Theory of (001) surface and bulk states in Y 1$_{−}$ $_{y}$Ca$_{y}$ Ba 2 Cu 3 O 7$_{−}$ δ K. Pasanai 1 , 2 and W. A. Atkinson 2 ∗ $^{1}$School of Physics, Institute of Science, Suranaree University of Technology, 111 University Ave., Muang District, Nakhon Ratchasima 30000, Thailand and $^{2}$Trent University, 1600 West Bank Dr., Peterborough ON, K9J 7B8, Canada (Dated: October 25, 2018) A self-consistent model is developed for the surface and bulk states of thin Y$_{1}$$_{−}$ y Ca y Ba$_{2}$Cu$_{3}$O$_{7}$$_{−}$$_{δ}$(YCBCO) films. The dispersions of the chain and plane layers are modelled by tight-binding bands, and the electronic structure is then calculated for a finite- thickness film. The dopant atoms are treated within a virtual crystal approximation. Because YCBCO is a polar material, self-consistent treatment of the long range Coulomb interaction leads to a transfer of charge between the film surfaces, and to the formation of surface states. The tight binding band parameters are constrained by the requirement that the calculated band structure of surface states at CuO$_{2}$-terminated surfaces be in agreement with photoemission experiments. The spectral function and density of states are calculated and compared with experiments. Unlike the case of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$, where the surfaces are believed to be representative of the bulk, the densities of states at the YCBCO surfaces are shown to be qualitatively different from the bulk, and are sensitive to doping. The calculated spectral function agrees closely with both bulk-sensitive and surface-sensitive photoemission results, while the calculated density of states for optimally-doped YCBCO agrees closely with tunneling experiments. We find that some density of states features previously ascribed to competing order can be understood as band structure effects. arXiv:1001.2695v1 [cond-mat.supr-con] 15 Jan 2010 I. INTRODUCTION The density of states (DOS) measured by tunneling experiments in the YBa$_{2}$Cu$_{3}$O$_{7}$$_{−}$$_{δ}$ (YBCO) family of high temperature superconductors is complicated. Sev- eral experiments on superconducting samples have mea- sured densities of states with multiple energy scales. 1–7 Some experiments find a subgap feature 3,4 while others do not, 2,5,6,8 and all experiments near optimal doping find satellite features at energies larger than the gap en- ergy. Many of these studies find that the spectra change qualitatively with doping and can vary significantly at different points on the sample surface. 4,6 Spectral fea- tures have been interpreted in terms of band structure, 4 competing order, 6,9 and coupling to bosonic modes. 5 In this work, we explore reasons why the YBCO single-particle spectrum is so complicated, particularly when compared to that of the related superconductor Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ (BSCCO). 10–12 In BSCCO, there is a clear d -wave-like gap in the density of states, and it has been possible to reproducibly extract detailed informa- tion about the band structure 7,13,14 and superconduct- ing state. 15,16 At present, there is no consensus on how to interpret the tunneling DOS in YBCO. In this work, we focus on two specific structural differences between BSCCO and YBCO, namely that YBCO is a polar ma- terial while BSCCO is not, and that YBCO has metallic one-dimensional CuO chains while BSCCO does not. We show that the confluence of these two factors explains some features of the experimentally-measured DOS. In particular, our results suggest that some features that were previously thought to indicate charge-ordering ac- tually come from peculiarities of the YBCO band struc- ture. The polarity of the YBCO unit cell is important for The second aspect of YBCO that makes it distinct from BSCCO is the presence of layers of one-dimensional CuO chains, in addition to the CuO$_{2}$ plane bilayers. The na¨ıve view is that these chains carry charge in parallel to the CuO$_{2}$ bilayers, but have little direct impact on the Surface charging in polar crystals has recently be- come prominent in the context of LaAlO$_{3}$/SrTiO$_{3}$ interfaces. 17,18 The essential idea is that, since the oppo- site faces of a polar unit cell have opposite charge, there is a potential difference between them. In a thin film, the potential difference between the top and bottom surfaces of the film is equal to the potential difference across a single unit cell times the number of unit cells spanning the film. The potential difference between the surfaces of the film is thus proportional to the film thickness, much like a parallel plate capacitor, and is typically large when the sample is more than a few unit cells thick. A “polar catastrophe” (i.e. a divergent electrostatic energy as the sample becomes macroscopically thick) is avoided by a transfer of charge between the two surfaces. This screen- ing charge eliminates the potential difference across the film, but changes the doping at the surfaces and leads to the formation of surface states. The existence of surface states in YBCO has recently been confirmed by angle resolved photoemission (ARPES) experiments. 19,20 several reasons. Unlike BSCCO, YBCO has no natural cleavage plane (i.e. no plane along which ionic forces van- ish), making it difficult to prepare surfaces that are clean enough for experiments. More importantly for this work, there is a charge transfer between YBCO surfaces as a result of the electric fields generated by the polar unit cell. This charge transfer leads to the formation of sur- face states that can differ significantly from states in the bulk. In contrast, it is widely believed that the surface layers in BSCCO are representative of the bulk.