Chiral Spin Mode on the Surface of a Topological Insulator

Using polarization-resolved resonant Raman spectroscopy, we explore collective spin excitations of the chiral surface states in a three dimensional topological insulator, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. 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.

Figure 1

(a) The band structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ around the Brillouin zone center, reconstructed from ARPES results in Refs. [39, 40]. The low-energy Dirac cones of surface states (SS1) are described by Eq. (1) with parameters $m^{*}\approx 0.066{\mathrm{eV}}^{-1}{\AA }^{-2}$, $v_1\approx 2.4\mathrm{eV}\AA$, and $v_{\mathrm{w}}\approx 25\mathrm{eV}{\AA }^3$ [41]. The upper and lower Dirac cones of opposite chirality are shown in red and blue, whereas the bulk bands are shown in gray. A pair of unoccupied surface Dirac cones (SS2) resides about 1.8 eV above SS1 [39]. The arrows illustrate a resonant Raman process stimulated by an incoming photon with energy ${\mathrm{\Omega }}_L$, in which an electron is promoted from the lower to the upper Dirac cone of SS1, through resonant scattering via the intermediate states, SS2. The energy difference between the excited and ground state defines the Raman shift, $\omega$. (b) Enlarged view of SS1 around $E_F$ with arrows showing the spin textures, where ${\omega }_{-}$ is the threshold energy for direct transitions between occupied states in the lower Dirac cone and available states in the upper cone. (c) Schematic dispersions of the particle-hole continua in the spin (red) and charge (blue) channels, Dirac plasmon [27, 29, 30] (blue line), and chiral spin modes [23] (red lines). The ${\omega }_{s,\perp }$ mode is observed in this work.

Figure 2

The intensities of secondary emission are measured in circular and linear (with respect to crystallographic axes) scattering polarization geometries, as shown pictorially next to the figure legend in panel (d) and also defined in the Supplemental Material [41], at 24 K with (a) 521 (2.38), (b) 647 (1.92), (c) 676 (1.83), and (d) 752 nm (1.65 eV) excitations. The vertical solid gray line indicates the sharp 150 meV peak, most pronounced for the incident photon energy of 1.83 eV, which is the closest to the energy difference between SS1 and SS2 [Fig. 1]. The dotted black line in (a) is a guide to the eye showing the energy threshold for the surface-to-bulk excitations [41]. The hatched areas in (b) and (c) are Lorentzian fits to the excitonic photoluminescence peaks.

Figure 3

Temperature dependence of the symmetry-separated Raman response $\mathcal{R}(\omega ,T)$ in the (a) $E_2$, (b) $A_1$, and (c) $A_2$ symmetry channels, measured with the near-resonant 676 nm (1.83 eV) excitation. The photoluminescence background was subtracted from the raw data [41]. The gray solid line locates the position of the sharp peak which is present only in the $A_2$ channel. (d) Calculated $\mathcal{R}(\omega ,T)$ in the $A_2$ channel with the dimensionless interaction coupling constant $u\approx 0.6$ and impurity broadening of $\mathrm{\Gamma }=8\mathrm{meV}$ was extracted form the data and used in the calculation. The lines are shifted vertically for clarity. Inset: Enlargement of the calculated $\mathcal{R}(\omega ,T)$ without vertical shift, showing the spin-flip continuum with a threshold energy of 260 meV.

Figure 4

(a) Raman vertex for resonant transitions from the lower cone in SS1 to the upper (left) and lower (right) cones in SS2. (b) The scattering cross section dressed with vertex corrections to account for many-body effects.