Tuning Dirac nodes with correlated d-electrons in BaCo_{1-x}Ni_{x}S_{2}

TL;DR

This study demonstrates how doping in BaCo_{1-x}Ni_{x}S_{2} can effectively tune Dirac states, their shape, position, and the material's metal-insulator transition, advancing control over topological electronic phases.

Contribution

It reveals the ability to manipulate Dirac lines and electronic phases in BaCo_{1-x}Ni_{x}S_{2} through doping, combining experimental and theoretical approaches.

Findings

Doping shifts Dirac lines in k-space along Gamma M.

Doping reshapes the Dirac band structure.

Doping controls the metal-insulator transition.

Abstract

Dirac fermions play a central role in the study of topological phases, for they can generate a variety of exotic states, such as Weyl semimetals and topological insulators. The control and manipulation of Dirac fermions constitute a fundamental step towards the realization of novel concepts of electronic devices and quantum computation. By means of ARPES experiments and ab initio simulations, here we show that Dirac states can be effectively tuned by doping a transition metal sulfide, BaNiS2, through Co/Ni substitution. The symmetry and chemical characteristics of this material, combined with the modification of the charge transfer gap of BaCo_{1-x}Ni_{x}S_{2} across its phase diagram, lead to the formation of Dirac lines whose position in k-space can be displaced along the Gamma M symmetry direction, and their form reshaped. Not only does the doping x tailor the location and shape of the Dirac bands, but it also controls the metal-insulator transition in the same compound, making BaCo_{1-x}Ni_{x}S_{2} a model system to functionalize Dirac materials by varying the strength of electron correlations.

Tables

Table 1. Table 1: Position of the Dirac point (DP) in energy and momentum space for different doping levels in the metallic phase.

Compound	EDP (eV)	$k_{DP}$(Å-1)
BaNiS2	0.03$\pm$ 0.01	0.52$\pm$ 0.01
BaCo0.25Ni0.75S2	0.19$\pm$ 0.01	0.49$\pm$ 0.01
BaCo0.7Ni0.3S2	0.37$\pm$ 0.02	0.39$\pm$ 0.02

Equations

$$E_{xc}^{HSE}=E_{xc}^{PBE}+\alpha \left(E_x^{HF,\text{screened}}(\omega )-E_x^{PBE,\text{screened}}(\omega )\right).$$

Full text

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In the vast domain of topological Dirac and Weyl materials (1, 2, 3, 4, 5, 6, 7, 8, 9), the study of various underlying mechanisms (10, 11, 12, 13, 14, 15) leading to the formation of non-trivial band structures is key to discover new topological electronic states (16, 17, 18, 19, 20, 21, 22, 23). A highly desirable feature of these materials is the tunability of the topological properties by an external parameter, which will make them suitable in view of technological applications, such as topological field effect transistors (24). While a thorough control of band topology can be achieved in principle in optical lattices (25) and photonic crystals (26) through the wandering, merging and reshaping of nodal points and lines in ${\bf k}$-space (27, 28), in solid state systems such a control is much harder to achieve. Proposals have been made by using optical cavities (29), twisted van der Waals heterostructures (30), intercalation (31), chemical deposition (32, 33), impurities (34), and magnetic and electric applied fields (35), both static (36) and time-periodic (37, 17). Here, we prove that it is possible to move and reshape Dirac nodal lines in reciprocal space by chemical substitution. Namely, by means of Angle Resolved Photo-Emission Spectroscopy (ARPES) experiments and ab initio simulations, we observe a sizable shift of robust massive Dirac nodes towards $\Gamma$ in BaCo1-xNixS2  as a function of doping $x$, obtained by replacing Ni with Co. At variance with previous attempts of controlling Dirac states by doping (38, 19), in our work we report both a reshape and a significant ${\bf k}$-displacement of the Dirac nodes.

