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Anisotropic superconducting energy gap studied by the dHvA effect in URu2Si2 and UPd2Al3
Physica B 281&282 (2000) 996}997
Anisotropic superconducting energy gap studied by the dHvA e!ect in URu Si and UPd Al 2 2 2 3 Y. Inada!,*, H. Ohkuni!, Y. Haga", Y. Tokiwa!, K. Sakurai!, T. Honma", E. Yamamoto", Y. O1 nuki!," !Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan "Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
Abstract One branch has been detected in the de Haas}van Alphen (dHvA) experiments in both the normal and superconducting mixed states of URu Si and UPd Al with an anisotropic superconducting energy gap. The corresponding Dingle 2 2 2 3 temperatures increase and the cyclotron e!ective masses decrease in the mixed states. An anisotropic energy gap with a line node has been discussed from the angular dependence of the dHvA amplitude in the mixed state. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: de Haas}van Alphen e!ect; Superconducting mixed state; UPd Al ; URu Si 2 3 2 2
Anisotropic non s-wave superconductivity is one of the interesting features in the strongly correlated electron systems. URu Si and UPd Al are considered to be 2 2 2 3 non-s-wave superconductors with an anisotropic energy gap from the NMR and speci"c heat measurements [1}4]. Recently, we have succeeded in observing clearly the de Haas}van Alphen (dHvA) oscillation in both the normal and superconducting mixed states of URu Si 2 2 [5] and UPd Al [6]. We have tried to determine the 2 3 position of the line node of the superconducting energy gap on the actual Fermi surface of these compounds of these compounds via the dHvA experiment. If the dHvA orbit, which circulates along a maximum (or minimum) cross-section, becomes a cross-section with the line node on the Fermi surface, the dHvA oscillation is expected to be observed as in the normal state. On the other hand, the density of states of Type II superconductors possesses a "nite value at the Fermi energy in magnetic "elds, forming a `gapless statea around the maximum cross-section [5,7]. It is not easy to distinguish the
* Corresponding author. Tel.: #81-6-6850-5371; fax: #816-6850-5372. E-mail address: [email protected] (Y. Inada)
line node from the latter `gapless statea based on the Cooper-pair breaking in magnetic "elds. It is thus necessary to deviate the dHvA orbit from the cross-section with the line node by tilting the "eld direction. In the present dHvA resutls of URu Si and 2 2 UPd Al , the dHvA frequency, which is proportional to 2 3 the maximum cross-sectional area, is found to be unchanged in both the normal and mixed states. The cyclotron mass decreases gradually and the Dingle temperature increases with decreasing the "eld below H . #2 First, we show the dHvA result of URu Si with the 2 2 tetragonal structure. The dHvA oscillation due to branch a is observed in the mixed state down to about 20 kOe, where the upper critical "eld is 30 kOe for HDD[0 0 1] and 130 kOe for [1 0 0]. Branch a corresponds to an almost spherical Fermi surface whose dHvA frequency for HDD[0 0 1] is 1.05]107 Oe. The cyclotron e!ective mass is 13 m for [0 0 1]. We show in Fig. 1 the angular depend0 ences of the dHvA amplitudes in both the normal and mixed states, which are shown by circles and squares, respectively. The e!ective "eld in the normal state is 46.9 kOe, while it is 23.7 kOe ("0.8H ) in the mixed #2 state for the magnetic "eld along [0 0 1]. The steep amplitude reduction at several "eld angles in the normal state is due to the spin-splitting zero. This is also re#ected even
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 8 1 1 - X
Y. Inada et al. / Physica B 281&282 (2000) 996}997
997
Fig. 1. Angular dependence of the dHvA amplitude in both the normal and mixed states for branch a in URu Si . 2 2
factor in the mixed state is unchanged against the "eld angle. This result implies that a line node does not exist on this Fermi surface and/or it is di$cult to distinguish experimentally the anisotropic line node with the `gapless statea based on the pair-breaking by applying the magnetic "eld. Next, we show the similar dHvA result of UPd Al 2 3 with the hexagonal structure. The angular dependence of the dHvA amplitude for branch a is shown in Fig. 2. Branch a corresponds to an ellipsoidal Fermi surface. Its dHvA frequencies are 2.6]106 Oe for HDD[0 0 0 1] and 6.9]106 Oe for [1 0 11 0], respectively. Cyclotron masses are 5.7m for [0 0 0 1] and 13.7m for [1 0 11 0], respective0 0 ly. The dHvA amplitudes were obtained in an e!ective "eld of 58.8 kOe in the normal state and 25.7 kOe ("0.7H ) in the mixed state. As shown in Fig. 2, the #2 dHvA oscillation is reduced in amplitude for the "eld direction tilted from [0 0 0 1] to [1 0 11 0]. It is caused due to an increase of mass and the curvature factor. A dotted line in Fig. 2 is the similar calculated curve at 25.7 kOe as in URu Si . The dHvA amplitudes in the mixed state, 2 2 shown by squares in Fig. 2, are reduced by half compared to the thick solid line. A dotted line, which is normalized at [0 0 0 1], is in good agreement with the dHvA data in the mixed state. Namely, the reduction factor in the mixed state is unchanged against the "eld angle. This result also implies that a line node in the anisotropic energy gap does not exist on this Fermi surface and/or it is di$cult to distinguish experimentally the line node from the `gapless statea based on the pair-breaking by applying the magnetic "eld.
