## What is/are Spin Momentum Locking?

Spin Momentum Locking - Photonic spin density (PSD) in the near-field gives rise to exotic phenomena such as photonic skyrmions, optical spin-momentum locking and unidirectional topological edge waves.^{[1]}Spin-momentum locking in the surface mode of topological insulators leads to the surface accumulations of spin-polarized electrons caused by bias currents through topological insulator samples.

^{[2]}The chiral interaction between light and matter is mainly caused by the spin-momentum locking and makes the chiral quantum optics enter a vigorous development stage.

^{[3]}Spin-orbit coupling in non-centrosymmetric crystals gives rise to interesting spin-momentum locking of the Fermi surface.

^{[4]}More importantly, the temperature-dependent harmonic measurement data can be divided into two categories, namely, the spin Hall effect of the TI bulk states gives rise to a relatively small spin Hall angle in the high-temperature region, whereas the spin-momentum locking nature of the interfacial Dirac fermions leads to the enhancement of the SOT strength once the topological surface states become the dominant conduction channel at deep cryogenic temperatures.

^{[5]}The spin of the eigenstates depends on the momentum in general, and a nontrivial spin-momentum locking arises for the case with no site inversion symmetry, without considering any spin-orbit couplings.

^{[6]}The helicity-dependent photocurrents, underpinned by spin-momentum locking of surface Dirac electrons, are weak and easily overshadowed by bulk contributions.

^{[7]}As such, by putting two topologically different systems together, the spin angular momentum dependent one-way interface modes can be selectively excited by acoustic spin sources, exhibiting robust transport protected by tight spin-momentum locking.

^{[8]}This effect can be observed in a variety of systems, for example in topological insulators where spin-momentum locking of the topologically protected surface states is the root cause for the effect and in magnetic systems where anisotropic magnetic ordering induces it.

^{[9]}In both systems, we observe the unidirectional routing of chiral elastic waves, characterize the different elastic spins along different directions, and demonstrate the spin-momentum locking in broad frequency ranges.

^{[10]}By utilizing the pseudospin-momentum locking in gapless graphene, two recent experiments demonstrate the measurement of the topological Berry phase by corresponding to the unique number of wavefront dislocations in Friedel oscillations.

^{[11]}We use this technique to image complete time sequences of propagating surface plasmons, demonstrating their spin-momentum locking, as well as plasmonic skyrmions on atomically flat single-crystalline gold films that have been patterned using gold ion beam lithography [1].

^{[12]}Indeed, three-dimensional TIs are predicted to host exotic properties like topologically protected surface states (TSS), which show Dirac-like band dispersion and spin-momentum locking [1].

^{[13]}For a near-field source with a specific spin, the guide mode with a fixed directional wave vector is excited due to spin-momentum locking.

^{[14]}However, one of the essential properties of a QSHI, spin-momentum locking of the helical edge states, has yet to be experimentally validated.

^{[15]}The spin-momentum locking due to the Rashba spin–orbit coupling is key to the formation of the Majorana state.

^{[16]}One of the most fundamental and exotic properties of three-dimensional (3D) topological insulators (TIs) is spin-momentum locking (SML) of their topological surface states (TSSs), promising for potential applications in future spintronics.

^{[17]}We report a unique type of gap in a magnetic Weyl semimetal ${\mathrm{Co}}_{3}{\mathrm{Sn}}_{2}{\mathrm{S}}_{2}$ where the electrons are spin polarized and preserve the spin-momentum locking feature of Weyl fermions.

^{[18]}The vortex generators in the form of whispering-gallery–mode microcavities can also achieve nondegenerate OAM modes from incident light with spin-momentum locking.

^{[19]}Patterning of topological insulator with mirror-symmetric forms of planar chiral design yields photogalvanic currents with opposite directions due to the interplay between the spin-momentum locking and polarization conversion in the pattern.

^{[20]}Usually, surface Dirac electrons are described by a linear-dispersion model, which exhibits various peculiar features originating from the spin-velocity equivalence or spin-momentum locking.

^{[21]}Here, we visualize the topological states of the WTI candidate ZrTe5 by spin and angle-resolved photoemission spectroscopy (ARPES): a quasi-1D band with spin-momentum locking was revealed on the side surface.

^{[22]}Topological Josephson plasmon modes appear at the interface between topological and trivial domains, carrying pseudospin dominant energy flows, which mimics the spin-momentum locking in the renowned quantum spin Hall effect.

^{[23]}FeTeSe enables Ising-type superconductivity, which involves spin-momentum locking.

^{[24]}Here we discover that more general spin symmetries in decoupled spin and crystal space categorize non-relativistic collinear magnetism in three phases: conventional ferromagnets and antiferromangets, and a third distinct phase combining zero net magnetization with an alternating spin-momentum locking in energy bands, which we dub "altermagnetic".

^{[25]}Photonic spin density (PSD) in the near-field gives rise to exotic phenomena such as photonic skyrmions, optical spin-momentum locking and unidirectional topological edge waves.

