Novel magnonic crystals and devices fabrication, static and dynamic behaviors

There are many challenges that need to be addressed before the full potential of MC based devices is realized, such as the lack of a systematic investigation of dynamic responses in tail

NOVEL MAGNONIC CRYSTALS AND DEVICES:

FABRICATION, STATIC AND DYNAMIC

BEHAVIORS

DING JUNJIA

(M Eng, Huazhong University of Science and Technology)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER

ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2013

DECLARATION

I hereby declare that this thesis is my original work and it has been written by

me in its entirety I have duly acknowledged all the sources of information

which have been used in the thesis

This thesis has also not been submitted for any degree in any university

previously

Ding Junjia

7 January 2014

Acknowledgements

The final outcome of this thesis received a lot of guidance and assistance from many people and I am extremely fortunate to have got this all along the completion of my PhD study While it is impossible to acknowledge all of those people here, I will always remember them I would like to acknowledge several people in particular

First and foremost, I would like to express my sincerest gratitude and appreciations to my supervisor Prof Adekunle O Adeyeye for giving me the opportunity to work on this topic Without his unwavering dedication, encouragement, support and guidance in all aspects varying from research to personal life, it is impossible for me to finish this thesis in four years Thanks Prof Adekunle for his time to read, modify and comment on all my previous research papers and several versions of this thesis

I would like to give special thanks to Prof Mikhail Kostylev from the University of Western Australia for his great help in the theory work of 1-Dimensional Magnonic Crystals and for his reading and comments on my thesis

I would also like to express my appreciation towards ISML lab supervisor Assoc Prof Vivian Ng, lab officers Ms Loh Fong Leong, Mr Alaric Wong and

Ms Xiao Yun for their help and support during my candidature

It has been a delight to work with the current and past members of Prof Adekunle’s group and ISML: Dr Shikha Jain for teaching me all the nanofabrication skills and helping in setting up the Ferromagnetic Resonance spectroscopy Dr Tripathy Debashish who taught me film deposition technique and helped me for the antidot papers Dr Navab Singh from the Institute of Microelectronics for providing the deep ultra violet resist patterns used in this thesis Dr Ren Yang and Mr Liu Xinming for their help in magnetooptical kerr effect measurement Mr Shimon Goei for his help in OOMMF simulation and

tasty coffee Dr Naganivetha Thiyagarajah and Dr Wu Baolei for their help in EBL process Dr Shyamsunder Regunathan for his help in SEM I would also like to thank Dr Xin Yi, Ms Ria, Ms Chen Ji, Dr Borja, Dr Dezheng, Dr Xuepeng, Dr Ajeesh, Dr Sankha, Mr Kaushik, Mr Sagaran, Mr Siddharth, Mr Jae-Hyun, Mr Wang Ying and Dr Lu Hui for all the enjoyable moments we have shared in ISML

In addition to the people already mentioned, friends and colleagues outside

of ISML have also made my time as a PhD candidate a rich and memorable one Thanks to all my friends for their help and encouragement

I would like to thank my entire family and all my friends in China for all their support, faith and advice during my stay in Singapore Lastly, but not least,

I would like to thank Ms Guo Li for her endless support and encouragement over the last two years

Table of Contents

Acknowledgements I!

Table of Contents III!

Summary VII!

List of Figures X!

List of Symbols and Abbreviations XIX!

Statement of Originality XXI!

Chapter 1 Introduction 1!

1.1! Background 1!

1.2! Motivation 3!

1.1.1.! 1-D MCs 4!

1.1.2.! 2-D MCs 5!

1.1.3.! Binary MCs 6!

1.1.4.! Applications of MCs 7!

1.3! Focus of Thesis 9!

1.4! Organization of Thesis 10!

Chapter 2 Theoretical Background 11!

2.1! Introduction 11!

2.2! Magnetization Reversal in Ferromagnetic Nanostructures 11!

2.2.1! Magnetic Energies in Nanostructures 12!

2.2.2! Magnetization Reversal in Ferromagnetic Nanowires 14!

2.2.3! Magnetization Reversal in a Ferromagnetic Antidot Array 16! 2.2.4! Magnetization Reversal in a Ferromagnetic Nanomagnet 19! 2.3! Ferromagnetic Resonance Phenomenon 21!

2.3.1! Theory of Ferromagnetic Resonance 21!

2.3.2! Dynamic Micromagnetism Simulation Method 25!

2.4! Summary 26!

Chapter 3 Experimental Details 28!

3.1! Introduction 28!

3.2! Patterning Techniques 28!

3.2.1! Ultraviolet (UV) Photolithography 28!

3.2.2! Deep Ultraviolet Lithography (DUL) 30!

3.2.3! Electron Beam Lithography (EBL) 32!

3.3! Deposition Techniques 35!

3.3.1! Electron-Beam Evaporation and Sputtering 35!

3.3.2! Self-aligned Shadow Deposition 37!

3.3.3! Lift-Off Process 42!

3.4! Characterization Techniques 42!

3.4.1! Scanning Electron Microscope 42!

3.4.2! Scanning Probing Microscope 44!

3.4.3! Magneto-Optical Kerr Effect 45!

3.4.4! FMR Spectroscopy 47!

Chapter 4 1-Dimensional Magnonic Crystals 50!

4.1! Introduction 50!

4.2! Homogeneous-width Nanowire Arrays 50!

4.2.1! Variation of the Width of Isolated Nanowires 52!

4.2.2! Homogeneous Width Arrays of Dipole-coupled Wires 59!

4.3! Alternating-width Nanowire Arrays 62!

4.3.1! Ferromagnetic Ground State 65!

4.3.2! Anti-ferromagnetic Ground State 67!

4.3.3! Tunable Disorder State 79!

4.4! Summary 87!

Chapter 5 2-Dimensional Magnonic Crystals 89!

5.1! Introduction 89!

5.2! Variation of Hole Diameter of Nanoscale Antidot Arrays 90!

5.3! Antidot Array with Alternating Hole Diameters 95!

5.4! Ni80Fe20 Anti-ring Nanostructures 106!

5.4.1! 30 nm-thick Anti-ring Array 108!

5.4.2! Effect of the Nanostructure Thickness 113!

5.5! Summary 123!

Chapter 6 Binary Magnonic Crystals 124!

6.1! Introduction 124!

6.2! Ni80Fe20 Nanomagnets 124!

6.2.1! Isolated Ni80Fe20 Nanomagnets 127!

6.2.2! 1-Dimensional Linear Chain of Ni80Fe20 Nanomagnets 130!

6.3! Binary Nanomagnets 132!

6.3.1! Static Magnetic Properties 133!

6.3.2! Effects of Magnetostatic Coupling 140!

6.3.3! Dynamic Properties 142!

6.4! Summary 146!

Chapter 7 Magnonic Logic Applications 147!

7.1! Introduction 147!

7.2! Magnetic Logic Based on a Meander-type Ni80Fe20 Nanowires Arrays 147!

7.2.1! Experimental Details 148!

7.2.2! Dynamic Response of the Device 150!

7.2.3! Realization of XOR and NOT Logic Operation 157!

7.3! Binary Nanomagnets for Logic Applications 159!

