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Anomalous upper critical field in ternary iron-silicide superconductor Lu2Fe3Si5
Physica C 469 (2009) 921–923
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Anomalous upper critical field in ternary iron-silicide superconductor Lu2Fe3Si5 Y. Nakajima a,*, H. Hidaka a, T. Tamegai a, T. Nishizaki b, T. Sasaki b, N. Kobayashi b a b
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan
a r t i c l e
i n f o
Article history: Available online 30 May 2009 PACS: 74.25.Op 74.70.Dd Keywords: Two-gap superconductor Upper critical field Lu2Fe3Si5 MgB2
a b s t r a c t We report the upper critical field Hc2 in a ternary iron-silicide superconductor Lu2Fe3Si5 with Tc 6 K obtained by the resistivity measurements. We find that Hc2 increases linearly with decreasing temperature down to Tc/3, and Hc2(T = 0) exceeds the orbital depairing field described by the Werthamer–Helfand–Hohenberg theory. We also find that the anisotropy of Hc2 is nearly independent of temperature and the angular dependence of Hc2 is well-described by the anisotropic Ginzburg–Landau model. These results strongly indicate the presence of two distinct superconducting gaps in Lu2Fe3Si5 although the behavior is slightly different from that of the typical two-gap superconductor MgB2. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. Experimental details
The discovery of the high temperature superconductivity of MgB2 with Tc 39 K has renewed interest in superconductors with multicomponent order parameters [1]. To shed light on universal properties of the multi-gap superconductivity, distinct and simple two-gap superconductors other than MgB2 have been sought for. Very recently, detailed study of specific heat in a ternary iron-silicide superconductor Lu2Fe3Si5 with Tc 6 K have revealed the definite two-gap superconductivity, which is similar to MgB2 [2]. This material can be a candidate for a canonical two-gap superconductor because of the simplicity of the multi-gap nature, i.e., the bands with the larger and smaller gaps have nearly the same contribution to the density of states, which is similar to MgB2. Therefore, the study of the superconducting properties in Lu2Fe3Si5 is suitable to clarify the details of two-gap superconductors. In this paper, to investigate the details of two-gap superconductivity, we address the issue of the upper critical field Hc2 of Lu2Fe3Si5 by measuring the temperature and angular dependence of the resistivity using the single crystal. We report that the unusual behavior of upper critical field can be described by the two-gap model. Similarity and difference in Hc2 between Lu2Fe3Si5 and MgB2 are discussed in detail.
Single crystals were obtained by the floating-zone method using an image furnace in Ar atmosphere at a rate of 2 mm/h. The starting polycrystalline rod was prepared by fusing several polycrystalline lumps, which were prepared by arc melting a 2:3:5 stoichiometric mixture of Lu, Fe, and Si in Ar atmosphere. The single crystals were annealed at 1250 °C for 5 days, followed by annealing at 800 °C for four weeks. Resistivity measurements were performed by the standard four contact method in a 3He cryostat with the rotating sample stage.
* Corresponding author. Address: Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. Tel./fax: +81 3 5841 6848. E-mail address: [email protected] (Y. Nakajima). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.099
3. Results and discussion Fig. 1 shows the temperature dependence of the upper critical field along with the a- and c-axis obtained from the resistivity measurements. The upper critical fields are determined by the midpoint of the resistive transition. The value of Hc2 increases almost linearly with decreasing temperature down to Tc/3, which is very different from that of the conventional type-II superconductors. It is well known that the temperature dependence of the upper critical field in conventional type-II superconductors is well-described by the Werthamer–Helfand–Hohenberg (WHH) theory [3], and there is a simple universal relation between the zero-temperature value Hc2(0) and the slope of Hc2 at T c; jdHc2 =dTjT¼T c j as
Hc2 ð0Þ ¼ 0:69T c jdHc2 dTjT¼T c j:
ð1Þ
922
Y. Nakajima et al. / Physica C 469 (2009) 921–923
Fig. 1. Temperature dependence of the upper critical fields along a- and c-axis for Lu2Fe3Si5 obtained from the resistivity measurements. Dashed lines represent linear fits to the data. Thin lines are the WHH curves. Thick lines represent the calculation by the two-gap model. From the fit, we obtained the parameters, k11 ¼ 0:202; k22 ¼ 0:103; k12 ¼ 0:059; k21 ¼ 0:053, and g = 0.19.
