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

Transcription

1 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

2 c 2010 Ritesh Kumar Das 2

3 I dedicate this to my parents and family for their active support. Without them it would have not been possible. 3

4 ACKNOWLEDGMENTS I am truly indebted to many 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 company. 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 enjoyed 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 N 2 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. 4

5 I am indebted to my parents for their support, encouragement and for always believing in me. I appreciate the warmth and affection of my sister Mridula. I could not have come this far without their blessings. 5

6 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 THEORY AND BACKGROUND Single Domain and Multi Domain Particles Hysteresis Loop of Single Domain Coherently Rotating Particles Hysteresis Loop of Multi Domain Particles Magnetization vs. Temperature Zero Field Cooled (ZFC) Magnetization Field Cooled (FC) Magnetization DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL SINGLE DOMAIN GRAIN SIZE OF NI IN LAYERED Ni/Al 2 O 3 COMPOSITES Abstract Introduction Experimental Details Data and Discussion Conclusion Methods Mathematical Analysis Basic Physical Understanding EFFECT OF DIPOLAR INTERACTION ON THE COERCIVE FIELD OF MAGNETIC NANOPARTICLES: EVIDENCE FOR COLLECTIVE DYNAMICS Abstract Introduction Results and Discussions Conclusions FINITE SIZE EFFECTS WITH VARIABLE RANGE EXCHANGE COUPLING IN THIN-FILM Pd/Fe/Pd TRILAYERS Abstract Introduction Experimental Details

7 4.4 Results and Discussion Conclusions TEMPERATURE DEPENDENCE OF COERCIVITY IN MULTI DOMAIN NI NANOPARTICLES, EVIDENCE OF STRONG DOMAIN WALL PINNING Abstract Introduction Results and Discussions Relation Between Micromagnetic Parameter and Magnetic Parameters Conclusions COERCIVE FIELD OF FE THIN FILMS AS THE FUNCTION OF TEMPERATURE AND FILM THICKNESS: EVIDENCE OF NEEL DISPERSE FIELD THEORY OF MAGNETIC DOMAINS Abstract Introduction Experimental Details Results and Discussion Conclusion SCALING COLLAPSE OF THE IRREVERSIBLE MAGNETIZATION OF FERROMAGNETIC THIN FILMS Abstract Introduction Experimental Results Conclusions Methods Ni Nanoparticle Gd Thin Film (La 1 y Pr y ) 0.67 Ca 0.33 MnO 3 (LPCMO) Thin Films Temperature Correction of Coercive Field REFERENCES BIOGRAPHICAL SKETCH

8 Table LIST OF TABLES page 1-1 H c vs. T

9 Figure LIST OF FIGURES page 1-1 SD and MD particle Coherent and incoherent rotation Single particle in magnetic field Two state energy Hysteresis of SD particle Diagram of a particle Thermal average of magnetization Flow diagram MH below T B MH below T B SD to MD transition and Hc Magnetization loop for MD particle Domain wall and Hc M vs. T for 3 nm Ni nanoparticles STEM image of Ni particle H c vs. d, different T d c vs. T H d and domain Sample MH loop Hc vs. d: dipolar interaction Dipolar interaction Physical and magnetic view of sample Saturation magnetization vs. x Coercive field vs. x

10 4-4 Curie temperature vs. x Three sets of sample MH loops of set H c vs. T 2/3 set 1 samples H c vs. T 2/3 set 2 samples H c vs. T 2/3 set 3 samples H c0 and E 0 of set TEM image of Fe film M-H loop of Fe film H c vs. T of Fe film H c vs. K of Fe film H c vs. d of Fe film Irreversible Magnetization Behavior of the M(H, T ) isotherms as the function of H and scaling collapse The anstz Scaling collapse of variety of ferromagnetic materials

