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Anisotropy in the upper and lower critical fields of MgB2 single crystals
Physica C 412–414 (2004) 258–261 www.elsevier.com/locate/physc
Anisotropy in the upper and lower critical fields of MgB2 single crystals H. Bando *, Y. Yamaguchi, N. Shirakawa, T. Yanagisawa National Institute of Advanced Industrial Science and Technology, AIST Tsukuba Central 2, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan Received 29 October 2003; accepted 5 January 2004 Available online 19 May 2004
Abstract MgB2 single crystals with a sharp transition width ðDTc 0:3 KÞ were grown by the ambient-pressure synthesis. The upper and lower critical fields, Hc2 and Hc1 were evaluated from the magnetization for the applied field l0 H 6 5 T parallel to the three crystallographic directions. The coherence length n and the penetration depth k for each direction were deduced from these critical fields with a little assumption for Hc1 . The anisotropy ratio cH ¼ nc =nab was estimated to be 6.5 at 2.5 K and 2.3 near Tc ( ¼ 37.6 K), and ck ¼ kc =kab ð¼ 3:0 0:2Þ to be almost constant over the range 2:5 K 6 T < Tc . 2004 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 74.60.Ec; 74.70.Ad Keywords: MgB2 ; Lower critical field; Upper critical field; Anisotropy
1. Introduction MgB2 attracts interest in its high superconducting transition temperature (Tc ) as metallic superconductor and its unique Fermi surface with two-gap property [1]. Effects of the two-band feature have been discussed in connection with anisotropy of the lower and the upper critical fields (Hc1 and Hc2 ) [2–8]. However, the discussion has been rather qualitative, partly because the experimental data are yet to be converged, especially
*
Corresponding author. Tel.: +81-29-861-5385; fax: +81-29861-5387. E-mail address: [email protected] (H. Bando).
concerning the anisotropy in the penetration depth which determines the Hc1 . In this paper, we report a detailed magnetization measurement on highquality single crystals of MgB2 .
2. Experimental Single crystals were grown from 99.9% boron with 99.99% magnesium sealed in a stainless tube whose inner side covered with molybdenum sheet [4]. The tube was kept for two weeks at 1120 C. Several pieces of crystal platelets with typical dimension of 500 · 400 · 60 lm3 (#A) were grown with many grains of smaller size. Laue-photographs of these crystals showed the 6/mmm Laue
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.055
H. Bando et al. / Physica C 412–414 (2004) 258–261
0.5
MgB2(#A) T=2.5 K
5 dM/dH//c
0.0
0
-0.5
-5
-1.0
M//a H//c H//a
-1.5
Hen//c
Hp//c
Hen//a
M//c
-3
-6
-2.0x10
0.00
-10
-15
Hp//a
-20x10 0.05
0.10
0.15
0.20
MgB2(#A)
µ0Hen (T)
0.25
This work
Hp//a Hp//c Hen//a Hen//c
0.20
0.15
Perkins et al.
Hen 0.10
0.05
0.00 0
10
20
30
40
T (K) Fig. 2. Temperature dependence of the observed Hpjjab , Hpjjc , Henjjc and Henjjab .
affected by the scan rate of H presumably because of negligible relaxation effect. Secondly, the Hp was almost free of sample-piece difference, while Hen heavily depended on the sample shape. For example, the ratios Hen =Hp for the sample #A were 0.9 and 0.85 under the field H jjab and H jjc, respectively, while for another sample #B (dimension of 600 · 400 · 60 lm3 ), respective ratios were 0.85 and 0.75, and for the sample #C (dimension of 600 · 200 · 40 lm3 ), 0.85 and 0.55. The samples with the shape closer to sphere tended to have the Hen =Hp ratio closer to unity. The values of Hp for both H jjab and H jjc are very close to Hen by Perkins et al. [5].
