Excitonic Aharonov–Bohm effect in QD-on-ring nanostructures

We firstly discuss the excitonic A–B effect in a 2D GaAs ring, where the motion of the electron and hole is assumed to be confined in the xy plane. With the assumption of a parabolic confinement potential, previous theoretical studies predicted a vanishing A–B effect for the exciton ground state of 2D rings due to the Coulomb interaction [18, 19]. But the parabolic confinement is not appropriate for the study of the A–B effect of quantum rings with a large width because of the loss of ring features due to the vanishing central hole. In our calculation, we applied a square potential-well confinement along the radial direction. In the following, the radius (R) and the width (W) of the ring are defined as the average value of the inner (R_i) and outer radius (R_o) of the ring and their difference.

The influence of the magnetic field on the single particle states was calculated by the perturbation theory [20]. The Hamiltonian of the single particle was expressed in the polar coordinates as

$$\begin{align} \displaystyle H=& H_{0} + H^{^{\prime}}(B), \ \nonumber \\ H_{0}=& -\frac{\hbar^{2}}{2m^{*}}\left[\frac{1}{r}\frac{\partial}{\partial r} \left(\frac{1}{r}\frac{\partial}{\partial r} \right)+ \frac{1}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}} \right] +V(r), \nonumber \\ H^{^{\prime}}=& -\frac{\hbar^{2}}{2m^{*}}\left[-\frac{{\rm i}qB}{\hbar} \frac{\partial}{\partial \theta} -\frac{(qBr)^{2}}{4\hbar^{2}} \right], \nonumber \end{align} \tag{ 2 }$$

where $H_{0}$ is the Hamiltonian for $B= 0$ and $H^{^{\prime}}(B)$ is the perturbative part. Symmetric gauge was applied for the vector potential. $V(r)$ is the confinement potential, which is equal to zero in the GaAs ring and equal to the band offset in the ${\rm Al}_{0.3}$${\rm Ga}_{0.7}$As barrier layer. $q= -e$ and e for electron and hole, respectively. $m^{*}$ is the effective mass for electron and hole, which is different between the GaAs ring and the ${\rm Al}_{0.3}$${\rm Ga}_{0.7}$As barrier layer. The calculation of the single particle state in a 2D ring without the magnetic field is shown in the appendix.

The second-order energy correction was much smaller than the first-order term, which was confirmed by our numerical calculation. In figure 2, the numerical results of the single particle energy calculated by the finite element method show a good agreement with the perturbative results, which confirms the validity of our calculation. In the first-order perturbation, we evaluated $\langle \varphi _{n l}\vert r^{2}/m^{*}\vert \varphi_{n l} \rangle$ by numerical integration using unperturbed wave function $\varphi _{n l}$ given in the appendix. The matrix element can be approximated by $\langle \varphi _{n l}\vert r^{2}\vert \varphi_{n l} \rangle / \overline{m^{*}}$, where $\overline{m^{*}}$ is $\langle \varphi _{n l}\vert m^{*}\vert \varphi_{n l} \rangle$, since the difference in $m^{*}$ between GaAs and ${\rm Al}_{0.3}$${\rm Ga}_{0.7}$As is small (less than $30\%$). Then, the single particle energy with the first-order correction is

$$\begin{align} \displaystyle \label{eqn3} E_{n l}(B)\cong & E_{n l}(0) -\frac{\hbar^{2}l^{2}}{2\overline{m^{*}} \langle \varphi _{n l}\vert r^{2}\vert \varphi_{n l} \rangle} \nonumber \\ & +\frac{\hbar^{2}}{2\overline{m^{*}} \langle \varphi_{n l}\vert r^{2}\vert \varphi_{n l} \rangle} \left(l-\frac{\phi^{^{\prime}}}{\phi_{0}^{^{\prime}}} \right)^{2}, \nonumber \end{align} \tag{ 3 }$$

where $E_{n l}(0)$ is the energy of the single particle state for $B = 0$, $\phi^{^{\prime}} = B \pi \langle \varphi_{n l}\vert r^{2}\vert \varphi_{n l} \rangle$, and $\phi_{0}^{^{\prime}} = 2\pi \hbar/ q$. The last term on the right-hand side of equation (3) is B-dependent and has a similar form as equation (1) of the 1D ring [3] except for three main differences: first, when $B= 0$, the energy of the lowest state is dependent on both the angular momentum and the confinement along the radial direction. Second, the fixed 1D ring radius is replaced with the average radius of the single particle that depends on the ring width and varies for different states. Third, because of the finite potential barrier height, the average value of the effective mass was used to take their r dependence into consideration. When the width of the 2D ring is infinitesimally small, the B dependence of equation (3) approaches equation (1).

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The average radius of the electron and hole in the 2D ring varies for different angular quantum numbers and different magnetic fields as shown in figure 2(c). Because the magnetic-field-induced potential pushes both the electron and hole towards the center of the structure, the average radius decreases with increasing magnetic field. The average radius reduction of the electron is faster than the hole due to the larger kinetic energy of the former. According to equation (3), the energy oscillation period of the lowest single particle state is determined by its average radius. Figure 2(c) shows that the low-energy electron and hole states have a very similar average radius when $B < 15$ T. Therefore, the energy oscillation periods of the lowest electron and hole states are close to each other, which is seen in figures 2(a) and (b).

