Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation

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Xinjie Wang,¹ Jonathan R. Yates,^2,3 Ivo Souza,^2,3 and David Vanderbilt¹
¹Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
²Department of Physics, University of California, Berkeley, California 94720, USA
³Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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Received 10 August 2006; published 21 November 2006

Theintrinsic anomalous Hall conductivity in ferromagnets depends on subtle spin-orbit-inducedeffects in the electronic structure, and recent ab initio studiesfound that it was necessary to sample the Brillouin zoneat millions of k-points to converge the calculation. We presentan efficient first-principles approach for computing this quantity. We startout by performing a conventional electronic-structure calculation including spin-orbit couplingon a uniform and relatively coarse k-point mesh. From theresulting Bloch states, maximally localized Wannier functions are constructed whichreproduce the ab initio states up to the Fermi level.The Hamiltonian and position-operator matrix elements, needed to represent theenergy bands and Berry curvatures, are then set up betweenthe Wannier orbitals. This completes the first stage of thecalculation, whereby the low-energy ab initio problem is transformed intoan effective tight-binding form. The second stage only involves Fouriertransforms and unitary transformations of the small matrices setup inthe first stage. With these inexpensive operations, the quantities ofinterest are interpolated onto a dense k-point mesh and usedto evaluate the anomalous Hall conductivity as a Brillouin zoneintegral. The present scheme, which also avoids the cumbersome summationover all unoccupied states in the Kubo formula, is appliedto bcc Fe, giving excellent agreement with conventional, less efficientfirst-principles calculations. Remarkably, we find that about 99% of theeffect can be recovered by keeping a set of termsdepending only on the Hamiltonian matrix elements, not on matrixelements of the position operator.

URL:	http://link.aps.org/doi/10.1103/PhysRevB.74.195118
DOI:	10.1103/PhysRevB.74.195118
PACS:	71.15.Dx; 71.70.Ej; 71.18.+y; 75.50.Bb 71.15.Dx Computational methodology (condensed matter electronic structure) including Brillouin zone sampling, iterative diagonalization, pseudopotential construction 71.70.Ej Spin–orbit coupling, Zeeman and Stark splitting, Jahn–Teller effect (condensed matter) 71.18.+y Fermi surface: calculations and measurements; effective mass, -g factor 75.50.Bb Ferromagnetism of Fe and its alloys YEAR: 2006
KEYWORDS:	iron, ferromagnetic materials, ab initio calculations, Hall effect, Wannier functions, spin-orbit interactions, Brillouin zones, Fermi level, Fourier transforms

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Erratum

Erratum: Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation [Phys. Rev. B 74, 195118 (2006)]
Xinjie Wang, Jonathan R. Yates, Ivo Souza et al.
Phys. Rev. B 76, 169902 (E) (2007)