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Magnetism in nanoscale materials, effect of finite size and dipolar interactions

Permanent Link: http://ufdc.ufl.edu/UFE0041972/00001

Material Information

Title: Magnetism in nanoscale materials, effect of finite size and dipolar interactions
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Das, Ritesh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: dipolar, magnetism, nanomagnetics, nanomagnetism
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Material physics is always motivated by the materials with exotic properties. It was a common belief that exotic properties are only associated with exotic materials. Now it is clear that geometrical confinement at nanoscale dimensions can give rise to exotic properties even in simple materials. Ferromagnetic materials in restricted dimensions are extremely interesting because of their potential applications as well as the rich fundamental science involved. Magnetic nanoparticles are useful in high density magnetic data storage devices, sensors, contrast agents in MRI, drug delivery, treating hyperthemia and many more. All the applications of nanomagnets are very crucial in modern day life. But most of the applications are restricted due to the limitations in the fundamental properties arises in nanoscale and also due to the technical limitations of controlling things at nanoscale. For example particles become superparamagnetic as the size is reduced below a certain value and the magnetization direction fluctuates randomly due to the thermal energy which limits the density of data storage. The promises of nanomagnets are huge and to really achieve the grand challenges in nanomagnetism, it is necessary to understand the basic sciences involve at small scales. In this present work, the magnetic properties of systems in nanoscale (nanoparticles and thin films) have been investigated. The effect of dipolar interactions, particle size, particle size distribution, temperature, magnetic field etc. on the magnetic properties have been studied.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ritesh Das.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Hebard, Arthur F.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041972:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041972/00001

Material Information

Title: Magnetism in nanoscale materials, effect of finite size and dipolar interactions
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Das, Ritesh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: dipolar, magnetism, nanomagnetics, nanomagnetism
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Material physics is always motivated by the materials with exotic properties. It was a common belief that exotic properties are only associated with exotic materials. Now it is clear that geometrical confinement at nanoscale dimensions can give rise to exotic properties even in simple materials. Ferromagnetic materials in restricted dimensions are extremely interesting because of their potential applications as well as the rich fundamental science involved. Magnetic nanoparticles are useful in high density magnetic data storage devices, sensors, contrast agents in MRI, drug delivery, treating hyperthemia and many more. All the applications of nanomagnets are very crucial in modern day life. But most of the applications are restricted due to the limitations in the fundamental properties arises in nanoscale and also due to the technical limitations of controlling things at nanoscale. For example particles become superparamagnetic as the size is reduced below a certain value and the magnetization direction fluctuates randomly due to the thermal energy which limits the density of data storage. The promises of nanomagnets are huge and to really achieve the grand challenges in nanomagnetism, it is necessary to understand the basic sciences involve at small scales. In this present work, the magnetic properties of systems in nanoscale (nanoparticles and thin films) have been investigated. The effect of dipolar interactions, particle size, particle size distribution, temperature, magnetic field etc. on the magnetic properties have been studied.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ritesh Das.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Hebard, Arthur F.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041972:00001


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MAGNETISM IN NANOSCALE MATERIALS, EFFECT OF FINITE SIZE AND
DIPOLAR INTERACTIONS



















By

RITESH KUMAR DAS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010


































2010 Ritesh Kumar Das

































I dedicate this to my parents and family for their active support. Without them it would
have not been possible.









ACKNOWLEDGMENTS

I am truly indebted to ri: irn: individuals who have contributed to the success of my

research work. Therefore, I express my sincerest regrets to any person not specifically

mentioned here.

First and foremost, I am thankful to my research advisor Prof. A. F. Hebard for

giving me the opportunity to work with him. It has been a great experience to work

under his supervision. His positive, open-minded attitude toward research creates a unique

laboratory environment full of encouragement. I have learned a lot from his unadulterated

enthusiasm, willingness to learn and elegant but simple approach to understanding

fundamental physics.

I would like to thank all the present and former lab members for their helps and

pleasant ( ".'r. ir:. I am grateful to John J. Kelly for teaching me many experimental

techniques when I joined the group. Thanks to all the lab members Patrick, Rajiv, Sef,

Siddhartha, Sanal, Xiaochang for their helps. I really enjoi, d working with you guys.

I would also like to acknowledge the staffs of machine shop and electric shop.

Specially cryogenic staffs, Greg and John, for their constant supply of liquid He and

N2 all year around 24/7. Thanks to Jay (really a nice guy) for looking after all the pumps

and chillers.

I would like to thank all of my committee members. I will specially thanks Prof.

Amlan Biswas. Though I did not have chance to collaborate with him, but his guidance

and support towards my degree have been very helpful. I am also greatly thankful to Prof.

D. Norton for the wonderful collaboration and for letting me use his lab facilities.

I am thankful to my collaborators D. Kumar and A. Gupta from NCA&T. I am also

very thankful to Matt, Patrick, Kyeong-Won from Prof. Norton's lab for their helps and

being good friends.









I am indebted to my parents for their support, encouragement and for alv--i-

believing in me. I appreciate the warmth and affection of my sister Mridula. I could not

have come this far without their blessings.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 8

LIST OF FIGURES .................................... 9

A BSTRA CT . . 11

CHAPTER

1 THEORY AND BACKGROUND ................... ..... 12

1.1 Single Domain and Multi Domain Particles ....... .......... 12
1.2 Hysteresis Loop of Single Domain Coherently Rotating Particles ...... 14
1.3 Hysteresis Loop of Multi Domain Particles ................. .. 24
1.4 Magnetization vs. Temperature .......... .. .. .. 29
1.4.1 Zero Field Cooled (ZFC) Magnetization 30
1.4.2 Field Cooled (FC) Magnetization ..... 31

2 DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL
SINGLE DOMAIN GRAIN SIZE OF NI IN LAYERED Ni/A120O COMPOSITES 32

2.1 Abstract .. .. .. .. ... .. .. .. ... .. .. .. .. ...... .. 32
2.2 Introduction .................. ................ .. 32
2.3 Experimental Details .................. ........... 33
2.4 Data and Discussion .................. ........... .. 36
2.5 Conclusion .. ..... .. 40
2.6 M ethods .................. .................. .. 41
2.6.1 Mathematical Analysis .............. .. 41
2.6.2 Basic Physical Understanding ............... .. .41

3 EFFECT OF DIPOLAR INTERACTION ON THE COERCIVE FIELD OF
MAGNETIC NANOPARTICLES: EVIDENCE FOR COLLECTIVE DYNAMICS 42

3.1 Abstract .................. .................. .. 42
3.2 Introduction .................. ................ .. 42
3.3 Results and Discussions .................. ......... .. 44
3.4 Conclusions .................. ................ .. 49

4 FINITE SIZE EFFECTS WITH VARIABLE RANGE EXCHANGE COUPLING
IN THIN-FILM Pd/Fe/Pd TRILAYERS .................. ..... 51

4.1 Abstract .................. .................. .. 51
4.2 Introduction .................. ................ .. 51
4.3 Experimental Details .................. ........... .. 52









4.4 Results and Discussion .................. ........ .. .. 54
4.5 Conclusions .................. ................ .. 58

5 TEMPERATURE DEPENDENCE OF COERCIVITY IN MULTI DOMAIN
NI NANOPARTICLES, EVIDENCE OF STRONG DOMAIN WALL PINNING 59

5.1 Abstract ...................... ............. 59
5.2 Introduction ...................... ........... 59
5.3 Results and Discussions ............................ 60
5.4 Relation Between Micromagnetic Parameter and Magnetic Parameters 65
5.5 Conclusions ...................... ........... 66

6 COERCIVE FIELD OF FE THIN FILMS AS THE FUNCTION OF TEMPERATURE
AND FILM THICKNESS: EVIDENCE OF NEEL DISPERSE FIELD THEORY
OF MAGNETIC DOMAINS ................... ....... 68

6.1 Abstract ...................... ............. 68
6.2 Introduction ...................... ........... 68
6.3 Experimental Details ................... ....... 69
6.4 Results and Discussion ................... ...... 70
6.5 Conclusion ...................... ........... 74

7 SCALING COLLAPSE OF THE IRREVERSIBLE MAGNETIZATION OF
FERROMAGNETIC THIN FILMS ............. ........... 75

7.1 Abstract ....................... ............ 75
7.2 Introduction ...................... ........... 75
7.3 Experimental Results .................. ........... .. 76
7.4 Conclusions .................. ................ .. 84
7.5 Methods ................... ............... 85
7.5.1 Ni N i'" .q. iticle. .................. .......... .. 85
7.5.2 Gd Thin Film. .. ..... .. .......... 86
7.5.3 (Lai_yPry)0.67Ca,, M!O3 (LPC'1O) Thin Films. ... 86
7.5.4 Temperature Correction of Coercive Field. ..... 86

REFERENCES ................... ............. ...... 87

BIOGRAPHICAL SKETCH ........... ........ ... 96









LIST OF TABLES


Table page

1-1 H, vs. T ...... ........... ............... .. .. 29









LIST OF FIGURES


Figure page

1-1 SD and MD particle ............... ............. 13

1-2 Coherent and incoherent rotation ... ............ ..... .. 14

1-3 Single particle in magnetic field ............... ........ ..15

1-4 Two state energy ............... ............ .. .. 16

1-5 Hysteresis of SD particle ............... ........... ..18

1-6 Diagram of a particle .................... 20

1-7 Thermal average of magnetization .......... .. 21

1-8 Flow diagram .................. .................. .. 21

1-9 MH below TB .. ..................... .. ..24

1-10 M H below TB ....... ....... .... ..... .. 25

1-11 SD to MD transition and He .................. .......... .. 26

1-12 Magnetization loop for MD particle .................. ..... .. 27

1-13 Domain wall and He .................. .............. .. 28

1-14 M vs. T for 3 nm Ni nanoparticles .................. ..... .. 30

2-1 STEM image of Ni particle .................. ........... .. 34

2-2 H, vs. d, different T .................. ............ .. .. 35

2-3 d, vs. T ................................ .. .... 37

2-4 Hd and domain ............... ............... .. 41

3-1 Sample ..... ........... ... .............. 45

3-2 MH loop. .................... ....... ...... .. ... 46

3-3 He vs. d: dipolar interaction .................. .......... .. 47

3-4 Dipolar interaction .................. .. 49

4-1 Physical and magnetic view of sample .................. ...... 53

4-2 Saturation magnetization vs. x .................. ........ .. 54

4-3 Coercive field vs. x .................. ............... .. 55









4-4 Curie temperature vs. x

5-1 Three sets of sample .

5-2 MH loops of set 2 .....

5-3 He vs. T2/3 set 1 samples

5-4 He vs. T2/3 set 2 samples

5-5 He vs. T2/3 set 3 samples

5-6 Hco and Eo of set 2 .

6-1 TEM image of Fe film .

6-2 M-H loop of Fe film ..

6-3 He vs. T of Fe film .

6-4 He vs. K of Fe film ..

6-5 He vs. d of Fe film .

7-1 Irreversible Magnetization

7-2 Behavior of the AM(H, T)

7-3 The anstz .. .......


isotherms


as the function of


7-4 Scaling collapse of variety of ferromagnetic materials .. ............


and


scaling


collapse









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Phil. .. hi,

MAGNETISM IN NANOSCALE MATERIALS, EFFECT OF FINITE SIZE AND
DIPOLAR INTERACTIONS

By

Ritesh Kumar Das

August 2010

C'!I ir: A. F. Hebard
Major: Physics

Material physics is ahv--, motivated by the materials with exotic properties. It was

a common belief that exotic properties are only associated with exotic materials. Now

it is clear that geometrical confinement at nanoscale dimensions can give rise to exotic

properties even in simple materials. Ferromagnetic materials in restricted dimensions

are extremely interesting because of their potential applications as well as the rich

fundamental science involved. Magnetic nanoparticles are useful in high density magnetic

data storage devices, sensors, contrast agents in MRI, drug delivery, treating hyperthemia

and many more. All the applications of nanomagnets are very crucial in modern d4v life.

But most of the applications are restricted due to the limitations in the fundamental

properties arises in nanoscale and also due to the technical limitations of controlling things

at nanoscale. For example particles become superparamagnetic as the size is reduced

below a certain value and the magnetization direction fluctuates randomly due to the

thermal energy which limits the density of data storage. The promises of nanomagnets

are huge and to really achieve the grand challenges in nanomagnetism, it is necessary to

understand the basic sciences involve at small scales. In this present work, the magnetic

properties of systems in nanoscale (nanoparticles and thin films) have been investigated.

The effect of dipolar interactions, particle size, particle size distribution, temperature,

magnetic field etc. on the magnetic properties have been studied.









CHAPTER 1
THEORY AND BACKGROUND



Ferromagnetism is known for more than 2500 years to man. The first magnetic

material discovered was magnetite (Fe2O3). The practical applications of ferromagnets

was recognized from very ancient time. The first use of magnetic material was as a

compass. According to the magnetic properties, materials can be divided into diamagnetic,

paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, spinglass etc. In this

present work ferromagnetism will be the main topic. C'! lpters followed by this chapter

will discuss the effect of finite size and dipolar interactions on the magnetic properties of

some materials with nanoscale structures. In this chapter a general theoretical background

will be given.

1.1 Single Domain and Multi Domain Particles

When the size of the particle is very small, it will contain only one magnetic

domain. 1-3 This is because the energy required to form a domain is larger than the

magnetostatic self energy. The magnetostatic self energy for a spherical particle is given

by1

Ema, ,1 2-V/12 (1-1)

where po is the free space permeability, i.. is the saturation magnetization and V is the

volume of the particle. The energy required to form a Bloch domain wall is1-3


Ed, 4= AKR2 (1-2)


where K is the anisotropy constant, A is the exchange stiffness and R is the radius of the

particle.

Note that Emag grows as R3 and Edw grows as R2. Domain formation is therefore

favorable for larger particles as the magnetostatic energy will be large compared to

the domain wall formation energy. The critical single domain radius (Rsd) where the









transition from single domain (SD) to multidomain (M\1)) occurs is given by1 3

36V/A
Rsd = (1-3)

The above equation is determined by solving the equation Ema, = Ed.1-3 Thus particles

having radius smaller than Rd are SD and particles having radius larger than Rd are MD

(see Fig. 1-1).









d < dc d >dc

Figure 1-1. Smaller (larger) particles are SD (\!1)) as the magnetostatic self energy is
smaller (higher) than the energy required to form domain. The critical size of
the particle where the single domain to multidomain transition occurs is given
by Eq. 1-3.


The magnetization dynamics for SD and MD particles are dramatically different.

SD particles reverse their magnetization by rotation only. MD particles reverse their

magnetization by domain wall motion and rotation. Rotation of magnetization for the SD

particles is mainly of two types: 1) coherent and 2) incoherent (Fig. 1-2). The exchange

length2-4

A (1-4)

is a measure of the distance over which the atomic exchange interactions dominate and all

the spins rotate coherently. Particles with size larger (smaller) than le rotate incoherently

(coherently). The exchange length is usually larger than Rd for soft ferromagnets where

the anisotropy energy is small. Thus in soft ferromagnets magnetization reversal occurs

either by coherent rotation (small particles) or by domain wall motion (large particles).









t tt t\ /
ttttt



Coherent rotation

/ttit 1t t

H H / t


Incoherent rotation

Figure 1-2. Coherent and incoherent rotation of the magnetization. In case of coherent
rotation all the spins rotate together and the whole particle can be considered
as a giant spin. Coherent rotation happens for SD particles with size smaller
than the exchange length lex

1.2 Hysteresis Loop of Single Domain Coherently Rotating Particles
The magnetization dynamics of the SD particles with R < lex will be coherent and the
particle can be treated as a giant single spin of value M MsV. When a magnetic field
(H) is applied along the easy axis of the magnetization (k) the energy of the particle is2-4

E(H) = KVsin20 i..VHcosO (1-5)

where 0 is the angle between the applied magnetic field and the direction of magnetization
as shown in the Fig. 1-3. The first term in Eq. 1-5 corresponds to the anisotropy energy
and the second term corresponds to the Zeeman energy. The energy, E(H), is shown
in Fig. 1-4 b) below as a function of 0 which shows two energy minima separated by
a barrier. The energy minima occur at 0 = 0 (corresponding to the magnetization
along the applied magnetic field or up direction) and 0 = 7 (corresponding to the
magnetization opposite to the applied magnetic field or down direction). The maximum

















H


Figure 1-3. A SD particle in an applied magnetic (H) field along the easy axis of
magnetization (k). 0 is the angle between the magnetization M and the easy
axis k.


of the energy occurs at 0 = 7/2 separating the two energy minima. Figurel-4 a) shows

the energy diagram at zero magnetic field as a function of 0. In this case the particle will

have magnetization parallel to the easy axis of magnetization since these correspond to

minimum energy states (up or down). Any other directions will cost some anisotropy

energy. The two states with minimum energy are separated by the anisotropy energy

barrier equal to KV. In an applied magnetic field along the easy axis the two energy

minima will be shifted due to the Zeeman energy (Fig. 1-4 b) ). Now the state along the

magnetic field will be most stable as the energy is lowered due to the Zeeman term. The

state with opposite direction of magnetization will be metastable. The magnetic field

dependent energy barrier for the spin up (E+(H)) and down (E_(H)) state is calculated

by Stoner and Wohlfarth to be,6

E(H)= KV 1- (1-6)

where E+(H) is the energy barrier seen by the up magnetized particles and E_(H) is the

energy barrier seen by the down magnetized particles and Ho = 2K/M, .. Derivation of the

Eq. 1-6 is given below.















E








Figure 1-4.


At t


Solu


0 7 0 7


Two state energy of a SD particle. Two energy minima correspond to the
direction of the easy axis of magnetization. a) At zero magnetic field the
particle will have magnetization along the easy axis of magnetization as those
correspond to minimum energy states (up and down). Up and down states are
separated by the energy barrier equal to KV. To reverse the magnetization
direction from up to down or vice versa the system has to overcome the energy
barrier.5 Brown proposed that this process requires a finite time given by Eq.
1-14.5 b) In an applied magnetic field along the easy axis, the two energy
minima will be shifted due to the Zeeman energy. Now the up state which is
along the applied magnetic field will be most stable and have the lowest
energy. The state with opposite direction (down state) of magnetization will
be metastable. The magnetic field dependent energy barrier for the spin down
state is calculated by Stoner-Wohlfarth (Eq. 1-6) 6


First order derivative of Eq. 1-5 with respect to 0 is

JE(H) 2KVsin0cos0 + i1 ..VHsinO
60

he maxima and minima


2KVsin0cos0 + i1 ..VHsinO 0


tions of the above equation are

sin0 0


cosO


(1-7)


(1-8)


(1-9)

(1-10)


i 2K
2K









Taking the second order derivative of Eq. 1-5, it can be shown that the Eq. 1-9 (Eq.

1-10) refers to minima (maximum) of the energy. Thus the energy minima are at 0 = 0

and 0 = 7 and maximum at when cosO=--. [. 1/2K (see Fig. 1-4). Energies correspond to

these extrema are

Emi, = -i [..VH (1-11)

Emin, = .VH (1-12)

Ema, = KV 1 + (f4[)] (1-13)
2K

where Emin, and Emin, corresponds to 0 = 0 (spin up) and 0 = 7 (spin down) respectively.

It is now easy to show that E+(H)= Emax Emin, and E_(H)= Emax Emin are given by

Eq. 1-6.

The energy barrier has to be overcome to reverse the magnetization direction from up

to down or vice versa. Brown proposed that this process requires a finite time5


r- oexp E ( ) (1-14)
( kBT )

where T is the temperature, 7-1 is the inverse attempt frequency of overcoming the energy

barrier and kB is the Boltzmann constant.

Figurel-5 shows the magnetization process when the magnetic field is swept from a

large positive value to a large negative value and again from a negative to positive value

to complete the magnetization loop. When magnetic field is large (scenario 1) all particles

will be magnetized along the magnetic field and a positive saturation magnetization is

achieved. As magnetic field is reduced to zero (scenario 2) the magnetization direction will

be trapped in the up direction as the temperature is not enough to overcome the energy

barrier. Now as the magnetic field is reversed the energy barrier, E_(H) will be reduced

according to Eq. 1-6 (scenario 3). But still the temperature is not enough to overcome

the energy barrier and the magnetization will still be trapped with a positive value. A

further increase in magnetic field in the opposite direction will keep lowering the energy









barrier until, at the coercive field, the energy barrier can be overcome by the thermal


energy and magnetization reversal will occur (scenario 4). When H


4 3


Figure 1-5.


He the energy


2 1


Hysteresis of a coherently rotating SD particle. Scenario 1) High positive
magnetic field is applied and saturation magnetization is observed. Scenario 2)
Magnetic field is reduced from positive value to zero. Magnetization is trapped
in the positive direction as the thermal energy is not enough to overcome the
energy barrier. Scenario 3) Magnetic field direction is reversed. Still the energy
barrier is large compared to the thermal energy and magnetization is trapped
in the positive direction. Scenario 4) Magnetic field equals to the coercive
field. Now the energy barrier can be overcome by thermal energy and
magnetization reversal occurs.


barrier E_ (H) is such that the relaxation time 7


Tm at the temperature T. Where Tm


is the experimental measurement time (around 100 sec for SQUID measurement). Thus

when H= H, magnetization reversal occurs. Combining Eq. 1-6 and 1-14, the coercivity

(He(T)) of the SD particle can be calculated as shown below.


7- = Tm-= oexp E_(H)BT
\kB-T


= E (H,)


kBTln T-
To









Now using the expression of E_(H) from Eq. 1-6 it is easy to show that6


H, Ho [ T 1) (1-15)


where TB = KV/kBln(Tm/l-o) is known as the blocking temperature. Below TB the

anisotropic energy barrier is larger than the thermal energy and magnetization is blocked

or trapped. Above TB the anisotropic energy barrier can be overcome easily by thermal

energy and the particles are called superparamagnetic as will be discussed later. It is clear

from the Eq. 1-15 that H, decreases with increasing temperature and above the blocking

temperature (T > TB) the particles lose their coercive field. Note that the origin of H,

in a SD particle is the finite time required to reverse the magnetization direction over the

anisotropy energy barrier.

The previous discussion is only true for an assembly of uniform size particles that

have easy axis of magnetization oriented along the same direction with magnetic field

applied along the easy axis. In real samples this is not the case since the easy axis of

magnetization is usually randomly oriented and the particle size is not uniform. A more

general case is shown in Fig. 1-6 below. Here a arbitrary angle between magnetic field

and the easy axis of magnetization (Q) is considered. The energy of the particle in this

case is7

(1 H

The magnetization of the particle at an applied magnetic field is given by McosO,in,

where O,i, is the angle corresponds to the minima of E(H). Note that here we have

not considered the effect of temperature on the magnetization. At finite temperature

other 0 values around the 0,i, will be occupied with a finite probability according to the

Boltzmann factor as shown in Fig. 1-7 below. Thus the average over all the occupied

direction with the occupation probability given by the Boltzmann factor will be the

thermal average of the magnetization for a fixed value of H and Q. The procedure should

be repeated for all values of H to get the M-H loop for a particular value of Q. Then the


































Figure 1-6. Single particle in an applied magnetic field (H). k is the direction of the easy
axis of magnetization. M and H are the magnetization and magnetic field
vectors respectively. Without loosing any generality M and H can be
considered in the same plane. The angle between M and H is 0. The angle
between H and k is Q. The energy barrier for this general configuration is
given by Eq. 1-16.7


M-H loops for all possible i should be calculated. Averaging over all these M-H loops will

give a magnetization loop at temperature T for a sample of uniform particle size and a

randomly-oriented easy axis of magnetization. All the above procedures should be done for

all possible particle sizes as the real samples usually have some particle size distribution.

The probability of a particular particle size can be modeled either as a lognormal or

gaussian distribution function. In this way the magnetization loop of a real sample with

nonuniform particle size and random orientation of the easy axis of magnetization can be

determined. If the all the above procedures are repeated for different temperatures then

the magnetization loop at different temperatures can be determined. Below we show a flow

diagram for the above process.79


























OT OT
Omin


Figure 1-7.


At finite temperature other 0 values around min, will be occupied with a finite
probability according to the Boltzmann factor and shown by the shaded
region. The thermal average of the magnetization will be the average of the all
magnetization directions over this shaded region.7 The probability of having
some magnetization direction will be determined by the Boltzmann factor.


1

Start with the energy of the single particle.

E(H) =2KV l(si 2 0) cose





2

Find the minima of E(H)

S..VCosOi, will be the magnetization at T=0 for the given value of H, and V


Figure 1-8. Flow diagram to show the process of calculating coercive field for real
nanoparticle samples with particle size distribution and random orientation of
the easy axis of magnetizations at finite temperature.


kBT
















Thermal average

(M(H, V))}T=


3

of the magnetization
SAE
f[2 Msvcosoexp 7BTde
AE
f02 exp ~WT d0


AE = Eo Eoin 01 and 02 are shown in Fig. 1-7

(M(H, p, V))T is the magnetization at temperature = T for the given value of H,

and V


Figure 1-8. continued


4

Step 1,2 and 3 should be repeated for different H. This will determine the M-H loop

for a given value of T, b and V


5

Step 1, 2, 3 and 4 should be repeated for all possible b and average of all those loops

will determine the M-H loop for a given value of T and V for an ensemble of particles

with random orientation of the easy axis of magnetization.










6

Step 1, 2, 3, 4 and 5 should be repeated for different particle size to determine the

M-H loop for a given value of T for a sample consisting of nonuniform particle size

and random orientation of the easy axis of magnetization. In real samples the particle

distribution function is usually lognormal or gaussian.9'10




7

Step 1, 2, 3, 4, 5 and 6 should be repeated for different T to determine the

temperature dependence of the M-H loop.7




8

Completion of step 7 will provide an opportunity to determine the temperature

dependence of the coercive field, remanent magnetization etc. Some of the temperature

dependent of coercive fields are listed in Tablel-1.


Figure 1-8. continued


Magnetization loops at different temperatures for a single l -.-r sample of Ni nanoparticles

of average diameter around 18 nm are shown in Fig. 1-9. The coercive field is determined

by the magnetic field where magnetization changes sign and passes through zero. It is

clear from the Fig. 1-9 that coercive fields decreases with increasing temperature as

discussed above. At temperatures high compared to the anisotropy energy KV, the

magnetization directions can rotate freely over the barrier and the particles become

superparamagnetic with H-=0. In this case the system can be treated similar to the case

of paramagnetism with each particle as a giant or super spin of value M1. V (thus called














4

2


E
-2


-4

-6E-5


-1500 -1000 -500 0 500 1000 1500

H (Oe)


Figure 1-9. Hysteresis loop of a Single 1~v. r Ni nanoparticles of 18 nm diameter embedded
in an Al203 matrix at temperature, T < TB. The loops show well defined
coercive field (where magnetization is zero) and decreases with increasing
temperatures.


superparamagnet). The magnetization for a collection of superparamagnetic particles is

given by the Langevin equation


MsVH kBT
M(H, T) V N .V[coth M] (117)
kBT 11.,VH

where N is the number of particles. Note that M is a function of H/T in the above Eq.

1-17. Thus if M is plotted as the function of H/T for different T, all the M-H loops will

fall on top of each other as shown in Fig. 1-10 for a single l v.-r sample of 12 nm Ni grains

in an A1203 host matrix.

1.3 Hysteresis Loop of Multi Domain Particles

In multidomain ferromagnetic system the origin of the hysteresis loop is dramatically

different than the SD case. Usually in soft ferromagnets (Rsd < le) the SD and MD

particles can be distinguished by the behavior of the coercive field as a function of

particle size. Figure 1-11 below is a schematic showing the behavior of coercive field as a


* 10 K
. 50 K ====
100 K *-*."u,,:
* 150K U/n i U
200 K = ,



,,,," -
/ U/

U
200' K E











4.5E-5
M275K
3 M300K

E2 M325K

0

-2


'3 At
-4.5E-5
-3 -2 -1 0 1 2 3
H/T(Oe/K)


Figure 1-10. Hysteresis loop of a SD coherently rotating particle at temperature (T > TB).
Sample shows zero coercive field as expected for superparamagnetic particles.
Note the H/T abscissa. Magnetization is plotted as a function of H/T for
three different temperatures as indicated in the legend. Loops at all different
temperatures fall on top of each other as predicted by the Langevin equation
for superparamagnetic particles.


function of particle size. For very small particles the coercive field is zero and particles are

superparamagnetic (SP) with magnetization determined by the Langevin function. As the

particle size is increased, the coercive field increases due to the fact that the energy barrier

increases. Particles with size larger than the critical single domain radius are multidomain

and the coercive field decreases with increasing particle size.2,3,9 This may be due to the

fact that as particle size increases the number of domains increases and thus it is easier to

have domain closure which decreases coercivity because there is less total magnetization.

The size dependence of the coercivity in MD region is experimentally found to be2


HcMD = a + b/d" (1-18)

where a, b are constants that depend on the real structure factor and materials, d is the

diameter of the particles and x has value around 1.2 There is no theoretical model that


























Particle diameter-


Figure 1-11. Coercive field plotted as a function of particle diameter. For very small
particles the coercive field is zero and the particles are known as
superparamagnetic particles (SP). As the particle size is increased the
coercive field increases due to the fact that the energy barrier increases.
Particles with size larger than the critical single domain radius are
multidomain and coercive field decreases with increasing particle size.


explains the behavior in Eq. 1-18. Thus the the peak in the coercive field when plotted

as the function of the particle size delineates the SD and MD behavior. In experiment we

have found the same behavior for both rmultil vr and single l -r samples of Ni particles

in A1203 matrix as will be discussed in detail in chapter 2.

Figurel-12 shows the possible domain wall configuration for different points in the

magnetization loop. Remember that compared to the SD case where the origin of the

hysteresis was the hopping over a energy barrier, in case of MD the origin of hysteresis is

irreversible domain wall motion. At very high magnetic field all the spins in the system

will be aligned along the magnetic field and positive saturation (. [.) will be achieved

(Fig. 1-12). As the magnetic field is reduced to zero a domain wall will be formed. Due

to the imperfections in the sample, the domain wall will be stuck in a position such

that the up domain is larger than the down domain and net magnetization or remanent








magnetization (Mr) will be seen at zero magnetic field. Reversing the magnetic field will


IH




II


Figure 1-12.


I I


Hysteresis loop of a MD system and possible domain wall configuration. At
very large positive magnetic field all the spins are aligned along the magnetic
field and saturation magnetization is achieved. When magnetic field is
reduced to zero, a domain wall forms. Due to the imperfections in the
sample, the domain wall will be stuck in a position such that the up domain
is larger than the down domain and remanent magnetization is measured. If
the direction of the magnetic field is reversed the domain wall will start to
move to the right and the down domain will grow. At a magnetic field equal
to the coercive field, the down and up domain will be equal in size and
magnetization will be zero. For a large negative magnetic field the domain
wall be moved to the right and all the spins will be in the direction of the
magnetic field and negative saturation will be reached.


move the domain wall to the right side and thus the down domain will start to grow and
magnetization will be reduced. When the negative magnetic field is equals to the coercive
field the up and down domain will have same size and magnetization will be zero. Further
increase in magnetic field in the negative direction will force the domain wall to move all
the way to right making all spins aligned along the magnetic field and negative saturation
will be reached.


* M


M S/









To derive the coercive field in MD domain case consider a simple case, as shown in

Fig. 1-13, where a single domain wall separates two domains. The right hand side is a spin

up domain and left hand side is a spin down domain. In an applied magnetic field, H,

















Figure 1-13. Single domain wall separating two magnetic domains. Right hand side is a
spin up domain and left hand side is a spin down domain. In an applied
magnetic field due to the Zeeman energy the domain wall will experience a
pressure and some work need to be done to move the wall against this
pressure. The origin of hysteresis in MD sample is the irreversible motion of
the domain wall.


along the spin up domain, the Zeeman energy of the up (down) domain will be -1 I[

(+M I) per unit volume. Thus the energy difference across the domain wall will be

-2HMs per unit volume. This energy difference can be considered as a pressure on the

wall and some work has to be done to move the domain wall against this pressure. The

work done to move the wall a distance dx is2,11


dW = 21. [Sdx (1-19)


where S the area of the domain wall. Thus the work done to move the wall by unit

distance is2,11

dW/dx = 21. IS (1-20)













where dw/dx can be thought of as the resistance of the domain wall motion. In real


samples due to the impurities, imperfections, strains etc, dW/dx passes through


maxima and minima. The wall motion over these maxima and minima is irreversible


in magnetic field and that is the origin of the hysteresis. The coercive field, the measure of


irreversibility, is usually given by2'11


1
H = (dW/dx)Tmax
211. 9


There are different theoretical models to calculate (dW/dx),ax for different imperfections


in the sample and the results for some of them are listed in Tablel-1.


