Details

We study the lowest single-electron energies and wavefunctions obeying the Schrödinger equation. The dimensionless external potential felt by an electron of charge and effective mass , confined between two in-plane gates is given by

,

where is the energy scale of the interlevel separation in the ring of radius , and is the electrostatic potential along the ring due to an individual negatively charged gate. We model the gates as charged wires, as in [1]. Thus the individual wire electrostatic potential is given by

.

Here labels the wires, is the linear charge density on a wire, and is its corresponding distance from the center of the ring. Zero potential is defined to be along the axis perpendicular to the reflection axis . For simplicity, equidistant gates are assumed. Making the approximation , we can expand up to second-order terms, which leads to the Hamiltonian

,

where

characterizes the global potential strength in terms of the contribution from gate . The relative contribution from gate is given by

.

We have accordingly shifted all calculated values by the arbitrary constant resulting from the expansion. Note that the approximated potential here is a corrected form of the potential used in [1–3] (which has a (harmless) error in the parameter in front of ). Solutions must be -periodic due to the ring periodic boundary conditions and are found as a sum of free particles on a ring basis function

.

The Demonstration plots the energy levels and wavefunctions for the Hamiltonian , as well as the approximate trigonometric potential for (red) and the full potential (dotted black), as functions of , and the ratio .

Snapshot 1: results for equivalent gate contributions

Snapshot 2: results for non-equivalent gate contributions and small global potential strength

Snapshot 3: results for deep global potential strength and non-equivalent gate contributions

References

[1] T. P. Collier, V. A. Saroka and M. E. Portnoi, "Tuning Terahertz Transitions in a Double-Gated Quantum Ring," Physical Review B, 96, 2017, 235430. doi:10.1103/PhysRevB.96.235430.

[2] T. P. Collier, V. A. Saroka and M. E. Portnoi, "Tuning THz Transitions in a Quantum Ring with Two Gates," Physics, Chemistry and Applications of Nanostructures (V. E. Borisenko, S. V. Gaponenko, V. S. Gurin and C. H. Kam, eds.), Singapore: World Scientific, 2017 pp. 172–175. doi:10.1142/9789813224537_0040.

[3] T. P. Collier, A. M. Alexeev, C. A. Downing, O. V. Kibis and M. E. Portnoi, "Terahertz Optoelectronics of Quantum Rings and Nanohelices," Semiconductors, 52(14), 2018 pp. 1813–1816. doi:10.1134/S1063782618140075.