Evidence for time-reversal symmetry breaking of the superconducting state near twin-boundary interfaces in FeSe
T. Watashige, Y. Tsutsumi, T. Hanaguri, Y. Kohsaka, S. Kasahara, A. Furusaki, M. Sigrist, C. Meingast, T. Wolf, H. v. Lohneysen, T. Shibauchi, Y. Matsuda
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Evidence for time-reversal symmetry breaking of the superconducting state neartwin-boundary interfaces in FeSe
T. Watashige,
1, 2
Y. Tsutsumi, T. Hanaguri, Y. Kohsaka, S. Kasahara, A. Furusaki,
2, 3
M. Sigrist, C. Meingast, T. Wolf, H. v. L¨ohneysen, T. Shibauchi,
6, 1 and Y. Matsuda Department of Physics, Kyoto University, Kyoto 606-8502, Japan RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Theoretische Physik, ETH Z¨urich, CH-8093 Z¨urich, Switzerland Institute of Solid State Physics (IFP), Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany Department of Advanced Materials Science, University of Tokyo, Chiba 277-8561, Japan (Dated: April 10, 2015)Junctions and interfaces consisting of unconventional superconductors provide an excellent exper-imental playground to study exotic phenomena related to the phase of the order parameter. Notonly the complex structure of unconventional order parameters have an impact on the Josephsoneffects, but also may profoundly alter the quasi-particle excitation spectrum near a junction. Here,by using spectroscopic-imaging scanning tunneling microscopy, we visualize the spatial evolution ofthe local density of states (LDOS) near twin boundaries (TBs) of the nodal superconductor FeSe.The π/ ξ ab . The modification ofthe low-energy states is even more prominent in the region between two neighboring TBs separatedby a distance ≈ ξ ab . In this region the spectral weight near the Fermi level ( ≈ ± I. INTRODUCTION
When two superconductors are in close proximity, theyare influenced by each other via the tunneling of Cooperpairs. The Cooper-pair tunneling results in the flow of asuperconducting Josephson current, which has been stud-ied for decades and is used in various superconductingquantum devices [1]. The Josephson current is governedby the phase difference of the order parameters of the twosuperconductors. Therefore, Josephson junctions con-sisting of unconventional superconductors, where the su-perconducting order parameter changes its sign depend-ing on the momentum direction, serve as a unique plat-form where novel phase-related phenomena, e.g., sponta-neous formation of half flux quanta in a tri-junction ofcuprate superconductors [2], take place. Compared withthe well-investigated Josephson currents, the spatial andenergy dependence of the superconducting order param-eter and quasiparticle states around these junctions re-main to be understood.Recent progress in scanning tunneling microscopy(STM) and spectroscopy (STS) technologies opens up away to directly visualize the spatial variation of the elec-tronic states in superconducting hetero-structures [3–5].However, STM/STS studies on superconducting junc-tions made of unconventional superconductors are stilldemanding. There are two reasons which make it diffi-cult to study unconventional junctions. First, it is often challenging to artificially fabricate well-defined junctionsof unconventional superconductors. Second, in most ofunconventional superconductors, surfaces are not elec-tronically neutral; the resultant charge accumulation atthe surfaces prevents STM/STS from accessing bulk su-perconducting properties. In this study, we solve theseproblems by inspecting the twin boundaries (TBs) in thenodal iron-based superconductor FeSe [6, 7].The TB is a crystallographic plane in a crystalshared by two neighboring domains with one beingthe mirror image of the other. The TBs are oftenformed by a tetragonal-to-orthorhombic structural phasetransition, which reduces the four-fold ( C ) symme-try at high temperature to two-fold ( C ) symmetryat low temperature. In such a case, the orthorhom-bic crystal may contain the TBs parallel to the (110)plane, which act as an atomically well-defined junction.Some unconventional-superconductor-related materials,such as YBa Cu O − δ , AE(Fe − x Co x ) As (AE: alkali-earth element) and NaFeAs, do form TBs upon thetetragonal-to-orthorhombic transition which were iden-tified by STM/STS measurements [8–10]. However,unavoidable surface state formation and/or insufficientamount of chemical doping prevent the STM/STS mea-surements to access superconductivity near TBs in thesematerials.FeSe (superconducting transition temperature T c ≈ c axis [11]. This guarantees the per-fect cleaved surface which is electronically neutral. Thetetragonal-to-orthorhombic structural phase transition,which is likely caused by the orbital ordering [12–17], oc-curs at T s ≈
