Dynamics of disordered type-II superconductors: Peak effect and the I-V curves

Dynamics of disordered type-II superconductors: Peak effect

and the I–V curves

G. Bel

, D.P. Li

, B. Rosenstein

, V. Vinokur

, V. Zhuravlev

a_{University of California, Department of Chemistry, Santa Barbara, CA, USA} b_{Beijing University, School of Physics, Beijing, China}

c

National Chiao Tung University, Department of Electrophysics, Hsinchu, Taiwan, ROC

d

Argonne National Laboratory, Material Science Division, Argonne, USA Available online 22 April 2007

Abstract

We quantitatively describe the competition between the thermal fluctuations and disorder by the Ginzburg–Landau approach using both the replica method in statics and the dynamical Martin–Siggia–Rose approach which allows generalization beyond linear response. The two methods are consistent in static, while the dynamical method allows calculation of the critical current as function of magnetic field and temperature. The surface in the J–B–T space defined by this function separates between a dissipative moving vortex matter regime and vortex glass. The non-Ohmic I–V curve is obtained.

Ó 2007 Elsevier B.V. All rights reserved.

Keywords: Type-II superconductor; Glass transition; Quenched disorder

1. Introduction

Calculation of the thermodynamic, magnetic and trans-port characteristics of the vortex matter in type-II super-conductors subject to both the quenched disorder and thermal fluctuations is a long standing problem. The main difficulty is to account for the ‘‘glassy’’ properties of the vortex matter. The vortex matter can be treated in various regions of the external parameters space (including mag-netic field H, temperature T, and electric field E in dynam-ics) either in London approximation (far from Hc2),

Ginzburg–Landau approximation (far from Hc1) or using

more phenomenological models of vortex lines. In this paper we use the time dependent GL equation and the dynamical Martin–Siggia–Rose approach. The obtained results are compared with that derived in the replica method.

2. The irreversibility line and peak effect critical current We obtain the following line separating the vortex liquid from the vortex glass:

ag_T¼ ð2rÞ2=3ð3  2=rÞ; ð1Þ where aT¼  2 5=3 ð2GiÞ1=3ðbtÞ2=3ð1  t  bÞ and r ¼ a2 h ppffiffiffiffiffi2Gitn are

determined by disorder parameter, n, proportional to den-sity of the pinning centers, Ginzburg number, Gi, and by dimensionless parameters t = T/Tc, b = H/Hc2(0).

Very similar line is obtained in the crystalline phase. The line is fitted to the experiment [1]on NbSe2inFig. 1. The

melting line calculated in[2]in this case is below the irre-versibility line unlike in BSCCO where they intersect [3]. This leads to the peak effect in magnetisation curve M(H), shown inFig. 2.

The critical current, Ic ¼ ðbtÞ2=3ðGi=rÞ1=3 28=3ð2pÞ2 1 t  b þ ð2GiÞ 1=3 ðrbtÞ2=3 32 r    

0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.04.169

*

Corresponding author.

E-mail address:vortexbar@yahoo.com(B. Rosenstein).

www.elsevier.com/locate/physc Physica C 460–462 (2007) 1213–1214

is defined as a current at which the glass is depinned and becomes a flow, neglecting exponentially small creep. It is found in a good agreement with transport data of[4](see also[5]).

3. The I–V curves

Contributions from the LLL and via first Landau level are: j_LLL¼ R0 32p 4t 2_Gi=b  1=3 E ð2Þ and j_d¼ 21=3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_ffiffiffi 2 p  1 =3pt r ðbtÞ2=3r1=6ð2GiÞ5=6; ð3Þ where dimensionless units for both, electrical current and field, have been used and the response function R0 is

obtained as a solution of the equation: 4ð1  rÞR3 0

aTR20þ 1 ¼ 0. The first contribution is dissipative and

con-sistent with Bardeen–Stephen, while the second contains

the persistent current. They are similar to that in [5,6]

(Fig. 3). 4. Summary

We present a quantitative theory of the vortex liquid to vortex glass transition with both thermal fluctuations and random quenched disorder effects and compare it to exper-iment on NbSe2.

It is shown that the static flux line lattice in type-II superconductors undergoes a transition into three disor-dered phases: vortex liquid (not pinned), homogeneous vortex glass (pinned) and crystalline Bragg glass (pinned) due to both thermal fluctuations and random quenched disorder. The location of the glass transition line is deter-mined and compared to experiments. The line is clearly dif-ferent from both the melting line and the second peak line describing the translational and rotational symmetry breaking at high and low temperatures, respectively. Acknowledgement

This research is supported by NSC of ROC, NSC#932112M009024 (B.R.,V.Z.).

References

[1] S.S. Banerjee et al., Physica C 355 (2001) 39.

[2] D.P. Li, B. Rosenstein, Phys. Rev. Lett. 90 (2003) 167004; D.P. Li, B. Rosenstein, Phys. Rev. B 70 (2004) 144521. [3] H. Beidenkopf et al., Phys. Rev. Lett. 95 (2005) 257004. [4] N. Kokubo et al., Phys. Rev. Lett. B 95 (2005) 177005. [5] O. Dogru et al., Phys. Rev. Lett. 90 (2000) 167004;

Y. Paltiel et al., Phys. Rev. Lett. 85 (2000) 3712. [6] A.D. Thakur et al., Phys. Rev. B 72 (2005) 134524;

A. Pautrat et al., Phys. Rev. B 71 (2005) 064517. Fig. 2. Magnetic field dependence of the critical current in the vortex glass

phase (solid line) and in the vortex crystalline phase (dash line) in comparison with experimental data[4](points).

Fig. 3. E–I curve for parameters above (1), below (3, 4), and on the glass transition line (2).

Fig. 1. H–T phase diagram with Hc2.(T) line (upper straight line), glass

line, Eq.(1), (middle curve), and melting line (down curve). Experimental values (points) for glass transition are taken from[1].