Magnetic force microscopy study on interacting rings

4.5.2 Vortex Chirality Controlling of Co/IrMn nanorings 64 5.3.4 Magnetization reversal process under field along minor axis 81 5.4 Magnetization reversal process of 15nm-thick Co nanori

MAGNETIC FORCE MICROSCOPY STUDY

ON INTERACTING RINGS

ZHANG XU

NATIONAL UNIVERSITY OF SINGAPORE

2008

MAGNETIC FORCE MICROSCOPY STUDY

ON INTERACTING RINGS

ZHANG XU (B.Sc Wuhan University)

A THEIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2008

Acknowledgements

First and foremost, I would like to express the most sincere thanks and gratitude

to my supervisor, Associate Professor Adekunle Adeyeye, for his invaluable guidance and encouragement throughout the course of my M Eng research His support, advice and working attitude helped me a lot in carrying out my research

I would like to acknowledge my supervisor, Professor Caroline Ross in MIT, who devoted considerable efforts to my research: providing professional instruction, sharing ideas without conservation, inspiring me with encouragement and more importantly teaching me how to conduct a good scientific research

I would like to give great appreciation and thanks Mr Navab Singh from Institude of Microelectronics for providing me with the deep ultra violet resist patterns used in this thesis I would like to give special thanks to Dr Wonjoon Jung and Dr Fernando Castaño from MIT to do the triode and ion beam sputtering deposition, and Dr Goolaup Sarjoosing from my lab to do the evaporation deposition

I am thankful to other research group members, Mr Gao Xinsen, Mr Wang Chenchen,

Mr Tripathy Debashish, and Ms Jain Shikha, and my colleges in MIT, Mr Lei Bi, Ms Vivian Peng-Wei Chuang, Mr Yeon Sik Jung, Mr Vikram Sivakumar, and Mr Filip Ilievski

Finally, I am also very grateful for the good friendship that developed with my laboratory mate including Dr Bi Jingfeng, Mr Chen Wei, Ms Wan Fang and Ms Ma Minjie And I am grateful for all my friends from Singapore-MIT Alliance I can not list all of their names because there are too many

Table of Contents

Acknowledgements i

Table of Contents ii

Summary v

List of Figures vii

List of Symbols and Abbreviations xvi

Chapter 1 Introduction

Chapter 2 Theoretical Background

2.4 Magnetization States in Ferromagnetic Meso-scopic rings 12

Chapter 3 Experimental Background

4.5.2 Vortex Chirality Controlling of Co/IrMn nanorings 64

5.3.4 Magnetization reversal process under field along minor axis 81 5.4 Magnetization reversal process of 15nm-thick Co nanoring pairs 83 5.4.1 Magnetization reversal process of 75nm nanoring pairs 84 5.4.2 Magnetization reversal process of 125nm nanoring pairs 89 5.4.3 Magnetization reversal process of 600nm nanoring pairs 92 5.5 Magnetization reversal process of 40nm-thick Co nanoring pairs 96

Chapter 6 Conclusion and Outlook 103

Summary

The magnetization reversal process and chirality controlling in patterned single and multilayer nanomagnets fabricated by deep ultra violet lithography and lift-off technique, has been systematically studied, as a function of various geometrical parameters, using a combination of magnetic force microscopy (MFM) and simulation tools (micromagnetic simulation, magnetic energy theory, and dipole approximation)

Firstly, magnetization reversal process of patterned single ferromagnetic structure, Co square rings, and exchange-biased structure, which is two layers Co/IrMn rings, is investigated And the inter-ring spacing is varied to investigate the collective effect It is relatively hard to reverse Co/IrMn nanorings due to the pinning effect of the anti-ferromagnetic material IrMn For widely-spaced and closely-packed

Co nanorings, and widely-spaced Co/IrMn nanorings, collective effect is not observable For closely-packed Co/IrMn nanorings, collective effect is apparent This

is modeled by dipole approximation that the inter-ring spacing and starting positions

of domain walls are important in determining the collective effect

Secondly, the vortex chirality on a large scale of arrays of Co and Co/IrMn rings

is investigated, based on the technique that individual rings can be selected and observed by scanning the patterns at the corner The chirality of the vortex states of

Co and Co/IrMn can be controlled by altering the direction of applied field For Co patterns, the clockwise (CW) chirality appears when the angle from major axis to

direction of field is negative; and the counterclockwise (CCW) is for the positive angle For Co/IrMn patterns, the transition angle of CW/CCW is near the angle of exchange field, due to the exchange bias effect The results are confirmed by the magnetic energy theory

