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Fig. 6, we see that the main effect on the CuO$_{2}$ surface layer ( i = 30) is indeed near the ( π, 0) point, where the spectral weight is reduced by Ca doping. 

In Fig. 6, the intensity of A$_{i}$ ( k , ω ) strongly depends on the amount of hybridization between chain and plane states. The Fermi surface at the CuO-terminated surface ( i = 1) is relatively undistorted, indicating weak chain- plane coupling. By contrast, the chain Fermi surface in the bulk is more strongly hybridized to the CuO$_{2}$ plane states and is correspondingly washed out. This is a pos- sible reason that the chain Fermi surface was imaged in surface-sensitive ARPES measurements 19,20 but not in bulk-sensitive measurements. 28 

We remark that the amount of hybridization between chains and planes is strongly k -dependent, even though the matrix element t$_{⊥}$$_{c}$ is independent of k . This is be- cause the hybridization at each k depends on the energy difference between the chain and plane bands at that value of k . Because these bands have different symme- tries, the energy difference (and thus the hybridization) is a strong function of k . It follows that the amount of hybridization at ε$_{F}$ is a strong function of the relative positions of chain and plane Fermi surfaces, with the hy- bridization being largest where the Fermi surfaces cross. This explains the difference between the surface ( i = 1) and bulk ( i = 13) spectral functions shown in Fig. 6. A more extensive discussion of this point can be found in Ref. 31. 

The degree to which the chain and plane Fermi sur- faces hybridize determines the size of the induced gap in the chain layers. The small gap at the CuO surface 

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(Fig. 5) is thus due to the weak hybridization of the sur- faces chains with the underlying CuO$_{2}$ planes. In the bulk, the chain DOS shown in Fig. 5 has a main gap of about 20 meV, and a small gap of about 4 meV. The small gap originates from sections of the chain Fermi sur- face that are only weakly coupled to the planes (namely, k$_{x}$ < ∼ 1) while the large gap comes from sections of the Fermi surface that are strongly coupled to the planes ( k$_{x}$ > ∼ 1). The small gap was discussed previously as a possible source for the subgap structure measured in some tunneling experiments. 4 

Having discussed general features of the DOS and spec- tral function, we now discuss how these are affected by changes in specific model parameters. First, we allow for the possibility that the surface layers of the YBCO thin film are doped by the adsorption of atoms or molecules onto the surfaces. Adsorption happens naturally, for ex- ample, when YBCO is exposed to air, and deliberate potassium adsorption has been used to control the elec- tron concentration in CuO$_{2}$ surface states. 37 In this work, we are interested in the possibility that the adsorbate layers partially screen the electric fields at the YBCO surfaces. We model this by assuming that the adsorbed layer has an average charge density of + en$_{s}$ ( − en$_{s}$ ) at the CuO$_{2}$ (CuO) surface, where n$_{s}$ > 0. In our self- consistent calculations, the effect of the adsorbate charge is to reduce the charge transfer between the chain and plane surfaces. 

Figure 7 shows the effect of n$_{s}$ on the density of states and spectral function. We see that even a relatively small adsorbate charge density has a significant effect on both ρ$_{i}$ ( ω ) and A$_{i}$ ( k , ω ) at the surfaces. In particular, ρ$_{i}$ ( ω ) and A$_{i}$ ( k , ω ) at the surfaces are increasingly similar to the bulk as n$_{s}$ increases. This follows directly from the 

