- Email: [email protected]
Nonlinear Meissner effect of the cuprate superconductors investigated by London penetration depth measurement
PHYSICA ELSEVIER
Physica C 263 (1996) 438-441
Nonlinear M e i s s n e r effect of the cuprate superconductors investigated by L o n d o n penetration depth m e a s u r e m e n t A. Maeda a,*, T. Hanaguri a, y. Iino a, S. Masuoka a, y. Matsushita b, M. Hasegawa b, H. Takei b a Departmentof Pure andApplied Sciences. The Universityof Tokyo, 3-8-1. Kornaba, Meguro-ku, Tokyo, Japan b Institute of Solid State Physics, The Universityof Tokyo, 7-22-1. Roppongi, Minato-ku, Tokyo, Japan
Abstract London penetration depth A was investigated as a function of magnetic field H in YBa2Cu3Oy and in Tl2Ba2CaCu2Oy. h changes linearly in H at low temperatures, which is consistent with the nonlinear Meissner effect in unconventional superconductors. Together with the published data in Bi2Sr2CaCu2Oy, it is suggested that the order parameter of the double layered cuprates has nodes in k space.
The symmetry of the order parameter of the highTc superconductors (HTSC) is one of the most important topics of current interest. Recently, it has been suggested theoretically that the magnetic-field dependence of the London penetration depth originating from the non-linear Meissner effect can be a new method to judge whether there are nodes in the superconducting gap or not. Namely, according to Yip and Sauls (YS) [1], the small variation of A as a function of magnetic field H is linear in H even at T = 0 K: AA(H)
[
Inl
~
(1)
where AA(H, T) = h ( H , T) - h(O, T) is the variation of h upon application of H, Ho(T) is a charac-
* Corresponding author. Fax: +81 3 3467 8945; e-mail: [email protected].
teristic field of the order of the thermodynamic critical field He(T), and a ( T ) is a temperature-dependent coefficient. At T = 0 K, (Xll= 2 / 3 and a .~ = ( 1 / v ~ ) ( 2 / 3 ) (the subscript indicates the direction of H with respect to the node direction). At finite temperatures, a ( T ) depends only weakly on temperature, because in a d-wave superconductor, a finite number of quasi-particles can be excited by a magnetic field even at T = 0 K. These predictions are in sharp contrast with that for an s-wave superconductor, for which A A changes quadratically with H:
X(H=O)--~(T) ~
.
(2)
The coefficient [3(T) decreases strongly when T decreases, simply because the number of quasi-particles decreases rapidly due to the complete gap opening. Thus, both the magnetic field dependence of the penetration depth in the Meissner state and the
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0921-4534(95)00759-8
A. Maeda et a l . / Physica C 263 (1996)438-441
temperature dependence of the coefficient make it possible to discuss whether there are nodes in the gap or not. It should be noted that there is a thermal smearing effect, which leads to a quadratic dependence of )t even for d-wave superconductors:
(3) below a crossover field H' given by H ' / H o ~ T/Tc. In this case, however, ~/(T) increases with decreasing T ( ~ 1 / T ) [1]. Thus, the H E term arising from the intrinsic s-wave nonlinear Meissner effect and that from the smearing effect can be distinguished from one other by investigating the temperature dependence of the prefactor. Prior to the YS prediction, the conventional H E dependence was reported in a single crystal of YBCO [2]. However, /3(T) in this study is independent of temperature down to l0 K ( ~ 0.109T~), which is contrary to the prediction for s-wave superconductors. Therefore, the nonlinear Meissner effect in HTSC deserves further detailed investigation. Recently, we have found that h of optimally doped Bi2SrECaCuEOy (BSCCO) samples changes linearly with H [3], which is in sharp contrast with the results obtained in conventional type-I [4] and type-II superconductors [5]. Together with the detailed temperature dependence of A(H), we have concluded that the superconducting gap of BSCCO has nodes, and that A(H) measurement is a useful tool to discuss the presence/absence of nodes in the gap. Thus, it is an interesting question whether the linear AA(H) is universal among various cuprate superconductors. In this paper, A(H, T) was investigated in other double-layered cuprates, YBaECU3Oy (YBCO) and T1EBaECaCuEOy ( T B C C O ) . It was found that both of them show the linear-H dependence of A, at least at low temperatures. Thus, we infer that the gap structure has nodes at least in the double-layered cuprates which have T~ of ~ 100 K. Single crystals of YBCO and TBCCO were grown by a flux method. Both of them are optimally doped crystals with a typical transition temperature Tc of 92-89 K (YBCO) and 115 K (TBCCO). Although the transition width of YBCO crystals (defined by the temperature dependence of the magnetization) is 0.5 K, TBCCO crystals show a broader transition
