Isospin phases of vertically coupled double quantum rings under the influence of perpendicular magnetic fields

Vertically coupled double quantum rings submitted to a perpendicular magnetic field $B$ are addressed within the local spin-density functional theory. We describe the structure of quantum ring molecules containing up to 40 electrons considering different inter-ring distances and intensities of the applied magnetic field. When the rings are quantum mechanically strongly coupled, only bonding states are occupied and the addition spectrum of the artificial molecules resembles that of a single quantum ring, with some small differences appearing as an effect of the magnetic field. Despite the latter has the tendency to flatten the spectra, in the strong coupling limit some clear peaks are still found even when $B\neq 0$ that can be interpretated from the single-particle energy levels analogously as at zero applied field, namely in terms of closed-shell and Hund's-rule configurations. Increasing the inter-ring distance, the occupation of the first antibonding orbitals washes out such structures and the addition spectra become flatter and irregular. In the weak coupling regime, numerous isospin oscillations are found as a function of $B$.


I. INTRODUCTION
Systems made of correlated electrons confined in semiconductor nanoscopic dot and ring structures, so-called quantum dots (QDs) and rings (QRs) respectively, have been the subject of intense theoretical and experimental research, see e.g. Refs. 1,2 and references therein.
From the latter point of view, for quantum dots it has been proved 3 the possibility to tune over a wide range the number of electrons contained in the system, as well as to control both the size and the shape of the dots by means of external gate voltages, goal that has not been achieved yet for ring geometries due to the higher complexity of their fabrication process, 4,5,6 which involves several experimental techniques such as atomic force microscopy, 7 strain-induced self-organization 4 or droplet molecular beam epitaxy. 8 The interest of QRs arises from their peculiar behavior in the presence of a perpendicularly applied magnetic field (B), which is very distinct from that observed in QDs and shows up as an oscillatory behavior of their energy levels as a function of B. This property, together with the fact that in narrow enough QRs the electrons experiment a nearly onedimensional Coulomb repulsion, leads to the integer and fractional Aharonov-Bohm effects, usually associated with the appearance of the so-called persistent currents in the ring. 9 These quantum-interference phenomena have been experimentally reported 10 and have motivated a series of theoretical works whose number is steadily increasing, see e.g., Refs. 11,12,13,14, 15,16 and references therein.
One of the most appealing possibilities offered by electron systems confined in semiconductor heterostructures is their ability to form coupled entities, usually referred to as "artificial molecules", in which the role of the constituent "atoms" is played by single quantum dots or rings and that have analogies with natural molecules such as the hybridization of the electronic states forming molecular-like orbitals. In addition, these artificially coupled systems present important advantages such as a tunable "interatomic" coupling by means, e.g., of the modification of the relative position/size of the constituents. This fact has, besides its intrinsic interest, potential relevance to quantum information processing schemes since basic quantum gate operations require controllable coupling between qubits.
In this sense, artificial molecules based on two coupled QDs, called quantum dot molecules (QDMs) have been proposed as scalable implementations for quantum computation purposes and have received great attention from the scientific community in the last years -see e.g. Refs. 17,18,19,20,21,22,23,24,25,26 and references therein.
Also, molecular-beam epitaxy techniques have recently allowed the synthesis of quantum ring molecules (QRMs) in the form of concentric double QRs 27,28 and vertically stacked layers of self-assembled QRs, 29,30 the optical and structural properties of the latters having also been characterized by photoluminescence spectroscopy and by atomic force microscopy, respectively. This has sparked theoretical studies on the structure and optical response of both vertically and concentrically coupled QRs of different complexity and scope, revealing properties different from those of their dot counterparts due to the non-simply connected ring topology. For instance, studies on the single-electron spectrum of vertical QRMs 31,32 have shown that the electronic structure of these systems is more sensitive to the inter-ring distance than that of coupled QDs. As a consequence, in ring molecules quantum tunneling effects are enhanced since less tunneling energy is required to enter the molecular-type phase. Also, the consideration of "heteronuclear" artificial molecules constituted by slightly different QRs offers the interesting possibility to control the effective coupling of directindirect excitons 33 by means of the application of a magnetic field and taking advantage of the fact that charge tunneling between states with distinct angular momentum is strongly suppressed by orbital selection rules. To this end, some authors have considered the case of QRMs made of strictly one-dimensional, zero-thickness QRs and have used diagonalization techniques to address the few-electron problem. 31,33,34,35 The simultaneous effect of both electric and magnetic fields applied to a single-electron QRM has also been studied 36 -see also Ref. 31-and the optical response of QRMs where the thickness of the constituent QRs is taken into account has been obtained. 37 In addition, the spatial correlation between electron pairs in vertically stacked QRs only electrostatically coupled has been shown to undergo oscillations as a function of the magnetic flux, with strongly correlated situations between ground states with odd angular momentum turning out to occur even at large inter-ring distances. 34 More recently, the structure of a QRM made of two vertically stacked quantum rings has been addressed at zero magnetic field for a few tens of electrons within the local spin-density functional theory (LSDFT) neglecting 38 and incorporating 39 the vertical thickness of the constituent QRs.
In this work we address the ground state (gs) of two thick, vertically coupled identical quantum rings forming "homonuclear" QRMs populated with up to 40 electrons and pierced by a perpendicularly applied magnetic field. We extend in this way our previous study, 39 addressing the appearance and physical interplay between the spin and isospin 23 degrees of freedom as a function of the variation of both the intensity of the magnetic field and the inter-ring separation. Modelling systems charged with such large number of electrons requires the employment of methodologies that minimize the computational cost. Here we have made use of the LSDFT, 13,15 whose accuracy for the considered values of the magnetic field has been assessed 24 by comparing the obtained results for a single QD with those given by the current-spin-density functional theory (CSDFT), 40 which in principle is better-suited for high magnetic fields, and also with exact results for artificial molecules. 41 This paper is organized as follows. In Sec. II we briefly introduce the LSDFT and the model used to represent the vertical QRMs. In Sec. III we discuss the obtained results for some selected configurations, and a summary is given in Sec. IV.

