Abstract
There has been increasing interest in materials where relativistic effects induce nontrivial electronic states with promise for spintronics applications. One example is the splitting of bands with opposite spin chirality produced by the Rashba spinorbit coupling in asymmetric potentials. Sizable splittings have been hitherto obtained using either heavy elements, where this coupling is intrinsically strong, or large surface electric fields. Here by means of angular resolved photoemission spectroscopy and firstprinciples calculations, we give evidence of a large Rashba coupling of 0.25 eV Å, leading to a remarkable band splitting up to 0.15 eV with hidden spinchiral polarization in centrosymmetric BaNiS_{2}. This is explained by a huge staggered crystal field of 1.4 V Å^{−1}, produced by a gliding plane symmetry, that breaks inversion symmetry at the Ni site. This unexpected result in the absence of heavy elements demonstrates an effective mechanism of Rashba coupling amplification that may foster spinorbit band engineering.
Introduction
The discovery of the integer quantum Hall effect in fieldeffect transistors^{1} initiated a novel approach to the electronic properties of crystalline solids based on the topological properties of Bloch states^{2}, which fostered the discovery of further remarkable effects, such as the fractional quantum Hall effect^{3}, the quantum spin Hall effect^{4} and charge antilocalization^{5,6}. These effects suggest novel device concepts for quantum computation and spintronics, where information is carried without dissipation by using spin, instead of charge, currents^{7}. In this context, there has been an increasing interest in systems where the spinorbit (SO) coupling leads to the opening of gaps at the Fermi surface and to spinpolarized bands^{8}. An attractive possibility is the splitting of electronic states with opposite spin chirality in nonmagnetic materials by the SO Rashba effect. The Rashba Hamiltonian, , where and β are constants and is the spin vector operator composed of the Pauli matrices , and , requires an electric field that breaks the inversion symmetry along the zdirection perpendicular to the plane containing the spin σ and the electron wave vector k.
It follows that typical Rashba systems are either bulk noncentrosymmetric crystals or surfaces, referred to as bulk or surface inversion asymmetry systems, respectively^{8}. Notable examples are the layered polar semiconductor BiTeI^{9} and the twodimensional electron gas formed at the surface of SrTiO_{3} (ref. 10), where Rashba splittings as large as meV have been reported. These large splittings are due to the presence of heavy elements (and thus to large SO couplings) in the former case and to a large electric field E_{z}≈10 mV Å^{−1} at the surface in the latter case. In fact, as noted recently by Zhang et al.^{11}, SO coupling effects are governed by the local symmetry of the potential felt by the electron, rather than by the symmetry of the bulk crystal. To the best of our knowledge, the first experimental evidence of this circumstance has been given by Riley et al.^{12} who observed spinpolarized bands in the layered system 2HWSe_{2}. This polarization, not expected in the centrosymmetric P6_{3}/mmc structure of 2HWSe_{2}, is explained by the confinement of the electron states within a single WSe_{2} layer with local inversion asymmetry (LIA)^{13}.
In the present work, we consider the quasitwodimensional semimetal BaNiS_{2} (ref. 14), precursor of the metal–insulator transition (MIT) observed in the BaCo_{1−x}Ni_{x}S_{2} system on the partial substitution of Ni for Co at x_{cr}=0.22 (ref. 15). This MIT has attracted a great deal of interest for the similarity of its temperaturedoping (Tx) phase diagram to that of cuprate^{16}, pnictide^{17} and heavyfermion^{18} superconductors, though it remains little studied. To the best of our knowledge, the experimental band structure is not available for BaNiS_{2} and firstprinciples calculations have been hitherto limited to a simple density functional theory (DFT) approach in the local density approximation neglecting the Hubbard repulsion term, U, and relativistic SO effects^{19,20}. The strong local Coulomb repulsion is expected to play a crucial role on the MIT, as indicated by dynamical mean field theory calculations on the insulating phase BaCoS_{2} (refs 21, 22). To elucidate these open issues, the objective of the present study is to provide a first thorough description of the electronic structure of BaNiS_{2}. Previous calculations show that one challenge is a complex multiband structure. On the other hand, a favourable condition of this study is the absence of disorder caused by chemical substitutions and a simple tetragonal structure. By means of angularresolved photoemission spectroscopy (ARPES) on highquality single crystals, we find a very large Rashba splitting up to Δɛ≈150 meV, which is unexpected in a centrosymmetric system without electronically active heavy elements. This observation is explained by a very effective LIA mechanism associated with a peculiar nonsymmorphic squarepyramidal structure with a gliding plane symmetry. Unlike WSe_{2}, in the present case, the large splittings observed are not due to the presence of heavy elements but to a huge staggered crystal field at the Ni site, E_{z}≈1.4 eV Å^{−1}, associated with the pyramidal coordination of the Ni ions. The mechanism unveiled by the present results, therefore, opens new possibilities to tune effectively the topological order of solids without the constraint of using either heavy elements or external fields.
