Chiral Spin Mode on the Surface of a Topological Insulator

TL;DR

This study uses polarization-resolved Raman spectroscopy to identify a transverse collective spin mode on the surface of a topological insulator, revealing a spin wave-like excitation influenced by spin-orbit coupling.

Contribution

First observation and identification of a transverse collective spin mode on a topological insulator surface using Raman spectroscopy.

Findings

Detected a sharp 150 meV peak in Raman spectra.

Identified the peak as a transverse collective spin mode.

Mode is analogous to a spin wave in a Fermi liquid.

Abstract

Using polarization-resolved resonant Raman spectroscopy, we explore collective spin excitations of the chiral surface states in a three dimensional topological insulator, Bi$_2$Se$_3$. We observe a sharp peak at 150 meV in the pseudovector $A_2$ symmetry channel of the Raman spectra. By comparing the data with calculations, we identify this peak as the transverse collective spin mode of surface Dirac fermions. This mode, unlike a Dirac plasmon or a surface plasmon in the charge sector of excitations, is analogous to a spin wave in a partially polarized Fermi liquid, with spin-orbit coupling playing the role of an effective magnetic field.

Equations

$$\hat{H}(\mathbf{k})=\frac{k^2}{2m^{\ast }}{\hat{\sigma }}_0+v_1\hat{\mathbf{\sigma }}\cdot \tilde{\mathbf{k}},$$

$$\gamma (\mathbf{k})=\frac{\left({\mathbf{e}}_S\cdot {\mathbf{p}}_{u_1{u}_2}\right)\left({\mathbf{e}}_I\cdot {\mathbf{p}}_{u_2{l}_1}\right)}{E_{u_2}(\mathbf{k})-E_{l_1}(\mathbf{k})-{\Omega }_L}+\frac{\left({\mathbf{e}}_S\cdot {\mathbf{p}}_{u_1{l}_2}\right)\left({\mathbf{e}}_I\cdot {\mathbf{p}}_{l_2{l}_1}\right)}{E_{l_2}(\mathbf{k})-E_{l_1}(\mathbf{k})-{\Omega }_L},$$

$$\mathcal{R}(\omega ,T)\propto {\chi }_{zz}^{\prime \prime }(\omega ,T)/(E_g-{\Omega }_L{)}^2,$$

$$\hat{\chi }(\omega ,T)=-\hat{\Pi }(\omega ,T){\left(1+\frac{U}{2}\hat{\Pi }(\omega ,T)\right)}^{-1}$$

$${\Pi }_{\alpha \beta }(i{\omega }_n)=T\sum _{{ϵ}_m}{\int }_{\mathbf{k}}\mathrm{Tr}\left[{\hat{\sigma }}_{\alpha }{\hat{G}}_{\mathbf{k}}(i{ϵ}_m+i{\omega }_n){\hat{\sigma }}_{\beta }{\hat{G}}_{\mathbf{k}}(i{ϵ}_m)\right].$$

$${\mathcal{R}}_{E_2}(\omega ,T)$$

$${\mathcal{R}}_{A_1}(\omega ,T)$$

$${\mathcal{R}}_{A_2}(\omega ,T)$$

$$E_{SS}(\mathbf{k})=$$

$$\approx$$

$$E_{CB}(\mathbf{k})=E_0+\frac{k_{||}^2}{2m_{||}^{\ast }}+\frac{k_{\perp }^2}{2m_{\perp }^{\ast }},$$

