Layer-Dependent Orbital Magnetization in Graphene-Haldane Heterostructures

Abstract

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 ($Δ\simeq -55$ 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.

Figures (6)

• Figure 1: (a)-(d) Low-energy bands along with schematic diagram of N-layer graphene (NLG) on a Haldane layer ($\tilde{G}$) at zero gate voltage in graphene layers. The blue-filled circles $H_1$ and $H_2$ in the schematic diagram represent the two distinct sublattices of the Haldane substrate layer. Intra-layer hopping within the Haldane layer is represented by $t^{\prime}_0$, and inter-layer hopping between the Haldane layer and the proximity layer of NLG is represented by $t^{\prime}_1$ and $t^{\prime}_4$. The next-nearest-neighbor (NNN) complex hopping process in the Haldane layer is represented by $t_2^{\prime}e^{i\phi}$. The gray-filled circles labeled $(A_i, B_i)$ represent the two sublattices of each graphene layer in NLG. Solid and dashed black lines represent the K and $K^{\prime}$ valley-resolved low-energy bands, respectively. The triangle, pentagon, circle, and square markers indicate the band edges of VB1 at $K^{\prime}$, VB1 at $K$, CB1 at $K^{\prime}$, and CB1 at $K$, respectively. The Chern number of the individual bands is also equal to the number of graphene layers in the system.
• Figure 2: Orbital magnetization in monolayer graphene (1LG) on a Haldane layer ($\tilde{G}$). The total orbital magnetization (black dashed line) is decomposed into contributions from self-rotation ($M_{\mathrm{SR}}$, blue line) and center-of-mass motion ($M_C$, red line). The star and dot mark the valence band maximum and conduction band minimum, respectively.
• Figure 3: Orbital magnetic moment distribution of the valence band at zero bias voltage at valleys $K$ and $K^{\prime}$ with increasing number of graphene layers (a) and with different interlayer potential bias at a $\rm{4LG/\tilde{G}}$ (b).
• Figure 4: Decomposition of orbital magnetization in (a) 2LG/$\tilde{\mathrm{G}}$, (b) 3LG/$\tilde{\mathrm{G}}$, (c) 4LG/$\tilde{\mathrm{G}}$. The self-rotation ($M_{\mathrm{SR}}$, blue), center-of-mass ($M_C$, red), and total ($M_z^{\mathrm{orb}}$, black) moments versus carrier density for biases of (i) $\Delta=-75$ meV, (ii) $\Delta=0$ meV, and (iii) $\Delta=60$ meV.
• Figure 5: Chern number variation of the low-energy valence band ($\mathrm{VB_1}$) with interlayer potential $\Delta$ (meV) in (a) 2LG/$\tilde{\mathrm{G}}$, (b) 3LG/$\tilde{\mathrm{G}}$, (c) 4LG/$\tilde{\mathrm{G}}$.