7.3.1! Experimental Details 159!

7.3.2! Magnetic Response of the Binary Nanomagnets 160!

7.3.3! Manipulating the Magnetic Ground States 166!

7.4! Summary 169!

Chapter 8 Conclusion 171!

8.1! Overview 171!

8.2! Summary of Results 171!

8.3! Future Work 174!

References 176!

Appendix 186!

Journal Publications 186!

Conference Proceedings 189!

Summary

In the last decade, magnonic crystals (MC), conceived as the magnetic analogue of photonic crystals, have attracted a lot of interest due to their potential use in a wide range of applications such as microwave resonators, filters and spin wave logic devices

There are many challenges that need to be addressed before the full potential of MC based devices is realized, such as the lack of a systematic investigation of dynamic responses in tailored ferromagnetic nanowire (NW) arrays (1-Diemsional MCs) and 2-Dimensional (2-D) MCs, the method of fabrication of bi-component magnonic crystals consisting of two contrasting ferromagnetic materials and the application of the MCs in logic schemes In this thesis, a comprehensive study of the static and dynamic magnetic properties of various types of MCs is presented

Firstly, the properties of tailored 1-D MCs consisting of NWs with different configurations have been systematically investigated Alternated arranged nanowires with two different widths have been introduced to control the magnetization ground state in the MCs By comparing to the normal nanowires array with a stripe width uniform across the whole array, a perfect antiparallel magnetization state has been realized in the presented engineered nanowires

We have imaged directly the parallel magnetization and anti-parallel magnetization ground states using magnetic force microscopy A simple analytical model has been suggested to explain the experimental data

Secondly, a systematic investigation of the static and dynamic response of 2-D MCs constituted by an antidot and an anti-ring array has been performed For a homogeneous antidot array with square lattice geometry, two main resonance modes were observed for the field applied along the lattice edge. It is also observed that the frequencies of all modes can be systematically tuned by

varying the antidot diameter A new design of antidot arrays with alternating

“hole” diameters has been introduced to further control the spin wave (SW) modes in the MCs The resonance modes and profiles are markedly modified due to the existence of modulated demagnetizing field distributions In anti-ring arrays, it was observed that the FMR response of the anti-rings is highly sensitive to the nanostructure magnetization state for a fixed film thickness The dynamic behavior of the surrounding rectangular antidot can be modified by controlling the magnetization state of the central elliptical nanomagnet It was also found that both static and dynamic responses of the structure are adjustable

by changing the film thickness The MOKE and MFM results show that the central nanomagnets remain in the saturated state for smaller sample thicknesses, while a multi-domain state or vortex state can be observed for thicker nanostructures

Thirdly, a “self-aligned shadow deposition” technique has been introduced

to fabricate bi-component MC consisting of two contrasting ferromagnetic materials (binary MC) High-quality Ni80Fe20/Ni80Fe20 and Ni/Ni80Fe20 binary elliptical  nanostructures  arranged  in  three  different  configurations were prepared using a simple self-aligned shadow deposition method We have also demonstrated that our technique can be applied to other structures, such as binary and thickness modulated nanowires The static and dynamic properties

of the binary MCs were investigated using a combination of MOKE and broadband FMR spectroscopy We showed that the magnetization reversal mechanism can be systematically controlled in the Ni80Fe20/Ni80Fe20 and Ni/Ni80Fe20 binary structures for tailor-made applications We directly confirmed the magnetization states of the structures at various field histories using the magnetic force microscopy Moreover, our micromagnetic simulations are in very good agreement with the experimental results

Lastly, this thesis proposes two logic designs based on nanoscale

reconfigurable MCs Multiple magnetic ground states can be achieved in one

MC by changing the amplitude and/or the angle of applied field The first design

is based on 1-D MCs; two logic states have been formed and detected in a meander-type ferromagnetic nanowires array using a microwave-DC hybrid system A multi-cluster magnetic ground state is formed when no current flows 

in the signal line, while a perfect AFM ground state is energetically preferable when the two input values are not same Functionalities of XOR and NOT gates have been demonstrated based on this phenomenon A method of detection of the logic state has  been  proposed  which  is  based  on  the  reconfigurable microwave  filter  capability  of  εC The second design is based on binary nanostructures We demonstrate the functionality of Ni80Fe20/Ni binary nanostructures cells fabricated using the self-aligned shadow deposition technique in logic applications Depending on the magnetic ordering of the cells, distinct dynamic states probed by broadband ferromagnetic resonance spectroscopy are realized We show that the magnetic ordering can be manipulated to achieve logic operations by controlling the amplitude and the orientation  of  reset  fields.  This proposed logic cell may be useful for downscaling magnonic logic devices

List of Figures

Fig 1.1 ! Typical SEM images of (a) 1-D, [2] (b) 2-D [3] and (c) 3-D [4] PCs

Typical SEM images of (e) 1-D [6] and (f) 2-D [7] MCs Typical band structures of PCs [5] and MCs [6] are shown in (d) and (g),

Fig 2.2 ! (a) The sketch of a nanowire with 10  m length (l), γ00 nm width (w) 

and 30 nm thickness (t) (b) The hysteresis loop of the nanowire when the field is applied along X-axis (c) the sketch of the magnetization

Fig 2.3 ! (a) The sketch of an antidot array (b) The simulated hysteresis loop

of the antidot array The magnetization states for H app = 1500 Oe,

1000 Oe, 500 Oe, 0 Oe, –250 Oe and –400 Oe are shown in (a), (c), (d), (e), (f), (g) and inset of (b), respectively 17!

Fig 2.4 ! (a)The simulated hysteresis loop of the isolated nanomagnet The

sketch of the nanomagnet is shown as the left inset The magnetization states for H app = 1500 Oe, 200 Oe, 0 Oe, –20 Oe, –

200 Oe and –300 Oe are shown as the right inset of (a), (b), (c), (d),

Fig 2.5 ! (a) The experimental (blue dots) and calculated (black line) field

dependence of FMR frequency of a 30 nm thick Ni 80 Fe 20 continuous film (b) the experimental (blue dots) and fitted (black line) field dependence of FMR frequency of a Ni 80 Fe 20 nanowire with 240 nm width, 10  m length and γ0 nm thickness 24!

Fig 2.6 ! (a) Excitation pulse field along Y-axis and (b) magnetization

response of the system along Z-axis as a function of time (c) Simulated FMR absorption of a Ni 80 Fe 20 triangular ring for H app = –1500 Oe The SEM image of the triangular ring is shown as the

Fig 3.1 ! The sketch of the UV photolithography process It includes the three

main steps: (I) photoresist coating, (II) exposure and (III)

Fig 3.2 ! Comparison of the normal mask and the alternating phase shift mask

Fig 3.3 ! The sketch of the Electron Beam Lithography (EBL) process Three

main steps: (I) EBL Resist Coating, (II) E-Beam Writing and (III)

Fig 3.4 ! Sketch of the E-beam evaporation and sputtering hybrid thin film

deposition system In the actual system, the six magnetron sputter sources are circularly distributed under the substrate with a small

Fig 3.5 ! SEM images of the surface profile of the patterned resist of an array

of ellipsoidal nanostructures: (a) configuration A, (CNF A) (b) configuration B (CNF B), and (c) configuration C (CNF C) 38!

