Predicting Unconventional High Temperature Superconductors in Trigonal Bipyramidal Coordinations

Cuprates and iron-based superconductors are two classes of unconventional high Tc superconductors based on 3d transition elements. Recently, two principles, correspondence principle and magnetic selective pairing rule, have been emerged to unify their high Tc superconducting mechanisms. These principles strongly regulate electronic structures that can host high Tc superconductivity. Guided by these principles, here we propose high Tc superconducting candidates that are formed by cation-anion trigonal bipyramidal complexes with a d^7 filling configuration on the cation ions. Their superconducting states are expected to be dominated by the d+id pairing symmetry.

two groups by crystal fields, t 2g and e g . The two orbitals in the e g group, d z 2 and d x 2 −y 2 , because of their strong couplings to the p-orbitals of the surrounding oxygen atoms, have higher energies.
However, only the d x 2 −y 2 orbitals have strong in-plane couplings to the p-orbitals. Therefore, following the above rule, only the electronic band attributed to the d x 2 −y 2 orbitals can support high T c superconductivity. In a two-dimensional layer structure, the d z 2 energy level is lowered due to the Jahn-Teller effect and the d x 2 −y 2 orbital is the single orbital at the highest energy as shown in Fig.1(a). Thus, it is easy to see that in this case, an electronic band structure for high T c superconductors can only be achieved under the 3d 9 (Cu 2+ ) configuration. In iron-based superconductors, the Fe atoms are in a tetrahedral complex. Compared with the octahedral environment, the energy levels of the t 2g and e g orbitals in the tetrahedral complex reverse. The t 2g orbitals have higher energy because of their strong couplings to the As/Se anions. If we further consider two molecular orbitals formed by d xz and d yz , one molecular orbital is strongly coupled to the e g orbitals and becomes inactive in supporting pairing. Thus, as shown in Fig.1(b), the 3d 6 (Fe 2+ ) configuration is the filling level to make the pure t 2g orbitals to dominate electronic band structures close to Fermi energy. The high T c superconductivity is thus only achieved under the 3d 6 configuration. From these understandings, we can see that the two principles fix the d-orbital filling configuration if a structure formed by a given cation-anion complex is a high T c superconductivity candidate. This result partially explains why high T c superconductivity appears to be such a rare phenomena.
If we compare all cation-anion complexes, the trigonal bipyramidal complex has slightly lower symmetry than the octahedral or tetrahedral complexes. Materials with layered structures have also been formed by trigonal bipyramidal complexes, such as Y MnO 3 [7,8] in which Mn atoms in a Mn-O hexagonal lattice form a triangular lattice through conner-shared MnO 5 complexes as shown in Fig.2. The d-orbitals in the trigonal bipyramidal complex are split into three groups as shown in Fig.1(c). The d z 2 orbital has the highest energy due to its strong couplings to apical anions. The degenerate d x 2 −y 2 and d xy orbitals are strongly coupled to the in-plane anions. The degenerate d xz and d yz orbitals have the lowest energy and are only weakly coupled to anions. Thus, one can guess that a 3d 6 or 3d 7 configuration may result in a possible band structure in which the d x 2 −y 2 and d xy orbitals dominate near Fermi surfaces. If we further consider two molecular orbitals formed by the d x 2 −y 2 and d xy orbitals, one of them can strongly couple to the d z 2 . As the d z 2 orbital has higher energy, the coupling lowers the energy level of this molecular orbital. Therefore, to form a band structure that is dominated by the pure d x 2 −y 2 and d xy orbitals near Fermi energy, the 3d 7 filling configuration is expected as shown in Fig.1(c). Both Co 2+ and Ni 3+ ions have a 3d 7 filling configuration. The MnO 3 layer in Y MnO 3 is the simplest prototype layer structure that can be formed by trigonal bipyramidal complexes without anion bonding. Here we focus on this prototype structure and check whether a desired electronic structure for high T c superconductivity exists. Fig.3(a) shows the electronic band structure of Y NiO 3 . The electronic structure is rather quasi-two dimensional and thus can be attributed to a single NiO 3 layer. In Fig.3(a), one band near the Fermi level, which has the largest dispersion and will be referred as the α band, is mainly attributed to the two d xy and d x 2 −y 2 orbitals. Another band that will be referred as the β band, contributes a small hole pocket at Γ point. The β band is resulted from the bonding between the d z 2 orbital and one d xy,x 2 −y 2 molecular orbital. Near Γ point, the orbital character of the β band is mainly d z 2 . The other band from the anti-bonding between d z 2 and d xy,x 2 −y 2 orbitals, which will be referred as γ band, stays at much higher energy and is mainly attributed to the d z 2 orbital character. The bands from d xz and d yz orbitals with much less dispersion are located below the Fermi level. Although it is possible that these bands may contribute a small hole pockets at K points, they can be assumed to be fully occupied. The porbitals of the oxygen atoms is far below the Fermi level. The large dispersion of the d xy and d x 2 −y 2 bands suggests a strong d-p hybridization. These features are consistent with the above crystal field analysis and suggest that the 3d 7 filling configuration in trigonal bipyramidal complexes is indeed a possible candidate for high temperature superconductivity. Neglecting the interlayer coupling, the electronic structure can be well described by a three-band tight binding (TB) model, including d xy , d x 2 −y 2 and d z 2 orbitals. Fig.3(b) shows that the band structure obtained from the TB model well captures the first principle calculation results. The corresponding hopping parameters are given in Table.I. It is worth to note that the signs of intraorbital hopping parameters for the d xy and d x 2 −y 2 orbitals also indicate that the hopping is caused by the oxygen atoms.