Table 1-1. Table here lists some known models along with the variation of coercive field

according to the model.

Theory He System References

Stoner-Wohlfarth H = 2K [1 (kBg I /KV)1/2] SD, CR nanoparticle with uniaxial


Stoner-Wohlfarth


Micromagnetic


Inclusion Theory






Inclusion Theory


Inclusion Theory


Inclusion Theory


Inclusion Theory


H, = [1 ( Ig
H = (kB

H, 2K
Ms


S/KV)3/4]

r -2/3


2/3
Hc =
M,/d





H, -- 1/2(ln2L)


H, ( )3/2,


H 2 = [0.386 + log 2 ]


H, = 3-ylo/Md2


anisotropy along the applied magnetic field

SD, CR nanoparticle with uniaxial
anisotropy randomly oriented

MD, 2 phase material, hard magnet, a3 is
the micromagnetic parameter and depends
on the K, Ms, A

MD system, d < 6, free pole energy is
ignored, coercivity is assumed to be equal
to the maximum pining field, d is the
diameter of the inclusion, 6 is the domain
wall thickness, 7 is the domain wall energy
per unit area, a is the volume fraction of
the inclusion

MD system, d > 6, free pole energy is
ignored, L is the linear dimension of the
sample

MD system, d < 6, free pole energy is ig-
nored

MD system, d < 6, free pole energy is con-
sidered

MD system, closer domain, large inclusion,
commonly seen in the case of Neel's spike, I
is the equilibrium length of the spike


1.4 Magnetization vs. Temperature


Until now we have been discussing the behavior of magnetization as a function of


magnetic field at a fixed temperature. Now we will discuss how magnetization changes


with the temperature at a fixed magnetic field. At small applied magnetic field, spins


are trapped in metastable energy minima separated by energy barriers from the global


(1-21)


7


12


11,13






11,14


11,14


15


16









minima. As the temperature is increased the spins can hop over the energy barrier to

reach the global minima. Due to this trapping of spins in local minima, magnetization

values depends strongly on the cooling protocol. There are mainly two different cooling

protocols, field cooled (FC) and zero field cooled (ZFC). The behavior of magnetization

as a function of temperature for the two protocols is shown in Fig. 1-14 below for the

sample of Ni nanoparticles of 3 nm diameter at an applied field of 20 Oe. The temperature

where the difference between FC and ZFC disappears is generally called the irreversible

temperature (Ti,). For nanoparticles Ti, is same as the blocking temperature (TB).2



1.5x10 -0 Im
FC H = 20 Oe

1.0x10 -
i/4

5.0x10^ -


ZFC
0.0

0 50 100 150 200 250 300
T(K)


Figure 1-14. Magnetization vs. temperature at an applied magnetic field of 20 Oe for the 3
nm diameter Ni nanoparticles. The red color is the field cooled (FC)
magnetization and the black one is the zero field cooled (ZFC) magnetization.


1.4.1 Zero Field Cooled (ZFC) Magnetization

Zero field cooled magnetization is measured by cooling the sample from high

temperature (temperature above the irreversible temperature (Tir)) without any applied

magnetic field. At low temperature a small magnetic field is applied and magnetization

is measured as a function of temperature during the warm up while keeping the magnetic

field on. Here we will discuss the shape of the ZFC magnetization in a qualitative manner.

In general the magnetic system can be treated as a two-state problem as shown previously









in Fig. 1-4 where spin up and down correspond to the energy minima separated by some

energy barrier. The origin of the energy barrier in the SD case is the anisotropy whereas

for the case of MD the origin is domain wall pinning at defects. At high temperature the

energy barrier is easily overcome due to the thermal energy and the spin up and down

states will be equally populated. Thus at high temperature above Ti,,, magnetization will

be zero. Now if the sample is cooled to a low temperature without any applied magnetic

field, then zero magnetization state will be blocked as the energy barrier is now large

compared to the thermal energy.2,3,6 If a small magnetic field is applied the change in

magnetization will occur only for the small energy barriers that can be overcome at that

temperature and a small magnetization will be achieved. As temperature is increased, the

probability of overcoming the larger barriers increases and magnetization increases. At

temperature Tir the probabilities to overcome the barrier for spin up and down become

nearly equal and the spin up and down mixing starts to happen and thus magnetization

decreases with further increase in temperature.

1.4.2 Field Cooled (FC) Magnetization

Field cooled magnetization is measured by cooling the sample from high temperature

to the low temperature in an applied magnetic field and magnetization is measured during

the warm up process.2 In this case at high temperature due to the applied magnetic field,

the spin up states are more populated than the spin down states. Cooling the sample at a

low temperature while keeping the field on will thus lock the system in magnetized state.

An increase in temperature will increase the probability of spin up and down mixing and

thus magnetization will gradually decrease.









CHAPTER 2
DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL SINGLE
DOMAIN GRAIN SIZE OF NI IN LAYERED NI/AL20O COMPOSITES

2.1 Abstract

Pulsed laser deposition has been used to fabricate Ni/Al2Os mruiltili, r composites in

which Ni nanoparticles with diameters in the range of 3-60 nm are embedded as 1l. -_i in

an insulating A1203 host. At fixed temperatures, the coercive fields plotted as a function

of particle size show well-defined peaks, which define a critical size that delineates a

crossover from coherently rotating single domain to multiple domain behavior. We observe

a shift in peak position to higher grain size as temperature increases and describe this

shift with theory that takes into account the decreasing influence of dipolar magnetic

interactions from thermally induced random orientations of neighboring grains.

2.2 Introduction

The magnetic properties of nanoparticles have been the focus of many recent

experimental and theoretical studies. Technological improvements have now made it

possible to reproducibly fabricate nanomagnetic particles with precise particle size

and interparticle distances. 1722 These controlled systems have enabled study of the

fundamental properties of single as well as interacting particles. Most applications require

that the particles be single domain with a uniform magnetization that remains stable with

a sufficiently large anisotropy energy to overcome thermal fluctuations,23 which establishes

a temperature-dependent lower bound to the particle size. These considerations must take

into account the effect of interactions on magnetic properties as is evident for high-density

recording media24 where particles are very close to each other. Considerable insight has

already been gained from experimental studies of the effect of dipolar interaction on

superparamagnetic relaxation time25-34 and blocking temperature.29 Less understood

however is the effect of dipolar interactions on the establishment of an upper bound to

particle size, which defines the crossover from single domain (SD) to multi domain (M!l))

behavior. In the following we show using coercivity measurements on Ni/A12Os composites









that with increasing temperature this upper bound to particle size increases and then

saturates due to attenuated dipolar interactions from thermally induced coherent motions

of the magnetization of the neighboring randomly oriented particles.

2.3 Experimental Details

The composite system studied in this paper comprises elongated and pi li-, iv-- H11i.'

Ni particles with diameters in the range of 3-60 nm embedded as lv-ir in an insulating

A1203 host. The muiltiliv- r samples were fabricated on Si(100) or sapphire (c-axis)

substrates using pulsed laser deposition from alumina and nickel targets. High purity

targets of Ni (99.9' '. ) and A12O3 (99.9'" .-) were alternately ablated for deposition. Before

deposition, the substrates were ultrasonically degreased and cleaned in acetone and

methanol each for 10 min and then etched in a I!' I'_ hydrofluoric acid (HF) solution to

remove the surface silicon dioxide l-v,_, thus forming hydrogen- terminated surfaces.35

The base pressure for all the depositions was of the order of 10-' Torr. After substrate

l., lii.- the pressure increased to the 10-Torr range. The substrate temperature was kept

at about 550" C during growth of the A1203 and Ni 1 .-ir-. The repetition rate of the laser

beam was 10 Hz and energy density used was ~ 2 Jcm-2 over a spot size 4 mm x 1.5 mm.

A 40 nm-thick buffer li;v-r of A12O3 was deposited initially on the Si or sapphire substrate

before the sequential growth of Ni and A1203. This procedure results in a very smooth

starting surface for growth of Ni as verified by high resolution scanning transmission

electron microscopy studies (Fig. 2-1). Multill ,-- samples were prepared having 5 1 ,-ris

of Ni nanoparticles spaced from each other by 3 nm-thick Al20s3 1I- ,r. A 3 nm-thick cap

l -,-r of A1203 was deposited to protect the topmost l1-*ir of Ni nanoparticles.

Shown in Fig. 2-1 is a cross-sectional TEM image from a multi-1 i,-, 1 (5 l-.. -rs)

Ni-A1203 sample grown on c-plane sapphire. The Ni particles have a size of 23 5

nm in width and 9 nm in height. The separation between neighboring particles is on

the order of 3 nm (measured as a projected distance in cross-sectional view), which

is comparable to the thickness of the A1203 spacer l iVt-. For the purposes of this





























Figure 2-1. Cross sectional dark field STEM image of a 5-1i- -r Ni-Al203 sample grown on
c-axis sapphire


experiment the grain size d, as measured by the amount of Ni deposited referenced to a

calibrated standard, represents the average size of the disk-shaped grains shown in the

figure. This calibration was obtained from cross-sectional TEM micrographs of single l1-.-r

- i1 !1. '' by comparing the average grain size with d. The TEM observation also shows

that the Al203 spacer l.r-i~ are partially crystallized. Due to the large surface energy

difference between Ni and A1203 Ni forms well-defined, separated islands within the

Al203 matrix.36 Previous studies on similarly-prepared samples using atomic number (Z)

contrast imaging in TEM together with electron energy loss spectroscopy (EELS) have

confirmed the absence of NiO at the Ni/A1203 interfaces.36 The Ni/A1203 interfaces were

chemically abrupt without an intermixing between Ni, Al and oxygen. In addition we did

not observe exchange-bias induced .*,-mmetric magnetization loops, thus lending support

to the conclusions of previous studies36 that antiferromagnetic NiO is absent in our 1 ,, i, t

Ni/A1203 system.

Previous TEM studies on single li--r samples have shown the particles to be

p&I, *1 ,-1 ilii,. For example, a three nm particle comprising three crystalline grains










has been observed.3 P li. i lii, i11 1!: particles will therefore have crystalline grains

oriented in different directions, thus tending to average any net crystalline anisotropy

to zero. Accordingly, temperature-independent shape anisotropy is dominant and

temperature-dependent (
important to note that the exchange length l,, = 14.6 nm for Ni,37 which is the length

scale below which atomic exchange interactions dominate over magnetostatic fields,

determines the critical radii (Rcoh) for coherent rotation: Rcoh ~ 51,, for spherical particles

and Rcoh ~ 3.51,, for nanowires.3 The particle sizes (1.5-30 nm in radius) that we have

investigated are thus smaller than the critical radius below which coherent rotation of Ni

prevails.


400


2001[


Figure 2-2.


0 20 40 60
Particle Size (nm)

Coercivity for 5-1,ir Ni/Al20 muiltil ,--r samples (5 repeated units) plotted
as a function of particle size (diameter) at the temperatures indicated in the
legend. The peak positions at d = d, for each isotherm, indicated by vertical
arrows, delineate the crossover from single domain (SD) to multiple domain
(Ml1)) behavior (d > de). Inset shows the behavior of H, as a function of 1/d
for the particles with d > de at 10 K. The linear dependence up to 24 nm
diameter particles with saturation at a constant value for large par i- !. is
consistent with the behavior expected for multidomain particles. Thus
particles on the right-hand side of the peak are multidomain.


In Fig. 2-2 we show plots of H, as a function of particle size d at each of the

temperatures indicated in the legend. Coercive fields were extracted from magnetization


SD < ---> MD 45 -- H js. Id
10 K


24..r.
002 0.04 m0.6 0.00
lid(nm
-----------l




S\100 K
Sr- -1o K
'/ --4- 15 K









loops measured by a Quantum Design superconducting quantum interference device

(SQUID) after subtracting out the diamagnetic contribution from the substrate. Magnetic

field was applied along the plane of the films. To obtain the magnetization loops, the

magnetic field was varied over the full range (5 T) while keeping temperature fixed.

The high magnetic field data show linear magnetization with magnetic field, which is

due to the diamagnetic contribution from the substrate (as signal from ferromagnetic Ni

particles saturates at high magnetic fields) and can thus be subtracted from the data.

The decrease of H, with increasing temperature for fixed d is clearly apparent and can

be understood as the effect of thermal fluctuations.2 For the low-temperature isotherms,

there are pronounced peaks which define a temperature-dependent critical particle size

dc delineating SD (d < dc ) behavior of coherently rotating particles from MD (d > d,)

behavior. 2,8,3945 The reason why there is a peak in H,(d) is explained in the introduction

chapter, page 29.

In the inset of Fig. 2-2 we have plotted H, versus 1/d for the particles of size d > d,

at 10 K. It is clear that H, behaves linearly with 1/d up to particle size of 24 nm and

then saturates. This behavior is consistent with the dependence expected for multidomain

particles. 3 Thus particles of size d > d, are multidomain and the peak defines the

crossover from SD to MD behavior. The formation of domain structure is driven by

the reduction of long range magnetostatic energy, which at equilibrium is balanced by

shorter range exchange and anisotropy energy costs associated with the spin orientations

within a domain wall. The purpose of this chapter is to show that this well-defined SD

region of coherently rotating particles extends over a larger range of grain sizes at higher

temperatures because of the diminishing influence of dipolar interactions from neighboring

grains.

2.4 Data and Discussion

The influence of dipolar interactions on the SD/\!I) crossover can be understood in a

qualitative way by considering the three randomly oriented particles shown schematically










16 Simulation
Experiment
14 *

S12

S10 I I

8-

6 I
0 50 100 150 200 250
T(K)

Figure 2-3. Peak position, d,, plotted as a function of temperature (red circles). The black
squares are the results derived from equation 2-5. The blue star represents the
observed value of dc for a series of single lw-r samples at 10 K. The inset, a
schematic of three neighboring particles oriented in different directions,
illustrates how the dipolar fields from particle 2 and 3 facilitate the formation
of domains in particle 1, as the dipolar magnetic fields are in different
directions.


in the inset of Fig. 2-3. Particle 1 experiences dipolar fields from particles 2 and 3, which

are not collinear for most orientations of a randomly oriented particle system. Because

dipolar fields decrease rapidly with interparticle separation, the dipolar field due to

particle 3 (2) will be stronger than particle 2 (3) on the left (right) side of the particle 1.

The separate and unequal influence of the neighboring particles thus favors the formation

of domains in particle 1.

.To make these notions more quantitative, we modify the treatment of Dormann

et a126 for interacting paramagnets to include the temperature region below the blocking

temperature TB and find the temperature-dependent dipolar magnetic field Hd arising

from temperature induced fluctuations in the magnetization of nearest neighbor nanometer

size particles to be,
Hd .a eC(1 e-) 1
47r v/-j(erfi(3) erfi(v -- T ))









where 'erfi' is the imaginary error function, i. ., is the saturation magnetization, /

KV/kBT and a = V(3cos2 l)/s3 is a dimensionless parameter with and s

corresponding respectively to an angle parameter and the separation between two .idi i:ent

particles each with volume V The parameter Q is alv--,v- greater than one for T < TB

where there is still coercivity; i.e., the magnetization is fluctuating but not going over

barriers. Then Eq. 2-1 has the limiting value at T -> 0 as given below.



Hd T T 0 (2-2)
47

The derivation of Eq. 2-1 includes averaging over the accessible directions of

magnetization weighted by a Boltzmann factor. Higher temperatures give smaller

magnetizations since the particles fluctuate over larger angles. Specifically, spin up

and down particles will be in energy minima separated by an anisotropy energy barrier.

At absolute zero temperature only the direction corresponding to the minima of the

energy will be occupied. At finite temperatures, according to the Boltzmann law, other

energy states will be occupied around this minimum and will have different directions

of magnetizations. Thus to obtain the actual magnetization, an average over all these

accessible directions is calculated, constrained by the fact that the probability of those

states to be occupied is given by the Boltzmann factor

OT exp E(0)
OTep[ ( ]'( '
fK exp[- f]dO

where at zero magnetic field E(O) =KVsin20 Thus Omin = 0 and OT is temperature

dependent, obeying the relation, sin20T = kBT/KV The parameter OT (see Fig. 1-7 on

page 21 of chapter) will be higher at higher temperatures and thus the thermal average of

the magnetization will diminish at higher temperatures. Using Eq. 2-3 one can determine

the temperature dependence of the dipolar magnetic field Hd as shown in Eq. 2-1 for

particles treated as simple dipoles.









.In the absence of interactions (Hd = 0) the condition for the SD to MD transition

is given for spherical particles with radius d/2 by, Ad = Bd2 where Ad' is the total

magnetostatic energy and Ed, = Bdc is the domain wall energy.46 We have absorbed

the factor of two, which relates diameter to radius, into the constants A and B In the

presence of the dipolar magnetic field Hd the formation of domain walls will be assisted

by a Zeeman term which is proportional to the volume of the affected particle. The

condition determining the SD to MD transition now becomes,


Ad = Bd 7ri.. .,1'/6 (2-4)


When the dipolar interaction is a small perturbation, i.e., 3 [f,/A < 1, Eq. 2-1 and

2-4 can be combined to give the relation,

-(1 e-()
dc = dco ddw 3( (2-5)
V/7P(erfi(3) erfi(vP -1 ))

where dco = B/A is the temperature-independent critical diameter in the absence of

interactions (high-temperature limit) and dd, = poBM2Tr/(72A2) for a = r/3. The second

term on the right-hand side of Eq. 2-5 thus becomes a temperature-dependent correction

to dc due to interactions from neighboring particles and decreases with increasing T .

Since the magnetic field due to the dipole-dipole interactions are weaker at higher

temperatures Eq. 2-1, the nanoparticles remain in the SD state to a larger size, which

by Eq. 2-5 results in a shift of dc towards higher values at higher temperatures. This is

indeed evident in Fig. 2-3, which shows the temperature dependence of dc as determined

from the data in Fig. 2-2. The black squares are the simulated data according to Eq. 2-5

using the two fitting parameters: dco and dd, Qualitatively, the data agree quite well

with the prediction of the theoretical model without taking into account the topology

and size distribution of the particles. We have found dco = 84 nm from our simulation

(Fig. 2-3, black squares) to be close to the value for a particle with shape anisotropy

constant Kshape = 3.1 x 104 J3 (do = 72AeK/,,,JV where Ae, is exchange stiffness,









K is anisotropy constant).3 Values of A (oc ,,,, 1' ) and B (oc A,,K ) have been found

to be 1.44 x 104Jm3 and 1.21 x 10lJm2 respectively. This value of A is very close to

the theoretical predicted value3 and the value of B is again consistent with the value

of the shape anisotropy. The value of the shape anisotropy can also be predicted from

the zero- temperature extrapolation H,, ~ K/M1. for randomly oriented particles.3 For

Shape = 3.1 x 104Jm3, H,, ~ 620 Oe. This is in good agreement with the 500 Oe coercive

field observed at 10 K for the 6 nm sample.

For a separate series of single liv--r samples the coercivities at 10 K peak at de

S14 nm as shown in Fig. 2-3 by the blue star. In the single l-v-r samples the peak

position occurs at higher particle size (14 nm) than muilti-liv r samples (8 nm). This

difference reinforces our interpretation and can be understood by realizing that the

dipolar interactions of the single 1 i-r samples are significantly reduced compared to the

rniitil i.r samples because of the smaller number of nearest neighbors.

2.5 Conclusion

In summary, we have fabricated magnetic nanoparticles in an insulating thin film

matrix with tunable properties achieved by varying particle size and temperature.

The peaks in the coercivity isotherms delineate a critical grain size de which identifies

the crossover from SD to MD behavior. The presence of dipolar interactions and

their diminishing influence with increasing temperature is responsible for the observed

dependence of de on temperature and is in good qualitative agreement with our modification

of present theory26 of interacting particles. The well-established influence of dipolar

interactions on superparamagnetic relaxation time25-34 together with the connection

between relaxation time r and coercivity H, si-l-. -1- that there is a concomitant influence

of dipolar interactions on the coercivity observed near the superparamagnetic limit where

He = 0. The work reported here extends this connection to the upper limits on the size

of SD particles by showing that dipolar interactions can facilitate the formation of multi

domain particles especially at low temperatures.









2.6 Methods


2.6.1 Mathematical Analysis

The Eq. 2-5 is self consistent (as the term / contains dc) and can not be solved


analytically. The equation, d dco + dd w (ei(1- ), ))
w3ir(erfi~jQ I I 3P1)


0, is solved by numerical


approach and simultaneously the solution is fitted to the experimental data according

to a nonlinear list square method. Mathematica, a commercial software, is used for this

purpose.

2.6.2 Basic Physical Understanding

A simplified physical understanding of the problem is shown in Fig. 2-4.


1 1 Hd
\i\~s~ wy


Hd'


Figure 2-4.


The net effect of dipolar magnetic field (Hd) is shown on the particle 1. As
particles are randomly oriented, Hd from particle 3 will be in different
direction than that from particle 2. As dipolar interaction decreases rapidly
with distance, particle 1 will experience local dipolar magnetic fields in
different directions from different neighboring particles and thus making it
easy to form domains.









CHAPTER 3
EFFECT OF DIPOLAR INTERACTION ON THE COERCIVE FIELD OF MAGNETIC
NANOPARTICLES: EVIDENCE FOR COLLECTIVE DYNAMICS

3.1 Abstract

The effect of dipolar interaction on the coercive field is discussed for the single

domain and coherently rotating Ni nanoparticles embedded in A1203 matrix. Results

for two sets of 5 li-v-r samples with different interlayer spacing and a set of single l1-ir

samples of Ni nanoparticles are compared. The dipolar interactions are strongest in the

samples with shorter interlayer distances and weakest for the single 1iv-r samples. In

this present study, the dipolar interaction is found to increase the coercive field. On the

other hand the critical single domain radius decreases due to the dipolar interactions.

These two behaviors together indicate that collective dynamics pl i', an important role in

understanding the origin of the coercive field.

3.2 Introduction

The origin of coercive field (H,) for coherently rotating ferromagnetic nanoparticles is

remarkably different than that of the bulk,47 where irreversible domain wall motion is the

dominant mechanism.4 In the case of nanoparticles, when the size of the particle is smaller

than a critical size (dc), the most favorable energy state is to have single magnetic domain

and particles are called single domain (SD) particles. When H, is plotted as a function of

particle diameter (d), there is a well defined peak at de. Particles with d < d, (d > d,) are

SD (multidomain (I\l))). 2,8,41,42,44,48 Kittel3,46 has shown that for a spherical particle, d, is

given by the relation (see Eq. 1-3 on page 13 of chapter)


dc 72-- (3-1)
S' ,, /.-'

where A is the exchange stiffness, K is the anisotropy constant, /o is the free space

permeability and [.. is the saturation magnetization. In SD particles there is no domain

wall. The origin of H, in this case is the finite time required to reverse the magnetization

direction over the magnetic field dependent anisotropy energy.47 The time required to









reverse the direction of the magnetization of a coherently rotating SD particle is given by

the relation2'5'49'50

T = Toexp (3-2)
kBT_
Here, To is the inverse of the attempt frequency to overcome the energy barrier, V is the

volume of the particle, kB is the Boltzmann constant and T is the temperature. Stoner

and Wohlfarth have calculated He for SD particles in the simple case when particles are

coherently rotating and the applied magnetic field is along the easy axis of magnetization

of the particles. The coercive field for a Stoner-Wohlfarth particle is given by

2K I (ntr1n (2f
Hesw =- 1 n ( (3-3)


where Tm is the time of measurement. From the simple Stoner-Wohlfarth model it is clear

that He for the nanoparticle can depend on many different factors. He increases with

decreasing T,, increasing T and increasing K.

In the presence of dipolar interactions the above equation will be modified. The

widely accepted modification is achieved by treating the dipolar interactions to result in

an effective anisotropy energy.51-61. Thus if due to the dipolar interactions K increases

(decreases) then '-, according to Eq. 3-2, will also increase (decrease) and as a net result

He will increase (decrease). A more familiar famous form of Eq. 3-3 is Hsw(T) =

2K/M..(1 (T/TB)1/2), where TB = KV/25kB is the blocking temperature. The factor 25

comes from the fact that Tm, 100 s is a typical measurement time and ro-l ~ 109 sec-1 is

a typical attempt rate.

The effect of dipolar interaction on the coercive field (He) has been investigated

extensively. The first theoretical treatment by Neel62 showed that He decreases with the

increase in the packing fraction (c) or the dipolar interaction as shown below in Eq. 3-4,

where the interaction effect has been introduced as an "Interaction Field" and shown to









lower the anisotropy energy.


H, H,. (t e) (3-4)

The "Interaction Field" is a function of the packing fraction (e). Later Wohlfarth51

showed that the effect of the interaction on the He can be increasing or decreasing

depending on the particle orientation as the dipolar interaction is direction dependent.

But all of those results have been constructed considering the fact that the anisotropy

constant, K, either increases or decreases due to the interactions. Previous theoretical and

experimental works have been reported either showing an increase or decrease in H, and

explained in terms of a corresponding increase or decrease in the anisotropy energy.5161,63

In this present experiment we find that an increase in the dipolar interaction increases

H, but decreases de. Equation 3-1 sil--. -1 that the decrease in dc may be due to a

decrease in the K. But a decrease in the K will also decrease r (Eq. 3-2) and thus will

decrease H, (Eq. 3-3) which is contradictory to the present experimental result. Thus the

change of K due to the dipolar interactions must not be applicable in the present case. As

any change in K will give rise to change in H, and d, both in the same direction (both

increase or decrease at the same time). Below, we show qualitatively that the increase

in the He can be realized in terms of the collective dynamics of the magnetization of the

particles and decrease in d, can be understood as discussed in reference48

3.3 Results and Discussions

Samples were grown using pulsed laser deposition technique.48 Base pressure of the

growth chamber was on the order of 10-' Torr and the growth temperature was around

550" C. Multilayer structure of Al203 and Ni nanoparticle were grown without breaking

the vacuum of the chamber. First a thick (40 nm) buffer li-v-r of Al203 is grown on top

of the substrate. The purpose of this buffer li-v-r is to prevent any diffusion of the Ni into

the substrate. Then Ni nanoparticles and A1203 are sequentially deposited on this buffer

l-?v-r (see Fig. 1). The top -1v.-r of A1203 acts as a capping l-,--r which prevents oxidation

of the nanoparticles.36 Three different sets of samples are grown. Set 1 and set 2 samples








consist of 5 1l- ri of Ni nanoparticles separated by A12O3 1-V-, r. For set 1 (set 2) the
A1203 separation is 3 nm (40 nm). Set 3 samples are single l-1v-r of Ni nanoparticles in
A1203 matrix. Dipolar interactions are strongest in set 1, moderate in set 2 and weakest
in set 3. The dipolar interactions are stronger in Set 1 compared to set 2 as the interlayer
separation of the Ni particles is smaller in set 1 compared to set 2. Set 3 consists of only
a single 1 v,- of Ni particles and thus the dipolar interactions are weakest. All sets of
samples consist of different samples with varying particle size from 3 nm to 60 nm.


C) 1 41 1 1 4 4 4






Estate


Multi Layer


Figure 3-1.


Fig la) shows the TEM image of a single 1i,--r sample with average particle
diameter of ~ 24 nm. Particles are well defined with inter particle distance of
around 4 nm. Ib) shows a schematic of the single lv,- r sample. A 40 nm thick
buffer 1-v,-.r of A1203 is first grown on top of substrate. Then the Ni
nanoparticles are grown on to of the buffer 1-V.-r. Finally a 3 nm thick capping
l-Iv,-r of A12O3 is grown to protect it from oxidation. ic) shows the schematic
of 5 li-v.-i~ of Ni nanoparticle sample.


Figure 3-la) shows the TEM image of the single lv,- r Ni particles with average
particle diameter of ~ 24 nm (set 3). The simplified schematic of the single and rmiltil iv-r
samples are shown in 3-1 b) and c).
Typical magnetization loops at three different temperatures are shown in Fig.
3-2a) for the sample with 3 nm A1203 spacer l-v1 -r (set 1) and 6 nm in diameter. The
coercive field H,(T) is determined from the loop as shown by the arrow. This procedure
to determine H, is repeated for all samples belonging to all three sets. At temperatures


a)


b) 4 44


statee


Single Layer











15- M10K
10- M50K
M100K,- /

0
5-


-5-
-10 H,
-15
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
H (KOe)


b)
10-

5-

0

S-5-
S-10-


a)


E


L


Figure 3-2. a) Magnetization loop of a sample from set 1 of average particle diameter of ~
6nm. Coercive field (H,) is determined from the loop as shown by the arrow.
H, decreases with increasing temperature and goes to zero above the blocking
temperature. b) Magnetization loops above blocking temperatures.
Magnetization is plotted as the function of H/T to show the
superparamagnetic behavior as expected for the SD particles above the
blocking temperature.


above the blocking temperatures (TB) SD samples behave as superparamagnetic particles.

Figure 3-2b) shows the superparamagnetic behavior of the set 1, 6 nm diameter sample.

Note the magnetization data fall on top of each other when plotted as a function of H/T.

This behavior is a direct consequence of the superparamagnetic behavior as expected from

the coherently rotating SD particles.

Figure 3-3 shows H, plotted as a function of d for the set 1, set 2 and set 3 samples.

The data that correspond to the different sample sets are indicated in the legends. The

peak in the He separates SD and MD particles. 2,8,41,42,44,48 It is clear from the data that

dc decreases with increasing dipolar interactions (de1 < d,2 < d). H, on the other hand

increases with the increasing dipolar interactions (vertical dotted arrow) in the SD region.

These two results can not be explained in terms of the commonly reported change in K

due to the dipolar interactions.5161 The decrease in d, due to the dipolar interactions has

been discussed elsewhere.48 In this present study, the collective dynamics of the particles


-6 -4 -2 0 2 4 6
H/T(Oe/K)


M250K
o M300K
M325K






' J ,










d
at \


10 K


3 nm separation (Set 1)
40 nm separation (Set 2)
Single layer (Set 3)


---


30 40


Figure 3-3.


Coercive field (He) as a function of particle diameter (d). The peak separates
the single domain (SD) and multidomain (\ l)) particles. Particles with
diameter higher (smaller) than the peak diameter (de) are MD (SD). Data for
the 3 different sample sets are shown and indicated in the legends. The critical
diameters d1,, dc2, d,3 are shown from the samples of set 1, set 2 and set 3
respectively. In the single domain region (below d,) the coercivity increases
with increasing dipolar interactions as shown by the vertical dotted arrow.


magnetization due to the dipolar interactions is found to be responsible for the increase in

He. These observations are shown in Fig. 3-3 and summarized in Fig. 3-4.

We first discuss the effect of dipolar interactions on He as presented in previous

investigations. 5161 The treatment begin by including the change in anisotropy energy Edrp,

due to dipolar interaction into the expression for r, as given by63


SKV + Ed ]
T7 Toexp L j-
kL IT


(3-5)


600-




400-


0
o


200-


0 10 20
d (nm)


d 0









Equation3-5 can be rewritten as shown in Eq. 3-6. Thus the effect of the dipolar

interactions is treated as either an increase (+ Edip) or decrease (- Edip) of anisotropy

energy.


7 0oexp K ) V (3 6)

The effect of dipolar interactions on the He can be explained according to Eq. 3-6. In

our case a + Edrp increases 7- and give rise to an increase in He with increasing dipolar

interactions (Eq. 3-2 and 3-3). If this is to be true in our case then according to Eq. 3-1,

de should also increase with increasing dipolar interactions. According to the previous

approach both He and de should change in the same way, both increase or both decrease.