90 K and the TBs are spontaneously formedin the orthorhombic phase as illustrated in Fig. 1(b).Band-structure calculations show that the Fermi sur-face of FeSe consists of hole cylinders around thezone center and compensating electron cylinders aroundthe zone corner [18, 19]. Several measurements, in-cluding penetration depth, quasiparticle interference,thermoelectric response [7], quantum oscillations [14,20, 21], and angle-resolved photoemission spectroscopy(ARPES) [12–15, 22] reveal that the Fermi surface in theorthorhombic phase consists of one hole and one (or two[14, 21]) electron bands, both of which have very lowcarrier densities. The tunneling spectrum [6], tempera-ture dependence of the penetration depth down to 80 mKand the residual thermal conductivity at T → ≈ . II. EXPERIMENTAL METHOD
STM/STS experiments were conducted in a constant-current mode with a commercial ultra-high vacuum very-low temperature STM (UNISOKU, USM-1300) modi-fied by ourselves [24]. The samples used in this studywere high-quality bulk single crystals grown using thevapor transport method [26]. Superconducting tran-sition temperature defined at zero resistance is about9 K. These crystals are undoped and stoichiometric, en-abling us to investigate uniform and clean TBs. Sam-ples were cleaved in-situ at liquid N temperature toprepare clean and flat (001) surfaces. Immediately af-ter cleaving, the samples were transferred to the STMunit kept below 10 K. We used electrochemically-etchedpolycrystalline tungsten wires for the scanning tips whichwere cleaned and sharpened in-situ by field evaporationusing field-ion microscopy. The tunneling conductance g ( r , E ) ≡ dI t /dV s ( r , E ) reflecting the local density ofstates (LDOS) at a position r and energy E , was ac-quired by standard lock-in technique. Here, I t and V s denote the tunneling current and the sample-bias volt- age, respectively. III. RESULTS AND DISCUSSIONA. Imaging the twin boundary in FeSe
Figure 1(c) depicts an STM image of the cleaved sur-face of an FeSe single crystal at temperature T = 1 . π/
2, indicating that the “groove” represents theTB. We also observed that the elongated vortex cores [6],which were imaged by mapping g ( r , E = 0) in a magneticfield, are rotated by π/ B. Local density of states across the twinboundaries
We examined the LDOS evolution across the TB bytaking a series of g ( r , E ) along the line indicated inFig. 2(a). Here and in the following, we are interestedonly in the evolution of g ( r , E ) along the x axis run-ning perpendicular to the TB leaving the y coordinateconstant, hence g ( x, E ). Figure 2(b) shows an intensityplot of g ( r , E ). Individual spectra taken at representa-tive points are depicted in Fig. 2(c). At the position faraway from the TB (I), g ( r , E ) exhibits a superconduct-ing gap with clear quasiparticle peaks at ≈ ± . ≈ ± . g ( r , E ) = 0 inan extended E region near E = 0, g ( r , E ) in FeSe ap-proaches zero only for E → E = 0 is negligi-bly small, indicating that the TB hardly gives rise to apair breaking effect. In the vicinity of the TB, the quasi- FIG. 1: (a) Crystal structure of FeSe visualized using the VESTA program [25]. (b) Schematic top view of the atomicarrangement near the TB of FeSe (not in scale). Green filled circle and orange open circle denote top-most Se and Fe atoms,respectively. Se atoms beneath the Fe layer are not shown. (c) A constant-current STM image of the cleaved (001) surfaceof FeSe at 1.5 K showing the TB running from bottom left to top right. Crystallographic axes parallel to the Fe-Fe directionare shown by white arrows ( b > a ). The two insets show a magnified image of the defect (8.8 nm × π/ V s = +95 mV and I t = 10 pA. (d) Zero-bias conductance image g ( r , E = 0) at 1.5 K showing vortices. A magnetic fieldof 1 T was applied along the c axis. The tip was stabilized at V s = +10 mV and I t = 100 pA. A bias modulation amplitude V mod = 0 .
