On the Ginzburg–Landau analysis of a mixed s–dx2−y2-wave superconducting mesoscopic square

SSC 5199

PERGAMON

Solid State Communications 114 (2000) 499–504 www.elsevier.com/locate/ssc

On the Ginzburg–Landau analysis of a mixed s–dx2 2y2 -wave superconducting mesoscopic square V.R. Misko 1,a, V.M. Fomin 2,a, J.T. Devreese a,*, V.V. Moshchalkov b b

a Departement Natuurkunde, Universiteit Antwerpen (UIA), Universiteitsplein I, B-2610 Antwerpen, Belgium Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium

Received 14 December 1999; accepted 16 December 1999 by D.E. Van Dyck; received in final form by the Publisher 3 March 2000

Abstract The self-consistent solution of the coupled Ginzburg–Landau equations for the s- and d-wave order parameters is obtained in a mesoscopic square of a mixed s–dx2 2y2 -wave superconductor which serves as a model of high-Tc cuprates. The spatial distributions of the order parameters in the square are shown to be fourfold symmetric. The shape of the calculated distributions reflects the internal symmetry of the d-wave electron pairing as well as the geometry of the mesoscopic square. The analysis of the phase and the amplitude of the d-wave order parameter for low applied magnetic fields reveals a specific behaviour of the current which appears in an external lead connected to the midpoints of the adjacent sides of the square. For a YBa2Cu3O7 square with a side length of 1 mm, the current reaches its maximal value ,2.2 mA at zero applied magnetic field. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. High-Tc superconductors; A. Nanostructures; D. Phase transitions

1. Introduction Since the discovery of the high-Tc superconductors, the symmetry of the pairing state and its underlying mechanism has been intensively discussed in the literature. Several theoretical and experimental studies have suggested that the high-Tc cuprates exhibit an unconventional pairing state, characterised by an anisotropic order parameter. The experiments presented in Refs. [1–3] are sensitive to the phase of the superconducting order parameter and allow for an unambiguous determination of the symmetry of the pairing state. These experiments give strong evidence for * Corresponding author. Also at: Universiteit Antwerpen (RUCA), Groenenborgerlaan 171, B-2020 Antwerpen, Belgie¨ and Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. 1 Permanent address: Department of Theory of Semiconductors and Quantum Electronics, Institute of Applied Physics, Academy of Sciences of Moldova, Str. Academiei 5, MD-2028 Kishinev, Republic of Moldova. 2 Permanent address: Department of Theoretical Physics, State University of Moldova, Str. A. Mateevici 60, MD-2009 Kishinev, Republic of Moldova. Also at: Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

pairing in the channel with d-wave symmetry in YBa2Cu3O7. In the theoretical works [4,5] coupled Ginzburg–Landau (GL) equations were derived for the s- and d-wave order parameters in a mixed s–dx2 2y2 -wave superconductor. Disregarding magnetic field effects, analytical solutions of these equations were obtained for an infinite plane in two limits, near the core of a vortex and far away from it. The obtained solutions possess fourfold symmetry. In Ref. [6] the structure of a vortex in the dx2 2y2 -wave superconductor is described on the basis of the solutions of the quasi-classical Eilenberger equations [7]. It was demonstrated that, for an infinite planar s–d mixed superconductor the order parameter of the s-wave is fourfold symmetric around the vortex and the order parameter of the d-wave possesses eightfold symmetry. In the present communication, the vortex state in a mesoscopic (high-Tc) superconducting square is treated on the basis of the GL equations for the mixed s–dx2 2y2 -wave superconductor [4,5]. The interplay between the internal symmetry of the superconductor due to the d-pairing and the symmetry imposed to the vortex by the geometry of the superconducting square is analysed. The letter is organised as follows. The GL equations for the s- and d-wave order parameters in a mesoscopic

