Gap symmetry of superconducting borocarbide YNi2B2C and skutterudite PrOs4Sb12

Abstract

Within the quasiclassical approximation we have studied the thermodynamics and the thermal conductivity in the vortex state in nodal superconductors (sc). Recent angle dependent magnetothermal conductivity results indicate a gap function delta(k) corresponding to f- wave sc in Sr2RuO4 and d- wave sc in CeCoIn5 and k-(ET)2Cu(NCS)2 respectively. More recently it is shown that delta(k) in both YNi2B2C and PrOs4Sb12 have point nodes described by hybrid s+g wave gap function.

Gap symmetry of superconducting borocarbide YNi₂B₂C and skutterudite PrOs₄Sb₁₂

H. Won^I; Q. Yuan^II; P. Thalmeier^III; K. Maki^IV

^IDepartment of Physics, Hallym University, Chunchon 200-702, South Korea

^IITexas Center for Superconductivity and Advanced Materials, University of Houston Houston TX 77204, USA

^IIIMax-Planck-Institute for the Chemical Physics of Solids, Nöthnitzer Str.40, 01187 Dresden, Germany

^IVDepartment of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA

ABSTRACT

Within the quasiclassical approximation we have studied the thermodynamics and the thermal conductivity in the vortex state in nodal superconductors (sc). Recent angle dependent magnetothermal conductivity results indicate a gap function D(k) corresponding to f- wave sc in Sr₂RuO₄ and d- wave sc in CeCoIn₅ and k-(ET)₂Cu(NCS)₂ respectively. More recently it is shown that D(k) in both YNi₂B₂C and PrOs₄Sb₁₂ have point nodes described by hybrid s+g wave gap function.

1 Introduction

The gap symmetry of superconductivity is the central issue since the appearance of many unconventional and/or nodal superconductors in heavy Fermion systems, organic superconductors, high T_c cuprate superconductors, Sr₂RuO₄ and borocarbides YNi₂B₂C and LuNi₂B₂C . Following the quasiclassical approach introduced by Volovik [2] it is possible to calculate both the thermodynamic and transport properties of the vortex state in nodal superconductors.

Especially in quasi 2D systems like high T_c cuprates, Sr₂RuO₄ , CeCoIn₅ and k-ET₂X with X=Cu(NCS)₄, Cu[N(CN)₂]Br and Cu[N(CN)₂]Cl the thermal conductivity in a magnetic field within the conducting plane provides the crucial insight on the nodal structure of D(k) [3,4]. Indeed Izawa et al have shown f-wave sc in Sr₂RuO₄ [5], d-wave sc in CeCoIn₅ [6] and k-(ET)₂Cu(NCS)₂ [7] through the angle dependent magnetothermal conductivity. More recently gap functions D(k) in borocarbide YNi₂B₂C and skutterudite PrOs₄Sb₁₂ are shown to have point nodes and described in terms of s+g-wave sc [8-12]. We show |D(k)| of these superconductors in Fig. 1.

2 The Volovik effect

For simplicity we shall consider four examples of nodal D(k) in quasi- two dimensional systems; D(k) =Df(k) with f = cos(2f), sin(2f), e^±ifcos(ck_z), e^±ifcos(k_z). They are wave, wave, f- wave and f'- wave. The last one is introduced by Zhitomirsky and Rice [13] (i.e. the multigap model) in order to account for the nodal superconductor seen in Sr₂RuO₄ through the specific heat [14] and the magnetic penetration depth [15]. Indeed all four states have the same quasiparticle density of states as d-wave superconductors [16]. As we shall see later, however, the multigap model is incompatible with the angle dependent magnetothermal conductivity observed by Izawa et al [5] and the universal heat conduction in k_zz observed by Suzuki et al [17] in Sr₂RuO₄ . On the other hand these thermal conductivity data are fully consistent with f-wave sc shown in Fig. 1.

