Dzyaloshinskii-Moriya interaction and chiral magnetism in $3d-5d$ zigzag chains: Tight-binding model and ab initio calculations

We investigate the chiral magnetic order in freestanding planar $3d-5d$ biatomic metallic chains $(3d$: Fe, Co; $5d$: Ir, Pt, Au) using first-principles calculations based on density functional theory. We find that the antisymmetric exchange interaction, commonly known as the Dzyaloshinskii-Moriya interaction (DMI), contributes significantly to the energetics of the magnetic structure. For the Fe-Pt and Co-Pt chains, the DMI can compete with the isotropic Heisenberg-type exchange interaction and the magnetocrystalline anisotropy energy, and for both cases a homogeneous left-rotating cycloidal chiral spin-spiral with a wavelength of 51 Å and 36 Å, respectively, was found. The sign of the DMI, which determines the handedness of the magnetic structure, changes in the sequence of the $5d$ atoms $\mathrm{Ir}(+)$, $\mathrm{Pt}(-)$, $\mathrm{Au}(+)$. We use the full-potential linearized augmented plane wave method and perform self-consistent calculations of homogeneous spin spirals, calculating the DMI by treating the effect of spin-orbit interaction in the basis of the spin-spiral states in first-order perturbation theory. To gain insight into the DMI results of our ab initio calculations, we develop a minimal tight-binding model of three atoms and four orbitals that contains all essential features: the spin canting between the magnetic $3d$ atoms, the spin-orbit interaction at the $5d$ atoms, and the structure inversion asymmetry facilitated by the triangular geometry. We find that spin canting can lead to spin-orbit active eigenstates that split in energy due to the spin-orbit interaction at the $5d$ atom. We show that the sign and strength of the hybridization, the bonding or antibonding character between $d$ orbitals of the magnetic and nonmagnetic sites, the bandwidth, and the energy difference between occupied and unoccupied states of different spin projection determine the sign and strength of the DMI. The key features observed in the trimer model are also found in the first-principles results.

Figure 1

Structure of the $3d-5d$ transition metal chains. The lattice parameter $a$ denotes the equilibrium bond length between two consecutive $3d$ $\left(5d\right)$ atoms. $d$ represents the distance between the $3d-5d$ atoms and $\alpha$ is the angle spanned by the $5d-3d-5d$ atoms.

Figure 2

Total energies (upper panel) and magnetic spin moments (lower panel) of Fe- and Co-$5d$ chains plotted as a function of lattice constants $a$ [i.e., $E(a,d_o)$ and $M(a,d_o)]$ for optimal $d_o$ at given $a$. The results for Fe-$5d$ chains are shown by red lines with triangles, whereas the blue lines with squares denote results for Co-$5d$ chains.

Figure 3

The dispersion energy of flat spirals for Fe-$5d$ (left) and Co-$5d$ chains (right) without SOI as a function of the spin-spiral vector $q$ calculated using the LDA-VWN functional. Because of the symmetry $E_0\left(q\right)=E_0(-q)$, we show only the dispersion energies for $q>0$. The inset shows the magnified region near the ferromagnetic state $\left(q=0\right)$.

Figure 4

The isotropic exchange-interaction energy $E_0\left(q\right)$ (red circles), the Dzyaloshinskii-Moriya interaction $E_{\mathrm{DM}}\left(q\right)$ (blue triangles), and the sum of both (black squares) for the various $3d-5d$ chains as a function of the spin-spiral vector $q$ is displayed, calculated using LDA-VWN functional. The inset in each figure shows the magnified view in the vicinity of ferromagnetic state.

Figure 5

The relativistic electronic band structures, i.e., calculated including the SOI, of biatomic (a) Fe-$5d$ chains and (b) Co-$5d$ chains are shown for a spin-spiral state with a wave vector of $q=0.15$. The blue (red) dots indicate a positive (negative) shift of the spectral energies $\delta {\epsilon }_{k\nu }$, i.e., towards higher (lower) single particle energies, with respect to the scalar-relativistic eigenvalues, i.e., calculated without SOI. For visual clarity, the radius of the dot represents the actual difference in eV magnified 140 times. The highlighted rectangle shows the area in the band structure exhibiting the maximum SOI effect. The Fermi energy was chosen as origin of the energy scale. The fourth panel shows the energy-resolved DMI contribution $e_{\mathrm{DM}}\left(\epsilon \right)$, separated into a curve originating from positive (blue) and negative (red) shifts only, for the $3d$-Au chains. In the fifth panel, the sum of the two and the effective energy-integrated DMI energy $E_{\mathrm{DM}}\left(\epsilon \right)$ is shown.

