Abstract
Complete theoretical understanding of the most complex superconductors requires a detailed knowledge of the symmetry of the superconducting energygap \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\), for all momenta k on the Fermi surface of every band α. While there are a variety of techniques for determining \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha \), no general method existed to measure the signed values of \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\). Recently, however, a technique based on phaseresolved visualization of superconducting quasiparticle interference (QPI) patterns, centered on a single nonmagnetic impurity atom, was introduced. In principle, energyresolved and phaseresolved Fourier analysis of these images identifies wavevectors connecting all kspace regions where \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\) has the same or opposite sign. But use of a single isolated 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 \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\) it generates to the \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\) determined from singleatom scattering in FeSe where s_{±} energygap symmetry is established. Finally, to exemplify utility, we use the multiatom technique on LiFeAs and find scattering interference between the holelike and electronlike pockets as predicted for \({\mathrm{{\Delta}}}_{\mathbf{k}}^\alpha\) of opposite sign.
Introduction
The macroscopic quantum condensate of electron pairs in a superconductor is represented by its orderparameter \({\Delta}_{\mathbf{k}}^\alpha \propto \langle c_{\mathbf{k}}^{\alpha \dagger }c_{  {\mathbf{k}}}^{\alpha \dagger }\rangle\), where \(c_{\mathbf{k}}^{\alpha \dagger }\) is the creation operator for an electron with momentum k on band α. 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 \({\Delta}_{\mathbf{k}}^\alpha\) that are critical for identification of the Cooper pairing mechanism^{1} and, moreover, for understanding the macroscopic phenomenology^{1}. While macroscopic techniques can reveal energygap symmetry for singleband systems^{2,3}, no general technique existed to determine the relative signs of \({\Delta}_{\mathbf{k}}^\alpha\) and \({\Delta}_{{\mathbf{k}}^\prime }^\beta\) between k_{α} and k_{β} for all Fermi surface (FS) momenta in an arbitrary superconductor.
In 2015, a conceptually simple and powerful technique for determining \({\Delta}_{\mathbf{k}}^\alpha\) symmetry was introduced^{4}, by Hirschfeld, Eremin, Altenfeld, and Mazin (HAEM). It is based on interference of weakly scattered quasiparticles at a single, nonmagnetic, impurity atom. Given a superconductor Hamiltonian
where H_{k} is the normalstate Hamiltonian and Δ_{k} the superconducting energy gap, a nonmagnetic impurity atom is modeled as a weak pointlike potential scatterer, with Hamiltonian \(H_{{\mathrm{imp}}} = V_0c_{\mathbf{r}}^\dagger c_{\mathbf{r}}\) centered at the origin of coordinates r = 0. Effects of scattering are then represented by a Tmatrix derived from the local Green’s function \(G_0\left( E \right) = \mathop {\sum}\nolimits_{\mathbf{k}} {G_{\mathbf{k}}^0} (E)\), where \(G_{\mathbf{k}}^0(E) = \left( {E + i0^ +  {\cal{H}}_{\mathbf{k}}} \right)^{  1}\). When the impurity potential is constant in kspace, the Green’s function becomes \(G_{{\mathbf{k}},{\mathbf{k}}^{\prime} }\left( E \right) = G_{{\mathbf{k}},{\mathbf{k}}^{\prime} }^0\left( E \right) + G_{\mathbf{k}}^0\left( E \right)T\left( E \right)G_{{\mathbf{k}}^{\prime} }^0\left( E \right)\), with the Tmatrix given by \(T\left( E \right) = \left[ {1  V_{{\mathrm{imp}}}G_0\left( E \right)} \right]^{  1}V_{{\mathrm{imp}}}\), where V_{imp} is the impurity matrix. From \(G_{{\mathbf{k}},{\mathbf{k}}^\prime }\left( E \right)\), the perturbations to the local densityofstates δN(r, E) are predicted surrounding the impurity atom, and its Fourier transform can be determined directly from Δ_{k} as
which is a purely real quantity because, in the theoretical calculation, the single impurity is exactly at the origin of coordinates. The authors of ref. 4 realized that the particlehole symmetry of Eq. (2) for scattering interference wavevector \({\mathbf{q}} = {\mathbf{k}}_{\mathrm{f}}^\beta  {\mathbf{k}}_{\mathrm{i}}^\alpha\), depends on the relative sign of the energygaps \({\Delta}_{{\mathbf{k}}_{\mathbf{i}}}^\alpha\) and \({\Delta}_{{\mathbf{k}}_{\mathbf{f}}}^\beta\) at these two momenta. Consequently, the experimentally accessible energyantisymmetrized function ρ^{−}(q, E) of phaseresolved Bogoliubov scattering interference amplitudes