439
with ATc of ~ 3 K, and it also shows a very slight mixing of a higher-Tc phase (2223 phase). Penetration depth was measured by an ordinary rf ( ~ 45 MHz) resonator method [6], where the change in frequency of the resonator circuit A f is proportional to the change in A (AA): Af = GAA, where G is a geometric factor. The resolution of A A in our o system is ~ 10 A, which is limited by the temperature stability of AT-- 10 mK. In this paper, we will present the data taken in the configuration where both the DC field and the AC field are perpendicular to the C u t 2 planes. For YBCO, we also performed the similar measurement in several different field configurations, and found that the essential features shown below were unchanged. Therefore, we believe that various effects which may arise from the demagnetization effect are not the dominant origin of the bahavior which we report here. Details will be discussed in a separate publication. Fig. 1 shows the field dependence of the penetraI
I
I
I
I A
YBCO
30K o
A
o
A
...._° o o o °
<~ ~***°°°
A ~
A I
0
A ,
I
,
I
,
I
10 20 30 DC Field (Oe)
,
I
40
Fig. 1. Magnetic-field dependence of h in an YBCO crystal. Solid lines are guides for the eye, and arrows indicate the crossover field H *.
440
A. Maeda et a l . / Physica C 263 (1996)438-441
(a) YBCO ~ ~ ~
o0°°_
o 9 ~ ^ 0 0 0
(b) YBCO oOO
° oo
On++
G oOoo88~ ~Oooo
~,
oo++
t=0.34
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1
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t=0.79
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ooo
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4
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,
,
,
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t
i
~
,
I
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,
~
,
,
15
Fig. 2. (a) Magnetic-field dependence of A in the same sample as shown in Fig. 1, in the normalized form at various temperatures; t =- T I T c. (b) the same data as a function o f ( H / H o ) 2.
tion depth of an YBCO crystal (Tc = 89.5 K). The total field-dependence behavior is similar to that in the previously published data [2]. We identify the field where there is a small kink as H " - H c l / ( 1 - v) (shown by arrows in Fig. 1), where v is the demagnetization factor. After a normalization as in Eqs. (1)-(3), a normalized change l - A A , b (H, T)/A,b(0, T) was obtained as functions of the
800
300
i
o BSCCO
600
<> YBCO 9.
400
\\,
200
o o
200
8 "" - - . . . .
0
'01
o
100 ¢ ....................~ct. "~o---~_ .............. I
0..
T,rr~
016
018
0
Fig. 3. Temperature dependence of the ceeffi¢iem y of YBCO (diamonds) and BSCCO (circles) crystal. The dashed curve is the theoretical prediction of the smearing which has a thermal origin for a d-wave superconductor.
normalized field h = H / H o and h 2, as shown in Figs. 2(a) and (b), respectively. At high temperatures, l behaves a s h 2 in the whole field region. On the other hand, at low temperatures, the h 2 behavior was observed only in a small field range around zero field. At higher fields, a crossover from the h 2 to an h-linear behavior was observed. Thus, the data for YBCO are consistent with the theoretical prediction for the superconductor with a gap having nodes. At higher temperatures, the quadratic dependence of A on H becomes prominent, which may come partly from the thermal effect and partly from the disorder-induced effect. In Fig. 3 we compare the smearing coefficient y between YBCO and BSCCO (Ref. [3]). In YBCO, the temperature dependence of y is well expressed by a 1 / T behavior, which is consistent with the thermal smearing effect [1]. On the other hand, in BSCCO, a saturation is observed at low temperatures. This saturation can be interpreted by the presence of a considerable disorder-induced pair-breaking effect, which also provides the H2-AA dependence, but with a temperature-independent coefficient. The existence of a considerable amount of pair-breaking in BSCCO is also reported in recent NMR measurements [7,8].