0038-1098/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(00)00090-9

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Fig. 1. Solutions of the GL equations for the dx2 2y2 -wave mesoscopic superconducting square with a vortex, L ˆ 1: The distributions of the phase, the squared amplitude of the order parameter for the s-wave (a, b) and d-wave (c, d) at zero applied magnetic field and T=Tc ˆ 0:6: We use the following mode of presentation of the phase of s- (d-) wave used in Ref. [6]: the total phase shift of 2pL …22pL†; which is acquired after one revolution around the vortex core in anti-clockwise direction, is represented by two jumps equal to p L …2pL† each. This is provided by cutting the surface representing f…x; y† …u…x; y†† along a straight line and shifting one of the parts of the surface by 2p…p†:

superconducting square based on Gor’kov’s theory of weakly coupled superconductors [4,5] are discussed in Section 2. In Section 3 we analyse solutions of the GL equations in a mixed s–dx2 2y2 -wave superconducting mesoscopic square at zero applied magnetic field. Section 4 is devoted to the discussion of the dependence of the vortex structure in a s–dx2 2y2 -wave superconducting mesoscopic square on the applied magnetic field. The calculation of the current in an external lead connected to the midpoints of adjacent sides of the square is presented in Section 5.

Refs. [4,5] based on the Gor’kov’s theory of weakly coupled superconductors: n 2as Dps 1 ald 12 v2F P 2 Dps 1 14 v2F …P x2 2 P y2 †Dpd o 1 2uDs u2 Dps 1 2uDd u2 Dps 1 …Dpd †2 Ds ˆ 0;

2ld Dpd

T ln c T

! 1 ald

n

1 2 2 p 4 vF P D d

1 2uDs u2 Dpd 1…Dps †2 Dd 1 2. The Ginzburg–Landau equations for a dx2 2y2 -wave superconductor In our analysis of the superconducting state in the presence of s- and the d-waves, we rely upon the coupled GL equations for a dx2 2y2 -wave superconductor derived in

3 4

1

2 1 2 4 v F …P x

…1†

2 P y2 †Dps

o uDd u2 Dpd ˆ 0; …2†

where D s and D d are the order parameters for the s- and dwaves, Dps and Dpd are their complex conjugate values, P ˆ 2i7 R 2 2eAR ; a and a s are material parameters, l d is a function of the density of states at the Fermi surface, vF is the Fermi velocity.

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Fig. 2. The structure of a vortex …L ˆ 1† as a function of the applied magnetic field. The distributions of the phase, the squared amplitude of the order parameter of s-wave at H0 ˆ 0 (a, b) and at H0 ˆ 0:08Hc …0† (c, d), of d-wave at H0 ˆ 0 (e, f) and at H0 ˆ 0:08Hc …0† (g, h), for T=Tc ˆ 0:96:

To obtain a dimensionless form of the GL equations, the following transformation is performed: r ! r; j0

D !D D0

…3†

Ds …x; y† ! c…x; y† e if…x;y† ; D d …x; y† ! x…x; y† eiu…x;y† :

…4†

The GL Eqs. (1) and (2) are solved with as the boundary condition the vanishing of the total superconducting current [4,5] through the boundaries of the mesoscopic square.

with p p j 0 ˆ avF =2; and D0 ˆ 4=3a

(Refs. [4,5]). The order parameters for the s- and d-waves are written as …5†

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3. Solutions of the Ginzburg–Landau equations for the sand d-wave order parameters When neglecting the magnetic field effects following [5], the GL Eqs. (1) and (2) become " # 1 22 22 p 2 p as Ds 2 7 D s 2 2 2 Dpd 2 2x2 2y 1  2ln 1

4 3

uDs u2 Dps 1 43 uDd u2 Dps 1

2 3

…Dpd †2 Ds ˆ 0;

…6†

" 2 #  Tc p 2 22 Dd 2 7 2 Dpd 2 2 Dps T 2x2 2y2 8 3

uDs u2 Dpd 1 43 …Dps †2 Dd 1 uDd u2 Dpd ˆ 0:

