Electronic Properties of Sr₄V₂O₆Fe₂As₂ Superconductor by Mean of DFT Investigations

Abstract

Based on the Full potential linearized augmented plane wave (FLAPW) method, the electronic properties of incommensurate anti-ferromagnetic Sr₄V₂O₆Fe₂As₂ with wave vector q = (0, 0, 0.306) have been investigated. Comparing total energy between those of non-magnetic and incommensurate anti-ferromagnetic states (i-AF) we suggest that the i-AF is the stable configuration in this superconductor. The density of states data exhibit strong spin-polarized effect on the V site and possible hybridization between V-3d and Fe-3d orbitals. We found as many as six bands crossing the Fermi level, indicating strong inter-band scattering. The analysis of Fermi surfaces reveals a multi-sheet character, which is compiled of several hole-type cylinders around the center and electron-type sheets at the corners of the Brillouin zone.

Fermi surfaces of Sr₂V₄O₆Fe₂As₂ show the nodal-gap structure.

1. Introduction

Since the high-Tc superconductivity was firstly discovered in La[O_1−xF_x]FeAs by Kamihara et al.,¹⁾ many higher Tc Fe-based superconductors have been identified with various unconventional properties.^2–8) These compounds are generally build up by Fe₂As₂ layers alternating with non-superconducting blocks. One of the Fe-based HTc-families of layered structure has the stoichiometry A₄M₂O₆Fe₂As₂ (A = Ca, Sr, Ba and M = V, Cr, Sc)^9,10) and is denoted in this were as 42622. The compounds of this family crystalize in the ZrSiCuAs-type tetragonal structure are characterized by the common FeAs layers and A₂MO₃ perovskite blocks.

Among them, Sr₄V₂O₆Fe₂As₂ has many interesting properties, e.g., Tc attains a high as 37 K, even up to 45 K under high pressure.¹¹⁾ In addition, there is the presence of magnetic V ions, which probably participate in the formation of the Cooper pairs. Up to now, although many theoretical^12–16) and experimental^17–19) efforts have been devoted, the magnetic state of Fe and V atoms as well as the influence of V orbitals on the superconductivity have not been clarified. Apparently, the discrepancy between the experimental, theoretical results of the system is still a compelling challenge to be solved. Remarkably, using ⁵⁷Fe Mössbauer spectroscopy measurement Cao et al. proved the non-magnetic (NM) arrangement in the Fe orbitals.¹⁷⁾ Moreover, Tegel et al. using the neutron-scattering experiment showed the incommensurate antiferromagnetic (i-AF) ordering of the V moment vector q = (0, 0, 0.306).¹⁶⁾ Unfortunately, these experimental results have not been reported by theoretical studies, where magnetic arrangements of both Fe and V moments have been considered.^12–16) Notably, two AF magnetic configurations: checkerboard and stripe structures of V have been embodied. Therefore, in order to examine the stability of AF configuration as well as to find out the role of V in the magnetism coexistent state of superconductivity, we investigate the electronic properties of NM and i-AF Sr₄V₂O₆Fe₂As₂. In this paper, the electronic properties of Sr₄V₂O₆Fe₂As₂ superconductor such as total density of states (DOS), partial density of states (PDOS), electronic band structures (EBS), Fermi surfaces (FS) will be conceptualized.

2. Computational details

The projector augmented wave (PAW) method implemented in the Vienna ab initio simulation package (VASP)²⁰⁾ was employed to optimize geometry. The input crystal parameters were taken from our experiment results such as the unit cell data are a = b = 3.9318 Å, c = 15.6910 Å and the atomic positions include z_Sr1 = 0.1903, z_Sr2 = 0.4145, z_V = 0.3081, z_O1 = 0.2922, z_O2 = 0.4318, z_Fe = 0.0, z_As = 0.0909.¹⁶⁾ The Monkhorst-Pack grid 20 × 20 × 5 size within 198 k-points were applied for the Brillouin zone. The self-consistent relaxation step was done with the trial-energy change around 5 × 10⁻⁵ eV and the force approximately 0.01 eV/Å.

