Higher Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures
We show theoretically that a heterostructure of monolayer FeTeSe - a superconducting quantum spin Hall material - with a monolayer of FeTe - a bicollinear antiferromagnet - realizes a higher order topological superconductor phase characterized by emergent Majorana zero modes pinned to the sample corners. We provide a minimal effective model for this system, analyze the origin of higher order topology, and fully characterize the topological phase diagram. Despite the conventional s-wave pairing, we find rather surprising emergence of a novel topological nodal superconductor in the phase diagram. Featured by edge-dependent Majorana flat bands, the topological nodal phase is protected by an antiferromagnetic chiral symmetry. We also discuss the experimental feasibility, the estimation of realistic model parameters, and the robustness of the Majorana corner modes against magnetic disorder. Our work provides a new experimentally feasible high-temperature platform for both higher order topology and non-Abelian Majorana physics.
Introduction - For the past decade iron-based superconductors (FeSCs) have been a central research theme in condensed matter physics, owing to their high superconducting (SC) transition temperature , rich phase diagrams, and in particular the puzzle of the origin of pairing Kamihara et al. (2008); Hanaguri et al. (2010); Paglione and Greene (2010); Wang and Lee (2011); Hirschfeld et al. (2011); Chubukov (2012); Chubukov and Hirschfeld (2015). While the underlying microscopic mechanisms of SC in both bulk and monolayer FeSCs remain controversial, remarkable progress has been made recently towards revealing their nontrivial topological properties Hao and Hu (2014); Wu et al. (2015, 2016); Xu et al. (2016); Zhang et al. (2018); Wang et al. (2018); Zhang et al. (2019); Hao and Hu (); Machida et al. (). As the prototypical example of topological FeSC, bulk FeTeSe (FTS) with hosts a helical Dirac surface state above , as confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements Zhang et al. (2018, 2019). Below , strong evidence for Majorana vortex bound states has been found reproducibly in several scanning tunneling microscopy (STM) experiments Wang et al. (2018); Machida et al. (); Kong et al. (2019); Zhu et al. (2019), following from the theoretical prediction of surface topological superconductivity developed via a “self-proximity” effect Fu and Kane (2008). Similar to its bulk counterpart, the normal-state band structure of monolayer FTS has been theoretically predicted to be topological Wu et al. (2016). This prediction is further supported by a recent systematic ARPES measurement of monolayer FTS Shi et al. (2017); Peng et al. (2019), clearly revealing a bulk topological phase transition (band gap closing at ) by continuously changing the value of . With the highest among FeSCs Qing-Yan et al. (2012); Wen-Hao et al. (2014); Ge et al. (2015), one might wonder whether FTS monolayer also offers a new high temperature platform for topological Majorana physics. It should be noted, however, that the coexistence of nontrivial band topology and SC does not guarantee topological superconductivity (TSC). In fact, two-dimensional (2d), time reversal invariant TSC requires very strict conditions for both the Fermi surface geometry and SC pairing symmetry Qi et al. (2009); Zhang et al. (2013a). With the puzzle of pairing symmetry unresolved Huang and Hoffman (2017), the question of TSC in monolayer FTS remains open, although the answer is very likely negative.
