Theoretical studies of superconductivity in doped BaCoSO
Shengshan Qin Yinxiang Li Qiang Zhang Congcong Le Jiangping Hu












































































































apsrev













c_k,σ^†    c_k,σ
ç_̧ķ↑̧   c_-k↓
c_k↑^†   c_-k↓^†
c_p↑ c_-p+q↓
c_p↑^† c_-p+q↓^†

#1⟨#1⟩



    1/2Phys. Rev. A Phys. Rev. B Phys. Rev. D Rev. Mod. Phys.  Phys. Rev. Lett. 
        Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaBeijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaDepartment of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USABeijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, Chinajphu@iphy.ac.cn Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaCollaborative Innovation Center of Quantum Matter, Beijing, ChinaKavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences,  Beijing 100049, China
        We investigate  superconductivity that may exist in the doped BaCoSO, a multi-orbital Mott insulator with a strong antiferromagnetic ground state.  The superconductivity is studied in both t-J type and Hubbard type multi-orbital models  by  mean field approach and random phase approximation (RPA) analysis. Even if there is no C_4 rotational symmetry, it is found that  the system still carries  a d-wave like pairing symmetry state with gapless nodes and  sign changed superconducting order parameters on Fermi surfaces.  The results are largely doping insensitive.   In this superconducting state, the three t_2g orbitals have very different superconducting form factors in momentum space.  In particular,    the intra-orbital pairing of the d_x^2-y^2  orbital has a s-wave like pairing form factor.  The two methods also  predict very different pairing strength on different parts of Fermi surfaces.These results suggest that BaCoSO and related materials can be a new ground to test and establish fundamental principles for unconventional  high temperature superconductivity.
        
        75.85.+t, 75.10.Hk, 71.70.Ej, 71.15.MbTheoretical studies of superconductivity in doped  BaCoSO
            Jiangping Hu
            December 30, 2023
        =========================================================
        
        
        
        § INTRODUCTION
        
        
        Since the discovery of cuprates<cit.> and the iron based high T_c superconductors<cit.> (SCs), intensive research efforts have been made to understand their superconducting pairing mechanism. In the past three decades, great progress has  been made both experimentally and theoretically. Various experimental techniques, such as angle-resolved photoemission spectroscopy <cit.> (ARPES), inelastic neutron scattering<cit.> (INS), scanning tunneling spectroscopy<cit.> (STS), etc. and many different theories have been developed in the research of high T_c SCs. However, due to the complexity of the problem, no consensus about the microscopic pairing mechanism has been reached.
        
        Recently, by comparing cuprates and the iron based SCs,  we have pointed out that  those d-orbitals that are responsible for the superexchange antiferromagnetic (AFM) interactions mediated through anions  are isolated near Fermi energy to generate superconductivity in both families of high T_c SCs<cit.>.   In this scenario,  the pairing symmetry can be simply determined through an emergent empirical principle,  the  Hu-Ding principle<cit.>.    More interestingly, this  electronic feature is largely absent in other correlated electron systems. Thus,  we have suggested that  this property can be the gene of unconventional high T_c SCs and   materials satisfying the condition can be  promising high T_c candidates.  Based on such an understanding, two families of materials<cit.> have been proposed to be promising high T_c SCs. However, the proposals have  not been tested until now because of the difficulty in synthesizing the proposed materials.
        
        However,  recently we have observed that an already-synthesized material BaCoSO<cit.> may give us a chance to test the theory. The lattice structure of BaCoSO is similar to the case in ref.<cit.>, but the tetrahedron environment around the Co atoms is broken because of the anion mixture of the O and S atoms. Though it is not the ideal structure to maintain the required electronic condition,  we have expected that the theory<cit.> is  suitable to BaCoSO  and superconductivity may arise in the doped BaCoSO <cit.> if the structure distortion and disorder induced by doping can be minimized.   Motivated by this, we carry out the theoretical investigation of the superconducting state in this type of electronic structures.
        
        The paper is organized as follows. In the first part, the electronic structure and magnetic property of BaCoSO are reviewed. In the second part, the superconducting pairing in doped BaCoSO is analyzed by the mean field theory based on a t-J type multi-orbital model.  In the third part, we study the superconducting state in a Hubbard type of model under the RPA approximation. Then, we come to our conclusion in the last part.
        
