[Paper Review] Microscopic understanding of magnetic interactions in bilayer CrI$_3$

Seung Woo Jang, Min Yong Jeong+3 more|arXiv (Cornell University)|Sep 5, 2018

2D Materials and ApplicationsMaterials Science22 citations

TL;DR

This study provides a microscopic explanation for the interlayer magnetic coupling in bilayer CrI₃ using first-principles calculations with various van der Waals functionals and magnetic force response. It identifies the second-neighbor e_g–t_2g interaction as the dominant ferromagnetic (FM) channel in low-temperature (LT) stacking, while high-temperature (HT) monoclinic stacking suppresses this interaction, enabling stabilization of antiferromagnetic (AFM) order.

ABSTRACT

We performed the detailed microscopic analysis of the inter-layer magnetic couplings for bilayer CrI$_3$. As the first step toward understanding the recent experimental observations and utilizing them for device applications, we estimated magnetic force response as well as total energy. Various van der Waals functionals equivocally point to the ferromagnetic ground state for the low-temperature structured bilayer CrI$_3$ which is further confirmed independently by magnetic force response calculations. The calculated orbital-dependent magnetic forces clearly show that $e_g$-$t_{2g}$ interaction is the key to stabilize this ferromagnetic order. By suppressing this ferromagnetic interaction and enhancing antiferromagnetic orbital channels of $e_g$-$e_g$ and $t_{2g}$-$t_{2g}$, one can realize the desirable antiferromagnetic order. We showed that high-temperature monoclinic stacking can be the case. Our results provide unique information and insight to understand the magnetism of multi-layer CrI$_3$ paving the way to utilize it for applications.

Motivation & Objective

• To resolve the discrepancy between experimental observations of interlayer antiferromagnetic (AFM) coupling and theoretical predictions of ferromagnetic (FM) ground state in bilayer CrI₃.
• To provide a microscopic understanding of the orbital-resolved magnetic interactions governing interlayer spin order.
• To identify the key electronic mechanisms responsible for stabilizing FM or AFM order in different stacking configurations.
• To offer design principles for tuning magnetic order in 2D van der Waals magnets via structural control.

Proposed method

• Employed state-of-the-art constrained random phase approximation (cRPA) to calculate orbital-decomposed magnetic exchange interactions.
• Used multiple van der Waals functionals to assess total energy differences between FM and AFM interlayer configurations.
• Applied magnetic force response calculations to directly probe spin-spin interactions independent of total energy.
• Performed maximally localized Wannier function analysis to visualize orbital hybridization and hopping pathways.
• Compared low-temperature (R̄3) and high-temperature (C2/m) stacking structures to evaluate differences in magnetic coupling pathways.
• Quantified orbital-specific exchange coupling constants (J^eg–tg, J^eg–eg, J^tg–tg) to isolate dominant interaction channels.

Experimental results

Research questions

• RQ1 What is the microscopic origin of the interlayer ferromagnetic coupling in low-temperature-stacked bilayer CrI₃?
• RQ2 Why do experimental measurements observe antiferromagnetic order despite theoretical predictions favoring ferromagnetism?
• RQ3 How does the stacking configuration (low-temperature vs. high-temperature) alter the orbital-resolved magnetic interactions?
• RQ4 Which specific orbital channels (e_g–t_2g, e_g–e_g, t_2g–t_2g) dominate the interlayer coupling and determine the ground state spin order?
• RQ5 Can the magnetic order be tuned by modifying the interlayer bonding geometry and orbital hybridization?

Key findings

• The second-neighbor e_g–t_2g interaction is the primary source of ferromagnetic coupling in low-temperature-stacked bilayer CrI₃.
• This e_g–t_2g interaction is significantly suppressed in high-temperature monoclinic stacking, decreasing to ~22% of its LT value due to a bond angle increase from 106° to 136°.
• The total interlayer exchange coupling becomes antiferromagnetic in the HT phase, with a net J_z value that is negative, consistent with total energy calculations.
• The third-neighbor coupling becomes sizable and antiferromagnetic in the HT phase, primarily due to enhanced e_g–e_g interaction.
• Magnetic force response calculations independently confirm the ferromagnetic ground state in LT stacking, validating the total energy results.
• The weak interlayer coupling (~0.1 meV) arises from a two-step hopping process via I p orbitals, which is much weaker than conventional superexchange.