Figure 6

Band structure without SOI (labeled as “scalar”) of the biatomic Fe-Au chain for $q=0.15$, and analyzed in terms of the orbital character on the Fe and Au site. The Fermi energy, $E_F$, is chosen as origin of energy scale. In this figure, we have shown only $s$- and $d$-projected orbitals. The majority (minority) states are highlighted by blue (red) dots. The radius of the dots at each point in the band structure $\left(k\nu \right)$ is proportional to the respective orbital character. Additionally, also the relativistic band structure including the shift of the spectral energies due to SOI is shown in the lower left panel. The highlighted rectangle shows the area in the band structure where maximum SOI effect is observed.

Figure 7

The structure of the trimer. It consists of two magnetic sites A and B represented by blue spheres with a $d_{xz}$ orbital and one nonmagnetic site C represented by a brown sphere with the $d_{xz}$ and $d_{yz}$ orbital. The magnetic moments of the magnetic sites are canted by an angle $\phi$ with respect to the $x$ axis, in which the angle for the magnetic moment of the first/second site is defined in a mathematically positive/negative rotation sense [see also Eq. (10)]. We denote this magnetic structure as right-rotating with a chirality vector $\mathit{c}={\mathit{S}}_{\mathbf{A}}\times {\mathit{S}}_{\mathbf{B}}=-{\mathit{e}}_{\mathbf{z}}$.

Figure 8

The site-, orbital- and spin-resolved density of states (DOS) of the trimer is displayed for two magnetic configurations (a) $\phi =0^{\circ }$ and (b) $\phi ={45}^{\circ }$. In each case, the 8 eigenenergies are displayed in an upper panel above the respective DOS. The upper (lower) panel of the spin-resolved DOS describes the majority (minority) states with respect to the spin-quantization axis chosen along the $x$-direction. The blue curve displays the DOS of the two magnetic sites in the $d_{xz}$-orbital (denoted as $d_{xz}^{\mathrm{A}}+d_{xz}^{\mathrm{B}})$ and the brown solid, dashed curve represents the DOS of the $d_{xz}$- and $d_{yz}$-orbital of the non-magnetic site (denoted as $d_{xz}^{\mathrm{C}}$ and $d_{yz}^{\mathrm{C}})$. Note that the DOS of the non-magnetic site has been enhanced by a factor of 5. In the case of $\phi ={45}^{\circ }$, $e_{\mathrm{DM}}$ (red solid curve) and the integrated value $E_{\mathrm{DM}}$ (black dashed curve) is displayed in the lower panel. More information on these two quantities and details about the two eigenenergies 1 and 2 and the corresponding DOS can be found in the text.

Figure 9

The site-resolved but not spin-resolved density of states (DOS) of the trimer within our tight-binding model is shown for two different on-site energy differences: (a) $E_{\mathrm{C}}-E_{\mathrm{A}}=3$ eV and (b) $E_{\mathrm{C}}-E_{\mathrm{A}}=0.5$ eV. The maximally canted case with $\phi ={45}^{\circ }$ resulting in a negative DMI energy has been considered. The 8 eigenenergies of the system are displayed in the upper panel for each case. The site-resolved DOS is displayed above with the blue curve showing the DOS of the magnetic sites (denoted as A,B) and the brown curve the DOS of the non-magnetic site (denoted as C). The energy-resolved first-order SOI contribution $e_{\mathrm{DM}}$ (red solid curve) and the integrated quantity $E_{\mathrm{DM}}$ (black dashed curve) are shown in the lower panels. The inset displays the on-site energy difference between the non-magnetic site and the magnetic sites (brown: C, blue: A and B). A reasonable bandwidth of about 1-2 eV is indicated by the width of the blue and brown boxes.