Full text

Magnets and partially spin-polarized Fermi liquids support collective spin excitations (spin waves), in which all electron spins respond coherently to external fields, and the “glue” that locks the phases of precessing spins is provided by the exchange interaction. In nonmagnetic materials where inversion symmetry is broken but time-reversal invariance remains intact, strong spin-orbit coupling (SOC) may play the role of an effective magnetic field, which locks electron spins and momenta into textures. This phenomenon is encountered in three-dimensional (3D) topological insulators (TIs), which harbor topologically protected surface states Fu et al. (2007); Zhang et al. (2009); Chen et al. (2009); Hsieh et al. (2008); Hasan and Moore (2011). These states have been a focus of intense studies, both from the fundamental point-of-view Bansil et al. (2016); Wang et al. (2013); Zhu et al. (2014); Wu et al. (2016); Jozwiak et al. (2016); Li et al. (2016); Shao et al. (2017) and for potential applications in spintronics devices Pesin and MacDonald (2012); Xu et al. (2014); Fan et al. (2016, 2016); Wang et al. (2015); Scharf et al. (2016); Kondou et al. (2016); Tian et al. (2017); Ando and Shiraishi (2017). However, the many-body interactions leading to collective effects in TIs remain largely unexplored. An essential aspect of this physics is an interplay between the Coulomb interaction and SOC, which is expected to give rise to a new type of collective spin excitations – chiral spin waves Shekhter et al. (2005); Ashrafi and Maslov (2012); Maiti et al. (2015, 2016); Maiti and Maslov (2017); Kumar and Maslov (2017). In the long wavelength ($q=0$) limit, these modes are completely decoupled from the charge channel and thus distinct from spin-plasmons Raghu et al. (2010); Kondo et al. (2013), Dirac plasmons Di Pietro et al. (2013); Politano et al. (2015), and surface plasmons in TIs Kogar et al. (2015); Politano et al. (2015).

In this Letter, we employ polarization-resolved resonant Raman spectroscopy, a technique of choice for probing the collective charge Cardona (1982); Devereaux and Hackl (2007), spin Cottam and Lockwood (1986); Gozar et al. (2004); Perez et al. (2016); Gretarsson et al. (2016) and orbital excitations Miyasaka et al. (2005), to study collective spin excitations of the chiral surface states in Bi2Se3. To enhance the signal from the surface states, we tune the energy of incoming photons in resonance with a transition between two sets of chiral surface states: near the Fermi energy and about $1.8$ eV above it [Fig. 1(a)] Sobota et al. (2013). We observe a long-lived excitation at 150 meV in the pseudovector symmetry channel of the Raman spectra, which is most pronounced at low temperatures but persists up to room temperature. By comparing the data with calculations, we identify this excitation as the transverse collective chiral spin mode supported by spin-polarized surface Dirac fermions. Such collective modes – first predicted for non-topological systems Shekhter et al. (2005); Ashrafi and Maslov (2012); Maiti et al. (2015, 2016); Perez et al. (2016); Maiti and Maslov (2017); Kumar and Maslov (2017) but hitherto unobserved – are “peeled off” from the continuum of particle-hole excitations by the exchange interaction.

Chiral surface states in a 3D TI are described by the Hamiltonian Fu (2009):

[TABLE]

where $m^{\ast}$ is the effective mass, $\bm{\hat{\sigma}}=(\hat{\sigma}_{1},\hat{\sigma}_{2},\hat{\sigma}_{3})$ are the Pauli matrices, $\hat{\sigma}_{0}$ is the 2$\times$2 unit matrix, and ${\bf\tilde{k}}=(k_{y},-k_{x},\frac{v_{\text{w}}}{v_{1}}[k_{+}^{3}+k_{-}^{3}])$ with $k_{\pm}\equiv k_{x}\pm ik_{y}$. The $z$ component of ${\bf\tilde{k}}$ describes hexagonal warping of the surface states away from the Dirac point Fu (2009). The spectrum of Eq. (1) consists of hexagonally warped electron- and hole-like Dirac cones of opposite chirality. A light-induced excitation from the occupied state in the hole cone to an empty state in the electron cone is accompanied by a spin-flip of the quasi-particle [Fig. 1(b)]. Such direct transitions form a continuum which starts at the threshold energy $\omega_{-}$ [Fig. 1(c)] Riccardi et al. (2016); Kashuba and Fal’ko (2009).