In this Letter, we provide an alternative pairing-symmetry-independent route to obtain Majorana bound states in monolayer FTS systems. We demonstrate that Majorana zero modes emerge at physical corners of a sample when a FeTe layer is deposited on top of the FTS monolayer, as shown in Fig. 1 (b). The bicollinear antiferromagentic order of FeTe Ma et al. (2009); Bao et al. (2009); Li et al. (2009); Manna et al. (2017) is the key enabling higher order topology Benalcazar et al. (2017a, b); Zhang et al. (2013b); Schindler et al. (2018); Langbehn et al. (2017); Khalaf (2018); Liu et al. (2018); Peng and Xu (); Shapourian et al. (2018); Volpez et al. (2019); Wang et al. (2018a); Yan et al. (2018); Wang et al. (2018b); Wu et al. (2019); Zhu (2018); Bultinck et al. (2019); Pan et al. (); Ghorashi et al. (2019) in this heterostructure binding localized Majorana zero modes, without relying on the choice of pairing symmetry. This heterostructure-based mechanism is essentially different from our earlier proposal for bulk FTS, where the higher order topology is enabled by unconventional pairing Zhang et al. (2019). We construct a minimal lattice model to explain the origin of higher order topology in this heterostructure and we also study the stability of Majorana corner modes with respect to finite chemical potential and disorder effects. In the large limit, the FeTe layer also enables a novel topological nodal SC phase with symmetry protected edge Majorana flat bands, even when the SC pairing is singlet s-wave.
Model Hamiltonian - The low-energy theory of monolayer FTS around the -point is a superconducting version of the Bernevig-Hughes-Zhang (BHZ) model Bernevig et al. (2006); Wu et al. (2016). While the pairing mechanism in monolayer iron chalcogenide systems is still under debate, a conventional s-wave singlet pairing will suffice for our purpose. The Hamiltonian for FTS is then
in terms of a choice of matrices
with and for . Here , and Pauli matrices denote spin, orbital and particle-hole degrees of freedom, respectively. When , the normal-state part of is topologically nontrivial and possesses helical edge modes. However, the s-wave SC pairing necessarily trivializes the band topology of the full BdG model and introduces a pairing gap on the edge.
Covering the FTS monolayer with a monolayer FeTe introduces an exchange coupling with the bicollinear AFM order of FeTe to the system. Unlike a conventional collinear AFM, the magnetic moments in the bicollinear AFM flip their orientation every two atoms along the diagonal direction (e.g. the  direction) Ma et al. (2009), as shown in Fig. 1 (a). As a result, the unit cell is enlarged to contain four inequivalent atoms, labeled by a sublattice index . The new unit cell is characterized by the lattice vectors , where are the lattice vectors of the original square lattice BHZ model. The crystal momenta in the folded Brillouin zone will be denoted as and . Then the full effective model for the FTS/FT heterostructure is
The matrices describe the sublattice degree of freedom; the hopping matrix elements are described by
and the onsite matrix elements are described by diagonal matrices and . Here deontes the interlayer exchange coupling between FTS and FT. In particular, describes opposite coupling for electrons with sublattice and , which captures the bicollinear AFM texture.
For convenience, we have assumed the alignment of magnetic moments of FeTe to be along the direction. The model parameter accounts for the possible difference in factors of -electrons and -electrons. Numerically, we find that changing gives little contribution to the physics we are interested in sup (), so without loss of generality we take in our discussion.
Majorana Corner Modes - To understand the emergence of topological Majorana zero modes in our system, it is instructive to first switch off superconductivity and study the topological consequence of the bicollinear AFM. Despite explicitly breaking the time reversal symmetry , introducing AFM to a QSH system does NOT necessarily destroy the helical edge states. These edge states are instead now protected by an effective TRS , which combines with a half-unit-cell translation along . As shown in Fig. 1 (a), swaps electrons of index with those of . Therefore, is a magnetic space group operation and has a crucial difference from the conventional TRS Mong et al. (2010); Liu (2013); Zhang and Liu (2015): the Kramers degeneracy of only arises at the high symmetry points with .
Due to the crystalline nature of , not every edge preserves and is capable of hosting helical edge states. In particular, as shown in Fig. 1 (a), the magnetic configuration of the edge (which is parallel to ) is ferromagnetic (FM), which locally breaks and produces an edge magnetic gap. This is in contrast to gapless edges (such as the edge) with AFM ordering and protection. To verify this picture, we have used the iterative Green function method to numerically calculate the edge dispersion with finite and zero for both and edges. As shown in Fig. 2 (a), the edge has a Kramers degeneracy at (with ) but not at (with ). Meanwhile, Fig. 2 (b) clearly shows the magnetic gap at on the edge, which follows our expectation.
