Towards a quantitative description of tunneling conductance of superconductors: Application to LiFeAs

Since the discovery of iron-based superconductors, a number of theories have been put forward to explain the qualitative origin of pairing, but there have been few attempts to make quantitative, material-specific comparisons to experimental results. The spin-fluctuation theory of electronic pairing, based on first-principles electronic structure calculations, makes predictions for the superconducting gap. Within the same framework, the surface wave functions may also be calculated, allowing, e.g., for detailed comparisons between theoretical results and measured scanning tunneling topographs and spectra. Here we present such a comparison between theory and experiment on the Fe-based superconductor LiFeAs. Results for the homogeneous surface as well as impurity states are presented as a benchmark test of the theory. For the homogeneous system, we argue that the maxima of topographic image intensity may be located at positions above either the As or Li atoms, depending on tip height and the setpoint current of the measurement. We further report the experimental observation of transitions between As- and Li-registered lattices as functions of both tip height and setpoint bias, in agreement with this prediction. Next, we give a detailed comparison between the simulated scanning tunneling microscopy images of transition-metal defects with experiment. Finally, we discuss possible extensions of the current framework to obtain a theory with true predictive power for scanning tunneling microscopy in Fe-based systems.

Figure 1

Isosurface plots of a Fe-$d_{xz}$ Wannier orbital in LiFeAs at two different values (a,c) and corresponding plots of the same Wannier function at two different heights (b,d). The green plane at height $z_0$ in (a,c) is inserted for visual clarity of the isosurfaces. Red and blue indicate the phase of the wave function, see scale. The arrows point to the maxima of the Wannier function that gives rise to the maxima in the conductance maps whenever the weight of the Fe-$d_{xz}$ Wannier function is large. Note that the maxima move from positions above the atomic As positions (open squares) at small heights (e.g., large isovalues) to positions close to the atomic Li positions (open circles) at larger heights (e.g., smaller isovalues of the wave function). Experimental STM topography showing a native defect exhibiting chiral nature (e).

Figure 2

Schematic illustration of the tunneling process indicating how tunneling into As states can lead to an intensity maximum above the Li positions: Interference between electrons in Wannier states of neighboring Fe atoms (phase of wave function symbolized by red/blue color) can give rise to maximal tunneling when the STM tip is located above the Li atom. A tip closer to the surface will shift the maximal tunneling positions towards the As atoms.

Figure 3

(a) Superconducting gap from the self-consistent BdG equation, plotted at the positions of the Fermi surface in momentum space. (b) For the $s_{++}$ gap function we use the same gap structure (thus giving rise to the same DOS), but without the sign change.

Figure 4

(a) Orbital resolved DOS of the homogeneous superconductor. The orbital content changes strongly between the energy scales 10 and $20\text{meV}$ due to the gap structure. Note renormalization of DFT energy scale by $Z$ as discussed in text. (b) Experimental STM spectrum showing the differential conductance of homogeneous LiFeAs. ($V_s=15\mathrm{mV},I=150\mathrm{pA}$,T = 1.6 K).

Figure 5

Continuum LDOS calculated at different heights [left column, (a,b): 340 pm; right column, (c,d): 440 pm] above the surface, energies as indicated on the plots ($\omega /Z=10\text{meV},\omega /Z=20\text{meV}$). While for the small tip height the positions of the maxima change from the surface As positions at high energies to the Li positions at small energies, there is no such change when the tip is at larger distances from the surface. All plots are with the same field of view covering an area of $1\times 1{\text{nm}}^2$ with the atomic positions of the surface atoms as marked in (e). Lines along the crystallographic axis in the calculated maps are artifacts from using a Wannier function that has a finite extension in the plane. These become stronger at larger heights because the Wannier function is less peaked, see Fig. 1.

Figure 6

Topographs of an area of $1\times 1{\text{nm}}^2$ calculated such that the closest point of the tip is fixed to $z_{\mathrm{down}}=440\text{pm}$ (measured from the Fe plane, see Fig. 2) for different bias voltages (a–c) as indicated. Note that for huge positive setpoint bias voltages, the maximal heights are above the As atoms in the lattice (c), whereas for smaller voltages the maximal heights are registered above the Li atoms (a,b). For even smaller setpoint bias voltages, e.g., within the large superconducting gap, the maxima are again found above the As positions, see Fig. 5. Artifacts of the finite range of the Wannier functions used in the calculations are present, mostly visible in (b). See Fig. 5 for the relative lattice position of the surface atoms in the field of view.

