Spin-fluctuation-mediated pairing in the heavy fermion superconductors without inversion symmetry CeRhSi3 and CeIrSi3

ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 69 (2008) 3341– 3344

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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs

Spin-fluctuation-mediated pairing in the heavy fermion superconductors without inversion symmetry CeRhSi3 and CeIrSi3 Yasuhiro Tada a,, Satoshi Fujimoto b, Norio Kawakami a,b a b

Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan

a r t i c l e in fo

Keywords: D. Superconductivity

abstract We study the pairing symmetry of the noncentrosymmetric heavy fermion superconductors CeRhSi3 and CeIrSi3 under pressures, which are both antiferromagnets at ambient pressure. We solve the Eliashberg equation by means of the random phase approximation and find that the mixing state of extended s-wave and p-wave rather than the d þ f wave state could be realized by enhanced antiferromagnetic spin fluctuations. The gap function has line nodes on the Fermi surface. We also examine the NMR relaxation rate in the superconducting state and find that 1=ðT 1 TÞ exhibits no coherence peak and behaves as 1=ðT 1 TÞ / T 2 . & 2008 Elsevier Ltd. All rights reserved.

1. Introduction Recent discoveries of heavy fermion superconductors without inversion symmetry have attracted much interest. CePt3Si [1] was first identified and accompanied by the discoveries of CeRhSi3 [2], CeIrSi3 [3], UIr [4] and CeCoGe3 [5]. Besides these heavy fermion systems, non-heavy fermion materials such as Li2Pd3B and Li2Pt3B were also found [6]. In all these materials, there exist strong anisotropic spin-orbit interactions due to lack of inversion symmetry, which leads to many interesting phenomena [7–12]. One of the outstanding properties is the parity mixing in superconducting states [10,11,13], i.e. the admixture of the spinsinglet and triplet states, which are both well defined in superconductors with inversion symmetry. Among this new kind of compounds, CeRhSi3 and CeIrSi3 have many similarities due to the same crystal structure: qualitatively similar pressure-temperature phase diagrams were indeed obtained from resistivity measurements [2,3]. They are both in antiferromagnetic (AF) ordered states at low pressures, which are driven to the superconducting states beyond the critical pressures where the Neel temperature rapidly decreases. The Neel temperature at ambient pressure is T N ¼ 1:6 K (5.0 K) for CeRhSi3 (CeIrSi3) and the superconductivity appears in wide pressure ranges with approximate maximal transition temperature 1.1 K at 2.6 GPa (1.6 K at 2.5 GPa), respectively. Moreover, the NMR measurements of the relaxation rate 1=T 1 for CeIrSi3 [16] suggest the existence of the AF spin fluctuations. Also, the neutron scattering experiments for CeRhSi3 [17] identified the AF ordering  Corresponding author.

E-mail address: [email protected] (Y. Tada). 0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.085

vectors as Q ¼ ð0:215; 0; 0:5Þ and it is concluded that the character of the AF is SDW-like. Besides these experiments, the recent band calculations elucidated that the above two compounds have very similar Fermi surfaces [14,15](FS). These similarities deduced both experimentally and theoretically motivate us to discuss the superconductivities of these two compounds in the same theoretical framework. In this paper, we study the noncentrosymmetric superconductors CeRhSi3 and CeIrSi3 with particular emphasis on the influence of AF fluctuations to identify the pairing symmetry realized in these systems. We also examine the NMR relaxation rate which characterizes the nodal structure of a gap function on the FS.

2. Formulation and calculation In CeRhSi3 and CeIrSi3, heavy 4f-electrons around the Fermi level play important roles for low energy phenomena, where 4f electrons come from Ce ions formig a body-centered tetragonal lattice [2]. For our analysis of the superconductivity, we start with the situation that heavy fermions have already been formed through hybridizations with conduction electrons and are described by an effective Hamiltonian. Although these materials have two kinds of the Fermi surfaces [14,15] apart from the splitting by the spin-orbit interaction, we focus on one of them that has a large weight of the total density of states. This enables us to simply describe the electrons in the materials by the following single band model, X X X y H¼ ek cyk ck þ U ni" ni# þ a ck ðL0 ðkÞ  rck , (1) k