3. Results and discussion dM/dH
M (emu)
dM/dH//a
0.30
µ0Hp ,
symmetry. Tc was observed to be 37.6 K with the transition width DTc 0:3 K (Fig. 3). Magnetization was measured using SQUID magnetometers, MPMS-1 and MPMS-5 produced by Quantum Design. Typical M–H curves near the lower critical field (Hc1 ) are shown in Fig. 1, with the corresponding dM=dH –H curves. The figure includes two sets of curves, i.e. magnetic field perpendicular to the plate ðH jjcÞ or parallel to the plate ðH jjabÞ. These curves are plotted with Heff ¼ Happ =ð1 N Þ corrected by the demagnetization factor N . Since Nc (N for H jjc) was large and uncertain, while Nab (N for H jjab) was small and evaluated rather precisely from the sample shape, the Nc ð¼ 0:79Þ was determined empirically to fit the slope of the M–Hjjab curve for the a-axis, where Hjjab ¼ Heff ¼ Happ =ð1 Nab Þ with Nab ¼ 0:1 (for sample #A). This procedure is the same as that by Perkins et al. [5]. We defined both Hp (peak field of the M–H curve) and Hen (field where the M–H curve deviates quadratically from the linear feature). The temperature dependence of the observed Hp and Hen for both the axes are shown in Fig. 2. In this work, we took Hp as a measure of Hc1 because of the following two reasons. Firstly, Hp was not
259
0.25
µ0Heff (T) Fig. 1. M–H curves and the corresponding dM=dH –H curves to define Hp and Hen . A dotted arrow shows the sweep direction.
In Figs. 3a and b, M–T curves under various applied field H are shown. The curve for l0 H ¼ 0:2 mT is shown as a curve to determine the critical temperature Tc ðH 0). None of the M–T curves, except for that at l0 H ¼ 0:2 T, had hysteresis against the sweep direction of temperature. The Tc ðH Þ was determined as the temperature where the extrapolation of the linear part of the M–T curve crosses the temperature axis. In Fig. 3a, two curves for l0 H ¼ 3 T, both H jja and H jjb, are shown to elucidate the negligible anisotropy in the ab-plane. The anisotropy was within the
260
H. Bando et al. / Physica C 412–414 (2004) 258–261 6
1
(a)
MgB2(#A)
0 5 Tc(3T)=29.3 K
-2 -3 -4 -5x10
(1T)
(0.5T) (0.2T)
// a-axis 0.2mT 0.2T 0.5T 1T 3T // b*(=aXc) 3T
MgB2(#A) H//ab
4
Mzfc Tc(0.2mT) =37.6 K
30
Hc2//ab
3 Lyard Lyard et et al al
2
-6
28
µ0Hc2 (T)
M (emu)
-1
Hc2 from M//ab-vs-T M//c-vs-T M-vs-H//c Lyard et al
32
34
36
Hc2//c
38
T (K)
1
5
M (emu)
0
0
(b)
0
-5 Tc(0.6T) =31.2 K
-10 -15
40
Tc(0.2mT) =37.6 K
Fig. 4. Temperature dependence of Hc2jjab for H jjab-plane and Hc2jjc for Hjjc-axis. Data from Ref. [6] are shown as dashed curves.
-6
30
32
34
36
38
T (K)
0
(c) M (emu)
30
MgB2(#A) H//c
-20 -25x10
20
T (K)
(0.5T) (0.4T) (0.3T) (0.2T) (0.1T)
Mzfc
10
Hc2(17.5K)
-10
15 K
12.5K 10K
Hc2(2.5K) =25.5T
-20 2.5 K 5K
-30 7.5 K
MgB2(#A) H//c
-6
-40x10
1.6
1.8
2.0
2.2
2.4
2.6
0H (T)
Fig. 3. Determination of (a,b) Hc2 from Ma;b –T and (c) Ma –H curves. Two Ma;b –T curves for 3 T in (a) show absence of anisotropy in the ab-plane.
