Determination of the Superconducting Order Parameter from Defect Bound State Quasiparticle Interference
Shun Chi, W. N. Hardy, Ruixing Liang, P. Dosanjh, Peter Wahl, S. A. Burke, D. A. Bonn
DDetermination of the Superconducting Order Parameter fromDefect Bound State Quasiparticle Interference
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, 2, ∗ 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) a r X i v : . [ c ond - m a t . s up r- c on ] O c t bstract The superconducting order parameter is directly related to the pairing interaction, with theamplitude determined by the interaction strength, while the phase reflects the spatial structureof the interaction. However, given the large variety of materials and their rich physical proper-ties within the iron-based high-Tc superconductors, the structure of the order parameter remainscontroversial in many cases. Here, we introduce Defect Bound State Quasi Particle Interference(DBS-QPI) as a new method to determine the superconducting order parameter. Using a lowtemperature scanning tunneling microscope, we image in-gap bound states in the stoichiometriciron-based superconductor LiFeAs and show that the bound states induced by defect scattering areformed from Bogoliubov quasiparticles that have significant spatial extent. Quasiparticle interfer-ence from these bound states has unique signatures from which one can determine the phase of theorder parameter as well as the nature of the defect, i.e. whether it is better described as a magneticvs a nonmagnetic scatterer. DBS-QPI provides an easy but general method to characterize thepairing symmetry of superconducting condensates.
2n superconductors, a pairing interaction binds electrons into Cooper pairs, condensingthem into a coherent ground state with an order parameter ∆ k = | ∆ k | e iφ k . Here, | ∆ k | isthe magnitude of the superconducting gap and φ k is the phase of the order parameter [1–3].Uncovering the complex order parameter is key to a determination of the pairing mechanism.In particular, the phase factor e iφ k gives insight into how two electrons overcome Coulombrepulsion and bind together. For conventional superconductors, attractive interactions me-diated by phonons result in an s -wave state in which ∆ k has the same sign everywhere inmomentum space [1]. The e iφ k factor changes sign only at higher energies where Coulombrepulsion dominates, and the chance two electrons come close to each other is reduced [1].In high-temperature cuprate superconductors, the electron-phonon interaction is believed tobe too weak to be responsible for pairing [3], and strong on-site Coulomb repulsion favours a d -wave order parameter with a sign-change (or phase shift by π ) in momentum space [2, 3].For the case of iron-based superconductors, which have multiple bands crossing the Fermienergy, the precise order parameters are still controversial and could differ between differentcompounds, but Coulomb repulsion may again favour a sign-changing order parameter [4].Many methods have been explored for determining superconducting order parameters,however most are sensitive only to | ∆ k | , which controls the gap in the density of states andcannot directly detect a sign-change. For order parameters with nodes, such as the d -wavestate in the single-band cuprates, a sign change can be inferred from the angular dependenceof the order parameter | ∆ k | . These methods were remarkably successful in studies of cuprateand heavy fermion superconductors [5–7]. However, a definitive identification of the phasestill required specialized experiments that were sensitive to the phase factor e iφ k . In cupratesuperconductors, whose sign change involves a broken rotational symmetry, this was achievedthrough measurements detecting the sign change associated with rotations by 90 ◦ , usingtunnel junctions, or through the detection of half flux quanta [8, 9]. In contrast to cuprates,most iron-based superconductors possess a nodeless order parameter, suggesting an s -wavepairing state. A vast amount of research has been undertaken to determine whether or notthere is a sign change in the order parameter between different sheets of the Fermi surface,designated either as an s ± or an s ++ order parameter. The lack of both broken rotationalsymmetry and absence of nodes mean that the techniques which have delivered definitiveevidence for the pairing symmetry in the cuprate superconductors are inconclusive for theiron-based superconductors [2]. 