Due to the Pauli exclusion principle, two electrons in the triplet state avoid each other, thus reducing the energy of the Coulomb repulsion. Therefore, the repulsive Coulomb interaction between electrons translates into an attractive exchange interaction between their spins, leading to bound states below the continuum of spin-flip excitation, i.e., chiral spin modes. In general, there are three such modes [red curves in Fig. 1(c)], which correspond to linearly polarized oscillations of the magnetic moment in the absence of the external magnetic field Maiti et al. (2015, 2016); Maiti and Maslov (2017). At $q=0$, there is a doubly degenerate mode with an in-plane magnetic moment and a transverse mode with an out-of-plane moment, with energies $\omega_{s,||}$ and $\omega_{s,\perp}$, respectively. Because the chiral-spin modes are below the continuum, they are expected to remain long-lived even at elevated temperatures Maiti and Maslov (2015). These modes are in essence (zero-field) spin waves that can be measured by resonant Raman scattering because they couple to the electromagnetic field through antisymmetric Raman tensors Maiti and Maslov (2017).

Bi2Se3 is an archetypical 3D TI, with a rhombohedral crystal structure with $D_{3d}^{5}$ group symmetry ($R\overline{3}m$) in the bulk. The crystal is composed of quintuple layers weakly bonded by van der Waals force, allowing easy cleavage of optically flat $ab$ surfaces with the symmorphic $P6mm$ wallpaper group symmetry (2D point group $C_{6v}$) Li et al. (2013); Slager et al. (2013). Bi2Se3 has a relatively simple band structure near the $\Gamma$-point [Fig. 1], with a pair of topologically protected surface Dirac cones, labeled SS1, and another pair at about 1.8 eV above SS1, labeled by SS2 [Fig. 1(a)]. As-grown Bi2Se3 is usually electron-doped due to naturally formed Se vacancies Huang et al. (2012). In this study, we use well-characterized samples with low concentration of impurities and crystalline defects Dai et al. (2016). All the bulk phonon modes in this crystal are sharp with no signatures of any impurity modes, and all the surface phonon modes are clearly observed Kung et al. (2017). The Fermi energy ($E_{F}$) was determined by scanning tunneling spectroscopy to be about 150 meV above the Dirac point of SS1 [Fig. 1(b)] Dai et al. (2016).

The polarized Raman spectra were acquired in a quasi-backscattering geometry from the $ab$ surface of Bi2Se3 single crystals grown by modified Bridgman method SM . We use 521, 647, 676 and 752 nm lines of a Kr+ laser for excitation, where the spot size is roughly $50\times 50\,\mu m^{2}$, and the power is about 10 mW. The scattered light is analyzed by a custom triple-grating spectrometer.

In Fig. 2, we show spectra of secondary emission for four scattering geometries employing both linear and circular polarizations, as defined in the Supplementary SM . Of the four excitations, the 521 nm (2.38 eV) one is the farthest from near-resonant transition between SS1 and SS2, while the 676 nm (1.83 eV) one is the closest. The spectra contain contributions from electronic Raman scattering and exciton photoluminescence. The latter is present for all polarizations and can be subtracted from the spectra SM . The signal below 50 meV is dominated by phonon modes which are discussed elsewhere Kung et al. (2017).

The spectrum for the non-resonant 521 nm (2.38 eV) excitation shows no sharp peaks but a broad feature SM . This is in stark contrast to the spectra of other three excitations, where a sharp peak around 150 meV is observed in the XY and RR geometries. The peak is the strongest for the 676 nm (1.83 eV) excitation, which is in near-resonance with the transition between SS1 and SS2, thus confirming the surface origin of the observed signal. In order to better understand the origin of the 150 meV peak, we subtract the photoluminescence contributions and then utilize the symmetry properties of the Raman tensors to separate the measured spectra into the $E_{2}$, $A_{1}$ and $A_{2}$ symmetry channels of $C_{6v}$ point group SM . In Fig. 3, we plot the temperature dependence of Raman response $\mathcal{R}(\omega,T)$ in three symmetry channels. It is clearly seen that the 150 meV peak is associated with the $A_{2}$ symmetry channel. The continuum broadens and becomes invisible above 150 K, but the peak is still well-defined even at $T=300$  K.

