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Multiple superconducting phases in skutterudite PrOs4Sb12
Physica C 408–410 (2004) 177–178 www.elsevier.com/locate/physc
Multiple superconducting phases in skutterudite PrOs4Sb12 Peter Thalmeier b
a,*
, Kazumi Maki b, Qingshan Yuan
c,d
a Max-Planck-Institute for the Chemical Physics of Solids, N€othnitzer Str. 40, 01187 Dresden, Germany Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA c Texas Center for Superconductivity, University of Houston, Houston, TX 77204, USA d Pohl Institute of Solid State Physics, Tongji University, Shanghai 200092, PR China
Abstract The skutterudite PrOs4 Sb12 is a Heavy Fermion (HF) superconductor (sc) below Tc ¼ 1:8 K [Phys. Rev. B 65 (2002) 100506(R)]. Specific heat measurements indicate two separate sc transitions. Recent angle dependent magnetothermal conductivity experiments [Phys. Rev. Lett. 90 (2003) 117001] have shown that PrOs4 Sb12 is another of the very few examples of multiphase-superconductivity. Two phases A and B with different node structures have been identified. We propose a model for their anisotropic gap function DðkÞ. While the high-field A-phase has fourfold planar symmetry (tetragonal s + g-wave) with four point nodes along two cubic axis, the low-field B-phase has lower symmetry with only two point nodes. We discuss the field-angle dependent magnetothermal conductivity and propose a microscopic pairing mechanism. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.Rp; 74.25.Fy; 74.70.Tx Keywords: Skutterudite; PrOs4 Sb12 ; Magnetothermal transport
1. Introduction Superconductivity in the Pr-HF compound [1] (c ¼ 313 mJ/mol K2 ) is of great interest because both the order parameter and the pairing mechanism may be unconventional. A phase diagram with two sc phases A (high-field) and B (low-field) with different nodal structure was concluded from magnetothermal conductivity jzz ðHÞ [2] and phenomenological models for the gap functions have been proposed [3,4]. The similarity of the Fermi surface in LaOs4 Sb12 and PrOs4 Sb12 suggests a mechanism for the heavy mass formation which is different from the sf-hybridisation or Kondo model invoked for Ce-compounds. Indeed one has a stable Pr3þ configuration possibly with nonmagnetic C1 -ground state and a C5 first excited state due to the tetrahedral CEF. Since at higher fields a quadrupolar order is induced in the C1 –C5 system it is suggestive that at zero field (off*
Corresponding author. Tel.: +49-35146462234; fax: +4935146463232. E-mail address: [email protected] (P. Thalmeier).
diagonal) quadrupolar fluctuations provide a novel mechanism for sc pair formation in PrOs4 Sb12 . 2. Order parameter models To account for the different nodal structure of A- and B-phase a model for the gap function DðkÞ has been proposed [3]. The cusp-like structure observed in angular dependence of jzz ðh; /Þ indicates the existence of point nodes rather than line nodes. This is reminiscent of the situation in borocarbide superconductors [5]. Assuming singlet pairing one has to invoke hybrid gap functions DðkÞ ¼ Df ðkÞ with form factors given by ðAÞ f ðkÞ ¼ 1 kx4 ky4 ðBÞ f ðkÞ ¼ 1 ky4
ð1Þ
to realise the node structure. The A-phase gap is of tetragonal s + g-wave type while the B-phase gap has only twofold symmetry. Both are threefold degenerate, specific domains (nodes in xy-plane and along y direction) have been chosen. These gap functions are shown in the inset of Fig. 1.
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.162
178
P. Thalmeier et al. / Physica C 408–410 (2004) 177–178 1
Here jn is the normal state p thermal conductivity. The ffiffiffiffi leading field dependence is H but terms H are also present. The angular dependences are shown in Fig. 1. The twofold (lower part, B-phase) and fourfold (upper part, A-phase) oscillations and the cusp-like minima when H sweeps over the node directions are clearly visible. Experimentally the amplitude of twofold and fourfold transition decrease or increase rather suddenly when H becomes bigger than a critical H ðT Þ. This defines the experimental phase boundary between A and B phases. Aside from thermal conductivity it has also been seen in an anomaly of the flux flow resistivity.
