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We suggest that spin-singlet pseudo-scalar s-wave superconducting pairing creates a two dimensional sea of Majorana fermions on the surface of three dimensional Dirac superconductors (3DDS). This pseudo-scalar superconducting order parameter Δ5, in competition with scalar Dirac mass m, leads to a topological phase transition due to band inversion. We find that a perfect Andreev-Klein reflection is guaranteed by presence of anomalous Andreev reflection along with the conventional one. This effect manifests itself in a resonant peak of the differential conductance. Furthermore, Josephson current of the Δ5|m|Δ5 junction in the presence of anomalous Andreev reflection is fractional with 4π period. Our finding suggests another search area for condensed matter realization of Majorana fermions which are beyond the vortex-core of p-wave superconductors. The required Δ5 pairing can be extrinsically induced by a conventional s-wave superconductor into a three dimensional Dirac material (3DDM).
Band topology in insulators and superconductors is connected with the change in the sign of the gap parameter which in turn creates zero energy states at the location of gap kink. This mechanism in the case of insulators leads to gapless surface modes protected by a topological invariant1–4. When the spectral gap is of the superconducting (pairing) nature, the sign change of the oder parameter gives rise to midgap states5, 6. These topologically protected modes will be Majorana zero modes, which are their own anti-particles7–10. To realize Majorana fermions (MFs) various scenarios have been proposed which involve closing and re-opening the superconducting gap in one way or another. Gaping chiral modes of topological insulators (TIs) by Zeeman field and superconducting pairing gives rise to MFs in the interface region where the strength of these two gapping mechanisms are equal11, 12. In one-dimensional nano-wires this can be achieved by the competition between a polarizing Zeeman field, and depolarizing spin-orbit interaction13, 14. In two-dimensions, the vortex core of a p-wave SC binds a MF15. The required p-wave SC can be engineered on the surface of a TI by proximity to a conventional s-wave SC11. The above scenarios are: (i) limited to low dimensions, (ii) require a TR breaking by a Zeeman field, and (iii) require p-wave superconductors which are not abundant in nature.
In this report we propose yet another three dimensional system that admits a two dimensional sheet of MFs without requiring a Zeeman field. As such the resulting MFs are not associated with a vortex core and hence no low-energy excitations other than MFs exist16, 17. We find that a peculiar pseudo-scalar superconducting order parameter, Δ5 can give rise to a two-dimensional sheet of MFs when it is interfaced with a 3DDM. To set the stages for our finding, let us start by noting that in a one-band situation described by a parabolic band dispersion, the strength of the gap is characterized by a (scalar) gap parameter. However in 3DDM where the relevant degrees of freedom are described by the Dirac equation18, new opportunities can arise. The low-energy effective Hamiltonian of 3DDM candidate compounds like Cux(Bi2Se3)1−x 19, 20, Na3Bi1−xSbx 21, 22 and Cd3(As1−xPx)2 21, 22 are described by a Dirac equation in the Γ point of their Brillouin zone. The material CuxBi2Se3 is a Dirac superconductor below 3.8K23 and hence is a 3DDS. For such a 3DDM, the low-energy degrees of freedom are described by 4-component Dirac spinor. The most general superconducting pairing is where ψ c satisfies the same Dirac equation as ψ, but with opposite charge and is covariant under Lorentz transformation. The 4×4 pairing matrix can be expanded in terms of a basis composed of 1, four γ μ, γ 5, four γ 5 γ μ and six anti-commutators σ μν=i[γ μ, γ ν]/224 as . Then the pairing Δs will be the conventional scalar pairing, while Δ5 will be a pseudo-scalar pairing under the Lorentz transformation. Similarly Δμ, Δ5μ, Δμν, will be vector, pseudo-vector and tensor superconducting pairings25. We find that only the pseudo-scalar superconducting gap Δ5 competes with scalar Dirac gap mγ 0 and leads to gap closing. Therefore placing a m dominated region (i.e. a 3DDM) next to a Δ5 dominated region denoted as m|Δ5 junction (Fig. 1), gives rise to a two-dimensional sheet of Majorana fermions. This mechanism, does not require p-wave pairing, nor a magnet to break time-reversal. Instead, it requires a peculiar form of pseudo-scalar superconducting order parameter that changes sign in the mirror. We then proceed to show that such a system belongs to the DIII topological class allowing Z-number classification which in turn guarantees the existence of two-dimensional Majorana sea (2DMS) on the region where the strength of m and Δ5 are equal. We further corroborate our results by showing an anomalous Andreev reflection which couples electron and holes with the same spin direction, genuinely creates a perfect Andreev-Klein effect independent of angle of incidence. This effect manifests itself in the robust zero-mode resonance peak in the differential conductance and fractional supercurrent in the Δ5|m|Δ5 Josephson junction.
