BaCuS2: a superconductor with moderate electron-electron correlation
BBaCuS : a superconductor with moderate electron-electron correlation Yuhao Gu, Xianxin Wu, Kun Jiang, and Jiangping Hu
1, 2, ∗ Beijing National Laboratory for Condensed Matter Physics,and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China CAS Center of Excellence in Topological Quantum Computation and Kavli Institute of Theoretical Sciences,University of Chinese Academy of Sciences, Beijing 100190, China
We show that the layered-structure BaCuS is a moderately correlated electron system in which the electronicstructure of the CuS layer bears a resemblance to those in both cuprates and iron-based superconductors. The-oretical calculations reveal that the in-plane d - p σ ∗ -bonding bands are isolated near the Fermi level. As theenergy separation between the d and p orbitals are much smaller than those in cuprates and iron-based super-conductors, BaCuS is expected to be moderately correlated. We suggest that this material is an ideal system tostudy the competitive/collaborative nature between two distinct superconducting pairing mechanisms, namelythe conventional BCS electron-phonon interaction and the electron-electron correlation, which may be helpfulto establish the elusive mechanism of unconventional high-temperature superconductivity. I. INTRODUCTION
From conventional BCS superconductors to unconventionalhigh T c superconductors, cuprates[1], it has been widely sug-gested that the superconducting mechanism changes fromphonon-mediated attraction to electron-electron correlationdriving pairing. However, the existence of such a difference isstill under intensive debate. For example, in the search of con-ventional BCS superconductors with relatively high transitiontemperature, a possible guiding principle is to find systemswith metallized σ -bonding electrons from light atoms so thatthe electron-phonon interaction can be maximized[2–4]. Sucha textbook example is MgB [2, 5, 6], in which the in-plane σ bands are formed by the p x and p y orbitals of B atoms. TheMg ions further lower the B π ( p z ) bands, which causes acharge transfer from σ to π bands and drives the self-dopingof the σ band. The 2D nature of σ bonds then leads to anextremely large deformation potential for the in-plane E g phonon mode, which greatly enhances the electron-phononcoupling [2, 6]. This principle was also argued to be valideven for cuprates, in which the d - p σ -bonding band is respon-sible for superconductivity [4, 7]. The argument has left aroom to discuss the likelihood of the electron-phonon mecha-nism in cuprates[8].However, the d - p bonding displays fundamental differ-ences from the p - p bonding because of the multiplicity andstrong localization of the d -orbitals. Such a simple exten-sion is highly questionable. For example, the absence ofclear isotope effect [9] , unconventional electronic proper-ties in normal states and strongly antiferromagnetic fluctua-tions in cuprates suggest the electron-electron correlation canbe responsible for high T c superconductivity[10, 11]. Fur-thermore, the discovery of high T c iron-based superconduc-tors(SCs) finishes another large part of the superconductingjigsaw puzzle. Much evidence has suggested that the super-conductivity of iron-based superconductors originates fromthe Fe-As/Te plane with strong electron-electron correlationand is very similar to cuprates[12–14]. On the other hand, ∗ [email protected] electron-phonon coupling also plays a non-negligible role iniron-based superconductors[15–17].Recently, focusing on the d -orbitals, emphasizing electron-electron interaction, we have suggested a new guiding princi-ple for the search of unconventional high T c superconductors:those d -orbitals with d - p σ -bondings must be isolated nearFermi energy. Under this principle, local cation complexes,the connection between the complexes, the electron filling fac-tor at transition metal atoms and lattice symmetries must col-laborate to fulfill the criteria[18–22]. This simple principlecan explain why cuprates and iron-based superconductors areso special as high T c superconductors.It is noticeable that the above mentioned two principles arelinked. While they emphasize different interactions, both ofthem are featured by σ -bonding. Thus, why are the two typesof σ -bonding fundamentally different? In order to answerthese two questions, we want to find a system with a mod-erate electron-electron correlation from the d - p σ -bonding sothat an explicit comparison between the electron-electron cor-relation and electron-phonon interaction can be examined.In this paper, we propose that a new material BaCuS canfulfill the above task. BaCuS is a moderately correlated elec-tron system in which the d - p σ ∗ -bonding bands solely controlthe electronic physics near Fermi energy. Similar to cuprates,iron pnictides and MgB , BaCuS also has a layered struc-ture, where the electronic structure is dominated by the CuS square layer. We demonstrate that the electron-electron cor-relation may drive superconductivity in BaCuS with a d xy -wave pairing symmetry, very similar to the superconductiv-ity in cuprates. While, if the superconductivity is causedby electron-phonon couplings, a conventional BCS s -wavestate is expected, similar to La N Ni B . According to first-principles calculations, we find that the BaCuS phase is ther-modynamically stable and has lower formation energy underpressure compared with other known phases, suggesting that itcan be synthesized in future experiments under external pres-sure. a r X i v : . [ c ond - m a t . s up r- c on ] S e p FIG. 1. (a) Crystal structure of BaCuS . Here S a represents theapical S atoms while S h represents the horizontal S atoms. (b-d) Crystal structures of other layered superconductors (b) CaCuO ,(c) FeSe and (d) La N Ni B . (e) The comparison of parametersin dp -models of BaCuS , CaCuO , FeSe and La N Ni B . Here | t dp | is the amplitude of the major hopping parameter in each com-pound ( | t Cu,d x − y − S h ,p x/y | in BaCuS , | t Cu,d x − y − O,p x/y | inCaCuO , | t Fe,d xz/yz − Se,p x/y | in FeSe and | t Ni,d xz/yz − B,p x/y | inLa N Ni B ) and ∆ dp is the corresponding on-site energy differ-ence. II. BaCuS ELECTRONIC STRUCTURE ANDCOMPARISON WITH OTHER SC MATERIALS
We start from the crystal structure and electronic struc-ture. The layered ternary transition metal sulfide BaCuS has a structure: the BaS layers alternate with Cu S layersand the transitional metal atom is in square pyramidal coor-dination, as shown in FIG.1(a). The upward-pointing squarepyramidals connect the downward-pointing ones by sharingedges, forming the glide symmetric Cu S plane. Similar toother layered SCs, the main electronic structure of BaCuS stems from its Cu S layer. To demonstrate it, we carriedout density functional theory calculations for BaCuS . Theelectronic structure and density of states (DOS) of BaCuS are plotted in FIG.2. The bands around the Fermi level areformed by the d - p valence manifold. Partially-filled d - p σ ∗ -bonding bands cross the Fermi level, where Cu d x − y or-bitals strongly hybridize with in-plane S p x/y orbitals and Cu d z orbitals strongly couple with apical S p z orbitals. Owingto the planar nature of Cu d x − y orbitals and S p x/y orbitals,these bands have a weak dispersion along the k z direction.In contrast, the bands from Cu d z orbitals and apical S p z orbitals exhibit a large dispersion. We expect that the SC ofBaCuS is mainly contributed from the 2D cylindrical Fermisurface, which is similar to the other layered SCs.It is interesting to compare the electronic structureof BaCuS with those of high- T c superconductors(CaCuO [23]; FeSe[24]) and known BCS superconduc-tor (La N Ni B [25]). These three materials are layeredtransition metal compounds, are shown in FIG.1(b-d). Their FIG. 2. The band structure and density of states for BaCuS fromdensity functional theory (DFT) calculation. The sizes of dots rep-resent the weights of the projection. Here S a represents the apical Satoms while S h represents the horizontal S atoms. electronic structures are mainly attributed to the squarelayers. In CaCuO and FeSe, the electron correlation plays animportant role in the unconventional superconductivity[19].From an electronic-structure perspective, this is consistentwith the fact that the d -orbitals dominate around the Fermisurfaces in these materials, giving rise to strong correlations.However, in La N Ni B , the extended s - p bands of anionsdominate Fermi surfaces and are featured by strong electron-phonon couplings via B’s high frequency A g phonons,while Ni’s d -orbitals play a less pronounced role[26, 27].As shown in FIG.2, in the d - p σ ∗ -bonding bands of BaCuS near Fermi energy, the weight of p -orbitals of the in-plane Satoms is much larger than those in cuprates and iron-basedsuperconductors. In contrast to La N Ni B , the weight of d x − y orbitals of Cu atoms is still