## Abstract

The recent discovery of nickel oxide superconductors have highlighted the importance of first-principles simulations for understanding the formation of the bound electrons at the core of superconductivity. Nevertheless, superconductivity in oxides is often ascribed to strong electronic correlation effects that density functional theory (DFT) cannot properly take into account, thereby disqualifying this technique. Being isostructural to nickel oxides, Sr_{1-x}K_{x}BiO_{3} superconductors form an ideal testbed for unveiling the lowest theory level needed to model complex superconductors and the underlying pairing mechanism yielding superconductivity. Here I show that parameter-free DFT simulations capture all the experimental features and related quantities of Sr_{1-x}K_{x}BiO_{3} superconductors, encompassing the prediction of an insulating to metal phase transition upon increasing the K doping content and of an electron-phonon coupling constant of 1.22 in sharp agreement with the experimental value of 1.3 ± 0.2. The proximity of a disproportionated phase is further demonstrated to be a prerequisite for superconductivity in bismuthates.

## Introduction

Superconductivity is a peculiar state of materials characterized by zero resistance to direct current and the expulsion of magnetic flux. It is explained by the formation of bound electrons called Cooper pairs^{1}. To date, the microscopic mechanism behind the Cooper pair formation is yet to be clarified and unified between all known superconductors. In simple elements, it is usually explained by the exchange of phonons. In high-temperature oxides superconductors, the proximity of a magnetic phase transition and/or a charge-ordered state is proposed to explain the formation of bound electrons. The discovery of superconductivity in nickel-based oxides R_{1-x}Sr_{x}NiO_{2} in 2019^{2} arouses the newest interest from solid-state scientists as it offers a new testbed for theories of superconductivity in complex oxides. It also highlights the importance of electronic structure calculations for understanding phenomena associated with superconductivity. In that regard, there is an established consensus that high critical temperature (*T*_{c}) reached in the oxide superconductors might be favored by strong correlation effects that have to be accounted for in electronic structure simulations^{3,4,5,6,7,8,9}.

Aiming at understanding the role of electronic correlations and the mechanism behind Cooper pairs formation, Sr_{1-x}K_{x}BiO_{3} and Ba_{1-x}K_{x}BiO_{3} sit as ideal compounds for testing our first-principles simulations techniques since these materials (i) host several complexities exhibited by oxides and (ii) belong to the few oxide superconductors adopting the simple ABO_{3} perovskite structure with SrTiO_{3-x} and Ba_{1-x}K_{x}SbO_{3}^{10,11,12,13,14}. In bulk, SrBiO_{3} and BaBiO_{3} are insulating with a band gap E_{g} estimated between 0.2 and 0.8 eV in BaBiO_{3}^{15,16,17}. They both adopt a monoclinic P2_{1}/n symmetry at low temperature^{10,18}, characterized by the usual a^{0}a^{0}c^{+} (\(\emptyset _z^ +\), irreps M_{2}^{+} with A cation sitting at the corner of the primitive *Pm-3m* cell) and a^{-}a^{-}c^{0} (\(\emptyset _{{{{\mathrm{xy}}}}}^ -\), irreps R_{5}^{−}) octahedra rotations in Glazer’s notations^{19} (Fig. 1a, b) induced by the A-to-B cation size mismatch—quantified by a Goldschmidt^{20} tolerance factor *t* = 0.88 in SrBiO_{3}. While the two O_{6} group rotations produce the usual orthorhombic Pbnm symmetry exhibited by most ABO_{3} perovskites, the P2_{1}/n phase is reached by the appearance of a breathing distortion B_{oc} (Fig. 1c)—also called bond disproportionation^{21}—producing a dimerization of the material along the [111] cubic direction. This lattice instability, appearing at the R point of the primitive Pm-3m, undistorted, cubic cell Brillouin zone (i.e., (1/2,1/2,1/2), irreps R_{2}^{-}), results in a rock-salt pattern of large and compressed O_{6} octahedra, splitting the Bi cations into Bi_{L} and Bi_{S,} respectively (Fig. 1c)^{22,23}. The breathing mode is associated with disproportionation of the unstable 4+ formal oxidation state (FOS) of Bi^{4+}-6s^{1} cations to more stable 3 + (6s^{2}) and 5 + (6s^{0}) FOS in the P2_{1}/n insulating phase^{16,21,22,24,25}. However, bismuthates fall within the negative charge transfer insulator regime^{26,27} and the Bi 6s band is localized well below the Fermi level and the O 2p band^{24,25,28}. While the disproportionation should result in the localization of spin-paired electrons and holes on the *s* states of Bi_{L} and Bi_{S} cations, respectively, O anions supply electrons to the depleted Bi_{S}^{5+} cations resulting in the localization of two holes on the surrounding O atoms and yielding a Bi_{L}–6s^{2} and Bi_{S}–6s^{2}__L__^{2} electronic configuration where the notation __L__ stands for a ligand hole^{21,24,25,28,29}.