The thin solid lines in Fig. 1 are the curves calculated by the WHH theory, which are apparently different from the present results. Surprisingly, the values of Hc2(0) along the c- and a-axis for Lu2Fe3Si5 are about 11 T and 6 T, respectively, which correspond to 0:9T c jdHc2 dTjT¼T c j and are much larger than the values expected from WHH theory. Fig. 2 shows the temperature dependence of the anisotropy c obtained by the ratio of the upper critical fields along the a- and c c-axis, Hab c2 =Hc2 . The anisotropy c is almost nearly independent of temperature. Its value is about 0.5, which reflects the weakly one-dimensional shape of the Fermi surface expected by the band structure calculation [2]. It should be noted that T-independent anisotropy of Hc2 for Lu2Fe3Si5 is very different from that for a typical two-gap superconductor MgB2 [4,5]. Fig. 3 depicts the upper critical field as a function of angle h between the magnetic field and the ab-plane at several temperatures. Reflecting the anisotropy, twofold symmetry is observed. In the simple anisotropic superconductors, the anisotropy of Hc2 is well-described by the anisotropic Ginzburg–Landau (GL) model written as,
Fig. 2. Temperature dependence of the anisotropy of the upper critical field c Hab c2 =Hc2 .
Fig. 3. Angular dependence of the upper critical field for Lu2Fe3Si5. Thin lines represent the curve calculated by the anisotropic GL model.
Hc2 ðhÞ ¼ Hab c2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 h þ c2 sin h:
ð2Þ
We note that the angular dependence of Hc2 in MgB2 does not obey the anisotropic GL model since r- and p-band responsible for two gaps have different anisotropy [5]. In order to check the validity of the anisotropic GL model for Lu2Fe3Si5, we fit the data by using Eq. (2). The solid lines shown in Fig. 3 represent the curves calculated by the anisotropic GL model. There is no explicit deviation from GL behavior, unlike the case of MgB2. To investigate the deviation from GL model more precisely, we plot the ratio h . i2 as a function of cos2 h as shown in Fig. 4. When Hc2 ðhÞ HGL c2 ðhÞ h . i2 should be indethe anisotropic GL model is valid, Hc2 ðhÞ HGL c2 ðhÞ pendent of cos2h as shown by the dashed line in Fig. 4. Apparently, the data is independent of both temperature and cos2 h, and coincide with the dashed line, which strongly indicates the anisotropic h . i2 for GL model is valid. Note that the values of Hc2 ðhÞ HGL c2 ðhÞ MgB2 definitely deviate from the anisotropic GL value and shows strongly temperature (not shown) and angular dependence as shown by open circles in Fig. 4 [5]. Summarizing anomalous properties of the upper critical field in dependence of Lu2Fe3Si5, we have found: (1) unusual temperature Hc2, (2) large values of Hc2 ð0Þ 0:9T c jdHc2 dTjT¼T c j; (3) T-independent anisotropy of Hc2, and (4) angular dependence of Hc2 well-described by anisotropic GL model, all of which are different from the behavior of Hc2 in MgB2. In order to clarify these anomalous behavior of upper critical field in Lu2Fe3Si5, we will analyze the present results using the model describing the upper critical field for
Fig. 4. Angular dependence of the upper critical field normalized by that expected from GL model for Lu2Fe3Si5. Dashed lines represent the anisotropic GL model. The open circles show the data of MgB2 at 25.0 K [5].
