Theoretical Study of Current-Induced Dynamics Of An Antiferromagnetic Skyrmionium

Theoretical Study of Current-Induced Dynamics Of An Antiferromagnetic Skyrmionium

ABSTRACT

In this thesis, we present a theoretical study on the current-induced nucleation and propagation of antiferromagnetic skyrmionium.

A skyrmionium, also known as 2π-skyrmion, is a vortex-like magnetic structure characterized by zero topological charge.

We show by means of micro-magnetic simulation that an antiferromagnetic skyrmionium can be nucleated via a local injection of spin current with toroidal distribution.

Our systematic study of the current-induced dynamics shows that a spatially uniform spin current induces propagation of antiferromagnetic skyrmionium with measurable distortion at high current densities.

We derive expression for the velocity of the antiferromagnetic skyrmionium based on the collective-coordinate model and obtain good agreement with our numerical results.

TABLE OF CONTENTS

Abstract         i
Dedication          ii
Acknowledgements            iii
List of Figures           viii

1 Introduction                                                                              2
1.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Non-collinear magnetic textures . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Antiferromagnetic textures . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Objectives and contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Background                                                               8
2.1 Magnetism in Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Magnetic Moment of an electron . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Ferromagnetic Materials (FM) . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Ferrimagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Antiferromagnetic Materials (AFM) . . . . . . . . . . . . . . . . . . 12
2.2.4 Paramagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.5 Diamagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Anisotropy energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Zeeman Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 Magnetostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.5 Spin Orbit Coupling (SOC) . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.6 Dzyaloshinskii – Moriya Interaction . . . . . . . . . . . . . . . . . . . 19
2.4.7 Effective Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Spin-transfer torque (STT) . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Spin Hall effect and Spin-orbit torques . . . . . . . . . . . . . . . . 25
2.6 Magnetic Skyrmion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.1 Skyrmion Dynamics and Topological Effect . . . . . . . . . . . . . . 27
2.6.2 Skyrmionium or 2π Skyrmion . . . . . . . . . . . . . . . . . . . . . . 31

3 Research Contribution                                                                  33
3.1 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Nucleation of Skyrmionium . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Propagation of Skyrmionium . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Small Ferromagnetic-Canting Approximation . . . . . . . . . . . . . 38
3.3.2 Collective-Coordinate Model . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Skyrmionium core displacement and velocity as a function of time. . 42
3.4.2 Skyrmionium velocity as a function of charge current jc . . . . . . . 44
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Future Outlook                                                                           47
APPENDICES                                                                                49
A Emergent Magnetic and Electric field . . . . . . . . . . . . . . . . . . . . . . 49
B Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
C Numerical Method and Fortran code . . . . . . . . . . . . . . . . . . . . . . 55
C.1 Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . 55
C.2 Forward, backward and central difference derivative method . . . . . 56
C.3 Algorithm of our Fortran code . . . . . . . . . . . . . . . . . . . . . 57
C.4 Fortran code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References                                                                              60

INTRODUCTION

Data storage which has to do with the collective ways and technologies involved in the capturing and retaining of digital information on optical, electromagnetic or silicon-based storage media is an important aspect of human lives, as we rely on it to preserve information that ranges from personal to critical business or general information.

Magnetic memory or magnetic data storage is an important component in computer operation. It depends on the ability of magnets to magnetize other materials and the ability of those materials to retain their magnetization until they are forced to change the direction of their magnetization under the influence of an external magnetic field.

Data are stored in storage devices as set of 0’s and 1’s or binary codes; this is achieved by using an electromagnetic (or the write head) to magnetize the magnetic materials in a specific direction (e.g up direction with bit registered as 1) as it moves over the magnetic material.

The direction of magnetization of the magnetic material can be flipped by changing the direction of the current flowing in the electromagnet coil.

Once written, the read head reads the encoded information or bits. The process of reading and writing information described above uses the principle of Faraday’s law of electromagnetic induction and Oersted field respectively.

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O. Tretiakov, “Antiferromagnetic skyrmions,” 2015. [Online]. Available: http://www.issp.u-tokyo.ac.jp/public/npsmp2015/w/w16-3.pdf

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A. Fert, V. Cros, and J. Sampaio, “Skyrmion on the track,” Nature Technology, vol. 8, no. 3, pp. 152–156, 2013. [Online]. Available: https://www.nature.com/articles/nnano.2013.29.pdf

Y. Y. Dai, H. Wang, P. Tao, T. Yang, W. J. Ren, and Z. D. Zhang, “Skyrmion ground state and gyration of skyrmions in magnetic nanodisks without the dzyaloshinsky-moriya interaction,” Phys. Rev. B, vol. 88, p. 054403, Aug 2013.[Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.88.054403