21 mV rms was used for spectroscopy. (e) A low-bias STM image at 1.5 K taken with V s = +20 mV and I t = 10 pA.The field of view for (c)-(e) is the same. (f) An atomic-resolution STM image near the TB which is running vertically in thecenter of the field of view. V s = +95 mV and I t = 100 pA. particle peak and the shoulder associated with the su-perconducting gap diminish, and instead, sharp particle-hole symmetric peaks appear at E ≈ ± . ξ ab ≈ H c ( k c ) ≈
15 T [7, 20]. These results suggest that the1.5 meV peak represents the bound state induced by theTB.Another interesting observation is that low-energyquasiparticle excitations are modified over a very longdistance from the TB. High-resolution g ( r , E ) spectra atthe positions of (I), (II) and (III) are plotted in Fig. 2(d).While the overall V-shaped behavior is maintained, theexact shape near the bottom of the gap depends on theposition. In order to examine this behavior, we fit anempirical power-law g ( r , E ) ∝ | E | α to the low-energy( | E | < . α as afunction of the distance from the TB at x = 0 [Fig. 2(e)].Except close to the TB ( | x | . ξ ab ) where the 1.5 meV peaks dominate, α increases gradually with decreasing x by about ≈ α continues to change even at | x | > ξ ab ( ≈
50 nm), indicating an unexpectedly long-distance in-fluence of the TB.The long-distance TB effect on the LDOS can be seenin a more dramatic way in two junctions in series formedby two TBs. As shown in Fig. 3(a) we find an area wheretwo TBs are running parallel to each other. The dis-tance between the TBs is 34 nm, which is about 7 timeslarger than ξ ab . Figure 3(b) shows the spatial evolutionof g ( r , E ) across the double TBs. Individual spectra atrepresentative points are plotted in Fig. 3(c). The overallspectral features, the 2.5 meV peak, the 3.5 meV shoul-der and the 1.5 meV peak observed near a single TB areall reproduced (positions I, II, and III). However, the low-energy spectrum taken inside the central domain (posi-tion IV) shows a striking anomaly which is absent in thecase of a single TB. Figure 3(d) depicts g ( r , E ) spectraat low energies. It is clear that, in between the doubleTBs, there is a finite energy range where g ( r , E ) is almost FIG. 2: (a) A constant-current STM image near a TB taken at V s = +95 mV and I t = 100 pA. (b) Intensity plot of g ( r , E )along the yellow broken line in (a). V s = +20 mV, I t = 100 pA and V mod = 0 .
05 mV rms . (c) Tunneling spectra at therepresentative points indicated in (a). Positions (II) and (II’) are symmetric about the TB. V s = +20 mV, I t = 100 pA and V mod = 0 .
05 mV rms . (d) High-resolution tunneling spectra at low E taken at the same positions as for (c). V s = +10 mV, I t = 100 pA and V mod = 0 .
025 mV rms . Open symbols and solid lines denote experimental data and fitted results, respectively.Spectra shown in (c) and (d) are shifted vertically for clarity. (e) The exponent α ( g ( r , E ) ∝ | E | α ) determined from the fit tothe experimental data in the range of | E | ≤ . completely zero. The noticeable difference of the gapstructure between inside and outside the central domainis clearly seen in Fig. 3(e), which shows the exponent α plotted as a function of the distance from one of the TBs; α is strongly enhanced in the central domain and peaksat the middle of the domain. The large power α ≈ ≈ x ,essentially indistinguishable from an exponential energydependence. This apparent large power again corrobo-rates the finite gap opening in the excitation spectrum ofquasiparticle. C. Possible time-reversal-symmetry-broken statenear the twin boundary
The above observations, the TB-induced bound statesat finite energies and the suppression of the low-energyquasiparticle excitations over a length scale much longerthan ξ ab , suggest a novel role of the TB in an unconven-tional superconductor. Before discussing the origin ofthese anomalies, we briefly review what can be expectedat a TB of FeSe. Recent high-resolution laser-ARPESmeasurements of FeSe indicate that the hole cylinder isfully gapped [27], implying that the line nodes are present on the electron cylinder, as has been also inferred fromvortex imaging [6]. Given this information, we considertwo possible phase structures for symmetry of the su-perconducting gap across a TB as illustrated in Fig. 4,where either the global phase of the superconducting gapis fixed to the crystallographic axis [Fig. 4(a)] or is flippedacross the TB [Fig. 4(b)]. It should be noted that the signof either the nodal gap or the nodeless gap is reversedbetween the two domains in Fig. 4(a) or Fig. 4(b), re-spectively. This means that the amplitude of at leastone of the gaps vanishes at the TB, giving rise to thezero-energy quasiparticle state that should appear as azero-energy peak in g ( r , E ). This argument applies notonly for the particular phase structure shown in Fig. 4but also for a general case in which nodal and nodelessgaps reside on multiple Fermi surfaces.The observed bound-state peak at 1.5 meV apparentlycontradicts this conjecture and suggests instead that theTB induces an additional gap component which shiftsthe position of a zero-energy peak to a finite energy. Wepoint out that, as long as the induced gap is real, a sumof the bulk gap and the TB-induced gap reverses its signat a finite distance from the TB and still gives rise toa zero-energy peak. However, as shown in Fig. 2(b), wedid not observe a zero-energy peak in g ( r , E ) over more FIG. 3: (a) Constant-current STM image with double TBs taken at V s = +95 mV and I t = 10 pA. (b) Intensity plot of g ( r , E ) along the yellow broken line in (a). Positions of the TBs are indicated by broken lines. A low-conductance positionat ≈ -47 nm is due to a point defect nearby. V s = +20 mV, I t = 100 pA and V mod = 0 .