Thirdly, interaction between ferromagnetic nanostructures is investigated by examining the magnetization reversal process of units which are made up of a pair of circular Co rings Small, closely spaced groups of rings may show collective magnetic configurations that are stabilised by magnetostatic interactions For 25nm-thick patterns, the ring size is unchanged, but the inter-ring spacing is varied in order to investigate the effect of magnetostatic interactions on the collective behavior of the ring pair The different micromagnetic configurations have been explored as a function of the applied field and edge-to-edge spacing The switching field between different magnetic states is described for ring pairs of different spacing and for fields applied along both in-plane directions It gives evidence that the stray field from one onion state is 1/r3 decay with the distance which is consistent with the dipole approximation The stable range of vortex/vortex state increases as the inter-ring spacing increases The results are compared with a micromagnetic model

Finally, the magnetization reversal process of pairs of Co thin-film circular rings with different thickness has been explored For thin patterns (15nm thick), nano ring pairs show various metastable states which is hard to be analyzed; for thick patterns (40nm thick), it is hard to get the whole magnetization reverse process since the coercivity is large The results are compared with a micromagnetc simulations

List of Figures

Fig 1.1 A designed sandwich-type nano-ring MTJ structure Free layer is a

ferromagnetic layer; reference layer, generally, consists of a ferromagnetic/ anti-ferromagnetic double layer, which shows exchange bias effect Three stable magnetization patterns of vortex, symmetric onion, or asymmetric onion states (magnetic states will be introduced in chapter 2) can be respected to exist in each ring-shaped free layer at zero magnetic field (b) A prototype 2×2 MRAM demo device based on one NR-MTJ and one transistor structure Here the word line, also as the gate line, plays the role of addressing each bit together with the cross bit line Adopted from ref [13]

4

Fig 2.1 Illustration for the exchange interactions between two neighboring

spins Blue daggers are spins; red wave denotes the exchange interaction

9

Fig 2.2 Schematic illustration of the break up of magnetisation from single

domain into closure domains The reason to form closure domain is

to minimize magnetostatic energy

10

Fig 2.3 Illustration showing (a) a Bloch wall and (b) Neel wall In a Bloch

wall the magnetization rotates out of plane, and in a Neel wall the magnetization rotates in-plane

12

Fig 2.4 Demonstration of hysteresis loop of the magnetic ring array, and the

corresponding magnetic states

13

Fig 2.5 Typical transition from onion to vortex state Black circles are the

boundaries of rings; blue lines are the transverse domain wall; red arrows indicate the direction of magnetization

14

Fig 2.6 A schematic flowchart of the OOMMF program The path outlined

in red is the function performed by iterations, and the path in green is the function performed by stages, from [27]

16

Fig 2.7 A graphic representation of the Landau-Lifshitz-Gilbert equation

The magnetization vector M precesses around the effective applied field, H, and it also tends to align itself with H The damping coefficient, α, determines how quickly the magnetization lines up with the effective applied field

18

Fig 2.8 Scheme of the situation that dipole is parallel to the line which

connects the two centers of dipole

19

Fig 2.9 Scheme of the situation that dipole is perpendicular to the line which

connects the two centers of dipole

20

Fig 3.1 Working of alternating phase shift mask 27

Fig 3.2 Schematic diagram for the evaporator systems 28

Fig 3.3 Schematic diagram for the ion beam sputtering systems 29

Fig 3.4 Schematic diagram of the electron and sample interaction 31

Fig 3.5 Inter-molecular force as a function of the separation 32

Fig 3.6 A scheme of AFM with the major components (some components

such as data pick-up components and servo components are not shown here) Cantilever is typically silicon (for tapping mode) or silicon nitride (for contact mode) Tip radius curvature is on the order of nanometer In some AFM equipment, the 3D piezoelectric scanner is in contact with cantilever [11], not the sample stage

33

Fig 3.7 Left: the procedure of fabricating the tip Right: SEM images of the

tip during the fabrication, from [15]

35

Fig 3.8 Two-pass method of MFM Tapping scan is used to get the

topography, which is indicated as first pass While the lift scan

follows the topography with a lift height zΔ , and that generates the magnetic contrast

36

Fig 3.9 AC detection techniques, amplitude of cantilever oscillation

Cantilever is driven by a fixed frequencyϖ1, a cantilever resonance frequency shift causes the cantilever oscillation amplitude and phase

to change The amplitude or phase change can be interpreted as a magnetic signal

37

Fig 3.10 Scheme of the interaction between sample and tip 38

Fig 4.1 The SEM images of the patterns (a) closely-packed Co patterns; (b)

widely-spaced Co patterns; (c) closely packed Co/IrMn patterns; (d) closely packed Co/IrMn patterns Edge-to edge spacing for closely packed rings is 50nm both along minor and major axis; while for widely-spaced ones, it is 500nm along minor axis and 350nm along major axis

