Introduction

The discovery of high–temperature superconductivity in cuprates1 initiated the quest for exploring superconductivity in other transition-metal compounds. Ag2+ is isoelectronic with Cu2+ (d9 configuration). F− and O2− are also isoelectronic ions, closed–shell species, moreover both F− and O2− are weak–field ligands. Given the existence of the superconducting cuprates, one might be naturally interested in searching the superconductivity in Ag2+–F− solids. Theoretical study2,3,4 of Grochala and Hoffmann has also suggested that properly hole–or electron–doped AgII fluorides might be good superconductors, due to similarity in structure and properties between the AgII fluorides and the cuprate superconductors. Particularly, Jaron and Grochala have predicted that the high pressure σ form of AgF2 compound (>15 GPa) is a layered material with antiferromagnetic (AFM) ordering4, as seen for the parent compounds of high–temperature cuprate superconductors5.

It is well known that [CuO2]∞ plane with tetragonal tetra coordination of Cu (weak apical Cu–O bonds), is an essential structural element for superconductivity in cuprates. The basic band structure of the doped cuprates is a single 2D band deviating from half filled. In this situation, AFM fluctuations prevail in the undoped parent compounds and are often believed to mediate the superconductivity. The Fermi surface (FS) from this EquationSource math mrow msub mid mrow msup mix mn2 mo- msup miy mn2 band has been observed in many overdoped cuprates and agrees with the density functional theory based band structure calculations5,6.

As Grochala and Hoffmann pointed out in their review paper2, analogous [AgF2]∞ plane with tetracoordinate Ag has not been reported experimentally. Noteworthy recent development of heterostructure interface technology, superlattices (SLs) containing AgIIF2 square lattices can be prepared by using appropriate synthetic techniques to incorporate alternating layers of different transition metal compounds7,8,9,10,11,12,13,14, even technically a single atomic layer11. Here, interface can be used to modulate electronic structure for manipulating physical properties and generating novel phases which are not present in the bulk constituents. Whereby the quest for AgII superconductors could be achieved by the aforementioned novel paradigm on designing and fabricating two dimensional materials15,16,17,18. In our paper, our research focuses on artificial superlattice materials design and their electronic properties and doping effects, different from research on real bulk AgII fluorides materials2,3,4.

To identify possible superconductivity in 2D Ag2+–F− square sheet, we investigate electronic structures, magnetic states, model hamiltonian parameters and effective FSs for three proposed superlattices: SrTiO3/AgF2 (STO/AgF2), BaTiO3/AgF2 (BTO/AgF2) and SrTiO3/CsAgF3 (STO/CsAgF3), as illustrated in the top panels of Fig. 1 and compare these with corresponding properties of the cuprate superconductors. We find cuprate-like band structures and strong AFM fluctuations in all these SLs. More importantly, large cation size increases cation–anion polarization strength and corresponding apical Ag–O/F distance and makes oxygen band edge shift above the Fermi level to exchange charge with band, leading to out–of–plane electronic reconstruction. Here, similar to charge transfer in cuprates and recently studied La2CuO4–related heterostructure18, the TiO2 plane actually serves as a charge reservoir block and transfer electrons to the AgF2 plane. As a result, AFM state presents an unstable trend. Finally, the applied in–plane tensile strain drives a novel phase transition from the AFM metallic state to a stable nonmagnetic (NM) metallic state in STO/CsAgF3 SL. Model hamiltonian parameters and FS character of STO/CsAgF3 are extracted and compared to La2CuO4 (LCO) and HgBa2CuO4 (HBCO), indicating that STO/CsAgF3 is a promising candidate for AgII superconductivity.

Figure 1
figure 1

Schematic geometrical structures and GGA bandstructures of bulk LCO, bulk HBCO, STO/AgF2, BTO/AgF2 and STO/CsAgF3 SLs from left to right.

The Fermi level εF is set at zero. Dark cyan and orange fatbands represent contribution of and orbitals respectively.

