Introduction

The effect of reduced dimensionality on the electronic structure, especially on the ordered phases, has been one of the key issues in the condensed matter physics. For a one-dimension (1D) system with short-range interactions, the thermal and quantum fluctuations prevent the development of a long-range order at finite temperatures1,2, unless finite transverse coupling is present, i.e. the system becomes quasi-1D3. The scenario is exemplified with the discovery of the quasi-1D superconductor including the Bechgaard salts (TMTSF)2X4, M2Mo6Se6 (M = Tl, In)5, Li0.9Mo6O176,7, etc. In general, the Bechgaard salts exhibits charge-density-wave or spin-density-wave instabilities and superconductivity emerges once the instability is suppressed by applying external pressure. For the molybdenum compounds, although the density-wave instabilities were absent in the superconducting samples, they were observed in the closely related A2Mo6Se6 and A0.9Mo6O17 (A = K, Rb, etc). Therefore, a “universal” phase diagram8,9 suggests the superconductivity is in close proximity to the density-wave and most probably originated from non-phonon pairing mechanism. However, the determination of the pairing symmetry for these compound was difficult due to the competition between several energetically close ordering parameters10.

Recently, a new family of superconducting compounds A2Cr3As3 (A = K, Rb, Cs) was reported11,12,13,14. The crystal of the representing compound K2Cr3As3 (Tc = 6.1K) consists of K atom separated (Cr3As3)2− nanotubes (Fig. 1(a,b)) that are structurally very close to (Mo6Se6)2− in the M2Mo6Se6 (Mo-266) compounds. However, unlike the Mo-266 superconductors, K2Cr3As3 exhibits peculiar properties at both the normal and the superconducting states. It shows a linear temperature dependent resistivity within a very wide temperature range from 7 to 300 K at its normal state; and most interestingly, its upper critical field (Hc2) steeply increases with respect to decreasing temperature, with extrapolation to 44.7 T at 0 K, almost four times the Pauli limit. In comparison, the upper critical field of Tl2Mo6Se6 is 5.9 T and 0.47 T parallel and perpendicular to the quasi-1D direction15, respectively. Such unusually high Hc2 is beyond the explanation of multiband effect and spin-orbit scattering and usually implies possible spin triplet pairing. It is therefore quite essential to study the electronic structure of this compound, in particular its ground state, the spin-orbit coupling effect, as well as the possible spin fluctuations.

Figure 1
figure 1

(a,b) Crystal structure of K2Cr3As3 and (c–f) the considered magnetic states. The blue atoms are K, red atoms Cr, green atoms As. In panel (b), the larger spheres denote atoms at z = 0.5 (CrII, AsII and KII) and the smaller ones are at z = 0.0 (CrI, AsI and KI). In panels (c–f), only Cr atoms are shown for a clear view. IAF, NCL, IOP and CLK refer to interlayer antiferromagnetism, noncollinear antiferromagnetism, in-out state antiferromagnetism and chiral-like antiferromagnetism, respectively.

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In this paper, we present our latest first-principles results on the K2Cr3As3 electronic structure. We show that the ground state of K2Cr3As3 is paramagnetic. The anti-symmetric spin-orbit coupling (ASOC) effect is negligible far away from the Fermi level, but sizable around it. Combining with the noncentral symmetric nature of the crystal, the sizable ASOC effect around EF enables spin-triplet pairing mechanism16,17. The Fermi surfaces of K2Cr3As3 consist of one 3D sheet and two quasi-1D sheets and the Cr-3d orbitals, in particular the , dxy and orbitals dominate the electronic states around EF.

