Helical Majorana fermions at the interface of Weyl semimetal and d-wave superconductor: Application to Iridates and high-Tc Cuprates
Majorana bound states exist inside a core of half-quantum vortex in spin-triplet superconductors. Despite intense efforts, they have not been discovered in any spin-triplet superconductor candidate materials. After the success of topological insulator, another route to achieve Majorana fermions has been suggested: heterostructures of -wave superconductors and topological insulator or semimetal with strong spin-orbit coupling provide an effective spinless pairing which supports Majorana bound states in a single vortex. This theoretical observation has stimulated both experimental and theoretical communities to search for Majorana fermions, and recently a localized Majorana state at the end of one-dimensional wire has been reported. Here we study the two-dimensional interface of Weyl semimetal and -wave superconductor which promotes a pair of helical Majorana fermions propagating along the edge of the interface. We suggest that Iridium oxide layer IrO classified as two-dimensional Weyl semimetal in close proximity to high temperature Cuprates would be a best example to explore these helical Majorana modes.
Introduction. Self-conjugate fermions called Majorana fermions (MFs)Majorana1937 () occur inside a core of half-quantum vortex in spin-triplet with equal-spin pairing or in a single vortex of spinless pairing states.Volovik1976 (); Kopnin1991 (); Ivanov2001 () It is worthwhile to note that this -wave superconductor (SC) belongs to the same topological class as the Pfaffian quantum Hall state where the excitations are half-quantum vortices with non-Abelian statistics.Moore1991 (); Read2000 () These non-Abelian statistics of MFs promote their potential use as a topological quantum qubitTewari2007 (), and the search for MFs has regained some interest. Recently, discoveries of MFs were claimed: localized Majorana bound states were theoretically proposed at both ends of quantum wire LutchynPRL2010 (); OregPRL2010 (); FlensbergPRL2015 () and further reported at the edge of topological superconducting wire made of ferromagnetic atom chain on the surface of superconducting lead.Mourik2012 (); Rokhinson2012 (); Lee2014 (); Yazdani2014 (); Xu2014 () The experimental set-up was motivated by the theoretical finding that the proximity-induced s-wave superconducting pairing on the spin-momentum locked surface states of a topological insulator (TI) is effectively spinless .FuPRL2008 ()
The success of TI has not only lead to a discovery of MFs, but also extended classes of MFs beyond localized bound states. Analog to the edge states of two-dimensional (2D) quantum spin Hall insulator, a pair of counter-propagating MFs protected by time reversal symmetry (TRS) could occur along the edge of 2D systems. These are called helical MFs. It was suggested that helical MFs exist at a one-dimensional (1D) line junction of -wave SC, TI, and -wave SC where the SC pairing should change its sign across the junction FuPRL2008 (). They have been also proposed to emerge in time-reversal invariant topological SC QiPRL2009 (); HaimPRB2014 (); TewariPRB2013 (); SunSciRep2016 (); NakosaiPRL2012 (), and a 2D Rashba layer proximity to -wave superconductor FanPRL2013 (). These proposals require either bulk time-reversal invariant topological SC, or the sign change of -wave pairing potential across the junction or inside the SC.
In this paper, we suggest a different mechanism to generate a pair of helical MFs. We prove that a pair of helical MFs appear at the interface of a 2D Weyl semimetal and d-wave SC. Here the d-waveness naturally provides the sign change of the pairing potential across the node and the edge states manifest topological nature of underlying Weyl semimetal. Since the interface breaks the inversion symmetry, a 2D Dirac semimetal in lieu of a 2D Weyl semimetal also supports the same helical MFs. Our theory is generally applicable to the interface made of d-wave SCs and Weyl semimetals, but we focus on the most promising candidate: IrO layer classified as Weyl semimetal with effective pseudospin (J=1/2)Carter2012 (); William2013 (); YChen14 (); Rau2016 () in proximity to d-wave high temperature (high ) Cuprates such as hole-doped LaSrCuO (LSCO) or YBaCuO(YBCO). This also offers a way to induce high temperature SCs in Iridates, which has been a long sought in the community of correlated Mott physics with strong spin-orbit coupling (SOC).Fa2011 (); KimNP2016 (); JKimPRL2012 ()
Model. The superlattice structure shown in Fig. 1(a) can be experimentally fabricated by growing SrIrO film on high Tc Cuprates using molecular beam epitaxy or pulsed laser deposition techniques Matsuno2015 (). The Hamiltonian for IrO layer, including the Rashba SOC and proximity effects due to high SC, is given by :
Here where denotes the electronic creation/annihilation operator to create/destroy an electron with crystal momentum at site and the pseudospin in J basis. , and are Pauli matrices for J, sublattice, and particle-hole subspace, respectively. The 2D crystal momentum defined in the pseudo-square lattice. The Bravais lattice vectors and due to the staggered rotations of IrO as shown in Fig. 1(b). The nearest neighbour (NN) intra, inter-orbital, and next-nearest neighbour (NNN) hopping terms with strength , , and are given by , , , respectively. The NN and NNN Rashba SOC terms occur as and , respectively. These are due to the broken inversion symmetry along -direction, and can be further enhanced by external electric field. The Cooper pairing potential induced by the proximity effect of high Tc Cuprates manifests -wave pairing: where is the strength of pairing.