Results
Ab initio calculations
Before presenting the ARPES results and the evidence of Rashba effect, we recall the salient features of the electronic structure of BaNiS_{2} that we recalculated in the generalized gradient approximation (GGA) of DFT supplemented by the Hubbard repulsion, U, and by the Hund coupling, J. In Fig. 1a–c, we show the crystal structure, the calculated Fermi surface, the band structure and the total and projected density of states (DOS) for each dorbital of Ni. The parameters of the local interaction matrix (U=3 eV, J=0.95 eV) were estimated from absorption data^{23}. In the band structure of Fig. 1c, false colours represent the dorbital component of each band. We use a reference system for the dorbitals where the x (y) axis coincides with the a (b) axis, while the z axis coincides with the c axis and points towards the apical sulphur atom. All dorbitals, except the lowlying xy one, contribute to the DOS at the Fermi level, ɛ_{F}, and hybridize strongly with the 3p orbitals of the sulphur atoms. In Fig. 1c, the relevant bands near ɛ_{F} are labelled in Schönflies notation for the irreducible representations (IR) of the little point group of the corresponding kvector^{24}. We first note two degenerate holelike bands with E_{g} (d_{xz} and d_{yz}) character at the D_{4h} symmetry Γ point and with E character along the Λ direction of C_{4v} symmetry; along the Δ (or U) and Σ (or S) directions of lower C_{2v} symmetry, these bands are split into B_{1} and B_{2} bands. Second, at X (or R) two bands with A_{g} character carrying and components split into two A_{1} bands along Δ (or U). Finally, along Σ (or S), two A_{1} electronlike bands carrying and components, respectively, display a linear dispersion and cross at a point denoted Q, located almost exactly at ɛ_{F} between the Γ and M points (or between Z and A).
The resulting Fermi surface consists of four sheets. The E_{g} band forms a hole pocket centred at Γ with a peculiar shape similar to that of a carambola fruit. As a result of the k_{z} dispersion along Λ, this band forms an electron pocket centred at Z with squareshaped section. The A_{g} bands form a strongly oblated electron pocket centred at R. Finally, the two linearly dispersing A_{1} bands with and component form a Diraclike cone with ellipticlike section and hole character. The origin Q of the cone is located near k_{z}=0. All pockets are small, which leads to semimetallic properties, as apparent from the pronounced dip of the DOS at ɛ_{F}, and to enhanced electronic correlations. Each band is at least twofold degenerate in the orbital sector since the glide reflection plane of the P4/nmm symmetry generates two Ni positions, r_{1}=(1/4,1/4,z) and r_{2}=(3/4,3/4,−z), in the unit cell.
Angular resolved photoemission spectroscopy
To obtain highquality spectra, we optimized the growth of BaNiS_{2} single crystals as described in the experimental methods section. We successfully measured ARPES spectra on several highpurity single crystals cleaved in situ at low temperature within the ab plane, which allowed us to measure the inplane band dispersion, ɛ_{k}. As an indication of the improved purity of these crystals, resistivity measurements (see Methods section) yielded residual resistivity ratios RRR≈12–17 significantly larger than those ∼4 reported previously^{14,25}. In Fig. 2, we show the 26 eV ARPES bands measured in both p and spolarizations along the Δ (or U) and Σ (or S) symmetry directions. Consistent with the orbital character of the bands calculated without SO coupling (Fig. 1c), due to the different parity of the dorbitals with respect to the scattering plane, the states are observed only in ppolarization along Σ, the state in ppolarization along Δ and in spolarization along Σ and the d_{yz} and d_{xz} states in both polarizations along Δ.
