Ab initio study of spin-spiral noncollinear magnetism in a free-standing Fe(110) monolayer under in-plane strain

We investigate the magnetic phase transition from collinear ferromagnetic (FM) ordering to noncollinear spin-spiral (SS) ordering in an Fe(110) monolayer under in-plane strain by performing fully unconstrained first-principles spin-density-functional calculations. The FM Fe(110) monolayer undergoes a FM-SS phase transition on the application of in-plane compression, whereas the application of tension keeps the system FM. The stability and wavelength of the excited SS state are further increased by compressive strains, especially along [$\overline{1}$10]. The FM-SS transition in the isotropically strained monolayer is dominated by competing exchange interactions between the ferromagnetically coupled first neighbor and the antiferromagnetically coupled second neighbor; the third neighbor also contributes to the transition under anisotropic strain. In addition, we demonstrate the stabilization mechanism of SS noncollinear magnetism from the electronic band structures: The noncollinear SS state is stabilized by a remarkable interband repulsion between the majority and minority spins, which occurs under in-plane compression.

Figure 1

(a) Simulation model of an Fe(110) monolayer. The solid black box indicates the simulation cell. (b) First Brillouin zone of an Fe(110) monolayer.

Figure 2

SS excitation energy, $\Delta E=E_{\mathit{SS}}-E_{\mathit{FM}}$, as a function of SS wave vector $\mathbf{q}$, along the $\Gamma$-S-H-$\Gamma$-N path in the Brillouin zone. Under various isotropic in-plane strains, ${\varepsilon }_{xx}={\varepsilon }_{yy}$.

Figure 3

(a) SS wave excitation energy, $\Delta E=E_{\min }-E_{\mathit{FM}}$, (b) SS wave vector $q_x$, and (c) magnitude of magnetic moment of Fe atom $m$, as functions of in-plane strains (${\varepsilon }_{xx}$, ${\varepsilon }_{yy}$). The white lines indicate the FM-SS magnetic phase transition. For $\Delta E$, the minimum energy in the Brillouin zone is taken at each strain.

Figure 4

Exchange interaction parameter of the $j$th neighbor, $J_{0j}$, in the Fe(110) monolayer as a function of isotropic in-plane strain, ${\varepsilon }_{xx}={\varepsilon }_{yy}$.

Figure 5

Exchange interaction parameters (a) $J_{01}$, (b) $J_{02}$, and (c) $J_{03}$ as functions of in-plane strains (${\varepsilon }_{xx}$, ${\varepsilon }_{yy}$). The yellow dashed lines indicate the FM-SS magnetic phase transition. Note that the contour of $J_{03}$ is confined within the smaller range than those of $J_{01}$ and $J_{02}$.

Figure 6

(a) FM-SS magnetic phase transition line calculated using the exchange parameters $J_{01}$ and $J_{02}$ (solid red line) and $J_{01}$, $J_{02}$, and $J_{03}$ (solid blue line). For comparison, the black dashed line indicates the FM-SS transition directly obtained from ab initio total-energy calculations. (b) Magnetic phase diagram based on the Heisenberg model considering $J_{01}$, $J_{02}$, and $J_{03}$. The red line indicates the critical condition of the FM-SS phase transition.

Figure 7

Schematic illustration of difference in the majority and minority spin band structures of the FM and SS states.

Figure 8

Electronic band structures of the Fe(110) monolayer with different SS wave vectors $q_x$ under in-plane strains of $({\varepsilon }_{xx}$, ${\varepsilon }_{yy})=\left(\mathrm{a}\right)$ (0.0, 0.0), (b) ($-$0.2, $-$0.2), (c) ($-$0.2, 0.0), and (d) (0.0, $-$0.2). The red and blue lines, respectively, indicate majority-spin and minority-spin bands in the collinear FM state ($q_x=0.0$).

Figure 9

Electronic band structures of the ferromagnetic Fe(110) monolayer under the in-plane strains of $({\varepsilon }_{xx}$, ${\varepsilon }_{yy})=(0.0,0.0)$ and (0.2, 0.2). The red and blue lines indicate the majority-spin and minority-spin bands, respectively.