Fig 3.6 ! Sketch of the modified E-beam evaporation system with a tilt-table

Fig 3.7 ! (a-b) schematics of the self-aligned shadow deposition method (c)

Fig 3.8 ! SEM images of the (a) isolated Ni/Ni 80 Fe 20 binary structures and

magnetically coupled Ni/Ni 80 Fe 20 binary structures with (b) CNF B and (c) CNF C The magnified SEM images of (a) and (b) are shown

in (d) and (e), respectively Binary nanostructure with big overlay area for CNF A and CNF B are shown in (f) and (g),

Fig 3.9 ! The SEM micrographs of the (a) normal NWs, (b) binary NWs and

(c) thickness modulated NWs, respectively The corresponding schematic illustration of the NW structures are shown in (d-f)

Fig 3.11 !sketch of the working principle of a SPM system 45!

Fig 3.12 !Sketch of a longitudinal MOKE setup 46!

Fig 3.13 !Sketch of the FMR spectroscopy with a microstrip board 48!

Fig 3.14 !Sketch of the FMR spectroscopy with a coplanar wave guide 49!

Fig 4.1 ! (a) SEM image of the CPW (the SEM of 30-nm-thick Ni 80 Fe 20 NWs

with width w = 120 nm and interwire spacing s = 180 nm is shown

as an inset) SEM images of homogenous NWs with (b) w = 240 nm;

s = 360 nm, (c) w = 380 nm; s = 570 nm, (d) w = 540 nm; s = 810

nm, (e) w = 540 nm; s = 120 nm, and (f) w = 540 nm; s = 80

Fig 4.2 ! (a) FMR spectra for sparse homogeneous width nanowire arrays

with w = 120 nm, 240 nm and 540 nm and s = 1.5×w at remanence (H app = 0) (b) FMR frequency at remanence as a function of the wire

Fig 4.3 ! (a) The respective field dependencies of the FMR frequency for w =

120 nm, 240 nm and 540 nm (b) Effective demagnetizing factors extracted experiment and calculation (c) The experimental and calculated switching field of the nanowire arrays as a function of the

Fig 4.4 ! 2-D absorption spectra of homogeneous-width nanowire arrays

Wire width is 540 nm Wire separations are: (a) s = 810 nm, (b) s =

120 nm and (c) s = 80 nm The MOKE results for s = 80 nm is shown

in (d) The representative FMR spectra for this geometry are shown for H app = 0 Oe in (e) and for H app = 120 Oe in (f) Filled squares: calculation with the values of effective demagnetizing factors

Fig 4.5 ! SEM images of coupled alternating NWs arrays with (a) w 0 = 200

nm; w a = 240 nm, (b) w 0 = 200 nm; w a = 380 nm, and (c) w 0 = 200

Fig 4.6 ! 2-D absorption spectra and MOKE results for alternating-width

nanowire arrays with different differences in width between the wide and narrow wires: (a) Fw = 340 nm, (b) Fw = 180 nm and (c) Fw =

40 nm Shown in (d) are the sketches of different magnetization states for NWs corresponding to the field range shown in (a) The MFM images of the FM and AFM ground states are shown in (e) and (f),

Fig 4.7 ! (a) Field dependence of FMR frequency for alternating-width

nanowire arrays for the ferromagnetic magnetic ground state as a function of Fw (b) FMR spectra of FM ground state for coupled

homogeneous width NW array with w = 160 nm, 200 nm, 240 nm, alternating (AW) NW arrays with Fw = /40 nm and 40 nm for H app

Fig 4.8 ! Minor-loop (backward half) absorption spectra of alternating

nanowire arrays with (a) Fw = 340 nm, and (b) Fw = 40 nm (c)

Field dependence of FMR frequency for alternating nanowire arrays with AFM order ground state as a function of Fw 68!

Fig 4.9 ! (a) Minor-loop absorption spectra of alternating nanowire array

with Fw = 0 nm (b) The MFM images of the NWarray at remanent

Fig 4.10 !(a) FMR spectra of AW NW arrays ( Fw = 40 nm) with FM order

ground state (f 0,FM , blue triangles) and AFM ground state (f 0,AFM , red dots) (b) f 0,FM , f 0,AFM and Ff 0 = f 0,FM - f 0,AFM as a function of

Fig 4.11 !The black triangles and the blue circles are the experimental FMR

dispersion of isolated NW array with w = 240 nm and 200 nm respectively T he thin extended lines are the fittings using Kittel’s  equation The red squares are minor-loop experimental results for

an array of dipole-coupled alternating-width nanowire array consisting of wires of the same width (w 1 = 240 nm and w 2 = 200 nm) The red lines are the calculated dispersion for the FM state (thin solid line) and for the AFM one (thick solid line is for the acoustic mode, thick dash line is for the optical mode) The thick and thin red dash-dotted lines are calculations for v 12 = v 21 = 0 74!

Fig 4.12 !(a) SEM image of the alternating width NW array (w 1 = 260 nm, w 2

= 220 nm and inter-wire spacing s = 60 nm) (b): Full loop 2D FMR absorption spectra for the array (c): Normalized M-H loop for the

Fig 4.13 !FMR absorption spectra inside the minor loops with (a) H max = 128

Oe, (b) 163 Oe, (c) 177 Oe, (d) 192 Oe, (e) 199 Oe and (f) 220 Oe Inset to (d): example of 1D simulation 81!

Fig 4.14 !MFM images for the (a) H max = 128 Oe, (b) 163 Oe, (c) 177 Oe, (d)

192 Oe, (e) 199 Oe and (f) 220 Oe at remanence The Fourier transforms of the MFM data is shown on the right side of

Fig 4.15 !Frequency of the fundamental mode at remanence (a) and the ratio

Fig 4.16 !(a) –(c): The magnetic ground states for H max = 163 Oe, 177 Oe and

192 Oe, respectively (d)–(f): the respective calculated profiles  of  dynamic magnetization Red solid line: AFM mode; blue dashed line:

FM mode A 1D numerical model has been used in this

calculation 85!

Fig 5.1 ! SEM images of homogeneous antidot array (a) D1, (b) D2, (c) D3,

and (d) D4 in which the pitch is fixed at 415 nm and diameters are  varied as d D1 = 265 nm, d D2 = 220 nm, d D3 = 185 nm, and d D4 = 145

Fig 5.2 ! FMR spectra of D1, D2, D3 and D4 for H app = 2000 Oe The satellite

peaks are indicated as dashed arrows The higher frequency mode for D4 is splits into two modes as indicated by the solid arrows 91!

Fig 5.3 ! (a) Magnetic field dispersions of FMR frequency of D1, Dβ and Dγ. 

