Electron spin and charge switching in a coupled quantum-dot–quantum ring system

Few-electron systems confined in a quantum dot laterally coupled to a surrounding quantum ring in the presence of an external magnetic field are studied by exact diagonalization. The distribution of electrons between the dot and the ring is influenced by the relative strength of the dot and ring confinement, and the magnetic field which induces transitions of electrons between the two parts of the system. These transitions are accompanied by changes in the periodicity of the Aharonov-Bohm oscillations of the ground-state angular momentum. The singlet-triplet splitting for a two electron system with one electron confined in the dot and the other in the ring exhibits piecewise linear dependence on the external field due to the Aharonov-Bohm effect for the ring-confined electron, in contrast to smooth oscillatory dependence of the exchange energy for laterally coupled dots in the side-by-side geometry.

Figure 1

Confinement potential [cf. Eq. (3) for $\hslash {\omega }_i=5\mathrm{meV}$, $\hslash {\omega }_o=10\mathrm{meV}$, $V_0=-5\mathrm{meV}$, $b=30\mathrm{nm}$, and the $\mathrm{GaAs}$ effective mass $m^{*}∕m_0=0.067$. The dot oscillator length $l_i=\sqrt{2\hslash ∕m{\omega }_i}$ is equal to $21.33\mathrm{nm}$ and the oscillator length for the ring $l_o=15.08\mathrm{nm}$ which gives the ring radius $R=66.4\mathrm{nm}$.

Figure 2

Single-electron spectrum for $\hslash {\omega }_i=6\mathrm{meV}$, $\hslash {\omega }_o=11\mathrm{meV}$, $V_0=0$, and $b=30\mathrm{nm}$ $(R=63.85\text{nm)}$. The solid lines correspond to states localized in the ring and the dashed lines to states localized in the dot. Lowest of the dashed lines corresponds to the $s$ state and the two higher to $p$ states. Inset shows the low-field and low-energy part of the spectrum-enlargement of the fragment surrounded by thin solid lines corresponding to anticrossing of 0 angular momentum dot- and ring-confined energy levels. Dotted lines in (b) and (c) shows the confinement potential (left scale) for the parameters applied in (a). Solid and dashed curves in (b) show the radial probability density $\rho {\mid \psi \mid }^2$ and in (c) the wave functions of the lowest states of $s$ and $p$ symmetry, respectively.

Figure 3

Single-electron spectrum for $\hslash {\omega }_i=20\mathrm{meV}$, $\hslash {\omega }_o=11\mathrm{meV}$, $V_0=-14\mathrm{meV}$, and $d=30\mathrm{nm}$ (potential is plotted in the inset). The solid lines correspond to states localized in the ring and the dashed line to the lowest-energy state localized in the dot.

Figure 4

Charge accumulated in the dot as a function of $\hslash {\omega }_o$ for different values of the barrier thickness and $V_0=0$. Inset in (a) shows the charge accumulated in the dot as function of $V_0$ for $b=30\mathrm{nm}$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=20\mathrm{meV}$. (b) Phase diagram for the distribution of two electrons for $B=0$, $V_0=0$, and $b=30\mathrm{nm}$. Solid lines in (b) divide regions of different electron localization in the two-electron system. Above the dotted line the ground state of a single electron is localized within the dot.

Figure 5

Two-electron energy spectrum (left scale) for $b=30\mathrm{nm}$, $V_0=0$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=14\mathrm{meV}$ (spin Zeeman effect neglected). Singlets (triplets) plotted with solid (dashed) lines. Numbers close to extrema of the lines denote the absolute values of the angular momentum in $\hslash$ units. The dotted line shows the absolute value of the angular momentum of the ground state of a single electron confined in the ring (right scale). (b) Same as (a) but with spin Zeeman effect included. Only the lowest energy level of the split spin triplet is plotted.

Figure 6

Exchange energy, i.e., the energy difference of the lowest triplet and the lowest singlet energy levels for two electrons and $V_0=0$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=18\mathrm{meV}$ for different values of the barrier width and spin Zeeman splitting is neglected. The dash-dotted line shows the Zeeman splitting between states with $S_z=0$ and $\hslash$.

Figure 7

Energy spectrum of two electrons for $b=30\mathrm{nm}$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=26\mathrm{meV}$ (spin Zeeman effect neglected). The energy levels of states in which both (one) electrons are localized in the dot are plotted with dashed (solid) lines. The inset shows the ground state angular momentum. The dotted line corresponds to twice the lowest Landau level energy.

Figure 8

Two-electron ground-state energy (left scale) for $b=30\mathrm{nm}$, $V_0=0$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=13.65\mathrm{meV}$ (spin Zeeman effect neglected). Energy of states corresponding to one electron in the dot and the other in the ring plotted with solid line. Energy of states in which both the electrons are localized in the ring are plotted by the dotted curve. The thin solid step-like line gives the total angular momentum which is referred to the right axis.

Figure 9

Solid line (left scale): the external potential $\hslash {\omega }_i=50\mathrm{meV}$ and $\hslash {\omega }_o=6\mathrm{meV}$, $b=20\mathrm{nm}$, $V_0=-66\mathrm{meV}$. The ground state for $B=0$ corresponds to the $L=0$ singlet with one electron in the dot and the other in the ring. Radial density of this state plotted with dotted line (right scale). The first excited $s$ singlet state corresponds to both electrons in the dot (dashed line).

Figure 10

Two-electron energy spectrum for the potential parameters of Fig. 9. Dotted lines show the energy levels of $s$ singlets. The dashed line corresponds to the $s$ triplet.

Figure 11

Phase diagram for the electron distribution in the $(\hslash {\omega }_i$, $\hslash {\omega }_o)$ plane for $V_0=0$ and $b=30\mathrm{nm}$ in the absence of the magnetic field. Solid lines separate regions of different electron distributions. Numbers denote the ground-state total spin and total angular momentum quantum number $(S,L)$.

Figure 12

Energy spectrum for $N=3$, $V_0=0$, $b=30\mathrm{nm}$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=37\mathrm{meV}$. The solid (dashed) lines show the lowest energy levels with the two dot-confined electrons with $S=1∕2$ and opposite ($S=3∕2$ and the same) spin and one electron in the ring. The states with the two dot-confined electrons of parallel spins are almost degenerate with respect to the spin orientation of the electron in the ring. The only exception is the state with $L=2$. The lower of the dash-dotted line shows this state for $S=1∕2$ and the upper for $S=3∕2$. Dotted lines correspond to all electrons confined in the dot. Quantum numbers $(L,S)$ of these states are indicated in the figure. Only nonpositive angular momenta are shown. The thick solid line in the lower part of the figure shows the the ground-state angular momentum quantum number (right scale). The panel above the upper axis shows the number of electrons in the dot $n_d$ and $S$ for the ground state in format $n_d\left(S\right)$, “deg” stands for degeneracy of the $S=1∕2$ and $3∕2$ states.

Figure 13

Ground-state energy (left scale) and the absolute value of the ground-state angular momentum (right scale) for $N=3$, $V_0=0$, $b=30\mathrm{nm}$, $\hslash {\omega }_i=6\mathrm{meV}$, and $\hslash {\omega }_o=27\mathrm{meV}$. The dotted vertical line marks the magnetic field for which the electron distribution is changed. The vertical arrows on the $L$ staircase correspond to triplet state of the ring subsystem.