sizable.To quantitatively confirm this point, we construct tight-binding models including the d -orbitals of transition-metalatoms and the p -orbitals of coordinated anions to analyze theirelectronic structures by calculating the maximally localizedWannier functions (MLWFs)[28]. Our Wannierization resultssuccessfully reproduce the band structures from DFT calcula-tionsp, as shown in appendix (FIG.S3). Then, we extract hop-ping parameters and on-site energies from our Wannierizationresults and display the representative parameters in appendix(TABLE.S1). In cuprate CaCuO and iron-based supercon-ductor FeSe, the electronic physics is dominated by d -orbitalsnear the Fermi level, whose on-site energies are much higher(about 2 eV) that those of coupled p -orbitals. Nevertheless, inBCS-type superconductor La N Ni B , where the NiB layeris isostructural to the FeSe layer, the electronic physics is quitedistinct: the on-site energy of B- p x/y orbital is even higherthan that of Ni- d xz/yz orbital. This is consistent with pre-vious studies[26, 27]: multiple components cross the Fermilevel, showing that La N Ni B is a good metal. The scenarioof BaCuS is different from above examples: the on-site en- FIG. 3. (color online) (a) Lattice structure of Cu layer in the tight-binding model. In each unit cell, there are two Cu atoms sitting aboveand below the xy plane. We label these two sublattices by A and B,respectively. The conventional crystal structure direction is definedfor Cu A to Cu A direction, labeled as X − Y . To simplify our modelusing glide symmetry, we rotate the global axis to Cu A to Cu B direc-tion, labeled as x - y . (b) The two-Cu tight-binding model for BaCuS in Eq.(1) (black lines). The blue dash lines are the folded energybands with a folding vector Q= ( π, π ) . ergies of d -orbitals are still higher than the p -orbitals, but theenergy difference is only about 1 eV, much less than those inhigh- T c superconductors. It suggests that the antiferromag-netic (AFM) order may not be stabilized in BaCuS due to theweak superexchange coupling [29]. Moreover, the differencein hopping parameters between partially-filled d -orbitals andcoupled p -orbital is not so significant, as shown in FIG.1.(b).The above analysis in BaCuS is consistent with the ab-sence of any magnetically ordered states in our calculation.Therefore, BaCuS tends to be a material with a moderateelectron-electron correlation.As mentioned previously, the guiding principle for search-ing high transition temperature BCS superconductors are lightatoms and metallized σ -bonding electrons[2, 3]. In principle,in BaCuS , the d - p σ ∗ -bonding bands cross the Fermi level,where those metallic σ -bonding electrons can support theBCS-type superconductivity. Thus, we calculate the electron-phonon coupling (EPC) properties of BaCuS [28]. The EPCstrength λ is about 0.59 in BaCuS , which is lower than thatin La N Ni B ( λ ∼ . , T c ∼ K[30]) but slightly higherthan that in LaNiBN ( λ ∼ . , T c ∼ . K[31]). However,due to the heavy mass of Cu and S atoms, we find that theelectron-phonon coupling in BaCuS can only induce inducesuperconductivity of T c less than 4 K.[28]. TABLE I. The optimized structural parameters for BaCuS (spacegroup P /nmm ). Here S a represents the apical S atoms while S h represents the horizontal S atoms.System a( ˚A) c( ˚A) Cu-S h ( ˚A) Cu-S a ( ˚A) Cu-S h -Cu( ◦ )BaCuS III. THE EFFECTIVE TWO BAND MODEL AND RPARESULTS
To study the correlation effect, we first construct an effec-tive minimal model by Wannierization based on the d x − y -like and d z -like MLWFs on Cu sites in BaCuS [28, 32, 33].The d - p σ ∗ -bonding bands are obviously more delocalizedin BaCuS than that in CaCuO (More details can be foundin appendix FIG.S4.(c-e)). As a consequence, the correla-tion strength in BaCuS should be weaker. Then by fittingthe Wannierization results[28], we arrive at an effective tight-binding (TB) model in the basis of d x − y orbital and d z orbital.There are two Cu atoms in each unit cell, as shown inFIG.3(a), which indicates that the minimal model for BaCuS contains four bands. Similar to FeSe, the space group ofBaCuS is P /nmm . There is a glide symmetry whichconsists of a translation Cu A to Cu B and a mirror reflec-tion perpendicular to the S plane. Then, using glide sym-metry, one can unfold the band structure into the Brillouinzone of one-Cu unit cell and write down a two-band modelfor BaCuS . Note that, the conventional crystal structure