BaCo1-xNixS2 is a prototypical transition metal system with a simple square lattice (39). In BaCo1-xNixS2  the same doping parameter $x$ that tunes the position of the Dirac nodes also controls the electronic phase diagram, which features a first-order metal-insulator transition (MIT) at a critical substitution level, $x_{cr}\sim$ 0.22 (40, 41), as shown in Fig. 1(a). The Co-rich side ($x=0$) is an insulator with collinear magnetic order and with local moments in a high-spin (S=3/2) configuration (42). Both electron correlation strength and charge-transfer gap $\Delta_{CT}$ increase with decreasing $x$, as typically found in the late transition metal series. The MIT at $x=0.22$ is of interest because it is driven by electron correlations (43) and is associated with a competition between an insulating antiferromagnetic phase and an unconventional paramagnetic semi-metal (44), where the Dirac nodes are found at the Fermi level. We show that a distinctive feature of these Dirac states is their dominant $d$-orbital character and that the underlying band inversion mechanism is driven by a large $d-p$ hybridization combined with the non-symmorphic symmetry (NSS) of the crystal (see Fig. 1(b)). It follows that an essential role in controlling the properties of Dirac states is played by electron correlations and by the charge-transfer gap (Fig. 1 (c)), as they have a direct impact on the hybridization strength. This results into an effective tunability of shape, energy and wave vector of the Dirac lines in the proximity of the Fermi level. Specifically, the present ARPES study unveils Dirac bands moving from $M$ to $\Gamma$ with decreasing $x$. The bands are well explained quantitatively by ab initio calculations, in a hybrid density functional approximation suitable for including non-local correlations of screened-exchange type, which affect the hybridization between the $d$ and $p$ states. The same functional is able to describe the insulating spin-density wave (SDW) phase at $x=0$, driven by local correlations, upon increase of the optimal screened-exchange fraction. These calculations confirm that the Dirac nodes mobility in ${\bf k}$-space stems directly from the evolution of the charge transfer gap, i.e. the relative position between $d$ and $p$ on-site energies. These results clearly suggest that BaCo1-xNixS2 is a model system to tailor Dirac states and, more generally, that two archetypal features of correlated systems such as the hybrid $d-p$ bands and the charge-transfer gap constitute a promising playground to engineer Dirac and topological materials using chemical substitution and other macroscopic control parameters.

Observation of Dirac states in BaNiS2

We begin with the undoped sample BaNiS2. In Fig. 1(d), we represent a three dimensional ARPES map of the Brillouin zone (BZ) for the high symmetry directions. Along $\Gamma-M$, we observe linearly dispersing bands and -within ARPES resolution- gapless nodes at the Fermi level $E_{F}$. The Fermi surface reveals two pairs of such Dirac-like crossings related to each other by the time-reversal and by the two-fold rotation axis $C_{2}$ of the $C_{2v}$ little group for the ${\bf k}$-vectors along $\Gamma-M$. The Dirac nodes lie on the $\sigma_{d}$ reflection planes and extend along the $k_{z}$ direction piercing the whole BZ, unlike other topological node-line semimetals known to date, like Cu3NPd (46, 47), Ca3P2 (48) and ZrSiS (49), where the nodal lines form closed areas around high-symmetry points.

As one can see in Fig. 1(e), along the $\Gamma-M$ direction, the linearly dispersing bands remain isolated up to about $E-E_{F}=-0.35$ eV. These bands create an oval-shaped section on the constant energy maps near the Fermi level (see Fig. 1(d)). This asymmetry, clearly visible in Fig. 1(e), arises from the tilted type-I nature of the Dirac cone. The model Hamiltonian explaining the low-energy spectrum of the linearly dispersing bands observed experimentally is described in the Supporting Information (SI), c.f. Sec. S1. The linear bands present no $k_{z}$ dispersion, as shown in the SI (Sec. S2 and Fig. S2). The absence of dispersion is an indication of the 2D nature of the Dirac cones. The Dirac point remains pinned almost at the Fermi level - about 30 meV above - and its wave vector is fixed along the $\Gamma(Z)-M(A)$ direction. Here, the Dirac point position is obtained by extrapolating the ARPES data. These values are in perfect agreement with those directly measured in a recent pump-probe experiment (50).

