Theory of (001) surface and bulk states in ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$

A self-consistent model is developed for the surface and bulk states of thin ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }\left(\text{YCBCO}\right)$ films. The dispersions of the chain and plane layers are modeled 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 to 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 ${\text{CuO}}_2$-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 ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$, 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.

Figure 1

Model of a Ca-doped YBCO thin film. The film is $N_c$ unit cells thick, and each unit cell contains two superconducting ${\text{CuO}}_2$ and one conducting CuO layer, for a total of $N=3N_c$ conducting layers. The conducting layers are labeled $i=1,\dots ,N$, with $i=1$ corresponding to the chain-terminated surface, and $i=N$ corresponding to the ${\text{CuO}}_2$ plane-terminated surface. The ${\text{CuO}}_2$ and CuO layers are described by two- and one-dimensional dispersions ${\xi }_{ip}\left(\mathbf{k}\right)$ and ${\xi }_{ic}\left(\mathbf{k}\right)$, respectively, where $\mathbf{k}=\left(k_x,k_y\right)$ and are coupled by interlayer hopping matrix elements $t_{\perp p}$ (plane-plane) and $t_{\perp c}$ (chain-plane). The nonconducting layers are not included in the band structure calculations, although the Y/Ca layers are included in calculations of the electrostatic potential. The BaO layers are electrically neutral and are ignored. Results in this work are shown for $N_c=10$ unit cells.

Figure 2

(Color online) Comparison of the self-consistently calculated spectral function at ${\epsilon }_F$ to the experimentally measured Fermi surface (circles). Results are for (a) surface states (data from Ref. 19) and (b) bulk states (data from Ref. 28).

Figure 3

(Color online) Self-consistent solutions for the charge density and electrostatic potential in ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Charge density is shown (a) for all CuO chain and ${\text{CuO}}_2$ plane layers in the thin film as a function of the layer index $i$, and (b) at the chain $\left(i=1\right)$ and plane $\left(i=30\right)$ surfaces as a function of $y$. The potential difference $\Delta {\Phi }_{ip\left(c\right)}={\Phi }_i-{\Phi }_{p\left(c\right)}$, between the potential in layer $i$ and the potential in the bulk planes (chains), is shown in (c) and (d).

Figure 4

(Color online) Superconducting order parameter as a function of layer index for ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. The figure shows the $d$-wave component of the order parameter, which is nonzero in the ${\text{CuO}}_2$ layers only. The magnitude of ${\Delta }_{id}$ is given by Eq. (6). Inset: superconducting order parameter near the plane surface $\left(i=29\right)$ as a function of Ca doping. Note that ${\Delta }_{30d}=0$.

Figure 5

(Color online) Results for films with identical surfaces. (a) Self-consistent charge density for a film whose terminating surfaces are both ${\text{CuO}}_2$ planes. Note that the surfaces should be Mott insulating based on their charge density. (b) Self-consistent charge density and (c) surface DOS for a film with two CuO-terminated surfaces.

Figure 6

(Color online) Surface and bulk density of states for ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Results are shown for $y=0.0$ [(a) and (b)] and for $y=0.05$ [(c) and (d)]. DOS is shown for ${\text{CuO}}_2$ planes [(a) and (c)] and CuO chains [(b) and (d)]. Model parameters are given in Table . The bulk DOS ($i=13$ and $i=15$) is essentially the same as that of a uniform three-dimensional system with the same band parameters. Results are for $N_c=10$ and we have checked that the DOS is essentially unchanged for larger $N_c$.

Figure 7

Spectral function at the surfaces and in the bulk of ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Columns show $A\left(\mathbf{k},{\epsilon }_F\right)$ for $y=0$ for (a) planes and (b) chains, and for $y=0.05$ for (c) planes and (d) chains. Rows (top to bottom) show layers $i=30$, 29, 27, 15 (planes) and $i=1$, 4, 7, 13 (chains); thus, surface states are shown in the top row, states deep in the bulk are shown in the bottom row.

Figure 8

Effects of an adsorbed surface layer on the surface states of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Columns are for (a) plane and (b) chain DOS, (c) plane and (d) chain spectral function. The adsorbate layer has a 2D charge density of $+e{n}_s$ for the ${\text{CuO}}_2$ surface and $-e{n}_s$ for the CuO surface. Rows correspond to $n_s=0.0$, 0.1, 0.15, and 0.2 (top to bottom).

Figure 9

Effects of chain-plane coupling on the surface states of ${\text{YB}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Columns are for (a) plane and (b) chain DOS and (c) plane and (d) chain spectral function. The adsorbate charge density is $n_s=0.15$. Rows correspond to $t_{\perp c}=0.0,0.8t_{\perp p},1.1t_{\perp p},1.3t_{\perp p}$ (top to bottom).

Figure 10

Effects of Ca doping on the density of states of ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$ at the surfaces. Columns are for (a) plane and (b) chain surfaces with $n_s=0.0$, and (c) plane and (d) chain surfaces with $n_s=0.15$. Rows correspond to $y=0.0,0.05,0.10,0.2$ (top to bottom). Arrows in (d) indicate the locations of satellite features discussed in the text.

Figure 11

Effect of Ca doping on the spectral function at the surfaces of ${\text{Y}}_{1-y}{\text{Ca}}_y{\text{Ba}}_2{\text{Cu}}_3{\text{O}}_{6.92}$. Columns are for $n_s=0.0$ for (a) planes and (b) chains, and for $n_s=0.15$ for (c) planes and (d) chains. Rows are for $y=0.0,0.05,0.10,0.2$ (top to bottom).