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Quasiparticle transport in vortex state of borocarbides YNi2B2C
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 160–161
Quasiparticle transport in vortex state of borocarbides YNi2B2 C$ K. Makia, P. Thalmeierb, H. Wonc,* a
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA b Max-Planck Institute for the Physics of Complex Systems, Nothinitzer Str. 38, 01187 Dresden, Germany . c Department of Physics, Hallym University Chuncheon, Chunchon 200-702, South Korea
Abstract It has recently been shown that the nodal superconductor in borocarbides YNi2 B2 C and LuNi2 B2 C are well described by s+g-wave model. Here we shall describe the thermal conductivity of s+g-wave model in the presence of impurity scattering. The effect of impurity scattering is completely different from other nodal superconductors like d-wave and f-wave ones. Also the present result should be applicable to skutterudite superconductor PrOs4 Sb12 if it is spin singlet. r 2003 Elsevier B.V. All rights reserved. PACS: 74.60Ec; 74.25.Fy; 74.70.Dd; 74.70.Tx Keywords: YNi2 B2 C; PrOs4 Sb12 ; Vortex state; Thermal conductivity
There are many experiments indicating the superconducting order parameter in borocarbides YNi2 B2 C and LuNi2 B2 C are hybrid s+g-wave superconductors [1–3] D DðkÞ ¼ ð1 sin4 y cos ð4fÞÞ; 2
ð1Þ
where y and f describe the orientation of the quasiparticle wave vector k: A more recent magneto-specific heat data [4] is also consistent with the nodal structure given in Eq. (1). On the other hand the recent magneto-thermal conductivity data [5] of YNi2x Ptx B2 C for x ¼ 0:05 indicate the extreme sensitivity of the nodal structure to the impurity scattering. As shown in Fig. 1 the four-fold term in the $ This work was supported by Korean Science Research Foundation through Grant No. 1999-2-114-005-5. We thank K. Kamata for allowing us to present her data in this work. We thank Q. Yuan, K. Izawa and Y. Matsuda for enlightening discussions on new superconductors in borocarbides and skutterudites. *Corresponding author. E-mail address: [email protected] (H. Won).
thermal conductivity kzz completely disappeared for the Pt-substituted sample. As is discussed elsewhere [6] the effect of impurity scattering is quite different from the one in usual nodal superconductors. For example the resonant scattering is absent in s+g-wave superconductors. So it is adequate to consider the impurity scattering in the Born limit. Making use of Tc ¼ 13:1 K of the Pt substituted sample, we can extract G ¼ 23:8 K where we used Tc0 ¼ 15:5 K: Therefore we do not expect any nodal excitations in the x ¼ 0:05 sample. As has been shown already [1,3] the quasiparticle density of states in the vortex state with magnetic field H in an arbitrary direction is given by pffiffiffiffiffiffiffi NðE ¼ 0; HÞ v* eH gðHÞ ¼ Iðy; fÞ; ð2Þ ¼ N0 2D pffiffiffiffiffiffiffiffi where v* ¼ va vc and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Iðy; fÞ ¼ 1 sin2 y sin2 f 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ þ 1 sin2 y cos2 f ;
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.066
ARTICLE IN PRESS K. Maki et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 160–161
161
and 1.00
kxx =kn ¼ κzz(φ) /κzz(45)
x=0.05 0.99
0.98
T=0.8 K
0.97
µ0H=1 T Y(Ni1-xPtx)2B2C
x=0.0 0.96 -45
0
45
90
135
φ (degree) Fig. 1. The magneto-thermal conductivity data [5] of YNi2x Ptx B2 C for x ¼ 0:05:
1
θ = π /4 θ = π /3 θ = π /2
I(θ,ϕ)
0.9
ð5Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 0 where x ¼ ð*v eH =2DÞIðy; fÞ; y ¼ G=ðxDÞ; x ¼ ð*v eH = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DÞ 1 sin2 y cos2 f; y0 ¼ G=ðx0 DÞ; and the corresponding thermal conductivity in the normal state kn ¼ p2 =3nT=Gm: These expressions are somewhat different from the ones given earlier [3], since here we consider only the Born limit. Although the angular dependence is practically the same as in Ref. [3], the pffiffiffiffiffi thermal conductivity kzz increases as H ; while kxx is independent of H: Indeed such a dependence has been noted in Ref. [2]. The present analysis is readily extended to skutterudite superconductor PrOs4 Sb12 [8,9]. There are two superconducting phases A and B in PrOs4 Sb12 : We assume [8] DðkÞ ¼ 32 Dð1 kx4 ky4 Þ
