## Topological surface superconductivity in FeSe0.45Te0.55

### Theoretical model

Our starting point for the investigation of topological superconductivity on the surface of FeSe_{0.45}Te_{0.55} is the experimental observation that superconductivity in the iron-based superconductors emerges form the pairing of electrons in the *α*-, *β*-, and *γ*-bands^{33}. ARPES experiments on the parent compound FeSe have shown that these bands are quasi two-dimensional, exhibiting only a weak dispersion along *k*_{z}^{31,32,33}. The smooth (and weak) dependence of *T*_{c} with Te-doping provides strong evidence that these bands remain quasi two-dimensional with Te-doping in FeSe_{1−x}Te_{x}^{44}. We, therefore, employ a 2D 5-orbital model^{45} that has previously been employed to successfully describe the 2D superconducting properties of the iron-based superconductors. The hopping parameters for this model were extracted from a fit to ARPES experiments on FeSe_{0.42}Te_{0.58}^{46} and scanning tunneling spectroscopy (STS) experiments on FeSe_{0.45}Te_{0.55}^{42}. To describe the emergence of topological superconductivity on the surface of FeSe_{0.45}Te_{0.55}, we include in this model (a) the experimentally observed surface ferromagnetism^{13,14,15,25} through an exchange field, and (b) a RSO interaction that arises from the breaking of the inversion symmetry on the surface (for detail, see Supplementary Note 1). The resulting Hamiltonian in real space is given by

$${H}_{0}= -\mathop{\sum }\limits_{a,b=1}^{5}\mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}},{{{{{{{\bf{{r}}}}}}}^{\prime}}},\sigma }{t}_{{{{{{{{\bf{r,{r}}}}}}}^{\prime}}}}^{ab}{c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{{r}}}}}}}^{\prime}}},b,\sigma }-\mathop{\sum }\limits_{a=1}^{5}\mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}},\sigma }{\mu }_{a}{c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }\\ +i\alpha \mathop{\sum }\limits_{a=1}^{5}\mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}},{{{{{{{\boldsymbol{\delta }}}}}}}},\sigma ,\sigma ^{\prime} }{c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }^{{{{\dagger}}} }{\left({{{{{{{\boldsymbol{\delta }}}}}}}}\times {{{{{{{\boldsymbol{\sigma }}}}}}}}\right)}_{\sigma \sigma ^{\prime} }^{z}{c}_{{{{{{{{\bf{r}}}}}}}}+{{{{{{{\boldsymbol{\delta }}}}}}}},a,\sigma ^{\prime} }^{}+J\mathop{\sum }\limits_{a=1}^{5}\mathop{\sum}\limits_{{{{{{{{\bf{r}}}}}}}},\sigma ,{\sigma }^{\prime}}{{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{\bf{r}}}}}}}}}\cdot {c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{\sigma {\sigma }^{\prime}}{c}_{{{{{{{{\bf{r}}}}}}}},a,{\sigma }^{\prime}}^{}\\ +\mathop{\sum }\limits_{a=1}^{5}\mathop{\sum}\limits_{\langle \langle {{{{{{{\bf{r}}}}}}}},{{{{{{{\bf{{r}}}}}}}^{\prime}}}\rangle \rangle }{{{\Delta }}}_{{{{{{{{\bf{r}}}}}}}}{{{{{{{{\bf{r}}}}}}}}}^{\prime}}^{a}{c}_{{{{{{{{\bf{r}}}}}}}},a,\uparrow }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{{r}}}}}}}^{\prime}}},a,\downarrow }^{{{{\dagger}}} }+{{{{{{{\rm{H.c.}}}}}}}}$$

(1)

Here *a*, *b* = 1, . . . , 5 are the orbital indices corresponding to the *d*_{xz}-, *d*_{yz}-, \({d}_{{x}^{2}-{y}^{2}}\)-, *d*_{xy}-, and \({d}_{3{z}^{2}-{r}^{2}}\)-orbitals, respectively, \(-{t}_{{{{{{{{\bf{r{r}}}}}}}^{\prime}}}}^{ab}\) represents the electronic hopping amplitude between orbital *a* at site **r** and orbital *b* at site \({{{{{{{\bf{{r}}}}}}}^{\prime}}}\) on a 2D square lattice, *μ*_{a} is the on-site energy in orbital *a*, \({c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma }^{{{{\dagger}}} }({c}_{{{{{{{{\bf{r}}}}}}}},a,\sigma })\) creates (annihilates) an electron with spin *σ* at site **r** in orbital *a*, and ** σ** is the vector of spin Pauli matrices. The superconducting order parameter \({{{\Delta }}}_{{{{{{{{\bf{r}}}}}}}}{{{{{{{\bf{r}}}}}}}}^{\prime} }^{a}\) represents intra-orbital pairing between next-nearest neighbor Fe sites