Upon hole doping by substituting the divalent Sr or Ba cations with the monovalent K ion, Sr_{1-x}K_{x}BiO_{3} and Ba_{1-x}K_{x}BiO_{3} show superconductivity for doping contents ranging from *x* = 0.45–0.6 and 0.3–0.45, respectively, with a critical temperature *T*_{c} measured between 5 to 30 K, depending on the doping content as well as on the O stoichiometry of the samples^{10,11,12}. Superconductivity is explained by an electron-phonon coupling (EPC) with a constant *λ* = 1.3 ± 0.2^{4}. It is proposed on the basis of first-principles simulations that *λ* is enhanced by strong electronic correlations^{3,4} but also by octahedral rotations^{30}. At moderate hole doping content, these compounds exhibit a semiconducting state, explained by the appearance of localized states in the gap and trapped holes in the lattice (i.e., a hole bipolaronic state)^{29,31}.

Several theoretical studies have tried to address the physics of the bismuthates and their related superconducting phase^{3,21,22,23,24,25,28,29,30,31,32,33,34,35,36,37}. Most of these studies point out that density functional theory (DFT) is unable to account for the insulating character or appearance of the breathing mode B_{oc} in BaBiO_{3} within the usual Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) exchange-correlation functionals^{3,22,23,38} thereby hindering its doping effect study. This is also true for the extraction of superconducting quantities such as electron-phonon matrix elements and coupling strength that are underestimated by LDA or GGA with respect to experiments (*λ* = 0.34^{3,23} or 0.48^{30} with LDA and GGA, respectively, instead of 1.3 ± 0.2 with angular resolved photoemission spectroscopy experiments^{4}). Both the band gap and superconducting fundamental quantities can be improved by more sophisticated, but prohibitive, DFT hybrid functional and/or Green Functions and screened Coulomb interaction (GW) calculations^{3,29,31}. Nevertheless, including all degrees of freedom (e.g., structural such as symmetry lowering events and cation disorder appearing in alloys, electronic instabilities lifting degeneracies, local spin formation, and long-range spin orders…) of complex oxides for different doping contents are not affordable for these techniques, that are way too demanding in terms of computational resources since one has to deal with very large supercells—hybrid DFT and GW calculations are usually restricted to very small cell size preventing symmetry lowering events. Thus, affordable but still predictable DFT calculations are required for understanding trends in the doping of complex oxides and superconductors.

Recently, the band gap opening of ABO_{3} oxides perovskites with a *3d* element has been addressed by DFT simulations and revealed to originate from four simple modalities rather than the usual explanation based on strong dynamical correlation effects codified by the Hubbard model:^{39} (i) the octahedral crystal field splitting the *d* states and Hund’s rule; (ii) symmetry lowering events such as octahedral rotation further lifting orbital degeneracies; intrinsic electronic instabilities yielding (iii) a Jahn-Teller effect and removing orbital degeneracies, or (iv) disproportionation effects of unstable formal oxidation state (FOS) to more stable FOS and resulting in a double local environment for the B cations. These results have been ratified by using appropriate DFT exchange-correlation functionals properly amending self-interaction errors inherent to the implementation of DFT and by supplying enough flexibility to the simulation (e.g., local motif, symmetry lowering events, spin polarization…)^{21,27,39,40,41}. Furthermore, these conclusions are supported by DFT simulations using the Strongly Constrained and Appropriately Normalized (SCAN)^{42} exchange-correlation functional, but without any empirical parameter U such as in DFT + U, that properly account for the bulk perovskite oxide properties^{27,40} and the trends in doping effects of oxide insulators such as rare-earth nickelates^{43} RNiO_{3} or copper oxides^{44} such as La_{2}CuO_{4}. One may thus question “*what is the lowest DFT exchange-correlation functional needed for capturing trends in doping effects in complex oxides superconductors*”. Furthermore, an open issue in the physics of SrBiO_{3} (and BaBiO_{3}) is the superconducting transition upon doping, that is up to now largely elusive. Studies have been performed for selected K doping contents in Ba_{1-x}K_{x}BiO_{3} for instance, but getting a global trend on structural and electronic properties as a function of *x* remains theoretically ignored.

