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Study of pseudogap phenomena by STM and other probes
Journal of Physics and Chemistry of Solids 62 (2001) 65±68
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Study of pseudogap phenomena by STM and other probes A. Matsuda a,b,*, S. Sugita a, T. Fujii b, T. Watanabe a a NTT Basic Research Laboratories, 3-1 Morionsato Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan Department of Applied Physics, Faculty of Science, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
b
Abstract The temperature and doping dependence of the tunneling spectrum of Bi2.1Sr1.9CaCu2Ox single crystals was obtained by using a low-temperature scanning tunneling microscope. Above Tc, for x # 8.27, the tunneling density of states shows a clear gap-like feature with a larger gap value than the superconducting one, while for x . 8.27, it shows the feature expected from the conventional superconducting ¯uctuation (SCF). We determined the SCF component in the static susceptibility (x ). By subtracting the effect of the SCF from x , we con®rmed that the two pseudogap phase boundaries determined by the tunneling and x coincide with each other. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Superconductors; C. Scanning tunneling microscopy; D. Electrical properties; D. Magnetic properties
Recently, the pseudogap phenomenon has been recognized as one of the fundamental features characterizing high-Tc superconductor in the underdoped region. However, the origin of the pseudogap and its relation to the mechanism of high-Tc superconductivity remains unknown. Many experiments have con®rmed the existence of the pseudogap [1±8]. Among spectroscopic probes, electron tunneling has proven to be a powerful tool, because it directly provides the electronic density of states (DOS) both above and below the Fermi level. Renner et al. [9] and present authors [10] reported the observation of the pseudogap using a scanning tunneling microscope (STM). However, the experimentally deduced characteristics of the pseudogap rather strongly depend on the experimental methods, which might prevent us from complete understanding of the pseudogap phenomenon. For example, static susceptibility measurements [11,12] place the pseudogap phase boundary in a higher temperature range than those obtained by other methods. Besides this kind of fundamental discrepancy, there might be others that should be attributed to experimental problems. In this report, we ®rst reconcile our previous experiment [10] and provide new set of doping and temperature dependence data for the Bi-2212 DOS, in which the problem has been almost eliminated. Then, we compare our results with the static susceptibility (x ) measurements [12]. Through an * Corresponding author. Tel.: 181-46-240-3525; fax: 181-46240-4722. E-mail address: [email protected] (A. Matsuda).
analysis of the anisotropy of x , we were able to derive the diamagnetic component of the superconducting ¯uctuation (SCF), which exists for all doping levels and shows identical temperature dependence. Considering the effect of the SCF, we show that the pseudogap phase boundaries obtained by the STM and x measurements coincide with each other. This, in turn, provides a view that the SCF and the pseudogap are independent phenomena. In the tunneling and susceptibility measurements, we used Bi2.1Sr1.9CaCu2Ox (Bi-2212) single crystals with 8:22 , x , 8:30: Single crystals were grown using a travelling solvent ¯oating zone (TSFZ) method. The oxygen content x for 8:22 , x , 8:28 was adjusted by using an improved annealing method, where the equilibrium oxygen pressures are maintained down to suf®ciently low temperature [13]. A heavily overdoped sample with Tc < 60 K was made by high-O2 pressure (400 atm) annealing at 5008C for 50 h, using a hot isostatic pressing (HIP) furnace. Its oxygen content was estimated to about 8.30. (The relation between x and Tc is shown in Fig. 5.) For the tunneling experiments, we used a newly designed low-temperature UHV STM, which is capable of in situ sample and tip exchange. In our previous experiment [10], we cleaved a crystal at room temperature. We found that this causes oxygen depletion from the topmost surface, and hence a doping level shift. To avoid this problem, in the present experiment, we cleaved samples at a temperature below 84 K in an UHV condition. Then, the samples were transferred to the STM head, which had been precooled to
0022-3697/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(00)00102-5
66
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
about 80 K. Immediately after the transfer, the head was cooled to the 10 K range within several hours. We believe this procedure preserves the oxygen content of the topmost surface. The tip was mechanically cut Pr±Ir wire. The lowtemperature measurements were mostly done with the pressure in the mid 10 29 Torr range. We could acquire a good Bi±O lattice image with the conditions described above. I±V characteristics were obtained by ramping the bias voltage while the feedback loop was cut. Then the DOS was calculated by numerically differentiating I±V characteristics. The gap value showed more or less a spatial dependence in a reproducible manner, which we now consider as an inherent feature of a Bi-2212 cleaved surface. The spatial dependence was stronger when the doping level was reduced. The details will be published elsewhere. To avoid uncertainty from the spatial nonuniformity, we took I±V characteristics where the distribution is small and averaged many I±Vs taken along a line that covers several periods of the distribution. Fig. 1 shows examples of the temperature dependence of the tunneling DOS for: (a) an overdoped sample (x 8.28); and (b) a slightly underdoped (x 8.24) one. Unlike in our previous study [10], the overdoped sample did not show any strong evidence of pseudogap opening. However, for x # 8.26, a clear pseudogap feature appeared (Fig. 1 (b)). As in the previous experiments [9,10], the gapped DOS above Tc lost its sharp peak structure at gap
Fig. 1. Temperature dependence of the tunneling DOS for: (a) x 8.28; (b) x 8.24.
