3 perimentally, it is known that YBCO cleaves at the BaO layer, so that the top conducting layer may be either a CuO chain layer or a CuO$_{2}$ plane layer. To study both cases, we assume that the first layer ( i = 1) is composed of CuO chains, and the last layer ( i = N ) is a CuO$_{2}$ plane. Of the three nonconducting layers, two (the BaO lay- ers) are nominally neutral and are not explicitly consid- ered in the model. The yttrium and calcium atoms are nominally in the Y 3+ and Ca 2+ states and are retained in calculations of the long-ranged Coulomb potential. Of the three conducting layers, it is assumed that only the CuO$_{2}$ planes are intrinsically superconduct- ing, and that the chains are superconducting because of their proximity to the planes. The proximity model for chain superconductivity has been discussed at length elsewhere. 29–32 The essential idea is that single-electron hopping between the chain and plane layers leads to mix- ing of the electronic wavefunctions, and to induced pair- ing in the chain layer. The mean-field Hamiltonian of the system is where ˆ C † k = [ˆ c † 1 k $_{↑}$, ˆ c$_{1}$$_{−}$$_{k}$$_{↓}$, . . . , ˆ c † N k $_{↑}$, ˆ c$_{N}$$_{−}$$_{k}$$_{↓}$ ] and ˆ c † i k σ is the creation operator for a spin- σ electron in the conducting layer i with 2D wave vector k = ( k$_{x}$, k$_{y}$ ). H$_{k}$ is given by where ξ$_{ic}$$_{(}$$_{p}$$_{)}$ ( k ) is the 2D chain (plane) energy disper- sion in conducting layer i . The matrix elements t$_{⊥}$$_{c}$ and t$_{⊥}$$_{p}$ are for plane-chain and plane-plane hopping respec- tively. We remark that, in BSCCO, the plane-plane hop- ping matrix element has a strong k -dependence; ARPES experiments 28 and DFT calculations 33 agree that the k - dependence in YBCO is weaker, and we have neglected it here as a way of reducing the number of fitting param- eters in our model. For the intralayer dispersions, we use tight-binding models with nearest neighbor and third-nearest neighbor hopping for the chains and planes respectivley: 15% of the d-wave component, and we assume that this ratio holds at both the surfaces and in the bulk. Denot- ing the d-wave and s-wave components in layer i by Δ$_{id}$ and Δ$_{is}$ respectively, we have Note that, although the order parameter is zero in the chains, superconductivity is induced by the proximity ef- fect. We adopt the phenomenological expression for Δ$_{id}$ in the CuO$_{2}$ layers where the values of t$_{c}$, t$_{p}$, t $^{′}$, t$^{′′}$, t$^{′′′}$, and t$_{⊥}$$_{p}$ are shown in table I. The potential Φ$_{i}$ includes short and long range Coulomb interactions, as well as offset potentials for the plane and chain layers, and the chemical potential. The method of calculating Φ$_{i}$ is discussed below. It is known that the superconducting order parame- ter in the CuO$_{2}$ planes has d$_{x}$ $^{2}$− y 2 + s symmetry 34 as a result of the orthorhombic crystal structure. From experiments, 35 the s-wave component is approximately where n$_{i}$ is the charge density in plane i and n$_{min}$ = 0 . 7. We expect (6) to be valid on the overdoped side of the phase diagram. At optimal doping (with planar charge density n$_{p}$ ≈ 0 . 84 in the bulk), this gives a d -wave order parameter of magnitude 46 meV, which is close to values inferred from recent ARPES measurements. 28 Equation (6) also implies that the gap vanishes when the electron concentration is less than n$_{min}$ . The value of n$_{min}$ is not well-known and is probably material-dependent (depend- ing, for example, on the level of doping-related disorder); the canonical form for La$_{2}$$_{−}$$_{x}$Sr$_{x}$CuO$_{4}$ gives n$_{min}$ = 0 . 73, but tunneling experiments 4 on Y$_{1}$$_{−}$$_{y}$Ca$_{y}$Ba$_{2}$Cu$_{3}$O$_{7}$$_{−}$$_{δ}$ suggest that superconductivity is still present at this dop- Δ$_{i}$$_{k}$ = { Δ id 2 [cos( k$_{x}$ ) − cos( k$_{y}$ )] + Δ$_{is}$ , i ∈ plane 0 , i ∈ chain (5) Δ$_{id}$ = 100[ n$_{i}$ − n$_{min}$ ] 1 − n$_{min}$ meV , (6) ξ$_{ic}$ ( k ) = − 2 t$_{c}$ cos k$_{y}$ + Φ$_{i}$ (3) ξ$_{ip}$ ( k ) = − 2 t$_{p}$ [cos k$_{x}$ + cos k$_{y}$ + 2 t $^{′}$cos k$_{x}$ cos k$_{y}$ + t $^{′′}$(cos 2 k$_{x}$ + cos 2 k$_{y}$ ) + 2 t $^{′′′}$(cos 2 k$_{x}$ cos k$_{y}$ + cos 2 k$_{y}$ cos k$_{x}$ )] + Φ$_{i}$ (4) H$_{k}$ =      ξ$_{1}$$_{c}$ ( k ) 0 t$_{⊥}$$_{c}$ 0 − ξ$_{1}$$_{c}$ ( k ) 0 − t$_{⊥}$$_{c}$ t$_{⊥}$$_{c}$ 0 ξ$_{2}$$_{p}$ ( k ) Δ$_{2}$$_{k}$ t$_{⊥}$$_{p}$ − t$_{⊥}$$_{c}$ Δ$_{2}$$_{k}$ − ξ$_{2}$$_{p}$ ( k ) 0 − t$_{⊥}$$_{p}$ 0 t$_{⊥}$$_{p}$ 0 ξ$_{3}$$_{p}$ ( k ) Δ$_{3}$$_{k}$ − t$_{⊥}$$_{p}$ Δ$_{3}$$_{k}$ − ξ$_{3}$$_{p}$ ( k ) 0 . . . − t$_{⊥}$$_{p}$ ξ$_{Np}$ ( k ) Δ$_{N}$$_{k}$ − t$_{⊥}$$_{p}$ Δ$_{N}$$_{k}$ − ξ$_{Np}$ ( k )      (2) H = ∑ k ˆ C † $_{k}$H$_{k}$ ˆ C$_{k}$ (1)