9 both n$_{s}$ = 0 and n$_{s}$ = 0 . 15. The corresponding spectral functions are shown in Fig. 10. As discussed above, the DOS exhibits subgap, main gap, and satellite features. In Ngai et al, the three fea- tures were attributed to superconductivity on different regions of the Fermi surface. This is generally consistent with our findings here although we have found it diffi- cult to attribute some satellite peaks to specific Fermi surface elements. Experimentally, the ratio between the satellite, main gap, and subgap energies was found to be approximately constant as a function of Ca doping, and this was argued to show that there is a common pairing mechanism for all three features. In our model the en- ergies of the satellite and subgap features in Fig. 9 also scale with the main gap. In some cases, our calculations reproduce the detailed structure of the experimental spectra. For y = 0 . 05, Ngai et al showed two types of spectrum. The first is remark- ably similar to that shown for y = 0 . 05 in Fig. 9(c), having pronounced coherence peaks and satellite features resembling shoulders. Spectra of this type are also mea- sured in optimally-doped ( y = 0) samples. 2,4,6,8 The good agreement between our calculations and the measured spectra suggests that our model captures the essential physics of the surface states in optimally-doped YBCO. The second type of measured spectrum qualitatively resembles that shown for y = 0 . 0 in Fig. 9(d), having weak coherence peaks, a subgap feature, and satellite peaks (indicated by arrows in Fig. 9). However, there is a discrepancy between our calculations and the exper- iments; at higher Ca doping levels, the measured spectra continue to exhibit three sets of peaks while the satellite peaks in our calculations become less prominent. Given the sensitivity of the DOS to small changes in the model parameters, it is plausible that this discrepancy can be corrected by small changes to the model. It is also possi- ble that extrinsic effects not considered here, for example tunneling matrix elements that emphasize the ( π, 0) and (0 , π ) regions of the Brillouin zone, 38,39 could increase the prominence of the satellite features in tunneling experi- ments. However, we also cannot rule out the possibility that our simple model for Ca substitution is overly na¨ıve. Another feature of the DOS that is not explained by our model is the large residual DOS measured experi- mentally; the DOS is never seen to vanish at ε$_{F}$ in the superconducting state, and is often 50% of the normal state DOS. It is not clear whether the residual DOS is intrinsic (for example, due to pair breaking at the sur- face) or extrinsic (coming from surface states in an ad- sorbate layer). It is possible that a full description of the YBCO surface states will require a proper accounting of this residual DOS. We have studied the surface and bulk states of Y$_{1}$$_{−}$$_{y}$Ca$_{y}$Ba$_{2}$Cu$_{3}$O$_{6}$$_{.}$$_{92}$ with 0 < y < 0 . 2 within a tight- We would like to acknowledge helpful coversations with J. Mannhart, T. Kopp, J. Wei, and J. Ngai. K.P. thanks The Commission on Higher Education Grant, Thailand (Grant No.12/2548) for financial support. This work was supported by NSERC of Canada and by SFB 484 from the DFG. This work was, in part, made possible by the facilities of the Shared Hierarchi- cal Academic Research Computing Network (SHARC- NET:www.sharcnet.ca) and Compute/Calcul Canada. IV. CONCLUSIONS Acknowledgments • our model produces results which are in simultane- ous agreement with surface and bulk ARPES mea- surements. This supports two key assumptions of the model, that the surface and bulk bands are con- nected by a simple chemical potential shift and that the Coulomb potential can be treated in a planar approximation. • the DOS in optimally-doped YBCO can be quan- titatively explained by our model. In particular, shoulders measured in the density of states that were previously attributed to pairing or to compet- ing order are found to be band structure effects. • the superconducting DOS is sensitive to small changes in the model parameters. This suggests that, in materials with CuO chains, small changes in doping of the surface states can have a qualita- tive effect on the density of states, purely as a result of changes to the band structure. This should be contrasted with BSCCO where it is believed that the doping dependence of the DOS is primarily due to strong correlations. • we can understand some features of Ca-doped YBCO; for example, our calculations find densities of states at the chain surface with subgap, main gap, and satellite features similar to experiments. However, we have not understood the dependence of this spectrum on Ca concentration. binding model. The model parameters for the CuO$_{2}$ planes are extracted from photoemission experiments, and self-consistent calculations are used to relate the sur- face and bulk states. We have calculated the density of states ρ$_{i}$ ( ω ) and spectral function A$_{i}$ ( k , ε$_{F}$ ) as functions of adsorbed surface charge density, chain-plane coupling, and Ca doping. Our main findings are that