Superconductivity and the gauge invariance of the Ginzburg–Landau equations1

International Journal of Engineering Science 37 (1999) 1487±1494

Letters in Applied and Engineering Sciences

Superconductivity and the gauge invariance of the Ginzburg±Landau equations 1 Mauro Fabrizio

*

Department of Mathematics, University of Bologna, Piazza S. Donato, 5, 40127 Bologna, Italy Received 16 June 1998; accepted 18 August 1998 _ (Communicated by E.S. S ß UHUBI)

Abstract A summary is given of pertinent parts of the London theory of superconductivity as revised in (M. Fabrizio, G. Gentili, B. Lazzari, Mathematical Models and Methods in Applied Sciences 7 (1997) 345±362). In this framework, the free energy of the Ginzburg±Landau theory can be represented in terms of two observable quantities, the magnetic ®eld and the superconducting charge density. An alternative gaugeinvariant form of the Ginzburg±Landau equations is then derived. Ó 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The phenomenon of superconductivity is described within the London theory by specifying two constitutive equations for the superconducting current Js , when the temperature is constant and below the critical value Tc (see Refs. [1±6]). The ®rst London equation explains the Meissner e€ect, or perfect diamagnetism, and the second accounts for the absence of electric resistance. However it has been shown in Refs. [7,8], that the absence of resistance can be accounted for solely by the ®rst London equation. For this and other reasons, this paper works within the theory obtained by requiring that only the ®rst London equation for Js holds at constant temperatures below Tc . The main aim of this paper is to show that if the free energy of the Ginzburg±Landau theory is considered within this modi®ed London theory, we can derive a pair of equations which are equivalent to the usual Ginzburg±Landau equations. These alternative equations involve only * 1

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0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 1 2 9 - 3

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M. Fabrizio / International Journal of Engineering Science 37 (1999) 1487±1494

the modulus jwj of the complex order parameter w and the magnetic ®eld H, since the free energy can be expressed only as a function to H and jwj, whereas usually it is written as a function of w and the vector potential A. Consequently, the theory must automatically be gauge-invariant. 2. London theory Consider a domain X in R3 , which is not necessarily simply connected but has a boundary oX of class C 1 . Suppose that X is ®lled with a superconducting material. The evolution in time of the electromagnetic ®eld is described by Maxwell equations: _ ˆ r  H ÿ J; D

…1†

B_ ˆ ÿr  E;

…2†

where E is the electric ®eld, H the magnetic ®eld, D the electric displacement, B the magnetic induction and J the electric current. The nature of the material is expressed by suitable constitutive equations. Firstly it is supposed that D…x; t† ˆ …x†E…x; t†;

…3†

B…x; t† ˆ l…x†H…x; t†;

…4†

where  and l are the dielectric constant and the magnetic permeability, respectively. The constitutive properties of the electric current in a superconductor are described through the London model, called ``two ¯uid model'', where the electric current is the sum of a supercurrent Js and a normal current Jn , such that J ˆ Js ‡ J n ;

Jn ˆ rE;

r  KJs ˆ ÿB;

KJ_s ˆ E;

…5† …6†

where r…x† is the electric conductivity and K is a new material coecient. The equations in (6) are called the ®rst and second London equations. Superconductors exhibit two physical phenomena. One, which are discovered by H.K. Ones in 1908, is the the absence of electric resistance whenever the temperature falls under the critical temperature Tc . The other is the Meissner e€ect which was discovered in 1933, and it consists in the expulsion of the magnetic ®eld from the superconductor whenever the temperature falls under the value T < Tc . The London theory accounts for the absence of electrical resistance with (6)2 and the Meissner e€ect using (6)1 . There are good reasons however for not including (6)2 as a constitutive equation. There is the observation that there are more mathematical equations than physical variables, suggesting that the London theory is inconsistent. Also Eq. (6)2 is used to model of perfect conductivity, a quite di€erent physical phenomenon from superconductivity (see Ref. [1,2,6,9].) Starting from these