A. Maeda et al. / Physica C 263 (1996) 438-441 1
I
I
I
T12Ba2CaCu2Oy
. ....
oo 10K 20K
"*I
oO°°:~'~ oa"
^o~a ~.000000~
,,,0
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° nr' ~
I
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o v
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--
. O~
o<'
00(>0
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AA a
t~ A
"~'ooO ,x~tx
o
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on
0 u
behavior for the nonlinear Meissner effect in unconventional superconductors. Together with the published data in Bi2Sr2CaCu:Oy, it is inferred that the order parameter of the double-layered cuprates has nodes in k space. In the above, we have shown that the study of the magnetic-field dependence of A can be a new and good method to discuss the presence/absence of nodes. If the experimental devices are sufficiently improved to be able to distinguish the difference of coefficients by 1/ ~ " between the two different field orientations, this method will be able to distinguish the location of the nodes in the k space.
U°
an °a
~aa~
~
O00~A,~ ~ n °
0000
00
•
U A~O
_o °
441
Acknowledgements
]0.02
..0..0 O O o D D l
0
,
I
1
,
I
1
2
3
DC Field (Oe)
,
I
4
Fig. 4. Change in the resonating frequency A f / f o as a function of magnetic field in TBCCO.
Fig. 4 shows Af/fo as a function of magnetic field in TBCCO. In the data below 50 K, a crossover from nearly flat Af/fo to an almost linear Af/fo can be seen. In this case, however, the characteristic field H * cannot be definitely identified, probably because of the pinning. Detailed analysis of the data is in progress. Even so, it can be safely said that the gap in TBCCO has nodes. Therefore, from the results discussed above, it can be inferred that the gap structure is the same in double-layered cuprate superconductors. In conclusion, the London penetration depth h was investigated as a function of magnetic field H in YBa2Cu3Oy and in Tl2Ba2CaCu2Oy. A changes linearly in H at low temperatures, and the results of detailed analyses are consistent with the expected
We appreciate helpful discussions with many researchers. Discussions with E. Zeldov of the Weizmann Institute and J. Sauls and S. Yip of Northwestern University were particularly stimulating and encouraging.
References [1] S.K. Yip and J. Sauls, Phys. Rev. Lett. 69 (1992) 2264; D. Xu, S.K. Yip and J.A. Sauls, Phys. Rev. B 51 (1995) 16233. [2] S. Sridhar, D.H. Wu and W. Kennedy, Phys. Rev. Lett. 63 (1989) 1873. [3] A. Maeda, Y. Iino, T. Hanaguri, N. Motohira, K. Kishio and T. Fukase, Phys. Rev. Lett. 74 (1995) 1202. [4] S. Sridhar and J.E. Mercereau, Phys. Rev. B 34 (1986) 203. [5] T. Hanaguri, Y. lino, A. Maeda and T. Fukase, Physica C 246 (1995) 223. [6] A.L. Schawlow and G.E. Devlin, Phys. Rev. 113 (1959) 120; A.J. Slavin, Cryogenics 12 (1972) 121. [7] K. Ishida, Y. Kitaoka, K. Asayama, K. Kadowaki and T. Mochiku, J. Phys. Soc. Jim. 63 (1994) 1104. [8] M. Takigawa and D.B. Mitzi, Phys. Rev. Lett. 73 (1994) 1287.
