Environment-dependent noncollinear magnetic orders and spin-wave spectra of Fe chains and stripes

The ground-state magnetic properties of free-standing Fe wires having linear, zigzag, and rectangular geometries are investigated in the framework of density-functional theory. The stability of various wires geometries and types of magnetic orders is analyzed as a function of the spin-density-wave (SDW) number $q$. In particular, we consider ferromagnetic (FM) and antiferromagnetic (AF) collinear states as well as periodic and alternating noncollinear (NC) states. The relaxed interatomic bond lengths $d$, frozen-magnon dispersion relations $\Delta E\left(q\right)$, local magnetic moments ${\mu }_i$, spin-polarized electronic densities of states ${\rho }_{d\sigma }\left(\varepsilon \right)$, and effective exchange couplings $J_{ij}$ are determined. Iron zigzag chains show FM order ($q=0$) at the equilibrium lattice parameter $a$ with $\Delta E\left(q\right)$ increasing monotonously with $q$. In contrast, biatomic rectangular ladders (RLs) develop spontaneously a spiral SDWs along the wire, keeping FM dimer couplings in the perpendicular direction. This NC order is further stabilized by compressing the wire. However, stretched RLs show FM order. In all cases, strong ${\mu }_i$ are found within the Wigner-Seitz spheres, which decrease moderately with increasing $q$. The electronic structures of collinear and NC states are analyzed by comparing the density of states ${\rho }_{d\sigma }\left(\varepsilon \right)$ for different representative values of $q$. Effective exchange-interaction parameters $J_{ij}$ between the local moments ${\mu }_i$ and ${\mu }_j$ are derived by fitting the ab initio magnon dispersion relation $\Delta E\left(q\right)$ to a classical Heisenberg model. The interplay between the various competing $J_{ij}$ is shown to depend decisively on the lattice parameter and wire geometry. Comparing the ab initio results within a model phase diagram provides useful insights on the magnetic order of Fe wires from a local perspective.

Figure 1

Illustration of the wire structures and magnetic orders corresponding to a spin-wave vector $\vec{q}=(0,0,\pi /4a)$: (a) linear chain (LC), (b) zigzag chain with periodic noncollinear (PNC) order, (c) zigzag chain with alternating noncollinear (ANC) order, and (d) rectangular-ladder (RL) with parallel moments in the direction perpendicular to the chain. In (a) and (b), the simplest PNC spin-density wave having ${\widehat{\mu }}_i·{\widehat{\mu }}_{i+1}=\cos \left(qa\right)$ is shown, while (c) and (d) correspond to the so-called ANC order having ${\widehat{\mu }}_i·{\widehat{\mu }}_{i+1}=\cos \left(qa\right)$ for $i$ even and ${\widehat{\mu }}_i·{\widehat{\mu }}_{i+1}=1$ for $i$ odd. The lattice parameter $a$, nearest-neighbor distance $d$, and bond angle $\theta$ are indicated.

Figure 2

(a) Binding energy of zigzag (ZZ) Fe chains as a function of the lattice parameter $a$ [see Fig. 1b]. Different SDW vectors $q$ are considered as indicated in the inset. Filled symbols correspond to the periodic noncollinear order (PNC) order [see Fig. 1b], while open symbols correspond to the alternating noncollinear (ANC) order [see Fig. 1c]. (b) Binding energy of Fe rectangular ladders (RLs) as a function of the lattice parameter $a$ [see Fig. 1d].

Figure 3

Frozen-magnon dispersion relation $▵E\left(q\right)=E\left(q\right)-E\left(0\right)$ (top) and local magnetic moments $\mu$ within the Winger-Seitz (WS) cells (bottom) of Fe wires as a function of the SDW vector $q$: (a) zigzag (ZZ) chain with periodic noncollinear order (PNC), (b) ZZ chain with alternating noncollinear magnetic (ANC) order, and (c) rectangular ladders (RLs) with parallel moments in the direction perpendicular to the wire [see Fig. 1]. Results are given for representative values of the lattice parameter $a$ as indicated.

Figure 4

Local $d$-electron density of states (DOS) of free-standing zigzag (ZZ) Fe chains with periodic noncollinear (PNC) order at the equilibrium lattice parameter $a=2.52$ Å for representative SDW numbers $q$. Results are given for the majority-spin (dotted) and minority-spin (dashed) components along the direction of a local magnetic moment ${\vec{\mu }}_i$. The thin full lines show the total $d$-DOS ${\rho }_d={\rho }_{d\uparrow }+{\rho }_{d\downarrow }$. The Fermi energy ${\varepsilon }_F$ is indicated by the vertical dashed lines.

Figure 5

Local $d$-electron density of states (DOS) of free-standing zigzag (ZZ) Fe chains with alternating noncollinear (ANC) order at the equilibrium lattice parameter $a=2.52$ Å for representative SDW numbers $q$. Results are given for the majority-spin (dotted) and minority-spin (dashed) components along the direction of a local magnetic moment ${\vec{\mu }}_i$. The thin full lines show the total $d$-DOS ${\rho }_d={\rho }_{d\uparrow }+{\rho }_{d\downarrow }$. Notice that for $q=0$ the PNC and ANC orders coincide. The Fermi energy ${\varepsilon }_F$ is indicated by the vertical dashed lines.

Figure 6

Local $d$-electron density of states (DOS) of free-standing Fe rectangular ladders (RLs) at the equilibrium lattice parameter $a=2.4$ Å for representative SDW numbers $q$. Results are given for the majority-spin (dotted) and minority-spin (dashed) components along the direction of a local magnetic moment ${\vec{\mu }}_i$. The thin full lines show the total 3$d$-DOS ${\rho }_d={\rho }_{d\uparrow }+{\rho }_{d\downarrow }$. The Fermi energy ${\varepsilon }_F$ is indicated by the vertical dashed lines.

Figure 7

Effective exchange interactions $J_{0\delta }$ in Fe wires between a local moment ${\widehat{\mu }}_0$ its $\delta$th nearest neighbors ${\widehat{\mu }}_{\delta }$ as a function of $\delta$ (left column). The corresponding magnon dispersion relations $\Delta E\left(q\right)$ are shown in the right column, where the curves refer to the classical Heisenberg model fitted to the ab initio calculations (symbols). Results are shown for (a) zigzag (ZZ) chains with periodic noncollinear (PNC) order, (b) ZZ chains with alternating noncollinear (ANC) order, and (c) rectangular ladders (RL) at the indicated lattice parameters $a$ [see Fig. 1].

Figure 8

Magnetic phase diagram of the classical one-dimensional Heisenberg model with first and second NN interactions $J_{01}$ and $J_{02}$. The regions of stable antiferromagnetic (AF), ferromagnetic (FM), and spiral spin-density wave (SDW) orders are shown. The symbols indicate the effective exchange couplings in Fe wires as derived from ab initio calculations for linear monoatomic chains (dots), zigzag chains with PNC (triangles), zigzag chains with ANC order (cross), and rectangular ladders (squares). The corresponding lattice parameters $a$ are given in Å, where the equilibrium value is underlined.