Spin-filtering efficiency of ferrimagnetic spinels CoFe${}_2$O${}_4$ and NiFe${}_2$O${}_4$

We assess the potential of the ferrimagnetic spinel ferrites CoFe${}_2$O${}_4$ and NiFe${}_2$O${}_4$ to act as spin filtering barriers in magnetic tunnel junctions. Our study is based on the electronic structure calculated by means of first-principles density functional theory within different approximations for the exchange correlation energy. We show that, in agreement with previous calculations, the density of states suggests a lower tunneling barrier for minority spin electrons, and thus a negative spin-filter effect. However, a more detailed analysis based on the complex band structure reveals that both signs for the spin-filtering efficiency are possible, depending on the band alignment between the electrode and the barrier materials and depending on the specific wave-function symmetry of the relevant bands within the electrode.

Figure 1

Total and projected DOS per formula unit for CoFe${}_2$O${}_4$ (left) and NiFe${}_2$O${}_4$ (right) calculated with different exchange-correlation potentials (from left to right: $\mathrm{GGA}+U$, HSE, and ASIC). The $t_{2g}$ and $e_g$ states of Fe, Co, and Ni on the $O_h$ sites and the $e$ and $t_2$ states of Fe on the $T_d$ sites are shown as black (blue) and dark grey (red) lines, respectively. The shaded grey area in all panels depicts the total DOS. Minority spin projections are shown using negative values. The zero energy is set to the middle of the band gap.

Figure 2

Band structures for energies around the band gap of CoFe${}_2$O${}_4$ [top (a) and (b)] and NiFe${}_2$O${}_4$ [bottom (c) and (d)] calculated by using the GGA$+U$ exchange-correlation functional [left (a) and (c)] and the ASIC scheme [right (b) and (d)]. Majority- and minority-spin bands are shown as full (black) and dashed (red) lines.

Figure 3

Minimal value of $\kappa$ at different energies (indicated at the top left in each graph) within the gap for CoFe${}_2$O${}_4$ (a)–(c) and NiFe${}_2$O${}_4$ (d)–(f) along different transport directions, calculated within the ASIC approach. Zero energy corresponds to the middle of the band gap.

Figure 4

The complex band structure corresponding to $k_x=k_y=0$ for NiFe${}_2$O${}_4$ (upper two panels) and CoFe${}_2$O${}_4$ (lower two panels) along [001] and [111], calculated within ASIC for the $Imma$ ionic configuration. The up and down arrows indicate the spin character of the lowest lying complex bands. The vertical dashed lines indicate the energies that were used for the $k_x$-$k_y$ plots in Fig. 3.

Figure 5

The complex band structure corresponding to $k_x=k_y=0$ for NiFe${}_2$O${}_4$ (upper panel) and CoFe${}_2$O${}_4$ (lower panel) along [001] for the $P{4}_{1}22$ configuration, calculated within ASIC. The up and down arrows indicate the spin-character for some of the lowest lying complex bands.