experimental errors of 0.1 K, which corresponds to 1% of Hc2 Since the experimental errors in Tc ðH Þ became large at temperatures below 10 K, the M–H curves were used to determine the Hc2 ðT Þ. As shown in Fig. 3c, M–H curves exhibited a small dip just below Hc2 which became obvious at lower temperatures. These M–H curves including the dip feature were reversible against the field sweep. Temperature dependence of Hc2jjab for H jjab and Hc2jjc for H jjc are shown in Fig. 4. The present measurement was limited in magnetic field less
than 5 T, no data of Hc2jjab below 25 K were obtained. However, the values are very close to those by Lyard et al. [6] for 25 K 6 T 6 Tc . In the following analysis, therefore, we used their Hc2jjab data for T 6 20 K. Moreover, a parabolic extrapolation was used for T ¼ 2:5 K. The coherence lengths, nc and nab , and their anisotropy cH ð¼ nab =nc Þ are easily evaluated from the values of Hc2 as Hc2jjc ¼ U0 =2pn2ab and Hc2jjab ¼ U0 =2pnc nab . Meanwhile, evaluation of the penetration depths is a little cumbersome. For H jjc, Hc1jjc ¼ ðU0 =4pk2ab ÞLðjc Þ, where jc ¼ kab =nab . Asymptotically, Lðjc Þ ¼ lnðjc Þ in the limit of large jc , but jc is not always large. Referring p the ffiffiffi numerical solutions [9,10] of arbitrary jc > 1= 2, we adopted the form: LðjÞ ¼ lnðjÞ þ ð0:25j þ 1 0:6Þ . For H jjab, the flux is elongated along the ab-plane. We simply extended the idea for H jjc and applied Hc1jjab ¼ ðU0 =4pkc kab ÞLðjab Þ, where 1=2 jab ¼ ðkc kab =nc nab Þ . These formulae are slightly different from those by Zehetmayer et al., [7] where they implicitly assumed that cH is equal to ck ¼ kc =kab . The evaluated values of the parameters are shown in Figs. 5a and b. The anisotropy ratio cH ¼ nc =nab ¼ Hc2jjab =Hc2jjc was 6.5 at 2.5 K and 2.3 near Tc ð¼ 37:6 KÞ. The deduced ratio ck ¼ kc = kab ð¼ 3:0 0:2Þ was almost constant over the observed range 2:5K 6 T < Tc while the ratio cH c1 ¼ Hc1jjc = Hc1jjab was slightly temperature dependent
H. Bando et al. / Physica C 412–414 (2004) 258–261
λ, ξ (nm)
1000
λc ξc
(a)
261
Bud’ko predicted that ck ð0Þ 1 and ck ðTc Þ ¼ cH ðTc Þ 2:5 [2]. Present result of ck is considerably larger than theirs and almost temperature independent. It must be carefully investigated whether the procedure to deduce kc and kab are appropriate, however, if ck 3 is to be accepted, the way k’s are determined from hmjj2 i’s of individual Fermi surfaces should be reexamined. 1
λab ξab
100
10
Tc=37.6 K
4. Summary
1 0
10
20
30
40
T (K)
In conclusion, we evaluated the temperature dependence of the anisotropy parameter ck , and confirmed a large temperature dependence of cH . The value ck ffi 3 should be investigated based on the two band nature of MgB2 . Negligibly small anisotropy within the ab-plane was also verified. Note added in proof––After the symposium the authors reevaluated Hpjjc taking Heff ¼ Happ 4pNc Mc , to obtain Hpjjc nearly 13% smaller than those in the text, which leads to the ratio ck ð¼ 2:8 0:2Þ nearly 8% smaller.
10 1.0
2
λ (2.5K)/λ (T)
(b)
0.6 0.4
Anisotropy
2
8
λc λab
0.8
6
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
T/Tc
4
γH= ξab/ξc
2
γλ= λc/λab 0 0.0
0.2
0.4
0.6
References 0.8
1.0
T/Tc
Fig. 5. Temperature dependence of (a) kc ; kab ; nc ; nab and (b) the anisotropy ratios cH ¼ Hc2jjc =Hc2jjab and ck ¼ kc kab .
(1.93 ± 0.03 at low temperature region and 2.16 ± 0.03 near Tc ). For anisotropic superconductors with the gap varying on the Fermi surface it has been theoretically predicted that the ratios cH and ck are different in both the magnitude and temperature dependence [2,3]. Consensus is almost established about cH ðT Þ. As for ck ðT Þ, Kogan and
[1] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Nature 418 (2002) 758. [2] V.G. Kogan, S.L. Bud’ko, Physica C 385 (2003) 131. [3] A.A. Golubov, A. Brinkman, Phys. Rev. B 66 (2002) 054524. [4] Y. Machida et al., Phys. Rev. B 67 (2003) 094507. [5] G.K. Perkins et al., Supercond. Sci. Technol. 15 (2002) 1156. [6] L. Lyard et al., Phys. Rev. B 66 (2002) 180502. [7] M. Zehetmayer et al., Phys. Rev. B 66 (2002) 052505. [8] R. Cubitt et al., Phys. Rev. Lett. 90 (2003) 157002. [9] J. Matricon, in: R.D. Parks (Ed.), Superconductivity, vol. 2, Marcel Dekker, New York, 1969, pp. 861–863 (figures reprinted). [10] J.L. Harden, V. Arp, Cryogenics 3 (1963) 105.