di-rection of BaCuS is defined for the Cu A to Cu A direction,labeled as X − Y . Similar to iron based superconductors, wedefine a new coordinate system with the x and y axes alignedto the Cu A to Cu B direction, labeled as x - y [34, 35].The two-band model in one-Cu unit cell in the basis ψ k =( d z ( k ) , d X − Y ( k )) (the spin index is omitted here) can bewritten as H = (cid:88) k ψ † k ˆ H k ψ k (1)where the 2 by 2 matrix H k and more details are providedin appendix. The band structure in the original unit cell canbe obtained by the folding the band structures of the Hamil-tonian H k , as plotted in FIG.3(b), where the blue dash linesare folded bands with a folding vector Q = ( π, π ) . The corre-sponding FSs are displayed in Fig.4(d) and the large oval FSsaround ( π ,0) or (0, π ) are mainly attributed to d X − Y orbitalwhile the smaller circular FS around the M point is mainlyattributed to d z orbitals.Using the above two-band model, we can apply the standardapproach to investigate the intrinsic spin fluctuations by carry-ing out RPA calculation[36–40]. We adopt the general multi-orbital Coulomb interactions, including on-site Hubbard intra-and inter-orbital repulsion U/U (cid:48) , Hund’s coupling J and pair-hopping interactions J (cid:48) , H int = U (cid:88) iα n iα ↑ n iα ↓ + U (cid:48) (cid:88) i,α<β n iα n iβ + J (cid:88) i,α<β,σσ (cid:48) c † iασ c † iβσ (cid:48) c iασ (cid:48) c iβσ + J (cid:48) (cid:88) i,α (cid:54) = β c † iα ↑ c † iα ↓ c iβ ↓ c iβ ↑ , (2)where n iα = n iα ↑ + n iα ↓ . The distribution of the largesteigenvalues of bare susceptibility matrices is displayed inFig.4(a). There is a prominent peak at q , close to ( π , ). k x k y χ R P A ( q ) λ U(eV) B B A k y k x (a) (b)(c) (d) q q q q (0,0) (π,0) (π,π) (0,0)π-π-π 00π -π0π π-π 0 αβ FIG. 4. (color online) (a) Distribution of the largest eigenvalues forbare susceptibility matrices χ ( k ) at n = 2 . . (b) Largest eigenval-ues of RPA susceptibility with U = 0 . eV and J/U = 0 . . (c)Pairing strength eigenvalues for the leading states as a function of in-teraction U with J/U = 0 . . (d) Gap function of the dominant B g state ( d xy -wave pairing). In addition, the bare susceptibility shows a broad peak at q .The former is attributed to the inter-pocket nesting between α and β and the latter is contributed by intra pocket nest-ing in α Fermi surface. From the RPA spin susceptibilityalong high-symmetry lines shown in Fig.4(b), we find thatthese peaks get significantly enhanced when interactions areincluded. All peaks in the susceptibility are far away from the Γ point, indicating intrinsic antiferromagnetic fluctuations inthe system. To investigate the pairing symmetry, we calculatethe pairing strengths as a function of Coulomb interaction U with J/U = 0 . . The dominant pairing has a B g symmetry,whose gap function is shown in Fig.4(d). Each pocket has a d xy -wave gap but the intrapocket nesting in α pockets inducesadditional sign changes in the corresponding gap functions.Moreover, there is a sign change between the gap functionson α and β pockets, which is determined by inter pocket nest-ing. The gap functions on these Fermi surfaces can be qualita-tively described by a form factor sink x sink y ( cosk x + cosk y ) ,which is classified as the d xy ( B g ) pairing symmetry. In thissuperconducting state, there are gapless nodes on high sym-metry lines as well as nodes on the original BZ boundary.The d xy pairing symmetry is quite robust here. In fact, ifwe consider the system in a strong electron-electron correla-tion region in which a short AFM interaction can be producedthrough the superexchange mechanism. We can easily arguethat the pairing symmetry in this limit is still d xy based onthe Hu-Ding principle[41] which states the pairing symme-try is selected by the momentum space form factor of AFMexchange couplings that produce the largest weight on Fermisurfaces. In this case, we would expect the gap function isproportional to sink x sink y , in which the nodal points at the TABLE II. The optimized volumes of BaCuS , Ba Cu S and theirdecomposition phases under different pressures. All the data are nor-malized according to the formula. The unit of volume is ˚ A here.0 GPa 5 GPa 10 GPa 15 GPa 20 GPaBaCuS Cu S BZ boundary in the above RPA calculations will not appear.
IV. DISCUSSION AND SUMMARY
FIG. 5. (a) Formation energies of BaCuS and Ba Cu S under dif-ferent pressures. (b) The band structure of BaCuS with an externalpressure P =
