Extracting phase information about the superconducting order parameter from defect bound states
Shun Chi, W. N. Hardy, Ruixing Liang, P. Dosanjh, Peter Wahl, S. A. Burke, D. A. Bonn
EExtracting phase information about the superconducting orderparameter from defect bound states
Shun Chi,
1, 2
W. N. Hardy,
1, 2
Ruixing Liang,
1, 2
P. Dosanjh,
1, 2
Peter Wahl,
3, 4
S. A. Burke,
1, 2, 5 and D. A. Bonn
1, 21
Department of Physics and Astronomy,University of British Columbia, Vancouver BC, Canada V6T 1Z1 Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver BC, Canada V6T 1Z4 SUPA, School of Physics and Astronomy,University of St. Andrews, North Haugh,St. Andrews, Fife, KY16 9SS, United Kingdom Max-Planck-Institut f¨ur Festk¨orperforschung,Heisenbergstr. 1, D-70569 Stuttgart, Germany Department of Chemistry, University of BritishColumbia, Vancouver BC, Canada V6T 1Z1 (Dated: October 26, 2017)
Abstract
Impurity bound states and quasi-particle scattering from these can serve as sensitive probes foridentifying the pairing state of a superconducting condensate. We introduce and discuss defectbound state quasi-particle interference (DBS-QPI) imaging as a tool to extract information aboutthe symmetry of the order parameter from spatial maps of the density of states around magneticand non-magnetic impurities. We show that the phase information contained in the scatteringpatterns around impurities can provide valuable information beyond what is obtained throughconventional QPI imaging. Keeping track of phase, rather than just magnitudes, in the Fouriertransforms is achieved through phase-referenced Fourier transforms that preserve both real andimaginary parts of the QPI images. We further compare DBS-QPI to other approaches which havebeen proposed to use either QPI or defect scattering to distinguish different symmetries of theorder parameter. a r X i v : . [ c ond - m a t . s up r- c on ] O c t . INTRODUCTION The symmetry of the order parameter in a superconductor is key information required tocharacterize and understand superconductivity in a material. In conventional superconduc-tors, where the pairing interaction is due to electron-phonon coupling, the pairing symmetryand hence the symmetry of the superconducting order parameter is s -wave. For unconven-tional superconductors, the pairing interaction is believed to be governed by electron-electroninteractions, for example, through spin-fluctuation mediated pairing. In this case the strongCoulomb repulsion between the electrons means that non- s -wave pairing is favoured, leadingto sign-changing order parameters. While for some unconventional superconductors, suchas the cuprates or some heavy fermion materials, there is well-established experimental ev-idence for a specific symmetry of the order parameter, in many cases this is not a settledissue. For the iron-based superconductors, there are only a few compounds where there isconsensus that the pairing is of the s ± type, whereas for others this remains an open ques-tion. One reason for uncertainty about the superconducting order parameter is that whilemany experimental probes are sensitive to the magnitude of the order parameter, there areonly very few experimental techniques which can probe the phase and hence completelyconstrain identification of the symmetry of the order parameter.Quasi-particle interference (QPI) imaging, enabled through spectroscopic mapping in ascanning tunneling microscope (STM), has in recent years been established as a powerfultool to characterize electronic states in superconductors as well as a wide range of othermaterials . Its ability to image electron scattering both in the occupied and unoccupiedstates with an energy resolution limited only by the temperature of the experiment (fornormal metal tips) provides sufficient resolution to map out the structure of the supercon-ducting order parameter . While the majority of works concentrated on determining themagnitude and k -space structure of the superconducting gap , QPI imaging carriedout with and without magnetic field has been shown to provide phase sensitive informationabout the superconducting order parameter . The magnetic field produces vortices thatact as additional scatterers contributing to the QPI, which increase the signal of certainscattering wave vectors. For the cuprate superconductors, relating these changes in thescattering intensity to the symmetry of the superconducting order parameter has workedquite well , however for the iron chalcogenides, the interpretation of field-dependent QPI2xperiments has been disputed . More recently, it has been argued that the interpre-tation of magnetic field dependent QPI data is not straightforward , in particular in caseswhere the vortex cores are more spatially extended objects, as happens in lower temperaturesuperconductors. With a question mark on the general applicability of field-dependent QPIto study the symmetry of the superconducting condensate, new methods to determine thesymmetry of the order parameter based on characterizing the scattering phase at individualdefects have been proposed .While QPI has typically been analyzed in a way that discards the phase, the scatteringphase does encode important information about both the scatterer and the superconductingcondensate. For a superconductor with normal s ++ symmetry, the phase shift between thescattering