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remarks, it has been proven in Refs. [7,8] that the global absence of electric resistance in a superconductor is a consequence of (6)1 alone and of a global dissipation principle for electromagnetic energy. In contrast, there is no local electric resistance if (6)2 also holds. The arguments of [7,8] are now reviewed. The global principle of dissipation of electromagnetic energy asserts in every cyclic process of period T , the following inequality holds: IT Z n o _ _ D…x; t†
E…x; t† ‡ B…x; t†
H…x; t† ‡ J…x; t†
E…x; t† dx dt P 0

…7†

0 X

In order to prove the absence of electric dissipation due to the superconducting current on the grounds of the sole equation (6)1 , it is necessary to put such equation into the form KJ_s ˆ E ‡ ru:

…8†

In fact, taking the curl of Eq. (8) yields (6)1 . Conversely, taking the time derivative of (6)1 and using Eq. (1) yields r  KJ_s ˆ ÿr  E hence Eq. (8) follows, where u is an arbitrary suciently smooth function. Furthermore, we recall that the properties of the current Js require that r
Js ˆ 0 Js
njoX ˆ 0:

…9†

Considering stationary processes, by Eqs. (7)±(9) we get: IT Z 0 X

 
dÿ
dÿ
1 dÿ 2 2 2 …x†E …x; t† ‡ l…x†H …x; t† ‡ K…x†Js …x; t† dx dt 2 dt dt dt IT Z

‡

  Js …x; t†
ru…x; t†‡r…x†E2 …x; t† dx dt ˆ

0 X

IT Z u…x; t†Js …x; t†
n…x†dr dt 0 oX

IT Z IT Z   2 ‡ÿ u…x; t†r
Js …x; t† ÿ r…x†E …x; t† dx dt ˆ r…x†E2 …x; t† dx dt P 0: 0 X

…10†

0 X

Thus all global dissipation is due to the normal current Jn , and none to Js . Let us emphasize that this has been possible only because we used a global formulation of the Dissipation principle. We can thus claim that the two main phenomena in superconductivity are both explained by Eq. (6)1 alone.

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3. Non-local nature of modi®ed London theory It is brie¯y pointed out in this section that a theory of superconductivity which supposes that Eq. (6)1 is the only constitutive equation satis®ed by the supercurrent, has a nonlocal nature. Let us consider a simply connected domain 2. Superconductivity is described by r  KJs ˆ ÿB;

r
Js ˆ 0;

Js
njoX ˆ 0:

…11†

Moreover, since r
B ˆ 0, there exists a vector ®eld hH 1 …X† such that r  h ˆ ÿB;

r
h ˆ 0;

h  njoX ˆ 0;

…12†

so that, comparing Eqs. (11)1;2 and (12)1;2 , there exists a scalar ®eld uH 2 …X† such that KJs ˆ h ‡ ru: The ®eld u must therefore satisfy the boundary value problem r
ru ˆ 0;

ru
njoX ˆ ÿh
njoX :

…13†

Problem (12), since r
h ˆ 0, can be rewritten using a vector ®eld KH 2 …X† such that r  K ˆ h;

r
K ˆ 0:

The new problem takes the form r  r  K ˆ ÿB;

r
K ˆ 0;

…r  K†  njoX ˆ 0:

Let us consider the corresponding problem for the tensor Green's function C…x; x0 † r0  r0  C…x; x0 † ˆ ÿd…x-x0 †I; …r0  C…x; x0 ††  n…x0 † oX ˆ 0;

…14†

…15†

where r0 ˆ rx0 . By Eqs. (14) and (15) we get Z K…x† ˆ C…x; x0 †B…x0 † dx0 ; X

hence Z h…x† ˆ

r  C…x; x0 †B…x0 † dx0 :

X

Similarly, considering problem (13) and its Green's function G…x; x0 † we obtain

2

By a simply connected domain X  R3 we mean a suciently regular set such that every closed surface wholly contained in X can be continuously shrunk to a point.

