High Landau levels of two-dimensional electrons near the topological transition caused by interplay of spin-orbit and Zeeman energy shifts

In the presence of spin-orbit coupling two branches of the energy spectrum of two-dimensional electrons get shifted in the momentum space. Application of an in-plane magnetic field causes the splitting of the branches in energy. When both spin-orbit coupling and Zeeman splitting are present, the branches of the energy spectrum cross at a certain energy. Near this energy, the Landau quantization becomes peculiar since semiclassical trajectories, corresponding to individual branches, get coupled. We study this coupling as a function of the proximity to the topological transition. Remarkably, the dependence on the proximity is strongly asymmetric, reflecting the specifics of the behavior of the trajectories near the crossing. Equally remarkable, on one side of the transition, the magnitude of coupling is an oscillating function of this proximity. These oscillations can be interpreted in terms of the Stückelberg interference. Scaling of characteristic detuning with magnetic length is also unusual. This unusual behavior cannot be captured by simply linearizing the Fermi contours near the crossing point.

Figure 1

Evolution of the Fermi contours, defined by Eq. (12), with the ratio, $\nu$, of the spin-orbit and Zeeman energy shifts [Eq. (13)]: (a) $\nu =2$, (b) $\nu =1.5$, (c) $\nu =1$, and (d) $\nu =0.5$. As $\nu$ decreases, the inner contour grows. The Fermi energy is chosen to be $\mathcal{E}=1.1{\mathcal{E}}_0$ in all panels. Thus, strictly speaking, the separation between the inner and the outer contours is finite at $q_y=0$. This separation can be distinguished in (a) and (b) but cannot be distinguished in (c) and (d).

Figure 2

Evolution of the Fermi contours, defined by Eq. (12), with energy. Energies: (a) $\mathcal{E}=1.1{\mathcal{E}}_0$, (b) $\mathcal{E}=1.5{\mathcal{E}}_0$, (c) $\mathcal{E}=1.75{\mathcal{E}}_0$, and (d) $\mathcal{E}=2{\mathcal{E}}_0$. We chose $\nu =1$ in all panels. In (c) the outer contour is vertical at $q_y=0$, in accordance with Eq. (18).

Figure 3

Without the right-hand sides, Eqs. (23) and (24) are decoupled and describe the electron motion in parabolic potentials (blue and red, respectively). The potentials cross at $u=u_c$. Without coupling, the incident wave, $i$ (see inset) is fully reflected into the wave, $r_1$, propagating in the red parabola. With right-hand sides caused by spin-orbit coupling, another channel of reflection into the wave $r_2$ propagating in the blue parabola emerges. The corresponding reflection probability should be identified with the transmission probability $|{\mathcal{T}}_E{|}^2$.

Figure 4

Illustration of the asymmetry of the Fermi contours with respect to the sign of detuning, $\delta$. In the left panel the contours are shown for $\delta =0.3$; in the right panel, for $\delta =-0.3$. Parameter $\nu$ is chosen to be 0.4 in both panels. Despite the seemingly small difference between the two panels, the transmission coefficient, $|{\mathcal{T}}_E{|}^2$, in the right panel is much smaller than in the left panel.

Figure 5

Numerical solution, $|{\rho }_2{\left(z\right)|}^2$, of the system, Eq. (20), is plotted for two values of detuning, (a) $\delta =3$ and (b) $\delta =-3$, and for $G=1$. The black line in (a) and the green line in (b) are the envelopes plotted from Eq. (42). We see that the agreement with theory is much better in (b), since Eq. (42) neglects interference.

Figure 6

Numerical solution, $|{\rho }_2{\left(z\right)|}^2$, of the system, Eq. (20), is plotted for two values of detuning, (a) $\delta =0.5$ and (b) $\delta =-0.5$, and for $G=0.7$. The black line in (a) and the green line in (b) are the envelopes plotted from Eq. (42). We see that $|{\rho }_2{\left(z\right)|}^2$ approaches finite values at large $z$. This approach is accompanied by huge oscillations which complicate the determination of the transmission coefficient.

Figure 7

The blue curve, which shows the transmission coefficient obtained numerically, is plotted versus the dimensionless magnetic field, $G$, directly at the topological transition $\delta =0$. The black curve, which shows the theoretical prediction for the transmission coefficient, is plotted from Eq. (43). Inset: The prediction, ${sin}^2\left(\frac{8}{3G^2}\right)$, based on the heuristic argument given in Sec. V.

Figure 8

The transmission coefficient (blue curve) obtained numerically is plotted versus the detuning, $\delta$, for the dimensionless magnetic field $G=1.5$. The theoretical prediction for the transmission coefficient (black curve) is plotted from Eq. (40). Small wiggles in the numerical curves are an artifact of the averaging procedure. For negative detunings the agreement with theory is good. For positive detunings, theory predicts oscillations with a magnitude larger than 1, while in the numerical curve these oscillations, manifesting the Stückelberg interference [21], slowly decrease with $\delta$.