ο
IA (θ,φ)
θ = 45 0.8
60
0.6
90
ο
ο
A 0.4 IB (θ,φ)
B 0.2 0 90
45
0 φ [deg.]
45
90
Fig. 1. Calculated angular variation of jzz ðh; /Þ from Eq. (2) exhibiting fourfold (A) and twofold (B) oscillations in / due to the different node structure of DA;B ðkÞ. Their polar plots (inset) show nodes along two (A) or only one (B) cubic axis. Here h and / are polar and azimuthal angle of the field H with respect to [0 0 1]. Possibly nodes along [0 0 1] also exist.
3. Magnetothermal conductivity The conjecture of sc gap nodes in PrOs4 Sb12 has been obtained from the field dependence of jzz ðHÞ which is approximately linear in H at low temperatures [2] for Hk½1 0 0. This nonexponential behaviour is a strong indication of the presence of itinerant gapless quasiparticle excitations in the intervortex region which can carry a heat current perpendicular to the vortex lines. This is due to the Doppler shift (Volovik effect) of quasiparticle (momentum q) energies given by x ¼ v q=D which results from the supercurrent velocity field v around the vortex lines. Further evidence comes from the jzz -dependence on field polar (h) and azimuthal (/) angles when H is swept around the [0 0 1] axis for constant field strength at low temperature. Doppler shift and induced DOS depends on field orientation relative to the fixed nodal positions, therefore thermodynamic and transport properties show typical oscillation as function of h; / leading to maxima and minima for H along antinodal or nodal pffiffiffiffiffiffi ffi directions respectively. In the limit C < T v eH Dð0Þ (C ¼ scatt. rate) one obtains for the thermal conductivity ði ¼ A; BÞ pffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffi 2 v eH 87 v eH Ii ðh; /Þ 1 þ Ii ðh; /Þ p 125 D D 1 C 1 pffiffiffiffiffiffiffi Ii ðh; /Þ v eH i 1 1 1h IA ðh; /Þ ¼ ð1 sin2 h sin2 /Þ2 þ ð1 sin2 h cos2 /Þ2 2 1 1 IB ðh; /Þ ¼ ð1 sin2 h sin2 /Þ2 4 ð2Þ jzz ¼ jA
4. Quadrupole fluctuations and pair potential The C1 –C5 (DCEF ) low lying CEF level scheme does not lead to exchange scattering of conduction electrons below DCEF ’ 6 K. Therefore heavy mass generation cannot be due to the Kondo mechanism and also Cooper pair formation must have a different source than exchange of spin fluctuations commonly invoked for unconventional HF superconductors. Due to the vicinity of the field induced quadrupolar phase, mass renormalisation and pair formation via exchange of quadrupolar fluctuation seems possible. It relies on the aspherical Coulomb scattering of conduction electrons due to the nondiagonal C1 –C5 quadrupole moment. Elimination of local 4f-excitations via Schrieffer–Wolff transformation then leads to an effective quadrupole fluctuation mediated quasiparticle interaction which in RPA is given by (cq ¼ cubic structure factor, j2 ¼ Bessel function, vQ ð0Þ ¼ single ion quadrupole susceptibility for x ¼ 0) KðqÞ ¼ G2 ðqÞvQ ð0Þ½1 cq vQ ð0Þ1 q2z þ ^ q2x þ ^ q2y G2 ðqÞ hj2 ðqrÞiq2 ½^ q2y ^ q2z ^ q2x ^
ð3Þ
The q dependence of the interaction is mainly due to the quadrupolar form factor G2 ðqÞ contrary to spin fluctuation theory where it is a constant. The form factor and hence KðqÞ is small for q along the cubic axes which is in qualitative agreement with the assumed gap node structure.
References [1] [2] [3] [4] [5]
E.D. Bauer et al., Phys. Rev. B 65 (2002) 100506(R). K. Izawa et al., Phys. Rev. Lett. 90 (2003) 117001. K. Maki et al., Europhys. Lett. 64 (2003) 496. J. Goryo, Phys. Rev. B 67 (2003) 184511. K. Izawa et al., Phys. Rev. Lett. 89 (2002) 137006.
