In this work, we choose Nb3Al/Nb3Sn as a new test case for flat/steep band model of superconductivity. Based on the density functional theory in the generalized gradient approximation, the electronic structure of Nb3Al/Nb3Sn has been studied. The obtained results agree well with those of the earlier studies and show clearly flat bands around the Fermi level. The steep bands as characterized in this work locate around the M point in the first Brillouin zone. The obtained results reveal that Nb3Al/Nb3Sn fits more to the “Flat/steep” band model than to the van-Hove singularity scenario. The flat/steep band condition for superconductivity implies a different thermodynamic behavior of superconductors other than that predicted from the conventional BCS theory. This observation sets up an indicator for selecting a suitable superconductor when its large-scale industrial use is needed, for example, in superconducting maglev system or ITER project.
1 Introduction
Superconductors have found widespread applications ranging from civil to military uses. The superconductor-based Magnetic Resonance Imaging (MRI) technique used in many hospitals, the superconducting super-collider project in Texas USA, the Large Hadron Collider (LHC) in CERN, the ITER project, the Experimental Advanced Superconducting Tokomak (EAST) in China and so on are all examples where superconductors are necessary materials. Another important application is the superconducting magnetic levitation system, which not only saves much of the electrical energy but also saves the space and weight of the vehicle and dramatically increases its speed [1]. Such important applications require deeper and exact understanding of the properties, e.g. mechanical and critical ones, of a superconductor.
The flat/steep band model [2, 3] for superconductivity has been proposed over many years in attempting to understand superconductivity from a chemistry point of view. The previous studies [4] indicate that the electronic structures of conventional elemental and complex superconductors show good agreement with this model. However, this model met a non-trivial challenge in the case of Calcium, although it finally proved to be the shortcoming of the first-principle methods instead of that of the flat/steep band model [5]. Nb3Al/Nb3Sn is a typical A15 superconductor, which though has been intensively studied, respectively, remains a challenge to the existing theories regarding their strain and scale-dependent critical properties such as the upper critical magnetic field and current density. On the other hand, Nb3Al and Nb3Sn have very important applications on the thermonuclear fusion experimental reactor as magnetic plasma-control materials for their overall merit factors. The results of earlier studies on the preparations, crystal structures, electronic structures, and various properties of A15 compounds have been systematically reviewed by many authors [6‐9]. With respect to the electronic structures of A15 compounds, the earlier studies based on the first-principle methods [10‐12] have revealed very flat bands around the Fermi level, which implies a van-Hove singularity in A15 system. However, the earlier experimental studies have never observed any Peierls-like structural instability in a wide temperature range. This fact itself indicates that either the flat band feature is over estimated as in the case of Calcium [5] or there are some other factors which counteract the effect of the “flat/bands” or both reasons may play a role. In this work, we study the electronic structure of Nb3Al/Nb3Sn using the state-of-the-art first-principle method.
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2 Computational details
The electronic structure of Nb3Al/Nb3Sn has been calculated using the DFT theory as implemented in the Vienna ab initio simulation package (VASP) [13, 14]. The exchange–correlation potential has been calculated at the generalized gradient approximation (gga) level using the Perdew, Burke and Ernzerhof (PBE) formalism [15]. The 10 × 10 × 10 and 20 × 20 × 20 Monkhorst grids have been used to sample the first BZ for structural relaxation and self-consistent calculations, respectively. The cutoff energy for the plane wave expansion is chosen as 320 eV.
3 Results and discussion
3.1 Crystal structure
Nb3Al/Nb3Sn is a typical A15 compound with a space group \( Pm\bar{3}n \). The unit cell structure of the compound is shown in Fig. 1. The structure belongs to a simple cubic crystal system and contains six Nb atoms and 2 Al or Sn atoms in one conventional crystallographic unit cell. Each unit cell contains six A-type atoms which lie on the surface of the unit cell and form chains along the axis directions, respectively. On each chain, the A-type atoms distribute equidistantly. The two B-type atoms occupy the corners and the center of a unit cell. The A-type and B-type atoms occupy the 6c and 2a Wyckoff positions of the space group, respectively.
Fig. 1
The crystal structure of an A15 compound
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3.2 Electronic structure
The electronic density of states (DOS) of Nb3Al and Nb3Sn is shown in Fig. 2. As can be seen from Fig. 2, except for tiny shifts and changes of the peak positions and heights, the overall agreement of the DOS curves of Nb3Al and Nb3Sn is very good. The DOS around the Fermi level has a sharp peak, which implies a dramatic change of physical properties if the Fermi level is changed by whatsoever means, e.g. doping or stress. By decomposing the electronic states into atomic centered states, we have found that the states around the Fermi level are essentially contributed by the Nb d-states. It is thus obvious that the superconducting properties are crucially decided by the Nb atoms. The energetically lowest peaks at −5.6 and −7.9 eV for Nb3Al and Nb3Sn, respectively, are due to the s-states of the B-type atoms. These results agree generally with those of the earlier studies [12, 16].
Fig. 2
Density of states of Nb3Al and Nb3Sn
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The first BZ for Nb3Al and Nb3Sn is shown in Fig. 3 where the symmetry lines and points are indicated. The coordinates of the symmetry points are (0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), for Γ, X, M, and R, respectively. The calculated band structures for Nb3Al and Nb3Sn are shown in Figs. 4 and 5, respectively. As for the DOS curves, the band structures of Nb3Al and Nb3Sn are quite similar. In both cases, the lowest bands are from the s, p states of the B-type atoms. The bands lying in the range of −3.2 to 5 eV for Nb3Sn and those between −3 and 4.5 eV for Nb3Al are contributed by the d-states of Nb. These bands split into the bonding and anti-bonding groups. As shown in Figs. 4 and 5, the bonding bands are flatter than the anti-bonding bands.
Fig. 3
The first BZ for Nb3Al and Nb3Sn structures
Fig. 4
Electronic band structure of Nb3Al along symmetry directions
Fig. 5
Electronic band structure of Nb3Sn along symmetry directions
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As revealed in earlier studies [10‐12], the bands around the Fermi level of Nb3Al and Nb3Sn appear rather flat which is, however, someway an artificial fact due to the chosen large energy scale in representing the band structure. Moreover, the earlier results are not consistent with each other with respect to the accurate position of the flat bands relative to the Fermi level [10‐12]. Our investigations on the bands around the Fermi level indicate that the energy bands nearby Γ and R points are significantly flatter than some other bands, while those around the M point are obviously steeper. These steep bands play an important role in counteracting the possible structural instability caused by the high DOS (EF). In this sense, Nb3Al and Nb3Sn fit more to the “flat/steep” band model than to the van-Hove singularity scenario. It is evident that more quantitative experimental data are needed to judge the theoretical results. In particular, the metric details of the Fermi surface are necessary to draw a solid conclusion.
4 Ending remarks
Our studies indicate that the importance of the steep band was not realized in earlier work. The steep band character in the A15 system was neglected due to the use of large energy scale. Our studies indicate clearly that the steep bands locate around the M point in the first BZ (Fig. 4). The obtained results reveal that Nb3Al/Nb3Sn fits more to the “flat/steep” band model than to the van-Hove scenario.
Acknowledgements
This work is financially supported by the Science Foundation for International Cooperation of Sichuan Province (2014HH0016), the Fundamental Research Funds for the Central Universities (SWJTU2014: A0920502051113-10000), National Magnetic Confinement Fusion Science Program (2011GB112001).
This article is published under license to BioMed Central Ltd. Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
