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This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

https://iopscience.iop.org/article/10.1088/1361-6633/aa6ac7/pdf

Topological superconductors: a review

A possible sighting of Majorana states Nearly 80 years ago, the Italian physicist Ettore Majorana proposed the existence of an unusual type of particle that is its own antiparticle, the so-called Majorana fermion. The search for a free Majorana fermion has so far been unsuccessful, but bound Majorana-like collective excitations may exist in certain exotic superconductors. Nadj-Perge et al. created such a topological superconductor by depositing iron atoms onto the surface of superconducting lead, forming atomic chains (see the Perspective by Lee). They then used a scanning tunneling microscope to observe enhanced conductance at the ends of these chains at zero energy, where theory predicts Majorana states should appear. Science, this issue p. 602; see also p. 547 Scanning tunneling microscopy is used to observe signatures of Majorana states at the ends of iron atom chains. [Also see Perspective by Lee] Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides strong evidence, corroborated by other observations, for the formation of a topological phase and edge-bound Majorana fermions in our atomic chains.

/pdf/observation-of-majorana-fermions-in-ferromagnetic-atomic-5451hode2h.pdf

Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

A topological superconductor A promising path toward topological quantum computing involves exotic quasiparticles called the Majorana bound states (MBSs). MBSs have been observed in heterostructures that require careful nanofabrication, but the complexity of such systems makes further progress tricky. Zhang et al. identified a topological superconductor in which MBSs may be observed in a simpler way by looking into the cores of vortices induced by an external magnetic field. Using angle-resolved photoemission, the researchers found that the surface of the iron superconductor FeTe0.55Se0.45 satisfies the required conditions for topological superconductivity. Science, this issue p. 182 Angle-resolved photoemission spectroscopy indicates that FeTe0.55Se0.45 harbors Dirac-cone–type spin-helical surface states. Topological superconductors are predicted to host exotic Majorana states that obey non-Abelian statistics and can be used to implement a topological quantum computer. Most of the proposed topological superconductors are realized in difficult-to-fabricate heterostructures at very low temperatures. By using high-resolution spin-resolved and angle-resolved photoelectron spectroscopy, we find that the iron-based superconductor FeTe1–xSex (x = 0.45; superconducting transition temperature Tc = 14.5 kelvin) hosts Dirac-cone–type spin-helical surface states at the Fermi level; the surface states exhibit an s-wave superconducting gap below Tc. Our study shows that the surface states of FeTe0.55Se0.45 are topologically superconducting, providing a simple and possibly high-temperature platform for realizing Majorana states.

Observation of topological superconductivity on the surface of iron-based superconductor

We study theoretically the superconductivity in doped Weyl semimetals with an inversion symmetry based on the Bogoliubov-de Gennes equations. In principle, the two superconducting states, i.e., the zero momentum BCS-like pairing and the finite momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing are competing in this kind of systems. Our self-consistent calculation indicates that the BCS-type state may be the ground state. The competition between these two pairing states is studied in detail through normal state Fermi surface and the finite energy spectral functions. Generally, the Fermi surface topology supports the FFLO pairing while the finite energy band structure favors the BCS-type pairing. We also study the physical properties and address the Majorana Fermions excitation in these two superconducting state respectively.

/pdf/superconductivity-in-doped-inversion-symmetric-weyl-3css2vqzkp.pdf

Superconductivity in doped inversion-symmetric Weyl semimetals

Applying a three-band model and the random phase approximation, we theoretically study the spin excitations in nickelate superconductors, which have been newly discovered. The spin excitations were found to be incommensurate in the low energy region. The spin resonance phenomenon emerged as the excitation energy increased. The intensity can be maximized at the incommensurate or commensurate momentum, depending on the out-of-plane momentum. The spin excitations reverted to incommensurate at higher energies. We also discuss the similarities and differences in the spin excitations of nickelate and cuprate superconductors. Our predicted results can be later validated in inelastic neutron scattering experiments.

Spin excitations in nickelate superconductors

Starting from a two-orbital model and based on the random phase approximation, spin excitations in the superconducting state of the newly discovered BiS2 superconductors with three possible pairing symmetries are studied theoretically. It is found that the spin response is uniquely determined by the pairing symmetry. Possible spin resonance excitations may occur at an incommensurate momentum about (0.7π,0.7π) for the d-wave symmetry, while the transverse spin excitation near (0,0) is enhanced for the p-wave symmetry and no spin resonance signature is seen for the s-wave pairing symmetry. These distinct features may be used for probing or determining the pairing symmetry in this newly discovered compound.

/pdf/probing-the-superconducting-pairing-symmetry-from-spin-1zdemywsym.pdf

Probing the Superconducting Pairing Symmetry from Spin Excitations in BiS2 Based Superconductors

The quasiparticle states around a nonmagnetic impurity in electron-doped iron-based superconductors with spin-density-wave (SDW) order are investigated as a function of doping and impurity scattering strength. In the undoped sample, where a pure SDW state exists, two impurity-induced resonance peaks are observed around the impurity site and they are shifted to higher (lower) energies as the strength of the positive (negative) scattering potential (SP) is increased. For the doped samples, where the SDW order and the superconducting order coexist, the main feature is the existence of sharp in-gap resonance peaks, the positions and intensity of which depend on the strength of the SP and the doping concentration. In all cases, the local density of states exhibits clear ${C}_{2}$ symmetry. We also note that, in the doped cases, the impurity will divide the system into two sublattices with distinct values of magnetic order. Here, we use the band structure of a two-orbital model, which considers the asymmetry of the As atoms above and below the Fe-Fe plane. This model is suitable to study the properties of the surface layers in the iron pnictides and should be more appropriate to describe the scanning tunneling microscopy experiments.

Quasiparticle states around a nonmagnetic impurity in electron-doped iron-based superconductors with spin-density-wave order

We study theoretically the coexistence of the spin-density-wave (SDW) and superconductivity in electron-doped iron-pnictide superconductors based on the two orbital model and Bogoliubov-de Gennes equations. The phase diagram is mapped out and the evolution of the Fermi surface as the doping varies is presented. The local density of states (LDOS) has also been calculated from low to high doping. We show that the strength of the superconducting coherent peak at the positive energy gets enhanced and the one at the negative energy is suppressed by the SDW order in the underdoped region. Several features of our results are in good agreement with the experiments.

Tao Zhou

Papers

Superconductivity in doped inversion-symmetric Weyl semimetals

Spin excitations in nickelate superconductors

Probing the Superconducting Pairing Symmetry from Spin Excitations in BiS2 Based Superconductors

Quasiparticle states around a nonmagnetic impurity in electron-doped iron-based superconductors with spin-density-wave order

Spin-density wave and asymmetry of coherence peaks in iron pnictide superconductors from a two-orbital model