- Email: [email protected]

Basic properties of a vortex in a noncentrosymmetric superconductor N. Hayashi a

a,*

, Y. Kato b, P.A. Frigeri a, K. Wakabayashi a, M. Sigrist

a

Institut fu¨r Theoretische Physik, ETH-Ho¨nggerberg, CH-8093 Zu¨rich, Switzerland b Department of Basic Science, University of Tokyo, Tokyo 153-8902, Japan Available online 3 February 2006

Abstract We numerically study the vortex core structure in a noncentrosymmetric superconductor such as CePt3Si without mirror symmetry about the xy plane. A single vortex along the z axis and a mixed singlet–triplet Cooper pairing model are considered. The spatial proﬁles of the pair potential, local density of states, supercurrent density, and radially-textured magnetic moment density around the vortex are obtained in the clean limit on the basis of the quasiclassical theory of superconductivity. Ó 2005 Elsevier B.V. All rights reserved. PACS: 74.20.Rp; 74.70.Tx; 74.25.Op Keywords: CePt3Si; Unconventional superconductivity; Vortex core; Local density of states; Core magnetization; Broken inversion symmetry

Much attention has been focused on the heavy fermion superconductor CePt3Si, which has a noncentrosymmetric crystal structure without mirror symmetry about the xy plane [1]. CePt3Si is an extreme type-II superconductor and the vortex structure of the mixed state in this system was recently studied by Kaur et al. [2] and Yip [3] on the basis of the Ginzburg–Landau theory and the London theory. In this paper, we investigate the vortex core structure on the basis of the quasiclassical theory of superconductivity [4], which enables us to calculate more microscopically the physical quantities such as the pair potential, local density of states, supercurrent density, and magnetic moment density. We consider a single vortex along the z axis in the clean limit. The noncentrosymmetricity (or the lack of inversion symmetry) leads to the mixture of Cooper pairing channels of diﬀerent parity [5]. We consider the following superconducting order parameter in a singlet–triplet mixing form: ^ k ¼ ðW^ ^Þi^ ^x þ ~k x r ^y Þi^ D r0 þ d k r ry ¼ ½W^ r0 þ Dð~k y r ry , with the s-wave pairing component W and the d vector d k ¼ Dð~k y ; ~k x ; 0Þ. This s + p-wave pairing state is pro*

Corresponding author. E-mail address: [email protected] (N. Hayashi).

0921-4534/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.12.048

^y ; r ^z Þ are the Pauli rx ; r posed for CePt3Si in Ref. [6]. Here, ð^ ^0 is the unit matrix, and matrices in the spin space, r ~ ¼ ð~k x ; ~k y ; ~k z Þ ¼ ðcos / sin h; sin / sin h; cos hÞ. k The lack of inversion symmetry here is incorporated through a Rashba-type spin–orbit coupling with a form proposed in Ref. [7]. It splits the Fermi surface into two ones (I and II) by lifting the spin degeneracy [6]. From the original Eilenberger equation for noncentrosymmetric superconductivity [8], we obtain two equations corresponding to these split Fermi surfaces I and II in the case of the above s + p-wave pairing state [9], I;II ; gI;II ¼ 0; ivI;II $ gI;II þ ½ixns3 D

ð1Þ

I;II ¼ ½ðs1 þ is2 ÞDI;II ðs1 is2 ÞD =2, DI,II = W ± where D I;II Dsin h are the order parameters on the Fermi surfaces I and II, ðs1 ; s2 ; s3 Þ are the Pauli matrices in the particle-hole space, and the commutator ½a; b ¼ ab ba. We neglect the vector potential in Eq. (1) assuming the extreme type-II superconductivity. We use units in which h = kB = 1. The Green functions gI;II on the Fermi surfaces I and II are written as a matrix in the particle-hole space ! gI;II if I;II ~ . ð2Þ gI;II ðr; k; ixn Þ ¼ ip if I;II gI;II

N. Hayashi et al. / Physica C 437–438 (2006) 96–99

The regular Green function g^ as a matrix in the spin space is given by [6,8,9] ! gI þ gII k 0þ ðgI gII Þ 1 ^I þ gII r ^II ¼ g^ ¼ gI r ; 2 k 0 ðgI gII Þ gI þ gII