Symmetry analysis of the electronic bands: mechanism of band inversion and formation of Dirac states

To unveil the physical mechanism responsible for the formation of Dirac cones in BaNiS2 we performed a detailed theoretical analysis of the symmetry of the electronic bands. We carried out density functional theory (DFT) calculations, by employing a modified Heyd–Scuseria–Ernzerhof (HSE) functional. The details of the band structure are presented in Methods and SI (see SI Sec. S3, where we also discuss how the inclusion of the spin-orbit coupling (SOC) affects the topological properties). The use of the HSE functional is dictated by non-local correlation effects present in this material. Indeed, a hybrid HSE functional with the optimal screened-exchange fraction $\alpha=7\%$ (see Eq. 1) is needed to account for the Fermi surface renormalization of BaNiS2 seen in quantum oscillations (51). Previous theoretical calculations (52, 41, 53) have shown that both S 3$p$- and Ni 3$d$-orbitals contribute to the Bloch functions near the Fermi level. We ascribe the electronic states close to the Fermi level mainly to the Ni $3d$-orbitals hybridized with the S $3p$-orbitals. In this situation, the exchange contribution to the hybridization with the ligands plays a crucial role in determining the topology of the Fermi surface (Fig. S6(b) illustrates the electronic structure dependence upon $\alpha$). Hereafter, we consider a Cartesian reference frame where the $x$- and $y$-axis are parallel to the Ni-S bonds in the tetragonal $ab$-plane. Neighbouring Ni ions are aligned along the diagonal $xy$ direction (Fig. 1(b)). In this frame, at the crossing points, located along the $(u,u,v)$ directions, the bands have dominant $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ character. This multi-orbital nature was confirmed by a polarization dependent laser-ARPES study (see SI, Sec. S4).

As sketched in Fig. 2(a), the crystal structure of BaNiS2 is made of square-lattice layers of staggered, edge-sharing NiS5 pyramids pointing along the out-of-plane [001] $c$-axis direction (40). The Ni atoms inside the S pyramids probe a crystal field that splits the atomic $d$-shell into the following levels (in descending energy order): $d_{x^{2}-y^{2}}$, $d_{z^{2}}$, the degenerate doublet $(d_{xz},d_{yz})$ and $d_{xy}$. Due to the $3d^{8}4s^{0}$ electronic configuration of the Ni2+ ion, we expect all $d$-orbitals to be filled, except the two highest ones, $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$, which are nearly half-filled assuming that the Hund’s exchange is sufficiently strong.

The puckering of the BaNiS2 layers gives rise to a tetragonal nonsymmorphic $P$4/$nmm$ structure characterized by a horizontal gliding plane which generates two Ni and two apical S positions at (1/4,1/4,$z$) and (3/4,3/4,$-z$), separated by a fractional f=(1/2,1/2,0) translation in the plane, Fig. 2(a). The two Ni atoms occupy Wyckoff position $2c$, corresponding to the M symmetry, while the two planar S are at the $2a$ site corresponding to the $\Gamma$ symmetry.

