Electron correlations in charge coupled vertically stacked quantum rings

We study spatial correlations for two electrostatically coupled electrons in a pair of vertically stacked quantum rings with configuration-interaction approach. We demonstrate that for distant quantum rings, the correlations undergo abrupt oscillations with the external magnetic field, which are strong for odd multiples of the flux quantum for which the electrons distinctly avoid being localized one above the other and are negligible for other values of the flux. The oscillations of the correlation strength vanish in the single-ring limit for which the electrons form a Wigner molecule. The wave function of the Hartree approximation is frustrated between reproducing the exact angular momentum eigenstate or the correlation between the electrons. We show that for distant rings the mean-field solution breaks the rotational symmetry of the system in a reentrant manner near odd multiples of half of the flux quantum to account for the correlation appearing in the exact solution.

Figure 1

(Color online) Exact energy levels of given angular momentum $L$ for $R=13.2\mathrm{nm}$ and $Z=3R$ are plotted in solid lines. The solid line at the bottom of the plot shows the expectation value of the electron-electron interaction energy calculated for the exact ground state. Dotted lines show the results of the Hartree approximation for the total ground-state energy (the higher line, very close to the exact energy) and the interaction energy (the lower line). The inset shows the zoom of the region of the near-threefold degeneracy at $\Phi ={\Phi }_0∕2$.

Figure 2

(Color online) (a) The exact pair-correlation function plotted for another electron fixed in the lower ring at the angle $\varphi =\pi$ as function of the magnetic flux for $R=13.2\mathrm{nm}$ and $Z=3R$. (b) Hartree charge density of the electron localized in the upper ring as obtained in the Hartree solution. (c) Same as (b) but for the electron localized in the lower ring.

Figure 3

(Color online) Same as Fig. 1 but for $Z=R$. The dashed curve marked by “e” is the mean-field single-electron energy, the eigenvalue of Eqs. (12, 13).

Figure 4

(Color online) [(a), (c), and (e)] Contour plots of the exact pair-correlation function for an electron position fixed in the lower ring at the angle $\varphi =\pi$. The numbers inside the plots give the ground-state angular momentum. The scale for PCF multiplied by ${\left(2\pi \right)}^2$ is given by the horizontal bar at the bottom of the plot. [(b), (d), and (f)] Hartree charge density of the electron localized in the upper ring. In all the plots $R=13.2\mathrm{nm}$, and the lightest shade corresponds to the minimal value of the presented functions.

Figure 5

(Color online) [(a)–(d)] Contour plots of the exact pair-correlation function for an electron position fixed in the lower ring at the angle $\varphi =\pi$ as function of the interdot distance for [(a) and (b)] $R=13.2\mathrm{nm}$ and [(c) and (d)] $R=132\mathrm{nm}$. Plots (a) and (c) correspond to the even $L$ states, and (b) and (d) to the odd $L$ states. [(e)–(g)] Contour plots of the ground-state Hartree charge density of the electron in the upper ring as function of the interdot distance for [(e) and (f)] $R=13.2\mathrm{nm}$ and [(g) and (h)] $R=132\mathrm{nm}$. Plots (e) and (g) correspond to fluxes equal to even multiples of ${\Phi }_0∕2$, and (f) and (h) to odd multiples of ${\Phi }_0∕2$. Note the inversion of the vertical axis direction in upper and lower panels.

Figure 6

(Color online) (Solid lines) The pair-correlation function for equal localization angles of both electrons as function of the interdot distance for the exact ground state of even $L$ [cross section of Figs. 5a, 5c at $\varphi =\pi$]. (Dotted lines) The minimal value of the Hartree charge density calculated for integer fluxes [cross sections of Figs. 5e, 5g].

Figure 7

(Color online) Results of the mean-field approach for $R=13.2\mathrm{nm}$ as function of the external flux and the interring separation: (a) Deviation of the single-electron charge density from the uniform value Eq. (16), (b) interaction energy, (c) single-electron energy, and (d) the total energy.

Figure 8

The mean-field energy contributions for $R=132\mathrm{nm}$ at integer flux quanta. The inset shows the total energy overestimated by the Hartree approach.

Figure 9

(Color online) The exact ground-state angular momentum (solid lines) and the mean-field expectation value (dotted line) for $R=13.2\mathrm{nm}$ and (a) $Z=1.5R$, (b) $Z=0.7R$, and (c) $Z=0.1R$.