Results

We first examine the ground state of K2Cr3As3. Figure 1(a,b) shows the crystal structure of K2Cr3As3. The Cr atoms form a sub-nanotube along the c-axis, surrounded by another sub-nanotube formed by As atoms. The CrI-CrI and CrII-CrII bond lengths are respectively 2.61 Å and 2.69 Å, while the CrI-CrII bond length is 2.61 Å. Therefore the six Cr atoms in each unit cell (as well as the six Cr atoms from any two adjacent layers) form a slightly distorted octahedron and become non-centrosymmetric. Since the Cr sub-lattice is magnetically strongly frustruated, we considered several different possible magnetic states in our current study: the ferromagnetic state (FM), the inter-layer antiferromagnetic state (IAF)(Fig. 1c), the noncollinear antiferromagnetic state (NCL)(Fig. 1d), the noncollinear in-out state (IOP) (Fig. 1e), the noncollinear chiral-like state (CLK) (Fig. 1f) and the nonmagnetic state (NM). The total energy calculation results are shown in Table 1. Without structural optimization, all the magnetically stable states (FM, IAF, IOP and CLK) are energetically degenerate within the DFT errorbar. The presence of multiple nearly degenerate magnetic configurations indicates that the ground state is indeed paramagnetic phase. It is important to notice that the converged IAF state yields −0.40 μB and 0.58 μB magnetic moment for CrI and CrII, respectively; although it is initially constructed with zero total magnetization per unit cell. Such difference leads to a non-zero total magnetization in one unit cell, therefore the IAF state is indeed a ferrimagnetic state. This phase cannot be stabilized in DFT unless the effective inter-cell magnetic interaction is ferromagnetic. With structural optimization, the IAF state will eventually converge to FM state which is also energetically degenerate with NM state within the DFT errorbar. The above facts suggest that the system is close to magnetic instabilities but long range order cannot be established. Therefore, we argue that the ground state of K2Cr3As3 is paramagnetic (PM) state.

Table 1 Total energies (in meV/Cr) and Cr magnetic moment (in μB) of considered magnetic states at different optimization levels.
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We have also performed GGA + U calculations with U = 1.0 and 2.0 eV, J = 0.7 eV for the compound. The inclusion of on-site U strengthened the magnetic interactions, leading to more stable magnetic states. As a result, all the non-collinear phases became energetically stable upon structural optimization even with a small U = 1.0 eV. Nevertheless, these states are still energetically degenerate within LDA errorbar, leaving the above conclusion unaltered. However, the staggered moment per Cr atom become ~30 times the experimental value with U = 2.0 eV. Thus the GGA + U method significantly overestimates the magnetism in this system, suggesting that the GGA functional better suits the current study.

With closer examination of Table 1, it is important to notice that after full structural optimization (where both lattice constants and atomic coordinates are allowed to relax), the resulting a is 1.2% larger than the experimental value, but c is 2% shorter. Similar situation was also observed in the studies of iron-based superconductors18 and was due to the fact that the NM state in LDA is only a very rough approximation to the actual PM state without considering any magnetic fluctuations. As the electronic structure close to EF seems to be sensitive to the structural changes, all the rest reported results are based on the experimental geometry unless specified, following the examples of iron-pnictides/chalcogenides.

