Band-filling effect on magnetic anisotropy using a Green's function method

We use an analytical model to describe the magnetocrystalline anisotropy energy (MAE) in solids as a function of band filling. The MAE is evaluated in second-order perturbation theory, which makes it possible to decompose the MAE into a sum of transitions between occupied and unoccupied pairs. The model enables us to characterize the MAE as a sum of contributions from different, often competing terms. The nitridometalates ${\mathrm{Li}}_2\left[\left({\mathrm{Li}}_{1-x}{T}_x\right)\mathrm{N}\right]$, with $T=$ Mn, Fe, Co, Ni, provide a system where the model is very effective because atomiclike orbital characters are preserved and the decomposition is fairly clean. Model results are also compared against MAE evaluated directly from first-principles calculations for this system. Good qualitative agreement is found.

Figure 1

Illustration of the dependence of the easy-axis direction on the orbital quantum numbers $(m,m^{\prime })$ and the spin quantum numbers $(\sigma ,{\sigma }^{\prime })$ of two subbands. Configurations (a) and (d) favor uniaxial anisotropy, while (b) and (c) favor easy-plane anisotropy. The vertical dotted line corresponds to the Fermi energy, $E_F$. The horizontal line separates the majority (up) and minority (down) spin channels. Occupied states with different $\left|m\right|$ numbers are filled with different colors.

Figure 2

(a) Schematic Lorentzian-shape densities of states for subbands $m$ and $m^{\prime }$. (b) ${\chi }_{m{m}^{\prime }}^{\epsilon }$ and its four spin components as functions of Fermi energy. The amplitudes of ${\chi }_{m{m}^{\prime }}^{\epsilon }$ with (c) the maximum at ${\varepsilon }_F^{(1,3)}$ and (d) the minimum at ${\varepsilon }_F^{\left(2\right)}$ as functions of spin splitting $\mathrm{\Delta }s$ and crystal-field splitting $\mathrm{\Delta }c$.

Figure 3

Schematic representation of the supercell used in the DFT calculation for ${\mathrm{Li}}_2[$(${\mathrm{Li}}_{1-x}{T}_x\left)\mathrm{N}\right]$ with $x=0.5$. Both $T_{1a}$ and $T_{2d}$ sites are derived from the ${\mathrm{Li}}_I$ ($1b)$ site in the original $\alpha -{\mathrm{Li}}_3\mathrm{N}$ structure, while other Li atoms, which form coplanar hexagons, correspond to ${\mathrm{Li}}_{II}$ ($2c)$ sites in the original $\alpha -{\mathrm{Li}}_3\mathrm{N}$ structure.

Figure 4

Partial densities of states projected on the $3d$ states of the $T_{1a}$ site in the $hex2$ structure in ${\mathrm{Li}}_2[$(${\mathrm{Li}}_{1-x}{T}_x\left)\mathrm{N}\right]$, where $x=0.5$ and $T$ is (a) Mn, (b) Fe, (c) Co, and (d) Ni. The vertical dotted line corresponds to the Fermi energy, $E_F$. The horizontal dotted line separates the majority (up) and minority (down) spin channels. Calculation is within LDA, without spin-orbit coupling included.

Figure 5

(a) Schematic partial densities of states projected on the $3d$ states of ${\mathrm{Fe}}_{1a}$ sites. Orbital transitions and the sign of their contributions to the MAE are also shown. Solid line indicates positive contribution (easy axis) and the dashed line indicates negative contribution (easy plane) to the easy-axis anisotropy. (b) Scaled MAE $4K/{\xi }^2$ from $T_{1a}$ site and its decomposition into orbital susceptibilities as functions of band filling. (c) Magnetic anisotropy energy $K$ from $T_{1a}$ site as a function of $T$. Different sets of electronic structure parameters $\mathrm{\Delta }s, \mathrm{\Delta }c$, and $w$ are used to represent the DFT PDOS on $T_{1a}$ sites in ${\mathrm{Li}}_2[$(${\mathrm{Li}}_{0.5}{T}_{0.5}\left)\mathrm{N}\right]$ for different $T$ elements.