Berry phase and Rashba fields in quantum rings in tilted magnetic field

We study the role of different orientations of an applied magnetic field as well as the interplay of structural asymmetries on the characteristics of eigenstates in a quantum ring system. We use a nearly analytical model description of the quantum ring, which allows for a thorough study of elliptical deformations and their influence on the spin content and Berry phase of different quantum states. The diamagnetic shift and Zeeman interaction compete with the Rashba spin-orbit interaction, induced by confinement asymmetries and external electric fields, to change spin textures of the different states. Smooth variations in the Berry phase are observed for symmetric quantum rings as a function of applied magnetic fields. Interestingly, we find that asymmetries induce nontrivial Berry phases, suggesting that defects in realistic structures would facilitate the observation of geometric phases.

Figure 1

(a) Magnetic field orientation and coordinate system. Potential profile maps for (b) the circularly symmetric and (c) the elliptically deformed ring; purple regions show lowest energy values of the confinement potential. Electronic orbital for an excited state of an elliptical ring ($\delta =2$ meV), in the presence of a Rashba field ($F=100$ kV/cm) in a magnetic field ($B=2.375$ T) at different tilt angles, (d) $\theta =0^{\circ }$ and (e) $\theta ={60}^{\circ }$.

Figure 2

Electronic structure for quantum rings in a magnetic field at fixed tilt angle, $\theta ={60}^{\circ }$, and Rashba field $F=100$ kV/cm, as a function of the total magnetic field strength for (a) symmetric $\left(\delta =0\right)$ and (i) asymmetric ($\delta =2$ meV) rings. The Berry phase for different levels for $\delta =0$ is shown in panels (b)–(g) in the left column; and for $\delta =2$ meV in panels (j)–(o) in the right column. The cumulative Berry phase for different occupation numbers is shown in panel (h) for the symmetric ring, and panel (p) for the asymmetric ring.

Figure 3

Electronic structure for quantum rings under fixed magnetic and Rashba fields, $B=6.625$ T and $F=100$ kV/cm, as a function of the magnetic field tilt angle $\theta$ for (a) symmetric $\left(\delta =0\right)$ and (i) asymmetric ($\delta =2$ meV) rings. Berry phases for different states in both rings are shown in the panels below. The cumulative Berry phase for different occupations is shown in panels (h) and (p), for the symmetric and asymmetric rings, respectively.

Figure 4

Panels in the left column show expansion coefficients for the four lowest states of an asymmetric ring ($\delta =2$ meV), and fixed Rashba field $F=100$ kV/cm, and magnetic field tilt angle $\theta ={60}^{\circ }$, as functions of magnetic field intensity. Level admixtures clearly evolve with sudden switches at level anticrossings. The right column shows spin density vector maps along the ring ($z$-integrated) for the four lowest states at a field $B=2.375$ T (indicated by the vertical line in the left panels). Blue arrows have a positive projection along $z$, while for red arrows the projection is negative. Notice nearly parallel vectors in level 1 result in a null Berry phase; in contrast, canting of vectors in level 2 contribute to a Berry phase of $\simeq -\pi$ [see Fig. 2].