Quantum condensation in superconductivity

Downloaded from rspa.royalsocietypublishing.org on March 6, 2013

Quantum condensation in superconductivity perceived in its space −time aspects Taizo Masumi Proc. R. Soc. Lond. A 2003 459, doi: 10.1098/rspa.2002.1108, published 8 October 2003

Email alerting service

Receive free email alerts when new articles cite this article sign up in the box at the top right-hand corner of the article or click here

To subscribe to Proc. R. Soc. Lond. A go to: http://rspa.royalsocietypublishing.org/subscriptions


10.1098/ rspa.2002.1108

Quantum condensation in superconductivity perceived in its space-time aspects By Taizo M a s u m iy National Institute for Materials Science, Nano-Materials Laboratory, 3-13 Sakura, Tsukuba, Ibaraki, 305-0003 Japan and Department of Pure and Applied Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902 Japan ([email protected]) Received 5 February 2002; revised 8 October 2002; accepted 26 November 2002; published online 28 August 2003

Since the work of F. and H. London in 1935, the important question of explaining the relation between the static and uniform electric and magnetic responses of superconductors, namely, the zero-resistivity E = 0 and the perfect diamagnetism B = 0, seems to have long been forgotten. London & London postulated their famous macroscopic equation ¤ cJ = ¡ A, where ¤ = m=nS e2 . A logical gap [¬ ], however, has been clearly admitted for a long time in the argument used to obtain B = 0 from dB=dt = 0. Here, we point out that there exists another hidden logical gap [ ] in the argument used to obtain E = 0 from curl E = 0. Microscopically, the Bardeen{ Cooper{Schrie®er (BCS) theory was constructed with the London equation in mind, and the concept of Josephson’s phase locking in the macroscopic wave function ª m acro was established later. Quite recently (in 2001), we successfully clari¯ed a substantial problem in superconductivity, unsolved and forgotten for a long time, in a stabilized form. Here, in particular, we must point out that, in order to remove logical gaps [¬ ] and [ ], we must simultaneously account for the zero-resistivity, ¿ (R) = 0 at ! = 0, and the perfect diamagnetism, [~K ¡ (q=c)A(R)] = 0 at q = 0, equivalent to the second London equation, as a set of equally fundamental inherent properties of pure superconductivity at T ’ 0 K. We further clarify why and how the BCS theory must be extended to the (1+3)-dimensional Minkowski space-time on the basis of the concept of coherence in the macroscopic wave function, ª m acro, as an inevitable consequence of the gauge ¯eld theory. Keywords: pure superconductivity; zero-resistivity; Meissner e®ect; London equation; quantum coherent condensation; Minkowski space-time

1. Introduction A set of experimental observations on the (i) electric and (ii) magnetic responses of superconductors was originally discovered for both static and uniform ¯elds by Kamerlingh Onnes (1911a{c) and by Meissner & Ochsenfeld (1933). y Permanent address: 4-55-10 Utsukushigaoka, Aoba-ku, Yokohama, Kanagawa 225-0002, Japan. Proc. R. Soc. Lond. A (2003) 459, 2643{2661

2643

° c 2003 The Royal Society


2644

T. Masumi

Since London & London’s (1935) paper, the di±cult question of explaining the relation between the electromagnetic responses of superconductivity, namely, (i) the zero-resistivity E = 0 and (ii) the perfect diamagnetism B = 0, in terms of phenomenological theory has been potentially left unanswered, perhaps long forgotten. In fact, it was unfortunate that we did not obtain a satisfactory set of explanations of (i) electrostatic and (ii) uniform magnetostatic responses of superconductors during the 20th century. It is widely known that response (i) is not su± cient to derive (ii). London & London (1935) ¯rst intended abandoning the àcceleration theory’, ¤

dJ = E; dt

¤ =

me2 ; nS

(I)

but they then postulated their `fundamental equation’, µ ¶ 1 H; curl ¤ J = ¡ c

and eventually reached the so-called London equation, ¤ cJ = ¡ A;

¤ c2 » = ¡ ¿ :

(II)

Surprisingly, even Landau & Lifshitz (1960) state that `this absence of electrical resistivity is not the most fundamental property of a superconductor. The transition involves profound changes in the magnetic properties of the metal and, as we shall see, the change in its electric properties is a necessary consequence.’ Kittel (1996) states that `the argument based on dB=dt = 0 is not entirely transparent’ to obtain B = 0. One is thus apt to believe that (ii) is even more substantial than (i) in superconductivity. This is not a self-apparent issue, but it clearly raises a serious question. The dramatic discovery of the isotope e®ect (Frohlich 1950; Maxwell 1950a; b) emerged simultaneously with the publication of a monumental book by London (1950). The success of the Bardeen{Cooper{Schrie®er (1957) (BCS) theory de¯nitely supported the London equation [II] that described the condensation of conduction electrons in terms of Cooper pairs, q = (¡ 2e), on the basis of the phonon-mechanism in momentum space. These facts support the view that (ii) the Meissner e®ect is more important than (i) the zero-resistivity. Such arguments may have been regarded as reasonable but are limited to the spatial aspect of the subject. They lack a temporal aspect. Later, Josephson (1962) further clari¯ed the concept of the phase in the macroscopic wave function for an ensemble of Cooper pairs. Feynman (1965) pedagogically expressed it as a quasi-Boson condensation in the macroscopic wave function ª m acro. Langenberg et al . (1966) gave a simpli¯ed explanation of (i) the zero-resistivity on the basis of BCS theory as an introduction to the Josephson e®ects, but not of (ii) the Meissner e®ect. Only Ashcroft & Mermin (1976) explicitly remarked that `the property for which superconductors are named is unfortunately one of the most di±cult to extract from the microscopic theory’. They noticed `long-range phase coherence’ in space. Superconductivity as such remained a largely forgotten enigma for a long time. Quite recently (in 2001) we recognized that, instead of (i) E = 0 and (ii) B = 0, we can simultaneously account for both (i) zero-resistivity ¿ (R) = 0 and (ii) the Proc. R. Soc. Lond. A (2003)


Quantum condensation perceived in space-time (a)

(b) voltmeter

2645

surface of superconductor

V = 0 (normal conductor)

resistor

an external field B (magnetic flux density)

V = 0 (superconductor)

ammeter

shielding current

vector potential A

coherence length penetration depth

battery

Figure 1. (a) Zero-resistivity ¿ ( ) = 0 of an in¯nitely long rod-like superconductor (Masumi 1986, 2001); (b) perfect diamagnetism [~ ¡ (q=c) ( )] = 0 of a semi-in¯nitely deep superconductor (Masumi 1986, 2001).

perfect diamagnetism [~K ¡ (q=c)A(R)] = 0 equivalent to the London equations, as illustrated in ¯gure 1 (Masumi 2001). Both (i) and (ii) should not have been postulated but rather derived inevitably in terms of gauge ¯eld theory as a set of equally fundamental inherent character of the pure superconductive component ns (T ’ 0 K) resulting from the coherence of ª m acro in its space-time aspects. Here, we add a further remark on a perspective view of superconductivity resulting from reconsideration of the electromagnetic responses of superconductors. In particular, we describe (i) the existence of the logical gaps [¬ ] and [ ]; (ii) the quantum phase locking, modi¯cation not only in space but also in time; (iii) how quantum condensations occur, starting from the microscopic scale, that is, the molecular level, up to the macroscopic scale, as in superconductors; and (iv) small corrections to a previous paper (Masumi 2001). We do not consider subjects involving the normal component nN (T ), such as the two-°uid model and the time-dependent Ginzburg{ Landau theory at 0 < T 6 Tc .