II. DENSITY FUNCTIONAL CALCULATION FOR MANY-ELECTRON VERTI-CAL QUANTUM RING HOMONUCLEAR MOLECULES
The axial symmetry of the system allows one to work in cylindrical coordinates. The confining potential V cf (r, z) has been taken parabolic in the xy-plane with a repulsive core around the origin, plus a symmetric double quantum well in the z-direction, each one with width w, depth V 0 , and separated by a distance d. To improve on the convergence of the calculations, the double-well profile has been slightly rounded off, as illustrated in Fig. 2 of Ref. 24. The potential thus reads V cf (r, z) = V r (r) + V z (z), where with σ = 2 × 10 −3 nm, and Θ(x) = 1 if x > 0 and zero otherwise. The convenience of using a hard-wall confining potential to describe the effect of the inner core in QRs is endorsed by several works in the literature. 42 We have taken R 0 = 10 nm, V 0 = 350 meV,hω 0 = 6 meV and w = 5 nm. These parameters determine the confinement for the electrons together with the distance between the constituent quantum wells that is varied to study QRMs in different inter-ring coupling regimes.
For small electron numbers (N), it is justified to take ω 0 to be N-independent. However, in a more realistic scheme its value should be tuned according to the number of electrons contained in the system, relaxing the confinement as the latter is increased. In the case of quantum dots it has oftenly been used a N −1/4 -dependence that arises from the r-expansion near the origin of the Coulomb potential created by a two-dimensional uniform positive charge distribution -jellium model-and that it is generalized to the case of quantum dot molecules as ω 0 = κN −1/4 B , N B being the number of electrons filling bonding orbitals -see below. The rationale for this generalization is given in Ref. 25. It is clear that the mentioned N-dependence would be harder to justify for QRs, and in fact no alternative law is known for a single QR that could be generalized to the case of QRMs. For this reason, in this work we have taken ω 0 to be N-independent, which is to some extend less realistic for the largest values of N we have considered.
Considering the N-electron system placed in a magnetic field parallel to the z-axis, within the LSDFT in the effective mass, dielectric constant approximation, the Kohn-Sham where the single-particle (sp) wave functions have been taken to be of the form φ nlσ (r, z, θ, σ) = u nlσ (r, z)e −ılθ χ σ with n = 0, 1, 2, . . ., l = 0, ±1, ±2, . . ., −l being the projection of the single-particle orbital angular momentum on the symmetry axis, and σ=↑(↓) of effective atomic unitsh = e 2 /ǫ = m =1, where ǫ is the dielectric constant and m the electron effective mass. In units of the bare electron mass m e one has m = m * m e , the length unit being the effective Bohr radius a * 0 = a 0 ǫ/m * and the energy unit the effective Hartree H * = Hm * /ǫ 2 . In the numerical applications we have considered GaAs quantum rings, for which we have taken ǫ = 12.4, and m * = 0.067; this yields a * 0 ∼ 97.9Å and H * ∼ 11.9 meV, the effective gyromagnetic constant being g * = −0.44.
To label the gs configurations ("phases") we use an adapted version of the ordinary spectroscopy notation, 41 namely 2S+1 L ± g,u , where S and L are the total |S z | and |L z |, respectively. The superscript +(−) corresponds to symmetric (antisymmetric) states under reflection with respect to the z = 0 plane bisecting the QRMs, and the subscript g(u) refers to positive(negative) parity states. All these are good quantum numbers even in the presence of an axial magnetic field. By analogy with natural molecules, symmetric and antisymmetric states are referred to as bonding (B) and antibonding (AB) orbitals, respectively. We have defined the "isospin" quantum number I z -bond order in Molecular Physics-as 22,24,41 being the number of occupied bonding(antibonding) sp states.