The energy of the non-interacting pair state $E_{\rm ps}(B)$ ($= E_{\rm g} + E_{n l}^{(e)}(B) + E_{n^{^{\prime}} l^{^{\prime}}}^{(h)}(B)$, E_g: band gap of GaAs) is shown in figure 3(a). For the lowest pair state, the oscillating energy is observed with increasing B. By magnifying the region surrounded by the green square in figure 3(a), the total angular momentum transition from $L= 0$ to 1 for the lowest-energy pair state is found in a narrow magnetic field range as shown in figure 3(b), which is attributed to the similar energy oscillation periods for the electron and hole states.

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The exciton energy of the 2D ring was calculated by the configuration interaction method as shown in figure 3(c). The A–B oscillation was not observed for the exciton ground state, but was found in the excited states, which agrees with [18]. By the configuration interaction method, we can also analyze the contribution of each pair state to the exciton ground state. We found that the wave functions of the pair states with the same L were mixed by the Coulomb interaction, and the crossing behavior between their energies was replaced by an anti-crossing behavior. So, the oscillation feature of the magnetic field dependence of the lowest pair state energy was smoothed, and the narrow B range for the non-zero L vanished. These resulted in the disappearance of the excitonic A–B effect, and the lowest exciton was always a bright one with $L= 0$. Note that the electric-dipole transition between the exciton state and the ground state is allowed only for excitons with $L=0$ due to the conservation of angular momentum [21]. As an evidence of the Coulomb mixing, we found that as the magnetic field increased, the angular momentum of the largest contributing pair state for the exciton ground state changed from ($l_{e}=0, l_{h}=0$) to ($l_{e}=-1, l_{h}=+1$) and ($l_{e}=-2, l_{h}=+2$) for $B= 10,$ and 20 T, respectively. Our analysis showed that the Coulomb interaction was the reason for the vanishing A–B effect in the 2D ring.

To avoid the Coulomb mixing among pair states, we may increase their kinetic energy interval by decreasing the width of the ring. But this method will simultaneously decrease the difference between the electron and hole average radii, which is undesirable for the A–B effect. If a ring with a small radius is used alternatively, the energy oscillation and the L transition of the exciton ground state may occur for a large magnetic field, which causes difficulty for experimental studies. Thus, in previous theoretical and experimental investigations, the visibility of the excitonic A–B effect was not clear for 3D rings [6, 22].

For a clear excitonic A–B effect in 3D nanostructures, we propose the use of the QD-on-ring nanostructure with an applied electric field (E) along the growth direction. The advantages of the coupled nanostructure stated in section 1 are also the conditions for the realization of a clear A–B effect. The recently reported QD-on-rings fulfill these conditions [11], thus the effect of the magnetic field on the exciton states of the QD-on-ring was numerically investigated to verify the appearance of the A–B effect.

Figure 4(a) shows the exciton energy of the QD-on-ring as a function of the magnetic field for $E= 0$. Neither the energy oscillation nor the L transition was observed for the exciton ground state because the electron and hole were strongly bound (binding energy of 27 meV) in the QD part as can be seen from their average radii in figure 5(a).

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When an electric field is applied, the excitons have lower energy than those for $E= 0$ due to the Stark shift [21] as shown in figure 4. When $B= 0$, the largest contributing pair state for the lowest exciton state with $L= 0$ is ($l_{e}=0, n_{e}=1$, $l_{h}=0, n_{h}=1$), in which the electron is redistributed from the QD to the ring with the increasing electric field and the hole remains in the QD.

Figure 6(a) shows the pair state energy of the QD-on-ring for $E= 120$ kV ${\rm cm}^{-1}$ as an example. When $B= 0$, the lowest electron and hole states are separated in the ring and QD, respectively. According to equation (1), the kinetic energy of single particles is inversely proportional to the square of its trajectory radius, thus the energy difference between the electron states (in the ring) with $l_{e}= 0$ and $l_{e}= \pm 1$ is smaller than that between the hole states (in the QD) with $l_{h}= 0$ and $l_{h}= \pm 1$ for $E= 120$ kV ${\rm cm}^{-1}$. This leads to a large energy difference between the lowest pair states (with $L= 0, -1$) and the corresponding higher energy pair states with nonzero-$l_{h}$ hole involved as shown in figure 6(a). By the charge separation, the Coulomb binding energy (7 meV) of the lowest exciton is smaller than this energy difference, which avoids the Coulomb mixing. At the same time, the magnetic field response of the lowest pair states with $L= 0, -1$ is dominated by that of the electron which is localized in the ring and has a larger trajectory radius, so a clear L transition is observed in figure 6(a). When Coulomb interaction is taken into consideration, the electron and hole remain spatially separated with a big difference in their average radii as shown in figure 5(a), which results in a large magnetic flux between their trajectories. So a clear A–B effect is observed for $E= 120$ kV ${\rm cm}^{-1}$ as shown in figure 6(b).

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The magnetic-field-induced potential leads to the decreasing average radii of both the electron and hole as shown in figure 5(a). Furthermore, the squared overlap integral of the lowest exciton ($L=0$) increases with the magnetic field as shown in figure 5(b). This leads to an increasing binding energy and the effect of the Coulomb mixing becomes obvious, shown as the anti-crossing behavior between the first and second lowest excitons (L = 0) for $B= 12$ T in figure 6(b). The influence of the magnetic field on the distribution of the low-energy single particles partly cancels for the electric-field-induced charge separation.

The lowest exciton for $E= 0$ is always a bright exciton. When the L transition occurs for the exciton ground state at $B= 4$ T for $E= 120$ kV ${\rm cm}^{-1}$, this bright exciton with a non-negligible squared overlap integral turns dark. Therefore, we propose an excitonic A–B effect with the moderate external fields, which may be verified by the experimental observation of the emission spectra by tuning the applied electric field.