In ferromagnetic bcc Fe, the intrinsic anomalous Hall conductivity calculated from the Berry curvature of Bloch states (about 751 S/cm within the generalized gradient approximation) differs noticeably from values extracted from transport measurements (about 1032 S/cm for iron whiskers at room temperature), and the balance between intrinsic and disorder-induced extrinsic (skew-scattering and side-jump) mechanisms in the low-temperature, high-conductivity regime was left unsettled by the original 2004 calculation. The problem is to give a predictive, parameter-free ab initio account of the anomalous Hall conductivity of bcc Fe in which the exchange-correlation treatment of the band structure, the impurity-scattering contributions, and finite-temperature effects are all handled on the same footing and validated against well-characterized experiments, thereby closing the residual quantitative discrepancy and fixing the mechanism decomposition.

Background

In a ferromagnet the Hall resistivity contains, besides the ordinary field-proportional term, an anomalous contribution proportional to the magnetization: ρH=R0B+Rs4πM\rho_H = R_0 B + R_s 4\pi M. The anomalous Hall effect (AHE) arises from spin-orbit coupling in a magnetized crystal and has been discussed since Karplus and Luttinger, who attributed it to an intrinsic, scattering-independent transverse current of Bloch electrons. Smit later argued that a periodic lattice cannot produce a Hall current and proposed the extrinsic skew-scattering mechanism, in which spin-orbit coupling makes impurities scatter spin-polarized electrons preferentially to one side, giving ρxy∝ρxx\rho_{xy}\propto\rho_{xx}; Berger proposed the side-jump mechanism, a lateral displacement of scattered electron trajectories at impurity sites, giving ρxy∝ρxx2\rho_{xy}\propto\rho_{xx}^2. The modern semiclassical framework resolves this debate: the wave-packet group velocity acquires an anomalous term −k×Ωn(k)-\mathbf{k}\times\mathbf{\Omega}_n(\mathbf{k}), where Ωn(k)=−Im ⟨∇kunk∣×∣∇kunk⟩\mathbf{\Omega}_n(\mathbf{k}) = -\mathrm{Im}\,\langle\nabla_\mathbf{k} u_{n\mathbf{k}}|\times|\nabla_\mathbf{k} u_{n\mathbf{k}}\rangle is the Berry curvature of the Bloch state, and integrating Ωn\mathbf{\Omega}_n over the occupied states of a perfect crystal yields the intrinsic anomalous Hall conductivity (AHC) σxyint\sigma_{xy}^{\mathrm{int}} (the Karplus-Luttinger term, now computable from first principles), while skew scattering and side jump appear as distinct disorder corrections to the Boltzmann transport solution.

