Searching for Majorana quasiparticles at vortex cores in iron-based superconductors

The unambiguous detection of the Majorana zero mode (MZM), which is essential for future topological quantum computing, has been a challenge in recent condensed matter experiments. The MZM is expected to emerge at the vortex core of topological superconductors as a zero-energy vortex bound state (ZVBS), amenable to detection using scanning tunneling microscopy/spectroscopy (STM/STS). However, the typical energy resolution of STM/STS has made it challenging to distinguish the MZM from the low-lying trivial vortex bound states. Here, we review the recent high-energy-resolution STM/STS experiments on the vortex cores of Fe(Se,Te), where the MZM is expected to emerge, and the energy of the lowest trivial bound states is reasonably high. Tunneling spectra taken at the vortex cores exhibit a ZVBS well below any possible trivial state, suggesting its MZM origin. However, it should be noted that ZVBS is a necessary but not sufficient condition for the MZM; a qualitative feature unique to the MZM needs to be explored. We discuss the current status and issues in the pursuit of such Majorananess, namely the level sequence of the vortex bound states and the conductance plateau of the ZVBS. We also argue for future experiments to confirm the Majorananess, such as the detection of the doubling of the shot noise intensity and spin polarization of the MZM.

A vortex core, which is introduced by an applied magnetic field to a superconductor, can serve as such a boundary accommodating the MZM when the host superconductor is topologically nontrivial 14,15 . In general, at the vortex core, the superconducting gap ∆ locally vanishes, resulting in the formation of vortex bound states. In an ordinary s-wave superconductor, the energies of these bound states, known as the Caroli-de Gennes-Matricon (CdGM) states 16 , are represented by the half-odd-integer level sequence E = µ∆ 2 /ϵ F (µ = ± 1/2, ±3/2, ±5/2, · · · )(ϵ F : Fermi energy) [ Fig. 1(a)]. It should be noted that there is no state at exactly zero energy. In contrast, for a chiral p-wave superconductor, which is a representative of the topological superconductor, the vortex bound states follow the integer level sequence (µ = ± 0, ±1, ±2, · · · ), because of the additional angular momentum of the superconducting order parameter. Hence, an exactly zero energy state appears for µ = 0 14 [ Fig. 1(b)]. This µ = 0 state is nothing but the MZM.
Obviously, a key signature of the putative vortex MZM is the zero-energy vortex bound state (ZVBS), which can be detected by STM/STS as a peak in the local density-of-states spectrum at the vortex core. However, this experiment is challenging because, for typical superconductors with ∆ ∼ 1 meV and ϵ F ∼ 1 eV, the energy spacing between the bound states δE ∼ ∆ 2 /ϵ F is as small as 1 µeV, while the currently available energy resolution of STM/STS is limited to be tens of µeV at the best. In such a situation, many vortex bound states crowd inside the superconducting gap, forming a broad peak at zero energy in the tunneling spectrum. This peak does not represent the ZVBS associated with the MZM but is a resolution-smeared bundle of multiple CdGM states, which should appear irrespective of the topological nature of superconductivity. Therefore, the experimental identification of the MZM demands the following conditions: (i) the superconductor should be the chiral Half-odd integer level sequence   Besides the topological nature, large energy separation δE between the vortex bound states is expected in Fe(Se,Te) because it is in the crossover regime between the Bardeen-Cooper-Schrieffer superconductivity and Bose-Einstein condensation where ϵ F is comparable to ∆. According to ARPES and STM results, ϵ F is estimated to be 5 ∼ 20 meV 38 , and the smallest ∆ is about 1.5 meV 50 . Therefore, the energy of the lowest trivial CdGM state E µ=1/2 ∼ ∆ 2 /2ϵ F = 60 ∼ 220 µeV. This is two orders of magnitude larger than those in usual superconductors, providing an opportunity to resolve the discrete vortex bound states individually. Nevertheless, the energy resolution of STM/STS must be better than a few 6 tens of µeV to clearly distinguish the MZM from the lowest CdGM states that may appear as low as 60 µeV. Such a high energy resolution is only possible at ultra-low temperatures below 100 mK achieved by a dilution refrigerator. The energy resolution of 20 ∼ 30 µeV has been achieved by a recently developed dilution-refrigerator-based STM/STS system, of which the attainable electron temperature is as low as 90 mK 26 . were performed using 3 He-refrigerator-based STM/STS systems with an energy resolution of ∼ 250 µeV, which is insufficient to resolve the individual vortex bound states. Therefore, the observed peaks in the spectrum may consist of multiple bound states. We emphasize here that it is easy for the ZVBS to be misinterpreted as the finite-energy peak and for the finite-energy peak to be misinterpreted as the ZVBS, if the energy resolution of the STM/STS is worse than δE.
As mentioned above, the energy-resolution issue can be solved by adopting the dilution-   It is indispensable to perform the level-sequence analyses for a large number of vortices with higher energy resolutions. Figures 6(a) to (c) show the results of such analyses using the same data set for Fig. 5. As shown in Fig. 6(a), there are multiple low-lying states even below the lowest-energy state in the previous work 65 . We have indexed them from low to high energy with the ZVBS to be n = 0 if it exists. We then normalized the energy of the n-th peak E n by the energy of the lowest finite energy peak E 1 for more than 400 vortices to make histograms of distribution probabilities shown in Fig. 6(b) and (c). For the integer and half-odd integer sequences, the peaks should appear at |E n /E 1 | = n (solid line) and |E n /E 1 | = 2n − 1 (dashed line), respectively. The histograms are broad, with little differences between the two types of vortices with and without the ZVBS. Apparently, the level-sequence analyses do not work properly for high-energy-resolution data.
There may be a couple of reasons that make the level-sequence analysis challenging.
First, the vortex bound states originating from the bulk bands may also be detectable at the observe the conductance plateau, one has to stabilize the tip above the surface by a feedback loop that keeps the tunneling current constant to be typically ∼ 1 nA at a typical bias voltage of ∼ 1 mV. Since the putative conductance plateau may appear at a current 1 or 2 orders of magnitude larger, the tip-sample distance has to be reduced by 100 ∼ 200 pm for typical tunneling conditions with the effective work function of a few eV. However, as long as the tip and sample are both reasonably clean, the tunneling condition should break down above ∼ 10 nA, resulting in a situation where the tip and sample are in contact [76][77][78] . We found that this jump-in-contact phenomenon occurs in Fe(Se,Te) as shown in Fig. 7(c). It is difficult to control the junction in such a contact regime because the conditions are sensitive to the actual shape of the tip apex that deforms in an uncontrollable manner after the contact.
Therefore, the tip-sample distance dependence of the tunneling spectrum and the height of the ZVBS do not show systematic changes in the contact regime as shown in Fig. 7(d) and (e). For the putative conductance plateau, it is necessary to guarantee that the tunneling current flows only through the MZM even in the contact regime.