20 GPa from DFT calculation. The sizes of dots rep-resent the weights of the projection.
In summary, we propose a new superconducting mate-rial BaCuS . By comparing the electronic structures withCaCuO , FeSe and La N Ni B , we find that BaCuS shouldbe a moderately correlated electron system with strong p − d hybridization. The calculations based conventional BCSelectron-phonon coupling suggests that it is a standard s -wavesuperconductor with T c < K, while the electron-electroncorrelation results in an unconventional B g -wave supercon-ductor and possibly much higher T c .The structure of the material is in a highly stable phaseaccording to our theoretical calculations. In particular, wefind that the structure has much lower formation energy thanother known structures under external pressure. We cal-culate the formation energy of BaCuS and its sister com-pound Ba Cu S [42] under different pressures, as shown inFIG.5.(a). The volume of BaCuS /Ba Cu S is remarkablyless than that of BaS+CuS/3BaS+2CuS, as shown in TA-BLE.II. The main electronic physics does not vary much un-der pressure as shown in FiG.5.(b), in which the band structureof BaCuS under 20 GPa is plotted. Therefore, it is promis-ing that BaCuS can be synthesized in future experiments andstudying its superconductivity can help us reveal the relation-ship between conventional and unconventional superconduct-ing mechanisms. Acknowledgement:
We thank the useful discussions withJianfeng Zhang and Yuechao Wang. This work is supportedby the Ministry of Science and Technology of China 973 pro- gram (No. 2017YFA0303100), National Science Foundationof China (Grant No. NSFC11888101), and the Strategic Pri-ority Research Program of CAS (Grant No.XDB28000000). [1] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17(2006).[2] J. M. An and W. E. Pickett, Phys. Rev. Lett. , 4366 (2001).[3] M. Gao, Z.-Y. Lu, and T. Xiang, Physical Review B , 045132(2015).[4] M. Gao, Z.-Y. Lu, and T. Xiang, PHYSICS (in Chinese) ,421 (2015).[5] J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P. Antropov, andL. L. Boyer, Phys. Rev. Lett. , 4656 (2001).[6] X. X. Xi, Rep. Prog. Phys. , 116501 (2008).[7] F. Zhang and T. Rice, Physical Review B , 3759 (1988).[8] Y. Zhong, Y. Wang, S. Han, Y.-F. Lv, W.-L. Wang, D. Zhang,H. Ding, Y.-M. Zhang, L. Wang, K. He, et al. , Science Bulletin , 1239 (2016).[9] J. 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Our electronic structure calculations employ the Vienna abinitio simulation package (VASP) code[43] with the projec-tor augmented wave (PAW) method[44]. The Perdew-Burke-Ernzerhof (PBE)[45] exchange-correlation functional is usedin our calculations. The kinetic energy cutoff is set to be600 eV for the expanding the wave functions into a plane-wave basis in VASP calcuations. For body centered tetrag-onal La N Ni B and Ba Cu S , we employ their primitivecells to perform calculations. In the calculations of the forma-tion energy, the energy convergence criterion is − eV andthe force convergence criterion is 0.01 eV/ ˚A. The Γ -centered k -meshes are × × , × × , × × , × × , × × , × × and × × for BaCuS , Ba Cu S ,BaS, CuS, CaCuO , FeSe and La N Ni B , respectively.We employ Wannier90[32, 33] to calculate maximally lo-calized Wannier functions in BaCuS , CaCuO , FeSe andLa N Ni B . In the calculations of the d - p models, the initialprojectors are transition metal atoms’ d -orbitals and anions’ p -orbitals in BaCuS , CaCuO and FeSe. For La N Ni B ,the Ni( d )-B( p ) valence manifold strongly entangles with otherbands, so La’s d -orbitals and B’s s -orbitals are added in its ini-tial projectors to reproduce DFT-calculated band structures. Inthe calculation of the d - p σ ∗ MLWFs, the initial projectors areCu’s d x − y + d z orbitals in BaCuO and Cu’s d x − y orbitalin CaCuO , respectively.We employ EPW package[46] to calculate the electron-phonon coupling properties of BaCuS . The MLWFs arecalculated by Wannier90[32, 33] interfacing with QuantumESPRESSO[47]. We take the × × k -mesh and × × q -mesh as coarse grids and then interpolate to the × × k -mesh and × × q -mesh. The kinetic energy cutoff isset to 80 Ry. The Gaussian smearing method with the widthof 0.005 Ry is used for the Fermi surface broadening. Theenergy convergence criterion is − eV. In the highly accu-rate structural optimization, the lattice constants and atomiccoordinates are relaxed and the force convergence criterionis 0.000001 Ry/Bohr. The exchange-correlation functional isalso PBE and the norm-conserving SG15 pseudopotentials areused[48–50]. Appendix B: electron-phonon properties of BaCuS The phonon density of states F ( ω ) and the correspond-ing Eliashberg spectral function α F ( ω ) are plotted inFIG.S1.(a). By intergating α F ( ω ) , we get a moderateEPC strength λ = 0 . . We estimate the superconduct-ing transition temperature T c with the McMillan-Allen-Dynesformula[51, 52], T c = ω log . (cid:20) − . λ ) λ (1 − . µ ∗ ) − µ ∗ (cid:21) , (B1)where µ ∗ is the effective screened Coulomb repulsion con-stant and the logarithmic average of the Eliashberg spectral FIG. S1. (a) Eliashberg spectral function α F ( ω ) (red line) andPhonon density of states F ( ω ) (black line) for BaCuS . (b) Eval-uated T c as a function of µ ∗ for BaCuS . function ω log is denfined as ω log = exp (cid:20) λ (cid:90) dωω α F ( ω ) ln( ω ) (cid:21) . (B2)As µ ∗ is an input parameter, we plot T c as a function of µ ∗ inFIG.S1.(b). The phonon-mediated T c for BaCuS should beless than 4 K. Appendix C: Ba Cu S : separation by three rock salt-type BaSlayers As shown in FIG.S2, the crystal structure of Ba Cu S issimilar to that of BaCuS : The inverse α -PbO-type Cu S layer is separated by 3 rock salt-type BaS layers in Ba Cu S but separated by 2 BaS layers in BaCuS (Ba Cu S ). It alsoshares a similar electronic structure with BaCuS , as shown inFIG.S2.(b). Ba Cu S is not thermodynamically stable, but itis possible to synthesized Ba Cu S under external pressuredue to Cu’s five-coordination, as shown in FIG.5.(a). FIG. S2. (a) The crystal structure of Ba Cu S . (b) The band struc-ture of BaCuS with its primitive cell from DFT calculation. Thesizes of dots represent the weights of the projection. Here S a repre-sents the apical S atoms while S h represents the horizontal S atoms.The choice of the k -path is same as the literature’s[27]. Appendix D: Wannierization projected by d -orbitals and p -orbitals Our Wannierization results successfully reproduce DFT-calculated band structures, as shown in FIG.S3. The rele-vant representative hopping parameters and on-site energiesare listed in TABLE.S1. Here we use the conventional nota-tions of the local crystal field coordinations.
TABLE S1. The hopping parameters and on-site energies forBaCuS , CaCuO , FeSe and La N Ni B . Here S a represents theapical S atoms while S h represents the horizontal S atoms.BaCuS ε Cu,d z -2.21 ε Cu,d x − y -2.23 ε Cu,d xz/yz -2.32 ε Cu,d xy -2.48 ε S h ,p z -2.58 ε S h ,p x/y -3.08 ε S a ,p z -2.17 ε S a ,p x/y -1.40 | t Cu,d x − y − S h ,p z | | t Cu,d x − y − S h ,p x/y | | t Cu,d z − S h ,p z | | t Cu,d z − S h ,p x/y | | t Cu,d z − S a ,p z | | t S a ,p x − S a ,p x | ε Cu,d z -2.42 ε Cu,d x − y -1.92 ε O,p z -2.58 ε O,p x/y -3.83 | t Cu,d x − y − O,p x/y | ε Fe,d x − y -0.88 ε Fe,d xz/yz -0.78 ε Se,p z -3.07 ε Se,p x/y -3.09 | t Fe,d x − y − Se,p x/y | | t Fe,d x − y − Se,p z | | t Fe,d xz/yz − Se,p x/y | | t Fe,d x − y − Se,p z | N Ni B ε Ni,d z -2.11 ε Ni,d x − y -2.26 ε Ni,d xz/yz -2.11 ε Ni,d xy -2.24 ε B,s ε B,p z ε B,p x/y | t Ni,d x − y − Ni,p x/y | | t Ni,d x − y − Ni,p z | | t Ni,d xz/yz − Ni,p x/y | | t Ni,d x − y − Ni,p z | , (b) CaCuO ,(c) FeSe and (d) La N Ni B . The red/blue lines representDFT/Wannierization results, respectively. The choice of the k -pathin (d) is same as the literature’s[27]. Appendix E: Wannierization of the d - p σ ∗ -bonding bands andthe effective tight-binding model As mentioned above, the in-plane d - p σ ∗ -bonding bandsare isolated around the Fermi surface. In order to con-struct the effective minimal model to describe the in-planeelectronic physics in BaCuS , we downfold the full d − p model into an effective minimal model[53] by only Wannier-izing the d X − Y -like and d z -like MLWFs in BaCuS witha smaller energy window. Our Wannierization results cap- ture the main characters of BaCuS ’s electronic structure, asshown in FIG.S4.