Symmetry of Iridium oxide layer. To determine the existence of symmetry protected edge states, let us first investigate the symmetry of IrO layer itself. When , exhibits symmetry enforced 2D Weyl semimetal KanePRB2016 (). Unlike their three-dimensional partner, these are protected by time-reversal and crystalline symmetries KanePRB2016 (): there are two glides perpendicular to (or ) directions, which protect the pair of Weyl nodes together with TRS. The Rashba SOC is necessary to generate the Weyl nodes. Without such inversion breaking term, the Weyl nodes collapse into time-reversal invariant momentum (TRIM) points, and it becomes a Dirac semimetal. The energy dispersion is displayed in Fig. 2. Note that, there are two Weyl points at along / high-symmetry line. In the rest of the paper, we use the reciprocal lattice vectors and to represent the 2D crystal momentum. Under this notation, TRIM points and (see the inset of Fig. 2).
The Bloch bands are doubly degenerate along both and protected by a product of TRS and glide symmetries. In other words, the product of glide, (or ), and time-reversal operator, (or ) becomes the antiunitary operator along the (or ) line, which ensure the Kramer degeneracy . Besides, a four-fold rotational axis along -direction guarantees that the Fermi surface (FS) is symmetric respect to the interchange of and and thus we show only 1/4 of the FS in the inset of Fig. 2. When chemical potential (half filling), the FS made of consists of two hole-pockets around and , which are related to the 2D Weyl nodes, and electron-pockets near . footnote17 ().
One-dimensional Edge states and Majorana Kramer Pairs. Let us investigate the proximity effects of d-wave SC on this 2D symmetry enforced Weyl semimetal of Iridium oxide layer. The bulk band structure of remains gapless at two points and along / direction (highlighted with green color in Fig. 3(b) and (d)) because the d-wave node is along and in the reduced Brillouin zone (BZ). The symmetry properties associated with the bulk Bloch states hint various exotic excitations on the boundary of the system, which we will discuss in details later. Below we first show the MFs on the 1D boundary using the slab geometry of superlattice.
The slab geometry of superlattice has an open-boundary along direction but is periodic along -axis, and thus the crystal momentum parallel to the sample edge is a good quantum number. The gapless bulk excitations, mentioned above are projected onto the edge momentum at and , respectively. When the system is at half filling, there exists a pair of propagating MFs bounded between and : see the linear dispersion centered at with red color in Fig. 3(a). The existence of these helical MFs depends on the relative strength of and of NN Rashba SOC term. Note that the Weyl nodes at and occur only with a finite , which simultaneously generates a band gap at (and )-points. When the chemical potential lies inside the band gap at -point, the helical MFs are emerged. Mathematically, when the chemical potential is within the range , corresponding to the direct gap of at TRIM points, a pair of helical MFs centered at appears at the boundary of system.
This should be captured by checking the Berry phase of bulk spectrum and corresponding topological invariance. The FS along of bulk layer forms a closed loop, enclosing the underlying Weyl node, and its Berry phase is . On the other hand, the two FSs, enclosing point, have -Berry phase. In the presence of the time-reversal invariant SC pairing, it was shown that the topological invariance is determined by the matrix element of SC pairing operator between the Kramer pair of Bloch states, i.e., . Qi2010 () The time-reversal invariant path , displayed as the dotted line in Fig. 3(b), is picked so that it’s invariant under and moves from . Note that the 1D Hamiltonian on the path has both time-reversal () and particle-hole () symmetries, hence belongs to the topological class , which supports a classification in 1D. Then a 1D topological invariant is defined as Qi2010 ()
where the product of is over all the Fermi points at which the FS meets the momentum along the contour . As shown in Fig. 3(b), is positive at two s of the FS along displayed by sign, but it has negative sign at one of the two different FS enclosing point. This confirms that , which suggests there exists edge states. Due to the nature of Weyl metal, a pair of edge states are helical MFs within the projected regime of path onto -direction. A proof of TRS protected MFs at is presented in Supplemental Materials.