The basis functions of the $A_{2}$ representation of $C_{6v}$ transform as the $z$ component of the angular momentum, which is a pseudovector Ovander (1960); Koster (1963). This suggests that the observed peak in the $A_{2}$ channel corresponds to a spin mode [marked by $\omega_{s,\perp}$ in Fig. 1(c)] with an out-of-plane magnetic moment (also a pseudovector).

To quantify the assignment of the 150 meV peak to the out-of-plane chiral spin mode, we calculate the Raman response of surface chiral states. We are interested in spin-flip resonant Raman processes between states near the Fermi level in SS1 and the states in SS2. Two resonance transitions are possible: an electron from the lower cone of SS1 can be transferred into either the lower or upper cones of SS2 and come back to the upper cone of SS1, producing a particle-hole pair. The diagrams for the corresponding Raman vertex and scattering cross-section are shown in Figs. 4 (a) and (b), correspondingly. Since the shift in the photon energy $\approx 150$ meV is much smaller than energy difference between the Dirac points $E_{g}\approx 1.8$ eV, the resonant part of the Raman vertex can be written as

[TABLE]

where $l_{1,2}$ and $u_{1,2}$ refer to the lower and upper cones of SS1 (SS2), ${\bf p}_{ab}$ is the matrix element of a dipole transition between states $a$ and $b$, and ${\bf e}_{I,S}$ are the polarizations of incident and scattered photons, correspondingly. Furthermore, $E_{u_{1},l_{1}}=\varepsilon^{(1)}_{\pm}({\bf k})$ and $E_{u_{2},l_{2}}=E_{g}+\varepsilon^{(2)}_{\pm}({\bf k})$, where $\varepsilon^{(1)}_{\pm}({\bf k})=k^{2}/2m^{*}\pm v_{1}\tilde{k}$ are the eigenergies of Eq. (1), and $\varepsilon^{(2)}_{\pm}({\bf k})$ are the eigenenergies of a similar Hamiltonian for SS2. The scattering cross-section contains $|\gamma({\bf k})|^{2}$, integrated over $k\geq k_{F}$. However, if trigonal warping is neglected, the denominators of the first (second) terms in Eq. (2) become $E_{g}+(v_{2}\pm v_{1})k-\Omega_{L}$, where $v_{1,2}$ are the velocities of Dirac fermions in SS1 and SS2. A characteristic feature of Bi2Se3 is that the lower cones of SS1 and SS2 are almost perfectly nested: a fit to the ARPES data in Ref. Nomura et al. (2014); Sobota et al. (2013) gives $v_{1}=2.4$ eVÅ  and $v_{2}=2.0$ eVÅ . Therefore, the second (hole-to-hole) term in $\gamma$ is essentially dispersionless, and the corresponding transition probability is enhanced by a factor of $\approx 1/(E_{g}-\Omega_{L})^{2}$, whereas the first (hole-to-electron) term is small 111If $\gamma$ disperses with $k$, the Raman response does not reduce to the spin susceptibility. However, the pole corresponding to the collective mode in the Raman response is still the same as in the spin susceptibility.. This explains why only one resonance is observed in the experiment.

Since the initial and final states of the Raman vertex form a $2\times 2$ space, $\hat{\gamma}$ can be expanded over a complete set of Pauli matrices as $\hat{\gamma}=c\hat{\sigma}_{0}+{\bf s}\cdot\bm{\hat{\sigma}}$. In the $XY$ scattering geometry, which contains the $A_{2}$ symmetry channel of $C_{6v}$ group, ${\bf s}={\bf e}_{I}\times{\bf e}_{S}={\bf e}_{z}$ and thus $\hat{\gamma}\propto\hat{\sigma}_{z}$. The Raman response function can then be written as

[TABLE]

where $\chi^{\prime\prime}_{\alpha\beta}$ is the imaginary part of the $\alpha\beta^{\text{th}}$ component of the spin susceptibility tensor. The many-body interactions are accounted for within the Random Phase Approximation (RPA) with a Hubbard-like interaction ($U$) in the spin channel Maiti and Maslov (2017):

