Role of Dzyaloshinskii-Moriya interaction for magnetism in transition-metal chains at Pt step edges

We explore the emergence of chiral magnetism in one-dimensional monatomic Mn, Fe, and Co chains deposited at the Pt(664) step edge carrying out an ab initio study based on density functional theory (DFT). The results are analyzed employing several models: (i) a micromagnetic model, which takes into account the Dzyaloshinskii-Moriya interaction (DMI) besides the spin stiffness and the magnetic anisotropy energy, and (ii) the Fert-Levy model of the DMI for diluted magnetic impurities in metals. Due to the step-edge geometry, the direction of the Dzyaloshinskii vector ($\mathrm{D}$ vector) is not predetermined by symmetry and points in an off-symmetry direction. For the Mn chain we predict a long-period cycloidal spin-spiral ground state of unique rotational sense on top of an otherwise atomic-scale antiferromagnetic phase. The spins rotate in a plane that is tilted relative to the Pt surface by ${62}^{\circ }$ towards the upper step of the surface. The Fe and Co chains show a ferromagnetic ground state since the DMI is too weak to overcome their respective magnetic anisotropy barriers. An analysis of domain walls within the latter two systems reveals a preference for a Bloch wall for the Fe chain and a Néel wall of unique rotational sense for the Co chain in a plane tilted by ${29}^{\circ }$ towards the lower step. Although the atomic structure is the same for all three systems, not only the size but also the direction of their effective $\mathrm{D}$ vectors differ from system to system. The latter is in contradiction to the Fert-Levy model. Due to the considered step-edge structure, this work provides also insight into the effect of roughness on DMI at surfaces and interfaces of magnets. Beyond the discussion of the monatomic chains we provide general expressions relating ab initio results to realistic model parameters that occur in a spin-lattice or in a micromagnetic model. We prove that a planar homogeneous spiral of classical spins with a given wave vector rotating in a plane whose normal is parallel to the $\mathrm{D}$ vector is an exact stationary state solution of a spin-lattice model for a periodic solid that includes Heisenberg exchange and DMI. In the vicinity of a collinear magnetic state, assuming that the DMI is much smaller than the exchange interaction, the curvature and slope of the stationary energy curve of the spiral as a function of the wave vector provide directly the values of the spin stiffness and the spiralization required in micromagnetic models. The validity of the Fert-Levy model for the evaluation of micromagnetic DMI parameters and for the analysis of ab initio calculations is explored for chains. The results suggest that some care has to be taken when applying the model to infinite periodic one-dimensional systems.

Figure 1

Step-edge structure, unit vectors, and parameters used in the text with respect to the Cartesian coordinate system $\left(x,y,z\right)$: the $\mathrm{D}$ vector, being orthogonal to the $y$ axis, points along ${\widehat{\mathrm{e}}}_{\mathrm{DM}}$ and encloses an angle ${\vartheta }_D$ with the $z$ axis. The pairwise orthogonal principal axes of the anisotropy tensor $\mathcal{K},{\widehat{\mathrm{e}}}_1,{\widehat{\mathrm{e}}}_2$, and ${\widehat{\mathrm{e}}}_3$, are associated with $K_1,K_2$, and $K_3$, respectively, where ${\widehat{\mathrm{e}}}_3$ encloses an angle ${\vartheta }_K$ with the $z$ axis. The rotation axis ${\widehat{\mathrm{e}}}_{\mathrm{rot}}$ is perpendicular to the $y$ axis and encloses an angle ${\vartheta }_r$ with the $z$ axis. $D_r$ denotes the projection of the $\mathrm{D}$ vector onto ${\widehat{\mathrm{e}}}_{\mathrm{rot}}$. ${\widehat{\mathrm{e}}}_{\parallel }$ and ${\widehat{\mathrm{e}}}_{\perp }$ are parallel and perpendicular to the $y$ axis and are associated with $K_{\parallel }$ and $K_{\perp }$, respectively, the anisotropy components within the rotation plane perpendicular to ${\widehat{\mathrm{e}}}_{\mathrm{rot}}$. The magnetization density $\mathrm{m}\left(y\right)$ varies as a function of distance along the step edge ($y$ axis) within the rotation plane (see semitransparent orange area) and encloses the spin-spiral rotation angle $\phi$ with ${\widehat{\mathrm{e}}}_{\perp }$. Note that the angles ${\vartheta }_K,{\vartheta }_D$, and ${\vartheta }_r$ are positive (negative) when pointing towards the lower (upper) terrace of the step edge.

Figure 2

Sketch of the unit cell, a slab of a (664) vicinal surface decorated with a monatomic chain along the edge, and its repetition within the $x^{\prime }y$ plane. The dark blue spheres correspond to the transition metals (Co, Fe, Mn) and the bright gray spheres represent the substrate atoms (Pt). The chain-to-chain distance is 13.24 Å and the nearest-neighbor distance within the chain is 2.82 Å. The inset in the lower left illustrates the use of the coordinates within the text. Note that the structure is periodic with respect to the $x^{\prime }y$ plane. Although in the actual calculation all quantities are referenced with respect to $\left(x^{\prime }y{z}^{\prime }\right)$, throughout this paper they are given with respect to $\left(xyz\right)$, in accordance with Fig. 1.