1 Formally, ck values estimated ignoring the r-band or p-band are 0.9 or 6.8, respectively.
Anisotropy in the upper and lower critical fields of MgB2 single crystals H. Bando *, Y. Yamaguchi, N. Shirakawa, T. Yanagisawa National Institute of Advanced Industrial Science and Technology, AIST Tsukuba Central 2, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan Received 29 October 2003; accepted 5 January 2004 Available online 19 May 2004
Abstract MgB2 single crystals with a sharp transition width ðDTc 0:3 KÞ were grown by the ambient-pressure synthesis. The upper and lower critical fields, Hc2 and Hc1 were evaluated from the magnetization for the applied field l0 H 6 5 T parallel to the three crystallographic directions. The coherence length n and the penetration depth k for each direction were deduced from these critical fields with a little assumption for Hc1 . The anisotropy ratio cH ¼ nc =nab was estimated to be 6.5 at 2.5 K and 2.3 near Tc ( ¼ 37.6 K), and ck ¼ kc =kab ð¼ 3:0 0:2Þ to be almost constant over the range 2:5 K 6 T < Tc . 2004 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 74.60.Ec; 74.70.Ad Keywords: MgB2 ; Lower critical field; Upper critical field; Anisotropy
1. Introduction MgB2 attracts interest in its high superconducting transition temperature (Tc ) as metallic superconductor and its unique Fermi surface with two-gap property [1]. Effects of the two-band feature have been discussed in connection with anisotropy of the lower and the upper critical fields (Hc1 and Hc2 ) [2–8]. However, the discussion has been rather qualitative, partly because the experimental data are yet to be converged, especially
*
Corresponding author. Tel.: +81-29-861-5385; fax: +81-29861-5387. E-mail address: [email protected] (H. Bando).
concerning the anisotropy in the penetration depth which determines the Hc1 . In this paper, we report a detailed magnetization measurement on highquality single crystals of MgB2 .
2. Experimental Single crystals were grown from 99.9% boron with 99.99% magnesium sealed in a stainless tube whose inner side covered with molybdenum sheet [4]. The tube was kept for two weeks at 1120 C. Several pieces of crystal platelets with typical dimension of 500 · 400 · 60 lm3 (#A) were grown with many grains of smaller size. Laue-photographs of these crystals showed the 6/mmm Laue
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.055
H. Bando et al. / Physica C 412–414 (2004) 258–261
0.5
MgB2(#A) T=2.5 K
5 dM/dH//c
0.0
0
-0.5
-5
-1.0
M//a H//c H//a
-1.5
Hen//c
Hp//c
Hen//a
M//c
-3
-6
-2.0x10
0.00
-10
-15
Hp//a
-20x10 0.05
0.10
0.15
0.20
MgB2(#A)
µ0Hen (T)
0.25
This work
Hp//a Hp//c Hen//a Hen//c
0.20
0.15
Perkins et al.
Hen 0.10
0.05
0.00 0
10
20
30
40
T (K) Fig. 2. Temperature dependence of the observed Hpjjab , Hpjjc , Henjjc and Henjjab .
affected by the scan rate of H presumably because of negligible relaxation effect. Secondly, the Hp was almost free of sample-piece difference, while Hen heavily depended on the sample shape. For example, the ratios Hen =Hp for the sample #A were 0.9 and 0.85 under the field H jjab and H jjc, respectively, while for another sample #B (dimension of 600 · 400 · 60 lm3 ), respective ratios were 0.85 and 0.75, and for the sample #C (dimension of 600 · 200 · 40 lm3 ), 0.85 and 0.55. The samples with the shape closer to sphere tended to have the Hen =Hp ratio closer to unity. The values of Hp for both H jjab and H jjc are very close to Hen by Perkins et al. [5].