ð3Þ ^Þ=2 and gk ¼ ðk y ; k x ; 0Þ. Here, ^I;II ¼ ð^ r0 gk r with r ¼ ðk x ; k y ; 0Þ ¼ ðcos /; sin /; 0Þ. k 0 ¼ k y ik x and k We consider a single vortex which has a form, DI,II (r, /r;h) = [W(r) ± D(r) sin h]exp(i/r). Here, the real-space coordinates r = (rcos/r, rsin/r, 0), and the vortex center is situated at r = 0. The Fermi surface is assumed to be spherical, and the diﬀerences of the density of states and the Fermi velocity vI,II between the two Fermi surfaces I and II are assumed to be small and are ignored. The results in this paper depend predominantly on the spin structure [Eq. (3)] and the gap structure on the 3D Fermi surfaces, and the Fermi surface anisotropy would not lead to qualitatively diﬀerent results as long as the spin and gap topologies are not altered. We numerically solve the gap equations given in Ref. [6,8,9] and the Eilenberger equations in Eq. (1) self-consistently as in Ref. [10]. When solving the gap equations, we adopt the same values of parameters as used in Ref. [8]. Thus, both the pair potentials D and W are real and positive, and jDj > jWj [8]. From now on, Tc is the superconducting critical temperature and n0 = vF/Tc is the coherence length at zero temperature (vF = jvFj is the Fermi velocity). In Fig. 1, we show the spatial proﬁles of the pair potentials D (p-wave component) and W (s-wave one) around the vortex for several temperatures T. It is noticed that while the amplitude is diﬀerent between D and W, the characteristic recovery length (namely, the core radius) is the same for both. The local density of states (per spin) is calculated by N0 RehTr½^ gðixn ! E þ igÞi 2 N0 RehgI þ gII i; ¼ ð4Þ 2 where h i denotes the average over the Fermi surface, N0 is the density of states per spin at the Fermi level, and g

N ðE; rÞ ¼

1.6

(>0) is the energy smearing factor. Before going into the vortex bound states, let us see in Fig. 2 the density of states in the bulk without vortices. There are four gap edges (solid line). The system has two split Fermi surfaces I and II [8,9]. The two of the gap edges originate from the fully-gapped Fermi surface I (dashed line), and the other two originate from the line-node-gap Fermi surface II (dash–dotted line). In Fig. 3, we show the local density of states inside the vortex core. There are four branches of peaks, which are related to the vortex bound states. The outer (inner) two branches originate from the vortex bound states of the quasiparticles on the Fermi surface I (II). Thus, the present spectra inside the vortex core in the clean limit possess the same structure as those in a two-gap superconductor. In Fig. 4, we plot the supercurrent density jjj, which is calculated by X N 0 hvF Tr½^ r0 ðip^ gÞi j ¼ eT xn

¼ ipeT

where lB is the magnetic moment of the quasiparticle. Substituting Eq. (3) into this, we obtain X M x ¼ iplB T N 0 hðk y ÞðgI gII Þi; ð7Þ xn

M y ¼ iplB T

X

N 0 hk x ðgI gII Þi;

ð8Þ

xn

M z ¼ 0.

ð9Þ

0.8 0.1

0.7 0.6

1

0.7

/ Tc

/ Tc

ð5Þ

xn

0.8

0.8 0.6

0.5

0.7

0.4

0.8

0.3 T / Tc = 0.9

0.4

T / Tc = 0.9

0.2

0.2

a

N 0 hvF ðgI þ gII Þi;

where e is the electric charge of the quasiparticle. We have conﬁrmed numerically that jjj decays as 1/r far away from the core. jjj exhibits essentially the same structure as that in usual s-wave superconductors. Finally, we investigate the magnetic moment density M. The vortex-core magnetization in the present noncentrosymmetric system has been reported by Kaur et al. [2] and Yip [3]. Here, we calculate it to obtain numeric results at various temperatures by means of a more microscopic derivation. M is calculated by X M ¼ lB T N 0 hTr½^ rðip^ gÞi; ð6Þ

1.2

0

X xn

0.1

1.4

97

0.1 0

0.2

0.4

0.6

r/

0.8 0

1

1.2

0

1.4

b

0

0.2

0.4

0.6

r/

0.8

1

1.2

1.4

0

Fig. 1. Spatial proﬁles of the pair potentials. T/Tc = 0.1–0.9 from top to bottom by 0.1 step. (a) the p-wave component D, and (b) the s-wave one W.

N. Hayashi et al. / Physica C 437–438 (2006) 96–99 2.5 total FS-I FS-II

1.5

M

DOS [N0]

2

1 0.5 0 -3

-2

-1

0

1

2

3

0 -0.1 -0.2 -0.3 0.4 -0.4 -0.5 0.3 -0.6 0.2 -0.7 -0.8 -0.9 T / Tc = 0.1 -1 0 0.2 0.4

Fig. 2. The density of states in the bulk without vortices. T/Tc = 0.1 and g = 0.01Tc.

Fig. 3. The local density of states N(E, r) inside the vortex core in units of N0. T/Tc = 0.1 and g = 0.05Tc. Large zero-bias peak at (E, r) = (0, 0) is truncated in the ﬁgure.