2. Evolution of theoretical concepts of superconductivity One usually starts a theoretical consideration of the basis of superconductivity with use of the Newtonian equation of motion, F = m¬ ;

(2.1)

and the Maxwell equations (normally of the Heaviside{Hertz form): 1 @B ; c @t @D 4º ; J+ curl H = @t c curl E = ¡

Proc. R. Soc. Lond. A (2003)

(2.2 a)

(2.2 b)


2646

T. Masumi div D = 4º » ;

(2.2 c)

div B = 0;

(2.2 d)

D = "E;

(2.2 e)

B = · H:

(2.2 f )

Alternatively, we may use a set of expressions ¿ (r; t), A(r; t), where @A(r; t) ; @t Namely, we may start from the Maxwell{Lorentz ¸ · 1 @ 2 (L) 2 r ¡ 2 2 ¿ =¡ c @t ¸ · 1 @2 2 r ¡ 2 2 A(L) = ¡ c @t E(r; t) = ¡ grad ¿ (r; t) ¡

B(r; t) = curl A(r; t):

(2.3)

form:

4º » ; "0

(2.4 a)

4º · c

(2.4 b)

0J ;

1 @¿ (L) = 0: (2.4 c) c2 @t It is possible to express ¿ (r; t), A(r; t) in terms of ¿ (!; q), A(!; q) using Fourier transformations. But we focus on the situation in which ! = 0 and q = 0. Seitz (2001) made the noteworthy remark that James Clerk Maxwell preferred to work with the potentials rather than the equations describing the interrelation of the electric and magnetic ¯elds, a practice that has found its place in modern ¯eld theory. J. J. Thomson, who edited the third edition of Maxwell’s treatise, placed more emphasis on the latter in his own version of the work. Much of the notation Maxwell introduced in the course of his analysis remains in use today. div A(L) +

(a) Logical gap [¬ ] London & London (1935) and London (1950) described a set of basic equations on superconductivity as follows, using (2.1), (2.2 a) and (2.2 b), with unwarranted initial conditions, ¸ · nS e 2 dJS E; JS ; = (¡ e)nS vS : (2.5) = [I] m dt ¸ · nS e2 A(r); (2.6) [II] JS (r)= ¡ mc

with the gauge div A(r) = 0 and the boundary condition An = 0 normal to the surface as if r2 À (r) = 0 in A0 (r) = A(r) + grad À (r). London (1950) tried to justify the gauge div A(r) = 0 by further emphasizing the `rigidity’ of the many-body wave function in his microscopic program. Here, the London equations [I] and [II] correspond to (i) the zero-resistivity E(r) = 0 and (ii) the perfect diamagnetism B(r) = 0, respectively. Nevertheless, there remained an unresolved problem in relation to (i) and (ii): how to unambiguously derive B = 0 from dB=dt = 0. Such arguments necessarily produce a logical gap [¬ ], which we will subsequently use, for example, to explain the existence of hysteresis in B during the freezing-in of B in superconductors while cooling in the presence of a ¯eld. Proc. R. Soc. Lond. A (2003)


Quantum condensation perceived in space-time

2647

(b) Logical gap [ ] The vanishing of resistivity in superconductors was observed by Kamerlingh Onnes in 1911 prior to the perfect diamagnetism. The Meissner{Ochsenfeld e®ect was discovered in 1933. If the history of superconductivity had developed chronologically in an inverse order, we would have had the perfect diamagnetism ¯rst. In this event the Meissner e®ect B(r) = 0 would have led us to the relations curl E(r) = 0, E(r) = ¡ grad ¿ (r), but it would not necessarily imply zero-resistivity E(r) = 0. It is not apparent that one can simply derive E(r) = 0 from curl E(r) = 0. One must face another hidden logical gap [ ]. If the experiments had occurred in reverse order, London & London would have faced a di®erent problem. They may not have reached their famous equation JS (r) = ¡ [nS (r)e2 =mc]A(r) in a framework of classical electrodynamics. London & London (1935; see also London 1950) did not adequately use the set of Maxwell equations, passing over analyses with the logical gap [¬ ] and not even noting the logical gap [ ]. However, once one uses a set of the Maxwell equations, one must be consistent and use all of the set without arbitrariness in initial and boundary conditions when treating superconductivity as an electrodynamical phenomenon. Schrie®er (1964, 1999) noticed how to understand the relation between the two London equations. He noted that in postulating the origin of (2.6) London & London added the all-important restriction B = 0 inside the superconductor regardless of its history, which is the essence of the Meissner e®ect. With the publication of London’s (1950) book, however, a series of dramatic discoveries of the isotope e® ect (Frohlich 1950; Maxwell 1950a; b; Reynolds et al. 1950, 1951) simultaneously emerged both theoretically and experimentally. While the discovery provided a major breakthrough in the history of superconductivity, the timing was bad; circumstances changed and the logical gap [¬ ] was virtually ignored, the logical gap [ ] not even being noted, for several decades afterwards. The brilliant achievement of the BCS theory in 1957 de¯nitively established the microscopic theory of superconductivity based on the phonon-mechanism and gave emphasis to the London equations, especially [II]. Bogoliubov et al. (1959)y introduced a set of linear transformations of fermion operators to derive the energy gap and excitation spectra needed to decompose Cooper pairs. More surprisingly, in their book and a related paper, Landau & Lifshitz (1960) were not careful and overlooked the existence of the logical gap [¬ ]. Moreover, they did not notice the potential existence of the logical gap [ ], though it must de¯nitely underlie any consideration of superconductivity. London (1950) considered the electrodynamics of a superconducting state in depth. He also provided a subsection (x 8 (e)) on a four-dimensional approach to the superconductivity in his derivation of the London equation in the framework of classical electrodynamics, where the London gauge div A(r) = 0 with r2 À (r) = 0 in A(r) = A0 (r) + rÀ (r) was employed. In his quantum-mechanical program, the London gauge div A(r) = 0 with r2 À (r) = 0 was not explicitly required. He also considered a single `rigid’ state, which might subconsciously have signi¯ed `coherence’ in later, modern terminology. Unfortunately he was not explicitly aware of y The concept of bogolons played a substantial role in providing a clearer image of the coherent ground state ªm acro assumed at ¯rst to underlie the phenomena. It assured the existence of ª (n; T ) at T = 0 K, equivalent to ©G in superconductors. Proc. R. Soc. Lond. A (2003)


2648

T. Masumi (b)

current, I

(a)

superconducting Josephson states (Cooper pairs)

quasi-particles excitations (normal states)

current

J0 VG

B=0 voltage, Vdc

ti m

e,

t

0

current

B = B0/2

Figure 2. Coherence prevents control of phase in ª m acro : (a) DC Josephson characteristics in a static electric ¯eld = ¡ grad ¿ (Masumi 1986); (b) static magnetic ¯eld = curl (Langenberg et al . 1966).

the `coherent phase of material waves’, later introduced by Josephson (1962), which is nearly equivalent to ª m acro, as denoted by Feynman (1965, 1972) and recently reconsidered (Masumi 1986, 2001).

