Study of Magnetic Domain Wall Dynamics for Memory and Logic Implementation.

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ABSTRACT

JUMADE, RAGHAV JAYANT. Study of Magnetic Domain Wall Dynamics for Memory and Logic Implementation. (Under the direction of Dr. Veena Misra.)

With conventional CMOS scaling approaching its end, implementations of logic

and memory in non-conventional technologies have become focus of research. Logic

schemes based on motion of magnetic domains in ultra-thin ferromagnetic films have been

anticipated to offer probable low-power, non-volatile alternative to CMOS design. Speed

of magnetic domain walls has been found to be low and unlike motion of electrons (or

holes), the motion of magnetic domain walls is not very well understood. This report

presents scrutiny of dynamics of magnetic domain walls in order to bring out design goals,

limitations and various trade-offs in this technology. Motion of magnetic domain walls in

Permalloy (N i80F e20) nanowires caused by application of both external magnetic field and

electric current is the focus of this thesis. In this document, effect of material parameters

like saturation magnetization (MS) and Gilbert damping factor (α) on velocities of

magnetic domain walls has been illustrated. These parameters also affect the Walker

Breakdown process and this development has also been presented. Similarly, effect of

nanowire dimensions on their speed characteristics has also been analyzed. In addition

to these, simulations showing consequence of increasing amounts of non-adiabatic spin

torques are also performed. Using each of these simulations, pattern between variation

of Walker Breakdown point, critical current density (or field), mobility, etc. and the

controlling parameter (magnetization, damping, nanowire dimension or non-adiabaticity)

can be developed. These simulations are also extended to nanowires with Perpendicular

Magnetic Anisotropy (PMA). The relationships obtained in simulations described above

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logic gates using Co/Pt multi-layer PMA nanowires. All nanowire devices are simulated

using nmag-0.1 and rings are simulated using OOMMF. Simulations are used to assess

performance of designed memory and logic devices and compared to existing CMOS

technology. These comparisons based on our simulations reveal that logic and memory

implementations based on motion of magnetic domain walls provide higher functional

density and lower power consumption than CMOS implementations. These advantages

are obtained at the cost of speed of operation. Low power and higher density along with

non-volatility imply that this technology can be a great aid for embedded data processing

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c

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Study of Magnetic Domain Wall Dynamics for Memory and Logic Implementation

by

Raghav Jayant Jumade

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Master of Science

Electrical Engineering

Raleigh, North Carolina

2010

APPROVED BY:

Dr. John Muth Dr. Salah Bedair

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DEDICATION

To

Aai and Baba

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BIOGRAPHY

The author was born in a well educated family on 18 July 1986. He completed his

Bachelor’s degree in Electronics and Communication Engineering from Visvesvaraya

National Institute of Technology. He then worked in Dr. Krishnapura’s group at Indian

Institute of Technology Madras. In August 2008, Raghav started his Masters at North

Carolina State University where he worked in Dr. Misra’s group, implementing logic

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ACKNOWLEDGEMENTS

Infinite stars in the sky, so few that catch the eye...

Firstly, I must thank my advisor Dr. Veena Misra for creating the opportunity for me

to work on this project. I am grateful to her for her unrelenting support, interest and

critique throughout this work. I am also honored to have Dr. Muth and Dr. Bedair on

my graduate committee. I greatly value their suggestions which helped me improve this

work.

My group members helped me sustain the most barren of times and to enjoy the

packets of success. I appreciate the help I received from Rebecca, Rahul, Steven, Srikant,

Smita, Xiangyu, Jason and Casey. The discussions and debates I had with them were

most educating and entertaining.

I must also thank my room-mates Nikhil and Bhushan for putting up with my quirks.

If not for their culinary skills, American fast-food would have claimed another soul. I

am indebted to my friends - Saket, Nandu and Prashanth, for directing me towards the

decisions I made.

Most importantly, I grab this opportunity to thank my loving parents, Vandana

and Jayant, for maintaining faith in me and supporting me throughout. I run short of

words and action to express my gratitude towards them. I am also thankful to my little

sister Anushree for being my eyes and ears back home. I thank my family for being the

ubiquitous guiding force in my life.

I extend my gratitude and sincere apologies to all others, and there are many, who

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LIST OF TABLES

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LIST OF FIGURES

Figure 2.1 Structure of (a) Bloch Wall and (b) Neel Wall. . . 13 Figure 2.2 Difference between in-plane walls. (a) Transverse Wall and (b)

Vortex Wall. . . 13 Figure 2.3 Changing shape of the domain wall as width of the nanowire is

increased. Nanowire widths for (a)-(d) are 10nm, 30nm, 50nm and 100nm respectively. . . 15 Figure 2.4 Variation of length of domain wall (∆) with width of nanowire for

various nanowire thicknesses. . . 17 Figure 2.5 Variation of length of domain wall (∆) with thickness of nanowire

(Width = 100nm). . . 17 Figure 2.6 Device geometry used to obtain domain wall velocity from

magneti-zation curve. . . 19 Figure 2.7 Change in net magnetization of a nanowire due to motion of

mag-netic domain wall. . . 19 Figure 2.8 Typical DW Speed v/s Applied Field Curve: Figure shows the

nature of domain wall in pre and post Walker Breakdown regions. 21 Figure 2.9 Magnetization curves of nanowires with different cross-sections

under external applied field of 10Oe: (a)N W1 is 10×10nm2 while (b) N W2 is 20×10nm2. . . 23 Figure 2.10 Plot showing change in average domain wall velocity with applied

magnetic for nanowires having thickness 10nmand different widths. 25 Figure 2.11 Plot showing the effect of width of nanowire (thickness = 10nm)

on the field at which Walker Breakdown occurs in it. Figure also shows the effect of width of nanowire on maximum velocity reached before Walker Breakdown. . . 25 Figure 2.12 Variation in domain wall velocity - magnetic field curves of nanowire

(50×10nm2 in cross-section) with change in damping coefficient of the material. . . 27 Figure 2.13 Variation of peak velocity with Gilbert damping of the nanowire.