We consider a Dirac material with a single Dirac cone,
We use the representation γ0 = τ3 ⊗ σ0 and γj = τ1 ⊗ iσj for the Clifford algebra where σ j and τ j are Pauli matrices acting on spin and band spaces, respectively. Also, σ 0 and τ 0 are the 2×2 unit matrices. The k is the wave vector of excitations. In four space-time dimensions one can also construct γ 5=iγ 0 γ 1 γ 2 γ 3 which will be very essential for our discussion in this paper. In Eq. (1), the mass term is of the ordinary mγ 0 nature and is responsible for the band gap, and v F is the Fermi velocity. The pairing Hamiltonian for such a system is:
corresponding to which the Dirac-Bogoliubov-deGennes (DBdG) equation in k-space is,
where ε is the energy of eigenstate with respect to the chemical potential E F. Here ϕ is the macroscopic phase of superconductor, u(v) is the electron (hole) part of BdG wave function in Nambu space. The anti-unitary operator 𝒞 = γ2γ0K is the charge-conjugation of Eq. (1), where K is the complex-conjugation: This means that Lorentz transformation for is the inverse of the Lorentz transformation for ψ. Therefore requiring the superconducting pairing to be Lorentz invariant24, 26, the scalar (conventional) superconducting pairing is given by . Similarly pseudo-scalar pairing is , etc. We have examined all of the above 16 possible superconducting pairing channels. We find that the Δ5 parameter gives rise to a gap closing when both m and Δ5 are present. To see this in Δ5 channel, let us for clarity of notation set ħ and v F to 1 which gives the eigenvalues of Eq. (3) as , where and . When the Dirac mass m dominates the spectral gap (Δ5→0) the above eigenvalues reduce to . In this case each eigenvalue is doubly (spin) degenerate27. Deep in the 3DDS region where Δ5 dominates the eigenvalues acquire the following structure which corresponds to a pairing gap at the Fermi level. In a region where both m and Δ5 are non-zero, the nature of gap can be more clearly seen if we look at E F=0 where dispersion becomes and the band gap is determined by Δ±=(m±Δ5) which clearly indicates the competition between the Dirac mass m and the pseudo-scalar superconducting parameter Δ5. Figure 2 summarizes the closing and reopening of the spectral gap as one moves from m dominated region to Δ5 dominated region. The closing of the Dirac gap and re-opening of it in the form of a pseudo-scalar superconducting gap in this system does not require any magnetic field. This implies that when a Δ5 3DDS is brought next to a normal 3DDM with gap parameter m, such that Δ5>m, the induced Δ5 on the normal 3DDM side decays towards the bulk of 3DDM and crosses m somewhere in the interface where the excitations become gapless.
Let us now prove that when Δ5=m a two dimensional Majorana sea appears. The pseudo-scalar character of Δ5 pairing can be interpreted as a spin- singlet superconductor whose mirror image has an opposite sign. To understand further the properties of such a pairing, let us now explicitly construct the anti-unitary operators corresponding to particle-hole, and time-reversal transformation for Eq. (3): Let η j be set of Pauli matrices in the Nambu space. Then the particle-hole and time-reversal symmetries in this space can be defined as PH = iη2 ⊗ γ0γ5γ2K and TR = η0 ⊗ γ0γ1γ3K, respectively. Owing to PH 2=1 and TR 2=−1, the chiral symmetry SL = PH ∗ TR satisfies SL 2=1 which places the present system in the DIII class of topological superconductors which can be classified with winding number. We explicitly obtain the topological charge27,
To further corroborate our central result concerning a sea of Majorana fermions in the present system, let us now focus on the transport properties arising from 2DMS which can be directly accessed in experiments. Let us begin by looking into a single m|Δ5 junction. The lateral coordinates in the plane of junction are (x, y) confined within a lateral dimensions (W x, W y), which means the corresponding components of wave vector in these directions are quantized as . Each mode can be identified with a set of quantum number n=(n x, n y). In the 3DDM side of the junction (z<0), there are eight components of wave functions, where the indices e (h) characterize electron (hole)-like quasi-particles, ↑ (↓) denotes its spin direction with respect to z-axis and ± indicates whether they are right- or left-movers27. The interface between m (z<0) and Δ5 (z>0) regions can reflect an incident electron as a hole. In this Andreev reflection process, the missing charge of 2e is absorbed by superconductor as Cooper pair30. Typically the s-wave nature of superconductivity imposes that the reflected hole from an ↑-spin incident electron must be in a ↓-spin state and vice versa. However, due to strong spin-orbit coupling encoded in the Dirac Hamiltonian of a 3DDM, when an electron in a given spin direction hits a 3DDM, the spin of the electron transmitted into 3DDM can be flipped31. Therefore in 3DDM|3DDS junction we also take into account the possibility of an anomalous –i.e. spin flipped–Andreev reflection of the reflected hole. The boundary condition for an ↑ spin electron hitting the interface is given by,
The left hand side is related to the wave function in 3DDM side, and the right hand side describes the wave equation in 3DDS side. Here r N,↑ and r N,↓ are the amplitudes of conventional and spin-flipped normal reflection, respectively. The r A,↓ and r A,↑ are the amplitudes of conventional and anomalous Andreev reflection, respectively. Similar processes take place for ↓ spin and hole carriers as well27. The probability of these reflections vs. θ, the polar angle of incidence, are depicted in Fig. 3. Because of the conservation of parallel component of wave vector, , at the scattering process, the angle of propagation for reflected hole (θ′) has a critical value beyond which the reflected hole can not contribute to transport. For zero modes, ε=0, the conventional and spin-flipped normal reflections would disappear and we are left with the conventional and anomalous Andreev reflection given by,
where α=arctan(k y/k x) is azimuthal angle. From Eq. (6), it is obvious that for a zero-energy incident electron at any angle of propagation we have a perfect Andreev reflection, |rA,↓|2 + |rA,↑|2 = 1. This is a transport signature of Majorana fermions9, 32. This effect is robust against changing the chemical potential and angles of incidence (α, θ)33.
