Optimizing the tight-binding parametrization of the quasi-one-dimensional superconductor K2Cr3As3

We study the tight-binding dispersion of the recently discovered superconductor K2Cr3As3, obtained from Wannier projection of Density Functional Theory (DFT) results. In order to establish quantitatively the actual degree of quasi-one dimensionality of this compound, we analyze the electronic band structure for two reduced sets of hopping parameters: one restricted to the Cr-As tubes and another one retaining a minimal number of in-plane hoppings. The corresponding total and local density of states of the compound are also computed with the aim of assessing the tight-binding results with respect to the DFT ones. We find a quite good agreement with the DFT results for the more extended set of hopping parameters, especially for what concerns the orbitals that dominate at the Fermi level. Accordingly, we conclude that one cannot avoid taking into account in-plane hoppings up to the next-nearest-neighbors cells even only to describe correctly the Fermi surface cuts and the populations along the kz direction. Such a choice of a minimal number of hopping parameters directly reflects in the possibility of correctly describing correlations and magnetic interactions.


I. INTRODUCTION
Systems such as heavy fermion compounds 1-3 , high transition-temperature cuprate superconductors [4][5][6][7][8][9][10][11][12][13][14] , strontium ruthenate superconductors [15][16][17][18][19][20][21][22][23] and iron-pnictide superconductors 24,25 have been investigated very thoroughly due to their unconventional properties determined by the existence of a superconductive phase that appears close to a magnetic one when the latter is suppressed by applying an external agent, such as for instance pressure. In recent years, this type of investigation has been extended to the search of superconductivity in Cr pnictides, although chromium is one of the few metallic elements that doesn't superconduct even under high pressures 26 . Accordingly, the discovery of superconductivity in chromium arsenide (CrAs) was only recent and rather unexpected. CrAs exhibits a superconducting transition of T c ≈ 2 K for a critical pressure of P c ≈ 8 Kbar, where its magnetically ordered phase is completely suppressed [27][28][29] . Then, another family of superconducting CrAs-based compounds has been discovered, that of A 2 Cr 3 As 3 , where A is K 26 , Rb 30 or Cs 31 .
It is worth noting that while CrAs is a 3D compound and superconducts only under a sizable pressure, A 2 Cr 3 As 3 are considered quasi-1D superconductors at ambient pressure. In particular, K 2 Cr 3 As 3 has a T C ≈ 6.1 K, an hexagonal crystal structure at room temperature with a=9.9832 Å and c=4.2304 Å, and [(Cr 3 As 3 ) 2-] ∞ double-walled subnanotubes separated by columns of K + ions 26 . This material is very peculiar because it is considered quasi-one dimensional and, at the same time, it is a superconductor: superconductivity is not so frequent in quasi-one dimensional compounds due to the Peierls instability 32,33 . As regards K 2 Cr 3 As 3 , a large specific heat coa) Corresponding author: Adolfo Avella (avella@sa.infn.it) efficient of 70-75 mJ K -2 mol -1 indicates strong electron correlations, while a linear temperature dependence of the resistivity from 7 to 300 K supports the hypothesis of a Tomonaga-Luttinger liquid behavior 26 , as foreseeable for fermions confined in 1D. On the other hand, Kong et al. report a T 3 dependence of resistivity from 10 to 40 K 34 and therefore the degree of unidimensionality of the system is still under current debate.
All experiments point to an unconventional type of superconductivity: an upper critical field that is 3-4 times the Pauli limit 26 , the absence of the Hebel-Schlichter coherence peak in 1/T 1 just below T C 35 and a linear temperature dependence of the penetration depth for T T c 36 . Muon spin rotation measurements show results consistent with a d-wave model with line nodes. The weak evidence of a spontaneous appearance of an internal magnetic field below the transition temperature is probably another proof that the superconducting state is not conventional 37 . The Debye temperature is found to be 220 K, the specific heat jump at the superconducting transition is 2.2γT c and the upper critical field is anisotropic with different amplitudes between fields parallel and perpendicular to the rodlike crystals 34 . NMR measurements on the compound show evidence for a strong enhancement of Cr spin fluctuations above T c in the subnanotubes based on the nuclear spinlattice relaxation rate 1/T 1 35 .
From a theoretical point of view, first-principles calculations indicate that the ground state is paramagnetic and that the Cr-d xy , d x 2 −y 2 , d z 2 orbitals dominate close to the Fermi level, with three bands crossing E F to form one 3D Fermi surface sheet and two quasi 1D-sheets 38 . Wu et al predict that both K 2 Cr 3 As 3 and Rb 2 Cr 3 As 3 exhibit strong frustrated magnetic fluctuations and are close to a novel in-out coplanar magnetic ground state 39 . They also report that frustrated magnetism is very sensitive to the c-axis lattice constant and can thus be suppressed by increasing pressure 39 . Moreover, threeand six-band models have been built and they show in both set primitive cells (n1, n2, n3) weak and strong coupling limits that a triplet p z -wave pairing is the leading pairing symmetry for physically realistic parameters 40,41 . Zhou et al have obtained, for A 2 Cr 3 As 3 (A=K, Rb or Cs), that at small U the pairing comes from 3D γ band and has spatial symmetry f y(3x 2 −y 2 ) , with line nodes in the gap function, while at large U a fully gapped p-wave state, p zẑ dominates at the quasi-1D α-band 42 .
In this paper, we study the electronic properties of K 2 Cr 3 As 3 by implementing a tight-binding reduction procedure onto state-of-the-art DFT calculations in order to contribute to clarify the issue of the quasi-one-dimensionality of the compound. We obtain the band structure and the total and local density of states (DOS) considering two reduced sets of hopping parameters and compare them with the DFT results to establish the minimal number of in-plane hopping amplitudes necessary to get a reasonable agreement and an adequate description of the properties of the material. The hopping parameters have been extracted by means of the Wannier procedure applied to the DFT results. The paper is organized as follows: in the next section, we describe our tight-binding procedure and show the band structure obtained considering the two reduced sets of hopping parameters. In Sec. III, we focus on the total and the local DOS to check the reliability of the results from the previous section with respect to the DFT ones. In Sec. IV, we draw our conclusions and make some final remarks.

