$k$-asymmetric spin splitting at the interface between transition metal ferromagnets and heavy metals

We systematically investigate the spin-orbit coupling-induced band splitting originating from inversion symmetry breaking at the interface between a Co monolayer and $4d$ (Tc, Ru, Rh, Pd, and Ag) or $5d$ (Re, Os, Ir, Pt, and Au) transition metals. In spite of the complex band structure of these systems, the odd-in-$k$ spin splitting of the bands displays striking similarities with the much simpler Rashba spin-orbit coupling picture. We establish a clear connection between the overall strength of the odd-in-$k$ spin splitting of the bands and the charge transfer between the $d$ orbitals at the interface. Furthermore, we show that the spin splitting of the Fermi surface scales with the induced orbital moment, weighted by the spin-orbit coupling.

Figure 1

(a) Unit cell of the $X$/Co interfaces $(X=4d$ metals $=$ Tc, Ru, Rh, Pd, Ag; and $X=5d$ metals $=$ Re, Os, Ir, Pt, Au) and (b) their electronic configurations used in this work. Numbers in red and gray stand, respectively, for the strength of spin-orbit coupling [44] (in meV) and lattice parameters (in pm) obtained in this work. Numbers in brackets are the out-of-plane lattice constants for the hcp structures.

Figure 2

Projected density of states (DOS) for the interfacial $X$ and Co layers at the $X$/Co interfaces, where $X$ is a heavy metal, as indicated in the figure. The amount of hybridization between the $3d$ orbitals of Co and 4(5)$d$ orbitals of the heavy metal decreases when increasing the $d$-orbital filling of the heavy metal. The densities of states for $X=$ Tc, Ru, Re, and Os are not represented as they show a qualitatively similar behavior as heavy metals with partially filled $d$ shells.

Figure 3

Layer-resolved magnetic properties of the $X$/Co interfaces, where $X$ is a heavy metal. (a) Spin and (b) orbital contributions to the magnetic moment for a magnetization pointing along x. The left (right) panels represent the $4d$ $\left(5d\right)$ metals, as indicated in the figure. The solid (dashed) line indicates the value for the bulk (freestanding layer of) Co.

Figure 4

Central panel: two-dimensional Fermi surface of Ir/Co interface in $\left(k_x,k_y\right)$ plane, when the magnetization direction is along $+\mathbf{x}$ (red lines) or $-\mathbf{x}$ (blue lines). Right panel: Band structure of Ir/Co interfaces calculated along the $y$ direction in the Brillouin zone. Top panel: Band structure of Ir/Co interfaces calculated along the $x$ direction in the Brillouin zone. A clear distortion of the band structure is observed when projected perpendicular to the magnetization direction (right panel), while no such modification appears when the band structure is projected along the magnetization direction (top panel).

Figure 5

(a) Brillouin zone and schematic representation of the odd-in-$k$ splitting along the $\beta$ direction. ${\mathcal{A}}_n$ is the spanned area. (b) $(x,y)$ projection of the $X$/Co interface. $k_{\beta }$ are examples of the electron propagation directions with respect to the crystal structure. (c)–(e) $\mathcal{A}=\mathcal{A}\left(\beta \right)$, for the magnetization along the $+\mathbf{x},+\mathbf{y}$, and $+\mathbf{z}$ directions, respectively. Small and large polar coordinates have the same scale and refer to the $4d$-metal/Co and $5d$-metal/Co interfaces, respectively. For the sake of readability, the values of $\mathcal{A}$ for Tc, Ru, Os, and Re are omitted in this figure.

Figure 6

(a) Angular dependence of the total band splitting $\mathcal{A}\left(\beta \right)$ calculated by first principles (vasp and fleur) and by the standard Rashba model $(H_{\mathrm{R}}\propto sin\beta$; see main text) for the ${\mathrm{Pt}}_n$/Co interfaces $(n$ is a number of Pt layers). Magnetization is along $\mathbf{x}$. (b) Dependence of $\mathcal{A}(\beta =0^{\circ })$ on the number of substrate layers for Ir, Pt, and Au. Magnetization is along $\mathbf{y}$.

Figure 7

Correlation between odd-in-$k$ spin splitting and interfacial electronics properties as a function of substrate. (a) Spanned area $\mathcal{A}(\beta ={90}^{\circ })$ (blue symbols) and charge transfer from the $d$ orbitals of substrate to the $d$ orbitals of Co (red symbols). (b) Effective momentum shift $\langle k_y^{\mathrm{F}}\rangle$ (blue symbols) and summation of the induced orbital moments of each atom weighted by their spin-orbit coupling constant, ${\sum }_i{\xi }_i{L}_i^{\mathrm{ind}}$ (red symbols).

Figure 8

Induced orbital moment weighted by spin-orbit coupling for (a) $5d$-metal/Co and (b) $4d$-metal/Co interfaces. Induced orbital moment for each $i$ is $L_i^{\mathrm{ind}}=L_i^{X/\mathrm{Co}}-L_i^{\mathrm{bulk}}$.