Abstract
The electronic structure of quasionedimensional superconductor K_{2}Cr_{3}As_{3} is studied through systematic firstprinciples calculations. The ground state of K_{2}Cr_{3}As_{3} is paramagnetic. Close to the Fermi level, the , d_{xy}, and orbitals dominate the electronic states, and three bands cross E_{F} to form one 3D Fermi surface sheet and two quasi1D sheets. The electronic DOS at E_{F} is less than 1/3 of the experimental value, indicating a large electron renormalization factor around E_{F}. Despite of the relatively small atomic numbers, the antisymmetric spinorbit coupling splitting is sizable (≈60 meV) on the 3D Fermi surface sheet as well as on one of the quasi1D sheets. Finally, the imaginary part of bare electron susceptibility shows large peaks at Γ, suggesting the presence of large ferromagnetic spin fluctuation in the compound.
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 onedimension (1D) system with shortrange interactions, the thermal and quantum fluctuations prevent the development of a longrange order at finite temperatures^{1,2}, unless finite transverse coupling is present, i.e. the system becomes quasi1D^{3}. The scenario is exemplified with the discovery of the quasi1D superconductor including the Bechgaard salts (TMTSF)_{2}X^{4}, M_{2}Mo_{6}Se_{6} (M = Tl, In)^{5}, Li_{0.9}Mo_{6}O_{17}^{6,7}, etc. In general, the Bechgaard salts exhibits chargedensitywave or spindensitywave instabilities and superconductivity emerges once the instability is suppressed by applying external pressure. For the molybdenum compounds, although the densitywave instabilities were absent in the superconducting samples, they were observed in the closely related A_{2}Mo_{6}Se_{6} and A_{0.9}Mo_{6}O_{17} (A = K, Rb, etc). Therefore, a “universal” phase diagram^{8,9} suggests the superconductivity is in close proximity to the densitywave and most probably originated from nonphonon pairing mechanism. However, the determination of the pairing symmetry for these compound was difficult due to the competition between several energetically close ordering parameters^{10}.
Recently, a new family of superconducting compounds A_{2}Cr_{3}As_{3} (A = K, Rb, Cs) was reported^{11,12,13,14}. The crystal of the representing compound K_{2}Cr_{3}As_{3} (T_{c} = 6.1K) consists of K atom separated (Cr_{3}As_{3})^{2−} nanotubes (Fig. 1(a,b)) that are structurally very close to (Mo_{6}Se_{6})^{2−} in the M_{2}Mo_{6}Se_{6} (Mo266) compounds. However, unlike the Mo266 superconductors, K_{2}Cr_{3}As_{3} 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 (H_{c2}) 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 Tl_{2}Mo_{6}Se_{6} is 5.9 T and 0.47 T parallel and perpendicular to the quasi1D direction^{15}, respectively. Such unusually high H_{c2} is beyond the explanation of multiband effect and spinorbit 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 spinorbit coupling effect, as well as the possible spin fluctuations.
In this paper, we present our latest firstprinciples results on the K_{2}Cr_{3}As_{3} electronic structure. We show that the ground state of K_{2}Cr_{3}As_{3} is paramagnetic. The antisymmetric spinorbit 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 E_{F} enables spintriplet pairing mechanism^{16,17}. The Fermi surfaces of K_{2}Cr_{3}As_{3} consist of one 3D sheet and two quasi1D sheets, and the Cr3d orbitals, in particular the , d_{xy}, and orbitals dominate the electronic states around E_{F}.
Results
We first examine the ground state of K_{2}Cr_{3}As_{3}. Figure 1(a,b) shows the crystal structure of K_{2}Cr_{3}As_{3}. The Cr atoms form a subnanotube along the caxis, surrounded by another subnanotube formed by As atoms. The Cr^{I}Cr^{I} and Cr^{II}Cr^{II} bond lengths are respectively 2.61 Å and 2.69 Å, while the Cr^{I}Cr^{II} 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 noncentrosymmetric. Since the Cr sublattice is magnetically strongly frustruated, we considered several different possible magnetic states in our current study: the ferromagnetic state (FM), the interlayer antiferromagnetic state (IAF)(Fig. 1c), the noncollinear antiferromagnetic state (NCL)(Fig. 1d), the noncollinear inout state (IOP) (Fig. 1e), the noncollinear chirallike 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 μ_{B} and 0.58 μ_{B} magnetic moment for Cr^{I} and Cr^{II}, respectively; although it is initially constructed with zero total magnetization per unit cell. Such difference leads to a nonzero 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 intercell 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 K_{2}Cr_{3}As_{3} is paramagnetic (PM) state.
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 onsite U strengthened the magnetic interactions, leading to more stable magnetic states. As a result, all the noncollinear 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 ironbased superconductors^{18}, 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 E_{F} 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 ironpnictides/chalcogenides.
