Topological superconductor candidates: materials and devices

Scope

Topological superconductivity concerns the paired quasiparticle state, not merely normal-state band topology next to a superconducting transition. The target and dimensionality must be stated before interpreting a probe:

This comparison separates mathematical classification, probe evidence and proposed operations. The classification reference treats non-interacting or mean-field Hamiltonians and discusses stability to interactions small compared with the gap; it is not a complete classification of strongly interacting or nodal superconductivity. R1 R2 R3 R8

For a fully gapped Bogoliubov–de Gennes description, topology depends on dimensionality, symmetries and the occupied quasiparticle states. This restricted framework does not exhaust interacting or nodal superconductivity. A local zero-bias peak is neither a universal requirement for every topological superconductor nor sufficient proof of a Majorana mode. Demonstrating non-Abelian operations is a further question; it is not the definition of every bulk topological superconducting phase. R1 R4 R5 R11

Mathematical assumptions

One clean single-subband wire model uses $H(k)=\xi_k\tau_z+\alpha k\tau_z\sigma_y+V_Z\sigma_x+\Delta\tau_x$, where $\xi_k=\hbar^2k^2/(2m)-\mu$, $\sigma$ acts on spin and $\tau$ on particle–hole space. Here $\Delta$ is real induced pairing, spin–orbit and Zeeman axes are perpendicular, and $V_Z=|g\mu_B B|/2$ is an energy rather than a magnetic field. With $a=\alpha k$, diagonalization yields $E_\pm^2=\xi_k^2+a^2+V_Z^2+\Delta^2\pm2\sqrt{V_Z^2(\xi_k^2+\Delta^2)+\xi_k^2a^2}$. R4 R5

The zero-momentum gap closes at $V_Z^2=\mu^2+\Delta^2$. In the stated gapped wire model, the side with $V_Z^2>\mu^2+\Delta^2$ supports the topological phase. The inequality alone is not a material diagnostic: setting spin–orbit coupling to zero can leave finite-momentum gapless excitations even when it holds. Independent checks compare 54 matrix spectra, three closure points and that counterexample. They do not simulate multiband disorder, orbital depairing, interfaces or a complete material invariant. R4 R5

The Fu–Berg criterion assumes a three-dimensional, fully gapped, time-reversal-invariant BdG state with inversion symmetry and odd-parity pairing. Its normal-state Fermi-surface shortcut also assumes weak pairing and bands separated from the Fermi energy at the time-reversal-invariant momenta (TRIM).

• Use the BdG inversion operator: odd-parity pairing requires $\widetilde P=P\tau_z$. The invariant multiplies its negative-energy eigenvalues at the eight TRIM, taking one member of each Kramers pair. Multiplying both members would square each sign and lose the parity information.
• Read the result as a sufficient test: the parity $\nu=n\bmod2$ is only part of the three-dimensional integer classification. An odd result establishes nonzero $n$ within these assumptions. An even result does not by itself establish $n=0$; an even, nonzero invariant is not excluded. Counting enclosed TRIM neither measures the pairing parity nor certifies a doped specimen.
• Keep surface and vortex states separate: Fu–Kane's uniform proximitized helical surface has a gapped spectrum and preserves time reversal, despite its projected analogy with spinless $p_x+ip_y$ pairing. A vortex introduces a winding pairing phase, $\Delta(r,\theta)=\Delta_0(r)e^{\pm i\theta}$, and admits a localized zero mode in that ideal model. The vortex calculation is not an observation of a zero mode on every uniform surface, nor a demonstration of braiding.

These are different theoretical constructions. Applying either to a material requires checking its own symmetry, gap, geometry and disorder conditions; neither theorem replaces those measurements. R2 R3

Experimental criteria

Hor's Cu-intercalated Bi2Se3 study reports a transition near 3.8 K. In the checked author manuscript, Cu0.12Bi2Se3 has a magnetic response roughly one-fifth of ideal diamagnetism at the lowest accessible temperature, which the authors treat as a lower bound because the transition is incomplete. The resistance does not reach zero. This establishes why sample preparation and connected superconducting paths matter; neither a universal superconducting volume nor topological pairing can be inferred from those numbers alone. R6

Butch's YPtBi crystals show a resistive transition near 0.77 K with an AC diamagnetic response. The critical-field curve uses a 50% resistive criterion, with 10–90% bounds, and field perpendicular to [001]. The often-quoted 1.5 T is a zero-temperature extrapolation. Converting it with $B_{c2}=\Phi_0/(2\pi\xi^2)$ gives about 14.8 nm as a consistency estimate. Superconductivity in a candidate topological semimetal does not itself identify its pairing topology or establish a calibrated superconducting volume. R7