05 mV rms . (c) Tunneling spectra atthe representative points I to IV indicated in (a). V s = +20 mV, I t = 100 pA and V mod = 0 .
05 mV rms . (d) High-resolutiontunneling spectra at low E taken at the same positions as for (c). V s = +10 mV, I t = 100 pA and V mod = 0 .
025 mV rms .Symbols and solid lines denote experimental data and fitted results, respectively. Spectra shown in (c) and (d) are shiftedvertically for clarity. (e) The exponent, ( g ( r , E ) ∝ | E | α ) determined by the fitting in the range of | E | ≤ . than 100 nm from the TB. Thus, we speculate that theinduced gap has an imaginary component, which meansthat time reversal symmetry is locally broken near theTB. In such a case, bound state peaks are located atfinite energies E = ± ∆ cos( δϕ/
2) because the phase shift δϕ on the TB is reduced from π [28, 29]. Here, ∆ is theamplitude of the superconducting gap. The possibilityof the TB-induced time-reversal-symmetry-broken statehas been argued theoretically in d -wave YBa Cu O − δ with a small s -wave component [30], but the experimentalobservation is still lacking.In order to substantiate the relevance of this scenario,we have calculated the spatial evolution of the LDOSfor a model order parameter with broken time-reversalsymmetry near TBs. The C -symmetric order parameteris represented by a sum of the isotropic component ∆ iso and the four-fold nodal component ∆ φ sin(2 φ ),∆( x ) = ∆ iso + ∆ φ ( x ) sin(2 φ ) , (1)where φ is the azimuthal angle in the momentum space;see Fig. 4. We assume that the global phase of the orderparameter is fixed to the crystallographic axis, that is,the nodal component changes its sign across a TB as shown in Fig. 4(a). The spatial variation of ∆ φ arounda TB at x = x is modeled by the form∆ φ ( x ) = ∆ bulk4 φ { tanh[( x − x ) /ξ ] cos θ ( x ) + i sin θ ( x ) } , (2)where the x axis is taken to be perpendicular to the TB,and ∆ bulk4 φ is the amplitude of ∆ φ in the bulk. The phase ϕ of ∆ φ ( x ) equals θ ( x ) for x − x ≫ ξ and π − θ ( x ) for − ( x − x ) ≫ ξ . The phase variable θ ( x ) is assumed totake a nonvanishing value near the TB and exponentiallydecay with another length scale ˜ ξ . It is important tonote that the characteristic length ˜ ξ for the local time-reversal symmetry breaking can be much longer than thecoherence length ξ [30]. (The derivation of the length˜ ξ is given in Appendix B.) To account for low-energyexcitations near the nodes observed in the LDOS, wefocus on the electron cylinder with nodal gaps by settingthe parameters ∆ iso = 0 . and ∆ bulk4 φ = 0 . .As a model order parameter with a TB at x = x = 0,we take θ ( x ) = ( π/ x/ ˜ ξ ) with ˜ ξ = 5 ξ , which gives∆ φ ( x = 0) = ( i/ bulk4 φ . The order parameter ∆ φ ( x )is shown in Fig. 5(a), where | ∆ φ | changes with the lengthscale ξ while Im(∆ φ ) decays with the longer length scale FIG. 4: Schematic illustration of the phases of the superconducting gaps across the TB shown by the red line. Top panelrepresents the iron lattice near the TB together with the momentum-space phase structure of the superconducting gaps openingat multiple Fermi cylinders, a hole cylinder at the center and electron cylinders at the corner of the Brillouin zone (black brokensquare). Different colors (red and blue) denote different signs of the phase. We assume that the gap node exists on the electroncylinder and the sign reversal is between the main lobe of the gap on the electron cylinder and the gap on the hole cylindersbut the argument given in the text applies not only for this particular case but also for other cases. There are two possibilities:either the phase structure is fixed to the lattice (a) or is flipped across the TB (b). In the former case, the nodal component∆ φ should change its sign across the TB, whereas the sign of the isotropic component ∆ iso (either due to the fully gappedFermi cylinder or associated with the C symmetry of the nodal gap) should be reversed in the latter case. ˜ ξ . The phase ϕ abruptly changes near the TB and grad-ually approaches 0 or π . Using this order parameter,we calculate the spatial dependence of the LDOS withinthe quasi-classical approximation [31]. Figure 5(b) showsthe global peak structure of the LDOS at representativepoints calculated with energy smearing of η = 0 . .Far from the