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Results

For the in–plane lattice constant a of SLs, we first took that of STO (3.905 Å), often used as the substrate. The lattice constant c and atomic z coordinates were fully relaxed. The main effect of the relaxation, is to make the negatively charged O/F and positively charged cations displaced relative to each other in SrO, BaO and CsF atomic layers and thereby polarize the cation and anion planes. Cation size affects significantly polarization strength and corresponding apical Ag–O(F) bond length. AgF2 layer acts as the mirror plane of whole unit cell. In STO/AgF2 and BTO/AgF2 SLs, oxygen atoms move symmetrically towards and against AgF2 plane by 0.077 Å and 0.064 Å respectively. As a result, apical Ag–O bond length in the BTO/AgF2 is slightly larger than that in the STO/AgF2 by 0.143 Å, due to the larger size of the Ba2+ cation. The largest cation–anion polarization occurs in CsF plane in the STO/CsAgF3 and is 0.623 Å against AgF2 plane. This polarization distortion produces a local ionic dipole moment and together with in–plane strain it also leads to a larger increment in the apical Ag–F distance () compared to apical Ag–O distance () in other two SLs.

Local ionic dipole moment perturbs electrostatic potential and changes band positions around the Fermi Level. An evolution of Ag-eg states with structural chemical modification can be clearly observed in band structures. Spin–polarized GGA calculations give paramagnetic ground state for all these superlattices. Fig. 1 shows energy bands of STO/AgF2, BTO/AgF2 and STO/CsAgF3 SLs in a 13 eV region around the Fermi level εF ≡ 0 and along the symmetry–lines . The energy bands of bulk LCO and HBCO are also plotted in Fig. 1 for comparison. For three SLs, electronic properties around εF are still mainly controlled by Ag-eg bands, which are above the filled O/F-2p and Ag-t2g bands and below the empty Ti-3d bands. We plot (dark cyan) and (orange) fatbands around εF to disclose their orbital contribution. Ag-eg antibonding bands have similar band width in superlattice configuration STO/AgF2 and BTO/AgF2 and resemble that of LCO. Note that oxygen 2p bands become closer to Fermi Level in BTO/AgF2 due to atomic polarization distortion in z direction. However, in STO/CsAgF3 case, the antibonding band between and F-p states disappears due to the weak mixing of Ag-3d and F-p states in z direction. As a result, eg bands from −4 to 2 eV looks more like that of HBCO with a larger apical Cu-O distance of 2.784 Å. Most importantly, oxygen 2p band edge of TiO2 plane upshifts eventually above the Fermi level and exchanges charge with band, as seen in layer–projected density of states in the left panel of Fig. 2.

Figure 2
figure 2

Layer–projected density of states (left panel) of STO/CsAgF3 SL and corresponding partial charge density isosurfaces (right panel) for the unoccupied bands from Fermi level to 0.25 eV.

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In order to investigate the microscopic orbital physics of polarization–induced electron doping, we plot the typical partial charge density of the unoccupied bands of STO/CsAgF3 from Fermi level to 0.25 eV in the right panel of Fig. 2. Obviously, electron doping originates from charge transfer between O-px,py orbitals at TiO2 interface and orbital. Here, has considerable covalent hybridization with in–plane fluorine atoms' px, py orbitals, similar to Cu2+–O2− bonds. Interestingly, O-px,py state around the Fermi level is mainly located at TiO2 interface and far away from the AgF2 plane, as illustrated in Fig. 2, which favors the realization of possible superconductivity in 2D AgIIF2 plane.