In the present experiment we find however that He increases and de decreases due to

dipolar interactions (see Fig. 3-4) and strongly -ii--.- -1- an alternative approach to the

problem. The effect of dipolar interactions on d, is discussed in reference48, where it

has been shown that the local dipolar magnetic field from the nearby randomly oriented

particles try to align the magnetization direction of the particle in different directions and

thus favoring domain formation. The effect of dipolar interactions on He will be discussed

below in terms of collective dynamics.

It is well known that the magnetization dynamics can be collective in nature due to

the interactions between the particles and the relaxation time (7*) in this case is given

by 64,65


T T > T, (3-7)

where T is the relaxation time of the single non interacting particle (Eq. 3-2), T

1 ,, f2/47kBr3 is the critical temperature and depends on the interparticle distance

and particle magnetization and z is a critical exponent. The above equation clearly

-i -I- -i that the relaxation time will be larger in the presence of dipolar interactions

and thus according to Eq. 3-3 He will be larger, and thus agreeing with our experimental












14 ---- Hc
-4
Set 2
12 0
E *
2
o 10- t
10 Set 2

Set 1
8-
Set 3 0

Dipolar interaction strength


Figure 3-4. Coercive field (He) and critical diameter (de) as the function of the increasing
dipolar interaction. H, (de) increases (decreases) with increasing dipolar
interaction. The opposite behavior of He and d, -, r--.- -I that the collective
dynamics and the critical slowdown is responsible for the increase in H, due to
the dipolar interactions. The decrease in d, is discussed elsewhere.48

observations (Fig. 3-4). Note that in this case the anisotropy energy is unaffected by

dipolar interactions and the increase in relaxation time is due to the fact that the reversal

of magnetization is collective in nature.64,65

3.4 Conclusions

A study of dipolar interactions is presented for the single and muiiltil i-- structure of

Ni nanoparticles. The coercive field has been found to increase with increasing dipolar

interactions and can be understood qualitatively in terms of collective dynamics.

Three sets of samples are investigated. Each set consists of samples having particle

size varying from 3 nm to 60 nm in diameter. Dipolar interactions are stronger in set 1

and decreases for set 2 and set 3. Behavior of coercive field and critical single domain

radius are observed. Coercive field increases and critical single domain radius decreases


Set 1


Set 3


-*---dc









with increasing dipolar interactions. These two behaviors together si-'-- I a collective

dynamics of the magnetization reversal process in the SD region in the presence of dipolar

interactions. To our knowledge, this is the first time that the effect of collective dynamics

on a coercive field of the nanoparticle system has been observed.









CHAPTER 4
FINITE SIZE EFFECTS WITH VARIABLE RANGE EXCHANGE COUPLING IN
THIN-FILM PD/FE/PD TRILAYERS

4.1 Abstract

The magnetic properties of thin-film Pd/Fe/Pd trilayers in which an embedded

~1.5A-thick ultra thin 1. -r of Fe induces ferromagnetism in the surrounding Pd have

been investigated. The thickness of the ferromagnetic trilayer is controlled by varying

the thickness of the top Pd 1v,-r over a range from 8 A to 56 A. As the thickness of the

top Pd 1- -r decreases, or equivalently as the embedded Fe 1,- -r moves closer to the

top surface, the saturated magnetization normalized to area and the Curie temperature

decrease whereas the coercivity increases. These thickness-dependent observations for

proximity-polarized thin-film Pd are qualitatively consistent with finite size effects that

are well known for regular thin-film ferromagnets. The functional forms for the thickness

dependence, which are strongly modified by the nonuniform exchange interaction in

the polarized Pd, provide important new insights to understanding nanomagnetism in

two-dimensions.

4.2 Introduction

The presence of 3d magnetic transition metal ions in palladium (Pd) gives rise to

giant moments thus significantly enhancing the net magnetization 6670. Pd is known to

be in the verge of ferromagnetism because of its strong exchange enhancement with a

Stoner enhancement factor of ~ 1071. The magnetic impurities induce small moments on

nearby Pd host atoms thereby creating a cloud of polarization with an associated giant

moment71'72. Neutron scattering experiments show that the cloud of induced moments can

include ~200 host atoms with a spatial extent in the range 10 to 50 A72,73. Thus a thin

1I,-- of Fe encapsulated within Pd will be sandwiched between two ..11i i'.ent thin l,'- is of

ferromagnetic Pd with nonuniform magnetization and a total thickness in the range 20 to

100 A.









We have investigated thin-film Pd/Fe/Pd trial -'r;i in which the thickness dFe of the

Fe is held constant near 1.5A and the thickness of the polarized ferromagnetic Pd is

varied by changing the top Pd 1-v-r thickness x. The magnetic properties are studied as

a function of x. Our experiments are motivated by the recognition that ferromagnetism

in restricted dimensions has attracted significant research interest74 78. For example,

the coercive field He increases as the thickness of the ferromagnetic film is decreased

toward a thickness comparable to the width of a typical domain wall79'80. Moreover, the

Curie temperature T, decreases as the thickness of the ferromagnetic film is decreased

toward a thickness comparable to the spin-spin correlation length8183. We will show

below that similar phenomenology applies to ferromagnetically polarized Pd films, albeit

with different functional dependence arising from the fact that exchange coupling, which

decays with distance from the ferromagnetic impurity84, is not uniform throughout the

film.

4.3 Experimental Details

The samples were grown on glass substrate by RF magnetron sputtering. The base

pressure of the growth chamber was of the order of 10-9 Torr. First a thick li-v-r of Pd of

thickness 200 A is grown on top of the substrate. The root mean square surface roughness

of this Pd l-1 -r was measured by atomic force microscopy to be ~ 6 A. Then a very thin

(1.5 A as recorded by a quartz i -I I1 monitor) l-Iv-r of Fe is deposited on top of the

first Pd 1 i.-r. Finally a top 1-V.-r of Pd with thickness x is grown to complete the trili v-r

structure shown schematically in Fig. 4-la. We discuss six different samples with the

top Pd 1-~vr having a thickness x varying from 8 to 56 A. The total thickness y of the

polarized Pd (see Fig. 4-1b) can range from 20 to 100 A72,73. Thus for x < y/2, changes

in x will give rise to changes in y. Auger electron spectroscopy (AES) was used to verify

the presence of a well defined Fe l.- -r. The AES measurements were performed in a

10-10 Torr vacuum at sequential intervals following removal of sub angstrom amounts

of Pd using an Argon etch. The depth profile of the high intensity Fe3 (703.0 eV) LMM




























Figure 4-1.


14.0k-
S12.0k-
- 10.0k-
3 8.0k-
6.0k-
| 4.0k-
S2.0k-
0.0-


a) Physical structure


Paramagnetic Pd
Poromngmnetic Pd



Paramagnetic Pd
b) Magnetic structure


a) Multil]i r structure of a Pd/Fe/Pd trilayer. The bottom l i-r of Pd is
200 A thick. The thickness of the Fe 1.- r is 1.5 A as recorded by the quartz
crystal monitor. The thickness x of the top 1I-,r of Pd is varied from 8 to
56 A. b) Magnetic structure of the sample. The total thickness y of polarized
Pd is in the range 20 to 100 A (shaded red area). Thus by varying x, it is
possible to vary the thickness y of the polarized ferromagnetic Pd li. r. c)
Intensity of Fe3 (703.0 eV) LMM Auger electron peak plotted as a function of
material removed by argon sputtering. The data (solid black circles) are fit to
a Gaussian distribution (red line). The full width half maximum value of
1.85 A is consistent with crystal monitor measurements


Auger electron peak of Fig. 4-1c shows that the Fe is embedded in the Pd as a distinct

2D l ,--r with a FWHM thickness of 1.8 A. All of these steps were performed without

breaking vacuum. Measurements of the magnetization M (Fig. 4-2) were performed

using a Quantum Design MPMS system. The magnetic field H was along the plane of the

substrate. Since the magnetization measurements were ex situ, x was constrained to be

greater than 8 A; otherwise the exposure of the sample to air caused unwanted oxidation

of the Fe. The magnetic parameters H((x) (Fig. 4-3) and T((x) (Fig. 4-4) are calculated

respectively from magnetization loops taken at 10 K (see inset of Fig. 4-3) and linear

extrapolations of the temperature-dependent magnetization taken at H = 20 Oe (see

inset of Fig. 4-4). The magnetic contribution from the bottom ferromagnetic Pd li, r is

independent of x, since y/2 < 200A, the constant thickness of the bottom 1- VT.


Data
- Gaussian fit
FWHM is i.asA







0 10 20 30 40 50 60
Thickness of the film from the top (A)
c) Auger electron spectroscopy














*4

o10 -
E


5)
< 8- MsA (Fe)= 2.63x105 emu/cm
I



6 I

0 10 20 30 40 50 60
x(A)


Figure 4-2. The saturation magnetization normalized to the area of the sample if i shows
a smooth increase with increasing thickness x. The experimental data are
shown as solid black circles and the dashed black line is a guide to the e,-
Saturation to a constant value occurs near 30A (vertical arrow).


4.4 Results and Discussion

For large values of x, the thickness y of the combined polarized ferromagnetic Pd

1~.-.-i~ and the associated saturated magnetization M = I ., will reach a constant value.

This expectation is borne out in Fig. 4-2 which shows the x-dependence of saturated

magnetization i. i normalized to sample area. We note that this normalized saturated

magnetization i [. (x) increases with increasing x as the total amount of polarized Pd

increases. The onset of saturation, near x = 30 A indicates that the polarization cloud

including the embedded Fe li---r is ~ 60 A thick. This value is consistent with previous

observation73. The increase of if.. with x shown in Fig. 4-2 is thus straightforward to

understand. As x increases the thickness of the top polarized ferromagnetic Pd l-v1-r

increases with a concomitant increase of magnetic material in the system. Variation of x

clearly controls the thickness of the polarized ferromagnetic Pd l1~V-r. When normalized to




















o 40 -2-

-400 -200 0 200 400
20 H(Oe)


10 20 30 40 50 60
x(A)


Figure 4-3. The coercive field He shows a strong increase as the thickness x of the top
-V,- r of the Pd decreases. The data are shown as solid black circles and the
black solid line is a power law fit with exponent Tl 2.3(0.1). The inset
shows magnetization loops at T 10 K for x 8A (solid black squares) and
x 56A (solid red circles).

the number of Fe atoms present, the saturated magnetization i = 1.1 x 10-4emu/cm2

corresponds to 9.2 fi per Fe atom, in close agreement with previous observations

of the giant moment of Fe in Pd to be near 10 PB72. Modeling the x dependence of

i (x) shown in Fig. 4-2 for our Pd/Fe/Pd trilhv'-ir is not straightforward. For regular

ferromagnets with if., uniform throughout the thickness, we would expect if. ,(x) to be

linear in x; clearly it is not. A reasonable model will incorporate an exchange interaction

J that decays radially with the distance from the point ferromagnetic impurity84. This

complication requires modeling J as a function of distance x from the plane of impurity. A

starting point would be to write the magnetization M is a function of J4,

M(H, T, x) i= 1. 8, PBH + 2pMJ(x) (4-1)
\kBT














200- --------
/ 3-

180- % Mvs T(x = 56A)at200e
2-
U E


0-- T ............
140- c
i 100 200 300
T(K)
0 20 40 60
x (A)


Figure 4-4. The Curie temperature T, rapidly increases with increasing x. Data are shown
as solid black circles and the dashed black line is a guide to the ev.
Saturation to a constant value occurs near 20A (vertical arrow) The inset with
T, indicated by the vertical arrow shows the temperature-dependent
magnetization taken in a field H = 20 Oe.


where B8 is the Brillouin function and p is the number of the nearest neighbors beyond

which J is zero. In principle the experimentally determined values of M(H, T, x) can be

fit to Eq. 4-1 to find the best fit values of J(x) for different values of the parameter p. We

have not performed such an ,in !1i-

Fig. 4-3 shows the behavior of the coercivity H,(x) as a function of x (solid black

circles). The data are well described by a power-law dependence (solid black line),

H,(x) oc x-", where the exponent r = 2.3(0.1) is close to the ratio 7/3. Similar

power-law behavior reveals itself in regular ferromagnetic thin films where q has a

somewhat smaller value varying from 0.3 to 1.576. Because rl depends strongly on strain,

roughness, impurity, and the nature of the domain wall (Bloch or Neel type)76, it is not

surprising to see a wide variation in Tl. Neel predicted for example that for Bloch domain

walls, He of a ferromagnetic thin film should vary as x-4/3 when the thickness x of the film









is comparable to the domain wall thickness w79. For the case of Neel walls, He depends

only on the roughness of the film and does not depend on film thickness77. The variation

of Hc(x) becomes particularly pronounced when the film thickness becomes comparable to

w.

A qualitative understanding of the steeper H((x) dependence becomes evident by

recognizing that the formation of domain structure is driven by the reduction of long

range magnetostatic energy which at equilibrium is balanced by shorter range exchange

and anisotropy energy costs associated with the spin orientations within a Bloch or Neel

domain wall. Domain wall thickness is given by w = /A/K3'82 where K is the i--i 1 H!iii

anisotropy constant and A is the exchange stiffness, proportional to the exchange energy,

Jss. The domain wall size w increases for decreasing K and increasing J. If K, which

depends on the relatively constant spin-orbit interaction4 within the Pd component of the

Pd/Fe/Pd trili- r-is, remains constant, then variations in w are dominated by variations

in J. Thus as x decreases toward zero, the increase in J84 gives rise to an increase in

w which in turn gives rise to a more rapid increase in He than would be seen in regular

ferromagnets with constant J. As discussed above, this rapid variation with q ~ 7/3 is

observed experimentally.

The data in Fig. 4-4 show that T, increases as x increases and reaches a relatively

constant value near x = 20 A. The dashed black line is a guide to the eye and is

qualitatively similar to the behavior of [. !(x) shown in Fig. 4-2 which saturates at a

larger value near 30 A. These observations are again qualitatively consistent with the finite

size effect associated with critical phenomena in ferrei,, iii, i-" "-. Although the data are

not of sufficient quality to distinguish the power-law behavior that is predicted for finite

size eff. I i1-83, we expect that the dependence is further complicated by the previously

discussed dependence of J on x in polarized ferromagnetic Pd. The behavior of Tc(x)

-Ii--,- -i that Pd/Fe/Pd tril-i'- -r should be treated as a single li-v-r with a well defined

spin-spin correlation length. If the Pd l zV--i~ are treated separately, then the bottom 1 rl.r









with fixed thickness y/2 would have a Tc equal to the highest T, of the top li-v-r. In this

case the overall measurement would not show a strong change in T, as a function of x,

since the T, of the bottom l-?v-r would dominate for all x.

We note that for our planar geometry, Tc decreases with decreasing thickness as has

also been shown for thin-film Nisl and epitaxial thin-film structures based on Ni, Co

and Fe82. On the other hand T, increases with decreasing size of ferrimagnetic MnFe204

nanoscale particles with diameters in the range 5-26 nm83. This increase of T, with

decreasing size is attributed to finite size scaling in three dimensions where all three

dimensions simultaneously collapse83. In our two-dimensional planar thin films only one of

the dimensions, the thickness, collapses and T, decreases rather than increases in accord

with the observations of previous studies81'82

4.5 Conclusions

In conclusion, we have characterized the magnetic properties of thin-film Pd/Fe/Pd

tril1v--is and determined that critical size effects apply to I! iiii ;iwtic" Pd where the

ferromagnetism is induced by proximity to an underlying ultra thin Fe film. The critical

size, or equivalently the critical thickness, is controlled by varying the thickness x of

the top Pd l~.v-r. The dependence on film thickness of the coercive field He and the

Curie temperature T, are in qualitative agreement with finite size effects seen in regular

ferromagnetic films where the exchange coupling J is constant throughout the film. The

results presented here increase our understanding of nanomagnetism in ultra thin systems

by showing that the spatial variations of J in the proximity coupled Pd have a pronounced

influence on the form of thickness-induced variations, namely: a nonlinear dependence

of I.[. (x), an unusually strong power-law dependence of H (x) and a dependence of

T,(x) which indicates that the trilayer acts as a single l1v.r that necessarily includes the

constant thickness Pd l-?v-r serving as a substrate for the Fe 1l.--r.









CHAPTER 5
TEMPERATURE DEPENDENCE OF COERCIVITY IN MULTI DOMAIN NI
NANOPARTICLES, EVIDENCE OF STRONG DOMAIN WALL PINNING

5.1 Abstract

The temperature dependence of the coercivity of the single and 5 liv r samples of

Ni nanoparticles in A1203 matrix is studied. A linear T2/3 dependence of coercivity over

a wide range of temperature (10 K to 350 K) is observed. All the samples consists of

particles with multiple magnetic domains as the size of the particles are larger than the

critical single domain size (see Eq. 1-3 on page 13 of chapter 1 and Fig. 3-3 on page 47

of chapter 3). The experimental results are understood in terms of strong domain wall

pinning.

5.2 Introduction

The temperature dependence of the extrinsic magnetic properties, for example

coercive field (H,(T)), arise from two mechanisms. The first mechanism is, due to

the temperature dependence of the intrinsic magnetic properties 11,15'86'7 such as

saturation magnetization (. [.), magnetic anisotropy (K) and exchange stiffness (A)

and will be discussed in chapter 6. The second mechanism is, due to the thermally

activated hoping of the metastable states over some energy barrier.2-4,6 From the

magnetization loops at different temperatures (Fig. 5-2) we have found that [., does

not change with temperature. All the samples comprise pcl i, i- I l 11iiw. par- !I. and

thus magne'- i, --1 iii.w anisotropy can be neglected and temperature independent shape

anisotropy is dominant.48 The experimental temperature range is 10 K to 300 K which is

much smaller than the curie temperature of Ni (630 K)3 and A can be considered constant

over this temperature range."8 In this chapter we will discuss the second mechanism as the

origin of the temperature dependence of the coercive field (11.,, K and A are temperature

independent). To understand the temperature dependence of the He due to the thermally

activated hopping over metastable energy minima separated by some energy barrier, it is

necessary to find out the magnetic field dependence of the energy barrier. A commonly









used phenomenological energy barrier is2,3,47,89


AE =Eo[1 H/Ho] (5-1)


where Eo is the energy barrier at zero magnetic field and energy barrier vanishes at

H = H0 at T = 0. At H = H,, thermal energy, kBT, is sufficiently high to cause most

of the moments to be thermally activated over the barrier. For example for the case of

Stoner-Wohlfarth particles m = 2, Eo = KV and Ho = 2K/M... For Stoner-Wohlfarth

particles the scenario is very simple and the Eq. 5-1 can be derived analytically (see

Eq. 1-6 on page 15 of chapter 1). Remember that if AE(H) is known, it is possible to

calculate H,(T). In this chapter we will discuss how to derive AE(H) (Eq. 5-1) for the

MD nanoparticles and will compare H,(T) with the model.

5.3 Results and Discussions

The sample preparation technique is discussed in chapter 2 and chapter 3. Three

different sets of samples are investigated. Set 1 consists of single 1-i-r Ni particles in

an A1203 matrix. Set 2 and Set 3 consists of 5 1-. -is, of Ni particles separated by A1203

L-i-,-. The interlayer separation in Set 2 and Set 3 are 3 nm and 40 nm respectively. The

schematic of all three sets of samples are shown in Fig. 5-1 below.

In this chapter we will focus on the temperature dependence of H, for the MD Ni

nanoparticles. A total of 15 samples are studied, 5 samples from each set. Magnetization

loops are measured for every sample for seven (on average) different temperatures. This

means a total of around 105 magnetization loops have been measured for the present

study. Magnetization loops for the sample of average particle diameter of 12 nm of set

2 at different temperatures (indicated in the legends) are shown in Fig. 5-2. The arrow

shows H, at 10 K. Note that H, decreases with increasing temperature. The temperature

dependence of H, normalized to Ho for five different samples belonging to set 1 is shown

in Fig. 5-3. The particle diameters are indicated in the legends. Note T2/3 in x axis. All

the data follow a linear T2/3 dependence. To understand the above data, we will start










Alumina


Ni particles


Substrate

Set 1 Set 2 Set 3


Figure 5-1. Schematic of three sets of samples. Set 1 comprises a single lI--r of Ni
particles embedded in an A1203 matrix. Set 2 and Set 3 comprises of 5 1l.ri
of Ni particles separated by different distances in an A1203 matrix. The
interlayer distances in Set 2 and Set 3 are 3 nm and 40 nm respectively.


with a general magnetic energy landscape of the system written as a polynomial expansion

of the domain wall position (x) around a strong pinning center.2-4,6


E(x) = ao + aix + a2x2 + :3 boHx (5-2)
2 3

where ao, al, a2, a3 and bo are micromagnetic parameters that depend on the magnetic

parameters K, ., and A. For the strong pinning center the x3 term is included as the

effect of the pinning center is long distance compared to the weak pinning center where

the x3 term is neglected.3 The relation between micromagnetic and magnetic parameter

can be determined from the particular model used. Note that the micromagnetic

parameters are temperature independent in our case as they only depend on the

temperature independent magnetic parameters. First, we will derive the energy barrier

separating the metastable minima from the global minima. The maxima or minima of

E(x) are determined by setting the first order derivative to zero.

E
= at + a2x + a3X 2 boH = 0 (5-3)
6x


4w4w4w
IM IMIM
40 4040












8.0xl104 -

-4
6.0x104 -

4.0x104 -

2.0x104 -

0.0-

-2.0x104 -

-4.0x104 -

-6.0x104 -

-8.0x104 -


e


500 1000 1500


H (Oe)


Figure 5-2. Magnetization loops for the sample of average particle diameter of 12 nm of
set 2 at different temperatures (indicated in the legends). The coercive field
(He) at 10 K is indicated by the arrow. He decreases with increasing
temperature. Saturation magnetization (3[.,) is constant at different
temperatures.


The two solutions for the above equations are


a2 + 4/3(aj boH)
2a3

a2 V 4a3(a bo0H)


(5-4)


(5-5)


Taking the second derivative of E(x) with respect to x it is easy to show that

62E/6X2 Ix,> 0 (62E/6X2 IX2< 0) and corresponds to the maximum (minimum). Thus the

energy barrier is


(a2 4ala3 + 4a3boH)3/2
AE(H) = E(xi) E(x2) a
6a3


-o- MsublOK
-- Msub50K
MsublOOK
--- Msubl50K
--- Msub200K
--- Msub250K
--- Msub300K
--- Msub325K


24 nm Ni/AIl20
5 Layer

l)

11111J11


-1500 -1000 -500


__





(5-6)






















O 0.4-
I


0.2-


0.0 .
0 5 10 15 20 25 30 35 40 45 50

T2/3 (K2/3)


Figure 5-3. Coercive field (He) vs. T2/3 for five different samples of set 1. The linear
behavior is observed for samples with particle size from 18 nm to 42 nm in
diameter.


For the reverse field, ie H -


AE(H)


H the above equation reduces to

(a 4a3(al + boH))3/2


(a 4ala3)3/2
6aj 1


which is in the same form of Eq. 5-1, where

(a 4ala3)3/2
Eo6a


a 4ala3
4a3bo


From Eq. 5-7 it is clear that AE(H) decreases with increasing H and when H


energy barrier can be overcome by thermal energy (definition of the coercive field). Thus


H
a -481a3
4a3bo /


(5-7)


(5-8)


(5-9)


He the










at H = H, the Eq. 5-7 can be rewritten as


kBT (a


4aia3)3,
6a


H )3/2
H)
ao-4ala3
4a360bo


The above equation can be solved for He


H Ho 1 \ ( 2/3


where Hco and Eo are given by Eq. 5-8 and 5-9. This temperature dependence of He is

consistent with the experimental results shown in Fig. 5-3, 5-4 and 5-5.



1.0

0.9- Set 2
12 nm
0.8- 18 nm
A 24 nm
0.7- v 42 nm
4 60 nm
0 0.6

o 0.5
I
0.4

0.3

0.2

0 5 10 15 20 25 30 35 40 45 50

T2/3 (K2/3)



Figure 5-4. Coercive field (He) vs. T2/3 for five different samples of set 2. The linear
behavior is observed for samples with particle size from 12 nm to 60 nm in
diameter.


2 (1


(5-10)


(5-11)




















O
0 0.5

0.4-

0.3-

0.2-

0.1

0.0
0 10 20 30 40 50

T2/3 (K2/3)



Figure 5-5. Coercive field (He) vs. T2/3 for five different samples of set 3. The linear
behavior is observed for samples with particle size from 16 nm to 44 nm in
diameter.


5.4 Relation Between Micromagnetic Parameter and Magnetic Parameters

Here we will outline a road-map to relate the micromagnetic parameters ao, al, a2,

a3 and bo to the magnetic parameters K, ., and A. To do that we will start with the

magnetic energy expression,3


E(x) = A (v ( ))2 K(x)(k.M())2 oM(x).H M(x).Hd(M) dV

(5-12)

where the first term corresponds to the exchange energy cost due to the spin misalignment,

the second term is the anisotropy energy, the third term is the Zeeman energy and the

fourth term is the magnetostatic self energy. The position of the domain wall is given by x

and k is the unit vector along the easy axis. The above equation should be solved for real










samples while taking into account real structure and imperfections. The real structure and

imperfections are responsible for the x dependence of the magnetic parameters (3 ..(x),

A(x), K(x)). After solving Eq. 5-12 and by comparing the coefficients of the different

power of x, it is possible to find out the micromagnetic parameters in terms of magnetic

parameters. The behaviors of H o and Eo/kB are shown in the figure below.


500-



450-

0
O 400-

350

350-


10 20 30 40 50 60
d (nm)


700



600



500 c
C)
LU

400



300


Figure 5-6. The behaviors of H o and Eo on particle diameter are shown for set 2 samples.
Ho decreases and Eo/kB increases with increasing particle size.


The increasing behavior of Eo and decreasing behavior of Hco are consistent with

the literature.3 The actual behavior can be very complicated as it depends on the real

structure factors and imperfections in the material.3

5.5 Conclusions

We have investigated the temperature dependence of the coercive field of MD Ni

nanoparticles in A1203 matrix. He decreases linearly with the T2/3. This behavior can be

understood according to the strong domain wall pinning. We show that the general energy


Set 2
--- Hco

-e- EO/kB









barrier that arises due to strong domain wall pinning depends on the magnetic field with a

power of 3/2 and is responsible for the temperature dependence of the H,.









CHAPTER 6
COERCIVE FIELD OF FE THIN FILMS AS THE FUNCTION OF TEMPERATURE
AND FILM THICKNESS: EVIDENCE OF NEEL DISPERSE FIELD THEORY OF
MAGNETIC DOMAINS

6.1 Abstract

The temperature dependence of the coercive field of Fe thin films has been investigated.

Three different samples of different thickness are studied. The coercive field decreases with

temperature and follows the same temperature dependence as the first order anisotropy

constant. This behavior is consistent with the theoretical prediction made by Neel15

based on the disperse field theory of magnetic domain which takes in to account the effect

of free poles on the coercive field that occurs at small inclusions. The value of coercive

field increases with decreasing film thickness. This behavior is expected for multi domain

ferromagnetic systems at nanoscale where the domain wall thickness is comparable to or

larger than the film thickness.

6.2 Introduction

The most interesting aspect of ferromagnetism is the hysteresis loop,90 which refers

to the history dependent behavior of magnetization with applied magnetic field (Fig. 6-2).

Hysteresis is a complex nonlinear, nonequilibrium and nonlocal phenomenon, reflecting

the existence of anisotropy-related metastable energy minima separated by field-dependent

energy barriers.3 An extrinsic property of crucial importance in permanent magnetism is

the coercive field, the magnetic field where magnetization changes sign as it passes through

zero. The coercive field basically describes the stability of the remnant state and is a

very important concept for most practical applications.9199 Coercivity in ferromagnets

is known from very long time. 90 But, due to the complex nature, the origin of coercive

field is still a subject of study. In this present work the behavior of coercive field of three

different iron thin films with different thicknesses has been investigated. The temperature

dependence of the coercive field agrees well with the theory of domain wall pinning arising









from small inclusions (for example

free pole is not negligible. 15


impurity or vacancy defects) where the energy of the


6.3 Experimental Details


Figure 6-1. TEM image of Fe thin film of thickness 9 nm.


Thin films of Fe were fabricated on Si(100) and sapphire (c-axis) substrates using

pulsed laser deposition from alumina and iron targets. High purity targets of Fe (99.9' .)

and A1203 (99.9'-.) were alternately ablated for deposition. Before deposition, the

substrates were ultrasonically degreased and cleaned in acetone and methanol each for 10

min and then etched in a 49'. hydrofluoric acid (HF) solution to remove the surface silicon

dioxide l -ir (for the Si substrates only), thus forming hydrogen- terminated surfaces.35

The base pressure for all the depositions was of the order of 107- Torr. After substrate

1i. lii.- the pressure increased to the 10-Torr range. The substrate temperature was

kept at about 550" C during growth of the A1203 and Fe lv ir-. The repetition rate of the

laser beam was 10 Hz and energy density used was ~ 2 Jcm-2 over a spot size 4 mm x

1.5 mm. A 40 nm-thick buffer liv-r of A12O3 was deposited initially on the Si or sapphire

substrate before the sequential growth of Fe and A1203. This procedure results in a very










smooth substrate independent starting surface for the growth of Fe, as verified by high

resolution scanning transmission electron microscopy studies (Fig. 2-1). A 40 nm-thick

cap 1-.- r of A1203 was deposited to protect the Fe 1-v-r from oxidation. Three different

samples with thickness of 9 nm, 21 nm and 30 nm were prepared for the present study.

Magnetization measurements are performed in a quantum design Magnetic Property

Measurement System (\!PMS). The magnetic field was along the plane of the films. The

dependence of temperature and film thickness on coercive field is investigated.

6.4 Results and Discussion





4.0x10- O --M10K
-m- M50K
2.0x10 M100K
? M150K /
E 0M.0-
5 -2.0x10 M250K
M300K
-4.0x104- M325
-6.0x10- |0
-8.0x1 0 -
-600 -400 -200 0 200 400 600
H (Oe)


Figure 6-2. M-H loop of Fe film of thickness 21 nm at different temperatures as indicated
in the legend. The coercive field defined at M=0, decreases with increasing
temperature.



Shown in Fig. 6-1 is the TEM image of the 9 nm thick Fe sample. Due to the

low surface energy difference between Fe and A1203 (650 mJ/m2) and high adhesion

energy (1205 mJ/m2) between Fe and Al203,100 Fe wets the surface of Al203 and thus

a continuous thin film is grown instead of grains as is the case for the Ni/AO203 system

discussed in chapter 2 and 3. Three different samples with thickness 9 nm, 21 nm and

30 nm were grown. The magnetic hysteresis loops at different temperatures from 10 K

to 350 K are measured with the applied magnetic field along the plane of the thin film.

M-H loops for 21 nm sample are shown in Fig. 6-2. The magnetic field is swept from 5










T to -5 T and again from -5 T to 5 T to complete the loop. At high magnetic fields the

magnetization from the Fe saturates and the magnetic field dependence arises due to the

diamagnetic contribution from the substrate. The diamagnetic contribution from the Fe

film is negligible as the thickness of the film is very small compared to the thickness of the

substrate. Thus the high magnetic field data is linear in magnetic field with a negative

slope. The slope of the line is determined from the linear fit of the high field data and

subtracted from the raw data to extract the ferromagnetic signal due to the Fe thin film.