45

Fig 4.2 Hysteresis loop of single layer and exchange bias films 46

Fig 4.3 Magnetic states evolution of widely-spaced Co ring arrays From (a)

to (e), field of 0, 100, 200, 300, 400, 500 Oe were applied before the image is taken; under each Image, it is the sketch of the majority state for corresponding image The caption of sketch is the description of the state The arrows in sketch indicate the magnetization direction in the pattern Transverse line in the pattern indicated the transverse domain wall Arrow outside of pattern indicates the direction of external magnetic field These will be illustrated in the following figures

47

Fig 4.4 Magnetic states evolution of widely-spaced Co/IrMn ring arrays

From (a) to (f), field of 0, 100, 200, 300, 400, 500 Oe were applied before the image is taken; under each Image, it is the sketch of the majority state for corresponding image

48

Fig 4.5 State Percentage of widely-spaced (a) Co ring arrays and (b)

Co/IrMn ring arrays as a function of field

50

Fig 4.6 Magnetic states evolution of closely-packed Co ring arrays All the

images, sketches, icons are similar to Fig 4.3

51

Fig 4.7 magnetic states evolution of closely-packed Co ring arrays All the

images, sketches, icons are similar to fig 4.4

52

Fig 4.8 State Percentage of closely-packed (a) Co ring arrays and (b)

Co/IrMn ring arrays as a function of field

53

Fig 4.9 Model for the interaction of the domain walls in Co rings In a ring

with onion state, the two domain walls are located at position 1 or 2 and 3 or 4 Each domain wall can be treated as a dipole Daggers denote the direction of the dipole

55

Fig 4.10 Model for the interaction of the domain walls in Co/IrMn rings In a

ring with onion state, the two domain walls are located at 2 and 4 Each domain wall can be treated as a dipole Daggers denote the direction of the dipole In ring A, red dagger means the direction of domain wall movement, the dot dagger at position 3 is dipole after the movement

57

Fig 4.11 Magnetic states evolution of widely-spaced Co/IrMn ring arrays at

the corner From (a) to (f), field of 0, 100, 200, 300, 400, 500 Oe were applied before the image is taken Under each Image, it is the sketch of the majority state for corresponding image The caption of sketch is the description of the state The arrows in sketch indicate the magnetization direction in the pattern Transverse line in the pattern indicated the transverse domain wall Arrow outside of pattern indicates the direction of external magnetic field

59

Fig 4.12 Magnetic states evolution of ring 1-1of widely-spaced exchange

biased ring arrays The image is cut from Fig 4.11

60

Fig 4.13 Magnetic states evolution of ring 1-4 of widely-spaced exchange

biased ring arrays The image is cut from Fig 4.11

60

Fig 4.14 Chirality controlling of Co pattern The MFM images are taken (a)

after saturation, (b) after -30o reverse field of 100 Oe, (c) after saturation, (d) after +30o reverse field of 100 Oe

61

Fig 4.15 After cutting and zooming from in Fig 4.14 Angle alpha is from

major axis to applied field H After applying reverse field at an angle

of -30o, the chirality is clockwise; while after applying +30o, the chirality is CCW

62

Fig 4.16 Models for the two chirality cases in Co rings 62

Fig 4.17 Zeeman energy versus y, when the applied field is -30o The solid

red curve is for domain wall 1; the dashed green curve is for domain wall 2 After directed observation, domain 1 to rotate CW is the most energy favorable, which is observed in the first two cases of Fig 4.13

64

Fig 4.18 Chirality controlling of Co pattern The MFM images are taken (a)

after saturation, (b) after -30o reverse field of 100 Oe, (c) after saturation, (d) after +30o reverse field of 100 Oe, (e) after saturation, (f) after +60o reverse field of 100 Oe

65

Fig 4.19 Exchange-biased ring, after cutting and zooming in from the

previous images, angle beta is from major axis to exchange bias direction

66

Fig 4.20 Zeeman energy VS y-axis displacement for Co/IrMn elliptical ring 67

Fig 5.1 SEM images of ring pairs The inter-ring spacing is (a) 75nm, (b)

125 nm, and (c) 600 nm

72

Fig 5.2 (a) to (d) are OOMMF simulation of the signle ring which have the

same geometry as our rings in units, under a reverse field of 0, 400,

600 and 1400 Oe respectively; From left to right, the upper images

of (e) to (i) are OO, OV, VV, VR, and RR states imaged by magnetic force microscopy, in 75nm-spacing pairs The sketches correspond to the upper images Black lines are the boundary of the ring; blue lines indicate the traverse domain walls (the domain wall is traverse after OOMMF simulation, which will be shown later.), and red dashed arrows show the magnetization direction