Z

M. Fabrizio / International Journal of Engineering Science 37 (1999) 1487±1494

u…x† ˆ ÿ

1491

h…x0 †
n…x0 †G…x; x0 † dr0 ;

oX

so that 0 1 Z Z Js …x† ˆ Kÿ1 …x†@ r  C…x; x0 †B…x0 † dx0 ÿ f …x0 †r0
G…x; x0 † dr0 A; X

…16†

oX

where 0 1 Z f …x† ˆ @ r0  C…x; x0 †B…x0 † dx0 A
n…x†: X

It follows that the theory based on system (11) be typically non-local and analogous to that proposed by Pippard in Ref. [10] The expression (17)1 is analogous to the representation of Pippard [10] (see also Ref. [1,2].) Therefore the London model, whenever restricted to equation (6)1 , is non-local. This is the reason why a global formulation of the Dissipation Principle, as given in Section 2, is necessary. 4. Ginzburg±Landau Theory System (11) describes well the phenomenon of superconductivity for processes that are spatially homogeneous, so that the whole medium is in the superconducting phase. Ginzburg±Landau (GL) theory 3 deals with the case of nonhomogeneous phase, in which some of the medium is in superconducting state and some in the normal state. Since B is solenoidal, it is possible to introduce a vector potential A satisfying r  A ˆ B;

r
A ˆ 0:

…17†

Comparison with Eq. (11) that KJs ˆ ÿA ‡ ru;

…18†

where the scalar function u must satisfy r2 u ˆ 0: Clearly (11)1 is equivalent to Eq. (18). In Ginzburg±Landau theory Eq. (18) is favored over (11)1 . This compels to make the results invariant under gauge transforms

3

The papers [2], [11], are valuable references for the subject and the bibliography.

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M. Fabrizio / International Journal of Engineering Science 37 (1999) 1487±1494

A0 ˆ A ‡ ru:

…19†

It is possible to uniquely determine the gauge A by imposing the boundary condition A
njoX ˆ 0: In fact, if the domain X is simply connected 4, u must satisfy r2 u ˆ 0; ru
njoX ˆ 0 which implies ru ˆ 0 in Eq. (18). Within GL theory the phase transition between normal and superconducting state is investigated by introducing a complex parameter w ˆ feih ;

…20†

whose squared modulus f 2 ˆ ns represents the density of superconducting electrons of mass m and charge e , while h is related to u appearing in Eq. (18). Clearly the medium is in the normal state (T < Tc ) when f ˆ 0, while for f ˆ 1 the medium is in the state of perfect superconductivity (T ˆ 0.) Within this theory a major role is played by the free energy of X  Z Z  1 1 2 2 4 2  …21† e…w; A† dx ˆ ÿ ajwj ‡ bjwj ‡ lH ‡  j…ÿihr ÿ e A†wj dx; 2 2m X

X

where the a and b coecients depend on temperature and h is Planck's constant. The search for a minimum of Eq. (21) leads to the so called Ginzburg±Landau equations: 1 …ÿi hr ÿ e A†2 w ÿ aw ‡ 2bjwj2 w ˆ 0; 2m

…22†

2

Js ˆ ÿ

i he  …e †  …w rw ÿ wrw † ÿ jwj2 A: 2m m

…23†

These equations are obtained calculating the variation of Eq. (21) with respect to w and A respectively, in quasi-static processes, so that J ˆ Js ˆ r  H, hence r
Js ˆ 0. Eq. (23) can be written   h  …24† K…f †Js ˆ ÿ  rh ‡ A ; e where K…f † ˆ …m =ns e2 †. Therefore Eq. (24) becomes identical to Eq. (18) if we set u ˆ ÿ…h=e †h. Finally, taking the curl of Eq. (24), we get (11)1 . The use of the vector potential A as independent variable both in the expression of the free energy and in the constitutive relation for Js implies the need for such quantities to be invariant under gauge transforms of the form (19). It is natural to ask whether the free energy and the

~ because it All that is said here for a simply connected domain X can be extended to a multiply connected domain X, is always possible to cut X so as to make it simply connected. In such cases it is necessary to specify boundary ~ conditions for A over the whole of oX. 4