11 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 Philosophy MAGNETISM IN NANOSCALE MATERIALS, EFFECT OF FINITE SIZE AND DIPOLAR INTERACTIONS Chair: A. F. Hebard Major: Physics By Ritesh Kumar Das August 2010 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. 11

12 CHAPTER 1 THEORY AND BACKGROUND Ferromagnetism is known for more than 2500 years to man. The first magnetic material discovered was magnetite (Fe 2 O 3 ). 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. Chapters 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 by 1 Emag = µ0m 2 s V/12 (1 1) where µ 0 is the free space permeability, M s is the saturation magnetization and V is the volume of the particle. The energy required to form a Bloch domain wall is 1 3 E dw = 4π AKR 2 (1 2) where K is the anisotropy constant, A is the exchange stiffness and R is the radius of the particle. Note that E mag grows as R 3 and E dw grows as R 2. 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 (R sd ) where the 12

13 transition from single domain (SD) to multidomain (MD) occurs is given by 1 3 R sd = 36 AK µ 0 M 2 s (1 3) The above equation is determined by solving the equation E mag = E dw. 1 3 Thus particles having radius smaller than R sd are SD and particles having radius larger than R sd are MD (see Fig. 1-1). Figure 1-1. Smaller (larger) particles are SD (MD) 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 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 length 2 4 A l ex = (1 4) µ 0 Ms 2 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 l ex rotate incoherently (coherently). The exchange length is usually larger than R sd 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). 13

14 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 l ex 1.2 Hysteresis Loop of Single Domain Coherently Rotating Particles The magnetization dynamics of the SD particles with R < l ex will be coherent and the particle can be treated as a giant single spin of value M =M s V. When a magnetic field (H) is applied along the easy axis of the magnetization (k) the energy of the particle is 2 4 E(H) = KV sin 2 θ M s V Hcosθ (1 5) where θ is the angle between the applied magnetic field and the direction of magnetization as shown in the Fig 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 θ which shows two energy minima separated by a barrier. The energy minima occur at θ = 0 (corresponding to the magnetization along the applied magnetic field or up direction) and θ = π (corresponding to the magnetization opposite to the applied magnetic field or down direction). The maximum 14

15 Figure 1-3. A SD particle in an applied magnetic (H ) field along the easy axis of magnetization (k). θ is the angle between the magnetization M and the easy axis k. of the energy occurs at θ = π/2 separating the two energy minima. Figure1-4 a) shows the energy diagram at zero magnetic field as a function of θ. 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 ± H H co ] 2 (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 H c0 = 2K/M s. Derivation of the Eq. 1 6 is given below. 15

16 a) b) k B T k B T E H=0 E H KV E - =KV(1-H/H co ) 2 E + =KV(1+H/H co ) 2 0 H = 0 0 H Figure 1-4. 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 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 θ is δe(h) δθ = 2KV sinθcosθ + M s V Hsinθ (1 7) At the maxima and minima 2KV sinθcosθ + M s V Hsinθ = 0 (1 8) Solutions of the above equation are sinθ = 0 (1 9) cosθ = M sh 2K (1 10) 16

17 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 θ = 0 and θ = π and maximum at when cosθ=-m s H/2K (see Fig. 1-4). Energies correspond to these extrema are E min+ = M s V H (1 11) E min = M s V H (1 12) [ ( ) ] 2 Ms H E max = KV 1 + (1 13) 2K where E min+ and E min corresponds to θ = 0 (spin up) and θ = π (spin down) respectively. It is now easy to show that E + (H)=E max - E min+ and E (H)=E max - E min are given by Eq 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 time 5 ( ) E± (H) τ ± = τ 0 exp k B T (1 14) where T is the temperature, τ 1 0 is the inverse attempt frequency of overcoming the energy barrier and k B is the Boltzmann constant. Figure1-5 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 17

