6 FIG. 4: (Color online) Superconducting order parameter as a function of layer index for Y$_{1}$$_{−}$ y Ca y Ba$_{2}$Cu$_{3}$O$_{6}$ . $_{92}$. The figure shows the d -wave component of the order parameter, which is nonzero in the CuO$_{2}$ layers only. The magnitude of Δ$_{id}$ is given by Eq. (6). Inset: Superconducting order parameter near the plane surface ( i = 29) as a function of Ca doping. Note that Δ$_{30}$$_{d}$ = 0. FIG. 5: (Color online) Surface and bulk density of states for Y$_{1}$$_{−}$ y Ca y Ba$_{2}$Cu$_{3}$O$_{6}$ . $_{92}$. Results are shown for y = 0.0 [(a) and (b)] and for y = 0.05 [(c) and (d)]. DOS is shown for CuO$_{2}$ planes [(a) and (c)] and [(b) and (d)] CuO chains. Model parameters are given in Table I. B. Density of states and spectral function size of the superconducting gap at the CuO$_{2}$ surface is sensitive to how the surface is prepared. The main results reported in this work are for the layer- dependent density of states and spectral function. The density of states at energy ω in layer i is given by Note that the eigenstates Ψ$_{α,}$$_{k}$( i, σ ) and eigenenergies E$_{α,}$$_{k}$ describe electrons ( σ = ↑ ) or holes ( σ = ↓ ) depending on the spin index. In this work, all results for A$_{i}$ ( k , ω ) are shown for the normal state at ω = ε$_{F}$ , while results for ρ$_{i}$ ( ω ) are shown in the superconducting state. Results for Y$_{1}$$_{−}$$_{y}$Ca$_{y}$Ba$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{92}$ with y = 0 and y = 0 . 05 are shown in Fig. 5. This figure shows that the density of states at the surfaces is very different from in the bulk, for both the CuO$_{2}$ planes and CuO chains. Notably, the d -wave gap is significantly smaller at the surfaces than in the bulk. In the CuO$_{2}$ layers, this re- flects the suppression of Δ$_{id}$ near the surface due to the decreased electron density in the surface layers. This is consistent with bulk-sensitive ARPES experiments that found a gap of order 42 meV, 28 and surface-sensitive tun- neling experiments, which consistently find that the gap is < ∼ 25 meV. 1–7 In the chain layers, the reduced gap at the surface reflects the reduced proximity coupling at the chain surface, in part because the chain has only one nearest neighbor CuO$_{2}$ plane. Our calculation shows that the DOS obtains its bulk value within a few layers of ei- ther surface. We note that, relative to the conventional model of a single-layer d -wave superconductor, the DOS in Fig. 5 shows a lot of structure. In particular, the CuO$_{2}$ surface (for y = 0) has a main gap of about 10 meV, whose co- herence peaks are split into pairs of closely-spaced peaks, and a satellite “shoulder” at about 20 meV. Shoulder fea- tures have been commonly observed in tunneling experi- ments, and have been attributed to pairing, 4 and to com- peting order. 6 In our calculations, this structure comes from the interplay of pairing and band structure effects, namely the mixing of chain and plane states resulting in an orthorhombic distortion of the Fermi surfaces. This is illustrated in Fig. 6, which shows the layer-resolved spectral function A$_{i}$ ( k , ε$_{F}$ ) at the Fermi energy. We see from this figure that chain-plane coupling strongly dis- torts the Fermi surfaces in the ( π, 0) region of the Bril- louin zone. This distortion is particularly important for the CuO$_{2}$ layers, because there is a van Hove singularity near the ( π, 0) point in the undistorted spectrum. The DOS is therefore sensitive to small changes in the Fermi surface shape, caused either by chain-plane coupling, by doping, or by band-bending (changes in the electrostatic potential) near the surfaces. Thus, the addition of 5% Ca changes the electron density by only ∼ 0 . 02 electrons per CuO$_{2}$ plaquette but qualitatively changes the shape of the coherence peaks [compare Fig. 5(a) and (c)]; in where N$_{k}$ is the total number of k -points, and A$_{i}$ ( k , ω ) is spectral function in layer i , ρ$_{i}$ ( ω ) = 1 N$_{k}$ ∑ k A$_{i}$ ( k , ω ) (14) A$_{i}$ ( k , ω ) = 2 N ∑ α =1 [ | Ψ$_{α,}$$_{k}$( i, ↑ ) | $^{2}$δ ( ω − E$_{α,}$$_{k}$ ) + | Ψ$_{α,}$$_{k}$( i, ↓ ) | $^{2}$δ ( ω + E$_{α,}$$_{k}$ )] . (15) 0 5 10 15 20 25 30 layer index i 0 10 20 30 40 50 60 70 80 90 100 Δ $^{id}$  (meV) y = 0.0 y = 0.05 y = 0.1 y = 0.2 0 0.05 0.1 0.15 0.2 y 0 5 10 15 20 25 29d  (meV) 0 1 2 3 4 5 i = 30 i = 29 i = 27 i = 15 0 0.2 0.4 0.6 0.8 1 1.2 i = 1 i = 4 i = 7 i = 13 -60 -40 -20 0 20 40 60 (meV) 0 1 2 3 4 5 [eV -1 .a 0 -2 ] -60 -40 -20 0 20 40 60 0 0.2 0.4 0.6 0.8 1 1.2 (a) (c) (b) (d)