Red line indicates the fitting with curve of the form y=A/(x−x0). 27 Figure 2.14 Variation of Magnetization curves with change in saturation

magne-tization MS of the nanowire (50×10nm2 in cross-section) material. Inset shows variation of the point of Walker Breakdown due to change in saturation magnetization. . . 29 Figure 2.15 Definition of edge roughness in a Permalloy nanowire. Depth ’d’ in

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Figure 2.16 Magnetization curves of rough permalloy nanowire of cross-section 20nm×10nm. Different traces refer to different seeds used for ran-domization. (a) has higher magnitude of roughness (d0 = 0.25nm) compared to (b) withd0 = 0.1nm. . . 32 Figure 2.17 Variation of speed of magnetic firld driven domain wall motion in

rough nanowires. Each curve relates to different amount of added roughness. . . 34 Figure 2.18 Figure shows variation of Walker Breakdown field HW B and speed

vW B with magnitude of roughness added to a smooth nanowire. . 35 Figure 2.19 Figure showing domain speedv v/s external longitudinal fieldHlong

for increasing amounts of transverse magnetic fields. For all of these, transverse field is in the same direction as the domain wall. . . 36 Figure 2.20 Variation of the change in point of Walker Breakdown (δHW B,δvW B)

with applied positive transverse field. This plot can be divided into three distinct zones HT rans < 60Oe, 60Oe < HT rans < 150Oe,

HT rans >150Oe. . . 36 Figure 2.21 Variation of the change in point of Walker Breakdown (δHW B,

δvW B) with transverse field applied opposite to the direction of domain wall. . . 38 Figure 2.22 Variation of the change in point of Walker Breakdown (δHW B,

δvW B) with transverse field applied out of plane of the nanowire. Figure makes it clear that amount of improvement obtained is lower than the situation where transverse field is in plane of the nanowire. 39

Figure 3.1 The relative orientation of spin-torque and damnping force which leads to precession of atomic magnetic moments. . . 46 Figure 3.2 Phenomenon of Adiabatic Spin Torque Transfer: Figure shows the

change spin of an electron when it crosses the domain wall. . . 47 Figure 3.3 Process of Non-adiabatic Spin Torque Transfer: Figure shows

re-flection of a spin polarized electron at the magnetic domain wall. . 50 Figure 3.4 Figure shows change in domain wall speed with respect to Spin

velocity for nanowires of varying widths. Spin velocity is the measure of current passing through the nanowires. ’u’ in these figures can be replaced by ‘J’ using scaling factor of 1.38×1010. . . 51 Figure 3.5 Variation of critical spin velocity with respect to width of the

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Figure 3.6 Variation of the point of Walker Breakdown with width of the wire. ‘uW B’ represents the maximum current that the nanowire can carry

in absence of external magnetic field before experiencing sharp degradation of mobility. Similarly, ‘vW B’ is the maximum velocity of domain wall motion before Walker Breakdown occurs. . . 53 Figure 3.7 Effect of changing Gilbert Damping on Adiabatic Current Driven

Domain Wall Motion in a nanowire of cross-section 50×10nm2_{. .} ₅₄ Figure 3.8 Linear relationship between critical current density (represented by

critical spin velocity) and Gilbert Damping of the nanowire. . . . 55 Figure 3.9 Variation of the point of Walker Breakdown with damping factor of

material of the wire. ‘uW B’ represents the maximum current that the nanowire can carry in absence of external magnetic field before experiencing sharp degradation of mobility. Similarly, ‘vW B’ is the maximum velocity of domain wall motion before Walker Breakdown occurs. . . 56 Figure 3.10 Domain Wall Speed versus Spin Velocity (Electric Current

Den-sity) curves of permalloy nanowire (cross-section 50×10nm2_{) with} different values of saturation magnetization. . . 57 Figure 3.11 Variation in key features of domain wall motion with various values

of saturation magnetization MS. . . 57 Figure 3.12 Domain Wall speed plots for permalloy nanowires of cross-section

50×10nm2 and damping coefficient 0.05 at various values of non-adiabaticities. . . 59 Figure 3.13 (a) Variation of maximum electric current that can be sustained

before Walker breakdown occurs. Each curve represents different value of Gilbert damping. (b) Variation of maximum speed of domain walls reached before Walker Breakdown occurs for given value of damping coefficient. . . 61 Figure 3.14 Consequence of applying transverse field along the direction of

magnetic domain wall on its motion. Figure shows the effects of transverse field on features of domain wall motion like critical current density and Walker Breakdown. . . 64 Figure 3.15 Consequence of applying transverse field opposite to the direction

of magnetic domain wall on its motion. Figure shows the effects of transverse field on features of domain wall motion like critical current density and Walker Breakdown. . . 64 Figure 3.16 Effect of applying out-of-plane transverse magnetic field on current

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Figure 3.17 Each of the features of current driven domain wall motion plotted to a proper scale to analyze the effect of magnitude of out of plane field on these parameters. . . 66

Figure 4.1 Comparison of Domain Wall dynamics in materials with in-plane and perpendicular magnetic anisotropies. PMA type nanowires show much lower critical spin velocity. . . 70 Figure 4.2 Variation of Domain Wall velocity characteristics in materials with

PMA. Simulations were done for wires with MS = 800kA/m and

α= 0.03. . . 72 Figure 4.3 Variation of features of domain wall motion, viz, critical spin velocity,

Breakdown spin velocity and domain wall speed at Walker point in materials with PMA. Simulations were done for wires withMS = 800kA/mandα= 0.03. Hollow symbols represent the resuls for IMA. 72 Figure 4.4 Impact of saturation magnetization on the characteristics of

mag-netic domain wall motion in PMA type nanowires of cross-section 50×10nm2 _and _α _{= 0}_._{03. Hollow symbols represent the resuls for} IMA. . . 74 Figure 4.5 Variation in critical spin velocity required to start adiabatic domain

wall motion in PMA type nanowires. Hollow symbols represent the results for IMA. . . 75 Figure 4.6 Variation of Walker Breakdown point with change in saturation

mag-netization. For comparison results from IMA simulations denoted by hollow symbols are also presented. . . 75 Figure 4.7 Variation in critical spin velocity required to start adiabatic domain

wall motion in PMA type nanowires as a function of Gilbert damping. Hollow symbols represent the results for IMA. . . 76 Figure 4.8 Variation of Walker Breakdown point with change in Gilbert

Damp-ing. For comparison results from IMA simulations denoted by hollow symbols are also presented. . . 76 Figure 4.9 Simulation results for non-adiabatic domain wall motion in nanowires

with PMA. Results can be compared to those from IMA type wires (hollow symbols). . . 78 Figure 4.10 Variation of current at Walker Breakdown point (represented by

uW B) with change in ration β/α. Results can be compared to those for IMA using traces with hollow symbols. . . 79 Figure 4.11 Variation of domain wall speed at Walker Breakdown point

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Figure 4.12 Comparing the maximum domain wall speed obtained before Walker Breakdown. The figure shows that under adiabatic region CDDWM in IMA type wires is slower compared to both field driven and CDDWM PMA wires. . . 82 Figure 4.13 Comparison showing that the fastest motion of domain walls is

obtained in PMA nanowires under non-adiabatic situations. . . . 82 Figure 4.14 A nanowire containing more than two domains. (a) shows the way

the nanowire is defined. It also shows the definition of the term bit-length. (b) shows the domain structure of the nanowire after relaxation. Domain walls have grown to their stable sizes. . . 83 Figure 4.15 Magnetization characteristics of the nanowire when the magnetic

domain engulfed by the two domain walls remains unharmed till the end of simulation. MX is replaced by MZ in case of PMA type wires. . . 85 Figure 4.16 Magnetization characteristics of the nanowire when the magnetic

domain engulfed by the two domain walls gets destroyed during motion of domain walls. MX is replaced by MZ in case of PMA type wires. . . 85 Figure 4.17 Variation of minimum bit-length required to ensure non-destructive

transmission of data as a function of applied current density (repre-sented by uC) and Gilbert Damping. These simulations are for the case of IMA type wires. . . 86 Figure 4.18 Variation of minimum bit-length required to ensure non-destructive

transmission of data as a function of applied current density (repre-sented by uC) and Gilbert Damping. These simulations are for the case of PMA type wires. . . 87