A quantitative comparison between experimental and calculated bands is shown in Fig. 3a, where the previous GGA+U bands are plotted together with the full ARPES band structure obtained by merging the p and spolarized spectra of Fig. 2. We noticed that the inclusion of the Hubbard repulsion U term in the GGA calculations improves significantly the agreement with the experiment, especially concerning the energy position of the B_{1g} band at Γ (or Z) and the dispersion of the B_{1} band along Γ–X (or Z–R). To estimate the k_{z} value, we calculated the band dispersion at different k_{z} values, k_{z}=0, (in crystal units of ), and best agreement with the experiment is obtained for . In Fig. 3a–c, the corresponding symmetry points and lines are labelled using primed letters.
As predicted by the calculations for , the experimental Fermi surface of Fig. 3b consists of the four hole pockets of the Diraclike bands along S' and of the squareshaped electron pocket centred around Z'. A slight discrepancy between experiment and calculations concerns the hole pocket centred at R', not seen experimentally. According to the calculations, this may suggest that the k_{z} value probed by the 26 eV photons rather lies in the k_{z} range. In fact, in the calculated Fermi surface of Fig. 3d, both hole pockets at Z and at R shrink rapidly with decreasing k_{z}, so an accurate determination of the relative k_{z} position of the bottom of these pockets goes beyond the capabilities of the present DFT method. In summary, the experimental data are explained by a pocket at R located at a higher k_{z} value than calculated. This explanation is supported by the fact that the 100 ev ARPES data do probe the bottom of the pocket, as seen in Fig. 4b. We conclude that the 26 eV data of Figs 2, 3 and the 100 eV data of Fig. 4b probe the bands in the vicinity of and of , respectively.
Discussion
We are now in the position of analysing one of the main features of the band structure, that is, the large ≈50 meV splitting of the E_{g} bands at Z, which suggests the importance of SO coupling effects not included in the calculations. A straightforward symmetry analysis^{24,26} substantiates this hypothesis. As far as the SO band splittings is concerned, we consider the kpoints with , such as Z, on the same footing as the k′ points with , such as Z'. The reason is that the predictions of splitting for the little groups of the former points are valid for the little groups of the latter points as well, as are subgroups of . Thus, the symmetry analysis of the SO band splitting is the same for both 26 eV and 100 eV ARPES bands shown in Figs 3a and 4b, respectively. This analysis shows that, while the E_{g} bands are degenerate in the non relativistic calculations, the product of their IR Z_{5} of the D_{4h} little group for the Z symmetry point with the D_{1/2} spin IR yields the following sum of IR's:
We have then recalculated the band structure including the SO coupling. Remarkably, the result shown in Figs 3a and 4a is fully consistent with the above band splitting. The SO coupling also accounts for the experimental observation of a mixing of the B_{1g} band below the Fermi level with the lower split E_{g} band, clearly visible in the data at Z (Fig. 3a). Indeed, by denoting Z_{3} the IR of the B_{1g} band at Z, one obtains , so the two bands are described by the same Z_{12} symmetry in the presence of SO coupling.
