Abstract
Superconductivity in crystals without inversion symmetry has received extensive attention due to its unconventional pairing and possible nontrivial topological properties. Using first-principles calculations, we systemically study the electronic structure of noncentrosymmetric superconductors A2Cr3As3 (A = Na, K, Rb, and Cs). Topologically protected triply degenerate points connected by one-dimensional arcs appear along the C3 axis, coexisting with strong ferromagnetic (FM) fluctuations in the non-superconducting state. Within random phase approximation, our calculations show that strong enhancements of spin fluctuations are present in K2Cr3As3 and Rb2Cr3As3 and are substantially reduced in Na2Cr3As3 and Cs2Cr3As3. Symmetry analysis of pairing gap Δ(k) and spin–orbit coupling gk suggest that the arc surface states may also exist in the superconducting state, giving rise to possible nontrivial topological properties.
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Introduction
Materials with nontrivial topological properties have been extensively studied over the past two decades. Initially, comprehensive attentions were paid to topological insulators (TIs)1,2,3, which have an insulating gap in the bulk and metallic surface states at the boundary. Fully gapped superconductors with the topological protected gapless surface mode, a close analogy with TI, are regarded as promising candidates for hosting Majorana fermions. Since the discovery of Weyl, Dirac, nodal line semimetals, and triply degenerate point (TP) topological metals4,5,6,7,8,9,10,11,12,13,14,15, it was shown that gapless systems can possess novel topology as well. Simultaneously, it is also possible for superconductors with nodes (e.g., CePt3Si, UPt3) to have topologically protected edge states, which are guaranteed by momentum-dependent topological numbers16,17,18. Among these exotic superconductors, nodal noncentrosymmetric superconductors (NCSs) with topological stable nodes have fascinating properties, i.e., the zero-energy boundary modes19,20. These zero-energy boundary modes are believed to be closely associated with the nodal gap structure via the so-called bulk–boundary correspondence21.
The Cr-based arsenides A2Cr3As3 (A = Na, K, Rb, and Cs) are of great interest in terms of low dimensionality, strong ferromagnetic (FM) fluctuations, and noncentrosymmetric superconductivity. Their nodal, unconventional superconductivity was suggested by London penetration depth, nuclear magnetic resonance (NMR), muon spin spectroscopy, and specific-heat measurements22,23,24,25,26,27,28,29. Besides, spin–orbit coupling (SOC) has a remarkable effect on β and γ bands in K2Cr3As3 with a band spin-splitting much larger than the superconducting gap30. The coalescence of considerable SOC effect and strong FM fluctuations in A2Cr3As3 NCSs is crucial to the superconducting pairing symmetry, leading to a predominant spin–triplet component, which is distinguished from the pairing in the isotropic channel of an usual s-wave superconductor. More recently, NMR experiments on A2Cr3As3 suggest that the compounds in this family are possibly a solid-state analog of superfluid 3He, implying that the unconventional superconductor A2Cr3As3 may host nontrivial topological properties31. In addition, the NMR measurements of Cs2Cr3As3 were distinct from that of K2Cr3As3 (Rb2Cr3As3), which displayed suppression of FM fluctuations in the former32. Therefore, a systematic study of the SOC effect, the topological properties as well as the FM fluctuations, and a thorough comparison among the family is in need.
In this article, we report our latest first-principles results on the A2Cr3As3 family. Our results show: (1) The variations in band structures and Fermi surfaces (FSs) due to alkaline element substitution do not show apparent systematic behavior; (2) The anti-symmetric spin–orbit coupling (ASOC) splitting is the largest in K2Cr3As3, but its effect is most significant in Cs2Cr3As3 and enhances its one dimensionality; (3) All compounds of this family host TPs along Γ–A and the surface states emerging from the TPs form one-dimensional (1D) Fermi arcs; (4) The magnetic susceptibility spectrum exhibits a strong peak of the spin susceptibilities at the Γ point in K2Cr3As3, followed by Rb2Cr3As3, while in Na2Cr3As3 and Cs2Cr3As3 the enhancement of spin susceptibilities at the Γ point is not obvious. Inclusion of dynamic self-energy due to spin fluctuation does not alter the topology of electronic spectrum, and thus the existence of TPs is robust against the dynamic spin fluctuation at random phase approximation (RPA) level. Finally, we discuss the possibility of the existence of topologically stable arc states in the superconducting phase.