0.6 T / Tc = 0.1

0.6

0.4

|j|

0.2

|j|

0.3 0.2

0.3 0.3

total FS-I FS-II

0.5

0.4

0.1 0 0

0.4

0.2

0.4

0.6

r/

0.8

1

1.2

1.4

0

0.2

0 -0.4 -0.6 -0.8 -1 -1.2 0

0.6

0.2

0.4

0.8

0.6

r/

0.8

total FS-I FS-II 1 1.2

1.4

0

1

1.2

1.4

0

Fig. 5. The radial component of the magnetic moment density M around the vortex in units of lBN0Tc. T/Tc = 0.1–0.9 from bottom to top by 0.1 step. Inset: M at T/Tc = 0.1 (solid line), the contribution of the Fermi surface I (dashed line), and that of the Fermi surface II (dash–dotted line). The minus value means that the sense of M is toward the vortex center.

In conclusion, we calculated the vortex core structure in the noncentrosymmetric superconductor. We found that the singlet and triplet components of the order parameter have the same characteristic length (core radius) as each other. The local density of states exhibits the two-gap property in the clean limit, which is expected to be observed by STM [11,12] in order to clarify the validity of the present mixed-parity pairing model. We mention here that impurity eﬀects in the noncentrosymmetric system can be diﬀerent from those in simple two-gap systems in general, therefore interesting phenomena in the density of states are expected when the system deviates from the clean limit. Detailed investigations on such impurity eﬀects are left for future studies. The radially-textured magnetic moment appears around the vortex, which is expected to be observed directly by spin-polarized STM [13] in order to clarify the electronic structure [Eq. (3)] speciﬁc to the noncentrosymmetric superconductivity with the rotating spin on the Fermi surfaces as shown in a picture in Ref. [14]. Acknowledgements

0.1 0 0

0.2 -0.2

r/

E / Tc

0.5

0.4

M

98

0.2

0.4

0.6

r/

0.8

1

1.2

1.4

0

Fig. 4. Spatial proﬁles of the supercurrent density jjj in units of 2jejvF N0Tc. T/Tc = 0.1–0.9 from top to bottom by 0.1 step. Inset: jjj at T/ Tc = 0.1 (solid line), the contribution of the Fermi surface I (dashed line), and that of the Fermi surface II (dash–dotted line).

We thank D.F. Agterberg for stimulating discussion. We are grateful for ﬁnancial support from the Swiss Nationalfonds and the NCCR MaNEP. One of us (N.H.) is also supported by 2003 JSPS Postdoctoral Fellowships for Research Abroad. References

We have conﬁrmed numerically that the azimuthal component of M is zero around the vortex, namely M is aligned in the radial direction in the xy plane. In Fig. 5, we show the spatial proﬁles of the radial component of M. We checked that jMj decays as 1/r far away from the core. The spatial proﬁles of jMj are similar to those of jjj, but there is a difference because of their microscopic origin. While jjj is composed of gI + gII in Eq. (5), jMj is composed of gI gII in Eqs. (7) and (8) (see also the insets of Figs. 4 and 5).

[1] E. Bauer et al., Phys. Rev. Lett. 92 (2004) 027003. [2] R.P. Kaur, D.F. Agterberg, M. Sigrist, Phys. Rev. Lett. 94 (2005) 137002. [3] S.K. Yip, J. Low, Temp. Phys. 140 (2005) 67. [4] J.W. Serene, D. Rainer, Phys. Rep. 101 (1983) 221, and references therein. [5] L.P. Gor’kov, E.I. Rashba, Phys. Rev. Lett. 87 (2001) 037004. [6] P.A. Frigeri, D.F. Agterberg, I. Milat, M. Sigrist, cond-mat/0505108. [7] P.A. Frigeri, D.F. Agterberg, A. Koga, M. Sigrist, Phys. Rev. Lett. 92 (2004) 097001.

N. Hayashi et al. / Physica C 437–438 (2006) 96–99 [8] N. Hayashi, K. Wakabayashi, P.A. Frigeri, M. Sigrist, cond-mat/ 0504176. [9] N. Hayashi, K. Wakabayashi, P.A. Frigeri, M. Sigrist, Phys. Rev. B 73 (2006) 024504. [10] N. Hayashi, Y. Kato, M. Sigrist, J. Low Temp. Phys. 139 (2005) 79.

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[11] H.F. Hess, R.B. Robinson, J.V. Waszczak, Phys. Rev. Lett. 64 (1990) 2711. [12] H. Nishimori et al., J. Phys. Soc. Jpn. 73 (2004) 3247. [13] A. Wachowiak et al., Science 298 (2002) 577. [14] S.S. Saxena, P. Monthoux, Nature 427 (2004) 799.

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