Interpretation of Scanning Tunneling Quasiparticle Interference and Impurity States in Cuprates

We apply a recently developed method combining first principles based Wannier functions with solutions to the Bogoliubov–de Gennes equations to the problem of interpreting STM data in cuprate superconductors. We show that the observed images of Zn on the surface of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ can only be understood by accounting for the tails of the Cu Wannier functions, which include significant weight on apical O sites in neighboring unit cells. This calculation thus puts earlier crude “filter” theories on a microscopic foundation and solves a long-standing puzzle. We then study quasiparticle interference phenomena induced by out-of-plane weak potential scatterers, and show how patterns long observed in cuprates can be understood in terms of the interference of Wannier functions above the surface. Our results show excellent agreement with experiment and enable a better understanding of novel phenomena in the cuprates via STM imaging.

Figure 1

(a) Resonant state real-space BdG patterns at ${\mathrm{\Omega }}_0=-3.6\mathrm{meV}$ as obtained from conventional BdG calculations in logarithmic scale, (b) $xy$ cut through continuous 3D LDOS $(x,y,z\approx 5\AA ;{\mathrm{\Omega }}_0)$ at ${\mathrm{\Omega }}_0=-3.6\mathrm{meV}$ showing strong similarity to the measured conductance maps (c) reproduced from Ref. [8] rotated to match the orientation in (a) and (b) and cropped to $11\times 11$ elementary cells with the impurity located at the center.

Figure 2

(a) Elementary cell used in first principles calculation to obtain the electronic structure on the ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ surface. Isosurface plots of the Cu-$d_{x^2-y^2}$ Wannier function at (b) 0.05, (c) 0.005, and (d) $0.0002{\mathrm{bohr}}^{-3/2}$. Arrows indicate nearest-neighbor apical oxygen tails and red and blue indicate the sign of the Wannier function.

Figure 3

(a) Continuum LDOS at 5 Å above the BiO surface in the superconducting state with a single Zn impurity modeled by an onsite $-5\mathrm{eV}$ potential. Shown are positions directly above Cu atoms far from the impurity (black), at the impurity position (black, dashed), on the nearest neighbor position (red [light gray]), and on the next-nearest neighbor position (blue [dark gray]), calculated using $20\times 20$ supercells with broadening of 1 meV and (b) for a weak impurity scatterer with $V_{\mathrm{imp}}=0.3\mathrm{eV}$ as used for the QPI analysis. In (c) we compare the spectral properties of the lattice density of states (red [light gray], dashed) with the continuum LDOS above a Cu atom calculated using the diagonal terms of the lattice Greens function $G_{\sigma }(\mathbf{R},\mathbf{R};\omega )$ only (black) and the full Green’s function as given in Eq. (2) (blue [gray], dash dotted); all of them calculated for a homogeneous superconductor and scale adjusted such that two curves (black and red [light gray], dashed) exactly overlay.

Figure 4

Simulated QPI pattern from conductance maps: (a) Fourier transform of the lattice LDOS (BdG) and (b) Fourier transform of the continuum LDOS ($\mathrm{BdG}+\mathrm{W}$) at the same energy $\omega =24\mathrm{meV}$. Impurity potential for the weak scatterer $V_{\mathrm{imp}}=0.3\mathrm{eV}$. The red open symbols indicate the expected positions of the spots from the octet model.

Figure 5

(a) QPI $Z$ map obtained from the Fourier transform of the real conventional space BdG patterns at $\omega =24\mathrm{meV}$, (b) QPI $Z$ map obtained from the Fourier transform of the continuous 3D LDOS $(x,y,z\approx 5\AA ;\omega )$ at $\omega =24\mathrm{meV}$ showing strong similarity to (c) the experimental results reproduced from Ref. [14] and rotated to match the orientation in (a) and (b). For the theoretical calculations a weak impurity scatterer of $V_{\mathrm{imp}}=0.3\mathrm{eV}$ was used. (d) $T$-matrix scattering interference simulation for $\mathrm{\Lambda }(q)$ from Ref. [14]; (e) the same obtained from conventional BdG calculations; (f) $\mathrm{\Lambda }$ map obtained from the Fourier transform of the continuous 3D LDOS showing strong similarity to the experimental results (g) reproduced from Ref. [14]. All $\mathbf{q}$ maps are from $[-2\pi /a\dots 2\pi /a]$.