Here I show that the SCAN functional is sufficient to capture the trends in insulating to the metal character of SrBiO_{3} upon hole doping and to reveal the mechanism and prerequisites behind the appearance of superconductivity. By mapping the first-principles DFT results on a Landau model involving the relevant lattice distortions, the insulating phase is shown to be reached by disproportionation effects associated with an intrinsic instability of Bi^{4+} cations to disproportionation to Bi^{3+}/Bi^{5+} cations in the bulk ground state. This is accompanied by a breathing mode distortion B_{oc} whose amplitude is further enhanced by the octahedra rotations. At weak doping content (*x* = 0.0625 to 0.125), holes are trapped on the lattice and intermediate states are localized in the band gap, ultimately resulting in a semiconducting behavior. At intermediate doping content (*x* = 0.1875 to 0.375), a metallic phase is reached but the breathing mode is still present in the ground state due to its coupling with octahedral rotations, despite the fact that the structural distortion alone is not willing to spontaneously pop up in the material. The presence of this mode induces small gaps in the bands dispersing around the Fermi level. No gaps are anymore identified in the band structure at *x* = 0.4375, a doping content reminiscent of the superconducting phase reported experimentally (*x* = 0.45–0.6). Around *x* ≥ 0.4375, the breathing mode is found on the verge of becoming stable in the material due to octahedral rotations and thus its vibration can form spin-paired electrons and holes in the material, i.e., Cooper pairs. These results thus suggest that the proximity of a lattice instability producing spin-paired electrons and holes is a prerequisite for superconductivity in the bismuthates, in sharp agreement with the bounded doping content observed experimentally for the superconducting phase. Within the superconducting phase at *x* = 0.4375, an electron-phonon coupling constant *λ* associated with the breathing mode B_{oc} of 1.22 and a B_{oc} frequency of 66 meV are extracted from the simulations, in sharp agreement with the experimental values (*λ* = 1.3 ± 0.2 and *ω* = 62 meV, respectively). At larger doping content, the breathing mode frequency becomes harder and the density of states at the Fermi level decreases but a slightly increased reduced electron-phonon matrix element with increasing x preserves a non-zero *T*_{c} up to *x* = 0.625. This study thus (i) validates the use of SCAN-DFT for studying doping effects in complex oxide superconductors and (ii) calls for inspection of disproportionation effects in superconducting nickelates and other oxide superconductors to see if the identified mechanism in bismuthates is also relevant for these newly identified oxide superconductors.

## Results

### The bulk material

#### DFT ground states properties

The bulk P2_{1}*/n* structure experimentally observed at low temperatures for SrBiO_{3} is first relaxed in order to identify the ground state with our DFT functional. Key structural and electronic properties of the bulk material are reported in Table 1. SCAN-DFT predicts that SrBiO_{3} is an insulator in agreement with experiments, with a band gap amplitude improved with respect to GGA calculations (0.3 eV in ref. ^{25}. instead of 0.48 in the present study). In terms of structural parameters, the computed lattice parameters are in close agreement with experiments^{10}, with less than 1% of an error on the volume of the unit cell. Regarding the key lattice distortion amplitude exhibited by the compound, the optimized structure in DFT is also in agreement with the experimental structure although the amplitude of the breathing mode B_{oc} is underestimated by ~10% with respect to the experiment. This mode produces the rock-salt pattern between the compressed and contracted octahedra (see Fig. 1c). It results in a clear cut of the electronic structure between the two types of Bi cations as inferred by the projected density of states of Fig. 2a. Nevertheless, the Bi-s states are well below the O-p states, the band gap is mostly formed between occupied and unoccupied O-p states (see Fig. 2a) and SrBiO_{3} falls within the negative charge transfer insulator regime. Using the partial charge density maps of states just above the Fermi level (Fig. 2b), the unoccupied O-p bands have an s-like orbital character centered on Bi_{s} site—in agreement with the Wannier functions analysis presented in ref. ^{25}—hinting at the fact that the O anions supply electrons to the depleted Bi_{S} cation and bear the two holes that are centered at Bi_{S} sites.

#### The origin of the insulating phase

Figure 2c–e report the potential energy surface associated with the relevant distortion modes identified in the ground state of SrBiO_{3} when starting from a perfectly undistorted Pm-3m cubic cell (see method). As one can see, octahedral rotations show a double-well potential whose minimum is located at non-zero amplitudes *Q* of the distortions. It indicates that these modes are unstable and are willing to spontaneously appear in the ground state due to steric effects. The breathing mode B_{oc} also exhibits a double-well potential, albeit with smaller energy gains with respect to octahedral rotations. (Fig. 2d) Thus, the material spontaneously wants to get rid of the unstable 4+ formal oxidation state (FOS) of Bi cations and adopt the more stable 3+/5+ FOS. This is in agreement with previous DFT work on BaBiO_{3} performed by Tonhauser and Rabe in Ref. ^{22}. Nevertheless, the breathing mode would exhibit only 50% of its amplitude appearing in the P2_{1}/n ground state if considered alone.

