Orbital polarization effects on the magnetic anisotropy and orbital magnetism of clusters, films, and surfaces: A comparative study within tight-binding theory

The effects of orbital polarizations on the magnetic properties of transition-metal nanostructures are investigated in the framework of a self-consistent tight-binding theory. Three different approximations to the intra-atomic two-center Coulomb interactions are considered: (i) full orbital dependence of the direct and exchange Coulomb interactions $U_{m{m}^{\prime }}$ and $J_{m{m}^{\prime }}$ as given by atomic symmetry, (ii) orbital independent interactions $U=\overline{U_{m{m}^{\prime }}}$ and $J=\overline{J_{m{m}^{\prime }}}$, and (iii) orbital polarization (OP) approximation of the form $H_{\mathrm{OP}}=-(B∕2){\sum }_i{L}_i^2$, where $L_i$ refers to the orbital momentum operator at atom $i$ and $B$ to the Racah coefficient. Results are given for the local orbital magnetic moments $⟨L_{i\delta }⟩$ along high-symmetry magnetization directions $\delta$ and for the corresponding magnetic anisotropy energies $\mathrm{\Delta }{E}_{\delta \gamma }$ of surfaces, films, and clusters of Fe, Co, and Ni. The quantitative differences between the approximations allow us to quantify the effects of orbital polarizations on $⟨L_{i\delta }⟩$ and $\mathrm{\Delta }{E}_{\delta \gamma }$. One observes that, with an appropriate choice of $B$, the OP ansatz yields a very good agreement with the rigorous orbital dependent calculations. The simplest orbital independent approach underestimates $⟨L_{i\delta }⟩$ and $\mathrm{\Delta }{E}_{\delta \gamma }$ systematically. However, it provides a good qualitative description of the main general trends as a function of dimensionality, local environment, and $d$-band filling. Advantages and limitations of the various approaches are discussed.

Figure 1

Local orbital magnetic moments $⟨L_{i\delta }⟩$ at the $\left(0001\right)$ hcp-Co surface as a function of layer number $i$, where $i=1$ refers to the surface layer. The magnetization directions are (a) the $\left(1\overline{1}0\right)$ in-plane direction $\delta =x$, and (b) the surface normal $\delta =z$. Dots refer to the full orbital-dependent calculations [Eq. (4)], crosses to the orbital polarization approximation [Eq. (8)], and open circles to the orbital-independent approach [Eq. (6)].

Figure 2

Local orbital magnetic moments $⟨L_{i\delta }⟩$ at the $\left(001\right)$ bcc-Fe surface as a function of layer number $i$, where $i=1$ refers to the surface layer. The magnetization directions are (a) the $\left(100\right)$ in-plane direction $\delta =x$, and (b) the surface normal $\delta =z$.

Figure 3

Orbital magnetic moment per atom $⟨L_{\delta }⟩$ in an fcc(111)-monolayer as a function of $d$-band filling $n_d$. (a) In-plane magnetization $\delta =x$ along $\left(1\overline{1}0\right)$, and (b) perpendicular magnetization $\delta =z$. Dots refer to the full orbital-dependent calculations [Eq. (4)], crosses to the orbital polarization approximation [Eq. (8)], and open circles to the orbital-independent approach [Eq. (6)].

Figure 4

(a) MAE per atom $\mathrm{\Delta }{E}_{xz}=E_x-E_z$ and (b) anisotropy of the orbital magnetic moments $\mathrm{\Delta }{E}_{zx}=L_z-L_x$ in an fcc(111) monolayer as a function of the $d$-band filling $n_d$. The considered magnetization directions and model interactions are the same as in Fig. 3.

Figure 5

(a) MAE per atom $\mathrm{\Delta }{E}_{xz}=E_x-E_z$ and (b) orbital magnetic moments $L_x$ (dashed curves) and $L_z$ (full curves) of a Co fcc(111) monolayer as a function of the spin-orbit coupling constant $\xi$. Dots refer to the full orbital-dependent calculations [Eq. (4)], crosses to the orbital polarization approximation [Eq. (8)], and open circles to the orbital-independent approach [Eq. (6)]. The $L_x$ curve for $B=0$ has been omitted since it is barely distinguishable from the $L_z$ curve for the orbital-dependent approximations.

Figure 6

Average orbital moment per atom $⟨L_{\delta }⟩$ in a seven-atom cluster with a pentagonal bipyramid structure as a function of $d$-band filling $n_d$. For $\delta =z$ the magnetization is along the principal $C_5$ symmetry axis, and for $\delta =x$ it is perpendicular to $z$ and to one of the nearest neighbor bonds in the pentagonal ring. Dots refer to the full orbital-dependent calculations [Eq. (4)], crosses to the orbital polarization approximation [Eq. (8)], and open circles to the orbital-independent approach [Eq. (6)].

Figure 7

(a) Magnetic anisotropy energy per atom $\mathrm{\Delta }{E}_{xz}=E_x-E_z$ and (b) anisotropy of the orbital magnetic moments per atom $\mathrm{\Delta }{L}_{zx}=⟨L_z⟩-⟨L_x⟩$ in a seven-atom cluster as a function of $d$-band filling $n_d$. The cluster structure and magnetization directions are the same as in Fig. 6.

Figure 8

(a) MAE per atom $\mathrm{\Delta }{E}_{xz}=E_x-E_z$ and (b) orbital magnetic moments $L_x$ (dashed curves) and $L_z$ (full curves) in a pentagonal bipyramid ${\mathrm{Co}}_7$ cluster as a function of the spin-orbit coupling constant $\xi$. Dots refer to the full orbital-dependent calculations [Eq. (4)], crosses to the orbital polarization approximation [Eq. (8)], and open circles to the orbital-independent approach [Eq. (6)].

Figure 9

Local orbital moments $⟨L_{i\delta }⟩$ along the easy magnetization direction $\delta$ in fcc-like ${\mathrm{Ni}}_N$ clusters averaged at each NN shell $i$ surrounding the central atom $i=1$.