Fig. 1. Fe-based superconductors (FeSCs) as a better Majorana platform: (a) The original idea for searching Majorana zero mode (MZM) in FeSCs; (b) the typical band structure of FeSCs, the orbital characters of each band are as follows: α (d_xz); β (d_yz); γ (d_xy); η (d_xy); δ (d_xz); ω (p_z); (c) the typical Fermi surface and superconducting order parameters of FeSCs^[74,76]; (d) the mass renormalization among different compounds, indicating strong electron-electron interactions in FeSCs^[81]; (e) band structure near the Γ point of Fe(Te, Se) single crystals measured by ARPES^[82].

Fig. 2. The mechanism of topological band structure and band inversion of Fe(Te, Se): (a) First-principle calculation of band structure of FeSe (without SOC), the size of red circles represents the components of p_z orbital^[87]; (b) crystal structure of Fe(Te, Se)^[87]; (c) band inversion mechanism and orbital overlapping in Fe(Te, Se)^[87]; (d) first-principle calculation of band structure of FeTe_0.5Se_0.5 (without SOC)^[87]; (e) experimental band structure around Γ in FeTe_0.55Se_0.45 measured by ultra-high resolution laser ARPES^[101]; (f) realistic topological band structure in FeTe_0.55Se_0.45 (with SOC). TDS stands for topological Dirac semimetal, TI stands for topological insulator^[101].

Fig. 3. Experimental observation of the linear dispersion of Dirac surface states in FeTe_0.55Se_0.45^[100]: (a) The matrix element effect which defines the selection rule of ARPES intensity, depending on relationship between photon polarization and electron orbitals; (b) the Dirac surface states observed under p-polarization; (c) the d_xz bulk bands observed under s-polarization; (d) orbital characters around Γ in FeTe_0.55Se_0.45; (e), (f) orbital character determined by the matrix element analysis under p- and s-polarization, respectively. The orbital characters marked at the bottom represent the active orbitals under certain polarization and momentum. The dashed parts in the band structure represent the intensity suppressed by the selection rule.

Fig. 4. Spin-momentum locking and isotropic superconducting gap on the Dirac surface state: (a) The spin-momentum locking feature in FeTe_0.55Se_0.45 single crystal^[100]; (b), (c) spin-resolved ARPES data measured along Cut 1 and Cut 2 in panel (a), respectively^[100]; (d) temperature dependent energy distribution curves measured at k_F of the Dirac surface state indicating a superconducting gap of the Dirac surface state opens below 14.5 K, which is the bulk T_c^[100]; (e) isotropic superconducting gap on the Dirac surface state^[100]; (f) summary of the main observations of the Dirac surface state on FeTe_0.55Se_0.45 single crystal, i.e. topological band inversion, linear dispersion, spin-momentum locking, large superconducting gap, small Fermi energy^[102]; (g) the Dirac surface state of FeTe_0.55Se_0.45 single crystal acquires an effective spinless pairing due to the proximity effective from s ±-wave bulk superconductivity, which satisfies all of the requirements of Fu-Kane model^[100].

Fig. 5. Evidence of Dirac semimetal phase in FeTe_0.55Se_0.45 single crystal^[101]: (a), (b) The spin-integrated and spin-resolved ARPES spectrum around Γ respectively; (c) the projected band structure on the (001) surface of a C₄ symmetry protected Dirac semimetal. The spin polarized surface states are mixing with the bulk bands; (d), (e) spin polarization of the d_yz bulk band measured on four representative k_F around the Fermi surface (as indicates in the insert). It is clear that the d_yz bulk band has the helical spin texture; (f) summary of the topological band structure along the in-plane momentum. There are a strong topological insulator phase around the Fermi level and a topological Dirac semimetal phase above it; (g) the transport evidence of Dirac semimetal phase in FeTe_0.55Se_0.45 single crystal. The linear transverse magnetoresistance indicates the incorporation of bulk Dirac electrons. The transport experiments were carried at 16 K. The PMF and SMF represent pulse and static magnetic field respectively.