Physica C 263 (1996) 438-441
Nonlinear M e i s s n e r effect of the cuprate superconductors investigated by L o n d o n penetration depth m e a s u r e m e n t A. Maeda a,*, T. Hanaguri a, y. Iino a, S. Masuoka a, y. Matsushita b, M. Hasegawa b, H. Takei b a Departmentof Pure andApplied Sciences. The Universityof Tokyo, 3-8-1. Kornaba, Meguro-ku, Tokyo, Japan b Institute of Solid State Physics, The Universityof Tokyo, 7-22-1. Roppongi, Minato-ku, Tokyo, Japan
Abstract London penetration depth A was investigated as a function of magnetic field H in YBa2Cu3Oy and in Tl2Ba2CaCu2Oy. h changes linearly in H at low temperatures, which is consistent with the nonlinear Meissner effect in unconventional superconductors. Together with the published data in Bi2Sr2CaCu2Oy, it is suggested that the order parameter of the double layered cuprates has nodes in k space.
The symmetry of the order parameter of the highTc superconductors (HTSC) is one of the most important topics of current interest. Recently, it has been suggested theoretically that the magnetic-field dependence of the London penetration depth originating from the non-linear Meissner effect can be a new method to judge whether there are nodes in the superconducting gap or not. Namely, according to Yip and Sauls (YS) [1], the small variation of A as a function of magnetic field H is linear in H even at T = 0 K: AA(H)
[
Inl
~
(1)
where AA(H, T) = h ( H , T) - h(O, T) is the variation of h upon application of H, Ho(T) is a charac-
* Corresponding author. Fax: +81 3 3467 8945; e-mail: [email protected].
teristic field of the order of the thermodynamic critical field He(T), and a ( T ) is a temperature-dependent coefficient. At T = 0 K, (Xll= 2 / 3 and a .~ = ( 1 / v ~ ) ( 2 / 3 ) (the subscript indicates the direction of H with respect to the node direction). At finite temperatures, a ( T ) depends only weakly on temperature, because in a d-wave superconductor, a finite number of quasi-particles can be excited by a magnetic field even at T = 0 K. These predictions are in sharp contrast with that for an s-wave superconductor, for which A A changes quadratically with H:
X(H=O)--~(T) ~
.
(2)
The coefficient [3(T) decreases strongly when T decreases, simply because the number of quasi-particles decreases rapidly due to the complete gap opening. Thus, both the magnetic field dependence of the penetration depth in the Meissner state and the
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0921-4534(95)00759-8
A. Maeda et a l . / Physica C 263 (1996)438-441
temperature dependence of the coefficient make it possible to discuss whether there are nodes in the gap or not. It should be noted that there is a thermal smearing effect, which leads to a quadratic dependence of )t even for d-wave superconductors:
(3) below a crossover field H' given by H ' / H o ~ T/Tc. In this case, however, ~/(T) increases with decreasing T ( ~ 1 / T ) [1]. Thus, the H E term arising from the intrinsic s-wave nonlinear Meissner effect and that from the smearing effect can be distinguished from one other by investigating the temperature dependence of the prefactor. Prior to the YS prediction, the conventional H E dependence was reported in a single crystal of YBCO [2]. However, /3(T) in this study is independent of temperature down to l0 K ( ~ 0.109T~), which is contrary to the prediction for s-wave superconductors. Therefore, the nonlinear Meissner effect in HTSC deserves further detailed investigation. Recently, we have found that h of optimally doped Bi2SrECaCuEOy (BSCCO) samples changes linearly with H [3], which is in sharp contrast with the results obtained in conventional type-I [4] and type-II superconductors [5]. Together with the detailed temperature dependence of A(H), we have concluded that the superconducting gap of BSCCO has nodes, and that A(H) measurement is a useful tool to discuss the presence/absence of nodes in the gap. Thus, it is an interesting question whether the linear AA(H) is universal among various cuprate superconductors. In this paper, A(H, T) was investigated in other double-layered cuprates, YBaECU3Oy (YBCO) and T1EBaECaCuEOy ( T B C C O ) . It was found that both of them show the linear-H dependence of A, at least at low temperatures. Thus, we infer that the gap structure has nodes at least in the double-layered cuprates which have T~ of ~ 100 K. Single crystals of YBCO and TBCCO were grown by a flux method. Both of them are optimally doped crystals with a typical transition temperature Tc of 92-89 K (YBCO) and 115 K (TBCCO). Although the transition width of YBCO crystals (defined by the temperature dependence of the magnetization) is 0.5 K, TBCCO crystals show a broader transition