edges and showed a larger gap value compared to the superconducting one. The effect of the low-temperature cleaving seems to simply be a shift of the doping level to a higher one. Fig. 2 shows the doping dependence of the tunneling DOS taken at slightly above Tc. The DOS at these temperatures should be predominantly governed by a band structure. Two characteristic structures can be seen in this ®gure. In the heavily overdoped sample with x 8.3, the DOS shows a clear peak at the Fermi level. Then the peak shifts toward lower energy and rapidly broadens with decreasing the doping level. We identify this structure as a van Hove singularity in the two-dimensional Fermi surface. Details will be published elsewhere. For the samples with x # 8.27, the pseudogap develops as a dip structure at the Fermi level. The dip structure is thought not to have a band origin, because of its strong doping and temperature dependence. Fig. 3 shows the temperature and doping dependence of the DOS peak voltage, which coincides to a superconducting gap value in the lowest temperature range. Here, we only used a positive peak, which can be clearly de®ned even above Tc in the pseudogap state. It can be seen that the gap of the x 8.3 sample follows a BCS-like temperature dependence, and no pseudogap is observed. This tendency continues down to x 8.27. Then a pseudogap suddenly develops for x # 8.26, accompanied by a steep increase in the peak voltage. The ®gure shows that the pseudogap value is larger than the superconducting one and the superconducting gap structure is rather rapidly replaced by the pseudogap structure above Tc. The analysis in our previous work [10] suggests that the pseudogap value is larger even after the effects of thermal smearing and inelastic scattering are incorporated in the present case. The dip structure gradually disappears with increasing temperature and becomes indistinguishable from the background structure at a certain p temperature, which we de®ne as Ttun ; the pseudogap
Fig. 2. Doping dependence of the tunneling DOS in the normal state.
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
67
Fig. 3. Temperature and doping dependence of the DOS peak voltage.
opening temperature. When x $ 8.27, the gap-like structure was observed above Tc in a limited temperature range. We consider that this is evident of the conventional SCF effect, which can be distinguished from the pseudogap because the gap value does not largely exceed the superconducting one. p In our previous studies [10,14], we compared Ttun ; with various T p s obtained by other probes, such as the transport property and susceptibility x . Recently, we measured the temperature and doping dependence of the anisotropic x of Bi-2212 in detail [12] and obtained the corresponding T p Txp : Here, by shedding a new light on the role of SCF, we reconsider the comparison using the new STM data. Fig. 4(a) shows the temperature dependence of the ab-axis susceptibility x ab for various doping levels. c-axis x also showed the similar temperature dependence. The Txp is de®ned as the temperature at which x starts to deviate from its high-temperature behavior. In our study [12], the anisotropy was analyzed using x ab-x c plots with temperature as an implicit parameter. The result revealed that, for all doping levels, anisotropy could be interpreted in terms of a slightly doping dependent g-factor of the Cu 3d electron. However, this interpretation breaks down in a certain temperature range above Tc, which we ascribe to the SCF effect. Assuming that the diamagnetic component of the SCF (dx dia) only contributes to x c, and the anisotropy of the spin component of the SCF can again be determined by the g-factor, we can estimate dx dia as xc 2 gc =gab 2 xab 2 x0 ; where (gc /gab) 2 and x 0 are the slope and intercept of the linear part of the x ab 2 x c plot. Fig. 4(b) shows the 1 ln T=T c dependence of dx dia. All dx dias approximately follow the power law 1 22.3 and give the same magnitude. These observations justify our SCF interpretation of this component, although the 2d BCS theory predicts an 1 21 dependence. One important point is that irrespective of the existence of the pseudogap, the SCF exists in a universal
Fig. 4. (a) Temperature and doping dependence of the static susceptibility with H==ab. (b) Diamagnetic ¯uctuation susceptibility as a function of ln(T/Tc).