3 more broadly applicable method to probe the phase exploits quantum interferencebetween the quasiparticle wavefunctions. Quasiparticle interference (QPI), measured viascanning tunneling microscopy (STM), detects the oscillating pattern resulting from inter-ference of quasiparticles scattered by defects, and hence is sensitive to the phase differencebetween the initial and final states. This technique has been successfully applied to a num-ber of cuprate superconductors[10, 11]. To extract information about the phase of the orderparameter, QPI due to scattering by vortices has been analyzed to detect signatures ofthe sign-changing order parameter [12–14]. However, application to the iron-based super-conductors has encountered difficulties: the QPI intensities are located near Bragg peaks,making it hard to discern the true QPI signal [15, 16]. Also vortices are spatially moreextended in iron-based superconductors due to longer superconducting coherence lengths,which complicates comparison with theoretical calculations that assume point-like scatteringpotentials.[17]Here we will show that this difficulty can be overcome by studying QPI between well-defined Bogoliubov quasiparticles which inherit the phase of the order parameter [18]. In-gapbound states, the excitations of Cooper pairs due to defect scattering, are excellent sources ofBogoliubov quasiparticles. Within the gap, the density of states of the clean superconductoris zero, but near defects there is a contribution from the Bogoliubov quasiparticles whichmake up the bound states. These bound states are confined to the vicinity of defects inreal space, but have sufficient spatial extent that they can be relatively well localized inmomentum space [19]. The screening of the defect potentials by these in-gap Bogoliubovquasiparticles leads to defect bound state QPI (DBS-QPI) . Defects play a vital role in thisdistinct form of QPI, both as the source of the Bogoliubov quasiparticles and as the scatteringcentre that leads to interference effects. Characterization of this DBS-QPI provides a directphase-sensitive measurement of the superconducting order parameter.We study DBS-QPI in LiFeAs, one of the stoichiometric iron-based superconductors,whose surface after cleaving is ideal for STM study [13, 20, 21]. By comparing experimen-tal data with theoretical simulations that use a realistic band structure, we provide solidevidence for a sign change in the superconducting order parameter between the hole andelectron bands. In turn, this significantly constrains the form of the pairing interaction.Fig. 1a schematically shows the scattering from a defect present in the lattice. A quasi-particle travels in the lattice in the state Ψ k ,σ ( r ), where k and σ are the momentum and the4 π , π ) (0, π )030-30 E ( m e V ) e D O S Δ Δ ad b + E (meV) -E B1 -E B2 E B1 E B2 Ψ ( r ) k , σ Ψ ( r )e k+q , σ ' i δ q ( π , ) ( π , π ) c E = 1.3 Δ ( π , π ) ( π , ) H i gh L o w f E = - E B1 h h h e -E B1 -E B2 ( π , π )(0,0) q h -h q e-e q h -h q h-e FIG. 1.
Theory: a
Schematic of the quasiparticle scattering process. b Fermi Surface of LiFeAsderived from a five-orbital model. The red and purple arrows indicate possible momentum transfersof the scattering process depicted in a ). The three red arrows are intra-band scattering processesand the purple arrow is an inter-band scattering vector. c The simulated QPI for an energyabove the superconducting gaps ( E = 1 . ) with the QPI features indicated by q h − h , q h − h , q e − e , and q h − e , respectively. q h − h corresponds to the ring-like feature in the center and is toosmall to be labeled. d LDOS for an s ± order parameter: the unperturbed LDOS (black) showstwo superconducting gaps, and the DOS on a nonmagnetic defect (red) gives two sets of in-gapbound states. e The defect-induced change of the DOS δρ ( k , E ) is shown in k -space, revealing theadditional states due to defect bound states at ± E B1 , . f Simulated magnitude of the DBS-QPIat − E B1 , showing intra-band and inter-band QPI features. spin quantum numbers. When the quasiparticle encounters a defect, it scatters elasticallywith a certain probability to a final state Ψ k + q ,σ (cid:48) ( r ) e iδ q . The square modulus of the sumof the wavefunctions of all the possible scattering events produces spatial modulations inthe local density of states (LDOS), which are referred to as QPI. The allowed wave vectors q connect the available states at a given energy. In LiFeAs, three hole bands ( h , h , h )and two electron bands ( e ) cross the Fermi energy. Four examples of q at the Fermi energyare shown in Fig. 1b, with the red and purple arrows indicating intra-band and inter-bandscatterings, respectively. Fourier transformation (FT) of the real-space oscillations in theLDOS yields the QPI features in q -space, with maxima in the amplitude corresponding to5he dominant q vectors. Fig. 1c shows the calculated