pattern in the occupied and unoccupied states can be used to extract informationabout the scattering strength of a magnetic defect . However, for a superconductor witha sign-changing order parameter, there can be an additional contribution to the phase if aquasiparticle scatters between states with a different sign. Here we present a detailed accountof defect-bound state QPI (DBS-QPI), a new probe for the symmetry of the superconductingorder parameter. DBS-QPI uses spatially extended impurity bound states which reside insidethe superconducting gap. We demonstrate its application to LiFeAs , and discuss here indetail its robustness for different models and its relation to other methods based on usingdifferential conductance maps to determine the phase of the order parameter. To this end,we will use simulated scattering patterns for magnetic and non-magnetic impurities in s ++ and s ± superconductors, comparing between different models and data.We will first introduce the theoretical framework to study the impact of defect scatteringon the density of states, as well as the physical observables we will use to study the scatteringphase (section II). This includes a discussion of the phase-referenced QPI which we haveintroduced as a means to analyze the DBS-QPI. We will then use these to simulate theQPI patterns from defect bound states, in particular considering the sign (phase) of theQPI amplitude, and demonstrate that consistent results are obtained for different modelsfor the band structure (section III and Appendix A). In section IV, we compare the ratiomap QPI, phase referenced DBS-QPI, and the HAEM method to analyze scattering patternsnear defect bound states, discussing the strengths and weaknesses of each of these methods.3 I. SCATTERING THEORY AND PHASE IN QPIA. T -matrix approximation To study scattering by a defect, we use the following Hamiltonian: H = H + H imp , (1)where H is the bare Hamiltonian including the superconducting BCS term, H = (cid:15) ( k ) ∆( k )∆ † ( k ) − (cid:15) T ( − k ) , (2)with (cid:15) ( k ) representing the normal state band structure, and ∆( k ) the superconducting orderparameter. The impurity Hamiltonian for point-like defects, H imp , is given by H imp = V (cid:88) µσ c † µσ c µσ . (3)It describes the scattering for orbital µ and spin σ at site r = (0, 0) with potential V . Onlyintra-orbital scattering is considered in the calculation because it leads to the dominantcomponents in the scattering matrix . An equal strength of the scattering potential isassumed for all orbitals. In momentum space the potential is constant, given by V ( k , k (cid:48) ) ≡ V = 1 N V I ∓ I , (4)where N is the system size, and the choice of the sign, - or +, is for nonmagnetic andmagnetic defects, respectively.Using the T -matrix approximation , the Green’s function in the presence of a defect isgiven by G ( k , k (cid:48) , ω ) = G ( k , ω ) δ k , k (cid:48) + G ( k , ω ) T k , k (cid:48) ( ω ) G ( k (cid:48) , ω ) , (5)where G = [( ω + i η ) I − H ] − and the T -matrix is given by T k , k (cid:48) ( ω ) ≡ T ( ω ) = [ I − V g ( ω )] − V, (6)with g ( ω ) = (cid:80) k G ( k , ω ). The total density of states (DOS) ˜ ρ ( k , ω ) is obtained from the4maginary part of the Green’s function˜ ρ ( k , ω ) = − π Im G ( k , k , ω )= − π Im G ( k , ω ) − π Im G ( k , ω ) T ( ω ) G ( k , ω ) ≡ ˜ ρ ( k , ω ) + δ ˜ ρ ( k , ω ) , (7)where ˜ ρ ( k , ω ) is the bare DOS resulting from H , and δ ˜ ρ ( k , ω ) is the perturbation of theDOS due to the presence of a defect.QPI is calculated from the Fourier transform of the local density of states (LDOS) in realspace. Given a defect at site r = (0 ,
0) the Green’s function in real space can be obtainedfrom the T -matrix G ( r , r (cid:48) , ω ) = G ( r , r (cid:48) , ω ) + N G ( r , , ω ) T ( ω ) G ( , r (cid:48) , ω ) , (8)so that the LDOS is given by ρ ( r , ω ) = − π Im G ( r , r , ω )= − π Im G ( r , r , ω ) − π Im N G ( r , , ω ) T ( ω ) G ( , r , ω )= ρ ( r , ω ) + δρ ( r , ω ) . (9)Finally, the QPI map is given by δ ˜ ρ ( q , ω ) = 1 N (cid:88) r e − i q · r δρ ( r , ω ) . (10)In experiments, one measures the tunneling conductance map, g ( r , ω ), which, if matrixelement effects can be neglected, is proportional to ρ ( r , ω ). B. The phase in Fourier transformed QPI
The Fourier transform of g ( r , ω ) gives a map of complex values, ˜ g ( q , ω ), with real andimaginary parts. Typically, only the modulus of ˜ g ( q , ω ) is analyzed and the phase is ignoredbecause the main interest is in identifying dominant scattering vectors q , i.e. where | ˜ g ( q , ω ) | has maxima. However, for the present purpose, the phase contains important informationto discriminate the in-phase and antiphase signals between two energies. For a particularscattering vector q at two energies ω and ω , in-phase refers to ˜ g ( q , ω ) ∝ ˜ g ( q , ω ) e nπ i =5 g ( q , ω ), and anti-phase refers to ˜ g ( q , ω ) ∝ ˜ g ( q , ω ) e (2 n +1) π i = − ˜ g ( q , ω ). In this study, thephase information ˜ g ( q , ω ) is preserved, allowing for the study of effects of different orderparameters and defects of different nature.Defect apparent shape and size can affect the phase of the Fourier transform. First, thesymmetry determines the signal weighting between the real and imagine parts. It is oftenassumed that defects have point symmetry, namely g ( r , ω ) = g ( − r , ω ) for a defect at theorigin. With point symmetry of the scattering pattern, it can be shown that ˜ g ( q , ω ) is real,because for the complex conjugate ˜ g † ( q , ω ), we have˜ g † ( q , ω ) = 1 N (cid:88) r e i q · r g ( r , ω )= 1 N (cid:88) r e − i q · r g ( − r , ω )= 1 N (cid:88) r e − i q · r g ( r , ω )= ˜ g ( q , ω ) . (11)Therefore, the QPI signals are all in the real part of ˜ g ( q , ω ). For defects without pointsymmetry, QPI signals are shared between the real part Re[˜ g ( q , ω )] and the imaginary partImag[˜ g ( q , ω )]. Second, the spatial shape of a defect can also affect the phase. Real spaceLDOS oscillations are shifted out from the defect center by the size of the defect, whichcontributes an additional phase shift in the Fourier transform.In theoretical calculations, the δ scattering potential is typically assumed to have pointsymmetry and hence the simulated QPI signals only exist in Re[˜ g ( q , ω )]. It is also anideal potential with zero size in r -space. In experimental measurements, real defects havefinite sizes and various shapes, which complicates the direct comparison between theory andexperiments. Below, we discuss a new approach that can be taken to deal with phase in theFourier transform of the tunneling conductance. C. Phase-referenced QPI
In addition to complications in the phase arising from the symmetry, shape and size ofa defect, the Fourier transform also includes an overall phase factor related to the defectpositions in an image. Unless one is dealing with an ideal point scatterer situated at thecentre of an image, all of these factors come into play in the complex Fourier transform6f the tunneling conductance. Phase-referenced QPI has been introduced as a way to dealwith this by zeroing the phase at positive energies and then applying this adjustment atnegative energies. . Since the phase contributions coming from the nature and placementof the defects will generally contribute in the same way at positive and negative energies, thisreferencing of the phase leaves the contrast associated with a sign-changing order parameterin the negative energy images. This then makes it much more straightforward to compare therelative phase between QPI at positive and negative energies ± E . The Fourier transformof the tunneling conductance can be written as | ˜ g ( q , E ) | × e iθ q ,E , where | ˜ g ( q , E ) | is theintensity and θ q ,E is the phase at wave vector q and energy E . A phase-referenced Fouriertransform (PRFT) is obtained by taking the Fourier transform of g ( r , E ) at positive energy E , obtaining the phase factor e iθ q ,E , and then using that as a reference for the Fouriertransform at negative energy − E . The PRFT of the tunneling conductance ˜ g c ( q , ± E ) for E > g c ( q , E ) = Re( | ˜ g ( q , E ) | × e iθ q ,E ) × e − iθ q ,E = | ˜ g ( q , E ) | (12)˜ g c ( q , − E ) = Re ( | ˜ g ( q , − E ) | × e iθ q , − E ) × e − iθ q ,E = | ˜ g ( q , − E ) | × Re( e i ( θ q , − E − θ q ,E ) ) . (13)The phase factor Re( e i ( θ q , − E − θ q ,E ) ) of the PRFT is +1 for in-phase oscillations, and − III. SIMULATIONS OF DBS-QPI
In this section, we present and discuss simulated DBS-QPI maps to demonstrate howinformation about the scatterer and the superconducting order parameter can be extracted.We use the five-orbital tight-binding model for LiFeAs from Ref. 23 for simulating QPI. Thecorresponding Fermi surface is shown in Fig. 1b. For the superconducting gap amplitude weuse ∆( k ) = ∆ cos( k x ) cos( k y ) for the s ± order parameter, and | ∆( k ) | = | ∆ cos( k x ) cos( k y ) | for the s ++ order parameter, respectively. Here we set ∆ = 0 .
016 eV which is consistentwith both the gap amplitude obtained for spin-fluctuation mediated pairing , and with theexperimental gap amplitudes using a band renormalization factor of 2-4 . Both non-7agnetic and magnetic defects can generate in-gap bound states for an s ± order parameter.However, only a magnetic defect can give rise to in-gap bound states in a superconductorwith an s ++ order parameter. Here we have calculated DBS-QPI for all three scenarios thatgenerate in-gap bound states. A. Identification of QPI vectors ( π , π ) ( π , ) a ( π , π ) ( π , ) d ( π , π ) ( π , ) b ( π , π ) ( π , ) e (0, 0) ( π , ) c ( π , π ) ( π , ) f h e e h h e e e e q h -e q e - e q h - h q h -e q e - e q h - h q h -e q h - h l o w h i g h Figure 1 . Identification of QPI vectors in LiFeAs. ( a )-( c ) The isolated Fermi surfaces for[ h , e ], [ h , e ], and [ h , e ], respectively. In ( c ), the Brillouin zone is centered at k = ( π, π ). ( d )-( e )The autocorrelation is given for each isolated Fermi surface of the upper panel. For the case of five bands crossing E F , considerable care is required to identify intra-bandand inter-band QPI features among all the available states. In the five-orbital model, wehave separated the bands at E F into three sets, with one hole band per set, and plottedtheir autocorrelations in Figure 1. In this way, the origin of the intra-band and inter-bandQPI features can be identified with confidence. In particular, from Figure 1d-f, we are ableto unambiguously separate the QPI features for the inter-band scattering between h − e , h − e and h − e . Here e and e are identical except for being in orthogonal directions,hence the QPI features are identical apart from a rotation by π/
