4 TABLE I: Parameters for the tight binding model for YBCO. ing level in YBCO. Our choice seems reasonably consis- tent with these experiments. The potentials in (3) and (4) are calculated from a self-consistent mean-field treatment. We write where φ$_{i}$ is the electrostatic potential, and ϵ$_{c}$$_{(}$$_{p}$$_{)}$ includes the chemical potential and the energy of the chain (plane) tight binding orbitals. We determine ϵ$_{p}$ and ϵ$_{c}$ by spec- ifying the bulk plane and chain charge densities at opti- mal doping ( δ = 0 . 08), which we take to be n$_{p}$ = 0 . 84 and n$_{c}$ = 0 . 48 electrons per 2D unit cell respectively. Note that, once n$_{p}$ is chosen, n$_{c}$ is set by the constraint of charge neutrality, given by Eq. (11) below. Our self- consistent calculations then find ϵ$_{c}$ − ϵ$_{p}$ = 3 . 176 eV at optimal doping. We assume that the tight binding or- bitals are not modified by doping, so that ϵ$_{c}$ − ϵ$_{p}$ is held constant througout this work. In our calculations chemi- cal doping modifies the band structure only through the electrostatic potential φ$_{i}$ . The Coulomb potential φ$_{i}$ is then calculated self- consistently within the Hartree approximation, under the assumption that the charge is uniformly distributed within each layer. For the planar geometry shown in Fig. 1, where the Y$_{1}$$_{−}$$_{y}$Ca$_{y}$ layers are implicitly included in the sum over j and In (8), U is the intraorbital Coulomb potential for the CuO$_{2}$ layers, which we take to be 4 eV. The total 2D charge density in layer i is σ$_{i}$ = Z$_{i}$ − n$_{i}$ , where Z$_{i}$ is the charge density of the ionic cores. Here, n$_{i}$ is measured relative to the Cu 3+ and O 2 − states, so that All charge densities are in units of e/a 2 $_{0}$, where a$_{0}$ ≈ 4 ˚ A is the 2D lattice constant. Electrical neutrality requires that In Eq. (8), z$_{i}$ is the z -coordinate of layer i , in units of the c -axis lattice constant d$_{z}$ ≈ 12 ˚ A. Within a unit cell, the layers are at z = 0 (chain), z = 0 . 354 (first plane), z = 0 . 5 (Y layer), and z = 0 . 646 (second plane). 36 The weak doping dependence of these values is ignored here. The dielectric constant ϵ in (8) is not well known, but is believed to be around ϵ = 20, which is the value taken here. For a given potential Φ$_{i}$, the charge density in layer i is found from the eigenvalues E$_{α,}$$_{k}$ and eigenstates Ψ$_{α,}$$_{k}$( i, σ ) of H$_{k}$ via where f ( x ) is the Fermi function, the factor of 2 is for spin, and N$_{k}$ is the number of k-points in the sum. The band index α ranges from 1 to 2 N because the number of bands in the film is equal to twice (including spin) the number of conducting layers. The updated charge density is used to re-calculate Φ$_{i}$, which is then used in the next iteration for n$_{i}$ . The iterations proceed until the difference between n$_{i}$ in two consecutive iterations is less than 10 − $^{5}$. To reduce the computational workload, Φ$_{i}$ is calculated in the normal state. We finish this section with a brief discussion of the fit- ting procedure used to get the model parameters shown in Table I. As discussed above, we fit the 2D disper- sions for the CuO$_{2}$ layers to the surface states mea- sured in Ref. 19. In order to avoid complications from the chains, we fit the energy spectrum for an isolated CuO$_{2}$ bilayer to the measured bands in regions of the Brillouin zone far from the chain Fermi surface. The model bilayer has bonding and antibonding bands, with energies ξ $^{±}$( k ) = ξ$_{p}$ ( k ) ± t$_{⊥}$$_{p}$ , which allows us to de- termine t$_{p}$, t $^{′}$, t $^{′′}$, t $^{′′′}$, and t$_{⊥}$$_{p}$ . One potential diffi- culty with this fitting process is that it assumes that the electrostatic potential is the same in the top two CuO$_{2}$ layers, meaning that we attribute the experimental bi- layer splitting entirely to t$_{⊥}$$_{p}$ . If we allow for a poten- tial difference Δ φ between the CuO$_{2}$ layers making up the bilayer (due to band bending at the surfaces), then ξ $^{±}$( k ) = ξ$_{p}$ ( k ) ± √ (Δ φ/ 2) 2 + t 2 ⊥ $_{p}$. Self-consistent cal- culations reported in the next section suggest Δ φ ∼ 10 meV while the measured t$_{⊥}$$_{p}$ ≈ 60 meV; it follows that Δ φ modifies the band energies by less than 1% and can safely be neglected. There are, at present, no reliable measurements of the chain band structure. We therefore assume that strong correlations are not significant in the CuO chains (which are roughly quarter-filled), and that the chain bandwidth n$_{c}$ + 2 n$_{p}$ = 2 + 2 δ − y. (11) σ$_{j}$ =    1 + 2 δ − n$_{j}$, j ∈ CuO$_{1}$$_{−}$$_{δ}$ chain − 1 − n$_{i}$, j ∈ CuO$_{2}$ plane 3 − y, j ∈ Y 3+ 1 − $_{y}$Ca 2+ y layer (10) κ = 2 πe $^{2}$d$_{z}$ ϵa 2 0 . (9) φ$_{i}$ = { − κ ∑ j σ$_{j}$ | z$_{i}$ − z$_{j}$ | + U n i 2 , i ∈ plane − κ j σ$_{j}$ | z$_{i}$ − z$_{j}$ | , i ∈ chain (8) n$_{i}$ = 2 N$_{k}$ ∑ k 2 N ∑ α =1 | Ψ$_{α,}$$_{k}$( i, ↑ ) | $^{2}$f ( E$_{α,}$$_{k}$ ) (12) Φ$_{i}$ = { φ$_{i}$ + ϵ$_{p}$, i ∈ plane φ$_{i}$ + ϵ$_{c}$, i ∈ chain (7) Parameter Value t$_{p}$ 105 meV t ′ -0.277 t ′′ 0.234 t ′′′ -0.042 t$_{c}$ 500 meV t$_{⊥}$$_{p}$ 61 meV t$_{⊥}$$_{c}$ 1.1 t$_{⊥}$$_{p}$