Orbital Hall conductivity in a Graphene Haldane and Haldane Haldane bilayers

We investigate the orbital Hall conductivity in bilayer graphene (G/G) by modifying one or both the layers as Haldane type ([Formula: see text] : Graphene/Haldane and [Formula: see text] : Haldane/Haldane) with the inclusion of next nearest neighbour (NNN) hopping strength ([Formula: see text]) and flux (ϕ). It is observed that the low energy bands of [Formula: see text] and [Formula: see text] are isolated with a gap at charge neutrality with the next nearest neighbour (NNN) hopping term [Formula: see text]. The time reversal (TR) symmetry breaking with [Formula: see text] induces a large orbital magnetic moment ([Formula: see text]) for the [Formula: see text] band in [Formula: see text] and [Formula: see text] bilayers. This TR symmetry breaking, modulated by the [Formula: see text] strength, leads to the emergence of Orbital Ferromagnetism and Valley Orbital Magnetism within the BZ for the Haldane single layer as well for both [Formula: see text] and [Formula: see text]. We show that for the applied longitudinal electric fields, the intrinsic angular momentum ([Formula: see text]) gives the orbital current ([Formula: see text]) 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 Orbital Chern Insulator. Interestingly, the orbital Hall conductivities are finite and exhibit a large plateau in the gap over the occupied bands. Moreover, the accumulation of orbital magnetic moment of the bands in Haldane graphene bilayer shows Orbital Hall Insulator and Orbital Chern Insulators with large plateaus. 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 orbital Hall phases.