Results
Electronic structure and topological properties
The crystal structure and Brillouin zone of K2Cr3As3 are illustrated in Fig. 1a, b. For the compounds of this family, each primitive unit cell consists of [(Cr6As6)]∞ sub-nanotube along the c axis forming a quasi-one-dimensional (Q1D) structure29,33,34,35,36. In contrast to ACr3As3 (A = K, Rb)37,38,39,40, the A+ ions around the [(Cr6As6)]∞ sub-nanotube break the inversion symmetry, rendering A2Cr3As3 noncentrosymmetric (with symmetry D3h, space group 187). From Na2Cr3As3, K2Cr3As3, Rb2Cr3As3 to Cs2Cr3As3, the lattice expands in-plane while the average Cr-Cr bond length barely changes, implying increased distances between the [(Cr6As6)]∞ tubes.
Despite this systematic structural variation, the band structures (Fig. 2a–d) of A2Cr3As3 around the Fermi level resemble each other and do not exhibit apparent change except for Cs2Cr3As3. In the absence of SOC, the α and β bands cross the Fermi level along Γ–A, forming two Q1D FSs (see Supplementary Fig. 1 for details). In contrast, the γ band forms one three-dimensional (3D) FS around the Γ point (Fig. 3a–c) for Na2Cr3As3 and Rb2Cr3As3, similar to K2Cr3As330,41. For Cs2Cr3As3, it is worth noting that the γ band do not cross the Fermi level in the kz = 0 plane. As a result, this band forms one deformed Q1D FS. In addition, a fourth band (\(\gamma ^{\prime}\)) around the Γ point emerges, creating a new 3D FS (Fig. 3d). Once the SOC effect is included, for all members in this family, the β and γ bands further split due to the ASOC effect.
Remarkably, the compounds of A2Cr3As3 host TPs along Γ–A. In particular, the TPs in Na2Cr3As3 (8 meV above the Fermi level ϵF) and K2Cr3As3 (0.6 meV below ϵF) around the Γ point are very close to the Fermi level. Considering the presence of time reversal symmetry T and mirror symmetry σh (orthogonal to the C3 axis) in A2Cr3As3, the TP fermions in current compounds belong to type A14, which is accompanied by one Weyl nodal line instead of four in type B15,42. Including SOC, along the C3 rotation axis, the singly degenerate band (Λ5 or Λ6 state) belongs to the 1D representation of C3v symmetry, while the doubly degenerate bands (Λ4 state) form the two-dimensional representation, and these TPs along kz are due to band inversion between the Λ5/Λ6 state and the Λ4 state. Around the Fermi level, band inversion occurs for α, β, and γ bands in Na2Cr3As3 (Fig. 4a) as well as α and β in K2Cr3As3 (see Supplementary Fig. 2), and these TPs are protected by the C3v symmetry. We compare the band structures of A2Cr3As3 along kz (Supplementary Fig. 2) as well as TPs and (010) surface states (see Supplementary Fig. 3a–c). Owing to the overwhelming bulk states, only the surface states in Na2Cr3As3 can be clearly distinguished from the bulk band continuum, resulting in clear Fermi arc structures on the (010) surface (Fig. 4b, c). We also performed calculations using the modified Becke–Johnson (mBJ) potentials43 and obtained results similar to that of Perdew, Burke, and Ernzerhoff (PBE) for Na2Cr3As3 (for details, see “Methods” below). In K2Cr3As3, however, the Λ4 state near ϵF, which is lower than the Λ5 and Λ6 states in PBE calculations (see Supplementary Fig. 2b), is now elevated higher around Γ points, leading to extra band inversions between β and γ band and hence two new TPs (located at ϵF+90 and ϵF-22 meV), as shown in Fig. 4d. Figure 4b, c, e, f show the surface states in Na2Cr3As3 (PBE results) and K2Cr3As3 (mBJ results), respectively. For both of these two compounds, two surface states, SS1 and SS2, emerge from two TPs (TP1 and TP2) near the Γ point, respectively, while the surface states from the TPs (TP3 and TP4 in Na2Cr3As3, TP3 in K2Cr3As3) away from Γ point are mixed with surrounding bulk states and cannot be easily distinguished. The iso-energy surface states at TP1 in Na2Cr3As3 (ϵF+8 meV) and K2Cr3As3 (ϵF+90 meV) are shown in Fig. 4c, f. Akin to ZrTe family of compounds, the TPs (TP1 in Na2Cr3As3 and K2Cr3As3) marked as blue (Fig. 4c) and red (Fig. 4f) dots are connected by double 1D Fermi arcs on the (010) surface. We note that, for K2Cr3As3, the number of TPs are different in PBE and mBJ calculations. Nevertheless, as long as the band inversions between the Λ4 and Λ5 (Λ6) exist, these intrinsic TPs will be protected by the extra global symmetry (C3v). In the next section, we will further discuss the dynamic correlation effect due to the spin fluctuation on these TPs.