**r**and \({{{{{{{\bf{r}}}}}}}}^{\prime}\) (in the 1 Fe unit cell), yielding a superconducting

*s*

_{±}-wave symmetry

^{42}. Moreover,

*α*denotes the RSO interaction arising from the breaking of the inversion symmetry at the surface

^{3}with

**being the vector connecting nearest neighbor sites. Due to the full superconducting gap, which suppresses Kondo screening, we consider the magnetic moments to be static in nature, such that**

*δ***S**

_{r}is a classical vector representing the direction of a surface atom’s spin located at

**r**, and

*J*is its exchange coupling with the conduction electron spin. We here assume that the magnetic moment couples with equal strength to all 5 orbitals. The experimentally observed opening of a gap at the Dirac point

^{13,14,15}implies an ordering of the magnetic moments perpendicular to the surface, as also confirmed by quantum sensing experiments

^{25}, such that \(\langle {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{\bf{r}}}}}}}}}\rangle =S\hat{{{{{{{{\bf{z}}}}}}}}}\) with

*S*being the ordered spin moment on the surface. In the normal state, the above Hamiltonian yields Fermi surfaces

^{42}that are consistent with those reported by ARPES experiments on FeSe

_{0.42}Te

_{0.58}

^{46}(see Supplementary Note 1 and Supplementary Fig. 1). Due to the orbital character of the Fermi surfaces, the superconducting order parameter is only non-zero in the

*d*

_{xz}-,

*d*

_{yz}-, and

*d*

_{xy}-orbitals

^{42}. The local density of states (LDOS) resulting form the above Hamiltonian in the superconducting state reproduces all salient features of the differential conductance, d

*I*/d

*V*, measured via STS (see Supplementary Note 1 and Supplementary Fig. 1), and in particular shows the existence of several superconducting gaps ranging from 1.6 meV to 2.4 meV. As noted in ref.

^{15}, the electronic structure of FeSe

_{0.45}Te

_{0.55}obtained from DFT calculations

^{22}shows significant differences to those observed via ARPES experiments

^{46}, such that the former cannot be used as a reliable starting point for our calculations. Below, we assume that the topological superconducting state arising from the 3DTI mechanism is destroyed

^{26,27}by the experimentally observed surface ferromagnetism

^{13,14,15,25}—a destruction which is achieved already for rather weak exchange coupling

*J*

*S*(see “Discussion” below)—and thus neglect the bulk bands and Dirac cone associated with the 3DTI mechanism.

We note that due to the particle-hole symmetry of the superconducting state, and the broken time-reversal symmetry arising from the presence of magnetic moments, FeSe_{0.45}Te_{0.55} belongs to the topological class D^{47,48}. For a 2D system, the topological invariant is therefore given by the Chern number, which can be computed via^{49}

$$C = \frac{1}{2\pi i}\int_{{{{{{{{\rm{BZ}}}}}}}}}{{{{{{\rm{d}}}}}}}^{2}k{{{{{{{\rm{Tr}}}}}}}}({P}_{{{{{{{{\bf{k}}}}}}}}}[{\partial }_{{k}_{x}}{P}_{{{{{{{{\bf{k}}}}}}}}},{\partial }_{{k}_{y}}{P}_{{{{{{{{\bf{k}}}}}}}}}])\\ {P}_{{{{{{{{\bf{k}}}}}}}}} = \mathop{\sum}\limits_{{E}_{n}({{{{{{{\bf{k}}}}}}}}) < 0}\left|{{{\Psi }}}_{n}({{{{{{{\bf{k}}}}}}}})\right\rangle \left\langle {{{\Psi }}}_{n}({{{{{{{\bf{k}}}}}}}})\right|$$

(2)

where *E*_{n}(**k**) and \(\left|{{{\Psi }}}_{n}({{{{{{{\bf{k}}}}}}}})\right\rangle\) are the eigenenergies and the eigenvectors of the Hamiltonian in Eq. (1), with *n* being a band index, and the trace is taken over Nambu and spin space.

### Topological phase diagram

The existence of a hard superconducting gap, a RSO interaction, and an out-of-plane ferromagnetic order are in general sufficient requirements for the emergence of topological surface superconductivity^{34,35,36,37,38,39,40,41}. To demonstrate that this emergence is a robust phenomenon in FeSe_{0.45}Te_{0.55} within the proposed model, we present in Fig. 1 the topological phase diagram—in terms of the Chern number *C*—computed from Eq. (2) as a function of the effective magnetic exchange strength *J**S*, and of a shift of the chemical potential, Δ*μ*, from its value extracted in ref. ^{42}. We find that already for rather weak magnetism, as reflected in a magnetic exchange coupling *J**S* of the order of a few meV, the system undergoes transitions into topological superconducting phases.