At M, the energy hierarchy of the atomic orbitals follows closely the crystal field splitting (Fig. 2(b)). The little group admits the following four 2D irreducible representations (irreps) $E_{Mi=1,\dots,4}$ (54), each originating from the same orbitals of the two inequivalent Ni. However, the levels stacking at $\Gamma$, whose little group is isomorphic to $D_{4h}$, differs from that predicted by the crystal field. This is due to the sizable hybridization of Ni $d$-orbitals with the S $p$ ligands (see SI, Sec. S5). Owing to the NSS, each Bloch eigenfunction at $\Gamma$ is either even or odd upon exchanging the inequivalent Ni and S within each unit cell. Even and odd combinations of identical $d$-orbitals belonging to inequivalent Ni atoms split in energy since they hybridize differently with the ligands. The even combination of the $d_{x^{2}-y^{2}}$ Ni orbitals is weakly hybridized with the $p_{z}$-orbitals of the planar S, since the two Ni atoms are out of the basal plane. On the other hand, the odd combination is non-bonding. It follows that the $B_{1g}$ even combination shifts up in energy with respect to the $B_{2u}$ odd one. Seemingly, the $A_{2u}$ odd combination of the $d_{z^{2}}$-orbitals hybridizes substantially with the $p_{z}$-orbitals of the planar and apical S, thus increasing significantly the energy of the odd combination. Eventually, its energy raises above the $B_{1g}$ and $B_{2u}$ levels, as well as the $A_{1g}$ state (even combination of $d_{z^{2}}$-orbitals). This leads to a reverse of the crystal field order as reported in Fig. 2(c).

Because the irreps at the A and Z ${\bf k}$-points are equivalent to those at M and $\Gamma$ (54), respectively, the orbital hierarchy found at M and $\Gamma$ must be preserved along the $M-A$ and $\Gamma-Z$ directions. Thus, for any $v$ along the $(0,0,v)\rightarrow(1/2,1/2,v)$ path, a band inversion between bands with predominant $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ characters must occur. Therefore, band crossing is allowed without SOC, and leads to two Dirac points at a given $k_{z}$ right at the Fermi energy for $k_{z}=0$. Indeed, the crossing bands transform like different irreps of the little group, which is isomorphic to $C_{2v}$ for a ${\bf k}$-point $(u,u,v)$ with $v=0,1/2$, and to $C_{s}$ with $v\in\leavevmode\nobreak\ ]0,1/2[$. These Dirac nodes are massive as a consequence of the SOC, which makes the material a weak topological insulator. The SOC gap is however very small (about 18 meV), and below ARPES resolution. Nevertheless, the focus of the present work is not on these very-low-energy features, but rather on the tunability of the whole Dirac nodal structure. In the family of weak topological insulators having the same $P4/nmm$ space group and showing SOC gapped Dirac cones along the $\Gamma-M$ direction (such as ZrSiS, for instance), BaCo1-xNixS2 is a peculiar member. Indeed, the strong local Hund’s exchange coupling favors nearly half-filled $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals, that explains the proximity of the Dirac nodes to the Fermi level for $x=1$, in accordance also to Luttinger’s theorem (see SI, Sec. S6). This is another signature of the relevance of electron correlations in this transition metal compound, which manifest themselves in both local and non-local contributions, the former leading eventually to the insulating phase at the Co side of BaCo1-xNixS2, the latter affecting the variation of $\Delta_{CT}$ across the series.

ARPES evidence of Dirac states tuned by doping, $x$

We now turn our attention to the effect of the Co/Ni substitution on the evolution of the band structure, notably the Dirac states. According to the BaCo1-xNixS2 phase diagram, this substitution modifies the strength of the electron-electron correlations and the amplitude of $\Delta_{CT}$. A series of ARPES spectra are given for the $x=0.75$ and $x=0.3$ compositions. In Fig. 3, we display the evolution of the Fermi surface and the electronic band structure along $\Gamma-M$ with $x$. For $x=0.75$, the Fermi surface is composed of a four-leaf feature at the $\Gamma$ point and four hole-like pockets along the $\Gamma-M$, Fig. 3(c). These pockets originate from the Dirac states crossing the Fermi level. The Dirac cone is shown in Fig. 3(d) along and perpendicular to the $\Gamma-M$ direction. At higher substitution levels, for $x=0.30$, the Dirac states shift up to lower binding energies, so the size of the hole-like pockets in the $k_{x}-k_{y}$ plane is increased (see Fig. 3(e,f)). The ARPES signal is also broader: since our structural study indicates that the crystalline quality is not affected by Co/Ni substitution (see Sec. S7 and Tab. S2 in SI), this broadening is consistent with the increase in electron-electron correlations while approaching the metal-insulator transition. (52, 39, 43). On the theoretical ground, this is expected because Co-substitution brings the whole $d$-manifold closer to fillings where local correlation effects are enhanced, according to the Hund’s metals picture (55). Fig. 4(a) schematically illustrates the evolution of the Dirac cone with $x$; in Table 1 we give the position of the Dirac points determined by extrapolating the band dispersion. In summary, one notes that the Co-substitution moves the Dirac points further beyond the Fermi level and reduces its wave vector.

