Distorted $3Q$ state driven by topological-chiral magnetic interactions

We predict the occurrence of unexpected magnetic ground states in ultrathin Mn films due to the frustration of higher-order interactions. Based on density functional theory we show that significant chiral-chiral interactions occur in hexagonal Mn monolayers due to large topological orbital moments which interact with the emergent magnetic field. Due to the competition with biquadratic and four-spin interactions superposition states of spin spirals such as the $2Q$ state or a distorted $3Q$ state arise. Simulations of spin-polarized scanning tunneling microscopy images suggest that the distorted $3Q$ state could be the magnetic ground state of a Mn monolayer on Re(0001).

Figure 1

Spin structures for a hexagonal Mn monolayer along the continuous path given by Eq. (1) from (a) the $2Q$ state via (d) the $3Q$ to (h) the $1Q$ state. Yellow arrows denote magnetic moments which rotate in the $xy$ plane (shown in the coordinate system above), red arrows denote moments in the positive $z$ direction, green arrows in the negative $z$ direction. The angle $\alpha$ is applied to rotate $\pm z$ spins into the $-x$ direction and $\pm y$ spins to the $+x$ direction. (a) $2Q$ state with $\alpha =0^{\circ }$, (b) $\alpha ={12}^{\circ }$, (c) $\alpha ={24}^{\circ }$, (d) $3Q$ state with $\alpha \sim {35}^{\circ }$, (e) $\alpha ={50}^{\circ }$, (f) $\alpha \sim {60}^{\circ }$, (g) $\alpha ={75}^{\circ }$, (h) $1Q$ state (RW-AFM) with $\alpha ={90}^{\circ }$.

Figure 2

(a) Energy along the $2Q\text{−}3Q\text{−}1Q$ path given by Eq. (1) in hexagonal Mn unsupported monolayers (UMLs) for different in-plane lattice constants. Symbols denote total energies calculated from DFT using the fleur code, the lines show the fit to higher-order exchange interaction terms. The two insets show the contribution of the fourth and sixth order terms to the fit. Red color shows the result for $a=4.6$ a.u., orange and blue color for the theoretical and experimental in-plane lattice constant of Cu(111), respectively, and green color the values for the theoretical in-plane lattice constant of Re(0001) [27]. (b) Absolute value of the orbital moment per Mn atom along the direction perpendicular to the Mn UML. Symbols denote DFT values and lines the fit to the scalar spin chirality (see text for details). (c) Orbital moment of the $3Q$ state vs the absolute value of the sixth-order energy contributions at the $1Q$ with respect to the $3Q$ state [cf. lower inset of (a)].

Figure 3

Energy along the continuous $2Q\text{−}3Q\text{−}1Q$ path given by Eq. (1) for (a) Mn/Cu(111) and (b) hcp-Mn/Re(0001). Filled circles are energies calculated from DFT, the lines show the fit to higher-order-exchange interaction (insets show fourth and sixth order terms). For Mn/Cu(111) DFT calculations were performed using the vasp code and for hcp-Mn/Re(0001) both the fleur (squares) and the vasp codes (circles) were applied. (c) Absolute value of the orbital moment per Mn atom directed perpendicular to the surface. Symbols denote DFT values and lines the fit to the scalar spin chirality.

Figure 4

Simulated spin-polarized scanning tunneling microscopy images at constant height (0.8 nm) for the $2Q$, the $3Q$, and the distorted $3Q$ (${60}^{\circ }$) state (left, middle, and right row, respectively) for three different tip magnetization directions as indicated in the upper right corner of every image. The simulations have been performed using the model described in Ref. [34]. All panels have the same color scale of 1.5 pm from black to white.