The exciting feature of BaNiS_{2} revealed by the present work is that, in addition to the ordinary SO splittings discussed above, both the experimental bands and those calculated with SO coupling display Rashba splittings, as shown in Fig. 4a. We focus on the splitting of the electronlike band with A_{1} character with and components located just above ɛ_{F} near R along T (or near X along Y) and of the holelike band located in the same k region ≈0.4 eV below ɛ_{F}. As seen in Fig. 1c, near R the latter band displays B_{1} and B_{2} character with d_{xz} and d_{yz} components, respectively, that progressively evolves into A_{1} character with component moving to A. A closeup of the ARPES bands in Fig. 4b shows that only the bottom of the electronlike band is observed, as mentioned above; so the prediction of Rashba effect cannot be verified experimentally for this band but a slight electron doping may be sufficient to probe it. On the other hand, the predicted Rashba splitting is clear in the holelike band. The splittings in wave vector and in energy are as large as Δk=0.2 Å^{−1} and Δɛ≈150 meV, respectively. The latter value is even larger than in BiTeI ^{9} or SrTiO_{3} (ref. 10), in spite of the absence of heavy elements or surfaces.
Both electronlike and holelike bands are k split along R–A by the same Rashba interaction , where k_{R}=k–k^{0} and k^{0} is the wave vector of the R point. While the electronlike upper band would be centred at R in the absence of SO correction, the holelike band would have a maximum located between R and A. As shown by the present calculations, this difference leads to two different shapes of the spinor band structure, illustrated in Figs 4b and 5a, respectively. In Fig. 5a, the splitting of the electronlike band is decribed by the standard Mexicanhat shape. In Fig. 4b, the calculations predict a displaced split maximum for the holelike band along the same direction, in agreement with the ARPES data. We fit the latter band using the band dispersion obtained by diagonalising the above Rashba Hamiltonian and find α_{R}=0.26 eV Å from ARPES, in excellent agreement with the calculated value 0.28 eV Å. The difference between experimental and calculated splitting is due to the limited accuracy of the GGA+U approximation in calculating the unperturbed (that is, without SO) band dispersion and its mass renormalization. We computed the Rashba parameters also for the Mexicanhat splitting of the electronlike band, barely visible experimentally at ɛ_{F}. For this band, we obtained α_{R}=0.19 eV Å, k_{R}=0.04 Å^{−1} for the position of the minimum of the band with respect to R, and ɛ_{R}=11 meV for the depth of the Mexican hat (Fig. 5a).
Contrary to the usual Rashba case, here the spin degeneracy is not lifted, even for the electronlike Mexicanhat band, because every energy level is at least twofold degenerate due to the twofold multiplicity of the Ni site. It follows that the Rashba Hamiltonian involves two—instead of one—spinors. Our calculations show (Fig. 5c) that the corresponding Bloch states, and , are strongly localized either at r_{1} or at r_{2}, where the dipolar crystal field (and the NiS_{5} pyramid) has opposite orientation, ±E_{z}, along the z direction. Owing to this localization, the Hamiltonian splits into two independent Rashba Hamiltonians, one for each position, with a staggered distribution of Rashba coupling α_{R} forming a Rashba crystal in a square lattice. This situation corresponds to a LIA, as discussed by Zhang et al.^{11} who called hidden the resulting compensated spin polarization to distinguish it from that of the usual bulk inversion asymmetry and surface inversion asymmetry cases.
In the present case, the spin chirality at the two Ni positions, r_{1} and r_{2}, is opposite, so the net spin polarization of the unit cell is zero. For the Bloch state at r_{1}, where E_{z}>0, the solution for the Rashba eigenspinor at a given wave vector k reads
where the superscript ± denotes the upper and lower energy branch, respectively, and . Since E_{z} has opposite sign at r_{2}, it follows that ; hence, each Rashba branch is made of two opposite spinors whose spatial part is assigned to one of the two Ni positions. This is illustrated in Fig. 5c, where the spatial and spin parts of both Bloch states are plotted in real space at k= (crystal units), where the splitting is largest. At this k point, φ=0, so, according to equation (2), the Rashba spin moment is oriented along the y direction with . This picture is nicely confirmed by our ab initio calculations. The Bloch states projected on a given Ni position yield a local spin moment for the two branches, while . The slight reduction of s_{y} with respect to the expected value for the electron is due to the almost complete localization of each Bloch state around one Ni position (Fig. 5c). Finally, one notes that spinchiral polarized states can be formed by applying a magnetic or electric field.