439
with ATc of ~ 3 K, and it also shows a very slight mixing of a higher-Tc phase (2223 phase). Penetration depth was measured by an ordinary rf ( ~ 45 MHz) resonator method [6], where the change in frequency of the resonator circuit A f is proportional to the change in A (AA): Af = GAA, where G is a geometric factor. The resolution of A A in our o system is ~ 10 A, which is limited by the temperature stability of AT-- 10 mK. In this paper, we will present the data taken in the configuration where both the DC field and the AC field are perpendicular to the C u t 2 planes. For YBCO, we also performed the similar measurement in several different field configurations, and found that the essential features shown below were unchanged. Therefore, we believe that various effects which may arise from the demagnetization effect are not the dominant origin of the bahavior which we report here. Details will be discussed in a separate publication. Fig. 1 shows the field dependence of the penetraI
I
I
I
I A
YBCO
30K o
A
o
A
...._° o o o °
<~ ~***°°°
A ~
A I
0
A ,
I
,
I
,
I
10 20 30 DC Field (Oe)
,
I
40
Fig. 1. Magnetic-field dependence of h in an YBCO crystal. Solid lines are guides for the eye, and arrows indicate the crossover field H *.
440
A. Maeda et a l . / Physica C 263 (1996)438-441
(a) YBCO ~ ~ ~
o0°°_
o 9 ~ ^ 0 0 0
(b) YBCO oOO
° oo
On++
G oOoo88~ ~Oooo
~,
oo++
t=0.34
~°°°°
o o o
'~
= ~u~.~-+ zs + 7,,~. +++ ,
1
,
,
I
l
0
1
[ 0.02 l I
I
I
I
1
1
o o
° oo
t=0.34
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o
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oO
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t=0.56 t=0.67
ct
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1
o
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1
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°
o°°°
t--0.13 t=0.22
o t 9.7 9,
1
o
~ ~,'~"o o ooooOO t=0.22
oB ,÷ + " o v ¢
t=0.79
°
t=0.67
t=0.56
B~°++
o~,~fr++
ooo
c~ooooOO°° ~,~:x, oooOO
4
0
,
,
,
I
t
i
~
,
I
5 10 (H/Ho)2 (10 -4)
,
~
,
,
15
Fig. 2. (a) Magnetic-field dependence of A in the same sample as shown in Fig. 1, in the normalized form at various temperatures; t =- T I T c. (b) the same data as a function o f ( H / H o ) 2.
tion depth of an YBCO crystal (Tc = 89.5 K). The total field-dependence behavior is similar to that in the previously published data [2]. We identify the field where there is a small kink as H " - H c l / ( 1 - v) (shown by arrows in Fig. 1), where v is the demagnetization factor. After a normalization as in Eqs. (1)-(3), a normalized change l - A A , b (H, T)/A,b(0, T) was obtained as functions of the
800
300
i
o BSCCO
600
<> YBCO 9.
400
\\,
200
o o
200
8 "" - - . . . .
0
'01
o
100 ¢ ....................~ct. "~o---~_ .............. I
0..
T,rr~
016
018
0
Fig. 3. Temperature dependence of the ceeffi¢iem y of YBCO (diamonds) and BSCCO (circles) crystal. The dashed curve is the theoretical prediction of the smearing which has a thermal origin for a d-wave superconductor.
normalized field h = H / H o and h 2, as shown in Figs. 2(a) and (b), respectively. At high temperatures, l behaves a s h 2 in the whole field region. On the other hand, at low temperatures, the h 2 behavior was observed only in a small field range around zero field. At higher fields, a crossover from the h 2 to an h-linear behavior was observed. Thus, the data for YBCO are consistent with the theoretical prediction for the superconductor with a gap having nodes. At higher temperatures, the quadratic dependence of A on H becomes prominent, which may come partly from the thermal effect and partly from the disorder-induced effect. In Fig. 3 we compare the smearing coefficient y between YBCO and BSCCO (Ref. [3]). In YBCO, the temperature dependence of y is well expressed by a 1 / T behavior, which is consistent with the thermal smearing effect [1]. On the other hand, in BSCCO, a saturation is observed at low temperatures. This saturation can be interpreted by the presence of a considerable disorder-induced pair-breaking effect, which also provides the H2-AA dependence, but with a temperature-independent coefficient. The existence of a considerable amount of pair-breaking in BSCCO is also reported in recent NMR measurements [7,8].
