motions of individual electrons are corre- lated. According to the theory developed by John Bardeen, Leon Cooper, and Robert Schrieffer(BCS theory) nis correlation takes place when two electrons couple to form a Cooper pair For our purposes, we may therefore consider the electrical charge carriers in a superconductor to be Cooper pairs(or more colloquially, superelec rons)with a mass m* and charge twice those of normal electrons. The average distance between the two electrons in a Cooper pair is known as the coherence length, s. Both the coherence length and the binding energy of two electrons in a Cooper pair, 24, depend upon the particular superconducting material. Typically, the coherence length is many times larger than the interatomic spacing of a solid, and so we should not think of Cooper pairs as tightly bound electron molecules. Instead, there are many other electrons between those of a specific Cooper pair allowing for the paired electrons to change partners on a time scale of h/(2A)where h Planck's constant If we prevent the Cooper pairs from forming by ensuring that all the electrons are at an energy greater than the binding energy, we can destroy the superconducting phenomenon. This can be accomplished, for example, with thermal energy. In fact, according to the BCS theory, the critical temperature, T. associated with this 2△ ≈3.5 kaT where kg is Boltzmanns constant. For low critical temperature(conventional)superconductors, 24 is typically on the order of 1 meV, and we see that these materials must be kept below temperatures of about 10 K to exhibit their unique behavior. High critical temperature superconductors, in contrast, will superconduct up to temperatures of about 100 K, which is attractive from a practical view because the materials can be coole cheaply using liquid nitrogen. Other types of depairing energy are kinetic, resulting in a critical current density J e, and magnetic, resulting in a critical field He. To summarize, a superconductor must be maintained under the appropriate temperature, electrical current density, and magnetic field conditions to exhibit its speci properties. An example of this phase space is shown in Fig. 53.1 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 53 Superconductivity 53.1 Introduction 53.2 General Electromagnetic Properties 53.3 Superconducting Electronics 53.4 Types of Superconductors 53.1 Introduction The fundamental idea behind all of a superconductor’s unique properties is thatsuperconductivity is a quantum mechanical phenomenon on a macroscopic scale created when the motions of individual electrons are correlated. According to the theory developed by John Bardeen, Leon Cooper, and Robert Schrieffer (BCS theory), this correlation takes place when two electrons couple to form a Cooper pair. For our purposes, we may therefore consider the electrical charge carriers in a superconductor to be Cooper pairs (or more colloquially, superelectrons) with a mass m* and charge q* twice those of normal electrons. The average distance between the two electrons in a Cooper pair is known as the coherence length, j. Both the coherence length and the binding energy of two electrons in a Cooper pair, 2D, depend upon the particular superconducting material. Typically, the coherence length is many times larger than the interatomic spacing of a solid, and so we should not think of Cooper pairs as tightly bound electron molecules. Instead, there are many other electrons between those of a specific Cooper pair allowing for the paired electrons to change partners on a time scale of h/(2D) where h is Planck’s constant. If we prevent the Cooper pairs from forming by ensuring that all the electrons are at an energy greater than the binding energy, we can destroy the superconducting phenomenon. This can be accomplished, for example, with thermal energy. In fact, according to the BCS theory, the critical temperature, Tc , associated with this energy is (53.1) where kB is Boltzmann’s constant. For low critical temperature (conventional) superconductors, 2D is typically on the order of 1 meV, and we see that these materials must be kept below temperatures of about 10 K to exhibit their unique behavior. High critical temperature superconductors, in contrast, will superconduct up to temperatures of about 100 K, which is attractive from a practical view because the materials can be cooled cheaply using liquid nitrogen. Other types of depairing energy are kinetic, resulting in a critical current density Jc , and magnetic, resulting in a critical field Hc. To summarize, a superconductor must be maintained under the appropriate temperature, electrical current density, and magnetic field conditions to exhibit its special properties. An example of this phase space is shown in Fig. 53.1. 2 3 5 D k TB c ª . Kevin A. Delin Jet Propulsion Laboratory Terry P. Orlando Massachusetts Institute of Technology
