Orbital Hall Conductivity in a Graphene/Haldane and Haldane/Haldane Bilayers

Abstract

We investigate the orbital Hall conductivity in bilayer graphene (G/G) by modifying one or both the layers as Haldane type ($G/\tilde{\mathrm{G}}$ : Graphene/Haldane and $\tilde{\mathrm{G}}/\tilde{\mathrm{G}}$ : Haldane/Haldane) with the inclusion of next nearest neighbour (NNN) hopping strength ($t_2$) and flux ($\varphi$). It is observed that the low energy bands of $G/\tilde{\mathrm{G}}$ and $\tilde{\mathrm{G}}/\tilde{\mathrm{G}}$ are isolated with a gap at charge neutrality with the next nearest neighbour (NNN) hopping term $t_2{e}^{\pm i\varphi }$. The time reversal (\textit{TR}) symmetry breaking with $t_2{e}^{\pm i\varphi }$ induces a large orbital magnetic moment ($\vb{m}_n(\vb{k})$) for the $n^{th}$ band in $G/\tilde{\mathrm{G}}$ and $\tilde{\mathrm{G}}/\tilde{\mathrm{G}}$ bilayers. This \textit{TR} symmetry breaking, modulated by the $t_2$ strength, leads to the emergence of {\it Orbital Ferromagnetism} and {\it Valley Orbital Magnetism} within the BZ for the Haldane single layer as well for both $G/\tilde{\mathrm{G}}$ and $\tilde{\mathrm{G}}/\tilde{\mathrm{G}}$. We show that for the applied longitudinal electric fields, the intrinsic angular momentum ($L^z$) gives the orbital current (${\mathcal{J}}^{z,orb}$) along a transverse direction and generates the orbital Hall conductivity (OHC). We further show that the orbital magnetic polarity leads the Haldane single layer to {\it Orbital Chern Insulator} with the quantized OHC in the gap over the occupied bands. Moreover, the accumulation of orbital magnetic moment of the bands in Haldane graphene bilayer shows {\it Quantum Orbital Hall Insulator} and {\it Orbital Chern Insulators}. Similarly, we show that in the hetero-bilayers, one of the layers of the Haldane type generates the orbital magnetism and induces the OHC. We conclude that the isolated bands in Haldane graphene bilayers with external stimuli are of an orbital nature and have various quantum orbital Hall phases.

References in corpus (23)

• The electronic properties of graphene
• Graphene Bilayers with a Twist
• Orbital magnetization in periodic insulators
• Numerical studies of confined states in rotated bilayers of graphene
• Quantum Theory of Orbital Magnetization and its Generalization to Interacting Systems
• Orbitronics: Orbital Currents in Solids
• Flatbands in twisted double bilayer graphene
• Orbital Hall effect as an alternative to valley Hall effect in gapped graphene
• Imaging Chern mosaic and Berry-curvature magnetism in magic-angle graphene
• Large Quantum Anomalous Hall Effect in Spin-Orbit Proximitized Rhombohedral Graphene
• Intrinsic orbital moment and prediction of a large orbital Hall effect in the 2D transition metal dichalcogenides
• Tunable quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene
• Orbital Hall effect in bilayer transition metal dichalcogenides: From the intra-atomic approximation to the Bloch states orbital magnetic moment approach
• Evidence of orbital ferromagnetism in twisted bilayer graphene aligned to hexagonal boron nitride
• Gate tunable topological flat bands in twisted monolayer-bilayer graphene
• Orbital Hall physics in two-dimensional Dirac materials
• Topological flat bands in rhombohedral tetralayer and multilayer graphene on hexagonal boron nitride moire superlattices
• Bulk valley transport and Berry curvature spreading at the edge of flat bands
• Electron-hole asymmetry and band gaps of commensurate double moire patterns in twisted bilayer graphene on hexagonal boron nitride
• Stacking and gate tunable topological flat bands, gaps and anisotropic strip patterns in twisted trilayer graphene
• Nearly flat bands in twisted triple bilayer graphene
• Topological phases in N-layer ABC-graphene boron-nitride moire superlattices
• Electrically tunable and reversible magnetoelectric coupling in strained bilayer graphene

Cited by in corpus (1)

• Gate and Carriers tunable Valley Imbalance in Topological Proximitized Rhombohedral Trilayer Graphene