(b) Comparison of experimental and simulated field dispersions of  FMR frequency of D3 The inset in (a) is the simulated magnetization state at remanence for D3 The top and bottom left insets in (b) are the simulated spin precession amplitudes of mode A and mode B at

1000 Oe, respectively. The bright area reflects a high spin precession  amplitude, whereas the dark area corresponds to zero amplitude The bottom right inset in (b) is the simulated FMR spectra of D3 for

H app = 2000 Oe The satellite peaks are indicated by the dashed

Fig 5.4 ! (a) SEM image of the alternating antidot array with d 1 = 300 nm

and d 2 = 150 nm, and a 425 nm center-to-center spacing between two adjacent “holes”. (b) the sketch of the sample with a integrated 

Fig 5.5 ! FMR absorption traces of antidots with varying H app Four different

modes can be observed in the curves as indicated by the A, B, C and

Fig 5.6 ! (a) Magnetization state and (b) divergence of magnetization

distribution of the engineered antidot array obtained from

Fig 5.7 ! (a) Simulated demagnetizing field distribution of the antidot array

for H app = /1000 Oe. Demagnetizing field profiles along (b) line “I” 

Fig 5.8 ! (a) Magnetization state and (b) divergence of magnetization

distribution of the engineered antidot array obtained from

Fig 5.9 ! (a) Simulated demagnetizing field distribution of the antidot array

for H app = /100 Oe. Demagnetizing field profiles along (b) line “I” 

Fig 5.10 !The simulated FMR spectra of the engineered antidot structure are

shown in (a) and (b) for H app = /1000 Oe and –100 Oe, respectively

The spatial distribution of spin precession amplitudes of different modes are shown as insets in (a) and (b) 102!

Fig 5.11 !The expeimental and simulated 2-D absorption spectra are shown in

(c) and (d), respectively The position of H app = –1000 Oe and –100

Oe are indicated by the two dashed lines in the figures 103!

Fig 5.12 !Simulated magnetization states of homogeneous antidot array (a)

without the small “holes” and (b) with reduced big “holes” for H app

= –1000 Oe The corresponding demagnetizing field distributions are shown in (c-d) The simulated FMR spectra are shown in (e) and (f) for H app = –1000 Oe for the two homogeneous antidot structures, respectively The spatial distribution of spin precession amplitudes

of different modes are shown as insets in (e) and (f) 104!

Fig 5.13 !SEM image of 30 nm thick Ni 80 Fe 20 anti-ring arrays 107!

Fig 5.14 !The measured (a) and simulated (b) M-H loops for 30 nm thick

Ni 80 Fe 20 anti-ring array The saturated magnetization state for H app

= /1000 Oe and 1000 Oe are shown as the left and right insets of

(b), respectively The simulated magnetization states for H app = –300

Oe, 0 Oe, 200 Oe and 300 Oe are shown in (c) 108!

Fig 5.15 !The experimental (a) and simulated (c) FMR absorption traces of

the anti-ring array with varying H app The experimental (b) and simulated (d) 2-D absorption spectra (e) The simulated spatial distributions of spin precession amplitudes of Modes A, A’, B and C.  The distributions for Modes A, B and C are shown for H app = /1000

Oe (first three figures) and for Mode A’ for H app = 0 Oe (the fourth

Fig 5.16 !Experimental (a) and Simulated (b) M-H loops for the anti-ring

Fig 5.17 !The MFM images of the remanent magnetization state of the

anti-ring arrays with differerent film thickness: (a) 8 nm, (b) 23 nm, (c)

Fig 5.18 !(a) FMR spectra for anti-ring arrays with t = 8 nm, 15 nm, 23 nm,

30 nm and 40 nm at H app = /1000 Oe (b) The extracted

experimental and simulated resonance frequencies for different modes as a function of thickness at H app = /1000 Oe 118!

Fig 5.19 !(a) FMR spectra for anti-ring arrays with t = 8 nm, 15 nm, 23 nm,

30 nm and 40 nm at H app = 2" Oe (b) The spatial distribution of spin

precession amplitudes for Modes B and A’ at H app = 2" Oe 119!

Fig 5.20 !The experimental 2-D absorption spectra of anti-ring arrays with t

Fig 5.21 !Simulated Magnetization state of the (a)anti-ring, (b)anti-rectangle

and (c)ellipse for 30 nm film thickness when H app = –1000 Oe (d) The internal field value in different areas of anti-ring and anti- rectangle structure as a function of the film thickness for H app = –

1000 Oe (e) The stray field in different areas of the ellipse array as

a function of the film thickness for H app = –1000 Oe 121!

Fig 6.1 ! (a) Structure of the sample and field configuration of the

measurement  for    =  0°.  Representative  microwave  absorption  curves measured on (b) continuous film, (c) isolated elements and (d)

Fig 6.2 ! Field dependence of FMR frequency on isolated ellipsoidal

nanomagnets  for  (a)    =  0°  and  (c)    =  λ0°.  The  corresponding  hysteresis loops of isolated elements are shown i n (b) and (d) for   

Fig 6.3 ! Field dependence of FMR frequency for coupled ellipsoidal

nanomagnets for (a)   = 0° and (c) 

 = λ0°. The corresponding M-H loops for coupled elements are shown in (b) and (d) for   = 0° and 

Fig 6.4 ! SEM images of the resulting Ni 80 Fe 20 nanostructures for the three

Fig 6.5 ! Hysteresis loops of (a) Ni 80 Fe 20 nanostructures, (b)

Ni 80 Fe 20 /Ni 80 Fe 20 binary structures and (c) Ni/Ni 80 Fe 20 binary

Fig 6.6 ! The MFM image of the remnant state (a) Ni 80 Fe 20 nanostructures,

(b) Ni 80 Fe 20 /Ni 80 Fe 20 binary structures and (c) Ni/Ni 80 Fe 20 binary structures (d)MFM image of the Ni/Ni 80 Fe 20 binary structure for

Fig 6.7 ! The simulated hysteresis loops for individual (a) Ni 80 Fe 20

nanostructure, (b) Ni 80 Fe 20 /Ni 80 Fe 20 binary structure and (c) Ni/Ni 80 Fe 20 binary structure The simulated magnetization states corresponding to positions (i-iii) on the M-H for the three structures are shown in (d), (e) and (f) respectively 137!

Fig 6.8 ! Simulated M-H loop for isolated Ni sub-element, isolated Ni 80 Fe 20

sub-element, Ni/Ni 80 Fe 20 nanostructure as a function of a gap size

Fig 6.9 ! A comparison of the M-H loops for Ni 80 Fe 20 nanostructure,

Ni 80 Fe 20 /Ni 80 Fe 20 and Ni/Ni 80 Fe 20 binary nanostructures for configurations (a-c) CNF B and (d-f) CNF C when the field is

Fig 6.10 !Representative FMR absorption traces of isolated (a) Ni 80 Fe 20

nanostructures, (b) Ni 80 Fe 20 /Ni 80 Fe 20 binary nanostructures and (c) Ni/Ni 80 Fe 20 binary nanostructures for varying H app values The corresponding 2-D absorption spectra for the three structures are shown in (d), (e) and (f), respectively 143!

Fig 7.1 ! (a) SEM image of the CPW line and of two meander-type nanowires

arrays (inset) (b) Blow-up SEM image of the structure (width w =

160 nm, edge-to-edge separation g = 80 nm, length l = 10 µm) (c) The sketch of the AC-DC hybrid measurement system 149!

