Magnetic materials have been successfully studied for decades
based on the Heisenberg isotropic exchange, the magnetocrystalline
anisotropy, and the shape anisotropy. These interactions accurately
describe the magnetic properties of centrosymmetric systems, i.e.
systems having structural inversion symmetry, including the great
majority of bulk metals. In non-centrosymmetric structures, on the
other hand, an additional interaction arises, known as the
Dzyaloshinskii-Moriya interaction (DMI), also referred to as
anisotropic exchange. It is a relativistic effect originated by the
interplay of spin-orbit coupling and lack of structural inversion
symmetry that can induce non-collinear magnetic order of a specific
chirality, thus breaking the mirror symmetry.
Although nanostructure surfaces intrinsically lack inversion symmetry
due to the vacuum interface, the DMI is typically neglected in their
description and its strength and actual relevance are so far unexplored.
The recent demonstration that the DMI can induce complex non-collinear
magnetic order at magnetic surfaces calls for the capability of its
estimation from first principles [1,2].
Here we present a theoretical scheme for the calculation of the DMI
within the framework of density functional theory, using the
full-potential linearized augmented plane wave method (FLAPW) [3].
It is based on the evaluation of the energy dispersion of homogeneous
spin spirals in the case that spin-orbit coupling is included in the
Hamiltonian. Assuming a small deviation from the collinear magnetic
state due to the presence of the DMI, the results are discussed in terms
of a continuum model [4], allowing to estimate the strength of DMI,
spin stiffness, and magnetic anisotropy and predict the magnetic ground
state among a huge variety of complex three-dimensional non-collinear
configurations. Explicit calculations show that DMI is sufficiently strong to
induce new magnetic phases on ultrathin films.
[1] U.K. Roessler, A.N. Bogdanov, and C. Pfleiderer, Nature 442, 797 (2006).
[2] M. Bode et al., submitted.
[3] M. Heide, et al., Psi-k, Scientific Highlight of the Month, 12-2006.
[4] I.E. Dzyaloshinskii, Sov. Phys. JETP 20, 665 (1965). |