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Study of dHvA effect in the superconducting mixed state of CeRu2
ELSEVIER
Physica B 237-238 (1997) 215-217
Study of dHvA effect in the superconducting mixed state of CeRu2 M. H e d o a'*, Y. I n a d a a, T. I s h i d a a, E. Y a m a m o t o b, Y. H a g a b, Y. O n u k i a'b, M . H i g u c h i b, A. H a s e g a w a c aDepartment of Physics, Faculty of Science, Osaka University, Toyonaka 560, Japan bAdvanced Science Research Center, Japan Atomic Energy Research Institute, TokaL Ibaraki 319-11, Japan CFaculty of Science, Niigata University, Niigata 950-21, Japan
Abstract
We have succeeded in growing high-quality single crystals of CeRu2 and have observed dHvA oscillations both in the normal and superconducting mixed states. The detected dHvA frequencies, i.e. extremal cross-sectional areas of Fermi surfaces have been well explained by band structure calculations. The dHvA frequency between the normal and superconducting states does not change in magnitude, while the Dingle temperature increases in the mixed state. The cyclotron mass is found to become a little smaller in the mixed state than in the normal state. Keywords: CeRu2; Superconductivity; de Haas-van Alphen effect; Fermi surface
CeRu2 with the cubic Laves-phase structure has attracted attention for its superconducting properties [1]. Recently, we have observed the de Haasvan Alphen (dHvA) oscillation both in the normal and superconducting mixed states, and the detected branches have been well explained by the 4f itinerant band model [2]. A long time ago, the dHvA oscillation in the mixed state was observed by Graebner and Robbins in the layered compound NbSe2 by means of a magnetothermal measurement [3]. (3nuki et al. reconfirmed this oscillation by the standard fieldmodulation AC susceptibility [4]. So far, it has been observed for several compounds such as V3Si, Nb3Sn and YNi2B2C [5,6]. The dHvA oscillation in CeRu2 is distinct compared to those of the other compounds because the assignment of the detected branches in CeRu2 is clear from the result of band calculations. Two branches * Corresponding author.
denoted by 61 and/31,2,3 are observed both in the normal state and the superconducting mixed state. We have continued to carry out the dHvA experiments to understand the damping mechanism of the dHvA amplitude and the reason why the dHvA oscillation is observed in the superconducting mixed state. Fig. 1 shows a typical dHvA oscillation under three different conditions for the field along the (111) direction. The upper part in Fig. 1 shows the dHvA signal in the sample #4 at 0.45 K. Here, the sample #4 possesses the residual resistivity ratio (RRR = PRT/PO) of 60. On the other hand, the sample #7 is a high-quality sample with RRR of 270. The middle and bottom figures show those for the sample #7 at 0.47 K and at 35 mK, respectively. As for the sample #4, two dHvA branches, denoted by el,2,3 and 61, are observed both in the normal and mixed states. Branch el,2,3 corresponds to the band 22-electron pocket Fermi surface, while t~ 1 a n arm of the band 20-hole Fermi surface. Branch et,2,3 is
0921-4526/97/$17.00 © 1997 Elsevier ScienceB.V. All rights reserved PH S092 1-4526(97)00105-1
M. Hedo et al./Physica B 237 238 (1997) 215~17
216
Normal Mixed state 100 50
20 kOe i
RRR=60 0.45 K
0.47 K
I
2
I
I
3 4 1/I--I(xl02 kOe-~)
I
--
5
Fig. 1. T y p i c a l d H v A oscillation f o r the s a m p l e s #4 a n d #7 in CeRu2.
