The anomalous Hall effect for a mixed s- and d-wave symmetry superconductor

The anomalous Hall effect for a mixed s- and d-wave symmetry

superconductor

M.C. Dai, T.J. Yang*

Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan

Received 7 December 1998; received in revised form 15 February 1999; accepted 19 February 1999 by J. Kuhl

Abstract

By taking account of the complex relaxation times for s- and d-wave order parameters in the time-dependent Ginzburg– Landau equations, our results indicate that the imaginary parts of the relaxation times can change the sign of the Hall angle for mixed s- and d-wave superconductors. The effect of the concentration of the nonmagnetic impurities on the Hall angle is investigated, and it is found that the concentration of the nonmagnetic impurities can affect the parameters in the Ginzburg– Landau free energy and the relaxation times. The results of the anomalous Hall effect arising from the nonmagnetic impurities are discussed.䉷 1999 Elsevier Science Ltd. All rights reserved.

Keywords: A. High-Tcsuperconductors; D. Electronic transport; D. Galvanomagnetic effects

The pairing symmetry of the order parameter in high-Tcsuperconductors (HTS) attracts attention and

a number of experimental observations strongly support the d-wave pairing symmetry [1–3]. Most of the theories proposed that the superconducting state might be relevant to the heavy fermion and the high-Tc

superconductors [4–6]. Based on group theory argu-ments, the phenomenological Ginzburg–Landau (GL) theory with many unknown parameters has been studied [4,5,7]. Ren et al. [8,9] microscopically derived the GL equations for the pure d-wave as well as mixed s- and d-wave superconductors in the framework of the weak coupling theory.

Recently attention has been focussed on the anom-alous behavior of the Hall effect, which appears to undergo a sign change in the superconducting mixed state [10–15]. The phenomenological theories about

the vortex motion have been proposed to investigate the anomalous Hall effect of superconductors [16– 20]. Dorsey [20] introduced a complex relaxation time in the time-dependent Ginzburg–Landau (TDGL) equation and then derived the equation of motion for a single vortex

Vs1× ^z ˆa1VL⫹a2VL× ^z; …1† where (a1anda2are functions of the parameters that

appear in the TDGL equation. The results showed that, if a2 ⬍ 0, the Hall effect in the vortex state

will change its sign, which is opposite to the sign of the normal-state Hall effect. Experimental measure-ments in Tl2Ba2CaCuO8 systems also showed a

consistent description of the behavior of the imagin-ary part of the order parameter relaxation time [21]. However, the problems of the Hall effect for d-wave superconductors have not been fully understood. Following the generalized London theory, derived by Affleck et al. [22], Alvarez, Domı´nguz and Bleeder Solid State Communications 110 (1999) 425–430

0038-1098/99/$ - see front matter䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 9 9 ) 0 0 0 9 2 - 7

PERGAMON

* Corresponding author. Tel.:⫹886-3-5712-121 ex56081; fax: ⫹886-3-5725-230.

studied the dynamics of vortices in d-wave supercon-ductors. They found that an intrinsic Hall effect depended on sin…4j† with an anglej with respect to the b crystal axes [23]. Dai and Yang [19] have eluci-dated that both the imaginary parts of the relaxation times for the order parameters and the parameters in the GL free energy functional can affect the anoma-lous Hall effect for a pure d-wave superconductor. Here we will investigate the Hall effect for mixed s-and d-wave superconductors from the TDGL equa-tions.

The phenomenological TDGL equations in the dimensionless form for dx2_⫺y2 superconductors can be expressed as hd…2t⫹ i ~f†d ˆadd⫹ 8 3 b2 b1   兩d兩2 d⫹ 4 3 b3 b1   兩s兩2 d ⫹ 8 3 b4 b1   d*s2⫹ 2 gd gs   P02_d ⫹ 2 gn gs   …P02 x ⫺P02y†s; …2† hs…2t⫹ i ~f†s ˆass⫹ 8 3兩s兩 2 s⫹ 4 3 b3 b1   兩d兩2 s ⫹ 8 3 b4 b1   s*d2⫹ 2P02s⫹ 2 gn gs    …P02 x ⫺P02y†d; …3†

where P0ˆ ⫺i7=k⫺ A; ~f ˆ 2ef; s and d are the order parameters,asandad depend on the tempera-ture, andb1,b2,b3,b4,gs,gdandgnare assumed to be positive. One should treat the order parameters not as wave functions but as a thermodynamic variable like the magnetic moment in an Ising magnet or like the volume in an expanding gas. The parametersgare related to the effective masses withgiˆ ប

