< Back to previous page

Project

Development of magnetic domain wall based logic and memory concepts

 

A ferromagnetic material consists of several uniformly magnetised regions, which are called magnetic domains and are separated by magnetic domain walls (DWs). The magnetisation inside these magnetic domains has a fixed orientation, which makes ferromagnetism attractive for data storage applications. Indeed, this orientation of the magnetisation can be used to store data and the stored data can be retained without the need for an external power source. In this doctoral thesis, the origin and the different types of magnetism, including ferromagnetism, as well as domains and DWs are discussed. In addition, an overview of several applications based on ferromagnetism is given, thereby emphasising magnetic data storage. A well-estabilished type of magnetic non-volatile memory is the hard disk drive, where the data are stored in local magnetic domains on a platter, which is a flat circular disk coated with magnetic material. In 2004, a new concept of magnetic memory has been proposed, called racetrack memory. Unlike the hard disk drive, this memory has the potential to be a universal memory that combines various advantages of other memories, like the low cost of the hard disk drive as well as the high performance and reliability of solid-state memory. The racetrack is a ferromagnetic nanostrip with a portion of the strip used to store the data. Due to the small cross sectional dimensions of the nanostrip, the magnetisation direction of the domains can only have two states corresponding to bit values 0 or 1. To move the bits of the racetrack past reading and writing elements, current pulses are applied at the ends of the nanostrip. Next to data storage, both field-driven and current-driven motion of DWs in ferromagnetic nanostrips are at the basis of some logic and sensing devices. There is already a commercially available multi-turn rotary sensor (RSM 2800 Series sensor) designed by Novotechnik and now produced by Sensitec. The advantages of the DW motion based technologies aroused the interest to better understand field-driven and current-driven magnetic DW motion. In this doctoral thesis, both the qualitative and the quantitative descriptions of magnetic DW motion are based on the micromagnetic theory. This theory bridges the gap between the quantum mechanical description of magnetism and the constitutive relation between the magnetic induction, the magnetic field strength and the magnetisation used in combination with MaxwellU+2019 macroscopic equations. The magnetisation dynamics is described at the nanometer length scale and the picosecond time scale, thereby approximating the magnetisation as a continuous vector field. This vector field is subject to several torques in the ferromagnetic material. In this thesis, we explain the micromagnetic theory, which is based on the minimisation of the magnetic Gibbs free energy present in the ferromagnetic material. After explaining the different energy terms, we use static micromagnetism (the system is in an energy minimum) to understand the shape of static DWs. In addition, we focus on dynamic micromagnetism (the system is evolving towards an energy minimum), which is governed by the Landau-Lifshitz-Gilbert (LLG) equation. Using micromagnetic simulations, we can describe theDWdynamics most accurately. Therefore, the nanostrip is integrally and spatially divided into non overlapping cuboid shaped cells and the LLG equation is subsequently solved in each cell at every timestep, giving rise to a large number of degrees of freedom. Although this results in the most accurate description of DW dynamics, more insight in DW dynamics is provided by theoretical models which describe the DW in terms of a limited number of variables, also called collective coordinates. In this thesis, we use two types of models: the already wellestablished analytical Lagrangian-based collective coordinate models (CCMs) and the during this PhD developed semi-analytical models. The Lagrangian-based CCMs describe the DW motion by a subset of four collective coordinates, i.e. the DW position, the magnetisation angle inside the DW, the DW width and the geometrical tilting angle. The main advantage of these Lagrangian-based CCMs is that they are computationally very efficient: the equations of motion can be solved purely analytically since there is one equation of motion for every collective coordinate. Moreover, they are very useful for conceptual studies or back-of-the-envelope calculations. However, since these Lagrangian-based collective coordinate models rely on a simplified ansatz for the DW profile, they are not always accurate or applicable. Therefore, we developed the semi-analytical approach which does not need an ansatz for the DW profile and is based on averaging the LLG equation over the DW volume. The DW velocity and the dynamics of the magnetisation angle inside the DW are defined in a form that can be linked to the averaged LLG equation. By combining these definitions with the averaged LLG equation, we derive semi-analytical equations of motion for the DW velocity and the dynamics of the magnetisation angle inside the DW as a function of the energetic interactions, just like the equations of motion in the Lagrangianbased CCMs. Subsequently, several DW variables can be identified. This is contrary to the Lagrangian-based CCMs where all the DW variables are predefined. The DW dynamics is decribed by sixteen DW variables: equivalents of the four analytical DW variables, five DW variables that account for asymmetry within the DW and variables corresponding to a constant or zero in the Lagrangian-based CCMs. Contrary to the Lagrangian-based CCMs, there are more DW variables than equations of motion, so the micromagnetic simulations are necessary to solve the equations of motion. Although the need for micromagnetic simulations makes the semi-analytical approach computationally more intensive than the Lagrangian-based CCMs, it enables a more in-depth investigation of the DW dynamics as well as a detailed analysis of the limitations of the Lagrangian-based CCMs. For both field-driven and current-drivenDWdynamics in nanostrips, we consider two regimes ofDWmotion. The regime of theDWmotion is determined by the excitation strength. When the excitation strength is below the so-called Walker Breakdown (WB), the internal degrees of freedom of the DW are able to adapt to the external field or the current. After that, the DW moves at a constant speed. In contrast, when the excitation strength is above the WB, the DW is no longer able to adapt to the external field or the current and both the DW