The procedure is repeated for all samples and for all temperatures. The coercive field

is found to decrease with increasing temperatures. The results for all three samples are

presented in Fig. 6-3. Figure 6-4 shows the relation between K(T)s7 and H,(T) where

T is the implicit variable."7 The linear behavior of H, and K -i r---, -I that the coercivity

mechanism is similar to the disperse field theory of magnetic domain which takes in to

account the effect of free poles on the coercive field that occurs at small inclusions. 15


600-
S 99 nm
550- --
500
450
8 400 -
350- 21 nm
300
250-
25 30 nm
200 ------ .
0 50 100 150 200 250 300
T (K)


Figure 6-3. H, vs. T of Fe films of thickness 9 nm, 21 nm and 30 nm. The temperature
dependence of all three samples is similar, which ir- -.-- -i that the origin of the
coercive field depends on the intrinsic property of the iron, which in our case is
the magnetocrystalline anisotropy. This is true as for the case of extrinsic
origin the energy barrier depends on the size of the sample as discussed in
chapter 5













56000- K
-Linear Fit of Data4 C
54000

52000-

50000

48000
380 400 420 440
H (Oe)


Figure 6-4. H, vs. K of Fe film of thickness 9 nm. The linear behavior -i-i- -I the
validation of Neel's disperse field theory of magnetic domains which takes in to
account the effect of free poles on the coercive field that occurs at small
inclusions (Eq. 6-3). The similar behavior is observed for other two samples
which is expected as they have similar temperature dependence.


Theories concerned with the coercivity of the multi domain ferromagnetic materials

begin with the consideration of the change in magnetic energy across a domain wall.2

Since the magnetization changes from [ ., to -I .. across a 180 domain wall, the effective

pressure on the wall will be the difference in the energy across the wall per unit volume.

At an applied magnetic field H, the pressure will be 2H.1 .. Thus the work done to move

a domain wall of area s to a distance dx will be2


dW 2H.1[.sdx (6-1)


The above equation can be generalized for any angle (not only 180") domain wall.2

The term dW/dx, which is basically the energy required to move a domain wall to unit

distance, is the measure of the resistance of domain wall motion.2 In real samples due to

impurities, imperfections or irregularities in crystal, dW/dx passes through maximum and

minimum. The origin of reversibility in the magnetization loop is the irreversible motion of

domains through these maximum and minimum. 2 The coercive field is usually calculated

using the equation below.2












2 1. -

Different theoretical approaches concentrate on the derivation of (dW/dx)max.

Substantial wall motion may require fields of tens or hundreds of Oersted. Evidently

real materials contain crystal imperfections of one sort or another which hinder the easy

motion of domain walls. These hindrances are generally of two kinds: inclusions and

residual microstress.2

From a magnetic point of view, an "inclusion" in a domain is a region which has a

different spontaneous magnetization from the surrounding material, or no magnetization

at all. According to the result obtain by Neel in his disperse field theory (which basically

calculates the term (dW/dx)max), coercivity is given by Eq. 6-3 below.15

1 A2a2 Kv2
H, = v + (63)
4 K, 1. 11.

The first term is due to the residual microstress and vl is the volume fraction of the free

pole at the residual microstress and A, a, K and i.., are the saturation magnetostriction,

internal stress, magnet' i-l ii.1_,i. anisotropy constant and saturation magnetization

respectively. The second term in Eq. 6-4 originates due to the poles at the inclusions of

volume fraction v2. For iron, i1., = 1743 emu/cm3,2 K=8x106 ergs/cm3,101 A 10xt0-6102

and a=1010 dynes/cm2.103 Putting these values in the Eq. 6-3 we obtain,


H, 0.18vl + 46002 Oe (6-4)


Thus in our case the coercive field will be dominated by the 2nd term in Eq. 6-3,

which is proportional to K and agrees well with the experimental results as shown in Fig.

6-3 and Fig. 6-4. Remember that i.., is constant.

In Fig. 6-5 the variation of H,(d) is shown. The increase in H, with decreasing film

thickness, d, could be due to the enhancement of K with decreasing d.101,104,105 The

increase in coercive field with decreasing thickness is well known for the multi-domain












600- \ H
S Linear Fit of Data7_B

500 -


0) 400-
0

300 -

200- U

5 10 15 20 25 30
d(nm)


Figure 6-5. He vs. d of Fe films at 10 K. Coercive field increases with decreasing d. This
behavior is due to fact that K increases with decreasing d which is very
common in multidomain thin films when the domain wall thickness is
comparable or larger than the film thickness. The domain wall thickness of
iron is about 60 nm,3 which is larger than the thickness of the films studied
here.


magnetic thin films when the domain wall thickness is comparable to or higher than the

film thickness.7 The domain wall thickness of iron is about 60 nm,3 which is larger than

the thickness of the film studied here. Note that we have used the value of K in Eq. 6-3,

which is larger than the bulk value. From Eq. 6-4, we find that v2 ~ 0.12.

6.5 Conclusion

The coercive field of multidomain Fe thin films has been investigated. The temperature

and thickness dependence of the coercive field agrees well with the prediction made by

Neel, which takes in to account the effect of free poles on the coercive field that occurs

at the small inclusions. The contribution of strain to the coercivity is very small for the

case of Fe and the dominant contribution comes from the free poles at the inclusions. The

volume fraction of inclusions has been estimated from the coercive field data to be around

0.12.









CHAPTER 7
SCALING COLLAPSE OF THE IRREVERSIBLE MAGNETIZATION OF
FERROMAGNETIC THIN FILMS

7.1 Abstract

In ferromagnetic materials, hysteresis, or equivalently the history dependent behavior

of the magnetization, reflects complex nonlinear and non equilibrium phenomenology that

has been recognized for many years5'6'106-108. Hysteresis depends strongly on materials

properties such as structural length scales spanning the nanometer to micrometer range3

and manifests complex behaviors including magnetic relaxation with aging dependence109

dimensionally dependent dipole-dipole interactions48'110, spin-glass like memory effects 1

and supermagnetism112. Here we show that the irreversible magnetization, defined as

the difference between field-cooled and zero-field cooled magnetization, has a striking

similarity for a wide variety of ferromagnetic materials. This similarity becomes apparent

when the irreversible magnetization is normalized to its maximum value and plotted with

respect to a temperature dependent renormalized field. The collapse of the irreversible

magnetization onto a single curve for a given system implies an underlying symmetry to

hysteresis that is not captured by previous analytical3,'5'6'48,106,107'109112 and computational

treatments108s113 and thus provides a unifying theme that embraces a broad range of

complex hysteretic behavior.

7.2 Introduction

In general, hysteresis is a complex nonlinear non equilibrium phenomenon which

reflects the presence of field-dependent energy barriers between anisotropy-dependent

metastable minima. Accordingly, hysteresis is affected by a combination of intrin-

sic properties such as magnetocrystalline anisotropy which depends on crystal field

energy and spin-orbit coupling and extrinsic properties such as sample shape, grain

boundaries, disorder and imperfections. For example, in bulk ferromagnets hysteresis is

often described as a superimposition of domain wall motion and domain rotation with

energy barriers related to magneto' i- i 1iiw., anisotropy together with imperfections









and/or impurities in the material107. For the simplest case of single domain coherently

rotating nanoparticles, the origin of hysteresis is the finite time scale for magnetization

reversal as the magnetization overcomes a magnetic field dependent energy barrier by

thermal activation and rotates from one easy axis direction to another5'6'106. Hysteresis

and associated magnetization reversals p1 i, an important role in applications such as

magnetic data storage devices91 93, GMR94'95 or MRAM96 devices, magnetic -, i -i-.'' and

motors98, generators99 etc.

7.3 Experimental Results

Measurements of irreversible magnetization are usually accomplished by one or both

of two techniques illustrated respectively in panels a) and b) of Fig. 7-1. The sample

under investigation in this figure (hereafter referred to a sample A) is a 20 A-thick pulsed

laser deposited thin film comprising five l~V -iS of 3 nm elongated Ni particles embedded

in an insulating Al203 host [see Methods]. The Ni particles are small enough to be in the

single domain (SD) regime where all the spins are aligned in the same direction and rotate

coherently together in response to a changing magnetic field. In the first technique (panel

a), the sample is field cooled (FC) in a field H = 20 Oe (black squares) from 300 K to 5 K

and then zero field cooled (ZFC). The irreversible magnetization (AM(H, T)), which is a

function of magnetic field (H) and temperature (T), is shown by the two-headed dashed

vertical arrow. In the second technique, the sample is held at fixed temperature T and

magnetization M loops obtained by repeatedly cycling the applied field H about H = 0

between two symmetric limits. The history dependent trajectories form closed loops shown

schematically in the insets of panel (b). These loops, which can be acquired at different

temperatures, are each characterized by a coercive field H,(T), a saturated moment .[.,(T)

and a remnant magnetization Mr(T). The coercive field H,(T), plotted versus T1/2 in

panel b for sample A discussed above, is shown as a blue line connecting the starred data

points. The absence of a T1/2 dependence for coherently rotating SD particles with easy

axes oriented along the field6 will be discussed below. While both AM(H, T) and H,(T)























0 50 100 150 200 250 300
T(K)


b)

300


S200
U

100


0


4 5 6 7
T'2K)


Figure 7-1.


Irreversible magnetization AM(H, T) defined as the difference of the FC and
ZFC magnetizations is a quantitative measure of hysteresis. a, Black squares
and red circles represent respectively the FC and ZFC temperature-dependent
magnetizations for sample A in a 20 Oe field applied parallel to the film
surface. The irreversible magnetization AM(H, T) is shown by the two-headed
vertical dotted arrow. Inset: Large thermally blocked magnetic particles (1
and 3) with respective vertical and horizontal easy axes of magnetization
indicated by the arrows. Particle 2 is small enough to be superparamagnetic
(thermally unblocked). For magnetic fields applied along the vertical direction,
particles 2 and 3 do not contribute to AM(H, T). b, The coercive field (He)
for the same sample shown as a function of T1/2 (Blue stars) does not show
the linear behavior expected for ideal Stoner-Wohlfarth particles6 where all the
particles are uniform size and aligned with the applied field (particle 1). The
solid blue line is a fit to the data using a lognormal distribution of particle size
together with a random orientation of the easy axis of the magnetizations.
Inset: Schematic magnetization curves for the cases where (bl) only particles
1 and 2 and (bl) only particles 1 and 3 are present. The resultant
magnetization loops (black curves) for the two cases show the pronounced
effects of particle size and easy axis orientation on the determination of H,.


are commonly accepted measures of hysteresis, the underlying phenomenology for each

is considerably different. For example in nanoparticle magnetic systems such as shown

in Fig. 7-1, AM(H, T) and Hc(T) are sensitive in different v--o- to both the presence of

superparamagnetic particles and the random orientation of the easy axis of magnetization

of each particle. The insets of Fig. 7-1 a) and b) schematically illustrate these differences

using three particles: particles 1 (red) and 3 (blue) with easy axes of magnetization


S-5


0


- -m-FC
-*-ZFC


I I I









respectively along the vertical (H) and perpendicular directions and large enough so that

at the temperature of measurement, changes in magnetization are blocked by energy

barriers that cannot be overcome by thermal activation, and particle 2 (green) with easy

axis along the vertical axis and small enough so that it is superparamagnetic with a

non-hysteretic magnetization depending only on H and T (i.e., unblocked). Consider the

case where only particles 1 and 2 are present. Since the superparamagnetic particle 2 will

have zero AM(H, T), the total AM(H, T) value will be only due to the blocked particle

1. On the other hand, the M-H loop will be the summed contributions of the hysteretic

loop for particle 1 and the reversible loop for particle 2 as shown in inset bl of Fig. 7-lb.

Thus for this case AM(H, T) is not affected by superparamagnetic particles but He(T)

is. Consider now the case where only particles 1 and 3 are present, i.e., both particles are

blocked but with different (parallel and perpendicular) easy axis orientations with respect

to the applied field. Simple theory for coherently rotating SD particles shows that upon

reversing the field particle 1 must surmount a H-dependent energy barrier whereas particle

3 can line up without having to overcome an energy barrier3. In like manner to the

previous case, The M-H loop for particle 1 shows hysteresis but particle 3 does not (inset

b2 of Fig. 7-1b). Hence H,(T) is affected by the presence of particles with perpendicular

orientation but AM(H, T) is not.

The above arguments can be generalized for SD magnetic nanoparticles with a

broad size distribution and a random orientation of the easy axes of magnetization.

Particles with an easy axis making an arbitrary angle with the applied magnetic field

will alv-wb contribute less to hysteresis than aligned particles. Because the perpendicular

component does not contribute, the correction is a straightforward integration over angle.

Accordingly. the measured value of AM(H, T) will be only due to blocked particles and

dominated by particles with easy axis of the magnetization along the applied magnetic

field. On the other hand, H,(T) will be strongly affected by the random orientation of the

easy axis of blocked particles and the presence of the particles which are small enough to









be superparamagnetic. For an ideal Stoner-Wohlfarth particle system6 in which all the

particles are aligned along the H direction, a linear square root temperature dependence

of H,(T) is expected. For the considerably more complicated case of randomly oriented

SD particles with a size distribution in which some of the particles are superparamagnetic

(e.g., sample A), H,(T) can be described (solid line of Fig. 7-1b) using a model with a log
normal size distribution and three fitting parameters9'10 (see Methods). The H-dependent

behavior of AM(H, T) for sample A is shown in Fig. 7-2a for the temperatures indicated

in the legend. The isotherms show peaks, AMma,(T), at magnetic fields, H,(T), indicated

by the vertical arrows. These peaks are expected, since at H = 0 the FC and ZFC

measurements are equivalent and the difference in magnetization should be zero, whereas

at high H both FC and ZFC magnetizations saturate to the same value and again the

difference should go to zero. The similarities in the AM(H, T) isotherms are -,i-'-. -.1i'.-

and become manifest as an unexpected data collapse onto a single curve when the reduced

irreversible magnetization, AM(H, T)/AMmax(T), is plotted as a function of reduced

magnetic field, h(T) = H/H,(T), as done in Fig. 7-2b. The characteristic field H,(T)

deviates significantly below Hc(T) for T < 25 K (inset of Fig. 7-2b) and identifies the

T-dependent field where irreversibility is at a maximum.

An important physical insight into the scaling collapse shown in Fig. 7-2b is gained

by plotting AMmax(T) as a function of T1/2. The observed linear behavior shown in

Fig. 7-3a is identical to the predicted temperature-dependent coercivity Hs W(T)

of Stoner-Wohlfarth (SW) particles6 mentioned above. Guided by this similarity,

we make the ansatz: AMl a( T) oc H'w. This ansatz is physically reasonable

since as shown above, AM(H, T) measurements are not effected by the presence of

superparamagnetic particles, and in addition the contribution from particles with easy

axis of magnetization along the magnetic field is dominant. As all real samples comprise

particle size distributions and random easy axis orientations, the conventional M-H loops

from which coercivities are extracted are markedly different than would occur for idealized


























H(Oe)


0 2 4
H/Hm


Figure 7-2.


All of the data for AM(T, H) of sample A can, with proper normalization, be
made to collapse onto a single curve, a, Isotherms of AM plotted as a function
of H show self-similar behavior with maxima AMma,x(T) occurring at
characteristic fields Hm(T) marked by the vertical arrows. With decreasing
temperature AMmax(T) increases and H,(T) moves to higher fields. b, The
scaling collapse occurs when AM(H, T)/AMma,(T) is plotted against the
normalized field H/Hm(T). Inset: Plots of H, (squares) and H,(T) (stars) as
a function of T. The data and fit for H, are the same as shown in Fig. Ib, but
plotted with respect to T rather than T1/2. The solid line for Hm(T) is a guide
to the cv


SW behavior. The insensitivity of AM(H, T) measurements to superparamagnetic

particles and perpendicular orientations of blocked particles together with our ansatz

imply that more useful information about the magnetization reversal process is obtained

from AM(H, T) measurements than H,(T) measurements.


In
0



o



b) 1.0


0.9
0.8



































0 100 200 300 400
H,(T) (From MH) (Oe)


Figure 7-3.


For single domain particles the ansatz AMmx(T) oc H,w is verified, a, The
linear dependence of AMmax(T) on T1/2 is in accordance with the
Stoner-Wohlfarth theoretical prediction that Hc(T) oc T1/2 for the coherently
rotating single domain particles of sample A. The use of AMmax(T) rather
than H,(T) as a measure of hysteresis removes the effects of nonuniform
particle size and random orientation. b, The values of H,"I(T), computed for
uniform size FePt particles (sample B) from the ansatz-derived Eq. 2 at the
indicated temperatures, scale linearly with HSW(T) which is determined from
the coercive fields of M H loops after correction for the random orientation
of the easy axis of magnetization (see Methods).


The scaling collapse behavior plotted for sample A in Fig. 7-2b si-. -,--- that AM has

the form,


AM(H, T) AM,,(T)F (H/H,(T)) ,


(7-1)


where F is an unknown function with the property F(1)


1. Taking the second partial


derivative of the both sides of this equation and solving for AMmax(T) gives the result:


(92F(h)F2 /h 2) 1 h 1H, T)a2 AM(H, T)/aH2 ,H(T) evaluated at the
hl 1mT


Slope 1.09(4)
10K



20 K


30 K


500 600


AMmC(T)









maximum where h(T) = H/Hm(T) = 1. The first term, (2F(h)/0h2) 1 h must be

a constant because of the scaling collapse. The remaining two terms have the same form

as the expected value of an effective anisotropy field114. Thus according to our ansatz,

AMmax(T) oc Hf we can write the relation

Hw(T) CH, (T)a2AM(H, T)/aH2 = CHf'(T), (7-2)

where C is a constant and Ha(T) = H2 (T)2 AM(H, T)/OH2 In (Tis computed from

experimental data.

To test the result expressed in Eq. 2, we use magnetization data on a system of
a;,,' .rm size (6 nm diameter) FePt nanoparticles (sample B, see Methods) synthesized

via thermal decomposition of Fe(CO)5 and reduction of Pt(acac)2115. The measured

coercive field HKH (T) is obtained from M-H loops at different temperatures. Assuming

that the particles easy axes of magnetization are randomly oriented, a temperature

dependent correction to HfH(T) must be made so that HSW(T) can be inferred. This

correction is needed (see Methods) since the magnetization of randomly oriented uniform

size particles shows a T3/4 dependence7 compared to the T1/2 dependence expected for

an idealized sample in which all the particles are aligned along the applied magnetic

field. The plot of Fig. 7-3 includes this correction and shows a linear dependence of

Hcal( T) on HKH(T) as would be expected for a system of coherently rotating SD uniform
size FePt nanoparticles. The linear behavior with slope of 1.09(4) confirms the validity

of our model as expressed by Eq. 2 with C ~ 1. In Fig. 7-4 the reduced irreversible

magnetization, AM(H, T)/AMm,a(T), is plotted as a function of reduced magnetic

field, h(T) = H/Hm(T) for a wide variety of thin-film magnetic materials labeled in

the inset and described in Methods. Unexpectedly, when plotted in this manner, the

magnetization data for each materials system collapse onto single curves which have

similar shapes described by a unique function F(H/Hm(T)) (see Eq. 1) for each system.

This data collapse is quite remarkable considering the variety of mechanisms responsible















V VMT Sr .y FePt nanoparticles (6 nm)






0 2 4 6
H/Hm


Figure 7-4. Scaling collapse describes irreversible magnetization (hysteresis) in a wide
variety of ferromagnetic materials. a, Plots of the reduced irreversible
magnetization AM(H, T)/AMlma(T) as a function of reduced magnetic field
H/H,(T) for the six different magnetic materials labeling each curve. The
implicit temperature variable increases from left to right and each color on a
given curve represents a different AM(H, T) isotherm. For clarity, the ordinate
values have been shifted and the solid lines have been added as guides to the
eye. The bottommost curve for single crystal spin-glass Cu:Mn (at. 1.5 .)
material is taken from the literature6


for magnetic ordering together with a wide range of materials properties. The scaling

collapse applies equally well: to single domain (SD) coherently rotating Ni nanoparticles

with average particle diameter of 3nm embedded in an insulating matrix; to multidomain

(l 1)) incoherently rotating Ni nanoparticles with average particle diameter of 12 nm also

embedded in an insulating matrix; to coherently rotating uniform size FePt nanoparticles

(6 nm diameter); to continuous metallic Gd thin films with magnetization derived from

local moments supplemented by band structure exhibiting some itinerant character; and

to mixed-phase manganite (LPC'\ 10) where the Mn spins order by a double exchange

mechanism in an environment where charge-ordered and paramagnetic insulating phases

compete with a ferromagnetic metallic phase. We have also included a spin glass material,

single crystal Cu:Mn (1.5 at -), described in the literature1l6. The collapse occurs for

nanoparticle systems which include SD, MD and superparamagnetic particles with both









broad and uniform size distributions and magnetic moment reorientations arising from

complex superimpositions of domain wall motion and domain rotation which can be very

different in continuous films compared to nanoparticles where the extent of the domain

wall is comparable to the size of the particle. Thus the materials measured here are very

much different in terms of the origin of the irreversible magnetization.

In conclusion, we have presented a heretofore unreported phenomenological result

showing that the temperature and field dependent hysteresis of at least six distinctly

different magnetic systems can be collapsed onto single curves (Fig. 7-4) using the

particularly simple functional form expressed by Eq. 1. Our finding that such a scaling

collapse applies to magnetic systems totally different than the spin glass for which similar

scaling has been previously noted116'117 (the lowest curve in Fig. 7-4) -t-'- -I-- that

explanations116 relying on spin glass phenomenology are too narrow. Thus a more general

theory is needed to explain the scaling collapse. This collapse must imply an underlying

symmetry that is not captured by previous analytical and computational treatments and

may be a crucial clue to understand the complex history dependent magnetization process.

The similarity with the spin glass material is interesting and may be because of the fact

that all ZFC magnetization is a metastable state of the system and shows properties

varying with time. Thus the dynamics of the spins may pl. i a very important role for

the scaling collapse. The behavior of A/Mmax(T) is investigated for the 3 nm Ni particles.

We -it'-'- -1 a new approach to investigate the magnetization reversal process from the

AAM,,x(T) measurement. We have successfully applied the method for the coherently

rotating SD particles.

7.4 Conclusions

In conclusion, we have presented a surprisingly general and unrecognized phenomenological

result showing that the temperature and field dependent hysteresis of at least six

distinctly different magnetic systems can be collapsed onto single curves (Fig. 7-4)

using the particularly simple functional form expressed by Eq. 1. We have not yet found









any exceptions. For the particularly simple case of coherently rotating single domain

particles (samples A and B), our analysis of scaling collapse bypasses the complications of

nonuniform size distributions and random easy axis orientation, unveiling an underlying

Stoner-Wohlfarth behavior6. Our finding that the same scaling collapse more generally

applies to magnetic systems with a wide variety of mechanisms giving rise to hysteresis,

~i--.-' -I that explanations116 relying on spin glass phenomenology are too narrow.

Thus a more general theory is needed to explain the scaling collapse. This collapse

must imply an underlying symmetry that is not captured by previous analytical and

computational treatments and may be a crucial clue to understand complex history

dependent magnetization processes. The similarity with the spin glass material116

is -, .-.,~ -li' .- and may be related to the fact that all ZFC magnetizations represent

metastable states of the system, which if given sufficient time would relax toward the

field-cooled equilibrium state. Accordingly, the dynamics of the spins may pl iv a very

important role in understanding the scaling collapse.

7.5 Methods

7.5.1 Ni Nanoparticle.

Composite films comprising magnetic Ni nanoparticles embedded in an Al203 host

matrix were synthesized by pulsed laser deposition (PLD)36. High purity targets of Ni

(99.9' .- ) and Al203 (99.99 .) were alternately ablated in the same deposition run. The

base pressure of the deposition was on the order of 10-' Torr. The substrate temperature

was maintained near 550C during the growth. The repetition rate of the laser beam was

10 Hz and energy density used was 2 J cm-2 over a spot size of 4 mm x 1.5 mm. A 40

nm-thick buffer l-v-r of Al203 was deposited initially on the sapphire substrate before

the sequential growth of Ni and A1203. This procedure results in a very smooth starting

surface of growth of Ni as verified by the high resolution scanning transmission electron

microscopy (STEM) studies. Samples consists of 5 l' ,-irs of Ni and Al203. A cap l-ivCr

of A1203 was alv--i-i used to protect the sample from oxidation. Z-contrast STEM image









verifies the absence of the Ni oxide. Samples studied here consists of Ni particle size of 3

nm and 12 nm in diameter.

7.5.2 Gd Thin Film.

Gd thin films were deposited on Si substrates by DC magnetron sputtering. The base

pressure of the chamber was on the order of 5 x 10-7 Torr. The samples are continuous

with thickness near 100 nm.

7.5.3 (LalyPry)o.67Cao.33MnO3 (LPCMO) Thin Films.

Phase separated manganite (LalyPry)o.67Can, _,u03 (LPC'\O) films were grown

using pulsed laser deposition (PLD) at a rate of 0.05 nm/s on NdGaO3 (NGO) (110)

substrates kept at 820C in an oxygen atmosphere of 420 mTorr11

7.5.4 Temperature Correction of Coercive Field.

For ideal Stoner-Wohlfarth particles the coercive field is given by, Hw = Ho,(T)(1 -

(T/TB)1/2). For the case of randomly oriented coherently rotating particles all of which
have the same size, the coercive field is given by, H,"(T) = 0.48H,,(T)(1 (T/TB)3/4).

Since the FePt particles of sample B all have the same size, we can write H,"a(T)

HNH(T) where HNH(T) is the coercive field extracted from the magnetization loops. It is

then straightforward to show that, HW(T) = 2H H(T)(1 (T/TB)/2)(1 (TTB)3/4).

Using the fact that at low fields the zero-field-cooled peak temperature can be identified

as the blocking temperature119, we find that TB = 60 K for the FePt nanoparticles.

Accordingly HSW(T), the abscissa in Fig. 7-3b, can be calculated in a straightforward

manner from the measured values of TB and H7H(T).









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BIOGRAPHICAL SKETCH

Ritesh Kumar Das was born on 1981 in a very small village, Mohanbati, located east

of India. At Haripal Guru Dayal high school, Ritesh was introduced to physics and fell in

love with the subject. Ritesh got interested in science subjects during his high school di-,

where he found wonderful teachers, Mr. Robin C'i i 1. ijee and Mr. Uttam Saha. After

completing the high school, Ritesh enrolled in the Ramakrishna Mission Vidyamandira

(Belur) in August 1999 and graduated with a Bachelor of Science degree with honors in

physics on August 2002. During this time Ritesh met his favorite teacher Dr. Deepak

Ghsosh, who p1 i,- d a very crucial role to make Ritesh interested in experimental physics.

After completing the Bachelor of Science degree, Ritesh enrolled in Master of Science

degree at Indian Institute of Technology (Kanpur) and completed the degree on August

2004. After this Ritesh joined University of Florida to pursue his Ph.D. degree. Ritesh

was ahv--, interested in the properties of materials at nanoscale and found that Prof.

A. F. Hebard's lab to be the perfect place for the research in the field of thin films and

nanoparticles. Ritesh graduated in August 2010 with a Doctor of Philosophy degree in

physics.





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MAGNETISMINNANOSCALEMATERIALS,EFFECTOFFINITESIZEANDDIPOLARINTERACTIONSByRITESHKUMARDASADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 1

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c2010RiteshKumarDas 2

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Idedicatethistomyparentsandfamilyfortheiractivesupport.Withoutthemitwouldhavenotbeenpossible. 3

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ACKNOWLEDGMENTS Iamtrulyindebtedtomanyindividualswhohavecontributedtothesuccessofmyresearchwork.Therefore,Iexpressmysincerestregretstoanypersonnotspecicallymentionedhere.Firstandforemost,IamthankfultomyresearchadvisorProf.A.F.Hebardforgivingmetheopportunitytoworkwithhim.Ithasbeenagreatexperiencetoworkunderhissupervision.Hispositive,open-mindedattitudetowardresearchcreatesauniquelaboratoryenvironmentfullofencouragement.Ihavelearnedalotfromhisunadulteratedenthusiasm,willingnesstolearnandelegantbutsimpleapproachtounderstandingfundamentalphysics.Iwouldliketothankallthepresentandformerlabmembersfortheirhelpsandpleasantcompany.IamgratefultoJohnJ.KellyforteachingmemanyexperimentaltechniqueswhenIjoinedthegroup.ThankstoallthelabmembersPatrick,Rajiv,Sef,Siddhartha,Sanal,Xiaochangfortheirhelps.Ireallyenjoyedworkingwithyouguys.Iwouldalsoliketoacknowledgethestasofmachineshopandelectricshop.Speciallycryogenicstas,GregandJohn,fortheirconstantsupplyofliquidHeandN2allyeararound24/7.ThankstoJay(reallyaniceguy)forlookingafterallthepumpsandchillers.Iwouldliketothankallofmycommitteemembers.IwillspeciallythanksProf.AmlanBiswas.ThoughIdidnothavechancetocollaboratewithhim,buthisguidanceandsupporttowardsmydegreehavebeenveryhelpful.IamalsogreatlythankfultoProf.D.Nortonforthewonderfulcollaborationandforlettingmeusehislabfacilities.IamthankfultomycollaboratorsD.KumarandA.GuptafromNCA&T.IamalsoverythankfultoMatt,Patrick,Kyeong-WonfromProf.Norton'slabfortheirhelpsandbeinggoodfriends. 4

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Iamindebtedtomyparentsfortheirsupport,encouragementandforalwaysbelievinginme.IappreciatethewarmthandaectionofmysisterMridula.Icouldnothavecomethisfarwithouttheirblessings. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1THEORYANDBACKGROUND .......................... 12 1.1SingleDomainandMultiDomainParticles .................. 12 1.2HysteresisLoopofSingleDomainCoherentlyRotatingParticles ...... 14 1.3HysteresisLoopofMultiDomainParticles .................. 24 1.4Magnetizationvs.Temperature ........................ 29 1.4.1ZeroFieldCooled(ZFC)Magnetization ................ 30 1.4.2FieldCooled(FC)Magnetization ................... 31 2DIPOLARINTERACTIONSANDTHEIRINFLUENCEONTHECRITICALSINGLEDOMAINGRAINSIZEOFNIINLAYEREDNi/Al2O3COMPOSITES 32 2.1Abstract ..................................... 32 2.2Introduction ................................... 32 2.3ExperimentalDetails .............................. 33 2.4DataandDiscussion .............................. 36 2.5Conclusion .................................... 40 2.6Methods ..................................... 41 2.6.1MathematicalAnalysis ......................... 41 2.6.2BasicPhysicalUnderstanding ..................... 41 3EFFECTOFDIPOLARINTERACTIONONTHECOERCIVEFIELDOFMAGNETICNANOPARTICLES:EVIDENCEFORCOLLECTIVEDYNAMICS 42 3.1Abstract ..................................... 42 3.2Introduction ................................... 42 3.3ResultsandDiscussions ............................ 44 3.4Conclusions ................................... 49 4FINITESIZEEFFECTSWITHVARIABLERANGEEXCHANGECOUPLINGINTHIN-FILMPd/Fe/PdTRILAYERS ...................... 51 4.1Abstract ..................................... 51 4.2Introduction ................................... 51 4.3ExperimentalDetails .............................. 52 6

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4.4ResultsandDiscussion ............................. 54 4.5Conclusions ................................... 58 5TEMPERATUREDEPENDENCEOFCOERCIVITYINMULTIDOMAINNINANOPARTICLES,EVIDENCEOFSTRONGDOMAINWALLPINNING 59 5.1Abstract ..................................... 59 5.2Introduction ................................... 59 5.3ResultsandDiscussions ............................ 60 5.4RelationBetweenMicromagneticParameterandMagneticParameters .. 65 5.5Conclusions ................................... 66 6COERCIVEFIELDOFFETHINFILMSASTHEFUNCTIONOFTEMPERATUREANDFILMTHICKNESS:EVIDENCEOFNEELDISPERSEFIELDTHEORYOFMAGNETICDOMAINS ............................. 68 6.1Abstract ..................................... 68 6.2Introduction ................................... 68 6.3ExperimentalDetails .............................. 69 6.4ResultsandDiscussion ............................. 70 6.5Conclusion .................................... 74 7SCALINGCOLLAPSEOFTHEIRREVERSIBLEMAGNETIZATIONOFFERROMAGNETICTHINFILMS ......................... 75 7.1Abstract ..................................... 75 7.2Introduction ................................... 75 7.3ExperimentalResults .............................. 76 7.4Conclusions ................................... 84 7.5Methods ..................................... 85 7.5.1NiNanoparticle. ............................. 85 7.5.2GdThinFilm. .............................. 86 7.5.3(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)ThinFilms. .......... 86 7.5.4TemperatureCorrectionofCoerciveField. .............. 86 REFERENCES ....................................... 87 BIOGRAPHICALSKETCH ................................ 96 7