Spin fluctuations and multiorbital susceptibilities
The imaginary part of the bare electron susceptibility χ0 of K2Cr3As3 exhibited a strong peak at the Γ point30. In order to investigate the variation of electron susceptibility, we also calculate susceptibility χ0 using the Lindhard response function for all compounds of this family. As shown in Fig. 5a, b, the bare susceptibilities of Na2Cr3As3, Rb2Cr3As3, and Cs2Cr3As3 resemble that of K2Cr3As3. Specifically, the real part \(\chi ^{\prime}_0\) is relatively featureless, while the imaginary part \(\chi ^{\prime \prime}_0\) is dominated by a resonance peak at the Γ point. As reported in previous NMR studies24,25,32, there were clear differences in the Knight shift and NMR relaxation rate (1/T1T) of K2Cr3As3 (Rb2Cr3As3) and Cs2Cr3As3, implying that the spin susceptibility of K2Cr3As3 (Rb2Cr3As3) strikingly differs from that of Cs2Cr3As3. Therefore, electron–electron interactions must be taken into consideration to explain the aforementioned NMR experiments. We calculate both spin and charge susceptibilities within RPA with intra-orbital Coulomb (U), inter-orbital Coulomb (\(U^{\prime}\)), Hund’s coupling (J), and pair-hopping (\(J^{\prime}\)) interactions involved (see Supplementary Method for details). In Fig. 5c–f, the real part of charge (spin) susceptibility \(\chi ^{\prime}_c\) (\(\chi ^{\prime}_s\)) is presented for U = 2 eV and J = 0.3 eV in comparison to that of bare ones. For all members in this family, the spin susceptibilities are significantly enhanced, while the charge susceptibilities are suppressed. The sharp peak present at the Γ point of K2Cr3As3 indicates that it is very close to the critical point with certain values of U and J. Naively, using the simplest approximation, the enhancement of the imaginary part χ″ can be written as \(\chi ^{\prime\prime} ({\bf{q}},\omega )\approx {\chi }_{0}({\bf{q}},\omega )/{\left[1-\bar{U}{\chi ^{\prime} }_{0}({\bf{q}},\omega )\right]}^{2}\) 44,45, where \(\overline{U}\) is the Stoner factor. Applying this approximation, the imaginary part χ″ of K2Cr3As3 is also expected to be strongly enhanced at the Γ point when \(1-U{\chi }_{0}^{\prime}({\bf{q}},\omega )\) approaches zero. The scattering peak of \(\chi ^{\prime}\) in Rb2Cr3As3 still locates at the Γ point but is lower and broader compared to K2Cr3As3. In the case of Cs2Cr3As3 (Na2Cr3As3), in contrast, only a broadened and plateau-like structure can be found around the M(π, 0, 0) and \(K\left(\frac{2}{3}\pi ,\frac{2}{3}\pi ,0\right)\) points. Interestingly, there exists another broad hump around Q* = (0, 0, 0.6π) in Cs2Cr3As3, which is possibly due to intraband scattering within the γ band. Similar to previous report in iron-based superconductor LaOFeAs46, such a broad hump might be related to spin–density–wave (SDW) fluctuations. In K2Cr3As3 (Rb2Cr3As3), the SDW hump at Q* also exists but not apparent, overwhelmed by the large peak located at q(0, 0, 0). It is worth noting that the strong peak of \(\chi ^{\prime}\) at the Γ point of K2Cr3As3 (Rb2Cr3As3) is robust against the Hund’s coupling (J). It will also be present even if the Hund’s coupling is completely turned off (J = 0). In addition, the spin susceptibility appears to be insensitive to either the inter-orbital Coulomb repulsion (\(U^{\prime}\)) or the pair-hopping interaction (\(J^{\prime}\)).