3. Phase locking and modi¯cations of superconductivity Josephson (1962, 1965) ¯rst clari¯ed the concept of phase in the wave function in quantum mechanics for an ensemble of Cooper pairs in the tunnelling e®ect. He also indicated how to modify the phase with a static electric ¯eld or a uniform static magnetic ¯eld located at the tunnelling junction. His results are as follows (hereafter, we use ¿ (r; t), A(r; t), (2.3) and (2.4)): ¸ · 2eV 2eV ; t ; J = J0 sin ¯ 0 § (3.1) 2º f = ! = ~ ~ ¸ · ¸ · Z I hc e : A ds ; © = A ds = n J = 2J0 sin ¯ 0 § (3.2) 2e ~c

However, the introduction of Josephson’s new concept of phase locking gave profound signi¯cance not only to the tunnelling of Cooper pairs at the junction when ¯ 0 6= 0 but also to conditions in the interior of the quantum condensate itself, even in bulk, where ¯ 0 = 0. It is di±cult to destroy or modify phase locking in bulk superconductors. Figure 2 displays the di±culty of breaking phase locking in the quantum condensate discussed in the text by a simple electromagnetic excitation at a junction, ¯ 0 6= 0. In principle, phase breaking could be produced by either (a) an electrostatic ¯eld in time or (b) a uniform magnetostatic ¯eld in space. Proc. R. Soc. Lond. A (2003)



2649

crack junction

BY BY

CO

CO

Figure 3. Schematic of the crack junction (after Tsai et al . 1987).

Anderson & Rowell (1963) subsequently provided excellent experimental evidence for Josephson’s prediction at the junction shown in ¯gure 2. They employed an electric ¯eld E through its dependence on J when ¯ 0 6= 0 at the junction, as well as magnetic ¯eld B. The ¯elds E and B display the rigidity of the `coherent quantum condensate’, an ensemble of Cooper pairs in superconductors at T = 0 K. It was more di±cult to destroy or modify the phase locking in a bulk superconductor when ¯ 0 = 0, although it was possible at the junction when ¯ 0 6= 0. Actually, Tsai et al . (1987) performed an experiment on a mechanically cracked YBaCuO/YBaCuO Josephson junction originally made of a single rod of ceramic YBaCuO, as illustrated in ¯gure 3. They con¯rmed that, with the junction ¯ssured at cryogenic temperatures, the critical current persists up to 90 K, so that decoherence of phase locking of such a quantum condensate is di±cult to achieve with creased ¯ssures even if ¯ 0 6= 0. However, phase can be altered by a simple electromagnetic excitation at a junction, ¯ 0 6= 0, by applying either an electrostatic ¯eld in time or a uniform magnetostatic ¯eld in space, as displayed in ¯gure 2a; b, respectively. Feynman (1965, 1972) constructed ª m acro for a quasi-Boson condensate. He gave an explanation for the importance of the phase of the wave function in the case of an ensemble of an enormous number of Cooper pairs in superconductors, mainly in space. His path-integral Lagrangian formulation of quantum mechanics is inherently founded on a clear recognition of the importance of phase in the probability amplitude even after integrating an action integral over time. However, on the basis of quasi-Bose condensation, he treated coherence using an expression which clearly separated contributions of an electric ¯eld potential V (r) and a magnetic vector potential A(r). He applied this to a coherent ensemble of the Cooper pairs in superconductors under static and uniform conditions in his space-time approach, but without explicit consciousness or a proof of the signi¯cance of time. Later, we con¯rmed that this type of expression is valid for static and uniform electric and magnetic ¯elds in superconductors (Masumi 2001). Langenberg et al . (1966) gave an intuitive and clever explanation of zero-resistivity alone, on the basis of the concept of the coherence of Cooper pairs for the temporal part of the phase factor in an electric ¯eld. However, they did not develop the idea further to include the spatial part of the phase factor in a magnetic ¯eld as well as the temporal part in an electric ¯eld. A substantial part of the original text by Proc. R. Soc. Lond. A (2003)


2650

T. Masumi

Langenberg et al . (1966) was supplemented in a complete form by Masumi (1983, 1986, 2001) (see below). Now, as we have mentioned, according to quantum mechanics the wave associated with an electron pair has a spatial oscillation whose wavelength is determined by the center-of-mass momentum of the pair. The wave also has a time oscillation whose period is determined by the energy of the electron pair, so that the phase of the pair depends on the pair’s energy as well as on its center-of-mass momentum. As we have seen in a bulk superconductor the attractive interaction of electrons, the large number of bound electron pairs and the exclusion principle conspire to make the pair phases the same. These two facts lead to a simple explanation of the zero-resistance of the superconducting state: if a voltage di®erence V existed between the two ends of the current-carrying bar shown in the illustration : : :, the energy of an electron pair in one end would be greater by 2eV than the energy of an electron pair in the other end (e is the charge of an electron). Then, as time passed, a phase di®erence would develop between the electron pairs in the two ends of the bar. Rather than go to the higher energy state that this breakdown of the pair phase-locking would produce, the superconductor carries the current without allowing a voltage di®erence to appear. Langenberg et al. (1966, p. 33) Here, we suggest adding the following content or an equivalent part. Usually, we have a phase factor in the space-time description of propagation of classical waves Á = C exp i[!t ¡ K ¢ R]. In the case of de Broglie waves (a wave of probability amplitude or the wave function Á(r; t) in quantum mechanics), the material waves representing a Cooper pair of charged particles (¡ 2e) moving in an electrostatic ¯eld E = ¡ grad V (R) and a magnetostatic ¯eld (induction) B = curl A(R), we have additional parts due to the scalar potential V (R) and vector potential A(R) in the phase factor of waves (Masumi 1983, 1986, 2001). The former directly varies values of ! by (1=~)(¡ 2e)V (R), whereas the latter potentially contributes the phase factor as noted below. A charged particle, such as an electron pair (¡ 2e) moving in a magnetic ¯eld, involves two parts in the dynamical momentum P . Namely, it involves ¯rst the kinetic momentum ~K relevant to the velocity V and second the ¯eld momentum ¡ (¡ 2e=c)A(R). Thus, the phase factor in a wave function can be in°uenced not only by (¡ 2e)V (R) but also by the ¯eld momentum, ¡ (¡ 2e=c)A(R), as well as the kinetic momentum ~K. Thus, if [~K ¡ (¡ 2e=c)A(R)] were to be ¯nite inside of a superconductor, the phase factor of momenta of Cooper pairs would vary from position to position, and this breakdown would violate the phase-locking of pairs in space. Therefore, the superconductor repels an externally applied magnetic ¯eld or even excludes previously applied ¯eld out of inside without allowing any penetration of the magnetic ¯eld by sustaining the condition [~K ¡ (¡ 2e=c)A(R)] = 0. This eventually leads us to the London Proc. R. Soc. Lond. A (2003)