        
        
        
        § ELECTRONIC STRUCTURE AND MAGNETIC PROPERTY
        
        BaCoSO has an orthorhombic layered lattice structure in which each CoSO layer is constructed by vertex sharing mixed-anion tetrahedron complexes CoS_2O_2. In this layer structure, the Co chains along y-direction are connected through S atoms and the staggered Co chains along x-direction are linked through O atoms. Compared with a perfect tetrahedron environment in which  the crystal field splits the five d-orbitals into two groups, t_2g and e_g,  the  crystal field  here breaks  the degeneracy of the three t_2g orbitals as well.  However, we have shown that this breaking is relatively small and the three t_2g orbitals still control the major electronic physics.  Without doping, BaCoSO has been confirmed to be an antiferromagnetic(AFM)  Mott insulator both theoretically<cit.> and experimentally<cit.>.   The  AFM  order is G-type, similar to those of cuprates.
        
        
        
        
        
        
        
        With doping, the electronic structure in the normal state has been calculated in ref.<cit.>.  It has been shown that  the three t_2g orbitals dominate near the Fermi level. In this paper, we investigate the possible superconducting state of this system and  perform calculations based on the three band model in the unfolded Brillouin zone (BZ) constructed by the three t_2g orbitals (d_xz,d_yz,d_x^2-y^2) in ref.<cit.>
            H_0^11   =   ϵ_1+2t^11_xxcos(2k_x) +2t^11_ycos(k_y)+4t^11_yycos(2k_y) 
                  -2t^11_xcos(k_x)-4t^11_xycos(k_x)cos(k_y), 
            
            H_0^12   =   4t^12_xysin(k_x)sin(k_y), 
            
            H_0^13   =    2it^13_xxsin(2k_x)+2it^13_xsin(k_x)+4it^13_xysin(k_x)cos(k_y), 
            
            H_0^22   =   ϵ_2+2t^22_xxcos(2k_x)+2t^22_ycos(k_y)+2t^22_yycos(2k_y)
                  -2t^22_xcos(k_x)-4t^22_xycos(k_x)cos(k_y), 
            
            H_0^23   =    2it^23_ysin(k_y)+2it^23_yysin(2k_y)+4it^23_xycos(k_x)sin(k_y), 
            
            H_0^33   =   ϵ_3+2t^33_xxcos(2k_x)+2t^33_ycos(k_y)+2t^33_yycos(2k_y)
                  +2t^33_xcos(k_x)+4t^33_xycos(k_x)cos(k_y),
        
        
        the corresponding tight binding parameters are shown in Table.<ref>.
        The three bands model captures  the major electronic structure around the Fermi energy.  In Fig.<ref>, we plot Fermi surfaces and their corresponding orbital characters at two different electron doping levels. For the case of 0.6 electron doping away from the half filling,  the Fermi surfaces (FSs) as shown in Fig.<ref>(a) are composed of three pockets. A small hole pocket at the BZ center Γ point  is attributed to the d_x^2-y^2 and d_xz orbitals, so do the two electron pockets around  the BZ  boundary Y point.  A large open hole pocket around the BZ center is attributed to the d_yz orbital.  The hybridization of  the d_x^2-y^2 and d_xz orbitals stems from the the Zig-Zag Co chain structure along the x direction.  With heavy electron doping, the small hole pocket at  the BZ center can sink below the Fermi energy as shown in Fig.<ref>(b).
        
        
        
        
        
        
        The above electronic structure resembles those of iron-based SCs. We can make a good comparison between them. In  iron-based SCs, typically there are  also three types of pockets,   two hole pockets from d_xz,yz  and  one hole pocket from d_xy at the BZ center Γ point, and two electron pockets around the BZ corner at M point that are mixed with d_xy and d_xz,yz<cit.>.  The hole pockets at Γ point in iron-based SCs can also sink below FSs by electron doping<cit.>. The role of different pockets and the interactions among them have been the central debate in iron-based SCs<cit.>. Therefore, the study of this new material can provide much deep understanding on these issues.
        