A phase;
¼ Dð1 ky4 Þ B phase
ð6Þ
for A and B phases, respectively. Then the quasiparticle density of states and kzz of A phase is practically the same as in YNi2 B2 C: For B phase, on the other hand both of them are proportional to ð1 sin2 y sin2 fÞ1=2 : These angular dependencies are consistent with data frompPrOs ffiffiffiffiffi 4 Sb12 [7]. Also as in YN2 B2 C kzz increases like H in both A and B phase of PrOs4 Sb12 ; while kxx in A phase is independent of H:
0.8
0.7
0.6
0.5 0
π /4
π /2
3 π /4
π
ϕ Fig. 2. ffiffiffiffiffiffi Iðy; p ffi fÞ; the scaled density of states NðE ¼ 0; HÞ by N0 v* eH =2D:
where y and f are the angles describing the orientation of H: The function Iðy; fÞ is shown in Fig. 2. Now in the presence pffiffiffiffiffiffiffi of the impurity pffiffiffiffiffiffiffiscattering Eq. (3) is valid for v* eH bG: For G > v* eH there will be no quasiparticle around. Similarly, then thermal conductivity tensor are given by [7] kzz =kn ¼
x0 2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3 1 0 0 ðcos y y 1 y02 Þ yð1 y0 Þ; 2lnð2=x0 Þ x
pffiffiffiffiffiffiffiffiffiffiffiffiffi x 3y ð1 y2 Þ3=2 ðcos1 y y 1 y2 Þ yð1 yÞ 2lnð2=xÞ 2
ð4Þ
References [1] K. Maki, P. Thalmeier, H. Won, Phys. Rev. B 65 (2002) 140502. [2] K. Izawa, et al., Phys. Rev. Lett. 89 (2002) 137006. [3] P. Thalmeier, K. Maki, Acta Phys. Pol. B 34 (2003) 557. [4] T. Park, M.B. Salamon, E.M. Choi, H.J. Kim, S.I. Lee, Phys. Rev. Lett. 90 (2003) 177001. [5] K. Kamata, Master’s Thesis, University of Tokyo, 2003. [6] S. Lee, H. Won, H.-Y. Chen, Q. Yuan, K. Maki, P. Thalmeier, J. Magn. Magn. Mater., these proceedings, doi:10.1016/j.jmmm.2003.11.389. [7] H. Won, et al., Brazilian J. Phys. 33 (2003) 675; K. Maki, H. Won, S. Haas, Phys. Rev. B 69 (2004) 012502. [8] K. Izawa, Y. Nakajima, J. Goryo, Y. Matsuda, S. Osaki, H. Sugawara, H. Sato, P. Thalmeier, K. Maki, Phys. Rev. Lett. 90 (2003) 117001. [9] K. Maki, P. Thalmeier, Q. Yuan, K. Izawa, Y. Matsuda, Europhys. Lett. 64 (2003) 496.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 160–161
Quasiparticle transport in vortex state of borocarbides YNi2B2 C$ K. Makia, P. Thalmeierb, H. Wonc,* a
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA b Max-Planck Institute for the Physics of Complex Systems, Nothinitzer Str. 38, 01187 Dresden, Germany . c Department of Physics, Hallym University Chuncheon, Chunchon 200-702, South Korea
Abstract It has recently been shown that the nodal superconductor in borocarbides YNi2 B2 C and LuNi2 B2 C are well described by s+g-wave model. Here we shall describe the thermal conductivity of s+g-wave model in the presence of impurity scattering. The effect of impurity scattering is completely different from other nodal superconductors like d-wave and f-wave ones. Also the present result should be applicable to skutterudite superconductor PrOs4 Sb12 if it is spin singlet. r 2003 Elsevier B.V. All rights reserved. PACS: 74.60Ec; 74.25.Fy; 74.70.Dd; 74.70.Tx Keywords: YNi2 B2 C; PrOs4 Sb12 ; Vortex state; Thermal conductivity
There are many experiments indicating the superconducting order parameter in borocarbides YNi2 B2 C and LuNi2 B2 C are hybrid s+g-wave superconductors [1–3] D DðkÞ ¼ ð1 sin4 y cos ð4fÞÞ; 2
ð1Þ
where y and f describe the orientation of the quasiparticle wave vector k: A more recent magneto-specific heat data [4] is also consistent with the nodal structure given in Eq. (1). On the other hand the recent magneto-thermal conductivity data [5] of YNi2x Ptx B2 C for x ¼ 0:05 indicate the extreme sensitivity of the nodal structure to the impurity scattering. As shown in Fig. 1 the four-fold term in the $ This work was supported by Korean Science Research Foundation through Grant No. 1999-2-114-005-5. We thank K. Kamata for allowing us to present her data in this work. We thank Q. Yuan, K. Izawa and Y. Matsuda for enlightening discussions on new superconductors in borocarbides and skutterudites. *Corresponding author. E-mail address: [email protected] (H. Won).