2651

Quantum condensation perceived in space-time (a)

(-e) m

[p /a]

-[p /a]

(c)

(b)

(-e) m

[p /a]

-[p /a] (d ) (-2e) 2m

(-2e) 2m

dK

-[p /a]

[p /a]

-[p /a]

[p /a]

Figure 4. Image of an ensemble of the conduction electrons [ ; ¡ ] and the Cooper pairs in the -space of superconductors at static (a), (c) electric ((i) zero-resistivity) and uniform = 0; (b) = 0; (c) ¿ ( ) = 0; (d) (b), (d) magnetic ¯elds ((ii) perfect diamagnetism): (a) [~ ¡ (¡2e=c) ( )] = 0. There exists not only (a) an energy gap near the Fermi surface with a shift ¯ in the single conduction electron scheme as fermions, (¡e), m, in the upper row, but also (c) the coherence in the Cooper pairs, = (¡2e), 2m, in a quasi-Boson scheme with a shift . No relaxation exists at ¯ and at ¯ = [ ; 0 ], once they have been established, apart from ¯ in both (a) and (c). In (b), (d) the static magnetic ¯eld just the existence of the energy gap ¢ rotates the system.

equation, JS (R) = ¡ [nS (R)e2 =mc]A(R), even on the basis of the Drude model (1900), which implicitly assumed the in-phase motions of all electrons or Cooper pairs (Masumi 2001). We recognize that the Meissner e®ect is not a simple result of the combination of a current due to the magnetic induction with zero resistance. (Masumi 1983, 1986, 1997, 2001) Ziman (1965) explained zero-resistivity of superconductors solely on the basis of the existence of an energy gap ¢ k and the assumption that annihilation of Cooper pairs near the Fermi surface can be decomposed quasi-particles (Bogolon creation). Normal current occurs only when (¯ k)2 =2m > ¢ k , as denoted in ¯gures 2a and 4a. Besides, there exists another substantial reason to sustain supercurrent as a result of the coherence in an ensemble of Cooper pairs in superconductors, as noted in ¯gures 2a and 4a. A static magnetic ¯eld must be repelled or excluded as in ¯gure 2b, which is due to a rotation of the system shown in ¯gure 4b; d. Proc. R. Soc. Lond. A (2003)


2652

T. Masumi z

y

x t

Figure 5. Coherent world-lines for an ensemble of coherent Cooper pairs denote the macroscopic wave function ª m acro ° ying in Minkowski space-time.

Ashcroft & Mermin (1976) explicitly remarked that `the property for which superconductors are named is unfortunately one of the most di±cult to extract from the microscopic theory’. They recognized the existence of `long-range phase coherence’, but only in space. Another aspect of the enigma of superconductivity was neglected for a long time. It was implicitly noted by Langenberg et al . (1966). Later it was explicitly recognized in terms of ª m acro (Masumi 1986, 2001), as illustrated in ¯gure 5 here in the form of four [1+3]-dimensional Minkowski space-time aspects.y We point out that the Josephson part of the current is not the result of an in¯nite collision time of independent particles but rather of an in¯nite relaxation time due to the phase locking of an ensemble of coherent Cooper pairs as a whole, which must yield (i) the zero-resistivity at T = 0 K (Masumi 2001). Once a ¯nite value of K occurs, it never decays, because of the rigid coherence of Cooper pairs in superconductors. That is, the source of (i) the persistant current.

4. Perspective view of superconductivity in space-time Starting from the ideas of Heitler & London (1927), London extended his studies to the diamagnetism of aromatic compounds and regarded a superconductor to be a macromolecule (London 1950). Even in a hydrogen molecule, the eliminated part of exp[¡ i(E=~)t] in solutions of the steady-state Schrodinger equation must have potential signi¯cance: a system made of two electrons is condensed coherently into a stable state by lowering the energy by a de¯nite amount in an in¯nite time domain and within a ¯nite spatial domain. Nakamura et al . (1999) studied the coherent control of macroscopic quantum states in a single-Cooper-pair box. A nanometre-scale superconducting electrode connected y Note, however, that the magnitude of relativistic e®ects is very small because of small values of 10¡ 4 {10 ¡ 5 . Proc. R. Soc. Lond. A (2003)



2653

to a reservoir via a Josephson junction constitutes an arti¯cial two-level electronic system: a single Cooper pair in a nanoscale box. The two levels consist of charge states (di®ering by 2e, where e is the electronic charge) that are coupled by tunnelling of Cooper pairs through the junction. Although the two-level system is macroscopic, containing a large number of electrons, the two charge states can be coherently superposed. The Cooper-pair box has been suggested as a candidate for a quantum bit or `qubit’: the basic component of a quantum computer. Their results demonstrated the electrically coherent control of a `qubit ’ in a quantum computer, a solid-state electronic device, while the coherence of a large number of Cooper pairs is sustained. Conversely, this coherent control of macroscopic quantum states exempli¯ed with a single-Cooper-pair box signi¯es that we may be able to construct a macroscopic quantum condensate, starting from a single pair of electrons and adding one by one successively in space and extending the system up to a macroscopic scale in a stable state in time. We illustrate the authentic coherent part of the superconducting current, the Josephson part, conceptually in the Minkowski space in ¯gure 5. When an externally applied weak static ¯eld ¿ (t; r) and uniform A(t; r) is present or when ¿ (!; q), A(!; q) exists independently of t or averaged over the diameter of a pair » , a phase factor of the wave function for a set of Cooper pairs (R; K) at the centre of mass of two charged particles, q = (¡ 2e), ª m acro;G ! exc: in the ground state does tend to vary. Or it may incline to be modi¯ed somewhat together with the scalar potential ¿ (R) or vector potential A(R). In such a case, ¿ (R) or A(R) with À (R) may be regarded as a perturbation in the energy or momentum. From equation (2.4 c), we can obtain div A(L) = 0 equivalent to the London gauge. Quite recently, we introduced the `space-time aspects of superconductivity’ in the (1+3)-dimensional Minkowski space for an ensemble of coherent Cooper pairs, using the terminology of the gauge ¯eld theory (Masumi 2001). In contrast to the London theory (London & London 1935; London 1950), this procedure does not require us to abandon the acceleration equation or to postulate the London equation. We have been able to derive (i) the zero-resistivity and (ii) the perfect diamagnetism, that is, the London equation, from the coherence of ª m acro when regarded in a consistent manner in space-time. We treated the macroscopic wave function ª m acro of coherent Cooper pairs at T = 0 K without employing the summation of K at least in the cases of small static electric and uniform magnetic ¯elds (Kamerlingh Onnes 1911a{c; Meissner & Ochsenfeld 1933). The treatment is valid when and where the phase factor can be decoupled in the space-time aspects (Masumi 1986, 2001). Accordingly, we may have to consider a superconductor inherently as a `quantummechanical macromolecule’, denoted by ª m acro, with external electric and magnetic ¯elds represented in terms of a gauge ¯eld [¿ (x); A(x)]