2_=2m

i* for

iˆ s; d;n. These parameters in the TDGL equations can determine the thermodynamic stability. The dimensionless order-parameter relaxation time hd andhs can be defined by

hd ⬅hd1⫹ ihd2; hs⬅hs1⫹ ihs2: …4†

The imaginary part of the complex relaxation time breaks the particle-hole symmetry in superconductors,

resulting in nonvanishing Hall current [10,20]. As these nonlinear TDGL equations are complicated, we shall derive an equation of motion for a single vortex in the limit h p Hc2 Namely, the magnetic fields are slightly above the lower critical field Hc1. According to this limit, the vortices are well separated and may be studied individually. The normal current

Jnand the supercurrent Jscan be rewritten in the form

Jnˆs…n†· ⫺ 1 k7f~ ⫺2tA   ; …5† Jsˆ 2s*…P0s† ⫹ 2 gd gs   d*…P0d† ⫹ 2 gn gs   {^x‰s*…P0xd† ⫹d…P0 xs†*Š ⫺ ^y‰s*…P0yd† ⫹ d…P0ys†*Š} ⫹ h:c: …6†

The Hall conductivity s…n†xy of the normal-state conductivity tensor produces a Hall effect due to the transverse response of the normal fluid to the electric fields, generated in the vortex core for type-II super-conductors [20].

First, we express the complex order parameters d and s in terms of an amplitude and a phase, d…r; t† ˆ

f…r; t†eiud…r;t† _{and s}…r; t† ˆ g…r; t†eius…r;t†; to discuss an isolated vortex. In a superconductor the gradient of the phases can determine the observable quantity Js.

Affleck et al. [22] have derived the supercurrent of a d-wave superconductor by way of the generalized London theory. If g_nˆ 0; these TDGL equations have the same solutions as those presented in conven-tional superconductors. This case is not interesting to us. We attempt to solve these equations forg_n 苷 0: By substituting the complex order parameters and Eq. (4) for Eqs. (2), (3), (5) and (6), we can separate the real and imaginary parts from the TDGL equations. In order to solve these nonlinear TDGL equations, three essential steps are applied. First, we assume that the vortices translate uniformly. Therefore all the quanti-ties, which characterize the vortex system, are func-tions of the quantity r⫺ VLt with VL the vortex line velocity and can be expanded in first order of VL c…r; t† ˆc…0†_{r ⫺ V}

Lt† ⫹c…1†…r ⫺ VLt†: …7† Herein the correction termsc…1†are small relative to the velocity of motion VL. Next, the equations can be expanded in powers of VL. The terms of order of unit and order of VL correspond to the equilibrium GL

M.C. Dai, T.J. Yang / Solid State Communications 110 (1999) 425–430 426

equations and a set of inhomogeneous differential equations, respectively. The equilibrium GL equa-tions have been done well [8,9,24]. We need to solve the O…VL† equations. Following [25] the time-independent GL equations possess translational invar-iance. As the magnetic field can be neglected in the large-klimit, the distributions of the field and current can be investigated far from and close to the vortex cores, respectively (see Ref. [19] for a detailed deri-vation). Finally, we obtain an equation of motion for the vortices by deriving a solvability condition

1 2k Z dS·‰Js…1†u…`†d ⫹ J…1†s u…`†s ⫺ Js…`†u…1†d ⫺ Js…`†u…1†s Š ˆZd2r{hd1fvf`⫹hd2f0f`Pd⫹hs1gvg` ⫹hs2g0g`Ps⫹S2…VL† ⫺ ‰hd1f 2 0Pd⫺hd2f0fv ⫹hs1g20Ps⫺hs2g0gv⫺S1…VL†Š × …u…`†d ⫹u…`†s †}; …8†

where ` is an infinitesimal translation, fv⬅ VL·7f0,

gv⬅ VL·7g0, f`⬅ `·7f0; g`⬅ `·7g0,

u`

d ⬅ `·7u…0†d , and u `

s ⬅ `·7u…0†s :S1…VL† and S2…VL† come from the imaginary and real parts of the TDGL equations, respectively, and include multitudes of functions, which consist of the characteristic func-tions. This expression, satisfied by the first order in

VL, is equal in effect to derive an equation of motion for the vortices. The inhomogeneous equations are solvable while this solubility condition for steady vortex motion in Eq. (8) holds.