(a). This is an analogy to the Zhang-Ricesinglet in cuprates[7], so we also calculate the d X − Y -likeMLWF in CaCuO for comparison, as shown in FIG.S4.(b).We construct the effective tight-binding (TB) model in thebasis of d X − Y orbital and d z orbital to describe the in-plane electronic physics. Since there are two Cu atoms in oneunit cell, the TB model can be written as a × Hermitianmatrix: H = H = ε + 2 t x ( cos ( k x ) + cos ( k y )) + 2 t xx ( cos (2 k x ) + cos (2 k y )) + 4 t xxyy ( cos ( k x ) cos ( k y )) ,H = H = 2 t x ( cos ( k x ) − cos ( k y )) + 2 t xx ( cos (2 k x ) − cos (2 k y )) ,H = 4 t xy cos ( k x / cos ( k y /
2) + 4 t xxy ( cos ( k x / ∗ cos (3 k y /
2) + cos (3 k x / cos ( k y / ,H = H = 4 t xxy ( cos (3 k x / cos ( k y / − cos ( k x / cos (3 k y / ,H = H = ε + 2 t x ( cos ( k x ) + cos ( k y )) + 2 t xx ( cos (2 k x ) + cos (2 k y )) + 4 t xxyy ( cos ( k x ) cos ( k y )) ,H = 4 t xy cos ( k x / cos ( k y /
2) + 4 t xxy ( cos ( k x / ∗ cos (3 k y /
2) + cos (3 k x / cos ( k y / . (E1)The hopping parameters are truncated to the fifth-nearest-neighbour site. We get hopping parameters and on-site en-ergies by fitting to the Wannierization result in k z = 0 plane,as shown in FIG.S6. The corresponding parameters and theirnotations are listed in TABLE.S2. The major hopping parame-ter is t x , the intra-orbital hopping between two SNN d X − Y orbital, which is in the same energy scale with the dominat-ing intra-orbital hopping between two NN d x − y orbital incuprates ( t NNd x − y is about -0.47 in CaCuO ).As mentioned in our main text, we can transfer the × TBmodel into a block-diagonalized matrix with using the glidesymmetry: H eff ( k ) = (cid:18) H k H k + Q (cid:19) , (E2)here H k is the effective two-band model in our main text and Q = ( π, π ) . The explict form of H k is H k = (cid:18) H + H H + H H + H H + H (cid:19) , (E3)where H αβ are matrix elements in Eq.E1.Visually, we plot these d - p σ ∗ Wannier functions in BaCuS and CaCuO , as shown in FIG.S4.(c-e). These Wannier func-tions are composed of Cu’s d -orbitals and coordinated S/O’s p -orbitals symmetrically. As the isovalues of isosurfaces inFIG.S4.(c-e) are same, the d - p σ ∗ -bonding bands are moredelocalized in BaCuS than that in CaCuO . As a result, thecorrelation strength in BaCuS should be weaker. FIG. S4. (a-b) The band structures of (a) BaCuS and (b) CaCuO calculated by DFT (gray lines) and Wannierizaiton (red/blue dots).The sizes of dots represent the weights of the projection of the d - pσ ∗ Wannier functions. (c-d) The isosurface of (c) the d X − Y -likeMLWF and (d) the d z -like MLWF in BaCuS . (e) The isosurface ofthe d x − y -like MLWF in CaCuO . FIG. S5. The Fermi surfaces of BaCuS by Wannier fitting with 4MLWFs from (a) top view and (b) oblique view.FIG. S6. Comparision of the band structures of BaCuS by Wannier-izaiton (red lines) and fitted TB model (blue lines).TABLE S2. The hopping parameters and on-site energies of in-plane TB model for BaCuS . The energy unit is eV. Here superscript x labels the hopping between two second-nearest-neighbour (SNN)sites along X direction, superscript xx labels the hopping betweentwo forth-nearest-neighbour sites along X direction, superscript xy labels the hopping between two nearest-neighbour (NN) sites along Y = X direction, superscript xxyy labels the hopping between twothird-nearest-neighbour (TNN) sites along Y = X direction, super-script xxy labels the hopping between two fifth-nearest-neighboursites along Y = X/ direction; subscript 1-4 represent Cu A ’s d X − Y orbital, Cu A ’s d z orbital, Cu B ’s d X − Y orbital and Cu B ’s d z orbital, respectively. ε ε t x t xx t xxyy -0.31 -0.82 -0.28 -0.07 0.15 t x t xx t xxyy t x t xx t xy t xxy t xy t xxy t xxy Appendix F:
U/J parameters calculated by Local screenedCoulomb correction (LSCC) approach
The
