Doubling of the Bands in Overdoped ${{\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_{2}O}_{8+\mathit{\delta }}$: Evidence for $\mathit{c}$-Axis Bilayer Coupling

We present high resolution angle resolved photoemission data of the bilayer superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) showing a clear doubling of the near $E_F$ bands. This splitting approaches zero along the $(0,0)\to (\pi ,\pi )$ nodal line and is not observed in single layer ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ (Bi2201), indicating that the splitting is due to the long sought after bilayer splitting effect. The splitting has a magnitude of approximately 75 meV near the middle of the zone, extrapolating to about 110 meV near the $(\pi ,0\mathrm{})$ point. The existence of these two bands also helps to clear up the recent controversy concerning the topology of the Fermi surface.