Spin transfer torques and spin-dependent transport in a metallic F/AF/N tunneling junction

We study spin-dependent electron transport through a ferromagnetic-antiferromagnetic-normal metal tunneling junction subject to a voltage or temperature bias, in the absence of spin-orbit coupling. We derive microscopic formulas for various types of spin torque acting on the antiferromagnet as well as for charge and spin currents flowing through the junction. The obtained results are applicable in the limit of slow magnetization dynamics. We identify a parameter regime in which an unconventional dampinglike torque can become comparable in magnitude to the equivalent of the conventional Slonczewski's torque generalized to antiferromagnets. Moreover, we show that the antiferromagnetic sublattice structure opens up a channel of electron transport which does not have a ferromagnetic analog and that this mechanism leads to a pronounced fieldlike torque. Both charge conductance and spin current transmission through the junction depend on the relative orientation of the ferromagnetic and the antiferromagnetic vectors (order parameters). The obtained formulas for charge and spin currents allow us to identify the microscopic mechanisms responsible for this angular dependence and to assess the efficiency of an antiferromagnetic metal acting as a spin current polarizer.

Figure 1

Schematic description of the theoretical model considered in the present work. The ferromagnetic (F) and normal metal (N) leads are assumed to have fixed electron distribution functions $n_{F,N}\left(\epsilon \right)$, respectively. A finite difference $n_F-n_N$ drives the antiferromagnet (AF) out of equilibrium and it eventually settles down to a dynamical stationary state. The layers are separated by barriers with tunneling amplitudes $W_{F,N}$.

Figure 2

The choice of coordinate axes and the definitions of the azimuthal and polar angles $\theta ,\varphi$ seen in the ferromagnetic frame spanned by ${\widehat{\mathit{x}}}_F,{\widehat{\mathit{y}}}_F,{\widehat{\mathit{z}}}_F$ (left) and in the antiferromagnetic frame spanned by $\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}}$ (right). The Néel vector in the ferromagnetic frame is parameterized by the usual spherical coordinates. The incoming spin current, or equivalently the ferromagnetic moment, in the antiferromagnetic frame has $x$ and $z$ components only, ${\widehat{\mathit{z}}}_F=\widehat{\mathit{z}}cos\theta -\widehat{\mathit{x}}sin\theta$.

Figure 3

Spatial variation of wave functions (left) and the corresponding energy spectrum (right) (a) with neither intersublattice overlap nor tunneling, (b) including intersublattice overlap without tunneling, and (c) for full electron eigenstates of the structure under consideration. $\mathrm{t}$ and $\mathrm{b}$ denote the top and bottom bands split by the large gap $2{\mathrm{\Delta }}_l$. Spin is a good quantum number in an isolated bipartite antiferromagnet as for (a) and (b). Since the mixing with ferromagnetic electrons breaks rotational symmetry completely unless ${\widehat{\mathit{z}}}_F\parallel \widehat{\mathit{z}}$, spin cannot be used to label the states in (c). The ferromagnet also breaks the symmetry between $A$ and $B$ sites, reflected on the asymmetry in the wave functions.

Figure 4

Four types of spin torque. They are classified according to the effective field orientations illustrated by the electron spin accumulations on the two sublattices. If the effective field lies in the plane spanned by the Néel vector $\mathit{n}$ and the spin polarization ${\widehat{\mathit{z}}}_F$, the torque is fieldlike. An out-of-plane field represents an anti-damping-like torque. Each category has two staggered and nonstaggered varieties based on the relative sign between the effective fields at the two sublattice sites. The directions of the torques themselves are indicated by planer arrows.

Figure 5

Illustration of dephasing processes through $A$ and $B$ sublattices. An electron tunnels into a superposition of up and down states due to the difference in the quantization axes in F and AF. The dephasing leads to precession of the electron spin, whose chirality is opposite for the two sublattices. This intrasublattice process is the only channel of electron transport through AF if there is no intersublattice overlap $t_l=0$.

Figure 6

Two mechanisms of transverse spin conservation in AF. (Left) When $t_l\ne 0$, a ferromagnetic electron may propagate through AF as a superposition of up and down states that have exactly the same energy and wavelength. In the perturbative picture, an electron that tunneled into the $A$ site is initially a superposition of up and down with different wavelength. Subsequently, one of the electron states may hop onto a state in the $B$ site via $t_l$, which has the same wavelength as the state remaining at $A$. The phases of the two states evolve at the same rate, thus avoiding dephasing. (Right) The tilt of the quantization axes. At the first order in ${\delta }_l/{\mathrm{\Delta }}_l$, the axes for both $A$ and $B$ sites change by the same amount. At each sublattice, the $z_{A,B}^{\prime }$ component of the spin is conserved, which has a finite $x$ component.

Figure 7

Mechanism of the magnetoresistance. In the perturbation in the influence of F on AF ${\sigma }_l^R/{\mathrm{\Delta }}_l, 1/{\tau }_{l\pm }^F$ can be considered to be the rate of particle exchange between F and AF for up and down electrons, respectively. At the leading order, the charge current is proportional to $1/{\tau }_{l+}^F+1/{\tau }_{l-}^F$ and independent of $\theta$. The second-order effect is a backflow proportional to $-1/{\left({\tau }_{l+}^F\right)}^2-1/{\left({\tau }_{l-}^F\right)}^2$. The sum of the squares is $\theta$-dependent and it is greatest for the collinear configuration (left) and smallest for the perpendicular configuration (right). The backflow is accordingly strongest for the collinear configuration.

Figure 8

Dependence of the spin current transmission on the mutual orientation of the incoming spin current polarization and the Néel vector. (Left) Operation as a spin current polarizer. When ${\widehat{\mathit{z}}}_F·\mathit{N}\ne 0$, the injected spin current $J_F^{z_F}$ is projected onto the direction of the Néel vector $\widehat{\mathit{z}}$. The transverse ($x,y$) components are projected out and the outgoing spin current is mainly polarized along the Néel vector; ${\mathit{J}}_N\approx J_N^z\widehat{\mathit{z}}$. (Right) Operation as a spin current filter. When ${\widehat{\mathit{z}}}_F$ and $\mathit{N}$ are perpendicular, the output spin current is effectively zero. This is a direct consequence of transverse spin absorption in AF by the dephasing mechanism.