Visualization of atomic-scale phenomena in superconductors: Application to FeSe

We propose a simple method of calculating inhomogeneous, atomic-scale phenomena in superconductors which makes use of the wave function information traditionally discarded in the construction of tight-binding models used in the Bogoliubov–de Gennes equations. The method uses symmetry-based first-principles Wannier functions to visualize the effects of superconducting pairing on the distribution of electronic states over atoms within a crystal unit cell. Local symmetries lower than the global lattice symmetry can thus be exhibited as well, rendering theoretical comparisons with scanning tunneling spectroscopy data much more useful. As a simple example, we discuss the geometric dimer states observed near defects in superconducting FeSe.

Figure 1

Isosurface plots of Fe-$d$ Wannier orbitals in FeSe at $0.03{\mathrm{bohr}}^{-3/2}$; red and blue indicate phase of the wave function. (a) Top: Top view of unit cell containing two Fe atoms (red) and two Se atoms (yellow), plus four Se exterior to unit cell; bottom: side view of same cell. (b) Top: Top view of $d_{xy}$ Wannier orbital on Fe(I) site; bottom: side view of same orbital. (c) Same as (b) but on Fe(II) site. Images were produced with XCRYSDEN [22].

Figure 2

(a) Density of states in SC state (inset: without impurity), far from impurity (black), at impurity site (black, dashed), on nearest-neighbor site [orange (light gray)], and on next-nearest neighbor site [blue (dark gray)], calculated with a tetrahedron method using $40\times 40$ supercells. (b) Resonant-state real space BdG patterns at $\omega =-2\mathrm{meV}$ and (c) $\omega =2\mathrm{meV}$.

Figure 3

xy cuts through continuous 3D $\mathrm{LDOS}(x,y,z;\omega )$ in ${\left(\mathrm{eV}{\mathrm{bohr}}^3\right)}^{-1}$ at the surface, $z=0.45c$, and different energies (a) $\omega =-2\mathrm{meV}$, (b) $2\mathrm{meV}$, (c) $30\mathrm{meV}$, and heights at (close to) the Fe plane $z=0c$ (d), $z=0.05c$ [(e), (f)] and the same energy $2\mathrm{meV}$, where the $c$-axis lattice constant is $c=10.44\mathrm{bohrs}$. All maps are calculated from Eq. (5) except for (e), which includes only the local $\left(\mathbf{R}={\mathbf{R}}^{\prime },\mu =\nu \right)$ contributions. The schematic side views of the unit cell indicate the $z$ value of the cut (green line) relat-ive to the Fe (red circles) and Se (yellow triangle) positions. The cartoon left of (a) and (b) explains the LDOS patterns in terms of the resonant nearest-neighbor Fe Wannier orbitals [compare Figs. 2 and 2] and their upper Se tails. Thin red lines are directed along Fe-Fe bonds through the central impurity site, and the black border indicates the extent of the $5\times 5$ Fe region.

Figure 4

Topography of impurity state: comparison of topograph with set-point bias $+6\mathrm{mV}$ for (a) BdG calculation of impurity state as in Fig. 2; and (b) BdG-Wannier calculation of same impurity state. In (c) we reproduce the experimental topograph from Ref. [20], rotated such that the directions of the lattice vectors match in all images.