        
        
        § MEAN FIELD ANALYSIS
        
        
        We start with an effective t-J type Hamiltonian for BaCoSO, which is generally written as
        
            H= H̃_0+∑_<ij>,α,a,b (J^α_abS⃗_ia·S⃗_jb-1/4n_ian_jb),
        
        where H̃_0 is  the three bands Hamiltonian in Eq.<ref><cit.> subject to a projection to non-double occupant orbital state due to the onsite Hubbard interaction, <ij> labels the two nearest neighbour (NN) sites, α=x,y labels direction and a, b are the orbital indexes.  In local atomic orbital approximation, the strength of the AFM interaction can be roughly estimated from the superexchange process.  J^α_ab takes the form
        
            J^α_ab=(t^α_ab)^2(1/U_d+1/U_d+Δ_pd),
        
        where t^α_ab is the effective hoping parameter between the NN d-orbitals at Co sites, U_d is the Coulomb interaction for the d-orbitals and Δ_pd is defined as the energy difference between the d-orbital and p-orbital at O and S atoms. With the parameters in ref.<cit.>, we can get the AFM interaction strength in BaCoSO as follows: J^x_xz=0.20eV, J^x_yz=0.09eV, J^x_x^2-y^2=0.15eV and J^y_xz=0.09eV, J^y_yz=0.36eV, J^y_x^2-y^2=0.09eV.  It is important to note that this estimation is entirely based on atomic orbitals. As the p-orbitals of O and S atoms are very different  and the latter are more extended than the former,   the effective AFM coupling through S atoms is expected to be smaller than the estimated values. For this reason, we set a variable β on the value of J̃^̃ỹ=β J^y.
        
        
        
        
        
        
        Because of the space anisotropy of the two d-orbitals, the AFM interactions  for the  d_xz and d_yz orbitals  and those for the d_x^2-y^2 orbital along the two different directions have significant anisotropy. Such a large anisotropy suggests that the electronic physics here is rather nematic.
        
        
        
        
        
         In the mean field calculation, we  approximate   the projection as an overall renormalization factor of the bare Hamiltonian, namely H̃_0=γ H_0, where γ is the renormalization factor<cit.>.  γ generally is doping dependent and can be measured experimentally.  By rescaling the energy,  it is also equivalent to absorb the renormalization factor into the interaction parameters so that we can simply treat  H̃_0=  H_0 in the mean field calculation.  Combining with the estimated bare AFM interaction parameters, the mean field calculation is performed by setting AFM interaction in the unit of eV for the corresponding orbitals to be [0.20, 0.09, 0.15]× J  along the x-direction and [0.09, 0.36, 0.09]×β J along the y-direction.  We report the phase diagram of the superconducting  state with respect to  β and J.
        
        
        
        
        
        
        
        First, we set β=1 and report results as a function of J  for two different doping levels. The FSs is shown in Fig.<ref>(a) when the doping level is 0.6 electron per site, and the corresponding mean field results are shown in Fig.<ref>. The significant anisotropy of the AFM interaction for different orbitals leads to that, the pairing in the y-direction is dominated by d_yz orbital while in the x-direction d_xz and d_x^2-y^2 orbitals are dominant. As is shown in Fig.<ref>(c), the superconducting order parameter for d_x^2-y^2 is s-wave like, while it is d-wave like for both d_xz and d_yz orbitals. Furthermore, the following relationship is satisfied in the main area of the  phase diagram
        
        
            sign(Δ^x_xz)=-sign(Δ^y_yz),
            
            sign(Δ^x_xz)=sign(Δ^x_x^2-y^2),
        
        
        
        
        
        The results of the mean field approach can be well understood within the Hu-Ding principle<cit.>, the superconducting ground state always tends to open the largest superconducting gap on the FSs. For the sake that only the NN AFM interaction is considered, the superconducting order parameter takes a form factor in momentum space as Δ^xcos k_x+Δ^ycos k_y and Δ^α is proportional to J^α. Meanwhile, the three t_2g orbitals hybridize only at several small area on the FSs, as is shown in Fig.<ref>(a). Therefore, it is easy to see that   a s-wave like form factor cos k_x+cos k_y  for the d_x^2-y^2 orbital and a  d-wave like pairing on d_xz and d_yz orbitals can open the largest gaps on the FSs. The phase relationship of the order parameters between different orbitals can be also determined at the area where different orbitals hybridize on the FSs. To achieve  larger superconducting gaps, the order parameters of these orbitals tend to have the same phase. Specifically, for 0.6 electron doped BaCoSO, because the d_yz orbital and d_x^2-y^2 orbital hybridize strongly at the smaller FS near the Γ point, Δ^x_xz tends to have the same phase with Δ^x_x^2-y^2.
        
        
        
        
        
        
        
        
        A similar mean field analysis is also done when the electron doping level is about 1.0 per site. The corresponding FSs and mean field results are shown in Fig.<ref>(b) and Fig.<ref>, respectively. The mean field results here are  similar to those of  the 0.6 electron doped case. Furthermore,  we set J=2.0 and report results as a function of β. As shown in Fig. <ref>, the results are  also similar.   The qualitative results on  the superconducting order parameters  are very robust against β.
        