75

Fig 5.3 MFM images of 125nm-spacing units taken at remanence after

applying fields of (a) 100 Oe and (b) 493 Oe The arrow indicates the applied field direction before taken the MFM image We counted the number of different states and recorded the dominant state(s) in the phase diagram

76

Fig 5.4 (a) Phase diagram of the dominant states seen in ring pairs as a

function of ring spacing and reverse field (b) The simulated M-H values under reverse field

77

Fig 5.5 (a) 1/r fitting of the transition field from OO to OV state versus

spacing (b)OOMMF simulation result of 75nm unit after saturation and applied a reverse field of 243 Oe The upper ring shows a twisted state and the lower ring shows an onion state As we declared, it is an OV state, which is corresponding to the first data point in (a) (c) 600nm unit under a reverse field of 122.5 Oe, it is also an OV state which corresponding to the last data point in (a)

79

Fig 5.6 A plot of the switching field as a function of inter-ring spacing 81

Fig 5.7 (a) MFM image of 75nm and (b) 600nm units after saturation at 3

kOe Daggers are the direction of applied field before taken MFM images From (a), there is almost no unit show OO state along short axis For (b), it is just can be treated as isolated rings arrays (c) OOMMF simulation result of 75nm and (d) 600nm

82

Fig 5.8 Magnetization reversal process of 75nm spacing, pair 1 Arrow in

MFM images is the direction of applied magnetic field before taking the MFM images at remanence From second image, all the direction

of field is the same Number in MFM images are the magnitude of field in Oe Black lines are the boundary of the ring; blue line is the domain wall; red line is the magnetization direction

84

Fig 5.9 Magnetization reversal process of 75nm spacing, pair 2 85

Fig 5.10 Magnetization reversal process of 75nm spacing, pair 3 87

Fig 5.11 Micromagnetic simulation of nano pairs in this section Black square

dots show magnetizations of the circular rings pair Red circular dots show magnetizations of the elongated rings pair (along the major axis, the outer diameter is 562.5nm rather than 560nm) Insets are the simulated magnetic state under the indicated field

88

Fig 5.12 Simulated states of the two pairs under 660 Oe 89

Fig 5.13 Magnetization reversal process of 125nm spacing, pair 1 90

Fig 5.14 Magnetization reversal process of 125nm spacing, pair 2 91

Fig 5.15 Magnetization reversal process of 125nm spacing, pair 3 92

Fig 5.16 Magnetization reversal process of 600nm spacing, pair 1 93

Fig 5.17 Magnetization reversal process of 600nm spacing, pair 2 94

Fig 5.18 Magnetization reversal process of 600nm spacing, pair 3 95

Fig 5.19 Magnetization reversal process of 40nm-thick 75nm spacing pairs

The scale bar is only shown in first picture, and is the same for the rest Arrow in MFM images is the direction of applied magnetic field before taking the MFM images at remanence From second image, all the direction of field is the same

98

Fig 5.20 Comparison of magnetization vs field among 15nm, 25nm, and

40nm thick pairs For 40nm case, field range for VV state is widest, and the reversal field of the RR state is the largest

99

List of Symbols and Abbreviations

DUV Deep ultraviolet

EBL Electron beam lithography

FM Ferromagnetic

IrMn Iridium Manganese

KrF Krypton fluoride

LLG Landau Lifshitz Gilbert

MFM Magnetic force microscopy

MRAM Magnetic random access memory

MQCA Magnetic quantum cellular automata

MTJ Magnetic tunneling junction

OOMMF Object oriented micromagnetic framework

SEM Scanning electron microscopy

SPM Scanning probe microscope

Ta Tantalum

W Tungsten

Chapter 1

Introduction

1.1 Background

In the last few years, there has been growing interest in the magnetic nanostructure

due to advances in nanofabrication methods, and characterization techniques Aided

with the improved processing abilities of computers, the micromagnetics of

nanomagnets are now solvable, thus allowing for direct comparison with experimental

results Sub-micron to nanoscale magnets draw a lot of attention both from a

fundamental and application perspective

Fundamentally, novel properties emerge as the lateral dimensions of the magnets

becomes comparable to or smaller than certain characteristic length scales, such as

spin diffusion length, domain wall width and carrier mean free path Due to energy

minimization, the magnetization state of a bulk magnetic material is usually

magnetically divided into domains and the exact domain configuration is

unpredictable However, when the magnetic material is patterned down to the nano

scale, the number, size and orientation of the domain becomes walls well defined and

predictable On this length scale it becomes energetically favorable for a magnet to

behave like a ‘giant’ dipole [1] The lateral confinement results in both novel

magnetization reversal behaviors and also new transport properties in the structures

