Bilayer splitting and coherence effects in optimal and underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$

We have carried out extensive high-resolution angle-resolved photoemission (ARPES) experiments on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ samples, covering the entire doping range from the overdoped to the optimally and underdoped regimes in the normal state. Our focus is on delineating the doping dependence of the bilayer splitting which is associated with the intracell coupling of electrons between the two ${\mathrm{CuO}}_2$ planes. We exploit the photon energy of 47 eV, where strong ARPES matrix element effects are found to provide a tremendous enhancement of the antibonding to bonding component of the bilayer split bands near $(\pi ,0),$ in good agreement with the predictions of corresponding first-principles simulations. Our detailed analysis indicates that the size of the bilayer splitting is only weakly dependent on the doping level, implying that electronic excitations continue to maintain some degree of coherence even in the underdoped regime.