Critica ensity(1 FIGURE 53.1 The phase space for the superconducting alloy niobium-titanium. The material is superconducting inside the volume of phase space indicated. 53.2 General Electromagnetic Properties The hallmark electromagnetic properties of a superconductor are its ability to carry a static current without any resistance and its ability to exclude a static magnetic flux from its interior. It is this second property, known the Meissner effect, that distinguishes a superconductor from merely being a perfect conductor (which conserves the magnetic flux in its interior). Although superconductivity is a manifestly quantum mechanical phenomenon, a useful classical model can be constructed around these two properties. In this section, we will outline the rationale for this classical model, which is useful in engineering applications such as waveguides The zero dc resistance criterion implies that the superelectrons move unimpeded. The electromagnetic energy density, w, stored in a superconductor is therefore μlH 2 where the first two terms are the familiar electric and magnetic energy densities, respectively. (Our electromag netic notation is standard: E is the permittivity, u, is the permeability, E is the electric field, and the magnetic lux density, B, is related to the magnetic field, H, via the constitutive law B=H, H )The last term represents the kinetic energy associated with the undamped superelectrons' motion(n* and v, are the superelectrons density and velocity, respectively). Because the supercurrent density, J,, is related to the superelectron velocity y J, = nav,, the kinetic energy term can be rewritten J where a is defined as (534) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 53.2 General Electromagnetic Properties The hallmark electromagnetic properties of a superconductor are its ability to carry a static current without any resistance and its ability to exclude a static magnetic flux from its interior. It is this second property, known as the Meissner effect, that distinguishes a superconductor from merely being a perfect conductor (which conserves the magnetic flux in its interior). Although superconductivity is a manifestly quantum mechanical phenomenon, a useful classical model can be constructed around these two properties. In this section, we will outline the rationale for this classical model, which is useful in engineering applications such as waveguides and high-field magnets. The zero dc resistance criterion implies that the superelectrons move unimpeded. The electromagnetic energy density, w, stored in a superconductor is therefore (53.2) where the first two terms are the familiar electric and magnetic energy densities, respectively. (Our electromagnetic notation is standard: e is the permittivity, mo is the permeability, E is the electric field, and the magnetic flux density, B, is related to the magnetic field, H, via the constitutive law B = moH.) The last term represents the kinetic energy associated with the undamped superelectrons’ motion (n* and vs are the superelectrons’ density and velocity, respectively). Because the supercurrent density, Js , is related to the superelectron velocity by Js = n*q*vs , the kinetic energy term can be rewritten (53.3) where L is defined as (53.4) FIGURE 53.1 The phase space for the superconducting alloy niobium–titanium. The material is superconducting inside the volume of phase space indicated. w n m v o s = + m + 1 2 1 2 2 2 2 eE H2 * * n m s s * * 1 2 1 2 2 2 v Ê Ë Á ˆ ¯ ˜ = LJ L = ( ) m n q * * * 2
Assuming that all the charge carriers are superelectrons, there is no power dissipation inside the supercon ductor, and so Poynting s theorem over a volume V may be written x hdv (53.5) where the left side of the expression is the power flowing into the region. By taking the time derivative of the energy density and appealing to Faraday's and Ampere's laws to find the time derivatives of the field quantities, ve find that the only way for Poynting's theorem to be satisfied is if (53.6) This relation, known as the first London equation(after the London brothers, Heinz and Fritz), is thus necessary if the superelectrons have no resistance to their motion Equation (53. 6)reveals that the superelectrons' inertia creates a lag between their motion and that of an pplied electric field. As a result, a superconductor will support a time-varying voltage drop. The impedance associated with the supercurrent is therefore inductive and it will be useful to think of A as a kinetic inductance created by the correlated motion of the Cooper pairs If the first London equation is substituted into Faraday's law,VxE=-(aB/ar), and integrated with respect to time, the second London equation results V×(AJ.)