Figure 5.1 B-H Loop of a bulk ferromagnetic maerial. Figure indicates co-ercivity and remanence of the material. Remanence is directly proportional to saturation magnetization of the material. . . 92 Figure 5.2 Magnetization Characteristics of circular permalloy rings of inner

diameter 50nm and outer diameters varying between 100nm and 200nm. . . 95 Figure 5.3 Magnetization Characteristics of circular permalloy rings of outer

diameter 200nm and inner diameters varying between 0nm and 150nm. The ring with zero inner diameter is actually a disk and therefore shows peculiar magnetization curve . . . 96 Figure 5.4 Figure explaining the magnetization states of the ring at different

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Figure 5.5 Variation in magnetization curves of elliptical rings of various widths and major axes. For comparison, magnetization curves of corre-sponding circular rings are also given. . . 98 Figure 5.6 Variation in magnetization curves of rhombus shaped rings of various

widths and major axes. It clearly shows the onion rotation in the case of ring with Major Axis (diagonal) 200nm and width 25nm. . 99 Figure 5.7 Figure shows the actual implementation of memory using ring

shaped ferromagnetic films. The Magnetic Tunnel Junction shows a high TMR ration and hence large output swings are obtained. . 101 Figure 5.8 Figure shows the PMA inverter of [10]. . . 104 Figure 5.9 Working of the PMA inverter is explained in this figure. Figure

depicts the concept of dipolar field and its effect on magnetization of the output wire. . . 105 Figure 5.10 Figure shows variation of field required for nucleation of domain

wall in a nanowire as a function of its saturation magnetization. . 106 Figure 5.11 Variation of nucleation field with variation in thickness of the nanowire.106 Figure 5.12 Figure shows variation of field required for nucleation of domain

wall in a nanowire as a function of its saturation magnetization. . 108 Figure 5.13 Variation of nucleation field with variation in thickness of the nanowire.108 Figure 5.14 Figure shows proposed improvements in the PMA inverter. All

numbers indicate distances in nanometer units. Nanowires in blue indicate higher MS while those in green indicate lower MS. . . 110 Figure 5.15 Transient Response of the inverter. Values of magnetization at

various points on the inverter in Figure 5.14 are shown in this figure. ‘X’ is the input while ‘Y’ is the final output. ‘Y0’ is an intermediate

node. . . 112 Figure 5.16 Simplest implementation of NAND Gate using PMA. . . 113 Figure 5.17 Proposal for AND gate using PMA and current driven domain wall

logic. Numbers indicate lengths in nanometer units. Thickness of all nanowires is kept constant at 20nm. . . 114 Figure 5.18 Transient response of proposed AND Gate. Figure shows the change

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CHAPTER

ONE

Introduction

1.1 Background

In the last few decades, growth of electronics industry has been governed by scaling of

CMOS devices. Over the last few years, the size of CMOS devices has become so small that

degradation of device performance due to various quantum effects has become a limiting

factor [1]. The era of CMOS scaling is on the brink of coming to its end. Therefore,

novel techniques to drive scaling of devices are being actively sought. Conventional

electronics is based completely on the flow of electronic charge. Apart from their charge,

electrons possess another feature called ‘spin’. Spin of the electrons has remained a

very scarcely tapped resource throughout the history of development of electronics. In

recent times, interest has grown, mainly among the community of physicists, to exploit

this quantum feature to process information. This branch of research and technology is

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using magnetic properties of materials. Interest in magnetic materials for making memory

devices is quite logical because of the fact that ferromagnets have inherent property of

magnetic hysteresis. ‘Magnetic Bubble Memory’ had been proposed by Bobeck as early

as 1960s [2]. It fell off the agenda because Hard Disk Drives (HDDs) became cheaper

and denser due to the discovery of Giant Magneto-Resistance (GMR) [3]. Lately, the

interest in magnetic memories has shifted towards Non-Volatile Random Access Memories

(NVRAMs) called Magnetic Random Access Memory (MRAM) [4]. This new technology

is a resistive memory which is proposed to be highly scalable and thus is a contender for

becoming a Universal memory. Proposals for Racetrack memory have also been advanced

in recent years. It uses motion of stored magnetic domains in a ferromagnetic racetrack

to organize very densely packed memory [5].

With the magnetic memory devices advancing towards commercial production, interest

is growing towards making logic devices based on spintronics. Magnetic Quantum Cellular

Automata (MQCA) has been under investigation for over two decades [6, 7, 8]. Recently,

a novel logic scheme using motion of domain walls has been proposed. Domain Wall logic

[9] uses motion of magnetic domain walls under the influence of elliptically rotating field

to implement logic functions like NOT, AND and pass-over. Similarly, there have been

few more proposals that show implementation of inverter using various arrangements

of ferromagnetic wires and dots [10, 11, 12, 13]. In most of these schemes, however, a

full logic set has not been demonstrated. The most eye-catching techniques other than

Domain Wall Logic have been the inverter implemented using Perpendicular Magnetic

Anisotropy (PMA) [14] and logic scheme using Anisotropic Magneto-Resistance (AMR).

All these developments have motion of magnetic domain walls at their heart. It is

therefore that a lot of research effort has been focused on understanding the dynamics of

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and developed in early 1960s[15, 16, 17]. However, there is still lack of understanding

and consensus regarding various features of domain wall dynamics [18, 19, 20]. In recent

times, various studies have been carried out to study the features of domain wall motion.

This thesis is dedicated to achieve intuitive understanding of these domain wall dynamics.

1.2 Concepts and Definitions

To proceed forward, it is now important to introduce basic concepts of magnetic

materials and magnetic theory. In the next section, various terms used throughout this

thesis are defined. Similarly, effort has been made to explain various concepts of magnetic

theory.

1.2.1 Magnetic Materials

Magnetic materials are those in which the electronic configuration is such that there

exist cites of strong magnetic moment in their crystal structure. These atomic magnetic

moments may be aligned together or may be randomly scattered. The nature of these

moments helps to classify materials into various types. The material types of primary

interest for developing spintronic devices are the ferromagnets, anti-ferromagnets and

diluted magnetic semiconductors. Ferromagnetic materials are distinguished from rest of the magnetic materials on account of the strong coherence in their atomic magnets.

