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Magnetic response of field-cooled type-I superconducting hollow cylinders
Physica B 284}288 (2000) 801}802
Magnetic response of "eld-cooled type-I superconducting hollow cylinders Roberto De Luca* INFM, Dipartimento di Fisica, Universita% degli Studi di Salerno, 84081 Baronissi (SA), Italy
Abstract A type-I superconducting hollow cylinder may present positive "eld-cooled susceptibility in a very narrow applied "eld interval. This e!ect may be detected for ratios of the inner to outer radius of the cylinder greater than a limiting value (a "1/(2, below which the magnetic response is diamagnetic for all "elds. 2000 Elsevier Science B.V. All rights reserved. Keywords: Field-cooled susceptibility; Multiply connected superconductors
In the quest for the symmetry of the superconducting state in high-¹ materials, much attention has been paid to the Wohlleben e!ect [1]. This phenomenon has been considered as evidence of d-wave pairing in high-¹ compounds by some authors [2]. On the other hand, other researchers, who have shown that positive "eldcooled susceptibility may be observed in some Nb samples [3], consider this e!ect as not directly related to the problem of the symmetry of the superconducting order parameter of cuprates. A long hollow cylinder is the simplest multiply-connected system to allow a rigorous elementary thermodynamic analysis, which may shed some light on the fundamental aspects of the magnetic behavior of multiply connected superconductors cooled below the transition temperature in the presence of a small constant magnetic "eld H. Surprisingly, one "nds that multiply connected type-I superconductors may present positive "eld-cooled susceptibility in a "nite applied "eld interval. Let us then consider a very long hollow cylinder of a type-I superconductor with wall thickness t such that t
* Corresponding author. E-mail address: [email protected] (R. De Luca)
given by [4] G"G
U!U , U
(1)
where U is the elementary #ux quantum, U is the trap ped #ux and U "k HS , S being the cross-section of the hole, is the appied #ux. Fluxoid quantization implies:
!
A ) dl"U"nU ,
(2)
where C is a closed path well inside the superconducting region encircling the hole and n is an integer. The meaning of the integer n has been clari"ed by the experiment carried out by Goodman and Deaver [5] in 1970. In this experiment a tin hollow cylinder was cooled below its critical temperature ¹ in the presence of an external axial "eld H. The trapped #uxon number n in the cylindrical hole was found to satisfy the following simple non-linear relation: n"X
U , U
(3)
where the function X, when applied to a real number x, gives the closest integer n to x. In this way, neglecting V #ux penetration inside the cylinder wall, we can assume that the following "eld distribution is present in the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 1 0 1 - 8
802
R. De Luca / Physica B 284}288 (2000) 801}802
Fig. 1. Field-cooled susceptibility as a function of the normalized external applied #ux U /U for a microcylinder with R "20 lm and a". G
specimen: U X(U /U ) 0)r(R , G k h" S (4) 0 R (r(R . G M Assuming now that the measuring instrument is designed to detect the average "eld 1h2 in the specimen, we can de"ne the "eld-cooled (FC) susceptibility as follows:
U U s "a X !1, (5) $! U U where a"S /S, S being the total cross-section of the cylinder. The "rst addendum in Eq. (5) may result greater than 1 for a', thus giving positive values of s . A graph of $! the FC susceptibility as a function of k H is given in Fig. 1 for a microcylinder with R "20 lm and a". We see G that the "eld interval in which a paramagnetic signal can be detected is the following: 1U 2U (k H( , 2S 3S
(6)
where U /S "1.64 lT. The amplitude k *H of such interval is thus k *H"U /6S "0.27 lT. In conclusion, by adopting an elementary thermodynamic approach, the study of the magnetic response of hollow cylinders of type-I superconductors after "eldcooling predicts that these systems may show paramagnetic response, for a (S /S(1, with a ", in a narrow "eld interval. The amplitude *H of this interval depends inversely on the cross-section S of the hole. However, the response of the system is diamagnetic for all "elds if S /S)a . For micro cylinders of total cross-section S"6p;10\ m this e!ect may be observed for "elds H such that 0.82 lT( k H(1.09 lT. This example has been proposed to show that topological properties of the specimen may drastically a!ect the FC magnetic response of type-I superconductors. Finally, even though this analysis cannot be readily transferred to the case of high-¹ super conductors, it certainly sheds some light on the question of positive "eld-cooled susceptibility in traditional superconducting systems.
References [1] W. Braunish, N. Knauf, G. Bauer, A. Kock, A. Becker, B. Freitag, A. GruK tz, V. Kataev, S. Neuhausen, B. Roden, D. Khomskii, D. Wohlleben, Phys. Rev. B 48 (1993) 4030. [2] M. Sigrist, T.M. Rice, Rev. Mod. Phys. 67 (1995) 503. [3] P. KosticH , B. Veal, A.P. Paulikas, U. Welp, V.R. Todt, C. Gu, U. Geiser, J.M. Williams, K.D. Carlson, R.A. Klemm, Phys. Rev. B 53 (1996) 791. [4] A. Barone, G. PaternoH , Physics and Applications of the Josephson E!ect, Wiley, New York, 1982. [5] W.L. Goodman, B.S. Deaver, Phys. Rev. Lett. 24 (1970) 870.