with x = [t; x]:

We considered this issue ¯rst with ª m acro and [¿ (x); A(x)] in the Minkowski spacetime and in terms of the gauge ¯eld theory, as illustrated in ¯gure 6. In extending the BCS theory, which is based on the coherence of electrons and eventually of Cooper pairs in space, we here explicitly preserve the phase factor in time, which inherently exists but is usually eliminated. Thus, we expand a general Proc. R. Soc. Lond. A (2003)


T. Masumi 2654

1. Introduction

(Classical Physics)

E=0

=

-[nSe2/mc] A(r)

automatically

Logical gap [ ]

Bardeen–Cooper–Schrieffer Theory (1957)

Maxwell’s equations (Heaviside–Hertz)

q = -2e

(Relativistic scheme) with b ~ 0

(i) Zero-resistivity (ii) Perfect diamagnetism curl E = 0 automatically? B=0

Logical gap [ b ]

JS(r)

London equation

2. Theoretical background 2.1. Macroscopic theory to molecular theory

2.2. Microscopic (Quantum Mechanics)

(i) BCS ground state: F BCS,G ~ Y macro,G Ginzburg–Landau,Y (T = 0 K, nS = N0)

(ii) Quasi-particle excitation: (Bogoliubov et al. 1959)

Aµ[t, R]

Gauge Theory: Schrödinger equation in gauge fields

– . (iii) Quantum phases: Ymacro,G = S K=[k,k’]CK=[k,k’]exp(i/h)[E Kt - K R] (Josephson 1962, 1965)

Space-Time Aspect (Maxwell–Lorentz)

3. A set of electrostatic and uniform magnetic responses of superconductors (Masumi 1983, 1986, 1996, 1997, 2001)

(Gauge Theory) 4. Discussion 5. Conclusion

Phase independent of t and R

– - (q/c)A(R)} . R] – Ymacro,exc = CK=[k,k’]exp(i/h)[{-q f (R)} . t - {hK

Statistical Physics

Thermodynamics

Phase Transitions

Statistical Mechanics

Ginzburg–Landau Y (T ¹ 0 K, nS(T))

Simultaneously

(i) Zero-resistivity: f (R) = 0

(ii) Perfect diamagnetism: – - (q/c)A(R)] = 0 [hK

Figure 6. Logical block diagram of the space-time aspects of superconductivity.



Quantum condensation perceived in space-time form of the wave function ª ª

m acro;G!

=

X

m acro

2655

to cover low-lying excited states for pairs, if possible,

exc

CK =

K = [k;k0 ]

·½ £ EK ¡

[k ;k 0 ] exp(i=~)

µ

q ¿ (t; R) ¡

@À (t; R) @t

¶¾ t¡

f~K ¡

¸

(q=c)[A(R) + rÀ (t; R)]g ¢ R ; (4.1)

P

with K = [k;k 0 ] using the separated form of ¿ (R), A(R) and À (t; R) for static and uniform conditions of electric and magnetic ¯elds [! = 0; q = 0]. It is possible that, since all of the electrons form a coherent state of pairs in a cooperative way, only a single phase factor is left. `Coherence’ signi¯es the phase locking or ¯xing of the phase factor by À (R) over all space, which removes the summations over K; it also exists in time as À (t; R). However, since ¢N ¢À ¹ ~, we conclude that ¢À y is small and r2 À (t; R) ¹ 0. As noted from the Lorentz condition, equation (2.4 c), with q = 0 and w = 0, the gauge invariance can be restored, as anticipated by London (1950). As a second substantial step, once we recognizeP the coherence among Cooper pairs, we may rewrite (4.1) without the summation of K = [k;k 0 ] as ª

m acro;G!

exc

= CK =

[k ;k 0

µ ¶·½ i Ek ¡ ] exp ~

µ

q ¿ (R) ¡

@À (t; R) @t

¶¾ t

¸ ¡ f~K ¡ (q=c)[A(R) + rÀ (t; R)]g ¢ R ; (4.2)

as if the system were regressing to the ¯rst quantization scheme. Thus, we can divide the phase factor in ª m acro;G! exc into two independent parts by decoupling ¿ (t; R) and A(t; R), rather than in their Fourier transforms ¿ (!; q) ! V (R) and A(!; q) ! A(R) in the limits of ! ! 0 and q ! 0 or, to be more precise, when ! = 0 and q = 0. Because the value of rÀ (R) is ¯xed independently of t and R due to the `coherence’ found in macroscopic quantum-mechanical phenomena, we may treat it as a common constant factor that can be left aside. The ground state ª m acro;G! exc may be considered to be excited to low-lying excited states. After all, we can express possible low-lying excited states in the energy spectrum of elementary excitations with the bias given above as ª

m acro;G!

exc

= CK =

[k;k 0 ] exp(i=~)[fEK

¡ q¿ (R)gt ¡ f~K ¡ (q=c)A(R)g ¢ R]: (4.3)

y We must comment on the huge values, not only of N = N0 =2, but also of ¢N between T = 0 K and T ’ Tc needed to guarantee ¢Â » = 0 as a result of the Heisenberg uncertainty relation and the coherent state of Cooper pairs in ªm acro at T = 0 K. Here, the concepts found in statistical physics have been degenerated. We pay attention only to the pure superconductive component ns (T ’ 0 K) of conduction electrons. Thus, the two-°uid model in superconductors and the temperature dependence of speci¯c heat, etc., in ¯nite temperature regions are not of primary interest. The BSC theory gives results for these problems. We consider situations near to T ’ 0 K rather close to those in nuclear or particle physics at stable state. Proc. R. Soc. Lond. A (2003)


