Multipole modes and spin features in the Raman spectrum of nanoscopic quantum rings

We present a systematic study of ground state and spectroscopic properties of many-electron nanoscopic quantum rings. Addition energies at zero magnetic field (B) and electro-chemical potentials as a function of $B$ are given for a ring hosting up to 24 electrons. We find discontinuities in the excitation energies of multipole spin and charge density modes, and a coupling between the charge and spin density responses that allow to identify the formation of ferromagnetic ground states in narrow magnetic field regions. These effects can be observed in Raman experiments, and are related to the fractional Aharonov-Bohm oscillations of the energy and of the persistent current in the ring.


I. INTRODUCTION
Very recently, quantum rings in InAs-GaAs heterostructures have been fabricated in the nanometer scale [1][2][3] , and the capacitance and far-infrared (FIR) response have been measured for the one-and two-electron quantum rings. Previously, the FIR response 4 had been measured for mesoscopic rings in GaAs-Ga x Al 1−x As heterostructures, for which a description based on classical and hydrodynamical models works fairly well 5,6 . In nanoscopic rings quantum effects are important and for this reason theoretical studies at a more microscopic level have been undertaken [7][8][9][10][11][12][13][14][15][16][17][18][19] .
In some of these calculations a major emphasis has been put on describing the Aharonov-Bohm (AB) quantum effect which manifests in the presence of an external magnetic field (B) as the existence of a persistent current, and leads to periodic oscillations in the energy spectrum and the persistent current as a function of B. These AB oscillations have been observed in mesoscopic rings in a GaAlAs/GaAs heterostructure by measuring the conductance across the ring 20 .
The experimental results for one-and two-electron nanoscopic rings have been theoretically analyzed 21 using the current density (CDFT) and time-dependent local-spin densityfunctional (TDLSDFT) theories, and a good agreement between theory and experiment is found (see also Refs. 16 and 19). Motivated by this success, in this work we extend this approach to a systematic study of ground state (gs) and spectroscopic properties of nanoscopic rings containing up to N = 24 electrons. Our aim is to show the physical appearance of quantities that could be measured in the future if the problem of fabricating many-electron nanoscopic rings is eventually solved. Specifically, we have obtained electro-chemical potentials µ(N) = E(N) − E(N − 1), where E(N) is the total energy of the N-electron ring, addition energies ∆ 2 (N) = µ(N + 1) − µ(N) = E(N + 1) − 2E(N) + E(N − 1), and the spin and charge density responses at finite on-plane transferred wave-vector, which are relevant for the analysis of Raman spectra. All these quantities have been measured in N-electron quantum dots, see for example Refs. 22-27 and references therein. Particular emphasis is given to spectroscopic results. We show that the collective excitation energies exhibit discontinuities in their B dependence, which are a manifestation of changes in the spin configuration of the ground state, and that in a ring-wire would be related to the fractional Aharonov-Bohm effect.
The structure of the ring gs has been obtained within the CDFT as described in Refs. 28 and 29. To obtain the charge and spin density responses we have used TDLSDFT as described in Refs. 30 and 31. Essentially, the method implies that to obtain the ground state and excited modes of the system, besides the direct electron-electron interaction, exchange and correlation effects have been included in a local density approximation. We refer the reader to these references for a comprehensive exposure of the CDFT and TDLSDFT, of direct applicability here changing the shape of the confining potential from a dotlike potential to a ringlike potential.