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LISTOFTABLES Table page 1-1Hcvs.T ........................................ 29 8

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LISTOFFIGURES Figure page 1-1SDandMDparticle ................................. 13 1-2Coherentandincoherentrotation .......................... 14 1-3Singleparticleinmagneticeld ........................... 15 1-4Twostateenergy ................................... 16 1-5HysteresisofSDparticle ............................... 18 1-6Diagramofaparticle ................................. 20 1-7Thermalaverageofmagnetization .......................... 21 1-8Flowdiagram ..................................... 21 1-9MHbelowTB ..................................... 24 1-10MHbelowTB ..................................... 25 1-11SDtoMDtransitionandHc ............................. 26 1-12MagnetizationloopforMDparticle ......................... 27 1-13DomainwallandHc ................................. 28 1-14Mvs.Tfor3nmNinanoparticles ......................... 30 2-1STEMimageofNiparticle .............................. 34 2-2Hcvs.d,dierentT ................................. 35 2-3dcvs.T ........................................ 37 2-4Hdanddomain .................................... 41 3-1Sample ......................................... 45 3-2MHloop ........................................ 46 3-3Hcvs.d:dipolarinteraction ............................. 47 3-4Dipolarinteraction .................................. 49 4-1Physicalandmagneticviewofsample ....................... 53 4-2Saturationmagnetizationvs.x ........................... 54 4-3Coerciveeldvs.x .................................. 55 9

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4-4Curietemperaturevs.x ............................... 56 5-1Threesetsofsample ................................. 61 5-2MHloopsofset2 ................................... 62 5-3Hcvs.T2=3set1samples .............................. 63 5-4Hcvs.T2=3set2samples .............................. 64 5-5Hcvs.T2=3set3samples .............................. 65 5-6Hc0andE0ofset2 .................................. 66 6-1TEMimageofFelm ................................ 69 6-2M-HloopofFelm .................................. 70 6-3Hcvs.TofFelm .................................. 71 6-4Hcvs.KofFelm .................................. 72 6-5Hcvs.dofFelm .................................. 74 7-1IrreversibleMagnetization .............................. 77 7-2BehavioroftheM(H;T)isothermsasthefunctionofHandscalingcollapse 80 7-3Theanstz ....................................... 81 7-4Scalingcollapseofvarietyofferromagneticmaterials ............... 83 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMAGNETISMINNANOSCALEMATERIALS,EFFECTOFFINITESIZEANDDIPOLARINTERACTIONSByRiteshKumarDasAugust2010Chair:A.F.HebardMajor:Physics Materialphysicsisalwaysmotivatedbythematerialswithexoticproperties.Itwasacommonbeliefthatexoticpropertiesareonlyassociatedwithexoticmaterials.Nowitisclearthatgeometricalconnementatnanoscaledimensionscangiverisetoexoticpropertieseveninsimplematerials.Ferromagneticmaterialsinrestricteddimensionsareextremelyinterestingbecauseoftheirpotentialapplicationsaswellastherichfundamentalscienceinvolved.Magneticnanoparticlesareusefulinhighdensitymagneticdatastoragedevices,sensors,contrastagentsinMRI,drugdelivery,treatinghyperthemiaandmanymore.Alltheapplicationsofnanomagnetsareverycrucialinmoderndaylife.Butmostoftheapplicationsarerestrictedduetothelimitationsinthefundamentalpropertiesarisesinnanoscaleandalsoduetothetechnicallimitationsofcontrollingthingsatnanoscale.Forexampleparticlesbecomesuperparamagneticasthesizeisreducedbelowacertainvalueandthemagnetizationdirectionuctuatesrandomlyduetothethermalenergywhichlimitsthedensityofdatastorage.Thepromisesofnanomagnetsarehugeandtoreallyachievethegrandchallengesinnanomagnetism,itisnecessarytounderstandthebasicsciencesinvolveatsmallscales.Inthispresentwork,themagneticpropertiesofsystemsinnanoscale(nanoparticlesandthinlms)havebeeninvestigated.Theeectofdipolarinteractions,particlesize,particlesizedistribution,temperature,magneticeldetc.onthemagneticpropertieshavebeenstudied. 11

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CHAPTER1THEORYANDBACKGROUND Ferromagnetismisknownformorethan2500yearstoman.Therstmagneticmaterialdiscoveredwasmagnetite(Fe2O3).Thepracticalapplicationsofferromagnetswasrecognizedfromveryancienttime.Therstuseofmagneticmaterialwasasacompass.Accordingtothemagneticproperties,materialscanbedividedintodiamagnetic,paramagnetic,ferromagnetic,antiferromagnetic,ferrimagnetic,spinglassetc.Inthispresentworkferromagnetismwillbethemaintopic.Chaptersfollowedbythischapterwilldiscusstheeectofnitesizeanddipolarinteractionsonthemagneticpropertiesofsomematerialswithnanoscalestructures.Inthischapterageneraltheoreticalbackgroundwillbegiven. 1.1SingleDomainandMultiDomainParticles Whenthesizeoftheparticleisverysmall,itwillcontainonlyonemagneticdomain. 1 { 3 Thisisbecausetheenergyrequiredtoformadomainislargerthanthemagnetostaticselfenergy.Themagnetostaticselfenergyforasphericalparticleisgivenby 1 Emag=0M2sV=12(1{1) where0isthefreespacepermeability,MsisthesaturationmagnetizationandVisthevolumeoftheparticle.TheenergyrequiredtoformaBlochdomainwallis 1 { 3 Edw=4p AKR2(1{2) whereKistheanisotropyconstant,AistheexchangestinessandRistheradiusoftheparticle. NotethatEmaggrowsasR3andEdwgrowsasR2.Domainformationisthereforefavorableforlargerparticlesasthemagnetostaticenergywillbelargecomparedtothedomainwallformationenergy.Thecriticalsingledomainradius(Rsd)wherethe 12

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transitionfromsingledomain(SD)tomultidomain(MD)occursisgivenby 1 { 3 Rsd=36p AK 0M2s(1{3) TheaboveequationisdeterminedbysolvingtheequationEmag=Edw. 1 { 3 ThusparticleshavingradiussmallerthanRsdareSDandparticleshavingradiuslargerthanRsdareMD(seeFig. 1-1 ). Figure1-1. Smaller(larger)particlesareSD(MD)asthemagnetostaticselfenergyissmaller(higher)thantheenergyrequiredtoformdomain.ThecriticalsizeoftheparticlewherethesingledomaintomultidomaintransitionoccursisgivenbyEq. 1{3 ThemagnetizationdynamicsforSDandMDparticlesaredramaticallydierent.SDparticlesreversetheirmagnetizationbyrotationonly.MDparticlesreversetheirmagnetizationbydomainwallmotionandrotation.RotationofmagnetizationfortheSDparticlesismainlyoftwotypes:1)coherentand2)incoherent(Fig. 1-2 ).Theexchangelength 2 { 4 lex=s A 0M2s(1{4) isameasureofthedistanceoverwhichtheatomicexchangeinteractionsdominateandallthespinsrotatecoherently.Particleswithsizelarger(smaller)thanlexrotateincoherently(coherently).TheexchangelengthisusuallylargerthanRsdforsoftferromagnetswheretheanisotropyenergyissmall.Thusinsoftferromagnetsmagnetizationreversaloccurseitherbycoherentrotation(smallparticles)orbydomainwallmotion(largeparticles). 13

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Figure1-2. Coherentandincoherentrotationofthemagnetization.Incaseofcoherentrotationallthespinsrotatetogetherandthewholeparticlecanbeconsideredasagiantspin.CoherentrotationhappensforSDparticleswithsizesmallerthantheexchangelengthlex 1.2HysteresisLoopofSingleDomainCoherentlyRotatingParticles ThemagnetizationdynamicsoftheSDparticleswithR
PAGE 15

Figure1-3. ASDparticleinanappliedmagnetic(H)eldalongtheeasyaxisofmagnetization(k).istheanglebetweenthemagnetizationMandtheeasyaxisk. oftheenergyoccursat==2separatingthetwoenergyminima.Figure 1-4 a)showstheenergydiagramatzeromagneticeldasafunctionof.Inthiscasetheparticlewillhavemagnetizationparalleltotheeasyaxisofmagnetizationsincethesecorrespondtominimumenergystates(upordown).Anyotherdirectionswillcostsomeanisotropyenergy.ThetwostateswithminimumenergyareseparatedbytheanisotropyenergybarrierequaltoKV.InanappliedmagneticeldalongtheeasyaxisthetwoenergyminimawillbeshiftedduetotheZeemanenergy(Fig. 1-4 b)).NowthestatealongthemagneticeldwillbemoststableastheenergyisloweredduetotheZeemanterm.Thestatewithoppositedirectionofmagnetizationwillbemetastable.Themagneticelddependentenergybarrierforthespinup(E+(H))anddown(E)]TJ /F1 11.955 Tf 7.08 1.79 Td[((H))stateiscalculatedbyStonerandWohlfarthtobe, 6 E(H)=KV1H Hco2(1{6) whereE+(H)istheenergybarrierseenbytheupmagnetizedparticlesandE)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)istheenergybarrierseenbythedownmagnetizedparticlesandHc0=2K=Ms.DerivationoftheEq. 1{6 isgivenbelow. 15

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Figure1-4. TwostateenergyofaSDparticle.Twoenergyminimacorrespondtothedirectionoftheeasyaxisofmagnetization.a)Atzeromagneticeldtheparticlewillhavemagnetizationalongtheeasyaxisofmagnetizationasthosecorrespondtominimumenergystates(upanddown).UpanddownstatesareseparatedbytheenergybarrierequaltoKV.Toreversethemagnetizationdirectionfromuptodownorviceversathesystemhastoovercometheenergybarrier. 5 BrownproposedthatthisprocessrequiresanitetimegivenbyEq. 1{14 5 b)Inanappliedmagneticeldalongtheeasyaxis,thetwoenergyminimawillbeshiftedduetotheZeemanenergy.Nowtheupstatewhichisalongtheappliedmagneticeldwillbemoststableandhavethelowestenergy.Thestatewithoppositedirection(downstate)ofmagnetizationwillbemetastable.ThemagneticelddependentenergybarrierforthespindownstateiscalculatedbyStoner-Wohlfarth(Eq. 1{6 ) 6 FirstorderderivativeofEq. 1{5 withrespecttois E(H) =2KVsincos+MsVHsin(1{7) Atthemaximaandminima 2KVsincos+MsVHsin=0(1{8) Solutionsoftheaboveequationare sin=0(1{9) cos=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(MsH 2K(1{10) 16

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TakingthesecondorderderivativeofEq. 1{5 ,itcanbeshownthattheEq. 1{9 (Eq. 1{10 )referstominima(maximum)oftheenergy.Thustheenergyminimaareat=0and=andmaximumatwhencos=-MsH=2K(seeFig. 1-4 ).Energiescorrespondtotheseextremaare Emin+=)]TJ /F6 11.955 Tf 9.3 0 Td[(MsVH (1{11) Emin)]TJ /F1 11.955 Tf 16.71 2.87 Td[(=MsVH (1{12) Emax=KV"1+MsH 2K2# (1{13) whereEmin+andEmin)]TJ /F1 11.955 Tf 10.66 2.87 Td[(correspondsto=0(spinup)and=(spindown)respectively.ItisnoweasytoshowthatE+(H)=Emax-Emin+andE)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)=Emax-Emin)]TJ /F1 11.955 Tf 10.66 2.87 Td[(aregivenbyEq. 1{6 Theenergybarrierhastobeovercometoreversethemagnetizationdirectionfromuptodownorviceversa.Brownproposedthatthisprocessrequiresanitetime 5 =0expE(H) kBT(1{14) whereTisthetemperature,)]TJ /F3 7.97 Tf 6.58 0 Td[(10istheinverseattemptfrequencyofovercomingtheenergybarrierandkBistheBoltzmannconstant. Figure 1-5 showsthemagnetizationprocesswhenthemagneticeldissweptfromalargepositivevaluetoalargenegativevalueandagainfromanegativetopositivevaluetocompletethemagnetizationloop.Whenmagneticeldislarge(scenario1)allparticleswillbemagnetizedalongthemagneticeldandapositivesaturationmagnetizationisachieved.Asmagneticeldisreducedtozero(scenario2)themagnetizationdirectionwillbetrappedintheupdirectionasthetemperatureisnotenoughtoovercometheenergybarrier.Nowasthemagneticeldisreversedtheenergybarrier,E)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)willbereducedaccordingtoEq. 1{6 (scenario3).Butstillthetemperatureisnotenoughtoovercometheenergybarrierandthemagnetizationwillstillbetrappedwithapositivevalue.Afurtherincreaseinmagneticeldintheoppositedirectionwillkeeploweringtheenergy 17

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barrieruntil,atthecoerciveeld,theenergybarriercanbeovercomebythethermalenergyandmagnetizationreversalwilloccur(scenario4).WhenH=Hctheenergy Figure1-5. HysteresisofacoherentlyrotatingSDparticle.Scenario1)Highpositivemagneticeldisappliedandsaturationmagnetizationisobserved.Scenario2)Magneticeldisreducedfrompositivevaluetozero.Magnetizationistrappedinthepositivedirectionasthethermalenergyisnotenoughtoovercometheenergybarrier.Scenario3)Magneticelddirectionisreversed.Stilltheenergybarrierislargecomparedtothethermalenergyandmagnetizationistrappedinthepositivedirection.Scenario4)Magneticeldequalstothecoerciveeld.Nowtheenergybarriercanbeovercomebythermalenergyandmagnetizationreversaloccurs. barrierE)]TJ /F1 11.955 Tf 7.09 1.79 Td[((Hc)issuchthattherelaxationtime)]TJ /F1 11.955 Tf 10.99 1.79 Td[(=matthetemperatureT.Wheremistheexperimentalmeasurementtime(around100secforSQUIDmeasurement).ThuswhenH=Hc,magnetizationreversaloccurs.CombiningEq. 1{6 and 1{14 ,thecoercivity(Hc(T))oftheSDparticlecanbecalculatedasshownbelow. )]TJ /F1 11.955 Tf 17.05 1.79 Td[(=m=0expE)]TJ /F1 11.955 Tf 7.08 1.79 Td[((Hc) kBT)E)]TJ /F1 11.955 Tf 7.09 1.79 Td[((Hc)=kBTlnm 0 18

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NowusingtheexpressionofE)]TJ /F1 11.955 Tf 7.09 1.79 Td[((H)fromEq. 1{6 itiseasytoshowthat 6 Hc=Hc0"1)]TJ /F8 11.955 Tf 11.96 16.86 Td[(T TB1 2#(1{15) whereTB=KV=kBln(m=0)isknownastheblockingtemperature.BelowTBtheanisotropicenergybarrierislargerthanthethermalenergyandmagnetizationisblockedortrapped.AboveTBtheanisotropicenergybarriercanbeovercomeeasilybythermalenergyandtheparticlesarecalledsuperparamagneticaswillbediscussedlater.ItisclearfromtheEq. 1{15 thatHcdecreaseswithincreasingtemperatureandabovetheblockingtemperature(T>TB)theparticleslosetheircoerciveeld.NotethattheoriginofHcinaSDparticleisthenitetimerequiredtoreversethemagnetizationdirectionovertheanisotropyenergybarrier. Thepreviousdiscussionisonlytrueforanassemblyofuniformsizeparticlesthathaveeasyaxisofmagnetizationorientedalongthesamedirectionwithmagneticeldappliedalongtheeasyaxis.Inrealsamplesthisisnotthecasesincetheeasyaxisofmagnetizationisusuallyrandomlyorientedandtheparticlesizeisnotuniform.AmoregeneralcaseisshowninFig. 1-6 below.Hereaarbitraryanglebetweenmagneticeldandtheeasyaxisofmagnetization( )isconsidered.Theenergyoftheparticleinthiscaseis 7 E(H)=2KV1 2sin2( )]TJ /F6 11.955 Tf 11.95 0 Td[())]TJ /F6 11.955 Tf 16.89 8.09 Td[(H Hc0cos(1{16) ThemagnetizationoftheparticleatanappliedmagneticeldisgivenbyMcosmin,whereministheanglecorrespondstotheminimaofE(H).Notethatherewehavenotconsideredtheeectoftemperatureonthemagnetization.AtnitetemperatureothervaluesaroundtheminwillbeoccupiedwithaniteprobabilityaccordingtotheBoltzmannfactorasshowninFig. 1-7 below.ThustheaverageoveralltheoccupieddirectionwiththeoccupationprobabilitygivenbytheBoltzmannfactorwillbethethermalaverageofthemagnetizationforaxedvalueofHand .TheprocedureshouldberepeatedforallvaluesofHtogettheM-Hloopforaparticularvalueof .Thenthe 19

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Figure1-6. Singleparticleinanappliedmagneticeld(H).kisthedirectionoftheeasyaxisofmagnetization.MandHarethemagnetizationandmagneticeldvectorsrespectively.WithoutloosinganygeneralityMandHcanbeconsideredinthesameplane.TheanglebetweenMandHis.TheanglebetweenHandkis .TheenergybarrierforthisgeneralcongurationisgivenbyEq. 1{16 7 M-Hloopsforallpossible shouldbecalculated.AveragingoveralltheseM-HloopswillgiveamagnetizationloopattemperatureTforasampleofuniformparticlesizeandarandomly-orientedeasyaxisofmagnetization.Alltheaboveproceduresshouldbedoneforallpossibleparticlesizesastherealsamplesusuallyhavesomeparticlesizedistribution.Theprobabilityofaparticularparticlesizecanbemodeledeitherasalognormalorgaussiandistributionfunction.Inthiswaythemagnetizationloopofarealsamplewithnonuniformparticlesizeandrandomorientationoftheeasyaxisofmagnetizationcanbedetermined.Ifthealltheaboveproceduresarerepeatedfordierenttemperaturesthenthemagnetizationloopatdierenttemperaturescanbedetermined.Belowweshowaowdiagramfortheaboveprocess. 7 { 9 20

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Figure1-7. AtnitetemperatureothervaluesaroundminwillbeoccupiedwithaniteprobabilityaccordingtotheBoltzmannfactorandshownbytheshadedregion.Thethermalaverageofthemagnetizationwillbetheaverageoftheallmagnetizationdirectionsoverthisshadedregion. 7 TheprobabilityofhavingsomemagnetizationdirectionwillbedeterminedbytheBoltzmannfactor. 1 Startwiththeenergyofthesingleparticle. E(H)=2KV1 2sin2( )]TJ /F6 11.955 Tf 11.96 0 Td[())]TJ /F5 7.97 Tf 16.59 4.7 Td[(H Hc0cos + 2 FindtheminimaofE(H) MsVCosminwillbethemagnetizationatT=0forthegivenvalueofH, andV Figure1-8. Flowdiagramtoshowtheprocessofcalculatingcoerciveeldforrealnanoparticlesampleswithparticlesizedistributionandrandomorientationoftheeasyaxisofmagnetizationsatnitetemperature. 21

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+ 3 Thermalaverageofthemagnetization hM(H; ;V)iT=R21MsVcosexp)]TJ /F11 5.978 Tf 8.92 2.35 Td[(E kBTd R21exp)]TJ /F11 5.978 Tf 8.92 2.34 Td[(E kBTd E=E)]TJ /F6 11.955 Tf 11.96 0 Td[(Emin1and2areshowninFig. 1-7 hM(H; ;V)iTisthemagnetizationattemperature=TforthegivenvalueofH, andV + 4 Step1,2and3shouldberepeatedfordierentH.ThiswilldeterminetheM-HloopforagivenvalueofT, andV + 5 Step1,2,3and4shouldberepeatedforallpossible andaverageofallthoseloopswilldeterminetheM-HloopforagivenvalueofTandVforanensembleofparticleswithrandomorientationoftheeasyaxisofmagnetization. + Figure1-8. continued 22

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6 Step1,2,3,4and5shouldberepeatedfordierentparticlesizetodeterminetheM-HloopforagivenvalueofTforasampleconsistingofnonuniformparticlesizeandrandomorientationoftheeasyaxisofmagnetization.Inrealsamplestheparticledistributionfunctionisusuallylognormalorgaussian. 9 10 + 7 Step1,2,3,4,5and6shouldberepeatedfordierentTtodeterminethetemperaturedependenceoftheM-Hloop. 7 + 8 Completionofstep7willprovideanopportunitytodeterminethetemperaturedependenceofthecoerciveeld,remanentmagnetizationetc.SomeofthetemperaturedependentofcoerciveeldsarelistedinTable 1-1 Figure1-8. continued MagnetizationloopsatdierenttemperaturesforasinglelayersampleofNinanoparticlesofaveragediameteraround18nmareshowninFig. 1-9 .Thecoerciveeldisdeterminedbythemagneticeldwheremagnetizationchangessignandpassesthroughzero.ItisclearfromtheFig. 1-9 thatcoerciveeldsdecreaseswithincreasingtemperatureasdiscussedabove.AttemperatureshighcomparedtotheanisotropyenergyKV,themagnetizationdirectionscanrotatefreelyoverthebarrierandtheparticlesbecomesuperparamagneticwithHc=0.InthiscasethesystemcanbetreatedsimilartothecaseofparamagnetismwitheachparticleasagiantorsuperspinofvalueMsV(thuscalled 23

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Figure1-9. HysteresisloopofaSinglelayerNinanoparticlesof18nmdiameterembeddedinanAl2O3matrixattemperature,T
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Figure1-10. HysteresisloopofaSDcoherentlyrotatingparticleattemperature(T>TB).Sampleshowszerocoerciveeldasexpectedforsuperparamagneticparticles.NotetheH=Tabscissa.MagnetizationisplottedasafunctionofH=Tforthreedierenttemperaturesasindicatedinthelegend.LoopsatalldierenttemperaturesfallontopofeachotheraspredictedbytheLangevinequationforsuperparamagneticparticles. functionofparticlesize.Forverysmallparticlesthecoerciveeldiszeroandparticlesaresuperparamagnetic(SP)withmagnetizationdeterminedbytheLangevinfunction.Astheparticlesizeisincreased,thecoerciveeldincreasesduetothefactthattheenergybarrierincreases.Particleswithsizelargerthanthecriticalsingledomainradiusaremultidomainandthecoerciveelddecreaseswithincreasingparticlesize. 2 3 9 Thismaybeduetothefactthatasparticlesizeincreasesthenumberofdomainsincreasesandthusitiseasiertohavedomainclosurewhichdecreasescoercivitybecausethereislesstotalmagnetization. ThesizedependenceofthecoercivityinMDregionisexperimentallyfoundtobe 2 HcMD=a+b=dx(1{18) wherea,bareconstantsthatdependontherealstructurefactorandmaterials,disthediameteroftheparticlesandxhasvaluearound1. 2 Thereisnotheoreticalmodelthat 25

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Figure1-11. Coerciveeldplottedasafunctionofparticlediameter.Forverysmallparticlesthecoerciveeldiszeroandtheparticlesareknownassuperparamagneticparticles(SP).Astheparticlesizeisincreasedthecoerciveeldincreasesduetothefactthattheenergybarrierincreases.Particleswithsizelargerthanthecriticalsingledomainradiusaremultidomainandcoerciveelddecreaseswithincreasingparticlesize. explainsthebehaviorinEq. 1{18 .ThusthethepeakinthecoerciveeldwhenplottedasthefunctionoftheparticlesizedelineatestheSDandMDbehavior.InexperimentwehavefoundthesamebehaviorforbothmultilayerandsinglelayersamplesofNiparticlesinAl2O3matrixaswillbediscussedindetailinchapter 2 Figure 1-12 showsthepossibledomainwallcongurationfordierentpointsinthemagnetizationloop.RememberthatcomparedtotheSDcasewheretheoriginofthehysteresiswasthehoppingoveraenergybarrier,incaseofMDtheoriginofhysteresisisirreversibledomainwallmotion.Atveryhighmagneticeldallthespinsinthesystemwillbealignedalongthemagneticeldandpositivesaturation(Ms)willbeachieved(Fig. 1-12 ).Asthemagneticeldisreducedtozeroadomainwallwillbeformed.Duetotheimperfectionsinthesample,thedomainwallwillbestuckinapositionsuchthattheupdomainislargerthanthedowndomainandnetmagnetizationorremanent 26

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magnetization(Mr)willbeseenatzeromagneticeld.Reversingthemagneticeldwill Figure1-12. HysteresisloopofaMDsystemandpossibledomainwallconguration.Atverylargepositivemagneticeldallthespinsarealignedalongthemagneticeldandsaturationmagnetizationisachieved.Whenmagneticeldisreducedtozero,adomainwallforms.Duetotheimperfectionsinthesample,thedomainwallwillbestuckinapositionsuchthattheupdomainislargerthanthedowndomainandremanentmagnetizationismeasured.Ifthedirectionofthemagneticeldisreversedthedomainwallwillstarttomovetotherightandthedowndomainwillgrow.Atamagneticeldequaltothecoerciveeld,thedownandupdomainwillbeequalinsizeandmagnetizationwillbezero.Foralargenegativemagneticeldthedomainwallbemovedtotherightandallthespinswillbeinthedirectionofthemagneticeldandnegativesaturationwillbereached. movethedomainwalltotherightsideandthusthedowndomainwillstarttogrowandmagnetizationwillbereduced.Whenthenegativemagneticeldisequalstothecoerciveeldtheupanddowndomainwillhavesamesizeandmagnetizationwillbezero.Furtherincreaseinmagneticeldinthenegativedirectionwillforcethedomainwalltomoveallthewaytorightmakingallspinsalignedalongthemagneticeldandnegativesaturationwillbereached. 27

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ToderivethecoerciveeldinMDdomaincaseconsiderasimplecase,asshowninFig. 1-13 ,whereasingledomainwallseparatestwodomains.Therighthandsideisaspinupdomainandlefthandsideisaspindowndomain.Inanappliedmagneticeld,H, Figure1-13. Singledomainwallseparatingtwomagneticdomains.Righthandsideisaspinupdomainandlefthandsideisaspindowndomain.InanappliedmagneticeldduetotheZeemanenergythedomainwallwillexperienceapressureandsomeworkneedtobedonetomovethewallagainstthispressure.TheoriginofhysteresisinMDsampleistheirreversiblemotionofthedomainwall. alongthespinupdomain,theZeemanenergyoftheup(down)domainwillbe)]TJ /F6 11.955 Tf 9.3 0 Td[(MsH(+MsH)perunitvolume.Thustheenergydierenceacrossthedomainwallwillbe)]TJ /F1 11.955 Tf 9.3 0 Td[(2HMSperunitvolume.Thisenergydierencecanbeconsideredasapressureonthewallandsomeworkhastobedonetomovethedomainwallagainstthispressure.Theworkdonetomovethewalladistancedxis 2 11 dW=2MsHSdx(1{19) whereStheareaofthedomainwall.Thustheworkdonetomovethewallbyunitdistanceis 2 11 dW=dx=2MsHS(1{20) 28

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wheredw=dxcanbethoughtofastheresistanceofthedomainwallmotion.Inrealsamplesduetotheimpurities,imperfections,strainsetc,dW=dxpassesthroughmaximaandminima.Thewallmotionoverthesemaximaandminimaisirreversibleinmagneticeldandthatistheoriginofthehysteresis.Thecoerciveeld,themeasureofirreversibility,isusuallygivenby 2 11 Hc=1 2MsS(dW=dx)max(1{21) Therearedierenttheoreticalmodelstocalculate(dW=dx)maxfordierentimperfectionsinthesampleandtheresultsforsomeofthemarelistedinTable 1-1 Table1-1. Tableherelistssomeknownmodelsalongwiththevariationofcoerciveeldaccordingtothemodel. TheoryHc SystemReferences Stoner-WohlfarthHc=2K Ms[1)]TJ /F11 5.978 Tf 7.45 0 Td[((kBTlnm 0=KV)1=2] SD,CRnanoparticlewithuniaxialanisotropyalongtheappliedmagneticeld 2 6 Stoner-WohlfarthHc=0:96K Ms[1)]TJ /F11 5.978 Tf 7.45 0 Td[((kBTlnm 0=KV)3=4] SD,CRnanoparticlewithuniaxialanisotropyrandomlyoriented 7 MicromagneticHc=2K Ms)]TJ /F11 5.978 Tf 7.45 0 Td[([3p ja3j 4kBTlnm 0]2=3 MD,2phasematerial,hardmagnet,a3isthemicromagneticparameteranddependsontheK,Ms,A 12 InclusionTheoryHc=2=3 Msd MDsystem,d<,freepoleenergyisignored,coercivityisassumedtobeequaltothemaximumpiningeld,disthediameteroftheinclusion,isthedomainwallthickness,isthedomainwallenergyperunitarea,isthevolumefractionoftheinclusion 11 13 InclusionTheoryHc=1:751=2 MsL(ln2L d) MDsystem,d>,freepoleenergyisignored,Listhelineardimensionofthesample 11 14 InclusionTheoryHc=2:81=2 MsL(d )3=2(ln2L ) MDsystem,d<,freepoleenergyisig-nored 11 14 InclusionTheoryHc=2K Ms[0.386+logr 2M2s K] MDsystem,d<,freepoleenergyiscon-sidered 15 InclusionTheoryHc=3l=Msd2 MDsystem,closerdomain,largeinclusion,commonlyseeninthecaseofNeel'sspike,listheequilibriumlengthofthespike 16 1.4Magnetizationvs.Temperature Untilnowwehavebeendiscussingthebehaviorofmagnetizationasafunctionofmagneticeldataxedtemperature.Nowwewilldiscusshowmagnetizationchangeswiththetemperatureataxedmagneticeld.Atsmallappliedmagneticeld,spinsaretrappedinmetastableenergyminimaseparatedbyenergybarriersfromtheglobal 29

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minima.Asthetemperatureisincreasedthespinscanhopovertheenergybarriertoreachtheglobalminima.Duetothistrappingofspinsinlocalminima,magnetizationvaluesdependsstronglyonthecoolingprotocol.Therearemainlytwodierentcoolingprotocols,eldcooled(FC)andzeroeldcooled(ZFC).ThebehaviorofmagnetizationasafunctionoftemperatureforthetwoprotocolsisshowninFig. 1-14 belowforthesampleofNinanoparticlesof3nmdiameteratanappliedeldof20Oe.ThetemperaturewherethedierencebetweenFCandZFCdisappearsisgenerallycalledtheirreversibletemperature(Tirr).FornanoparticlesTirrissameastheblockingtemperature(TB). 2 Figure1-14. Magnetizationvs.temperatureatanappliedmagneticeldof20Oeforthe3nmdiameterNinanoparticles.Theredcoloristheeldcooled(FC)magnetizationandtheblackoneisthezeroeldcooled(ZFC)magnetization. 1.4.1ZeroFieldCooled(ZFC)Magnetization Zeroeldcooledmagnetizationismeasuredbycoolingthesamplefromhightemperature(temperatureabovetheirreversibletemperature(Tirr))withoutanyappliedmagneticeld.Atlowtemperatureasmallmagneticeldisappliedandmagnetizationismeasuredasafunctionoftemperatureduringthewarmupwhilekeepingthemagneticeldon.HerewewilldiscusstheshapeoftheZFCmagnetizationinaqualitativemanner.Ingeneralthemagneticsystemcanbetreatedasatwo-stateproblemasshownpreviously 30

