Quantum noise in the spin transfer torque effect

Abstract

The spin transfer torque (STT) is one of the most studied spintronics effect, in particular due to its applications in storage devices, since it allows the flip of the bits implemented by the magnetization direction of ferromagnetic layers. In a typical STT device a polarized current exerts a torque on the magnetization of a ferromagnet. The dynamics of the macroscopic magnetization in presence of a magnetic field is usually described by the Landau-Lifshitz-Gilbert (LLG) equation, which can be introduced by phenomenological arguments. In the LLG equation, one usually considers two types of terms. The first type includes the torque exerted by the total effective magnetic field (the torque is perpendicular to both the magnetization and the magnetic field), whereas the second type takes care of damping effects (the torque is perpendicular to both the magnetization and to its time derivative). In the context of the STT literature, polarized currents appear as additional terms, which may have both a field-like or a damping-like character, depending whether they act as the torques of the first or second type. In hybrid ferromagnetic-metal systems, the coupling of the electrical current to the macroscopic degrees of freedom of the magnetization is obtained by an exchange interaction. Furthermore, the interface between the ferromagnet and the metal is described by an effective spin-mixing conductance. Within such approach, the dynamics of the magnetization remains purely quasiclassical and the focus is on the diffusive aspects of the charge and spin dynamics of the free carriers. In recent years, the advances in fast time-resolved measurements have showed that the magnetization dynamics of a nanomagnet crossed by a polarized current presents a stochastic behavior at short time interval. If we want fast and small devices this stochastic behavior becomes important and we need to study it because: we do not want that the noise disturbs the device working; we can engineer the noise to help the magnet switching, without increasing the current which would imply dissipative effects and the heating up of the device. In order to take into account the noise effects and the quantum aspects of the magnetization dynamics, relevant in small magnets, one needs to develop a fully microscopic derivation of both the magnetization dynamics and of the transport properties of the carriers interacting with it. Our aim in this work is: to derive microscopically an equation that could describe the magnetization dynamics in the presence of a polarized current; to describe the thermal and the quantum noise induced in the magnet motion by the electric current; to employ a formalism that is easily generalizable. In order to carry out this program, we consider a greatly simplified microscopic model, which though still captures the key aspects of the problem at hand: the magnet is described as a single quantum spin of microscopically large size. The carriers are assumed to propagate ballistically along a single quantum channel. The coupling between the carriers and the magnet is introduced via the exchange interaction between the carriers' spin and the quantum spin. In recent literature this model was studied in the context of a scattering matrix approach. Here instead, with the purpose of possible future extensions to more complicated models (one might want to include the electron-electron interaction, the presence and interaction of several magnets, etc.), we employ the Keldysh formalism, which allows for a systematic perturbative expansion about the semiclassical limit described by the LLG equation. We find that the quantum noise affects both the filed-like and the damping-like terms in the dynamic equation of the nano-magnet and the spin-mixing conductance structure appears clearly. In the first part of the thesis, we review some theories that describe the interaction between ferromagnetic layers and polarized currents and the Keldysh technique is explained; in the second part, we derive our results.