II. GROUND STATE RESULTS
Following Ref. 7, we have modeled the ring confining potential by a parabola with R 0 = 20 nm and ω 0 = 12 meV. These values are close to the ones used 21 to describe the rings studied by Lorke et al 3 . The electron effective mass m * = 0.063 (we write m = m * m e with m e being the physical electron mass) and effective gyromagnetic factor g * = −0.43 have been taken from the experiments [32][33][34] , and the value of the dielectric constant has been taken to be ǫ = 12.4. The model is strictly two-dimensional, and as a consequence of circular symmetry, the single particle (sp) wave functions are eigenstates of the orbital angular momentum l z and can be written as u nlσ (r)e −ilθ with l = 0, ±1, ±2 . . . Notice that with this convention, the sp orbital angular momentum is −l.
Some times we have used effective atomic units, defined byh = e 2 /ǫ = m =1. In this system of units, the length unit is the effective Bohr radius a * 0 = a 0 ǫ/m * , and the energy unit is the effective Hartree H * = Hm * /ǫ 2 . This yields a * 0 ∼ 10.4 nm and H * ∼ 11.15 meV. The Bohr magneton is defined as µ B =he/2m e c. Fig. 1 shows the addition energies ∆ 2 (N) at zero magnetic field. Saw-tooth structures and large peaks at magic numbers N = 2, 6, 10, 16 and 24 that correspond to close-shell rings are clearly seen. Had we not taken into account electron-electron interactions (singleparticle picture), the magic numbers would have been N = 2, 6, 10, 16, 20 and 24. The effect of interaction is to weaken some shell closures, for example that at N = 20 in the These quasi-periodic oscillations are absent in quantum dots, 24 and are a characteristic inherent to quantum rings. We recall that for a one-electron ring-wire of radius R 0 the energies are given by where Φ/Φ 0 is the number of flux quanta penetrating the ring, Φ 0 = hc/e, and are periodic in the flux Φ = πR 2 0 B with periodicity Φ 0 . The persistent current I l associated with the l-state has the same periodicity. As B increases the occupied l-state changes and gives rise to discontinuities in the current I l (the AB effect), which have been experimentally observed in a mesoscopic loop in a GaAs heterojunction 20 .
We have plotted in Figs. 3 and 4 the total energy E and the persistent current determined where J p (r) = −ê θ l[u nlσ (r)] 2 /r, beingê θ the azimuthal vector unit, is the paramagnetic current that for many-electron rings is obtained adding the sp paramagnetic currents, 37 and the second term is the diamagnetic current, being n(r) = [u nlσ (r)] 2 and A(r) the electron density and external vector potential, respectively.
In a many-electron nanoscopic ring-wire, electron-electron interactions give rise to changes in the gs configuration that decreases the period of the AB oscillations 38 . This effect has been explained in Ref. 11 as a consequence of Hund's first rule which favors the occurence of ferromagnetic S z = 1 phases. A ring with a low degree of one-dimensionality tends to loose the periodic features of the ideal ring-wire, being somehow an intermediate case between a ring-wire and a quantum dot, and for these systems the effect is not so marked, 21 appearing as a small amplitude oscillation whose B-width coincides with that of the ferromagnetic phase (see Fig. 3). The crucial role played by the degree of onedimensionality in the appearance of the AB oscillations can also be appraised comparing the results for two and four electron rings obtained in Refs. 11,13. In the case of an S z = 0 ring as for N = 10, we have found a period halving. For an S z = 1 2 ring like that with N = 9, the oscillations due to ST and TS transitions are absent but there still is a period halving which in this case can be traced back to that arising for non-interacting spin-1 2 fermions. 11, 38 We have estimated that to have Φ = Φ 0 for a ring-wire of radius equal to the maximum electronic density radius in  31,41,42 in the charge (CDE) S nn (q, ω) and the spin density channel (SDE) S mm (q, ω): where ω = ω i − ω s is the energy difference of the incoming and scattered photon. Using the above expressions one assumes that only conduction band electron are involved, and only off-resonance Raman peaks excited by laser energies above the valence-conduction band gap can be described.
To obtain S nn (q, ω) and S mm (q, ω) within the TDLSDFT, we have calculated the response to operators whose spatial dependence in the on-plane wave vector q is a plane wave e i q r .
We refer the reader to Refs. 30 and 31 for a detailed description of the longitudinal response at q ≈ 0, and at finite q.
as many multipole terms as needed to exhaust the f -sum rule 31 : We infer from this figure that the modes have no appreciable wave-vector dispersion, a clear signature of the 'zero-dimensionality' of the ring. A similar conclusion was drawn for dots, 31 in agreement with experiment. 26 It can also be observed from this figure that a few number of multipoles (up to L = 2 for the largest q value) is enough to yield the plane wave response, and that in the q ≈ 0 long wave-length limit only the dipole L = 1 mode contributes.
The B dispersion of the dipole mode for the N = 10 ring in the long wave-length limit is shown in the top panel of Fig. 3. It is worth to see that edge modes have a smooth B dispersion except in the regions where the ring is in the ferromagnetic S z = 1 phase. The discontinuities in the magnetoplasmon and spin modes, or equivalently these appearing in the absorption spectrum, 11 are features that could be detected in optical-absorption experiments, making observable the ST and TS transitions and the ferromagnetic phases.
In the one-electron ring-wire case, the energies of the dipole modes are given by These excitation energies show AB oscillations with the same period as gs energy and persistent current oscillations. Furthermore, they are discontinuous at the B values corresponding to level crossings. The B dependence of the absorption spectrum of a quasi ring-wire with two electrons was obtained 11 by an exact diagonalization calculation, and it was shown to display fractionary AB oscillations of period Φ 0 /2. The ring-wire character of the system is crucial for it to present distinct AB oscillations.
Neither spin nor charge density modes with L = 1 can be detected in FIR spectroscopy.
In contradistinction, Raman spectroscopy in QD's has proved to be able to disentangle spin from density modes, and to identify non-dipole charge and spin density modes. 27

IV. SUMMARY
In this work we have studied physical aspects of charge and spin density responses of nanoscopic rings that might be detected by Raman spectroscopy. We have considered rings whose morphology is close to that of the systems recently fabricated, excluding for this reason the interesting case of nanoscopic ring-wires that can also be addressed using the same method. We have taken as case of study a ring with 10 electrons and have seen that monopole to quadrupole modes clearly appear in the spin, and especially in the charge density channel. Both channels are coupled if the gs is ferromagnetic, and the ST and TS transition points may be identified as discontinuities in the B dispersion of the multipole modes.
We have also discussed the addition spectrum of rings hosting up to 24 electrons, and have found that their shell structure is similar to that of quantum dots up to N = 6. For larger N values this is not so; the rings present shell closures at N = 2, 6, 10, 16, and 24.
Hund's first rule is satisfied up the fourth shell for the assumed ring geometry.
The non one-dimensional character of these rings hinders the existence of fractionary Other gs's with a spin value S z = 1 2 (grey circles) and S z = 1 (black circles) may be identified.