With an ensemble of such elements, the interaction among them could lead to new

collective magnetic properties, which is different from the isolated elements Arrays

of nanomagnets are a good system for studying micromagnetics and exploring new

physics

From an application viewpoint, nano magnets are the fundamental building blocks for various spintronics devices In data storage applications, as the conventional recording media are rapidly approaching the superparamagnetic limit, data storage is unstable due to thermal fluctuations Patterned media consisting of arrays of nanomagnets have been viewed as a candidate for ultra-high density storage of at least 1 Terabit/in2[2-4]

The spin-dependent transport properties of nanomagnets have also attracted a lot of attention due to the new idea to use the ‘spin’ angular momentum of electron Magnetic random access memory (MRAM) is one such device that makes use of the

‘spin’ of electron, to store information [5-20] Normally, MRAM element consists of two magnetic layers with different switching field, and a layer separating the two magnetic layers The states are determined by gauging the resistance of the element Typically if the two layers have the same polarity this is considered as state "0", while

if the two layers are of opposite polarity the resistance will be higher and is state "1" The nonvolatile characteristics of the MRAM are attractive not only to military and space applications, but also to the commercial world when it can be made dense, reliable, fast, and cost effective MRAM is able to replace not only DRAM and SRAM, but also able to replace disk drives It could truly be the technology enabling the entire memory system of computer to be made on a single chip

In addition to data storage, magnetic nanostructures could interact to perform some

kind of computation, resulting in a completely magnetic computer Networks of interacting nanomagnets have been used to perform logic operations and propagate information [21-24] These kinds of systems are under further research

1.2 Why Magnetic Rings

In MRAM, the non-repeatable of the switching associated with the ends of the linear elements, have long been a problem in MRAM design [9] Such problem is eliminated,

if the magnetization orientation in a memory element is circular, and the magnetization flux forms a closure in the circular mode To form a circular magnetization mode, the memory element can be either a ring or a disk However, for the disk shaped memory element, the center of the disk element provides a dominant contribution of the exchange energy And the circular magnetization configuration can only be maintained when the disk diameter is sufficiently large When the disk diameter becomes smaller, the exchange energy in the element becomes increasingly comparable to the magneto-static energy, and the circular magnetization mode can no longer be maintained

After removing the inner core, the ring shaped element results in significantly lower exchange energy than that of the disk with the same outer diameter Thus, the ring shaped element narrows switching distribution The reason is that, once it has been formed, the energy cost of moving a vortex is very small, so the vortex tends to seek out the best (most energetically favorable) path out of the particle This searching process causes a broadening in the switching field distribution The introduction of

the inner edge reinforces the circular magnetization configuration

Furthermore, rings also have potential for use in MRAM drives [16, 17], as shown in Fig 1.1 To simultaneously improve the power consumption and the thermal stability for developing the MRAM, it is desirable to fabricate ring-shaped magnetic tunneling junctions (MTJs) whose magnetization directions can be directly controlled by the spin-polarized current and spin transfer torque effect [18] With a ring-shaped magnetic monolayer and multilayers, it can offer a significant improvement in terms

of eliminating the stray field and enhancing the thermal stability since the magnetization will form a vortex structure free of magnetic poles [19, 20]

(a) (b)

Fig 1.1 (a) A designed sandwich-type nano-ring MTJ structure Free layer is a ferromagnetic layer; reference layer, generally, consists of a ferromagnetic/anti-ferromagnetic double layer, which shows exchange bias effect Three stable magnetization patterns of vortex, symmetric onion,

or asymmetric onion states (magnetic states will be introduced in chapter 2) can be respected to exist in each ring-shaped free layer at zero magnetic field (b) A prototype 2×2 MRAM demo device based on one NR-MTJ and one transistor structure Here the word line, also as the gate line, plays the role of addressing each bit together with the cross bit line Adopted from ref [13]

Among ferromagnetic nanorings, circular rings are generally used Square rings have shape anisotropy, thus the domain wall can be pinned at the corners after magnetization They are also quite useful in the research as well as in real application

1.3 Aim of the thesis

Despite the extensive research on ring-shaped nanomagnets, a direct mapping of the spin configuration during the reversal process is still lacking In addition, the effect of magnetostatic interaction in densely packed arrays of rings, which is vital in MRAM designs, is still not well understood In this thesis, a systematic study of the spin state evolution in single and bilayer ring-shaped nanomagnets has been carried out using magnetic force microscopy (MFM) measurement

