Topological orders competing for the Dirac surface state in FeSeTe surfaces
TTopological orders competing for the Dirac surface state in FeSeTe surfaces
Xianxin Wu,
1, 2
Suk Bum Chung,
3, 4, 5
Chaoxing Liu, and Eun-Ah Kim ∗ Department of Physics, the Pennsylvania State University, University Park, PA, 16802 Beijing National Laboratory for Condensed Matter Physics,and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, University of Seoul, Seoul 02504, Korea Natural Science Research Institute, University of Seoul, Seoul 02504, Korea School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea Department of Physics, Cornell University, Ithaca, New York 14853, USA
FeSeTe has recently emerged as a leading candidate material for the two-dimensional topologicalsuperconductivity (TSC). Two reasons for the excitement are the high T c of the system and the factthat the Majorana zero modes (MZMs) inside the vortex cores live on the exposed surface rather thanat the interface of a heterostructure as in the proximitized topological insulators. However, the recentscanning tunneling spectroscopy data have shown that, contrary to the theoretical expectation,the MZM does not exist inside every vortex core. Hence there are “full” vortices with MZMs and“empty” vortices without MZMs. Moreover the fraction of “empty” vortices increase with an increasein the magnetic field. We propose the possibility of two distinct gapped states competing for thetopological surface states in FeSeTe: the TSC and half quantum anomalous Hall (hQAH). The latteris promoted by magnetic field through the alignment of magnetic impurities such as Fe interstitials.When hQAH takes over the topological surface state, the surface will become transparent to scanningtunneling microscopy and the nature of the vortex in such region will appear identical to what isexpected of the vortices in the bulk, i.e., empty. Unmistakable signature of the proposed mechanismfor empty vortices will be the existance of chiral Majorana modes(CMM) at the domain wall betweena hQAH region and a TSC region. Such CMM should be observable by observing local density ofstates along a line connecting an empty vortex to a nearby full vortex. Introduction –
One particularly exciting feature of thetopological insulator (TI) its potential to host the Ma-jorana zero mode (MZM), which has led to many pro-posals [1–4] and attempts [5–9] to realize MZM throughintroducing superconducting gap to the TI surface state.Early works focused on introducing topological super-conductivity (TSC) through proximity effect [3, 10–12].More recently, the prospect of FeSeTe possessing at itssurface the equivalent of TI surface state with supercon-ducting gap proximity induced by the high T c intrinsicbulk superconductivity raised much enthusiasm [13–16].More recently it has been recognized that such state pos-sesses a higher order topology [17–20].Intensive experimental investigations of FeSeTe con-firmed the existence of Dirac surface state in the nor-mal state above T c [21]. The predicted evidence for theMZM in the vortex core of superconducting state was thezero-bias peak in scanning tunneling microscopy (STM).Indeed, the STM is a particularly suitable probe for theMZM in this material as it would exist at the surface[11, 12]. Despite several observations of a zero-bias peakin cores of some vortices [22–25], an apparent contradic-tion to the prediction has also been observed in the in-creasing fraction of “empty” vortices without a zero-biaspeak upon the increase in magnetic field [26, 27]. A care-ful study [26] revealed that the “empty” vortices cannotbe accounted for by a simple picture of pair-wise annihi-lation of MZM between two near-by vortices. AlthoughRef. [28] showed that a model allowing for long-range in-teraction among MZM’s far separated can in principle explain the “empty” vortices, an alternative explanationwith simpler