Impurity center in a semiconductor quantum ring in the presence of a radial electric field

The problem of an impurity electron in a quantum ring (QR) in the presence of a radially directed strong external electric field is investigated in detail. Both an analytical and a numerical approach to the problem are developed. The analytical investigation focuses on the regime of a strong wire-electric field compared to the electric field due to the impurity. An adiabatic and quasiclassical approximation is employed. The explicit dependences of the binding energy of the impurity electron on the electric field strength, parameters of the QR, and position of the impurity within the QR are obtained. Numerical calculations of the binding energy based on a finite-difference method in two and three dimensions are performed for arbitrary strengths of the electric field. It is shown that the binding energy of the impurity electron exhibits a maximum as a function of the radial position of the impurity that can be shifted arbitrarily by applying a corresponding wire-electric field. The maximal binding energy monotonically increases with increasing electric field strength. The inversion effect of the electric field is found to occur. An increase of the longitudinal displacement of the impurity typically leads to a decrease of the binding energy. Results for both low- and high-quantum rings are derived and discussed. Suggestions for an experimentally accessible setup associated with the $\mathrm{GaAs}∕\mathrm{GaAlAs}$ QR are provided.

Figure 1

Energy of the lateral motion a free electron in a quantum ring as a function of the charge $\lambda$ of the central wire. Effective atomic units are used.

Figure 2

Binding energy $E_b$ as a function of the position of the impurity center for quantum ring (A) $\lambda =4.3$ of the central wire. Effective atomic units are used.

Figure 3

Binding energy $E_b$ as a function of the radial position of the impurity center ${\rho }_0$ for quantum ring (A) for $z_0=0$ for several different values of the linear charge density $\lambda$ of the central wire. Effective atomic units are used.

Figure 4

Binding energy $E_b$ as a function of the radial position of the impurity center ${\rho }_0$ for quantum ring (A) for $\lambda =0$ and for different displacements $z_0=0,z_0=d∕2$ and the corresponding 2D quantum ring $(d=0)$. Effective atomic units are used.

Figure 5

Quantum ring (B): same as Fig. 4 and a 3D curve for $\lambda =10,z_0=0$. Effective atomic units are used.

Figure 6

Binding energy $E_b$ as a function of the central wire charge $\lambda$ for the different impurity positions ${\rho }_0,z_0$ in quantum ring (A). Effective atomic units are used.

Figure 7

Energy of a free electron in three different two-dimensional quantum rings as a function of the positive charge of the central wire $\lambda$. Numerical results and analytical estimations are shown. Effective atomic units are used.

Figure 8

Binding energy of an electron to an impurity center in a two-dimensional quantum ring as a function of the charge of the central wire. Numerical results and analytical estimates are shown. Effective atomic units are used.