Magnetic structure of monatomic Fe chains on Re(0001): Emergence of chiral multispin interactions

We present results of first-principles calculations of the magnetic properties of Fe chains deposited on the Re(0001) surface. By increasing the length of the chain, a transition is found from an almost collinear antiferromagnetic state for a five-atom-long chain to a spin spiral state with the rotational plane slightly tilted from the surface of the substrate for the 15-atom-long chain. It is shown that a classical spin model derived from the ab initio calculations containing only two-spin interactions supports opposite chirality of the spin spiral compared to a direct optimization of the spin configuration within the ab initio method. The differences between the results of the two methods can be understood by introducing chiral four-spin interactions in the spin model.

Figure 1

Nearest-neighbor (NN) and next-nearest-neighbor (NNN) isotropic interactions $J_{ij}^I$ in the Fe chains of three different lengths. The interaction strengths between pairs of atoms connected by bonds are presented in the yellow boxes in units of meV. Positive and negative signs correspond to FM and AFM couplings, respectively.

Figure 2

(a) Isotropic couplings in the 15-atom-long Fe chain between the atom at the middle of the chain indexed by 8 (see the bottom panel of Fig. 1) and the Fe atoms at positions $8+j$ ($j=2,\cdots ,7$). (b) Fourier transform of the isotropic couplings, Eq. (13), where $k$ labels the number of neighbors taken into account in the sum.

Figure 3

Side view of the NN and NNN DM vectors ${\vec{D}}_{ij}$ from Eq. (6) in the Fe chains of three different lengths. With respect to the direction parallel to the chains, the arrows are placed at the centers of the lines between the corresponding pairs of Fe atoms, where at the left side of the line is the atom number $i$ and at the right side is the atom number $j$ in the vector ${\vec{D}}_{ij}$. The lengths of the arrows are scaled according to the magnitude of the DM vectors. The numerical values of the components of the DM vectors for the 15-atom-long chain can be found in Table 1.

Figure 4

Site-resolved anisotropy vectors according to Eq. (10). The length of the arrows scales with the magnitude of the anisotropy vectors. The largest magnitude of ${\vec{k}}_i$ is 7.79, 9.03, and 8.92 meV for the five-, 10-, and 15-atom-long chains, respectively.

Figure 5

Top view of the ground state spin configurations of the chains obtained for different chain lengths and by using different calculation methods: (a) 5 Fe, spin model, (b) 10 Fe, spin model, (c) 15 Fe, spin model, (d) 15 Fe, homogeneous spin spirals, and (e) 15 Fe, ab initio spin dynamics. The $z$ component of the normalized spin vectors is visualized by color coding according to the color bar below the figures.

Figure 6

(a) The angles between NN spins ${\phi }_i$, as well as (b) the $y$ and (c) the $z$ components of the chirality vectors ${\vec{\chi }}_i$ in the 15-atom-long Fe chain for the three different calculation methods; blue circles: spin model, green triangles: ab initio spin dynamics, red horizontal lines: homogeneous spin spirals.

Figure 7

Energy per atom of different spin spiral configurations according to Eq. (16), calculated within the MFT for the 15-atom-long chain. $\delta$ denotes the NN angle of the spiral with respect to the collinear AFM state and $\alpha$ is the tilting angle from the $xy$ plane. The minimum, set as the zero level of the energy, is found at $\delta ={24}^{\circ }$, $\alpha ={12}^{\circ }$.

Figure 8

Illustration of the states for determining chiral interactions from MFT calculations. In each configuration $C_1\text{−}{C}_4$, a global rotation of the spins around the $y$ direction is performed, described by the vectors $\vec{e}=\left(sin\theta ,0,cos\theta \right)$ and ${\vec{e}}_{\perp }=\left(cos\theta ,0,-sin\theta \right)$.

Figure 9

Illustration of the rotation of the spin direction ${\vec{e}}_i$ around the perpendicular vectors ${\vec{e}}_{1i}$ and ${\vec{e}}_{2i}$ by angles ${\beta }_{1i}$ and ${\beta }_{2i}$, used in the torque method to calculate derivatives of the grand potential.