That is why bulk ferromagnets usually in a magnetized state. These materials are

strongly attracted (or repelled) by other magnets. It is these ferromagnets that are

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any net magnetization. But these materials interact with ferromagnetic materials in

order to fix their magnetization. That is why; anti-ferromagnets are commonly used

in devices requiring a ferromagnet as reference. Other common magnetic materials are

paramagnetic materials which have a weak interaction among their atomic magnets. Hence these get magnetized weakly under influence of magnetic field and lose their

magnetism on removal of the applied field. Similarly, diamagnetic materials oppose the magnetic field. Superconductors are a good example of diamagnetic materials. These

materials are generally used to sense the strength of magnetic fields.

A new generation of materials called Diluted Magnetic Semiconductors (DMS) have been a research interest. With these materials, it might become possible to control

the magnet-electronic properties of materials by controlling the magnetic doping and

charge concentration in these materials. However, it has been to achieve ferromagnetism

in these materials at room temperature.

1.2.2 Characteristic Features of Magnetic Materials

In rest of this thesis, only ferromagnetic materials have been used. There, here

forth the term ‘magnetic materials is used interchangeably with ferromagnetic materials.

Ferromagnetic materials can be completely described macroscopically using following

properties.

Saturation Magnetization (MS) : The saturation magnetization is the maximum induced magnetic moment that can be obtained in a magnetic field; beyond this field no

further increase in magnetization occurs. It is thus the measure of density of magnetic

moments that can be aligned parallel to one another. It determines the strength of

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Gilbert Damping (α): Gilbert Damping factor is analogous to the factor by which a spin wave gets damped in a material. It determines the opposition to any change in

direction of the atomic magnetic moments.

Exchange Coefficient (A): It determines the amount of coupling between neigh-boring atomic moments. It thus determines the effect of change of one atomic magnet

on its neighbor. It is the property of charged particles to share the overlap of their

wave-functions.

Anisotropy Constant (K): Anisotropy is the property of material to choose a particular direction. Depending on the anisotropy of the material, the preferred axis of

magnetization may be a particular crystal direction or one of the coordinate directions

dependent on shape of the device.

1.2.3 Magnetic Domain Walls

In order to define domain walls it is imperative to define magnetic domains. Magnetic

domain is a region of the magnetic material, where all the atomic magnetic moments are

aligned in the same direction. In paramagnetic materials, these zones are very small in

size whereas in ferromagnetic materials these domains can be very large. The regions that

separate adjacent magnetic domains are called magnetic domain walls. In the devices

that this thesis is associated with, thin narrow films of ferromagnetic materials are used.

Therefore limited variations of domains are possible. In such cases, the domain walls

can be considered as domains in their own right. However domain walls are different in

the way that they can exist only between opposing domains. In the absence of opposing

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1.2.4 Theory of Magnetism

All the magnetic properties can be derived from quantum mechanics by simply solving

spin dependent Schrodinger equation. However, studying dynamics of magnetic domain

walls using quantum theory can be excessively intensive computationally and therefore

very cumbersome. Therefore, micro-magnetic theory which is slightly more macroscopic

in nature is used to describe magnetization dynamics of thin films of ferromagnets. The

micro-magnetic theory is based on using overall magnetic parameters of materials defined

above to obtain the orientation of moments. In micro-magnetic theory, magnetization

of the moments remains unfazed in strength but may rotate in any possible direction

thus changing the effective magnetization of the material. The basis function for

micro-magnetic theory is called the Landau-Lifshitz-Gilbert equation given in Eq. 1.1 [21]. The

LLG is solved to get magnetization profile and the solution is used to calculate energy of

the system. The best solution is the one which allows minimum Eigen value. Thus, the

exact magnetization of the system can be derived using micro-magnetic theory.

∂m

∂t =γH×m+αm× ∂m

∂t (1.1)

1.2.5 Energies in Magnetic Theory

Anisotropy Energy: Anisotropy energy is the internal energy of the magnetic system on account of the direction of its magnetic moment. It is given by Eq. 1.2.

EAnis =K·sin2(θ) (1.2)

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magnetic moment and Hext is the externally applied field.

EZeeman =−M ·Hext (1.3)

Demagnetization Energy: Magnetization of a material leads to surface poles which introduces demagnetization field internally. This field opposes the magnetization and

increases the demagnetizing energy of the system given by Eq. 1.4 where HD is the

demagnetizing field.

EDemag = 1

2HD·M (1.4)

Exchange Energy: Exchange energy is the increase in internal energy of a magnetic system on account of friction between neighboring magnetic moments. It is given by

Eq. 1.5 where A is the exchange coefficient.

EEx =A(∇ ·M)2 (1.5)

1.3 Objective

Till now, the existing status of the study of microelectronic research has been presented.

The main aim of this thesis is to identify the design parameters which can be used to

effectively implement logic and memory structures using various shapes of ferromagnetic

thin films. It is known that the features of magnetic materials that can be controlled

by designers are the device geometries, the saturation magnetization and the damping

factor. Another option available to engineers is to choose the input used to manipulate

magnetization of the thin films. In this research work, the effect of various parameters

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studies are used to effectively design a memory scheme using rings of ferromagnetic

materials. Knowledge from the same simulations is used to design a functionally complete

set of logic gates. The organization of this thesis is as follows. In Chapter-2, motion of

magnetic domain walls using external magnetic field is studied in detail. In Chapter-3,

current driven domain wall motion is examined for use in place of field driven domain

wall motion. In Chapter-4, the effect of adding perpendicular magnetic anisotropy to

current driven domain wall motion is explored. In Chapter-5, use of ferromagnetic rings

for use in magnetic memory is explored. At the same time NAND/NOR gate and an

inverter are designed using perpendicular magnetic anisotropy. Finally, in Chapter-6, the

contributions of this thesis are summarized. This technology is then compared to the

existing CMOS technology in respects of speed of operation and power consumption. In

the end, proposal for future research is presented.

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[18] K. Fukumoto, W. Kuch, J. Vogel, F. Romanens, S. Pizzini, J. Camarero, M. Bonfim, and J. Kirschner, “Dynamics of magnetic domain wall motion after nucleation: Dependence on the wall energy,” Phys. Rev. Lett., vol. 96, p. 097204, Mar 2006.

[19] J. C. Slonczewski, “Dynamics of magnetic domains and walls,” Journal of Applied Physics, vol. 69, no. 8, pp. 4590–4592, 1991.

[20] G. Tatara and H. Kohno, “Theory of current-driven domain wall motion: Spin transfer versus momentum transfer,” Phys. Rev. Lett., vol. 92, p. 086601, Feb 2004.

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CHAPTER

TWO

Domain Wall Motion under Magnetic Field

Motion of magnetic domain walls (DW) in thin nanowires is sought to actualize logic

and memory structures based on magnetic states of solid state materials. The most

elementary cause for movement of a domain wall is the application of magnetic field.

Applied magnetic field leads to increase in energy of the system, if magnetization of the

nano-structure is not aligned with it. An existing domain wall moves in such a way that

the domain of nanowire which is aligned to the external field is enlarged; simultaneously

shrinking the domain opposite to applied field. The Zeeman energy of the nanowire is

thus reduced, while there is little effect on the Exchange and Anisotropy energies. The

speed at which the domain wall traverses through the nanowire is dependent on strength

of the applied field as well as the magnetic properties of the nanowire. The cross-section

of the nano-stripe determines the nature of the domain which is integral to its motion.