M. Fabrizio / International Journal of Engineering Science 37 (1999) 1487±1494

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constitutive relation for Js may be made independent of A, but dependent on the ®eld H and on the function f . It is possible to show (see [11,12]) that h2 …rf †2 ‡ f 2 … hrh ‡ e A†2 ˆ h2 …rf †2 ‡ KJs2 ; hrw ‡ e Awj2 ˆ  ji so that the free energy can be written as  Z Z  1 1 2 2 2 4 2 2 e~…w; A† dx ˆ ÿ af ‡ bf ÿ lH ‡  h …rf † ‡ K…f †Js dx; 2 2m X

…25†

…26†

X

where the term in …rf †2 represents the surface energy between the superconducting and the normal state, while …1=2m †KJs2 can be interpreted as kinetic energy associated with the superconducting current. The variation of the functional (26) with respect to f and H yields the equations: 2 2 h m 2 r f ÿ J ‡ af ÿ 2bf 3 ˆ 0; 2m 2e f 3 s

…27†

ÿ lH ˆ r  K…f †Js :

…28†

on X, and the boundary conditions rf
n ˆ 0

…29†

on oX. It is possible to show that Eq. (28) is equivalent to the GL Eq. (23). It is enough to divide Eq. (23) by K…f † ˆ …m =e f 2 † and then take the curl. In order to see the correspondence between Eqs. (27) and (22) recall that Eq. (22) is the variation of functional (21) taken with respect to w . This variation can be found by ®rst varying jwj ˆ f and then the argument h in Eq. (21) written in the form: Z Z   1 1  2 2 2 2 4 2 2  dx: …30† e…w; A† dx ˆ ÿ af ‡ bf ‡ lH ‡  h …rf † ‡ f … hrh ‡ e A† 2 2m X

X

We thus obtain ÿ

2 2 h e r f ‡  f K2 Js2 ÿ af ‡ 2bf 3 ˆ 0;  2m 2m 

r


  e 2  h f rh ‡ A ˆ 0: m e

Eq. (31) is the same as Eq. (27), while Eq. (32) is equivalent to the restriction on Js r
Js ˆ 0

…31†

…32†

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taking the variation with respect to h does not give a new equation, but rather a restriction already implicitly contained in (28). This con®rms that it is not essential to introduce the parameter h, since its use does not carry any actual increase of information about the system. 5. Conclusions Our research is motivated by the observation that the vector potential A and the phase h, are macroscopically not observable variables. The possibility of representing the free energy in terms of observable variables like the magnetic ®eld H and the superconducting charge density f allows the number of GL equations to be reduced from 5 to only 4. This new set of equations are of course gauge-invariant. References [1] J. Bardeen, Theory of superconductivity, in: S. Flugge (Ed.), Handbuch der Physik, Springer, Berlin, 1956, pp. 274±369. [2] B. Chandrasekhar, Early experiments and phenomenological theories, in: R. Parks (Ed.), Superconductivity, Dekker, 1969. [3] F. London, Super¯uids, vol. 1, Wiley, New York, 1950. [4] H. London, Proc. Roy. Soc. A 155 (1936) 102±110. [5] D. Shoenberger, Superconductivity, Cambridge University Press, Cambridge, 1952. [6] H. Thomas, Some remarks in the history of superconductivity, in: J. Bednorz, K. Muller (Eds.), Earlier and Recent Aspects of Superconductivity, Springer, Berlin, 1990, pp. 2±44. [7] M. Fabrizio, G. Gentili, B. Lazzari, Mat. Models and Methods in Appl. Scie. 7 (1997) 345±362. [8] M. Fabrizio, C. Giorgi, Riv. Mat. Univ. Parma. 4 (1984) 415±430. [9] B. Serin, Superconductivity, experimental part, in: S. Flugge (Ed.), Handbuch der Physik, Springer, Berlin, 1956, pp. 210±273. [10] A. Pippard, Proc. Roy. Soc. A 216 (1953) 547±568. [11] S. Chapman, S. Howison, J. Ockendon, SIAM Review 34 (1992) 529±560.  [12] L. Gorkov, G. Eliashberg, Soviet Phys. JETP 27 (1968) 328±334.