Fig 7.2 ! FMR measurement results of the structure with (a) full loop

measurement when In A = In B = ‘0’; (b) minor-loop (backward half) measurement when In A = In B = ‘0’; (c) MFM image results of the structure taken at remanence after saturating the structure in the field H app = –1000 Oe when both In A and In B are set as ‘0’. The MFM  images of the multi-cluster ground state of the structure for (d) area

A and (e) area B (f) Blow-up image of area B 151!

Fig 7.3 ! (a) full loop measurement when In A = ‘1’, In B = ‘0’; (b) minor-loop

(backward half) measurement when In A =  ‘1’,  In B =  ‘0’ The spectrum for H app = 284 Oe are shown in the right-hand part of the full loop measurement results The spectrum for H app = 0 Oe are shown in the right-hand part of the minor-loop measurement results (c) The MFM image of perfect AFM ground state of the structure for

Fig 7.4 ! Simulation results of the magnetization ground state of the structure

(a) H app = 0 Oe, H y = 0 Oe; (b) H app = 400 Oe, H y = 0 Oe; (c) H app

= 0 Oe, H y = 50 Oe; (d) H app = 400 Oe, H y = 50 Oe The structure was first saturated along –X direction A four-color color scheme is used to represent different magnetization directions 155!

Fig 7.5 ! (a) XOR truth table (b) The normalized spectrum for H app = 284 Oe

with different input values (c) The interpretation of the measurement results agrees with the XOR truth table (a) 157!

Fig 7.6 ! SEM image of the binary nanostructure array 160!

Fig 7.7 ! (a) Experimental M- H loops of the binary for   = 0º. MFM images 

of the array for saturated state (along –X direction) (b) and parallel magnetic state (c) (d) The simulated full hysteresis loop of the Ni/Ni 80 Fe 20 with a 5 nm gap (solid line) and without the gap (dashed line) separating the elements The simulated magnetization states corresponding to positions (I-III) on the hysteresis loop are

Fig 7.8 ! (a) Representative absorption curves of the Ni/Ni 80 Fe 20

nanostructure as a function of H app for saturated state (b) Experimental 2- D full loop absorption spectra of the binary for   =  0º (c) Representative absorption curves of the Ni/Ni 80 Fe 20

nanostructure as a function of H app for anti-parallel magnetic state (d) backward half of the minor loop FMR measurement results 164!

Fig 7.9 ! (a-d) The remanence resonance frequency versus reset field

orientation s as a function of H re amplitude (e) Sketches of different magnetization states for binary corresponding to the reset field orientation shown in (a-d) (f) The simulated angular dependent remanence magnetization for H re = –1400 Oe 167!

Fig 7.10 !Simulated |M x | when H app is swept from –1500 Oe to 1500 Oe for   

= 90º with (a) H x = 0 Oe and (b) 50 Oe The simulated remanence magnetization for   = λ0º with (c) H x = 0 Oe and (d) 50 Oe 168!

Fig 8.1 ! Sketches of the normal nanomagnet, Ni/Fe binary nanomagnet and

the proposed compositional gradient nanostructure 174!

List of Symbols and Abbreviations

ALT PSM alternating phase shift mask

MOSFET metal-oxide-semiconductor field effect transistor

Ms saturation magnetization

N x , N y and N z demagnetizing factors

gyromagnetic ratio

[1] J Ding,  ε.  Kostylev  and  A.  O.  Adeyeye.  “εagnonic  Crystal  as  a 

εedium  with  Tunable  Disorder  on  a  Periodical  δattice”  Physical

Review Letters, 107, 047205 (2011)

[2] J Ding,  ε.  Kostylev  and  A.  O.  Adeyeye.  “εagnetic  Hysteresis  of Dynamic Response of One-Dimensional Magnonic Crystals Consisting of Homogenous and Alternating Width Nanowires

Observed  with  Broadband  Ferromagnetic  Resonance”  Physical

submitted to Physical Review B, 88, 014301 (2013)

‚ Development  of  a  novel  “self-aligned  shadow  deposition”  technique  to 

fabricate bi-component MC consisting of two contrasting ferromagnetic materials High-quality nanostructures consisting of one material and bi-component (binary) nanomagnet have been fabricated and systematically investigated

[6] J Ding, S. Jain and A. O. Adeyeye, “Static and Dynamic Properties of One-Dimensional δinear Chain of Nanomagnets” Journal of Applied

Physics, 109, 07D301 (2011)

[7] J Ding and A O Adeyeye “Binary  Ferromagnetic  Nanostructures: 

Fabrication,  Static  and  Dynamic  Properties”  Advanced Functional

Materials, 23, 1684 (2013)

‚ Experimental demonstration of magnetic logic based on reconfigurable MCs Microwave signal has been used to probe the logic states of the devices and nanostructures

[8] J Ding, ε. Kostylev and A. O. Adeyeye. “Realization of a εesoscopic Reprogrammable Magnetic Logic Based on a Nanoscale

Reconfigurable  εagnonic  Crystal”  Applied Physics Letters, 100,

073114 (2012)

[9] J Ding and  A.  O.  Adeyeye.  “Ni80Fe20/Ni Binary Nanomagnets for

δogic Applications” Applied Physics Letters, 100, 073114 (2012)

‚ Investigation of the dynamic behavior of triangular ring nanostructures [10] J Ding, ε. Kostylev and A. O. Adeyeye. “Broadband Ferromagnetic

Resonance Spectroscopy of Permalloy Triangular Nanorings” Applied

Physics Letters, 100, 062401 (2012)

Chapter 1 Introduction

1.1 Background

Artificial ferromagnetic nanostructures with periodic lateral contrasts in magnetization are  known  as  “magnonic  crystals”  (εCs),  conceived  as  the magnetic analogue of photonic crystals (PCs) [1] In PCs, the propagations of light is manipulated by forming periodical dielectric constant variations along different dimensions Shown in Fig, 1.1(a-c) are SEM images of typical 1-dimension (1-D), [2] 2-dimension (2-D) [3] and 3-dimension (3-D) [4] PCs The manipulation of the light can be described by the band structures similar to the one shown in Fig 1.1(d) [5] Similar principle is also available for MCs The propagation of spin waves (SW) can be manipulated by introducing periodical magnetization variation along different dimensions in MCs Fig 1.1(e) and (f) shows the SEM images of typical 1-D [6] and 2-D [7] MCs The band structure has also been observed in these structures Like PCs, magnonic ones are expected to possess special and interesting properties arising from their frequency band gaps as shown in Fig 1.1(g) [6]

Fig 1.1 Typical SEM images of (a) 1-D, [2] (b) 2-D [3] and (c) 3-D [4] PCs Typical SEM images of (e) 1-D [6] and (f) 2-D [7] MCs Typical band structures of PCs [5] and MCs [6] are shown in (d) and (g), respectively

Recently, there has been a growing interest in the fundamental understanding of the SW propagation in MCs because of their potential use in a wide range of applications such as microwave resonators, filters [8] and spin wave logic devices [9]