Magnetic properties of the material like Saturation magnetization (MS), Gilbert Damping

(29)

In this chapter, evolution of the domain wall with dimensions of the nanowire is

studied. It is shown that, size and nature of domain wall play the most important role

in motion of domain wall. Based on these results, conclusions can be drawn about ideal

nanowire structure. Further, mobility of magnetic domain walls in thin nanowires with

increasing strength of applied field is investigated. The analysis of Walker Breakdown is

essential to complete the commentary on domain wall motion. Walker Breakdown, being

the single most damaging phenomenon affecting domain wall motion, must be avoided for

as long as possible. Effectiveness of some of the proposed techniques to subdue Walker

Breakdown is evaluated in the final section. To conclude this chapter, general inferences

from this chapter are asserted and critical problems with magnetic field driven domain

wall motion are discussed.

2.1 Nature of Domain Walls in Nanowires

In ferromagnetic materials, the well defined region separating neighboring magnetic

domains is known as a domain wall. Therefore, any attempt to generate a magnetic

domain in conflict with the existing magnetization leads to formation of a magnetic

domain wall. Various techniques to nucleate domain walls in thin ferromagnetic films

have been proposed [1, 2, 3]. In very thin films, nucleation of domain wall and its motion

is the most dominant mode of magnetization switching. Therefore, motion of domain

walls is at the heart of discrete magnetic logic. The velocity of motion of magnetic domain

wall depends on the type and size of the domain wall. There are two main types of

magnetic domain walls, viz, Bloch Wall and Neel Wall. In case of the Neel wall, there

is an in-plane one dimensional rotation of magnetization in going from one domain to

(30)

Figure 2.1: Structure of (a) Bloch Wall and (b) Neel Wall.

(a)

(b)

(31)

easy plane of magnetization [4]. These two types of magnetic domain walls are shown

in Figure 2.1. Neel wall dominates over Bloch wall as the thickness of the films goes on

reducing and therefore bears more significance for this work. Again both these types of

walls can be further divided into two sub-types transverse and vortex. Figure 2.2 shows

the distinction between transverse and vortex walls [5]. More complicated domain wall

structures also arise depending on various anisotropies of the nanowire[6]. In general, the

nature of domain walls depends on the size and shape of the nanowire. In this section,

effects of nanowire dimensions on the structure of the domain wall are studied.

Domain walls are normally nucleated in a nanowire by applying opposite magnetic field

of sufficient strength. For the purpose of simulation studies, however a different approach

is used. The nanowire is laid out with its easy axis in the X-direction. Therefore, the

preferred domain directions are ±X. Half of the nanowire is predefined to be magnetized

in the +X direction while other half of the nanowire is defined to be magnetized in X

direction. When a nanowire with such initial conditions is allowed to relax under zero

external fields, a domain wall is formed in the middle of the wire. To reduce the time

required for these simulations, a very high value of damping factor is chosen. Domain

wall structure is obtained only by minimization of exchange energy and is therefore

independent of the Gilbert damping (α). Damping coefficient (α) only determines the

transient response of the relaxation process and therefore, the time in which convergence

is obtained.

In the first set of simulations, the width of the nanowire is changed keeping its thickness

constant. The nanowire is allowed to relax and nature of the resulting domain wall is

analyzed. Figure 2.3 shows the pictures of domain walls in nanowires with thickness

10nm and varying widths. Sample (a) of Figure 2.3 shows domain wall in a nanowire

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(a)

(b)

(c)

(d)

(33)

that there is no shape anisotropy at the point of the domain wall. In other words, the

domain wall can either be aligned in Y or Z directions as long as it is normal to the easy

axis of the nanowire (i.e. the X axis). Due to such circumstances, the domain wall is

Bloch Wall with a complete rotation in the Y-Z plane. Such nature of the domain wall

has profound impact on its movements, as will be discussed later. Figure 2.3(b)-(d) show

perfect in-plane transverse walls. This is because widths of the nanowires are larger than

their thickness. It can be seen that length of the domain wall goes on increasing with

width of the nanowire.

Figure 2.4 shows the variation of length of domain wall with changing nanowire widths.

Cases (a)-(c) in Figure 2.4 correspond to thicknesses 10nm, 20nm, and 50nm. It can be

seen width has greater impact on domain wall length as the thickness of the nanowire

is increased. From the same data, Figure 2.5 is compiled to show variation of nanowire

length with thickness of the nanowire. From these results, it can be inferred that it is the

cross section of the nanowire that determines length of the domain walls. The nature of

these curves clearly indicates saturation at higher nanowire widths. Above critical width

of nanowires, the theoretical value given by Eq. 2.1.

∆ =π

r

A

K (2.1)

This study of the nature of domain wall is essential to the analysis of domain wall

motion in nanowires. Firstly, length of the domain wall is a determining factor for

transverse domain wall velocity as shown in Eq. 2.2 [7].

v = γ∆

α · · · Bef ore W B

= γ∆

(34)

0 20 40 60 80 100 120 10

20 30 40 50 60 70

Length of Domain Wall (nm)

Width of nanowire (nm)

T=10nm

T=20nm T=30nm T=50nm

Figure 2.4: Variation of length of domain wall (∆) with width of nanowire for various nanowire thicknesses.

0 10 20 30 40 50 60

45 50 55 60 65 70

Length of Domain Wall (nm)

Thickness of nanowire (nm)

W=100nm

(35)

Secondly, Walker Breakdown [8], which leads to drastic reduction in both mobility and

average speed of domain wall motion, is nothing but change in the structure of the domain

wall at high applied fields. Study of domain walls will thus help the understanding of

these phenomena as discussed in following sections.

2.2 Velocity of Domain Wall Motion

Time-dependent magnetization curves of nanowires are used to determine the velocity

of domain wall motion. The domain wall is already present in the nanowire because

of the relaxation process described in previous section. Because the domain wall is

present at exactly the center of the nanowire, the net magnetization in X-direction ‘MX’

is zero initially. Depending on the nature of domain wall, the other two components

of magnetization ‘MY’ and ‘MZ’ have small non-zero values. When external magnetic

field is applied, the domain parallel to the direction of magnetic field expands while the

domain aligned anti-parallel shrinks leading to domain wall motion. As a result of such

domain wall motion, MX changes from zero to MS in finite time. Typical magnetization

curve delineating development of magnetization with time is shown in Figure 2.7. Using

the device geometry and co-ordinates shown in Figure 2.6, it is found that the distance

travelled by magnetic DW is given by Eq. 2.3. Similarly, the domain wall velocity can be

obtained from rate of change of magnetization using Eq. 2.4.

x(t) = b·Mx(t)

MS

(2.3)

v = ∂x(t)

∂t = b MS

· dMx(t)

(36)

Domain Wa

ll

a

-a b

-b 0

x(t)

m

ain

W

all

x(t)

b+x

-(b-x) -a a

Do

m _W

0

H H_ext

Figure 2.6: Device geometry used to obtain domain wall velocity from magnetization curve.