Y. Nakajima et al. / Physica C 469 (2009) 921–923
two-gap superconductors [6,7]. In this model, using the reduced temperature t ¼ T=T c , reduced magnetic field h ¼ Hc2 D1 =2/0 T, and the ratio of diffusivities g ¼ D2 =D1 , where Di is a diffusivity for the band i, the upper critical field is determined by the following equations:
a0 ½ln t þ UðhÞ½ln t þ UðghÞ þ a1 ½ln t þ UðhÞ þ a2 ½ln t þ UðghÞ ¼ 0; UðxÞ ¼ wðx þ 1=2Þ wð1=2Þ;
ð3Þ ð4Þ
where w is a di-gamma function. Parameters a0, a1, and a2 are given 1 k =k0 , respectively, with as a0 ¼ 2w=k0 ; a1 ¼ 1 þ k =k0 , and a2 ¼ 1=2 , and k ¼ k11 k22 . Here, w ¼ k11 k22 k12 k21 ; k0 ¼ k2 þ 4k12 k21 kii and kij are intra- and inter-band coupling constants, respectively, with indices 1 and 2 corresponding to the bands with large and small superconducting gaps. For equal diffusivities, g = 1, Eq. (3) reduces to the equation ln t þ UðhÞ ¼ 0 for Hc2 in one gap superconductors [3,8,9]. We plot the curves obtained from Eq. (3) in Fig. 1 with the parameter k11 ¼ 0:202; k22 ¼ 0:103; k12 ¼ 0:059; k21 ¼ 0:053, and g = 0.19, which reproduces the present result. Here, we assume the band 1 is anisotropic because band structure calculation indicates that Fermi surfaces have weakly one-dimensional anisotropic parts and three-dimensional isotropic parts [2]. This assumption is necessary to explain the temperature dependence of anisotropy, which will be shown later. These parameters also reproduce the transition temperature by using the equation [10],
T c ¼ 1:14hxD exp½ðkþ k0 Þ=2w:
ð5Þ
It should be noted that the parameters of MgB2 obtained from ab-initio calculation are k11 ¼ 0:81; k22 ¼ 0:278; k12 ¼ 0:115; k21 ¼ 0:091, and g = 5.0 [11]. According to the theory, Hc2 in two-gap superconductors can be enhanced at low temperatures for g < 1. In Lu2Fe3Si5, D2 is about one fifth of D1. Then, Hc2 can be enhanced and exceeds the single-band WHH values of Hc2 ð0Þ ¼ 0:69T c jdHc2 dTjT¼T c j. In two-band superconductivity, the angular dependence of Hc2 is written as,
Hc2 ðhÞ ¼ 8/0 ðT c TÞ=p½a1 D1 ðhÞ þ a2 D2 ðhÞ;
ð6Þ
h i1=2 2 ðaÞ2 ðaÞ ðcÞ where, Di ðhÞ ¼ Di cos2 h þ Di Di sin h . The sum of the two angular-dependent diffusivities D1(h) and D2(h) in the denominator of Eq. (6) can result in a deviation of Hc2(h) from anisotropic GL model, such in the case of MgB2. However, in Lu2Fe3Si5, by using
923
the parameters obtained from the fit to the Hc2 data, we can obtain a2 D2 =a2 D1 1. Then, Eq. (6) reduces the one-gap anisotropic GL dependence dominated only by the anisotropy of the band 1 with the larger gap. Therefore, we can explain the present results, assuming that band 1 has weakly one-dimensional anisotropic Fermi surfaces. 4. Summary We have prepared the single crystals of ternary iron-silicide superconductor Lu2Fe3Si5 and studied the details of upper critical field in the entire temperature range. The measurements reveal a linear increase of Hc2 with decreasing temperature down to Tc/3 and striking excess of Hc2(0) over the WHH value 0:69T c jdHc2 =dTjT¼T c j. We also find that the anisotropy of Hc2 is nearly independent of temperature and well-described by the anisotropic GL model. These results can be well-described by the phenomenological two-gap model with the assumption of the weakly one-dimensional band with large superconducting gap and the three-dimensional band with small gap, which are expected from the band structure calculation. Acknowledgement This work is partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology.
References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature (London) 410 (2001) 63. [2] Y. Nakajima, T. Nakagawa, T. Tamegai, H. Harima, Phys. Rev. Lett. 100 (2008) 157001. [3] N.R. Werthamer, E. Helfand, P.C. Hohenberg, Phys. Rev. 147 (1966) 295. [4] L. Lyard, P. Szabo, T. Klein, J. Marcus, C. Marcenat, K.H. Kim, B.W. Kang, H.-S. Lee, S.-I. Lee, Phys. Rev. Lett. 92 (2004) 057001. [5] A. Rydh, U. Welp, A.E. Koshelev, W.K. Kwok, G.W. Crabtree, R. Brusetti, L. Lyard, T. Klein, C. Marcenat, B. Kang, K.H. Kim, K.H.P. Kim, H.-S. Lee, S.-I. Lee, Phys. Rev. B 70 (2004) 1325503. [6] A. Gurevich, Phys. Rev. B 67 (2003) 184515. [7] A. Gurevich, Physica C 456 (2007) 160. [8] P.G. de Gennes, Phys. Kondens. Mater 3 (1964) 79. [9] K. Maki, Phys. Rev. 148 (1966) 362. [10] H. Suhl, B.T. Matthias, L.R. Walker, Phys. Rev. Lett. 3 (1959) 552. [11] A.A. Golubov, J. Kortus, O.V. Dolgov, O. Jepsen, Y. Kong, O.K. Andersen, B.J. Gibson, K. Ahn, R.K. Kremer, J. Phys.: Condens. Matter 14 (2002) 1353.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Anomalous upper critical field in ternary iron-silicide superconductor Lu2Fe3Si5 Y. Nakajima a,*, H. Hidaka a, T. Tamegai a, T. Nishizaki b, T. Sasaki b, N. Kobayashi b a b