#### The importance of octahedral rotations for the stabilization of the breathing mode B_{oc}

Mercy et al. identified in rare-earth nickelates RNiO_{3} (R = Lu-Pr, Y)^{45} that the breathing mode B_{oc} amplitude *Q*_{Boc} is directly connected to the octahedra rotation amplitudes \(Q_{\emptyset _{{{{\mathrm{xy}}}}}^ - }\) and \(Q_{\emptyset _{{{\mathrm{Z}}}}^ + }\), respectively, through the lattice mode couplings allowed in the free energy expansion *F* displayed in Eq. 1:

where a, b, and c are coefficients. It follows that rotations can renormalize the coefficient in front of \(Q_{{{{\mathrm{Boc}}}}}^2\) thereby acting on the possibility to stabilize a finite amplitude of B_{oc} (*a*_{eff} < *0*) or not (*a*_{eff} > *0*) in the material. Figure 2e displays the potential energy surface as a function of the B_{oc} mode amplitude *Q*_{Boc} but at fixed octahedral rotation amplitudes \(Q_{\emptyset _{{{{\mathrm{xy}}}}}^ - }\) and \(Q_{\emptyset _{{{{\mathrm{xy}}}}}^ - }\). The stabilization of the breathing mode is enhanced by the O_{6} group rotations, increasing the amplitude of the B_{oc} mode appearing in the material and associated energy gain. Obviously, even though the breathing mode B_{oc} alone would not be intrinsically unstable (*a* > *0* in Eq.1), it can still be forced to appear in the ground state due to its coupling with finite octahedra rotation amplitude yielding *a*_{eff} < *0* in Eq.1. Thus, neglecting octahedral rotations by using smaller, but more convenient unit cells, is not indicated as it can ultimately act on the presence of B_{oc} and related electronic features.

### Polaronic state formation upon hole doping

#### Intermediate states in the band gap

Following the understanding of the bulk properties, DFT calculations at low hole doping content have been performed in order to check whether or not polaronic states can be trapped on the lattice and yield a semiconducting behavior. To that end, a Sr_{1-x}K_{x}BiO_{3} solid solution with *x* = 0.0625 using a 32 f.u. supercell is considered (see method). Then, 2 Sr^{2+} cations are substituted by 2 K^{+} cations in the 32 f.u. supercell, yielding the release of two holes in the material. After the structural relaxation, a semiconducting state with a band gap of 0.25 eV is identified in the material. By inspecting the projected density of states presented in Fig. 3a, a split-off band localized in the band gap mostly formed by O-p and Bi_{L}-s states, very similar to the character of the valence band maximum, is revealed. An acceptor state is thus created in the material. The charge density maps associated with this intermediate state presented in Fig. 3b confirms that the holes are localized on a Bi_{L} cation that is occupied by two spin-paired electrons in the pristine material.

#### Existence of a trapped hole bipolaronic state

This is locally accompanied by a modification of the lattice with the average “Bi_{L}-O bond length” of 2.16 Å on this hole site while Bi_{L}-O bond lengths are roughly 2.30 Å for the other Bi_{L} sites in the material. It is in fact very similar to a Bi_{S} cation that indeed shows an average Bi_{S}-O bond length of 2.15 Å. In conclusion, a hole bipolaronic state is trapped in the material and tends to transform Bi_{L} cations into Bi_{S} cations, i.e., locally annihilating the breathing distortion mode B_{oc}. This is in sharp agreement with previous DFT predictions in hole-doped BaBiO_{3} performed with higher DFT.

### Trends in insulating-to-metal transition and structural properties upon hole doping

#### Limited miscibility of K within SrBiO_{3}

Before inspecting the trend in electronic and structural properties with hole doping SrBiO_{3}, the possibility of inserting the K cations within the SrBiO_{3} P2_{1}/n structure has been checked. In bulk, KBiO_{3} is not willing to adopt a P2_{1}/n cell based on corner-sharing octahedra as in SrBiO_{3}, but it prefers to crystallize within an edge-sharing octahedra network with a Pn-3 cell. This symmetry is more stable than the P2_{1}/n symmetry by 290 meV/f.u. By inspecting the insertion of Sr within the Pn-3 and P2_{1}/n cells starting from KBiO_{3}, the doped Pn-3 cell remains more stable than the doped P2_{1}/n cell for *x* = 0.9–1 in K_{x}Sr_{1-x}BiO_{3} (Fig. 4). However, by comparing the total energy of single phase solution Sr_{1-x}K_{x}BiO_{3} either adopting the P2_{1}/n or the Pn-3 symmetries with that of a biphasic solid solution (i.e., \({{{\mathrm{E}}}}_{{{{\mathrm{biphase}}}}} = {{{\mathrm{E}}}}_{{{{\mathrm{SBO}}}}} + {{{\mathrm{x}}}}({{{\mathrm{E}}}}_{{{{\mathrm{KBO}}}}} - {{{\mathrm{E}}}}_{{{{\mathrm{SBO}}}}})\), one observes that the biphasic solid solution is more stable than any single phase solid solutions for 0.65 ≤ *x* <1 (Fig. 4). Thus, there is a limited miscibility of K within the SrBiO_{3} structure. These results are in sharp agreement with the impossibility to synthesize Sr_{1-x}K_{x}BiO_{3} for *x* > 0.6 experimentally^{10}. The present study will thus be limited to doping effects up to *x* = 0.625. Finally, one notices a strong stability of the monophasic solid solution Sr_{1-x}K_{x}BiO_{3} for 0.4 < *x* < 0.6.