Fig. 2. Angular dependence of the dHvA amplitude in both the normal and mixed states for branch a in UPd Al . 2 3
Acknowledgements
in the mixed state. A thick solid line in Fig. 1 is a curve obtained at 23.7 kOe under the assumption that values of the cyclotron mass and the Dingle temperature in the normal state are unchanged in the mixed state. The dHvA amplitudes in the mixed state, shown by squares, are reduced by one-tenth compared to the thick solid line. This reduction is due to a decrease of the lifetime of the quasiparticles in the mixed state. If calculated result for HDD[0 0 1] is "tted to the dHvA experimental amplitude for HDD[0 0 1] in the mixed state, the solid line is changed into a dotted line. The dotted line is close to the dHvA data in the mixed state. Namely, the reduction
This work was "nancially supported by COE Research (10CE 2004) of the Ministry of Education, Science, Sports and Culture.
References [1] [2] [3] [4] [5] [6] [7]
J.P. Brison et al., Physica B 199 & 200 (1994) 70. K. Matsuda et al., J. Phys. Soc. Japan 65 (1996) 679. H. Tou et al., J. Phys. Soc. Japan 64 (1995) 725. C. Geibel et al., Z. Phys. B 84 (1996) 1. H. Ohkuni et al., Philos. Mag. B 79 (1999) 1045. Y. Haga et al., J. Phys. Soc. Japan 68 (1999) 342. M. Hedo et al., Philos. Mag. B. 77 (1998) 975.
Anisotropic superconducting energy gap studied by the dHvA e!ect in URu Si and UPd Al 2 2 2 3 Y. Inada!,*, H. Ohkuni!, Y. Haga", Y. Tokiwa!, K. Sakurai!, T. Honma", E. Yamamoto", Y. O1 nuki!," !Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan "Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
Abstract One branch has been detected in the de Haas}van Alphen (dHvA) experiments in both the normal and superconducting mixed states of URu Si and UPd Al with an anisotropic superconducting energy gap. The corresponding Dingle 2 2 2 3 temperatures increase and the cyclotron e!ective masses decrease in the mixed states. An anisotropic energy gap with a line node has been discussed from the angular dependence of the dHvA amplitude in the mixed state. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: de Haas}van Alphen e!ect; Superconducting mixed state; UPd Al ; URu Si 2 3 2 2
Anisotropic non s-wave superconductivity is one of the interesting features in the strongly correlated electron systems. URu Si and UPd Al are considered to be 2 2 2 3 non-s-wave superconductors with an anisotropic energy gap from the NMR and speci"c heat measurements [1}4]. Recently, we have succeeded in observing clearly the de Haas}van Alphen (dHvA) oscillation in both the normal and superconducting mixed states of URu Si 2 2 [5] and UPd Al [6]. We have tried to determine the 2 3 position of the line node of the superconducting energy gap on the actual Fermi surface of these compounds of these compounds via the dHvA experiment. If the dHvA orbit, which circulates along a maximum (or minimum) cross-section, becomes a cross-section with the line node on the Fermi surface, the dHvA oscillation is expected to be observed as in the normal state. On the other hand, the density of states of Type II superconductors possesses a "nite value at the Fermi energy in magnetic "elds, forming a `gapless statea around the maximum cross-section [5,7]. It is not easy to distinguish the
* Corresponding author. Tel.: #81-6-6850-5371; fax: #816-6850-5372. E-mail address: [email protected] (Y. Inada)