starting point and a falsifiable prediction isdesirable.Here we provide an alternative interpretation of theobserved ”empty” vortices based on the role of the mag-netic field on aligning local moments of Fe-interstitials.Our main physical picture is summarized in Fig. 1a-d. Asit is known from the study of magnetic dopants added toTI surface states, the exchange field from magnetic impu-rities also gap the TI surface state to form the half quan-tum anomalous Hall (hQAH) state with the half-integerquantization of Hall conductivity [2, 29–31] (Fig. 1a -1b) Uneven distribution of interstitials can nucleate thehQAH regions on the surface of FeSeTe when the mo-ments get aligned with magnetic fields (Fig. 1c), prevent-ing TSC to form in that very region. Such hQAH surfacestate will reveal the bulk superconductivity to STM andthe vortices penetrating hQAH surface will show proper-ties of the bulk superconducting state with topologicallytrivial the s ± pairing [32, 33], i.e., becoming “empty”.With increasing magnetic fields, more hQAH regions arenucleated on the surface of FeSeTe, thus providing a nat-ural explanation of the increasing faction of empty vor-tices observed in experiments. Interestingly, it has beenknown that a boundary between hQAH and TSC shouldhost a chiral Majorana mode (CMM) [4, 34–36]. Henceour key prediction is that the MZM that would have beenin the vortex core transforms into the CMM located atthe boundary between the hQAH and TSC on the sur-face of FeTeSe(Fig. 1d). In the rest of this Letter, we first a r X i v : . [ c ond - m a t . s up r- c on ] A p r (a) (b)(c) (d) FIG. 1. (a) Gapless Dirac surface state with random mag-netic moments of Fe interstitials and (b) the gaped surfacestate when magnetic moments are aligned by external mag-netic fields. (c) Domain wall between the TSC region (withdominant superconducting gap) and the hQAH region (withdominant magnetic gap) on the surface of FeTeSe. (d) MZMexists at the vortex core (the red spots) in the TSC region, butnot in the hQAH region. The CMM exists at the boundarybetween the TSC and hQAH regions. present our proposal using a low energy effective theoryand then support it with a numerical simulation on amicroscopic model.
Exchange field and low energy effective theory –
Con-sider the low energy effective theory for the topologicalDirac surface state in FeSeTe. As noted by Jiang et al. [37], the interstitial Fe atoms can provide magnetic im-purities in Fe(Te . Se . ). Although the impurity mo-ments will point in random direction at zero-field (Fig1a), the external field applied to create vortices wouldalign the impurity moments (Fig 1b). In the regions withhigher concentration of aligned impurity moments, theexchange field generated by these moments would coupleto the topological surface state as in magnetically dopedTI [38–43]. Such exchange coupling can be captured by H ex ( r ) = −J (cid:80) i S i · s δ ( r − r i ), where s = (cid:126) σ is thesurface state electron spin, S i and r i are the spin andlocation, respectively, of the Fe interstitial and J is thecoupling constant. This exchange field will be heteroge-neous depending on the distribution of the interstitials.We consider the mean field approximation for the ex-change field, leading to the form H ex ( r ) = − I ex ( r ) · σ ,where I ex ( r ) = J (cid:126) (cid:80) i (cid:104) S i δ ( r − r i ) (cid:105) local is a smoothlyvarying field with (cid:104) ... (cid:105) local representing the average overa small region for Fe moments. In an ordinary topolog-ical insulator, such heterogeneous exchange field shouldresult in hQAH effect with spatially varying gaps for theDirac surface state [44]. However, non-topological bandscrossing the Fermi Surface will mask hQAH states in thenormal state of Fe(Te . Se . ).Once the system develops superconductivity, the hQAH and TSC can compete as the two possible ways ofgapping the Dirac