…7†

The self-consistent numerical solutions of the Eqs. (6) and (7) are presented in Fig. 1 for temperature T=Tc ˆ 0:6 and for the orbital quantum number L ˆ 1 (L denotes the number of magnetic flux quanta captured by the system). Throughout the present communication, the parameters of YBa2Cu3O7 are used: j 0 ˆ 1:8 nm; k ˆ 95 [8]; as ˆ 0:2: Fig. 1a shows the distribution of the phase of the order parameter for the s-wave, f…x; y†: This phase distribution resembles those obtained in Refs. [9,10] for a square loop of a pure s-type superconductor. The corresponding distribution of the squared amplitude of the order parameter, uc…x; y†u 2 ; is shown in Fig. 1b. The contour plots in Fig. 1b show that the distribution is close to cylindrically symmetric in the region near the vortex core. Away from the core, the shape of the distribution is determined by the geometry of the square. For the case of the d-wave, the distributions of the phase and the squared amplitude of the order parameter are shown in Fig. 1c and d. There are two features of d-wave phase u…x; y† (Fig. 1c) that distinguish it from the s-wave phase f…x; y† (cf. Fig. 1a): (i) f…x; y† increases when passing around the centre of the square in anti-clockwise direction, while u…x; y† decreases; and (ii) f…x; y† changes smoothly everywhere, while u…x; y† reveals regions of slow changes and fast changes (see Fig. 1c). The self-consistent solution of Eqs. (6) and (7) leads to the distribution of the squared amplitude of the order parameter for the d-wave, ux…x; y†u2 ; that has fourfold symmetry (Fig. 1d, see the contour plot). 4. Dependence of the order parameters for the s- and d-waves on the applied magnetic field

equations have a pronounced T-dependence. They are modified at T=Tc ˆ 0:96 as compared to T=Tc ˆ 0:6 (cf. Fig. 1) even at zero applied magnetic field. In contrast to the situation shown in Fig. 1b, the squared amplitude of the order parameter has a four-leaf shape (Fig. 2b) typical of the dwave (Fig. 1d). This result is in agreement with the conclusions of Refs. [5,6] concerning the induced s-component of the order parameter in a mixed s–dx2 2y2 -wave superconductor. Further, our calculations show that this specific shape of the order parameter distribution is manifested most clearly for states close to the …H–T† phase boundary. Though the distributions of the phase and the squared amplitude of the order parameter for the d-wave maintain fourfold symmetry, they substantially differ from the corresponding distributions for the s-wave. The phase of the order parameter of the d-wave shown in Fig. 2e has a “wavy” structure. This wavy structure leads to the appearance of additional maxima in the distribution of the squared amplitude of the order parameter (Fig. 2f). As shown in Ref. [6], in a bulk mixed s–dx2 2y2 wave superconductor the order parameter can possess eightfold symmetry. In the case of a mesoscopic square, the eightfold symmetry of the order parameter is broken by the presence of the boundaries. This leads to fourfold symmetry, although the maxima at the midpoints of the sides of the square are reminiscent of broken eightfold symmetry (see Fig. 2f). For nonzero applied magnetic field, “grooves” in the distribution of the square amplitude of the order parameter for the s-wave (Fig. 2d) become more pronounced than at the zero magnetic field (cf. Fig. 2b). The wavy structure of the phase and the squared amplitude of the order parameter of the d-wave displayed in the absence of a magnetic field (Fig. 2e and f), disappears gradually as the applied magnetic field takes increasing values (Fig. 2g and h). The order parameter distribution (Fig. 2h) has C4 symmetry. Its shape correlates with the geometry of the mesoscopic square. Thus, at T=Tc ˆ 0:96; L ˆ 1 and H0 ˆ 0:08Hc …0† the fourfold symmetry of the order parameter of the dwave is determined not by the internal symmetry of the dx2 2y2 -state, but by the confinement. This behaviour of the d-wave order parameter is specific for mesoscopic structures.

