Hole doping in YBa$_2$Cu$_3$O$_{6+\delta}$ from polarization dependent x-ray absorption spectroscopy

We present detailed polarization dependent x-ray absorption spectroscopy (XAS) measurements of the cuprate YBa$_2$Cu$_3$O$_{6+\delta}$ as a function of oxygen concentration for (0 $\leq \delta \leq$ 1). By adding O into the chain layer of YBCO, holes are doped into both the CuO$_2$ planes and CuO$_3$ chains. The presence of O induced states is directly observed by measurements of the O K edge and Cu L edge XAS, which probes unoccupied states in the O $2p$ and Cu $3d$ orbitals respectively. Owing to the different symmetry of the Cu d orbitals in the planes ($d_ {x^2-y^2}$) and chains ($d_{y^2-z^2}$), the contribution to the total XAS from the chains and planes is clearly separated by measuring the polarization dependence of the x-ray absorption. Comparison to LDA calculations of the unoccupied density of states are used to obtain a quantitative measure of the hole doping in the planes and chains as a function of O concentration. In addition, the doping of holes into the CuO$_2$ planes as a function of degree of oxygen order in the chains is observed by measuring YBa$_2$Cu$_3$O$_{6.5}$ in both the chain ordered (ortho II) and chain disordered phases. This provides direct evidence for the role of chain ordering on hole doping in YBCO.