i

k

ARTICLE IN PRESS 3342

Y. Tada et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3341–3344

L0 ðkÞ ¼ ðsin ky ;  sin kx ; 0Þ,

(2)

ek ¼  2t1 ðcos kx þ cos ky Þ þ 4t2 cos kx cos ky  8t 3 cosðkx =2Þ cosðky =2Þ cos kz  m,

w0s1 s2 s3 s4 ðqÞ ¼  (3)

tðyÞ

cðyÞ k

are expressed as,

¼ ðck" ; ck# Þ are the annihilation (creation) operators where of the Kramers doublet. L0 ðkÞ is the Rashba spin-orbit interaction due to the lack of inversion symmetry and a is its coupling constant which is estimated to be less than 0:1ekF (ekF is the Fermi energy) according to the band calculation [14]. In the following, we fix a ¼ 0:2t 1 for simplicity. With a natural assumption deduced from the above-mentioned experimental and theoretical results that the AF in these materials are driven by the nesting of the Fermi surfaces, we choose the parameters in the above Hamiltonian t 1 ; t 2 ; t 3 and filling n so that our model is consistent with the band calculation and the neutron scattering experiment; ðt 1 ; t 2 ; t 3 ; nÞ ¼ ð1:0; 0:475; 0:3; 1:055Þ. Here, we define t 1 as the energy unit. For these fixed parameters, the split Fermi surfaces with spin-orbit splitting 2a are shown in Fig. 1. The matrix elements of the momentum-dependent susceptibility w^ 0 ðqÞ at U ¼ 0

TX 0 G ðq þ kÞG0s4 s3 ðkÞ N k s2 s1

(4)

in terms of the bare Green’s functions with spin indices, where we have introduced k ¼ ðion ; kÞ. The above susceptibility has peaks around Q1 ðp=2; 0; p=2Þ, Q2 ð0; p=2; p=2Þ as shown in Fig. 2. Here, we show w0 ðqÞ with a ¼ 0 because there is only one nonzero matrix element with a ¼ 0 and the properties of w^ 0 ðqÞ are little changed with small a. The peak structure of w^ 0 ðqÞ is approximately in agreement with the neutron scattering experiment [17], although its profile in momentum space is not so sharp in our model. First, we study the pairing symmetry by solving Eliashberg equations for all the representations of point group C4v . It is legitimate to use the random phase approximation (RPA) because there exist large AF fluctuations at the pressures near the AF critical point and we neglect the normal self energy, which has little influence on determining the pairing symmetry [18]. The linearized Eliashberg equation is, with the eigenvalue l,

lDs1 s2 ðkÞ ¼ 

T N

X

V s1 s2 s3 s4 ðk; k ÞG0s1 s3 ðk ÞG0s2 s4 ðk ÞDs1 s2 ðk Þ. 0

0

0

0

0

k s3 s4 s1 s2

(5)

π

kz = 0

kz = π/2

The effective pairing interaction V evaluated within RPA consists of three parts, lad V s1 s2 s3 s4 ðk; k Þ ¼ U ds1 s3 ds2 s4 ds1 s¯2 þ V bub s1 s2 s3 s4 ðk; k Þ þ V s1 s2 s3 s4 ðk; k Þ, 0

0

0

bub

(6)

lad

ky

are where U is the bare Hubbard repulsion and V , V calculated by collecting bubble and ladder diagrams, respectively. The bubble terms are FS+

bub bub V bub ssss ðk; k Þ ¼ vssss ðk  k Þ  vssss ðk þ k Þ, 0

FS-

0

bub V bub s¯ss¯s ðk; k Þ ¼ vs¯ss¯s ðk  k Þ, 0

0

0

π

0

where

kx kz = 3π/4

kz = π

2 0 vbub ssss ðqÞ ¼ U ws¯ s¯ s¯ s¯ ðqÞ=Dbub ðqÞ, 2 0 0 0 0 0 vbub s¯ss¯s ðqÞ ¼ U ðws¯ ss¯s ðqÞ  U ws¯ ss¯s ðqÞws¯ss¯ s ðqÞ þ U ws¯ s¯ s¯ s¯ ðqÞwssss ðqÞÞ=Dbub ðqÞ,

Dbub ðqÞ ¼ ð1 þ U w0#""# ðqÞÞð1 þ U w0"##" ðqÞÞ  U 2 w0"""" ðqÞw0#### ðqÞ and V bub s1 s2 s3 s4 with other spin indices are zero. The ladder terms are lad lad V lad ss¯ss¯ ðk; k Þ ¼ vss¯ss¯ ðk  k Þ  vss¯ss¯ ðk þ k Þ, 0

0

0

lad V lad s¯ss¯s ðk; k Þ ¼ vs¯ss¯s ðk  k Þ, 0

0

where Fig. 1. Fermi surfaces in our model. The cross sections at kz ¼ 0, kz ¼ p=2, kz ¼ 3p=4 and kz ¼ p are shown.