Figure 2 shows the band structure and density of states (DOS) of K2Cr3As3. The band structure (Fig. 2(a)) shows flat bands at kz = 0.0 and kz = 0.5 around −0.93 eV, −0.86 eV, −0.64 eV, −0.56 eV, −0.37 eV, 0.51 eV and 0.95 eV relative to EF, forming quasi-1D van Hove singularities in the DOS plot at corresponding energies. All bands including the flat bands mentioned above disperse significantly along kz, which clearly indicates the quasi-1D nature of the compound. From the band structure plot, it is apparent that there are three bands cross the Fermi level, which are denoted as α, β and γ bands as shown in Fig. 2(a), respectively. The α and β bands are degenerate along Γ-A in the non-relativistic calculations and they do not cross the Fermi level in either the kz = 0 plane nor the kz = 0.5 plane, thus are likely to be quasi-1D bands. Instead, the γ band crosses the Fermi level not only along Γ-A, but also in the kz = 0 plane, therefore it is likely to be a 3D band. The total DOS at EF is calculated to be N(EF) = 8.58 states/(eV · f.u.) under non-relativistic calculations, with approximately 8%, 17% and 75% contribution from α, β and γ bands, respectively. Experimentally, the measured specific heat data yields γ = 70 mJ/(K2 ⋅ mol), which corresponds to N(EF) ≈ 29.7 states/(eV ⋅ f.u.), suggesting an electron mass renormalization factor ≈3.5, comparable to the one in the iron-pnictide case. The projected density of states (PDOS) (Fig. 2(b)) shows that the Cr-3d orbitals dominate the electronic states from EF-1.0 eV to EF + 0.5 eV and more importantly, the states from EF − 0.36 eV to EF + 0.26 eV are almost exclusively from the , dxy and orbitals. It is also instructive to notice that the five bands near EF has a band width of only ≈0.6 eV at the kz = 0 plane in our current calculation and would be ≈0.2 eV if the electron mass renormalization factor is considered. Thus these bands are extremely narrow at kz = 0 plane, consistent with the quasi-1D nature of the compound.

Figure 2
figure 2

(a)Electronic band structure of K2Cr3As3. The inset shows comparison between relativistic (magenta solid lines) and non-relativistic (grey solid lines) results. The size of the red, blue and green circles are proportional to the contributions from the , and dxy orbitals, respectively. (b) Electronic density of state of K2Cr3As3 from the non-relativistic calculation. Upper panel shows the total DOS (black line) and PDOS from As-4p (green line); middle and bottom panel shows the PDOS from all five-orbitals of Cr-3d (grey lines in the middle panel), the PDOS of Cr-3dzx and Cr-3dzy (red line in the middle panel) and the PDOS of (light blue line in the bottom panel), and Cr-3dxy (blue line in the bottom panel).

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We have also performed relativistic calculations to identify the spin-orbit coupling (SOC) effect. Since the magnitude of SOC effect is proportional to the square root of the atomic number, the SOC effect is initially expected to be negligible for this compound. Indeed, the SOC effect on the α band is negligible. However, it is notable that the SOC effect lifted the degeneracy between α and β bands at Γ, as well as along Γ-A. More importantly, sizable (≈60 meV) ASOC splitting can be identified on β and γ bands around K close to EF (insets of Fig. 2(a)). Considering that the crystal structure lacks of an inversion center, the presence of ASOC splitting facilitates the spin-triplet pairing, which is a natural and reasonable explanation of its unusually high Hc216,17.

We plot the Fermi surface sheets of K2Cr3As3 in Fig. 3. The non-relativistic Fermi surface of K2Cr3As3 consists of one 3D sheet formed by the γ band and two quasi-1D sheets formed by the α and β bands. These sheets cut kz axis at  =  = 0.30 Å−1 and  = 0.27 Å−1, respectively. Should the SOC effect taken into consideration, the γ band is mostly affected and its Fermi surface sheet splits into two (Fig. 3g). Nevertheless the two splitted γ sheets cut kz axis at the same point at  = 0.25 Å−1. For the β band, the SOC effect causes the splitting of its Fermi surface sheet, forming two adjacent quasi-1D sheets who cut kz axis at  = 0.31 Å−1 and  = 0.33 Å−1, respectively. The SOC effect on α band is negligible, thus the α band and its Fermi surface remain mostly unaltered and remains unchanged.

Figure 3
figure 3

Fermi surface sheets of K2Cr3As3.

(a–f) Non-relativistic results (SOC effect not included). (g) Relativistic result for γ band (with SOC effect). The irreducibie Brillouin zone and high symmetry points are indicated in (c). In all figures, Γ point is at the zone center.