thermal conductivity kzz completely disappeared for the Pt-substituted sample. As is discussed elsewhere [6] the effect of impurity scattering is quite different from the one in usual nodal superconductors. For example the resonant scattering is absent in s+g-wave superconductors. So it is adequate to consider the impurity scattering in the Born limit. Making use of Tc ¼ 13:1 K of the Pt substituted sample, we can extract G ¼ 23:8 K where we used Tc0 ¼ 15:5 K: Therefore we do not expect any nodal excitations in the x ¼ 0:05 sample. As has been shown already [1,3] the quasiparticle density of states in the vortex state with magnetic field H in an arbitrary direction is given by pffiffiffiffiffiffiffi NðE ¼ 0; HÞ v* eH gðHÞ ¼ Iðy; fÞ; ð2Þ ¼ N0 2D pffiffiffiffiffiffiffiffi where v* ¼ va vc and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Iðy; fÞ ¼ 1 sin2 y sin2 f 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ þ 1 sin2 y cos2 f ;
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.066
ARTICLE IN PRESS K. Maki et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 160–161
161
and 1.00
kxx =kn ¼ κzz(φ) /κzz(45)
x=0.05 0.99
0.98
T=0.8 K
0.97
µ0H=1 T Y(Ni1-xPtx)2B2C
x=0.0 0.96 -45
0
45
90
135
φ (degree) Fig. 1. The magneto-thermal conductivity data [5] of YNi2x Ptx B2 C for x ¼ 0:05:
1
θ = π /4 θ = π /3 θ = π /2
I(θ,ϕ)
0.9
ð5Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 0 where x ¼ ð*v eH =2DÞIðy; fÞ; y ¼ G=ðxDÞ; x ¼ ð*v eH = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DÞ 1 sin2 y cos2 f; y0 ¼ G=ðx0 DÞ; and the corresponding thermal conductivity in the normal state kn ¼ p2 =3nT=Gm: These expressions are somewhat different from the ones given earlier [3], since here we consider only the Born limit. Although the angular dependence is practically the same as in Ref. [3], the pffiffiffiffiffi thermal conductivity kzz increases as H ; while kxx is independent of H: Indeed such a dependence has been noted in Ref. [2]. The present analysis is readily extended to skutterudite superconductor PrOs4 Sb12 [8,9]. There are two superconducting phases A and B in PrOs4 Sb12 : We assume [8] DðkÞ ¼ 32 Dð1 kx4 ky4 Þ
A phase;
¼ Dð1 ky4 Þ B phase
ð6Þ
for A and B phases, respectively. Then the quasiparticle density of states and kzz of A phase is practically the same as in YNi2 B2 C: For B phase, on the other hand both of them are proportional to ð1 sin2 y sin2 fÞ1=2 : These angular dependencies are consistent with data frompPrOs ffiffiffiffiffi 4 Sb12 [7]. Also as in YN2 B2 C kzz increases like H in both A and B phase of PrOs4 Sb12 ; while kxx in A phase is independent of H:
0.8
0.7
0.6
0.5 0
π /4
π /2
3 π /4
π
ϕ Fig. 2. ffiffiffiffiffiffi Iðy; p ffi fÞ; the scaled density of states NðE ¼ 0; HÞ by N0 v* eH =2D:
where y and f are the angles describing the orientation of H: The function Iðy; fÞ is shown in Fig. 2. Now in the presence pffiffiffiffiffiffiffi of the impurity pffiffiffiffiffiffiffiscattering Eq. (3) is valid for v* eH bG: For G > v* eH there will be no quasiparticle around. Similarly, then thermal conductivity tensor are given by [7] kzz =kn ¼
x0 2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3 1 0 0 ðcos y y 1 y02 Þ yð1 y0 Þ; 2lnð2=x0 Þ x
pffiffiffiffiffiffiffiffiffiffiffiffiffi x 3y ð1 y2 Þ3=2 ðcos1 y y 1 y2 Þ yð1 yÞ 2lnð2=xÞ 2
ð4Þ
References [1] K. Maki, P. Thalmeier, H. Won, Phys. Rev. B 65 (2002) 140502. [2] K. Izawa, et al., Phys. Rev. Lett. 89 (2002) 137006. [3] P. Thalmeier, K. Maki, Acta Phys. Pol. B 34 (2003) 557. [4] T. Park, M.B. Salamon, E.M. Choi, H.J. Kim, S.I. Lee, Phys. Rev. Lett. 90 (2003) 177001. [5] K. Kamata, Master’s Thesis, University of Tokyo, 2003. [6] S. Lee, H. Won, H.-Y. Chen, Q. Yuan, K. Maki, P. Thalmeier, J. Magn. Magn. Mater., these proceedings, doi:10.1016/j.jmmm.2003.11.389. [7] H. Won, et al., Brazilian J. Phys. 33 (2003) 675; K. Maki, H. Won, S. Haas, Phys. Rev. B 69 (2004) 012502. [8] K. Izawa, Y. Nakajima, J. Goryo, Y. Matsuda, S. Osaki, H. Sugawara, H. Sato, P. Thalmeier, K. Maki, Phys. Rev. Lett. 90 (2003) 117001. [9] K. Maki, P. Thalmeier, Q. Yuan, K. Izawa, Y. Matsuda, Europhys. Lett. 64 (2003) 496.