3. Results and discussion dM/dH
M (emu)
dM/dH//a
0.30
µ0Hp ,
symmetry. Tc was observed to be 37.6 K with the transition width DTc 0:3 K (Fig. 3). Magnetization was measured using SQUID magnetometers, MPMS-1 and MPMS-5 produced by Quantum Design. Typical M–H curves near the lower critical field (Hc1 ) are shown in Fig. 1, with the corresponding dM=dH –H curves. The figure includes two sets of curves, i.e. magnetic field perpendicular to the plate ðH jjcÞ or parallel to the plate ðH jjabÞ. These curves are plotted with Heff ¼ Happ =ð1 N Þ corrected by the demagnetization factor N . Since Nc (N for H jjc) was large and uncertain, while Nab (N for H jjab) was small and evaluated rather precisely from the sample shape, the Nc ð¼ 0:79Þ was determined empirically to fit the slope of the M–Hjjab curve for the a-axis, where Hjjab ¼ Heff ¼ Happ =ð1 Nab Þ with Nab ¼ 0:1 (for sample #A). This procedure is the same as that by Perkins et al. [5]. We defined both Hp (peak field of the M–H curve) and Hen (field where the M–H curve deviates quadratically from the linear feature). The temperature dependence of the observed Hp and Hen for both the axes are shown in Fig. 2. In this work, we took Hp as a measure of Hc1 because of the following two reasons. Firstly, Hp was not
259
0.25
µ0Heff (T) Fig. 1. M–H curves and the corresponding dM=dH –H curves to define Hp and Hen . A dotted arrow shows the sweep direction.
In Figs. 3a and b, M–T curves under various applied field H are shown. The curve for l0 H ¼ 0:2 mT is shown as a curve to determine the critical temperature Tc ðH 0). None of the M–T curves, except for that at l0 H ¼ 0:2 T, had hysteresis against the sweep direction of temperature. The Tc ðH Þ was determined as the temperature where the extrapolation of the linear part of the M–T curve crosses the temperature axis. In Fig. 3a, two curves for l0 H ¼ 3 T, both H jja and H jjb, are shown to elucidate the negligible anisotropy in the ab-plane. The anisotropy was within the
260
H. Bando et al. / Physica C 412–414 (2004) 258–261 6
1
(a)
MgB2(#A)
0 5 Tc(3T)=29.3 K
-2 -3 -4 -5x10
(1T)
(0.5T) (0.2T)
// a-axis 0.2mT 0.2T 0.5T 1T 3T // b*(=aXc) 3T
MgB2(#A) H//ab
4
Mzfc Tc(0.2mT) =37.6 K
30
Hc2//ab
3 Lyard Lyard et et al al
2
-6
28
µ0Hc2 (T)
M (emu)
-1
Hc2 from M//ab-vs-T M//c-vs-T M-vs-H//c Lyard et al
32
34
36
Hc2//c
38
T (K)
1
5
M (emu)
0
0
(b)
0
-5 Tc(0.6T) =31.2 K
-10 -15
40
Tc(0.2mT) =37.6 K
Fig. 4. Temperature dependence of Hc2jjab for H jjab-plane and Hc2jjc for Hjjc-axis. Data from Ref. [6] are shown as dashed curves.
-6
30
32
34
36
38
T (K)
0
(c) M (emu)
30
MgB2(#A) H//c
-20 -25x10
20
T (K)
(0.5T) (0.4T) (0.3T) (0.2T) (0.1T)
Mzfc
10
Hc2(17.5K)
-10
15 K
12.5K 10K
Hc2(2.5K) =25.5T
-20 2.5 K 5K
-30 7.5 K
MgB2(#A) H//c
-6
-40x10
1.6
1.8
2.0
2.2
2.4
2.6
0H (T)
Fig. 3. Determination of (a,b) Hc2 from Ma;b –T and (c) Ma –H curves. Two Ma;b –T curves for 3 T in (a) show absence of anisotropy in the ab-plane.