Using a pointlike model for the S anion and for the Ba and Ni cations, the dipolar crystal field at the Ni site is estimated to be E_{z}≈1.4 V Å^{−1}. This huge value accounts for the large Rashba effect observed and for the stability of the structure. Indeed, the staggered orientation of the NiS_{5} pyramids minimizes the Coulomb repulsion between apical sulphur atoms. The resulting staggered field configuration turns out to be a very effective LIA mechanism. Thus, the magnitude of the Rashba effect observed reflects the magnitude of the cohesive energy of the crystal. To justify quantitatively this picture, using the GGA–DFT method, we calculated the dependence of the Rashba coupling α_{R} at R and the SO splitting Δɛ_{Z} at Z as a function of the Ni position, z, along the c axis within the NiS_{5} pyramid. The result reported in Fig. 5b shows that the maximum values for both α_{R} and Δɛ_{z} are obtained close to the equilibrium position of Ni, which means that the most stable structure corresponds to the largest Rashba splitting. This observation suggests an analogy with the JahnTeller systems where a gain of electronic energy drives a strain field that splits degenerate electronic states; here, the crystal field is the cause of the splitting.
Following Zhang et al.^{11}, the present observation of Rashba effect in a centrosymmetric crystal arises from the LIA C_{4v} at the Ni site caused by the above staggered field, which contrasts the usual case of Rashba band splitting at Γ in noncentrosymmetric crystals. The availability of a huge crystal field at the electronically active Ni site extends the possibilities of Rashba band engineering to light elements. The Rashba couplings obtained here could be further enhanced by substituting Ni for heavier 4d or 5d ions. To verify this expectation, we carried out preliminary GGA–DFT ab initio calculations on the hypothetical compounds BaPdS_{2} and BaPtS_{2} assuming that the crystal structure would be similar to that of BaNiS_{2}. Our expectation is confirmed by a significant enhancement of the Rashba coupling for the electronlike band; namely, we obtain α_{R}=0.35 and 0.73 eV Å for the relaxed structure of the Pd and Pt compound, respectively. The corresponding band splittings are also dramatically enhanced up to Δɛ≈ 0.5 eV in both compounds, which would enable a band structure engineering of solids in view of spintronics applications.
Methods
Singlecrystal growth
Highpurity BaNiS_{2} single crystals were synthesized by a selfflux method, as described elsewhere^{27,28}. In brief, we first prepared a slightly offstoichiometry mixture of finely grinded highpurity BaS (99.9%), Ni (99.999%) and S (99.9995%) powders in a graphite crucible. The chosen Ba:Ni:S=0.10:0.425:0.475 composition ensures the crystal growth through the liquidus line of the pseudobinary phase diagram BaNiS_{2}2/3 Ni_{3}S_{2+δ} system as shown in ref. 27. The crucible was then sealed under high vacuum at typical pressures P∼10^{−5} mbar in a silica tube, heated to 1,000 °C during 2 days and then quenched into water. The product was then grinded and pressed into a pellet for a second time by adding a 10% molar excess of sulphur and the aforementioned heat treatment was repeated. The silica tube was then slowly cooled from 1,000 °C down to 800 °C at a rate of 0.5–1 °C h^{−1} and then quenched into water. The asprepared melt contains plateletlike shiny black crystals of typical size 1 × 1 × 0.1 mm^{3} that can be mechanically removed from the melt. The crystalline quality, chemical composition and impurity concentration were checked by means of singlecrystal Xray diffraction, energy dispersive Xray analysis and resistivity measurements in the van der Pauw configuration.
Angular resolved photoemission spectroscopy
The ARPES experiment was carried out on crystals freshly cleaved in situ within the abplane at 100 K in ultrahigh vacuum, at pressures of 10^{−11} mbar or better, at the BaDElPh beamline of the Elettra synchrotron in Trieste, Italy, and at the Cassiopée beamline of the SOLEIL synchrotron in SaintAubin, France. In the former experiment, we used an incident photon energy hv=26 eV with a Δɛ=5 meV photon energy resolution. To determine the parity of the bands, the experiment was performed in both p and s photon polarizations. In the former case, the incoming electric field, perpendicular to the photon momentum, lies within the scattering plane defined by the photon momentum and by the photoelectron momentum. In the latter case, the electric field is perpendicular to the scattering plane. In our geometry, the p polarization forms an angle of 45° with the abplane when at normal emission, while the s polarization is parallel to this plane. In the experiment conducted at SOLEIL, the photon energy was 100 eV and the photon energy resolution was Δɛ=10 meV.