5. Experimental determination of d-pairing in a mesoscopic square 5.1. Anisotropy of D d

In order to study the dependence of the order parameter on the applied magnetic field, the complete set of GL Eqs. (1) and (2) is solved with the appropriate boundary conditions. The distributions of the phase and the squared amplitude of the order parameter of the s- and d-waves are shown in Fig. 2 for T=T c ˆ 0:96; L ˆ 1; H0 ˆ 0 and H0 ˆ 0:08Hc …0†: Our analysis indicates that the solutions of the GL

Experiments indicate the existence of a phase difference between the midpoints of the adjacent sides of a macroscopic square plate (with a side length of ,100 mm) fabricated with dx2 2y2 -wave superconductors [1–3,11]. This phase shift, attributed to the internal symmetry of the dwave, causes a current to flow in an external lead connected to the centres of the adjacent sides of the superconducting

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square, even in the absence of a magnetic field. To interpret this phase shift, the total order parameter [5] should be employed,

D…R; k† ˆ Ds …R† 1 Dd …R†…k^2x 2 k^2y †:

…8†

It should be noted that, as our analysis in the previous section shows, the order parameters for the s- and d-waves D s(R) and D d(R) are anisotropic themselves. But it is the factor …k^2x 2 k^2y † in Eq. (8), which explains the phase difference that is important for the experimental determination of the type of the superconducting electron pairing. It is useful to redefine Dd …R† ! D 0d …R† with the anisotropy factor …k^2x 2 k^2y † included. Then, in real space, the order parameter acquires the angular dependence [6,11]:

D 0d …r; w† / cos…2w†;

…9†

where w is the polar angle. To clarify the influence of the anisotropy factor cos(2w ) on the amplitude and the phase of the d-wave order parameter, we turn to the complex form of representation of the order parameters (Eq. (5)). Then D 0d …x; y† can be written as

D 0d …x; y† ˆ x 0 …x; y† eiu…x;y† ;

…10†

or as 0

D 0d …x; y† ˆ x…x; y† eiu …x;y† ;

…11†

with the anisotropy factor cos(2w ) included to the amplitude x 0 …x; y† or to the phase u 0 …x; y†; correspondingly. A sign of cos(2w ) can be represented as sign‰cos…2w†Š ˆ ^1 ; eipn

…12†

with n ˆ 0 or n ˆ 1 for sign “1” or “2”, respectively. Consequently, the sign of the amplitude x 0 …x; y† in Eq. (10) becomes angle-dependent, and the phase u 0 …x; y† in Eq. (11) acquires a shift of p , when ‰cos…2w†Š , 0: Therefore, the anisotropy factor can be assigned either to the amplitude or to the phase of the order parameter. Effectively this leads to an angle-dependent sign of the amplitude of the order parameter or, alternatively, to a phase difference of p between the midpoints of any of the adjacent sides of the square [11]. 5.2. Current in an external lead: dependence on the applied magnetic field

Fig. 3. (a) The distributions of the squared amplitude of the order parameter for the d-wave are plotted for several values of the applied magnetic field. (b) The current in the external lead, connected to the midpoints of the adjacent sides of a meoscopic superconducting square, as a function of the applied magnetic field H0 :

Owing to the internal phase difference, discussed in the previous section, any points at the adjacent sides of a s–dwave superconductor square are out of phase [1,11]. In what follows, we will consider the midpoints the adjacent sides of the square. If we connect these points to each other by a lead, a circulating current will appear in the lead. This current can be measured in the experiment. It should be noted, that for macroscopic samples the phase difference under discussion has been detected in experiments [1–3] by measuring the critical current. It has been shown that the critical current has a minimum at zero

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magnetic field. Instead, the circulating current has a maximum at this point [1,11]. In this letter, we report the calculations of the circulating current through a lead connecting the midpoints of the adjacent sides of the mesoscopic square. The lead is supposed to be made of the same material as the square and narrow enough in order to neglect the influence of this lead on the distribution of the order parameter in the square. A lead with a cross-sectional area of Scs ˆ 2 × 10211 cm2 is treated (the width of the lead with a square cross section is approximately 45 nm and is much less than a side length of the square equal to 1000 nm). The current through the lead for the d-wave can be represented in the form Id ˆ I0 x2 …x; y†…u2 2 u1 †;

