Spin transfer torque in antiferromagnetic spin valves: From clean to disordered regimes

Current-driven spin torques in metallic spin valves composed of antiferromagnets are theoretically studied using the nonequilibrium Green's function method implemented on a tight-binding model. We focus our attention on G-type and L-type antiferromagnets in both clean and disordered regimes. In such structures, spin torques can either rotate the magnetic order parameter coherently (coherent torque) or compete with the internal antiferromagnetic exchange (exchange torque). We show that, depending on the symmetry of the spin valve, the coherent and exchange torques can either be in the plane, $\propto \mathbf{n}\times (\mathbf{q}\times \mathbf{n})$ or out of the plane $\propto \mathbf{n}\times \mathbf{q}$, where $\mathbf{q}$ and $\mathbf{n}$ are the directions of the order parameter of the polarizer and the free antiferromagnetic layers, respectively. Although disorder conserves the symmetry of the torques, it strongly reduces the torque magnitude, pointing out the need for momentum conservation to ensure strong spin torque in antiferromagnetic spin valves.

Figure 1

Square lattice representation of an antiferromagnetic spin valve, as implemented in the tight-binding code. The systems studied in this paper are composed of a combination of G-type and L-type antiferromagnets separated by a metallic spacer and connected to two external leads.

Figure 2

Spin density profile generated by itinerant electrons flowing in (a) a ferromagnet, (b) a G-type antiferromagnet, and (c) an L-type antiferromagnet. The left (right) hand side plots show the reflected (transmitted) spin densities. The calculations were done with the parameters: $\epsilon =-3$ eV and $\Delta /t=1$.

Figure 3

Schematics of the coherent (a) and exchange torques (b) (green arrows) applied on the spin of two neighboring sites A and B (red and blue arrows, respectively). In the case of coherent torque, the torques exerted on the two neighbors are opposite of each other, which results in a coherent rotation of the order parameter. In the case of exchange torque, the torques exerted on the two neighbors are in the same direction, which results in breaking the collinear ordering of the spins.

Figure 4

Spatial profile of the three components of the nonequilibrium spin density obtained for a symmetric G-type spin valve. The left AF layer is oriented along z whereas its right AF layer is oriented along x. The parameters are the same as in Fig. 2.

Figure 5

Dependence of the efficiency of the spin torque exerted on the right layer when electrons are flowing from left to right as a function of the relative angle between the two layer's order parameter directions. The structures calculated are symmetric antiferromagnetic spin valves composed of (a) G-type and (b) L-type antiferromagnets, as well as asymmetric spin valves composed of L-type/G-type (c) and G-type/L-type (d). In-plane ($\square$) and perpendicular components ($◯$) of the coherent torque as well as in-plane ($▵$) and perpendicular components ($▽$) of the exchange torques are represented. The solid line in (a) presents the in-plane coherent torque calculated in conventional symmetric ferromagnetic spin valves shown for reference. The parameters are the same as in Fig. 2.

Figure 6

Illustration of an asymmetric antiferromagnetic spin valve composed of L-type and G-type layers stacked along the $x$ axis. (a) The order parameter of the G-type antiferromagnet is aligned along the $z$ axis, while the order parameter of the L-type antiferromagnet is rotated by an angle $\theta$ around the $x$ axis. (b) The rotation of the magnetic moments of both antiferromagnets by an angle $\pi$ around the $z$ axis is equivalent to rotating the order parameter of the L-type antiferromagnet by an angle $\theta$ around the $x$ axis.

Figure 7

The efficiency of the spin torque calculated for different spacer thicknesses in the case of a symmetric G-type antiferromagnetic spin valve ($\square$) and in the case of a symmetric ferromagnetic spin valve ($◯$). The relative angle between the order parameters (magnetizations) of the two antiferromagnetic (ferromagnetic) layers equals $\pi /2$. The calculations are performed in the absence of disorder, and the thicknesses vary from one to 40 layers. The parameters are the same as in Fig. 2.

Figure 8

Schematics of the ballistic transport occurring in an antiferromagnetic spin valve in the absence (a) and presence (b) of disorder. In the former case, the direction of the momentum is not modified in the spacer. In the latter case, disorder scattering alters the linear momentum direction so that the staggered spin density built in the left antiferromagnet is redistributed over the right antiferromagnet, resulting in a dramatic reduction of the spin torque magnitude.

Figure 9

Resistance of a metallic layer as a function of the width of the layer for different disorder strengths W. The approximate linear relationship between the resistance and the width allows for extracting the effective mean free path of the metallic layer.

Figure 10

Spin torques as function of the reduced mean free path $\lambda /L$ calculated in fully disordered ($\square$) and spacer-only disordered ($◯$) systems. The relative angle between the order parameters of the two antiferromagnetic layers equals $\pi /2$. The inset shows the corresponding conductances. The calculation is averaged over 2500 configurations and the parameters are the same as in Fig. 2. The logarithmic scale is used for both axes.

Figure 11

Normalized torques as function of the reduced mean free path $\lambda /L$ calculated respectively in ferromagnet (red), G-type antiferromagnet with different thicknesses [black ($L_0$), magenta ($0.5L_0$) and green ($2.5L_0$)], and the L-type type (open circles) for a relative angle of $\pi /2$. The inset shows the corresponding conductances. The parameters are the same as in Fig. 2 and we use the logarithmic scale for both axes.