The main objectives of this thesis are listed as follows:

(a) Investigation of the magnetostatic interaction in nano-ring pairs as a function of different geometrical parameters

(b) Direct observation of the exchange bias effect in bilayer FM/AFM rings

(c) Controlling the chirality of ring-shaped nanomagnets

1.4 Organization of the thesis

In chapter 2 and chapter 3, related theoretical and experimental background of our research is presented; in chapter 4, patterned single-layer and exchange-biased ferromagnetic nanostructures are investigated The magnetic states evolution of the patterns and individual rings are presented A method to control the vortex chirality with both experimental and theoretical study is introduced; in chapter 5, periodic arrays of pairs of rings are presented, with the special periodic configuration of pairs other than arrays of rings and isolated rings, the units show different properties; in chapter 6, conclusions are presented and possible future work is postulated

References

[1] S Collins, M Sc thesis, Mcgill University (2004)

[2] C A Ross, Annu Rev Mater Res., 31, 203 (2001)

[3] A Moser, K Takano, D T Margulies, M Albrecht, Y Sonobe, K Ikeda, S Sun and E E Fullerton, J phys D: Appl Phys 35, R157 (2002)

[4] T Kimura, Y Otani, H Masaki, T Ishida, R Antos and J Shibata, Appl Phys Lett 90, 132501 (2007)

[5] J M Daughton, Thin Solid Films 216, 162 (1992)

[6] W J Gallagher, S S P Parkin, Yu Lu, X P Bian, A Marley, K P Roche, R

A Altman, S A Rishton, C Jahnes, T M Shaw, and Gang Xiao., J Appl

[9] J G Zhu, Y Zheng and G A Prinz, J Appl Phys 87, 6668 (2000)

[10] G Prinz, U.S Patent No 5477482 (1995)

[11] H Meng, J Wang, Z Diao and J P Wang, J Appl Phys 97, 10C926 (2005) [12] J Z Sun, D J Monsma, T S Kuan, M J Rooks, D W Abraham, B Oezyilmaz, A D Kent, and R H Koch, J Appl Phys 93, 6859 (2003)

[13] Y Huai, F Albert, P Nguyen, M Pakala, and T Valet, Appl Phys Lett 84,

[18] J C Slonczewski, J Magn Magn Mater 159, L1 (1996)

[19] K Bussmann, G A Prinz, R Bass, and J.-G Zhu, Appl Phys Lett 78, 2029 (2001)

[20] F Q Zhu, G W Chern, O Tchernyshyov, X C Zhu, J G Zhu, and C L Chien, Phys Rev Lett 96, 027205 (2006)

[21] R R Schaller IEEE Spectrum, 34(6), 52 (1997)

[22] M Gardner Scientific American, 223, 120 (1970)

[23] R P Cowburn and M E Welland Science, 287, 1466 (2000)

[24] I Amlani, A O Orlov, G Toth, G H Bernstein, C S Lent, and G L Snider Science, 284, 289 (1999)

Chapter 2

Theoretical Background

2.1 Introduction

This thesis focuses on submicron patterned ferromagnetic ring structure In the

research, magnetic energies, micro-magnetic simulation and some classic magnetic

theories such as dipole approximation are used In the following parts, magnetic

energies and domains theory are presented; then, one important concept which will be

used in the following chapters, the magnetic states in mesoscopic rings are introduced;

finally, other useful concepts such as micro-magnetic simulation and dipole

interaction are introduced

2.2 Magnetic Energies

Micro-magnetism deals with the interactions between magnetic moments on

sub-micrometer length scales These are governed by several competing energy terms

The magnetic system adopts configurations that minimize the total energy Generally,

the energy density can be described as:

E=E +E +E +E +E (2.1)

The last energy term is often neglected due to their small contributions Where Eex is

the exchange energy; Ek is the magnetocrystalline anisotropy energy; ED is the

magnetostatic energy; EH is the Zeeman energy; and Es is the magnetostriction energy

(also known as magnetoelastic energy) The total free energy F is given as:

3

F =∫d r E +E +E +E (2.2)

Exchange energy aligns the magnetic moments in the immediate surrounding space parallel to one another (if the material is ferromagnetic) or antiparallel to one another (if antiferromagnetic) Anisotropy energy is the favorability for moments to align along a particular crystal direction Magnetostatic energy is essentially the energy associated with sources of internal or external fields, which causes magnets to align north to south pole Zeeman energy is resulting from an externally applied field, which is at its lowest when magnetic moments lie parallel to an external magnetic field