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan
a r t i c l e
i n f o
Article history: Available online 30 May 2009 PACS: 74.25.Op 74.70.Dd Keywords: Two-gap superconductor Upper critical field Lu2Fe3Si5 MgB2
a b s t r a c t We report the upper critical field Hc2 in a ternary iron-silicide superconductor Lu2Fe3Si5 with Tc 6 K obtained by the resistivity measurements. We find that Hc2 increases linearly with decreasing temperature down to Tc/3, and Hc2(T = 0) exceeds the orbital depairing field described by the Werthamer–Helfand–Hohenberg theory. We also find that the anisotropy of Hc2 is nearly independent of temperature and the angular dependence of Hc2 is well-described by the anisotropic Ginzburg–Landau model. These results strongly indicate the presence of two distinct superconducting gaps in Lu2Fe3Si5 although the behavior is slightly different from that of the typical two-gap superconductor MgB2. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. Experimental details
The discovery of the high temperature superconductivity of MgB2 with Tc 39 K has renewed interest in superconductors with multicomponent order parameters [1]. To shed light on universal properties of the multi-gap superconductivity, distinct and simple two-gap superconductors other than MgB2 have been sought for. Very recently, detailed study of specific heat in a ternary iron-silicide superconductor Lu2Fe3Si5 with Tc 6 K have revealed the definite two-gap superconductivity, which is similar to MgB2 [2]. This material can be a candidate for a canonical two-gap superconductor because of the simplicity of the multi-gap nature, i.e., the bands with the larger and smaller gaps have nearly the same contribution to the density of states, which is similar to MgB2. Therefore, the study of the superconducting properties in Lu2Fe3Si5 is suitable to clarify the details of two-gap superconductors. In this paper, to investigate the details of two-gap superconductivity, we address the issue of the upper critical field Hc2 of Lu2Fe3Si5 by measuring the temperature and angular dependence of the resistivity using the single crystal. We report that the unusual behavior of upper critical field can be described by the two-gap model. Similarity and difference in Hc2 between Lu2Fe3Si5 and MgB2 are discussed in detail.
Single crystals were obtained by the floating-zone method using an image furnace in Ar atmosphere at a rate of 2 mm/h. The starting polycrystalline rod was prepared by fusing several polycrystalline lumps, which were prepared by arc melting a 2:3:5 stoichiometric mixture of Lu, Fe, and Si in Ar atmosphere. The single crystals were annealed at 1250 °C for 5 days, followed by annealing at 800 °C for four weeks. Resistivity measurements were performed by the standard four contact method in a 3He cryostat with the rotating sample stage.
* Corresponding author. Address: Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. Tel./fax: +81 3 5841 6848. E-mail address: [email protected] (Y. Nakajima). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.099
3. Results and discussion Fig. 1 shows the temperature dependence of the upper critical field along with the a- and c-axis obtained from the resistivity measurements. The upper critical fields are determined by the midpoint of the resistive transition. The value of Hc2 increases almost linearly with decreasing temperature down to Tc/3, which is very different from that of the conventional type-II superconductors. It is well known that the temperature dependence of the upper critical field in conventional type-II superconductors is well-described by the Werthamer–Helfand–Hohenberg (WHH) theory [3], and there is a simple universal relation between the zero-temperature value Hc2(0) and the slope of Hc2 at T c; jdHc2 =dTjT¼T c j as
Hc2 ð0Þ ¼ 0:69T c jdHc2 dTjT¼T c j:
ð1Þ
922
Y. Nakajima et al. / Physica C 469 (2009) 921–923
Fig. 1. Temperature dependence of the upper critical fields along a- and c-axis for Lu2Fe3Si5 obtained from the resistivity measurements. Dashed lines represent linear fits to the data. Thin lines are the WHH curves. Thick lines represent the calculation by the two-gap model. From the fit, we obtained the parameters, k11 ¼ 0:202; k22 ¼ 0:103; k12 ¼ 0:059; k21 ¼ 0:053, and g = 0.19.