Evolution of Dirac states with doping

In order to account for the tunability of the Dirac cones detected by ARPES, we carried out extensive ab initio DFT-HSE calculations as a function of the screened-exchange fraction $\alpha$, which controls the correlation strength in the modified hybrid functional framework. To explicitly include the charge transfer $\Delta_{CT}$ variation led by chemical substitution, we computed the two end-members of the BaCo1-xNixS2 series, namely $x=1$ (BaNiS2) and $x=0$ (BaCoS2). For $x=1$ the optimal $\alpha=7\%$, since it reproduces the frequencies of quantum oscillations in BaNiS2 (51). In order to fix such percentage for $x=0$, we performed ab initio calculations assuming the collinear SDW observed experimentally (42), by means of both HSE and the generalized gradient approximation supplemented by local Hubbard interactions (GGA+U). The strength of the Hubbard repulsion $U=3.0\leavevmode\nobreak\ \text{eV}$ and local Hund’s coupling $J=0.9\leavevmode\nobreak\ \text{eV}$, included in GGA+U, was estimated from first principles within the constrained random phase approximation (43). GGA+U correctly predicts an insulating state (Fig. 4(e)). By varying the percentage $\alpha$ of screened exchange in HSE, we find that, while $\alpha=7\%$ gives a metal, $\alpha\simeq 19\%$ reproduces the main peaks across gap obtained by GGA+U (Fig. 4(e)). This result suggests that HSE can describe BaCo1-xNixS2 only if the percentage of screened exchange $\alpha$ increases from 7% up to around 19% with decreasing $x$ from 1 to 0. Starting from the most correlated Co side, the reduction of the Hubbard repulsion upon electron doping, implied by the $\alpha$ dependence on $x$, has been found in other strongly correlated compounds, such as La-doped Sr2IrO4 (56).

In BaCoS2, beside the SDW solution compatible with the observed low-temperature state, it is possible to obtain another one once magnetism is not allowed, namely forcing spin $SU(2)$ symmetry. This paramagnetic metallic (PM) phase is metastable at low temperature, and adiabatically connected with the metallic solution at $x=1$. Therefore, it hosts Dirac cones; it is metallic and separated by an energy barrier from the stable insulating SDW phase. In Fig. 4(d), we plot the distance of the Dirac node (${\bf k}_{\textrm{Dirac}}$) from the $\Gamma$ point as a function of $\alpha$, for $x=1$ and the metallic solution at $x=0$. ${\bf k}_{\textrm{Dirac}}$ strongly depends on both $x$ and $\alpha$ (See sec. S8, Figs. S6(a) and S6(b) plot the band structures where the ${\bf k}_{\textrm{Dirac}}$ values have been extracted from). By taking the optimal $\alpha$’s for each $x$, the Dirac node is predicted to drift from ${\bf k}_{\textrm{Dirac}}\simeq 0.52\,\text{\AA}^{-1}$ at $x=1$ down to ${\bf k}_{\textrm{Dirac}}\simeq 0.38\,\text{\AA}^{-1}$ at $x=0$, covering the colored $y$-axis range in Fig. 4(d), in agreement with the range of variation seen in experiment.