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inFig. 1-4 wherespinupanddowncorrespondtotheenergyminimaseparatedbysomeenergybarrier.TheoriginoftheenergybarrierintheSDcaseistheanisotropywhereasforthecaseofMDtheoriginisdomainwallpinningatdefects.Athightemperaturetheenergybarrieriseasilyovercomeduetothethermalenergyandthespinupanddownstateswillbeequallypopulated.ThusathightemperatureaboveTirr,magnetizationwillbezero.Nowifthesampleiscooledtoalowtemperaturewithoutanyappliedmagneticeld,thenzeromagnetizationstatewillbeblockedastheenergybarrierisnowlargecomparedtothethermalenergy. 2 3 6 Ifasmallmagneticeldisappliedthechangeinmagnetizationwilloccuronlyforthesmallenergybarriersthatcanbeovercomeatthattemperatureandasmallmagnetizationwillbeachieved.Astemperatureisincreased,theprobabilityofovercomingthelargerbarriersincreasesandmagnetizationincreases.AttemperatureTirrtheprobabilitiestoovercomethebarrierforspinupanddownbecomenearlyequalandthespinupanddownmixingstartstohappenandthusmagnetizationdecreaseswithfurtherincreaseintemperature. 1.4.2FieldCooled(FC)Magnetization Fieldcooledmagnetizationismeasuredbycoolingthesamplefromhightemperaturetothelowtemperatureinanappliedmagneticeldandmagnetizationismeasuredduringthewarmupprocess. 2 Inthiscaseathightemperatureduetotheappliedmagneticeld,thespinupstatesaremorepopulatedthanthespindownstates.Coolingthesampleatalowtemperaturewhilekeepingtheeldonwillthuslockthesysteminmagnetizedstate.Anincreaseintemperaturewillincreasetheprobabilityofspinupanddownmixingandthusmagnetizationwillgraduallydecrease. 31

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CHAPTER2DIPOLARINTERACTIONSANDTHEIRINFLUENCEONTHECRITICALSINGLEDOMAINGRAINSIZEOFNIINLAYEREDNI/AL2O3COMPOSITES 2.1Abstract PulsedlaserdepositionhasbeenusedtofabricateNi/Al2O3multilayercompositesinwhichNinanoparticleswithdiametersintherangeof3-60nmareembeddedaslayersinaninsulatingAl2O3host.Atxedtemperatures,thecoerciveeldsplottedasafunctionofparticlesizeshowwell-denedpeaks,whichdeneacriticalsizethatdelineatesacrossoverfromcoherentlyrotatingsingledomaintomultipledomainbehavior.Weobserveashiftinpeakpositiontohighergrainsizeastemperatureincreasesanddescribethisshiftwiththeorythattakesintoaccountthedecreasinginuenceofdipolarmagneticinteractionsfromthermallyinducedrandomorientationsofneighboringgrains. 2.2Introduction Themagneticpropertiesofnanoparticleshavebeenthefocusofmanyrecentexperimentalandtheoreticalstudies.Technologicalimprovementshavenowmadeitpossibletoreproduciblyfabricatenanomagneticparticleswithpreciseparticlesizeandinterparticledistances. 17 { 22 Thesecontrolledsystemshaveenabledstudyofthefundamentalpropertiesofsingleaswellasinteractingparticles.Mostapplicationsrequirethattheparticlesbesingledomainwithauniformmagnetizationthatremainsstablewithasucientlylargeanisotropyenergytoovercomethermaluctuations, 23 whichestablishesatemperature-dependentlowerboundtotheparticlesize.Theseconsiderationsmusttakeintoaccounttheeectofinteractionsonmagneticpropertiesasisevidentforhigh-densityrecordingmedia 24 whereparticlesareveryclosetoeachother.Considerableinsighthasalreadybeengainedfromexperimentalstudiesoftheeectofdipolarinteractiononsuperparamagneticrelaxationtime 25 { 34 andblockingtemperature. 29 Lessunderstoodhoweveristheeectofdipolarinteractionsontheestablishmentofanupperboundtoparticlesize,whichdenesthecrossoverfromsingledomain(SD)tomultidomain(MD)behavior.InthefollowingweshowusingcoercivitymeasurementsonNi/Al2O3composites 32

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thatwithincreasingtemperaturethisupperboundtoparticlesizeincreasesandthensaturatesduetoattenuateddipolarinteractionsfromthermallyinducedcoherentmotionsofthemagnetizationoftheneighboringrandomlyorientedparticles. 2.3ExperimentalDetails ThecompositesystemstudiedinthispapercompriseselongatedandpolycrystallineNiparticleswithdiametersintherangeof3-60nmembeddedaslayersinaninsulatingAl2O3host.ThemultilayersampleswerefabricatedonSi(100)orsapphire(c-axis)substratesusingpulsedlaserdepositionfromaluminaandnickeltargets.HighpuritytargetsofNi(99.99%)andAl2O3(99.99%)werealternatelyablatedfordeposition.Beforedeposition,thesubstrateswereultrasonicallydegreasedandcleanedinacetoneandmethanoleachfor10minandthenetchedina49%hydrouoricacid(HF)solutiontoremovethesurfacesilicondioxidelayer,thusforminghydrogen-terminatedsurfaces. 35 Thebasepressureforallthedepositionswasoftheorderof10)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Aftersubstrateheating,thepressureincreasedtothe10)]TJ /F3 7.97 Tf 6.59 0 Td[(6Torrrange.Thesubstratetemperaturewaskeptatabout550oCduringgrowthoftheAl2O3andNilayers.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.59 0 Td[(2overaspotsize4mm1.5mm.A40nm-thickbuerlayerofAl2O3wasdepositedinitiallyontheSiorsapphiresubstratebeforethesequentialgrowthofNiandAl2O3.ThisprocedureresultsinaverysmoothstartingsurfaceforgrowthofNiasveriedbyhighresolutionscanningtransmissionelectronmicroscopystudies(Fig. 2-1 ).Multilayersampleswerepreparedhaving5layersofNinanoparticlesspacedfromeachotherby3nm-thickAl2O3layers.A3nm-thickcaplayerofAl2O3wasdepositedtoprotectthetopmostlayerofNinanoparticles. ShowninFig. 2-1 isacross-sectionalTEMimagefromamulti-layered(5layers)Ni-Al2O3samplegrownonc-planesapphire.TheNiparticleshaveasizeof235nminwidthand9nminheight.Theseparationbetweenneighboringparticlesisontheorderof3nm(measuredasaprojecteddistanceincross-sectionalview),whichiscomparabletothethicknessoftheAl2O3spacerlayers.Forthepurposesofthis 33

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Figure2-1. CrosssectionaldarkeldSTEMimageofa5-layerNi-Al2O3samplegrownonc-axissapphire experimentthegrainsized,asmeasuredbytheamountofNidepositedreferencedtoacalibratedstandard,representstheaveragesizeofthedisk-shapedgrainsshowninthegure.Thiscalibrationwasobtainedfromcross-sectionalTEMmicrographsofsinglelayersamples 36 bycomparingtheaveragegrainsizewithd.TheTEMobservationalsoshowsthattheAl2O3spacerlayersarepartiallycrystallized.DuetothelargesurfaceenergydierencebetweenNiandAl2O3,Niformswell-dened,separatedislandswithintheAl2O3matrix. 36 Previousstudiesonsimilarly-preparedsamplesusingatomicnumber(Z)contrastimaginginTEMtogetherwithelectronenergylossspectroscopy(EELS)haveconrmedtheabsenceofNiOattheNi/Al2O3interfaces. 36 TheNi/Al2O3interfaceswerechemicallyabruptwithoutanintermixingbetweenNi,Alandoxygen.Inadditionwedidnotobserveexchange-biasinducedasymmetricmagnetizationloops,thuslendingsupporttotheconclusionsofpreviousstudies 36 thatantiferromagneticNiOisabsentinourlayeredNi/Al2O3system. PreviousTEMstudiesonsinglelayersampleshaveshowntheparticlestobepolycrystalline.Forexample,athreenmparticlecomprisingthreecrystallinegrains 34

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hasbeenobserved. 36 Polycrystallineparticleswillthereforehavecrystallinegrainsorientedindierentdirections,thustendingtoaverageanynetcrystallineanisotropytozero.Accordingly,temperature-independentshapeanisotropyisdominantandtemperature-dependentcrystallineanisotropycanbeneglected.Inaddition,itisalsoimportanttonotethattheexchangelengthlex=14.6nmforNi, 37 whichisthelengthscalebelowwhichatomicexchangeinteractionsdominateovermagnetostaticelds,determinesthecriticalradii(Rcoh)forcoherentrotation:Rcoh5lexforsphericalparticlesandRcoh3:5lexfornanowires. 3 Theparticlesizes(1.5-30nminradius)thatwehaveinvestigatedarethussmallerthanthecriticalradiusbelowwhichcoherentrotationofNiprevails. Figure2-2. Coercivityfor5-layerNi/Al2O3multilayersamples(5repeatedunits)plottedasafunctionofparticlesize(diameter)atthetemperaturesindicatedinthelegend.Thepeakpositionsatd=dcforeachisotherm,indicatedbyverticalarrows,delineatethecrossoverfromsingledomain(SD)tomultipledomain(MD)behavior(d>dc).InsetshowsthebehaviorofHcasafunctionof1=dfortheparticleswithd>dcat10K.Thelineardependenceupto24nmdiameterparticleswithsaturationataconstantvalueforlargeparticles 38 isconsistentwiththebehaviorexpectedformultidomainparticles.Thusparticlesontheright-handsideofthepeakaremultidomain. InFig. 2-2 weshowplotsofHcasafunctionofparticlesizedateachofthetemperaturesindicatedinthelegend.Coerciveeldswereextractedfrommagnetization 35

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loopsmeasuredbyaQuantumDesignsuperconductingquantuminterferencedevice(SQUID)aftersubtractingoutthediamagneticcontributionfromthesubstrate.Magneticeldwasappliedalongtheplaneofthelms.Toobtainthemagnetizationloops,themagneticeldwasvariedoverthefullrange(5T)whilekeepingtemperaturexed.Thehighmagneticelddatashowlinearmagnetizationwithmagneticeld,whichisduetothediamagneticcontributionfromthesubstrate(assignalfromferromagneticNiparticlessaturatesathighmagneticelds)andcanthusbesubtractedfromthedata.ThedecreaseofHcwithincreasingtemperatureforxeddisclearlyapparentandcanbeunderstoodastheeectofthermaluctuations. 2 Forthelow-temperatureisotherms,therearepronouncedpeakswhichdeneatemperature-dependentcriticalparticlesizedcdelineatingSD(ddc)behavior. 2 8 39 { 45 ThereasonwhythereisapeakinHc(d)isexplainedintheintroductionchapter,page 29 IntheinsetofFig. 2-2 wehaveplottedHcversus1/dfortheparticlesofsized>dcat10K.ItisclearthatHcbehaveslinearlywith1/duptoparticlesizeof24nmandthensaturates.Thisbehaviorisconsistentwiththedependenceexpectedformultidomainparticles. 38 Thusparticlesofsized>dcaremultidomainandthepeakdenesthecrossoverfromSDtoMDbehavior.Theformationofdomainstructureisdrivenbythereductionoflongrangemagnetostaticenergy,whichatequilibriumisbalancedbyshorterrangeexchangeandanisotropyenergycostsassociatedwiththespinorientationswithinadomainwall.Thepurposeofthischapteristoshowthatthiswell-denedSDregionofcoherentlyrotatingparticlesextendsoveralargerrangeofgrainsizesathighertemperaturesbecauseofthediminishinginuenceofdipolarinteractionsfromneighboringgrains. 2.4DataandDiscussion TheinuenceofdipolarinteractionsontheSD/MDcrossovercanbeunderstoodinaqualitativewaybyconsideringthethreerandomlyorientedparticlesshownschematically 36

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Figure2-3. Peakposition,dc,plottedasafunctionoftemperature(redcircles).Theblacksquaresaretheresultsderivedfromequation2-5.Thebluestarrepresentstheobservedvalueofdcforaseriesofsinglelayersamplesat10K.Theinset,aschematicofthreeneighboringparticlesorientedindierentdirections,illustrateshowthedipolareldsfromparticle2and3facilitatetheformationofdomainsinparticle1,asthedipolarmagneticeldsareindierentdirections. intheinsetofFig. 2-3 .Particle1experiencesdipolareldsfromparticles2and3,whicharenotcollinearformostorientationsofarandomlyorientedparticlesystem.Becausedipolareldsdecreaserapidlywithinterparticleseparation,thedipolareldduetoparticle3(2)willbestrongerthanparticle2(3)ontheleft(right)sideoftheparticle1.Theseparateandunequalinuenceoftheneighboringparticlesthusfavorstheformationofdomainsinparticle1. .Tomakethesenotionsmorequantitative,wemodifythetreatmentofDormannetal 26 forinteractingparamagnetstoincludethetemperatureregionbelowtheblockingtemperatureTBandndthetemperature-dependentdipolarmagneticeldHdarisingfromtemperatureinduceductuationsinthemagnetizationofnearestneighbornanometersizeparticlestobe, Hd=0Msa 4e(1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F3 7.97 Tf 6.59 0 Td[(1) p (er())]TJ /F1 11.955 Tf 11.96 0 Td[(er(p )]TJ /F1 11.955 Tf 11.96 0 Td[(1))(2{1) 37

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where'er'istheimaginaryerrorfunction,Msisthesaturationmagnetization,=KV=kBT,anda=V(3cos2)]TJ /F1 11.955 Tf 13.36 0 Td[(1)=s3isadimensionlessparameterwithandscorrespondingrespectivelytoanangleparameterandtheseparationbetweentwoadjacentparticleseachwithvolumeV.TheparameterisalwaysgreaterthanoneforT
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.Intheabsenceofinteractions(Hd=0)theconditionfortheSDtoMDtransitionisgivenforsphericalparticleswithradiusd/2by,Ad3c=Bd2c,whereAd3cisthetotalmagnetostaticenergyandEdw=Bdcisthedomainwallenergy. 46 Wehaveabsorbedthefactoroftwo,whichrelatesdiametertoradius,intotheconstantsAandB.InthepresenceofthedipolarmagneticeldHd,theformationofdomainwallswillbeassistedbyaZeemantermwhichisproportionaltothevolumeoftheaectedparticle.TheconditiondeterminingtheSDtoMDtransitionnowbecomes, Ad3c=Bd2c)]TJ /F6 11.955 Tf 11.96 0 Td[(MsHdd3c=6(2{4) Whenthedipolarinteractionisasmallperturbation,i.e.,MsHd=A1,Eq. 2{1 and 2{4 canbecombinedtogivetherelation, dc=dc0)]TJ /F6 11.955 Tf 11.96 0 Td[(ddwe(1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F3 7.97 Tf 6.58 0 Td[(1) p (er())]TJ /F1 11.955 Tf 11.95 0 Td[(er(p )]TJ /F1 11.955 Tf 11.96 0 Td[(1))(2{5) wheredc0=B=Aisthetemperature-independentcriticaldiameterintheabsenceofinteractions(high-temperaturelimit)andddw=0BM2s=(72A2)fora==3.Thesecondtermontheright-handsideofEq. 2{5 thusbecomesatemperature-dependentcorrectiontodcduetointeractionsfromneighboringparticlesanddecreaseswithincreasingT. Sincethemagneticeldduetothedipole-dipoleinteractionsareweakerathighertemperaturesEq. 2{1 ,thenanoparticlesremainintheSDstatetoalargersize,whichbyEq. 2{5 resultsinashiftofdctowardshighervaluesathighertemperatures.ThisisindeedevidentinFig. 2-3 ,whichshowsthetemperaturedependenceofdcasdeterminedfromthedatainFig. 2-2 .TheblacksquaresarethesimulateddataaccordingtoEq. 2{5 usingthetwottingparameters:dc0andddw.Qualitatively,thedataagreequitewellwiththepredictionofthetheoreticalmodelwithouttakingintoaccountthetopologyandsizedistributionoftheparticles.Wehavefounddc0=84nmfromoursimulation(Fig. 2-3 ,blacksquares)tobeclosetothevalueforaparticlewithshapeanisotropyconstantKshape=3:1104Jm3(dc0=72AexK=0M2s,whereAexisexchangestiness, 39

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Kisanisotropyconstant). 3 ValuesofA(/0M2s)andB(/AexK)havebeenfoundtobe1.44104Jm3and1.21103Jm2respectively.ThisvalueofAisveryclosetothetheoreticalpredictedvalue 3 andthevalueofBisagainconsistentwiththevalueoftheshapeanisotropy.Thevalueoftheshapeanisotropycanalsobepredictedfromthezero-temperatureextrapolationHcoK=Msforrandomlyorientedparticles. 3 ForKshape=3:1104Jm3;Hco620Oe.Thisisingoodagreementwiththe500Oecoerciveeldobservedat10Kforthe6nmsample. Foraseparateseriesofsinglelayersamplesthecoercivitiesat10Kpeakatdc=14nmasshowninFig. 2-3 bythebluestar.Inthesinglelayersamplesthepeakpositionoccursathigherparticlesize(14nm)thanmultilayersamples(8nm).Thisdierencereinforcesourinterpretationandcanbeunderstoodbyrealizingthatthedipolarinteractionsofthesinglelayersamplesaresignicantlyreducedcomparedtothemultilayersamplesbecauseofthesmallernumberofnearestneighbors. 2.5Conclusion Insummary,wehavefabricatedmagneticnanoparticlesinaninsulatingthinlmmatrixwithtunablepropertiesachievedbyvaryingparticlesizeandtemperature.ThepeaksinthecoercivityisothermsdelineateacriticalgrainsizedcwhichidentiesthecrossoverfromSDtoMDbehavior.Thepresenceofdipolarinteractionsandtheirdiminishinginuencewithincreasingtemperatureisresponsiblefortheobserveddependenceofdcontemperatureandisingoodqualitativeagreementwithourmodicationofpresenttheory 26 ofinteractingparticles.Thewell-establishedinuenceofdipolarinteractionsonsuperparamagneticrelaxationtime 25 { 34 togetherwiththeconnectionbetweenrelaxationtimeandcoercivityHcsuggeststhatthereisaconcomitantinuenceofdipolarinteractionsonthecoercivityobservednearthesuperparamagneticlimitwhereHc=0.TheworkreportedhereextendsthisconnectiontotheupperlimitsonthesizeofSDparticlesbyshowingthatdipolarinteractionscanfacilitatetheformationofmultidomainparticlesespeciallyatlowtemperatures. 40

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2.6Methods 2.6.1MathematicalAnalysis TheEq. 2{5 isselfconsistent(asthetermcontainsdc)andcannotbesolvedanalytically.Theequation,dc)]TJ /F6 11.955 Tf 12.13 0 Td[(dc0+ddwe(1)]TJ /F5 7.97 Tf 6.58 0 Td[(e)]TJ /F11 5.978 Tf 5.76 0 Td[(1) p (erfi())]TJ /F5 7.97 Tf 6.58 0 Td[(erfi(p )]TJ /F3 7.97 Tf 6.59 0 Td[(1))=0,issolvedbynumericalapproachandsimultaneouslythesolutionisttedtotheexperimentaldataaccordingtoanonlinearlistsquaremethod.Mathematica,acommercialsoftware,isusedforthispurpose. 2.6.2BasicPhysicalUnderstanding AsimpliedphysicalunderstandingoftheproblemisshowninFig. 2-4 Figure2-4. Theneteectofdipolarmagneticeld(Hd)isshownontheparticle1.Asparticlesarerandomlyoriented,Hdfromparticle3willbeindierentdirectionthanthatfromparticle2.Asdipolarinteractiondecreasesrapidlywithdistance,particle1willexperiencelocaldipolarmagneticeldsindierentdirectionsfromdierentneighboringparticlesandthusmakingiteasytoformdomains. 41

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CHAPTER3EFFECTOFDIPOLARINTERACTIONONTHECOERCIVEFIELDOFMAGNETICNANOPARTICLES:EVIDENCEFORCOLLECTIVEDYNAMICS 3.1Abstract TheeectofdipolarinteractiononthecoerciveeldisdiscussedforthesingledomainandcoherentlyrotatingNinanoparticlesembeddedinAl2O3matrix.Resultsfortwosetsof5layersampleswithdierentinterlayerspacingandasetofsinglelayersamplesofNinanoparticlesarecompared.Thedipolarinteractionsarestrongestinthesampleswithshorterinterlayerdistancesandweakestforthesinglelayersamples.Inthispresentstudy,thedipolarinteractionisfoundtoincreasethecoerciveeld.Ontheotherhandthecriticalsingledomainradiusdecreasesduetothedipolarinteractions.Thesetwobehaviorstogetherindicatethatcollectivedynamicsplaysanimportantroleinunderstandingtheoriginofthecoerciveeld. 3.2Introduction Theoriginofcoerciveeld(Hc)forcoherentlyrotatingferromagneticnanoparticlesisremarkablydierentthanthatofthebulk, 47 whereirreversibledomainwallmotionisthedominantmechanism. 4 Inthecaseofnanoparticles,whenthesizeoftheparticleissmallerthanacriticalsize(dc),themostfavorableenergystateistohavesinglemagneticdomainandparticlesarecalledsingledomain(SD)particles.WhenHcisplottedasafunctionofparticlediameter(d),thereisawelldenedpeakatdc.Particleswithddc)areSD(multidomain(MD)). 2 8 41 42 44 48 Kittel 3 46 hasshownthatforasphericalparticle,dcisgivenbytherelation(seeEq. 1{3 onpage 13 ofchapter 1 ) dc=72p AK 0M2s(3{1) whereAistheexchangestiness,Kistheanisotropyconstant,0isthefreespacepermeabilityandMsisthesaturationmagnetization.InSDparticlesthereisnodomainwall.TheoriginofHcinthiscaseisthenitetimerequiredtoreversethemagnetizationdirectionoverthemagneticelddependentanisotropyenergy. 47 Thetimerequiredto 42

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reversethedirectionofthemagnetizationofacoherentlyrotatingSDparticleisgivenbytherelation 2 5 49 50 =0expKV kBT:(3{2) Here,0istheinverseoftheattemptfrequencytoovercometheenergybarrier,Visthevolumeoftheparticle,kBistheBoltzmannconstantandTisthetemperature.StonerandWohlfarthhavecalculatedHcforSDparticlesinthesimplecasewhenparticlesarecoherentlyrotatingandtheappliedmagneticeldisalongtheeasyaxisofmagnetizationoftheparticles.ThecoerciveeldforaStoner-Wohlfarthparticleisgivenby HcSW=2K Ms241)]TJ /F8 11.955 Tf 11.95 20.44 Td[( ln(m 0) ln( 0)!1 235;(3{3) wheremisthetimeofmeasurement.FromthesimpleStoner-WohlfarthmodelitisclearthatHcforthenanoparticlecandependonmanydierentfactors.Hcincreaseswithdecreasingm,increasingandincreasingK. Inthepresenceofdipolarinteractionstheaboveequationwillbemodied.Thewidelyacceptedmodicationisachievedbytreatingthedipolarinteractionstoresultinaneectiveanisotropyenergy. 51 { 61 .ThusifduetothedipolarinteractionsKincreases(decreases)then,accordingtoEq. 3{2 ,willalsoincrease(decrease)andasanetresultHcwillincrease(decrease).AmorefamiliarfamousformofEq. 3{3 isHcSW(T)=2K=Ms(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)1=2),whereTB=KV=25kBistheblockingtemperature.Thefactor25comesfromthefactthatm100sisatypicalmeasurementtimeand)]TJ /F3 7.97 Tf 6.58 0 Td[(10109sec)]TJ /F3 7.97 Tf 6.58 0 Td[(1isatypicalattemptrate. Theeectofdipolarinteractiononthecoerciveeld(Hc)hasbeeninvestigatedextensively.ThersttheoreticaltreatmentbyNeel 62 showedthatHcdecreaseswiththeincreaseinthepackingfraction()orthedipolarinteractionasshownbelowinEq. 3{4 ,wheretheinteractioneecthasbeenintroducedasan"InteractionField"andshownto 43

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lowertheanisotropyenergy. Hc=Hc1(1)]TJ /F6 11.955 Tf 11.95 0 Td[()(3{4) The"InteractionField"isafunctionofthepackingfraction().LaterWohlfarth 51 showedthattheeectoftheinteractionontheHccanbeincreasingordecreasingdependingontheparticleorientationasthedipolarinteractionisdirectiondependent.Butallofthoseresultshavebeenconstructedconsideringthefactthattheanisotropyconstant,K,eitherincreasesordecreasesduetotheinteractions.PrevioustheoreticalandexperimentalworkshavebeenreportedeithershowinganincreaseordecreaseinHcandexplainedintermsofacorrespondingincreaseordecreaseintheanisotropyenergy. 51 { 61 63 InthispresentexperimentwendthatanincreaseinthedipolarinteractionincreasesHcbutdecreasesdc.Equation 3{1 suggeststhatthedecreaseindcmaybeduetoadecreaseintheK.ButadecreaseintheKwillalsodecrease(Eq. 3{2 )andthuswilldecreaseHc(Eq. 3{3 )whichiscontradictorytothepresentexperimentalresult.ThusthechangeofKduetothedipolarinteractionsmustnotbeapplicableinthepresentcase.AsanychangeinKwillgiverisetochangeinHcanddcbothinthesamedirection(bothincreaseordecreaseatthesametime).Below,weshowqualitativelythattheincreaseintheHccanberealizedintermsofthecollectivedynamicsofthemagnetizationoftheparticlesanddecreaseindccanbeunderstoodasdiscussedinreference 48 3.3ResultsandDiscussions Samplesweregrownusingpulsedlaserdepositiontechnique. 48 Basepressureofthegrowthchamberwasontheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(7Torrandthegrowthtemperaturewasaround550oC.MultilayerstructureofAl2O3andNinanoparticleweregrownwithoutbreakingthevacuumofthechamber.Firstathick(40nm)buerlayerofAl2O3isgrownontopofthesubstrate.ThepurposeofthisbuerlayeristopreventanydiusionoftheNiintothesubstrate.ThenNinanoparticlesandAl2O3aresequentiallydepositedonthisbuerlayer(seeFig.1).ThetoplayerofAl2O3actsasacappinglayerwhichpreventsoxidationofthenanoparticles. 36 Threedierentsetsofsamplesaregrown.Set1andset2samples 44

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consistof5layersofNinanoparticlesseparatedbyAl2O3layers.Forset1(set2)theAl2O3separationis3nm(40nm).Set3samplesaresinglelayerofNinanoparticlesinAl2O3matrix.Dipolarinteractionsarestrongestinset1,moderateinset2andweakestinset3.ThedipolarinteractionsarestrongerinSet1comparedtoset2astheinterlayerseparationoftheNiparticlesissmallerinset1comparedtoset2.Set3consistsofonlyasinglelayerofNiparticlesandthusthedipolarinteractionsareweakest.Allsetsofsamplesconsistofdierentsampleswithvaryingparticlesizefrom3nmto60nm. Figure3-1. Fig1a)showstheTEMimageofasinglelayersamplewithaverageparticlediameterof24nm.Particlesarewelldenedwithinterparticledistanceofaround4nm.1b)showsaschematicofthesinglelayersample.A40nmthickbuerlayerofAl2O3isrstgrownontopofsubstrate.ThentheNinanoparticlesaregrownontoofthebuerlayer.Finallya3nmthickcappinglayerofAl2O3isgrowntoprotectitfromoxidation.1c)showstheschematicof5layersofNinanoparticlesample. Figure 3-1 a)showstheTEMimageofthesinglelayerNiparticleswithaverageparticlediameterof24nm(set3).Thesimpliedschematicofthesingleandmultilayersamplesareshownin 3-1 b)andc). TypicalmagnetizationloopsatthreedierenttemperaturesareshowninFig. 3-2 a)forthesamplewith3nmAl2O3spacerlayer(set1)and6nmindiameter.ThecoerciveeldHc(T)isdeterminedfromtheloopasshownbythearrow.ThisproceduretodetermineHcisrepeatedforallsamplesbelongingtoallthreesets.Attemperatures 45

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Figure3-2. a)Magnetizationloopofasamplefromset1ofaverageparticlediameterof6nm.Coerciveeld(Hc)isdeterminedfromtheloopasshownbythearrow.Hcdecreaseswithincreasingtemperatureandgoestozeroabovetheblockingtemperature.b)Magnetizationloopsaboveblockingtemperatures.MagnetizationisplottedasthefunctionofH=TtoshowthesuperparamagneticbehaviorasexpectedfortheSDparticlesabovetheblockingtemperature. abovetheblockingtemperatures(TB)SDsamplesbehaveassuperparamagneticparticles.Figure 3-2 b)showsthesuperparamagneticbehavioroftheset1,6nmdiametersample.NotethemagnetizationdatafallontopofeachotherwhenplottedasafunctionofH=T.ThisbehaviorisadirectconsequenceofthesuperparamagneticbehaviorasexpectedfromthecoherentlyrotatingSDparticles. Figure 3-3 showsHcplottedasafunctionofdfortheset1,set2andset3samples.Thedatathatcorrespondtothedierentsamplesetsareindicatedinthelegends.ThepeakintheHcseparatesSDandMDparticles. 2 8 41 42 44 48 Itisclearfromthedatathatdcdecreaseswithincreasingdipolarinteractions(dc1
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Figure3-3. Coerciveeld(Hc)asafunctionofparticlediameter(d).Thepeakseparatesthesingledomain(SD)andmultidomain(MD)particles.Particleswithdiameterhigher(smaller)thanthepeakdiameter(dc)areMD(SD).Dataforthe3dierentsamplesetsareshownandindicatedinthelegends.Thecriticaldiametersdc1,dc2,dc3areshownfromthesamplesofset1,set2andset3respectively.Inthesingledomainregion(belowdc)thecoercivityincreaseswithincreasingdipolarinteractionsasshownbytheverticaldottedarrow. magnetizationduetothedipolarinteractionsisfoundtoberesponsiblefortheincreaseinHc.TheseobservationsareshowninFig. 3-3 andsummarizedinFig. 3-4 WerstdiscusstheeectofdipolarinteractionsonHcaspresentedinpreviousinvestigations. 51 { 61 ThetreatmentbeginbyincludingthechangeinanisotropyenergyEdip,duetodipolarinteractionintotheexpressionfor,asgivenby 63 =0expKVEdip kBT(3{5) 47

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Equation 3{5 canberewrittenasshowninEq. 3{6 .Thustheeectofthedipolarinteractionsistreatedaseitheranincrease(+Edip)ordecrease(-Edip)ofanisotropyenergy. =0expK(eff)V kBT(3{6) TheeectofdipolarinteractionsontheHccanbeexplainedaccordingtoEq. 3{6 .Inourcasea+Edipincreases+andgiverisetoanincreaseinHcwithincreasingdipolarinteractions(Eq. 3{2 and 3{3 ).IfthisistobetrueinourcasethenaccordingtoEq. 3{1 ,dcshouldalsoincreasewithincreasingdipolarinteractions.AccordingtothepreviousapproachbothHcanddcshouldchangeinthesameway,bothincreaseorbothdecrease.InthepresentexperimentwendhoweverthatHcincreasesanddcdecreasesduetodipolarinteractions(seeFig. 3-4 )andstronglysuggestsanalternativeapproachtotheproblem.Theeectofdipolarinteractionsondcisdiscussedinreference 48 ,whereithasbeenshownthatthelocaldipolarmagneticeldfromthenearbyrandomlyorientedparticlestrytoalignthemagnetizationdirectionoftheparticleindierentdirectionsandthusfavoringdomainformation.TheeectofdipolarinteractionsonHcwillbediscussedbelowintermsofcollectivedynamics. Itiswellknownthatthemagnetizationdynamicscanbecollectiveinnatureduetotheinteractionsbetweentheparticlesandtherelaxationtime()inthiscaseisgivenby 64 65 =T Tg)]TJ /F1 11.955 Tf 11.95 0 Td[(1)]TJ /F5 7.97 Tf 6.58 0 Td[(z;T>Tg(3{7) whereistherelaxationtimeofthesinglenoninteractingparticle(Eq. 3{2 ),Tg=0M2=4kBr3isthecriticaltemperatureanddependsontheinterparticledistanceandparticlemagnetizationandzisacriticalexponent.TheaboveequationclearlysuggeststhattherelaxationtimewillbelargerinthepresenceofdipolarinteractionsandthusaccordingtoEq. 3{3 Hcwillbelarger,andthusagreeingwithourexperimental 48