form. This indicates that the SCF and the pseudogap are mutually independent and actually additive in the pseudogap region. Since, the tunneling DOS is insensitive to the diamagnetic SCF signal, Txpab should be compared with the Txpab deterp p mined from x ab. In Fig. 5, Ttun ; Tab ; and Tc are plotted as a p represents the temperature where dxdia , function of x. Tscf xc : For the comparison, the Txpc ; which represents T p deterp mined from x c, and the Ttun from our previous paper [10] are also plotted. The apparent difference between the present p and previous Ttun illustrates the effect of oxygen depletion from the surface. As can be seen from the ®gure, the present p Ttun well coincides with the Txpab ; especially in the pseudogap region. Thus, we con®rmed that the dip structure in the tunneling DOS well explains the gradual decrease in x as expected. The discrepancy in the SCF dominated region can be ascribed to two factors: (1) uncertainty in determining p Ttun due to the strongly energy-dependent DOS (van Hovelike singularity); (2) the difference in the way the SCF affects the DOS and spin susceptibility. In summary, temperature and doping dependence of the tunneling spectrum of Bi2.1Sr1.9CaCu2Ox single crystals was obtained by using a low-temperature STM and lowtemperature cleavage. Above Tc, for x # 8.27, the tunneling DOS shows a clear gap-like feature with a larger gap value
68
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
showing that the gradual decrease in x is a result of the DOS reduction as detected by the present STM experiment. This analysis also showed that the SCF in x has a universal form irrespective to the existence of the pseudogap.
References
Fig. 5. Doping dependence of various characteristic temperatures.
than the superconducting one, while for x . 8.27, it shows the feature expected from the conventional superconducting ¯uctuation (SCF). We compared the results with the static susceptibility (x ). By subtracting the effect of the SCF in x , we con®rmed that the two pseudogap boundaries, determined by the tunneling and x coincide with each other,
[1] H. Yasuoka, et al., Correlation and Superconductivity, Springer, Berlin, 1989 (p. 254). [2] W.W. Warren, et al., Phys. Rev. Lett. 62 (1989) 1193. [3] J. Rossat-Mignot, et al., Physica B 180/181 (1992) 383. [4] J.W. Loram, et al., Phys. Rev. Lett. 71 (1993) 1740. [5] T. Ito, et al., Phys. Rev. Lett. 70 (1993) 3995. [6] D.N. Basov, et al., Phys. Rev. Lett. 77 (1996) 4090. [7] H. Ding, et al., Nature 382 (1996) 51. [8] D.S. Marshall, et al., Phys. Rev. Lett. 76 (1996) 4841. [9] C. Renner, et al., Phys. Rev. Lett. 80 (1998) 149. [10] A. Matsuda, et al., Phys. Rev. B 60 (1999) 1377. [11] M. Oda, et al., Physica C 281 (1997) 135. [12] T. Watanabe, et al., Phys. Rev. Lett. 84 (2000) 5848. [13] T. Watanabe, et al., Phys. Rev. Lett. 79 (1997) 2113. [14] A. Matsuda, et al., Advances in Superconductivity, vol. XI, Springer, Tokyo, 1999 (p. 151).
