Layer-Dependent Orbital Magnetization in Graphene-Haldane Heterostructures

Rhombohedral multilayer graphene (RMG) proximity-coupled to a Haldane substrate provides a platform to investigate the interplay between band topology, layer number, and electric-field control of orbital magnetism. Using a tight-binding model and the modern theory of orbital magnetization, we study the layer-dependent magnetic response in bilayer, trilayer, and tetralayer graphene under Haldane proximity. While monolayer graphene develops a global topological gap with quantized magnetization slope, multilayer systems remain metallic due to protected low-energy bands associated with unperturbed sublattices. Despite the absence of a global gap, finite valley-contrasting Berry curvature produces non-trivial layer-dependent Chern numbers. We decompose the total orbital magnetization into self-rotation ($M_{\mathrm{SR}}$) and center-of-mass ($M_C$) contributions, revealing their distinct behaviors across doping and applied interlayer bias. In bilayer graphene, magnetization remains negative and monotonic. Remarkably, trilayer and tetralayer graphene display a bias-induced sign reversal of orbital magnetization beyond critical thresholds ($\Delta \simeq -55$ meV for 3LG, $-50$ meV for 4LG) in the hole-doped regime, a feature completely absent in the bilayer. The effect persists across both hole and electron doping, demonstrating that layer count serves as a key tuning parameter for orbital magnetism. Our findings establish topologically proximitized multilayer graphene as a versatile platform for electric-field-manipulable orbitronic and valleytronic devices.