#### The insulator to metal transition upon K substitutions

The projected density of states on Bi_{L} and Bi_{S}-s states as a function of various K doping content *x* are reported in Fig. 5a. At moderate doping content (*x* = 0.0625 and *x* = 0.125), intermediate states are formed in the gap and yield a semiconducting behavior. At *x* = 0.1875 up to *x* = 0.375, the material is found metallic but still with a clear asymmetry of electronic structures between Bi_{L} and Bi_{S} cations. This hints at the fact that parts of disproportionation may still be present in the material. Finally, at *x* = 0.4375 up to *x* = 0.625, no asymmetries between Bi_{S} and Bi_{L} electronic structures are observed suggesting that disproportionation effects have disappeared and the two Bi sites become equivalent in the material.

#### Structural distortions upon K substitutions

The amplitude of distortions associated with the octahedral rotations \(\phi _{{{{\mathrm{xy}}}}}^ -\) and \(\phi _{{{\mathrm{z}}}}^ +\) as well as the breathing mode B_{oc} at different doping contents are reported in Fig. 5b. The two octahedral rotations' amplitude decrease with increasing the K doping content, albeit not disappearing at all in the ground state structure. This fact is due to the A-to-B cation size mismatch that are altered by the introduction of K atoms (KBiO_{3} adopting a cubic Pm-3m cell would have a tolerance factor *t* = 1.01 that favors a cubic cell). The behavior of the breathing mode B_{oc} as a function of the hole doping content is different with respect to rotations: (i) at moderate doping content (*x* = 0 to 0.375), the B_{oc} mode amplitude diminishes until (ii) it totally vanishes at *x* = 0.4375 up to *x* = 0.625. The absence of the breathing mode for 0.375 ≤ *x* ≤ 0.625 is in line with the similar electronic structures observed for Bi_{S} and Bi_{L} cations using the projected density of states presented in Fig. 5a. Thus, Bi cations become all equivalent in the material.

#### Doping effects suppress the electronic instability toward disproportionation of the formal oxidation state

Potential energy surfaces associated with the breathing mode starting from a perfectly cubic cell for different x values are displayed in Fig. 5c. Whatever *x*, the B_{oc} mode is associated with a single well potential (*a* > *0* in Eq. 1). It follows that the instability toward disproportionation of the unstable 4+ FOS of cations to more stable 3+/5+ FOS identified in the pristine compound is suppressed by the introduction of K atoms in the material. Furthermore, one can identify that the curvature of the total energy as a function the B_{oc} mode amplitude becomes steeper signaling that the breathing mode hardens with increasing the Sr substitutions. Thus considered alone, the B_{oc} mode possesses an *a* coefficient in Eq. 1 that becomes positive and larger when *x* increases.

#### Persistence of the breathing mode due to octahedral rotations

The coupling of B_{oc} with the two rotations observed in bulk may still renormalize the effective coefficient *a*_{eff} in Eq. 1 by supplying sufficiently large and negative *b* and *c* coefficient contributions counterbalancing the hardening of the B_{oc} mode alone. This is confirmed by the first-principles calculations of the potential energy surfaces associated with the B_{oc} mode but at fixed O_{6} group rotation amplitudes presented in Fig. 4d. At *x* = 0.25 or *x* = 0.3125, the B_{oc} mode indeed presents a double-well potential (i.e*., a*_{eff} < 0 in Eq. 1) while at *x* = 0.4375 or *x* = 0.5, it is associated with a single well potential whose minimum is at 0 (i.e*., a*_{eff} > 0). Thus, up to *x* = 0.375, rotations are sufficiently large to produce a negative effective coefficient *a*_{eff}. At *x* ≥ 0.4375, rotation amplitude are not large enough thereby resulting in a positive effective coefficient *a*_{eff} in front of \(Q_{{{{\mathrm{Boc}}}}}^2\) in Equation 1.