QPI pattern for the superconductingstate using a five-orbital model [22] and an s ± order parameter with a nonmagnetic scat-tering potential [23]. At E = 1 . , the QPI features are essentially identical to those inthe normal state [23]. The scattering vectors of Fig. 1b which have a significant degree ofnesting are easily identified as the more prominent QPI features in Fig. 1c.In QPI, the relative phase term e iδ q is primarily determined by two factors: the nature ofthe defect, nonmagnetic vs magnetic and weak vs unitary, which causes a phase shift duringscattering; and the intrinsic phase difference between the states before and after scattering.In a superconductor, the spontaneous gauge symmetry breaking sets the phase of Cooperpairs in momentum space, which is the phase of the superconducting order parameter.Scatterers play a second role in a superconductor, providing a distinct form of QPI. Theinterplay of the superconducting order parameter and the nature of defects yields boundstates inside the superconducting gaps [18]. In the case of a conventional order parameterwithout a sign change, only a magnetic defect can give rise to in-gap bound states. Witha sign-changing order parameter, both magnetic and nonmagnetic defects can induce in-gap bound states. These bound states are excitations of the superconducting ground state,namely Bogoliubov quasiparticles that are produced by defect scattering. Bogoliubov quasi-particles inherit the phases of Cooper pairs at different momentum states [24]. By studyingthe relative phase term e iδ q in QPI, one is able to decode both the superconducting orderparameter and the nature of the defects.Fig. 1d shows the calculated LDOS of LiFeAs at a defect-free site (in black) and anonmagnetic defect site (in red), assuming an s ± order parameter. There are two sets ofimpurity bound states inside the gaps. To better resolve the origin of the bound states, thedefect-induced change of the density of states (DOS) δρ ( k , ω ) is shown in Fig. 1e. The outerset at energies ± E B1 are the in-gap bound states for the large gaps in bands h , h and e ,and the inner set at energies ± E B2 are the in-gap bound states for the small gaps in band h and e . In the cases of a magnetic defect with s ++ and s ± order parameters, they yield verysimilar results (see section III.D of Ref. 23). These states consist of Bogoliubov quasiparticlesthat are relatively localized in energy but follow the dispersion of the bandstructure near E F . This means the wavefunctions of the superconducting bound states are delocalized in r -space. Thus, defect-bound state QPI (DBS-QPI) can be generated from these bound stateswith scattering vectors similar to conventional QPI from states above the gaps. Fig. 1f6hows the calculated magnitude of the DBS-QPI in q -space at − E B1 . While all DBS-QPIfeatures are generally consistent with the QPI seen outside the superconducting gaps inFig. 1c, there are some important differences. First, the DBS-QPI features are broadenedbecause the momentum distribution of the bound state is wider due to their confinement tothe general vicinity of the defect in real space. Second, and more interestingly, the DBS-QPIfeatures due to inter-band scattering processes q h − e , indicated by the purple arrow, becomesignificantly enhanced. This is an interference effect involving the interplay between thephase of the order parameter and the nature of the defect, and it is this effect that can beused to identify the sign-changing nature of the order parameter. a b E B1 -E B1 -E B1 E B1 c d +- s ± s ± s ++ s ++ Energye Δ Δ - Δ - Δ s ± s i gna l ( a . u . ) E B1 E B2 -E B1 -E B2 s ++ q h -e q e-e q h -h q h -h FIG. 2.
Theoretical results of phase-referenced DBS-QPI. a, b, c, and d The simulatedphase-referenced DBS-QPI at ± E B1 for s ± with a nonmagnetic defect and s ++ with a magneticdefect, respectively. e The integrated h - e DBS-QPI signal. The integration region is shown in a as a shaded area. DBS-QPI distinguishes s ± and s ++ through the phase of the Fourier transform. Here wedefine a phase-referenced Fourier transformation (PRFT) to clarify this difference, which is at7he basis of DBS-QPI. The experimentally measured quantity is the tunneling conductancemap g ( r , E ), which is proportional to the LDOS ρ ( r , E ). The FT of g ( r , E ) is | ˜ g ( q , E ) | × e iθ q ,E , where | ˜ g ( q , E ) | is the intensity and θ q ,E is the phase of the Fourier component atwave vector q and energy E . Conventionally, the phase is ignored, and only the intensityof the FT is analyzed. However, the phase is closely related to the scattering process andthe interference of the quasiparticle wavefunctions. Analysis of the