observed up to a low field of 20 kOe. On the other hand, branch 61 is barely observed around 40kOe. This branch is, however, detected clearly even in the mixed state for a better sample #7. Moreover, it reaches a lower field of 20 kOe at a lower temperature of 35 mK. We have done analyses of the dHvA oscillation to clarify three properties such as the dHvA frequency, the cyclotron mass and the Dingle temperature. First, the dHvA frequency, which is proportional to the cross-sectional area of the Fermi surface SF, is found to be unchanged both in the normal and mixed states. Next, we have determined the cyclotron mass me from the temperature dependence of the dHvA amplitude. The mass is found to be a little smaller in the mixed state than in the normal state. The mass in the normal (mixed) state is 0.61m0 (0.45mo) for el,2,3 and 1.56m0 (1.00m0) for 61. Lastly, we have determined the Dingle temperature To (=h/2rCkBZ), where ~ is the scattering lifetime of the conduction electron. The Dingle temperature in the normal (mixed) state is 1.2K (2.8 K) for t;1,2,3 and 1.0K (2.8 K) for 61 for the sample #4. As for the sample #7, it is 0.45K (2.1K) for el,2,3 and
0.50K (2.4K) for 61. From these values, we have found that the following simple relation holds between the Dingle temperature in the mixed state, Tt)(mixed) and that in the normal state, TD(normal); TD(mixed)=TD(normal)+ATD, where the additional Dingle temperature, ATD is 1.6K for ei,2,3 and 1.8 K for 61. Using the following three formulae of SF = rtk 2, hkF=m*vF and l=vrv, where kF is half of the caliper length of the Fermi surface and vv is Fermi velocity, we can determine the mean free path l. Its value in the normal (mixed) state is 920 A(530 A) for el,2,3 and 1000 A(550 A) for 61 in sample #4, while 2400 A(710 A) for ek2,3 and 2000 A(650 A) for 61 in sample #7. The dHvA oscillation in the normal state is continuously observed even in the mixed state, although the amplitude is reduced in general. Here we discuss the damping mechanism of the dHvA amplitude and the reason why the dHvA oscillation is observed in the mixed state. The dHvA oscillation is caused when the Landau levels cross the Fermi energy when increasing the field. There might exist two scenarios to explain the dHvA oscillation in the mixed state. The dHvA oscillation is caused either by quasi-particles or the Cooper-pair electrons. It is not certain whether each electron in the Cooper-pair could be quantized in energy in fields, holding the Cooper-pair with (k, T) and ( - k , ~) at OK. On the other hand, the unpairing electrons due to fields, namely quasi-particles can be quantized into Landau levels. Maki [7], and Wasserman and Springford [8] have theoretically discussed the dHvA oscillation in the mixed state on the basis of the quasi-particles. They claim that the dHvA frequency is unchanged for the normal state, but the amplitude is reduced by an additional quasi-particle scattering rate depending on the field and temperature. The present experiments for CeRu2 indicate that the additional scattering rate or the Dingle temperature does not depend on the field, namely constant in the field range of H/Hc2= 0.85 to 0.35. Much lower fields are necessary to clarify the field dependence of the Dingle temperature. Here we note the field dependence of the carrier concentration of de-pairing electrons. The density of states D(E,H = 0) in the superconducting state has an energy gap A at the Fermi energy EF. When the field
M. Hedo et al./ Physica B 237-238 (1997) 215-217
is applied, the density of states at EF becomes a finite value, almost linearly proportional to the field. The type-II superconductor becomes thus gapless in fields where we have observed the dHvA oscillation. This is most likely the reason why the dHvA oscillation is observed in the mixed state. Lastly, we discuss the damping mechanism. There exist three reduction factors in the dHvA oscillation. One is the field dependence of the carrier concentration of quasi-particles mentioned above. The second is an inhomogeneous field due to vortices. The last is the electron scattering at the boundary between the normal region and the superconducting one, called Andreev reflection. These are closely related to the additional Dingle temperature in the mixed state, as mentioned above. Qualitative analyses are left for a future study. This work was supported by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture.
217
References [1] R. Modler, P. Gegenwart, M. Lang, M. Deppe, M. Weiden, T. Lfihmann, C. Geibel, F. Steglich, C. Paulsen, J.L. Tholence, N. Sato, T. Komatsubara, Y. Onuki, M. Tachiki and S. Takahashi, Phys. Rev. Lett. 76 (1996) 1292. [2] M. Hedo, Y. lnada, T. Ishida, E. Yamamoto, Y. Haga, Y. Onuki, M. Higuchi and A, Hasegawa, J. Phys. Soc. Japan 64 (1995) 4535. [3] J.E. Graebner and M. Robbins, Phys. Rev. Lett. 36 (1976) 422. [4] Y. Onuki, I. Umehara, T. Ebihara, N. Nagai and K. Takita, J. Phys. Soc. Japan 61 (1992) 692. [5] R. Corcoran, N. Harrison, C.J. Haworth, S.M. Hayden, P. Meeson, M. Springford and P.J. van der Wel, Physica B 206&207 (1995) 534. [6] T. Terashima, H. Takeya, S. Uji, K. Kadowaki and H. Aoki, Solid State Commun. 96 (1995) 459. [7] K. Maki, Phys. Rev. B 44 (1991) 2861. [8] A. Wasserman and M. Springford, Physica B 194-196 (1994) 1801. [9] K. Maki, Private Communication.
