LDA energy bands, low-energy hamiltonians, t′, t″, t⊥ ( k), andJ⊥

Abstract

We describe the LDA band structre of YBa 2Cu 3O 7 in the ϵ F ± 2 eV range using orbital projections and compare with YBa 2Cu 4O 8. Then, the high-energy and chain-related degrees of freedom are integrated out and we arrive at two, nearest-neighbor, orthogonal, two-center, 8-band Hamiltonians, H 8 + and H 8 −, for respectively the even and odd bands of the bi-layer. Of the 8 orbitals, Cu x 2 − y 2 , O2 x , O3 y , and Cu s have σ character and Cu xz , Cu yz , O2 z , and O3 z have π character. The roles of the Cu s orbital, which has some Cu 3 z 2 − 1 character, and the four π orbitals are as follows: Cu s provides 2nd- and 3rd-nearest-neighbor ( t′ and t″) intra-plane hopping, as well as hopping between planes ( t ⊥). The π-orbitals are responsible for bifurcation of the saddle-points for dimpled planes. The 4-σ-band Hamiltonian is generic for flat CuO 2 planes and we use it for analytical studies. The k ∥-dependence is expressed as one on u ≡ (cos bk y + cos ak x) 2 and one on v ≡ (cos bk y − cos ak x) 2 . The latter arises solely through the influence of Cu s . The reduction of the σ-Hamiltonian to 3- and 1-band Hamiltonians is explicitly discussed and we point out that, in addition to the hoppings commonly included in many-body calculations, the 3-band Hamiltonian should include hopping between all 2nd-nearest-neighbor oxygens and that the 1-band Hamiltonian should include 3rd-nearest-neighbor hoppings. We calculate the single-particle hopping between the planes of a bi-layer and show that it is generically: t⊥ (k ∥) ≈ 0.25 eV · v 2 (1 − 2ut′ t ) −2 . The hopping through insulating spacers such as (BaO)Hg(BaO) is an order of magnitude smaller, but seems to have the same k∥-dependence. We show that the inclusion of t′ is crucial for understanding ARPES for the anti-ferromagnetic insulator Sr 2CuO 2Cl 2. Finally, we estimate the value of the inter-plane exchange constant J ⊥ for an un-doped bi-layer in meanfield theory using different single-particle Hamiltonians, the LDA for YBa 2Cu 3O 6, the eight- and four-band Hamiltonians, as well as an analytical calculation for the latter. We conclude that J⊥ ~ − 20 meV.