Next, we discuss the stability of magnetic states in three superlattices under GGA + Ud scheme. AFM band structures with Ag–eg orbitals character (see Fig. 1 of supplementary materials) indicate that STO/AgF2 SL presents an AFM insulating ground state with a energy gap of 0.45 eV. While in BTO/AgF2 electronic correlation drives a weak electron doping in state, which is absent in GGA bands, leading to an insulator–metal transition. For STO/CsAgF3, an AFM metallic ground state is obtained. By analyzing layer–projected density of states of STO/CsAgF3 from GGA + Ud calculations and corresponding partial charge density isosurfaces for the bands around Fermi level (see Fig. 2 of supplementary materials), we find that the obtained AFM metallic ground state is aroused by charge transfer between O-px,py orbitals in TiO2 plane and covalent hybrid orbitals of and F-px,py in AgF2 plane. In all three superlattices, FM state falls in between AFM and NM states in energy. In Table I, we summarize in–plane and apical bond lengths and , energy difference ENM-EAFM and magnetic moment on Ag/Cu atom in AFM state. With the increasing apical Ag-O/F distance, ENM-EAFM value decreases gradually from 91.160 meV/Ag for STO/AgF2 to 34.525 meV/Ag for BTO/AgF2, similar to the trend for cuprates (e.g. from 206.51 meV/Cu for LCO to 117.47 meV/Cu for HBCO in Table I) and finally to a much smaller value 4.105 meV/Ag in STO/CsAgF3. The process of AFM state instability is also companied by reduction of magnetic moment on Ag/Cu atoms. For superlattice structures, electron doping of AgF2 plane emerges with the change of apical Ag-O/F distance, as we discussed above, which is an important derivation of AFM state instability, especially in STO/CsAgF3 with much smaller ENM-EAFM value and magnetic moment of 0.184 μB/Ag.

Table 1 The in–plane and apical bond length and in Å, energy differences ENM-EAFM in meV/Ag(Cu) and Ag/Cu atom's magnetic moment of AFM state in μB/Ag(Cu), for LCO, HBCO, STO/AgF2, BTO/AgF2 and STO/CsAgF3 without (top subtable) and with (bottom subtable) in–plane strain
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Furthermore, we investigate the effect of in–plane strain on magnetic state, because electronic properties are subject to electron– and orbital–lattice couplings in perovskite–like materials. Similar calculations are made for STO/CsAgF3 with three additional in–plane lattice constants of 3.790, 4.005 and 4.105 Å. We find that compressive strain can effectively increase ENM-EAFM to 7.595 meV/Ag, but tensile strain decreases it to 0.335 meV/Ag for a = 4.005 Å. Extraordinarily, STO/CsAgF3 SL goes through phase transition to a stable NM metallic ground state under tensile strain a = 4.105 Å, suggesting that the tuning of in–plane lattice constant can serve as an effective tool to modulate magnetic properties and even superconductivity.

We know that effective low–energy hamiltonians containing the minimal set of bands are important tools for understanding chemical trends. Based on the aboved GGA simulations, we extract model hamiltonian parameters by MLWFs downfolding technique. In this work, we choose to downfold to a 6-band hamiltonian describing the in-plane , px, py orbitals and out–of–plane , two pz orbitals (see Fig. 3a). In particular, six parameters capture the essential physics: the eg crystal field splitting energy , the in–plane charge–transfer energy , the direct in–plane and out–of–plane Ag-F (Cu-O) hopping tpd and and the two shortest–ranged O-O hoppings tpp and . The extracted values are tabulated in Table II and corresponding interpolated band structure are shown in Fig. 3 of supplementary materials.

Table 2 Tight–binding parameters of the six–band p-d model, containing the in–plane , px, py orbitals and out–of–plane , pz orbitals for LCO, HBCO, STO/AgF2, BTO/AgF2 and STO/CsAgF3 without (top subtable) and with (bottom subtable) in–plane strain. Parameters include eg crystal field splitting energies , charge–transfer energies , the three nearest–neighbor (intra–cell) hoppings tpd, , tpp and the inter–cell oxygen–oxygen hopping . The in–plane and apical bond lengths and in Å are also listed
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Figure 3
figure 3

(a) Parameters of the six–band p-d model for the CuO and AgF(O) octahedral in cuprate superconductors and the proposed AgF2–related superlattices; (b) Localized Wannier functions of and orbitals in STO/CsAgF3; (c) Effective Fermi surface centered at Γ point in first Brillouin Zone from band for bulk LCO, bulk HBCO, STO/AgF2, BTO/AgF2 and STO/CsAgF3 SLs.

