Complete report - Donostia International Physics Center - Euskal ...

2011 DIPC Highlight
Strongly anisotropic anomalous Hall
effect in FePt ordered alloys
H. Zhang, F. Freimuth, S. Blugel, Y. Mokrousov, and I. Souza
Physical Review Letters 106, 117202 (2011)
Ferromagnets display a spontaneous Hall effect even in the absence of an applied magnetic field.
We have studied this "anomalous" Hall effect in FePt ordered alloys by means of first-principles
calculations, finding that its magnitude varies strongly with the orientation of the ferromagnetic
moment relative to the crystal axes. The calculations reveal that the strong anisotropy is caused
by the off-diagonal part of the spin-orbit interaction, which couples electron states with antiparallel
spins.
Energy (eV)
1
0.5
0
-0.5
-1
-1.5
-2
Γ
Figure 1. In the absence of spin-orbit, the energy bands in a ferromagnet (here, bcc
Fe) can be classified as purely up-spin (blue) or down-spin (red). The spin-orbit
interaction mixes the up- and down-spin states, leading to the rainbow-colored
states with intermediate values of the spin projection along the quantization axis.
H
The Hall effect has played a central role throughout the development of condensed-matter physics. In
its simplest form, it consists of a transverse voltage which develops in a current-carrying sample in crossed
electric and magnetic fields. This "ordinary" Hall effect occurs in all metals as a result of the Lorentz force
acting on the conduction electrons. In ferro- and ferrimagnetic metals, a Hall effect is observed even in
the absence of a magnetic field. The origin of this anomalous Hall effect (AHE) is much more subtle, and
has been a source of controversy for many decades. First-principles calculations carried out over the last
ten years have helped clarify the microscopic origin of the AHE. It is now understood that the effect can
be accounted for by a subtle modification to the velocity of the conduction electrons as they move
through the periodic crystal in the presence of the relativistic spin-orbit interaction (SOI).
Figure 2. The uniaxial crystal structure of the ordered
FePt alloy. The atoms form a face-centred cubic lattice,
with alternating layers of Fe and Pt atoms stacked in the
vertical direction.
In previous theoretical studies it was commonly assumed that most of the AHE could be ascribed to the
"spin-conserving" part of the SOI, which couples the components of the spin and orbital angular momenta
along the magnetization direction. However, in reality the electron eigenstates in the crystal are not purely
spin-up or spin-down: the nominally "up" states have in general a small "down" component, and vice-versa,
as a result of the off-diagonal ("spin-flip") part of the SOI. The spin mixing can become particularly strong
when two bands of opposite spin character cross and mix (Fig. 1). We have found that this mixed-spin
character of the electron states leads to surprising changes in the measured AHE in uniaxial crystals.
We calculated the AHC in ordered FePt alloys (Fig. 2), where the presence of the heavy Pt ions induces a
strong SOI. We first considered the lowest-energy ground state configuration, with the spontaneous magnetization
pointing along the [001] easy-axis direction. We then recalculated the AHC with the magnetization
along [100], finding that it was reduced by a factor of two. In order to understand the origin of this
large anisotropy, we redid the calculations after selectively turning off the spin-flip and spin-conserving
parts of the SOI. The surprising result was that the anisotropy is caused mainly by the spin-flip part of the
SOI, while the spin-conserving part leads to an almost perfectly isotropic Hall effect.
The spin-orbit mixing between up- and down-spin tates
in a uniaxial ferromagnet can lead to a strong dependence of the
anomalous Hall conductivity on the magnetization direction.
72 DIPC 10/11
DIPC 10/11 73