Ground state and infrared response of triple concentric quantum ring structures

Within local-spin density-functional theory, we study the ground state and infrared response of two-dimensional, triple concentric quantum ring nanostructures. Changes in their physical properties are presented as a function of the number of electrons or the intensity of a perpendicularly applied magnetic field. We discuss the addition spectrum of few-electron triple quantum rings at zero magnetic field, as well as the physical appearance of the ground state and dipole response of selected systems containing up to 50 electrons. We also investigate the ground state, persistent currents, and charge- and spin-density responses of a system made of 30 electrons.

Figure 1

(Color online) The solid line displays the TCQR confining potential (meV) as a function of the radial distance $r$ (nm). The electron density $\rho \left(r\right)$ (dashed line) and the spin magnetization $m\left(r\right)$ (dotted line) in dot units are also shown for $N=20$ at $B=0$ and $T=0.1\text{K}$.

Figure 2

Top left panel: single-particle energies as a function of minus the sp orbital angular momentum $\ell$ for the $N=20$ TCQR at $B=0$ and $T=0.1\text{K}$. Up (down) triangles represent spin-up (down) states. The horizontal thin solid line is the chemical potential. From the top right panel, clockwise, the other three panels represent the same sp energies split into the three constituent rings $R_1$, $R_2$, and $R_3$.

Figure 3

Addition energy ${\Delta }_2$ (meV) at $T=0.1\text{K}$ as a function of $N$ for $N=1–12$. The lines have been drawn as a guide to the eyes.

Figure 4

(Color online) Charge dipole strength (solid lines, arbitrary units), spin dipole strength (dashed lines, arbitrary units), and free-particle dipole strength (dotted lines, arbitrary units) for TCQR with $N=10$, 20, 30, 40, and 50 electrons as a function of the excitation energy (meV) at $B=0$ and $T=0.5\text{K}$. The scales are fixed in such a way that, for a given $N$, the most intense peaks of all channels roughly have the same height.

Figure 5

(Color online) Bottom panel: charge dipole strength (solid lines), spin dipole strength (dashed lines), and free-particle dipole strength (dotted lines) for a TCQR with $N=50$ at $B=0$, $T=0.5\text{K}$. The other panels display the strengths arising from the $R_i$ rings, obtained as indicated in the text. The scales are the same for every channel in all panels, and they are fixed in the bottom panel, so that the most intense peaks of each channel display roughly the same height. The very different strengths of the channels in the different panels can be explained from the $f$-sum rules of the corresponding operators and (sub)system since the $f$-sum rules are proportional to the number of electrons hosted.

Figure 6

Single-particle energies as a function of minus the sp orbital angular momentum $\ell$ for the $N=30$ TCQR at $T=0.5\text{K}$ and selected $B$ values: 0, 0.5, 1, and 2.5 T. Up (down) triangles represent spin-up (down) states. Horizontal, thin solid lines are displayed at the energies of the respective chemical potentials.

Figure 7

(Color online) Low-energy charge (solid lines, arbitrary units) and spin dipole strength (dashed lines, arbitrary units) for $N=30$ at $T=0.5\text{K}$ as a function of the excitation energy (meV). The scales are fixed in such a way that, for a given $B$, the most intense peaks in all channels roughly have the same height. The $(+)$ and $(-)$ symbols near the main charge-density peaks indicate their circular polarization. Some structures are superpositions of peaks with different polarizations; they are denoted by the compound symbol $\left(-/+\right)$, since in all cases the $(-)$ polarization is the dominant one, eventually.

Figure 8

Intensity of the persistent current of the TCQR as a function of the applied magnetic field, computed using the LSDA. The intensity starts from zero in the absence of magnetic field and displays main oscillations with amplitudes of $\Delta B\simeq 0.85\text{T}$.

Figure 9

Intensity of the persistent currents of the system (top panel) and each of the constituting rings (rest of panels) for the independent particle model of one-dimensional rings. The oscillations of the total current are dominated by that of the inner ring.