experimental errors of 0.1 K, which corresponds to 1% of Hc2 Since the experimental errors in Tc ðH Þ became large at temperatures below 10 K, the M–H curves were used to determine the Hc2 ðT Þ. As shown in Fig. 3c, M–H curves exhibited a small dip just below Hc2 which became obvious at lower temperatures. These M–H curves including the dip feature were reversible against the field sweep. Temperature dependence of Hc2jjab for H jjab and Hc2jjc for H jjc are shown in Fig. 4. The present measurement was limited in magnetic field less
than 5 T, no data of Hc2jjab below 25 K were obtained. However, the values are very close to those by Lyard et al. [6] for 25 K 6 T 6 Tc . In the following analysis, therefore, we used their Hc2jjab data for T 6 20 K. Moreover, a parabolic extrapolation was used for T ¼ 2:5 K. The coherence lengths, nc and nab , and their anisotropy cH ð¼ nab =nc Þ are easily evaluated from the values of Hc2 as Hc2jjc ¼ U0 =2pn2ab and Hc2jjab ¼ U0 =2pnc nab . Meanwhile, evaluation of the penetration depths is a little cumbersome. For H jjc, Hc1jjc ¼ ðU0 =4pk2ab ÞLðjc Þ, where jc ¼ kab =nab . Asymptotically, Lðjc Þ ¼ lnðjc Þ in the limit of large jc , but jc is not always large. Referring p the ffiffiffi numerical solutions [9,10] of arbitrary jc > 1= 2, we adopted the form: LðjÞ ¼ lnðjÞ þ ð0:25j þ 1 0:6Þ . For H jjab, the flux is elongated along the ab-plane. We simply extended the idea for H jjc and applied Hc1jjab ¼ ðU0 =4pkc kab ÞLðjab Þ, where 1=2 jab ¼ ðkc kab =nc nab Þ . These formulae are slightly different from those by Zehetmayer et al., [7] where they implicitly assumed that cH is equal to ck ¼ kc =kab . The evaluated values of the parameters are shown in Figs. 5a and b. The anisotropy ratio cH ¼ nc =nab ¼ Hc2jjab =Hc2jjc was 6.5 at 2.5 K and 2.3 near Tc ð¼ 37:6 KÞ. The deduced ratio ck ¼ kc = kab ð¼ 3:0 0:2Þ was almost constant over the observed range 2:5K 6 T < Tc while the ratio cH c1 ¼ Hc1jjc = Hc1jjab was slightly temperature dependent
H. Bando et al. / Physica C 412–414 (2004) 258–261
λ, ξ (nm)
1000
λc ξc
(a)
261
Bud’ko predicted that ck ð0Þ 1 and ck ðTc Þ ¼ cH ðTc Þ 2:5 [2]. Present result of ck is considerably larger than theirs and almost temperature independent. It must be carefully investigated whether the procedure to deduce kc and kab are appropriate, however, if ck 3 is to be accepted, the way k’s are determined from hmjj2 i’s of individual Fermi surfaces should be reexamined. 1
λab ξab
100
10
Tc=37.6 K
4. Summary
1 0
10
20
30
40
T (K)
In conclusion, we evaluated the temperature dependence of the anisotropy parameter ck , and confirmed a large temperature dependence of cH . The value ck ffi 3 should be investigated based on the two band nature of MgB2 . Negligibly small anisotropy within the ab-plane was also verified. Note added in proof––After the symposium the authors reevaluated Hpjjc taking Heff ¼ Happ 4pNc Mc , to obtain Hpjjc nearly 13% smaller than those in the text, which leads to the ratio ck ð¼ 2:8 0:2Þ nearly 8% smaller.
10 1.0
2
λ (2.5K)/λ (T)
(b)
0.6 0.4
Anisotropy
2
8
λc λab
0.8
6
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
T/Tc
4
γH= ξab/ξc
2
γλ= λc/λab 0 0.0
0.2
0.4
0.6
References 0.8
1.0
T/Tc
Fig. 5. Temperature dependence of (a) kc ; kab ; nc ; nab and (b) the anisotropy ratios cH ¼ Hc2jjc =Hc2jjab and ck ¼ kc kab .
(1.93 ± 0.03 at low temperature region and 2.16 ± 0.03 near Tc ). For anisotropic superconductors with the gap varying on the Fermi surface it has been theoretically predicted that the ratios cH and ck are different in both the magnitude and temperature dependence [2,3]. Consensus is almost established about cH ðT Þ. As for ck ðT Þ, Kogan and
[1] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Nature 418 (2002) 758. [2] V.G. Kogan, S.L. Bud’ko, Physica C 385 (2003) 131. [3] A.A. Golubov, A. Brinkman, Phys. Rev. B 66 (2002) 054524. [4] Y. Machida et al., Phys. Rev. B 67 (2003) 094507. [5] G.K. Perkins et al., Supercond. Sci. Technol. 15 (2002) 1156. [6] L. Lyard et al., Phys. Rev. B 66 (2002) 180502. [7] M. Zehetmayer et al., Phys. Rev. B 66 (2002) 052505. [8] R. Cubitt et al., Phys. Rev. Lett. 90 (2003) 157002. [9] J. Matricon, in: R.D. Parks (Ed.), Superconductivity, vol. 2, Marcel Dekker, New York, 1969, pp. 861–863 (figures reprinted). [10] J.L. Harden, V. Arp, Cryogenics 3 (1963) 105.
1 Formally, ck values estimated ignoring the r-band or p-band are 0.9 or 6.8, respectively.