[TABLE]

where $\hat{\Pi}(\omega,T)$ is obtained by analytic continuation of

[TABLE]

Here, $\int_{\bf k}\equiv\int\frac{d^{2}k}{(2\pi)^{2}}$, $\hat{G}^{-1}_{\bf k}(i\epsilon_{m})=i\varepsilon_{m}-\hat{H}({\bf k})+E_{F}+i\text{sgn}(\epsilon_{m})\Gamma/2$, $\hat{H}({\bf k})$ is given by Eq. (1), and $\Gamma$ is impurity broadening 222For calculation purposes, a momentum cutoff of $\Lambda_{k}=0.3$ Å*-1* was chosen SM .. $\chi_{zz}(\omega,T)$ has a continuum of spin-flip excitations and a pole which corresponds to the transverse collective mode. A simple result for the frequency of this mode can be obtained if one neglects hexagonal warping and considers the weak-coupling limit. In this case, $\omega_{s,\perp}=2E_{F}\left[1-2\exp(-4/u)\right]$, where $u\equiv UE_{F}/2\pi\hbar v_{1}^{2}\ll 1$ is the dimensionless coupling constant.

For a more general case, which includes the realistic band structure and finite temperature, the Raman response has to be evaluated numerically. The results of this calculation are shown in Fig. 3(d). With the band structure parameters obtained from ARPES measurements Nomura et al. (2014), the only two fitting parameters are the exchange coupling constant, fixed at $u\approx 0.6$ to reproduce the mode frequency at 15 K, and the impurity scattering rate chosen as $\Gamma=8$ meV. Comparison of the measured and computed spectra [Fig. 3(c) and (d), correspondingly] shows that the model describes well not only the overall shape of the signal but also its evolution with temperature SM . In particular, the theory reproduces the observed decrease in the peak position with increasing temperature, which can be ascribed to thermal smearing of the continuum boundary. For the well characterized samples studied here, the threshold of the spin-flip continuum is expected at $\omega_{-}\approx 260$ meV SM . However, the onset of this continuum is difficult to observe because its spectral weight is transferred into the collective mode. In the inset of Fig. 3(d), we show a zoom into the computed crossover region between the collective mode and the continuum. A quantitative agreement between the theory and experiment gives us confidence in that the observed 150 meV sharp peak in the $A_{2}$ symmetry channel is indeed a transverse chiral spin mode.

In conclusion, our results provide strong evidence for a new collective mode – the transverse chiral spin wave – in a time-reversal invariant system, a 3D TI. Strong spin-orbit coupling plays the role of a very high effective magnetic field, which protects the long-lived spin excitation. Such a robust collective spin mode may have potential applications in spintronics Najmaie et al. (2005); Dumas and Akerman (2014), magnonics Kruglyak et al. (2010); Chumak et al. (2015); Urazhdin et al. (2014), optoelectronics Politano et al. (2017) and quantum computing Fu and Kane (2008); Nayak et al. (2008); Alicea (2012). Moreover, the present results demonstrate an efficient way of probing the dynamical response of Dirac fermions and their collective modes through optical measurement. The methods we use here pave a new route for discriminating bulk excitations from the surface modes and for exploring collective properties of chiral fermions.

Supplemental Material for:

Chiral Spin Mode on the Surface of a Topological Insulator

.1 I. Material & Methods

.1.1 Material preparation

The single crystals measured in this spectroscopic study were grown by modified Bridgman method. Mixtures of high-purity bismuth (99.999%) and selenium (99.999%) with the mole ratio $\rm Bi:Se=2:3$ were heated up to 870 ∘C in sealed vacuum quartz tubes for 10 hours, and then slowly cooled to 200 ∘C with rate 3 ∘C/h, followed by furnace cooling to room temperature.