can be used to determine the relative sign of the superconducting energygaps connected by \({\mathbf{q}} = {\mathbf{k}}_{\mathrm{f}}^\beta  {\mathbf{k}}_{\mathrm{i}}^\alpha\). In the simplest case with two isotropic gaps Δ^{α} and Δ^{β} on distinct bands, it was demonstrated that
where \(E_ + = E + i0^ +\), so that the functional form of \(\rho ^  \left( {{\mathbf{q}},E} \right)\) is very different when the product Δ^{α}Δ^{β} is positive or negative. An elementary implication of Eq. (4) is that, when the order parameter has opposite signs on the two bands so that Δ^{α}Δ^{β} < 0, ρ^{−}(q, E) does not change sign and exhibits pronounced maxima or minima near E ≈ Δ^{α,β} whereas if the order parameter has the same sign so that Δ^{α}Δ^{β} > 0, ρ^{−}(q, E) exhibits weak maxima or minima near E ≈ Δ^{α,β} with a sign of change in between. More generally, especially with multiple bands and anisotropic gaps, HAEM requires that ρ^{−}(q, E) be predicted in detail for a specific H_{k} and Δ_{k} in Eq. (1) and then compared with quasiparticle interference imaging^{5} in which the scanning tunneling microscope (STM) differential electron tunneling conductance, \(g\left( {{\mathbf{r}},E} \right) \propto \delta N\left( {{\mathbf{r}},E} \right)\) is visualized.
This singleatom phaseresolved HAEM method has been established experimentally^{6,7}. For example, in the case of the multiband s_{±} superconductor FeSe, the complete energy and wavevector dependence of ρ^{−}(q, E) was used to determine that the kspace structure including relative sign of \({\Delta}_{\mathbf{k}}^\alpha\) and \({\Delta}_{\mathbf{k}}^\beta\), for all k_{α} and all k_{β} 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 ReδN(q, E) of Eq. (3) is then properly defined. This was critical because, an error of on the order of ~1% of a crystal unit cell in the coordinate of the origin (at the impurity atom) produces significant errors in ReδN(q, E) and ImδN(q, E) (Supplementary Note 1 and Fig. S1). Moreover, single impurity atombased measurements limit the kspace resolution because the field of view (FOV) is typically restricted in size, making them unsuitable for superconductors with large impurityatom densities. This provides the motivation for a variety of approaches to \({\Delta}_{\mathbf{k}}^\alpha\) determination beyond singleatom HAEM. One is to study Bogoliubov boundstates at individual impurity atoms^{8,9,10}, although this has proven problematic because the elementary HAEM concept (Eq. (3)) is only valid in the weak scattering range, i.e. well below the scattering strength sufficient to generate Bogoliubov bound states^{11}. Another approach is to use sparse blind deconvolution^{12} to analyze images of scattering interference at multiple atoms, yielding the phaseresolved real space structure of δN(r, E) although not the ρ^{−}(q, E) of Eq. (3). Overall, therefore, widespread application of the HAEM technique (Eq. (3)) as a general tool for \({\Delta}_{\mathbf{k}}^\alpha\) determination remains a challenge.
Here, we introduce a practical technique for determining ρ^{−}(q, E) of Eq. (3) from multiple impurity atoms in a large FOV. To understand this approach, consider the key issue of phase analysis as depicted in Fig. 1, a schematic simulation of Friedel oscillations \(\delta N\left( {\mathbf{r}} \right) = I_0\mathop {\sum}\nolimits_{{\mathbf{R}}_i} {\cos } \left( {2{\mathbf{k}}_{\mathrm{F}} \cdot ({\mathbf{r}}  {\mathbf{R}}_i) + \vartheta } \right)/\left {{\mathbf{r}}  {\mathbf{R}}_i} \right^2\) from multiple atoms at random locations R_{i}. The Fourier transform components of this δN(r) are shown in the top two panels of Fig. 1b. Obviously, the ReδN(q) required for the HAEM technique in Eq. (3), is weak, does not have a clear sign, and is indistinguishable from ImδN(q). Such effects occur because the spatial phases of all the individual Friedel oscillations at R_{i} are being added at random. The consequence is most obvious in the azimuthally integrated ReδN(q) shown in Fig. 1f where the phase information of singleatom Friedel oscillation is completely scrambled and the HAEM technique of Eq. (3) thereby rendered inoperable.