(a) Exchange Energy

The exchange interaction describes the force between two electrons, which depends only on their relative spin orientations [1]:

S i

S j

Fig 2.1 Illustration for the exchange interactions between two neighboring spins Daggers are spins; wave denotes the exchange interaction

It follows from Eq 2.3 that if the exchange energy is to be negative (that is, for ferromagnetism to be energetically favorable), the cos( )φ term must be positive That implies that the electron spins must be parallel rather than antiparallel for ferromagnetism to occur A more rigorous treatment of the exchange interaction can

be found in [2]

(b) Magnetostatic Energy

The magnetostatic energy, also called the stray field energy, is the energy of the magnetic field produced by the particle itself originated from the classical dipolar interaction It is given by:

is the local magnetization and HJJGd

is the demagnetizing field generated

by the sample The effect of magnetostatic energy is to minimize the surface magnetic charges, which are responsible for the subdivision of magnetic materials into domains,

as illustrated in fig 2.2

Fig 2.2 Schematic illustration of the break up of magnetisation from single domain into closure domains The reason to form closure domain is to minimize magnetostatic energy

(c) Crystalline Anisotropy Energy

Crystalline anisotropy energy arises from the orientation of individual spins relative to the crystal lattice, and to neighboring spins The origin of anisotropy energy is spin-orbit coupling

2.3 Domain Wall Configurations

Magnetic domains in ferromagnetic materials are a result of the minimization of the energy in Eq 2.1 The exchange energy is minimized if the spins are aligned parallel However, the magnetostatic energy is minimized by orienting spins in an anti-parallel configuration The compromise is the formation of domains The spins rotate from one direction to another to minimize the magnetostatic energy, but they do so gradually, to keep the energy cost (from the increase in exchange energy), as small as possible This region of gradual transition is known as a domain wall The width of a domain wall in the absence of an external field is given by [3]

and a is the unit cell dimension of the sample Domain wall widths are usually of the order of hundreds of nanometers [4, 5], but this width is dependent on the anisotropy energy of the sample The two most common types of domain walls are Bloch walls and Neel walls, which are illustrated in Figure 2.3 In a Bloch wall the magnetization rotates out of the plane occupied by the initial and final domain magnetizations Conversely, the magnetization in a Neel wall in the plane is occupied by the initial and final domain magnetizations Neel walls are found only in very thin films, where the thickness of the film is much less than the domain wall width [4], and occasionally

in samples that are in applied fields [6]

(a)

(b) Fig 2.3 Illustration showing (a) a Bloch wall and (b) Neel wall In a Bloch wall the magnetization rotates out of plane, and in a Neel wall the magnetization rotates in-plane

2.4 Magnetization States in Ferromagnetic Meso-scopic rings

Brown's fundamental theorem [7] states that if a ferromagnetic particle has dimensions of the order of domain wall width, the energy cost of incorporating a domain wall will be too large, and the particle will adopt a single domain configuration This condition is highly dependent on particle shape and composition, but this size is generally between several tens to a few hundred nanometers In this

arrangement all dipoles in the particle are pointed in the same direction, effectively creating a single spin comprised of many thousands of atoms If the anisotropy is uniaxial, these particles are binary: the atomic spins can either align all in the ‘up’ configuration or all in the ‘down’ configuration, and can therefore be used to store one bit (binary digit) of information

In a ring-shaped ferromagnetic mesostructure, due to the confinement of the boundary, the domain wall will just move along the direction of the tangent of the ring In the process of the moving of wall(s), there are different combinations of the positions of walls Thus, different magnetic states form

Circular and square rings are for our research Here, the hysteresis loop of nanorings which follows a two-step switching process is presented [8], as shown in Fig 2.4 From right to left, the insets show the corresponding states which are onion, vortex and reverse onion

Fig 2.4 Demonstration of hysteresis loop of the magnetic ring array, and the corresponding

magnetic states

Two step switching is not necessarily the only mechanism in the magnetization reversal process In an ideal circular ring, the pinning effect is negligible Thus, only one transition was observed [9], namely, directly from onion to reverse onion state without going through vortex state, which is called single switching In this case, two walls begin to move simultaneously, thus, the two walls keep their distance unchanged However, if the asymmetries are introduced in the rings, one wall is pinned less strongly than the other, so it moves at a lower field than the other In this case, there is some metastable state such as twisted state, as shown in Fig 2.5

Fig 2.5 Typical transition from onion to vortex state Black circles are the boundaries of rings; blue lines are the transverse domain wall; red arrows indicate the direction of magnetization

