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

,

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

[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.