2. Therefore, the labels for8lectron bands have been omitted.
B. Comparison between QPI with and without superconductivity at | E | > | ∆ | a b Figure 2 . QPI: superconducting state vs. normal state at E = 1 . . ( a ) QPI in thesuperconducting state. ( b ) QPI in the normal state. When | E | > | ∆ | , the superconducting coherence factors u ( k , E ) and v ( k , E ) approachtheir normal state values. Therefore, the QPI features in the superconducting state and thenormal-state should be almost identical to each other. Figure 2 shows the QPI intensitymaps at E = 1 . in both the superconducting and the normal states. The two QPI mapsagree very well on a quantitative level. Thus, QPI in the superconducting state at | E | > | ∆ | is a good representation of QPI in the normal state. C. DOS in momentum space
From Equation 7, we can calculate the bare DOS ρ ( k , ω ) and the change in DOS δρ ( k , ω )due to defect scattering. As shown in Figure 3(a), the superconducting gaps open at E F ,with the large gaps in the bands of h , h , and e along the (0 , π )-( π, π ) direction and the smallgaps in the bands of h and e along the (0 , π )-(0 ,
0) direction, consistent with experimentalobservations .Figure 3(b)-(d) shows δρ ( k , ω ) under three conditions that allow in-gap bound states.Scattering potentials were chosen to generate bound states with the bound state energy E B1 close to the gap edge of the small gap, to imitate the measured bound state on an Fe-D defect . The scattering potentials are − . s ± with a nonmagnetic defect, − . s ± with a magnetic defect, and − .
35 eV for s ++ with a nonmagnetic defect. The potential9 E ( m e V ) E ( m e V ) - + E ( m e V ) π , π ) (0, π ) E ( m e V ) s ++ magnetic ( k x , k y ) ( Å -1 ) s ± nonmagnetic s ± magnetic abcd ( π , π )(0,0) h h h e -E B1 -E B2 -E B1 -E B2 -E B1 -E B2 Figure 3 . DOS in k-space ( a ) Superconducting bare DOS, ρ ( k , ω ), along with overlay ofband dispersion (dashed line). ( b )-( d ) δρ ( k , ω ) for s ± with nonmagnetic and magnetic defects and s ++ with a magnetic defect, respectively. strength for s ± with a nonmagnetic defect is consistent with the theoretically estimated valuefor native defects . δρ ( k , ω ) has both positive and negative values with E > ∆ i , stemmingfrom the slight spectral weight shift attributed to the defect potentials. However, δρ ( k , ω )is orders of magnitude smaller than ρ ( k , ω ) and hence the total DOS, ρ ( k , ω ), is positivefor all states. For all three parameters in the calculation, bound states at large energy E B1 are associated with the large gaps and bound states at small energy E B2 are associated withthe small gaps. In addition, all bound states follow the band dispersion and are confinedin small momentum regions. These bound states consist of Bogoliubov quasiparticles, theexcitations of Cooper pairs, and QPI can be measured from these Bogoliubov quasiparticlesat the bound state energies. 10 E B1 E B1 s ± , nonmagnetic s ± , magnetic s ++ , magnetic q h -e q h -h q h -h +- ab cd ef Figure 4 . DBS-QPI at ± E B1 . Re[˜ g ( q , ± E B1 )] are shown for s ± with a nonmagnetic potentialand a magnetic potential, and for s ++ with a magnetic potential only. As opposed to | ˜ g ( q , ± E B1 ) | ,Re[˜ g ( q , ± E B1 )] reveals the phase of QPI features. D. DBS-QPI without phase referencing
While the phase information in experimental data always suffers from the effects of globalphase factors and details of the scatterers’ size and symmetry, simulations using a pointscatterer can yield meaningful phase information for DBS-QPI without any need for phasereferencing. To highlight the effects that can appear in DBS-QPI due to a sign-changingsuperconducting order parameter, we discuss here the DBS-QPI without phase reference.Calculations employing the full phase-referenced DBS-QPI are shown and discussed in detailin Ref. 22 and are summarized below in section IV.Figure 4 shows the simulated DBS-QPI at E B1 for all three combinations of order pa-rameters and types of defects. ˜ g ( q , ± E B1 ), which is proportional to ˜ δρ ( q , ω ), was calculatedaccording to Eq. 10. Because an isotropic point-like potential at the origin is used in thecalculation, all QPI signals are in the real part of ˜ g ( q , ω ) and the imaginary part is zero.Here, we show Re[˜ g ( q , ± E B1 )] instead of the absolute values | ˜ g ( q , ± E B1 ) | in order to retainthe phase information of the QPI oscillations after Fourier transformation. Comparing toFigure 1, DBS-QPI at E B1 mostly involves scattering within and between h , and e bands11s expected for the in-gap bound state of the large gaps. For all three cases, QPI signalsare broadened at the bound states and the relative strength of the inter-band q h , − e sig-nal becomes enhanced (compare to Figure 2). Thus, simply comparing the absolute value, | ˜ g ( q , ± E B1 ) | , is not adequate to distinguish between the order parameters s ± and s ++ . How-ever, there are clear qualitative differences in the phase, i.e. in the regions of positive signal(blue) vs negative signal (red). For the phase-referenced DBS-QPI at E B1 , there is a signinversion between ± E B1 for the majority of the q h , − e signal in the cases of s ± pairing withboth nonmagnetic and magnetic scattering potentials. In contrast, the sign of q h , − e scat-tering is mostly preserved between the positive and negative bound state energies. Thereare subtle quantitative distinctions between DBS-QPI of nonmagnetic and magnetic de-fects with the s ± order parameter. However, the experimental data does not have adequateresolution to quantify these differences from the bound-state QPI measured at E B1 . - E B2 E B2 s ± , nonmagnetic s ± , magnetic s ++ , magnetic q h -e q h -h - E B2 E B2 s ± , nonmagnetic s ± , magnetic s ++ , magnetic q h -e q h -h + -+ - ab cd ef Figure 5 . DBS-QPI at ± E B2 . DBS-QPI maps Re[˜ g ( q , ± E B2 )] are shown for s ± with non-magnetic and magnetic defects, and s ++ with a magnetic defect. The small q h − e arc right near( π,
0) is inward because the calculation is performed in the 1st Brillouin zone. It appears outwardin the autocorrelation (see Figure 1f) and experimental data (compare Figs. 4b and 4f of Ref. 22and Figure 7a).