II. BAND STRUCTURE
To study the electronic properties of the material, we start from the following tight-binding Hamiltonian in which i and j indicate the positions of Cr or As atoms in the crystal, σ is the spin, α and β are the orbital indices. The two terms take into account the on-site energies and the hopping processes between Cr and As orbitals, respectively.  TABLE II. DOS at the chemical potential ρ # ν (0) (states/eV) and related contribution to the total DOS at the chemical potential ρ # ν (0) = aν bν ρ # ν (0) /ρ # (0) for the different classes of orbitals (ν = 1: Cr dxy, d x 2 −y 2 , ν = 2: Cr d z 2 , ν = 3: Cr dyz, dxz, 4: As px, py, pz) present in the system and different sets of hoppings # = A, B, C where C stands for the DFT results. aν = (2, 1, 2, 3) is the number of orbitals in the class, while bν = 6 is the number of related atoms in the unit cell.
The Hamiltonian corresponds to a 48 × 48 matrix because we have to consider the d orbitals of the Chromium and the p orbitals of the Arsenic and the primitive cell contains six Cr and six As atoms. As we have already discussed in the introduction, the hopping parameters are obtained by the DFT overlap integrals of the orbitals. Diagonalizing the Hamiltonian, we obtain the corresponding band structure, whose behavior will be discussed referring to the high symmetry path in the hexagonal Brillouin zone chosen according to the notation of Ref. 43 (see Fig. 1). In what follows, we use the notation R = n 1 a 1 + n 2 a 2 + n 3 a 3 where R is a lattice vector in the direct space.
We analyze two cases that differ only for the chosen set of hopping parameters with the aim of assessing the degree of quasi-one dimensionality of the material. In the first case, we take into account only the hoppings between the orbitals of the atoms that lie within a single sub-nanotube extending along the z axis. Accordingly, we limit ourselves to consider the hoppings between the primitive cells in set A of Tab. I. In Fig. 2, we report the comparison between the band structure obtained in this case and the DFT spectrum. We can clearly see that this choice of hoppings not only misses completely the in-plane band structure, but it does not even allow to describe correctly the band structure along the z axis. This unveils an overlooked relevance of in-plane virtual processes that affect the z-axis physics.
Accordingly, in the second case we have also taken into account the in-plane hoppings between primitive cells in set B of Tab. I. In Fig. 3, we show how the agreement improves substantially not only for the in-plane paths, which was expected, but also for the Γ-A line of the Brillouin zone, that is for the k z direction. This demonstrates that it is necessary to take into account not only the z-axis hopping processes, but also several in-plane ones to get a qualitatively good description of the band structure.
In the next section, we will check whether this reduced set of hoppings (set B) is also sufficient to obtain a qualitatively correct description, that is in agreement with the DFT results, of the orbital populations close to the Fermi surface.

