Theory of non-resonant Raman scattering from electrons in nodal and flat bands

TL;DR

The paper develops a comprehensive, beyond–effective-mass theory of electronic Raman scattering for nodal semimetals with Weyl and quadratic-band-touching spectra. It shows that Weyl nodes yield a universal $R(\Omega) \propto \Omega^{2}$ above a node-related threshold and a polarization dependence captured by analytic expressions, while tilt splits the onset into two thresholds $\Omega_{0}$ and $\Omega_{1}$ and broadens with finite lifetimes. The QBT (Luttinger) case gives $R(\Omega) \propto \sqrt{|\Omega|}$ above threshold, with polarization tied to cubic symmetry; a Dirac-plus-flat-band scenario reveals an hourglass-feature around the flat-band energy $\epsilon$. Collectively, these results provide practical RDF templates to extract chemical potential, velocity scales, tilt, and lifetimes from Raman spectra and offer a diagnostic toolkit for topological nodal materials such as Nd$_2$Ir$_2$O$_7$ and related systems.

Abstract

Raman scattering is emerging as a surprising probe of electron topology in quantum materials. It has been used recently to detect and characterize a topological phase transition that accompanies the magnetic transition in Nd$_2$Ir$_2$O$_7$. Here we present a theory of Raman scattering from nodal electrons with Weyl and quadratic band touching spectra, which has to reach beyond the standard effective mass approximation. After reviewing and providing the details of our previous theory development, we discuss several new results. We show that the light-polarization dependence of Raman scattering is universal in the case of Weyl electrons and given by an analytic expression, while it contains symmetry-protected features in the case of quadratic band-touching nodes. We also analyze modifications of the Raman signal due to the ubiquitous tilting of the Weyl spectrum, and argue that universality is lost only in a finite frequency range that springs out of the threshold frequency for untilted nodes. Finally, we explore the frequency dependence of Raman scattering for the case of Dirac electrons coexisting with a flat band in the same region of the first Brillouin zone, which is inspired by the material V$_{1/3}$NbS$_2$.

Figures (7)

• Figure 1: (a) Electron transitions induced by Raman scattering. $E_\alpha$, $E_\beta$ are the final and initial electron states respectively, $E_\gamma$ is an intermediate state; $\omega_{\textrm{i}}$, $\omega_{\textrm{s}}$ are the incoming and scattered photons' frequencies respectively, and their difference $\Omega=\omega_{\textrm{i}}-\omega_{\textrm{s}}$ is known as Raman shift frequency. (b,c) Feynman diagrams for photon-electron scattering that impacts the Raman cross-section. Wavy lines represent photons and solid lines with arrows represent electrons. (b) A process that involves a virtual intermediate state in general non-resonant Raman scattering (present in both relativistic and non-relativistic theories). (c) A "diamagnetic" process is present only in non-relativistic theories and not relevant for Raman scattering on Weyl/Dirac electrons with infinite lifetime.
• Figure 2: (a) Illustration of the non-resonant Raman scattering from Weyl electrons with infinite lifetime and a spherically-symmetric spectrum. Incoming (i) and scattered (s) photons, represented by wavy lines, transfer energy $\Omega$ and negligible momentum to an electron-hole excitation. If the Fermi level is at energy $\mu$ relative to the node, than the minimum possible energy transfer with zero momentum transfer is the threshold frequency $\Omega=2|\mu|$ for the Raman scattering. The physical process is depicted in Fig.\ref{['RamanProcess']}, but the intermediate state is virtual and lives within the low-energy Weyl spectrum, hence cannot be handled by the effective mass approximation. (b) An illustration of Raman scattering from a "tilted" type-I Weyl node.
• Figure 3: Interband Raman scattering rate from quasiparticles with a tilted Weyl spectrum and infinite lifetime. The plots are parametrized by the tilt $u/v$, starting from zero (red curve) and growing in increments $0.1$ until $0.9$ (blue curves). The unique threshold frequency at $u/v=0$ splits into a lower threshold frequency $\Omega_0$ (solid circles) below which the Raman response vanishes, and an upper threshold frequency $\Omega_1$ (open circles) above which the Raman response falls back to the most universal form obtained in the absence of tilt. The polarization vectors $\hat{\bf e}_{\textrm{i}}(\theta_{\textrm{i}},\phi_{\textrm{i}})$ and $\hat{\bf e}_{\textrm{s}}(\theta_{\textrm{s}},\phi_{\textrm{s}})$ for this calculation were $\theta_{\textrm{i}}=0.14$, $\phi_{\textrm{i}}=2.87$, and $\theta_{\textrm{s}}=0.40$, $\phi_{\textrm{s}}=2.34$ respectively.
• Figure 4: Interband Raman scattering rate from spherically-symmetric Weyl quasiparticles parametrized by the quasiparticle lifetime $\tau$. The Raman shift frequency $\Omega$ and the scattering rate $R'$ are shown in arbitrary units. The red curve corresponds to infinite lifetime -- scattering occurs only above a threshold frequency $\Omega>|2\mu|$, which is determined by the chemical potential $\mu$ relative to the node energy, and proceeds as $R'\propto\Omega^2$. The blue curves illustrate the evolution of Raman scattering as the quasiparticle "decay rate" $\Gamma\propto\tau^{-1}$ increases from $0.05$ in steps $0.1$ (expressed with the same energy units as $\Omega$). The scattering below the "threshold" quickly fills up with a linear-looking dependence on $\Omega$.
• Figure 5: The light polarization dependence $I_{\textrm{L}}(\hat{\bf e}_{\textrm{s}},\hat{\bf e}_{\textrm{i}})$ of the Raman scattering from nodal electrons with quadratic band touching. The incident light is assumed to have linear polarization in the direction (a,b) $\hat{\bf e}_{\textrm{i}}=(0,0,1)$, (c,d) $\hat{\bf e}_{\textrm{i}}=(1,1,0)/\sqrt{2}$ or (e,f) $\hat{\bf e}_{\textrm{i}}=(1,1,1)/\sqrt{3}$. The left column shows spherical plots of the Raman intensity in arbitrary units as a function of the direction of the scattered light polarization vector $\hat{\bf e}_{\textrm{s}}=(x,y,z)/\sqrt{x^2+y^2+z^2}$ for the given fixed incident light polarization $\hat{\bf e}_{\textrm{i}}$. The obtained surfaces reflect the cubic lattice symmetry and the directional bias set by the fixed $\hat{\bf e}_{\textrm{i}}$. The right column shows spherical plots of the difference $|I_{\textrm{L}}-r|$ between the Raman intensity $I_{\textrm{L}}$ and its best spherical fit with radius $r$ for each given incident polarization (color coding is red for $I_{\textrm{L}}>r$ and blue for $I_{\textrm{L}}<r$). This depicts the symmetries and spherical components: (a,b) $I_{\textrm{L}}\propto 1s+0.24d_{z^2}$, (c,d) $I_{\textrm{L}}\propto 1s-0.12d_{z^2}-0.01d_{xy}$, (e,f) $I_{\textrm{L}}\propto 1s-0.01(d_{xy}+d_{yz}+d_{zx})$, for the model parameters $m'/M=0.2$ and $M_{\textrm{c}}/M=0.6$ in this calculation. Note that the composition of non-zero spherical harmonics is qualitatively the same as in the case of Weyl electrons (Eq.\ref{['PolarI1']}) for each considered $\hat{\bf e}_{\textrm{i}}$, but the relative signed amplitudes of different harmonics are different within symmetry restrictions and depend here on the model parameters.