Multi-Atom Quasiparticle Scattering Interference for Superconductor Energy-Gap Symmetry Determination

Complete theoretical understanding of the most complex superconductors requires a detailed knowledge of the symmetry of the superconducting energy-gap $\Delta_\mathbf{k}^\alpha$, for all momenta $\mathbf{k}$ on the Fermi surface of every band $\alpha$. While there are a variety of techniques for determining $|\Delta_\mathbf{k}^\alpha|$, no general method existed to measure the signed values of $\Delta_\mathbf{k}^\alpha$. Recently, however, a new technique based on phase-resolved visualization of superconducting quasiparticle interference (QPI) patterns centered on a single non-magnetic impurity atom, was introduced. In principle, energy-resolved and phase-resolved Fourier analysis of these images identifies wavevectors connecting all k-space regions where $\Delta_\mathbf{k}^\alpha$ has the same or opposite sign. But use of a single impurity atom, from whose precise location the spatial phase of the scattering interference pattern must be measured is technically difficult. Here we introduce a generalization of this approach for use with multiple impurity atoms, and demonstrate its validity by comparing the $\Delta_\mathbf{k}^\alpha$ it generates to the $\Delta_\mathbf{k}^\alpha$ determined from single-atom scattering in FeSe where $s_{\pm}$ energy-gap symmetry is established. Finally, to exemplify utility, we use the multi-atom technique on LiFeAs and find scattering interference between the hole-like and electron-like pockets as predicted for $\Delta_\mathbf{k}^\alpha$ of opposite sign.

scattering interference between the hole-like and electron-like pockets as predicted for Δ of opposite sign.
The macroscopic quantum condensate of electron pairs in a superconductor is represented by its order-parameter Δ ≡< † − † > where † is the creation operator for an electron with momentum on band . But electron pair formation can occur through a wide variety of different mechanisms and in states with many possible symmetries 1 . Thus, it is the symmetry properties of Δ that are critical for identification of the Cooper pairing mechanism 1 and, moreover, for understanding the macroscopic phenomenology 1 . While macroscopic techniques can reveal energy-gap symmetry for single-band systems 2,3 , no general technique existed to determine the relative signs of Δ and Δ ′ between and ′ for all Fermi surface momenta in an arbitrary superconductor.
In 2015 a conceptually simple and powerful new technique for determining Δ symmetry was introduced 4 , by Hirschfeld, Eremin, Altenfeld and Mazin (HAEM). It is based on interference of weakly scattered quasiparticles at a single, non-magnetic, impurity atom.

Given a superconductor Hamiltonian
where is the normal-state Hamiltonian and the superconducting energy gap, a weak non-magnetic impurity atom is modeled as a weak point-like potential scatterer, with  11 (2) which is a purely real quantity because in the theoretical calculation, the single impurity is exactly located in the center of the FOV. The authors of Ref. 4 realized that the particle-hole symmetry of Eqn. 2 for scattering interference wavevector = − , depends on the relative sign of the energy-gaps Δ and Δ at these two momenta.
Consequently, the experimentally accessible energy-antisymmetrized function where + = + 0 + , so that the functional form of − ( , | |) is very different when the product Δ Δ is positive or negative. An elementary implication of Eqn. 4 is that, when order parameter has opposite signs on the two bands so thatΔ Δ < 0, − ( , ) does not change sign and exhibits pronounced maxima or minima near ≈ Δ , whereas if the order parameter has the same sign so that Δ Δ > 0, − ( , ) exhibits weak maxima or minima near ≈ Δ , with a sign of change in between. More generally, especially with multiple bands and anisotropic gaps, HAEM requires that − ( , ) be predicted in detail for a specific and Δ in Eqn. (1) and then compared with quasiparticle interference imaging 5 in which the STM differential electron tunneling conductance, ( , ) ∝ ( , ) is visualized.
This single-atom phase-resolved HAEM method has been established experimentally 6,7 . For example, in the case of the multiband ± superconductor FeSe, the complete energy and wavevector dependence of − ( , ) was used to determine that the kspace structure including relative sign of Δ and Δ , for all and all on two different bands. But this result required that the impurity atom be highly isolated from other impurities and centered precisely at the origin of coordinates, with respect to which the Re ( , ) of Eqn. 3 is then properly defined. This was critical because, on the scale of a crystal unit-cell, a small error in the coordinate of the origin (at the impurity atom) produces major errors in Re ( , ) and Im ( , ) (SI section I). Moreover, single impurity atom based measurements limit the k-space resolution because the FOV is typically restricted in size, making them unsuitable for superconductors with large impurity-atom densities. This provides the motivation for a variety of new approaches beyond single-atom HAEM. One is to study Bogoliubov bound-states at individual impurity atoms 8,9,10 This 'shift theorem' shows how the correctly phase-resolved Fourier transform of a ( ) oscillation centered on an atom located at = ( , ), can be determined using where ( ) is the Fourier transform using the same arbitrary origin as determines the .
Thus we may define a multi-atom algorithm whose consequences are illustrated in the bottom half of Fig. 1b. The real part Re ( ) now becomes well-defined and the overall magnitude is also strongly enhanced compared to Re ( ). Moreover, the azimuthally integrated Re ( ) plotted in Fig. 1d shows that the sign of Re ( ) changes for = 0 and = as expected. Here it is essential that the impurity atom coordinates are determined accurately so that the phase is well-defined.
We therefore employ a picometer-scale transformation 13 This procedure adds all the individual − ( , ) signals from every impurity atom at Ri inphase, while effectively averaging out the random phase variations due to both locating the origin and the contributions of all other scatterers. We designate this procedure multi-atom HAEM (MAHAEM).
Determination of the magnitude of superconducting energy gaps |Δ | has long been achieved 16,17,18,19,20,21,22,23 using quasiparticle scattering interference (QPI). MAHAEM is the most recent advance of the QPI technique, and to test it we consider FeSe where the single impurity atom − ( , ) for determining Δ was first established experimentally. We measure the differential tunneling conductance ( , ) ≡ / ( , ) in a 30 nm FOV at T=280mK, followed by determination of = ( , ) for 17 impurity sites (SI Section IV), some of which are shown in the FOV in Fig. 2a. We then use Eqn. 9 to calculate − ( , ). A representative layer − ( , = 1.05 ) is shown in Fig. 2c, where the scattering feature at vector 1 is marked with a circle. We then sum over the encircled q-region to get − ( ) for this scattering feature which is shown as black dots in Fig. 2d. Results from our MAHAEM measurements agree very well with the experimental results using a single impurity atom − (black crosses) and the theoretically predicted curve for ± − ℎ (solid, black) in FeSe.
This demonstrates the validity and utility of the multi-atom HAEM technique.
Next we consider LiFeAs, a complex iron-based superconductor that is a focus of contemporary physics interest 24,25,26 , particularly the relative sign of Δ between all five bands. Fig. 3b shows the Fermi surface of LiFeAs calculated using a tight-binding fit 36 it was pointed out that, under these conditions, several s-wave channels are nearly degenerate. These channels include the conventional ± state where the signs on all hole pockets are the same 35,36 , so-called "orbital antiphase state" that occurs when the interaction is diagonal in orbital space 24 , and a distinct sign structure obtained when vertex corrections were included 37 . Ref. 38 considered the question of whether these various proposed phases could be distinguished using HAEM based on Eqn. 3 and concluded that it would be challenging.
Here we examine the relative signs of Δ in LiFeAs by using MAHAEM. Figure 3a shows Of note in Fig. 4(b) is the variety of structures at | | ≪ | ℎ |, which are challenging to understand. The thin outer blue ring is located at a radius in q-space that corresponds well to the expected intraband scattering within pocket h3. Furthermore, much of the q-space within this ring is blue and of rather high intensity for 1meV<|E|<6meV. This false-color, indicating sign-preserving scattering, is consistent with the conventional ± picture within a HAEM scenario, but the high intensity is not. As discussed in SI Section V there   c. d. c. d.  c.