2 0 vlad ss¯ss¯ ðqÞ ¼ U ws¯ss¯s ðqÞ=Dlad ðqÞ,

Fig. 2. Bare susceptibility w0 ðqÞ at o ¼ 0 with a ¼ 0. This result is obtained for T ¼ 0:04. The left panel is w0 ðqÞ on the xy plane at kz ¼ p=2 and the right panel is w0 ðqÞ on the xz plane at ky ¼ p=2.

ARTICLE IN PRESS Y. Tada et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3341–3344 2 0 0 0 0 0 vlad s¯ss¯s ðqÞ ¼ U ðwss¯ss¯ ðqÞ  U ws¯ s¯ ss ðqÞwss¯ss¯ ðqÞ þ U ws¯ss¯s ðqÞws¯ s¯ss ðqÞÞ=Dlad ðqÞ,

Dlad ðqÞ ¼ ð1  U w0##"" ðqÞÞð1  U w0""## ðqÞÞ  U 2 w0"#"# ðqÞw0#"#" ðqÞ and others are zero. Within RPA, only the V lad ss¯ss¯ terms do not conserve the spins of two particles before and after scattering and parity mixing is driven only by two Green’s functions which connect the 4-point vertex part with the gap function in Eq. (5). Thus, the parity mixing effect may not be strongly enhanced by U. In the numerical calculations, we do not take into account the lad antisymmetrization of V bub ssss and V ss¯ss¯ , because the singlet pairing states are dominant rather than the triplet pairing states in our model. Generally, the gap function Ds1 s2 ðkÞ is expressed as,

DðkÞ ¼ ðDs ðion Þd0 ðkÞs0 þ Dt ðion ÞdðkÞ  rÞis2 ,

(7)

where Ds d0 , Dt d are the order parameters for singlet and triplet states and they can have non-zero values simultaneously because of the Rashba spin-orbit interaction. We denote ðdm Þm¼03 ¼ ðd0 ; dÞ hereafter. We can determine, by solving the above eigenvalue equation (5), the symmetry of the gap function and the transition temperature T c at which the maximum eigenvalue lmax reaches unity. G The pairing symmetries of the singlet states fd0 ðkÞgG for five irreducible representations of C4v are listed in Table 1. Here, we G choose fd G ðkÞgG as d G ðkÞ ¼ d0 ðkÞL0 ðkÞ for each representation which is considered to be most stable in the superconductors with D5a [10,12]. The harmonic wave functions in Table 1 give the largest contributions among the functions which belong to a given symmetry, mainly because of the factor 12 in the propagating vectors Q1;2 . In the calculation, the first Brillouin zone is divided into 16  16  16 meshes and the number of the Matsubara frequencies is 512. We have checked that the following results are qualitatively unchanged for ð32Þ3 k-meshes. We solve the Eliashberg equation for all symmetries and trace G each maximum eigenvalue fl gG with decreasing T. Here, we fix U ¼ 3:52 which corresponds to the situation that the system is near the AF critical point. Among five symmetries, we find that A A only l 1 for ðd0 1 ; d A1 Þ ¼ ðcos 2kz ; cos 2kz L0 Þ can reach unity with the transition temperature T c 0:04. At this transition temperaG ture, l for other symmetries are l t0:3. Generally, this A1-symmetric gap function has line nodes perpendicular to c-axis A on the Fermi surface, but the averaged value over the FS hdm1 iFS has a non-zero value. In our model with this gap function, there exist line nodes at kz ¼ p=4; 3p=4 on the FS. Regarding the A nodal structure, our A1 gap function dm1 is very similar to that for the usual d-wave superconductivity. In our numerical study, the ratio Dt =Ds is less than 0.01. Thus, the admixture of the singlet and triplet components is very small. We also study the pairing symmetry with another set of parameters ðt 1 ; t 2 ; t 3 ; nÞ ¼ ð1:0; 0:4; 0:3; 0:98Þ at which w0 ðqÞ gives maximum values at Qð0:4p; 0:4p; 0:5pÞ. This is because the neutron scattering experiment [17] was performed at ambient