line node from the latter `gapless statea based on the Cooper-pair breaking in magnetic "elds. It is thus necessary to deviate the dHvA orbit from the cross-section with the line node by tilting the "eld direction. In the present dHvA resutls of URu Si and 2 2 UPd Al , the dHvA frequency, which is proportional to 2 3 the maximum cross-sectional area, is found to be unchanged in both the normal and mixed states. The cyclotron mass decreases gradually and the Dingle temperature increases with decreasing the "eld below H . #2 First, we show the dHvA result of URu Si with the 2 2 tetragonal structure. The dHvA oscillation due to branch a is observed in the mixed state down to about 20 kOe, where the upper critical "eld is 30 kOe for HDD[0 0 1] and 130 kOe for [1 0 0]. Branch a corresponds to an almost spherical Fermi surface whose dHvA frequency for HDD[0 0 1] is 1.05]107 Oe. The cyclotron e!ective mass is 13 m for [0 0 1]. We show in Fig. 1 the angular depend0 ences of the dHvA amplitudes in both the normal and mixed states, which are shown by circles and squares, respectively. The e!ective "eld in the normal state is 46.9 kOe, while it is 23.7 kOe ("0.8H ) in the mixed #2 state for the magnetic "eld along [0 0 1]. The steep amplitude reduction at several "eld angles in the normal state is due to the spin-splitting zero. This is also re#ected even
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 8 1 1 - X
Y. Inada et al. / Physica B 281&282 (2000) 996}997
997
Fig. 1. Angular dependence of the dHvA amplitude in both the normal and mixed states for branch a in URu Si . 2 2
factor in the mixed state is unchanged against the "eld angle. This result implies that a line node does not exist on this Fermi surface and/or it is di$cult to distinguish experimentally the anisotropic line node with the `gapless statea based on the pair-breaking by applying the magnetic "eld. Next, we show the similar dHvA result of UPd Al 2 3 with the hexagonal structure. The angular dependence of the dHvA amplitude for branch a is shown in Fig. 2. Branch a corresponds to an ellipsoidal Fermi surface. Its dHvA frequencies are 2.6]106 Oe for HDD[0 0 0 1] and 6.9]106 Oe for [1 0 11 0], respectively. Cyclotron masses are 5.7m for [0 0 0 1] and 13.7m for [1 0 11 0], respective0 0 ly. The dHvA amplitudes were obtained in an e!ective "eld of 58.8 kOe in the normal state and 25.7 kOe ("0.7H ) in the mixed state. As shown in Fig. 2, the #2 dHvA oscillation is reduced in amplitude for the "eld direction tilted from [0 0 0 1] to [1 0 11 0]. It is caused due to an increase of mass and the curvature factor. A dotted line in Fig. 2 is the similar calculated curve at 25.7 kOe as in URu Si . The dHvA amplitudes in the mixed state, 2 2 shown by squares in Fig. 2, are reduced by half compared to the thick solid line. A dotted line, which is normalized at [0 0 0 1], is in good agreement with the dHvA data in the mixed state. Namely, the reduction factor in the mixed state is unchanged against the "eld angle. This result also implies that a line node in the anisotropic energy gap does not exist on this Fermi surface and/or it is di$cult to distinguish experimentally the line node from the `gapless statea based on the pair-breaking by applying the magnetic "eld.
Fig. 2. Angular dependence of the dHvA amplitude in both the normal and mixed states for branch a in UPd Al . 2 3
Acknowledgements
in the mixed state. A thick solid line in Fig. 1 is a curve obtained at 23.7 kOe under the assumption that values of the cyclotron mass and the Dingle temperature in the normal state are unchanged in the mixed state. The dHvA amplitudes in the mixed state, shown by squares, are reduced by one-tenth compared to the thick solid line. This reduction is due to a decrease of the lifetime of the quasiparticles in the mixed state. If calculated result for HDD[0 0 1] is "tted to the dHvA experimental amplitude for HDD[0 0 1] in the mixed state, the solid line is changed into a dotted line. The dotted line is close to the dHvA data in the mixed state. Namely, the reduction
This work was "nancially supported by COE Research (10CE 2004) of the Ministry of Education, Science, Sports and Culture.
References [1] [2] [3] [4] [5] [6] [7]
J.P. Brison et al., Physica B 199 & 200 (1994) 70. K. Matsuda et al., J. Phys. Soc. Japan 65 (1996) 679. H. Tou et al., J. Phys. Soc. Japan 64 (1995) 725. C. Geibel et al., Z. Phys. B 84 (1996) 1. H. Ohkuni et al., Philos. Mag. B 79 (1999) 1045. Y. Haga et al., J. Phys. Soc. Japan 68 (1999) 342. M. Hedo et al., Philos. Mag. B. 77 (1998) 975.