[Paper Review] Unconventional Metallic Magnetism: Non-analyticity and Sign-changing Behavior of Orbital Magnetization in ABC Trilayer Graphene

Mainak Das, Chunli Huang|arXiv (Cornell University)|Aug 3, 2023

Graphene research and applicationsMaterials Science32 references3 citations

TL;DR

This paper reveals unconventional metallic ferromagnetism in rhombohedral trilayer graphene, where orbital magnetization dominates over spin magnetization and exhibits non-analytic behavior and two sign changes due to topological Lifshitz transitions. Despite a fixed ground-state valley and spin polarization, orbital magnetization varies dramatically across the $n_e$-$U$ parameter space, driven by Berry curvature near band edges and a transition from simply connected to annular Fermi surfaces.

ABSTRACT

We study an unique form of metallic ferromagnetism in which orbital moments surpasses the role of spin moments in shaping the overall magnetization. This system emerges naturally upon doping a topologically non-trivial Chern band in the recently identified quarter metal phase of rhombohedral trilayer graphene. Our comprehensive scan of the density-interlayer potential parameter space reveals an unexpected landscape of orbital magnetization marked by two sign changes and a line of singularities. The sign change originates from an intense Berry curvature concentrated close to the band-edge, and the singularity arises from a topological Lifshitz transition that transform a simply connected Fermi sea into an annular Fermi sea. Importantly, these variations occur while the groundstate order-parameter (i.e. valley and spin polarization) remains unchanged. This unconventional relationship between the order parameter and magnetization markedly contrasts traditional spin ferromagnets, where spin magnetization is simply proportional to the groundstate spin polarization via the gyromagnetic ratio. We compute energy and magnetization curves as functions of collective valley rotation to shed light on magnetization dynamics and to expand the Stoner-Wohlfarth magnetization reversal model. We provide predictions on the magnetic coercive field that can be readily tested in experiments. Our results challenge established perceptions of magnetism, emphasising the important role of orbital moments in two-dimensional materials such as graphene and transition metal dichalcogenides, and in turn, expand our understanding and potential manipulation of magnetic behaviors in these systems.

Motivation & Objective

• To investigate the role of orbital magnetization in metallic ferromagnetism within the quarter metal phase of rhombohedral trilayer graphene.
• To explore how orbital magnetization evolves across the electron density-interlayer potential ($n_e$-$U$) parameter space while the ground-state order parameter remains unchanged.
• To identify the origin of non-analytic behavior and sign changes in orbital magnetization, linking them to topological transitions and Berry curvature.
• To extend the Stoner-Wohlfarth model to include collective valley rotation and predict magnetic coercive fields for experimental validation.

Proposed method

• Employing self-consistent mean-field theory with a Hamiltonian incorporating Slonczewski-Weiss-McClure band parameters and gate-screened Coulomb interactions.
• Using Fock self-energy to model electron-electron interactions via Fourier components $V_{\mathbf{q}} = 2\pi k_e \tanh(|\mathbf{q}|d)/(͑_r |\mathbf{q}|)$.
• Computing orbital magnetization via the Kubo formula, decomposed into bulk and edge contributions for each valley ($M_B^\tau$, $M_E^\tau$).
• Mapping the $n_e$-$U$ phase space to identify regions with simply connected Fermi seas (SFS) and annular Fermi seas (AFS), signaling Lifshitz transitions.
• Introducing a Lagrange multiplier $\hat{H}_l$ to stabilize excited states with different Fermi surface topologies and probe the first-order nature of the transition.
• Analyzing energy and magnetization curves under collective valley rotation to model magnetization reversal dynamics and extract coercive fields.

Figure 1: The orbital magnetization, $M$ , of the quarter metal displays a rich landscape across the $n_{e}-U$ parameter space, even as the ground state order parameter (i.e. valley polarization) remains unchanged. In the white region where $M$ vanishes, the groundstate is unresponsive to weak orbit

Experimental results

Research questions

• RQ1 What drives the non-analytic behavior and sign changes in orbital magnetization in the metallic phase of rhombohedral trilayer graphene?
• RQ2 How does orbital magnetization vary independently of the ground-state valley and spin polarization across the $n_e$-$U$ parameter space?
• RQ3 What is the topological origin of the observed singularities in orbital magnetization, and how is it linked to the Fermi surface topology?
• RQ4 Can the dynamics of orbital magnetization be modeled using an extended Stoner-Wohlfarth framework, and what predictions does it yield for coercive fields?
• RQ5 How does the range of Coulomb interactions influence the nature of the Lifshitz transition and the abruptness of the Fermi surface change?

Key findings

• Orbital magnetization in the quarter metal phase of rhombohedral trilayer graphene exhibits two sign changes and a line of non-analyticity in the $n_e$-$U$ parameter space, despite a fixed ground-state valley and spin polarization.
• The non-analytic behavior arises from a topological Lifshitz transition where the Fermi surface transforms from simply connected to annular, marked by a discontinuity in the chemical potential and inverse compressibility at $n_{\text{ALT}} = -3.4 \times 10^{11} \, \text{cm}^{-2}$.
• The sign change in orbital magnetization is driven by intense Berry curvature concentrated near the band edge, particularly in the minority valley, with opposing contributions from bulk and edge channels.
• The total orbital magnetization is determined by the competition between bulk and edge contributions, with the dominant term dictating the overall sign, even as the valley polarization remains constant.
• The Lifshitz transition is first-order, with a critical area for the emergence of the electron-like Fermi pocket, confirmed by energy crossing in the $\lambda = \pm 10$ state trajectories.
• Magnetic coercive fields are predicted based on valley rotation dynamics, offering a direct experimental test for the model’s validity in future transport and magnetotransport measurements.

Figure 2: $\bm{a)}$ Inverse compressibility $\partial\mu/\partial n_{e}$ v.s. $n_{e}$ exhibits a line of singularities where annular Lifshitz transition (ALT) happens. $\bm{b)}$ $\mu$ vs $n_{e}$ along the $U=43\leavevmode\nobreak\ meV$ shows a singularity at $n_{ALT}=-3.4\times 10^{11}cm^{-2}$ . $\b