The thin solid lines in Fig. 1 are the curves calculated by the WHH theory, which are apparently different from the present results. Surprisingly, the values of Hc2(0) along the c- and a-axis for Lu2Fe3Si5 are about 11 T and 6 T, respectively, which correspond to 0:9T c jdHc2 dTjT¼T c j and are much larger than the values expected from WHH theory. Fig. 2 shows the temperature dependence of the anisotropy c obtained by the ratio of the upper critical fields along the a- and c c-axis, Hab c2 =Hc2 . The anisotropy c is almost nearly independent of temperature. Its value is about 0.5, which reflects the weakly one-dimensional shape of the Fermi surface expected by the band structure calculation [2]. It should be noted that T-independent anisotropy of Hc2 for Lu2Fe3Si5 is very different from that for a typical two-gap superconductor MgB2 [4,5]. Fig. 3 depicts the upper critical field as a function of angle h between the magnetic field and the ab-plane at several temperatures. Reflecting the anisotropy, twofold symmetry is observed. In the simple anisotropic superconductors, the anisotropy of Hc2 is well-described by the anisotropic Ginzburg–Landau (GL) model written as,
Fig. 2. Temperature dependence of the anisotropy of the upper critical field c Hab c2 =Hc2 .
Fig. 3. Angular dependence of the upper critical field for Lu2Fe3Si5. Thin lines represent the curve calculated by the anisotropic GL model.
Hc2 ðhÞ ¼ Hab c2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 h þ c2 sin h:
ð2Þ
We note that the angular dependence of Hc2 in MgB2 does not obey the anisotropic GL model since r- and p-band responsible for two gaps have different anisotropy [5]. In order to check the validity of the anisotropic GL model for Lu2Fe3Si5, we fit the data by using Eq. (2). The solid lines shown in Fig. 3 represent the curves calculated by the anisotropic GL model. There is no explicit deviation from GL behavior, unlike the case of MgB2. To investigate the deviation from GL model more precisely, we plot the ratio h . i2 as a function of cos2 h as shown in Fig. 4. When Hc2 ðhÞ HGL c2 ðhÞ h . i2 should be indethe anisotropic GL model is valid, Hc2 ðhÞ HGL c2 ðhÞ pendent of cos2h as shown by the dashed line in Fig. 4. Apparently, the data is independent of both temperature and cos2 h, and coincide with the dashed line, which strongly indicates the anisotropic h . i2 for GL model is valid. Note that the values of Hc2 ðhÞ HGL c2 ðhÞ MgB2 definitely deviate from the anisotropic GL value and shows strongly temperature (not shown) and angular dependence as shown by open circles in Fig. 4 [5]. Summarizing anomalous properties of the upper critical field in dependence of Lu2Fe3Si5, we have found: (1) unusual temperature Hc2, (2) large values of Hc2 ð0Þ 0:9T c jdHc2 dTjT¼T c j; (3) T-independent anisotropy of Hc2, and (4) angular dependence of Hc2 well-described by anisotropic GL model, all of which are different from the behavior of Hc2 in MgB2. In order to clarify these anomalous behavior of upper critical field in Lu2Fe3Si5, we will analyze the present results using the model describing the upper critical field for
Fig. 4. Angular dependence of the upper critical field normalized by that expected from GL model for Lu2Fe3Si5. Dashed lines represent the anisotropic GL model. The open circles show the data of MgB2 at 25.0 K [5].