This problem could be mitigated if the Fourier transform of the scattering interference pattern surrounding each R_{i} were evaluated as if it were at the zero of coordinates. In this regard, consider the Fourier transform of a scattering interference surrounding a single impurity atom at R_{i} = (x_{i}, y_{i}),
This “shift theorem” shows how the correctly phaseresolved Fourier transform of a δN_{i}(r) oscillation centered on an atom located at R_{i} = (x_{i}, y_{i}), can be determined using
where δN(q) is the Fourier transform using the same arbitrary origin as determines the R_{i}. Thus we may define a multiatom phasepreserving algorithm for QPI
The consequences of Eq. (7) are illustrated in Fig. 1d, e (Supplementary Note 3 and Supplementary Fig. S4). The real part ReδN_{MA}(q) now becomes welldefined and the overall magnitude is also strongly enhanced compared to ReδN(q). Moreover, the azimuthally integrated ReδN_{MA}(q) plotted in Fig. 1g shows that the sign of ReδN_{MA}(q) changes for \(\vartheta = 0\) and \(\vartheta = \pi\) as expected. Here it is essential that the impurity atom coordinates R_{i} be determined accurately so that the phase is welldefined. We therefore employ a picometerscale transformation^{13,14,15} which renders topographic images T(r) perfectly periodic with the lattice, and then use the same transformation on the simultaneously recorded g(r,E) to register all the scattering interference oscillations precisely to the crystal lattice (Supplementary Note 2).
Equation 7 then allows to correctly define the quantities in Eq. (3) for arbitrarily large numbers of scattering atoms. By using the analog of Eq. (6) for \(g\left( {{\mathbf{r}},E} \right) \propto \delta N({\mathbf{r}},E)\), ρ^{−}(q, E) for each impurity atom is determined from
while from Eq. (7) the sum over these \(\rho _i^  \left( {{\mathbf{q}},E} \right)\) yields
This procedure adds all the individual \(\rho _i^  \left( {{\mathbf{q}},E} \right)\) signals from every impurity atom at R_{i} inphase, while effectively averaging out the random phase variations due to both locating the origin and the contributions of all other scatterers (Supplementary Fig. S5). We designate this procedure multiatom HAEM (MAHAEM).
Results and discussions
Multiatom quasiparticle interference for \({\Delta}_{\mathbf{k}}^\alpha\) determination
Determination of the magnitude of superconducting energy gaps \({\Delta}_{\mathbf{k}}^\alpha \) 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 HAEM technique for determining \({\Delta}_{\mathbf{k}}^\alpha\) was established experimentally^{6}. We measure the differential tunneling conductance \(g({\mathbf{r}},E) \equiv {\mathrm{d}}I/{\mathrm{d}}V({\mathbf{r}},E)\) in a 30 nm FOV at T = 280 mK, followed by determination of R_{i} = (x_{i}, y_{i}) for 17 scattering sites (Supplementary Note 3), some of which are shown in the FOV in Fig. 2a (Supplementary Fig. S2 shows all the sites). These sites are wellknown Featom vacancies identified by their crystal locations, and are nonmagnetic^{6}; their empirical identicality is confirmed by highresolution electronic structure imaging. We then use Eq. (9) to calculate \(\rho _{{\mathrm{MA}}}^  ({\mathbf{q}},E)\). Figure 2b shows the FeSe FS with the holepocket α around Γpoint and electron pockets ε(δ) around X(Y) points. Scattering between α and ε at wavevector p_{1} was studied. A representative layer \(\rho _{{\mathrm{MA}}}^  ({\mathbf{q}},E = 1.05{\,\mathrm{meV}})\) is shown in Fig. 2c, where the scattering feature at vector p_{1} is marked with a circle. We then sum over the encircled qregion to get \(\rho _{{\mathrm{MA}}}^  (E)\) 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 \(\rho _{{\mathrm{Single}}}^{  {\mathrm{Exp}}}\) (black crosses) and the theoretically predicted curve for \(\rho _{{\mathrm{s}}_ \pm }^{  {\mathrm{Th}}}\) (solid, black) in FeSe. This demonstrates the validity and utility of the multiatom HAEM technique.