Multiple superconducting phases in skutterudite PrOs4Sb12 Peter Thalmeier b
a,*
, Kazumi Maki b, Qingshan Yuan
c,d
a Max-Planck-Institute for the Chemical Physics of Solids, N€othnitzer Str. 40, 01187 Dresden, Germany Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA c Texas Center for Superconductivity, University of Houston, Houston, TX 77204, USA d Pohl Institute of Solid State Physics, Tongji University, Shanghai 200092, PR China
Abstract The skutterudite PrOs4 Sb12 is a Heavy Fermion (HF) superconductor (sc) below Tc ¼ 1:8 K [Phys. Rev. B 65 (2002) 100506(R)]. Specific heat measurements indicate two separate sc transitions. Recent angle dependent magnetothermal conductivity experiments [Phys. Rev. Lett. 90 (2003) 117001] have shown that PrOs4 Sb12 is another of the very few examples of multiphase-superconductivity. Two phases A and B with different node structures have been identified. We propose a model for their anisotropic gap function DðkÞ. While the high-field A-phase has fourfold planar symmetry (tetragonal s + g-wave) with four point nodes along two cubic axis, the low-field B-phase has lower symmetry with only two point nodes. We discuss the field-angle dependent magnetothermal conductivity and propose a microscopic pairing mechanism. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.Rp; 74.25.Fy; 74.70.Tx Keywords: Skutterudite; PrOs4 Sb12 ; Magnetothermal transport
1. Introduction Superconductivity in the Pr-HF compound [1] (c ¼ 313 mJ/mol K2 ) is of great interest because both the order parameter and the pairing mechanism may be unconventional. A phase diagram with two sc phases A (high-field) and B (low-field) with different nodal structure was concluded from magnetothermal conductivity jzz ðHÞ [2] and phenomenological models for the gap functions have been proposed [3,4]. The similarity of the Fermi surface in LaOs4 Sb12 and PrOs4 Sb12 suggests a mechanism for the heavy mass formation which is different from the sf-hybridisation or Kondo model invoked for Ce-compounds. Indeed one has a stable Pr3þ configuration possibly with nonmagnetic C1 -ground state and a C5 first excited state due to the tetrahedral CEF. Since at higher fields a quadrupolar order is induced in the C1 –C5 system it is suggestive that at zero field (off*
Corresponding author. Tel.: +49-35146462234; fax: +4935146463232. E-mail address: [email protected] (P. Thalmeier).
diagonal) quadrupolar fluctuations provide a novel mechanism for sc pair formation in PrOs4 Sb12 . 2. Order parameter models To account for the different nodal structure of A- and B-phase a model for the gap function DðkÞ has been proposed [3]. The cusp-like structure observed in angular dependence of jzz ðh; /Þ indicates the existence of point nodes rather than line nodes. This is reminiscent of the situation in borocarbide superconductors [5]. Assuming singlet pairing one has to invoke hybrid gap functions DðkÞ ¼ Df ðkÞ with form factors given by ðAÞ f ðkÞ ¼ 1 kx4 ky4 ðBÞ f ðkÞ ¼ 1 ky4
ð1Þ
to realise the node structure. The A-phase gap is of tetragonal s + g-wave type while the B-phase gap has only twofold symmetry. Both are threefold degenerate, specific domains (nodes in xy-plane and along y direction) have been chosen. These gap functions are shown in the inset of Fig. 1.
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.162
178
P. Thalmeier et al. / Physica C 408–410 (2004) 177–178 1
Here jn is the normal state p thermal conductivity. The ffiffiffiffi leading field dependence is H but terms H are also present. The angular dependences are shown in Fig. 1. The twofold (lower part, B-phase) and fourfold (upper part, A-phase) oscillations and the cusp-like minima when H sweeps over the node directions are clearly visible. Experimentally the amplitude of twofold and fourfold transition decrease or increase rather suddenly when H becomes bigger than a critical H ðT Þ. This defines the experimental phase boundary between A and B phases. Aside from thermal conductivity it has also been seen in an anomaly of the flux flow resistivity.