DBS-QPI at E B2 are shown in Figure 5. Only QPI features within and between h and e bands appear for this set of bound states, consistent with the bound states being attributed12o the bands with small gaps. In the Re[˜ g ( q , ± E B2 )] images, the relative signal for inter-band scattering q h − e is strong for s ± with a nonmagnetic defect, while the relative signalfor intra-band scattering at q h − h is strong for s ± with a magnetic defect, and s ++ witha magnetic defect. Moreover, at the wave vector q h − e , the signal changes sign between+ E B2 and − E B2 in the case of s ± with a nonmagnetic defect. However, it only partiallychanges sign and mostly keeps the same phase between ± E B2 in the other two cases. Thistrend is opposite at the vector q h − h . By looking at the ring corresponding to q h − h andits extension outward, it has negative signal (red) for both polarities at E B2 for s ± with anonmagnetic defect, while it has the opposite sign between ± E B2 for the other two cases.Thus, by combining both sets of bound states, one is able to distinguish not only the orderparameter ( s ± ) but also the nature of the defect (non-magnetic). These effects in DBS-QPIhave also been tested using a two orbital model (Appendix A), which gives qualitativelyconsistent results, so the phase variation appears to be independent of the details of theband-structure model. IV. THREE APPROACHES TO PHASE INFORMATION IN QPI DATA
We apply three different approaches to extracting phase information from real QPI takenon LiFeAs (details of the experiment given in Appendix B). The data is the same as thatused in Ref. 22, where the phase-referenced approach was shown in detail. Here we contrastthat approach with two other techniques: the ratio-map QPI introduced by Hanaguri etal. , and the method developed by Hirschfeld, Altenfeld, Eremin and Mazin . A. Ratio-map DBS-QPI: Comparison between experimental data and calculations
One method to analyze QPI data (and to some degree obtain information about thephase of QPI) is to use ratio-map QPI. This was initially introduced to reduce the in-phasesystematic errors in conductance maps due to the setpoint effect. The ratio map in realspace is defined as: Z ( r , ω ) ≡ g ( r , ω ) g ( r , − ω ) = ρ ( r , ω ) ρ ( r , − ω ) . (14)The ratio-map QPI, | ˜ Z ( q , ω ) | , is obtained by taking the absolute value of Fourier transfor-mation of Z ( r , ω ). In the ratio-map QPI, the in-phase scattering components at q occuring13t + ω and − ω are suppressed while antiphase components are enhanced. Therefore, theratio-map QPI is a good way to examine the in-phase and anti-phase signals between twoenergies. Care is needed in interpreting it, since the ratio map can also provide misleadinginformation about the phase change of the QPI signal. For example, if a modulation inthe local density of states is in-phase but significantly stronger at ω than at − ω , the ratiomap barely cancels the in-phase part and enhances the anti-phase part. However, with thiscaveat in mind, ratio-map QPI is a complementary technique that can be used to enhancescattering signals which are out-of-phase as identified by phase-referenced QPI. Similar tothe phase-referenced QPI method, it also is only applicable if the change of scattering vec-tors due to the normal state band dispersion is negligible, which is indeed the case for theenergy range of interest in LiFeAs. a bc d E B - E B ratio maplow high q h -e s ++ magnetic s ± nonmagnetic magnetic e s ± lowhigh Figure 6 . Ratio-map DBS-QPI associated with the large gaps. ( a ) The QPI os-cillations near an Fe-D defect at the bound state energies and the ratio map Z ( r , E B1 ) = g ( r , + E B1 ) /g ( r , − E B1 ). ( b ) The ratio-map DBS-QPI | ˜ Z ( q , E B1 ) | obtained by Fourier transformingthe ratio map Z ( r , E B1 ). The q h , − e QPI features (the ovals centered at (0 , π )) is enhanced andthe rest of QPI features are suppressed. ( c )-( e ) The simulated ratio DBS-QPI with the settings of s ++ and a magnetic potential, s ± and a nonmagnetic potential, and s ± and a magnetic potential. Figure 6a shows the r -space LDOS oscillations near an Fe-D defect at ± E B1 and theirratio map, Z ( r , E B1 ). Oscillations can be clearly seen near the defect in the ratio map,14onfirming the existence of strong anti-phase signals between ± E B . By Fourier transforming Z ( r , E B1 ) for the large scale image shown in Appendix B, | ˜ Z ( q , E B1 ) | is obtained and is shownin Figure 6b. Comparing to QPI at the bound state energies (Fig. 3c and 3d in Ref. 22), the q h , − e features (highlighted by the purple ovals) become enhanced relative to the other QPIfeatures, consistent with the sign change of the order parameter for inter-band scatteringwavevectors q h , − e in the phase-referenced DBS-QPI. Figures 6c-e show the simulations ofratio-map DBS-QPI in the three cases that produce in-gap bound states. The simulationsbased on s ± agree best with experimental observations, consistent with the results fromphase-referenced DBS-QPI. E = E B2 a b c d q h -e q h -h s ++ magnetic s ± nonmagnetic magnetic s ± Figure 7 . Ratio-map DBS-QPI associated with the small gaps. ( a ) Measured | ˜ Z ( q , E B2 ) | with E B2 = 1 . , π ) are highlighted by the arc and ovals.The position of h - h scattering is indicated by the red arrow where signal is minimal. ( b )-( d )Simulated | ˜ Z ( q , E B2 ) | . The q h − e and q h − h QPI features are indicated by the purple shapes andthe red arrows, respectively.