phase is complicated bythe fact that it contains an arbitrary global phase factor related to the defect positions anddetails of defect apparent size and symmetry [23]. Therefore, in order to extract meaningfulinformation about the phase relation, we use the PRFT, which reveals the relative phasebetween QPI at positive and negative energies ± E . We extract the PRFT via the followingsteps: we first Fourier transform g ( r , E ) at positive energy E , obtaining the phase factor e iθ q ,E which we use as reference for the Fourier transform at negative energy − E . The PRFTof the tunneling conductance ˜ g c ( q , ± E ) for E > g c ( q , E ) = | ˜ g ( q , E ) | × e iθ q ,E × e − iθ q ,E = | ˜ g ( q , E ) | (1)˜ 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 ) ) (2)where Re means the real part. The phase factor Re( e i ( θ q , − E − θ q ,E ) ) of the PRFT is +1 forin-phase oscillations, and − ± E B1 . The major DBS-QPIfeatures seen here correspond to scattering vectors connecting bands with the large gaps, q h , − h , , q e − e ,and q h , − e , as expected for the bound states associated with the large gaps. q h − h is still present because the small gap of the h band is not fully open at E B1 (seeFig. 1d), but its strength is noticeably weaker than the other DBS-QPI features. The keydifference between s ± and s ++ is a sign change in the superconducting order parameterbetween hole and electron bands. Accordingly our analysis focuses on the inter-band DBS-QPI features, q h , − e , as indicated by purple arrows. For s ± with a nonmagnetic defect, themajority of the q h , − e signal has the opposite sign between ± E B1 , as shown in Fig. 2a and2b (blue for positive and red for negative). For the case of s ± with a magnetic defect, theresults are similar. In contrast, for s ++ with a magnetic defect, the sign of the q h , − e signalis mostly the same between ± E B1 , as shown in Fig. 2c and 2d. The inter-band DBS-QPI8eatures in ˜ g c ( q , E ) is integrated over an area indicated in Fig. 2a. The DBS-QPI signals asa function of energy is peaked at the bound state energies, as shown in Fig. 2e. In particular,the signals show the opposite sign between the positive and negative bound state energiesin the simulation with the s ± order parameter. On the other hand, the simulation with the s ++ order parameter preserves the sign of the signal at ± E B1 . a b ( π , π ) ( , π ) E = 1.3 Δ g ( V ) ( a . u . ) Sample Bias (mV) −
10 0 10 Δ Δ E B1 -E B2 * -E B1 Exp. E B1 Exp. c d +- e q h -h q h -h q h -e Δ Δ - Δ - Δ -5-100510 s i gna l ( a . u . ) Energy (meV) E B1 E B1 - FIG. 3.
Experimental results of phase-referenced DBS-QPI. a
The tunneling conductance g ( V ) obtained on a clean area (black) and on an Fe-D defect (red) ( T = 1 . defect ( V = 25 mV, I = 50 pA). The red dot indicates the location foracquiring the spectrum. b QPI outside the gaps ( E = 1 . = 7 . q h , , − h , , , centered at (0 , q h − e , centered at (0 , π ). c and d are the phase-referenced Fouriertransform ˜ g c ( q , ± E B ). e The integrated h - e DBS-QPI signals from experimental data with theintegration area indicated in c . Here sample bias (mV) is converted to energy (meV). Next, we show the application of this technique to study the measured DBS-QPI forLiFeAs. Single crystals of LiFeAs ( T c = 17.2 K) were grown using a self-flux technique [20].For DBS-QPI measurements, a sample of LiFeAs was cleaved in-situ at a temperature below90 K and inserted into a Createc scanning tunneling microscope (STM). DBS-QPI data wereacquired at a base temperature of 4.2 K by taking I - V spectra at each pixel in a grid of400 × I - V spectra to produce tunneling conductancemaps. A home-built low temperature STM operating at temperatures down to 1.5 K wasused to measure point spectra on native defects [25]. The lower base temperature enables usto better resolve the bound states inside the superconducting gaps, pinpointing the energiesto focus on in the DBS-QPI analysis. In as-grown LiFeAs, the most common native defect isthe Fe-D defect whose topography is shown in the insert of Fig. 3a[26]. Measured at 1.5 K,the tunneling spectra taken at a defect-free area (black) show two superconducting gaps with∆ = 6 meV and ∆ = 3 meV, consistent with previous reports[13, 20, 21]. The spectrumon an Fe-D defect shows a prominent bound state inside the large gap and close to the edgeof the small gap E B1 ∼ E ∗ B2 ∼ . defect requires higher energy resolution. Fig. 3bshows QPI measured outside the gaps ( E = 1 . ). The QPI features in the experimentaldata agree very well with the simulation (see Fig. 1c) except for the absence of q e − e whichis consistent with previous reports [21, 29] and is probably due to a tunneling matrix effect.Fig. 3c and 3d show the phase-referenced DBS-QPI at ± E B1 = ± bound state at E B produces QPI that resmbles the features seen in Fig. 2, confirming thestates at ± E B1 are the in-gap bound states of the large gap. A sign inversion in the q h , − e signal occurs between positive and negative bound state energies, as indicated by the purplearrows. This sign inversion is further confirmed by integrating the inter-band DBS-QPI (seeFig. 