        Overall, the mean field theory gives a rather robust superconducting state: a s-wave like order parameter for d_x^2-y^2 orbital, a d-wave like order parameter for both d_xz,yz   orbitals, and totally a d-wave like pairing symmetry on the FSs with nodes near (π/2,π/2).
        
        
        
        
        
        
        § RANDOM PHASE APPROXIMATION ANALYSIS
        
        
        Based on the three bands model above, the RPA analysis is carried out for BaCoSO in this section with onsite repulsive interactions. The total Hamiltonian   is given by
        
            H=   H_0+U∑_i,αn_iα↑n_iα↓+U^'∑_i,α<βn_iαn_iβ
               +J∑_i,α<β,σσ^'c^†_iασc^†_iβσ^,c_iασ^'c_iβσ
               +J^'∑_i,α≠βc^†_iα↑c^†_iα↓c_iβ↓c_iβ↑,
        
        where  n_i,α=n_i,α,↑+n_i,α,↓. For other indexes, we adopt the parameter notations given in ref.<cit.>. In the RPA approximation, the pairing vertex is
        
        
        
        
            Γ_ij(k,k^')=   Re[∑_l_1l_2l_3l_4a^l_2,∗_v_i(k)a^l_3,∗_v_i(-k)
               ×Γ_l_1l_2l_3l_4(k,k^',ω=0)a^l_1_v_j(k^')a^l_4_v_j(-k^')],
        
        where the momenta k and k^' is restricted to different FSs within an energy cutoff Λ,  with k∈ C_i and k^'∈ C_j. a^l_v(orbital index l and band index v) is the component of the eigenvectors of the three-orbitals tight binding Hamiltonian.
        The singlet channel of orbital vertex function Γ_l_1l_2l_3l_4 in RPA is given by
        
        
            Γ_l_1l_2l_3l_4(k,k^',ω)=   [3/2U̅^sχ^RPA_1(k-k^',ω)U̅^s+1/2U̅^s
               -1/2U̅^cχ^RPA_0(k-k^',ω)U̅^c+1/2U̅^c]_l_3l_4l_1l_2,
        
        where χ^RPA_1 and χ^RPA_0 are the spin and charge fluctuation RPA susceptibility, respectively. The spin and charge interaction matrix(U̅^s, U̅^c) are the same as in ref.<cit.>.
        The pairing strength function is
        
        
            λ[g(k)]=-∑_ij∮_C_idk_/v_F(k)∮_C_jdk^'_/v_F(k^')g(k)Γ_ij(k,k^')g(k^')/(2π)^2∑_i∮_C_idk_/v_F(k)[g(k)]^2,
        
        
        
        
        
        where v_F(k)=|∇_kE_i(k)| is the Fermi velocity on a given Fermi surface sheet C_i.
        The calculation is carried out in the spin-rotational invariance case meaning U̅^̅'̅=U̅-2J and J=J^'.
        
        
        
        
        
        
        First, we calculate  the bare  and RPA spin susceptibilities for BaCoSO at different doping levels as shown in Fig.<ref>.  The RPA spin susceptibility has a sharp peak near the wavevector (π,π) in  both doping levels.
        The peak mainly stems from the interaction between  the smaller hole pocket   near the Γ point and the electron pocket near the M point.  This situation is very similar to the case in iron-pnictides. The interaction between these two pockets are responsible for the superconducting pairing as well.  This is because the points on  the smaller hole pocket near the Γ point and the electron pocket near the M point contribute the largest density of states(DOS)  near the Fermi level. The pairing strength on the hole pocket attributed to the d_yz orbital is always small because of its large band dispersion.
        
        The  RPA results in the superconducting state are reported in Fig.<ref>. For the 0.6 electron doping, similar to the mean field results, the leading superconducting instability turns out to have a d-wave like pairing symmetry, as shown in Fig.<ref>(a)(c).    In the 1.0  heavy electron doping case, as shown in Fig.<ref>(b)(d) and (e), there are two leading superconducting instabilities which  are nearly degenerate. Both of them have many nodes on the FSs and the gap function is more complex compared to the 0.6 electron doped case.  The superconducting pairing strength is also much weaker than those with the 0.6 electron doping case.
        