Ab initio calculations
The ab initio calculations have been carried out in the GGA–DFT framework with the Perdew–Burke–Ernzerhof functional, implemented in the Quantum ESPRESSO package^{29}. In the GGA+U calculations, we used the rotationally invariant formalism^{30} with the Hubbard matrix correlating the Ni 3d orbitals and fulfilling the atomic spherical symmetries. The Hubbard parameters are U=3 eV, J=0.95 eV, and the F_{4}/F_{2} Slater integral ratio is taken equal to the atomic value. Ni, Ba and S atoms are described by normconserving pseudopotentials. The Ni pseudopotential has 10 valence electrons (4s^{2} 3d^{8}) and non linear core corrections. For Ba, the semicore states have been explicitely included in the calculations. The S pseudopotential is constructed with the 3s^{2} 3p^{4} invalence configuration. In the GGA+U calculations with SO coupling, the Ni pseudopotential is fully relativistic and a noncollinear spin (spinor) formalism is used in the DFT framework. The geometry of the cell and the internal coordinates are taken from the experimental values reported previously^{14}. The planewave cutoff is 120 Ry for the wave function and 500 Ry for the charge. A 8 × 8 × 8 electronmomentum grid and a Methfessel–Paxton smearing of 0.01 Ry are used in the electronic integration during the selfconsistent loop. Further calculations with a 16 × 16 × 16 electronmomentum grid and tetrahedra interpolation have been performed for a more accurate determination of the Fermi level starting from a previously converged 8 × 8 × 8 selfconsistent electron density. For band structure and Fermi surface calculations, we carried out a Wannier interpolation of the ab initio band structure by means of the Wannier90 (ref. 31) program on a N_{w}=4 × 4 × 4 electronmomentum mesh by including all Ni d and S p states in a 11 eV energy window, which allows to obtain highly accurate interpolated bands.
Additional information
How to cite this article: SantosCottin, D. et al. Rashba coupling amplification by a staggered crystal field. Nat. Commun. 7:11258 doi: 10.1038/ncomms11258 (2016).
References
Klitzing, K. V., Dorda, G. & Pepper, M. New method for highaccuracy determination of the finestructure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Wen, X.G. Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995).
Stormer, H. L., Tsui, D. C. & Gossard, A. C. The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999).
Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
Chen, J. et al. Gatevoltage control of chemical potential and weak antilocalization in Bi2Se3 . Phys. Rev. Lett. 105, 176602 (2010).
Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–U5 (2008).
Winkler, R. Spinorbit Coupling Effects in TwoDimensional Electron and Hole Systems. Springer Tracts in Modern Physics, Vol. 191, (Springer Verlag (2003).
Ishizaka, K. et al. Giant Rashbatype spin splitting in bulk BiTeI. Nat. Mater. 10, 521–526 (2011).
SantanderSyro, A. F. et al. Giant spin splitting of the twodimensional electron gas at the surface of SrTiO3 . Nat. Mater. 13, 1085–1090 (2014).
Zhang, X., Liu, Q., Luo, J.W., Freeman, A. J. & Zunger, A. Hidden spin polarization in inversionsymmetric bulk crystals. Nat. Phys. 10, 387–393 (2014).
Riley, J. M. et al. Direct observation of spinpolarized bulk bands in an inversionsymmetric semiconductor. Nat. Phys. 10, 835–839 (2014).
Schaibley, J. & Xu, X. Spintronics: a lucky break. Nat. Phys. 10, 798–799 (2014).
Grey, I. E. & Steinfink, H. Crystal structure and properties of barium nickel sulfide, a squarepyramidal nickel(II) compound. J. Am. Chem. Soc. 92, 5093–5095 (1970).
