Benchmarking First-Principles Approaches for Extracting Magnetic Exchange Interactions

TL;DR

The paper addresses reliable extraction of magnetic exchange parameters from first-principles for antiferromagnetic materials. It benchmarks three widely used approaches—Least-Squares Total Energy (LSTE), Four-State Total Energy (FSTE), and the Green's function-based LKAG method—across thirteen compounds and introduces an automated minimal-supercell framework for FSTE. The key findings show that LSTE and FSTE produce nearly identical dominant exchanges, while LKAG captures the main interactions but with quantitative deviations; TB2J reproduces trends but can diverge in magnitude for some cases. Overall, LSTE offers the best balance of accuracy and efficiency, FSTE provides a transparent route for targeted interactions, and the study delivers practical guidance and workflow for robust Heisenberg mapping in complex materials.

Abstract

Magnetic exchange interactions govern the macroscopic magnetic behavior of solids and underpin both fundamental spin phenomena and emerging technologies. The accurate and efficient determination of these interactions is therefore critical for predictive modeling of magnetic materials. Here we present a systematic first-principles comparison of three widely used approaches-the Least-Squares Total Energy (LSTE), the Four-State Total Energy (FSTE), and the Green's function-based Liechtenstein \textit{et al.} (LKAG) methods-applied to thirteen antiferromagnetic compounds. We introduce an framework for identifying the minimal supercells required for an accurate exchange parameter extraction in the FSTE method, significantly reducing computational cost while preserving precision. Our results show that LSTE and FSTE yield nearly identical exchange parameters, whereas the LKAG method reproduces the dominant exchange interactions but exhibits quantitative deviations. A detailed analysis of computational efficiency versus accuracy reveals that the LSTE scheme offers the most favorable balance, establishing a general, reproducible, and scalable workflow for Heisenberg mapping, while the FSTE approach remains the most straightforward for extracting specific exchange interactions.

Figures (3)

• Figure 1: Crystal structure of MgCr$_2$O$_4$, with Cr atoms labeled by numbers. (a) The conventional cell, in which only the $J_2$ exchange interaction can be calculated. Other interactions cannot be determined in this cell because, for example, if one takes (Cr1, Cr14) as a first-neighbor pair, its periodic image (Cr1+image, Cr14) appears as a fourth neighbor as well. Similar occurs for the (Cr1, Cr16) pair, preventing calculation of the $J_{3a}$ and $J_{3b}$ interactions. (b) The supercell generated and selected using the updated SUPERHEX code, which enables the calculation of exchange interactions up to the fourth-nearest neighbors. The Cr–Cr pairs whose spin reversals are employed to extract the corresponding exchange parameters are highlighted.
• Figure 2: Comparison of the exchange parameter ratios relative to the LSTE method, for the largest and the second-largest $J$ values, in (a) GGA+$U$ (b) GGA approximation. Filled symbols: same $J$ interaction identified as top coupling in both methods. Empty symbols: different $J$ interactions ranked as top couplings. $J$ annotations show the specific exchange interaction being scaled for each data point, in meV.
• Figure 3: Convergence behavior of the $J_1$ exchange parameter in BiFeO$_3$, using LSTE method within the GGA approximation in a 50-atom supercell, as a function of the number of magnetic configurations. Dots represent results obtained using all possible magnetic configurations, while square symbols correspond to the data excluding configurations with $|E(\sigma \rightarrow 0) - E| > 0.01~\text{eV}$. Here, $\sigma$ denotes the electronic smearing width used in the total energy calculations.