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We have also calculated the bare electron susceptibility χ0 (Fig. 4). The real part of the susceptibility shows no prominent peak, consistent with our PM ground state result. However, the imaginary part of the susceptibility shows extremely strong peaks at Γ, which may be due to the large electron DOS at the Fermi level EF. Such a strong peak of imaginary part of susceptibility at Γ is usually an indication of large FM spin fluctuation when finite on-site repulsion is taken into consideration. Considering the fairly large ASOC effect, together with the non-central symmetric nature of the crystal structure, the large FM spin fluctuation may be the pairing mechanism and contribute to the spin-triplet pairing channel. It should be noted that the strong peak of susceptibility imaginary part at Γ is robust with respect to hole doping, as our rigid-band calculation of 0.2 hole doping shows the same feature.

Figure 4
figure 4

Static bare electron susceptibility χ0(ω = 0) of K2Cr3As3 (solid lines) and 0.2 hole doped K2Cr3As3 (K1.8Cr3As3) (dashed lines) calculated using (a) 5 molecular orbitals localized on Cr-octahedron and (b) 10 molecular orbitals localized on Cr-octahedron. The red lines denote the real part Re[χ0(ω = 0)]; and the blue lines are the imaginary part .

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Discussion

Our above studies are focused on calculations with experimental structure parameters. However, it is important to point out that the number of Fermi surface sheets and their respective dimensionality characters, the orbital characters of the bands close to EF, the SOC effect on these bands and the features of χ0 are insensitive with respect to the structural optimization; therefore the physics is robust against the structural change (Supplementary Information, Fig. SI-2, SI-4 and SI-6). Nevertheless, the detailed band structure of K2Cr3As3 is quite sensitive to the structure relaxation. In particular, the 3D γ band will submerge below EF at Γ and the 3D γ sheet will be topologically different from the one shown in Fig. 3 (Supplementary Information, Fig. SI-3 and SI-5). The quasi-1D α and β bands are much less affected, although they intersect with the kz axis at different kFs.

In conclusion, we have performed first-principles calculations on the quasi-1D superconductor K2Cr3As3. The ground state of K2Cr3As3 is PM. The electron states near the Fermi level are dominated by the Cr-3d orbitals, in particular the , dxy and orbitals from EF − 0.5 eV to EF + 0.5 eV. The electron DOS at EF is less than 1/3 of the experimental value, suggesting intermediate electron correlation effect may in place. Three bands cross EF to form two quasi-1D sheets in consistent with its quasi-1D feature, as well as one 3D sheet who is strongly affected by the ASOC splitting as large as 60 meV. Finally, the real part of the susceptibility is mostly featureless, consistent with our PM ground state result, but the imaginary part of the susceptibility shows large peaks at Γ, indicating large ferromagnetic spin fluctuation exists in the compound. Combined with the large ASOC effect and the lack of inversion center in the crystal structure, the experimentally observed abnormally large Hc2(0) may be due to a spin-triplet pairing mechanism.

Methods

First-principles calculations were done using VASP19 with projected augmented wave method20,21 and plane-wave. Perdew, Burke and Enzerhoff flavor of exchange-correlation functional22 was employed. The energy cut-off was chosen to be 540 eV and a 3 × 3 × 9 Γ-centered K-mesh was found sufficient to converge the calculations. The K-3p, Cr-3p and As-3d electrons are considered to be semi-core electrons in all calculations.

The Fermi surfaces were plotted by interpolating a five orbital tight-binding Hamiltonian fitted to LDA results using the maximally localized Wannier function (MLWF) method23,24. The same Hamiltonian were also employed in the calculation of the bare electron susceptibility χ0 using:

where εμk and f(εμk) are the band energy (measured from EF) and occupation number of , respectively; denotes the nth Wannier orbital; and Nk is the number of the k points used for the irreducible Brillouin zone (IBZ) integration. The value of χ0 at q = Γ was approximated by the value at q = (δ, 0, 0) with δ = 0.001.

Additional Information

How to cite this article: Jiang, H. et al. Electronic structure of quasi-one-dimensional superconductor K2Cr3As3 from first-principles calculations. Sci. Rep. 5, 16054; doi: 10.1038/srep16054 (2015).