Figure 2 shows the band structure and density of states (DOS) of K_{2}Cr_{3}As_{3}. The band structure (Fig. 2(a)) shows flat bands at k_{z} = 0.0 and k_{z} = 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 E_{F}, forming quasi1D van Hove singularities in the DOS plot at corresponding energies. All bands including the flat bands mentioned above disperse significantly along k_{z}, which clearly indicates the quasi1D 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 ΓA in the nonrelativistic calculations, and they do not cross the Fermi level in either the k_{z} = 0 plane nor the k_{z} = 0.5 plane, thus are likely to be quasi1D bands. Instead, the γ band crosses the Fermi level not only along ΓA, but also in the k_{z} = 0 plane, therefore it is likely to be a 3D band. The total DOS at E_{F} is calculated to be N(E_{F}) = 8.58 states/(eV · f.u.) under nonrelativistic 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/(K^{2} ⋅ mol), which corresponds to N(E_{F}) ≈ 29.7 states/(eV ⋅ f.u.), suggesting an electron mass renormalization factor ≈3.5, comparable to the one in the ironpnictide case. The projected density of states (PDOS) (Fig. 2(b)) shows that the Cr3d orbitals dominate the electronic states from E_{F}1.0 eV to E_{F} + 0.5 eV, and more importantly, the states from E_{F} − 0.36 eV to E_{F} + 0.26 eV are almost exclusively from the , d_{xy}, and orbitals. It is also instructive to notice that the five bands near E_{F} has a band width of only ≈0.6 eV at the k_{z} = 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 k_{z} = 0 plane, consistent with the quasi1D nature of the compound.
We have also performed relativistic calculations to identify the spinorbit 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 ΓA. More importantly, sizable (≈60 meV) ASOC splitting can be identified on β and γ bands around K close to E_{F} (insets of Fig. 2(a)). Considering that the crystal structure lacks of an inversion center, the presence of ASOC splitting facilitates the spintriplet pairing, which is a natural and reasonable explanation of its unusually high H_{c2}^{16,17}.
We plot the Fermi surface sheets of K_{2}Cr_{3}As_{3} in Fig. 3. The nonrelativistic Fermi surface of K_{2}Cr_{3}As_{3} consists of one 3D sheet formed by the γ band and two quasi1D sheets formed by the α and β bands. These sheets cut k_{z} 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 k_{z} 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 quasi1D sheets who cut k_{z} 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.
We have also calculated the bare electron susceptibility χ_{0} (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 E_{F}. Such a strong peak of imaginary part of susceptibility at Γ is usually an indication of large FM spin fluctuation when finite onsite repulsion is taken into consideration. Considering the fairly large ASOC effect, together with the noncentral symmetric nature of the crystal structure, the large FM spin fluctuation may be the pairing mechanism and contribute to the spintriplet 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 rigidband calculation of 0.2 hole doping shows the same feature.
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 E_{F}, the SOC effect on these bands, and the features of χ_{0} are insensitive with respect to the structural optimization; therefore the physics is robust against the structural change (Supplementary Information, Fig. SI2, SI4 and SI6). Nevertheless, the detailed band structure of K_{2}Cr_{3}As_{3} is quite sensitive to the structure relaxation. In particular, the 3D γ band will submerge below E_{F} at Γ and the 3D γ sheet will be topologically different from the one shown in Fig. 3 (Supplementary Information, Fig. SI3 and SI5). The quasi1D α and β bands are much less affected, although they intersect with the k_{z} axis at different k_{F}s.
In conclusion, we have performed firstprinciples calculations on the quasi1D superconductor K_{2}Cr_{3}As_{3}. The ground state of K_{2}Cr_{3}As_{3} is PM. The electron states near the Fermi level are dominated by the Cr3d orbitals, in particular the , d_{xy}, and orbitals from E_{F} − 0.5 eV to E_{F} + 0.5 eV. The electron DOS at E_{F} is less than 1/3 of the experimental value, suggesting intermediate electron correlation effect may in place. Three bands cross E_{F} to form two quasi1D sheets in consistent with its quasi1D 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 H_{c2}(0) may be due to a spintriplet pairing mechanism.
Methods
Firstprinciples calculations were done using VASP^{19} with projected augmented wave method^{20,21} and planewave. Perdew, Burke and Enzerhoff flavor of exchangecorrelation functional^{22} was employed. The energy cutoff was chosen to be 540 eV, and a 3 × 3 × 9 Γcentered Kmesh was found sufficient to converge the calculations. The K3p, Cr3p and As3d electrons are considered to be semicore electrons in all calculations.
The Fermi surfaces were plotted by interpolating a five orbital tightbinding Hamiltonian fitted to LDA results using the maximally localized Wannier function (MLWF) method^{23,24}. The same Hamiltonian were also employed in the calculation of the bare electron susceptibility χ_{0} using:
where ε_{μk} and f(ε_{μk}) are the band energy (measured from E_{F}) and occupation number of , respectively; denotes the n^{th} Wannier orbital; and N_{k} is the number of the k points used for the irreducible Brillouin zone (IBZ) integration. The value of χ_{0} 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 quasionedimensional superconductor K_{2}Cr_{3}As_{3} from firstprinciples calculations. Sci. Rep. 5, 16054; doi: 10.1038/srep16054 (2015).
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Acknowledgements
The authors would like to thank Fuchun Zhang, Zhuan Xu, Yi Zhou and Jianhui Dai for inspiring discussions. This work has been supported by the NSFC (No. 11274006 and No. 11190023), National Basic Research Program (No. 2014CB648400 and No. 2011CBA00103) and the NSF of Zhejiang Province (No. LR12A04003). All calculations were performed at the High Performance Computing Center of Hangzhou Normal University College of Science.
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Affiliations
Condensed Matter Group, Department of Physics, Hangzhou Normal University, Hangzhou 310036, P.R. China
 Chao Cao
Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China
 Hao Jiang
 & Guanghan Cao
Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, China
 Guanghan Cao
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Contributions
H.J. performed most of the calculations; G.H.C. and C.C. were responsible for the data analysis and interpretation; C.C. calculated the response functions and drafted the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Chao Cao.
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