The ratio-map method was applied to the bound states at E B2 , as shown in Figure 7. Inthe experimental data, three inter-band QPI signals dominate, as indicated by the purpleshapes. This agrees well with the sign change of q h − e observed in the phase-referencedDBS-QPI (see Fig. 4 in Ref. 22). The middle oval (dashed shape) is the q h , − e QPI featuresfrom E B1 and present here because of thermal broadening effects. The other two shapes are q h − e QPI features from scattering between in-gap bound states for the small gaps in h and e bands. QPI features for intra-band q h − h were not well resolved within our measurementresolution (red arrow in Figure 7a) due to the in-phase cancellation effect. The simulatedratio-map bound-state QPI are shown in Figure 7b-d for the three possible scenarios thatallow in-gap bound states. The q h − e signal is enhanced in the simulation using s ± with15 nonmagnetic defect. However, the other two cases have the q h − h signal enhanced asindicated by the red arrows. Therefore, only the simulation using s ± with a nonmagneticdefect reproduces the dominant signatures seen in experimental data. B. Phase information in the HAEM method
Hirschfeld, Altenfeld, Eremin, and Mazin (HAEM) proposed a method using QPI todetermine the order parameter of iron-based superconductors , and this has recently beenapplied to FeSe . In the HAEM method, the quantity considered is˜ g ± ( q , ω ) = Re [˜ g ( q , ω ) ± ˜ g ( q , − ω )] , (15)where ˜ g ( q , ω ) is the QPI signal amplitude at energy ω and scattering vector q . In fact,˜ g + ( q , ω ) enhances in-phase QPI features and suppresses anti-phase QPI features, whereas˜ g − ( q , ω ) enhances anti-phase QPI features and suppresses in-phase QPI features. For ex-ample, if there are anti-phase QPI features, one has g ( r , ω ) ≈ − g ( r , − ω ) . (16)The Fourier transform gives ˜ g ( q , ω ) ≈ − ˜ g ( q , − ω ) . (17)Therefore, the HAEM signal is given by˜ g + ( q , ω ) ≈ g − ( q , ω ) ≈ g ( q , ω ) . (19)As a result, ˜ g − doubles the signal with anti-phase oscillations and cancels the signal within-phase oscillations. Essentially, similar to ratio-map QPI, the HAEM method sets apartthe in-phase and anti-phase signals. It also suffers the same problem that ratio-map QPIhas, as discussed in the section above: in-phase signals that are very different in amplitudesat positive and negative energies can confuse the interpretation. In principle, the HAEMmethod should produce results that are consistent with the ratio-map QPI.Experimentally, there are a few advantages of phase-referenced QPI and ratio-map QPIover the HAEM method. First, phase-referenced QPI and the ratio maps take accountof all defects in the measured area, providing better signal-to-noise ratio. Second, signal16ntegration in the HAEM method is effective in theory using an ideal δ -potential, but inexperiments, defects have spatial form factors which shift oscillations. This leads to extracomplexity in the phase of QPI at different scattering vectors q after Fourier transformation.Integration over an area with signals of varying complex phase further reduces the signal-to-noise ratio, rendering comparison with theory difficult. B A
Re( g ) + Re( g )- g ( q , - E B1 ) g ( q , E B1 ) a bc d Energy (meV)
Re( g ) + area Aarea B Re( g )- ∼∼ area Aarea B H AE M s i gna l ( a . u . ) Δ Δ e ( π , π ) ( , π ) +- ∼ ∼∼ ∼ Figure 8 . HAEM signal for a single Fe-D defect. ( a ) ˜ g ( q , E B1 ) and ( b ) ˜ g ( q , − E B1 ) of theFe-D defect (after symmetrization and interpolation). ( c ) ˜ g − ( q , E B1 ) and ( d ) ˜ g + ( q , − E B1 ) of theFe-D defect. QPI maps in ( b )-( e ) are in the same color scale. ( e ) The integrated inter-band g − and g + for the two areas indicated in ( d ). Here sample bias (mV) is converted to energy (meV). We applied the HAEM method to our experimental data for an Fe-D defect. Onerelatively isolated defect was selected from the area in Figure 11 with a size of 6 . × . as indicated by the yellow square. The DBS-QPI of this defect is shown in Figure 8a and8b, and exhibits a complicated phase pattern after Fourier transformation. In the HAEM17ap, the anti-phase signal (˜ g − ) primarily corresponds to q h , − e (see Figure 8c) and thein-phase signal does not exhibit scattering of significant strength (see Figure 8d). ˜ g − ( ω ) and˜ g + ( ω ) for q h , − e was integrated over areas A and B, indicated in Figure 8c. For the twodifferent integration areas, the shapes of ˜ g − ( ω ) and ˜ g + ( ω ) change on a scale comparable tothe signal strengths themselves. This is because of the relatively complicated phase featuresfor the q h , − e QPI in LiFeAs, as well as the lower signal-to-noise ratio for a single defect.In contrast, the HAEM approach produced a much more robust result in work on FeSe Alikely source of the difference between the results on the two materials is that the apparentsize of the defects in the topography of FeSe is smaller, making the influence of the defectform factor smaller in the QPI results on this material.