A number of periodic structures have been identified as candidates for MCs including 1-D magnetostatically coupled nanowires (NW), [10, 11] periodically modulated yttrium iron garnet (YIG) films [12] and 2-D ferromagnetic antidot nanostructures [13-15] With advances in controlled nanofabrication techniques,

it is now possible to synthesize high-quality magnetic nanostructures with precisely controlled dimensions In this situation, synthetic 1-D bi-component MCs consisting of alternating Ni80Fe20 NWs in direct contact with Co NWs have been fabricated as a new kinds of MCs [6, 16] Recently, the band gap tunability has also been demonstrated in 2-D antidot structures in which the holes are filled with another ferromagnetic material [17] The tailoring of the dispersion and the dependence of the band-gap structure on crystal dimensions contribute to further development of the theory and applications of magnonics, a nascent field which holds great promise in technologies such as microwave communications Additionally, this kind of MCs are excellent candidates for the fabrication of nanoscale microwave devices as the wavelengths of magnons in MCs are orders

of magnitude shorter than those of photons, of the same frequency, in photonic crystals

It is imperative to have control over both static and dynamic magnetic responses of MCs Magneto-optical Kerr effect magnetometer (MOKE) and Magnetic force microscopy (MFM) have been proved as good methods to detect the magnetic ground state of magnetic materials Brillouin light scattering (BLS) and ferromagnetic resonance (FMR) have been used earlier to characterize the

dynamic properties of ferromagnetic nanostructures Tacchi et al [18] have

reported a good agreement between BLS and broadband FMR spectroscopy for

the dynamic response of such structures A vector network analyzer (VNA) is used in the broadband FMR spectroscopy technique to detect the microwave absorption caused by FMR Compared with the classical cavity FMR, frequency and magnetic field can both be varied continuously; thereby enabling systematic characterization of the reversal mechanism in magnetic nanostructures It is important to point out that FMR measurement is a powerful characterization tool to probe the dynamic properties of nanostructures, although it only provides information for standing spin wave mode

The MCs are attracting much attention due to the potential application in magnetic logic In recent years, implementation of a number of classes of logic operations based on a variety of magnetic phenomena has been proposed Examples of suitable magnetic phenomena include traveling SWs, domain wall motion, and physically coupled nanomagnets networks Magnetic quantum-dot cellular automata signal-processing approach in a coupled nanomagnets networks has been demonstrated as a suitable architecture for logic application Various logic gates have been realized in properly structured arrays of coupled nanomagnets A variety of methods have been proposed to control the switching

of each nanomagnets The logic states of the structures have been detected by using MOKE and MFM The development of MCs may open routes for logic applications in on-chip microwave device

A detailed discussion of previous and on-going research on this topic will

be given in the following section

1.2 Motivation

This section presents the main challenges in the field of MCs Previous works pertaining to the main research topics in this thesis have been reviewed The discussion is separated by four sections 1-D, 2-D and binary MCs is reviewed in the first three section The fourth section focuses on the applications

of the MCs

1.1.1 1-D MCs

The static and dynamic properties of ferromagnetic NWs of rectangular cross-section have attracted a lot of interest both from a fundamental viewpoint and because of their potential in a wide range of applications such as microwave devices [8, 19] and domain wall logic [20] They also represent model systems

to study the properties of SW excitations in laterally confined magnetic elements [10, 11, 21-28] The fundamental understanding of SW propagation in coupled periodic magnetic NWs, which is also called 1-D MCs, have also attracted much attention In this regards, both the static [29-31] and dynamic [19, 32-34] properties of NW arrays have been studied using various experimental techniques It has been shown in the static experiments that the magnetization reversal mechanism of NWs is strongly dependent on the geometrical dimensions of the NWs [30] and inter-wire spacing [31] The major challenge for technological applications utilizing MCs is the precise control of

the dimension and magnetic ground state of the 1-D structures J Topp et al

[22] have reported that a number of periodic magnetic configurations (magnetic ground states) can be achieved in one MCs with the same periodic geometry of material They proved that the material can be switched between three ground states, i e.: saturated state, multidomain state and demagnetized state Unfortunately, the controlling of the magnetic ground states is not perfect in the MCs consisting of homogeneous NW arrays Alternating width (AW) NW

arrays have been introduced by S Goolaup et al [29] as 1-D MCs with

controllable switching process For the AW arrays it has been established that the NWs of a larger width switch earlier than the narrower ones when the sample

is released from the magnetic saturation In this regard, it is possible to tune the magnetic ground state to obtain both ferromagnetic (FM, saturated) and

antiferromagnetic (AFM, demagnetized) ordering from the same AW NW array There have been very few experimental works reported on the precise control

of the magnetic ground state of coupled NW arrays, which will influence dynamic response of the nanostructure Such manipulation of the magnetic ground state in MCs is crucial to fully exploit potential of MCs in future applications A systematic investigation of dynamic responses in tailored ferromagnetic NW arrays consisting of homogenous NWs and AW NWs is essential for this purpose

1.1.2 2-D MCs

Magnetic antidots, [35] which are periodic arrays of holes, patterned into a thin  film  represent  a  β-D MC [13, 24, 36, 37] in which the magnonic band structures can be engineered by varying the periodicity of the arrays While the static properties of antidots have been extensively studied in recent years, [38-41] it is also imperative to have control over the dynamic magnetic response Brillouin light scattering (BLS) [36, 42] and ferromagnetic resonance (FMR) [39, 43-47] have been used earlier to characterize the dynamic properties of antidot structures Tacchi et al [18, 48] have reported a good agreement between BLS and FMR for the dynamic response of such structures Moreover, the dynamic response of antidots has also been investigated as a function of lattice symmetry [18] and orientation of an applied field [43, 46, 48, 49] and a strong dependence was observed It has been shown that multiple resonances occur in antidot arrays for in-plane  magnetic  fields To achieve a consistent and experimentally tunable dynamic response in antidot arrays in the nanoscale regime, it is thus critical to understand the effect of physical dimensions on the dynamic properties

Bi-component Ni80Fe20 antidot lattice with embedded Co nanodisks was recently introduced to further manipulate SW properties in 2-D structure [17]

D Tripathy ed al [41] have reported the static response of bi-component antidot

nanostructures by patterning the neighbouring antidots with alternated dimensions They demonstrate that the use of two antidot sublattices greatly enhances the parameters available for engineering the behaviour of antidot nanostructures This design is also a good candidate for 2-D MCs It is also necessary to investigate the dynamic response of such structures

1.1.3 Binary MCs

One direct way to form periodic magnetization state in an array is to fabricate synthetic nanostructured MCs comprising more than one magnetic material This kinds of MCs have been theoretically predicted and experimental demonstrated to further control of the static and dynamic response Z K Wang

et al. [6, 16] have designed and fabricated a 1-D MC in the form of a periodic array comprising alternating contacting Co and Ni80Fe20 nanostripes Band gap tenability have been observed in such kind of MCs Nanomagnets are another important research object for this kind of MCs They have attracted a lot of interest due to their potential application in magnetic logic [50-53] While the static [54, 55] properties of isolated and coupled nanomagnets have been extensively studied, it is also imperative to have control over the dynamic magnetic response, since reading and writing speeds in magnetic devices are getting closer to the time scale of spin dynamics [24, 42] The nanomagnets arrays can also be recognized as MCs due to the periodical magnetization distribution Multi-level electron beam lithographic approach has been included

to fabricate the two material MCs There are various limitations with the quality

of MCs produced with the approach described in Ref [6, 16] including the issue

of alignment of the two contrasting elements To fully exploit MCs in applications, the key challenge associated with the nano fabrication of tunable bi-components MCs consisting of two contrasting ferromagnetic materials must

be resolved With our proposed self-aligned shadow deposition technique, it is now possible to fabricate large area binary MCs without alignment required between the two FM layers To further exploit MCs in applications, it is crucial

to carry out a systematical investigation on the normal and binary nanomagnet arrays

