Optical response of two-dimensional few-electron concentric double quantum rings: A local-spin-density-functional theory study

We have investigated the dipole charge- and spin-density response of few-electron two-dimensional concentric nanorings as a function of the intensity of a perpendicularly applied magnetic field. We show that the dipole response displays signatures associated with the localization of electron states in the inner and outer ring favored by the perpendicularly applied magnetic field. Electron localization produces a more fragmented spectrum due to the appearance of additional edge excitations in the inner and outer ring.

Progress in nanofabrication technology has allowed to produce self-assembled, strainfree, nanometer-sized quantum complexes consisting of two concentric, well-defined GaAs/AlGaAs rings 1,2 whose theoretical study has recently attracted some interest. 3,4,5,6 Most of these works are concerned with the properties of their ground state, although the optical properties of one-and two-electron concentric double quantum rings (CDQR) at zero magnetic field have been addressed using a single-band effective-mass envelope model discarding Coulomb correlation effects. 2 The singular geometry of CDQR has been found to introduce characteristic features in the addition spectrum compared to that of other coupled nanoscopic quantum structures. As a function of the inter-ring distance, the localization of the electrons in either ring follows from the interplay between confining, Coulomb and centrifugal energies. Each of them prevail in a different range of inter-ring distances, affecting in a different way the CDQR addition spectrum. 6 It is thus quite natural to investigate whether and how localization effects may show up in the dipole response, using a perpendicularly applied magnetic field instead of the inter-ring distance to control the electron localization in either ring.
The aim of this paper is to use local-spin-density-functional theory (LSDFT) as described in detail in Ref. 7, to investigate the dipole longitudinal response of CDQR. The method has been used in the past to address the response of single quantum rings (see e.g. Ref. 8 and references therein). We address here the few-electron case, and consider the CDQR's as strictly two-dimensional systems.
Within LSDFT, the ground state of the system is obtained by solving the Kohn-Sham (KS) equations. The problem is simplified by the imposed circular symmetry around the z axis, which allows one to write the single particle (sp) wave functions as ϕ nlσ (r, σ) = u nlσ (r)e −ılθ χ σ , being −l the projection of the sp orbital angular momentum on the z axis.
The confining potential has been taken in a form that slightly generalizes that of Ref. 4: with R 1 = 20 nm, R 2 = 40 nm, ω 1 = 30 meV, and ω 2 = 40 meV.  confining potentials, better suited to model the experimental devices, 2 have been considered. 6 On the CDQR system it may act a constant magnetic field B in the z direction, to which a cyclotron frequency ω c = eB/mc is associated. We have taken for the dielectric constant, electron effective mass, and effective gyromagnetic factor, the values appropriate for GaAs, namely, ǫ = 12.4, m * = 0.067, and g * = −0.44, and have solved the KS equations for up to N = 6 electrons, and for B values up to 4-5 T, depending on N. In the following, we discuss some illustrative results. 3 It is known that for an N-electron single quantum ring, sp states with small l values become progressively empty as B increases. This can be intuitively understood as follows.
If only nodeless radial states are occupied, the Fock-Darwin wave functions are proportional to x |l| e −x 2 /4 , where x = r/a, being a = h/(2mΩ) with Ω = ω 2 0 + ω 2 c /4. Of course, this is so for a harmonic confining potential of frequency ω 0 , and not for the ring confining potential, but some of that wave function structure remains even in this case. These wave functions are peaked at r max ∼ 2|l| a, and consequently, as B increases, the |l| values corresponding to occupied levels must increase so that r max sensibly lies within the range of r values spanned by the ring morphology. The same appears to happen for the CDQR we have studied, as illustrated in Fig Yet, they can be localized in either ring if the resulting configuration has a lower energy.
It is relevant for the analysis of the dipole response to notice that electron localization is best achieved when, due to the double well structure of the confining CDQR potential, KS orbitals corresponding to an intermediate l value are not occupied. In the present case, it happens for l = 3. Intuitively, the missing l is the one whose KS orbit has a 'radius' similar to that of the maximum of the inter-ring barrier.
A full delocalization situation produces a very regular sp energy pattern since, due to it, electrons 'feel' simultaneously the confining potential produced by both constituent rings.
This happens e.g. for B = 0 and other low-B values. The situation may change at higher magnetic fields. Indeed, Fig. 2 shows that around B = 3 T, the two lowest parabolic-like bands tend to cross between l = 2 and 3. Roughly speaking, each parabolic-like band arises The present study can be extended to the case of many-electron CDQR and to other multipole excitations, or to incorporate on-plane wave-vector effects for the analysis of Ra-8 man experiments. 8 A more realistic description of the CDQR confining potential 6 demands a three-dimensional approach. 11 Work along this line is in progress.
We would like to thank Josep Planelles for useful discussions. This work has been performed under grants FIS2005-01414 and FIS2005-02796 from DGI (Spain) and