Ab-initio calculation of magnetic exchange interactions using the spin-spiral method in VASP: Self-consistent versus magnetic force theorem approaches

TL;DR

The paper tackles the problem of extracting magnetic exchange interactions from first-principles spin-spiral calculations by comparing fully self-consistent (SC) and magnetic force theorem (MFT) approaches in VASP, across Fe, Co, Ni and Mn-based Mn$_2$-type Heuslers. It maps itinerant magnetism to a classical Heisenberg model, derives the relation between spin-spiral energies, exchange parameters $J_{ij}$, and magnon dispersions $\omega(\mathbf{q})$, and estimates Curie temperatures with MFA and RPA, highlighting the limitations of MFT. The authors show that SC spin-spiral results reproduce known magnon spectra and $J_{ij}$ very well (in agreement with TDDFT and prior theory), while MFT yields large, moment-dependent inaccuracies, especially in high-moment and multisublattice systems. They further demonstrate reliable SC results for Mn-based full Heuslers, with reasonable agreement to experimental $T_c$ values where available. Overall, the work establishes self-consistency as essential for accurate spin interactions in first-principles studies and offers concrete benchmarks for future spin-dynamics investigations.

Abstract

We present an ab initio investigation of magnetic exchange interactions using the spin-spiral method implemented in the VASP code, with a comparative analysis of the self-consistent (SC) and magnetic force theorem (MFT) approaches. Using representative 3d ferromagnets (Fe, Co, Ni) and Mn-based full Heusler compounds, we compute magnon dispersion relations directly from spin-spiral total energies and extract real-space Heisenberg exchange parameters via Fourier transformation. Curie temperatures are subsequently estimated within both the mean-field and random-phase approximations. The SC spin-spiral calculations yield exchange parameters and magnon spectra in excellent agreement with previous theoretical data, confirming their quantitative reliability across different classes of magnetic systems. In contrast, the MFT approach exhibits systematic quantitative deviations: it overestimates spin-spiral energies and exchange couplings in high-moment systems such as bcc Fe and the Mn-based Heuslers, while underestimating them in low-moment fcc Ni. The magnitude of these discrepancies increases strongly with magnetic moment size, exceeding several hundred percent in the high-moment compounds. These findings underscore the decisive role of self-consistency in accurately determining magnetic exchange parameters and provide practical guidance for future first-principles studies of spin interactions and excitations using the spin-spiral technique.

Figures (4)

• Figure 1: Comparison of spin-spiral energies computed using the magnetic force theorem (MFT) and fully self-consistent (SC) calculations for bcc Fe, fcc Co, and fcc Ni. (a) Spin-spiral energy as a function of $\sin^2\theta$ at the wave vector $\mathbf{q} = (0,0,1)$, showing linear trends for both methods but substantial quantitative deviations, particularly for Fe. (b) Spin-spiral dispersion $E(\mathbf{q})$ along the high-symmetry $\Gamma$--N direction for bcc Fe and $\Gamma$--X for fcc Co and fcc Ni, calculated at a fixed cone angle of $\theta = 30^\circ$. The MFT results systematically deviate from the SC reference, with overestimations in Fe and Co and underestimations in Ni near the Brillouin zone boundary.
• Figure 2: Spin-wave dispersions along high-symmetry directions in the Brillouin zone for (a) bcc Fe, (b) fcc Co, and (c) fcc Ni. Red circles represent self-consistent spin-spiral (SS-SC) calculations from this work. These are compared with spin-spiral calculations using the magnetic force theorem (MFT) by Halilov et al.halilov1998adiabatic (solid green lines) and time-dependent density functional theory (TDDFT) results by Buczek et al.buczek2011different (blue circles).
• Figure 3: Heisenberg exchange parameters $J_{ij}$ as a function of interatomic distance $R$ (in units of the lattice constant $a$) for (a) bcc Fe, (b) fcc Co, and (c) fcc Ni. Results from our constrained self-consistent spin-spiral (SS-SC) calculations are compared with previous studies based on different methodologies: Pajda et al.pajda2001ab (real-space MFT using TB-LMTO), Jacobsson et al.jacobsson2017parameterisation (self-consistent spin-spiral or transverse-field method using the Fleur code), and Lezaic et al.levzaic2013exchange (spin-spiral method combined with the magnetic force theorem in Fleur).
• Figure 4: Heisenberg exchange parameters $J_{ij}$ as a function of interatomic distance $R$ (in units of the lattice constant $a$) for the Mn-based full Heusler compounds: (a) Cu$_2$MnAl, (b) Ni$_2$MnGa, (c) Ni$_2$MnSn, and (d) Pd$_2$MnSn. For Cu$_2$MnAl and Pd$_2$MnSn, only Mn--Mn exchange interactions are shown and compared with the results of Galanakis et al.galanakis2012ab. In the Ni-based compounds Ni$_2$MnGa and Ni$_2$MnSn, both Mn--Mn and Mn--Ni exchange parameters are displayed, with literature values taken from the multi-sublattice study of Şaşıoğlu et al.csacsiouglu2004first. All exchange interactions were obtained from our constrained self-consistent spin-spiral (SS-SC) calculations.