surface state. Moreover, the hQAHregion will reveal itself by leaving the bulk supercon-ductivity bare when the exchange gap dominates overthe superconducting gap. This can be captured bythe BdG Hamiltonian for the Dirac surface state withboth exchange field and the s -wave pairing in the basis( c k , ↑ , c k , ↓ , c †− k , ↓ , − c †− k , ↑ ) T : H BdG = ( v k · σ − µ ) τ z − I ex σ z + ∆ τ x , (1)where σ i and τ i are the Pauli matrices in the spin spaceand particle-hole space, respectively. Here we assumedan s -wave gap to be real and only consider exchange fieldalong the z direction. It is straightforward to find uponincrease in the exchange term, the superconducting gapfor BdG quasiparticles closes at the critical exchange fieldstrength of [45–47] I ex,c = | ∆ | + µ . (2)When | I ex | < | I ex,c | , the TSC dominates to supportthe vortex core MZM, which can be explicitly obtainedby choosing ∆ τ x → | ∆ | ( τ x cos θ − τ y sin θ ) substitution ( θ is the azimuthal angle), which places a superconductingvortex at the origin. The zero mode we obtain for | I ex | < | µ | [3], ψ ↑ ( r ) ψ ↓ ( r ) ψ †↓ ( r ) − ψ †↑ ( r ) = e − (cid:82) r dr (cid:48) | ∆ | (cid:126) v ( µ − I ex ) e − i π √ µ + I ex J (cid:18) √ µ − I ex (cid:126) v r (cid:19) e i π e iθ √ µ − I ex J (cid:18) √ µ − I ex (cid:126) v r (cid:19) e − i π e − iθ √ µ − I ex J (cid:18) √ µ − I ex (cid:126) v r (cid:19) − e i π √ µ + I ex J (cid:18) √ µ − I ex (cid:126) v r (cid:19) (3)where J l is the l -th Bessel function of the first type,reduces the Fu-Kane vortex zero mode by setting first I ex = 0 and then µ = 0 [3]. It can also be generalizedto | I ex | > | µ | using J l ( ix ) = i n I l ( x ) for x ∈ R , where I l is the l -th modified Bessel function of the first type,provided, however, that | ∆( r → ∞ ) | > (cid:112) I ex − µ , i.e. | I ex | < | I ex,c | , as can be seen from the asymptotic formsfor the large real arguments, I l ( x ) ∼ e x / √ πx .On the other hand, when | I ex | > | I ex,c | , hQAH dom-inates without the vortex core MZM. The domain wallCMM can be demonstrated by setting µ = 0 with thedomain wall at y = 0 arising from I ex ( y ) = I Θ( y ) and∆ = ∆ Θ( − y ) will be considered, i.e. H BdG = v k · σ τ z − I Θ( y ) σ z + ∆ Θ( − y ) τ x ; (4)for I = ∆ >
0, it is straightforward to show the exis-tence of the domain wall CMM ψ ↑ ( r ) ψ ↓ ( r ) ψ †↓ ( r ) − ψ †↑ ( r ) = (cid:114) ∆ vL x e ik x x e − ∆ | y | /v / / / − / (5)with the eigenenergy E k x = vk x . Microscopic model –
Next we will support our resultsby the numerical simulations on the bulk model of Fe-SeTe system. For FeSeTe bulk system, the topologicalphase is attributed to the band inversion between twostates with opposite parities at Z point. Taking | S + , + (cid:105) , | S + , − (cid:105) , | P − , (cid:105) and | P − , − (cid:105) as the basis at Z point,the topological electronic structure can be described bythe Hamiltonian in a 3D lattice H T I = (cid:80) k ψ † k H T I ( k ) ψ k and Hamiltonian matrix reads H T I ( k ) = η x d · σ + M k η z − µ, (6)where ψ † k = ( c † S k ↑ , c † S k ↓ , c † P k ↑ , c † P k ↓ ), M k = M + m z cos k z + m x (cos k x + cos k y ) and d i = 2 t i sin k i ( i = x, y, z ). Here η are Pauli matrices in the orbital space.The mass term at Γ and Z points are M + m z + 2 m x and M − m z + 2 m x . Let us take m = − m z = m x ,the above model describes a strong topological insulatorphase with a band inversion at Z point if − < M m < − H BdG = (cid:80) k Ψ † k H T IBdG ( k )Ψ k with Ψ k =[ ψ † k , ψ T − k ( − iσ y )] and the Hamiltonian matrix reads, H T IBdG ( k ) = (cid:18) H T I ( k ) − I ex σ z ∆ s ∆ † s −H T I ( k ) − I ex σ z (cid:19) , (7)where ∆ s is the intra-orbital spin singlet pairing. In theabsence of exchange field, the (001) surface states willbe gapped by superconductivity and form an effective p + ip pairing, where Majorana modes can be trapped ina vortex core