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Figure3-4. Coerciveeld(Hc)andcriticaldiameter(dc)asthefunctionoftheincreasingdipolarinteraction.Hc(dc)increases(decreases)withincreasingdipolarinteraction.TheoppositebehaviorofHcanddcsuggeststhatthecollectivedynamicsandthecriticalslowdownisresponsiblefortheincreaseinHcduetothedipolarinteractions.Thedecreaseindcisdiscussedelsewhere. 48 observations(Fig. 3-4 ).Notethatinthiscasetheanisotropyenergyisunaectedbydipolarinteractionsandtheincreaseinrelaxationtimeisduetothefactthatthereversalofmagnetizationiscollectiveinnature. 64 65 3.4Conclusions AstudyofdipolarinteractionsispresentedforthesingleandmultilayerstructureofNinanoparticles.Thecoerciveeldhasbeenfoundtoincreasewithincreasingdipolarinteractionsandcanbeunderstoodqualitativelyintermsofcollectivedynamics.Threesetsofsamplesareinvestigated.Eachsetconsistsofsampleshavingparticlesizevaryingfrom3nmto60nmindiameter.Dipolarinteractionsarestrongerinset1anddecreasesforset2andset3.Behaviorofcoerciveeldandcriticalsingledomainradiusareobserved.Coerciveeldincreasesandcriticalsingledomainradiusdecreases 49

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withincreasingdipolarinteractions.ThesetwobehaviorstogethersuggestacollectivedynamicsofthemagnetizationreversalprocessintheSDregioninthepresenceofdipolarinteractions.Toourknowledge,thisisthersttimethattheeectofcollectivedynamicsonacoerciveeldofthenanoparticlesystemhasbeenobserved. 50

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CHAPTER4FINITESIZEEFFECTSWITHVARIABLERANGEEXCHANGECOUPLINGINTHIN-FILMPD/FE/PDTRILAYERS 4.1Abstract Themagneticpropertiesofthin-lmPd=Fe=Pdtrilayersinwhichanembedded1.5A-thickultrathinlayerofFeinducesferromagnetisminthesurroundingPdhavebeeninvestigated.ThethicknessoftheferromagnetictrilayeriscontrolledbyvaryingthethicknessofthetopPdlayeroverarangefrom8Ato56A.AsthethicknessofthetopPdlayerdecreases,orequivalentlyastheembeddedFelayermovesclosertothetopsurface,thesaturatedmagnetizationnormalizedtoareaandtheCurietemperaturedecreasewhereasthecoercivityincreases.Thesethickness-dependentobservationsforproximity-polarizedthin-lmPdarequalitativelyconsistentwithnitesizeeectsthatarewellknownforregularthin-lmferromagnets.Thefunctionalformsforthethicknessdependences,whicharestronglymodiedbythenonuniformexchangeinteractioninthepolarizedPd,provideimportantnewinsightstounderstandingnanomagnetismintwo-dimensions. 4.2Introduction Thepresenceof3dmagnetictransitionmetalionsinpalladium(Pd)givesrisetogiantmomentsthussignicantlyenhancingthenetmagnetization 66 { 70 .PdisknowntobeinthevergeofferromagnetismbecauseofitsstrongexchangeenhancementwithaStonerenhancementfactorof10 71 .ThemagneticimpuritiesinducesmallmomentsonnearbyPdhostatomstherebycreatingacloudofpolarizationwithanassociatedgiantmoment 71 72 .Neutronscatteringexperimentsshowthatthecloudofinducedmomentscaninclude200hostatomswithaspatialextentintherange10to50A 72 73 .ThusathinlayerofFeencapsulatedwithinPdwillbesandwichedbetweentwoadjacentthinlayersofferromagneticPdwithnonuniformmagnetizationandatotalthicknessintherange20to100A. 51

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Wehaveinvestigatedthin-lmPd/Fe/PdtrilayersinwhichthethicknessdFeoftheFeisheldconstantnear1.5AandthethicknessofthepolarizedferromagneticPdisvariedbychangingthetopPdlayerthicknessx.Themagneticpropertiesarestudiedasafunctionofx.Ourexperimentsaremotivatedbytherecognitionthatferromagnetisminrestricteddimensionshasattractedsignicantresearchinterest 74 { 78 .Forexample,thecoerciveeldHcincreasesasthethicknessoftheferromagneticlmisdecreasedtowardathicknesscomparabletothewidthofatypicaldomainwall 79 80 .Moreover,theCurietemperatureTcdecreasesasthethicknessoftheferromagneticlmisdecreasedtowardathicknesscomparabletothespin-spincorrelationlength 81 { 83 .WewillshowbelowthatsimilarphenomenologyappliestoferromagneticallypolarizedPdlms,albeitwithdierentfunctionaldependencesarisingfromthefactthatexchangecoupling,whichdecayswithdistancefromtheferromagneticimpurity 84 ,isnotuniformthroughoutthelm. 4.3ExperimentalDetails ThesamplesweregrownonglasssubstratebyRFmagnetronsputtering.Thebasepressureofthegrowthchamberwasoftheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(9Torr.FirstathicklayerofPdofthickness200Aisgrownontopofthesubstrate.TherootmeansquaresurfaceroughnessofthisPdlayerwasmeasuredbyatomicforcemicroscopytobe6A.Thenaverythin(1.5Aasrecordedbyaquartzcrystalmonitor)layerofFeisdepositedontopoftherstPdlayer.FinallyatoplayerofPdwiththicknessxisgrowntocompletethetrilayerstructureshownschematicallyinFig. 4-1 a.WediscusssixdierentsampleswiththetopPdlayerhavingathicknessxvaryingfrom8to56A.ThetotalthicknessyofthepolarizedPd(seeFig. 4-1 b)canrangefrom20to100A 72 73 .Thusforx
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Figure4-1. a)MultilayerstructureofaPd/Fe/Pdtrilayer.ThebottomlayerofPdis200Athick.ThethicknessoftheFelayeris1.5Aasrecordedbythequartzcrystalmonitor.ThethicknessxofthetoplayerofPdisvariedfrom8to56A.b)Magneticstructureofthesample.ThetotalthicknessyofpolarizedPdisintherange20to100A(shadedredarea).Thusbyvaryingx,itispossibletovarythethicknessyofthepolarizedferromagneticPdlayer.c)IntensityofFe3(703.0eV)LMMAugerelectronpeakplottedasafunctionofmaterialremovedbyargonsputtering.Thedata(solidblackcircles)arettoaGaussiandistribution(redline).Thefullwidthhalfmaximumvalueof1.85Aisconsistentwithcrystalmonitormeasurements AugerelectronpeakofFig. 4-1 cshowsthattheFeisembeddedinthePdasadistinct2DlayerwithaFWHMthicknessof1.8A.Allofthesestepswereperformedwithoutbreakingvacuum.MeasurementsofthemagnetizationM(Fig. 4-2 )wereperformedusingaQuantumDesignMPMSsystem.ThemagneticeldHwasalongtheplaneofthesubstrate.Sincethemagnetizationmeasurementswereexsitu,xwasconstrainedtobegreaterthan8A;otherwisetheexposureofthesampletoaircausedunwantedoxidationoftheFe.ThemagneticparametersHc(x)(Fig. 4-3 )andTc(x)(Fig. 4-4 )arecalculatedrespectivelyfrommagnetizationloopstakenat10K(seeinsetofFig. 4-3 )andlinearextrapolationsofthetemperature-dependentmagnetizationtakenatH=20Oe(seeinsetofFig. 4-4 ).ThemagneticcontributionfromthebottomferromagneticPdlayerisindependentofx,sincey=2<200A,theconstantthicknessofthebottomlayer. 53

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Figure4-2. ThesaturationmagnetizationnormalizedtotheareaofthesampleMsAshowsasmoothincreasewithincreasingthicknessx.Theexperimentaldataareshownassolidblackcirclesandthedashedblacklineisaguidetotheeye.Saturationtoaconstantvalueoccursnear30A(verticalarrow). 4.4ResultsandDiscussion Forlargevaluesofx,thethicknessyofthecombinedpolarizedferromagneticPdlayersandtheassociatedsaturatedmagnetizationM=Mswillreachaconstantvalue.ThisexpectationisborneoutinFig. 4-2 whichshowsthex-dependenceofsaturatedmagnetizationMsAnormalizedtosamplearea.WenotethatthisnormalizedsaturatedmagnetizationMsA(x)increaseswithincreasingxasthetotalamountofpolarizedPdincreases.Theonsetofsaturation,nearx=30AindicatesthatthepolarizationcloudincludingtheembeddedFelayeris60Athick.Thisvalueisconsistentwithpreviousobservation 73 .TheincreaseofMsAwithxshowninFig. 4-2 isthusstraightforwardtounderstand.AsxincreasesthethicknessofthetoppolarizedferromagneticPdlayerincreaseswithaconcomitantincreaseofmagneticmaterialinthesystem.VariationofxclearlycontrolsthethicknessofthepolarizedferromagneticPdlayer.Whennormalizedto 54

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Figure4-3. ThecoerciveeldHcshowsastrongincreaseasthethicknessxofthetoplayerofthePddecreases.Thedataareshownassolidblackcirclesandtheblacksolidlineisapowerlawtwithexponent=2:3(0:1).TheinsetshowsmagnetizationloopsatT=10Kforx=8A(solidblacksquares)andx=56A(solidredcircles). thenumberofFeatomspresent,thesaturatedmagnetizationMsA=1:110)]TJ /F3 7.97 Tf 6.59 0 Td[(4emu/cm2correspondsto9.2BperFeatom,incloseagreementwithpreviousobservationsofthegiantmomentofFeinPdtobenear10B 72 .ModelingthexdependenceofMsA(x)showninFig. 4-2 forourPd=Fe=Pdtrilayersisnotstraightforward.ForregularferromagnetswithMsuniformthroughoutthethickness,wewouldexpectMsA(x)tobelinearinx;clearlyitisnot.AreasonablemodelwillincorporateanexchangeinteractionJthatdecaysradiallywiththedistancefromthepointferromagneticimpurity 84 .ThiscomplicationrequiresmodelingJasafunctionofdistancexfromtheplaneofimpurity.AstartingpointwouldbetowritethemagnetizationMisafunctionofJ 4 M(H;T;x)=MsBs Ms kBT"gBH+2pMJ(x)#!;(4{1) 55

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Figure4-4. TheCurietemperatureTcrapidlyincreaseswithincreasingx.Dataareshownassolidblackcirclesandthedashedblacklineisaguidetotheeye.Saturationtoaconstantvalueoccursnear20A(verticalarrow)TheinsetwithTcindicatedbytheverticalarrowshowsthetemperature-dependentmagnetizationtakeninaeldH=20Oe. whereBsistheBrillouinfunctionandpisthenumberofthenearestneighborsbeyondwhichJiszero.InprincipletheexperimentallydeterminedvaluesofM(H;T;x)canbettoEq. 4{1 tondthebesttvaluesofJ(x)fordierentvaluesoftheparameterp.Wehavenotperformedsuchananalysis. Fig. 4-3 showsthebehaviorofthecoercivityHc(x)asafunctionofx(solidblackcircles).Thedataarewelldescribedbyapower-lawdependence(solidblackline),Hc(x)/x)]TJ /F5 7.97 Tf 6.58 0 Td[(,wheretheexponent=2:3(0:1)isclosetotheratio7/3.Similarpower-lawbehaviorrevealsitselfinregularferromagneticthinlmswherehasasomewhatsmallervaluevaryingfrom0.3to1.5 76 .Becausedependsstronglyonstrain,roughness,impurity,andthenatureofthedomainwall(BlochorNeeltype) 76 ,itisnotsurprisingtoseeawidevariationin.NeelpredictedforexamplethatforBlochdomainwalls,Hcofaferromagneticthinlmshouldvaryasx)]TJ /F3 7.97 Tf 6.59 0 Td[(4=3whenthethicknessxofthelm 56

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iscomparabletothedomainwallthicknessw 79 .ForthecaseofNeelwalls,Hcdependsonlyontheroughnessofthelmanddoesnotdependonlmthickness 77 .ThevariationofHc(x)becomesparticularlypronouncedwhenthelmthicknessbecomescomparabletow. AqualitativeunderstandingofthesteeperHc(x)dependencebecomesevidentbyrecognizingthattheformationofdomainstructureisdrivenbythereductionoflongrangemagnetostaticenergywhichatequilibriumisbalancedbyshorterrangeexchangeandanisotropyenergycostsassociatedwiththespinorientationswithinaBlochorNeeldomainwall.Domainwallthicknessisgivenbyw=p A=K 3 82 whereKisthecrystallineanisotropyconstantandAistheexchangestiness,proportionaltotheexchangeenergy,J 85 .ThedomainwallsizewincreasesfordecreasingKandincreasingJ.IfK,whichdependsontherelativelyconstantspin-orbitinteraction 4 withinthePdcomponentofthePd=Fe=Pdtrilayers,remainsconstant,thenvariationsinwaredominatedbyvariationsinJ.Thusasxdecreasestowardzero,theincreaseinJ 84 givesrisetoanincreaseinwwhichinturngivesrisetoamorerapidincreaseinHcthanwouldbeseeninregularferromagnetswithconstantJ.Asdiscussedabove,thisrapidvariationwith7=3isobservedexperimentally. ThedatainFig. 4-4 showthatTcincreasesasxincreasesandreachesarelativelyconstantvaluenearx=20A.ThedashedblacklineisaguidetotheeyeandisqualitativelysimilartothebehaviorofMsA(x)showninFig. 4-2 whichsaturatesatalargervaluenear30A.Theseobservationsareagainqualitativelyconsistentwiththenitesizeeectassociatedwithcriticalphenomenainferromagnets 81 { 83 .Althoughthedataarenotofsucientqualitytodistinguishthepower-lawbehaviorthatispredictedfornitesizeeects 81 { 83 ,weexpectthatthedependenceisfurthercomplicatedbythepreviouslydiscusseddependenceofJonxinpolarizedferromagneticPd.ThebehaviorofTc(x)suggeststhatPd=Fe=Pdtrilayershouldbetreatedasasinglelayerwithawelldenedspin-spincorrelationlength.IfthePdlayersaretreatedseparately,thenthebottomlayer 57

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withxedthicknessy=2wouldhaveaTcequaltothehighestTcofthetoplayer.InthiscasetheoverallmeasurementwouldnotshowastrongchangeinTcasafunctionofx,sincetheTcofthebottomlayerwoulddominateforallx. Wenotethatforourplanargeometry,Tcdecreaseswithdecreasingthicknessashasalsobeenshownforthin-lmNi 81 andepitaxialthin-lmstructuresbasedonNi,CoandFe 82 .OntheotherhandTcincreaseswithdecreasingsizeofferrimagneticMnFe2O4nanoscaleparticleswithdiametersintherange5-26nm 83 .ThisincreaseofTcwithdecreasingsizeisattributedtonitesizescalinginthreedimensionswhereallthreedimensionssimultaneouslycollapse 83 .Inourtwo-dimensionalplanarthinlmsonlyoneofthedimensions,thethickness,collapsesandTcdecreasesratherthanincreasesinaccordwiththeobservationsofpreviousstudies 81 82 4.5Conclusions Inconclusion,wehavecharacterizedthemagneticpropertiesofthin-lmPd=Fe=Pdtrilayersanddeterminedthatcriticalsizeeectsapplyto\ferromagnetic"PdwheretheferromagnetismisinducedbyproximitytoanunderlyingultrathinFelm.Thecriticalsize,orequivalentlythecriticalthickness,iscontrolledbyvaryingthethicknessxofthetopPdlayer.ThedependencesonlmthicknessofthecoerciveeldHcandtheCurietemperatureTcareinqualitativeagreementwithnitesizeeectsseeninregularferromagneticlmswheretheexchangecouplingJisconstantthroughoutthelm.TheresultspresentedhereincreaseourunderstandingofnanomagnetisminultrathinsystemsbyshowingthatthespatialvariationsofJintheproximitycoupledPdhaveapronouncedinuenceontheformofthickness-inducedvariations,namely:anonlineardependenceofMsA(x),anunusuallystrongpower-lawdependenceofHc(x)andadependenceofTc(x)whichindicatesthatthetrilayeractsasasinglelayerthatnecessarilyincludestheconstantthicknessPdlayerservingasasubstratefortheFelayer. 58

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CHAPTER5TEMPERATUREDEPENDENCEOFCOERCIVITYINMULTIDOMAINNINANOPARTICLES,EVIDENCEOFSTRONGDOMAINWALLPINNING 5.1Abstract Thetemperaturedependenceofthecoercivityofthesingleand5layersamplesofNinanoparticlesinAl2O3matrixisstudied.AlinearT2=3dependenceofcoercivityoverawiderangeoftemperature(10Kto350K)isobserved.Allthesamplesconsistsofparticleswithmultiplemagneticdomainsasthesizeoftheparticlesarelargerthanthecriticalsingledomainsize(seeEq. 1{3 onpage 13 ofchapter 1 andFig. 3-3 onpage 47 ofchapter 3 ).Theexperimentalresultsareunderstoodintermsofstrongdomainwallpinning. 5.2Introduction Thetemperaturedependenceoftheextrinsicmagneticproperties,forexamplecoerciveeld(Hc(T)),arisefromtwomechanisms.Therstmechanismis,duetothetemperaturedependenceoftheintrinsicmagneticproperties 11 15 86 87 suchassaturationmagnetization(Ms),magneticanisotropy(K)andexchangestiness(A)andwillbediscussedinchapter 6 .Thesecondmechanismis,duetothethermallyactivatedhopingofthemetastablestatesoversomeenergybarrier. 2 { 4 6 Fromthemagnetizationloopsatdierenttemperatures(Fig. 5-2 )wehavefoundthatMsdoesnotchangewithtemperature.Allthesamplescomprisepolycrystallineparticles 36 andthusmagnetocrystallineanisotropycanbeneglectedandtemperatureindependentshapeanisotropyisdominant. 48 Theexperimentaltemperaturerangeis10Kto300KwhichismuchsmallerthanthecurietemperatureofNi(630K) 3 andAcanbeconsideredconstantoverthistemperaturerange. 88 Inthischapterwewilldiscussthesecondmechanismastheoriginofthetemperaturedependenceofthecoerciveeld(Ms,KandAaretemperatureindependent).TounderstandthetemperaturedependenceoftheHcduetothethermallyactivatedhoppingovermetastableenergyminimaseparatedbysomeenergybarrier,itisnecessarytondoutthemagneticelddependenceoftheenergybarrier.Acommonly 59

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usedphenomenologicalenergybarrieris 2 3 47 89 E=E0[1)]TJ /F6 11.955 Tf 11.96 0 Td[(H=Hc0]m(5{1) whereE0istheenergybarrieratzeromagneticeldandenergybarriervanishesatH=Hc0atT=0.AtH=Hc,thermalenergy,kBT,issucientlyhightocausemostofthemomentstobethermallyactivatedoverthebarrier.ForexampleforthecaseofStoner-Wohlfarthparticlesm=2,E0=KVandHc0=2K=Ms.ForStoner-WohlfarthparticlesthescenarioisverysimpleandtheEq. 5{1 canbederivedanalytically(seeEq. 1{6 onpage 15 ofchapter 1 ).RememberthatifE(H)isknown,itispossibletocalculateHc(T).InthischapterwewilldiscusshowtoderiveE(H)(Eq. 5{1 )fortheMDnanoparticlesandwillcompareHc(T)withthemodel. 5.3ResultsandDiscussions Thesamplepreparationtechniqueisdiscussedinchapter2andchapter3.Threedierentsetsofsamplesareinvestigated.Set1consistsofsinglelayerNiparticlesinanAl2O3matrix.Set2andSet3consistsof5layersofNiparticlesseparatedbyAl2O3layers.TheinterlayerseparationinSet2andSet3are3nmand40nmrespectively.TheschematicofallthreesetsofsamplesareshowninFig. 5-1 below. InthischapterwewillfocusonthetemperaturedependenceofHcfortheMDNinanoparticles.Atotalof15samplesarestudied,5samplesfromeachset.Magnetizationloopsaremeasuredforeverysampleforseven(onaverage)dierenttemperatures.Thismeansatotalofaround105magnetizationloopshavebeenmeasuredforthepresentstudy.Magnetizationloopsforthesampleofaverageparticlediameterof12nmofset2atdierenttemperatures(indicatedinthelegends)areshowninFig. 5-2 .ThearrowshowsHcat10K.NotethatHcdecreaseswithincreasingtemperature.ThetemperaturedependenceofHcnormalizedtoHcOforvedierentsamplesbelongingtoset1isshowninFig. 5-3 .Theparticlediametersareindicatedinthelegends.NoteT2=3inxaxis.AllthedatafollowalinearT2=3dependence.Tounderstandtheabovedata,wewillstart 60

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Figure5-1. Schematicofthreesetsofsamples.Set1comprisesasinglelayerofNiparticlesembeddedinanAl2O3matrix.Set2andSet3comprisesof5layersofNiparticlesseparatedbydierentdistancesinanAl2O3matrix.TheinterlayerdistancesinSet2andSet3are3nmand40nmrespectively. withageneralmagneticenergylandscapeofthesystemwrittenasapolynomialexpansionofthedomainwallposition(x)aroundastrongpinningcenter. 2 { 4 6 E(x)=a0+a1x+a2 2x2+a3 3x3)]TJ /F6 11.955 Tf 11.95 0 Td[(b0Hx(5{2) wherea0,a1,a2,a3andb0aremicromagneticparametersthatdependonthemagneticparametersK,MsandA.Forthestrongpinningcenterthex3termisincludedastheeectofthepinningcenterislongdistancecomparedtotheweakpinningcenterwherethex3termisneglected. 3 Therelationbetweenmicromagneticandmagneticparametercanbedeterminedfromtheparticularmodelused.Notethatthemicromagneticparametersaretemperatureindependentinourcaseastheyonlydependonthetemperatureindependentmagneticparameters.First,wewillderivetheenergybarrierseparatingthemetastableminimafromtheglobalminima.ThemaximaorminimaofE(x)aredeterminedbysettingtherstorderderivativetozero. E x=a1+a2x+a3x2)]TJ /F6 11.955 Tf 11.95 0 Td[(b0H=0(5{3) 61

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Figure5-2. Magnetizationloopsforthesampleofaverageparticlediameterof12nmofset2atdierenttemperatures(indicatedinthelegends).Thecoerciveeld(Hc)at10Kisindicatedbythearrow.Hcdecreaseswithincreasingtemperature.Saturationmagnetization(Ms)isconstantatdierenttemperatures. Thetwosolutionsfortheaboveequationsare x1=)]TJ /F6 11.955 Tf 9.3 0 Td[(a2+p a22)]TJ /F1 11.955 Tf 11.96 0 Td[(4a3(a1)]TJ /F6 11.955 Tf 11.96 0 Td[(boH) 2a3(5{4) x2=)]TJ /F6 11.955 Tf 9.3 0 Td[(a2)]TJ /F8 11.955 Tf 11.96 10.37 Td[(p a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a3(a1)]TJ /F6 11.955 Tf 11.96 0 Td[(boH) 2a3(5{5) TakingthesecondderivativeofE(x)withrespecttoxitiseasytoshowthat2E=x2jx1>0(2E=x2jx2<0)andcorrespondstothemaximum(minimum).Thustheenergybarrieris E(H)=E(x1))]TJ /F6 11.955 Tf 11.95 0 Td[(E(x2)=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3+4a3b0H)3=2 6a23(5{6) 62

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Figure5-3. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset1.Thelinearbehaviorisobservedforsampleswithparticlesizefrom18nmto42nmindiameter. Forthereverseeld,ieH=)]TJ /F6 11.955 Tf 9.3 0 Td[(Htheaboveequationreducesto E(H)=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a3(a1+b0H))3=2 6a23=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23 1)]TJ /F6 11.955 Tf 27.42 8.09 Td[(H a22)]TJ /F3 7.97 Tf 6.59 0 Td[(4a1a3 4a3b0!3=2 (5{7) whichisinthesameformofEq. 5{1 ,where E0=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23(5{8) Hc0=a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3 4a3b0(5{9) FromEq. 5{7 itisclearthatE(H)decreaseswithincreasingHandwhenH=Hctheenergybarriercanbeovercomebythermalenergy(denitionofthecoerciveeld).Thus 63

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atH=Hc,theEq. 5{7 canberewrittenas kBT=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23 1)]TJ /F6 11.955 Tf 25.8 8.09 Td[(Hc a22)]TJ /F3 7.97 Tf 6.59 0 Td[(4a1a3 4a3b0!3=2(5{10) TheaboveequationcanbesolvedforHc Hc=Hc0"1)]TJ /F8 11.955 Tf 11.95 16.86 Td[(kBT E02=3#(5{11) whereHc0andE0aregivenbyEq. 5{8 and 5{9 .ThistemperaturedependenceofHcisconsistentwiththeexperimentalresultsshowninFig. 5-3 5-4 and 5-5 Figure5-4. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset2.Thelinearbehaviorisobservedforsampleswithparticlesizefrom12nmto60nmindiameter. 64

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Figure5-5. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset3.Thelinearbehaviorisobservedforsampleswithparticlesizefrom16nmto44nmindiameter. 5.4RelationBetweenMicromagneticParameterandMagneticParameters Herewewilloutlinearoad-maptorelatethemicromagneticparametersa0,a1,a2,a3andb0tothemagneticparametersK,MsandA.Todothatwewillstartwiththemagneticenergyexpression, 3 E(x)=Z"ArM(x) Ms2)]TJ /F6 11.955 Tf 11.96 0 Td[(K(x)(k:M(x))2 M2s)]TJ /F6 11.955 Tf 11.96 0 Td[(0M(x):H)]TJ /F6 11.955 Tf 13.15 8.09 Td[(0 2M(x):Hd(M)#dV(5{12) wherethersttermcorrespondstotheexchangeenergycostduetothespinmisalignment,thesecondtermistheanisotropyenergy,thethirdtermistheZeemanenergyandthefourthtermisthemagnetostaticselfenergy.Thepositionofthedomainwallisgivenbyxandkistheunitvectoralongtheeasyaxis.Theaboveequationshouldbesolvedforreal 65

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sampleswhiletakingintoaccountrealstructureandimperfections.Therealstructureandimperfectionsareresponsibleforthexdependenceofthemagneticparameters(Ms(x),A(x),K(x)).AftersolvingEq. 5{12 andbycomparingthecoecientsofthedierentpowerofx,itispossibletondoutthemicromagneticparametersintermsofmagneticparameters.ThebehaviorsofHc0andE0=kBareshowninthegurebelow. Figure5-6. ThebehaviorsofHc0andE0onparticlediameterareshownforset2samples.Hc0decreasesandE0=kBincreaseswithincreasingparticlesize. TheincreasingbehaviorofE0anddecreasingbehaviorofHc0areconsistentwiththeliterature. 3 Theactualbehaviorcanbeverycomplicatedasitdependsontherealstructurefactorsandimperfectionsinthematerial. 3 5.5Conclusions WehaveinvestigatedthetemperaturedependenceofthecoerciveeldofMDNinanoparticlesinAl2O3matrix.HcdecreaseslinearlywiththeT2=3.Thisbehaviorcanbeunderstoodaccordingtothestrongdomainwallpinning.Weshowthatthegeneralenergy 66

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barrierthatarisesduetostrongdomainwallpinningdependsonthemagneticeldwithapowerof3/2andisresponsibleforthetemperaturedependenceoftheHc. 67

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CHAPTER6COERCIVEFIELDOFFETHINFILMSASTHEFUNCTIONOFTEMPERATUREANDFILMTHICKNESS:EVIDENCEOFNEELDISPERSEFIELDTHEORYOFMAGNETICDOMAINS 6.1Abstract ThetemperaturedependenceofthecoerciveeldofFethinlmshasbeeninvestigated.Threedierentsamplesofdierentthicknessarestudied.Thecoerciveelddecreaseswithtemperatureandfollowsthesametemperaturedependenceastherstorderanisotropyconstant.ThisbehaviorisconsistentwiththetheoreticalpredictionmadebyNeel 15 basedonthedisperseeldtheoryofmagneticdomainwhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions.Thevalueofcoerciveeldincreaseswithdecreasinglmthickness.Thisbehaviorisexpectedformultidomainferromagneticsystemsatnanoscalewherethedomainwallthicknessiscomparabletoorlargerthanthelmthickness. 6.2Introduction Themostinterestingaspectofferromagnetismisthehysteresisloop, 90 whichreferstothehistorydependentbehaviorofmagnetizationwithappliedmagneticeld(Fig. 6-2 ).Hysteresisisacomplexnonlinear,nonequilibriumandnonlocalphenomenon,reectingtheexistenceofanisotropy-relatedmetastableenergyminimaseparatedbyeld-dependentenergybarriers. 3 Anextrinsicpropertyofcrucialimportanceinpermanentmagnetismisthecoerciveeld,themagneticeldwheremagnetizationchangessignasitpassesthroughzero.Thecoerciveeldbasicallydescribesthestabilityoftheremnantstateandisaveryimportantconceptformostpracticalapplications. 91 { 99 Coercivityinferromagnetsisknownfromverylongtime. 90 But,duetothecomplexnature,theoriginofcoerciveeldisstillasubjectofstudy.Inthispresentworkthebehaviorofcoerciveeldofthreedierentironthinlmswithdierentthicknesseshasbeeninvestigated.Thetemperaturedependenceofthecoerciveeldagreeswellwiththetheoryofdomainwallpinningarising 68

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fromsmallinclusions(forexampleimpurityorvacancydefects)wheretheenergyofthefreepoleisnotnegligible. 15 6.3ExperimentalDetails Figure6-1. TEMimageofFethinlmofthickness9nm. ThinlmsofFewerefabricatedonSi(100)andsapphire(c-axis)substratesusingpulsedlaserdepositionfromaluminaandirontargets.HighpuritytargetsofFe(99.99%)andAl2O3(99.99%)werealternatelyablatedfordeposition.Beforedeposition,thesubstrateswereultrasonicallydegreasedandcleanedinacetoneandmethanoleachfor10minandthenetchedina49%hydrouoricacid(HF)solutiontoremovethesurfacesilicondioxidelayer(fortheSisubstratesonly),thusforminghydrogen-terminatedsurfaces. 35 Thebasepressureforallthedepositionswasoftheorderof10)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Aftersubstrateheating,thepressureincreasedtothe10)]TJ /F3 7.97 Tf 6.59 0 Td[(6Torrrange.Thesubstratetemperaturewaskeptatabout550oCduringgrowthoftheAl2O3andFelayers.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.58 0 Td[(2overaspotsize4mm1.5mm.A40nm-thickbuerlayerofAl2O3wasdepositedinitiallyontheSiorsapphiresubstratebeforethesequentialgrowthofFeandAl2O3.Thisprocedureresultsinavery 69

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smoothsubstrateindependentstartingsurfaceforthegrowthofFe,asveriedbyhighresolutionscanningtransmissionelectronmicroscopystudies(Fig. 2-1 ).A40nm-thickcaplayerofAl2O3wasdepositedtoprotecttheFelayerfromoxidation.Threedierentsampleswiththicknessof9nm,21nmand30nmwerepreparedforthepresentstudy.MagnetizationmeasurementsareperformedinaquantumdesignMagneticPropertyMeasurementSystem(MPMS).Themagneticeldwasalongtheplaneofthelms.Thedependenceoftemperatureandlmthicknessoncoerciveeldisinvestigated. 6.4ResultsandDiscussion Figure6-2. M-HloopofFelmofthickness21nmatdierenttemperaturesasindicatedinthelegend.ThecoerciveelddenedatM=0,decreaseswithincreasingtemperature. ShowninFig. 6-1 istheTEMimageofthe9nmthickFesample.DuetothelowsurfaceenergydierencebetweenFeandAl2O3(650mJ/m2)andhighadhesionenergy(1205mJ/m2)betweenFeandAl2O3, 100 FewetsthesurfaceofAl2O3andthusacontinuousthinlmisgrowninsteadofgrainsasisthecasefortheNi/Al2O3systemdiscussedinchapter 2 and 3 .Threedierentsampleswiththickness9nm,21nmand30nmweregrown.Themagnetichysteresisloopsatdierenttemperaturesfrom10Kto350Karemeasuredwiththeappliedmagneticeldalongtheplaneofthethinlm.M-Hloopsfor21nmsampleareshowninFig. 6-2 .Themagneticeldissweptfrom5 70