We note that the positions of the topological phase transitions are independent of the strength of the RSO interaction *α* (as long as *α* does not vanish), only the topological gap increases with increasing *α*. While the presence of topological phases is robust against shifts in the chemical potential, varying Δ*μ* induces transitions between topological phases characterized by different Chern numbers. With increasing *J**S* (keeping all other band parameters fixed), the Fermi surfaces eventually cross the nodal lines of the superconducting *s*_{±}-wave order parameter, and the system becomes gapless (as denoted by the gray area in Fig. 1), and hence topologically trivial. We note, however, that this gapless region of the phase diagram can be shifted to larger values of *J**S* by appropriately adjusting the band parameters in Eq. (1). The points in the (Δ*μ*, *J**S*)-plane, where the topological phase transitions occur, are determined by the closing of the superconducting gap. Analytical expressions for these phase transition lines can be obtained (see Supplementary Note 2) and are plotted as solid, dashed, and dotted lines in Fig. 1) representing the closing of the gap at the Γ-, *X*/*Y*-, and *M*-points in the Brillouin zone, respectively. The *X*/*Y*-, and *M*-points possess a multiplicity of *m* = 2 and *m* = 1, leading to a change in the Chern number by Δ*C* = +2 and Δ*C* = −1, respectively. In contrast, while the Γ-point possesses a multiplicity of *m* = 1, the vanishing of the RSO interaction at the Γ-point yields a simultaneous gap closing in two bands, leading to a change in the Chern number by Δ*C* = −2. We note that the inclusion of a conventional **L** ⋅ **S** spin–orbit coupling^{28,29} in the Hamiltonian does not qualitatively change the topological phase diagram shown in Fig. 1, but rather increases the parameter space in which topological superconducting phases can be found (see Supplementary Note 3 for a more detailed discussion).

Having established the presence of topological superconducting phases in FeSe_{0.45}Te_{0.55} arising from the proposed mechanism, we next turn to a discussion of their unique physical properties that have been observed experimentally.

### MZM in a vortex core

The experimental observation of MZMs localized in vortex cores^{16,17,18,19} represents a salient signature of the topological nature^{34,50,51,52,53} of the superconducting surface in FeSe_{0.45}Te_{0.55}. To investigate the emergence of a vortex core MZM within our model, we implement the magnetic field via the Peierls substitution and compute the spatial dependence of the superconducting order parameters in the *d*_{xz}-, *d*_{yz}-, and *d*_{xy}-orbitals self-consistently (for details, see Supplementary Note 4). The resulting spatial structure of the superconducting order parameter in the *d*_{xz}-orbital, which vanishes at the center of the vortex, in the topological *C* = −1 phase is shown in Fig. 2a (the analogous plots for the *d*_{yz}-, and *d*_{xy}-orbitals are shown in Supplementary Note 4). We find that all three superconducting order parameters possess the same spatial symmetry as the orbitals they arise from, i.e., a *C*_{2}-symmetry in the *d*_{xz}-, and *d*_{yz}-orbitals, and a *C*_{4}-symmetry in the *d*_{xy}-orbital. A linecut of the energy-resolved LDOS for the *d*_{xz}– (Fig. 2b) and *d*_{xy}-orbitals (Fig. 2c) through the vortex core reveals the existence of a MZM in the topological *C* = −1 phase, which is absent in the trivial *C* = 0 phase (Fig. 2d) (results for other orbitals are shown in Supplementary Note 4).