In the absence of a magnetic field the quasiparticle density of states (DOS) of these four superconductors is given by [16]

Where N₀ is the the normal state DOS and K(x) is the complete elliptic integral of the first kind. In particular for |x| 1

Now in a magnetic field H perpendicular to the conducting plane Eq. (2) is replaced by

Where v· q is the Doppler shift [18]; with v denoting the Fermi velocity and 2q the pair momentum describing the circulating superflow around vortices. Here á...ñ means average over the Fermi surface and over the unit cell of the vortex lattice. In the present configuration this is readily done [2,19]

where we have assumed a square vortex lattice [20]. Here d=. For a usual triangle lattice Eq. (4) will be multiplied by a factor x= 0.9306. . Then the specific heat, the spin susceptibility and the superfluid density at low temperature (i.e. T

D₀) are given by [21]

This dependence of the specific heat has been seen in YBCO[22,23], LSCO[24] and Sr₂RuO₄ [21]. Further the dependence of c_s and the H linear dependence of have been seen by NMR in a slightly underdoped Bi2212 [25]. Although the above expressions are obtained in the absence of impurity scattering, the Eqs. (4),(5) still hold in the superclean limit where

v with G denoting the quasiparticle scattering rate in the normal state [3,4]. There are only two energy scales in this limit, namely v and T. Therefore the thermodynamic functions obey the scaling law as proposed by Simon and Lee [26] and shown for a particular case by Kübert and Hirschfeld [19].

The thermal conductivity is treated similarly. In the absence of magnetic field k_xx exhibits the universal heat conduction [27,28] for all four superconductors.

This is also valid for k_zz except for the f'- wave sc of the multigap model where we have

Thus for f'- wave k_zz depends on the scattering rate G and never reaches the universality limit contrary to [17].

In a magnetic field H || c we obtain

in the superclean limit for all four superconductors. The H linear thermal conductivity has been seen in Sr₂RuO₄ [5].

3 Field angle dependent thermal conductivity

When the magnetic field lies in the a-b plane the Doppler shift generates the quasiparticles in the plane perpendicular to the magnetic field. When the plane cuts the nodal directions, there will be strong increase in the quasiparticle density. Therefore the thermal conductivity takes the maximum value when the magnetic field is perpendicular to the nodal directions [3,4]. In particular in the superclean limit we obtain

for i= 1,2,3,4 which corresponds to wave, wave, f- wave and f'- wave superconductors respectively. Here = v_av_c and f is the azimuthal angle of H with respect to the a axis. We obtain the following angular functions I_i(f) and J_i(f):

with a₁ = 0.256, a₂ = 0, a₃ = 0.124 and a₄ = 0.106. Their angular dependence is shown in Fig. 2. As is readily seen the f- dependence of the thermal conductivity is adequate to identify one of four D(k) gap functions we have considered sofar.

In this way D(k) in Sr₂RuO₄ [5], CeCoIn₅ [6] and k-(ET)₂Cu(NCS)₂ [7] are identified. On the other hand the f- dependence of k_xx and k_xy in YBCO so far determined [30,32,33] do not match the one for wave. This is because in the above analysis we have limited ourselves for T , while the experiments have been done for T [29].

4 Superconductivity in Borocarbides YNi₂B₂C and LuNi₂B₂C

So far we limit ourselves to the quasi- 2D systems. In 3D systems it is more expedient to consider the thermal conductivity as function of both q and f, which are the polar coordinates defining the field direction. In this respect the superconductivity in YNi₂B₂C and LuNi₂B₂C is of great interest in part due to their relatively high transition temperature 15.5 K and 16.5 K resepectively [34]. Although the dominance of s-wave component in D(k) has been established by substituting Ni by a small amount of Pt and subsequent opening of the energy gap [35,36], the superconductivity exhibits many characteristics of nodal superconductors like T³ behaviour of and the - dependence of the specific heat[38]. Due to their tetragonal space group I4/mmm the borocarbides have fourfold rotational symmetry within the a-b plane. The simplest D(k) which satisfies these constraints appears to be s+g-wave superconductor [8,9]

where J and j are polar coordinates describing k. We show the quasiparticle DOS in Fig. 3. For |E|/D 1, we obtain