Considering the striking difference in \(\chi ^{\prime}_s\) between K2Cr3As3 and Cs2Cr3As3, we analyze the dynamic spin susceptibility of these two compounds at the Γ, M, and Q* points as shown in Fig. 6a, b. The spin susceptibility exhibits a larger enhancement at the Γ point in K2Cr3As3, while in Cs2Cr3As3 the spin susceptibility seems to be more enhanced at a finite q. For all members in this family, we further compare the enhancement of spin susceptibility \(\chi ^{\prime}_s\) and \(\chi ^{\prime \prime}_s\) at q = (0,0,0) in Fig. 6c, d. The results show that the enhancement is most significant in K2Cr3As3, followed by Rb2Cr3As3 and Cs2Cr3As3. Such a systematic trend can also be found in the temperature-dependent spin susceptibility enhancement (Fig. 6e, f). Note that the enhancement of χ″ is always larger than that of \(\chi ^{\prime}\) for both the T- and U-dependent spin susceptibilities, indicating that the system is away from an FM ordered state.
We now consider the NMR relaxation rate 1/T1T. It is proportional to \(\frac{1}{N}{\sum }_{{\bf{q}}}\frac{A({\bf{q}}){\rm{Im}}[\chi ({\bf{q}},\,\omega,\,T)]}{\omega }\), where A(q) is a geometrical structure factor. We chose A(q) = 1 since the geometrical structure factor has little effect on 1/T1T47. The NMR frequency ω is close to zero, which is chosen to be 1 × 10−5 eV in our calculation. Intriguingly, if we assume that the NMR relaxation rate 1/T1T is dominated by the imaginary part of the spin susceptibility at q = 0 (or the so-called long wavelength approximation48,49), the calculated 1/T1T for Na2Cr3As3, K2Cr3As3, and Rb2Cr3As3 (Fig. 6e, f) are qualitatively similar to the NMR experiments of24,25,31, although such a Curie–Weiss-like temperature dependence is more accurate within a self-consistent renormalization theory50. Nevertheless, in contrast to K2Cr3As3 and Rb2Cr3As3, the enhancement of the spin susceptibility in Cs2Cr3As3 (Fig. 6e, f) exhibits a weak temperature dependence. This implies that the large enhancement of the spin susceptibility at the Γ point is substantially suppressed in Cs2Cr3As3, consistent with ref. 32. Similar T dependence of the spin susceptibility were also obtained by Graser et al. in LaFeAsO47 and 26% Co-doped BaFe2As251.
It is also informative to consider the impact of FM fluctuation on the electronic band structure. As the spin fluctuation is most significant for K2Cr3As3 in our calculations, we calculated its dynamic RPA self-energy Σ(ω) (see Supplementary Fig. 4) and obtained its dynamic correlated electronic spectrum. In Fig. 7, we show the renormalized quasi-particle energy spectrum. Away from the Fermi level, we see significantly increased scattering rate, most prominent close to A around EF − 0.4 eV. However, the renormalization to the quasi-particle states close to the Fermi level is negligible within RPA. Thus the TPs remain and are only slightly shifted in K2Cr3As3. Our results are in agreement with previous theoretical studies that correlation effect may be greatly reduced owing to the formation of molecular orbitals52,53,54,55 and are in line with the experimental observation that no appreciable magnetic local moment is found in these systems. In addition, the comparison between the RPA and PBE results also illustrates that the dynamic RPA self-energy has little effect on the existence of TPs in these systems. Despite of its moderate effect at normal state, the FM fluctuation would influence the topological properties in superconducting state by allowing for the nontrivial spin–triplet pairing (p- or f-wave), as has been conjectured theoretically52,53,54,55,56 and investigated experimentally22,23,24,25,57.
