Analytical description of ballistic spin currents and torques in magnetic tunnel junctions

In this work we demonstrate explicit analytical expressions for both charge and spin currents which constitute the $2\times 2$ spinor in magnetic tunnel junctions with noncollinear magnetizations under applied voltage. The calculations have been performed within the free electron model in the framework of the Keldysh formalism and WKB approximation. We demonstrate that spin/charge currents and spin transfer torques are all explicitly expressed through only three irreducible quantities, without further approximations. The conditions and mechanisms of deviation from the conventional sine angular dependence of both spin currents and torques are shown and discussed. It is shown in the thick barrier approximation that all tunneling transport quantities can be expressed in an extremely simplified form via Slonczewski spin polarizations and our effective spin averaged interfacial transmission probabilities and effective out-of-plane polarizations at both interfaces. It is proven that the latter plays a key role in the emergence of perpendicular spin torque as well as in the angular dependence character of all spin and charge transport considered. It is demonstrated directly also that for any applied voltage, the parallel component of spin current at the FM/I interface is expressed via collinear longitudinal spin current components. Finally, spin transfer torque behavior is analyzed in a view of transverse characteristic length scales for spin transport.

Figure 1

Top: Schematic structure of the MTJ, consisting of left and right semi-infinite FM leads separated by a thin nonmagnetic insulating barrier. The magnetization ${\mathbf{M}}_R$ of the right FM lead is along $z$, whereas the magnetization ${\mathbf{M}}_L$ of the left lead is rotated by an angle $\gamma$ around the $y$ axis with respect to ${\mathbf{M}}_R$. Bottom: Schematic illustration of the potential profile, where $U_{L\left(R\right)}^{\uparrow }, U_{L\left(R\right)}^{\downarrow }$, and $U_B$ are the potentials of the majority and minority bands in the left (right) FM leads, and the barrier, respectively. The lower dashed line indicates the Fermi level in equilibrium.

Figure 2

The out-of-plane $T_{\perp }$ and in-plane $T_{\parallel }$ torques as a function of applied bias in an asymmetric MTJ. $U_B=1$ eV, ${\mathrm{\Delta }}_0=2.62$ eV, $m_{\mathrm{eff}}=0.4, \gamma =\pi /2, d=7$ Å.

Figure 3

Angular dependence of a universal deviation function given by Eq. (7) for $T_{\perp }$ (top panels) and $T_{\parallel }$ (bottom panels) for different values of thickness $d$, exchange splittings ${\mathrm{\Delta }}_{L,R}$, and applied voltages $V$. The parameters are ${\mathrm{\Delta }}_0=2.62$ eV, $U_B=0.5$ eV, $m_{\mathrm{eff}}=0.4$.

Figure 4

Distribution of the voltage-induced in-plane and out-of-plane STT terms in the right FM electrode of a symmetric MTJ for (a) $V=+1$ V and (b) $V=-1$ V applied voltage. The parameters are ${\mathrm{\Delta }}_0=2.62$ eV, ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=1.96$ eV, $U_B=3$ eV, $m_{\mathrm{eff}}=0.4, \gamma =\pi /2, d=7$ Å. $T_{0\perp }$ indicates the out-of-plane torque at zero voltage. The inset shows a voltage dependence of the real and imaginary parts of $\mathrm{\Delta }{k}_R=k_R^{\uparrow }-k_R^{\downarrow }$.

Figure 5

The same as in Fig. 4 for case in which the right FM electrode is a half-metal. Exchange splittings are ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_R=2.62$ eV and ${\mathrm{\Delta }}_L=1.96$ eV while other parameters are unchanged.

Figure 6

Bias voltage dependence of the torque efficiency for an MTJ. $\mathrm{\Delta }{T}_{\perp }=T_{\perp }\left(V\right)-T_{\perp }\left(0\right)$. The parameters are ${\mathrm{\Delta }}_0=2.62$ eV, ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=1.96$ eV, $U_B=2$ eV, $m_{\mathrm{eff}}=0.4, \gamma =\pi /2, d=7$ Å.