Also there are similarity and difference between d-wave superconductors as seen in [39]. In a magnetic field with orientation defined by (q,f) the quasiparticle DOS is given by [10]

Then the specific heat, the spin susceptibility and the superfluid density for T D₀ are given by

We show in Fig. 4 the f dependence of I(q,f) for various polar field angles q. In particular for q = , I(q,f) develops cusps at f=0 and etc. They are a characteristic signature of the point nodes. We note that a recent specific heat measurement on YNi₂B₂C has excactly found these cusps[40].

In order to calculate the thermal conductivity an analysis of impurity scattering is necessary. Unlike in other nodal superconductors we find the Born limit and the unitarity limit gives practically the same result. There is no resonance scattering in s+g- wave superconductors [41]. Secondly the energy gap opens up immediately in the presence of impurity scattering. The quasiparticle DOS in the presence of impurity scattering is also shown in Fig. 3 for several G/D. In the absence of magnetic field the related equations are given by

where f=sin⁴ Jcos(4j). Then for w ® 0, we find ® 0 and ® D + G. In other words the energy gap opens up immediately with G. The energy gap w_g(G) visible in Fig. 3 is well approximated by w_g(G) = G/(1+).

This immediate opening of the energy gap is very different from the case of s+d- wave superconductors as discussed in [36]. This has immediate consequences: There are no nodal excitations for T < G. In this limit both the specific heat and the thermal conductivity decrease exponentially. Also there will be no universal heat conduction [27,28] in sharp contrast to usual nodal superconductors. From this we expect that also the thermal conductivity in the vortex state of the s+g wave superconductor will be quite different from the case of usual nodal superconductors. It is now given by

Here q_H is the Heaviside step function. Furthermore k_n refers to the normal state, and we have assumed G,T

D. In the other limit < G there will be no nodal excitations. We note that first of all the angular dependence of k_zz is given by I(q,f) which is fully confirmed in [8]. The details of the above expressions are different from those given previously [9,10] due to the present more realistic treatment of the impurity scattering in the s+g wave case. On the other hand the conclusion that a gap w_g G opens immediately for s+g wave order parameter makes it now hard to understand the H linear dependence of thermal conductivity at lowest temperatures reported in [42].

As is readily seen from Eq. (18) we predict that k_zz increases like while the dominant term in k_xx is independent of H. Indeed the dependence of k_zz is consistent with the experimental data in [8]. More recently the anglular dependence of the thermal conductivity in a single crystal of Y(Ni_1-xPt_x)₂B₂C with x = 0.05 has been studied[43]. It is shown in Fig. 5 in comparison to the pure YNi₂B₂C (x=0). Clearly no fourfold oscillation in k,f) survives in the 5% Pt- doped compound. From T_c = 13.1 K of the 5% Pt- substituted crystal and T_c = 15.5 K for YNi₂B₂C (x=0) we can estimate G using

where y(z) is the di-gamma function, f = sin⁴qcos(4f) and áf²ñ = 0.203. The resulting G = 23.8 K > T_c shows that we are already close to the limit l=/G x₀. Clearly the large G has eliminated the nodal excitations. Therefore it is of great interest to study the thermal conductivity of the Pt- substituted crystals with x < 0.05. It is also necessary to study theoretically the extremely dilute limit where eventually the gap induced by impurity scattering should become inhomogeneous and a crossover to a new regime should set in.