Yao et al. (Phys. Rev. Lett. 92, 037204 (2004)) evaluated this intrinsic conductivity for bcc Fe from spin-density-functional theory with the full-potential linearized augmented plane-wave (LAPW) method and the generalized gradient approximation (GGA) for the exchange-correlation functional, fully relativistically, at the experimental lattice constant. Because spin-orbit mixing of nearly degenerate states near the Fermi level produces extremely sharp, nearly cancelling Berry-curvature features, they needed more than two million k\mathbf{k}-points in the Brillouin zone with adaptive mesh refinement, obtaining σxyint=751 S/cm\sigma_{xy}^{\mathrm{int}}=751\,\mathrm{S/cm} (siemens per centimeter, the conventional unit of conductivity) at zero temperature and 734 S/cm734\,\mathrm{S/cm} at 300 K, against 1032 S/cm1032\,\mathrm{S/cm} extracted from Dheer's iron-whisker data at room temperature. They concluded that the AHC of bcc Fe, and presumably all transition-metal ferromagnets, is primarily intrinsic, except possibly at low temperature in highly conductive samples where skew scattering (which grows with the relaxation time) gains relative weight, and attributed the remaining discrepancy to shortcomings of the GGA, neglect of scattering effects, and experimental uncertainties. Subsequent work mapped the sharp Berry-curvature hot spots of bcc Fe to Weyl points (two-fold chiral band degeneracies that act as monopole sources of Berry curvature in the Brillouin zone) carrying integer Fermi-sheet Chern numbers (topological winding indices of the Fermi-surface sheets), explaining why the computed value is so sensitive to the fine placement of bands relative to the Fermi level. For context, a unified theory of the AHE in multiband ferromagnetic metals with dilute impurities predicts an extrinsic-to-intrinsic crossover when the scattering rate becomes comparable to the spin-orbit splitting at band anticrossings, and a major review (Nagaosa et al., 2010) concluded that the intrinsic Berry-phase mechanism dominates in metallic ferromagnets of moderate conductivity while skew scattering dominates in highly conductive ones. The terminology used below: GGA and LDA (the local-density approximation) are semilocal exchange-correlation approximations of density-functional theory, LSDA is its spin-polarized counterpart, GW is a many-body quasiparticle correction to them, Wannier interpolation is a technique that transfers an ab initio band structure into a tight-binding-like form so that Brillouin-zone integrals can be evaluated on very dense meshes at low cost, and the coherent-potential approximation (CPA) is a standard scheme for configurational averaging over substitutional disorder in alloys.

Problem Statement

Starting from the 2004 result that the intrinsic, Berry-curvature-based anomalous Hall conductivity of ferromagnetic bcc Fe is approximately 751 S/cm within GGA while transport measurements on iron give values of order 1030 S/cm at room temperature, construct a predictive, parameter-free ab initio account of the anomalous Hall effect in bcc Fe that jointly satisfies all four of the following requirements: (i) treats the exchange-correlation description of the spin-orbit-split band structure at a level of fidelity beyond semilocal GGA, since the computed conductivity is known to be very sensitive to band positions relative to the Fermi level; (ii) incorporates disorder-induced extrinsic contributions, both skew scattering and side jump, on the same first-principles footing as the intrinsic Berry-curvature term, with a theoretically consistent and unambiguous decomposition of the total σxy\sigma_{xy}; (iii) accounts for the measured temperature dependence, including phonon (and, if relevant, magnon) effects; and (iv) is validated against well-characterized experiments across the conductivity range, reproducing the measured anomalous Hall conductivity within the experimental uncertainties and thereby resolving the residual quantitative discrepancy between theory and experiment and determining the regime, in particular the low-temperature high-conductivity (skew-scattering-dominated) limit, in which extrinsic mechanisms become important. The associated computational requirement, efficient and robust Brillouin-zone integration of the sharp Berry-curvature features of Fe, is part of the same objective as an enabling capability, though Wannier-interpolation and adaptive-mesh techniques have largely met it for this material.