Physica B 237-238 (1997) 215-217
Study of dHvA effect in the superconducting mixed state of CeRu2 M. H e d o a'*, Y. I n a d a a, T. I s h i d a a, E. Y a m a m o t o b, Y. H a g a b, Y. O n u k i a'b, M . H i g u c h i b, A. H a s e g a w a c aDepartment of Physics, Faculty of Science, Osaka University, Toyonaka 560, Japan bAdvanced Science Research Center, Japan Atomic Energy Research Institute, TokaL Ibaraki 319-11, Japan CFaculty of Science, Niigata University, Niigata 950-21, Japan
Abstract
We have succeeded in growing high-quality single crystals of CeRu2 and have observed dHvA oscillations both in the normal and superconducting mixed states. The detected dHvA frequencies, i.e. extremal cross-sectional areas of Fermi surfaces have been well explained by band structure calculations. The dHvA frequency between the normal and superconducting states does not change in magnitude, while the Dingle temperature increases in the mixed state. The cyclotron mass is found to become a little smaller in the mixed state than in the normal state. Keywords: CeRu2; Superconductivity; de Haas-van Alphen effect; Fermi surface
CeRu2 with the cubic Laves-phase structure has attracted attention for its superconducting properties [1]. Recently, we have observed the de Haasvan Alphen (dHvA) oscillation both in the normal and superconducting mixed states, and the detected branches have been well explained by the 4f itinerant band model [2]. A long time ago, the dHvA oscillation in the mixed state was observed by Graebner and Robbins in the layered compound NbSe2 by means of a magnetothermal measurement [3]. (3nuki et al. reconfirmed this oscillation by the standard fieldmodulation AC susceptibility [4]. So far, it has been observed for several compounds such as V3Si, Nb3Sn and YNi2B2C [5,6]. The dHvA oscillation in CeRu2 is distinct compared to those of the other compounds because the assignment of the detected branches in CeRu2 is clear from the result of band calculations. Two branches * Corresponding author.
denoted by 61 and/31,2,3 are observed both in the normal state and the superconducting mixed state. We have continued to carry out the dHvA experiments to understand the damping mechanism of the dHvA amplitude and the reason why the dHvA oscillation is observed in the superconducting mixed state. Fig. 1 shows a typical dHvA oscillation under three different conditions for the field along the (111) direction. The upper part in Fig. 1 shows the dHvA signal in the sample #4 at 0.45 K. Here, the sample #4 possesses the residual resistivity ratio (RRR = PRT/PO) of 60. On the other hand, the sample #7 is a high-quality sample with RRR of 270. The middle and bottom figures show those for the sample #7 at 0.47 K and at 35 mK, respectively. As for the sample #4, two dHvA branches, denoted by el,2,3 and 61, are observed both in the normal and mixed states. Branch el,2,3 corresponds to the band 22-electron pocket Fermi surface, while t~ 1 a n arm of the band 20-hole Fermi surface. Branch et,2,3 is
0921-4526/97/$17.00 © 1997 Elsevier ScienceB.V. All rights reserved PH S092 1-4526(97)00105-1
M. Hedo et al./Physica B 237 238 (1997) 215~17
216
Normal Mixed state 100 50
20 kOe i
RRR=60 0.45 K
0.47 K
I
2
I
I
3 4 1/I--I(xl02 kOe-~)
I
--
5
Fig. 1. T y p i c a l d H v A oscillation f o r the s a m p l e s #4 a n d #7 in CeRu2.