Figure 3

Determination of the spin stiffness. (Left) The total energies relative to their respective lowest energy, $E_{\mathrm{SS}}$, are shown as functions of the length of the wave vector $q$ for Mn, Fe, and Co chains (magenta circles, red diamonds, and blue triangles, respectively). All systems show a collinear ground state, i.e., a ferromagnetic $\left(q=0\right)$ ground state for the Co and Fe chains and an antiferromagnetic ($q=0.5$ in units of $2\pi /a_y)$ ground state for the Mn chains. (Right) The panel shows the energy as function of ${\lambda }^{-2}$ in the linear regime with the corresponding linear fits. The slope represents the spin stiffness, $A$. Note that for the Mn chain, we consider the antiferromagnetic ordering vector, meaning that $\lambda \to \infty$ leads to the AFM spin alignment. The relative error due to the linear regression is in the order of 5% to 8% for the shown data range.

Figure 4

Mn/Pt(664): (Top) the Dzyaloshinskii-Moriya energy $E_{\mathrm{DM}}$ [see Eq. (23)] in the vicinity of the AFM state as a function of the inverse wave length for flat homogeneous spirals rotating in the $yz$ plane (magenta squares and solid line) and in the $xy$ plane (magenta spheres and dashed line) for a temperature broadening of $k_{\mathrm{B}}T=27.2\mathrm{meV}$. The slopes give the values for the components of the $\mathrm{D}$ vector, $D_x$ and $D_z$ [see Eq. (2)]. (Bottom) $D_x$ and $D_z$ as functions of a broadening temperature. Since for $k_{\mathrm{B}}T=27.2\mathrm{meV}$ (dotted vertical line) the values are converged and the error bars are still reasonably small (for this system as well as for the other two the relative error due to the linear regression is in the order of 5% to 10%), the corresponding values are considered in the following.

Figure 5

The atom resolved contributions $\left\{{\mathrm{D}}^{\mu }\right\}$ to the $\mathrm{D}$ vectors are shown (a) for the Mn chain, (b) for the Fe chain, and (c) for the Co chain, extracted from the performed ab initio calculations, as well as (d) for a TM chain when applying the Fert-Levy model (see Appendix pp4). For each of the four cases, these contributions are depicted twice. In the left image, a cross section of the step-edge structure is shown with these vectors $\left\{{\mathrm{D}}^{\mu }\right\}$, for convenience, located at the corresponding atom $\mu$ (in fact, these vectors act only on the TM atoms within the chain, represented by the light blue circles). In the right image, they are given with respect to the same origin. In addition, the resulting $\mathrm{D}$ vector, i.e., the sum over atoms $\mu ,\mathrm{D}={\sum }_{\mu }{\mathrm{D}}^{\mu }$, is printed in boldface.

Figure 6

Magnetic anisotropy energy for Mn/Pt(664). The energy is plotted for magnetic moments pointing along directions discretized by the angles $\phi$ and $\vartheta$ relative to the orientation in $z$ direction. The symbols represent ab initio calculated energy differences, whereas the fit functions correspond to Eq. (27). In the case of $\phi =0^{\circ }$ (solid red line), the $xz$ plane and in the case of $\phi ={90}^{\circ }$ (dashed blue line) the $yz$ plane is sampled.

Figure 7

In the upper panels we show the values of the functions ${\mathrm{f}}_{\mathrm{crit}}^{\mathrm{hom}}$ and ${\mathrm{f}}_{\mathrm{crit}}^{\mathrm{inh}}$ for the appearance of homogeneous and inhomogeneous spin spirals [cf. Eqs. (15) and (19), respectively] in (a) Mn, (b) Fe, and (c) Co chains as functions of ${\vartheta }_r$, the direction of the rotation axis, see Fig. 1. For each system the inset shows the relative orientation of the $\mathrm{D}$ vector with respect to the principal axes of the anisotropy tensor. In the lower panels, the corresponding parameters $D_r,K_{\parallel }$, and $K_{\perp }$ are plotted. In the last two systems, the critical threshold of 1 is missed by more than one order of magnitude. For the Mn chains, however, both criteria are fulfilled, and their respective curves reach their maxima for ${\vartheta }_r=-{62}^{\circ }$. This can be seen as a compromise between finding the largest DMI contribution (dashed green line) and having the smallest anisotropy energy $K_{\perp }$ (dashed brown line).

Figure 8

Schematic visualization of the energetically preferred rotational sense for (a) the ground-state of the Mn chain (left-rotating spin-spiral) and (b) the domain wall for the Co chain (right-rotating Néel wall).

Figure 9

Dependence of the DMI energy originating form a Pt atom, as a function of the distance $d$ of the Pt atom normal to the chain. The nearest-neighbor distance is denoted by $a_y$. For the magnetic chain, we distinguish spin spirals of ferromagnetic (FM-SS) or antiferromagnetic (AFM-SS) short-range order, evaluated at a fixed $q_0=0.05\frac{2\pi }{a_y}$. Points and squares highlight the actual positions of atoms in the Pt(664) unit cell. Two different geometries need to be considered. For a description of the geometries see text.