1.1.4 Applications of MCs

Microwave devices as one of the main application of MCs have been extensively investigated in the past several decades [56] A variety of devices such as resonators, filters, delay lines and directional couplers have been designed based on periodic YIG films, [57-59] which can be considered as a type of MCs YIG has been widely used in this type of applications due to the extremely small spatial damping of SWs in the film However, it is difficult to fabricate nanostructures using YIG And there is no low-loss YIG films with thickness in nanometer range have been demonstrated so far For this reason, this application is not the main point of this thesis

Magnonics is also a good candidate for the application in magnetic logic [37] Magnonic logic devices are introduced as one of the possible routes to scaling down the size of the metal-oxide-semiconductor field effect transistor (MOSFET) In recent years, a variety of magnetic phenomena has been proposed to achieve logic operations [9, 20, 50-52, 60-68] Examples of suitable magnetic phenomena include traveling SWs, [9, 60-64] domain wall motion, [20, 65-68] and physically coupled nanomagnets networks [50-52] Only a few designs are practical with functionality that could be demonstrated experimentally.  The  first  working  SW  based  logic  device  has  been 

experimentally demonstrated by Kostylev et al [36] The authors used the

Mach–Zehnder-type current-controlled SW interferometer to demonstrate output voltage modulation as a result of spin wave  interference.  This  first 

working prototype device was of considerable importance for the development

of magnonic logic devices The complete set of logic devices such as NOT, XNOR and AND based on Mach–Zehnder-type spin-wave interferometer devices has been proposed An obvious disadvantage of these designed devices

is their macroscopic size Utilizing a patterned Ni80Fe20film replacing the YIG stripe can significantly reduce the dimension There is a high demand in design

of new logic cells for the magnonic logic devices Nanomagnets, as one important candidate for the new logic cells, have attracted a lot of interest due

to their smaller size when compared with domain wall logic systems

Both staticand dynamic properties of isolated and coupled nanomagnets have been extensively studied Magnetic Quantum-dot cellular automata signal-processing approach has also been demonstrated as a suitable architecture for logic application [50, 51] Various logic gates have been realized in properly structured arrays of coupled nanomagnets A variety of methods have been proposed to control the switching of each nanomagnets In all of the designs, only two stable magnetic states (when the elements are saturated along the positive or negative directions) are available to store and transfer information Thanks to the advances in nanofabrication techniques, it is now possible to fabricate binary nanostructures from two distinct ferromagnetic materials [6, 16] The presence of two magnetic materials in this structures supplies new routes to downscaling of the magnetic logic devices Some complicated methods such as MOKE [50, 65-68] or MFM [20, 51, 52] are often employed

to detect the logic state of the structures It is necessary to explore a method to manipulate and detect the logic states of the devices directly The latest research suggest that the collective dynamics in MCs differs for various magnetic ground states It shows that FMR spectroscopy can be employed as a detection method

of the magnetic ground state The frequency bands of the microwave response can be controlled in MCs [22] This means that it is now possible to control and

detect the logic state of the structures using a microwave signal In this regards, the reconfigurable MCs may be used to implement logic gate functionality

We have reviewed most of the previous work related to the topics of the thesis The main focus and objectives of the thesis will be presented in the next section

1.3 Focus of Thesis

In the previous section, we have reviewed most of the works related to the topic of this thesis In this thesis, a comprehensive study of the static and dynamic magnetic properties of various kinds of MCs will be presented The research work presented in this thesis focus on four parts The first and second part of the thesis are devoted to the characterization of the properties of tailored 1-D MCs consisting of NWs with different configurations and 2-D MCs, respectively The third part of the thesis deals with the characterization of the bi-component MC consisting of one or two ferromagnetic materials (binary MC) The final part of this thesis proposes two logic designs based on nanoscale reconfigurable MCs Multiple magnetic ground states can be achieved in one

MC by changing the amplitude and/or the angle of the applied field

The main objectives of this thesis are:

(a) A comprehensive study of the dynamic behaviour of 1-D MCs consisting

of homogenous and alternating width Ni80Fe20 NW arrays

(b) A systematic investigation of the 2-D MCs constituted by antidot arrays with different configurations and anti-ring arrays

(c) Characterization of the properties of the binary MCs fabricated by the aligned shadow deposition” technique

“self-(d) Suggestion of new designs for logic applications based on the multiple magnetic ground states of MCs Development of a new method to detect the magnetic states of the logic units

1.4 Organization of Thesis

Chapter 1 discusses the background and motivation for the work presented

in this thesis Chapter 2 gives the basic theoretical concept of the ferromagnetic resonance The magnetization reversal process on nanostructures is discussed briefly in this chapter Chapter 3 presents various techniques utilized for fabrication and characterization of the MCs presented in this thesis The dynamic response of 1-D and 2-D MCs are discussed in Chapter 4 and Chapter

5, respectively In Chapter 6, the static and dynamic properties of binary MCs fabricated  by  the  “self-aligned  shadow  deposition”  technique  are presented Chapter 7 demonstrates the application in logic schemes of the reconfigurable MCs Finally, Chapter 8 summarizes the main experimental results presented in the thesis

Chapter 2 Theoretical Background

2.1 Introduction

This chapter presents the basic theoretical backgrounds for better understanding of the main topics of this thesis The magnetization reversal in ferromagnetic nanostructures is discussed briefly in Section 2.2 The switching process of ferromagnetic nanowires, antidots and nanomagnets is also presented Section 2.3 concentrates on the concept of the ferromagnetic resonance phenomenon The Kittel’s equation for FMR frequencies is used to describe the experimental results for continuous Ni80Fe20 films This is followed by Section 2.4 in which the basic theoretical concepts of the 1-D and 2-D MCs is presented

2.2 Magnetization Reversal in Ferromagnetic Nanostructures

During the last decade, there has been significant efforts aimed at fundamental understanding of the magnetic properties of ferromagnetic nanostructures because of their potential in a wide range of applications [37,

51, 69] It is necessary to fully understand the magnetization reversal process in ferromagnetic nanostructures before the dynamic properties are discussed The reversal process of a nanostructure can be determined by minimizing the total energy of the system This energy mainly consists of exchange energy, anisotropy energy, magnetostatic energy and Zeeman energy [70, 71] The details of these energies will be discussed in Section 2.2.1 This is followed by the description of the magnetization reversal process in ferromagnetic nanowires, andtidots and nanomagnets in Section 2.2.2, 2.2.3 and 2.2.4, respectively