3343

pressure, and the true propagating vector under pressures is unknown while the superconductivity occurs at high pressures. In these parameters, we again find that the A1-symmetric pairing state is most likely to appear, which asserts the robustness of the stability of the pairing state with the A1 symmetry against slight A change of the Fermi surface. This is because d0 1 ¼ cos 2kz depends only on kz and the changes in the x, y components of Q do not affect the main scattering process for the A1-symmetric superconductivity. Let us consider a possible explanation for the stability of the superconductivity with this symmetry. We think that, in our model, taking into account only the scattering processes with k  k0 ¼ Q1;2 gives us intuitive but restricted information, because the peak structures of w^ 0 ðqÞ are not so sharp and the shape of the FS is complicated in the 3D momentum space. Nevertheless we try to figure out how these scattering processes on the FS contribute to the realization of the superconductivity with the A1 symmetry. Fig. 3 shows the FS and the signs of the singlet A1 A gap function d0 1 ¼ cos 2kz ; the gray and the white regions A A correspond to d0 1 ¼ cos 2kz 40 and d0 1 o0, respectively. The FS has a cylinder-like shape along z-axis and there exist wide ranges of hot spots. Among these regions, the subsets of the FS, A A where d0 1 ðkÞ  d0 1 ðk  QÞ is negative and large, are important for the superconductivity. We think that such spots might be on the sides of the cylinder-like FS as shown in Fig. 3 (enclosed by contours) and the area of the spots could be large. Thus, we think that the scattering processes drawn with white arrows in Fig. 3 could mediate the superconductivity. We now turn to discussions on the NMR 1=T 1 T in the A superconducting states with A1 symmetry dm1 . It is expressed as [12], 1 / T1T

Z

do 1 ðjNn ðoÞj2 þ jNa ðoÞj2 Þ, 2p 2Tcosh2 o 2T

(8)

Table 1 The irreducible representations of C4v and the basis functions Irreducible representation

Basis function

A1 (extended s)

d0 1 ðkÞ ¼ cos 2kz

A2 ðg xyðx2 y2 Þ Þ

d0 2 ðkÞ ¼ sin 2kx sin 2ky ðcos 2kx  cos 2ky Þ

B1 (dx2 y2 )

d01 ðkÞ ¼ ðcos 2kx  cos 2ky Þ

A A B B

B2 ðdxy Þ

d02 ðkÞ ¼ sin 2kx sin 2ky

E (dxz )

d0 ðkÞ ¼ sin kx sin 2kz

Every representation

d G ðkÞ ¼ d0 ðkÞL0 ðkÞ

E

G

Fig. 3. Fermi surface for a ¼ 0. The gray regions and white regions correspond to A A d0 1 ¼ cos 2kz 40, d0 1 o0, respectively. The boundaries between the two regions are the line nodes of the A1 gap function. The white arrows connecting the areas enclosed by contours represent the scattering processes associated with the propagating vector Q1;2þ.

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(T1T)-1 / (T1T)-1c

1

ðDð0Þ; gÞ, we find that 1=T 1 T has no coherence peak and behaves as 1=ðT 1 TÞT 2 , which is characteristic of line-node superconductors. This behavior of the NMR 1=ðT 1 TÞ is usually typical for dominant d-wave superconductivity but not for dominant extended s-wave one. In the present system, however, the FS is highly anisotropic and the A1 gap function with line nodes could effectively behave like a d-wave gap function on the FS. According to recent NMR experiments for CeIrSi3 [16], 1=ðT 1 TÞ exhibits line-node behavior with no coherence peak, which seems not contradictory to our results.