III. RESULTS
Due to the large number of variables needed to characterize a given QRM configuration (electron number, magnetic field and inter-ring distance), we limit ourselves to present results in a limited range of values for such variables, aiming at discussing calculations that might illustrate the appearance of some properties of the systems under study. For the sake of comparison, we have also addressed one single QR symmetrically located with respect to the z = 0 plane with the same thickness (5 nm) and radial confinement as the coupled rings. Fig. 1 shows the Kohn-Sham sp levels for one single ring hosting N=40 electrons as a function of l for different values of the applied magnetic field. As it is well known, these levels are ±l-degenerate at B = 0. In this particular case, the gs has S z = 1, and it is made up of symmetric (with respect to z = 0) sp states with up to n = 3. In the non-interacting single-electron model, in which the Coulomb energy is not considered and consequently the sp wave functions factorize into a r-dependent and a z-dependent part with associated quantum numbers n r and n z , i.e., u nl (r, z) → U nr (r)Z nz (z), one would say that the gs is made up of sp states with n z = 0 and radial quantum numbers up to n r = 3.
When B = 0, the ±l-degeneracy is lifted and, on the other hand, the l < 0 sp levels become progressively depopulated in favor of those with l > 0 as the magnetic field increases until eventually -at about ∼ 4 T-only l > 0 orbitals are filled. At this point, only a few states with n = 2 are occupied, and the ring has S z = 0. From this value of B on, the simultaneous filling of increasingly higher-l states and those close to l = 0 gives rise to configurations containing only states with n = 1 and with large values of the total spin (e.g., S z = 9 for B = 8 T). Eventually, the system becomes fully spin-polarized at B ∼ 13.5 T. It is worth noticing the conspicuous bending of the "Landau bands" (sets of bonding or antibonding states characterized by the same n and spin, and different value of l), instead of displaying a fairly flat region, as it happens when the in-plane confinement is produced by a jellium-like potential, 13 but not with our present choice of a N-independent parabola. It is also worth to stress that, due to the much stronger confinement in the vertical direction as compared to that in the radial one, only symmetric states are occupied.   This also explains why in most of the cases, and especially for the highest magnetic fields, the total angular momentum of the QRMs in the weakest coupling regime is reduced: the filled antibonding orbitals have lower l's than the replaced bonding states.
We have determined the magnetic field that gives rise to ring molecules with fully spin- The spectrum corresponding to d = 6 nm is shown in the top panel of the same figure.
One can see that, although some of the marked peaks are preserved, in particular those at N = 2 and 8, the ones at N = 4, 6 no longer exist -notice that for 6 electrons the spectrum presents now a minimum and also that a new peak is found at N = 5. This intricate structure can be understood from the corresponding single-particle energy levels. Indeed, it appears that the QRMs with N ≤ 4 are made up of only bonding states, the first antibonding state being filled when N = 5. From N ≥ 7 on, the QRMs have always occupied both B and AB orbitals but, however, the intermediate 6-electron configuration has again only symmetric states. This alternate behavior evidences that 6 nm is not a separation large enough for the QRMs to be in the weak coupling limit, but rather corresponds to an intermediate regime.
Notice also that, from the results of Ref. 39, in the weak coupling limit one would expect to find clearly marked peaks at the same N values as for the single ring multiplied by two -i.e. at N = 4, 12 and 20, indicating that the rings are so apart that behave as isolated entities. We have checked that for our QRMs to present such spectrum, we should consider inter-ring distances of about 10 nm. The different spin values for d = 6 nm as compared to those in the strong coupling regime can also be explained from the sp levels. For example, the 2S z = 3 assignation of the QRM with N = 5 is due to the above-mentioned filling of an antibonding -spin-up with l = 0-orbital replacing the spin-down |l| = 1 state occupied for d = 2 and 4 nm. Analogously, the configuration with S z = 1 (instead of S z = 0) for N = 10 can also be explained from the sp levels: in the strong coupling limit, the QRM is formed by the spin-degenerated sp levels with l = 0, |1| and |2|, but this closed-shell configuration is prevented by the filling of the antisymmetric orbitals at d = 6 nm. Finally, the reverse situation occurs at N = 8, where the closing of the antibonding l = 0 and |1| shells contrasts with the Hund's-rule configurations found for the strongly coupled molecules.

IV. SUMMARY
Within the local spin-density functional theory, we have addressed the ground state of quantum ring molecules containing up to 40 electrons, with different inter-ring distances, and submitted to perpendicular magnetic fields. In the strong coupling regime the energy levels and the addition energies of the QRMs are similar to those of a single QR, although some differences are found due to the effect of the magnetic field, which has a tendency to wash out the clearly marked peaks characteristic of the B = 0 case as well as to yield flatter addition spectra. However, even at B = 0, some peaks are still present and they can be interpretated as at zero magnetic field.
When the ring separation is increased until the first antibonding orbitals are occupied, the addition spectra become irregular and the ring molecules are fully spin-polarized at from Generalitat de Catalunya.        Fig. 9 for B = 6 T.