2.2.1 Magnetic Energies in Nanostructures

The most important energy in magnetic materials is the exchange energy

E ex, which describes the exchange interaction between the two nearby electrons The exchange interaction is the origin of the magnetic field It reflects the Coulomb repulsion and Pauli principle, which forbids the two electrons to occupy the same quantum state Exchange Hamiltonian is used to describe the total energy in a many-electron atomic system: [71]

where S i and S j is the sum over all pairs of atoms on lattice sites i, j, respectively

J ij is the exchange integral between S i and S j The J ij can be simplified to a single

exchange constant J if only the nearest-neighbour interactions is considered When J is a positive value, (2.1) describes a ferromagnetic interaction in which the two spins tend to align parallel J < 0 indicates an antiferromagnetic

interaction, which tends to align the two spins antiparallel

The anisotropy energy E a is another important energy for determining the

magnetization state in ferromagnetic nanostructures It describes the tendency that the direction of magnetization is usually constrained to lie along the easy axis of the material This energy can also be called as magneto-crystalline anisotropy energy because it originates from the crystal-field interaction and spin-orbit coupling or other interatomic dipole-dipole interaction The expression for this energy are different for the material with different symmetries When the structure is from a ferromagnetic material with uniaxial anisotropy, it can be expressed as: [72]

in which K un (n = 1, β, γ,…) are the anisotropy constants,  is the angle between the magnetization direction and the easy axis In this thesis, all the presented thin films used to fabricate the nanostructures are polycrystalline due to the

electronic beam evaporation process The easy axis of the grains in such kinds

of thin film are randomly distributed In this situation, the anisotropy of the structure becomes very weak but it may still be observable in some samples

The magnetostatic energy E ms can be considered as the magnetic field

energy generated by the demagnetizing field in the sample [72] Thus the source

of this energy is the shape of the ferromagnetic structure In this regard, this energy can also be called as the self-energy or the energy of a magnet in its own field It can be described as:

In which H d is the demagnetizing field generated by the samples, the M is the

magnetization of the samples It is important to notice that the energy is minimal

when H d is parallel to M However, usually H d is antiparallel to M, thus the

magnetostatic energy increases the total energy of the magnetic material It is

difficult to evaluate the distribution of H d accurately The demagnetizing factor

N d has been introduced to correlate the H d and M:

N d is a tensor For an ellipsoid of revolution the tensor has only diagonal

components N x , N y and N z

Fig 2.1 Sketch of an ellipsoid

Note that Eq (2.3) is valid only for ellipsoid of revolution For example in

an ellipsoid as shown in Fig 2.1, the sum of the demagnetizing factors along

!

∃

the three axis is 1 In this case, the E ms can be described as:

in which and are the tilt angle of M away from X-Y plane and Y-Z plane,

respectively This model can be extended to ferromagnetic nanostructures with other kinds of geometry with different shape However, for the shapes different from ellipsoidal, the demagnetizing field is spatially nonuniform and the demagnetizing factors are only effective This shape anisotropy strongly influences the magnetization reversal process in ferromagnetic nanostructures, [73] such as nanowires, [29, 30, 74] nanomagnets, [54, 55, 75] nanodots [76, 77] and antidot [35, 41]

The Zeeman energy E z rises when the magnetic structure is placed in an

externally applied field H app It can be described as:

in which M is the local magnetization The Zeeman energy is minimized when the M is aligned with the H app

The minimization of the sum of these mentioned energies E total determins

the final magnetization state of the ferromagnetic nanostructure In the following sections of this chapter, the reversal process of the ferromagnetic nanowires, antidots and nanomagnets will be presented

2.2.2 Magnetization Reversal in Ferromagnetic Nanowires

As mentioned in the previous section, the magnetization in a ferromagnetic nanostructure can be determined by minimizing the total free energy This section concentrates on the magnetization reversal in ferromagnetic nanowires Two models have been used to describe the magnetization reversal modes, namely coherent rotation [30, 78] and curling [30, 79, 80] In coherent rotation model, all the magnetic dipole moments remain parallel to each other at all times

during the reversal process, while in the curling mode, neighboring magnetic dipole moments are not kept aligned parallel to each other The coherent rotation model suits for a nanowire with a small thickness to width ratio [30]

Shown in Fig 2.2(a) is the sketch of a Ni80Fe20 nanowire with the length l

=  10  m,  width  w = 300 nm and thickness t = 30 nm The external static magnetic field (H app ) is applied along the X axis as shown in the figure A clear

hysteresis phenomenon with one-step switching process can be observed in the typical hysteresis loop measurement on the nanowires as shown in Fig 2.2(b) Fig 2.2(c) shows the sketches of the magnetization state I and II of the nanowire This result can be explained based on the energy minimization principle

Fig 2.2 (a) The sketch of a nanowire with 10  m length (l), γ00 nm width  (w) and 30 nm thickness (t) (b) The hysteresis loop of the nanowire when the field is applied along X-axis (c) the sketch of the magnetization direction for

different states

The E a is negligible because of the polycrystalline nature of the Ni80Fe20

E ex is also minimized due to the fact that all the spins remain parallel to each other in nanowire with such a small thickness to width ratio (0.1) [30] This is also the cause of the single one-step switching process Then the total energy can be expressed as:

(2.7)

The E ms is determined by the magnetization state and the demagnetizing factor

for the structures The demagnetizing factor for long nanowires with a small

I II

thickness-width ratio can be approximately expressed as N x = 0, N y = w/(w+t)

and N z = t/(w+t) [81] It is obvious that the E ms is minimized when the structure

is magnetized along the X-axis of the wires, while the E ms becomes much higher

when the spins are along the Y-axis or Z-axis In order to switch the nanowires from state I to state II, a reversal field has to be large enough to overcome the

high value of E ms when the spins are not along the easy axis of the structure

This effect induces the hysteresis phenomenon as shown in the Fig 2.2(b) The switching process of a nanowire array will be different due to the dipole-dipole interaction between the surrounding wires [31] The detailed static and dynamic properties of nanowire arrays will be presented in Chapter 4 of the thesis

2.2.3 Magnetization Reversal in a Ferromagnetic Antidot Array

Ferromagnetic antidots, which are periodic arrays of holes patterned into a thin film, are another attractive nanostructure [35] They were proposed as a potential candidate for high density data storage [38] Recently, it has also been proved that the antidot structure is an important model to study the SW propagation [13, 18, 24, 36, 39, 42-48, 82-84] A deep understanding of the magnetization reversal of the antidot nanostructure is one of the preconditions

to carry on the further investigation of such structures In this section, the magnetization reversal process of an Ni80Fe20 antidot array with a β65 nm “hole” diameter in square lattice geometry with 415 nm lattice factor (as shown in Fig 2.3(a)) will be presented based on simulation results

For a complicated structure such as antidot array, it is difficult to explain the switching process with the simple energy equation as shown in Eq (2.7), although the energy minimization theory is still the main principle A micromagnetics simulation software LLG simulator [85] can be used to visualize the magnetization reversal process in the nanostructures This software calculates the magnetization states in a ferromagnetic nanostructure by solving