18 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 = H c the energy 4 3 Figure 1-5. 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. 2 1 barrier E (H c ) is such that the relaxation time τ = τ m at the temperature T. Where τ m is the experimental measurement time (around 100 sec for SQUID measurement). Thus when H=H c, magnetization reversal occurs. Combining Eq. 1 6 and 1 14, the coercivity (H c (T )) of the SD particle can be calculated as shown below. ( ) E (H c ) τ = τ m = τ 0 exp k B T E (H c ) = k B T ln τ m τ 0 18

19 Now using the expression of E (H) from Eq. 1 6 it is easy to show that 6 ( T H c = H c0 [1 T B ) ] 1 2 (1 15) where T B = KV/k B ln(τ m /τ 0 ) is known as the blocking temperature. Below T B the anisotropic energy barrier is larger than the thermal energy and magnetization is blocked or trapped. Above T B 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 that H c decreases with increasing temperature and above the blocking temperature (T > T B ) the particles lose their coercive field. Note that the origin of H c 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 (ψ) is considered. The energy of the particle in this case is 7 E(H) = 2KV ( 1 2 sin2 (ψ θ) H ) cosθ H c0 The magnetization of the particle at an applied magnetic field is given by Mcosθ min, where θ min 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 θ values around the θ min will be occupied with a finite probability according to the (1 16) 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 ψ. The procedure should be repeated for all values of H to get the M-H loop for a particular value of ψ. Then the 19

20 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 θ. The angle between H and k is ψ. The energy barrier for this general configuration is given by Eq M-H loops for all possible ψ 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

21 Figure 1-7. At finite temperature other θ values around θ 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. ( ) 1 E(H) = 2KV 2 sin2 (ψ θ) H H c0 cosθ 2 Find the minima of E(H) M s V Cosθ min will be the magnetization at T =0 for the given value of H, ψ 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. 21

22 3 Thermal average of the magnetization θ2 θ M s V M(H, ψ, V ) T = 1 cosθexp k E B T dθ θ2 θ 1 exp k E B T dθ E = E θ E θmin θ 1 and θ 2 are shown in Fig. 1-7 M(H, ψ, V ) T is the magnetization at temperature = T for the given value of H, ψ and V 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, ψ and V 5 Step 1, 2, 3 and 4 should be repeated for all possible ψ 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. Figure 1-8. continued 22

23 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 Table1-1. Figure 1-8. continued Magnetization loops at different temperatures for a single layer sample of Ni nanoparticles of average diameter around 18 nm are shown in Fig 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 c =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 M s V (thus called 23

24 M (emu) 6E K 50 K 100 K 150 K 200 K -6E H (Oe) Figure 1-9. Hysteresis loop of a Single layer Ni nanoparticles of 18 nm diameter embedded in an Al 2 O 3 matrix at temperature, T < T B. 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 M(H, T ) = NM s V [coth M sv H k B T k BT M s V H ] (1 17) where N is the number of particles. Note that M is a function of H/T in the above Eq 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 for a single layer sample of 12 nm Ni grains in an Al 2 O 3 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 (R sd < l ex ) 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 24

25 4.5E-5 M(emu) M275K M300K M325K E H/T(Oe/K) Figure Hysteresis loop of a SD coherently rotating particle at temperature (T > T B ). 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 be 2 H cmd = a + b/d x (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 25

26 Figure 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 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 multilayer and single layer samples of Ni particles in Al 2 O 3 matrix as will be discussed in detail in chapter 2. Figure1-12 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 (M s ) 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 26

27 magnetization (M r ) will be seen at zero magnetic field. Reversing the magnetic field will Figure 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. 27

28 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 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 M s H (+M s H) per unit volume. Thus the energy difference across the domain wall will be 2HM S 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 is 2,11 dw = 2M s HSdx (1 19) where S the area of the domain wall. Thus the work done to move the wall by unit distance is 2,11 dw/dx = 2MsHS (1 20) 28