0.0 500.0p 1.0n 1.5n 2.0n

0.0 200.0k 400.0k 600.0k 800.0k

Magnetization (A/m)

Time (s)

M _X M _Y M _Z

(37)

2.2.1 Typical Domain Wall Speed Characteristics

Domain wall velocity is a function of applied magnetic field as shown in Eq. 2.5 where

µis mobility of the domain wall,Hthresis the minimum external field that leads to motion

of domain wall and Hef f is the effective applied field. For perfect smooth nanowires,

the pinning potential acting on the nanowire is zero and therefore threshold field is zero.

Increase in applied field leads to increase in the Zeeman Energy of that part of nanowire

which has magnetization opposite to direction of external field. To accomodate this

increase in energy, the speed at which this domain shrinks increases. In other words, the

domain wall moves faster. If the appllied field is increased beyond a certain strength,

local Zeeman energy of the domain wall also comes into picture. When the localized

Zeeman and Exchange energies in the region of domain wall become comparable, the

domain wall structure becomes unstable. Anti-vortix formation and annihilation starts in

the domain wall. This phenomenon leads to oscillatory motion of the domain wall and

therefore there is a sudden drop in speed of domain wall motion. This phenomenon is

called Walker Breakdown. If applied field is increased further, the domain wall structure

stabilizes because the total energy of the domain wall is dominated by its Zeeman energy.

Therefore, the speed of domain wall motion restarts to increase linearly with increase in

applied field. But the change in domain wall structure leads to reduction in mobility of

domain walls, hence different slopes in different regions. This typical curve of domain

wall motion is shown in Figure 2.8.

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M

t

M

t

Tr

ans

ver

se

W

all

Vortex Wall HWB

vWB or vmax

Applied Field ‘H’ Domain Wall

Speed ‘v’

M

t

HC

Slope ‘μ’ Mobility

(39)

2.2.2 Effect of Domain Wall Structure

In this section, the effect of domain wall structure on motion of magnetic domain

walls is analyzed. For this purpose, we simulate nanowires with different cross-sections.

One nanowire (N W1) is 1000×10×10nm3 and other one (N W2) is 1000×20×10nm3.

As seen in Figure 2.3, N W1 has a Bloch wall at the center because of its symmetric

cross-section while N W2 possesses a transverse Neel wall. The motion of domain wall in

such pre-relaxed wires is studied by applying external magnetic field.

Figure 2.9 shows magnetization curves of the two nanowires. N W1 has a symmetric

cross section. Therefore, region of the domain wall sees very low shape anisotropy. Because

of this, the magnetic domain wall has a tendency to rotate in the Y −Z plane under

influence of applied field. This precession of the magnetic domain wall dissipates most of

the energy supplied by the applied field. This leads to reduction of velocity of domain

walls in wires with symmetric cross-sections. Another noticeable feature of motion of the

Bloch Wall is its back and forth motion. This can also be explained by the tendency

of the domain wall to rotate. Because of its rotation, length of the domain wall keeps

on oscillating periodically. The Zeeman energy (Eq. 1.3) also changes with the same

frequency. Therefore, the domain wall has to show negative motion in order to conserve

angular momentum of the system. This kind of instantaneous negative velocity will be

encountered at the time of Walker Breakdown as well. However, the phenomenon of

negative velocity in Walker Breakdown has different mechanism. Figure 2.9 also shows

smooth motion of the domain wall in N W2. The domain wall moves more swiftly in wider

wires because very small amount of energy is dissipated in rotation of the domain walls.

It is for this reason that ultra-thin nanowires have garnered a lot of interest for motion of

(40)

(a)

0.0 100.0n 200.0n 300.0n 400.0n

-100k 0 100k 200k 300k 400k 500k 600k 700k 800k

Magnetization (A/m)

Time (s)

M _X M _Y M _Z

(b)

0.0 2.0n 4.0n 6.0n 8.0n 10.0n

0 100k 200k 300k 400k 500k 600k 700k 800k

Magnetization (A/m)

Time (s)

M

X

M _Y M _Z

(41)

deal with nanowires with widths greater or equal to twice the thicknesses.

2.2.3 Effect of Nanowire Dimensions

In Section 2.1, the effect of nanowire dimensions on length of the domain wall has

already been presented. It has been found that domain wall length increases as width of

the nanowire increases. From Eq. 2.2, it is clear that velocity of domain walls also increases

with domain wall length. This is also demonstrated in Figure 2.10. Another important

observation to be made from Figure 2.10 is the field at which Walker Breakdown occurs

(HW B). It is seen that Walker Breakdown is delayed in wider wires. Wider wires have

longer domain walls because of their higher exchange energy. For the same reason, i.e.

the dominance of exchange energy in the total energy of the system, process of formation

of anti-vortices is inhibited. Hence, a stronger magnetic field is required to bring about

Walker Breakdown in wider wires. Figure 2.11 shows HW B for nanowires with different

widths. It can be clearly seen that as width of nanowire becomes large compared to its

thickness,HW B becomes almost constant. At the same time, Figure 2.11 shows almost

linear increase in speed of the domain wall at the point of Walker Breakdown. Thus,

increasing width of the nanowire does not provide improvement in terms of maximum

magnetic field that can be applied to it; but it linearly increases the maximum speed

achieved before Walker Breakdown. [9] however presents contradiction to our results

relating field of Walker Breakdown to nanowire dimensions.

2.2.4 Effect of Gilbert Damping Constant ‘

α

’

Gilbert damping constant for spin waves in ferromagnetic materials is analogous

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0 200 400 600 800 1000 0 200 400 600 800 1000

Domain Wall Speed (m/s)

Applied Magnetic Field (Oe) W=20nm

W=30nm W=50nm W=75nm W=100nm

Figure 2.10: Plot showing change in average domain wall velocity with applied magnetic for nanowires having thickness 10nm and different widths.

20 40 60 80 100

100 125 150 175 200 225 250 240 360 480 600 720 840 Domain Wall Speed (m/s)

Magnetic Field (Oe)

Width of Nanowire (nm) H

WB

v

WB

(43)

therefore determines the extent of excitation caused by a change in magnetic spin at a

particular point. It therefore plays an important role in determining speed of domain wall

propagation. One-dimensional solution LLG gives the dependence of domain wall velocity

as shown in Eq. 2.2. To assess the effect of α, permalloy nanowire with dimensions

1000×50×10nm3 _{is simulated. In each set of simulations, external applied field is}

changed from 0 to 1000 Oe and value of α is changed per set.

