Spins and Phonons in Graphene Nanostructures

Quantum information science relies on quantum systems with sufficiently long coherence times. A spintronics implementation in graphene would benefit from the low density of nuclear spins in carbon and small spin-orbit interactions. The coupling of the spin to lattice dynamics puts an upper bound on spin coherence times. In addition, phonons in graphene are interesting due to the monatomic thickness of the material, accompanied by the lowest surface mass density of all solid materials. The prospects for quantum information science, nanoelectromechanical systems, and for nanotechnology in general inspire us to study the theory of spins and phonons in graphene nanostructures. We give an introduction to graphene and its properties and also comment on the development of research related to graphene and other two-dimensional materials like hexagonal boron nitride (hBN). We review the basic electronic properties of monolayer graphene and hBN in detail. Klein’s paradox is most important for the confinement of charge carriers in graphene. Bilayer systems, in particular a graphene/hBN heterostructure, are also discussed. We use the continuum model to derive electron-phonon couplings, spin-phonon coupling, the change of the Fermi velocity under external loading, and the deformation of the Brillouin zone under uniform strain. Parts of our studies rely on the modern theory of polarization or on Peierls’ phase. To this end, we also demonstrate the necessary concepts of the geometric phase. We use the continuum model to derive the acoustic phonons in graphene nanoribbons for fixed as well as for free lateral boundaries. In-plane and out-of-plane deformations are treated separately. Fixed boundaries lead to gapped phonon dispersions and free boundaries to gapless ones. As expected, our results are in accordance with previous results for bulk graphene if the phonon wavenumber is much shorter than the ribbon width. Building on these results, we calculate the electron spin relaxation in armchair graphene nanoribbon quantum dots. The combination of spin-orbit interaction and electron-phonon coupling to in-plane phonons yields an effective spin-phonon coupling. Out-of-plane phonons are also considered but give no lowest order contribution to the spin relaxation rate T−1 1 . We find Van Vleck cancellation and interference effects of the deformation potential and the bond-length change. As a result, spin relaxation times can exceed the range of seconds, which would be most suitable for graphene-based quantum computing. For magnetic fields below B=0.5 T, free mechanic boundaries lead to a relaxation rate that scales as T−1 1 ∝B. For fixed boundaries, the gapped phonon spectrum bears the potential to suppress spin relaxation in lowest order. The bond-length change manifests itself in the low-energy description of hBN, as for graphene, via a strain-induced pseudomagnetic gauge field. We evolve the sublattice potential adiabatically from its value for graphene to that of boron nitride and use the modern theory of polarization to calculate the piezoelectric effect of hBN. We find that all symmetry constraints are met and we provide an estimate of 3 eV for the so far unknown coupling strength of the strain-induced gauge field in hBN. The resulting values