29 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 by 2,11 H c = 1 2M s S (dw/dx) max (1 21) There are different theoretical models to calculate (dw/dx) max for different imperfections in the sample and the results for some of them are listed in Table1-1. Table 1-1. Table here lists some known models along with the variation of coercive field according to the model. Theory H c System References Stoner-Wohlfarth H c = 2K Ms [1 (k BT ln τm τ 0 /KV ) 1/2 ] SD, CR nanoparticle with uniaxial anisotropy along the applied magnetic field 2,6 Stoner-Wohlfarth H c = 0.96K Ms [1 (k BT ln τ m τ0 /KV ) 3/4 ] SD, CR nanoparticle with uniaxial anisotropy randomly oriented 7 Micromagnetic H c = Ms 2K [ 3 a 3 k 4 B T ln τ m τ0 ] 2/3 MD, 2 phase material, hard magnet, a 3 is the micromagnetic parameter and depends on the K, M s, A 12 Inclusion Theory H c = γα2/3 Msd MD system, d < δ, free pole energy is ignored, coercivity is assumed to be equal to the maximum pining field, d is the diameter of the inclusion, δ is the domain wall thickness, γ is the domain wall energy per unit area, α is the volume fraction of the inclusion 11,13 Inclusion Theory H c = 1.75 γα1/2 MsL (ln 2L ) MD system, d > δ, free pole energy is d ignored, L is the linear dimension of the sample 11,14 Inclusion Theory Inclusion Theory H c = 2.8 γα1/2 MsL ( d δ )3/2 (ln 2L ) MD system, d δ nored < δ, free pole energy is ig- 11,14 H c = πms 2Kα [ log 2πMs 2 ] MD system, d < δ, free pole energy is considered K 15 Inclusion Theory H c = 3γlα/M s d 2 MD system, closer domain, large inclusion, commonly seen in the case of Neel s spike, l is the equilibrium length of the spike 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 29

30 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 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 (T irr ). For nanoparticles T irr is same as the blocking temperature (T B ) x10-5 FC H = 20 Oe 1.0x10-5 M(emu) 5.0x ZFC T (K) Figure 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 Zero Field Cooled (ZFC) Magnetization Zero field cooled magnetization is measured by cooling the sample from high temperature (temperature above the irreversible temperature (T irr )) 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 30

31 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 T irr, 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 T irr 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 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. 31

32 CHAPTER 2 DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL SINGLE DOMAIN GRAIN SIZE OF NI IN LAYERED NI/AL 2 O 3 COMPOSITES 2.1 Abstract Pulsed laser deposition has been used to fabricate Ni/Al 2 O 3 multilayer composites in which Ni nanoparticles with diameters in the range of 3-60 nm are embedded as layers in an insulating Al 2 O 3 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 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 media 24 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 time 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 (MD) behavior. In the following we show using coercivity measurements on Ni/Al 2 O 3 composites 32

33 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 polycrystalline Ni particles with diameters in the range of 3-60 nm embedded as layers in an insulating Al 2 O 3 host. The multilayer 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 (99.99%) and Al 2 O 3 (99.99%) 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 layer, thus forming hydrogen- terminated surfaces. 35 The base pressure for all the depositions was of the order of 10 7 Torr. After substrate heating, the pressure increased to the 10 6 Torr range. The substrate temperature was kept at about 550 o C during growth of the Al 2 O 3 and Ni layers. The repetition rate of the laser beam was 10 Hz and energy density used was 2 Jcm 2 over a spot size 4 mm 1.5 mm. A 40 nm-thick buffer layer of Al 2 O 3 was deposited initially on the Si or sapphire substrate before the sequential growth of Ni and Al 2 O 3. 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). Multilayer samples were prepared having 5 layers of Ni nanoparticles spaced from each other by 3 nm-thick Al 2 O 3 layers. A 3 nm-thick cap layer of Al 2 O 3 was deposited to protect the topmost layer of Ni nanoparticles. Shown in Fig. 2-1 is a cross-sectional TEM image from a multi-layered (5 layers) Ni-Al 2 O 3 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 Al 2 O 3 spacer layers. For the purposes of this 33

