Theory of resonant Raman scattering in one-dimensional electronic systems

A theory of resonant Raman scattering spectroscopy of one-dimensional electronic systems is developed on the assumptions that (i) the excitations of the one-dimensional electronic system are described by the Luttinger-liquid model, (ii) Raman processes involve virtual excitations from a filled valence band to an empty state of the one-dimensional electronic system, and (iii) excitonic interactions between the valence and conduction bands may be neglected. Closed form analytic expressions are obtained for the Raman scattering cross sections, and are evaluated analytically and numerically for scattering in the polarized channel, revealing a “double-peak” structure with the lower peak involving multispinon excitations with total spin $S=0$ and the higher peak being the conventional plasmon. A key feature of our results is a nontrivial power-law dependence, involving the Luttinger-liquid exponents, of the dependence of the Raman cross sections on the difference of the laser frequency from resonance. We find that near resonance the calculated ratio of intensity in the lower energy feature to the intensity in the higher energy feature saturates at a value of the order of unity (times a factor of the ratio of the velocities of the two modes). We explicate the differences between the “Luttinger-liquid” and “Fermi-liquid” calculations of resonant Raman spectroscopy (RRS) spectra and argue that excitonic effects, neglected in all treatments so far, are essential for explaining the intensity ratios observed in quantum wires. We also discuss other Luttinger-liquid features which may be observed in future RRS experiments.

Figure 1

(a) Schematic representation of RRS process in the direct gap two band model of electron doped GaAs nanostructures. ${\omega }_i$ and ${\omega }_f$ are the initial and final frequencies of the external photons. (b) and (c) are the three-leg and four-leg scattering vertices for electron-photon interaction.

Figure 2

Polarized RRS spectra calculated via the bosonic expansion method for various resonance conditions, $\tilde{\Omega }=\Omega -{\Omega }_{\mathit{rrs}}$. One- and two-boson contributions have been plotted separately in order to show their relative contributions (see text). Finite broadening factor $\gamma$ is introduced to express the $\delta$ function (Ref. 35). Note that the overall spectral weights decrease dramatically off-resonance, as indicated by the individual scale factors on the right-hand side of each plot.

Figure 3

Polarized RRS spectra calculated via bosonic expansion method for various resonance conditions in the LL model with long-ranged Coulomb interaction. One- and two-boson contributions have been plotted separately in order to show their relative contributions (see the text).

Figure 4

Logarithm of polarized-channel Raman scattering cross section (normalized to appropriate mode velocity), $I_{\mathit{RRS}}$, plotted against incident laser frequency (normalized to mode excitation frequency $\nu =v_{\rho ,\sigma }q$ and measured from the average of incident photon resonance energy and outgoing photon resonance energy with respect to the resonance energy ${\Omega }_{\mathit{rrs}}$), for two different values of the Luttinger exponent. $\alpha$. The plotted intensities are coefficients of $\delta$ functions describing coherent charge (CDE) and spin (SPE) final states and are computed via Luttinger-liquid methods as described in the text.

Figure 5

Comparison of ratio of $\delta$-coefficients for CDE and SPE absorption for Luttinger (solid and long-dashed lines) and RPA-Fermi liquid (dash-dot and short-dashed lines) model, normalized to appropriate mode frequencies and plotted against incident laser frequency (scale as in Fig. 4). Inset: same ratio, displayed in linear scale.

Figure 6

Feynman diagrams considered in evaluation of RPA to unpolarized RRS cross section. Diagram 1 gives the dominant contribution to the SPE absorption. Excitonic interactions (such as those shown as dash-dot lines in the lower left corner of this diagram) are not considered in this paper. Diagram 2 gives the dominant contribution to the CDE absorption.