Electrical spin manipulation in graphene nanostructures

We propose a mechanism to drive singlet-triplet spin transitions electrically in a wide class of graphene nanostructures that present pairs of in-gap zero modes, localized at opposite sublattices. Examples are rectangular nanographenes with short zigzag edges, armchair ribbon heterojunctions with topological in-gap states, and graphene islands with $s{p}^3$ functionalization. The interplay between the hybridization of zero modes and the Coulomb repulsion leads to symmetric exchange interaction that favors a singlet ground state. Application of an off-plane electric field to the graphene nanostructure generates an additional Rashba spin-orbit coupling, which results in antisymmetric exchange interaction that mixes $S=0$ and $S=1$ manifolds. We show that modulation in time of either the off-plane electric field or the applied magnetic field permits performing electrically driven spin resonance in a system with very long spin-relaxation times.

Figure 1

Two types of graphene nanostructures that host pairs of zero modes localized in opposite sublattices. (a) Rectangular graphene nanoribbons with short zigzag edges that host one unpaired electron each. (b) Armchair graphene heterojunctions hosting one zero mode at each interface [39]. In both cases, the green arrows represent the magnetization calculated with a mean-field Hubbard model.

Figure 2

(a) Graphene nanoribbon with $W=7$. (b) Sketch of the single-particle energy spectrum for the graphene nanostructures shown in Fig. 1. A gap, separating the doubly occupied states from the empty states, contains two in-gap states ${\psi }_{\pm }$ split by $\delta ={\epsilon }_{+}-{\epsilon }_{-}$. (c) Dependence of $\delta$ on the spatial separation $W$ of the zero modes. For zero-dimensional (0D) ribbons, $W$ stands for the width of the ribbon. For the heterojunctions, $W$ stands for the distance between the interfaces. The splitting arises from the hybridization of the zero modes ${\psi }_A$ and ${\psi }_B$. These are shown (d) and (e) both for the ribbons and (f) and (g) the heterojunctions [see Eq. (2)].

Figure 3

(a) and (b) Magnetization in the FM ferromagnetically aligned configuration as calculated within the mean-field approximation for a graphene ribbon and a heterojunction, respectively. (c) Dependence of the exchange energy, calculated within the mean-field Hubbard model $J_{MF}=E_{FM}-E_{AF}$ on the dimensions of the graphene nanostructure and (d) scaling of $J_{MF}$ with $\frac{{\tilde{t}}^2}{\tilde{U}}$, demonstrating kinetic exchange.

Figure 4

(a) Scheme of energy levels for the two-site Hubbard model with two electrons. (b) Evolution of four lowest-energy eigenstates of Eq. (5), including the Zeeman [Eq. (6)] for the ribbon with $W=10,U=t$, and Rashba for $E=10\mathrm{V}/\mathrm{nm}$. The effect of the Rashba interaction is only apparent in the anticrossing of the $S_z=-1$ and $S=0$ states, shown in the inset. (c) Zoom of the anticrossing. (d) Magnitude of the singlet-triplet anticrossing energy $\hslash \mathrm{\Omega }$ as a function of the electric field.

Figure 5

(a) Representation of $P_2$ and the weight of the double occupancy states on the ground-state wave function for graphene ribbons as a function of $W$ (for $t=2.7\mathrm{eV}$ and $U=t$). (b) $P_2$ as a function of $U/t$ for the ribbon with $W=10$.

Figure 6

Dipolar interaction as defined in Eq. (C2) for rectangular graphene nanoribbons as a function of ribbon size $W$.

Figure 7

Influence of second-neighbor hopping (top panels) and edge atom hoppings (bottom panels) for a rectangular graphene nanoribbon with $W=10$ and $U=t$. The left panels show the relative variation of the in-gap single-particle splitting. The right panels show the variation of the exchange energy, computed within mean-field theory. In both cases, quantities are normalized by the unperturbed energy scale and expressed in terms of a percentage.