Induced superconductivity in noncuprate layers of the Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ high-temperature superconductor: Modeling of scanning tunneling spectra

We analyze how the coherence peaks observed in scanning tunneling spectroscopy (STS) of cuprate high-temperature superconductors are transferred from the cuprate layer to the oxide layers adjacent to the STS microscope tip. For this purpose, we have carried out a realistic multiband calculation for the superconducting state of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ (Bi2212) assuming a short-range $d$-wave pairing interaction confined to the nearest-neighbor Cu $d_{x^2-y^2}$ orbitals. The resulting anomalous matrix elements of the Green’s function allow us to monitor how pairing is then induced not only within the cuprate bilayer but also within and across other layers and sites. The symmetry properties of the various anomalous matrix elements and the related selection rules are delineated.

Figure 1

Theoretical (green) STS spectrum normalized as shown in the figure is compared with the experimental (dashed blue) spectrum (Ref. 2) in optimally doped Bi2212. Regular (red) and anomalous (turquoise) parts of the computed spectrum are shown separately. All computations are based on Eq. (6). Coherence peaks arise only from the anomalous component of the Green’s function.

Figure 2

Imaginary part of the matrix elements of the anomalous Green’s function $F_{\alpha \beta }$ for various ($\alpha \beta$) pairs. (Recall that $\alpha$ and $\beta$ are composite indices denoting both the site and the orbital.) The meaning of values of $\alpha$ and $\beta$, which range from 0 to 9, is explained in Table . For example, index 0 refers to the $d_{x^2-y^2}$ orbital on the central Cu atom in the cuprate plane nearest to the STM tip, and index 1 refers to the $d_{x^2-y^2}$ orbital on the nearest-neighbor Cu atom in the same cuprate plane. The main matrix element between the two preceding orbitals, i.e., the (01) element, is shown by red lines for reference in all panels. Other matrix elements are shown scaled by factors ranging from 2 to 20, as indicated in the legends. Symmetries of the orbitals involved in various cases are shown schematically in the upper right-hand-side portions of the figures. Matrix elements compared with (01) (red curve) are as follows: (a) (02) (green curve), (34) (dashed blue curve), and (99) (dotted black curve) for pairing within a CuO${}_2$ bilayer; (b) (56) (green curve) and (78) (dashed blue curve) for intralayer pairing in SrO and BiO layers; and (c) (03) (dotted black curve), (05) (green curve), and (07) (dashed blue curve) for pairing of CuO${}_2$ along the line connecting the central Cu and the surface Bi.