Discussion
In the normal state, the TP topological metal has been regarded as the intermediate phase between Dirac and Weyl semimetals14. The TPs can split into Weyl points by breaking σv containing the C3 axis or merge into Dirac point after imposing inversion symmetry. Although the 233-type A2Cr3As3 is absent of an inversion center, the 133-type ACr3As3 is centrosymmetric. It is interesting to notice that a Dirac-like crossing point appears between the α and β bands along kz in the recently reported KHCr3As358,59. In Na2Cr3As3, the surface state lies only 8 meV above the Fermi level, and it is possible for this surface state to become superconducting through proximity effect. In experiment, such a surface superconducting state might have influence on the in-plane Hc260, although such a possibility has been ruled out in K2Cr3As357,61. One possible reason is that the surface state is far away from the Fermi level in stoichiometric K2Cr3As3.
In the superconducting state, lacking inversion symmetry gives rise to Rashba-type ASOC interactions so that the single-particle Hamiltonian takes the form h(k) = ξ(k)σ0 + gk⋅σ. With an extra parity-breaking term gk⋅σ, the mixture of singlet and triplet pairing is allowed and the order parameter takes the general form Δ(k) = Ψkσ0 + dk⋅σ, where Ψk = Ψ−k and dk = −d−k. The triplet pairing state can be stable as long as dk is parallel to gk62. In the following, we discuss the possible nontrivial topological property in superconducting state by the symmetry analysis of superconducting gap Δ(k) and the spin–orbit coupling gk. First, considering the gap symmetry, the large FM fluctuations favor the spin–triplet pairing component than the spin-singlet one. For the gap, symmetry belongs to the A″ representation of D3h group (e.g., pz-wave53), Δ(k) satisfies \({U}_{{\bf{k}}}({S}_{6}){\rm{\Delta }}({\bf{k}}){U}_{{\bf{-k}}}{({S}_{6})}^{T}={\chi }_{{S}_{6}}{{\rm{\Delta }}}_{{S}_{6}}({\bf{k}})\), where \({\chi }_{{S}_{6}}=-1\) and the sixfold rotoinversion symmetry S6 = iC6. We calculate the \({{\mathbb{Z}}}_{6}\) indexes (\(z^{\prime}_6\) and \(z^{\prime\prime}_6\)) associated with the S6 according to the symmetry indicators for topological superconductors defined in refs 63,64. The resulting \(z^{\prime}_6\) and \(z^{\prime\prime}_6\) are both nonzeros (\(z^{\prime}_6 =1\) and \(z^{\prime \prime}_6\) = −1) for mBJ results of Na2Cr3As3 and K2Cr3As3 (see Supplementary Table 1 for details). Second, considering the large 3D FS around the Γ point in K2Cr3As3 (Rb2Cr3As3), the corresponding representation of D3h is Γ7 (Γ8), yielding \({{\bf{g}}}_{{\bf{k}}}={\beta }_{1}{k}_{z}\left[\left({k}_{x}^{2}-{k}_{y}^{2}\right){\sigma }_{x}-2{k}_{x}{k}_{y}{\sigma }_{y}\right]+{\beta }_{2}{k}_{x}\left({k}_{x}^{2}-3{k}_{y}^{2}\right){\sigma }_{z}\)65, where β1 and β2 are linear combination coefficients. Moreover, the ASOC splitting around \(K\left(\frac{2}{3}\pi ,\frac{2}{3}\pi ,0\right)\) is roughly 60 meV30, while along A(0, 0, π)–H(π, 0, π) it is nearly negligible. Therefore the ASOC splitting should be dominated by the β2 term, i.e., \({{\bf{g}}}_{{\bf{k}}}={k}_{x}\left({k}_{x}^{2}-3{k}_{y}^{2}\right){\sigma }_{z}\), and thus it is invariant under \({{\bf{g}}}_{{{\bf{k}}}_{i},+{k}_{0}}={{\bf{g}}}_{{{\bf{k}}}_{i},-{k}_{0}}\). It has been demonstrated by Schnyder et al. 16 that the presence of such type of symmetry gives rise to topological nontrivial features of line nodes in the gap of NCSs. More importantly, the Majorana surface states can exist at time-reversal-invariant momenta of the surface Brillouin zone. Since in the non-superconducting state the 1D arc surface states connecting the TPs already exist, it will be more interesting to further investigate the topological properties in the superconducting state of A2Cr3As3.