Gilbert Damping has been identified as a design parameter because techniques to

modify its magnitude are available in literature. It has been shown in [10] that changing

concentration of rare-earth dopants changes the damping coefficient of ferromagnetic

films. [11, 12] show that damping factor of multi-layered films can be changed because

of the exchange bias between different stratas. In both the techniques mentioned above,

annealing has varying effects on Gilbert damping of the material. Therefore, study of

effect of α of materials on their domain wall dynamics is essential.

Figure 2.12 shows the magnetization curves of nanowire for different values of Gilbert

damping. Form these curves, it can be seen that α has absolutely no effect on the field

at which Walker Breakdown (WB) occurs. At the same time, it is evident that mobility

of magnetic domain walls is governed by the damping factor both before and after WB.

Using data from Figure 2.12, the peak velocity before WB is plotted versus damping factor

in Figure 2.13. Figure 2.13 shows that domain wall mobility and hence velocity decreases

with increasing α. The fitting curve also reveals that inverse proportionality predicted in

Eq. 2.2 is almost exactly matched although with an off-set. The inverse proportionality

holds only for very small values of α. Even the velocities after WB are almost linearly

proportional to damping factor. It is thus very clear that, ability to manipulate α of

a material provides additional control to enhance speed of devices based on motion of

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0 200 400 600 800 1000 0

200 400 600 800 1000 1200

Domain Wall Speed (m/s)

Applied Magnetic Field (Oe)

=0.05 =0.10 =0.20 =0.50

Figure 2.12: Variation in domain wall velocity - magnetic field curves of nanowire (50×10nm2 _{in cross-section) with change in damping coefficient of the material.}

0.0 0.1 0.2 0.3 0.4 0.5

0 200 400 600 800 1000 1200 1400

Maximum Velocity 'v

WB

' (m/s)

Gilbert Damping (



)

(45)

2.2.5 Effect of Saturation Magnetization ‘

M

S

’

Saturation Magnetization (MS) of a material is the measure of density of polarization

of that material. In other words, if MS is high it means that a greater number of atomic

magnets per unit volume tend to align themselves in a particular direction. MS determines

the energy required to switch the direction of magnetization of the material. This implies

that higher the saturation magnetization more is the energy required to reverse the

direction of magnetization and therefore lower is the speed of domain wall motion. This

trend is clearly revealed in Figure 2.14. From Figure 2.14, it is also clear that MS also

affects the point at which WB occurs. From the inset of Figure 2.14 it becomes obvious

that although the maximum velocity reached before WB (vW B) increases with decreasing

MS, the applied field at which walker breakdown occurs (HW B) decreases. Thus, changing

the saturation magnetization is not the best way to improve performance of devices based

on domain wall motion because it gives increase in speed at the cost of reduction in the

range of usable applied field.

At this point, it is important to explain the choice of magnetization as a design

parameter. Firstly, various materials show varying amounts of saturation magnetization.

This study therefore helps choose a material to be used in device design. Similarly,

techniques like exchange biasing using caping layers [13] and doping of magnetic films

with rare-earth metals [10] also affect magnetization of the materials. These methods are

also used to modify damping in such materials. This means that although it is possible to

change magnetization of a material, doing so without changing Gibert Damping is very

difficult. That is why, changes in saturation magnetization by orders of magnitude are

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0 200 400 600 800 1000 0 200 400 600 800 1000 1200

600 700 800 900 1000 100 150 200 250 300 300 400 500 600 700

Applied Magnetic Field (Oe)

M

S (A/m)

HWB vWB

Domain Wall Speed (m/s)

Applied Magnetic Field (Oe)

M S=600A/m M S=800A/m M S=1000A/m

Figure 2.14: Variation of Magnetization curves with change in saturation magnetization

MS of the nanowire (50×10nm2 in cross-section) material. Inset shows variation of the point of Walker Breakdown due to change in saturation magnetization.

2.3 Improving the performance of Nanowires

In the field of domain wall motion, performance of ferromagnetic nanowires is

determined not only by the maximum speed at which domain walls can move about

in the nanowire but also by its dependence on the applied magnetic field. To improve

performance of nanowires for the purpose of implementing magneto-electronic device would

mean achieving higher domain wall speeds at low applied fields. Similarly, preventing

Walker Breakdown also implies improvement in magneto-electronic performance of the

nanowire because it avoids to and fro motion of the domain wall making the process of

magnetization switching more uniform. Various schemes have been proposed to enhance

domain wall motion in ferromagnetic nanowires [14]. The most basic one is to change the

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d(x,y,z)

Figure 2.15: Definition of edge roughness in a Permalloy nanowire. Depth ’d’ in the figure is a random variable with mean and standard deviation of d0.

extraordinary techniques proposed is the addition of edge and surface roughness to the

nanowire[15]. Similarly, it has been shown that externally applied transverse field can

improve the speed of domain wall motion[16]. In this section, these two novel techniques

have been studied in detail.

2.3.1 Application of Roughness

Impact of added surface roughness on the motion of magnetic domain walls in

Permalloy nanowires is detailed following paragraphs. Previously, experimental evidence

of improvement in Walker performance of nanowires because of edge roughness was

presented in [17]. The simulations carried out are similar to those of the perfect (smooth)

nanowire as described in previous sections. However, nanowire is made rough by defining

it in such a way that cubes of random dimensions are removed from random locations on

the surface of the smooth nanowire as shown in Figure 2.15. The chunks are removed

from top surface and side edges of the nanowire. The bottom surface is deliberately

kept smooth because it is undesirable to modify this surface. Bottom surface forms the

interface other materials like insulators in case of tunnel junctions and non-magnetic

metals in case of GMR structures. These interfaces play an important role in determining

(48)

indentations and their magnitude is generated using pseudo-random number generator.

The magnitude ‘d in Figure 2.15 is a random variable which has normal distribution

with mean and standard deviation both set to d0. The controlling parameter in these

experiments isd0. It was observed that there is some correlation in a set of random numbers

generated. Therefore, different seeds were used with the random number generator. Rest

of the procedure followed is exactly same as the one described above.

In concept, the most obvious impact of addition of roughness is the addition of pinning

sites. By saying this, it is implied that localized sites are present in the nanowire where

the total cross-section is smaller than its neighboring areas. Spin therefore tends to

accumulate in these regions. This causes domain wall to get pinned in such areas. But if a

domain wall is already in motion, the pinning potential may not be enough to completely

nullify the spin momentum of the domain wall. In such case, the domain wall loses some

of its energy and is therefore slowed down. This effect is similar to friction added by

roughness in mechanical systems. In any case, addition of roughness implies that some

excess energy is required to start domain wall motion. In other words, domain wall

motion will not start before the applied field reaches certain magnitude HC. On the other

hand, the randomness acts against formation of vortices and therefore prevents Walker

Breakdown.