TB, namely in the bulk (I), the LDOShas peaks at | E | = ∆ bulk4 φ ± ∆ iso . On the TB (III),the peaks observed in the bulk are suppressed, and al-ternative peaks appear at E ≈ ± . , which corre-spond to the bound states whose energies are shifted from E = 0 due to the local time-reversal symmetry breakingin ∆ φ . The bound-state peaks disappear at x = 3 ξ (II)since their wave functions decay into the bulk with the length scale ξ . These features of the calculated LDOSarising from the bound states are consistent with theLDOS peaks observed at E ≈ ± . η = 0 . . The clear V-shaped LDOS in the bulk (I)changes to the U-shaped LDOS upon approaching theTB, in agreement with the increase of the exponent α evaluated from the experimental data [Fig. 2(e)]. Thelow-energy LDOS is finite at x = 0 (III) and x = 3 ξ (II)because low-energy quasiparticles with momenta alongthe nodal directions of the bulk gap can linger over longdistance and reach the TB, even though the local gap, | ∆( x ) | = q [∆ iso + Re(∆ φ ) sin(2 φ )] + [Im(∆ φ ) sin(2 φ )] , (3)does not vanish near the TB where Im(∆ φ ) = 0.We also calculate the LDOS for double TBs located at x = ± x = ± . ξ , taking the model order parameter of theform -1-0.5 0 0.5 1 0-1-0.5 0 0.5 1-20 -15 -10 -5 0 5 10 15 20 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5-1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 -0.1 0 0.1 (a)(d) (e) (f)(c)(b) (III) (II)(IV)(III) (II) (III)(II)(I)(III)(II)(I)(IV) (III)(II)(I)(III)(II)(I)(IV) FIG. 5: (a) A model order parameter ∆ φ ( x ) with a TB located at x = 0. LDOS’s in the bulk (I), at x = 3 ξ (II), and on a TB(III), which are calculated with an energy smearing of η = 0 . (b) and η = 0 . (c). The lines (I) and (II) have offsets g and 0 . g in (b) and 0 . g and 0 . g in (c), respectively, where g is the density of states in the normal state at the Fermienergy. (d) A model order parameter ∆ φ ( x ) with double TBs located at x = ± . ξ . The LDOS in the bulk (I), at x = 7 ξ (II),on a TB (III), and at the middle point between double TBs (IV), which are calculated with η = 0 . (e) and η = 0 . (f). The lines (I), (II), and (III) have offsets g , (2 / g , and (1 / g in (e) and 0 . g , 0 . g , and 0 . g , in (f) respectively. ∆ φ ( x ) = ∆ bulk4 φ { tanh[( x − x ) /ξ ] tanh[( x + x ) /ξ ] cos θ ( x ) + i sin θ ( x ) } , (4)shown in Fig. 5(d). We assume that the distance 2 x between the TBs is in the range ξ ≪ x . ˜ ξ . The phase θ ( x ) is an even function of x and takes a maximum valueat x = 0. The global peak structure of the LDOS andits low-energy blowup are shown for representative pointsalong the x direction in Fig. 5(e) and 5(f), respectively.The large peaks at | E | ≈ . on a TB (III) and thesmall peaks at | E | ≈ . at the middle point x = 0 be-tween the TBs (IV) in Fig. 5(e) originate from the samedispersive mode of bound states at a TB. The calculatedLDOS spectrum at x = 0 between the double TBs (IV) inFig. 5(f) exhibits a clear energy gap extending over theregion | E | . . , reflecting the existence of a largerlocal gap at x = 0, where the bulk low-energy quasi-particles cannot reach. We conclude that the local gapenhanced by the local time-reversal symmetry breakingnear TBs over the length scale ˜ ξ can explain the strongsuppression of the LDOS between the two TBs observedin our STM/STS experiments. IV. SUMMARY
We have reported on the visualization of the atomicscale variation of the quasiparticle states of the nodalsuperconductor FeSe near TBs that enforce a sign in-version at least one of the superconducting gaps open-ing on multiple Fermi cylinders. In contrast to the ex-pectation that the sign inversion generates a zero-energyquasiparticle bound state near the TB, the TB-inducedquasiparticle states are not at zero but at finite ener-gies E ≈ ± . ξ ab .An even more dramatic change in the low-energy spec-trum has been detected in the region between doubleTBs separated by a distance ≈ ξ ab , where the quasipar-ticle weight near the Fermi energy is almost completelyremoved in the energy range | E | . . Acknowledgments
This work has been supported by Japan – GermanyResearch Cooperative Program, KAKENHI from JSPSand Project No. 56393598 from DAAD, and the “Topo-logical Quantum Phenomena” (No. 25103713) KAK-ENHI on Innovative Areas from MEXT of Japan.
