Spin and charge ordering in hole-doped Cu-O single chain, double chain, and ladder

We have studied the electronic structure of a hole-doped Cu-O single chain, double chain, and ladder using model Hartree-Fock calculations. In the single chain, the hole doping destroys the antiferromagnetic and insulating state of the undoped Cu-O chain and stabilizes a charge-ordered and insulating (COI) state. The COI states are lower in energy than antiferromagnetic and metallic (AFM) states for the hole-doped Cu-O chain. The energy difference between the COI and AFM states takes a maximum around x=1/2, indicating that the 1/4-filled case of x=1/2 is the most favored, where x is the hole concentration. In the Cu-O double chain, the COI state at x=1/2 is unstable and the COI at x=1/2 is favored, which is related to the COI state in the triangular lattice. In the Cu-O ladder, the COI at x=1/2 is as stable as that in the Cu-O single chain. In the COI of x=1/2, two different COI states, namely metal-centered charge-ordered (MCO) and oxygen-centered charge-ordered (OCO) states, are stable and degenerate in energy. Some of the COI states obtained for various x can be viewed as a mixture of the MCO and OCO states. In the COI states, the in-gap band with spectral weight of 2x is formed between the remnants of the upper and lower Hubbard bands.