Multi-Atom Quasiparticle Scattering Interference for Superconductor Energy-Gap Symmetry Determination I) Phase Effects of Displacement of Impurity Atom from Origin
For an atomic impurity located at the origin of the FOV, the scattering interference

III) MAHAEM Analysis Procedures
In order to get optimal phase resolution, it is essential to determine the scattering center of the defect in the conductance map as precisely as possible. Thus if the requiredspace resolution allows to resolve atoms, we use the technique as mentioned in SI section II to register each atom in the recorded topograph ( ) map to a perfectly periodic lattice using an affine transformation at a picometer level precision 1,2,3 . Then we apply the same transformation to the recorded conductance maps ( , ) for each energy to register each scattering interference pattern perfectly to the crystal lattice as seen in the topograph.

IV) Robustness of MAHAEM in LiFeAs
The results in the main-text Fig. 4b. are 8-fold symmetrized along vertical, horizontal and diagonal axes for enhanced clarity. However, the symmetry is present in the raw data itself as can be seen in the raw data presented in Fig. S4 for a representative layer = 3.33 meV.
The scattering vector from small hole bands around Γ-point to the electron bands at X point leads to the horn-shaped feature as discussed in the main-text. In Fig. S4,

V) Low |q| MAHAEM in LiFeAs
As shown in the main text Fig. 4b and also in the raw data in Fig. S4, there is a large intensity in − ( ) at small q which additionally has a different overall sign than the MAHAEM signal used to determine the relative sign between the gap on the electron-and hole-like pockets. c) Antiphase hole pocket gaps. Large intensity at small q could be produced by gap structures with sign changes among the hole pockets, or with time reversal symmetry -breaking structures of the s+is type 5 provided the internal phases are not too close to zero. However, one must explain why the sign of  -is opposite that of the e-h processes. This could occur if the impurity potential were nearly diagonal in orbital space, as expected, but had an opposite sign for one orbital channel relative to the others. For example, if the xy pocket h3 had a gap opposite in sign to the inner xz/yz pockets, and the defect potential was + for xz/yz and -for xy, scattering between h3 and the xy states on the h1 and h2 pockets would appear as negative (blue) in  -. However, there is very little xy weight on h1 and h2. An alternative would be that the off-diagonal elements of the impurity potential might be significant, mixing xy and xz/yz states, and of opposite sign; these off-diagonal potentials have been found to be negligible in microscopic calculations, however 6 and effects from changes of hoppings in vicinity of the impurity would need to be considered. Somewhat stronger potentials can enhance this effect, since the impurity T-matrix will generically acquire significant offdiagonal components. c. d.