Orbital magnetic moments in insulating Dirac systems: Impact on magnetotransport in graphene van der Waals heterostructures

In honeycomb Dirac systems with broken inversion symmetry, orbital magnetic moments coupled to the valley degree of freedom arise due to the topology of the band structure, leading to valley-selective optical dichroism. On the other hand, in Dirac systems with prominent spin-orbit coupling, similar orbital magnetic moments emerge as well. These moments are coupled to spin, but otherwise have the same functional form as the moments stemming from spatial inversion breaking. After reviewing the basic properties of these moments, which are relevant for a whole set of newly discovered materials, such as silicene and germanene, we study the particular impact that these moments have on graphene nanoengineered barriers with artificially enhanced spin-orbit coupling. We examine transmission properties of such barriers in the presence of a magnetic field. The orbital moments are found to manifest in transport characteristics through spin-dependent transmission and conductance, making them directly accessible in experiments. Moreover, the Zeeman-type effects appear without explicitly incorporating the Zeeman term in the models, i.e., by using minimal coupling and Peierls substitution in continuum and the tight-binding methods, respectively. We find that a quasiclassical view is able to explain all the observed phenomena.

Figure 1

The orbital magnetic moments of the spin-up (spin-down) states shown by thick red (dashed blue) lines, and the corresponding low-energy band structure, shown in black, for (a) $\Delta =0$ and ${\Delta }_{\text{SO}}=30$ meV, and (b) $\Delta =30$ meV and ${\Delta }_{\text{SO}}=0$. Note that in (b) the orbital magnetic moments for the two spins are equal, due to the absence of SOC.

Figure 2

Several lowest Landau levels of all spin and valley flavors for (a) ${\Delta }_{\text{SO}}=30$ meV and $\Delta =0$, (b) ${\Delta }_{\text{SO}}=0$ and $\Delta =30$ meV, and (c) ${\Delta }_{\text{SO}}=\Delta =30$ meV. The $n=0$ Landau level is depicted by the horizontal solid black line. Also shown are the bulk bands as red (spin-up) and blue (spin-down) shaded regions, as well as the sketch of the corresponding orbital moments, with the length of the arrow being proportional to the intensity.

Figure 3

Contour plots of the transmission coefficient as a function of incident angle and energy for ${\Delta }_{\text{SO}}=30$ meV, $\Delta =0$, and $W=200$ nm. The magnetic field equals 0 T in (a) and (b), $0.1$ T in (c) and (d), $0.2$ T in (e) and (f), and $0.3$ T in (g) and (h). The results are shown for both spin orientations. The semiclassical critical boundaries ${\epsilon }_{\text{cr}0}$ and ${\epsilon }_{\text{cr}W}$ are depicted by dash-dotted and dotted lines, respectively.

Figure 4

(a) The regions with different ranges of turning points for $W=200$ nm, ${\Delta }_{\text{SO}}=30$ meV, $\Delta =0$, and $B=0.2$ T. Different classically allowed trajectories are found in differently shaded regions, demarcated by the two critical boundaries. (b) A family of four different classical trajectories, which correspond to the states labeled by numbered crosses in each region of (a).

Figure 5

The derivative of the conductance vs incident energy for (a) $0.1$ T, (b) $0.2$ T, and (c) $0.3$ T. All other parameters are the same as in Fig. 3. Insets show the variation of the conductance with incident energy for the corresponding magnetic field.

Figure 6

Transmission curves calculated using (a) the continuum and (b) the TB NEGF method. The parameters are $W=200$ nm, $B=0.3$ T, and $E=100$ meV.