Unveiling the Superconducting Mechanism of ${\mathrm{Ba}}_{0.51}{\mathrm{K}}_{0.49}{\mathrm{BiO}}_3$

The mechanism of high superconducting transition temperatures ($T_c$) in bismuthates remains under debate despite more than 30 years of extensive research. Our angle-resolved photoemission spectroscopy studies on ${\mathrm{Ba}}_{0.51}{\mathrm{K}}_{0.49}{\mathrm{BiO}}_3$ reveal an unexpectedly 34% larger bandwidth than in conventional density functional theory calculations. This can be reproduced by calculations that fully account for long-range Coulomb interactions—the first direct demonstration of bandwidth expansion due to the Fock exchange term, a long-accepted and yet uncorroborated fundamental effect in many body physics.Furthermore, we observe an isotropic superconducting gap with $2{\mathrm{\Delta }}_0/k_B{T}_c=3.51\pm 0.05$, and strong electron-phonon interactions with a coupling constant $\lambda \sim 1.3\pm 0.2$. These findings solve a long-standing mystery—${\mathrm{Ba}}_{0.51}{\mathrm{K}}_{0.49}{\mathrm{BiO}}_3$ is an extraordinary Bardeen-Cooper-Schrieffer superconductor, where long-range Coulomb interactions expand the bandwidth, enhance electron-phonon coupling, and generate the high $T_c$. Such effects will also be critical for finding new superconductors.

Figure 1

(a) Phase diagram of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{BiO}}_3$ according to [6, 22]. The abbreviations stand for charge density wave insulator (CDWI) and superconductor (SC). The red marker shows the doping and $T_c$ of our samples. The inset illustrates the perovskitelike crystal structure. (b) Scanning electron microscope image of the cleaved surface. A beam spot of $50\mu \mathrm{m}\times 25\mu \mathrm{m}$ is illustrated. (c) Temperature dependence of the zero field-cooled (ZFC) and field-cooled (FC) magnetic susceptibility of our ${\mathrm{Ba}}_{0.51}{\mathrm{K}}_{0.49}{\mathrm{BiO}}_3$ single crystal measured at 100 Oe.

Figure 2

(a) Photoemission intensity map in the $\mathrm{\Gamma }ZX$ plane integrated over an energy window of $E_F\pm 15\mathrm{meV}$, measured with 57 to 132 eV photons. (b) In-plane photoemission intensity map measured with $h\nu =73\mathrm{eV}$ photons, integrated over $E_F\pm 15\mathrm{meV}$. The inset illustrates the Brillouin zone of ${\mathrm{Ba}}_{0.51}{\mathrm{K}}_{0.49}{\mathrm{BiO}}_3$ and the momentum space sampled with $h\nu =73\mathrm{eV}$ photons. (c) Photoemission intensity along cut #1 in panel (b). The bands $\alpha$, $\beta$, and $\gamma$ are indicated on the right side by markers in red, blue, and green, respectively. (d) Energy distribution curves (EDCs) of data in the lower part of panel (c), showing the dispersions of bands $\beta$ and $\gamma$. (e) Momentum distribution curves (MDCs) of data in the upper part of panel (c), showing the dispersions of bands $\alpha$ and $\beta$. (f) Band dispersion extracted from the photoemission data (markers in red, blue, and green) plotted over density functional theory (DFT) calculations of ${\mathrm{BaKBi}}_2{\mathrm{O}}_6$, using generalized gradient approximation (GGA, blue curves) and Heyd-Scuseria-Ernzerhof hybrid functional calculations (HSE06, black curves).

Figure 3

(a) Temperature dependence of symmetrized photoemission spectra along the $\mathrm{\Gamma }$$X$ direction as shown by the photoemission cut in the right inset (blue arrow), measured in the $\mathrm{\Gamma }MX$ plane by 30 eV photons. The right insets illustrate the Brillouin zones (orange squares) and Fermi surfaces (gray). (b) Temperature dependence of the symmetrized EDCs (diamonds) integrated around $k_F$ [double arrows above (a)]. The EDCs are vertically offset for better visualization, with horizontal bars indicating their zero positions. The red solid curves are fits to resolution-convolved Dynes functions. (c) Temperature dependence of the superconducting gap (red) fit to the BCS function (black). The error bars are determined by combining the standard deviations of the Dynes fits (c) and the energy uncertainty. (d) Symmetrized EDCs at various $k_F\mathrm{s}$ (as shown in the right top inset of panel a) of band $\alpha$ in the $\mathrm{\Gamma }MX$ plane measured at 10 K. The EDCs are vertically offset, with horizontal bars indicating their zero positions. The top panel illustrates the Brillouin zone (orange square), Fermi surface cross section (gray), and the color-coded Fermi momenta (open circles). (e) Same as panel d but along $k_F\mathrm{s}$ in the $\mathrm{\Gamma }ZX$ plane [right bottom inset of panel (a)].

Figure 4

(a) ARPES spectrum overlaid with the Lorentz fitting on the MDCs taken along $\mathrm{\Gamma }$$X$ using 72 eV photons. (b) MDC-derived dispersion from the raw data (circles), linear fits to the high energy ($E-E_F<-80\mathrm{meV}$, $v_F^0$ fit, solid line) and low energy ($E-E_F>-50\mathrm{meV}$, $v_F$ fit, dashed line) dispersions. (c) Real part of the self-energy $\mathrm{Re}\mathrm{\Sigma }$ obtained from the difference between the band dispersion near $E_F$ and the polynomial fit approximating the bare band dispersion (red dots), and from the Kramers-Kronig transformation of the imaginary part of the self-energy $\mathrm{Im}\mathrm{\Sigma }$ (black curve). (d) Full-width at half-maximum (FWHM) of the MDCs, and $\mathrm{Im}\mathrm{\Sigma }$ computed from the FWHM. (e) Electron-phonon spectral function ${\alpha }^{2}F(\omega ,k)$ (black squares) calculated from the $\mathrm{Im}\mathrm{\Sigma }(\omega )$ and the phonon density of states determined by neutron diffraction [39]. To avoid unphysical values, we set negative data points of ${\alpha }^{2}F(\omega ,k)$ to zero.