Two iron-chalcogenide papers require separate attribution. Zhang's ARPES study resolves a spin-helical surface band and a superconducting surface gap around 1.8 meV, interpreted as surface topological superconductivity. Its low-temperature measurement at 2.4 K is not the reported 14.5 K transition temperature. The study's surface interpretation must not be replaced by a statement that all bulk pairing is topological. In its angular-gap figure, hollow mirrored markers are not additional independent measurements. R8

Wang's distinct vortex-STM paper studies FeTe0.55Se0.45. Figure 2 spectra at 0.55 K and 0.5 T parallel to c are normalized by integrated spectral area; they cannot be read as an absolute quantized conductance plateau. The approximately 0.3 meV peak width contains instrumental and thermal broadening. Barrier and temperature scans elsewhere use different fields. The zero-energy signal is interpreted as Majorana evidence, while the displayed braiding sequence is a proposal, not a performed operation. R9

Xu's Bi2Te3/NbSe2 heterostructure study compares vortex spectra across film thicknesses. The checked 2013 author version underlying the 2015 publication distinguishes where split peaks become experimentally resolvable from the extrapolated origin of their splitting. The former is around 20 nm even for thinner films; the latter changes near four quintuple layers and supports the authors' Majorana interpretation. Treating every unsplit interval as a directly measured Majorana radius would discard both the extrapolation and approximately 0.2 meV resolution. R10

Semiconductor readout advances also need their own evidence category. Aghaee's 2025 InAs–Al experiment measures single-shot parity interferometrically and explicitly retains an Andreev-state alternative. Van Loo's 2026 work demonstrates parity readout in a minimal Kitaev-chain device, with a two-site rather than asymptotically long-chain model. These are useful experimental capabilities; neither result, as scoped here, establishes braiding or fault-tolerant topological computation. Readout performance, spatial nonlocality and a protected bulk phase remain distinct questions. R14 R15

A 10 July 2026 preprint by Zatelli and colleagues reports coherent parity oscillations in coupled minimal Kitaev chains. The authors distinguish partial protection at the sweet spot from the unprotected degeneracy-lifting operations used for rotation. No journal acceptance was verified here; this is not a demonstrated braid or a certified gate fidelity. R16

Limits and disputes

A local zero-bias peak can arise from ordinary Andreev states, confinement or disorder. Pan and Das Sarma's simulations also show that end-to-end conductance correlations are not conclusive on their own: accidental correlations can occur in selected disordered examples, including long wires. Length must be considered relative to the relevant coherence scale; a fixed micrometre threshold is not a universal diagnostic. These model counterexamples do not identify the mechanism in every later device. Temperature, barrier, calibration, disorder and nonlocal controls must be interpreted together.

Retracted quantization claim: Quantized Majorana conductance (2018) was retracted on 8 March 2021. The author-institution copy of the notice reports:

• Previously undisclosed charge-jump corrections in Figure 2a and Extended Data Figure 4b, and a mislabelled axis in Figure 4b.
• Recalibration shifting the conductance plateaus by 8%, above $2e^2/h$; this is a reported calibration change, not an uncertainty interval.
• Replotting the full parameter range placing points outside the stated two-sigma error bars, so the authors withdrew the quantized-conductance conclusion.

The linked notice and raw-data archive document the correction trail; this review has not independently recalibrated the apparatus or reproduced that archive's analysis. The retracted result remains historical context, never positive confirmation of later devices. Nor does retracting this result decide the status of every other platform. R11 R12 R13

Complete scientific acceptance remains open. This article checks selected primary passages, figures and restricted models. Earlier author versions are identified; complete final-version comparisons, supplements, raw-data analysis, calibration and exclusion of competing interpretations remain unfinished. These limits concern the present review and should not be inflated into a claim that every platform has the same evidential status. R4 R5 R6 R7 R8 R9 R10

Discovery implications

Specify the target first: bulk odd-parity superconductivity, a proximitized topological surface, a wire end state or a controllable parity device. Then record composition, geometry, temperature, field, superconducting criterion and competing-state tests for that target. Compare complementary probes on characterized samples and preserve failures and retractions. Strong spin–orbit coupling or an attractive band calculation can motivate an experiment, but cannot replace identification of the actual paired phase and its protected observables. R2 R3 R5 R11 R13

Related topics

Topological superconductivity, Majorana zero modes, Heavy-fermion superconductors, and Spin–orbit coupling.

For a dated comparison of spectroscopy, control and exchange criteria, see the Majorana identification question.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.