Perpendicular magnetic anisotropy at the $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$ interface: Comparative first-principles study with Fe/MgO

We present a theoretical study on interfacial magnetocrystalline anisotropy for $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$. This system has a very small lattice mismatch at the interface and therefore is suitable for realizing a fully coherent ferromagnet/oxide interface for magnetic tunnel junctions. On the basis of density functional theory, we calculate the interfacial anisotropy constant $K_{\mathrm{i}}$ and show that this system has interfacial perpendicular magnetic anisotropy (PMA) with $K_{\mathrm{i}}\approx 1.2\mathrm{mJ}/{\mathrm{m}}^2$, which is a little bit smaller than that of Fe/MgO ($K_{\mathrm{i}}\approx 1.5–1.7$ $\mathrm{mJ}/{\mathrm{m}}^2$). Second-order perturbation analysis with respect to the spin-orbit interaction clarifies that the difference in $K_{\mathrm{i}}$ between $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$ and Fe/MgO originates from the difference in contributions from spin-flip scattering terms at the interface. We propose that the insertion of tungsten layers into the interface of $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$ is a promising way to obtain huge interfacial PMA with $K_{\mathrm{i}}\gtrsim 3\mathrm{mJ}/{\mathrm{m}}^2$.

Figure 1

Supercells of (a) Fe(5)/${\mathrm{MgAl}}_2{\mathrm{O}}_4$(9) and (b) Fe(5)/MgO(5).

Figure 2

The interfacial anisotropy constant $K_{\mathrm{i}}$ as a function of the number of in-plane k points $N$ ($=N_x=N_y$) in (a) $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$, (b) Fe/MgO ($a=a_{\mathrm{MgO}}/\sqrt{2}$), and (c) Fe/MgO ($a=a_{\mathrm{Fe}}$).

Figure 3

(a) Results of second-order perturbation analysis on the interfacial PMA in $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$. The vertical green, yellow, blue, red, and pink bars show the values of $\mathrm{\Delta }{E}_{\uparrow \Rightarrow \uparrow }^i,\mathrm{\Delta }{E}_{\downarrow \Rightarrow \downarrow }^i,\mathrm{\Delta }{E}_{\uparrow \Rightarrow \downarrow }^i,\mathrm{\Delta }{E}_{\downarrow \Rightarrow \uparrow }^i$, and the local anisotropy energy $E_{\mathrm{MCA}}^i$, respectively, at each Fe layer. [See Eq. (4) and the corresponding text for details.] (b) Projected LDOSs for Fe $3d$ states at the interface of $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$. In panel (b), positive and negative values indicate the majority- and minority-spin projected LDOSs, respectively. The inset of panel (b) shows a magnified view near the Fermi level.

Figure 4

The wave-vector-resolved information on the interfacial PMA in $\mathrm{Fe}/{\mathrm{MgAl}}_2{\mathrm{O}}_4$. (a) The in-plane wave-vector (${\mathbf{k}}_{\parallel }$) dependence of $\mathrm{\Delta }E\left({\mathbf{k}}_{\parallel }\right)\equiv E_{\left[100\right]}\left({\mathbf{k}}_{\parallel }\right)-E_{\left[001\right]}\left({\mathbf{k}}_{\parallel }\right)$. (b) and (c) The band structure of the supercell in the majority- and minority-spin states, respectively. In panels (b) and (c), orbital components of each band are indicated by colors.

Figure 5

The same as Fig. 3, but for Fe/MgO with $a=a_{\mathrm{MgO}}/\sqrt{2}$.

Figure 6

The same as Fig. 4, but for Fe/MgO with $a=a_{\mathrm{MgO}}/\sqrt{2}$.

Figure 7

The same as Fig. 3, but for Fe/MgO with $a=a_{\mathrm{Fe}}$.

Figure 8

The same as Fig. 4, but for Fe/MgO with $a=a_{\mathrm{Fe}}$.

Figure 9

(a) The supercell of Fe(5)/W(3)/${\mathrm{MgAl}}_2{\mathrm{O}}_4$(9) used in our calculation. (b) Values of $K_{\mathrm{i}}$ obtained for Fe(5)/W (3–5)/${\mathrm{MgAl}}_2{\mathrm{O}}_4$(9) and two types of Fe(5)/W(3–5)/MgO(5) with different in-plane lattice constants. The data indicated by W(0) are those when W layers are not inserted (the values of $K_{\mathrm{i}}$ were already shown in Table 1).

Figure 10

(a)–(d) Projected LDOSs for $3d$ states at the middle layer of the W atom in Fe(5)/W(3)/${\mathrm{MgAl}}_2{\mathrm{O}}_4$(9), Fe(5)/W(3)/MgO(5) ($a=a_{\mathrm{Fe}}$), and Fe(5)/W(3)/MgO(5) ($a=a_{\mathrm{MgO}}/\sqrt{2}$), where positive and negative values indicate the majority- and minority-spin projected LDOSs, respectively.