ο
IA (θ,φ)
θ = 45 0.8
60
0.6
90
ο
ο
A 0.4 IB (θ,φ)
B 0.2 0 90
45
0 φ [deg.]
45
90
Fig. 1. Calculated angular variation of jzz ðh; /Þ from Eq. (2) exhibiting fourfold (A) and twofold (B) oscillations in / due to the different node structure of DA;B ðkÞ. Their polar plots (inset) show nodes along two (A) or only one (B) cubic axis. Here h and / are polar and azimuthal angle of the field H with respect to [0 0 1]. Possibly nodes along [0 0 1] also exist.
3. Magnetothermal conductivity The conjecture of sc gap nodes in PrOs4 Sb12 has been obtained from the field dependence of jzz ðHÞ which is approximately linear in H at low temperatures [2] for Hk½1 0 0. This nonexponential behaviour is a strong indication of the presence of itinerant gapless quasiparticle excitations in the intervortex region which can carry a heat current perpendicular to the vortex lines. This is due to the Doppler shift (Volovik effect) of quasiparticle (momentum q) energies given by x ¼ v q=D which results from the supercurrent velocity field v around the vortex lines. Further evidence comes from the jzz -dependence on field polar (h) and azimuthal (/) angles when H is swept around the [0 0 1] axis for constant field strength at low temperature. Doppler shift and induced DOS depends on field orientation relative to the fixed nodal positions, therefore thermodynamic and transport properties show typical oscillation as function of h; / leading to maxima and minima for H along antinodal or nodal pffiffiffiffiffiffi ffi directions respectively. In the limit C < T v eH Dð0Þ (C ¼ scatt. rate) one obtains for the thermal conductivity ði ¼ A; BÞ pffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffi 2 v eH 87 v eH Ii ðh; /Þ 1 þ Ii ðh; /Þ p 125 D D 1 C 1 pffiffiffiffiffiffiffi Ii ðh; /Þ v eH i 1 1 1h IA ðh; /Þ ¼ ð1 sin2 h sin2 /Þ2 þ ð1 sin2 h cos2 /Þ2 2 1 1 IB ðh; /Þ ¼ ð1 sin2 h sin2 /Þ2 4 ð2Þ jzz ¼ jA
4. Quadrupole fluctuations and pair potential The C1 –C5 (DCEF ) low lying CEF level scheme does not lead to exchange scattering of conduction electrons below DCEF ’ 6 K. Therefore heavy mass generation cannot be due to the Kondo mechanism and also Cooper pair formation must have a different source than exchange of spin fluctuations commonly invoked for unconventional HF superconductors. Due to the vicinity of the field induced quadrupolar phase, mass renormalisation and pair formation via exchange of quadrupolar fluctuation seems possible. It relies on the aspherical Coulomb scattering of conduction electrons due to the nondiagonal C1 –C5 quadrupole moment. Elimination of local 4f-excitations via Schrieffer–Wolff transformation then leads to an effective quadrupole fluctuation mediated quasiparticle interaction which in RPA is given by (cq ¼ cubic structure factor, j2 ¼ Bessel function, vQ ð0Þ ¼ single ion quadrupole susceptibility for x ¼ 0) KðqÞ ¼ G2 ðqÞvQ ð0Þ½1 cq vQ ð0Þ1 q2z þ ^ q2x þ ^ q2y G2 ðqÞ hj2 ðqrÞiq2 ½^ q2y ^ q2z ^ q2x ^
ð3Þ
The q dependence of the interaction is mainly due to the quadrupolar form factor G2 ðqÞ contrary to spin fluctuation theory where it is a constant. The form factor and hence KðqÞ is small for q along the cubic axes which is in qualitative agreement with the assumed gap node structure.
References [1] [2] [3] [4] [5]
E.D. Bauer et al., Phys. Rev. B 65 (2002) 100506(R). K. Izawa et al., Phys. Rev. Lett. 90 (2003) 117001. K. Maki et al., Europhys. Lett. 64 (2003) 496. J. Goryo, Phys. Rev. B 67 (2003) 184511. K. Izawa et al., Phys. Rev. Lett. 89 (2002) 137006.