…13†

with I0 ˆ j0 Scs ; j0 is the critical current density. u 1 and u 2 denote the values of the phase at the contacts of the lead. We have calculated the current in the external lead as a function of the applied magnetic field for the fields ranged in H0 ˆ 0 to 0:48Hc …0†: For low values of the applied magnetic field, the system is in its lowest state with L ˆ 0: It is worth noting, that in extreme type-II superconductors (i.e. with the GL parameter k q 1† such as high-Tc materials a vortex structure can be considered neglecting the magnetic effects [4–6,11]. In the previous sections we analysed a vortex structure in the absence of a magnetic field and at low magnetic field …H0 ˆ 0:08Hc …0††: In this section, we analyse the superconducting stage in a mesoscopic high-Tc superconducting square and, consequently, the current in an external lead as a function of the applied magnetic field. For the considered magnetic fields the system is in a state with L ˆ 0: We discuss now how the changing distribution of the squared amplitude of the order parameter influences the current. The distributions of the squared amplitude of the order parameter x2 …x; y† for L ˆ 0 are represented in Fig. 3a for various values of the applied magnetic field. The current in the external lead as a function of the applied magnetic field is plotted in Fig. 3b. The current has a maximum at H0 ˆ 0 and monotonously decreases with increasing applied magnetic field. This behaviour of the current in a lead is in agreement with the conclusions of Refs. [1–3] concerning the circulating current in a circuit consisting of an s–dx2 2y2 wave superconductor and a lead. According to Ref. [12], the critical current density j0 in type II superconductors can be of the order of 10 6 A/cm 2. Choosing j0 ˆ 5 × 106 A=cm2 [12], we find that the

maximal current in the external lead considered here can be ,2.2 mA at H0 ˆ 0: 6. Conclusion The order parameters of the s- and d-wave in a mixed s–dx2 2y2 superconducting mesoscopic square are shown to possess fourfold symmetry. Contrary to the case of bulk superconductors, the order parameters are determined both by the geometry of the square and by the internal symmetry of the d-pairing. For a YBa2Cu3O7 mesoscopic square with a side length of 1 mm, the current in the external lead (with a cross-sectional area of Scs ˆ 2 × 10 211cm 2 connected to the midpoints of the adjacent sides of the square reaches a maximal value of ,2.2 mA at H0 ˆ 0: Acknowledgements We acknowledge useful discussions with V. Bruyndoncx and L. Van Look. This work has been supported by the Interuniversitaire Attractiepolen—Belgische Staat, Diensten van de Eerste Minister—Wetenschappelijke, Technische en Culturele Aangelegenheden, the F.W.O.—V. projects Nos. G.0287.95, G.0232.96, W.O.G. WO.025.99N (Belgium), and the ESF Programme VORTEX. References [1] D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [2] D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett, Physica B 194-196 (1994) 1669. [3] D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515. [4] Y. Ren, J.-H. Xu, C.S. Ting, Phys. Rev. Lett. 74 (1995) 3680. [5] J.-H. Xu, Y. Ren, C.S. Ting, Phys. Rev. B 52 (1995) 7663. [6] M. Ichioka, N. Enomoto, N. Hayashi, K. Machida, Phys. Rev. B 53 (1996) 2233. [7] G. Eilenberger, Z. Phys. 214 (1968) 195. [8] C.P. Poole Jr., H.A. Farach, R.J. Crewich, Superconductivity, Academic Press, San Diego, 1995. [9] V.M. Fomin, V.R. Misko, J.T. Devreese, V.V. Moshchalkov, Solid State Commun. 101 (1997) 303. [10] V.M. Fomin, V.R. Misko, J.T. Devreese, V.V. Moshchalkov, Phys. Rev. B 58 (1998) 11703. [11] M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, New York, 1996. [12] A.A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, Amsterdam, 1988.