Appendix A: Absence of lattice distortion inducedby the twin boundary
Although STM has a high spatial resolution, possi-ble creep in the piezoelectric scanner and/or the thermaldrift make it difficult to estimate the small distortionsin the topographic image. Here we utilize the so-calledLawler-Fujita algorithm [32] to deduce the lattice distor-tion and show that the TB-induced strain is negligiblysmall.First we briefly explain the principle of the methodol-ogy. The observed STM topography T ( r ), which mainlyrepresents the top-most Se lattice, can be expressed as T ( r ) = T [cos { q x · ( r − u ( r )) } + cos { q y · ( r − u ( r )) } ] + · · · . (A1)Here, T is the amplitude of the Se-lattice modulation, q x and q y are wave vectors for the Se lattice, and · · · represent all other modulations. The distortions from theperfect lattice is described by the displacement field u ( r )that can be regarded as a spatially varying phase of the q x and q y modulations. This approximation is justifiedas long as the length scale of distortions is much longerthan the Se-Se distance a Se . Standard phase-sensitivedetection scheme can be used to evaluate u ( r ). By mul-tiplying T ( r ) and the reference signal cos ( q x · r ), we get T ( r ) cos ( q x · r ) = T h cos ( q x · u ( r ))+ cos (2 q x · r − q x · u ( r ))+ cos (( q x + q y ) · r − q y · u ( r ))+ cos (( − q x + q y ) · r − q y · u ( r )) i + · · · . (A2)All terms except the first exhibit periodic spatial modula-tions, which can be removed by low-pass Fourier filteringLPF {· · · } .LPF { T ( r ) cos ( q x · r ) } = T q x · u ( r )) . (A3)By using the quadrature reference sin ( q x · r ), we getLPF { T ( r ) sin ( q x · r ) } = T q x · u ( r )) . (A4)Therefore, we obtain u x ( r ), the x component of u ( r ) as u x ( r ) = a Se π tan − LPF { T ( r ) sin ( q x · r ) } LPF { T ( r ) cos ( q x · r ) } . (A5)The y component u y ( r ) can also be deduced as u y ( r ) = a Se π tan − LPF { T ( r ) sin ( q y · r ) } LPF { T ( r ) cos ( q y · r ) } . (A6)A schematic model of atomic arrangement near the TBis shown in Fig. 6(a). We expect that the orthorhombicdistortion affects the atomic arrangement along the y di-rection across the TB, while the periodicity along the x direction remains intact. In order to verify this modeland to check if there is an additional lattice distortion, wecalculated u x ( r ) and u y ( r ) of the high-resolution STMimage containing a TB running along the y direction[Fig 6(b)]. Reference wave vectors q x and q y were ob-tained by the Fourier analysis in the left domain. For low-pass Fourier filtering, we picked up only long-wavelengthcomponents by using a Gaussian mask with half width atthe half maximum of 0 . π/a Se ). Since there is a trans-lational symmetry along the TB, we average u x ( r ) and u y ( r ) along the y direction, yielding u avg x ( x ) and u avg y ( x ),respectively. This significantly enhances the signal-to-noise ratio.Figures 6(c) and (d) show u avg x ( x ) and its x deriva-tive. There is no noticeable anomaly in both u avg x ( x ) and du avg x ( x ) /dx , except for the smooth background associ-ated with the creep of the piezoelectric scanner. By con-trast, u avg y ( x ) exhibits a sharp kink at the TB [Fig. 6(e)].These features are consistent with the model shown inFig. 6(a). It should be noted that du avg y ( x ) /dx shown inFig. 6(f) is almost completely constant in both domains, FIG. 6: (a) A schematic top view of the atomic arrangement near the TB of FeSe (not in scale). Note that an atomicperiodicity along the x direction is hardly affected by the TB. (b) A constant-current STM image taken over a field of view of180 nm ×