In summary, we have performed first-principles calculations on A2Cr3As3 (A = Na, K, Rb and Cs) and analyzed the systematic variations of electronic structures, FSs, topological properties, and magnetic spin susceptibilities of the compounds in this family. Using the surface Green’s function method, we calculate the (010) surface state and find that the TPs in Na2Cr3As3 and K2Cr3As3 are connected by double 1D Fermi arcs. To explore the behavior of magnetic spin response, charge and spin susceptibility are obtained by employing RPA calculations. We demonstrate that the strong enhancement of spin fluctuations is present at the Γ point in K2Cr3As3 and Rb2Cr3As3, while in Cs2Cr3As3 and Na2Cr3As3, the FM spin fluctuations are not apparently enhanced. The existence of TPs is robust against the FM spin fluctuation under RPA. Based on symmetry analysis, the A2Cr3As3 compounds are possible candidates for topological superconductor.
Note added
During the preparation of our manuscript, we became aware of a recent paper66. In addition to the scenario we discussed here, they have proposed that triplet pz-pairing can also be topological superconductors with weak SOC.
Methods
Calculation parameters
The calculations were carried out using density functional theory as implemented in the Vienna Abinitio Simulation Package67,68. The PBE parameterization of generalized gradient approximation to the exchange correlation functional was employed69. The energy cutoff of the plane-wave basis was up to 450 eV, and 6 × 6 × 12 Γ-centered Monkhorst–Pack70 k-point mesh was chosen to ensure that the total energy converges to 1 meV/cell. We also performed a comparison with the mBJ exchange potentials43 for Na2Cr3As3 and K2Cr3As3.
Structural parameters
The calculations of K2Cr3As3, Rb2Cr3As3, and Cs2Cr3As3 were performed with experimental crystal structure. For Na2Cr3As3, only the lattice constants are available from experiment, therefore we employed its lattice constants from experiment, while the atomic positions are fully relaxed with dynamical mean-field theory at 300 K71,72.
Tight binding (TB) Hamiltonian
The band structures obtained with the PBE and the mBJ methods were fitted to a TB model Hamiltonian with maximally projected Wannier function method73,74 using 48 atomic orbitals including Cr 3d and As 4p. The resulting Hamiltonian was then used to calculate the FS, charge (spin) susceptibility, self-energy, and surface state with surface Green’s function75.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The computer code to perform RPA calculations is available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Yi Zhou, Guanghan Cao, Jianhui Dai, Si-Qi Wu, and Xiang Lv for the inspiring discussions. The calculations were partly performed at the Tianhe-2 National Supercomputing Center in China and the HPC center at Hangzhou Normal University. This work has been supported by the NSFC (Nos. 11874137, 11574265, and 11774309) and the 973 project (Nos. 2014CB648400 and 2016YFA0300402).
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C.X. and C.C. designed research; C.X. performed the calculations; C.X., N.W., Q.C., and C.C. drafted the manuscript; C.X. and C.C. were responsible for the data analysis; all the authors participated in discussions.
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Xu, C., Wu, N., Zhi, GX. et al. Coexistence of nontrivial topological properties and strong ferromagnetic fluctuations in quasi-one-dimensional A2Cr3As3. npj Comput Mater 6, 30 (2020). https://doi.org/10.1038/s41524-020-0294-9
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DOI: https://doi.org/10.1038/s41524-020-0294-9