2656

T. Masumi

This expression for ª m acro;G! exc has a reasonably simpli¯ed form but has profound implications. Near the superconducting ground state, we may also suppose that EK = 2"k ¡ ¢ k = 0. This scheme can be justi¯ed for static and uniform electric and magnetic ¯elds in space-time. Space-time aspects of superconductivity in terms of the gauge ¯eld theory must be treated with the Lagrangian formalism in which the action integral S of the Lagrangian extends over time. Using the Hamilton variational principle and the Hamilton{Jacobi equation, we reach the Hamiltonian formalism. For ª m acro , we may use the Schrodinger equation for a single-particle-like coherent entity of Cooper pairs denoted by ª m acro with x = [t; R],y ¶2 µ ¶ µ @ @ ~2 ª m acro(x); ¡ qA· ¡ qA· ª m acro (x) = ¡ (4.4) i~ · 2m |@x· {z } } |@x {z · = 1;2;3

· = 0

which leads us to a solution in the form ª (x)m

acro

= CK =

with

[k;k 0 ] exp(i=~)

ª (x)m

acro

·

!ª

fEK ¡ |

qA¹ (x)gx· ¡ f~K ¡ } | {z · = 0

0 m acro (x)

¸

(q=c)A¹ (x)g ¢ x· ; (4.5) } {z · = 1;2;3

= exp[¡ i(q=~c)À (x)]ª

m acro (x);

(4.6)

À (x) ¹ 0. For static and uniform electric and and the concept of coherence gives magnetic ¯elds and with x = [t; R], the coherence in (4.5) leads to r·

ª (x)m

(q=c)A(R)g ¢ R]; (4.7) in agreement with (4.3). If the phase factor in (4.7) is sustained, ª m acro;G must be variables independent of t and of R with ¿ (R) = 0 and [~K ¡ (q=c)A(R)] = 0 as revealed in ¯gure 1a; b. It is su±cient and inevitable to sustain persistent current when we make a singly connected ring and we have a bulk pure superconductor to repel externally applied magnetic ¯eld or exclude the remaining internal magnetic ¯eld. We conclude that superconductivity consists of acro;G!

exc

= CK =

[k;k 0 ] exp(i=~)[fEK

¡

q¿ (R)gt ¡

(i) the zero-resistivity in time: (ii) the perfect diamagnetism in space:

f~K ¡

¿ (R) = 0 at ! = 0; (4.8) [~K ¡

(q=c)A(R)] = 0 at q = 0: (4.9)

We also conclude that the London equation [II] should not have been postulated but, rather, should be automatically obtained from (4.9), as in (2.6) using the Drude-like theory (von Drude 1900; Wilson 1953). It implicitly includes two types of `coherence’, that is in-phase motion of conduction electrons, namely, one type to form Cooper pairs, and another type to form (nS =2) Cooper pairs with charge ¡ 2e and mass 2m, as illustrated in ¯gure 4. Thus, equation (4.9) is equivalent to the London equation [II]. It is a relatively simple result as a consequence of the coherent system of Cooper y Here, we have not taken account of quantum electrodynamical considerations. Proc. R. Soc. Lond. A (2003)



2657

pairs in ª m acro in the space-time framework of the gauge ¯eld theory, particularly at ! = 0 and at q = 0 (see Masumi 2001). As noted earlier, only Ashcroft & Mermin (1976) explicitly remarked that `the property for which superconductors are named is unfortunately one of the most di±cult to extract from the microscopic theory’. Even the great BCS theory, based on a coherent ensemble of the Cooper pairs in space alone, uses a perturbational approach to obtain the London equation [II] via j(q) = ¡ (1=c¤ T )a(q) in the limit q ! 0. The validity of this method is not self-apparent, because it may lead to divergent singularities at q = 0.y As far as the term `space-time’ is concerned, we note the development of the timedependent Ginzburg{Landau (TDGL) theory. It treats the problem with the term ìn space-time’. Gorkov (1959) started a microscopic derivation of the GL equation in the theory of superconductivity by using the thermodynamic Green functions in ìmaginary time, ½ ’ (see footnote). Abrahams & Tsuneto (1966) used the Green-function formulation of the BCS theory to study TDGL equations in space-time employing the order parameter ª (t; r) in superconductors. They found that su±ciently slowly varying TDGL equations exist near T ’ 0 K and T ’ Tc . At T ’ 0 K, the equation has wave-like character, and it is of di®usion type at T ’ Tc , under the restriction that either the characteristic frequency of the time variation of ª is greater than the gap frequency or the ratio of the Fermi velocity to the product of the characteristic wavelength and frequency of the space-time variation of ª is greater than unity. They also discuss the in°uence of slowly varying time-dependent ¯elds and derive the dependence of charge and current densities on the variations of ª . Their investigation is broad. It is more or less consistent with our concept of ª m acro at T ’ 0 K. We treat only the case in which T ’ 0 K and where ! = 0 and q = 0 in accordance with simple experimental situations. Gorkov & Eliashberg (1968) developed a method of extending the GL theory in `space-real time t’ (see footnote) and applied it to the case of type-II superconductors of alloys containing paramagnetic impurities. Thompson & Hu (1971) and Hu & Thompson (1972) developed and applied their method to pursue dynamic structures of vortices in gapless superconductors. Actually, the TDGL theory, which employed space-time aspects, was primarily constructed to treat problems such as magnetic impurities (Tinkham 1996; Cyrot 1973). It is useful at ¯nite temperatures, especially near T ’ Tc and H ’ Hc to explain matters such as the dynamic structure of vortices in type-II superconductors and the °uctuation in ª m acro above Tc . There, it is likely that one is involved with time domains of the order of 10¡9 {10¡12 s at most. On the other hand, the time domain discussed in the present work for the persistent current in a ring, if formed, extends at least over 105 {106 years, as exhibited in NMR experiments. In examining the possibility of the decay of the persistent current in a type-I pure superconducting ring as a result of the DC zero-resistivity » = 0, it was estimated that the time domain would extend to at least 1018 s, which is longer than the age of the Universe (Kittel 1996, pp. 359{360). y One usually transforms the time-dependent Green function G( ; t0 ; ; t) into G( 0 ; " 0 ; ; " ) using Á( 0 ; ! 0 ; ; !), A( 0 ; ! 0 ; ; !) to solve the problem and eventually takes the limits for ! ! 0 and ! 0. This is not necessarily adequate and rather an unreasonable extension for static and uniform cases, where there may be troublesome divergent singularities at = 0 and ! = 0. Whether or not the zero-frequency limit ¾(!) for ! ! 0 gives ¾(0) in the AC Drude-like approach is not self-apparent either. Proc. R. Soc. Lond. A (2003)