90 nm on a grid of 4096 × V s = +95 mV and I t = 100 pA. Inset:Fourier-transformed STM image taken in the left domain of the main panel. A peak at q x is sharp and well isolated from otherfeatures, guaranteeing that q x · u ( r ) can be treated as a spatially varying phase of the q x -modulations. (The same is true for q y .) (c) The x component of u ( r ) averaged over the y axis. Thick solid line and thin dashed line denote the data taken by theforward (left to right) and backward (right to left) scans, respectively. The symmetric hysteretic behavior between the forwardand backward scans indicates that u avg x ( x ) is governed by the creep of the scanner. No anomaly is observed at the TB. (d)The x derivative of u avg x ( x ). Spike-like features are associated with the point defects in the image. (e) The y component of u ( r ) averaged over the y axis. Thick solid line and thin dashed line denote the data taken by the forward (left to right) andbackward (right to left) scans, respectively. Since the y direction is the slow-scan direction, the effect of the creep is small. Aclear kink is observed at the TB. (f) The x derivative of u avg y ( x ). indicating that the TB-induced strain to the lattice isnegligibly small.The observed value of du avg y ( x ) /dx ≈ − . × − in the right domain means that the angle β defined inFig. 6(a) is +0.63 ◦ . This means that orthorhombic dis-tortion ( b − a ) / ( b + a ) ≈ . × − , being consistentwith the X-ray result [33]. Even if there were an addi-tional lattice distortion associated with the TB, it shouldbe much smaller than this tiny orthorhombic distortionwhich we have clearly detected. Appendix B: Asymptotic forms of the orderparameter derived by the Ginzburg-Landau theory
We derive asymptotic forms of the order parameterfar from TBs using the Ginzburg-Landau (GL) theory.We consider the GL free-energy functional for tetragonalsymmetric systems [30] as an expansion in the isotropic s -wave component ∆ iso and the four-fold d -wave compo-nent ∆ φ of the order parameter: F GL [∆ iso , ∆ φ ] = Z dV X µ =iso , φ (cid:2) ˜ a µ ( T ) | ∆ µ | + b µ | ∆ µ | + K µ | ∇ ∆ µ | (cid:3) + γ | ∆ iso | | ∆ φ | + γ (cid:0) ∆ ∗ ∆ φ + ∆ ∆ ∗ φ (cid:1) + e K ∂ a ∆ iso ) ∗ ( ∂ a ∆ φ ) − ( ∂ b ∆ iso ) ∗ ( ∂ b ∆ φ ) + c . c . ] ) , (B1)0where we have neglected the vector potential as it doesnot play an important role in our discussion. The co-efficients b µ , K µ , and e K are positive and ˜ a µ ( T ) = a µ ( T /T c µ −
1) with positive a µ . The differential oper-ator ∇ = ( ∂ a , ∂ b ) is defined according to the crystal axes a and b . As in Ref. 30, we assume γ >
0, so that thefree energy is minimized at ϕ = ± π/
2, and the time-reversal-symmetry-broken s ± id state is stabilized whenboth ∆ iso and ∆ φ are finite.The effect of orthorhombic distortion is taken into ac-count by adding the following term to the free-energy functional [30]: F ǫ = cǫ Z dV (∆ ∗ iso ∆ φ + ∆ iso ∆ ∗ φ ) , (B2)where c is a positive parameter and ǫ = ǫ aa − ǫ bb is theparameter of the orthorhombic lattice distortion. Thetotal free energy for a uniform state in the bulk is thengiven by F GL + F ǫ V = X µ =iso , φ (cid:0) ˜ a µ | ∆ µ | + b µ | ∆ µ | (cid:1) + γ | ∆ iso | | ∆ φ | + γ | ∆ iso | | ∆ φ | cos(2 ϕ ) + 2 cǫ | ∆ iso || ∆ φ | cos ϕ, (B3)where ∆ µ = | ∆ µ | e iϕ µ and the relative phase ϕ = ϕ φ − ϕ iso . If c | ǫ | ≥ γ | ∆ iso || ∆ φ | , then the free energy is minimizedat ϕ = 0 for ǫ < ϕ = π for ǫ >