34 Figure 2-1. Cross sectional dark field STEM image of a 5-layer Ni-Al 2 O 3 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 layer samples 36 by comparing the average grain size with d. The TEM observation also shows that the Al 2 O 3 spacer layers are partially crystallized. Due to the large surface energy difference between Ni and Al 2 O 3, Ni forms well-defined, separated islands within the Al 2 O 3 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/Al 2 O 3 interfaces. 36 The Ni/Al 2 O 3 interfaces were chemically abrupt without an intermixing between Ni, Al and oxygen. In addition we did not observe exchange-bias induced asymmetric magnetization loops, thus lending support to the conclusions of previous studies 36 that antiferromagnetic NiO is absent in our layered Ni/Al 2 O 3 system. Previous TEM studies on single layer samples have shown the particles to be polycrystalline. For example, a three nm particle comprising three crystalline grains 34

35 has been observed. 36 Polycrystalline 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 crystalline anisotropy can be neglected. In addition, it is also important to note that the exchange length l ex = 14.6 nm for Ni, 37 which is the length scale below which atomic exchange interactions dominate over magnetostatic fields, determines the critical radii (R coh ) for coherent rotation: R coh 5l ex for spherical particles and R coh 3.5l ex for nanowires. 3 The particle sizes ( nm in radius) that we have investigated are thus smaller than the critical radius below which coherent rotation of Ni prevails. Figure 2-2. Coercivity for 5-layer Ni/Al 2 O 3 multilayer 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 c for each isotherm, indicated by vertical arrows, delineate the crossover from single domain (SD) to multiple domain (MD) behavior (d > d c ). Inset shows the behavior of H c as a function of 1/d for the particles with d > d c at 10 K. The linear dependence up to 24 nm diameter particles with saturation at a constant value for large particles 38 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 c as a function of particle size d at each of the temperatures indicated in the legend. Coercive fields were extracted from magnetization 35

36 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 c 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 d c delineating SD (d < d c ) behavior of coherently rotating particles from MD (d > d c ) behavior. 2,8,39 45 The reason why there is a peak in H c (d) is explained in the introduction chapter, page 29. In the inset of Fig. 2-2 we have plotted H c versus 1/d for the particles of size d > d c at 10 K. It is clear that H c 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. 38 Thus particles of size d > d c 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/MD crossover can be understood in a qualitative way by considering the three randomly oriented particles shown schematically 36

37 Figure 2-3. Peak position, d c, 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 d c for a series of single layer 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 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 al 26 for interacting paramagnets to include the temperature region below the blocking temperature T B and find the temperature-dependent dipolar magnetic field H d arising from temperature induced fluctuations in the magnetization of nearest neighbor nanometer size particles to be, H d = µ 0M s a 4π e β (1 e 1 ) (2 1) πβ(erfi(β) erfi( β 1)) 37

38 where erfi is the imaginary error function, M s is the saturation magnetization, β = KV/k B T, and a = V (3cos 2 ξ 1)/s 3 is a dimensionless parameter with ξ and s corresponding respectively to an angle parameter and the separation between two adjacent particles each with volume V. The parameter β is always greater than one for T < T B 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. H d = µ 0M s a 4π, T 0 (2 2). 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 θt θ M T = M s θt θ exp[ E(θ) k B T ]cosθdθ exp[ E(θ) k B T ]dθ (2 3) where at zero magnetic field E(θ) = KV sin 2 θ. Thus θ min = 0 and θ T is temperature dependent, obeying the relation, sin 2 θ T = k B T/KV. The parameter θ T (see Fig. 1-7 on page 21 of chapter1) 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 H d as shown in Eq. 2 1 for particles treated as simple dipoles. 38

