**Acknowledgement**

One of the authors (M. K.) thanks his former and present graduate students, M. Ako, M. Hirayama, S. Nakajima, H. Suematsu, T.Minamino, S. Tomita, Y. Niwa, and D. Fujibayashi for useful discussions. This work is partly supported by "The Faculty Innovation Research Project" of Osaka Prefecture University and it was supported by the CREST-JST project.

### **6. References**

340 Superconductors – Materials, Properties and Applications

κ ξ

<sup>2</sup> 22 0

1 2

1 2

 α λ

*Institute of Material Sciences, Tohoku University, Sendai, Japan* 

*CCSE, Japan Atomic Energy Agency, Tokyo, Japan* 

*i i d*

2 α

**Author details** 

Masaru Kato

Takekazu Ishida

Tomio Koyama

Masahiko Machida

**Acknowledgement** 

*Japan* 

*Japan* 

 α

2 α

For s-wave superconducting regions, they are defined similarly.

,

αα

2

κ ξ

<sup>3</sup> Im 2 Re

 *<sup>s</sup> i ii i di di si si i i d*

<sup>2</sup> <sup>3</sup> Im Im

λ

4 α

1 2 <sup>2</sup> 3 Re

2 4

α

=

The coefficients in Eqs. 17 and 18 in the d-wave superconducting region are given as,

( ) { } ( ) ( ) ( ) 1 2 1 2 1 2 1 2

( ) { } ( ) 1 2 1 2

λ

λ

( )( ) <sup>2</sup> <sup>3</sup> 22 0

*xy yx*

λ

λ

2

≡Δ−

( ) ( ) 1 2 1 2 1 2

( )( ) 1 2 1 2 1 2

0

Φ *Ui dd d HJi* ( )

*Department of Mathematical Sciences, Osaka Prefecture University1-1, Gakuencho, Sakai, Osaka,* 

*Department of Physics and Electronics, Osaka Prefecture University, 1-1, Gakuencho, Sakai, Osaka,* 

One of the authors (M. K.) thanks his former and present graduate students, M. Ako, M. Hirayama, S. Nakajima, H. Suematsu, T.Minamino, S. Tomita, Y. Niwa, and D. Fujibayashi for useful discussions. This work is partly supported by "The Faculty Innovation Research Project" of Osaka Prefecture University and it was supported by the CREST-JST project.

 α π

α

*T J* ( )

*T J* ( )

∗ ∗ <sup>≡</sup> ΔΔ + ΔΔ

∗ ∗ <sup>≡</sup> ΔΔ + ΔΔ *<sup>s</sup> i iii di si si di*

( )<sup>2</sup> 22 0

3 2

 ≡ Δ κ ξ

<sup>∗</sup> <sup>≡</sup> Δ Δ *s ij i i ij di si i i <sup>d</sup>*

*ij dd d ij i i ij di di si si i i x y <sup>d</sup>*

 ≡ Δ <sup>+</sup> ΔΔ + ΔΔ *<sup>s</sup>*

*R* **A** *K I* , (41)

3 3 2Re 4 Re

2

∗ ∗

λ

λ

*R I* **A** , (42)

*ij d d ij ij S K K* , (43)

α

α

= *x*, *y* , (44)

= *x*, *y* , (45)

= *x*, *y* . (46)


[20] Agterberg D F. Square vortex lattices for two-component superconducting order parameters. Phys. Rev. 1998; 58; 14484-14489.

**Chapter 8**

**Chapter 14**

*<sup>e</sup>* , (1)

*<sup>C</sup>* d**r** · **A**(**r**) =

**Flux-Periodicity Crossover from hc/e in Normal**

One of the most important properties of superconductors is their perfectly diamagnetic response to an external magnetic field, the Meissner-Ochsenfeld effect. It is a pure quantum effect and therefore reveals the existence of a macroscopic quantum state with a pair condensate. A special manifestation of the diamagnetic response is observed for superconducting rings threaded by a magnetic flux: flux quantization and a periodic current

Persistent currents and periodic flux dependence are also known in normal metal rings and best known in form of the Aharonov-Bohm effect predicted theoretically in 1959 [4]. Since the wavefunction of an electron moving on a ring must be single valued, the phase of the wave function acquired upon moving once around the ring is a integer multiple of 2*π*. A magnetic

(2*πe*/*hc*) Φ, where *C* is a closed path around the ring and **A**(**r**) the vector potential generating the magnetic flux Φ threading the ring. Here, *e* is the electron charge, *c* the velocity of light, and *h* is Planck's constant. Thus, the electron wave function is identical whenever *ϕ* has an integer value and therefore the system is periodic in the magnetic flux Φ with a periodicity of

<sup>Φ</sup><sup>0</sup> <sup>=</sup> *hc*

the flux quantum in a normal metal ring. In particular, the persistent current *J*(*ϕ*) induced by

The periodic response of a superconducting ring to a magnetic flux is of similar origin as in a normal metal ring, though the phase winding of the condensate wavefunction has to be reconsidered. On account of the macroscopic phase coherence of the condensate, flux oscillations must be more stable in superconductors, and London predicted their existence in superconducting loops already ten years before the work of Aharonov and

> ©2012 Loder et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Loder et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

flux threading the ring generates an additional phase difference 2*πϕ* = (*e*/*c*)

the magnetic flux is zero whenever *ϕ* = Φ/Φ<sup>0</sup> is an integer.

cited.

**Metallic to hc/2e in Superconducting Loops**

Florian Loder, Arno P. Kampf and Thilo Kopp

http://dx.doi.org/10.5772/50833

**1. Introduction**

response.

Additional information is available at the end of the chapter