In addition to the MZM, the LDOS for all orbitals reveals topologically trivial Caroli-de Gennes-Matricorn (CdGM) states^{54}, similar to results obtained in other iron-based superconductors^{55}. The radially integrated LDOS for the *d*_{xz}-orbital (see Fig. 2e) shows an equal energy spacing of the low-energy CdGM states described by *E*_{n} = *n**E*_{0} (see Fig. 2f). This integer spacing of the low-energy CdGM states was also observed in refs. ^{18,19}. Moreover, certain CdGM states possess a large spatial extent, exhibiting significant spectral weight up to a distance of *r* ≈ 20*a*_{0} from the vortex core (see red arrows in Fig. 2b). These spatially extended CdGM states, in turn, also exhibit an equal energy spacing \({E}_{p}^{\prime}=p{E}_{0}^{\prime}\), with \({E}_{0}^{\prime} > {E}_{0}\) (see filled red circles in Fig. 2f). While the existence of an MZM is topologically protected, we find that the spatial and energy structure of the trivial CdGM states can vary significantly within a given topological phase (see Supplementary Note 4). Indeed, the spatial structure of the CdGM states even varies significantly between orbitals (cf. the LDOS for the *d*_{xz}– and *d*_{xy}-orbitals shown in Fig. 2b and c, respectively) for a given point in the phase diagram. This might explain the observation that while different STS experiments have observed the existence of zero-energy vortex core bound states in FeSe_{0.45}Te_{0.55}, the observed spatial and energy structure of the trivial CdGM states varies greatly between experiments^{16,17,18,19}. While it will be of interest in the future to develop a more detailed understanding of how the spatial and energy structure of the CdGM states emerges from the multi-band electronic structure of FeSe_{0.45}Te_{0.55}, and how the observed differential conductance emerges from co-tunneling effects, our results demonstrate that the spatial and energy structure of the CdGM states bears no relevance for the existence of a topological phase, in agreement with the experimental observations^{16,17,18,19}.

Moreover, the experimental observation that only a fraction of vortices on the surface of FeSe_{0.45}Te_{0.55} possesses MZMs^{16,17} has remained an important unsolved problem. One possible explanation for this finding could arise from the experimental observation of strong disorder on the surface of FeSe_{0.45}Te_{0.55}^{56,57}, which is particular evident in the spatial variation of the chemical potential^{57}. Indeed, within the scenario proposed here, spatial variations in the chemical potential, as reflected in Δ*μ* in Fig. 1, could lead to topological or trivial domains on the surface of FeSe_{0.45}Te_{0.55}, in which case only those vortices located in the topological domains would exhibit a MZM. As trivial and topological domains need to be separated by a domain wall, which necessarily harbors a chiral Majorana mode (see “Discussion” below) the observation of such a mode in LDOS linecuts connecting trivial and topological vortices would be an important signature of this scenario^{26}.

### MZM at the end of line defects

The recent observation of MZMs at the end of line defects in monolayer FeSe_{0.5}Te_{0.5} deposited on a SrTiO_{3} substrate^{20} has raised the question of whether these MZMs are a characteristic feature of an underlying topological phase, similar to line defect MZMs predicted to occur in topological *p*_{x} + i*p*_{y}-wave superconductors^{58}, or are independent of it, simply utilizing the monolayer’s complex electronic structure to form a 1D topological superconductor as proposed in refs. ^{20,59,60}. While it is currently unknown whether the FeSe_{0.5}Te_{0.5}/SrTiO_{3} system is a topological superconductor, we note that the mechanism proposed here can in general give rise to topological superconductivity in monolayer systems (if ferromagnetism is present), while the 3DTI mechanism^{12,22,23,24} cannot. We therefore want to address two important questions: (i) can line defect MZMs emerge within the model of Eq. (1), and (ii) if so, are these MZMs directly tied to the existence of topological surface superconductivity, thus representing a sufficient condition for its existence.

To address these questions we represent the line defect for simplicity as a line of potential scatterers (though magnetic scatterers could also be realized^{59,60}) described by the Hamiltonian

$${H}_{def}={U}_{0}\mathop{\sum }\limits_{a=1}^{5}\mathop{\sum}\limits_{{{{{{{{\bf{R}}}}}}}},\sigma }{c}_{{{{{{{{\bf{R}}}}}}}},a,\sigma }^{{{{\dagger}}} }{c}_{{{{{{{{\bf{R}}}}}}}},a,\sigma },$$

(3)