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The hopping integrals tpd and tpp of LCO and HBCO are in good agreement with the 3–band model results by Weber et al.30. While Δ2 and are further corrected in our model by including three additional out–of–plane orbitals. From Table II, we find that cuprates and AgIIF2–related SLs share some common features. The larger or value leads to the increasing Δ1 and decreasing out–of–plane hopping respectively. And Δ2, in–plane hoppings tpd and tpp increase with the decreasing (see bottom subtable) or respectively. However, in–plane hopping is an exception and is affected considerably by out–of–plane Cu–O distance. From LCO to HBCO, the weakened electrostatic repulsion enhanced the hopping by 0.034 eV when the negatively-charged apical oxygen moves against the CuO plane, although increases from 1.894 to 1.941 Å at the same time. The similar feature can be seen more clearly in three SLs with the fixed of 1.953 Å. changes from 0.089 eV to 0.091 eV and finally 0.118 eV with the increasing apical Ag–O/F distance. Compared to cuprates, generally STO/CsAgF3 has relatively larger Δ1, Δ2 and in–plane Ag–F hopping tpd, while hopping , tpp and are smaller. After applying in–plane tensile strain , 2.053 Å (see bottom subtable), one can find that parameters evolve towards those in cuprates, except for tpp and .

In Fig. 3b, we plot the localized Wannier functions of and in STO/CsAgF3 SL. Both look like those of cuprates and have a strong p-d covalent hybridization characteristic. is more localized due to the big apical Ag–F distance (>3 Å). The FSs centered at Γ point for LCO and HBCO are shown in the first row of Fig. 3c. Compared to LCO (transition temperature Tc = 40 K), the FS of HBCO (Tc = 90 K) has the typical shape of high-Tc cuprates superconductor with constant–energy surface obviously bulging toward Γ point. The FS shape of STO/AgF2 and BTO/AgF2 (see the second row of Fig. 3c) is far away from that of HBCO or LCO. However, for STO/CsAgF3 (the third row of Fig. 3c) with polarized electron–doping in AgF2 plane, effective FSs from band presents the considerable similarity to that of HBCO.

Discussion

We analyze the cuprate-like electronic structures and strong AFM fluctuations in the proposed superlattice with 2D AgIIF2 square sheet. Atomic polarization induces out–of–plane electronic reconstruction occurring between O-px,py orbitals in TiO2 plane and covalent hybrid orbitals of and F-px,py in AgF2 plane, which is an important origin of AFM state instability. A stable NM metallic ground state emerges in STO/CsAgF3 SL subjected to in–plane tensile strain, meanwhile corresponding Wannier functions of , and FS shape present considerable similarity to those in cuprates with the high–Tc. Therefore, d9 AgIIF2–related superlattices are promising because their physics contains the main ingredients of high–temperature superconductivity.

Method

We carried out the numerical calculations using the Vienna ab initio Simulation Package (VASP)19,20,21,22 within the framework of the generalized gradient approximation (GGA) (Perdew-Burke-Ernzerhof exchange correlation functional)23 and recently developed maximally localized Wannier functions (MLWFs) downfolding technique24,25,26. The ion–electron interaction was modeled by the projector augmented wave (PAW) method27,28 with a uniform energy cutoff of 500 eV. Spacing between k points was 0.02 Å−1. The structures of the SLs were optimized by employing the conjugate gradient technique and in the final geometry, no force on the atoms exceeded 0.01 eV/Å. For magnetic states calculations, we used Ud = 7.5 eV and Jd = 0.98 eV for Ag-d and Cu-d states29.