C. Comparison to phase-referenced DBS-QPI
Figures 9a-b shows the DBS-QPI at ± E B1 after PRFT . The effect of the technique isapparent: the information in the form of sign contrast is moved to negative energies, and thesign difference between positive and negative energies for the interband scattering vectors ismaximized. The inter-band QPI signal is integrated over the same two areas as those usedabove for the HAEM method. As shown in Figure 9c, the result is relatively independentof the integration windows, and the inter-band QPI signal is strongest near the bound stateenergies. The slight shift of the peak near E B1 is due to thermal broadening at the 4 . E B2 for the larger integration window B can also be seen. This is becausewindow B covers all the QPI signals of q h , − e and q h − e , while window A mainly covers q h , − e . It only shows up at negative energies for two reasons. First, q h − e is stronger at thenegative energy (see Fig.4 in Ref. 22). Second, the positive energy is zero-phased, so hascontributions from the absolute values of both the QPI signal and the noise background.The weak signal at E B2 is merged in the background noise integration. However, at negativeenergies, the background noise has random phase, and therefore the integration cancels thebackground noise significantly. 18 B1 a b +- c g c ( q , E ) area Aarea B -10-505101520 E B1 E B1 - Δ Δ - Δ - Δ Energy (meV) s i gna l ( a . u . ) B A -E B1 ∼ E B2 - Figure 9 . The integrated signal of Phase-referenced Fourier transform QPI. ( a )˜ g c ( q , E B1 ) and ( b ) ˜ g c ( q , − E B1 ) including all defects. ( c ) The integrated inter-band g c ( q , E ) forthe two areas indicated in ( a ). V. CONCLUSION
We have shown that by using the phase information contained in the spatial modulationof defect bound states, whose contribution to the QPI signal is usually ignored, informationabout the phase of the superconducting order parameter as well as the properties of thescatterer can be extracted. The robustness of the DBS-QPI method is demonstrated bycomparison of a number of models against experimental data. To enable an experimentallyrobust analysis of the phase, we have introduced the phase-referenced DBS-QPI, which pro-vides a systematic framework in which the phase of defect scattering can be interpreted.Our work provides a direct link between the theoretically predicted phase shifts for quasi-particle scattering at defects and experimental results, and provides strong evidence for an s ± pairing state in LiFeAs. 19 I. ACKNOWLEDGEMENTS
The authors are grateful for helpful conversations with George Sawatzky, Mona Berciu,Peter Hirschfeld, Andreas Kreisel, and Steven Johnston. Research at UBC was supportedby the Natural Sciences and Engineering Research Council, the Canadian Institute for Ad-vanced Research, and the Canadian Foundation for Innovation. SAB was further supportedby the Canada Research Chairs program. PW acknowledges support from EPSRC grant noEP/I031014/1 and DFG SPP1458.
Appendix A: Simulations using a two orbital model
To test the robustness of the conclusions reached from calculations within the five orbitalmodel, we have tested the result using a minimal two-orbital model. We employ a model fromRaghu et al. to qualitatively compare bound state QPI between theory and experiment.Similar to the five-orbital model, the superconducting gap is ∆ k = ∆ cos( k x ) cos( k y ) for s ± order parameter and | ∆ k | for s ++ order parameter, respectively. ∆ = 0 .
09 eV is used in thesimulations; the precise value does not affect the qualitative result. In Figure 10(a) and (b),the Fermi surface of this model and its QPI for energies outside the superconducting gapsare shown. In this model, the DOS is dominated by the electron band, hence the prominentQPI intensities are intra-band q e − e and inter-band q h − e features. The intra-band q h − h QPI feature is very weak. In the superconducting state, two gaps open at E F as indicatedby the dashed lines in Figure 10(c). Both a nonmagnetic defect ( V = − . s ± order parameter and a magnetic defect ( V = − . s ++ order parameter producestrong in-gap bound states, as shown in Figure 10(c) and (e). In their ratio-map QPI (seeFigure 10(d) and (f)), the q h − e features are enhanced and q e − e features are suppressed inthe simulation using the s ± with a nonmagnetic defect. However, the opposite intensitychanges occur in the simulation using s ++ with a magnetic defect. These results agree verywell qualitatively with the results calculated from the five-orbital model. Appendix B: Supporting experimental data
Differential conductance maps were acquired at a temperature of 4.2 K in a commercialCreatec STM on single crystals of LiFeAs, cleaved at a temperature below 20 K . Two20 DosOnsite E (eV) D o s -0.15 -0.1 -0.05 0 0.05 0.101234567 DosOnsite E (eV) D o s E = -1.3 Δ e - e h-h h - e h - e h - e h - e lowhigh E B E B e - e a bc de f ( π , ) ( π , π ) E B ratio map E B ratio map h h e e Figure 10 . Bound state QPI from the two-orbital model. ( a ) The Fermi surface of thetwo-orbital model. ( b ) ˜ g ( q , E = − . ) QPI outside the superconducting gap. ( c ) LDOS withoutdefect (black), and on the defect site (red), calculated with the input of s ± and a nonmagneticscattering potential. ( d ) The ratio-map QPI ˜ Z ( q , E B ) of the bound state seen in ( c ). ( e ) LDOSwithout defect (black) and on the defect site (red) calculated with the input of s ++ and a magneticscattering potential. ( f ) The ratio-map QPI ˜ Z ( q , E B2 ) of the bound state seen in ( e ). conductance maps were measured for QPI. The results from the two datasets are consistentand the average of them is shown in this report. Figure 11 shows the topography of the areafor one dataset. Taking one dataset as an example, there are fourteen native defects in thearea of the sample: ten Fe-D defects, one As-D defect, two Li-D defects, and one Fe-C defect (Figure 11).Tunneling spectra obtained on four native defects are shown in Figure 12. The referencespectrum measured on a defect-free spot shows two superconducting gaps, ∆ = 6 meV and21 l o w h i gh Figure 11 . Topography of an area used for QPI measurements.