where *U*_{0} is the potential scattering strength, and the sum runs over all sites **R** of the line defect. In Fig. 3a, we present the energies of the three lowest energy states for a line defect of length *L* = 119 in the *C* = −1 phase as a function of *U*_{0}. One can clearly discern three regions of *U*_{0} (shown with a gray background in Fig. 3a) where the lowest energy state is essentially located at zero energy (with the LDOS exhibiting a full gap of 1.6 meV, see Supplementary Note 1), suggesting the existence of a MZM. At the boundary of the white and gray regions in Fig. 3a, the superconducting gap of an infinitely long line defect closes, consistent with the occurrence of a topological phase transition. Further evidence for the existence of Majorana modes is revealed by line-cuts of the LDOS along the line defect with *U*_{0} = 71 meV, located in the rightmost gray region in Fig. 3a, for the lowest and second lowest energy states with energies *E*_{1} and *E*_{2}, respectively (see Fig. 3b). The spectral weight of the lowest energy state at *E*_{1} is confined to the end of the line defect—with essentially no spectral weight located inside the defect line—consistent with the localized nature of an MZM. In contrast, the state at *E*_{2} exhibits considerable spectral weight along the entire length of the line defect. This result is also confirmed by a spatial plot of the LDOS at energies *E*_{1} and *E*_{2}, shown in Fig. 3c and d, respectively. These findings provide strong evidence that the zero energy states localized at the end of the line defects in Fig. 3c are MZMs. Line defect MZMs also exist in the *C* = −3 phase, but no zero-energy states occur at the end of a line defect when FeSe_{0.45}Te_{0.55} is in the topological trivial (*C* = 0) phase (see Supplementary Note 5). Within our model, the existence of line defect MZMs is therefore directly tied to and a sufficient condition for the existence of an underlying topological superconducting phase, while a failure to observe MZMs at line defects does not necessarily imply a trivial nature of the underlying superconducting phase. We thus predict that in addition to FeSe_{0.5}Te_{0.5}/SrTiO_{3}, MZMs should emerge when line defects are placed on the surface of bulk FeSe_{0.45}Te_{0.55}. Finally, we note that while topological superconductivity on the surface of FeSe_{0.45}Te_{0.55} is characterized by a *Z* topological invariant, the line defect as a one-dimensional system possesses a topological *Z*_{2} classification, similar to the Kitaev chain^{61}.

### Chiral Majorana modes along domain walls

The bulk-boundary correspondence dictates that Majorana edge modes need to arise along domain walls that separate regions of different Chern numbers^{61}. Indeed, the observation of a nearly constant LDOS at a domain wall in FeSe_{0.45}Te_{0.55}^{21} was recently interpreted as a signature of a Majorana edge mode (whether the observed mode is helical in nature as would arise from the 3DTI mechanism^{12,22,23,24,50}, or chiral, as in our model, is experimentally currently unclear).

This raises the intriguing question not only as to which types of physical domain walls can give rise to the emergence of Majorana modes, but also of how to distinguish Majorana modes from trivial, in-gap states. To address this question, we calculate the electronic structure near two different types of domain walls: a spin domain wall at which the magnetic moment is inverted, i.e., **S** → −**S**, and a *π*-phase domain wall, where the superconducting order parameter undergoes a *π*-phase shift, i.e., Δ → −Δ, for all electronic bands. As the spin domain wall separates regions with different Chern numbers (since **S** → −**S** implies *C* → −*C*), the bulk boundary correspondence requires the emergence of dispersive Majorana edge modes that traverse the superconducting gap, as shown in Fig. 4a where we present the system’s electronic band structure as a function of momentum *k*_{∥} along the domain wall (Majorana modes are shown as red lines). In addition, the system also exhibits trivial in-gap modes, which do not connect the upper and lower bands. Similar results also hold for a domain wall separating a topological from a trivial domain. In contrast, regions separated by a *π*-phase domain wall possess the same Chern number, and the electronic band structure therefore only exhibits trivial in-gap states, as shown in Fig. 4b. In Fig. 4c, d, we present the LDOS as a function of energy along a linecut perpendicular to the domain wall. In both cases, we find that the LDOS near the domain wall exhibits considerable spectral weight inside the superconducting gap, with the LDOS being nearly energy independent for the spin domain wall, but exhibiting a pronounced peak at zero-energy for the *π*-phase domain wall. A linecut of the zero-energy LDOS, shown in Fig. 4e, f, however, reveals that the zero-energy state is localized close to the domain wall in both cases. Thus, the differences in the LDOS between these two types of domain walls is quantitative rather than qualitative in nature, and STS measurements might therefore not be able to distinguish between topological Majorana edge modes, and trivial in-gap states. However, a qualitative difference between these domain walls can be identified when considering the spatial structure of the induced supercurrents (see Supplementary Note 6). For a spin domain wall, the chirality of the induced supercurrent, which is determined by the sign of the Chern number, changes between the two separated regions, implying that the supercurrents associated with each region flow in the same direction along the domain wall^{40}, as shown in Fig. 4g, yielding a non-vanishing net supercurrent. In contrast, for the *π*-phase domain wall, the chirality of the supercurrents in both regions is the same, implying that they flow in opposite directions along the domain wall (see Fig. 4h), yielding a vanishing net supercurrent. This qualitative difference, a non-zero net supercurrent for a spin domain wall and a vanishing net supercurrent for the *π*-phase domain wall, can be imaged using a SQUID, thus providing an unambiguous experimental signature to distinguish the existence of topological Majorana modes for a spin domain wall, from that of trivial in-gap states for a *π*-phase domain wall.