## Landauer's theory of charge and spin transport in magnetic multilayers

Human, Trevor (2014).
*Landauer's theory of charge and spin transport in magnetic multilayers*.
(Unpublished Doctoral thesis, City University London)

## Abstract

The nonequilibrium Keldysh formalism has been used to study the spin transport effects found in magnetic multi-layered nanostructures. We formulate a new methodology based on Landauer and show it to be in very good qualitative agreement with Keldysh. However, our theory provides more information regarding the physics of these effects because it allows us to calculate the contributions of individual electrons incident from either side of a junction as well as the contributions within a single layer that are incident on and reflected from an adjacent interface.

Chapter 1 provides a consolidated introduction to spintronics in magnetic multilayer nanostructures (the key focus of this thesis) including phenomena such as giant magnetoresistance (GMR), tunneling magnetoresistance (TMR) and current induced switching of magnetization. We then describe how to calculate the local charge and spin current in the direction perpendicular to the layers of an arbitrary magnetic layer structure using the nonnonequilibrium Keldysh formalism before introducing our Landauer approach to investigating the transport of charge and spin current in these magnetic multilayers using the simplest paraboilic band model for electrons in each layer.

In Chapter 2 we formulate our approach by defining the general solution to the wave equation for a given layer in a system in terms of the angle by which the spin polarization is rotated in-plane in that layer and the generalized wave vectors for each electron spin band. We determine a general transfer matrix that enables us to solve explicitly the coefficients of the wave functions in each layer of any general multi-layered system before defining an expression for the in-plane and out-of-plane spin current components in terms of these wave functions before detailing our Landauer formalism to calculating the local spin current in a realistic system consisting of ferromagnets with a finite exchange splitting and appropriate boundary conditions.

We apply our formulated approach in Chapter 3 to a set of collinear spin problems whereby the two magnetic layers in our general multilayer junction (consisting of two ferromagnets separated by a non-magnetic spacer layer) have their rotated magnetizations either ferromagnetically or anti-ferromagnetically aligned (parallel/anti-parallel to the net magnetization). Our analytical results provide the necessary conditions for optimising tunneling magnetoresistance (TMR) and show how a `switching' effect can be used to control it. We achieve this by calculating analytically in the ferromagnetic configuration the necessary conditions to support a 100% transmission success rate in one spin channel whilst making it very difficult for transmission to occur in the other spin channel. However, we show conclusively that re-aligning the magnetization to the anti-ferromagnetic configuration under the same conditions will make it very difficult for transmission to occur in either spin channel.

In Chapter 4 we investigate the spin current in a general five layer junction and show that a zero out-of-plane spin current in the nonmagnetic spacer exists only when perfect symmetry is introduced because the contribution from the left cancels exactly the contribution from the right. We identify a number of properties within the nonmagnetic layers and observe the effect of varying the angle of rotated magnetization and width of the polarizing magnet on the spin current components in the nonmagnetic layers.

In Chapter 5 we define the appropriate boundary conditions for our Landauer approach before investigating analytically the origin of out-of-plane spin current in the nonmagnetic spacer in Chapter 6 by looking specifically at an interface between a semi-infinite magnet and a semi-infinite nonmagnet and obtaining qualitative insight into the out-of-plane spin current found in a non-magnetic spacer sandwiched between two finite ferromagnets. In this final chapter we also calculate numerically the effect of an additional insulating barrier on our classical junction consisting of a nonmagnetic spacer sandwiched between two ferromagnets. We compare our results for the charge and spin current in the nonmagnetic spacer to those obtained previously using the Kelydysh formalism before showing for the first time the physical dependence on multiple magnetic interfaces of the out-of-plane spin current in a nonmagnetic spacer and how the out-of-plane spin current in the spacer can be large even when the charge current and the in-plane spin current are both negligibly small.

Publication Type: | Thesis (Doctoral) |
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Subjects: | Q Science > QA Mathematics |

Departments: | Doctoral Theses School of Science & Technology > School of Science & Technology Doctoral Theses School of Science & Technology > Mathematics |