New Selection Rules for Resonant Raman Scattering on Quantum Wires

– The bosonisation technique is used to calculate the resonant Raman spectrum of a quantum wire with two electronic sub-bands occupied. Close to resonance, the cross section at frequencies in the region of the inter sub-band transitions shows distinct peaks in parallel polarisation of the incident and scattered light that are signature of collective higher order spin density excitations. This is in striking contrast to the conventional selection rule for non-resonant Raman scattering according to which spin modes can appear only in perpendicular polarisation. We predict a new selection rule for the excitations observed near resonance, namely that, apart from charge density excitations, only spin modes with positive group velocities can appear as peaks in the spectra in parallel configuration close to resonance. The results are consistent with all of the presently available experimental data. Raman spectroscopy is one of the most powerful experimental techniques for investigating the dispersion relation of elementary excitations in condensed matter [1, 2]. Recently, the spectra of the strongly correlated electrons in quantum wires [3, 4, 5] have been the subject of considerable experimental effort. It has been suggested that intraband charge and spin density excitations (CDE and SDE) which are signature of Luttinger liquid behaviour [6] can be identified in parallel and perpendicular polarisations of incoming and scattered light, respectively. In addition, the peaks in the polarised resonant Raman spectra, when the energy of the incident photon approaches the energy gap, that were associated so far with “single particle excitations” (“SPE”) have been identified as collective, higher order spin density excitations in the region of the intraband modes [7]. In order to treat the interband spectra, the bosonisation approach has been generalised to include interband coupling terms [8]. It has been shown that the essential experimental features observed far from resonance in the () present address: I. Institut für Theoretische Physik, Universität Hamburg Jungiusstraße 9, D-20355 Hamburg, Germany. Typeset using EURO-TEX 2 EUROPHYSICS LETTERS polarised and depolarised configurations of the light beams, in the frequency region of the lowest interband CDE and SDE, can be recovered. In the present paper, we show that in the region of the interband transitions, higher-order SDE give rise to pronounced peaks in the resonant Raman spectra in parallel polarisation, in analogy with those found for the intraband modes (so-called “SPE”). We predict that these peaks are subject to a new selection rule: only interband SDE with a positive group velocity can appear in the resonant Raman spectra in the parallel configuration as sharp peaks. All of the presently available experimental data of quantum wires are completely consistent with our predictions. Nevertheless, more detailed experimental investigations, especially on quantum wires with two sub-bands, are necessary in order to confirm our results quantitatively. This would provide further strong evidence for the importance of the Coulomb interaction and the collective and bosonic nature of the electronic excitations in quantum wires which are signature of Luttinger liquid behavior [9]. We consider a quantum wire of length L (periodic boundary conditions) with two occupied conduction sub-bands with dispersion relations Ei(k) = εi + h̄ k2/2m (i = 1, 2) where ε1 = 0, ε2 = E0, E0 the intersub-band separation, and m the effective mass of an electron in the conduction band. The position of the Fermi level EF defines the Fermi velocities vF1,2 corresponding to the two sub-bands. We assume a three dimensional (3D) Coulomb interaction. When projected with the confining wave functions onto the sub-bands, this gives the effective 1D interaction between the electrons [7]. The Hamiltonian of the free electrons in terms of Fermion operators c†is(k), cis(k) (spin s and wave number k) is