39 . In the absence of interactions (H d = 0) the condition for the SD to MD transition is given for spherical particles with radius d/2 by, Ad 3 c = Bd 2 c, where Ad 3 c is the total magnetostatic energy and E dw = Bd c 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 H d, 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 3 c = Bd 2 c πm s H d d 3 c/6 (2 4) When the dipolar interaction is a small perturbation, i.e., M s H d /A 1, Eq. 2 1 and 2 4 can be combined to give the relation, d c = d c0 d dw e β (1 e 1 ) πβ(erfi(β) erfi( β 1)) (2 5) where d c0 = B/A is the temperature-independent critical diameter in the absence of interactions (high-temperature limit) and d dw = µ 0 BMs 2 π/(72a 2 ) for a = π/3. The second term on the right-hand side of Eq. 2 5 thus becomes a temperature-dependent correction to d c 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 d c towards higher values at higher temperatures. This is indeed evident in Fig. 2-3, which shows the temperature dependence of d c as determined from the data in Fig The black squares are the simulated data according to Eq. 2 5 using the two fitting parameters: d c0 and d dw. 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 d c0 = 84 nm from our simulation (Fig. 2-3, black squares) to be close to the value for a particle with shape anisotropy constant K shape = Jm 3 (d c0 = 72A ex K/µ 0 Ms 2, where A ex is exchange stiffness, 39

40 K is anisotropy constant). 3 Values of A ( µ 0 Ms 2 ) and B ( A ex K ) have been found to be Jm 3 and Jm 2 respectively. This value of A is very close to the theoretical predicted value 3 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 co K/M s for randomly oriented particles. 3 For K shape = Jm 3, H co 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 layer samples the coercivities at 10 K peak at d c = 14 nm as shown in Fig. 2-3 by the blue star. In the single layer samples the peak position occurs at higher particle size (14 nm) than multilayer samples (8 nm). This difference reinforces our interpretation and can be understood by realizing that the dipolar interactions of the single layer samples are significantly reduced compared to the multilayer 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 d c 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 d c on temperature and is in good qualitative agreement with our modification of present theory 26 of interacting particles. The well-established influence of dipolar interactions on superparamagnetic relaxation time together with the connection between relaxation time τ and coercivity H c suggests that there is a concomitant influence of dipolar interactions on the coercivity observed near the superparamagnetic limit where H c = 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. 40

41 2.6 Methods Mathematical Analysis The Eq. 2 5 is self consistent (as the term β contains d c ) and can not be solved analytically. The equation, d c d c0 + d dw e β (1 e 1 ) πβ(erfi(β) erfi( β 1)) = 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 Basic Physical Understanding A simplified physical understanding of the problem is shown in Fig Figure 2-4. The net effect of dipolar magnetic field (H d ) is shown on the particle 1. As particles are randomly oriented, H d 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. 41

42 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 Al 2 O 3 matrix. Results for two sets of 5 layer samples with different interlayer spacing and a set of single layer samples of Ni nanoparticles are compared. The dipolar interactions are strongest in the samples with shorter interlayer distances and weakest for the single layer 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 plays an important role in understanding the origin of the coercive field. 3.2 Introduction The origin of coercive field (H c ) 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 (d c ), the most favorable energy state is to have single magnetic domain and particles are called single domain (SD) particles. When H c is plotted as a function of particle diameter (d), there is a well defined peak at d c. Particles with d < d c (d > d c ) are SD (multidomain (MD)). 2,8,41,42,44,48 Kittel 3,46 has shown that for a spherical particle, d c is given by the relation (see Eq. 1 3 on page 13 of chapter1) d c = 72 AK µ 0 M 2 s (3 1) where A is the exchange stiffness, K is the anisotropy constant, µ 0 is the free space permeability and M s is the saturation magnetization. In SD particles there is no domain wall. The origin of H c 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 42

