Giant Spin-Orbit Interaction Due to Rotating Magnetic Fields in Graphene Nanoribbons

We study graphene nanoribbons theoretically in the presence of spatially varying magnetic fields produced, e.g., by nanomagnets. We show, both analytically and numerically, that an exceptionally large Rashba spin-orbit interaction of the order of 10 meV can be produced by a nonuniform magnetic field. As a consequence, helical modes exist in armchair nanoribbons that exhibit nearly perfect spin polarization and are robust against boundary defects. This result paves the way for realizing spin-filter devices in graphene nanoribbons in the temperature regime of a few kelvins. If a nanoribbon in the helical regime is in proximity contact to an $s$-wave superconductor, the nanoribbon can be tuned into a topological phase that sustains Majorana fermions.

Popular Summary

In the world of electronics, the quest for smaller and faster devices is a never-ending one. In that quest, one of the challenges is the problem of heating in such devices. The field of spintronics emerged as an answer to that challenge, which exploits the spins of electrons in addition to their charges in electronic transport. Spin-polarized currents—currents of electrons whose spins point in the same direction—and efficient spin filters are the foremost must-haves in spintronics. The so-called helical modes, which transport opposite spins in opposite directions, are the physical basis for spin filters. These modes have their origin in the spin-orbit interaction (SOI), a relativistic effect that couples the spin and orbital (motional) degrees of freedom of an electron together. For the modes to be observed and operative, pristine materials are needed to avoid mode mixing caused by impurities. The high purity of graphene makes the material an ideal candidate in this respect. However, its intrinsic SOI is known to be extremely small.

In the present work, we address this challenge and propose a novel way to generate a giant, effective SOI in graphene nanoribbons. The essential idea is to enhance the SOI intrinsic in a nanoribbon by applying to it spatially varying magnetic fields that can be produced by nanomagnets. As our work demonstrates, large values of SOI become possible and result in helical modes of nearly perfect spin polarization. Moreover, the high purity of graphene nanoribbons and their considerable sub-band splittings allow for great control of the number of helical transport channels at temperatures higher than what could be achieved with semiconducting nanowires. All these together make graphene nanoribbons the most promising family of candidates for spintronics effects in the temperature range of a few kelvins.

The potential of our proposal goes beyond the context of spintronics. The search for Majorana fermions, exotic particles that are their own antiparticles, is currently a white-hot topic in physics. Our proposal offers another material avenue for the search: Bringing a nanoribbon with helical modes into proximity to an $s$-wave superconductor would lead to a topological state that supports Majorana fermions.

Figure 1

An armchair GNR formed by a finite strip of graphene aligned along the $z$ axis and of the width $W=Na$ in the $x$ direction. The GNR is composed of two types of atoms, $A$ (blue dot) and $B$ (green dot), and is characterized by hexagons in real space with translation vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. The low-energy physics is determined by the momenta $\mathbf{k}=(k_x,k_z)$ around the two valleys $\mathbf{K}=-{\mathbf{K}}^{\prime }=(4\pi /3a,0)$. Nanomagnets (green slabs) placed with period ${\lambda }_n$ on the sides of the GNR provide a spatially varying magnetic field ${\mathbf{B}}_n$ (red arrows).

Figure 2

The spectrum of an armchair GNR obtained by numerical diagonalization of the tight-binding Hamiltonian ${\overline{H}}_0+{\overline{H}}_{\mathrm{SO}}+{\overline{H}}_Z$. The low-energy spectrum is linear for metallic [(a),(b) with $N=82$] and quadratic for semiconducting [(c),(d) with $N=81$] GNRs. The SOI [(b) ${\Delta }_{\mathrm{SO}}=1\mathrm{meV}$ and (d) ${\Delta }_{\mathrm{SO}}=5\mathrm{meV}$] lifts the spin degeneracy, so the spectrum consists of (b) two Dirac cones or (d) two parabolas shifted by $k_{\mathrm{SO}}$ from zero (shown by green lines). For a metallic GNR, each branch is characterized not only by the spin projection $s$, but also by the isospin $\gamma$. The solid (dashed) lines correspond to $\gamma =1$ ($\gamma =-1$) [see (b)]. Whereas for a semiconducting GNR, a magnetic field (${\Delta }_Z=0.1\mathrm{meV}$) alone opens a gap $2{\Delta }_g\approx 0.2\mathrm{meV}$ [(d)], we also need to include intervalley scattering (modeled by fluctuations in on-site energies) for the metallic GNR [(b)]. If the chemical potential $\mu$ is tuned inside the gap, the system is in the helical regime with nearly perfect polarization, $⟨s_x⟩\approx 0.99$, in both semiconducting and metallic cases.

Figure 3

Defects on the edges of a metallic armchair GNR ($N=82$), resulting in the opening of a gap at zero energy. In the numerical diagonalization, the defects are modeled by omitting two atoms on the edges, which is assumed to be periodic with period $l_d=5\sqrt{3}a$. We see that the spectrum changes only slightly, and the qualitative features of a metallic armchair GNR are maintained.