References

  1. Yugui Yao, L. Kleinman, A. H. MacDonald, Jairo Sinova, T. Jungwirth, Ding-sheng Wang, Enge Wang, and Qian Niu, 'First Principles Calculation of Anomalous Hall Conductivity in Ferromagnetic bcc Fe', Phys. Rev. Lett. 92, 037204 (2004). DOI: 10.1103/PhysRevLett.92.037204, https://doi.org/10.1103/PhysRevLett.92.037204 (arXiv: cond-mat/0307337)
  2. P. N. Dheer, 'Galvanomagnetic Effects in Iron Whiskers', Phys. Rev. 156, 637 (1967). DOI: 10.1103/PhysRev.156.637, https://doi.org/10.1103/PhysRev.156.637
  3. Xinjie Wang, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, 'Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation', Phys. Rev. B 74, 195118 (2006). DOI: 10.1103/PhysRevB.74.195118, https://doi.org/10.1103/PhysRevB.74.195118
  4. Shigeki Onoda, Naoyuki Sugimoto, and Naoto Nagaosa, 'Intrinsic Versus Extrinsic Anomalous Hall Effect in Ferromagnets', Phys. Rev. Lett. 97, 126602 (2006). DOI: 10.1103/PhysRevLett.97.126602, https://doi.org/10.1103/PhysRevLett.97.126602
  5. Yuan Tian, Li Ye, and Xiaofeng Jin, 'Proper Scaling of the Anomalous Hall Effect', Phys. Rev. Lett. 103, 087206 (2009). DOI: 10.1103/PhysRevLett.103.087206, https://doi.org/10.1103/PhysRevLett.103.087206
  6. Naoto Nagaosa, Jairo Sinova, Shigeki Onoda, A. H. MacDonald, and N. P. Ong, 'Anomalous Hall effect', Rev. Mod. Phys. 82, 1539 (2010). DOI: 10.1103/RevModPhys.82.1539, https://doi.org/10.1103/RevModPhys.82.1539
  7. S. Lowitzer, D. Ködderitzsch, and H. Ebert, 'Coherent Description of the Intrinsic and Extrinsic Anomalous Hall Effect in Disordered Alloys on an Ab Initio Level', Phys. Rev. Lett. 105, 266604 (2010). DOI: 10.1103/PhysRevLett.105.266604, https://doi.org/10.1103/PhysRevLett.105.266604 (arXiv: 1008.2640)
  8. Jürgen Weischenberg, Frank Freimuth, Jairo Sinova, Stefan Blügel, and Yuriy Mokrousov, 'Ab Initio Theory of the Scattering-Independent Anomalous Hall Effect', Phys. Rev. Lett. 107, 106601 (2011). DOI: 10.1103/PhysRevLett.107.106601, https://doi.org/10.1103/PhysRevLett.107.106601
  9. Diemo Ködderitzsch, Kristina Chadova, Ján Minár, and Hubert Ebert, 'Impact of finite temperatures and correlations on the anomalous Hall conductivity from ab initio theory', New J. Phys. 15, 053009 (2013). DOI: 10.1088/1367-2630/15/5/053009, https://doi.org/10.1088/1367-2630/15/5/053009 (arXiv: 1211.0702; the arXiv preprint version carries the title 'Impact of correlations and finite temperatures on the anomalous Hall conductivity of 3d-transition-metals')
  10. Daniel Gosálbez-Martínez, Ivo Souza, and David Vanderbilt, 'Chiral degeneracies and Fermi-surface Chern numbers in bcc Fe', Phys. Rev. B 92, 085138 (2015). DOI: 10.1103/PhysRevB.92.085138, https://doi.org/10.1103/PhysRevB.92.085138
  11. K. Hyodo, A. Sakuma, and Y. Kota, 'First-principles evaluation of intrinsic, side-jump, and skew-scattering parts of anomalous Hall conductivities in disordered alloys', Phys. Rev. B 94, 104404 (2016). DOI: 10.1103/PhysRevB.94.104404, https://doi.org/10.1103/PhysRevB.94.104404
  12. Lei Wang, Tai Min, and Ke Xia, 'First-principles study of the anomalous Hall effect based on exact muffin-tin orbitals', Phys. Rev. B 103, 054204 (2021). DOI: 10.1103/PhysRevB.103.054204, https://doi.org/10.1103/PhysRevB.103.054204
  13. E. Młyńczak, I. Aguilera, P. Gospodarič, T. Heider, M. Jugovac, G. Zamborlini, J.-P. Hanke, C. Friedrich, Y. Mokrousov, C. Tusche, S. Suga, V. Feyer, S. Blügel, L. Plucinski, and C. M. Schneider, 'Fe(001) angle-resolved photoemission and intrinsic anomalous Hall conductivity in Fe seen by different ab initio approaches: LDA and GGA versus GW', Phys. Rev. B 105, 115135 (2022). DOI: 10.1103/PhysRevB.105.115135, https://doi.org/10.1103/PhysRevB.105.115135