Relying on the relaxed crystallographic parameters (by VASP: a = b = 3.7937 Å, c = 14.5016 Å and z_Sr1 = 0.1768, z_Sr2 = 0.4121, z_V = 0.2944, z_O1 = 0.2876, z_O2 = 0.4268, z_Fe = 0.0, z_As = 0.0864; by ELK: a = b = 3.7019 Å, c = 13.8128 Å and z_Sr1 = 0.1764, z_Sr2 = 0.4125, z_V = 0.2888, z_O1 = 0.2871, z_O2 = 0.4259, z_Fe = 0.0, z_As = 0.0897) the electronic properties were calculated using the all-electron full-potential linearized augmented-plane wave (LAPW) method employed in the ELK code.²¹⁾ The self-consistent procedure finishes where the total energy attains less than 1 meV. The obtained muffin-tin radii of all atoms are as R_Sr = 2.600 (a.u.), R_V = 2.025 (a.u.), R_O = 1.152 (a.u.), R_Fe = R_As = 2.123 (a.u.). The Brillouin zone was chosen in 16 × 16 × 4 mesh with 144 k-points. In the Fermi surface calculations, we chose 60 × 60 × 60 k-points mesh. The electronic band structures were established along the high-symmetry line Z–R–X–M–Γ–Z–A–M–R–A. Additionally, the frequency distribution of phonons was investigated by the harmonic approximation and supercell method.²²⁾ A plane-wave energy cutoff of 500 eV and the energy convergence criterion of 10⁻⁸ eV were used. A 24 × 24 × 6 k-point sampling mesh and supercell with the dimension 3 × 3 × 1 of 144 atoms were utilized in the phonon calculations.

We applied the generalized gradient approximations (PBEsol-GGA),²³⁾ spin-orbit interaction, and spin-polarized effect in the whole study.

3. Results and Discussions

The phonon dispersion of Sr₄V₂O₆Fe₂As₂ is shown in Fig. 1 (Fig. 1 should be here). Apparently, the spectrum has negative frequencies around the Γ point, indicating a softness of phonon modes. There are several reasons for obtaining imaginary modes. For our considered here compound, the first reason could be the existence of some phonon-magnetic coupling. The incommensurate AF ordering, not only supplies magnetic interactions but certainly lowers the crystal structure, which was not taken into account in our calculations yet. Another reason may be that cooperative interactions, e.g., spin fluctuations, Jahn-Teller effect, Hubbard interaction, etc, allow for the electron-phonon coupling to preserve the softness of the electronic mode. In the superconducting Sr₄V₂O₆Fe₂As₂, the influence of electron-phonon on the phonon spectra is quite possible. In fact, the formation of cooper pairs in a high Tc superconductor has been linked to the mechanism based on the spin fluctuation model.^24,25) Moreover, the V³⁺ ions located in the perovskite Sr₄V₂O₆ block are mostly Jahn-Teller active. Therefore, further calculations and experiments are needed to delineate the appropriate mechanism of the imaginary frequencies.

Fig. 1

Phonon dispersion of Sr₄V₂O₆Fe₂As₂.

The self-consistent calculations yield the lower energy in i-AF phase of ΔE = −51.7 meV, relatively to those in NM. Therefore, we conclude that the i-AF is the more stable than NM. Consequently, we only consider the electronic properties of i-AF Sr₄V₂O₆Fe₂As₂ in this work.

The total and interstitial density of states (TDOS, IDOS) are shown in Fig. 2(a) (Fig. 2 should be here). Apparently, the DOS structures are formed mainly by three peak ranges. The first area −8 ÷ −3 eV is dominantly come from the As- and O-p orbitals, which do not participate in the magnetic properties as well as in superconductivity. The next area is of the range −3 ÷ 3 eV mostly contributed by both Fe- and V-3d orbitals, where electrons around Fermi level (E_F) play an important role in all the physical properties. Finally, the DOS located in the range 3 ÷ 8 eV is largely attributed by Sr-s orbitals. The density of states at Fermi level is N(E_F) = 2.827 st/(eV.mol) that is lower than that reported by Lee and Pickett results for nonmagnetic phase of 11.2 st/(eV.mol).¹²⁾ This result additionally supports for the stability of magnetic state of Sr₄V₂O₆Fe₂As₂. Based on the relation   

\begin{equation*} \gamma_{\textit{th}} = \frac{1}{3}\pi^{2}k_{B}^{2}N_{A}N(E_{F}), \end{equation*}

where N_A is Avogadro’s number and k_B is Boltzmann’s constant, we estimated Sommerfeld coefficient of γ_th = 6.66 mJ/(K².mol). Remarkably, we recognize the presence of DOS peak at −0.8 eV, which may be ascribed to the van Hove singularity, implying an unstable of electronic state. Usually, the van Hove singularity near E_F interacts with phonon modes and plays an important role in superconductivity.

