Electronic Raman scattering from 2D metals with broken inversion symmetry

Abstract

Lack of inversion symmetry in metals breaks SU(2) symmetry which results in spin-splitting of the electronic states at the Fermi level due to various types of spin-orbit coupling (SOC) such as Dresselhaus, Rashba, or Ising (also called valley-Zeeman). This splitting is known to enable both incoherent spin-flip excitations and coherent chiral-spin modes. Another effect of breaking of SU(2) is the introduction of a direct spin-photon interaction. We use this concept to formulate a theory of inelastic scattering of photons from the charge carriers of such a system [electronic Raman scattering (eRS)]. As a result of broken SU(2), we show that the eRS probe, unlike conventional theory of Raman scattering, couples to spin excitations even without tuning the laser to an internal resonance. We show that the spin dependent excitations induced by photon scattering are sensitive to the polarization geometries as well as to the spin structure of the Hilbert space of the low-energy states. As a concrete realization, we examine doped/gated graphene on substrates with strong SOC with various compositions of Rashba and valley-Zeeman SOC and compare their spectra with those for a model 2D electron gas (2DEG). The spectra are shown to have a resonant feature in select polarization geometries near the SOC-splitting energy and, importantly, is shown to be different in the two systems. The signal in graphene systems is shown to be stronger than that in a 2DEG by orders of magnitude owing to the large Dirac velocity. We also outline how the lineshapes from the spectra can be used to infer various components of SOC in the system.

Figures (6)

• Figure 1: Electronic structure and possible excitations for systems in which (a) Rashba SOC and (b) valley--Zeeman SOC dominate. The insets show the chiral/spin texture (l--left, r--right, i--in, o--out) at the Fermi surface. In systems with dominant Rashba SOC, both chirality-preserving excitations at $2\mu$ and chirality-flipping excitations at $\lambda_{\rm R}$, $2\mu-\lambda_{\rm R}$, and $2\mu+\lambda_{\rm R}$ are allowed Raman excitations. These excitations correspond respectively to the green, red, orange, and blue arrows in panel (a), with shaded regions of matching colors indicating the associated continua. In contrast, when valley--Zeeman SOC dominates, only spin-preserving excitations appear in Raman at energies $\ge 2\mu \pm \lambda_{\rm Z}$. The arrows mark the threshold energies, while the shaded regions denote the continua of the same type of excitations.
• Figure 2: Differential scattering cross-section, $\Sigma\equiv d^2\sigma/d\mathcal{O}d\Omega$, normalized to $\Sigma_0\equiv r_0^2 S(m_ev_F/\hbar)^2/\pi\Omega_{\rm I}$ for graphene with proximity induced Rashba SOC(red) for (a) XX, (b) XY, (c) RR, and (d) RL scattering geometries. For comparison, we also show the cross-section for graphene without SOC (light green). The XY geometry has spin split features but is dominated by the spin-blind leading contribution. The inset shows a zoom-in around the region $\hbar\Omega\approx \lambda_{\rm R}$. The height of the peak is similar to that in the XX polarization. The RR geometry does not couple to spin-flip excitations. Here, $\mu=2\lambda_{\rm R}, \Omega_{\rm I}=20\lambda_{\rm R}$. We included damping ($1/\tau$) to broaden the $\delta$-peak by setting $\lambda_{\rm R}\tau/\hbar=33$.
• Figure 3: Normalized differential scattering cross-section ($\Sigma/\Sigma_0$, see caption to Fig. \ref{['fig:diff_cross_G_Rashba']}) for graphene with proximity induced VZ SOC (dark green line) for (a) XX, (b) XY, (c) RR, and (d) RL. For comparison, we also show the cross-section for graphene without SOC in light green. All features arise from spin-non-flip excitations. Although the splitting is seen here at $2\mu\pm\lambda_{\rm Z}$, it is not due to spin-flip excitations but due to the shift in the occupied/unoccupied energies of $i,o$ bands. Here, $\mu=2\lambda_{\rm Z}, \Omega_{\rm I}=20\lambda_{\rm Z}$.
• Figure 4: Normalized differential scattering cross-section for graphene with both Rashba and VZ SOCs for (a) XX, (b) XY, (c)RR, and (d) RL scattering geometries. The splitting in the spectrum is $\lambda_{\rm SOC}=\sqrt{\lambda_{\rm R}^2+\lambda_{\rm Z}^2}$. The parameter $\phi\equiv\arctan(\lambda_{\rm Z}/\lambda_{\rm R})$ tracks the contribution to the splitting from the two SOCs. It is evident that the weight of the resonance at $\lambda_{\rm SOC}$ is controlled by $\lambda_{\rm R}$, vanishing when $\lambda_{\rm R}\rightarrow 0$. The step jump at $2\mu$ is also controlled by dominance of the $\lambda_{\rm R}$, while the splitting around $2\mu$ is controlled by $\lambda_{\rm Z}$. Here, $\mu=2\lambda_{\rm SOC}, \Omega_{\rm I}=20\lambda_{\rm SOC}$. We included a broadening by setting $\lambda_{\rm SOC}\tau/\hbar=20$. This leads to a tail in the leading term that extends to $\Omega\sim 0$ causing a background $\sim2\times10^{-3}$.
• Figure 5: Comparison of differential cross-sections for graphene with Rashba SOC calculated in the $4\times4$ Hilbert space (dark) and in the projected space (light) with the $\mu$ in the conduction bands. The inset of panel (b) uses a logarithmic scale on the y‑axis. The model in the projected space can only capture the sharp spin-flip excitations of the low energy Hilbert space, but it does so with a very weak signal strength. This indicates that models used to predict Raman signals will get the right resonant frequencies, but a significantly weaker signal strength as it misses significant spectral weight from the intermediate transitions to the off-shell (non-resonant) states from other bands. Here, $\mu=2\lambda_{\rm R}$, $\Omega_{\rm I}=20\lambda_{\rm R}$, and $\lambda_{\rm R}\tau/\hbar\sim 33$.