www.elsevier.nl/locate/jpcs
Study of pseudogap phenomena by STM and other probes A. Matsuda a,b,*, S. Sugita a, T. Fujii b, T. Watanabe a a NTT Basic Research Laboratories, 3-1 Morionsato Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan Department of Applied Physics, Faculty of Science, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
b
Abstract The temperature and doping dependence of the tunneling spectrum of Bi2.1Sr1.9CaCu2Ox single crystals was obtained by using a low-temperature scanning tunneling microscope. Above Tc, for x # 8.27, the tunneling density of states shows a clear gap-like feature with a larger gap value than the superconducting one, while for x . 8.27, it shows the feature expected from the conventional superconducting ¯uctuation (SCF). We determined the SCF component in the static susceptibility (x ). By subtracting the effect of the SCF from x , we con®rmed that the two pseudogap phase boundaries determined by the tunneling and x coincide with each other. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Superconductors; C. Scanning tunneling microscopy; D. Electrical properties; D. Magnetic properties
Recently, the pseudogap phenomenon has been recognized as one of the fundamental features characterizing high-Tc superconductor in the underdoped region. However, the origin of the pseudogap and its relation to the mechanism of high-Tc superconductivity remains unknown. Many experiments have con®rmed the existence of the pseudogap [1±8]. Among spectroscopic probes, electron tunneling has proven to be a powerful tool, because it directly provides the electronic density of states (DOS) both above and below the Fermi level. Renner et al. [9] and present authors [10] reported the observation of the pseudogap using a scanning tunneling microscope (STM). However, the experimentally deduced characteristics of the pseudogap rather strongly depend on the experimental methods, which might prevent us from complete understanding of the pseudogap phenomenon. For example, static susceptibility measurements [11,12] place the pseudogap phase boundary in a higher temperature range than those obtained by other methods. Besides this kind of fundamental discrepancy, there might be others that should be attributed to experimental problems. In this report, we ®rst reconcile our previous experiment [10] and provide new set of doping and temperature dependence data for the Bi-2212 DOS, in which the problem has been almost eliminated. Then, we compare our results with the static susceptibility (x ) measurements [12]. Through an * Corresponding author. Tel.: 181-46-240-3525; fax: 181-46240-4722. E-mail address: [email protected] (A. Matsuda).
analysis of the anisotropy of x , we were able to derive the diamagnetic component of the superconducting ¯uctuation (SCF), which exists for all doping levels and shows identical temperature dependence. Considering the effect of the SCF, we show that the pseudogap phase boundaries obtained by the STM and x measurements coincide with each other. This, in turn, provides a view that the SCF and the pseudogap are independent phenomena. In the tunneling and susceptibility measurements, we used Bi2.1Sr1.9CaCu2Ox (Bi-2212) single crystals with 8:22 , x , 8:30: Single crystals were grown using a travelling solvent ¯oating zone (TSFZ) method. The oxygen content x for 8:22 , x , 8:28 was adjusted by using an improved annealing method, where the equilibrium oxygen pressures are maintained down to suf®ciently low temperature [13]. A heavily overdoped sample with Tc < 60 K was made by high-O2 pressure (400 atm) annealing at 5008C for 50 h, using a hot isostatic pressing (HIP) furnace. Its oxygen content was estimated to about 8.30. (The relation between x and Tc is shown in Fig. 5.) For the tunneling experiments, we used a newly designed low-temperature UHV STM, which is capable of in situ sample and tip exchange. In our previous experiment [10], we cleaved a crystal at room temperature. We found that this causes oxygen depletion from the topmost surface, and hence a doping level shift. To avoid this problem, in the present experiment, we cleaved samples at a temperature below 84 K in an UHV condition. Then, the samples were transferred to the STM head, which had been precooled to
0022-3697/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(00)00102-5
66
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
about 80 K. Immediately after the transfer, the head was cooled to the 10 K range within several hours. We believe this procedure preserves the oxygen content of the topmost surface. The tip was mechanically cut Pr±Ir wire. The lowtemperature measurements were mostly done with the pressure in the mid 10 29 Torr range. We could acquire a good Bi±O lattice image with the conditions described above. I±V characteristics were obtained by ramping the bias voltage while the feedback loop was cut. Then the DOS was calculated by numerically differentiating I±V characteristics. The gap value showed more or less a spatial dependence in a reproducible manner, which we now consider as an inherent feature of a Bi-2212 cleaved surface. The spatial dependence was stronger when the doping level was reduced. The details will be published elsewhere. To avoid uncertainty from the spatial nonuniformity, we took I±V characteristics where the distribution is small and averaged many I±Vs taken along a line that covers several periods of the distribution. Fig. 1 shows examples of the temperature dependence of the tunneling DOS for: (a) an overdoped sample (x 8.28); and (b) a slightly underdoped (x 8.24) one. Unlike in our previous study [10], the overdoped sample did not show any strong evidence of pseudogap opening. However, for x # 8.26, a clear pseudogap feature appeared (Fig. 1 (b)). As in the previous experiments [9,10], the gapped DOS above Tc lost its sharp peak structure at gap
Fig. 1. Temperature dependence of the tunneling DOS for: (a) x 8.28; (b) x 8.24.
