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fact that the charge densities at the surfaces approach their bulk values as n$_{s}$ increases. In the CuO$_{2}$ layers, this results in a larger Δ$_{id}$ from Eq. (6), while in the CuO layers, this results in an increased hybridization between the surface chains and the adjacent CuO$_{2}$ plane. 

The CuO$_{2}$ spectrum for n$_{s}$ = 0 . 15 shown in Fig. 7 is consistent with existing tunneling experiments on optimally-doped YBCO. For this case, we obtain co- herence peaks at the CuO$_{2}$ surface at ± 20 meV and a satellite peak at ≈ − 30 meV, in approximate agreement with Refs. 2,6,8 (when comparing with experiments, re- call that a peak at negative energy in the DOS corre- sponds to a peak at positive voltage bias in a tunneling experiment). We note that having a charged adsorbate layer is not the only way to obtain agreement with exper- iments; a different value for the dielectric constant, for example, will affect the surface charge density, and conse- quently the DOS. Whether or not our model is correct in all details, it demonstrates that existing tunneling mea- surements on optimally-doped YBCO can be explained within a band picture. 

Throughout this work, we have assumed that the plane-chain coupling parameter is t$_{⊥}$$_{c}$ = 1 . 1 t$_{⊥}$$_{p}$ = 67 meV. This is the hardest of the model parameters to establish, and was chosen because it gives reasonable results for ρ$_{i}$ ( ω ) and A$_{i}$ ( k , ω ). Figure 8 shows the ef- fect of varying t$_{⊥}$$_{c}$ on the density of states and spectral function. Not surprisingly, chain-plane coupling has lit- tle effect on the DOS at the CuO$_{2}$ surface, apart from a shift of a weak negative-energy satellite peak away from the Fermi energy with increasing t$_{⊥}$$_{c}$ . In contrast, the chain surface is strongly influenced by coupling to the CuO$_{2}$ layer; the CuO chains are metallic when t$_{⊥}$$_{c}$ = 0, and an induced gap appears when t$_{⊥}$$_{c}$ is nonzero. The induced gap grows approximately linearly with t$_{⊥}$$_{c}$ and 

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has both a subgap and a main gap, as discussed earlier. The spectral function also changes with increasing t$_{⊥}$$_{c}$ , becoming increasingly distorted near ( π, 0). 

Finally, we look more closely at the effects of Ca doping. In Ngai et al, 4 samples with up to 20% Ca substitution for Y were studied, while in Yeh et al $^{3}$, 30% Ca-doped samples were studied. In Fig. 9, we show a series of calculations for the surface DOS of a Y$_{1}$$_{−}$$_{y}$Ca$_{y}$Ba$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{92}$ thin film. Results are shown for 