Next we consider LiFeAs, a complex ironbased superconductor that is a focus of contemporary physics interest^{24,25,26}, particularly the relative sign of \({\Delta}_{\mathbf{k}}^\alpha\) between all five bands. Figure 3b shows the FS of LiFeAs calculated using a tightbinding fit^{27,28} to the experimental data. It consists of three hole pockets h_{1}, h_{2}, and h_{3} around Γpoint and two electron pockets e_{1} and e_{2} around Xpoint. The hole pockets around Γ−point on the FS revealed by spectroscopic imaging STM (SISTM)^{18} and confirmed by angle resolved photoemission spectroscopy (ARPES)^{29,30}, are much smaller as compared to most other Febased superconductors. Local density approximation (LDA) and dynamical mean field theory (DMFT) calculations have attributed the small size of hole pocket to stronger electron–electron correlation in this material. The superconducting energygaps \({\Delta}_{\mathbf{k}}^\alpha\) are substantially anisotropic^{18}. Theoretically, in the case of \({\Delta}_{\mathbf{k}}^\alpha\) with s_{±} symmetry, if both electronlike and holelike pockets are present^{31,32}, the pairing arises from spinfluctuations which are enhanced by nesting between the electronlike and holelike pockets. But the presence of three hole pockets, combined with relatively weak spin fluctuations^{33}, allow for several possible competing ground states in the presence of repulsive interactions. In ref. ^{34}, it was pointed out that, under these conditions, several swave channels are nearly degenerate. These channels include the s_{±} state where the signs on all hole pockets are the same^{35,36} and opposite to the signs on the electron bands, socalled “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^{36}. Reference 37 considered the question of whether these various proposed phases could be distinguished using HAEM based on Eq. (3) and concluded that it would be challenging.
Here we examine the relative signs of \({\Delta}_{\mathbf{k}}^\alpha\) in LiFeAs by using MAHAEM. Figure 3a shows the typical cleaved surface of LiFeAs. The scattering sites used in our analysis are Featom vacancies which are nonmagnetic (Supplementary Fig. S3). The theoretical simulations for LiFeAs were performed from the experimentally fitted tight binding model^{27} and anisotropic gap magnitude structure^{18,30}. At wavevectors corresponding to electronhole scattering in qspace, a “hornshaped” feature in g(q, E) appears within which we focus on an exemplary scattering vector q_{eh} indicated by a dashed arrow in Fig. 3b. Figure 3d then shows the theoretical, singleatom ρ^{−Th}(q, E) integrated for the q in the brown oval in Fig. 3c for s_{±} and s_{++} gaps, where sign of the gap was imposed by hand. The sign of \(\rho _{{\mathrm{s}} \pm }^{  {\mathrm{Th}}}\) does not change for the energy values within the superconducting gap and its amplitude peaks at the energy \(E \approx {\Delta}^{{\mathrm{e}}_1}{\Delta}^{{\mathrm{h}}_1}\), both characteristics of a sign changing gap^{37}; contrariwise \({\uprho}_{{\mathrm{s}}_{ + + }}^{  {\mathrm{Th}}}\) changes sign indicative of same sign energy gaps throughout.
For comparison, differential conductance g(r, E) imaging of LiFeAs is performed at T = 1.2 K. The typical g(E) spectrum consists of two gaps corresponding to Δ_{1} = 5.3 meV and Δ_{2} = 2.6 meV. The measured g(q, E) are shown in Fig. 4a and the feature at q_{eh} expected from the theoretical model in Fig. 3c is indicated by a circle. We evaluate \(\rho _{{\mathrm{MA}}}^  ({\mathbf{q}},E)\) from Eq. (9) for N = 100 atomic scale Featom vacancy sites (Supplementary Note 4). The resulting image \(\rho _{{\mathrm{MA}}}^  ({\mathbf{q}},E)\) at a representative subgap energy E = 3.3 meV is shown in Fig. 4b.
Of note in Fig. 4b is the variety of structures at \(\left {\mathbf{q}} \right \ll {\mathbf{q}}_{{\mathrm{eh}}}\), which are challenging to understand. The thin outer blue ring (indicated by dashed light blue curve as guidetoeye) is located at a radius in qspace that corresponds well to the expected intraband scattering within pocket h_{3}. Furthermore, much of the qspace within this ring is blue and of rather high intensity for 1 meV < E < 6 meV (Supplementary Fig. S6a shows dashed contours for various possible interholeband scatterings overlaid on the unprocessed ρ^{−}(q, E)). The blue color, indicating signpreserving scattering, is consistent with the conventional s_{±} picture within a HAEM scenario, but the high intensity is not. As discussed in Supplementary Note 5 there are several possible explanations of these low q phenomena, including strong scattering, quasiparticle bound states, and antiphase holepocket gaps.
