5 FIG. 2: (Color online) Comparison of the self-consistently calculated spectral function at ε$_{F}$ to the experimentally mea- sured Fermi surface (circles). Results are for (a) surface states (data from Ref. 19) and (b) bulk states (data from Ref. 28). FIG. 3: (Color online) Self-consistent solutions for the charge density and electrostatic potential in Y$_{1}$$_{−}$ y Ca y Ba$_{2}$Cu$_{3}$O$_{6}$ . $_{92}$. Charge density is shown (a) for all CuO chain and CuO$_{2}$ plane layers in the thin film as a function of the layer index i , and (b) at the chain ( i = 1) and plane ( i = 30) surfaces as a function of y . The potential difference ΔΦ$_{ip}$$_{(}$$_{c}$$_{)}$ = Φ i − Φ$_{p}$$_{(}$$_{c}$$_{)}$, between the potential in layer i and the potential in the bulk planes (chains), is shown in (c) and (d). III. RESULTS AND DISCUSSION A. Self-consistent potential, charge density, and superconducting order parameter of ∼ 2 eV from DFT is reasonable. 33 This gives the intra- chain hopping matrix element t$_{c}$ = 500 meV. The hardest parameter to establish is t$_{⊥}$$_{c}$ , the chain-plane coupling. We show results for multiple values of t$_{⊥}$$_{c}$ in Sec.III B, and find that the experimental density of states is rea- sonably well fit for t$_{⊥}$$_{c}$ = 1 . 1 t$_{⊥}$$_{p}$ for YBa$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{92}$. We show the results of our fitting procedure in Fig. 2. In this figure, we compare calculated surface and bulk spectral functions at the Fermi energy with the surface and bulk spectral functions measured by ARPES. The calculations are in good agreement with the experiments, which suggests that the assumptions made in developing our model are reasonable. Figure 3 shows the results of self-consistent calcula- tions for the charge density and electrostatic potential. We take the specific case of Y$_{1}$$_{−}$$_{y}$Ca$_{y}$Ba$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{92}$, which corresponds to optimal doping when y = 0 and to over- doping when y > 0. The self-consistently determined charge density is shown in Fig. 3(a) as a function of layer index i . (Recall that the layer indices i = 1 and i = 30 label the CuO chain-terminated and the CuO$_{2}$ plane-terminated surfaces respectively.) As discussed in the introduction, Fig. 3(a) shows that there is charge transfer from the plane-terminated surface to the chain- terminated surface. This charge screens the electric field produced by the polar unit cells, so that the electric po- tential is constant across the thin film except near the surfaces. This is shown in Fig. 3(c) and (d). In these figures, we have plotted the difference between the potential in layer i and the potential for a plane (chain) in the bulk in order to make the comparison between different y values simpler. ΔΦ$_{ip}$$_{(}$$_{c}$$_{)}$ is nonzero within a screening length of the surfaces and is posi- tive (negative) at the plane (chain) surface. The charge density on the plane (chain) surface is correspondingly smaller (larger) than in the bulk as shown in Fig. 3(a). We note that ΔΦ$_{1}$$_{c}$, the potential shift at the chain- terminated surface, is roughly seven times larger than ΔΦ$_{30}$$_{p}$, the potential shift at the plane-terminated sur- face. This ratio is approximately the same as the ratio of the plane and chain densities of states (in the nor- mal state); because of the small density of states in the chains, a large chemical potential shift is required at the chain-terminated surface to accommodate the charge transferred from the plane-terminated surface. Figure 3 also shows the effect of Ca substitution. De- spite the proximity of the Ca ions to the CuO$_{2}$ layers, n$_{i}$ changes by roughly the same amount in both the CuO$_{2}$ and CuO layers. The one notable exception to this is at the chain surface, where the electron concentration changes by roughly half as much as in the bulk [Fig. 3(b)]. Figure 4 shows the d -wave superconducting order pa- rameter Δ$_{id}$ as a function of layer index i in the CuO$_{2}$ plane layers, and for different levels of Ca doping. Be- cause Δ$_{id}$ is calculated phenomenologically from Eq. (6), it follows n$_{i}$ . Thus, Δ$_{id}$ is larger than in the bulk near the chain surface and smaller than in the bulk near the plane surface. In our calculations, Δ$_{id}$ actually vanishes at the CuO$_{2}$ surface layer ( i = 30), although there is a spectral gap due to proximity coupling to the subsur- face layers. Note that recent ARPES experiments, 19,20 found that the surface is nonsuperconducting, but tun- neling experiments 1–7 found a clear superconducting gap at the surface. We will show in the next section that the ΔΦ$_{ip}$$_{(}$$_{c}$$_{)}$ = Φ$_{i}$ − Φ$_{p}$$_{(}$$_{c}$$_{)}$ , (13) 0 5 10 15 20 25 30 layer index i 0.4 0.6 0.8 1 1.2 n i y = 0.0 y = 0.05 y = 0.1 y = 0.2 0 0.05 0.1 0.15 0.2 y 0.55 0.6 0.65 0.7 0.75 n  (surface) 0 5 10 15 20 25 30 layer index i -20 0 20 40 60 ip  (meV) 0 5 10 15 20 25 30 layer index i -400 -300 -200 -100 0 ic  (meV) plane chain plane surface chain surface (a) (b) (c) (d)