III. DENSITY OF STATES
We obtain the total DOS from the usual definition: in which ε k is the energy dispersion of Hamiltonian (1) and the sum is carried out on the N values of momenta k in the Brillouin zone; our numerical grid contains 6 × 6 × 12 kpoints. The delta functions in Eq. (2) have been approximated by Gaussian functions with a variance of 0.025 eV.
In Fig. 4, we report the total DOS obtained taking hopping parameters related to set A and B of Tab. I, together with the DFT results: set B leads to a very good qualitative agreement with the latter, this showing that this set of hoppings manages to catch correctly also the overall populations close to the Fermi level, in contrast to set A, which exhibits evident limitations.
We can now check, by projecting the total DOS onto the Cr and As orbitals, if we obtain a correct description of the local orbital occupations close to the Fermi level. To this purpose, we have reported in Tab. II the values of the projected densities of states (PDOSs) at the Fermi level, and its contribution to the total DOS, related to (i) the Cr-d symmetrical orbitals with respect to the basal plane (d xy , d x 2 −y 2 , d z 2 ), (ii) the Cr-d antisymmetrical orbitals (d yz , d xz ) and (iii) the As-p orbitals, for the previous three cases: set A, set B and DFT. We notice that the value for each class, labelled by ν, has been obtained by averaging over all a ν = (2, 1, 2, 3) orbitals and b ν = (6, 6, 6, 6) atoms in the unit cell belonging to the class. We clearly find that the symmetrical orbitals with respect to the basal plane dominate, in agreement to what reported in Ref. 38, and that the values found for set B are evidently much closer to those coming from DFT (all discrepancies below 1%) with respect to those obtained for set A (discrepancies up to 6.5%). We conclude that set B is the minimal number of hoppings necessary to achieve a proper description of the band structure along the k z direction as well as of its orbital occupations.
It is worth noting that, as already reported in the literature 38 , the orbital characterization of the spectrum reveals a high degree of covalency between the Cr-d and the As-p orbitals, as it happens for the CrAs [44][45][46] . This opens up a new quest for the minimal number of effective bands necessary to describe qualitatively, but also quantitatively, the system, at least close to the Fermi level. We are currently investigating this aspect too.

IV. CONCLUSIONS
We have used a tight-binding model, obtained through the Wannier projection of DFT results, to study the electronic properties of K 2 Cr 3 As 3 in order to explore its degree of unidimensionality. We have then analyzed the band structure and the total and the local DOS of the compound taking into account not only the usually considered hoppings along the zaxis, but also those up to the next-nearest-neighbor primitive cells in the plane. Only adding these latter, we managed to achieve a good qualitative agreement with the DFT spectrum near the Fermi level, demonstrating that the system does not exhibit a genuine one-dimensional character. Consequently, this study gives clear hints on the minimal number of hopping parameters needed to perform further studies and understand the role of correlations, magnetic interactions and spin-orbit as well as the mechanism and the nature of the superconductivity in this material. A more detailed analysis of the electronic properties of the compound, aiming to derive a reliable minimal effective band model, is underway.