Nevertheless, when the high q scattering between holelike and electronlike pockets (Fig. 3b, c) is integrated within the qspace region shown by a brown circle on the \(\rho _{{\mathrm{MA}}}^  ({\mathbf{q}},E)\) of Fig. 4a, it yields \(\rho _{{\mathrm{MA}}}^  (E)\) as plotted in Fig. 4c. The theoretically predicted \(\rho _{\,}^  (E)\) curves are overlaid for comparison. It is clear that the experimental \(\rho _{{\mathrm{MA}}}^  (E)\) is consistent with the \(\rho _ \pm ^{  {\mathrm{Th}}}(E)\) theory because it does not change sign and exhibits a peak at \(E \approx 3.7\,{\mathrm{meV}} \approx \sqrt {{\Delta}_1{\Delta}_2}\). In this way, the MAHAEM technique efficiently demonstrates that \({\Delta}_{\mathbf{k}}^\alpha\) changes sign between electronlike and holelike bands of LiFeAs.
Conclusions
We report development and demonstration of an improved approach for signed \({\Delta}_{\mathbf{k}}^\alpha\) determination (Eq. (9)), but now for use with multiple impurity atoms or scattering centers. This MAHAEM technique for measuring ρ^{−}(q, E) is based on a combination of the Fourier shift theorem and highprecision registry of scatterer locations. It extends the original HAEM approach^{4} to more disordered superconductors (Figs. 2a, 3a), enables its application to far larger fields of view thereby enhancing qspace resolution (Fig. 4b), and greatly increases signaltonoise ratios (Figs. 1d, 4b) by suppressing phase randomization in multiatom scattering interference. Overall, MAHAEM now represents a powerful and general technique for \({\Delta}_{\mathbf{k}}^\alpha\) determination in complex superconductors.
Methods
Sample growth and preparation
FeSe samples with T_{c} ≈ 8.7 K were prepared using KCl_{3}/AlCl_{3} chemicalvapor transport and LiFeAs samples with T_{c} ≈ 15 K were grown using LiAs flux method. The highly reactive LiFeAs samples are prepared in a dry nitrogen atmosphere in a glove box.
SISTM measurements and analysis
All samples are cleaved in situ in our ultrahigh cryogenic vacuum STM at low temperature. The g(r, E) data were acquired with a ^{3}Herefrigeratorequipped STM. The picometer level atomic registration was performed before applying the HAEM technique as described in full detail in the Supplementary Note 2. Full details of the multiatom HAEM analysis are presented in detail in Supplementary Note 3. Theoretical predictions for ρ^{−}(E) curves were performed using the Tmatrix formalism with energy gap on each band and normal state tight binding parameters fitted to experiments.
Data availability
The datasets generated and/or analyzed during this study are available to qualified requestors from the corresponding author.
Code availability
The simulation code for Fig. 1 is provided as Supplemental material. All the other codes used during the current study are available to qualified requestors from the corresponding author.
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Acknowledgements
Work done by P.C.C. and A.E.B. was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering and was performed at the Ames Laboratory. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DEAC0207CH11358. R.S. acknowledges support from Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR1719875). The authors are thankful to M.A. Müller for the discussion of the QPI results in LiFeAs. P.J.H. and M.A.S. acknowledge support from NSFDMR1849751; H.E. acknowledges GrantinAid for Scientific Research on Innovative Areas “Quantum Liquid Crystals” (KAKENHI Grant No. JP19H05823) from JSPS of Japan. J.C.S.D. acknowledges support from the Moore Foundation’s EPiQS Initiative through Grant GBMF9457, from the Royal Society through Award R64897, from Science Foundation Ireland under Award SFI 17/RP/5445, and from the European Research Council (ERC) under Award DLV788932.
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R.S., A.Kr., and M.A.S. contributed to this project equally. P.O.S., R.S., P.J.H., and J.C.S.D. designed the project. P.O.S. and M.A.S. developed the phaseresolved multiatom averaging method; M.P.A., A.Ko., and P.O.S. carried out the experiments; R.S. and P.O.S. carried out the data analysis; A.Kr., M.A.S., J.B., P.J.H., and I.E. carried out the theoretical analysis. P.C.C. and A.E.B. synthesized singlecrystalline FeSe samples; H.E. synthesized singlecrystalline LiFeAs samples. J.C.S.D. and P.J.H. supervised the investigation and wrote the paper with key contributions from P.O.S., R.S., M.A.S., and A.Kr. The manuscript reflects the contributions of all authors.
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Sharma, R., Kreisel, A., Sulangi, M.A. et al. Multiatom quasiparticle scattering interference for superconductor energygap symmetry determination. npj Quantum Mater. 6, 7 (2021). https://doi.org/10.1038/s41535020003034
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DOI: https://doi.org/10.1038/s41535020003034
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