5 Skutterudite PrOs₄Sb₁₂

Skutterudites with rare earth atoms may exhibit Heavy Fermion behaviour and in addition magnetic, quadrupolar and superconducting phase transitions. Among them PrOs₄Sb₁₂ is a rather unique case [44,45,46]. Again the angular dependent thermal conductivity data [12,11] suggest i) the presence of two phases denoted A and B and ii) a gap D(k) with planar symmetry in A while one with axial symmetry in B phase (see Fig. 6). Also, the singlet pairing in this system is very likely though not confirmed sofar. In order to describe the angular dependent thermal conductivity we have proposed [12,11] the hybrid gap functions

Where the A- phase again corresponds to a (tetragonal) s+g wave gap. A and B- phases have point nodes in x,y and y directions respectively. We assume here the absence of nodes along the z- direction since they have not been confirmed sofar. Their corresponding quasiparticle DOS is shown in Fig. 7 In zero field these phases exhibit characteristics of nodal superconductors like those in YNi₂B₂C . Again impurity scattering immediately opens a gap. Also as before there is no resonance scattering and universal low temperature heat conduction since it vanishes like exp(-G/T). In the presence of a magnetic field the residual quasiparticle DOS is given by (v = v_a,c)

whith angular functions given by

We note that I_A(q,f) º I(q,f) given in Eq. (14). Despite the different functional form of |D(k)| these s+g wave sc phases have the same quasiparticle DOS. As discussed in Sect.(4) this expression is valid when G. Also the effect of a small G may be incorporated by changing vI_i(q,f) ® vI_i(q,f)-G. The specific heat etc. are expressed in terms of g(0,H) as before. Like the residual quasiparticle DOS the thermal conductivity k_zz(q,f) for the A- phase is given by the same expression Eq. (18) as for the YNi₂B₂C s+g- wave gap with I(q,f) ® I_i(q,f) (i=A,B). Its angular dependence is completely determined by I_i(q,f) and agrees with the experimentally observed fourfold and twofold oscillations observed in the A- and B- phases respectively in PrOs₄Sb₁₂ below T = 0.5 K. However experimental results at lower temperatures are highly desirable. At this preliminary stage the microscopic origin of the symmetry breaking into planar and axial symmetry seen in A and B- phases is not yet clear.

6 Concluding remarks

First we have reviewed our earlier work on the magnetothermal conductivity in quasi 2D nodal superconductors. These superconductors are described in the BCS context by Cooper pairs having non- zero angular momomentum. The effect of a magnetic field is incorporated within the quasiclassical approximation in the vortex phase as first done by Volovik[2]. In nodal superconductors it represents a simple and accurate description of quasiparticle excitations. The associated magnetothermal conductivity provides a unique window to investigate nodal structures of D(k) . In this way Izawa et al have succeeded in identifying the gap symmetry of Sr₂RuO₄ , CeCoIn₅ and k-(ET)₂Cu(NCS)₂ [5,6,7]. More recently the magnetothermal conductivity data from YNi₂B₂C and PrOs₄Sb₁₂ has revealed the presence of point nodes in D(k) [8,11]. These order parameters are described in terms of s+g wave gap functions[9,10,12]. This is the first time such hybrid gap functions consisting of a superposition of representations have been found. In addition, due to the presence of an s- wave component in D(k) , the effect of impurities is very different[41] from that in usual nodal superconductors whose gap functions have sign change and belong to a single nontrivial representation. For example there is no resonance scattering due to nonmagnetic impurities. Also the energy gap opens up immediately due to impurity scattering[41] which is completely different from the p- or d- wave superconductors. Therefore the hybrid nodal superconductors YNi₂B₂C and PrOs₄Sb₁₂ appear to open completely new vista in the rich field of unconventional superconductors.

Acknowledgement

We thank Konomi Kamata, Koichi Izawa and Yuji Matsuda for useful discussions, especially on ongoing experiments on YNi₂B₂C and PrOs₄Sb₁₂ . We thank also K. Kamata to allow us to use her unpublished data of the single crystal Y(Ni_1-xPt_x)₂B₂C with x = 0.05.

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[4] T. Dahm, K. Maki, and H. Won, cond-mat/006301

[12] K. Maki, W. Won', P. Thalmeier, Q. Yuan, K. Izawa, and Y. Matsuda, cond-mat/0212090

[39] H. Jang, H. Won and K. Maki, cond-mat/0302103.

[41] S. Lee et al, to be published.

Received on 23 May, 2003

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Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  23 May 2003