0.1

3. Conclusion

0.01 0.1

1 T/Tc

Fig. 4. NMR relaxation rate 1=T 1 T as a function of T. The amplitude of the gap function at T ¼ 0 and the quasiparticle damping factor are chosen as Ds ð0Þ ¼ 2:0T c , Dt ð0Þ=Ds ð0Þ ¼ 0:01 and g ¼ 0:01Ds ð0Þ.

where Nn ðoÞ ¼ 

X

Im G0R t ðo þ ig; kÞ,

kt

N a ð oÞ ¼ 

X

Im F 0R t ðo þ ig; kÞ.

kt

The normal and anomalous Green’s functions in the superconducting state are given by 0

Gt ðkÞ ¼ F 0t ðkÞ ¼

ion þ xkt ðion Þ2  E2kt

Dkt ðion Þ2  E2kt

,

,

with

xkt ¼ ek þ takL0 ðkÞk, kL0 ðkÞk ¼ Ekt ¼

We have studied the pairing symmetry and the nature of the gap function in the superconducting state in noncentrosymmetric heavy fermion superconductors CeRhSi3 and CeIrSi3. Solving the Eliashberg equation within RPA, we found that AF fluctuations could mediate the superconductivity with the parity mixing of the extended s-wave and p-wave states rather than the d þ f wave state through the Rashba spin-orbit interaction. In this superconducting state, the NMR relaxation rate exhibits 1=ðT 1 TÞ / T 2 with no coherence peak at T c as in the case of the usual d-wave superconductivity. Our results suggest a possible understanding of the recent NMR experiment within the extended s þ p wave state.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L20x þ L20y þ L20z ,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2kt þ D2kt ,

Dkt ¼ Ds ðTÞd0 ðkÞ þ tDt ðTÞkdðkÞk. We assume the T dependence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the order parameters as Dm ðTÞ ¼ Dm ð0Þ tanhð1:74 T c =T  1Þ, regarding Dm ð0Þ and the quasiparticle damping factor g as fitting parameters. Because 1=ðT 1 TÞ depends on N n ðoÞ and N a ðoÞ not only at o0 but also at oDð0Þ, existence of line nodes does not directly mean that 1=T 1 T in the A1 superconducting state is similar to that in d-wave states. Fig. 4 shows the temperature dependence of 1=ðT 1 TÞ normalized by 1=ðT 1 TÞc . By repeating similar calculations for several choices of

Acknowledgements We thank M. Sigrist, N. Kimura, H. Mukuda, H. Harima and T. Terashima for valuable discussions. Numerical calculations were partially carried out at the Yukawa Institute Computer Facility. References [1] E. Bauer, et al., Phys. Rev. Lett. 92 (2004) 3129. [2] N. Kimura, et al., Phys. Rev. Lett. 95 (2005) 247004; N. Kimura, et al., J. Phys. Soc. Jpn. 76 (2007) 033706. [3] I. Sugitani, et al., J. Phys. Soc. Jpn. 75 (2006) 043703. [4] T. Akazawa, et al., J. Phys. Soc. Jpn. 73 (2004) 3129. [5] R. Settai, et al., J. Mag. Mag. Matt. 310 (2007) 844. [6] K. Togano, et al., Phys. Rev. Lett. 93 (2004) 247004. [7] V.M. Edelstein, Sov. Phys. JETP. 68 (1989) 1244; V.M. Edelstein, Phys. Rev. Lett. 75 (1995) 2004. [8] L.P. Gorkov, et al., Phys. Rev. Lett. 87 (2001) 037004. [9] S.K. Yip, et al., Phys. Rev. B 65 (2002) 144508. [10] P.A. Frigeri, et al., Phys. Rev. Lett. 92 (2004) 097001. [11] M. Sigrist, et al., AIP Conference Proceedings 816 (2006) 124. [12] S. Fujimoto, Phys. Rev. B 72 (2005) 024515; S. Fujimoto, J. Phys. Soc. Jpn. 75 (2006) 083704; S. Fujimoto, J. Phys. Soc. Jpn. 76 (2007) 051008. [13] Y. Yanase, et al., J. Phys. Soc. Jpn. 76 (2007) 043712. [14] H. Harima, private communication. [15] T. Terashima, et al., Phys. Rev. B, to be published. [16] H. Mukuda et al., unpublished. [17] N. Aso, et al., J. Mag. Mag. Matt. 310 (2007) 602. [18] Y. Yanase, et al., Phys. Rep. 387 (2004) 1.