A. Maeda et al. / Physica C 263 (1996) 438-441 1
I
I
I
T12Ba2CaCu2Oy
. ....
oo 10K 20K
"*I
oO°°:~'~ oa"
^o~a ~.000000~
,,,0
o
° nr' ~
I
oo
o
0
oo
40K
oo v~ 50K oOvv v v
o v
oOvv
0<>
o 60K
.0.000 ° 0
oo°vv v ^ o ° ~ o 70K o v o " o -. ~ 80K oOv v ^0 0
~JO
V
vv~v
--
. O~
o<'
00(>0
.~..OO O 0
AA a
t~ A
"~'ooO ,x~tx
o
90K
on
0 u
behavior for the nonlinear Meissner effect in unconventional superconductors. Together with the published data in Bi2Sr2CaCu:Oy, it is inferred that the order parameter of the double-layered cuprates has nodes in k space. In the above, we have shown that the study of the magnetic-field dependence of A can be a new and good method to discuss the presence/absence of nodes. If the experimental devices are sufficiently improved to be able to distinguish the difference of coefficients by 1/ ~ " between the two different field orientations, this method will be able to distinguish the location of the nodes in the k space.
U°
an °a
~aa~
~
O00~A,~ ~ n °
0000
00
•
U A~O
_o °
441
Acknowledgements
]0.02
..0..0 O O o D D l
0
,
I
1
,
I
1
2
3
DC Field (Oe)
,
I
4
Fig. 4. Change in the resonating frequency A f / f o as a function of magnetic field in TBCCO.
Fig. 4 shows Af/fo as a function of magnetic field in TBCCO. In the data below 50 K, a crossover from nearly flat Af/fo to an almost linear Af/fo can be seen. In this case, however, the characteristic field H * cannot be definitely identified, probably because of the pinning. Detailed analysis of the data is in progress. Even so, it can be safely said that the gap in TBCCO has nodes. Therefore, from the results discussed above, it can be inferred that the gap structure is the same in double-layered cuprate superconductors. In conclusion, the London penetration depth h was investigated as a function of magnetic field H in YBa2Cu3Oy and in Tl2Ba2CaCu2Oy. A changes linearly in H at low temperatures, and the results of detailed analyses are consistent with the expected
We appreciate helpful discussions with many researchers. Discussions with E. Zeldov of the Weizmann Institute and J. Sauls and S. Yip of Northwestern University were particularly stimulating and encouraging.
References [1] S.K. Yip and J. Sauls, Phys. Rev. Lett. 69 (1992) 2264; D. Xu, S.K. Yip and J.A. Sauls, Phys. Rev. B 51 (1995) 16233. [2] S. Sridhar, D.H. Wu and W. Kennedy, Phys. Rev. Lett. 63 (1989) 1873. [3] A. Maeda, Y. Iino, T. Hanaguri, N. Motohira, K. Kishio and T. Fukase, Phys. Rev. Lett. 74 (1995) 1202. [4] S. Sridhar and J.E. Mercereau, Phys. Rev. B 34 (1986) 203. [5] T. Hanaguri, Y. lino, A. Maeda and T. Fukase, Physica C 246 (1995) 223. [6] A.L. Schawlow and G.E. Devlin, Phys. Rev. 113 (1959) 120; A.J. Slavin, Cryogenics 12 (1972) 121. [7] K. Ishida, Y. Kitaoka, K. Asayama, K. Kadowaki and T. Mochiku, J. Phys. Soc. Jim. 63 (1994) 1104. [8] M. Takigawa and D.B. Mitzi, Phys. Rev. Lett. 73 (1994) 1287.