Magnetic response of "eld-cooled type-I superconducting hollow cylinders Roberto De Luca* INFM, Dipartimento di Fisica, Universita% degli Studi di Salerno, 84081 Baronissi (SA), Italy
Abstract A type-I superconducting hollow cylinder may present positive "eld-cooled susceptibility in a very narrow applied "eld interval. This e!ect may be detected for ratios of the inner to outer radius of the cylinder greater than a limiting value (a "1/(2, below which the magnetic response is diamagnetic for all "elds. 2000 Elsevier Science B.V. All rights reserved. Keywords: Field-cooled susceptibility; Multiply connected superconductors
In the quest for the symmetry of the superconducting state in high-¹ materials, much attention has been paid to the Wohlleben e!ect [1]. This phenomenon has been considered as evidence of d-wave pairing in high-¹ compounds by some authors [2]. On the other hand, other researchers, who have shown that positive "eldcooled susceptibility may be observed in some Nb samples [3], consider this e!ect as not directly related to the problem of the symmetry of the superconducting order parameter of cuprates. A long hollow cylinder is the simplest multiply-connected system to allow a rigorous elementary thermodynamic analysis, which may shed some light on the fundamental aspects of the magnetic behavior of multiply connected superconductors cooled below the transition temperature in the presence of a small constant magnetic "eld H. Surprisingly, one "nds that multiply connected type-I superconductors may present positive "eld-cooled susceptibility in a "nite applied "eld interval. Let us then consider a very long hollow cylinder of a type-I superconductor with wall thickness t such that t
* Corresponding author. E-mail address: [email protected] (R. De Luca)
given by [4] G"G
U!U , U
(1)
where U is the elementary #ux quantum, U is the trap ped #ux and U "k HS , S being the cross-section of the hole, is the appied #ux. Fluxoid quantization implies:
!
A ) dl"U"nU ,
(2)
where C is a closed path well inside the superconducting region encircling the hole and n is an integer. The meaning of the integer n has been clari"ed by the experiment carried out by Goodman and Deaver [5] in 1970. In this experiment a tin hollow cylinder was cooled below its critical temperature ¹ in the presence of an external axial "eld H. The trapped #uxon number n in the cylindrical hole was found to satisfy the following simple non-linear relation: n"X
U , U
(3)
where the function X, when applied to a real number x, gives the closest integer n to x. In this way, neglecting V #ux penetration inside the cylinder wall, we can assume that the following "eld distribution is present in the
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 1 0 1 - 8
802
R. De Luca / Physica B 284}288 (2000) 801}802
Fig. 1. Field-cooled susceptibility as a function of the normalized external applied #ux U /U for a microcylinder with R "20 lm and a". G
specimen: U X(U /U ) 0)r(R , G k h" S (4) 0 R (r(R . G M Assuming now that the measuring instrument is designed to detect the average "eld 1h2 in the specimen, we can de"ne the "eld-cooled (FC) susceptibility as follows:
U U s "a X !1, (5) $! U U where a"S /S, S being the total cross-section of the cylinder. The "rst addendum in Eq. (5) may result greater than 1 for a', thus giving positive values of s . A graph of $! the FC susceptibility as a function of k H is given in Fig. 1 for a microcylinder with R "20 lm and a". We see G that the "eld interval in which a paramagnetic signal can be detected is the following: 1U 2U (k H( , 2S 3S
(6)
where U /S "1.64 lT. The amplitude k *H of such interval is thus k *H"U /6S "0.27 lT. In conclusion, by adopting an elementary thermodynamic approach, the study of the magnetic response of hollow cylinders of type-I superconductors after "eldcooling predicts that these systems may show paramagnetic response, for a (S /S(1, with a ", in a narrow "eld interval. The amplitude *H of this interval depends inversely on the cross-section S of the hole. However, the response of the system is diamagnetic for all "elds if S /S)a . For micro cylinders of total cross-section S"6p;10\ m this e!ect may be observed for "elds H such that 0.82 lT( k H(1.09 lT. This example has been proposed to show that topological properties of the specimen may drastically a!ect the FC magnetic response of type-I superconductors. Finally, even though this analysis cannot be readily transferred to the case of high-¹ super conductors, it certainly sheds some light on the question of positive "eld-cooled susceptibility in traditional superconducting systems.
References [1] W. Braunish, N. Knauf, G. Bauer, A. Kock, A. Becker, B. Freitag, A. GruK tz, V. Kataev, S. Neuhausen, B. Roden, D. Khomskii, D. Wohlleben, Phys. Rev. B 48 (1993) 4030. [2] M. Sigrist, T.M. Rice, Rev. Mod. Phys. 67 (1995) 503. [3] P. KosticH , B. Veal, A.P. Paulikas, U. Welp, V.R. Todt, C. Gu, U. Geiser, J.M. Williams, K.D. Carlson, R.A. Klemm, Phys. Rev. B 53 (1996) 791. [4] A. Barone, G. PaternoH , Physics and Applications of the Josephson E!ect, Wiley, New York, 1982. [5] W.L. Goodman, B.S. Deaver, Phys. Rev. Lett. 24 (1970) 870.