Martinson, L. S., Schweitzer, J. W. & Baenziger, N. C. Metalinsulator transitions in BaCo1xNixS2y . Phys. Rev. Lett. 71, 125–128 (1993).
Lee, P. A., Nagaosa, N. & Wen, X.G. Doping a Mott insulator: Physics of hightemperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Johnston, D. C. The puzzle of high temperature superconductivity in layered iron pnictides and chalcogenides. Adv. Phys. 59, 803–1061 (2010).
Joynt, R. & Taillefer, L. The superconducting phases of UPt3 . Rev. Mod. Phys. 74, 235–294 (2002).
Mattheiss, L. Electronic structure of quasitwodimensional BaNiS2 . Solid State Commun. 93, 879–883 (1995).
Hase, I., Shirakawa, N. & Nishihara, Y. Electronic structures of BaNiS2 and BaCoS2 . J. Phys. Soc. Jpn 64, 2533–2540 (1995).
Zainullina, V. M. & Korotin, M. A. Ground state of BaCoS2 as a set of energydegenerate orbitalordered configurations of Co^{2+} ions. Phys. Solid State 53, 978–984 (2011).
Zainullina, V. M., Skorikov, N. A. & Korotin, M. A. Description of the pressureinduced insulatormetal transition in BaCoS2 within the LDA+DMFT approach. Phys. Solid State 54, 1864–1869 (2012).
Krishnakumar, S. R., SahaDasgupta, T., Shanthi, N., Mahadevan, P. & Sarma, D. D. Electronic structure of and covalency driven metalinsulator transition in BaCo1xNixS2 . Phys. Rev. B 63, 045111 (2001).
Atkins, P., Child, M. & Phillips, C. Tables for Group Theory Oxford University Press (1970).
Martinson, L. S., Schweitzer, J. W. & Baenziger, N. C. Properties of the layered BaCo1xNixS2 alloy system. Phys. Rev. B 54, 11265–11270 (1996).
Miller, S. C. & Love, W. F. Tables of Irreducible Representations of Space Groups and CoRepresentations of Magnetic Space Groups Pruett Press (1967).
Shamoto, S., Tanaka, S., Ueda, E. & Sato, M. Single crystal growth of BaNiS2 . J. Cryst. Growth 154, 197–201 (1995).
Shamoto, S., Tanaka, S., Ueda, E. & Sato, M. Single crystal growth of BaCo1xNixS2 . Physica C 263, 550–553 (1996).
Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Densityfunctional theory and strong interactions: Orbital ordering in MottHubbard insulators. Phys. Rev. B 52, R5467–R5470 (1995).
Mostofi, A. A. et al. wannier90: A tool for obtaining maximallylocalized Wannier functions. Comp. Phys. Commun. 178, 685–699 (2008).
Acknowledgements
We gratefully acknowledge financial support provided by the University Pierre and Marie Curie under the ‘Programme émergence’, by the EU/FP7 under contract Go Fast (Grant No. 280555) and by ‘Investissement d'Avenir Labex PALM’ (ANR10LABX0039PALM). The computational resources used for this work have been provided by the GENCI under Project No. 096493. We are grateful to B. Baptiste for the singlecrystal Xray diffraction measurements.
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Contributions
D.SC. made and characterized the single crystals, led the ARPES data analysis and contributed in the figure preparation. M.C. carried out the firstprinciples calculations and led the theoretical analysis. G.L. led the ARPES data analysis and contributed in the figure preparation. G.L. L.P., P.L.F., G.L., F.B., E.P. and M.M. contributed to the ARPES experiments. Y.K. contributed to the growth and characterization of the single crystals. M.M. led the ARPES experiment. A.G. and M.C. led the analysis, wrote the paper and contributed in the figure preparation. A.G. proposed the research project.
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SantosCottin, D., Casula, M., Lantz, G. et al. Rashba coupling amplification by a staggered crystal field. Nat Commun 7, 11258 (2016). https://doi.org/10.1038/ncomms11258
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DOI: https://doi.org/10.1038/ncomms11258
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