Fig. 2

(a) The total (solid line) and interstitial (dashed line) density of states, and the partial density of states of (b) V and (c) Fe ions of i-AF Sr₄V₂O₆Fe₂As₂.

To determine the contribution of magnetic atoms, the partial density of states (PDOS) of V and Fe are shown in Fig. 1(b) and (c), respectively. Explicitly, the big difference of PDOS structures between two spin directions of V in Fig. 2(b) indicates the magnetic order of V moments. Otherwise, the PDOS in Fig. 2(c) shows no spin-polarized effect in Fe sites. Thus, the magnetic behavior in this superconductor is basically governed by V ions. Based on the TDOS and PDOS data, we suspect that the V and Fe ions give rise to the van Hove peak, and there is possible hybridization between Fe-3d and V-3d orbitals. The finding further suggests that not only Fe but also V maybe joined in the superconductivity in this material.

The electronic band structures (EBS) crossing E_F are exhibited in Fig. 3(a) (Fig. 3 should be here). There are totally six bands but the band number 1 and 2 supply a dominant contribution to DOS. A large number of bands overpassing E_F hints strong inter-band scattering, which favors the occurrences of superconductivity.

Fig. 3

(a) Electronic band structures, and the projected pDOS of 3d-orbitals of (b) V and (c) Fe ions in i-AF Sr₄V₂O₆Fe₂As₂.

The projected EBS of V-3d and Fe-3d orbitals are shown in Fig. 3(b) and (c), the band width corresponds to their weight. At the first sight, the electrons of both V and Fe atoms exhibit the same bandwidth. However, on close inspection it seems that the bandwidth of the Fe-3d orbital is a little wider, meaning a large weight to DOS.

Figure 3 shows the Fermi surfaces (FS) topologies in a) 2D and b) 2D-quasi viewed perpendicularly to 001 configuration. The Fermi surface sheets are drawn in the same color as shown in Fig. 3. (Fig. 4 should be here). We observe six hole-like cylinders coming from the whole crossing E_F bands around the Brillouin zone center. Further, two electron-like sheets, formed by band 1 and 2, situate at the corners of Brillouin zone. It is interesting we look on the 2D-quasi presentation: the FS of band 1(2) and 3(4) tend to touch together at some points. This observation reminds the nodal-gap structure that has been found in other iron-based materials.^26–28) Therefore, it is desired to perform further measurements to examine the gap structure in the Sr₄V₂O₆Fe₂As₂ superconductor.

Fig. 4

Fermi surfaces in 3D and the 2D-quasi configuration of i-AF Sr₄V₂O₆Fe₂As_2.

4. Conclusion

In summary, our DFT calculations show that the incommensurate anti-ferromagnetic Sr₄V₂O₆Fe₂As₂ is the more stable state, compared to that of non-magnetic one. The total and partial density of states studies for the i-AF state reveal that both Fe and V, forming the van Hove singularities and presumably are involved in the cooper pairing. Moreover, the DOS data point out that the magnetic property of Sr₄V₂O₆Fe₂As₂ superconductor is related to the moments of V. Besides, the electronic band structures are composed many bands crossing E_F, which would indicate strong inter-band scattering and approve high-Tc superconductivity. We found multi-sheet Fermi surfaces formed by several hole-like cylinders around the Γ point as well as electron-like at the Brillouin zone corners. We observed the existence of adjoining hole and electron Fermi surfaces, which may suggest the nodal structure. Additionally, we performed phonon calculations. Imaginary mode was found around the Γ point. In order to establish the appropriate mechanism of this behavior, further investigations will be required.

Acknowledgments

The financial support by the National Science Centre of Poland under the grant No. 2016/21/B/ST3/01366 is gratefully acknowledged.

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