In reality, Iridium layer in close proximity to Curpates is not at half-filling. Because of potential difference, IrO will be effectively hole doped. However, when , the above results hold. On the other hand, when the chemical potential (or doping effect) is bigger than the band gap at / point generated by the Rashba SOC, a pair of propagating MFs disappears, but the flat zero modes remain as shown in Fig. 3(c). The disappearance of MFs happens because the Bogoliubov quasi-particle at experiences gap closing and re-opening process, accompanying a jump of number. It corresponds to the case when the path intersects even times with the outermost FS (see Fig. 3(d)). Hence the system no longer supports a pair of helical MFs at the boundary, which is consistent with the slab calculation result shown in Fig. 3(c). The finding implies that the Iridium layer SC goes into a phase transition to another topological SC by changing , which can be further tuned by an external electric field and/or chemical doping.
Zero-energy flat bands In addition to the helical MFs, the edge spectrum exhibits zero-energy flat states within some regime of , localized on the boundary of the system. The existence of zero-energy flat modes is independent of the chemical potential and the size of Rashba SOC. In the regime (a) , there exist doubly degenerate zero-energy edge states at every . On the other hand, the zero-energy edge modes transit from double to single degeneracy in the region (b) .
The zero-energy flat bands, emerging within the range and are due to a chiral symmetry, and characterized by a different topological invariant from . Note that, the 2D Hamiltonian can be viewed as a collection of 1D Hamiltonian, parametrized by the edge momentum , i.e. . RyuPRL2002 () The topological class for such 1D Hamiltonian at is , and an integer 1D winding number is suitable to identify the bulk-edge correspondence:
with chiral symmetry operator and integrating along momentum. The 1D winding number corresponds the region, highlighted with green color in both Fig. 3(a) and (c), where only one zero-energy edge mode exists on each boundary. The regime of edge states with orange color in Fig. 3(a) and (c), on the other hand, acquire two zero-energy states, which is consistent with the 1D winding number . Fig. 4 summarizes the phase diagram by tuning the chemical potential. There exist two distinct topological SC states characterized by two different topological invariants. When the Iridium film is near half-filling, the system possesses both a pair of helical MFs and the zero-energy flat modes.
Discussions. In this Letter, we have explored the novel Majorana edge spectra of 2D Iridates proximal to -wave SC like high Tc Cuprates. We find that the 1D edge states exhibit multiple non-trivial features depending on various segment of edge momentum and the amount of doping. Most strikingly, a pair of helical MFs protected by TRS appears along the 1D boundary. Here the d-wave pairing potential vanishes along and changes the sign across these nodal lines. Since the Cooper pairing on the pseudospin-momentum locked Weyl states behaves as a pairing, such a change of pairing potential in d-wave acts like a line of single vortex, which supports a pair of helical MFs. One should note that the pairing is finite between the sublattice A and B, thus the MF operator is composed of four fermionic operators as shown in Supplemental Materials. The partners of MF pair are related by TRS to each other.
In addition to the helical MFs, the edge spectra in certain momentum regimes exhibit flat zero energy bands, which resembles the situation presented in 1D edge states of noncentrosymmetric d-wave superconductor. These flat edge states require the clean edge shape. For example, the current results were obtained by the translation invariant boundary along -axis. These edge modes are emerged due to the chiral symmetry, and thus they could acquire a slight dispersion when small but finite further neighbour hopping paths are taken into account. On the other hand, the helical MFs are not sensitive to either the boundary shape nor further neighbour hopping, because they are protected by the TRS. However, they disappear when the open boundary is along the or -direction. This is because the two Weyl states near and points are projected into the same boundary and the sign change of d-wave pairing occurs twice for this open boundary, which cancel out the nontrivial topological effects. These results are robust in the presence of a small isotropic (or anisotropic) -wave pairing potential as long as TRS is preserved. In this case, the weak -wave pairing potential slightly shifts the location of nodal lines, and thus it does not alter the existence of MFs.
There has been a long sought for a high SC in Iridates. Since high Tc Cuprates provide both hole doping and proximity to -wave SC, this interface also offers a way to generate a high SC in Iridates, where the amount of doping can be controlled by the thickness of insulating barrier to reduce the potential difference, but preserving a proximity effect. In summary, the helical MFs along the 1D boundary of Iridium oxide layer proximal to high Tc Cuprates exist due to a combination of the d-wave pairing and Weyl states at any open boundary except - or -direction. Our proposal also offers a way to induce high Tc SC in Iridates in addition to uncovering the helical MFs in correlated oxide systems.
Acknowledgement. This work was supported by the NSERC of Canada and the center for Quantum Materials at the University of Toronto.
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