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Tto-5Tandagainfrom-5Tto5Ttocompletetheloop.AthighmagneticeldsthemagnetizationfromtheFesaturatesandthemagneticelddependencearisesduetothediamagneticcontributionfromthesubstrate.ThediamagneticcontributionfromtheFelmisnegligibleasthethicknessofthelmisverysmallcomparedtothethicknessofthesubstrate.Thusthehighmagneticelddataislinearinmagneticeldwithanegativeslope.TheslopeofthelineisdeterminedfromthelineartofthehighelddataandsubtractedfromtherawdatatoextracttheferromagneticsignalduetotheFethinlm.Theprocedureisrepeatedforallsamplesandforalltemperatures.Thecoerciveeldisfoundtodecreasewithincreasingtemperatures.TheresultsforallthreesamplesarepresentedinFig. 6-3 .Figure 6-4 showstherelationbetweenK(T) 87 andHc(T)whereTistheimplicitvariable. 87 ThelinearbehaviorofHcandKsuggestthatthecoercivitymechanismissimilartothedisperseeldtheoryofmagneticdomainwhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions. 15 Figure6-3. Hcvs.TofFelmsofthickness9nm,21nmand30nm.Thetemperaturedependenceofallthreesamplesissimilar,whichsuggeststhattheoriginofthecoerciveelddependsontheintrinsicpropertyoftheiron,whichinourcaseisthemagnetocrystallineanisotropy.Thisistrueasforthecaseofextrinsicorigintheenergybarrierdependsonthesizeofthesampleasdiscussedinchapter 5 71

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Figure6-4. Hcvs.KofFelmofthickness9nm.ThelinearbehaviorsuggeststhevalidationofNeel'sdisperseeldtheoryofmagneticdomainswhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions(Eq. 6{3 ).Thesimilarbehaviorisobservedforothertwosampleswhichisexpectedastheyhavesimilartemperaturedependence. Theoriesconcernedwiththecoercivityofthemultidomainferromagneticmaterialsbeginwiththeconsiderationofthechangeinmagneticenergyacrossadomainwall. 2 SincethemagnetizationchangesfromMsto)]TJ /F6 11.955 Tf 9.3 0 Td[(Msacrossa180odomainwall,theeectivepressureonthewallwillbethedierenceintheenergyacrossthewallperunitvolume.AtanappliedmagneticeldH,thepressurewillbe2HMs.Thustheworkdonetomoveadomainwallofareastoadistancedxwillbe 2 dW=2HMssdx(6{1) Theaboveequationcanbegeneralizedforanyangle(notonly180o)domainwall. 2 ThetermdW=dx,whichisbasicallytheenergyrequiredtomoveadomainwalltounitdistance,isthemeasureoftheresistanceofdomainwallmotion. 2 Inrealsamplesduetoimpurities,imperfectionsorirregularitiesincrystal,dW=dxpassesthroughmaximumandminimum.Theoriginofreversibilityinthemagnetizationloopistheirreversiblemotionofdomainsthroughthesemaximumandminimum. 2 Thecoerciveeldisusuallycalculatedusingtheequationbelow. 2 72

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Hc=1 2Mss(dW=dx)max(6{2) Dierenttheoreticalapproachesconcentrateonthederivationof(dW=dx)max.SubstantialwallmotionmayrequireeldsoftensorhundredsofOersted.Evidentlyrealmaterialscontaincrystalimperfectionsofonesortoranotherwhichhindertheeasymotionofdomainwalls.Thesehindrancesaregenerallyoftwokinds:inclusionsandresidualmicrostress. 2 Fromamagneticpointofview,an"inclusion"inadomainisaregionwhichhasadierentspontaneousmagnetizationfromthesurroundingmaterial,ornomagnetizationatall.AccordingtotheresultobtainbyNeelinhisdisperseeldtheory(whichbasicallycalculatestheterm(dW=dx)max),coercivityisgivenbyEq. 6{3 below. 15 Hc=1 4v122 KMs+Kv2 Ms(6{3) Thersttermisduetotheresidualmicrostressandv1isthevolumefractionofthefreepoleattheresidualmicrostressand;;KandMsarethesaturationmagnetostriction,internalstress,magnetocrystallineanisotropyconstantandsaturationmagnetizationrespectively.ThesecondterminEq. 6{4 originatesduetothepolesattheinclusionsofvolumefractionv2.Foriron,Ms=1743emu/cm3, 2 K=8x106ergs/cm3, 101 =10x10)]TJ /F3 7.97 Tf 6.58 0 Td[(6 102 and=1010dynes/cm2. 103 PuttingthesevaluesintheEq. 6{3 weobtain, Hc'0:18v1+4600v2Oe(6{4) Thusinourcasethecoerciveeldwillbedominatedbythe2ndterminEq. 6{3 ,whichisproportionaltoKandagreeswellwiththeexperimentalresultsasshowninFig. 6-3 andFig. 6-4 .RememberthatMsisconstant. InFig. 6-5 thevariationofHc(d)isshown.TheincreaseinHcwithdecreasinglmthickness,d,couldbeduetotheenhancementofKwithdecreasingd. 101 104 105 Theincreaseincoerciveeldwithdecreasingthicknessiswellknownforthemulti-domain 73

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Figure6-5. Hcvs.dofFelmsat10K.Coerciveeldincreaseswithdecreasingd.ThisbehaviorisduetofactthatKincreaseswithdecreasingdwhichisverycommoninmultidomainthinlmswhenthedomainwallthicknessiscomparableorlargerthanthelmthickness.Thedomainwallthicknessofironisabout60nm, 3 whichislargerthanthethicknessofthelmsstudiedhere. magneticthinlmswhenthedomainwallthicknessiscomparabletoorhigherthanthelmthickness. 76 Thedomainwallthicknessofironisabout60nm, 3 whichislargerthanthethicknessofthelmstudiedhere.NotethatwehaveusedthevalueofKinEq. 6{3 ,whichislargerthanthebulkvalue.FromEq. 6{4 ,wendthatv20:12. 6.5Conclusion ThecoerciveeldofmultidomainFethinlmshasbeeninvestigated.ThetemperatureandthicknessdependenceofthecoerciveeldagreeswellwiththepredictionmadebyNeel,whichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatthesmallinclusions.ThecontributionofstraintothecoercivityisverysmallforthecaseofFeandthedominantcontributioncomesfromthefreepolesattheinclusions.Thevolumefractionofinclusionshasbeenestimatedfromthecoerciveelddatatobearound0.12. 74

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CHAPTER7SCALINGCOLLAPSEOFTHEIRREVERSIBLEMAGNETIZATIONOFFERROMAGNETICTHINFILMS 7.1Abstract Inferromagneticmaterials,hysteresis,orequivalentlythehistorydependentbehaviorofthemagnetization,reectscomplexnonlinearandnonequilibriumphenomenologythathasbeenrecognizedformanyyears 5 6 106 { 108 .Hysteresisdependsstronglyonmaterialspropertiessuchasstructurallengthscalesspanningthenanometertomicrometerrange 3 andmanifestscomplexbehaviorsincludingmagneticrelaxationwithagingdependence 109 ,dimensionallydependentdipole-dipoleinteractions 48 110 ,spin-glasslikememoryeects 111 andsupermagnetism 112 .Hereweshowthattheirreversiblemagnetization,denedasthedierencebetweeneld-cooledandzero-eldcooledmagnetization,hasastrikingsimilarityforawidevarietyofferromagneticmaterials.Thissimilaritybecomesapparentwhentheirreversiblemagnetizationisnormalizedtoitsmaximumvalueandplottedwithrespecttoatemperaturedependentrenormalizedeld.Thecollapseoftheirreversiblemagnetizationontoasinglecurveforagivensystemimpliesanunderlyingsymmetrytohysteresisthatisnotcapturedbypreviousanalytical 3 5 6 48 106 107 109 { 112 andcomputationaltreatments 108 113 andthusprovidesaunifyingthemethatembracesabroadrangeofcomplexhystereticbehavior. 7.2Introduction Ingeneral,hysteresisisacomplexnonlinearnonequilibriumphenomenonwhichreectsthepresenceofeld-dependentenergybarriersbetweenanisotropy-dependentmetastableminima.Accordingly,hysteresisisaectedbyacombinationofintrin-sicpropertiessuchasmagnetocrystallineanisotropywhichdependsoncrystaleldenergyandspin-orbitcouplingandextrinsicpropertiessuchassampleshape,grainboundaries,disorderandimperfections.Forexample,inbulkferromagnetshysteresisisoftendescribedasasuperimpositionofdomainwallmotionanddomainrotationwithenergybarriersrelatedtomagnetocrystallineanisotropytogetherwithimperfections 75

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and/orimpuritiesinthematerial 107 .Forthesimplestcaseofsingledomaincoherentlyrotatingnanoparticles,theoriginofhysteresisisthenitetimescaleformagnetizationreversalasthemagnetizationovercomesamagneticelddependentenergybarrierbythermalactivationandrotatesfromoneeasyaxisdirectiontoanother 5 6 106 .Hysteresisandassociatedmagnetizationreversalsplayanimportantroleinapplicationssuchasmagneticdatastoragedevices 91 { 93 ,GMR 94 95 orMRAM 96 devices,magneticsensors 97 andmotors 98 ,generators 99 etc. 7.3ExperimentalResults Measurementsofirreversiblemagnetizationareusuallyaccomplishedbyoneorbothoftwotechniquesillustratedrespectivelyinpanelsa)andb)ofFig. 7-1 .Thesampleunderinvestigationinthisgure(hereafterreferredtoasampleA)isa20A-thickpulsedlaserdepositedthinlmcomprisingvelayersof3nmelongatedNiparticlesembeddedinaninsulatingAl2O3host[seeMethods].TheNiparticlesaresmallenoughtobeinthesingledomain(SD)regimewhereallthespinsarealignedinthesamedirectionandrotatecoherentlytogetherinresponsetoachangingmagneticeld.Inthersttechnique(panela),thesampleiseldcooled(FC)inaeldH=20Oe(blacksquares)from300Kto5Kandthenzeroeldcooled(ZFC).Theirreversiblemagnetization(M(H;T)),whichisafunctionofmagneticeld(H)andtemperature(T),isshownbythetwo-headeddashedverticalarrow.Inthesecondtechnique,thesampleisheldatxedtemperatureTandmagnetizationMloopsobtainedbyrepeatedlycyclingtheappliedeldHaboutH=0betweentwosymmetriclimits.Thehistorydependenttrajectoriesformclosedloopsshownschematicallyintheinsetsofpanel(b).Theseloops,whichcanbeacquiredatdierenttemperatures,areeachcharacterizedbyacoerciveeldHc(T),asaturatedmomentMs(T)andaremnantmagnetizationMr(T).ThecoerciveeldHc(T),plottedversusT1=2inpanelbforsampleAdiscussedabove,isshownasabluelineconnectingthestarreddatapoints.TheabsenceofaT1=2dependenceforcoherentlyrotatingSDparticleswitheasyaxesorientedalongtheeld 6 willbediscussedbelow.WhilebothM(H;T)andHc(T) 76

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Figure7-1. IrreversiblemagnetizationM(H;T)denedasthedierenceoftheFCandZFCmagnetizationsisaquantitativemeasureofhysteresis.a,BlacksquaresandredcirclesrepresentrespectivelytheFCandZFCtemperature-dependentmagnetizationsforsampleAina20Oeeldappliedparalleltothelmsurface.TheirreversiblemagnetizationM(H;T)isshownbythetwo-headedverticaldottedarrow.Inset:Largethermallyblockedmagneticparticles(1and3)withrespectiveverticalandhorizontaleasyaxesofmagnetizationindicatedbythearrows.Particle2issmallenoughtobesuperparamagnetic(thermallyunblocked).Formagneticeldsappliedalongtheverticaldirection,particles2and3donotcontributetoM(H;T).b,Thecoerciveeld(Hc)forthesamesampleshownasafunctionofT1=2(Bluestars)doesnotshowthelinearbehaviorexpectedforidealStoner-Wohlfarthparticles 6 wherealltheparticlesareuniformsizeandalignedwiththeappliedeld(particle1).Thesolidbluelineisattothedatausingalognormaldistributionofparticlesizetogetherwitharandomorientationoftheeasyaxisofthemagnetizations.Inset:Schematicmagnetizationcurvesforthecaseswhere(b1)onlyparticles1and2and(b1)onlyparticles1and3arepresent.Theresultantmagnetizationloops(blackcurves)forthetwocasesshowthepronouncedeectsofparticlesizeandeasyaxisorientationonthedeterminationofHc. arecommonlyacceptedmeasuresofhysteresis,theunderlyingphenomenologyforeachisconsiderablydierent.ForexampleinnanoparticlemagneticsystemssuchasshowninFig. 7-1 ,M(H;T)andHc(T)aresensitiveindierentwaystoboththepresenceofsuperparamagneticparticlesandtherandomorientationoftheeasyaxisofmagnetizationofeachparticle.TheinsetsofFig. 7-1 a)andb)schematicallyillustratethesedierencesusingthreeparticles:particles1(red)and3(blue)witheasyaxesofmagnetization 77

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respectivelyalongthevertical(H)andperpendiculardirectionsandlargeenoughsothatatthetemperatureofmeasurement,changesinmagnetizationareblockedbyenergybarriersthatcannotbeovercomebythermalactivation,andparticle2(green)witheasyaxisalongtheverticalaxisandsmallenoughsothatitissuperparamagneticwithanon-hystereticmagnetizationdependingonlyonHandT(i.e.,unblocked).Considerthecasewhereonlyparticles1and2arepresent.Sincethesuperparamagneticparticle2willhavezeroM(H;T),thetotalM(H;T)valuewillbeonlyduetotheblockedparticle1.Ontheotherhand,theM-Hloopwillbethesummedcontributionsofthehystereticloopforparticle1andthereversibleloopforparticle2asshownininsetb1ofFig. 7-1 b.ThusforthiscaseM(H;T)isnotaectedbysuperparamagneticparticlesbutHc(T)is.Considernowthecasewhereonlyparticles1and3arepresent,i.e.,bothparticlesareblockedbutwithdierent(parallelandperpendicular)easyaxisorientationswithrespecttotheappliedeld.SimpletheoryforcoherentlyrotatingSDparticlesshowsthatuponreversingtheeldparticle1mustsurmountaH-dependentenergybarrierwhereasparticle3canlineupwithouthavingtoovercomeanenergybarrier 3 .Inlikemannertothepreviouscase,TheM-Hloopforparticle1showshysteresisbutparticle3doesnot(insetb2ofFig. 7-1 b).HenceHc(T)isaectedbythepresenceofparticleswithperpendicularorientationbutM(H;T)isnot. TheaboveargumentscanbegeneralizedforSDmagneticnanoparticleswithabroadsizedistributionandarandomorientationoftheeasyaxesofmagnetization.Particleswithaneasyaxismakinganarbitraryanglewiththeappliedmagneticeldwillalwayscontributelesstohysteresisthanalignedparticles.Becausetheperpendicularcomponentdoesnotcontribute,thecorrectionisastraightforwardintegrationoverangle 7 .Accordingly.themeasuredvalueofM(H;T)willbeonlyduetoblockedparticlesanddominatedbyparticleswitheasyaxisofthemagnetizationalongtheappliedmagneticeld.Ontheotherhand,Hc(T)willbestronglyaectedbytherandomorientationoftheeasyaxisofblockedparticlesandthepresenceoftheparticleswhicharesmallenoughto 78

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besuperparamagnetic.ForanidealStoner-Wohlfarthparticlesystem 6 inwhichalltheparticlesarealignedalongtheHdirection,alinearsquareroottemperaturedependenceofHc(T)isexpected.FortheconsiderablymorecomplicatedcaseofrandomlyorientedSDparticleswithasizedistributioninwhichsomeoftheparticlesaresuperparamagnetic(e.g.,sampleA),Hc(T)canbedescribed(solidlineofFig. 7-1 b)usingamodelwithalognormalsizedistributionandthreettingparameters 9 10 (seeMethods).TheH-dependentbehaviorofM(H;T)forsampleAisshowninFig. 7-2 aforthetemperaturesindicatedinthelegend.Theisothermsshowpeaks,Mmax(T),atmagneticelds,Hm(T),indicatedbytheverticalarrows.Thesepeaksareexpected,sinceatH=0theFCandZFCmeasurementsareequivalentandthedierenceinmagnetizationshouldbezero,whereasathighHbothFCandZFCmagnetizationssaturatetothesamevalueandagainthedierenceshouldgotozero.ThesimilaritiesintheM(H;T)isothermsaresuggestiveandbecomemanifestasanunexpecteddatacollapseontoasinglecurvewhenthereducedirreversiblemagnetization,M(H;T)=Mmax(T),isplottedasafunctionofreducedmagneticeld,h(T)=H=Hm(T),asdoneinFig. 7-2 b.ThecharacteristiceldHm(T)deviatessignicantlybelowHc(T)forT<25K(insetofFig. 7-2 b)andidentiestheT-dependenteldwhereirreversibilityisatamaximum. AnimportantphysicalinsightintothescalingcollapseshowninFig. 7-2 bisgainedbyplottingMmax(T)asafunctionofT1=2.TheobservedlinearbehaviorshowninFig. 7-3 aisidenticaltothepredictedtemperature-dependentcoercivityHSWc(T)ofStoner-Wohlfarth(SW)particles 6 mentionedabove.Guidedbythissimilarity,wemaketheansatz:Mmax(T)/HSWc.Thisansatzisphysicallyreasonablesinceasshownabove,M(H;T)measurementsarenoteectedbythepresenceofsuperparamagneticparticles,andinadditionthecontributionfromparticleswitheasyaxisofmagnetizationalongthemagneticeldisdominant.Asallrealsamplescompriseparticlesizedistributionsandrandomeasyaxisorientations,theconventionalM-Hloopsfromwhichcoercivitiesareextractedaremarkedlydierentthanwouldoccurforidealized 79

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Figure7-2. AllofthedataforM(T;H)ofsampleAcan,withpropernormalization,bemadetocollapseontoasinglecurve.a,IsothermsofMplottedasafunctionofHshowself-similarbehaviorwithmaximaMmax(T)occurringatcharacteristiceldsHm(T)markedbytheverticalarrows.WithdecreasingtemperatureMmax(T)increasesandHm(T)movestohigherelds.b,ThescalingcollapseoccurswhenM(H;T)=Mmax(T)isplottedagainstthenormalizedeldH=Hm(T).Inset:PlotsofHm(squares)andHc(T)(stars)asafunctionofT.ThedataandtforHcarethesameasshowninFig.1b,butplottedwithrespecttoTratherthanT1=2.ThesolidlineforHm(T)isaguidetotheeye. SWbehavior.TheinsensitivityofM(H;T)measurementstosuperparamagneticparticlesandperpendicularorientationsofblockedparticlestogetherwithouransatzimplythatmoreusefulinformationaboutthemagnetizationreversalprocessisobtainedfromM(H;T)measurementsthanHc(T)measurements. 80

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Figure7-3. ForsingledomainparticlestheansatzMmax(T)/HSWcisveried.a,ThelineardependenceofMmax(T)onT1=2isinaccordancewiththeStoner-WohlfarththeoreticalpredictionthatHc(T)/T1=2forthecoherentlyrotatingsingledomainparticlesofsampleA.TheuseofMmax(T)ratherthanHc(T)asameasureofhysteresisremovestheeectsofnonuniformparticlesizeandrandomorientation.b,ThevaluesofHcalc(T),computedforuniformsizeFePtparticles(sampleB)fromtheansatz-derivedEq.2attheindicatedtemperatures,scalelinearlywithHSWc(T)whichisdeterminedfromthecoerciveeldsofM)]TJ /F6 11.955 Tf 11.95 0 Td[(Hloopsaftercorrectionfortherandomorientationoftheeasyaxisofmagnetization(seeMethods). ThescalingcollapsebehaviorplottedforsampleAinFig. 7-2 bsuggeststhatMhastheform, M(H;T)=Mmax(T)F(H=Hm(T));(7{1) whereFisanunknownfunctionwiththepropertyF(1)=1.TakingthesecondpartialderivativeofthebothsidesofthisequationandsolvingforMmax(T)givestheresult:Mmax(T)=(@2F(h)=@h2))]TJ /F3 7.97 Tf 6.59 0 Td[(1h=1H2m(T)@2M(H;T)=@H2Hm(T),evaluatedatthe 81

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maximumwhereh(T)=H=Hm(T)=1.Therstterm,(@2F(h)=@h2))]TJ /F3 7.97 Tf 6.58 0 Td[(1h=1,mustbeaconstantbecauseofthescalingcollapse.Theremainingtwotermshavethesameformastheexpectedvalueofaneectiveanisotropyeld 114 .Thusaccordingtoouransatz,Mmax(T)/HSWc,wecanwritetherelation HSWc(T)=CH2m(T)@2M(H;T)=@H2Hm(T)=CHcalc(T);(7{2) whereCisaconstantandHcalc(T)=H2m(T)@2M(H;T)=@H2Hm(T)iscomputedfromexperimentaldata. TotesttheresultexpressedinEq.2,weusemagnetizationdataonasystemofuniformsize(6nmdiameter)FePtnanoparticles(sampleB,seeMethods)synthesizedviathermaldecompositionofFe(CO)5andreductionofPt(acac)2 115 .ThemeasuredcoerciveeldHMHc(T)isobtainedfromM-Hloopsatdierenttemperatures.Assumingthattheparticleseasyaxesofmagnetizationarerandomlyoriented,atemperaturedependentcorrectiontoHMHc(T)mustbemadesothatHSWc(T)canbeinferred.Thiscorrectionisneeded(seeMethods)sincethemagnetizationofrandomlyorienteduniformsizeparticlesshowsaT3=4dependence 7 comparedtotheT1=2dependenceexpectedforanidealizedsampleinwhichalltheparticlesarealignedalongtheappliedmagneticeld.TheplotofFig. 7-3 includesthiscorrectionandshowsalineardependenceofHcalc(T)onHMHc(T)aswouldbeexpectedforasystemofcoherentlyrotatingSDuniformsizeFePtnanoparticles.Thelinearbehaviorwithslopeof1.09(4)conrmsthevalidityofourmodelasexpressedbyEq.2withC1.InFig. 7-4 thereducedirreversiblemagnetization,M(H;T)=Mmax(T),isplottedasafunctionofreducedmagneticeld,h(T)=H=Hm(T)forawidevarietyofthin-lmmagneticmaterialslabeledintheinsetanddescribedinMethods.Unexpectedly,whenplottedinthismanner,themagnetizationdataforeachmaterialssystemcollapseontosinglecurveswhichhavesimilarshapesdescribedbyauniquefunctionF(H=Hm(T))(seeEq.1)foreachsystem.Thisdatacollapseisquiteremarkableconsideringthevarietyofmechanismsresponsible 82

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Figure7-4. Scalingcollapsedescribesirreversiblemagnetization(hysteresis)inawidevarietyofferromagneticmaterials.a,PlotsofthereducedirreversiblemagnetizationM(H;T)=Mmax(T)asafunctionofreducedmagneticeldH=Hm(T)forthesixdierentmagneticmaterialslabelingeachcurve.TheimplicittemperaturevariableincreasesfromlefttorightandeachcoloronagivencurverepresentsadierentM(H;T)isotherm.Forclarity,theordinatevalueshavebeenshiftedandthesolidlineshavebeenaddedasguidestotheeye.Thebottommostcurveforsinglecrystalspin-glassCu:Mn(at.1.5%)materialistakenfromtheliterature 116 formagneticorderingtogetherwithawiderangeofmaterialsproperties.Thescalingcollapseappliesequallywell:tosingledomain(SD)coherentlyrotatingNinanoparticleswithaverageparticlediameterof3nmembeddedinaninsulatingmatrix;tomultidomain(MD)incoherentlyrotatingNinanoparticleswithaverageparticlediameterof12nmalsoembeddedinaninsulatingmatrix;tocoherentlyrotatinguniformsizeFePtnanoparticles(6nmdiameter);tocontinuousmetallicGdthinlmswithmagnetizationderivedfromlocalmomentssupplementedbybandstructureexhibitingsomeitinerantcharacter;andtomixed-phasemanganite(LPCMO)wheretheMnspinsorderbyadoubleexchangemechanisminanenvironmentwherecharge-orderedandparamagneticinsulatingphasescompetewithaferromagneticmetallicphase.Wehavealsoincludedaspinglassmaterial,singlecrystalCu:Mn(1.5at%),describedintheliterature 116 .ThecollapseoccursfornanoparticlesystemswhichincludeSD,MDandsuperparamagneticparticleswithboth 83

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broadanduniformsizedistributionsandmagneticmomentreorientationsarisingfromcomplexsuperimpositionsofdomainwallmotionanddomainrotationwhichcanbeverydierentincontinuouslmscomparedtonanoparticleswheretheextentofthedomainwalliscomparabletothesizeoftheparticle.Thusthematerialsmeasuredhereareverymuchdierentintermsoftheoriginoftheirreversiblemagnetization. Inconclusion,wehavepresentedaheretoforeunreportedphenomenologicalresultshowingthatthetemperatureandelddependenthysteresisofatleastsixdistinctlydierentmagneticsystemscanbecollapsedontosinglecurves(Fig. 7-4 )usingtheparticularlysimplefunctionalformexpressedbyEq.1.Ourndingthatsuchascalingcollapseappliestomagneticsystemstotallydierentthanthespinglassforwhichsimilarscalinghasbeenpreviouslynoted 116 117 (thelowestcurveinFig. 7-4 )suggeststhatexplanations 116 relyingonspinglassphenomenologyaretoonarrow.Thusamoregeneraltheoryisneededtoexplainthescalingcollapse.Thiscollapsemustimplyanunderlyingsymmetrythatisnotcapturedbypreviousanalyticalandcomputationaltreatmentsandmaybeacrucialcluetounderstandthecomplexhistorydependentmagnetizationprocess.ThesimilaritywiththespinglassmaterialisinterestingandmaybebecauseofthefactthatallZFCmagnetizationisametastablestateofthesystemandshowspropertiesvaryingwithtime.Thusthedynamicsofthespinsmayplayaveryimportantroleforthescalingcollapse.ThebehaviorofMmax(T)isinvestigatedforthe3nmNiparticles.WesuggestanewapproachtoinvestigatethemagnetizationreversalprocessfromtheMmax(T)measurement.WehavesuccessfullyappliedthemethodforthecoherentlyrotatingSDparticles. 7.4Conclusions Inconclusion,wehavepresentedasurprisinglygeneralandunrecognizedphenomenologicalresultshowingthatthetemperatureandelddependenthysteresisofatleastsixdistinctlydierentmagneticsystemscanbecollapsedontosinglecurves(Fig. 7-4 )usingtheparticularlysimplefunctionalformexpressedbyEq.1.Wehavenotyetfound 84

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anyexceptions.Fortheparticularlysimplecaseofcoherentlyrotatingsingledomainparticles(samplesAandB),ouranalysisofscalingcollapsebypassesthecomplicationsofnonuniformsizedistributionsandrandomeasyaxisorientation,unveilinganunderlyingStoner-Wohlfarthbehavior 6 .Ourndingthatthesamescalingcollapsemoregenerallyappliestomagneticsystemswithawidevarietyofmechanismsgivingrisetohysteresis,suggeststhatexplanations 116 relyingonspinglassphenomenologyaretoonarrow.Thusamoregeneraltheoryisneededtoexplainthescalingcollapse.Thiscollapsemustimplyanunderlyingsymmetrythatisnotcapturedbypreviousanalyticalandcomputationaltreatmentsandmaybeacrucialcluetounderstandcomplexhistorydependentmagnetizationprocesses.Thesimilaritywiththespinglassmaterial 116 issuggestiveandmayberelatedtothefactthatallZFCmagnetizationsrepresentmetastablestatesofthesystem,whichifgivensucienttimewouldrelaxtowardtheeld-cooledequilibriumstate.Accordingly,thedynamicsofthespinsmayplayaveryimportantroleinunderstandingthescalingcollapse. 7.5Methods 7.5.1NiNanoparticle. CompositelmscomprisingmagneticNinanoparticlesembeddedinanAl2O3hostmatrixweresynthesizedbypulsedlaserdeposition(PLD) 36 .HighpuritytargetsofNi(99.99%)andAl2O3(99.99%)werealternatelyablatedinthesamedepositionrun.Thebasepressureofthedepositionwasontheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(7Torr.Thesubstratetemperaturewasmaintainednear550Cduringthegrowth.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.58 0 Td[(2overaspotsizeof4mm1.5mm.A40nm-thickbuerlayerofAl2O3wasdepositedinitiallyonthesapphiresubstratebeforethesequentialgrowthofNiandAl2O3.ThisprocedureresultsinaverysmoothstartingsurfaceofgrowthofNiasveriedbythehighresolutionscanningtransmissionelectronmicroscopy(STEM)studies.Samplesconsistsof5layersofNiandAl2O3.AcaplayerofAl2O3wasalwaysusedtoprotectthesamplefromoxidation.Z-contrastSTEMimage 85

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veriestheabsenceoftheNioxide.SamplesstudiedhereconsistsofNiparticlesizeof3nmand12nmindiameter. 7.5.2GdThinFilm. GdthinlmsweredepositedonSisubstratesbyDCmagnetronsputtering.Thebasepressureofthechamberwasontheorderof510)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Thesamplesarecontinuouswiththicknessnear100nm. 7.5.3(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)ThinFilms. Phaseseparatedmanganite(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)lmsweregrownusingpulsedlaserdeposition(PLD)atarateof0.05nm/sonNdGaO3(NGO)(110)substrateskeptat820Cinanoxygenatmosphereof420mTorr 118 7.5.4TemperatureCorrectionofCoerciveField. ForidealStoner-Wohlfarthparticlesthecoerciveeldisgivenby,HSWc=Hco(T)(1)]TJ /F1 11.955 Tf -453.9 -23.9 Td[((T=TB)1=2).Forthecaseofrandomlyorientedcoherentlyrotatingparticlesallofwhichhavethesamesize,thecoerciveeldisgivenby,Hranc(T)=0:48Hco(T)(1)]TJ /F1 11.955 Tf 12.36 0 Td[((T=TB)3=4).SincetheFePtparticlesofsampleBallhavethesamesize,wecanwriteHranc(T)=HMHc(T)whereHMHc(T)isthecoerciveeldextractedfromthemagnetizationloops.Itisthenstraightforwardtoshowthat,HSWc(T)=2HMHc(T)(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)1=2)=(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)3=4).Usingthefactthatatloweldsthezero-eld-cooledpeaktemperaturecanbeidentiedastheblockingtemperature 119 ,wendthatTB=60KfortheFePtnanoparticles.AccordinglyHSWc(T),theabscissainFig. 7-3 b,canbecalculatedinastraightforwardmannerfromthemeasuredvaluesofTBandHMHc(T). 86

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BIOGRAPHICALSKETCH RiteshKumarDaswasbornon1981inaverysmallvillage,Mohanbati,locatedeastofIndia.AtHaripalGuruDayalhighschool,Riteshwasintroducedtophysicsandfellinlovewiththesubject.Riteshgotinterestedinsciencesubjectsduringhishighschooldays,wherehefoundwonderfulteachers,Mr.RobinChatterjeeandMr.UttamSaha.Aftercompletingthehighschool,RiteshenrolledintheRamakrishnaMissionVidyamandira(Belur)inAugust1999andgraduatedwithaBachelorofSciencedegreewithhonorsinphysicsonAugust2002.DuringthistimeRiteshmethisfavoriteteacherDr.DeepakGhsosh,whoplayedaverycrucialroletomakeRiteshinterestedinexperimentalphysics.AftercompletingtheBachelorofSciencedegree,RiteshenrolledinMasterofSciencedegreeatIndianInstituteofTechnology(Kanpur)andcompletedthedegreeonAugust2004.AfterthisRiteshjoinedUniversityofFloridatopursuehisPh.D.degree.RiteshwasalwaysinterestedinthepropertiesofmaterialsatnanoscaleandfoundthatProf.A.F.Hebard'slabtobetheperfectplacefortheresearchintheeldofthinlmsandnanoparticles.RiteshgraduatedinAugust2010withaDoctorofPhilosophydegreeinphysics. 96