The types of nativedefects shown in this area are 1: Fe-D , 2: Fe-C , 3: Li-D , 4: As-D . The yellow square is a6 . × . area selected for the HAEM analysis. c b As-D −
10 0 10 Δ Δ -E B2 E B1 a Fe-C E B2 E B1 −
10 0 10 Δ Δ g ( V )( a . u . ) −
10 0 10 Δ Δ Sample Bias (mV) Sample Bias (mV) d g ( V )( a . u . ) Li-D E B2 E B1 −
10 0 10 Δ Δ Fe-D -E B2 E B1 * Figure 12 . Tunneling spectra g ( V ) on (a) Fe-D , (b) As-D , (c) Li-D , and (d) Fe-C defects. The black curve is the reference spectrum taken on a defect-free area. The locations fortaking the defect spectra are marked as blue dots in the inserts. All data in this figure were takenin a home-built low temperature STM at 1.6 K . ∆ = 3 meV, consistent with previous results . As shown in Figure 12(a), the Fe-D defect produces a strong bound state corresponding to the in-gap bound state for the large22ap. QPI of this bound state is used to determine the order parameter in LiFeAs (see Fig. 3of Ref. 22). The shallow shoulder feature at E ∗ B2 ∼ . . These strong bound states, labeled as E B2 , giverise to the strongest signal in DBS-QPI at 1.2 meV and are associated with the bands atthe small gap. Appendix C: Recovering phase information for scattering from multiple defects
In the case of an area with many defects, the actual phase of the QPI for individualdefects can be recovered . In the sparse case, a tunneling conductance map with multipledefects can be written as g ( r , ω ) = (cid:88) R i g S ( r − R i , ω ) , (C1)where R i is the location of the i -th defect and g S ( r , ω ) is the tunneling conductance mapfor a single defect at the origin. Then the Fourier transformation is˜ g ( q , ω ) = (cid:90) d r e − i qr g ( r , ω )= (cid:90) d r e − i qr (cid:88) R i g S ( r − R i , ω )= (cid:88) R i e i qR i (cid:90) d r e − i qr g S ( r , ω )= (cid:88) R i e i qR i ˜ g S ( q , ω ) , (C2)where ˜ g S ( q , ω ) is the QPI of a single defect in q -space. The prefactor (cid:80) R i e i qR i causes inter-ference effects between defects, reducing the signal strength. Its absolute value is generallyproportional to √ N instead of N , where N is the number of defects. From Equation C2, thesingle defect QPI, ˜ g S ( q , ω ), can be extracted using a multiple-defect-configuration correction˜ g S ( q , ω ) = ˜ g ( q , ω ) (cid:80) R i e i qR i . (C3)This method only applies to maps with primarily one type of defect, in which ˜ g S ( q , ω ) of alldefects are identical.We extracted ˜ g S ( q , ω ) for the Fe-D defect by identifying the positions of all Fe-D defectsand masking the signals from the other defects. Because of the D symmetry of the defect,23 - q h -e q h -h q h -h a b c d E B1 -E B1 Δ -1.3 Δ Figure 13 . QPI of all Fe-D defects with multiple-defect-configuration correction. ( a ) - ( d ) Re[˜ g S ( q , ω )] above the superconducting gap (1.3∆ ), at the bound state energies ( ± E B1 ),and below the superconducting gap (-1.3∆ ). signals exist only in the real part, Re[˜ g S ( q , ω )]. As shown in Figure 13, a map Re[˜ g S ( q , ω )]for a Fe-D was reconstructed from two measured maps which contain multiple defects inthe areas. The QPI features are consistent with the one obtained from a single defect (seeFigure 8b-c). However, multiple-defect-configuration correction gives enhanced q -space res-olution. The q h , − e QPI features is only enhanced at ± E B1 and has the sign-change betweenpositive and negative bound state energies, in agreement with theoretical calculations usingthe s ± order parameter.However, the experimental ˜ g S ( q , ω ) has a more complicated phase than from the theo-retical calculation using a δ -potential, hence a direct comparison between experimental dataand theoretical simulations is challenging. This is the reason that phase-referenced QPI andratio maps were employed to analyze the scattering pattern here and in Ref. 22. Phase-referenced QPI sets the zero phase for QPI at E and highlights the phase changes at energy − E with opposite polarity. Ratio-map QPI incorporates the phase contrast by taking theratio between the positive and negative energies. J. E. 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