Development of magnetization in roughed nanowires is shown in Figure 2.16 (a) and

(b). Figure in (a) shows magnetization-time curve of wire with cross section 20nm×10nm

with d0 = 0.25nmunder an applied longitudinal field of 75Oe. Different traces refer to

different seeds used for randomization. Figure (b) shows results of similar experiment

in whichd0 is changed to 0.1nm. Although traces from only three randomization seeds

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(a)

0.0 10.0n 20.0n 30.0n 40.0n 50.0n 60.0n

0 100k 200k 300k 400k 500k 600k

Magnetization (A/m)

Time (sec)

Seed=0 Seed=1 Seed=2

(b)

0.0 10.0n 20.0n 30.0n 40.0n 50.0n 60.0n

0.0 200.0k 400.0k 600.0k 800.0k

Magnetization (A/m)

Time (sec)

Seed=0 Seed=1 Seed=2 Linear Fit

Figure 2.16: Magnetization curves of rough permalloy nanowire of cross-section 20nm×

(50)

Figure 2.16 (a) it is seen that domain wall motion continues unhindered in first two cases,

the magnetization stops changing in the third case. It implies that there is a pinning site

in the nanowire large enough to completely dissipate the kinetic energy of the domain wall

in motion. In such a case where domain wall comes to halt for any of the randomization

seeds, it is considered that the field applied is not sufficient to cause domain wall motion

in the nanowire for that particular value ofd0. As in Figure 2.16 (b), if the magnetization

switches completely, linear fit of all the traces is used to determine slope for calculation

of average domain wall speed. Using data extracted from many such plots, variation

of domain wall speed with applied field is tracked. Also, the fact that instantaneous

velocities show a lot of variation without ever becoming negative is noteworthy.

Plot showing variation of domain wall speeds in rough nanowires with applied fields is

shown in figure Figure 2.17. Inset shows variation of critical field HC with the magnitude

of roughness. It is seen that this curve has hyperbolic nature. The plot of domain wall

velocities has error bars in the pre-Walker Breakdown regime. These error bars show the

maximum variation of instantaneous domain velocities when calculated every 2ns. As

expected, the magnitude of these error bars goes on increasing with increase in magnitude

of roughness. However, the most important conclusions to be drawn from this plot are

regarding the point of Walker Breakdown. Firstly, Walker Breakdown was found to

occur in a range of fields when different randomization seeds were used. This variation

of HW B is also clearly increasing with increase in magnitude of roughness. It is also

evident that HW B goes on increasing with increase in random roughness introduced in

the nanowire. As seen in Figure 2.18, there is a linear increase in Walker Breakdown field

with amount of roughness. On the other hand, it is also clear that the increase in HW B

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0 200 400 600 800 1000 0

100 200 300 400 500

0.0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100

HC

(Oe)

d(nm)

W=20nm, T=10nm

Domain Wall Velocity (m/s)

Applied Field (Oe)

Smooth d=0.05nm d=0.10nm d=0.25nm

Figure 2.17: Variation of speed of magnetic firld driven domain wall motion in rough nanowires. Each curve relates to different amount of added roughness.

throughout the nanowire. This also shown in Figure 2.18. Hence, it is safe to conclude

that introducing artificial roughness is a very inefficient way to obtain improvement in

domain wall speeds.

2.3.2 Effect of Transverse Magnetic Field

One of the proposed techniques to increase mobility of domain walls in nanowires is

the application of transverse magnetic field. This topic has been studied in some detail

[16, 18, 19, 20], but the literature does not provide any intuitive understanding of the

phenomenon. The aim of this study is to obtain such understanding of the process. For

this, simulations are run over smooth permalloy nanowires of cross-section 50nm×10nm.

However, in these set of simulations, external transverse field is applied simultaneously

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0.0 0.1 0.2 0.3 0.4 0.5 0 200 400 600 800 1000 1200 1400 1600 160 180 200 220 240 260 280 300 320

Magnetic Field (Oe)

d (nm)

H

WB

v

WB

Figure 2.18: Figure shows variation of Walker Breakdown fieldHW B and speed vW B with magnitude of roughness added to a smooth nanowire.

transverse field. A permalloy nanowire has an easy magnetization axis on account of

its shape anisotropy. In other words, it can carry stable domains only in the easy axis

directions. Any magnetic field applied normal to the easy axis is called the transverse field

HT rans. The transverse field can be in the plane of the nanowire or it can be perpendicular

to the plane of the nanowire or a linear combination of the two. This section is a detailed

study of the effect of these transverse fields on the motion of domain walls.

Firstly, it is important to study the effect of in-plane transverse field because it can

reveal the mechanism by which domain wall motion gets altered. The in-plane transverse

field can either be along the direction of the domain wall (transverse wall) or it can be

opposite to the direction of the domain wall. Figure 2.19 shows the variation of domain

wall speeds with applied longitudinal field under various transverse fields. Each of the

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0 200 400 600 800 1000 0 400 800 1200 1600 2000 2400

H

Long (Oe)

0Oe 20Oe 40Oe 60Oe 80Oe 100Oe 150Oe 200Oe 250Oe

Figure 2.19: Figure showing domain speed v v/s external longitudinal field Hlong for increasing amounts of transverse magnetic fields. For all of these, transverse field is in the same direction as the domain wall.

0 50 100 150 200 250

0 60 120 180 240 300 360 420 -300 0 300 600 900 1200 1500 1800

Domain Wall Speed (m/s)

Longitudinal Field (Oe)

In-Plane Transverse Field (Oe)

H

_WB

v

_WB

v

WB

Figure 2.20: Variation of the change in point of Walker Breakdown (δHW B, δvW B) with applied positive transverse field. This plot can be divided into three distinct zones

Study of Magnetic Domain Wall Dynamics for Memory and Logic Implementation.

ABSTRACT

DEDICATION

BIOGRAPHY

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

ONE

Introduction

1.1

Background

1.2

Concepts and Definitions

1.2.1

Magnetic Materials

1.2.2

Characteristic Features of Magnetic Materials

1.2.3

Magnetic Domain Walls

1.2.4

Theory of Magnetism

1.2.5

Energies in Magnetic Theory

1.3

Objective

References

TWO

Domain Wall Motion under Magnetic Field

2.1

Nature of Domain Walls in Nanowires

Length of Domain Wall (nm)

Width of nanowire (nm)

Length of Domain Wall (nm)

Thickness of nanowire (nm)

2.2

Velocity of Domain Wall Motion

2.2.1

Typical Domain Wall Speed Characteristics

2.2.2

Effect of Domain Wall Structure

2.2.3

Effect of Nanowire Dimensions

2.2.4

Effect of Gilbert Damping Constant ‘

α

’

Domain Wall Speed (m/s)

Applied Magnetic Field (Oe)

Maximum Velocity 'v

' (m/s)

Gilbert Damping (

)

2.2.5

Effect of Saturation Magnetization ‘

M

’

Domain Wall Speed (m/s)

Applied Magnetic Field (Oe)

2.3

Improving the performance of Nanowires

2.3.1

Application of Roughness

Domain Wall Velocity (m/s)

Applied Field (Oe)

2.3.2

Effect of Transverse Magnetic Field

Domain Wall Speed (m/s)

Longitudinal Field (Oe)

In-Plane Transverse Field (Oe)

H

v

v