0. In the following discussion we assume that this inequality is satisfied andthe time reversal symmetric s ± d state is realized in the bulk.Next, we consider a TB located at x = x along the y axis, where the x and y axes are rotated by 45 ◦ from thecrystalline axes, x = ( a − b ) / √ y = ( a + b ) / √
2. The orthorhombic lattice distortion parameter ǫ changes itssign across the TB. We assume ǫ ( x ) → ∓| ǫ | for x → ±∞ , so that the s ± d state is realized in x → ±∞ . Near the TBwhere ǫ ( x ) is small, the s ± id state is favored. Then, the area density of the total free energy is given by f GL + f ǫ = Z dx " X µ =iso , φ (cid:0) ˜ a µ | ∆ µ | + b µ | ∆ µ | + K µ | ∂ x ∆ µ | (cid:1) + γ | ∆ iso | | ∆ φ | + γ (cid:0) ∆ ∗ ∆ φ + ∆ ∆ ∗ φ (cid:1) + cǫ ( x )(∆ ∗ iso ∆ φ + ∆ iso ∆ ∗ φ ) . (B4)Let us first assume that ∆ iso is a real and uniform order parameter while ∆ φ changes its sign across the TB, asshown in Fig. 4(a). If we restrict ∆ φ to be real, then ∆ φ varies over the coherence length [30] ξ = s K φ ˜ a φ + 6 b φ | ∆ bulk4 φ | + ( γ + γ ) | ∆ iso | , (B5)where | ∆ bulk4 φ | is the amplitude of ∆ φ in the bulk. However, we expect that time-reversal symmetry should be locallybroken at the TB. Thus, we allow ∆ φ to be complex, ∆ φ ( x ) = | ∆ φ ( x ) | e iϕ ( x ) . With this order parameter, the totalfree energy is given by f GL + f ǫ = Z dx " ˜ a iso | ∆ iso | + b iso | ∆ iso | + ˜ a φ | ∆ φ | + b φ | ∆ φ | + γ | ∆ iso | | ∆ φ | + γ | ∆ iso | | ∆ φ | cos(2 ϕ ) + 2 cǫ ( x − x ) | ∆ iso || ∆ φ | cos ϕ + K φ (cid:2) ( ∂ x | ∆ φ | ) + | ∆ φ | ( ∂ x ϕ ) (cid:3) . (B6)In the bulk region ( x − x ≫ ξ ) where ∂ x | ∆ φ | = 0 and ǫ ( x ) = −| ǫ | , the GL differential equation to minimize f GL + f ǫ is K φ ∂ x ϕ = − | ∆ iso || ∆ bulk4 φ | (cid:2) γ | ∆ iso || ∆ bulk4 φ | sin(2 ϕ ) − c | ǫ | sin ϕ (cid:3) . (B7)1Since ϕ ≪ ϕ ∝ exp (cid:16) − x/ ˜ ξ (cid:17) with the characteristic length˜ ξ = vuut K φ | ∆ bulk4 φ || ∆ iso | ( c | ǫ | − γ | ∆ iso || ∆ bulk4 φ | ) . (B8)The characteristic length diverges when approaching thephase boundary, where c | ǫ | = 2 γ | ∆ iso || ∆ φ | , betweenthe time reversal symmetric s ± d state and the time-reversal-symmetry-broken s ± id state. Finally, we consider double TBs at x = ±| x | , where ξ ≪ | x | . ˜ ξ . We assume ǫ > ǫ < x = 0 between thedouble TBs, we can set | ∆ φ | = | ∆ bulk4 φ | because | x | ≫ ξ .With this approximation, the GL differential equation tominimize f GL + f ǫ for | x | ≪ | x | is K φ ∂ x ϕ = − | ∆ iso || ∆ bulk4 φ | (cid:2) γ | ∆ iso || ∆ bulk4 φ | sin(2 ϕ ) + c | ǫ | sin ϕ (cid:3) . (B9)Integration of the differential equation yields K φ ( ∂ x ϕ ) = | ∆ iso || ∆ bulk4 φ | [ γ | ∆ iso || ∆ bulk4 φ | cos(2 ϕ ) + 2 c | ǫ | cos ϕ − γ | ∆ iso || ∆ bulk4 φ | cos(2 ϕ ) − c | ǫ | cos ϕ ] , (B10)where the integration constant is determined from theconditions ∂ x ϕ ( x = 0) = 0 and ϕ ( x = 0) ≡ ϕ . Sincewe assume the distance between the TBs is in the range | x | . ˜ ξ , the relative phase does not reach π at x = 0,i.e., ϕ < π . For ϕ − ϕ ≪ x = 0, the differentialequation (B10) has the solution ϕ ( x ) = ϕ − (cid:18) x ˜ ξ (cid:19) . (B11) For the model order parameter shown in Fig. 5(d), wehave determined ϕ and ˜ ξ by the continuity condi-tion at x = ±| x | /
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