#### Band structure upon K substitutions and the absence of Peierls instability signatures in the superconducting state

Figure 6a–g displays the unfolded band structures in the primitive high symmetry Pm-3m cubic Brillouin zone of Sr_{1-x}K_{x}BiO_{3} for the different doping contents *x* tested in the simulations. Only bands dispersing around the Fermi level are reported. The pristine material is insulating with an indirect band gap of 0.48 eV. One can notice the existence of gaps along a specific path in the Brillouin zone, notably a gap of 1.2 eV halfway through the Γ-R path. The existence of this gap comes from the R-point lattice instability producing a dimerization along the [111] cubic direction and the rock-salt pattern of Bi_{L}^{3+} and Bi_{S}^{3+} cations in the insulating phase. At low doping content (*x* = 0.0625 and *x* = 0.125), the band structure is mainly not altered but intermediate states appear in the band gap. For doping contents x ranging from 0.1875 to 0.375, the band structure is altered notably the band gap along the Γ-R path progressively diminishing with increasing *x*. Although the material is metallic, energy gaps are still present in the band structure and the conduction and valence bands have not yet merged into a single “parabola” centered at the Γ point. This is due to the persistence of the B_{oc} lattice distortion discussed above. At *x* = 0.4375, no more gaps are visible in the band structure and the initial valence and conduction bands in the pristine material have now totally merged. One is then left with a single parabola centered at the Γ point and dispersing on a bandwidth of roughly 4.4 eV. This feature only appears once any finite B_{oc} mode amplitude has totally vanished in the ground state of the material.

#### The vicinity of a phase possessing a disproportionation instability is a prerequisite to superconductivity

The extinction of any type of disproportionation signatures (Bi_{L}-O and Bi_{S}-O bond lengths and Bi_{S} and Bi_{L} electronic structure asymmetries), as well as the observation of a single parabola centered at Γ and dispersing over a large energy range for doping contents *x* ranging from *x* = 0.4375 to *x* = 0.625 in the simulations, relates closely with the experimental doping content (*x* = 0.45–0.6) required to reach the superconducting state^{10}—one recalls here that the B_{oc} mode is underestimated by ~10% in the DFT bulk with respect to experiments thereby one may predict a superconducting transition at lower doping content with DFT simulations. It clearly suggests that the breathing mode B_{oc} is the key behind the superconducting phase. The latter is likely reached when the effective coefficient in front \(Q_{{{{\mathrm{Boc}}}}}^2\) in Eq. 1 is nearly zero or slightly positive. Lattice vibrations with the B_{oc} character potentially facilitated by O_{6} group rotation vibrations may produce spin-paired electrons and holes on the lattice, i.e., Cooper pairs—although the B_{oc} distortion does not stabilize any finite amplitude in the material. One indeed checks in the simulations that the B_{oc} mode at *x* = 0.4375 is strongly softened when rotation amplitudes are fixed to the pristine value (∅ = 1 in Fig. 5d) with respect to the situation for the ground state at *x* = 0.4375 (∅ = 0.66 in Fig. 5d).

#### K doping is mainly related to simple steric effects

Following the proposed mechanism, the role of the K doping for the appearance of superconductivity is rather indirect and related to a simple steric effect: (i) Sr substitutions by K thus increase the *t* factor of Sr_{1-x}K_{x}BiO_{3} thereby diminishing the rotations amplitude and (ii) in turn it suppresses the breathing mode stabilization in the material.

#### The proposed scenario is corroborated by experimental facts in BaBiO_{3}

Ba is larger than Sr atoms and hence the *t* factor of BaBiO_{3} is larger (*t* = 0.92) than the *t* factor of SrBiO_{3}. Octahedral rotations in BaBiO_{3} are ultimately smaller than in SrBiO_{3} and thus the expected K doping needed to suppress the stabilization of the B_{oc} distortion in the ground state is necessarily smaller in BaBiO_{3} than in SrBiO_{3} following the identified mechanism. This is verified experimentally:^{11} the superconducting phase in Ba_{1-x}Bi_{x}O_{3} is reached for *x* *≈* 0.30–0.45 while it is for *x* = 0.45–0.6 in SrBiO_{3}.

### Superconducting properties associated with the breathing distortion

#### The B_{oc} mode frequency in the superconducting state

In order to assess the frequency of the breathing mode B_{oc} in the superconducting phase, the potential energy surface associated with the B_{oc} mode is computed by freezing some displacements of the distortion in the 32 f.u. ground state structure (i.e., the relaxed DFT cell containing the octahedral rotations) for doping contents *x* of 0.4375 up to 0.625. By fitting the resulting energy curve (see method), the frequency *ω*_{Boc} for the breathing mode is computed to 66 meV at *x* = 0.4375. This value is in sharp agreement with the quantity extracted experimentally in the superconducting phase of BaBiO_{3} for this mode (*ω* = 62 meV in ref. ^{46}). As one can see from Fig. 7b, *ω*_{Boc} linearly hardens upon adding more K in Sr_{1-x}K_{x}BiO_{3}. This is in line with diminishing octahedral rotations induced by Sr substitutions that result in a positive and increasing *a*_{eff} in Eq. 1.