Fig. 6. The discovery of vortex Majorana zero mode in FeTe_0.55Se_0.45 single crystal: (a) The theoretical prediction of vortex MZMs in FeTe_0.55Se_0.45 single crystal^[100]; (b) STM topography of FeTe_0.55Se_0.45 single crystal^[102]; (c) zero-bias conductance map which shows vortex lattice^[102]; (d) a sharp zero-bias conductance peak measured at the center of a vortex. In order to make sure the observation is indeed a zero energy vortex bound state, three careful checks are listed as follows. First of all, to make sure that the signal measured is indeed from vortex bound state^[102]: (e) ZBC map after and before applying a magnetic field. It shows the local environment of the vortex is clean and free of impurities^[102]; (f) ZBCP is stable under different tunneling barriers. Secondly, to make sure that the observed ZBCP is truly a single peak^[102]; (g) FWHM of ZBCP measured under different tunneling barriers; (h), (i) the observed ZBCP is truly a zero mode^[102]. (h) is the simultaneous measured I(V) curve and dI/dV curve on the center of a vortex core^[102], (i) is the enlarged display of red box area in Fig. (h).

Fig. 7. The wavefunction of vortex Majorana zero mode^[102]: (a) A zero bias conductance map of a topological vortex; (b) a dI/dV(r, V) line-cut intensity plot along the black dashed line indicated in (a); (c) a waterfall-like plot of (b) with 65 spectra; (d) an overlapping display of eight dI/dV spectra selected from (c); (e) spatial dependence of the height (top) and FWHM (bottom) of the ZBCP; (f) comparison between ARPES and STS results, Δ₀ = 1.8 meV, E_F = 4.4 meV, ξ = ν_F/Δ₀ = 123 Å; (g) comparison between the measured ZBCP peak intensity with a theoretical calculation of MZM spatial profile with the parameters extracted from (f)

Fig. 8. Quasiparticle poisoning of vortex Majorana zero modes^[102]: (a) Three vortex Majorana zero modes measured on different locations, the FWHM of ZBCP at the center of the vortex core is larger when the SC gap around the vortex core is softer; (b) a zero bias conductance map of vortex and line-cut intensity plot of Majorana zero modes measured under 0.55 K (left) and 4.2 K (right), respectively; (c) temperature evolution of ZBCPs in a vortex core. The gray curves are numerically broadened 0.55 K data at each temperature; (d) amplitude of the ZBCPs of three vortex MZMs under different temperatures. The amplitude is defined as the peak-valley difference of the ZBCP; (e) left panel: C/T fitting of amplitude of Majorana ZBCPs under different temperatures. Right panel: summary on several temperature evolution measurements; (f) schematic explanation of the temperature effect on Majorana ZBCPs. The red line is the vortex MZM and the blue line is the bound state of body votex.

Fig. 9. Topological vortex phase transition in the three-dimensional vortex line model. The first line: Evolution of the band structure of a topological material by tuning the chemical potentials. The second line: Evolution of the low energy vortex bound state at k_z = 0 under different chemical potentials. The third line: The k_z dispersion of low energy vortex bound states. The fourth and fifth line: evolution of vortex Majorana zero modes under different chemical potentials in topological insulator and normal insulators, respectively. This figure is adapeted from Ref. [176], some features are added by us.