Figure 7

(a–f) Topographic experimental STM images taken with different bias voltages with the same current $I=2\mathrm{nA}$. All images show the same defect and have been taken with the same tip configuration. The small red dots label the same lattice positions relative to the impurity; for the theoretical calculations (g–l), these are the positions of the Li atoms. The black lines cross at the center line of the defect state (Fe position), the position of which is fixed and can be used as a reference. A phase shift is observed in the atomic lattice when the bias voltage and tip-sample distance is reduced, compare e.g., panels (a,f) to panels (b–e). A similar shift is found in the theoretical simulations within the same energy range where the cross also marks a Fe position. Note that for the theoretical calculations, the energy has been renormalized by a factor of $Z=1/2$ consistent with all other calculations in this and previous work [16].

Figure 8

Imaging the atomic lattice in STM with different tips; (a) STM topography showing a single Mn impurity with maxima imaged at the position of lithium (Li) atoms ($V_s=-25\mathrm{mV},I=100\mathrm{pA}$). The impurity here serves merely as a point of reference. (b) STM topography of the same Mn impurity obtained with a different apex of the tip, imaging maxima at the positions of arsenic (As) atoms. ($V_s=-50\mathrm{mV},I=300\mathrm{pA}$). Topographs are imaged in the normal state at $B=10\mathrm{T}$ and $T=12\mathrm{K}$. (c) Height profile of the Mn defect along the vertical dashed lines indicated in (a,b) and (d) height profile of the atomic modulation of the homogeneous system extracted along the horizontal lines in (a,b). The atomic resolution is found to be in antiphase between the two topographs.

Figure 9

(a) Lattice LDOS at $\omega /Z=-30\text{meV}$ around a Ni impurity showing a ${\mathrm{C}}_4$ symmetric impurity state and the positions of the surface As and Li atoms in the field of view. (b) Corresponding cLDOS with the same field of view of $1\times 1{\text{nm}}^2$ at $z=340\text{pm}$ reflecting the symmetry to be seen in conductance maps and topography close to a Fe centered impurity.

Figure 10

Lattice LDOS spectra close to a Ni impurity (potential renormalized down by a factor of 2.5) calculated in the lattice representation where $r=(x,y)$ refers to the distance in Fe bond lengths from the impurity. Panel (a) shows the result for the sign-changing gap function calculated from spin-fluctuation pairing theory, as plotted in the inset over the Fermi surface using colors blue=+, red=-, with (c) the associated difference LDOS with spectrum far from the impurity subtracted. Panel (b) is the corresponding LDOS result for an analogous gap structure where the signs have been artificially removed [see Fig. 3], showing no in-gap bound states. The associated spectra are quite particle-hole symmetric, as seen in the LDOS difference spectrum in (d).

Figure 11

Comparison of cLDOS results for a non-sign-changing order parameter (right) and a sign-changing order parameter (left): cLDOS close to a Ni impurity (top row) and the relative continuum LDOS (bottom row).

Figure 12

Continuum LDOS close to a Mn impurity (a), Co impurity (b), and Ni impurity (c) and the corresponding relative spectra (d–f) calculated for the sign-changing order parameter. All impurity potentials reduced by 2.5 relative to ab initio calculations.

Figure 13

Simulated topograph (a) for a Ni impurity at a bias of $30\text{mV}$ calculated using a shifted Wannier function at the impurity position to approximate the distortion close to the impurity (see text) and corresponding cut along the dashed lines in vertical and horizontal direction (b). (c) Experimental STM topograph of a single Ni impurity at $V_s=10\mathrm{mV},I=500\mathrm{pA}$. (d) Line cuts along and normal to the Ni impurity shown in (c).

Figure 14

Orbital character of the Fermi surface shown in $k$ space (set the lattice constants $a=c=1$) plotted in the unfolded (1-Fe) reciprocal unit cell, visualized with the summed-color method where the absolute value of the overlap is mapped to the RGB value of the color.

Figure 15

Converged superconducting gaps $\widehat{\mathrm{\Delta }}{\left(\mathbf{k}\right)}^{nm}$ (units in eV) from the BdG calculation at $T=1\text{meV}$ in orbital space (a) and transformed to band space (b). One $nm$ element in each square represents the 1-Fe Brillouin zone with the $\mathrm{\Gamma }$ point at the center. These gaps are calculated using the interaction parameters $U=0.9\text{eV}$ and $J=U/4$ and are used for all subsequent real-space calculations. The elements marked with a star $*$ are imaginary. In the band space representation, the corresponding Fermi surfaces are marked in black in part of the Brillouin zone, compare Fig. 14.