The BTK formula for the differential conductance of the junction34 will be,
where G 0=e 2/h is the quantum of conductance. Note that the summation over ν includes spin-flipped contributions as well. When the linear dimensions of the interface are much larger than the superconducting coherence length, ξ s=ħv F/Δ5, i.e. Wx, Wy ≫ ξs, the summation over mode indices n in Eq. (7) can be replaced by an integral. The conductance for various values of chemical potential has been plotted in Fig. 4 as a function of energy. Note that the height of the resonance peak at zero energy is pinned at 2G 0 which indeed arises from the 2DMS.
Now let us move to the next transport signature of 2DMS, namely the fractional Josephson current. Consider the geometry depicted in part(b) of Fig. 1 and assume that L is the length of junction and δϕ=ϕ R−ϕ L is the phase difference between the two Δ5 superconductors. In th short junction limit, L ≪ ξS, the Andreev bound states (ABS) responsible for carrying the super-current between two 3DDS region are given by refs 27 and 35,
where τ n is the normal transmission probability of the junction, , with . Despite that the Hamiltonian in Eq. (3) is invariant under 2π phase shift, ϕ→ϕ+2π, the ABS in Eq. (8) clearly has a 4π period. This has been depicted in Fig. 5. Using Eq. (8), the corresponding Josephson current becomes, I±(δφ) = (eΔ5/ℏ) ∑n ∂ε±,n (δφ)/∂δφ = ± Ic sin (δφ/2), where is the critical value of the Josephson current. In the absence of perturbations which violate fermion parity conservation, such a form of fractional Josephson current is a signature of 2DMS on the surface of 3DDS12.
We have shown that a 2DMS can be obtained from singlet pairing as well, provided the superconducting order is pseudo-scalar, i.e. it changes sign under mirror reflection. Such form of superconductivity among other forms of superconductivity can be induced into the 3DDM by proximity to a conventional s-wave superconductor25. The two dimensional Majorana sea can emerge if we can manage to change the sign of Δ5−m, i.e. to invert the gap parameter. If we are dealing with an intrinsic 3DDS for which Δ5>m, its natural surface with vacuum already is where 2DMS appears, because the vacuum itself can be thought of a trivial gapped state with m→∞ and Δ5=0. This leads to simpler experimental setting. However if the superconductivity is induced by proximity25, we only need to link a conventional superconductor to a 3DDM and the 2DMS will appear inside the 3DDM. This has been show in Fig. 6. A conventional superconductor (dark blue) to the left of a 3DDM (green) induces various forms of superconductivity into the 3DDM25. However, among them all, only the Δ5 superconductivity sets a battle against m to close the spectral gap. We have explicitely checked this by considering simultaneous precence of Δ5 and Δother where the later can be any of the 15 remaining order parameters. The surface at which the induced Δ5 pairing crosses the Dirac gap m (boundary between gray and green regions) hosts the two dimensional Majorana sea.
The present system is a singlet cousin in the family of odd-parity superconductors20. Our scenario for Majorana surface states does not require proximity to any magnet, as no triplet pairing is involved here. Advantages of our scenario over the existing p-wave superconductor+magnet scenarios is that
Perfect Andreev reflection and fractional Josephson current as two hallmarks of the ensuing Majorana sea leave clear transport footprints. The junction between such a superconductor and conventional s and d wave superconductors similar to the odd-parity p-wave superconductors may provide anomalous flux quantization in units of h/4e 20.
We have used the properties of the Lorentz group to identify the pseudo-scalar nature of the Δ5 superconductor considered here. The characterization of the topology of the pseudo-scalar superconducting state cosidered here requires the use of topology to calculate the winding number. In evaluation of the energy spectrum, and construction of symmetry operations we have used the standard linear algebra.
M.S. thanks M. Alidoust and R. Beiranvand for useful discussions. We are grateful to A. Kitaev and A. Furusaki for insightful comments. We thank Z. Faraei for many discussions.
S.A.J. conceived the project. Mathematical solution of the model was carried out by M.S., and cross-checked by S.A.J. Figures were perepared by M.S. Paper was jointly written by both authors. Both authors have equally contributed to critical discussions of the work.
The authors declare that they have no competing interests.
Morteza Salehi and S. A. Jafari contributed equally to this work.
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