Studies were conducted to investigate the phase diagram between the double switching and single switching regime Only for very thin rings, vortex state is suppressed; and it is pervasive over a large range of geometrical parameters [10-15] The onion to vortex switching field depends strongly on the ring width, but less so on the ring diameter and thickness In some wide rings, a triple step magnetization (onionÆvortexÆvortex coreÆreverse onion) process has been reported [16-19]

Besides the width and thickness, the switching process can also be engineered by modifying and controlling other intrinsic factors such as shape [20, 21] Domain wall pinning and controlled magnetic switching can be achieved in narrow ferromagnetic ring structures with defects such as notches [22] and intrusions [20] Furthermore, in-plane anisotropy can also be introduced by creating decentered [23], elliptical [24] and square ring [25] structures

Interring Spacing can be another important factor to influence the switching behavior The dipole interaction will be strong if the spacing is small while it will be weak if the spacing is large The interaction will change switching field and switching process [25]

Nanorings need to be characterized to determine what dimensions will yield single domain rings and at what applied fields the single domain rings will switch configurations Micromagnetic simulations were used to predict sample behavior

2.5 Micromagnetic Simulation

All micromagnetic simulations were performed using the Object Oriented MicroMagnetic Framework (OOMMF) code from the National Institute of Standards and Technology [26] OOMMF is portable, extensible and is in the public domain The Tcl/Tk scripting language is required to execute OOMMF

OOMMF runs simulations according to a sequence, as illustrated in Figure 2.6 The simulation as a whole is run in large increments called stages Individual stages are run in small increments called iterations or steps

Fig 2.6 A schematic flowchart of the OOMMF program The inner path is the function performed

by iterations, and the outer path is the function performed by stages, from [27]

Iterations are run by evolvers, which update the magnetic state of the sample from one step to the next The evolvers, in turn, are controlled by drivers Stages are run by drivers, which coordinate how the simulation evolves New conditions (the magnitude

of an applied magnetic field, for example) are introduced at the beginning of stages The evolvers then update magnetization of the sample The length of a stage is controlled by the user, who defines stopping criteria in the driver Once the stopping criteria have been met, a new stage begins Depending on the stopping criteria, a stage can have from as few as one iteration to as many as hundreds of thousands of iterations Therefore, choices for stopping criteria are critical

For a certain driver to be used its corresponding evolver must be used in conjunction There are two types of driver-evolver pairs The first type is the minimization driver-evolvers These locate local energy minima through direct minimization techniques For this study the minimization approach was inadequate since the global minimum was sought rather than local minima That is, the simulated particle often evolved into metastable states or unstable equilibria

The second pair is the time driver-evolver These track the time evolution of the magnetization according to the Landau-Lifshitz-Gilbert (LLG) equation:

= − × − × ×

JJG

JJG JJG JJG JJG JJG

(2.7) where MJJG

is the magnetization, HJJGeff

is the effective field, γ is the Landau-Lifshitz gyromagnetic ratio, and α is the damping constant

The LLG equation describes the general behavior of individual dipoles in an applied magnetic field See Figure 2.7 Use of the right hand rule shows the first term in the LLG equation describes the precession of the magnetization vector around the applied field vector The second term tends to align the magnetization vector with the applied field, when the right hand rule is used twice and the negative sign in front of the term included

Fig 2.7 A graphic representation of the Landau-Lifshitz-Gilbert equation The magnetization vector M precesses around the effective applied field, H, and it also tends to align itself with H The damping coefficient, α, determines how quickly the magnetization lines up with the effective applied field

The damping coefficient α determines how quickly the magnetization vector aligns itself with the applied field It differs between materials and must be user defined When the sample thickness become small this value becomes dependent on the thickness; as sample thicknesses decrease the damping parameter tends to increase [28] The default value for α set by OOMMF is 0.5, which is much larger than real values A large damping coefficient decreases computation times since individual dipoles align themselves with the applied field quickly

It needs to be emphasized that Micromagnetic modeling is only a computational method based on a phenomenological LLG equation 'Micromagnetism' (LLG computer model) is different from the 'physics of magnetism'

2.6 Dipole Interaction

Under an external magnetic field, ferromagnetic material will be magnetized The magnetized material can be considered as a dipole [29]

In a magnetic medium of the relative permeability μf, when the direction of magnetic dipole is parallel to the direction of line which connects the center of the two dipole ( rG lG

& , r is the distance between the two centers of dipoles, l is the length of a

dipole), as shown in fig 2.8

is magnetic charge

Under the field BG, qm will feel a force

0 3

Fig 2.9 Scheme of the situation that dipole is perpendicular to the line which connects the two centers of dipole

The magnetic field projected along direction of rG

The total force along the direction of rG

is twice as the upper value,