We choose a system of coordinates with the z direc-tion along the applied magnetic field, the applied transport current Jtto be in the x direction; the

direc-tion of the vortex in modirec-tion at an angleuHwith respect

to the ⫺ y direction and the origin of r–w plane (in the polar coordinates) at the center of a vortex. The displacement vector ` makes an anglexrespect to the

x-axis. Numerical and asymptotic solutions for the

single vortex have been carried out [8,9,24]. At a distance of several coherence lengths jd from the Fig. 1. The function…hd2=hd1†v0versus the reduced temperature tˆ T=Td0. The solid line represents the condition in the absence of impurities

core the s-wave order parameter is much less than d-wave order parameter so that the s–d coupling terms are weak enough to be neglected. For the purpose of investigating how the s–d coupling terms affect the sign of the Hall effect, we choose the region close to the vortex core. The interaction between two vortices can be neglected as r qldwithldthe penetration depth of the d-wave order parameter. The integration regions in Eq. (8), therefore, are cutoff atldand we obtain

tanuHˆv0⫹v0; …9†

where

Here the scalar potentials Pd1, Pd2, Ps1 and Ps2 satisfy a set of homogeneous equations related to O(VL) as r! 0.v0 is independent of the imaginary

parts of the relaxation times and positive due to the

large-k limit. Ifv0is negative and satisfies

兩v0_{兩 ⬎v} 0;

we can get the sign change of the Hall effect. Xu et al. [26,27] derived the Ginzburg–Landau equations for a mixed s⫹ d symmetry superconductor with nonmagnetic impurities and showed that the transition temperature for s-wave order parameter can only be affected by the magnetic impurity scatter-ing while the transition temperature for d-wave order parameter is dominantly affected by the nonmagnetic scattering. Hence, the parameters, as and ad, are given by asˆ ln T Ts0   ; adˆ ⫺ 1 2 ln Td0 T ⫹C 1 2   ⫺C 1₂ ⫹ ca pT     ; …11†

where Ts0 andTd0 are the critical temperatures of a

M.C. Dai, T.J. Yang / Solid State Communications 110 (1999) 425–430 428 v0ˆ ⫺8s…n†xx k2_l2 d p…1†_d2⫹ 8s …n† xy kl2 d h0 8s…n†xx k2_l2 d p…1†_d1⫹hd1c21⫹hs1c22⫺ 4 k2 b2c2⫹ gd gs   a2c1⫺ gn gs   …b2c1⫹ a2c2†   ; …10a† v0_ˆ ⫺hd2 c21⫹hs2c 2 2⫺ 4 k2 b1c2⫹ gd gs   a1c1⫺ gn gs   …b1c1⫹ a1c2†   8s…n†xx k2_l2 d p…1†_d1 ⫹hd1c21⫹hs1c22⫺ 4 k2 b2c2⫹ gd gs   a2c1⫺ gn gs   …b2c1⫹ a2c2†   ; …10b† a1 ˆk 2 6hd2c1⫺ 8 gn gs   hs2c2   =D0; a2 ˆk 2 _⫺6 hd1c1⫺ 8 gn gs   hs1c2   =D0; b1 ˆk 2 6 gd gs   hs2c2⫺ 8 gn gs   hd2c1   =D0; b2 ˆk 2 6 gd gs   hs1c2⫹ 8 gn gs   hd1c1   =D0; D0ˆ 36 gd gs   ⫺ 64 gn gs  2 ; c2ˆ 1 2 兩ad兩 as   _g n gd   c1: …10c†

clean superconductor, c is the concentration of the impurities, C(z) is the digamma function, and aˆ pU2=N0…0† and 1=pN0…0† for the Born limit and the

unitary limit, respectively. (U is the nonmagnetic potential due to the static impurities and N0(0) is the

normal state electron density of states at the Fermi level.) When the concentration of the impurities increases, the parameter ad will change from linear

T behavior to T2 behavior. The high scattering strength can also affect the ratio of the parameters g, related to the effective masses. In this limit, one can find that

gsqgd ⬎gn:

In order to illustrate the effect of nonmagnetic impurities on v0, we take gs=gd ˆ 2, Tc ˆ 0.95Td0 and Ts0 ˆ 0.5Td0. In Fig. 1 we plot the function

…hd2=hd1†v0versus t…ˆ T=Td0† in the absence of impu-rities and in the presence of the low impurity density doped. In Fig. 2 we plot the function …hs2=hs1†v0

versus t for the high scattering strength. Fig. 1 shows thatv0is negative near the critical temperature. v0_{⬍ 0 can lead to the sign change of the Hall angle.}

Also, the impurity concentrations increasing from low to high, the lowest Hall angle shifts to lower tempera-ture and becomes nonnegative. This implies that the anomalous Hall effect also depends on the concentra-tion of the impurities. This feature is consistent quali-tatively with the experimental results.