2658

T. Masumi

Under these circumstances, one may regard the situation we present here, for T = 0 K and H ½ Hc , to resemble the time-independent scheme but in space-time, and in terms of the gauge-theory. Accordingly, we must discriminate between the two limiting cases [! ! 0; q ! 0] and [! = 0; q = 0] on the basis of di®ering orders of magnitudes in the time domains as in a previous paper (Masumi 2001). Finally, we remark on the concepts which appear in condensed-matter physics and nuclear and particle physics. Nambu (1960) studied `quasi-particles and gauge invariance in the theory of superconductivity’. Nambu & Jona-Lasinio (1961a; b) extended the work beyond condensed-matter physics to nuclear and particle physics, especially to describe their dynamical model of elementary particles. They suggested that the nucleon mass arises largely as a self-energy of some primary fermion ¯eld as a result of a mechanism similar to that which produces the appearance of an energy gap in the BCS theory of superconductivity. On the basis of the similarity in form of q Ek = ¢ 2k + "2k ; (4.10)

with the energy gap equation ¢

k

=¡

Ep =

P

p

Vkk 0 ¢

k0

=2Ek 0 , and

m2 c 4 + c 2 p 2 ;

(4.11)

they revealed the creation of reasonable nucleon condensations in nuclear physics. These concepts are parallel to the concept of o®-diagonal long-range order (ODLRO) developed by Yang (1962), apart from a problem with the di®erence between the U(1) Abelian and SU(2), SU(3) non-Abelian gauge symmetries (Ziman 1969). Ideally, superconductivity occurs in a quantum condensate at T = 0 K. It is a universal topic, occurring when the force is either Coulombic, weak (with long decay time) or strong in either nuclear or particle physics. In typical textbooks of gauge ¯eld theory, such as Peshkin & Schroeder (1995), superconductivity is noted only for (ii) the Meissner e®ect. No explicit remark concerning (i) zero-resistivity occurs regardless of the space-time aspects. Scientists in nuclear and particle physics normally consider problems at T = 0 K, as if the concept of temperature is not signi¯cant in comparison with their basic energy scale (which is much much higher than MeV and GeV). At T = 0 K a superconductor may be considered to be a huge continuum limit of an ensemble of Cooper pairs in which the concept of the ODLRO developed by Yang (1962) is applied to a `macroscopic quantum condensate’ as if it were a single entity resembling an elementary particle. Therefore, we can say that a superconductor acts like an elementary particle composed of quarks. Actually, it acts like a `macromolecule’ for small perturbations such as a small electric ¯eld or magnetic ¯eld when T = 0 K. Like a huge tennis ball, all of the materials made of de Broglie waves may condense into an entity with harmonious `coherence’ in the phase factor in space-time to compose a ¯nite-size stable entity at various levels. London (1950) vaguely but intuitively must have perceived the situation correctly over half a century ago. There is no doubt that recognition of the isotope e®ect theoretically by Frohlich (1950, 1952) and experimentally by Maxwell (1950a; b) and Reynolds et al . (1950, 1951), as well as the thermodynamical considerations involved in the Ginzburg{Landau theory (Ginzburg & Landau 1950), especially the BCS theory (in 1957), and the Josephson (1962, 1965) conProc. R. Soc. Lond. A (2003)


Quantum condensation perceived in space-time cept on phase locking in ª superconductivity.

m acro

2659

were all historical steps in advancing the science of

5. Conclusion Kamerlingh Onnes (1911a{c) and Meissner & Ochsenfeld (1933) observed the fact that, for static and uniform electric and magnetic ¯elds, E = 0 if (i) holds and B = 0 if (ii) holds. However, a set of serious logical gaps, [¬ ] and [ ], have remained in the history of the physics of superconductivity. London & London’s (1935) research left unanswered the di±cult problem of determining the link between the responses of superconductors to electrostatic and uniform magnetostatic ¯elds even in singly connected superconductors at T = 0 K. The remarkable fact that (i) zero-resistivity E = 0 and (ii) perfect diamagnetism B = 0 coexist remained a neglected issue. We have forgotten that there is a logical gap [¬ ] in the argument used to obtain the condition B = 0 from dB=dt = 0. Here, we point out that there exists another hidden logical gap, [ ], in the argument used to obtain E = 0 from curl E = 0. London & London (1935) postulated their famous equation ¤ cJ = ¡ A, ¤ = m=nS e2 to escape from [¬ ] by assuming the constant of integration to be zero. Thus, microscopically, the BCS theory was constructed in response to the London equation [II]. Josephson’s concept of phase locking not only in the junction but also in bulk was de¯nitely a monumental contribution. Quite recently, we successfully clari¯ed in a stabilized form a substantial problem related to superconductivity that was unsolved and forgotten for a long time (Masumi 2001). Here, we provide supplements to recon¯rm our understanding of the phenomena by eliminating the existence of the logical gap [¬ ] and more importantly the other potential logical gap [ ] for static and uniform electric and magnetic ¯elds [! = 0; q = 0]. We recon¯rm with the use of ª m acro that, instead of (i) E = 0 and (ii) B = 0, we must simultaneously account for (i) the zero-resistivity in time: (ii) the perfect diamagnetism in space:

¿ (R) = 0 at ! = 0; (5.1) [~K ¡

(q=c)A(R)] = 0 at q = 0: (5.2)

Here, (i) corresponds to the abandoned London equation [I] and (ii) is equivalent to the postulated London equation [II] in a coherent system of Cooper pairs. It follows that the London equation [II] need not have been postulated. We de¯nitely have to discriminate between the conditions when ! = 0, q = 0 and those when q ! 0 and ! ! 0. The substantial signi¯cance of the experimental results obtained by Kamerlingh Onnes (1911a{c) and Meissner & Ochsenfeld (1933) at static and uniform electric and magnetic ¯elds should not have been considered only in the electrodynamical framework. We here clarify why and how the BCS theory should be extended further into the (1+3)-dimensional Minkowski space-time in order to obtain a better understanding of the fundamental nature of superconductivity inevitably only in terms of gauge ¯eld theory (Masumi 2001). The author expresses his sincere thanks to Dr Tsuyoshi Uda at the Joint Research Center for Atom Technology (JRCAT), National Institute for Advanced Interdisciplinary Research, Tsukuba, Japan, for his kind advice on publishing the concept of the space-time aspect of superconductivity, stimulating discussion and his careful reading of the manuscript. Dr Giyuu Kido and his group have been extremely friendly, understanding and encouraging about continuation of the author’ s work and have o® ered long-sustaining support for the author’ s research, Proc. R. Soc. Lond. A (2003)


2660

T. Masumi

for which the author is sincerely grateful. There is no doubt that this work would never have been completed without Dr Giyuu Kido’ s warm friendship and support for the COE (Centre of Excellence) Project at the Nano-Materials Laboratory, National Institute for Materials Science.

