[Paper Review] DFT calculations of magnetocrystalline anisotropy energy with fixed spin moment

Justyn Snarski-Adamski, Joanna Marciniak+3 more|arXiv (Cornell University)|Mar 10, 2026

Magnetic Properties of Alloys0 citations

TL;DR

The paper presents the fully relativistic fixed spin moment (FR-FSM) method to analyze magnetocrystalline anisotropy energy (MAE) and reconcile MAE results across exchange-correlation potentials, and to guide alloy design for permanent magnets.

ABSTRACT

The development of new-generation permanent magnets is based on experimental efforts and innovative theoretical tools for modeling magnetic properties. Magnetocrystalline anisotropy energy (MAE) - one of the main intrinsic properties of permanent magnets - can be calculated using density functional theory (DFT). However, MAEs determined with different exchange-correlation potentials can vary widely. We show how these seemingly contradictory results can be reconciled using the fully relativistic fixed spin moment (FR-FSM) method. This is because the equilibrium pairs [MAE, $m_s$] calculated with different exchange-correlation potentials overlap with the MAE($m_s$) curve determined from the FR-FSM method ($m_s$ denotes the spin magnetic moment). The FR-FSM method also enables the hypothetical maximum MAE value for a given material to be estimated. In the case of magnetic alloys, MAE(FSM) analysis allows the optimal alloying additions to be determined in order to improve the MAE value. Concluding, the framework we describe for MAE versus FSM calculations can be a useful tool in the design of new permanent magnets.

Motivation & Objective

• Motivate accurate MAE calculations for permanent magnets and address variability across exchange-correlation functionals.
• Introduce and apply the fully relativistic fixed spin moment (FR-FSM) method to MAE calculations.
• Show how MAE(FSM) curves reveal a functional-independent relation and a hypothetical maximum MAE for a material.
• Demonstrate applicability to alloy systems and discuss implications for designing hard magnetic materials.

Proposed method

• Implement FR-FSM within FPLO to fix the total spin magnetic moment while including full relativistic effects and spin–orbit coupling.
• Compute MAE as the energy difference between hard- and easy-m magnetization directions using the fixed-moment constraint.
• Compare MAE results obtained with different exchange-correlation potentials (BH, PW92, PBE, exchange-only, LDA variants).
• Use the tetrahedron method for Brillouin-zone integration and a fast two-step approach (scalar-relativistic self-consistent step followed by a single fully-relativistic iteration).
• Explore MAE as a function of fixed spin moment (MAE(FSM)) and as a function of alloy composition in Fe–Si–B and Fe–P–B systems via VCA.
• Discuss limitations related to spin moments across directions and computational demands of FR-FSM.

Figure 1: Overlap between MAE(FSM) results for various exchange-correlation potentials. Magnetocrystalline anisotropy energies of selected rare-earth-free magnets are denoted by red circles. We used exchange-correlation potentials in the form of: von Barth-Hedin (BH), Perdew-Wang (PW92), Perdew-Burk

Experimental results

Research questions

• RQ1 How does MAE depend on the fixed spin moment in a fully relativistic DFT framework?
• RQ2 Why do MAE values vary between different exchange-correlation functionals, and can FR-FSM reconcile these discrepancies?
• RQ3 Can MAE(FSM) define a hypothetical maximum MAE for a material and aid alloy design for enhanced magnetic hardness?
• RQ4 To what extent can MAE(FSM) analysis guide alloying strategies in Fe–Si–B and Fe–P–B systems?
• RQ5 What are the practical limitations of FR-FSM implementations across DFT codes?

Key findings

• MAE curves obtained with different xc potentials overlap when viewed as MAE(FSM) versus m_s, indicating a functional-independent trend.
• MAE values from various LDA/GGA functionals follow an equilibrium pattern but MAE can change sign and magnitude strongly with xc choice (e.g., CeFe12 example).
• MAE(FSM) analysis yields a broad, functional-independent MAE dependence that can be interpreted as a hypothetical maximum MAE for a given material.
• Alloying effects (Fe1−xCox)5SiB2 and (Fe1−xCox)5PB2 shift MAE via changes in electron count and magnetic moment; MAE(x,m) maps provide guidance for alloy optimization.
• Alignment between MAE(FSM) trends and changes due to volume and Co alloying suggests consistent qualitative behavior of MAE with magnetic moment and composition changes.
• MAE mappings can be correlated with temperature for some systems, reproducing experimental trends, and the FR-FSM method can aid understanding of MAE across temperature ranges.

Figure 2: Magnetocrystalline anisotropy energies (MAE) of (Fe 1-x Co x ) 5 SiB 2 and (Fe 1-x Co x ) 5 PB 2 systems as a function of Co concentration $x$ and total magnetic moment. Calculations were made with the FPLO14 code in the GGA-PBE approach. The Co concentration was modeled using the virtual