43 reverse the direction of the magnetization of a coherently rotating SD particle is given by the relation 2,5,49,50 [ ] KV τ = τ0exp. (3 2) k B T Here, τ 0 is the inverse of the attempt frequency to overcome the energy barrier, V is the volume of the particle, k B is the Boltzmann constant and T is the temperature. Stoner and Wohlfarth have calculated H c 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 ( ) 1 H csw = 2K ln( τ m 2 1 τ0 ) M s ln( τ, (3 3) τ 0 ) where τ m is the time of measurement. From the simple Stoner-Wohlfarth model it is clear that H c for the nanoparticle can depend on many different factors. H c increases with decreasing τ m, increasing τ 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 Thus if due to the dipolar interactions K increases (decreases) then τ, according to Eq. 3 2, will also increase (decrease) and as a net result H c will increase (decrease). A more familiar famous form of Eq. 3 3 is H csw (T ) = 2K/M s (1 (T/T B ) 1/2 ), where T B = KV/25k B is the blocking temperature. The factor 25 comes from the fact that τ m 100 s is a typical measurement time and τ sec 1 is a typical attempt rate. The effect of dipolar interaction on the coercive field (H c ) has been investigated extensively. The first theoretical treatment by Neel 62 showed that H c decreases with the increase in the packing fraction (ɛ) 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 43

44 lower the anisotropy energy. H c = H c (1 ɛ) (3 4) The Interaction Field is a function of the packing fraction (ɛ). Later Wohlfarth 51 showed that the effect of the interaction on the H c 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 c and explained in terms of a corresponding increase or decrease in the anisotropy energy ,63. In this present experiment we find that an increase in the dipolar interaction increases H c but decreases d c. Equation 3 1 suggests that the decrease in d c may be due to a decrease in the K. But a decrease in the K will also decrease τ (Eq. 3 2) and thus will decrease H c (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 c and d c both in the same direction (both increase or decrease at the same time). Below, we show qualitatively that the increase in the H c can be realized in terms of the collective dynamics of the magnetization of the particles and decrease in d c can be understood as discussed in reference Results and Discussions Samples were grown using pulsed laser deposition technique. 48 Base pressure of the growth chamber was on the order of 10 7 Torr and the growth temperature was around 550 o C. Multilayer structure of Al 2 O 3 and Ni nanoparticle were grown without breaking the vacuum of the chamber. First a thick (40 nm) buffer layer of Al 2 O 3 is grown on top of the substrate. The purpose of this buffer layer is to prevent any diffusion of the Ni into the substrate. Then Ni nanoparticles and Al 2 O 3 are sequentially deposited on this buffer layer (see Fig. 1). The top layer of Al 2 O 3 acts as a capping layer which prevents oxidation of the nanoparticles. 36 Three different sets of samples are grown. Set 1 and set 2 samples 44

45 consist of 5 layers of Ni nanoparticles separated by Al 2 O 3 layers. For set 1 (set 2) the Al 2 O 3 separation is 3 nm (40 nm). Set 3 samples are single layer of Ni nanoparticles in Al 2 O 3 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 layer 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. Figure 3-1. Fig 1a) shows the TEM image of a single layer sample with average particle diameter of 24 nm. Particles are well defined with inter particle distance of around 4 nm. 1b) shows a schematic of the single layer sample. A 40 nm thick buffer layer of Al 2 O 3 is first grown on top of substrate. Then the Ni nanoparticles are grown on to of the buffer layer. Finally a 3 nm thick capping layer of Al 2 O 3 is grown to protect it from oxidation. 1c) shows the schematic of 5 layers of Ni nanoparticle sample. Figure 3-1a) shows the TEM image of the single layer Ni particles with average particle diameter of 24 nm (set 3). The simplified schematic of the single and multilayer 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 Al 2 O 3 spacer layer (set 1) and 6 nm in diameter. The coercive field H c (T ) is determined from the loop as shown by the arrow. This procedure to determine H c is repeated for all samples belonging to all three sets. At temperatures 45

