A Holomorphic Scaling Function for YBa2Cu3O7-x and Hg(Xe) Superconductors

HAL Id: jpa-00249459

https://hal.archives-ouvertes.fr/jpa-00249459

Submitted on 1 Jan 1996

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Holomorphic Scaling Function for YBa2Cu3O7-x and Hg(Xe) Superconductors

P. Fang

To cite this version:

P. Fang. A Holomorphic Scaling Function for YBa2Cu3O7-x and Hg(Xe) Superconductors. Journal

de Physique III, EDP Sciences, 1996, 6 (3), pp.333-336. �10.1051/jp3:1996126�. �jpa-00249459�

A Holomorphic Scaling Function for YBa2Cu307-x and Hg(Xe) Superconductors

P-H- Fang (*)

F-S- Lab, 156 Common Street, Belmont, Massachusetts, 02178, U-S-A-

(Received 16 September1994, revised 10 November 1995, accepted 5 December 1995)

PACS.74.70.b Superconducting materials

Abstract. A single scaling function is presented, which describes the voltage-current relation in the whole temperature range, both above and below the superconducting temperature. This

function is applied to analyze the experimental data of YBa2Cu307-~ ceramic and Hg(Xe) film

superconductors.

List of Symbols

Y V/tH

V induced voltage by I, the transport current

t+ [T T~(0)]/T~(0), t-

-t+

T~ Temperature of phase transition

~1 scaling index

( scaling index from the slope of V

=

[I I~(T)]f [(T)

[T~(0) T]~ where 4l is power law index

b ~1/4l

z I/t~

Z~ [(T) /t~

Y+T > T~(0)

Y-T < T~(0)

The subscripts for other symbols have similar meanings.

In _a recent paper of Soret and Ammor [ii, based _on voltage-current data of YBa2Cu3 O

7-~, at the temperature region of 77.30 to 90.95 K, ^different scaling functions _are proposed, depending

on the temperature, either above, _or below the superconductivity temperature T~. In the present note, ^I will propose a single holomorphic scaling function with its asymptotics becoming

those of Soret and Ammor.

(*)Fax: (617) 489 0212

@ Les #ditions _{de Physique 1996}

334 JOURNAL DE PHYSIQUE III N°3

The salient results of reference iii to be discussed in the present note _are the following:

When T _> T~(0),

y+

~

z, (1)

at small

At large _z,

Y+

z~/~ (2)

When T < T~(0) and _{z <} 10,

Y-

lZ Zc)~'~ 13)

with _z~

=

1.6. At large _z,

Y-

r~

Y+. (4)

In above,

r~

implies _an asymptotic equality, _up to _a constant multiplier _{c [2].}

We found _that equations (i) to (4) _can be represented by _a single equation.

Y

c(z + z~)fz6-f (5)

where b

=

~1/4l ^{of reference} [ii, is the exponent ^of _z at large _{z [3].} The exponent parameter (

was introduced in reference [ii. The _constant parameter _za ^will be discussed later.

We _now analyze the asymptotics of Y according to equation (5). For this purpose, ( of reference [ii is generalized to (+ and (-. For Y+i, I-e- in T > T~(0),

Y+

c+z(+zl~~f+~ (i + (+z/z+ ₊ for small

(1')

=

c+z~(i +(+z+/z+.. for large _z (2')

where _we denote _za by _z+. For T < T~(0), _za is to be identified with _-z~ of reference [ii and Y-

c-z)~~f-)(z _{z~)f- when}

z

z~ (3')

=

c-z~(i _{(-z~/z + for large}

z (4')

The symbols _c- and (- _are for T _< T~(0) equivalent of _c+ and (+ ^{for T} > T~(0). The equations

are numbered such that the equivalencies between equations (i) and (i') etc. are transparent.

Based _on these relations, the exponents _can be calculated from reference iii,

b (+

i, b

=

i-S and (-

2.3, (6)

following values _are obtained:

(+

0.5 and b (-

=

-0.8. (7)

Independently, a set of numbers has been depicted from Figure 5 of reference [ii and the value of the exponents ^is determined based _on the _curve fitting. Within the precision which _can be read from the figure, similar values for the exponents were obtained. In addition, from the result of the _curve fitting, numerical values of other parameters _can be derived and _are listed in Table I.

With the _same procedure prescribed above, _we have analyzed another superconductor _sys- tem, ^the Hg(Xe) film. As in the _case of reference [ii, b, ~1and 4l _were given, ^other parameters

are determined from Figure 3 of reference [3], ^and are listed in Table1.

Table 1. Values of indices of the scaling function

ceramic film

i-S 3.0

~1(= fib) 2.7 5.28

4l(= fl) 1.8 1.76

~1- 4l(= _~y) 0.9 3.6

b (- -0.8 -0.7

(- 2.3 3.9

b (+ 1.0 1.0

(+ 0.5 2.0

log _c- -2.6 -6.2

log _c+ ^-2.5 -4.2

z-(= -z~) -1.6 -1.5 _x 10~

z+ 64 6.i x 10~

According to Table I, the important ^difference between the Y-Ba-Cu-O ceramic and the

Hg(Xe) film is the numerical value of b, being i-S for Y-Ba-Cu-O and 3.0 for Hg(Xe). This difference has been interpreted _{as a} corresponding difference between X Y system _[4] ^and

Kosterlitz-Thouless system _[5] by Peyral et al. [4].

Concerning the meaning of _za, which is clear in the _case of T _< T~(0): _za becomes _-z~,

corresponding to I

=

[. In the _case of T > T~(0), Y+ vanishes at _z

=

za < 0 _as shown in Table I. This virtual _zero has _no meaning ^{in the real domain.} However, (I) ^{because of their} roles in the symmetrization and completeness of equation (5), (it) a phenomenological necessity

in the representation of (log Y) _versus (log z) (Fig. 5 of Ref. [ii and Fig. 3 of Ref. _[3] ), and,, (iii)

the sign of _za which correctly reflects the convexity and concavity of the experimental data,

these _za parameters must be contained in the theoretical formulation of the scaling functions.

In conclusion, the scaling ^function proposed ^{in this} note demonstrates _an interdependence

among the exponents of the scaling ^functions through _a holomorphic formulation, thus provides

an exploration of the mathematical _structure of high temperature superconductors.

Acknowledgments

acknowledge Teresa G. Fang in the preparation of this _paper.

References

[ii Soret J-C- and Ammor L., Loi d'4chelle dans YBa2Cu307-~ cAramique, J. Phys. III France 2 (1992) 203.

[2] Stanley H-E-, Introduction to Transitions and Critical Phenomena (Oxford Univ. Press, 1971) _p. 42.

[3] Epstein K., Goldman A.M. and Kadin A-M-, Universal Current Scaling in Hg(Xe) Films,

Physica B 107 (1981) 323.

336 JOURNAL DE PHYSIQUE III N°3

[4] Peyral P. et al., Scaling in Superconducting Ceramics, J. Less Common Metals 151 (1989)

49.

[5] Kosterlitz J-M- and Thouless D-J-, Ordering, metastability and phase transition in two dimensional systems, ^J. Phys. C 6 (1971) i181.

[6] Wolf S-A-, Gubser D-U- and Imry Y., Universal Scaling in the Critical Region of _a Two-

Dimensional Superconducting Phase Transition, J. Phys. Rev. 42 (1979) 324.