Semirealistic tight-binding model for Dzyaloshinskii-Moriya interaction

In this work, we discuss the nature of Dzyaloshinskii-Moriya interaction (DMI) in transition metal heterostructures. We first derive the expression of DMI in the small-spatial-gradient limit using Keldysh formalism. This derivation provides us with a Green's function formula that is well adapted to tight-binding Hamiltonians. With this tool, we first uncover the role of orbital mixing: Using both a toy model and a realistic multiorbital Hamiltonian representing transition metal heterostructures, we show that symmetry breaking enables the onset of interfacial orbital momentum that is at the origin of the DMI. We then investigate the contribution of the different layers to the DMI and reveal that it can expand over several nonmagnetic metal layers depending on the Fermi energy, thereby revealing the complex orbital texture of the band structure. Finally, we examine the thickness dependence of DMI on both ferromagnetic and nonmagnetic metal thicknesses and we find that whereas the former remains very weak, the latter can be substantial.

Figure 1

Schematics of the two tight-binding models discussed in this work. (a) Two-orbital diatomic chain: The atoms of the bottom chain (gray) possess both $p_x$ and $p_z$ orbitals and spin-orbit coupling, while the atoms of the top chain (blue) has only $p_z$ orbitals and supports magnetism. (b) Multiorbital bilayer heterostructure: The heterostructure is composed of two bcc monoatomic thin films whose elements possess all five $d$ orbitals. The bottom layer (grey) is a nonmagnetic metal, whereas the top layer (blue) is magnetic. Both layers possess spin-orbit coupling.

Figure 2

Schematics of the spin momentum induced by mixing $d$ orbitals in the presence of spin-orbit coupling.

Figure 3

The spin-resolved density of states of FM(5)/NM(7) bilayer with $E_F=14$ eV. The blue shaded area corresponds to the contribution of the nonmagnetic metal, while the red shaded area corresponds to the ferromagnetic metal contribution. The vertical dotted line indicates $E_F=12.6$ eV.

Figure 4

Orbital-resolved band structure of FM(5)/NM(7) bilayer. The band structure is projected on (a) $d_{xy}$, (b) $d_{yz}$, (c) $d_{zx}$, (d) $d_{z^2}$, and (e) $d_{x^2-y^2}$. The scale spans from 0 (dark blue) to 1 (red). Close to Fermi level (white dashed line), one can see that $d_{xy}$, $d_{z^2}$, and $d_{x^2-y^2}$ contributions are isotropic in momentum, while $d_{yz}$ and $d_{zx}$ contributions are anisotropic.

Figure 5

Band structure of FM(5)/NM(7) bilayer projected on the three components of the orbital momentum. $L_x$ and $L_y$ are antisymmetric along $\overline{\mathrm{Y}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{Y}}$ and $\overline{\mathrm{X}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{X}}$, respectively, whereas $L_z$ remains even in momentum.

Figure 6

The different coefficients of the DMI tensor as a function of the Fermi energy, computed in FM(5)/NM(7) bilayer using our multiorbital tight-binding approach. In this calculation, the broadening is set to $\mathrm{\Gamma }=50$ meV and $E_F=14$ eV.

Figure 7

The DMI coefficient $D_{xy}$ as a function of the disorder broadening in FM(5)/NM(7) bilayer, for two values of the Fermi energy.

Figure 8

$D_{xy}$ as a function of the Fermi energy when turning on only specific coefficients of the spin-orbit coupling matrix, in FM(5)/NM(7) bilayer. Here the spin-orbit coupling of the ferromagnetic layer is turned off. The broadening is set to $\mathrm{\Gamma }=50$ meV and $E_F=14$ eV.

Figure 9

$D_{xy}$ as a function of the Fermi energy when turning on only specific coefficients of the spin-orbit coupling matrix, in FM(5)/NM(7) bilayer. The spin-orbit coupling of both the ferromagnetic and nonmagnetic layers is turned on. The broadening is set to $\mathrm{\Gamma }=50$ meV and $E_F=14$ eV.

Figure 10

$D_{xy}$ as a function of the Fermi energy when turning on the spin-orbit coupling only on specific layers. The vertical dashed and dotted lines indicate $E_F=12.6$ eV and $E_F=13.5$ eV, respectively.

Figure 11

Thickness dependence of $D_{xy}$ when (a) varying the nonmagnetic thickness and setting the ferromagnetic layer to three monolayers, and (b) varying the ferromagnetic thickness and setting the nonmagnetic layer to five monolayers. The calculations have been performed for $E_F=14$ eV (blue) and 12.6 eV (red).