U/J parameters represent the correlation strength inDFT + U calculations and are often chosen empirically. Herewe employ the first-principle LSCC approach[54] to calcu-late the U/J parameters in layered transition metal com-pounds CaCuO , FeSe, La N Ni B , BaNiS and BaCuS . InLSCC method, the local Coulomb interactions are calculatedby using the Yukawa potential, so the U/J should decreasewhen the system becomes more metallic. Since the
U/J isstrongly dependent on the muffin-tin radium R MT , we shouldonly compare the U/J with the same pseudopotential. Asshown in TABLE.S3, the
U/J is larger when AFM order ex-ists in CaCuO /FeSe. The U/J in BaNiS is larger that inLa N Ni B because the correlation effect are non-negligiblein BaNiS [55, 56] and La N Ni B is a typical metal. Fromthis point of view, our results also demonstrate that the corre-lation in BaCuS is weaker than in cuprate CaCuO . TABLE S3. The
U/J parameters and moments calculated by LSCCmethod.LSCC U (eV) J (eV) moment ( µ B )CaCuO (AFM) 5.78 1.16 0.478CaCuO (NM) 5.74 1.16 0FeSe(CAFM) 4.88 0.91 3.05FeSe(NM) 4.75 0.89 0La N Ni B Appendix G: method of RPA calculation
In this section, we explain the formalism of the multiorbitalRPA approach[36, 37, 39, 57, 58], adopted in the main text.The multi-orbital susceptibility is defined as, χ l l l l ( q , τ ) = 1 N (cid:88) kk (cid:48) (cid:104) T τ c † l σ ( k + q , τ ) (G1) c l σ ( k , τ ) c † l σ (cid:48) ( k (cid:48) − q , c l σ (cid:48) ( k (cid:48) , (cid:105) . In momentum-frequency space, the multi-orbital bare suscep-tibility is given by χ l l l l ( q , iω n ) = − N (cid:88) k µν a l µ ( k ) a l ∗ µ ( k ) a l ν ( k + q ) (G2) a l ∗ ν ( k + q ) n F ( E µ ( k )) − n F ( E ν ( k + q )) iω n + E µ ( k ) − E ν ( k + q ) , where µ and ν are the band indices, n F is the usual Fermidistribution, l i ( i = 1 , , , are the orbital indices, a l i µ ( k ) is the l i orbital component of the eigenvector for band µ result-ing from the diagonalization of the tight-binding Hamiltonian H and E µ ( k ) is the corresponding eigenvalue. With interac-tions, the RPA spin and charge susceptibilities are given by χ RP As ( q ) = χ ( q )[1 − ¯ U s χ ( q )] − ,χ RP Ac ( q ) = χ ( q )[1 + ¯ U c χ ( q )] − , (G3)where ¯ U s ( ¯ U c ) is the spin (charge) interaction matrix, ¯ U sl l l l ( q ) = U l = l = l = l ,U (cid:48) l = l (cid:54) = l = l ,J l = l (cid:54) = l = l ,J (cid:48) l = l (cid:54) = l = l , (G4) ¯ U cl l l l ( q ) = U l = l = l = l , − U (cid:48) + 2 J l = l (cid:54) = l = l , U (cid:48) − J l = l (cid:54) = l = l ,J (cid:48) l = l (cid:54) = l = l , . (G5)In the main text, we plot the largest eigenvalues of the suscep-tibility matrix χ l l l l ( q , and χ RP As,l l l l ( q , . Within RPAapproximation, the effective Cooper scattering interaction onFermi surfaces is, Γ ij ( k , k (cid:48) ) = (cid:88) l l l l a l , ∗ v i ( k ) a l , ∗ v i ( − k ) (G6) Re (cid:20) Γ l l l l ( k , k (cid:48) , ω = 0) (cid:21) a l v j ( k (cid:48) ) a l v j ( − k (cid:48) ) , where the momenta k and k (cid:48) is restricted to different FSs with k ∈ C i and k (cid:48) ∈ C j . The orbital vertex function Γ l l l l inspin singlet channel[59, 60] is Γ Sl l l l ( k , k (cid:48) , ω ) = (cid:20)
32 ¯ U s χ RPAs ( k − k (cid:48) , ω ) ¯ U s + 12 ¯ U s (G7) −
12 ¯ U c χ RPAc ( k − k (cid:48) , ω ) ¯ U c + 12 ¯ U c (cid:21) l l l l , where χ RP As and χ RP Ac are the RPA spin and charge sus-ceptibility, respectively. The pairing strength functional for aspecific pairing state is given by, λ (cid:2) g ( k ) (cid:3) = − (cid:80) ij (cid:72) C i d k (cid:107) v F ( k ) (cid:72) C j d k (cid:48)(cid:107) v F ( k (cid:48) ) g ( k )Γ ij ( k , k (cid:48) ) g ( k (cid:48) )(2 π ) (cid:80) i (cid:72) C i d k (cid:107) v F ( k ) (cid:2) g ( k ) (cid:3) , (G8)where v F ( k ) = |∇ k E i ( k ) | is the Fermi velocity on a givenFermi surface sheet C i . The pairing vertex function in spinsinglet and triplet channels are symmetric and antisymmetricparts of the interaction, that is, Γ S/Tij ( k , k (cid:48) ) = [Γ ij ( k , k (cid:48) ) ± Γ ij ( k , − k (cid:48) )])]