#### The reduced electron-phonon coupling matrix element

In order to extract the quantification of the REPME labeled *D* (see method), a B_{oc} displacement *u* is frozen in the ground state structure for *x* = 0.4375 to *x* = 0.625. It results in a gap opening of *∆E*_{g} = 1.23 eV along the *Γ*-R path for *x* = 0.4375 (Fig. 6h). It yields a reduced electron-phonon matrix element (REPME) *D* = 11.7 eV/Å (see method). This value is an improvement over classical LDA and GGA quantities (GGA yields *D* = 7.8 eV/Å) and hybrid DFT using the HSE06 functional (*D* = 11 eV/Å) and closely matches the REPME computed with quasi-particle Green’s function and screened Coulomb interaction (GW) technic that evaluates *D* to 13.7 eV/Å in optimally doped BaBiO_{3} (ref. ^{3}). Amazingly, *D* linearly increases with the doping content, suggesting that one cannot simply fix the *D* value obtained at one *x* for all other doping contents *x* (Fig. 7a)

#### The electron-phonon coupling constant and the critical temperature

Using the REPME, the computed frequency of the breathing mode and the density of states *N*(*E*_{F}) evaluated for all ground states for *x* = 0.4375 to *x* = 0.625 (Fig. 7c)—values in sharp agreement with *N*(*E*_{F}) evaluated experimentally to 0.225–0.335 in Ba_{1-x}K_{x}BiO_{3}, see ref. ^{11}—the electron-phonon coupling (EPC) constant *λ* associated with the breathing mode is evaluated between 1.03 to 1.24 for 0.4375 < *x* < 0.625 (Fig. 7d). These values are in agreement with the experimental value of *λ* = 1.3 ± 0.2 obtained in Ba_{0.51}K_{0.49}BiO_{3}^{4} and is an improvement over hybrid DFT calculation of ref. ^{3}. The discrepancy with more sophisticated DFT functionals may originate from the fact that important structural distortions and relevant lattice mode couplings were not considered in the ground state calculations of ref. ^{3}. The importance of octahedral rotations on the EPC was already suggested in ref. ^{30}.

Following the methodology proposed in ref. ^{3}, the logarithmic average frequency is identified by implementing the SCAN-DFT corrections to *λ* within the *ω*_{log} obtained by LDA simulations. Using the Mac Millan equation^{47} (Eq. 4), the critical temperature is found to be rather constant and between 25–33 K for screened Coulomb potentials *μ*^{*} of 0.15 (Fig. 7e). This is in very good agreement with the critical temperatures evaluated in potassium doped bismuthates reaching 34 K^{11}. The rather constant *T*_{c} computed for 0.4375 < *x* < 0.625 originates from compensating quantities: the diminishing *N*(*E*_{F}) and \(1/\omega _{{{{\mathrm{Boc}}}}}^2\) contributions are counterbalanced by an increasing *D* factor with increasing the doping content (Fig. 7a–c).

## Discussion

The DFT simulations reveal that hole doping has an indirect effect progressively suppressing the charge and bond disproportionation effects through the reduction of the octahedral rotations induced by simple steric effects. Once the disproportionation effects have vanished, no more energy gaps are identified in the band structure and the superconducting phase is reached. In this regime, the bond disproportionation vibration, that can be favored by its coupling with octahedral rotation vibrations, can produce spin-paired electrons and holes on the lattice, i.e., Copper pairs. The computed electron-phonon and related quantities are all in sharp agreement with experimental quantities obtained in these bismuthates. It is thus clear that the proximity of a charge and bond-ordered phase is a prerequisite to superconductivity. Numbers may of course be improved by involving a full electron-phonon calculation implying all phonon modes—this is however not affordable for a 32 f.u with cation disorder. However, the electron-phonon coupling constant and estimated *T*_{c} trend are already well captured by the SCAN functional and the simpler model, thereby suggesting that the breathing distortion phonon mode is dominating the superconducting properties of bismuthates. This would possibly be a common factor to several materials prone to exhibit charge orderings^{14,48,49,50,51,52}.

These results being ratified with parameter-free DFT calculations, further shows that DFT is sufficient to capture the physics of complex oxide superconductors, thereby showing that strong correlation effects may not be a universal explanation behind the high critical temperature observed in oxide superconductors. The “constant” *T*_{c} computed within the superconducting region finally appears fortuitous due to antagonist effects with increasing *D* that counterbalances the diminishing N(*E*_{F}) and \(1/\omega _{{{{\mathrm{Boc}}}}}^2\) factors. It thus suggest that computing the superconducting parameters at a fixed doping content and applying them at other doping content is not indicated as doping can act differently on the important quantities entering the Mac Millian’s equation (Eq. 4). Thus, properly modeling trends in electronic and structural properties versus doping effects may not be circumvented. The search for similar strongly coupled electron-phonon features such as disproportionation signatures, trends in doping effects, and lattice mode couplings between relevant distortions might be a key point for understanding pairing mechanisms in other oxide superconductors.

