Quantum rings are one of the most commonly used structures in the study of new spintronic devices. Some of these spintronic devices utilize the spin-orbit interaction which mixes the two spin states. The strength of the spin-orbit interaction can be controlled experimentally by an electric field, and so by applying an electric and a magnetic field on the ring its electronic characteristics can be manipulated.

One of the most important characteristics of spin based devices is the relaxation rate of the electron in a certain spin-state, which is generally slower than the relaxation rate in electronic devices. The inverse of this rate measures the lifetime of a certain spin state. Most of the studies in this field either treat the quantum ring as a quasi-one-dimensional object, [1] or treat the spin-orbit interaction as a perturbation. [2]

We study the motion of an electron in a two-dimesional ring threaded by a magnetic field and in which the Rashba spin-orbit interaction is active. The system is described by the Hamiltonian
H=(m/2)v²+α(v_xσ_y-v_yσ_x)+gμ_BBσ_x/2+V(r) ,
where v=[P-(e/c)A]/m, α is the strength of the spin-orbit interaction and V(r) is the confining potential. Following a similar solution for a quantum dot [3], we find an exact solution for the eigenenergies and wavefunctions, for general values of the ring's width and of the spin-orbit coupling.

Using these solutions we calculate the ring's magnetization and the electron's relaxation rate. We find that when applying a magnetic field parallel to the ring, the spin-orbit interaction increases the magnetization when the magnetic field is below some value which depends on the Fermi energy and the width. The spin-orbit interaction reduces the magnetization when the magnetic field is above that value. When the magnetic field is perpendicular to the ring, the spin-orbit interaction generally reduces the magnitude of the magnetization. This two effects happen due to the mixing of the spin polarizations by the spin-orbit interaction.

Regarding the relaxation rate, we find that by applying a magnetic field parallel to the ring, and keeping it below a certain width-dependent critical value, the relaxation rate has a maximum as a function of the spin-orbit strength. This critical value increases as the ring widens, and so if the magnetic field is above the critial value, it will remain so even when the ring's width is reduced. We also find that as long as the magnetic field is above that critical value, reducing the ring's width reduces the relaxation rate. When the magnetic field is perpendicular to the ring, the relaxation rate also has a maximum as a function of the spin-orbit strength. Even more interesting, below a certain value for the Zeeman coupling, the relaxation rate has a minimum as a function of the magnetic field. This effect can be attributed to the way the spin-orbit interaction splits the energy levels.

[1] J. Splettstoesser, M. Governale and U. Zulicke, Phys. Rev. B 68, 165341 (2003).

[2] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 (2000).

[3] E. Tsitsishvili, G.S. Lozano and A.O. Gogolin, Phys. Rev. B 70, 115316 (2004).