Spin exchange interaction with tunable range between graphene quantum dots

We study the spin exchange between two electrons localized in separate quantum dots in graphene. The electronic states in the conduction band are coupled indirectly by tunneling to a common continuum of delocalized states in the valence band. As a model, we use a two-impurity Anderson Hamiltonian which we subsequently transform into an effective spin Hamiltonian by way of a two-stage Schrieffer-Wolff transformation. We then compare our result to that from a Coqblin-Schrieffer approach as well as to fourth-order perturbation theory.

Figure 1

One-dimensional quantum dot array on an armchair graphene nanoribbon (drawing not to scale). Due to the ribbon structure, the dispersion relation of graphene can exhibit a gap, which scales inversely with the ribbon width. This gap allows for electrostatic confinement of electrons in quantum dots. As the band gap is small compared to regular semiconductors, the spin exchange mechanism between the quantum dots is not dominated by RKKY-type processes alone, and superexchange processes contribute significantly.

Figure 2

Several virtual tunnel processes contribute to the spin-spin interaction between the dots. These processes can be classified by the intermediate state of the system. While particle-hole excitation (a) leads to an RKKY-like interaction, processes (b–d) are usually summarized as superexchange.

Figure 3

If the quantum dot level lies close to the valence band, and the charging energy is lower than the band gap, direct tunneling processes will dominate the spin exchange between the quantum dots.