Magnetic structure map for face-centered tetragonal iron: Appearance of a collinear spin structure

For fcc and tetragonal distorted fct iron a large number of magnetic configurations as a function of crystal structural parameters were studied by means of density functional theory concepts. The stability of magnetic structures was defined by the magnetic reorientation energy $\Delta E_{\mathrm{reor}}^i$ as the difference of the total energy of configuration $i$ and that of the fcc ferromagnetic state. The cluster expansion technique was applied to six volumes deriving $\Delta E_{\mathrm{reor}}$ for more than 90 000 collinear spin structures at each volume. Structures with low $\Delta E_{\mathrm{reor}}$ were tetragonally distorted according to a two-dimensional mesh defined by volume per atom $V$ and $c/a$ ratio. At each mesh point $\Delta E_{\mathrm{reor}}$ for all collinear structures were compared to results for spin spirals (SSs) which were calculated on a grid of propagation directions, and then the lowest $\Delta E_{\mathrm{reor}}$ defined the magnetic structure map. Three local minima were identified and for each of the minima SSs were calculated on a fine grid of propagation vectors. At the minimum with $V=10.6 \AA$${}^3$ and $0.94\le c/a\le 1.01$ a hitherto unknown simple collinear spin structure with four atoms per fct unit cell was the most stable one. It consists of two atoms with antiferromagnetically ordered local moments of $\pm 1.8{\mu }_{\mathrm{B}}$ and of two atoms with zero local moment.

Figure 1

Upper panel: Cluster expansion for collinear magnetic structures on a parent fcc lattice for up to 16 atoms per unit cell for the volume per atom $V=10.81\AA$${}^3$. Magnetic reorientation energy vs total spin polarization (in percent), $\Delta E_{\mathrm{reor}}=E\left(V\right)-E_{\mathrm{FM}}\left(V_0\right)$, as defined by the total energies $E\left(V\right)$ of each spin configuration with respect to the total energy $E_{\mathrm{FM}}\left(V_0\right)$ of fcc ferromagnetic Fe at equilibrium volume $V_0=10.52\AA$. Conspicuously, the ground state line is almost horizontal up to $\pm 50$% indicating the stability of structures with L$1_2$-like admixtures. Center panel: Collinear L$1_2$-like magnetic configuration (left) with a cubic lattice. It is unstable under tetragonal distortion by which the AFM/NM configuration (right) for $c/a<1$ is stabilized (see Fig. 2). Black spheres: Atoms with zero local moment. Lowest panel: Spin-polarized total density of states (DOS). Positive/negative values: DOS of majority/minority spin states. The DOS for L$1_2$ is asymmetric because of its antiferrimagnetic ordering, whereas for AFM/NM it is perfectly symmetric.

Figure 2

Magnetic reorientation energy $\Delta E_{\mathrm{reor}}$ as a function of $c/a$ for a variety of collinear configurations and SSs. For each point, i.e., fixed $c/a,\Delta E_{\mathrm{reor}}$ is minimized with respect to volume $V$. Symbols refer to magnetic configurations as defined in Fig. 3.

Figure 3

Symbols denote the investigated collinear orderings and SSs (see text). Collinear AFM spin structures with tetragonal distortion applied perpendicular to the sequence of alternating spin up and down FM planes are ${\mathrm{AFM}}_X$, dl-${\mathrm{AFM}}_X$; parallel to FM planes: ${\mathrm{AFM}}_Z$, dl-${\mathrm{AFM}}_Z$. Upper panel: Three-dimensional magnetic structure map defined by the lowest $\Delta E_{\mathrm{reor}}$ as a function of $c/a$ and volume $V$. Lower panel: Two-dimensional representation of upper panel. Contour lines drawn in steps of 2.38 meV/atom. Local minima are denoted by $x_1,x_2$, and $x_3$ (see text).

Figure 4

Magnetization density ${\rho }_{\mathrm{mag}}={\rho }_{\mathrm{up}}-{\rho }_{\mathrm{down}}$ (${\rho }_{\mathrm{up}},{\rho }_{\mathrm{down}}$: spin up and down charge densities) of AFM/NM ordering. Left panel: Three-dimensional mantle; right panel: contour plots in planes as sketched in left panel. Figure created by vesta [45].