Now we include SC in our discussion. Through the self-proximity effect accompanying the development of bulk SC in the FTS layer, the gapless edge opens a SC gap. For the edge, however, there exists a competition between the edge FM gap and the edge SC gap. In particular, when the FM gap dominates the edge, the corner between and edges represents a zero-dimensional domain wall between SC and FM gaps, which necessarily binds a single Majorana zero mode to the corner Fu and Kane (2009); Alicea (2012), thus enabling higher order topology.
Therefore, the higher order topology in the heterostructure is controlled by the character of the edge gap where SC and FM compete. This motivates us to construct an effective theory of the edge that describes the competition between FM and SC,
Here is the effective exchange coupling on the edge, which orignates from the edge projection of the bulk AFM order . While it is generally difficult to analytically express in terms of , we numerically confirm a simple linear relation with . The linear coefficient depends on the details of hopping parameters and we find for our choice of parameters. For nonzero and , the edge topological phase transition occurs when the energy gap of closes. The topological condition of HOTSC is thus given by sup (),
when FM exceeds SC on the edge. To further confirm Eq. 6, we numerically map out the energy gap distribution in the parameter space spanned by and at a fixed . As shown in Fig. 2 (c), the topological phase transition predicted by Eq. 6 (white dashed line) agrees well with the colormap of the edge gap from a numerical nanoribbon calculation (where the gap closing regions are labeled in purple).
Following this topological criterion, we calculate the eigenvalues of on a open cluster by direct diagonalization. As shown in Fig. 2 (d), in the topological phase at , four Majorana zero modes are found to live inside the edge gap. We plot the combined spatial profile of these four zero modes in Fig. 2 (e), and additionally confirm that they are exponentially localized at the corners of the system. These corner localized 0d Majorana bound states are the hallmark of higher order topology in this 2d system.
Emergent Nodal TSC - In the small limit, the edge topological condition in Eq. 6 provides a simple analytical diagnostic for the appearance of Majorana corner modes. To explore the fate of higher order topology at finite , we fix the value of and numerically map out the topological phase diagram tuning and . In Fig. 2 (f), the HOTSC phase and the trivial phase are denoted by the red and white regions respectively. The phase boundary that separates the HOTSC and trivial SC corresponds to the gap closing of FM edge. In addition, an emergent nodal superconducting phase (blue region) is found to dominate the phase diagram when both and are large. Since only conventional s-wave singlet pairing is considered in our model, this nodal structure is unusual and emerges from the combined effects of AFM and SC.
The origin of the emergent nodal SC can be understood by projecting s-wave pairing onto the bulk Fermi surface. At zero , the effective pairing gap is always uniform on the Fermi surface, which simply signals the uniform, isotropic s-wave pairing. As is turned on from zero, the Fermi surface develops a spin texture such that becomes anisotropic in the Brillouin zone and thus develops momentum contours with . For example, the Fermi surface for is mapped out in Fig. 3 (a), which clearly shows the position of SC nodes in the spectrum. As a comparison, we plot the Fermi surface of the normal band structure alone in Fig. 3 (b), along with the calculated zero-pairing contours (white dashed lines). As expected, the BdG spectrum has nodal points where the zero-pairing contour intersects the normal state Fermi surface.
These emergent bulk SC nodes carry non-trivial topological charge and thus lead to interesting boundary Majorana physics. By combining the effective TRS and particle-hole symmetry , an AFM chiral symmetry operation is defined as
for every SC node.