The Bi2Se3 crystals used in this study were characterized by STM and phononic Raman scattering studies in Refs. Dai et al. (2016); Kung et al. (2017). The Fermi energy ($E_{F}$) is determined by scanning tunneling spectroscopy to be about 150 meV above the Dirac point of SS1 [Fig. 1(b) in Main Text] Dai et al. (2016). Characterization confirmed that the samples have low concentration of impurities, Se vacancies, or other crystalline defects. All the bulk phonon modes in this crystal are sharp with no signatures of impurity modes, and all the expected surface phonon modes are clearly observed Kung et al. (2017). All spectroscopic features we present in this study were reproducible for a series of cleaves, immediately observed for each cool down, and did not show any signatures of time-dependent contamination.

.1.2 Raman scattering

In this study, we used the 520.8, 647.1, 676.4 and 752.5 nm lines of a Kr+ ion laser to promote secondary emission from the Bi2Se3 crystals. The spectra were acquired in a quasi-backscattering geometry from the ab surfaces, cleaved and transferred into the cryostat in nitrogen environment immediately prior to each cool down. About 10 mW of the laser power was focused into $50\times 50\,\mu m^{2}$ laser spot. Scattered photons were collected and analyzed by a custom triple-grating spectrometer with a liquid nitrogen cooled charge-coupled device (CCD) detector. The secondary emission intensity, ${\rm I}_{\mu\nu}(\omega,T)$, was normalized to the laser power and corrected for the spectral response of the spectrometer and CCD, where $\mu$ ($\nu$) is the polarization of incident (collected) photon. In a Raman process, $\rm{I}_{\mu\nu}(\omega,T)$ is related to the Raman response function, $\mathcal{R}_{\mu\nu}(\omega,T)={\rm I}_{\mu\nu}(\omega,T)/[1+n(\omega,T)]$, where $n(\omega,T)$ is the Bose-Einstein coefficient, $\omega$ is Raman shift and $T$ is temperature.

The Raman response functions for given polarizations of incident and scattered photon are defined by the rank-2 Raman tensors, which can be symmetrized according to the irreducible representations of the crystal’s point group. The scattering geometries used in this experiment are denoted as $\mu\nu=$RR, RL, XX and XY, probing $A_{1}+A_{2}$, $2E_{2}$, $A_{1}+E_{2}$ and $A_{2}+E_{2}$ symmetries of the $C_{6v}$ group, respectively Cardona (1982); Ovander (1960). $\text{R}=\text{X}+i\text{Y}$ and $\text{L}=\text{X}-i\text{Y}$ denotes the right- and left-circular polarizations, respectively, where X (Y) denotes linear polarization parallel (perpendicular) to the plane of incidence.

After subtracting luminescence contributions (Sec. II), the measured spectra $\rm{I}_{\mu\nu}(\omega,T)$, are then decomposed into $E_{2}$, $A_{1}$ and $A_{2}$ symmetry channels as follows:

[TABLE]

.1.3 Computational details

In relation to Eq. (4) in the Main Text where we calculate the spin susceptibility, a momentum cutoff of $\Lambda_{k}=0.3$ Å*-1* was chosen. Any ambiguity that may arise due to the choice of cutoff can be subsumed into the interaction parameter $U$, thus making the physics of the appearance of the chiral spin collective modes universal. The threshold for the spin-flip continuum ($\omega_{-}$) is obtained by finding the smallest $\omega$ such that $\Pi^{\prime\prime}(\omega,T=0)\neq 0$. We find that $\omega_{-}\approx 260$ meV in the sample measured.

.2 II. Photoluminescence contribution removal

Figure S5 shows the intensity of secondary emission measured for RR and RL polarizations at 24 K for 647, 676 and 752 nm excitation wavelengths, plotted as function of emission photon energy. The exciton emission centers at 1.54 eV for 647 and 676 nm excitations, and has about the same intensity for both RR and RL scattering geometries. Another weaker emission peak is observed at 1.64 eV for both excitations. These peaks are absent for 752 nm excitation spectra, suggesting that the emission has a threshold of about 1.8 eV.

