Quantum spin Hall effect in graphene nanoribbons: Effect of edge geometry

Abstract

There has been tremendous recent progress in realizing topological insulator initiated by the proposal of Kane and Mele for the graphene system. They have suggested that the odd Z 2 index for the graphene manifests the spin filtered edge states for the graphene nanoribbons, which lead to the quantum spin Hall effect(QSHE). Here we investigate the role of the spin-orbit interaction both for the zigzag and armchair nanoribbons with special care in the edge geometry. For the pristine zigzag nanoribbons, we have shown that one of the σ edge bands located near E = 0 lifts up the energy of the spin filtered chiral edge states at the zone boundary by warping the π-edge bands, and hence the QSHE does not occur. Upon increasing the carrier density above a certain critical value, the spin filtered edge states are formed leading to the QSHE. We suggest that the hydrogen passivation on the edge can recover the original feature of the QSHE. For the armchair nanoribbon, the QSHE is shown to be stable. We have also derived the real space effective hamiltonian, which demonstrates that the on-site energy and the effective spin orbit coupling strength are strongly enhanced near the ribbon edges. We have shown that the steep rise of the confinement potential thus obtained is responsible for the warping of the π-edge bands.

Influence of electric and magnetic fields and $σ$-edge bands on the electronic and optical spectra of graphene nanoribbons

“…GNRs passivated with other atoms, such as hydrogen, oxygen, and metals on the edge, will alter their fundamental properties, as already demonstrated by various research groups 24,25 .…”

“…The parameters used in our calculations are optimized numerically, following Ref. [39], so as to reproduce the energy bands calculated previously by the first-principles method and TBM 25,26 . The application of an external perpendicular magnetic field is included in the calculations by adding an extra position-related Peierls phase in the nearest-neighbor hopping integral 28,40 .…”

“…So far, the fourorbital energy bands, including the π bands made of p z and σ bands made of (s, p x , p y ), have been reported in the literature. 25,26 In addition to the π bands, the σ bands located near E = 0 are also important in our understanding of the low-energy physics of GNRs. As far as we know, there is still insufficient study of the magnetic quantization and magneto-optical properties of four-orbital GNRs.…”

Intrinsic half-metallicity of the Stone-Wales edge reconstructed graphene nanoribbons

“…Upon taking into account the Coulomb interaction, ZGNR can also exhibit a variety of novel properties such as the edge ferromagnetism and the lateral bias‐controlled half‐metallicity. Their properties strongly depend on edge geometry and hence there have been extensive studies on the electronic and magnetic properties of GNR . The GNR had been first proposed theoretically to generate the finite band gap for graphene, which has been one of the main obstacles to the semiconductor applications of graphene.…”

Quantum spin Hall phase in multilayer graphene

“…When SOC is added to the model, we expect that this s-p z mixing naturally leads to spin-mixing terms in the Hamiltonian, which is indeed the case [39]. The presence of this spin-flip channel interaction casts doubt on the validity of the spin-conserving Kane-Mele model for multilayers [40,41] and motivates our choice of the standard [2,[42][43][44][45][46] four-orbital tight-binding calculations.…”