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

TL;DR

This paper derives explicit analytical formulas for charge and spin currents, as well as spin transfer torques, in magnetic tunnel junctions with noncollinear magnetizations, revealing key factors influencing their angular dependence and magnitude.

Contribution

It provides the first explicit analytical expressions for spin and charge currents and torques in magnetic tunnel junctions using the Keldysh formalism and WKB approximation, highlighting key parameters affecting their behavior.

Findings

Spin/charge currents are expressed through three key quantities.

Deviation mechanisms from sine angular dependence are identified.

Perpendicular spin torque is linked to effective interfacial polarizations.

Abstract

In this work we demonstrate explicit analytical expressions for both charge and spin currents which constitute the 2x2 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.