To remove photoluminescence background from the measured spectra, we fit the 1.54 and 1.64 eV exciton peaks with a Lorentzian function, as shown by the hatched peaks in Fig. 2 of Main Text. We also subtract a small constant background from all spectra to account for other photoluminescence contribution.

.3 III. Transitions between surface states and bulk bands

In relation to Fig. 2(a) in the Main Text, we present in this section an explanation of the spectroscopic features observed for the non-resonant 521 nm (2.38 eV) excitation.

Figure S6 shows the band structure of Bi2Se3 reconstructed from ARPES measurements Nomura et al. (2014). The dispersion of the surface states are Fu (2009):

[TABLE]

where $\pm$ denote the upper and lower Dirac cones and $\theta$ is the azimuth angle of momentum ${\bf k}$ with respect to the $x$ axis ($\Gamma-K$). Fitting data in Ref. Nomura et al. (2014) to Eq. .3 gives $m^{\ast}\approx 0.066\,\text{eV}^{-1}\text{\AA}^{-2}$, $v_{F}\approx 2.4$ eVÅ , and $v_{3}\approx 25$ eVÅ3. One can readily see that the energy of a direct transition from the lower to upper Dirac cone is $2\sqrt{v_{F}^{2}k^{2}+(\frac{2v_{3}}{v_{F}})^{2}k^{6}\cos^{2}(3\theta)}$. In samples measured, $\Delta$ is determined by tunneling spectroscopy Dai et al. (2016) to be about $-150$ meV, therefore the Fermi momentum $k_{F}\approx 0.054$ Å*-1* along $k_{x}$, thus resulting in a threshold energy $\omega_{-}\approx 260$ meV.

The direct transition energy between SS1 and the bulk conduction band is given by $\epsilon({\bf k})=E_{CB}({\bf k})-E_{SS}({\bf k})$, where $E_{SS}({\bf k})$ is given by Eq. (.3), and the bulk conduction band dispersion follows a quasi-2D parabolic model Lahoud et al. (2013):

[TABLE]

where $E_{0}\approx 130$ meV is determined by $E_{F}$ and the relative position between SS1 and bulk conduction band minimum Nomura et al. (2014); Dai et al. (2016), $m_{||}^{\ast}\approx 0.03\,\text{eV}^{-1}\text{\AA}^{-2}$ is the in-plane effective mass, determined from fitting the ARPES data in Ref. Nomura et al. (2014) to quadratic dispersion. In the measured sample where $k_{F}\approx 0.054$ Å*-1* along $k_{x}$, the threshold energy $E_{\min}\approx 180$ meV, similar to what was observed in Fig. 2(a).

.4 IV. Temperature dependence of the surface chiral spin mode

Figure S7 shows the temperature dependence of the Raman intensity [(a)], full width at half maximum [FWHM, (b)], and peak center energy [(c)] of the chiral spin mode. The parameters were obtained by fitting the data in Fig. 3(c) of the Main Text to a Lorentzian lineshape. FWHM is approximately independent of temperature for $T\leq 150$ K. This indicates that the main damping mechanism of spin waves for these temperatures is due to disorder via the D’yanokov-Perel’ mechanism D’yakonov and Perel’ (1971). This is in line with the theoretical predictions for damping of chiral spin waves Shekhter et al. (2005); Maiti and Maslov (2015). At higher temperatures, inelastic scattering mechanisms, e.g., electron-electron Bir et al. (1975); Maiti and Maslov (2015) or electron-phonon Huber and Van Vleck (1966) interactions, may also contribute to damping. However, we found that a model, which incorporates the finite-temperature effects only via thermal smearing of the Fermi functions and neglects inelastic damping mechanisms, describes the experiment rather well. The results of such model with a $T$-independent damping rate of $8$ meV (taken as 1/2 of FWHM at $T\to 0$) are shown in Fig. 3 (d) of the Main Text. On the other hand, the fact that the measured intensity decreases with increasing temperature faster than the calculated one is an indication of unaccounted spin decay channels at elevated temperatures, e.g., through interaction with surface phonons Kung et al. (2017).