of the surface (as described by Eq. 1 with I ex = 0). We then study the effect of exchange field onthe (001) surface states by adopting the above Hamilto-nian with open boundary condition along z direction.The microscopic model reproduces the topologicalphase transition of the low energy effective theory. Fig. 2demonstrates the existence of the topological phase tran-sition of the surface states by fixing the pairing potentialand increasing the exchange field strength. In Fig. 2a,with zero exchange field, the surface state is gappedby superconducting pairing. When the exchange fieldstrength reaches the critical strength which is equal tothe superconducting gap for µ = 0, the gap of the sur-face states closes (Fig. 2b), consistent with the conditionof Eq. 2. With further increasing exchange field, the sur-face state gap reopens and the system is driven into thehQAH state (Fig. 2c).Next we turn to how exchange field affects the vortexcore MZM in topological surface state superconductivity.We introduce a vortex located at the center of the sys-tem by setting ∆ s ( r ) = | ∆ s ( r ) | e iθ and adopt the Hamil-tonian with open boundary conditions along the x, y, z directions. A lattice size of 17 × ×
16 is chosen for the (a) (b) (c)
FIG. 2. Topological phase transition for (001) surface stateswith increasing exchange field I ex : (a) I ex = 0, (b) I ex = − . B z = − .
4. The blue and red dots represent surfacestate at top and bottom surfaces, respectively. The adoptedparameters are: t x,y = t z = 0 . M = 2 . m x = − m z = − .
0, ∆ = 0 . µ = 0. following numerical calculations. The exchange field isonly restricted to the top (001) surface of the system.With the above sample configuration, Figs. 3a and bshow the distribution of the zero-energy local density ofstates on the bottom and top surfaces, respectively, forthe exchange field exceeding the critical strength definedby Eq. 2. One can see that an “empty vortex” appearson the top surface in Fig. 3b, in sharp contrast to the“full vortex” on the bottom surface where there is no ex-change field in Fig 3a. At the core of a full vortex, thereis a well-defined MZM with zero-bias peak in the localdensity of states. On the other hand, the MZM is absentat the core of an empty vortex. Despite of the absenceof MZM in the vortex core, the edges of the top surfaceunder exchange field show a large amount of density ofstates that depict the presence of edge CMM. In Sup-plementary Materials, we study the profile evolution ofzero-energy local density of states on the top surface withincreasing magnetic fields, from which one find that thelocalized MZM gradually extends outside of the vortexand becomes localized on the the edges of (001) surface. Experimental prediction –
Based on our results thathave been well established by both the effective theoryand microscopic bulk model, a natural prediction is theexistence of the domain wall CMM between an emptyvortex and a full vortex. Consider an experimental setupshown in Fig. 1c, in which two vortices are located at theTSC and hQAH regions, respectively. Fig.4a displays thespatial profiles of zero-energy states in the vicinity of afull vortex (left) and an empty vortex (right) and Fig.4bshows the progrssion of the local density of states (LDOS)as a tip marches from a full vortex to an empty vortex.Experimentally, one can implement an STM measure-ment of LDOS along the line connecting a full vortex (in-dicating the TSC state) and an empty vortex (indicatingthe hQAH state). As shown in Fig. 4b, a zero-bias peakis expected to exist in the intermediate region withoutany vortex and can be attributed to the existence of theCMM at the domain wall between the hQAH and TSCregions. This chiral Majorana mode always possessing azero-energy state is distinct from a normal chiral mode (a) (b)
FIG. 3. The 3D profiles of MZMs for (001) surface on a 17 × ×
16 lattice with I ex = − .
