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 ($\rm G/ \tilde G$ : Graphene/Haldane and $\rm \tilde G/ \tilde G$ : Haldane/Haldane) with the inclusion of next nearest neighbour (NNN) hopping strength ($t_2$) and flux ($\phi$). It is observed that the low energy bands of $\rm G/ \tilde G$ and $\rm \tilde G/ \tilde G$ are isolated with a gap at charge neutrality with the next nearest neighbour (NNN) hopping term $t_2e^{\pm i\phi}$. The time reversal (\textit{TR}) symmetry breaking with $t_2e^{\pm i\phi}$ induces a large orbital magnetic moment ($\vb{m}_n(\vb{k})$) for the $n^{th}$ band in $\rm G/ \tilde G$ and $\rm \tilde G/ \tilde G$ 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 $\rm G/ \tilde G$ and $\rm \tilde G/ \tilde G$. 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.