        
        
        There are significant differences between the RPA and mean field results. First, the superconducting gap on the two electron pockets near Y point tends to have an uniform phase in the RPA results. Second,  the pairing strength on the FSs contributed by  the d_yz orbital is much weaker in the RPA analysis than in the mean field approach.  Finally, competing superconducting pairing states is much easier to appear   in the RPA analysis as well.  These differences can be well understood in the build-in structure of the  RPA  analysis as the interaction between  the smaller hole pocket   near the Γ point and the electron pocket near the M point becomes dominant. Moreover, according to Eq.<ref>, the large DOS leads to strong pairing strength, and to avoid repulsive interaction to save energy,  the pairings at these two areas tend to have a π phase difference.
        
        The importance of the interaction between the smaller hole pocket   near the Γ point and the electron pocket near the M point  in the RPA analysis can be demonstrated further.  By increasing the electron doping,  the smaller hole  pocket at γ goes  a Lifshitz transition. At the 1.0 doping, it  sinks just below Fermi level.  If the contribution from  the hole pocket is important,  the RPA calculation near the Lifshitz transition becomes very sensitive to the cutoff energy   Λ from the Fermi level. For the results in Fig.<ref>(b)(d)(e), Λ is taken to be 0.005eV. If we increase this cutoff to involve the hole pocket contribution, the leading superconducting instability at the 1.0 doping is expected to vary quickly. This expectation is demonstrated in  Fig.<ref> in which  the cutoff energy is increased to  0.01eV.  It is clear that the leading superconducting instability  becomes similar to the 0.6 electron doping case.
        
        
        
        
        
        § SUMMARY AND DISCUSSION
        
        
        In summary, we have carried out mean field and RPA calculation to analyze the possible superconducting ground state in the family of  materials similar to BaCoSO whose electronic structures are described by the three t_2g  orbitals.  It is found that a d-wave like superconducting state with gapless nodes is generally favored.
        
         The superconducting properties in this family of materials can help us to establish  fundamental principles regarding the emergence of superconductivity in unconventional high T_c SCs.
        
         First, the energy scale of the parameters in this  family of materials are similar to those of iron-based superconductors. Therefore, the maximum T_c  that can be achieved here should be close to the maximum T_c  in the bulk material of iron-based superconductors if they share the same superconducting mechanism, which has been assumed in this paper.
        
         Second,  in cuprates and iron-based SCs,  the pairing symmetries are classified by the D_4h group.  The superconducting states fall  into  specific irreducible representations of this high symmetry group.  It is difficult to mix different representations.  As a result,  a pure or close to a pure  d-wave  and  s-wave state have been realized in cuprates and iron-based supercnoductors respectively.   Here due to the absence of C_4 rotation system,  the superconducting state is classified by much lower symmetry group.  Thus, the superconducting state, in the term of the D_4h group, is a mixture of s-wave and d-wave state.  Our results on the role of different orbitals, the location of gapless nodes and the pairing strength on different parts of Fermi surfaces  thus can provide critical information about the validity of theoretical methods and test different pairing mechanisms.
        
          Finally, the interaction between the smaller hole pocket   near the Γ point and the electron pocket  near the M point   is very similar to the case in iron-pnictides which are also characterized with the interaction between the hole pockets at the  Γ point and the electron pockets at M point<cit.>.  Our results from the RPA analysis are consistent with those in iron-pnictides <cit.> . Both calculations suggest  that the interactions are responsible for superconductivity. However,  in iron-chalcogenides<cit.>,   the simple RPA result has been seriously challenged because the high T_c superconductivity can still be achieved in the absence of hole pockets.  As the hole pockets can also sink below Fermi level by doping in this family of materials,  the validity of the RPA analysis can be further tested.  For example,  the heavy electron doping may be achieved by substituting Co with Ni atoms. We want to mention that strong superconducting instability cannot  be obtained  by  the standard functional renormalization group (FRG) method <cit.> in the above model.  Combining all these results and  the fact that the FRG  is also only valid in the weak interaction region,  observing high T_c superconductivity in the family of materials may finally explore the limitation of these standard approaches.
        
        
        
        
        
        
        § ACKNOWLEDGEMENT
        
        This work is supported by the Ministry of Science and Technology of China 973 program(Grant No. 2015CB921300), National Science Foundation of China (Grant No. NSFC-1190020, 11534014, 11334012), and   the Strategic Priority Research Program of  CAS (Grant No. XDB07000000).
        
        
        
        
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