3e) using the integration area indicated by the shadowed area in Fig. 3c. Plotted asa function of energy in Fig. 3e, the q h , − e signal peaks at the bound state energies ± E B1 but with the opposite signs at the two energies. This is only consistent with the simulationusing the s ± order parameter (see Fig. 2). In the experimental data, the absence of featuresat ± E B2 in the integrated signals is due to thermal broadening at the temperature of themeasurement (4.2K) and the weakness of the signal compared to measurement noise.In addition to Fe-D defects, a few other native defects are present and give strong boundstates inside the small gap, for example the Li-D defect whose g ( V ) is shown in Fig. 4a. Thephase-referenced DBS-QPI was measured at E B2 ∼ . h -e q h -h E B2 Exp. -E B2 Exp. a Li-D g ( V ) ( a . u . ) −
10 0 10 Δ Δ E B2 E B1 Sample Bias (mV) b c d ef g h i +- nonmag. magnetic s ± magneticnonmag. s ± magnetic s ± s ++ magnetic s ± s ++ FIG. 4.
Phase-referenced DBS-QPI associated with the small gaps. a g ( V ) measured ona Li-D defect shows two sets of bound states. The strong bound state at ± E B2 is associated withthe small gaps. b and f Measured phase-referenced DBS-QPI at ± E B2 = ± . , π ) are highlighted by the arc and ovals, two of which are q h − e indicatedby purple arrows. The position of q h − h is also specified by a red arrow. c, d, e, g, h and i Calculated phase-referenced DBS-QPI for the three cases allowing in-gap bound states. as indicated by the purple shapes. The middle oval (dashed shape) is the q h , − e QPI featurefrom E B and present here because of thermal broadening. The other two shapes are QPIfeatures from scattering between in-gap bound states for the small gaps in h and e bands, q h − e . Consistent with the results above for q h , − e , q h − e has a sign inversion between E B2 and − E B2 whereas QPI features for intra-band q h − h scattering preserve the sign.The calculated phase-referenced DBS-QPI are shown in Fig. 4c-d and 4g-i for the threepossible scenarios that allow in-gap bound states: s ± with a nonmagnetic defect, s ± witha magnetic defect, and s ++ with a magnetic defect. With s ± and a nonmagnetic defect,the q h − e signal changes sign and the q h − h signal retains the same sign between E B2 and − E B2 . For the other two cases with a magnetic defect, the q h − h features dominates thesignal and changes sign between E B2 and − E B2 , as indicated by the red arrows. The onethat is in best agreement with the experimental data is the calculation using s ± with anonmagnetic defect. Hence QPI at the in-gap bound state for the small gaps further verifiesthe consistency between experiment and theoretical results assuming an s ± order parameter.In addition, it identifies the nonmagnetic nature of the native defects in LiFeAs since noneof the simulations with a magnetic defect fit the phase-referenced DBS-QPI at E B2 .11n LiFeAs, the sign change of the superconducting order parameter between hole andelectron bands indicates that electrons pair together between next-nearest-neighbor (NNN)sites. The most plausible interaction that is able to generate an attractive channel betweenNNN sites is due to stripe antiferromagnetic spin fluctuations [2, 3, 30].LiFeAs is not the only superconducting compound which shows DBS-QPI. The anti-phase oscillations had been predicted from theory for d -wave superconductors [18] and seenin experimental observations of the bound states in cuprate and heavy fermion superconduc-tors [7, 31]. DBS-QPI provides a robust method for revealing both the order parameter andthe nature of defects in superconductors with unconventional order parameters. Given thatthe typical apparent size of a defect is only a few lattice constants, a δ -function potential is agood approximation for theoretical modeling, which makes it easier to compare experimentwith theory than when using vortex cores. However, the analysis of experimental data con-taining contributions to phase from multiple defects of finite size and non-point symmetryis aided here by the use of phase-referenced Fourier transforms to help isolate phase changesdue to the order parameter. ACKNOWLEDGEMENTS
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