Next, we analyze the 22-bands full $d-p$ tight-binding model derived from the ab initio DFT-HSE for $x=1$ (with $\alpha=7\%$) and for $x=0$ (with $\alpha=19\%$), c.f. Sec. S9, and Fig. S8 The $x=0$ state has shifted Dirac cones in both ${\bf k}$ and energy position with respect to the BaNiS2 parent compound. To underpin the mechanism behind the evolution of the cones, we compared the two tight-binding Hamiltonians for $x=0$ and $x=1$ The main difference involves the on-site energies and, in particular, the relative position of the $p$ and $d$ states, i.e. the charge transfer gap $\Delta_{CT}$. This proves that the doping $x$ via chemical substitution is indeed an effective control parameter, as it alters the $d-p$ charge transfer gap $\Delta_{CT}$ together with the correlation strength and, consequently, the $d-p$ hybridization amplitude, which directly affects position and shape of the Dirac nodes.

In the following, we define $\Delta_{CT}$ as the energy difference between the average energy position of the full $d$ manifold and the average one of the $p$ manifold. According to our HSE calculations, $\Delta_{CT}$ varies from 1.1 eV ($x=1$) to 1.6 eV ($x=0$). Assuming a linear variation of $\Delta_{CT}$ and on-site energies upon Ni-content $x$, we are able to estimate $\Delta_{CT}=\Delta_{CT}(x)$ and, thus, predict the evolution of the band structure and Dirac states by interpolating between the BaCoS2 and BaNiS2 TB models. This evolution is reported in Fig. 4(b), while the actual Dirac states dynamics - represented by the behavior of both the ${\bf k}$ and energy position of the Dirac point as a function of $\Delta_{CT}$ - is plotted in Fig. 4(c). This shows that the tunability upon doping found experimentally does not merely consist of a rigid shift of the Dirac cones (19), but it involves the change of both their shape and ${\bf k}$-position (see also Fig. S8).

This theoretical prediction is in good agreement with the observed evolution of the Dirac cone with $x$, as apparent in Fig. 4(a). Such movable Dirac nodes in the ${\bf k}$-space have recently attracted a great deal of interest from theory (15, 28, 57), as well as in the context of optical lattices (25) and photonic crystals (26). The present system offers the opportunity of observing in a real material how a simple experimental parameter - chemical substitution - can be used to tune Dirac states.

Manipulating the shape and position of the Dirac cones is also expected in BaCo1-xNixS2 using pressure in bulk samples or strain in thin films. Specifically, strain can be used to distort the square lattice, thus breaking one of the symmetries that protect the fourfold Dirac nodal lines. Non-trivial phases, such as Weyl semimetals, could then be triggered by time-inversion breaking perturbations, like an external electromagnetic field. A further possibility is the creation of spin-chiral edge states thanks to the proximity of the material to a topological insulator.

Conclusion

In conclusion, we have shown that BaCo1-xNixS2 offers the opportunity of effectively tuning Dirac bands by exploiting a peculiar inversion mechanism of $d$-electron bands. Namely, the Co/Ni substitution has been found to alter both the charge transfer gap and the strength of the electron-electron correlations that control position and shape of the bands. Remarkably, the same Co/Ni substitution makes it possible to span the electronic phase diagram, with the Dirac states present across its metallic phase. We emphasize the applicability of the present approach to a wide class of materials described by the $d-p$ effective Hamiltonian, thus enabling to forge new Dirac states controlled by chemical substitutions. This opens the perspective of engineering Dirac states in correlated electronic systems by exploiting macroscopically tunable parameters.

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ARPES measurements

Single crystals of BaCo1-xNixS2 were cleaved in-situ, exposing the ab plane under UHV conditions (base pressure better than $10^{-11}$ mbar). Most of the synchrotron radiation ARPES measurements were performed on the Advanced Photoelectric Effect (APE) beamline at the Elettra light source, with linearly polarized beam and different photon energies. The sample temperature was 70 K. The data were collected with a VG-DA30 Scienta hemispherical analyzer that operates in deflection mode and provides high-resolution two-dimensional ${\bf k}$-space mapping while the sample geometry is fixed (58). The total measured energy resolution is $\sim$ 15 meV and the angular resolution is better than 0.2*∘. Some of the data were also acquired with a 6.2 eV laser source (59); and some at the Spectromicroscopy beamline (60): the end station hosts two exchangeable multilayer-coated Schwarzschild objectives (SO) designed to focus the radiation at 27 eV and 74 eV to a small spot ($\sim$600 nm). The photoelectrons are collected by an internal movable hemispherical electron energy analyzer that can perform polar and azimuthal angular scans in UHV. The energy and momentum resolutions are $\sim$33 meV and $\sim$0.03 Å-1*, respectively.