B. Other potential experiments to detect the Majorananess
Besides the level sequence and the quantized conductance plateau, more experiments have been considered to detect the "Majorananess". One of the ideas is to detect the unique shot-noise expected in the current associated with the MZM 79-81 . In general, the power of the shot-noise of the tunneling current S is proportional to the effective charge per tunneling event q eff : S = 2q eff |I|. Because of the complete Andreev reflection, the effective charge per tunneling event for the junction with the MZM should be q eff = 2e, whereas q eff = e in the case of the conventional single electron tunneling process. This stark contrast can be used to distinguish the MZM from the trivial bound states 79-81 . The shot-noise spectroscopy has been performed at the vortex core of Fe(Se,Te) using the recently-developed shot-noise STM 82 with a superconducting STM tip. Although the observed q eff was nearly e even at the vortex core with the ZVBS, it is still unclear whether the ZVBS is associated with the MZM or not because the energy resolution was limited to be ∼ 250 µeV 83 . Experiments at ultra-low temperatures with a higher energy resolution are anticipated.
Another possible experiment is the detection of the spin polarization expected in the vortex core with the MZM 84,85 . Since no spin polarization is expected in the trivial vortex core, a spin-sensitive experiment can provide an important clue. In principle, STM can be spin-sensitive if a magnetic metal is used for the scanning tip 86,87 . Since the principle of such a spin-polarized STM is the tunneling magneto-resistance effect, the larger the spin polarization of the tip, the better spin sensitivity is expected. The spin polarization of a conventional magnetic tip is limited to a few tens of %, and it has been difficult to estimate the absolute value of the spin polarization on the sample side. Recently, a technique to achieve a 100% spin-polarized tip has been developed, enabling us high-sensitivity and quantitative spin polarization measurements [88][89][90] . The idea is to use the Yu-Shiba-Rusinov (YSR) state, which is a 100% spin-polarized bound state created near a magnetic impurity in a superconductor. If a single magnetic atom is attached at the apex of the superconducting STM tip, the YSR state can act as a perfect spin filter for the tunneling current [88][89][90] .
Another important feature is that the narrowness of the YSR state in the spectrum is limited only by temperature 90 , enabling us sub-meV energy resolution. Therefore, spin-polarized spectroscopy using the YSR tip can be a promising tool to clarify the spin polarization of the ZVBS in Fe(Se,Te).
Besides the measurement techniques, more appropriate platforms that may host the MZM should be developed. Although Fe(Se,Te) has great potential as the MZM platform because it naturally realizes the Fu-Kane proposal 27 and has a large energy spacing between the vortex bound states, its chemical and electronic disorders make straightforward data interpretation difficult. Although the source of the disorder is unclear, the substitution of Te for Se may cause inhomogeneities in the Fe-chalcogen layer. Several other iron-based super-conductors with stoichiometric Fe-chalcogen or Fe-arsenic layer have been proposed to host the chiral p-wave superconductivity at the surface. These include (Li 1−x Fe x )OHFeSe 23 and CaKFe 4 As 4 24 . These compounds possess a T c more than twice as high as that of Fe(Se,Te), resulting in a larger energy separation of vortex bound states. Indeed, even by using 3 Herefrigerator STM/STS systems, the ZVBSs were successfully resolved from the lowest trivial bound states in these materials 23,24 . Although these materials are more homogeneous than Fe(Se,Te), the vortex core spectra still vary from vortex to vortex. In addition, these materials possess multiple cleaving planes, resulting in various surfaces that may exhibit different properties even if the MZM should be topologically protected. It is an important challenge to develop a platform without these uncertain factors.