Fig. 10. Resonance Andreev reflection induced Majorana quantum conductance: (a) Conventional electron resonance tunneling in a semiconductor heterostructure^[207]; (b) the wavefucntion of conventional resonance tunneling^[105]; (c) two tips cross-tunneling can be regarded as a replacement of semiconductor heterostructure for realizing semiconductor heterostructure under the condition of equal hopping amplitude around the two tips (t₁ = t₂)^[204]; (d) the Majorana induced resonance Andreev reflection (MIRAR) can be regarded as a superconducting version of the conventional resonance tunneling in the particle-hole Hilbert space. Here a single electrode plays both roles of electron and hole electrode^[204]. Due to the particle-hole equivalence property, Majorana modes couple to the incident electron and reflected hole with equal tunneling coupling strength, which satisfies the resonant condition ( ; Γ_t = 2πρ₀|t|², ρ₀ being the related density of states); (e), (f) the wavefucntion of Andreev reflection mediated by MZM and a conventional Andreev bound states, respectively^[105]; (g) the material setup used in Law-Lee-Ng model^[204]; (h) the theoretical calculated re-sonance quantum conductance of Majorana modes^[204]; (i) theoretical calculated Majorana conductance under finite temperature and poisoning rate.

Fig. 11. Variable-tunnel-coupling STM method and the observation of conductance plateau of vortex Majorana zero modes^[105]: (a) The tunnel coupling strength can be changed by the tip-sample separation distance under the effect of STM regulation loop; (b) a three-dimensional plot of tunnel coupling dependent measurement, dI/dV(E, G_N), which shows a zero bias conductance plateau; (c), (d) the general phenomena observed on Majorana conductance of FeTe_0.55Se_0.45, i.e. non-quantized plateau; (e), (f) the rare case of Majorana conductance of FeTe_0.55Se_0.45, i.e. quantized plateau; (g) a histogram of the plateau conductance (G_P) from 31 sets of data; (h)—(j) the conductance evolution under different tunnel couplings. It shows no plateau feature measured on finite energy CdGM states, the continuum outside the superconducting gap and zero filed superconducting state, respectively.

Fig. 12. Surface Dirac electron induced half-integer level shift of vortex bound states: (a) Half-odd-integer quantized level sequences of vortex bound states in a conventional s-wave superconductor. There are only parabolic bulk bands involved^[104]; (b) the quantum limit is difficult to reach in conventional s-wave superconductors, so that a large ZBCP observed in the center of vortex core is generally due to multiple overlapping of densely packed non-zero peaks^[231]; (c) integer quantized level sequences of the vortex bound state in Fu-Kane model. The intrinsic spin Berry phase carried by Dirac surface states induces the half-integer level shift^[104]; (d) the zero-doping limit is defined as the chemical potential is approaching the energy of the Dirac point. In this case, a vortex MZM is the only allowed subgap bound state^[104]; (e) the theoretical calculated angular momentum resolved wavefunction of BdG eigenstate, the blue and green curves are spin down and up components, respectively^[210]. Insert: The calculated spin-integrated 2 D local density of states of three lowest levels of vortex bound states in the case of (c) and (a), respectively^[104]; (f) theoretical calculated eigenvalue of BdG Hamiltonian near the zero chemical potential limit.

Fig. 13. Observation of integer quantized vortex bound states^[104]: (a) A dI/dV(r, V) line-cut intensity plot measured on a topological vortex #1. Integer quantized vortex bound states are clearly observed; (b) peak positions extracted from (a); (c) the comparison between experimental observed and theoretical calculated level energy in topological vortex #1; (d) same as (a), but measured on vortex #11, which is close to the zero chemical potential limit; (e) overlapping display of dI/dV spectra selected from (d); (f) same as (c), but shows the case of vortex #11; (g) the comparison of observed MZM line profile in topological vortices under integer quantization (open circles) and near the zero chemical potential limit (dark stars); (h) the calculated MZM wavefuction under different chemical potential by Fu-Kane model; (i) a histogram of averaged level energies normalized by the first level spacing, i.e. the ratio E_L/ΔE. The statistical analysis is performed among all the 35 topological vortices which show integer quantized CdGMs levels; (j) experimentally observed spatial pattern of the lowest three levels of vortex bound state in a topological vortex.