Ab initio calculations

We carried out ab initio DFT calculations in a modified HSE functional. It improves upon the Perdew, Burke, and Ernzerhof (PBE) (61, 62) exchange-correlation ($xc$) functional by the addition of a screened Fock term ($E_{x}^{HF,\textrm{screened}}$), such that the resulting functional reads as

[TABLE]

The screened interaction is written as: $V^{\textrm{screened}}(r)=\operatorname{erfc}(\omega r)/r$, where $\operatorname{erfc}$ is the complementary error function, and $\omega=0.108$ in atomic units, i.e. the HSE regular value. In this work, $\alpha$ is instead taken as an adjustable parameter, which depends on the correlation strength of the system.

We used the Quantum Espresso package (63, 64) to perform modified HSE calculations for BaNiS2 ($x=1$) and BaCoS2 ($x=0$) in a plane-waves (PW) basis set. The geometry of the cell and the internal coordinates are taken from experiment (45). We replaced the core electrons of the Ni, Co, Ba, and S atoms by norm-conserving pseudopotentials. For the Ni (Co) pseudopotential, we used both fully- and scalar-relativistic versions, with 10 (9) valence electrons and nonlinear core corrections. The Ba pseudopotential includes the semicore states, while the S pseudopotential has $3s^{2}3p^{4}$ in-valence electrons. We employed a $8\times 8\times 8$ electron-momentum grid and a Methfessel-Paxton smearing of 0.01 Ry for the ${\bf k}$-point integration. The PW cutoff is 60 Ry for the wave function. The non-local exchange terms of the HSE functional are computed through the fast implementation of the exact Fock energy (64), based on the adaptively compressed exchange scheme (65). In the non-local Fock operator evaluation, the integration over the ${\bf q}$-points is downsampled on a $8\times 8\times 2$ grid. We applied a half-a-grid shift in the $z$ direction to minimize the number of nonequivalent momenta in the ${\bf k}+\textbf{q}$ grid. By means of the Wannier90 code (66), we performed a Wannier interpolation of the ab initio bands for $x=1$ in the energy window spanned by the $d-p$ manifold, to accurately resolve the band structure, chemical potential, and Fermi surface, and to derive a minimal TB model.

To successfully deal with the most demanding simulations (HSE functional evaluated in a larger cell with spin resolved orbitals), we supplemented the Quantum Espresso calculations with some performed by means of the Crystal17 package(67), particularly suited to efficiently compute the exact exchange operator. In this framework, we used scalar-relativistic Hartree-Fock energy-consistent pseudopotentials by Burkatzki, Filippi, and Dolg(68), and an adapted VTZ Gaussian basis set, for both Ni and Co. In our Crystal17 calculations, the k-grid has been set to a $32\times 32\times 32$ dense mesh, with a Fermi smearing of 0.001 Hartree. We cross-checked the Crystal17 and Quantum Espresso band structures for the paramagnetic phase of BaNiS2 and BaCoS2, in order to verify the convergence of all relevant parameters in both PW and Gaussian DFT calculations.

\showmatmethods\acknow

This work was supported by "Investissement d’Avenir" Labex PALM (ANR-10-LABX-0039-PALM), by the Region Ile-de-France (DIM OxyMORE), and by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. We acknowledge Benoît Baptiste for XRD characterization and Imène Estève for her valuable assistance in the EDS study. M.C. is grateful to GENCI for the allocation of computer resources under the project N. 0906493. M.F. and A.A. acknowledge support by the European Union, under ERC AdG "FIRSTORM", contract N. 692670.

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