Y. Nakajima et al. / Physica C 469 (2009) 921–923
two-gap superconductors [6,7]. In this model, using the reduced temperature t ¼ T=T c , reduced magnetic field h ¼ Hc2 D1 =2/0 T, and the ratio of diffusivities g ¼ D2 =D1 , where Di is a diffusivity for the band i, the upper critical field is determined by the following equations:
a0 ½ln t þ UðhÞ½ln t þ UðghÞ þ a1 ½ln t þ UðhÞ þ a2 ½ln t þ UðghÞ ¼ 0; UðxÞ ¼ wðx þ 1=2Þ wð1=2Þ;
ð3Þ ð4Þ
where w is a di-gamma function. Parameters a0, a1, and a2 are given 1 k =k0 , respectively, with as a0 ¼ 2w=k0 ; a1 ¼ 1 þ k =k0 , and a2 ¼ 1=2 , and k ¼ k11 k22 . Here, w ¼ k11 k22 k12 k21 ; k0 ¼ k2 þ 4k12 k21 kii and kij are intra- and inter-band coupling constants, respectively, with indices 1 and 2 corresponding to the bands with large and small superconducting gaps. For equal diffusivities, g = 1, Eq. (3) reduces to the equation ln t þ UðhÞ ¼ 0 for Hc2 in one gap superconductors [3,8,9]. We plot the curves obtained from Eq. (3) in Fig. 1 with the parameter k11 ¼ 0:202; k22 ¼ 0:103; k12 ¼ 0:059; k21 ¼ 0:053, and g = 0.19, which reproduces the present result. Here, we assume the band 1 is anisotropic because band structure calculation indicates that Fermi surfaces have weakly one-dimensional anisotropic parts and three-dimensional isotropic parts [2]. This assumption is necessary to explain the temperature dependence of anisotropy, which will be shown later. These parameters also reproduce the transition temperature by using the equation [10],
T c ¼ 1:14hxD exp½ðkþ k0 Þ=2w:
ð5Þ
It should be noted that the parameters of MgB2 obtained from ab-initio calculation are k11 ¼ 0:81; k22 ¼ 0:278; k12 ¼ 0:115; k21 ¼ 0:091, and g = 5.0 [11]. According to the theory, Hc2 in two-gap superconductors can be enhanced at low temperatures for g < 1. In Lu2Fe3Si5, D2 is about one fifth of D1. Then, Hc2 can be enhanced and exceeds the single-band WHH values of Hc2 ð0Þ ¼ 0:69T c jdHc2 dTjT¼T c j. In two-band superconductivity, the angular dependence of Hc2 is written as,
Hc2 ðhÞ ¼ 8/0 ðT c TÞ=p½a1 D1 ðhÞ þ a2 D2 ðhÞ;
ð6Þ
h i1=2 2 ðaÞ2 ðaÞ ðcÞ where, Di ðhÞ ¼ Di cos2 h þ Di Di sin h . The sum of the two angular-dependent diffusivities D1(h) and D2(h) in the denominator of Eq. (6) can result in a deviation of Hc2(h) from anisotropic GL model, such in the case of MgB2. However, in Lu2Fe3Si5, by using
923
the parameters obtained from the fit to the Hc2 data, we can obtain a2 D2 =a2 D1 1. Then, Eq. (6) reduces the one-gap anisotropic GL dependence dominated only by the anisotropy of the band 1 with the larger gap. Therefore, we can explain the present results, assuming that band 1 has weakly one-dimensional anisotropic Fermi surfaces. 4. Summary We have prepared the single crystals of ternary iron-silicide superconductor Lu2Fe3Si5 and studied the details of upper critical field in the entire temperature range. The measurements reveal a linear increase of Hc2 with decreasing temperature down to Tc/3 and striking excess of Hc2(0) over the WHH value 0:69T c jdHc2 =dTjT¼T c j. We also find that the anisotropy of Hc2 is nearly independent of temperature and well-described by the anisotropic GL model. These results can be well-described by the phenomenological two-gap model with the assumption of the weakly one-dimensional band with large superconducting gap and the three-dimensional band with small gap, which are expected from the band structure calculation. Acknowledgement This work is partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology.
References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature (London) 410 (2001) 63. [2] Y. Nakajima, T. Nakagawa, T. Tamegai, H. Harima, Phys. Rev. Lett. 100 (2008) 157001. [3] N.R. Werthamer, E. Helfand, P.C. Hohenberg, Phys. Rev. 147 (1966) 295. [4] L. Lyard, P. Szabo, T. Klein, J. Marcus, C. Marcenat, K.H. Kim, B.W. Kang, H.-S. Lee, S.-I. Lee, Phys. Rev. Lett. 92 (2004) 057001. [5] A. Rydh, U. Welp, A.E. Koshelev, W.K. Kwok, G.W. Crabtree, R. Brusetti, L. Lyard, T. Klein, C. Marcenat, B. Kang, K.H. Kim, K.H.P. Kim, H.-S. Lee, S.-I. Lee, Phys. Rev. B 70 (2004) 1325503. [6] A. Gurevich, Phys. Rev. B 67 (2003) 184515. [7] A. Gurevich, Physica C 456 (2007) 160. [8] P.G. de Gennes, Phys. Kondens. Mater 3 (1964) 79. [9] K. Maki, Phys. Rev. 148 (1966) 362. [10] H. Suhl, B.T. Matthias, L.R. Walker, Phys. Rev. Lett. 3 (1959) 552. [11] A.A. Golubov, J. Kortus, O.V. Dolgov, O. Jepsen, Y. Kong, O.K. Andersen, B.J. Gibson, K. Ahn, R.K. Kremer, J. Phys.: Condens. Matter 14 (2002) 1353.