speed as the DW itself change continuously. Field-driven and current-driven DW dynamics in PMA nanostrips (perpendicularly magnetised domains) are investigated in the absence and in the presence of the Dzyaloshinskii-Moriya interaction (DMI), which is relevant for the development of racetrack memory. Apart from enhancing the understanding of DW motion in nanostrips beyond the limitations of the Lagrangian-based CCMs, the semi-analytical approach is also used to study the accuracy of the predictions of the Lagrangian-based CCMs. Despite the fact that Lagrangianbased CCMs describeDWmotion in PMA nanostrips quite well, we still identify several differences between these models and micromagnetic simulations.For PMA nanostrips without DMI, we also introduce the fingerprint method, which is based on the semi-analytical approach. Using this method, we demonstrate how one micromagnetic simulation at a fixed, but arbitrary excitation strength above the Walker breakdown is sufficient to characterise the DW motion for a given set of material and geometrical parameters. This demonstrates that the semi-analytical approach has predictive power and that it can be used as a tool to reduce the computational time needed to characterise the DW motion in such nanostrips. For PMA nanostrips with DMI, we use the semi-analytical approach to study the effect of the DMI and additional external in-plane fields on the DW motion instead. Both field-driven and current-driven DW dynamics are discussed in both narrow and wide Permalloy nanostrips (in-plane magnetised domains), which is relevant for some logic and sensing devices. Therefore, we mainly use the semi-analytical approach. Compared to PMA nanostrips, the DW dynamics in Permalloy nanostrips is much more complex and the DW profile of DWs in Permalloy nanostrips is not well described by the analytical ansatz. As expected, we then found that Lagrangian-based CCMs are significantly less accurate in their predictions for the DW dynamics in Permalloy nanostrips than in PMA nanostrips. A slightly modified semi-analytical approach combined with micromagnetic simulations proves to be very useful, enabling again an in-depth investigation of the DW dynamics beyond the limitations of Lagrangian-based CCMs. In the case of field-driven DW motion below the WB, we found that the DW asymmetry plays a much larger role in the DW dynamics in Permalloy nanostrips than in PMA nanostrips. In contrast, for current-driven DW motion below the WB, we found that the DW asymmetry does not directly affect the DW dynamics, just like in PMA nanostrips. Note that this is even the case when the DW is asymmetric in the absence of any excitation. We also identify the universal role of the out-of-plane magnetisation tilting, which varies locally and is only strong in a small part of the DW, on both field-driven and current-driven DW dynamics: the DW velocity is linearly proportional to the out-of-plane tilting; apart from a proportionality factor, a fixed out-of-plane tilting gives rise to identical DW velocities for field-driven and current-driven DW motion; at the WB field and the WB current, the outof- plane magnetisation tilting is identical. To study the DW dynamics at excitation strengths above the WB, a qualitative study based on the torques in the LLG equation combined with an analysis of the evolution of the semianalytical DW variables as a function of time provides a lot of insight. Furthermore, we also demonstrate how the fingerprint method is successfully adapted for DW motion in Permalloy nanostrips. Recently, the interest is aroused to better understand field-driven bubble wall (BW) motion in thin films. This BW motion is a two-dimensional DW motion. This is relevant for the magnetic bubblecade memory, which was introduced in 2015 as a new concept of magnetic memory. It is a new scheme for the unidirectional bubble motion based on magnetic bubble domains in PMA thin films in the presence of the DMI. Opposed to the original bubble memory, which was considered to be a contender for a universal memory in the early 1980s, the bubble domains can be easily displaced by an oscillating magnetic field without the need for patterned films. We extend the semi-analytical approach to investigate field-driven chiral bubble dynamics in the presence of the DMI. More specifically, we define semianalytical expressions for the bubble center coordinates. We also derive both global and local semi-analytical equations of motion for the bubble wall (BW), thereby taking its two-dimensional nature into account. These semi-analytical equations of motion also account for BW asymmetry and they look like the semi-analytical equations of motion for DWs in PMA nanostrips. Since the BW asymmetry is strongly related to the BW curvature, BW asymmetry is always present in BWs. This is in contrast to DW asymmetry in nanostrips. We investigate the BW motion quantitatively under a constant external magnetic field by (1) a case study for symmetric bubbles and by (2) a case study for asymmetric bubbles. Both expanding and shrinking bubble dynamics are investigated and an expression for the transition field between shrinkage and expansion is derived. When the excitation strength is not large enough to compensate for the BW curvature, the bubble will shrink, otherwise it will expand. In other words, the BW motion is determined by the imbalance between the external field strength and the BW curvature. External in-plane fields break the symmetry of the bubble form due to the interaction of these fields with the local in-plane BW magnetisation. They significantly affect the local BW width, the local in-plane magnetisation angle and the local velocity. Finally, using the semi-analytical approach we also show how the recently proposed magnetic bubblecade memory can operate in the flow regime in the presence of a tilted sinusoidal magnetic field and at greatly reduced bubble sizes compared to the original bubblecade memory. At the end, we make some general conclusions as well as perspectives for further research. Building on the work done for this thesis, we identify several possibilities for future research. One possibility is to use the fingerprint method for the investigation of the influence of geometrical and material parameters on the DW dynamics. Another possibility is the extension of the semi-analytical approach to account for additional current dependent spin orbit interactions present in PMA materials with DMI. One could also extend the semi-analytical framework to the description of other magnetic objects, like e.g. skyrmion motion in ferromagnetic nanostrips.

Date:1 Jan 2014 →  31 Dec 2017
Keywords:ferromagnetism
Disciplines:Electronics