Theory of the square to rhomb structural phase transitions in the vortex lattice of type-II superconductors

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Theory of the square to rhomb structural phase transitions

in the vortex lattice of type-II superconductors

B. Rosenstein

a

, B.Ya. Shapiro

b,*

, I. Shapiro

b

a_{National Chiao Tung University, Department of Electrophysics, Hsinchu, Taiwan, ROC} b_{Department of Physics, Bar-Ilan University, Ramat-Gan, Israel}

Available online 12 April 2007

Abstract

The theory of structural transformation of the vortex lattice in a fourfold symmetric in a–b plane type-II superconductors is con-structed using both the London and the lowest Landau level Ginzburg–Landau model. Thermal fluctuations and quenched disorder influence the location of the square to rhomb structural transition line. The self consistent harmonic approximation for lattice anharmo-nicity within the fourfold symmetric generalization of the London model is used. Without disorder in the T–B plane the slope of the line is generally negative: thermal fluctuations favour a more symmetric square lattice. The theoretical line’s location and slope are in good agreement with the experimental ‘‘second magnetization peak’’ line measured in both LaSCCO and YBCO in a wide range of doping. We find that, while the thermal fluctuations are negligible for the transitions in low Tcmaterials, disorder plays an important role and creates

Keywords: High temperature superconductor; Structural phase transition; Quenched disorder

1. Introduction

Structural phase transitions (SPT) between crystalline systems possessing different lattice symmetry is an old and still not a sufficiently well developed branch of the phase transitions physics. The simplest such transition is the square–rhomb transformation in the vortex lattice. In this case, single crystals of a tetragonal material like those of the borocarbide family are placed into an external mag-netic field oriented along ‘‘c’’ crystallographic axis to pre-serve fourfold symmetry in the basal plane.

The location in T–H plane of the critical line of the square–rhomb SPT in the vortex crystal at low temperatures is mapped precisely by various techniques and explained well by the nonlocal London (NLL) theory proposed by Kogan and collaborators[1]. The agreement with theoreti-cal predictions is readily understood conceptually because

the SPT in question occurs at low magnetic ﬁelds (much lower than Hc2), that is under the conditions when the

con-ventional London theory of the vortex lattice is known to be very reliable. It is assumed that vortices are well separated although the vortex core structure is also accounted for phe-nomenologically by a cut-off. The NLL theory then includes additionally the four derivative terms which bring in the anisotropy effects essential to trigger the SPT between the vortex lattice phases. The more symmetric square vortex crystal, stable at a stronger magnetic field (higher density of vortices), transforms into a less symmetric rhombic vor-tex crystal as the magnetic field weakens (density of vortices decreases). At higher temperatures and closer to the Hc2(T)

curve the detailed data became available only recently. A neutron scattering experiment of Eskildsen et al. [2] on LuNi2B2C suggests that the SPT line sharply bends upwards

and even become re-entrant in this region. This is very sur-prising if one assumes that fluctuations of some kind are responsible for such behaviour. Gurevich and Kogan [3] were first to consider the thermal fluctuations. They worked

*

Corresponding author.

E-mail address:[email protected](B.Ya. Shapiro).

www.elsevier.com/locate/physc Physica C 460–462 (2007) 1249–1250

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in the NLL framework and succeeded in producing transi-tion curves similar to the experimental ones. At a ﬁrst glance, the structural phase transition in such a system, even at ﬁnite temperature (below the melting temperature of course), is driven by interactions on scales smaller that C(L)a0, where C(L) = 0.1 is the Lindemann constant, and

consequently have nothing to do with anisotropy of the vor-tex core. However, it was claimed in a recent theory of ther-mal ﬂuctuations that the core anisotropy is crucial. Thus the problem should be considered from a more fundamental approach.

2. Theory

In this paper we show that the spatial disorder rather than thermal ﬂuctuations is responsible for SPT line upwards and re-entrant behavior. We are primarily inter-ested in the part of the phase diagram B Hc2(T), where

the vortex core size is much smaller than the distance a between vortices. A standard approach to the crystal struc-ture of point-like (or line-like) objects at finite tempera-ture requires a sufficiently comprehensive account of the lattice anharmonicity. The simplest version of such a the-ory takes into account the interacting phonon excitations self consistently (the self consistent harmonic approxima-tion – SCHA) using the microscopic derivaapproxima-tion of the vor-tex–vortex interaction for a d-wave superconductor by Yang. In our previous paper [4] we obtain a structural phase transition line with a negative slope in the B–T plane. In our theory, unlike the preceding ones, no cutoff is required. Here we expand our approach taking into account perturbatively the influence of quenched disorder on the structural phase transition line. A microscopic man-ifestation of the structural phase transition is the softening of the elastic squash modulus Csq= 2(C11+ C12) C66at

the transition line. Disorder introduces the random pinning potential U(ra) with UðrÞU ðr0Þ ¼ Kðr r0Þ: The energy

expansion in the main order in the displacements of vorti-ces uafrom their equilibrium square lattice position Raand

pinning potential reads E½uq ¼ E0þ X BZ Uqþ X BZ iqaUquaqþ 1 2 X BZ KabðqÞuaqu b q; E0¼ 1 2 X m;n wð~GmnÞ; ua a¼ X BZ ua qexpðiqRaÞ; UðrÞ ¼X BZ UqexpðiqRaÞ: ð1Þ

w(Gnm) is the Fourier transform of the fourfold

vortex–vor-tex interaction, Gnmare the reciprocal vectors. Elastic

mod-ules Cabcdof the clean sample are given by the expansion of

Kabto the second order of q: Cabcdqcqd, v(q) is the Fourier

transform of the isotropic pair vortex–vortex interaction

energy. For rhombic vortex lattice with the opening angle 2H where ~G¼ n~q1þ m~q2, with

~q1¼ ð2 tan HÞ1=2ð1; tan HÞ;

~q2¼ ð2 tan HÞ1=2ð1; tan HÞ: ð2Þ

For a given vortex structure one generally obtain the vor-tex displacement (strain) due to disorder ua

q¼ K

1 abiqbUq,

which when substituted into Eq.(1) gives after averaging over disorder: E2D¼ E0ðHÞ 1 2 X BZ K1_abðqÞqaqbKðqÞ: ð3Þ

Here KðqÞ ¼PBZKðqÞ expði~q~rÞ. The energy E2Dshould be

minimized to obtain the lattice with lowest energy. Result essentially depends on disorder amplitude, in-plane anisot-ropy, magnetic ﬁeld and the temperature as it presented in Fig. 1.

3. Summary

In low Tc superconductors the slope of the

rhomb-square coexistence line can be positive.

Quenched disorder rather than temperature ﬂuctuations are responsible for such behaviour. These results can be easily generalized to the case of three dimensional fourfold symmetric superconductor.

Acknowledgment

This research is supported by the Israel Science Founda-tion (ISF) (Grant No. 4/03-11.7).

References

[1] V.G. Kogan et al., Phys. Rev. B 54 (1996) 12386; Phys. Rev. B 55 (1997) R8693.

[2] M.R. Eskildsen et al., Phys. Rev. Lett. 82 (2001) 4082. [3] A. Gurevich, V.G. Kogan, Phys. Rev. Lett. 87 (2001) 177009. [4] B. Rosenstein et al., Phys. Rev. B 72 (2005) 144512.

Fig. 1. Square and rhombic vortex lattice structures under disorder (black squares are experimental data of a LuNi2ÆB2C superconductor).