edges and showed a larger gap value compared to the superconducting one. The effect of the low-temperature cleaving seems to simply be a shift of the doping level to a higher one. Fig. 2 shows the doping dependence of the tunneling DOS taken at slightly above Tc. The DOS at these temperatures should be predominantly governed by a band structure. Two characteristic structures can be seen in this ®gure. In the heavily overdoped sample with x 8.3, the DOS shows a clear peak at the Fermi level. Then the peak shifts toward lower energy and rapidly broadens with decreasing the doping level. We identify this structure as a van Hove singularity in the two-dimensional Fermi surface. Details will be published elsewhere. For the samples with x # 8.27, the pseudogap develops as a dip structure at the Fermi level. The dip structure is thought not to have a band origin, because of its strong doping and temperature dependence. Fig. 3 shows the temperature and doping dependence of the DOS peak voltage, which coincides to a superconducting gap value in the lowest temperature range. Here, we only used a positive peak, which can be clearly de®ned even above Tc in the pseudogap state. It can be seen that the gap of the x 8.3 sample follows a BCS-like temperature dependence, and no pseudogap is observed. This tendency continues down to x 8.27. Then a pseudogap suddenly develops for x # 8.26, accompanied by a steep increase in the peak voltage. The ®gure shows that the pseudogap value is larger than the superconducting one and the superconducting gap structure is rather rapidly replaced by the pseudogap structure above Tc. The analysis in our previous work [10] suggests that the pseudogap value is larger even after the effects of thermal smearing and inelastic scattering are incorporated in the present case. The dip structure gradually disappears with increasing temperature and becomes indistinguishable from the background structure at a certain p temperature, which we de®ne as Ttun ; the pseudogap
Fig. 2. Doping dependence of the tunneling DOS in the normal state.
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
67
Fig. 3. Temperature and doping dependence of the DOS peak voltage.
opening temperature. When x $ 8.27, the gap-like structure was observed above Tc in a limited temperature range. We consider that this is evident of the conventional SCF effect, which can be distinguished from the pseudogap because the gap value does not largely exceed the superconducting one. p In our previous studies [10,14], we compared Ttun ; with various T p s obtained by other probes, such as the transport property and susceptibility x . Recently, we measured the temperature and doping dependence of the anisotropic x of Bi-2212 in detail [12] and obtained the corresponding T p Txp : Here, by shedding a new light on the role of SCF, we reconsider the comparison using the new STM data. Fig. 4(a) shows the temperature dependence of the ab-axis susceptibility x ab for various doping levels. c-axis x also showed the similar temperature dependence. The Txp is de®ned as the temperature at which x starts to deviate from its high-temperature behavior. In our study [12], the anisotropy was analyzed using x ab-x c plots with temperature as an implicit parameter. The result revealed that, for all doping levels, anisotropy could be interpreted in terms of a slightly doping dependent g-factor of the Cu 3d electron. However, this interpretation breaks down in a certain temperature range above Tc, which we ascribe to the SCF effect. Assuming that the diamagnetic component of the SCF (dx dia) only contributes to x c, and the anisotropy of the spin component of the SCF can again be determined by the g-factor, we can estimate dx dia as xc 2 gc =gab 2 xab 2 x0 ; where (gc /gab) 2 and x 0 are the slope and intercept of the linear part of the x ab 2 x c plot. Fig. 4(b) shows the 1 ln T=T c dependence of dx dia. All dx dias approximately follow the power law 1 22.3 and give the same magnitude. These observations justify our SCF interpretation of this component, although the 2d BCS theory predicts an 1 21 dependence. One important point is that irrespective of the existence of the pseudogap, the SCF exists in a universal
Fig. 4. (a) Temperature and doping dependence of the static susceptibility with H==ab. (b) Diamagnetic ¯uctuation susceptibility as a function of ln(T/Tc).