4, ∆ = 0 . µ = 0. There isa localized Majorana in the vortex and chiral Majorana modeslocalized on edges on bottom and top surface, respectively. −1.5 −1 −0.5 0 0.5 1 1.500.511.522.5 d I / d V ( a . u . ) Energy/ ∆ (a)(b) ξ FIG. 4. The profiles of MZMs at top and bottom (001) sur-faces (a) and position-dependent local density of states be-tween a normal and an “empty” vortices (b). and the energy spectrum is related to the circumferenceof the region with Zeeman field (see SM). With a largethermal smearing in STM measurements, the LDOS atthe domain wall exhibits a broad peak around zero en-ergy. While the external magnetic field cannot gap outthe CMM, changing its magnitude will shift the locationof the CMM as the hQAH region expands while the TSCregion contracts or vice versa.
Conclusion–
To summarize, we proposed a new mech-anism by which magnetic field can increase the fractionof “empty” vortices without MZM in Fe(Te . Se . ).Our mechanism is purely local , i.e. a vortex is “empty”because of its intersecting the surface inside the hQAHdomain rather than the long-range MZM interaction ef-fects. We postulate that these hQAH domains arise fromthe alignment of local moments associated with Fe inter- stitial which produces heterogeneous exchange fields ex-ceeding the superconducting gap in isolated puddles. Ithas been known that there should be the CMM localizedat the domain wall between regions with dominant su-perconducting gap and regions with dominant exchangegap. Through an explicit calculation on a minimalisticlattice model of topological bands, we showed that MZMin the vortex core of topological superconductor trans-forms into the domain wall CMM upon increase in theexchange field on the region supporting the vortex.Our proposal is distinct from an earlier proposal inRef. [28] that relies on pair-wise extinction of MZM’sthrough tunneling between vortices. In our proposal, theMZM relocates and extends to the domain wall CMM in-stead of disappearing. A clear signature of the proposedmechanism will be the existence of the domain wall CMMbetween an “empty” vortex and a “full” vortex whichcan be detected through STM measurements along a lineconnecting an “empty” vortex to a nearby “full” vortex.Given the clear distinction between the domain wall andthe vortex core as shown in Fig. 4b, our proposal suggeststhat the CMM detection in Fe(Te . Se . ) through theSTM measurement may be relatively easy compared tothe recent transport experiments [48, 49]. Another pre-diction that should be easy to check is that we anticipatethe “full” vortices and “empty” vortices to segregate astheir segregation will represent the regions dominated byTS or by hQAH. Acknowledgements–
We thank Hai-Hu Wen, Tet-suo Hanaguri, and Vidya Madhavan for useful discus-sions. EAK was supported by National Science Founda-tion (Platform for the Accelerated Realization, Analysis,and Discovery of Interface Materials (PARADIM)) un-der Cooperative Agreement No. DMR-1539918. C.X.Lacknowledges the support of the Office of Naval Re-search (Grant No. N00014-18-1-2793), the U.S. Depart-ment of Energy (Grant No. DESC0019064) and Kauf-man New Initiative research grant KA2018-98553 of thePittsburgh Foundation. SBC acknowledges the supportof the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT) (No.2020R1A2C1007554). ∗ [email protected][1] L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007).[2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B , 195424 (2008).[3] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[4] L. Fu and C. L. Kane, Phys. Rev. Lett. , 216403(2009).[5] M.-X. Wang, C. Liu, J.-P. Xu, F. Yang, L. Miao, M.-Y.Yao, C. L. Gao, C. 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16 lattice).∆ = 0 . µ = 0 are adopted. Evolution of Majorana modes on top (001) surface
We include a Zeeman field on the (001) surface to investigate its effect the on Majorana states. With increasingmagnetic field, a topological phase transition on (001) surface states will occur, as shown in Fig.3 in the main text.If the magnetic field is large enough (larger than (cid:112) ∆ + µ ), the (001) surface becomes topologically trivial. As theother sides surface states are topologically nontrivial, chiral Majorana modes should occur. Fig.5 shows the profilesof Majorana modes as a function of Zeeman field I ex . With increasing Zeeman field, the localized Majorana modeat vortex core gradually becomes extended and finally transforms into a chiral Majorana mode (on a 17 × × Vortex states in the superconcuting 2D Dirac surface states