Bond versus charge disproportionation and nature of the holes in $s-p AB{X}_3$ perovskites

We use density functional theory methods to study the electronic structures of a series of $s\text{−}p$ cubic perovskites, $AB{X}_3$: the experimentally available $\mathrm{Sr}\mathrm{Bi}{\mathrm{O}}_3, \mathrm{Ba}\mathrm{Bi}{\mathrm{O}}_3, \mathrm{Ba}\mathrm{Sb}{\mathrm{O}}_3, \mathrm{Cs}\mathrm{Tl}{\mathrm{F}}_3$, and $\mathrm{Cs}\mathrm{Tl}{\mathrm{Cl}}_3$, as well as the hypothetical $\mathrm{Mg}{\mathrm{PO}}_3, \mathrm{Ca}\mathrm{As}{\mathrm{O}}_3, \mathrm{Sr}\mathrm{Sb}{\mathrm{O}}_3$, and $\mathrm{Ra}\mathrm{Mc}{\mathrm{O}}_3$. We use tight-binding modeling to calculate the interatomic hopping integrals $t_{sp\sigma }$ between the $Bs$ and $Xp$ atomic orbitals and charge-transfer energies $\mathrm{\Delta }$, which are the two most important parameters that determine the low-energy electron and hole states of these systems. Our calculations elucidate several trends in $t_{sp\sigma }$ and $\mathrm{\Delta }$ as one moves across the periodic table, such as the relativistic energy lowering of the $Bs$ orbital in heavy $B$ cations, leading to strongly negative $\mathrm{\Delta }$ values. Our results are discussed in connection with the general phase diagram for $s\text{−}p$ cubic perovskites proposed by Khazraie et al. [Phys. Rev. B 98, 205104 (2018)], who find the parent superconductors $\mathrm{Sr}\mathrm{Bi}{\mathrm{O}}_3$ and $\mathrm{Ba}\mathrm{Bi}{\mathrm{O}}_3$ to be in the regime of negative $\mathrm{\Delta }$ and large $t_{sp\sigma }$. Here, we explore this further and search for different materials with similar parameters, which could lead to the discovery of new superconductors. Also, some considerations are offered regarding a possible relation between the physical properties of a given $s\text{−}p$ compound (such as its tendency to bond disproportionate and the maximal achievable superconducting transition temperature) and its electronic structure.

Figure 1

(a) Schematic diagram of the Bi $6s$ and O $2p$ energy levels in $A\mathrm{Bi}{\mathrm{O}}_3$ before (top panel) and after (middle panel) hybridization. A and B denote antibonding and bonding bands, respectively. The bottom panel demonstrates the effect of Bi-O bond disproportionation, whereby the bonding and the antibonding bands are each split into two subbands associated with large and small $\mathrm{Bi}{\mathrm{O}}_6$ octahedra, denoted as B(L), B(S), A(L), and A(S), respectively, and a charge gap is opened as a result. (b) The six $\mathrm{O}2p_{\sigma }$ orbitals of an ${\mathrm{O}}_6$ octahedron and their molecular orbital combinations, $a_{1g}, t_{1u}$, and $e_g$, with their corresponding energies given in units of the nearest-neighbor hopping integral $-|t_{pp}|=\frac{1}{2}[\left(pp\sigma \right)+\left(pp\pi \right)]$, where $\left(pp\sigma \right)$ and $\left(pp\pi \right)$ are the Slater-Koster parameters.

Figure 2

The electronic band structures of (a) $\mathrm{Mg}{\mathrm{PO}}_3$, (b) $\mathrm{Ca}\mathrm{As}{\mathrm{O}}_3$, (c) $\mathrm{Sr}\mathrm{Sb}{\mathrm{O}}_3$, (d) $\mathrm{Ba}\mathrm{Bi}{\mathrm{O}}_3$, and (e) $\mathrm{Ra}\mathrm{Mc}{\mathrm{O}}_3$ calculated in GGA and plotted with black solid lines. The Fermi energy is set to zero and marked by a horizontal black dashed line. The red and yellow circles indicate, respectively, the presence of the O $a_{1g}$ and cation $B s$ orbital characters in a given Bloch eigenstate, with the amount of their contribution being proportional to the circles' radii. The dashed blue lines represent the eigenstates of the MLWF-based TB model. (f) and (g) use a two-dimensional (2D) analog of the cubic perovskite structure to explain the absence of hybridization between the O $2p_{\sigma }$ and $B s$ Bloch functions at the $\mathrm{\Gamma }$ point $[\mathit{k}=(0,0,0\left)\right]$ and its maximum strength at the $R$ point $[\mathit{k}=(\pi ,\pi ,\pi \left)\right]$, respectively. The dashed lines mark boundaries between 2D unit cells with the lattice constant $a$. (h) The amount of the O $a_{1g}$ molecular orbital character in the antibonding band of the $AB{\mathrm{O}}_3$ TB models, $|\langle a_{1g}\left(\mathit{k}\right)|{\psi }^{\text{TB}}\left(\mathit{k}\right)\rangle {|}^2$, at the $R$ point as a function of chemical composition.

Figure 3

The partial densities of states projected onto the $B s$ atomic orbital and the molecular combinations of O $2p_{\sigma }$ orbitals of (a) $\mathrm{Mg}{\mathrm{PO}}_3$, (b) $\mathrm{Ca}\mathrm{As}{\mathrm{O}}_3$, (c) $\mathrm{Sr}\mathrm{Sb}{\mathrm{O}}_3$, (d) $\mathrm{Ba}\mathrm{Bi}{\mathrm{O}}_3$, and (e) $\mathrm{Ra}\mathrm{Mc}{\mathrm{O}}_3$. The Fermi energy is set to zero and marked by a vertical black dashed line.

Figure 4

The electronic band structures of (a) $\mathrm{Cs}\mathrm{Tl}{\mathrm{F}}_3$ and (b) $\mathrm{Cs}\mathrm{Tl}{\mathrm{Cl}}_3$. Notations are similar to those in Fig. 2. Partial densities of states of (c) $\mathrm{Cs}\mathrm{Tl}{\mathrm{F}}_3$ and (d) $\mathrm{Cs}\mathrm{Tl}{\mathrm{Cl}}_3$, projected onto the Tl $6s$ atomic orbital and the molecular combinations of the F $2p_{\sigma }$ or Cl $3p_{\sigma }$ orbitals. The zero of the energy is at the Fermi energy.

Figure 5

The phase diagram of Ref. [26] representing the dominant character of holes as a function of the charge-transfer energy $\mathrm{\Delta }$ and hybridization $t_{sp\sigma }$. The $+$ and $\times$ symbols mark the parameters relevant to the hypothetical and existing cubic $s\text{−}pAB{X}_3$ perovskites, respectively. Green, yellow, and red colors represent the amount of $X{e}_g, X{a}_{1g}$, and $B$ cation $s$ orbital contributions to the holes' character, respectively.

Figure 6

The electronic band structures of (a) $\mathrm{Ba}\mathrm{Bi}{\mathrm{O}}_3$ and (b) $\mathrm{Ra}\mathrm{Mc}{\mathrm{O}}_3$ calculated within the scalar-relativistic approximation (SR) and using a second-variational method to account for spin-orbit interaction ($+\mathrm{SO}$). The zero of the energy is at the Fermi energy.