Exchange interaction function for spin-lattice coupling in bcc iron

Functional representations of the spin polarization and the exchange interaction in terms of the lattice configuration is necessary to model the dynamics of the coupled spin and lattice subsystems in large-scale atomistic simulation of magnetic materials. Data needed for this purpose have only existed in the regime of small displacements from the equilibrium perfect lattice configurations. In this paper, we report and discuss the results of our first-principles calculations for bcc iron over a wide range of lattice constants using the magnetic force theorem and the one-electron Green’s function. Despite the relatively complex functional form of the exchange interaction function for bcc iron our results show that it can be expressed as a superposition of Bethe-Slater-type curves representing interatomic exchange interaction of the $3d$ electrons.

Figure 1

(Color online) Total energy as a function of lattice constant calculated with different models.

Figure 2

(Color online) Calculated MM as a function of lattice constant.

Figure 3

(Color online) The partial DOS of bcc Fe using (a) LMTO-GF and (b) WIEN2K methods for different lattice constants: (i) 2.25 (ii), 2.45, and (iii) $2.88\text{Å}$. The $t_{2\text{g}}$ and $e_{\text{g}}$ components of the $3d$-DOS are shown in (c) and (d) panels, respectively.

Figure 4

(Color online) (a) Total and [(b) and (c)] partial $3d$-DOS of bcc Fe for different lattice constants. The origin is set to its Fermi level for each lattice constant as shown in the inset of (c). The arrows show the change trend of DOS at Fermi level for spin up and spin down as decreasing the lattice constant.

Figure 5

(Color online) Charge-density map on (110) plane in bcc Fe with different lattice constants for both spin up and spin down with directions of the lattice vectors $⟨001⟩$, $⟨111⟩$, and $⟨110⟩$ also shown.

Figure 6

(Color online) (a) The number of electrons on $s$, $p$, and $d$ orbitals for spin-up and spin-down subbands, and (b) the corresponding magnetic moment component as a function of lattice constant. Here, data are obtained using LMTO-GF method.

Figure 7

(Color online) (a) The effective exchange interaction parameter $J_{ij}^e$ from the first to the eighth neighbors as a function of the lattice constant $a$. Data from Morán et al. (Ref. 53) are also presented for comparison. (b) Components $J_1$ and $J_2$ plotted as a function of the interatomic distance with $R_1=\sqrt{3}a/2$ and $R_2=a$ being, respectively, the first- and second-nearest-neighbor distances. (c) Assuming a form for $J_1$: $J_1=J_1^{t_{2g}}+J_1^{e_g}$, where $J_1^{e_g}=A_1{J}_2\left({\alpha }_{1}R\right)$ with $A_1=1.3$ and ${\alpha }_1=1.12$, a very good representation of the relatively complex form of $J_1$ can be obtained.