References Abrahams, E. & Tsuneto, T. 1966 Time variation of the Ginzburg{Landau order parameter. Phys. Rev. 152, 416{432. Anderson, P. W. & Rowell, J. M. 1963 Probable observation of the Josephson superconducting tunneling e® ect. Phys. Rev. Lett. 10, 230{232. Ashcroft, N. W. & Mermin, N. D. 1976 Solid state physics, pp. 750{751. Tokyo: Holt Saunders International Editions. Bardeen, J., Cooper, L. N. & Schrie® er, J. R. 1957 Theory of superconductivity. Phys. Rev. 108, 1175{1204. Bogoliubov, N. N., Tolmachev, V. V. & Shirkov, D. V. 1959 A new method in the theory of superconductivity. New York: Consultants Bureau. Cyrot, M. 1973 Ginzburg{Landau theory for superconductors. Rep. Prog. Phys. 36, 103{158. Feynman, R. P. 1965 The Feynman lectures on physics, vol. III, ch. 21. Addison-Wesley. Feynman, R. P. 1972 Statistical mechanics: a set of lectures. Reading, MA: Benjamin. Frohlich, H. 1950 Theory of the superconducting state. I. The ground state at the absolute zero temperature. Phys. Rev. 79, 845{856. Frohlich, H. 1952 Interaction of electrons with lattice vibration. Proc. R. Soc. Lond. A 215, 291{298. Ginzburg, V. L. & Landau, L. D. 1950 Zh. Eksp. Teoret. Fiz. 20, 1064{1082. Gorkov, L. P. 1959 Microscopic derivation of the Ginzburg{Landau equations in the theory of superconductivity. Sov. Phys. JETP 9, 1364{1367. Gorkov, L. P. & Eliashberg, G. M. 1968 Generalization of the Ginzburg{Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov. Phys. JETP 27, 328{334. Heitler, W. & London, F. 1927 Wechselwirkung neutraler Atome und homoeopolare Binding nach der Quantenmechanik. Z. Phys. 44, 455{472. Hu, C.-R. & Thompson, R. S. 1972 Dynamic structure of vortices in superconductors. II. H ¿ H c2 . Phys. Rev. B 6, 110{120. Josephson, B. D. 1962 Possible new e® ects in superconductive tunneling. Phys. Lett. 1, 251{253. Josephson, B. D. 1965 Supercurrents through barriers. Adv. Phys. 14, 419{451. Kamerlingh Onnes, H. 1911a Further experiments with liquid helium. C. On the change of electrical resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures. Comm. Phys. Lab. Univ. Leiden 120b, 3{5. Kamerlingh Onnes, H. 1911b Further experiemnts with liquid helium. D. On the change of electrical resistance of pure metals at very low temperatures, etc. V. The disappearance of the resistance of mercury. Comm. Phys. Lab. Univ. Leiden 122b, 13{15. Kamerlingh Onnes, H. 1911c Further experiemnts with liquid helium. G. VI. On the sudden change in the rate at which the resistance of mercury disappears. Comm. Phys. Lab. Univ. Leiden 124c, 21{25. Kittel, C. 1996 Introduction to solid state physics, 7th edn. ch. 12, pp. 339, 359, 360. Wiley. Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of continuous media (transl. J. B. Sykes & J. S. Bell), ch. VI, p. 167. Oxford: Pergamon. Langenberg, D. N., Scalapino, D. J. & Taylor, B. N. 1966 The Josephson e® ects. Scient. Am. 214, 30{39. London, F. 1950 Super° uids: macroscopic theory of superconductivity, vol. 1. Wiley. Proc. R. Soc. Lond. A (2003)



2661

London, F. & London, H. 1935 The electromagnetic equations of the supraconductor. Proc. R. Soc. Lond. A 149, 71{88. Masumi, T. (ed.) 1983 The theory of superconductivity and magnetic ° ux quantization. Nikkei Sci. Suppl. Solid State Phys. 64, 166{176. Masumi, T. 1986 Quasi-particle physics in condensed matters, ch. 6 and 8. (In Japanese.) Tokyo: Baifukan. Masumi, T. (ed.) 1997 Theory of superconductivity and magnetic ° ux quantization. Nikkei Sci. Suppl. Phys. Cond. Matter 121, 114{123. Masumi, T. 2001 Space-time aspects of superconductivity. Phase Transitions 73, 447{482. Maxwell, E. 1950a Isotope e® ect in the superconductivity of mercury. Phys. Rev. 78, 477. Maxwell, E. 1950b Superconductivity of Sn124. Phys. Rev. 79, 173. Meissner, W. & Ochsenfeld, R. 1933 Ein neuer E® ekt bei Eintritt der Supraleitfaehigkeit. Naturwiss. 21, 787{788. Nakamura, Y., Pashkin, Y. A. & Tsai, J. S. 1999 Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature 398, 786{788. Nambu, Y. 1960 Quasi-particles and gauge invariance in the theory of superconductivity. Phys. Rev. 117, 648{663. Nambu, Y. & Jona-Lasinio, G. 1961a Dynamical model of elementary particles based on an analogy with superconductivity. I. Phys. Rev. 122, 345{358. Nambu, Y. & Jona-Lasinio, G. 1961b Dynamical model of elementary particles based on an analogy with superconductivity. II. Phys. Rev. 124, 246{254. Peshkin, M. E. & Schroeder, D. V. 1995 An introduction to quantum ¯eld theory, p. 692. The Advanced Book Program. Cambridge, MA: Perseus Books. Reynolds, C. A., Serin, B., Wright, W. H. & Nesbitt, L. B. 1950 Superconductivity of isotopes of mercury. Phys. Rev. 78, 487. Reynolds, C. A., Serin, B. & Nesbitt, L. B. 1951 The isotope e® ect in superconductivity. I. Mercury. Phys. Rev. 84, 691{694. Schrie® er, J. R. 1964 Theory of superconductivity, 1st edn. New York: W. A. Benjamin. Schrie® er, J. R. 1999 Theory of superconductivity, revised edn, ch. 1, 2 and 8. Reading, MA: Perseus Books. Seitz, F. 2001 James Clark Maxwell (1831{1879): member of APS 1875. Proc. Am. Phil. Soc. 175, 1{44. Thompson, R. S. & Hu, C.-R. 1971 Dynamic structure of vortices in superconductors. Phys. Rev. Lett. 27, 1352{1355. Tinkham, M. 1996 Introduction to superconductivity, 2nd edn. Singapore: McGraw-Hill. Tsai, J. S., Kubo, Y. & Tabuchi, J. 1987 All-ceramics Josephson junctions operative up to 90 K. Jpn. J. Appl. Phys. 26, L701{703. von Drude, P. 1900 Zur Elektronentheorie der Metalle. Annln Phys. Leipzig 1, 566{613. Wilson, A. H. 1953 The theory of metals, 2nd edn, ch. 1. Cambridge University Press. Yang, C. N. 1962 Concept of o® -diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys. 34, 694{704. Ziman, J. M. 1965 Principles of the theory of solids, ch. 11. Cambridge University Press. Ziman, J. M. 1969 Elements of advanced quantum theory, ch. 5{7. Cambridge University Press.