Following the second principle, the α-band from the d xy and d x 2 −y 2 orbitals can host high temperature superconductivity. We can check whether this structure also satisfies the HDDL principle.
Near 3d 7 filling configuration, this band is close to half-filling. The α band can be described by a simple one-dimensional effective Hubbard or t-J models in a two-dimensional triangle lattice. The dominant hopping parameter is the nearest neighbour (NN) hopping and the short range magnetic superexchange coupling is also the NN antiferromagnetic(AFM) exchange. In the supplementary, we also show that the AFM state has significantly lower energy than the paramagnetic state, which indicates the existence of the strong NN AFM exchange couplings in the parental compound Y NiO 3 . In a triangle lattice, the NN AFM exchange coupling can lead to two types of pairing symmetries: s-wave or d ± id-wave [5]. As the pairing should be dominated on the NN bonds, for the s-wave pairing, the form factor of the gap function in the momentum space is given by ∆ s ∝ cosk y + 2cos √ 3 2 k x cos 1 2 k y , and similarly for the d ± id-wave pairing, the factor is given by 2 k x sin 1 2 k y . Following ref. [5], we calculate the overlaps between the Fermi surfaces and the form factors. Fig.4 shows the overlaps for the α-band obtained in Y NiO 3 . It becomes obvious that the d ± id-wave form collaborates well with Fermi surfaces near half filling and its' overlap with the Fermi surfaces is much larger than the s-wave form. Therefore, the system is a good candidate to host a high T c superconducting state with a robust d ± id-wave pairing symmetry.
The α band is a rather robust electronic structure as long as the two dimensional triangle lattice is maintained. Without considering the lattice instability, we can extend the Y NiO 3 prototype to include many possible variations by choosing different valence anions and replacing the apical anions with different elements. In the supplementary, we provide a list of possible materials in estimation, we can compare the energy scales of the effective models with those of cuprates and iron-based superconductors. In cuprates, the NN effective hopping parameter induced through the d-p hybridization is about 0.43ev [11]. In iron-based superconductors, it is the next NN (NNN) effective hopping parameters induced primarily by the d-p hybridization. The values of the NNN hopping parameters range from 0.15ev to 0.25ev [12], depending on materials and orbitals. Thus the energy scale in iron-based superconductors is roughly half of the energy scale in cuprates. The highest T c in iron-based superconductors is also around the half of the value achieved in cuprates.
In the fitted TB model in Table.I, the NN hopping is about 0.31ev. Therefore, we expect that the highest T c here is at least comparable to those in iron-based superconductors. Namely, it should be over 50k. It is important to note that the above estimation is only for the possible maximum It is interesting to compare the proposed electronic structure with those of the layered sodium cobalt oxyhydrate, NaCoO 2 , which owns a triangular cobalt oxygen lattice [13]. However, the triangular cobalt lattice is built by edge-shared CoO 6 octahedral complexes. The NN hopping between two Co atoms stems from the d-d direct chemical bondings. Thus, even if the strong electron-electron correlation has been argued in this material [14], the material violates our basic principles so that it is not a candidate for high T c superconductivity.
We can also design the similar structure with 4d or 5d transition metal elements as cation atoms in the 4d 7 or 5d 7 filling configuration. In the supplementary, we provide the band structure of Pdbased materials in which Pd 3+ is in a 4d 7 filling configuration. The essential α band is very similar to the above results. Although the correlation effect is generally weakened in heavier transition metal systems, the robust α band suggests that the proposed class of high T c superconductors may include many series of materials.
In summary, we predict that high T c superconductivity exists in a triangle lattice formed by the cation-anion trigonal bipyramidal complexes close to a d 7 filling configuration on the cation ions. The predicted Co/Ni based superconductors or corresponding 4d/5d transition metal based superconductors should have a robust d xy ± id x 2 −y 2 pairing symmetry. If the prediction is verified, together with cuprates and iron-based superconductors, it can convincingly establishes the high T c superconducting mechanism and also paves a way to design and search new unconventional high T c superconductors.