Fig. 14. The inhomogeneity of material helps coexisting ordinary and topological vortices: (a) A dI/dV(r, V) line-cut intensity plot measured on ordinary vortex #8. Half-odd-integer quantized vortex bound states are clearly observed^[104]; (b) the comparison between experimental observed and theoretical calculated level energy in ordinary vortex #8^[104]; (c) a histogram of averaged level energies normalized by the first level spacing, i.e. the ratio E_L/ΔE. The statistical analysis is performed among all the 26 ordinary vortices which show half-odd-integer quantized CdGM levels^[104]; (d) surface disorder transforms the strong topological insulator to a normal insulator. The scattering potentials are gradually larger from left to right^[262]; (e) concentration of the dopants could drive a strong topological insulator to be a normal insulator or weak topological insulator in Fe(Te, Se). The bands in green (red) represent p_z (d_xz/d_yz) orbital with odd (even) parity^[201].

Fig. 15. Spatial distribution of the two classes of vortices^[104]: (a), (c), (e) Zero-bias conductance maps of three well-separated regions. The yellow solid circles mark the vortices with ZBCPs and integer quantized CdGM levels, yellow dashed circles mark the vortices with ZBCPs but its CdGM level sequences can not be fitted to integer quantization, blue solid circles mark the vortices without ZBCPs and half-integer quantized CdGM levels, and blue dashed circles mark the vortices without ZBCPs or half-integer quantized CBS levels. The green dashed lines encircle the same class of vortices. Topological vortices and ordinary vortices usually group together, which indicates topological region and trivial region coexist on the sample surface due to spatial inhomogeneity; (b), (d), (f) summary of the ratio of different types of vortices in the three regions, respectively. The data in the three regions are measured under 40 mK and 2.0 T.

Fig. 16. Mechanism of the presence or absence of MZMs in Fe(Te, Se)^[104]: (a) Fe(Te, Se) single crystals are intrinsically inhomogeneous. Disappearance of Dirac surface states is possible in some regions of the (001) surface (brown color). In the conventional regions, the corresponding bulk states can be normal insulators or weak topological insulators. Consequently, the Dirac surface state moves deeper into the bulk and go around the conventional region, as indicated by the gray boundary inside the crystal. In other topological regions (gray color), where Dirac surface states remain intact, the corresponding bulk states are still in the strong topological insulating phase; (b) a schematic phase diagram of vortex MZMs appearing in topological regions (topological vortices). The gradient blue areas in (b) and (c) indicate the phase sector that MZMs can be detected by STM/S experiments. In the dark blue sector, the Majorana wave function is more localized on the sample surface, while in brighter positions, the Majorana wave function strongly hybridizes with bulk quasiparticles and moves deeper beneath the surface, leading to weak ZBCP signal measured by STM/S. The vertical axis demonstrates MZMs evolution as a function of effective temperature which can be represented by extrinsic broadening of the observed ZBCPs. The horizontal axis demonstrates the MZMs evolution as a function of quantum parameters, e.g., chemical potential (μ) measured from the Dirac point. The black dots with an arrow indicate the quantum critical points in which a vortex phase transition happens. Across the critical point, the vortex line turns to be topological trivial and MZMs disappear in the topological region. The red dashed line indicates the achievable region in experiments; (c) a schematic phase diagram of vortex MZMs appearing in conventional regions (ordinary vortices). There are no MZMs in our measurements in those vortices. The observable MZMs can only exist above the critical points when the vortex phase transition turns the trivial vortex line into a 1D topological superconductor in the conventional region.

Fig. 17. Braiding vortex MZMs and topological quantum computing. Left-top panel: Surface effective spinless p-wave pairing induce by k-proximity effect from bulk bands in Fe(Te, Se)^[84]. Left-bottom panel: The pristine vortex MZM observed in Fe(Te, Se)^[103]. Middle panel: It is possible to use a STM tip to manipulate vortex MZMs on the surface of Fe(Te, Se)^[102]. Right panel: Topological qubit built by braiding four vortex MZMs^[3].