^{[26]}We demonstrate the emergence of flat bands featuring strong spin-momentum locking and the emergence of symmetry broken states with associated non-coplanar magnetization when interactions are included.

^{[27]}Transverse spin angular momentum of light is a key concept in recent nanophotonics to realize unidirectional light transport in waveguides by spin-momentum locking.

^{[28]}Two-dimensional (2D) noncentrosymmetric NbSe 2 is a promising candidate because its pair breaking is protected by the spin-momentum locking effect, resulting in a giant in-plane H c2 (~50 T).

^{[29]}This is similar to the process of spin accumulation in TIs, where a current corresponds to an effective spin due to spin-momentum locking [Qi11].

^{[30]}The frequency of the terahertz driving field sharply discriminates between HH generation from the bulk and from the topological surface, where the unique combination of long scattering times owing to spin-momentum locking17 and the quasi-relativistic dispersion enables unusually efficient HH generation.

^{[31]}Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta.

^{[32]}Spin-momentum locking of evanescent waves describes the relationship between the propagation constant of an evanescent mode and the polarization of its electromagnetic field, giving rise to applications in light nano-routing and polarimetry among many others.

^{[33]}Weyl fermions induce topologically protected spin-momentum locking, which is closely related to spin-wave gap formation in magnetic crystals.

^{[34]}They indicated significant spin-momentum locking due to the large Rashba effect.

^{[35]}This result indicates formation of the theoretically predicted radial spin-polarized texture in k-space of chiral systems owing to spin-momentum locking.

^{[36]}Spin-momentum locking is a universal wave phenomenon promising for applications in electronics and photonics.

^{[37]}Spin-momentum locking is essential to the spin-split Fermi surfaces of inversion-symmetry broken materials, which are caused by either Rashba-type or Zeeman-type spin-orbit coupling (SOC).

^{[38]}Based on the calculation of Wannier–Stark states, we investigate dynamical properties of BOs with the chiral character of spin-momentum locking.

^{[39]}Spin angular momentum (SAM) is an important feature for wave systems, prominent in various properties like spin-momentum locking of wave propagations.

^{[40]}The surface of a topological insulator hosts Dirac electronic states with the spin-momentum locking, which constrains spin orientation perpendicular to electron momentum.

^{[41]}Topological quantum materials, where spin-momentum locking is believed to lead to particularly long spin lifetimes, are regarded as a promising platform for such applications.

^{[42]}Topological insulators (TIs), such as Bi2Se3 [7–10], were reported to show even larger ξDL, which arises from the spin-momentum locking of their surface states.

^{[43]}The NPNE has a quantum origin arising from the conversion of a nonlinear transverse spin current to a charge current due to a joint result of hexagonal warping effect, spin-momentum locking, and the time-reversal symmetry breaking induced by the magnetic field.

^{[44]}We show that in a material with a given spin-momentum locking, the density of states plays a crucial role in determining the charge-spin interconversion efficiency, and a simple inverse relationship can be obtained.

^{[45]}Under circularly polarized light illumination, Dirac surface electrons in topological insulators show a natural chiral response due to spin-momentum locking, which can be interrogated by probing the dependence of photocurrent on the polarization of incident light.

^{[46]}Spin-momentum locking arising from strong spin-orbit coupling is one of the key natures of topological materials.

^{[47]}Topological insulators (TIs) have emerged as some of the most efficient spin-to-charge convertors because of their correlated spin-momentum locking at helical Dirac surface states.

^{[48]}Apart from the group-IV, III-V, and II-VI semiconductors and their nanostructures, spin injection and detection with 2D materials such as graphene, transition-metal dichalcogenides (TMDs) and topological insulators (TIs) has become a new trend and a particularly interesting topic due to either the long spin lifetime or strong spin-orbit coupling induced spin-momentum locking, which potentially leads to dissipationless electronic transport.

^{[49]}The surface state of a three dimensional strong topological insulator (TI) is well described in the independent particle picture (IPP) by an isotropic Dirac cone at the $\Gamma$-point and perpendicular spin-momentum locking.

^{[50]}

## spin orbit coupling

Spin-orbit coupling in non-centrosymmetric crystals gives rise to interesting spin-momentum locking of the Fermi surface.^{[1]}The spin of the eigenstates depends on the momentum in general, and a nontrivial spin-momentum locking arises for the case with no site inversion symmetry, without considering any spin-orbit couplings.

^{[2]}Spin-momentum locking is essential to the spin-split Fermi surfaces of inversion-symmetry broken materials, which are caused by either Rashba-type or Zeeman-type spin-orbit coupling (SOC).

^{[3]}Spin-momentum locking arising from strong spin-orbit coupling is one of the key natures of topological materials.

^{[4]}Apart from the group-IV, III-V, and II-VI semiconductors and their nanostructures, spin injection and detection with 2D materials such as graphene, transition-metal dichalcogenides (TMDs) and topological insulators (TIs) has become a new trend and a particularly interesting topic due to either the long spin lifetime or strong spin-orbit coupling induced spin-momentum locking, which potentially leads to dissipationless electronic transport.

^{[5]}