observed up to a low field of 20 kOe. On the other hand, branch 61 is barely observed around 40kOe. This branch is, however, detected clearly even in the mixed state for a better sample #7. Moreover, it reaches a lower field of 20 kOe at a lower temperature of 35 mK. We have done analyses of the dHvA oscillation to clarify three properties such as the dHvA frequency, the cyclotron mass and the Dingle temperature. First, the dHvA frequency, which is proportional to the cross-sectional area of the Fermi surface SF, is found to be unchanged both in the normal and mixed states. Next, we have determined the cyclotron mass me from the temperature dependence of the dHvA amplitude. The mass is found to be a little smaller in the mixed state than in the normal state. The mass in the normal (mixed) state is 0.61m0 (0.45mo) for el,2,3 and 1.56m0 (1.00m0) for 61. Lastly, we have determined the Dingle temperature To (=h/2rCkBZ), where ~ is the scattering lifetime of the conduction electron. The Dingle temperature in the normal (mixed) state is 1.2K (2.8 K) for t;1,2,3 and 1.0K (2.8 K) for 61 for the sample #4. As for the sample #7, it is 0.45K (2.1K) for el,2,3 and
0.50K (2.4K) for 61. From these values, we have found that the following simple relation holds between the Dingle temperature in the mixed state, Tt)(mixed) and that in the normal state, TD(normal); TD(mixed)=TD(normal)+ATD, where the additional Dingle temperature, ATD is 1.6K for ei,2,3 and 1.8 K for 61. Using the following three formulae of SF = rtk 2, hkF=m*vF and l=vrv, where kF is half of the caliper length of the Fermi surface and vv is Fermi velocity, we can determine the mean free path l. Its value in the normal (mixed) state is 920 A(530 A) for el,2,3 and 1000 A(550 A) for 61 in sample #4, while 2400 A(710 A) for ek2,3 and 2000 A(650 A) for 61 in sample #7. The dHvA oscillation in the normal state is continuously observed even in the mixed state, although the amplitude is reduced in general. Here we discuss the damping mechanism of the dHvA amplitude and the reason why the dHvA oscillation is observed in the mixed state. The dHvA oscillation is caused when the Landau levels cross the Fermi energy when increasing the field. There might exist two scenarios to explain the dHvA oscillation in the mixed state. The dHvA oscillation is caused either by quasi-particles or the Cooper-pair electrons. It is not certain whether each electron in the Cooper-pair could be quantized in energy in fields, holding the Cooper-pair with (k, T) and ( - k , ~) at OK. On the other hand, the unpairing electrons due to fields, namely quasi-particles can be quantized into Landau levels. Maki [7], and Wasserman and Springford [8] have theoretically discussed the dHvA oscillation in the mixed state on the basis of the quasi-particles. They claim that the dHvA frequency is unchanged for the normal state, but the amplitude is reduced by an additional quasi-particle scattering rate depending on the field and temperature. The present experiments for CeRu2 indicate that the additional scattering rate or the Dingle temperature does not depend on the field, namely constant in the field range of H/Hc2= 0.85 to 0.35. Much lower fields are necessary to clarify the field dependence of the Dingle temperature. Here we note the field dependence of the carrier concentration of de-pairing electrons. The density of states D(E,H = 0) in the superconducting state has an energy gap A at the Fermi energy EF. When the field
M. Hedo et al./ Physica B 237-238 (1997) 215-217
is applied, the density of states at EF becomes a finite value, almost linearly proportional to the field. The type-II superconductor becomes thus gapless in fields where we have observed the dHvA oscillation. This is most likely the reason why the dHvA oscillation is observed in the mixed state. Lastly, we discuss the damping mechanism. There exist three reduction factors in the dHvA oscillation. One is the field dependence of the carrier concentration of quasi-particles mentioned above. The second is an inhomogeneous field due to vortices. The last is the electron scattering at the boundary between the normal region and the superconducting one, called Andreev reflection. These are closely related to the additional Dingle temperature in the mixed state, as mentioned above. Qualitative analyses are left for a future study. This work was supported by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture.
217
References [1] R. Modler, P. Gegenwart, M. Lang, M. Deppe, M. Weiden, T. Lfihmann, C. Geibel, F. Steglich, C. Paulsen, J.L. Tholence, N. Sato, T. Komatsubara, Y. Onuki, M. Tachiki and S. Takahashi, Phys. Rev. Lett. 76 (1996) 1292. [2] M. Hedo, Y. lnada, T. Ishida, E. Yamamoto, Y. Haga, Y. Onuki, M. Higuchi and A, Hasegawa, J. Phys. Soc. Japan 64 (1995) 4535. [3] J.E. Graebner and M. Robbins, Phys. Rev. Lett. 36 (1976) 422. [4] Y. Onuki, I. Umehara, T. Ebihara, N. Nagai and K. Takita, J. Phys. Soc. Japan 61 (1992) 692. [5] R. Corcoran, N. Harrison, C.J. Haworth, S.M. Hayden, P. Meeson, M. Springford and P.J. van der Wel, Physica B 206&207 (1995) 534. [6] T. Terashima, H. Takeya, S. Uji, K. Kadowaki and H. Aoki, Solid State Commun. 96 (1995) 459. [7] K. Maki, Phys. Rev. B 44 (1991) 2861. [8] A. Wasserman and M. Springford, Physica B 194-196 (1994) 1801. [9] K. Maki, Private Communication.