=-B (53.7) where the constant of integration has been defined to be zero. This choice is made so that the second London equation is consistent with the Meissner effect as we now demonstrate. Taking the curl of the quasi-static form of Amperes law, VX H=J,, results in the expression V2B=-kV X J,, where a vector identity, VxVX C v(V. C)-V2C; the constitutive relation, B=kH; and Gauss's law, V.B=0, have been used By now appealing to the second London equation, we obtain the vector Helmholtz equatio V-B B=0 where the penetration depth is defined n*(q) From Eq. (53.8), we find that a flux density applied parallel to the surface of a semi-infinite superconductor will decay away exponentially from the surface on a spatial length scale of order A. In other words, a bulk superconductor will exclude an applied flux as predicted by the Meissner effect The London equations reveal that there is a characteristic length A over which electromagnetic fields can change inside a superconductor. This penetration depth is different from the more familiar skin depth of electromagnetic theory, the latter being a frequency-dependent quantity. Indeed, the penetration depth at zero temperature is a distinct material property of a particular superconductor Notice that A is sensitive to the number of correlated electrons( the superelectrons)in the material. As previously discussed, this number is a function of temperature and so only at t=o do all the electrons that usually conduct ohmically participate in the Cooper pairing. For intermediate temperatures, 0< T<T, there c2000 by CRC Press LLC
© 2000 by CRC Press LLC Assuming that all the charge carriers are superelectrons, there is no power dissipation inside the superconductor, and so Poynting’s theorem over a volume V may be written (53.5) where the left side of the expression is the power flowing into the region. By taking the time derivative of the energy density and appealing to Faraday’s and Ampère’s laws to find the time derivatives of the field quantities, we find that the only way for Poynting’s theorem to be satisfied is if (53.6) This relation, known as the first London equation (after the London brothers, Heinz and Fritz), is thus necessary if the superelectrons have no resistance to their motion. Equation (53.6) reveals that the superelectrons’ inertia creates a lag between their motion and that of an applied electric field. As a result, a superconductor will support a time-varying voltage drop. The impedance associated with the supercurrent is therefore inductive and it will be useful to think of L as a kinetic inductance created by the correlated motion of the Cooper pairs. If the first London equation is substituted into Faraday’s law, ¹ ¥ E = –(]B/]t), and integrated with respect to time, the second London equation results: (53.7) where the constant of integration has been defined to be zero. This choice is made so that the second London equation is consistent with the Meissner effect as we now demonstrate. Taking the curl of the quasi-static form of Ampère’s law, ¹ ¥ H = Js , results in the expression ¹2 B = –mo ¹ ¥ Js , where a vector identity, ¹ ¥ ¹ ¥ C = ¹(¹ • C) – ¹2 C; the constitutive relation, B = moH; and Gauss’s law, ¹ • B = 0, have been used. By now appealing to the second London equation, we obtain the vector Helmholtz equation (53.8) where the penetration depth is defined (53.9) From Eq. (53.8), we find that a flux density applied parallel to the surface of a semi-infinite superconductor will decay away exponentially from the surface on a spatial length scale of order l. In other words, a bulk superconductor will exclude an applied flux as predicted by the Meissner effect. The London equations reveal that there is a characteristic length l over which electromagnetic fields can change inside a superconductor. This penetration depth is different from the more familiar skin depth of electromagnetic theory, the latter being a frequency-dependent quantity. Indeed, the penetration depth at zero temperature is a distinct material property of a particular superconductor. Notice that l is sensitive to the number of correlated electrons (the superelectrons) in the material. As previously discussed, this number is a function of temperature and so only at T = 0 do all the electrons that usually conduct ohmically participate in the Cooper pairing. For intermediate temperatures, 0 < T < Tc , there - —× ¥ ( ) = Ú Ú E H dv w t dv V V ¶ ¶ E J = ( ) ¶ ¶t L s — ¥ (LJ B ) = - s —- = 2 2 1 B B 0 l l º m = ( ) m L o o m n q * * * 2
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 contributes 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 superconductors. 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