## Method

### The choice of the exchange-correlation functional

DFT simulations have been performed with the meta-GGA SCAN^{42} functional that improves the correction of self-interaction errors inherent to practice DFT over the classical LDA and GGA functionals and yields correct trends in lattice distortions and metal-insulator transitions as a function of A and B cations in bulk ABO_{3} perovskite oxides^{40}. This functional also has the advantage of being parameter-free and can therefore perfectly adapt to multiple formal oxidation states that an ion can develop in a doped material.

### The method to describe doping effects

Hole doping effects are modeled by substituting Sr^{2+} cations with the monovalent K^{+} cation. Special Quasi-random Structure (SQS) technique proposed in ref. ^{53} that allows to extract the cation arrangement maximizing the disorder characteristic of an alloy within a given supercell size are used to determine the best approximant of the cation disorder at each doping content. Thus, all possible local motifs for Bi cations in terms of surrounding Sr and K cations are available at each doping content, reminiscent of the situation in a real alloy. In order to allow enough flexibility for the material to develop the different lattice distortions exhibited by perovskites that can open the band gap^{39}, such as those displayed in Fig. 1, and to stabilize polaronic states, a 32-formula unit supercell that corresponds to a (2√2, 2√2, 4) supercell with respect to the primitive, undistorted, cubic Pm-3m cell is used.

### The used crystallographic cell

Only the low-temperature P2_{1}/n phase adopted by SrBiO_{3} is used throughout the study as a starting point.

### Potential energy surfaces and relevant phonon frequency calculation

Potential energy surfaces associated with lattice distortions are plotted starting from a perfectly undistorted Pm-3m cubic cell in which finite amplitudes of the different lattice distortions are condensed. In order to determine the frequencies *ω* of interesting lattice distortions in the ground state structure reached upon doping, a full phonon calculation is not affordable for a 32-formula unit supercell without any symmetry. Instead, some lattice mode amplitudes *Q* associated with a lattice distortion are frozen in the ground state structure and the associated potential energy surface is computed. Recalling that *E* = 1/2*Mω*^{2} *Q*^{2} for a harmonic oscillator with a frequency *ω* and where *M* is the mass of the moving atoms, the frequency is identified by fitting the potential energy surface with an expression, of the *E* = *aQ*^{2} + *bQ*^{4} where *a* and *b* are coefficients representing harmonic and anharmonic effects. One can then identify that \(\omega = \sqrt {\frac{{2a}}{M}}\).

#### Computing superconducting properties

The reduced electron-phonon coupling matrix element (REPME) associated with the breathing distortion B_{oc} are evaluated by freezing its atomic displacements in the relaxed ground state structure. Using the gap opening amplitude *∆E*_{g} appearing in the band structure due to the frozen phonon displacement, the REPME labeled *D* is computed by the following formula:

where *u* is the displacement of one oxygen atom. The electron-phonon coupling constant *λ* is evaluated by the following equation:

where *N*(*E*_{F}) is the density of states at the Fermi level per spin channel per formula unit, *M* is the mass of the displaced atoms and \(\omega _{{{{\mathrm{Boc}}}}}^2\) is the square of the computed B_{oc} frequency. The critical temperature *T*_{c} is computed by using the modified Mac Millan equation:^{47}

where *ω*_{log} is average logarithmic frequency and *μ*^{*} is the screened Coulomb potential with conventional values between 0.1 and 0.15.

#### Other technical details

DFT simulations are performed with the Vienna Ab initio Simulation package (VASP)^{54,55}. The energy cut-off is set to 650 eV and is accompanied by a 4 × 4 × 3 Gamma-centered kmesh for the 32 f.u. supercell. The kmesh is increased to 5 × 5 × 3 for density of states, potential energy surfaces, and frequency calculations. Projector augmented wave (PAW) potentials^{56} are used with Bi d states being treated as core states. Geometry relaxations (atomic positions plus lattice parameters) are performed until forces acting on each atom are lower than 0.05 eV/Å. The symmetry of the relaxed structures are extracted with the Findsym application^{57} and amplitudes of distortions are extracted using a symmetry mode analysis with respect to the primitive Pm-3m cell with the Isodistort tool from the Isotropy applications^{58,59}.

## Data availability

All data are available upon reasonable request to the author.

## Code availability

VASP DFT code license can be purchased from Vienna University.

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## Acknowledgements

J.V. acknowledges access granted to HPC resources of Criann through the projects 2020005 and 2007013 and of Cines through the DARI project A0080911453.

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The study was designed by J.V. J.V. performed all calculations and analysis of the results and wrote the manuscript.

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Varignon, J. Origin of superconductivity in hole doped SrBiO_{3} bismuth oxide perovskite from parameter-free first-principles simulations.
*npj Comput Mater* **9**, 30 (2023). https://doi.org/10.1038/s41524-023-00978-w

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DOI: https://doi.org/10.1038/s41524-023-00978-w