form. This indicates that the SCF and the pseudogap are mutually independent and actually additive in the pseudogap region. Since, the tunneling DOS is insensitive to the diamagnetic SCF signal, Txpab should be compared with the Txpab deterp p mined from x ab. In Fig. 5, Ttun ; Tab ; and Tc are plotted as a p represents the temperature where dxdia , function of x. Tscf xc : For the comparison, the Txpc ; which represents T p deterp mined from x c, and the Ttun from our previous paper [10] are also plotted. The apparent difference between the present p and previous Ttun illustrates the effect of oxygen depletion from the surface. As can be seen from the ®gure, the present p Ttun well coincides with the Txpab ; especially in the pseudogap region. Thus, we con®rmed that the dip structure in the tunneling DOS well explains the gradual decrease in x as expected. The discrepancy in the SCF dominated region can be ascribed to two factors: (1) uncertainty in determining p Ttun due to the strongly energy-dependent DOS (van Hovelike singularity); (2) the difference in the way the SCF affects the DOS and spin susceptibility. In summary, temperature and doping dependence of the tunneling spectrum of Bi2.1Sr1.9CaCu2Ox single crystals was obtained by using a low-temperature STM and lowtemperature cleavage. Above Tc, for x # 8.27, the tunneling DOS shows a clear gap-like feature with a larger gap value
68
A. Matsuda et al. / Journal of Physics and Chemistry of Solids 62 (2001) 65±68
showing that the gradual decrease in x is a result of the DOS reduction as detected by the present STM experiment. This analysis also showed that the SCF in x has a universal form irrespective to the existence of the pseudogap.
References
Fig. 5. Doping dependence of various characteristic temperatures.
than the superconducting one, while for x . 8.27, it shows the feature expected from the conventional superconducting ¯uctuation (SCF). We compared the results with the static susceptibility (x ). By subtracting the effect of the SCF in x , we con®rmed that the two pseudogap boundaries, determined by the tunneling and x coincide with each other,
[1] H. Yasuoka, et al., Correlation and Superconductivity, Springer, Berlin, 1989 (p. 254). [2] W.W. Warren, et al., Phys. Rev. Lett. 62 (1989) 1193. [3] J. Rossat-Mignot, et al., Physica B 180/181 (1992) 383. [4] J.W. Loram, et al., Phys. Rev. Lett. 71 (1993) 1740. [5] T. Ito, et al., Phys. Rev. Lett. 70 (1993) 3995. [6] D.N. Basov, et al., Phys. Rev. Lett. 77 (1996) 4090. [7] H. Ding, et al., Nature 382 (1996) 51. [8] D.S. Marshall, et al., Phys. Rev. Lett. 76 (1996) 4841. [9] C. Renner, et al., Phys. Rev. Lett. 80 (1998) 149. [10] A. Matsuda, et al., Phys. Rev. B 60 (1999) 1377. [11] M. Oda, et al., Physica C 281 (1997) 135. [12] T. Watanabe, et al., Phys. Rev. Lett. 84 (2000) 5848. [13] T. Watanabe, et al., Phys. Rev. Lett. 79 (1997) 2113. [14] A. Matsuda, et al., Advances in Superconductivity, vol. XI, Springer, Tokyo, 1999 (p. 151).