Interplay between magnetism and energetics in Fe-Cr alloys from a predictive noncollinear magnetic tight-binding model

Magnetism is a key driving force controlling several thermodynamic and kinetic properties of Fe-Cr systems. We present a tight-binding model for Fe-Cr, where magnetism is treated beyond the usual collinear approximation. A major advantage of this model consists in a rather simple fitting procedure. In particular, no specific property of the binary system is explicitly required in the fitting database. The present model is proved to be accurate and highly transferable for electronic, magnetic, and energetic properties of a large variety of structural and chemical environments: surfaces, interfaces, embedded clusters, and the whole compositional range of the binary alloy. The occurrence of noncollinear magnetic configurations caused by magnetic frustrations is successfully predicted. The present tight-binding approach can apply to other binary magnetic transition-metal alloys. It is expected to be particularly promising if the size difference between the alloying elements is rather small and the electronic properties prevail.

Figure 1

Total energy (per atom) as a function of the atomic volume for various crystallographic structure [body-centered cubic (bcc), face-centered cubic (fcc), and hexagonal closed pack (hcp)] of Fe (left) and Cr (right). Several magnetic solutions are considered: nonmagnetic (NM), ferromagnetic (FM), simple-layer antiferromagnetic (AF), double-layer antiferromagnetic (AFD).

Figure 2

Two-center Slater-Koster $d$ hopping (left) and overlap (right) integrals as a function of the interatomic distance $r$ between two Fe-Fe, Cr-Cr, and Fe-Cr atoms. The Fe-Cr hopping and overlap integrals are rescaled by a factor $1.023$ with respect to the arithmetic average.

Figure 3

Variation of the excess spin magnetization per atom (with respect to the bulk value) decomposed on successive atomic layers for the $\left(001\right)$ and $\left(110\right)$ surfaces of Fe.

Figure 4

Variation of the excess spin magnetization per atom (with respect to the bulk value) decomposed on successive atomic layers for the $\left(001\right)$ and $\left(110\right)$ surfaces of Cr. Note the particularly strong enhancement of magnetization at the outermost layer of $\mathrm{Cr}\left(001\right)$.

Figure 5

Excess local moments with respect to the corresponding bulk value ($M^{\text{bulk,Cr}}=\pm 1{\mu }_B$ and $M^{\text{bulk,Fe}}=2.44{\mu }_B$) across the Fe-Cr $\left(001\right)$ interface.

Figure 6

Top: noncollinear magnetic configuration in the vicinity of the Fe-Cr $\left(110\right)$ interface. Bottom: excess local moments with respect to the corresponding bulk value $(M^{\text{bulk,Cr}}=\pm 1{\mu }_B$ and $M^{\text{bulk,Fe}}=2.44{\mu }_B)$ across the Fe-Cr $\left(110\right)$ interface for a collinear and and noncollinear magnetic solution. We note a rather modest canting of the magnetic moments of chromium and iron in the vicinity of the interface.

Figure 7

Left: four-atom magnetic unit cell used for our TB calculations of the Fe-Cr B2 system. Right: local Fe and Cr moments versus the bcc lattice parameter $a_{\text{bcc}}$ for the various magnetic phases. For the FM/AF phase, the magnetization of chromium is switching sign at 2.88 Å.

Figure 8

Calculated (TB: left, siesta: right) total energies (per formula unit) versus the bcc lattice parameter $a_{\text{bcc}}$ for the various magnetic phases. From siesta calculations the AF-FMD structure cannot be obtained for lattice parameters smaller than 2.89 Å. For better comparison, the minimum of the FM/AM curve has been set to zero in both calculations (TB and siesta).

Figure 9

Calculated (TB: left, siesta: right) spin-polarized density of states of states projected on the chromium (red line) and iron (black line) atomic orbitals for the Fe-Cr B2 FM/AF structure at the equilibrium lattice parameter $2.86$ Å. Note that for this lattice constant, Fe and Cr both have positive magnetization.

Figure 10

Calculated (TB: left, siesta: right) spin-polarized band structure for the Fe-Cr B2 FM/AF structure at the equilibrium lattice parameter $2.86$ Å. Note that for this lattice constant, Fe and Cr both have positive magnetization.

Figure 11

Enthalpy of mixing as a function of the concentration $c$ of Cr for the ${\mathrm{Fe}}_{1-c}{\mathrm{Cr}}_c$ alloy evaluated using our TB model compared to siesta results. For each concentration $c$ we have used a sqs structure and only plotted the lowest-energy solution when several magnetic configurations were found (essentially in the high-concentration region). A few ordered structures were also calculated.