In conclusion, we show that the imaginary parts of the complex relaxation times can give rise to the anomalous Hall effect for mixed s- and d-wave symmetry superconductors. We decompose the Hall angle into two parts: v0 andv0. The former part is

independent of the imaginary parts of the relaxation times and positive in the large-klimit. The sign of the latter part can be affected by the relaxation times. If the real and imaginary parts of the relaxation times have the same sign,v0will be negative near the criti-cal temperature. We also show that the doped Fig. 2. The function…hs2=hs1†v0versus reduced the temperature tˆ T=Td0for high nonmagnetic impurity density doped. We choose Tcˆ

nonmagnetic impurities can influence the temperature range of the anomalous Hall effect. In addition, it is noteworthy that the concentration of the impurities can affect the Hall effect. When the impurity concen-trations are increasing from low to high, the depen-dence of the Hall angle changes from the d-component to the s-d-component.

Acknowledgements

We sincerely thank Prof. C.S. Ting for bringing his unpublished results to our knowledge. We also thank the National Science Council of the Republic of China to support this work under grant number NSC88-2112-M009-003.

References

[1] D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Gins-berg, A.J. Leggett, Phys. Rev. Lett. 71 (1993) 2134. [2] A. Mathai, Y. Gim, R.C. Black, A. Amar, F.C. Wellstoo, Phys.

Rev. Lett. 74 (1995) 4523.

[3] C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun, M.B. Ketchen, Phys. Rev. Lett. 73 (1994) 593.

[4] P. Kumar, P. Wolfle, Phys. Rev. Lett. 59 (1987) 1954. [5] R. Joynt, Phys. Rev. B 41 (1990) 4271.

[6] Q.P. Li, B.E.C. Koltenbah, R. Joynt, Phys. Rev. B 48 (1993) 437.

[7] M.M. Doria, H.R. Brand, H. Dleiner, Physica C 159 (1989) 46.

[8] Young Ren, J.-H. Xu, C.S. Ting, Phys. Rev. Lett. 74 (1995) 3680.

[9] Young Ren, J.-H. Xu, C.S. Ting, Phys. Rev. B 52 (1995) 7663. [10] A.T. Dorsey, M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 694. [11] J. Luo, T.P. Orlando, J.M. Graybeal, X.D. Wu, R.

Muenchau-sen, Phys. Rev. Lett. 68 (1992) 690.

[12] S.J. Hagen, C.J. Lobb, R.L. Greene, M. Eddy, Phys. Rev. B 43 (1991) 6246.

[13] T.R. Chien, T.W. Jing, N.P. Ong, Z.Z. Wang, Phys. Rev. Lett. 66 (1991) 3075.

[14] S.J. Hagen, C.J. Lobb, R.L. Greene, M.G. Forrester, J.H. Kang, Phys. Rev. B 41 (1990) 11630.

[15] Y. Iye, S. Nakamura, T. Tamegai, Physica C 159 (1989) 616. [16] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465.

[17] Z.D. Wang, Q. Wang, P.C.W. Fung, J. Supercond. Sci. Tech-nol. 9 (1996) 333.

[18] A. van Otterlo, M. Feigel’man, V. Geshkenbein, G. Blatter, Phys. Rev. Lett. 75 (1995) 3736.

[19] M.C. Dai, T.J. Yang, unpublished. [20] A.T. Dorsey, Phys. Rev. B 46 (1992) 8376.

[21] A.V. Samoilov, Z.G. Ivanov, L.G. Johansson, Phys. Rev. B 49 (1994) 3667.

[22] I. Affleck, M. Franz, M.H. Sharifzadeh Amin, Phys. Rev. B 55 (1997) R704.

[23] J.J. Vicente Alvarez, D. Dominquez, C.A. Balseriro, Phys. Rev. Lett. 79 (1997) 1373.

[24] M. Franz, C. Kallin, P.I. Soininen, A.J. Berlinsky, A.L. Fetter, Phys. Rev. B 53 (1996) 5795.

[25] L.P. Gor’kov, N.B. Kopnin, Sov. Phys Usp. 18 (1975) 496. [26] W. Xu, W. Kim. Y. Ren, C.S. Ting, Phys. Rev. B 54 (1996)

R12693.

[27] J.X. Zhu, W. Kim, C.S. Ting, C.-S. Hu, Phys. Rev. B 58 (1998) 15020.

M.C. Dai, T.J. Yang / Solid State Communications 110 (1999) 425–430 430