The bulk-boundary correspondence then implies the existence of edge Majorana flat bands between SC nodes with opposite topological charges in a nanoribbon geometry. In Fig. 3 (c) and (d), we show the calculated edge spectrum for the AFM edge and FM edge, respectively. As expected, the edge hosts zero-energy Majorana flat bands between the projections of the nodal points. These Majorana flat bands are doubly degenerate due to . On the edge, however, the AFM chiral symmetry is explicitly broken because of the absence of . Therefore, the inter-node edge modes are not protected by and therefore need not be pinned to zero energy. As expected, the edge modes in Fig. 3 (d) are found to hybridize with each other with a nonzero splitting which shifts the modes from zero energy. The edge-dependent Majorana flat bands are a unique feature of the AFM chiral symmetry-protected nodal TSC phase.
Feasibility of Experimental Realization - We now discuss the experimental feasibility of our proposal. We first notice that the fabrication techniques for iron chalcogenide heterostructures are well-developed Wen-Hao et al. (2014); Sun et al. (2014); Nabeshima et al. (2017). In particular, bilayers of different iron chalcogenide layers (for example, a FeSe layer and a FeTe layer) were found to be coherently constrained to each other Nabeshima et al. (2017), which should hold in our proposed FeTeSe/FeTe bilayer as well. The precise epitaxial lattice matching between different iron chalcogenide layers greatly facilitates the edge characterization and identification of corner Majorana signal in realistic materials.
We also attempt to make some realistic estimates on the energy scales of physical quantities involved in the topological condition of Eq. 6. ARPES studies on the monolayer FeSe system reveal a SC gap of about 10 meV Zhang et al. (2016). The magnetic structure of FeTe has been measured in Ref. Bao et al. (2009); Li et al. (2009), which leads to local magnetic moments about along b axis (parallel spin axis), where is the Bohr magneton. The local magnetic moment of FeTe layer induces a magnetic proximity effect through exchange coupling with the FTS layer. To evaluate the scale of the induced exchange coupling, we perform a DFT calculation of a bilayer FeSe system, introducing ferromagnetism to the top FeSe layer sup (). With a magnetic moment of , the proximity-induced exchange coupling of the bottom layer is around 100 meV. Thus, for the experimentally observed magnetic moment of in FeTe, the induced exchange coupling in the FTS layer is expected to be meV. Given that , the edge FM potential is still much greater than the edge SC potential and thus the topological condition for Majorana corner modes is always satisfied for a small chemical potential.
Conclusion and Discussion - We have established the higher order TSC phase with Majorana corner modes in monolayer Fe(Se,Te) heterostructures, together with an emergent symmetry-protected nodal TSC phase. In addition to our proposed heterostructure being feasible to create, both the Majorana corner modes and the dispersing Majorana edge flat bands exhibit distinct features in local spectroscopy and should therefore be experimentally visible using standard STM techniques. Our proposed realization of Majorana corner modes does not rely on fine-tuning of the chemical potential, nor does it require perfect ordering of moments in the FeTe layer. In the Supplemental Material sup (), we consider a disordered model with a fixed density of magnetic defects where , which explicitly breaks , and we find that the corner modes persist even for a substantial defect density. Finally, unlike most previous proposals for higher order TSC Yan et al. (2018); Wang et al. (2018a, b); Zhang et al. (2019), our setup does not require an unconventional pairing symmetry. Rather, the combination of AFM and conventional s-wave pairing effectively mimics anisotropic pairing leading to generic higher order topological SC. However, it would be interesting (and possibly experimentally relevant for FTS) to generalize to unconventional pairing. In fact, the full landscape of trivial vs. topological normal state band structure, uniform vs. nonuniform magnetism, and conventional vs. unconventional pairing symmetry appears to be quite rich and already at hand in iron-based materials.
Acknowledgment - R.-X.Z is indebted to Chao-Xing Liu, Jiabin Yu, Fengcheng Wu and Biao Lian for helpful discussions. This work is supported by Laboratory for Physical Sciences and Microsoft. R.-X.Z is supported by a JQI Postdoctoral Fellowship.
We have recently become aware of an upcoming related work on the monolayer FTS, where the higher order topology is driven by an in-plane magnetic field Wu et al. ().
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