46 a) M (10-5 emu) 15 M10K 10 M50K M100K H c H (KOe) M (10-5 emu) b) M250K M300K M325K H/T(Oe/K) Figure 3-2. a) Magnetization loop of a sample from set 1 of average particle diameter of 6nm. Coercive field (H c ) is determined from the loop as shown by the arrow. H c 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 (T B ) 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 c 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 H c separates SD and MD particles. 2,8,41,42,44,48 It is clear from the data that d c decreases with increasing dipolar interactions (d c1 < d c2 < d c3 ). H c 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 The decrease in d c due to the dipolar interactions has been discussed elsewhere. 48 In this present study, the collective dynamics of the particles 46

47 600 d c1 d c2 d c3 10 K 400 H c (Oe) nm separation (Set 1) 40 nm separation (Set 2) Single layer (Set 3) d (nm) Figure 3-3. Coercive field (H c ) as a function of particle diameter (d). The peak separates the single domain (SD) and multidomain (MD) particles. Particles with diameter higher (smaller) than the peak diameter (d c ) are MD (SD). Data for the 3 different sample sets are shown and indicated in the legends. The critical diameters d c1, d c2, d c3 are shown from the samples of set 1, set 2 and set 3 respectively. In the single domain region (below d c ) 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 H c. These observations are shown in Fig. 3-3 and summarized in Fig We first discuss the effect of dipolar interactions on H c as presented in previous investigations The treatment begin by including the change in anisotropy energy E dip, due to dipolar interaction into the expression for τ, as given by 63 [ ] KV ± Edip τ ± = τ 0 exp k B T (3 5) 47

48 Equation3 5 can be rewritten as shown in Eq Thus the effect of the dipolar interactions is treated as either an increase (+ E dip ) or decrease (- E dip ) of anisotropy energy. [ ] K(eff±) V τ ± = τ 0 exp k B T (3 6) The effect of dipolar interactions on the H c can be explained according to Eq In our case a + E dip increases τ + and give rise to an increase in H c 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, d c should also increase with increasing dipolar interactions. According to the previous approach both H c and d c should change in the same way, both increase or both decrease. In the present experiment we find however that H c increases and d c decreases due to dipolar interactions (see Fig. 3-4) and strongly suggests an alternative approach to the problem. The effect of dipolar interactions on d c is discussed in reference 48, 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 H c 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 (τ ) in this case is given by 64,65 τ = τ ( ) z T 1, T > T g (3 7) T g where τ is the relaxation time of the single non interacting particle (Eq. 3 2), T g = µ 0 M 2 /4πk B r 3 is the critical temperature and depends on the interparticle distance and particle magnetization and z is a critical exponent. The above equation clearly suggests that the relaxation time will be larger in the presence of dipolar interactions and thus according to Eq. 3 3 H c will be larger, and thus agreeing with our experimental 48

49 dc (nm) Set 3 Set 2 Set 2 dc Hc Set 1 4 Hc (100 Oe) 2 8 Set 3 Set 1 0 Dipolar interaction strength Figure 3-4. Coercive field (H c ) and critical diameter (d c ) as the function of the increasing dipolar interaction. H c (d c ) increases (decreases) with increasing dipolar interaction. The opposite behavior of H c and d c suggests that the collective dynamics and the critical slowdown is responsible for the increase in H c due to the dipolar interactions. The decrease in d c 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, Conclusions A study of dipolar interactions is presented for the single and multilayer 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 49

50 with increasing dipolar interactions. These two behaviors together suggest 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. 50

51 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 1.5Å-thick ultra thin layer 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 layer over a range from 8 Å to 56 Å. As the thickness of the top Pd layer decreases, or equivalently as the embedded Fe layer 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 dependences, 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 Pd is known to be in the verge of ferromagnetism because of its strong exchange enhancement with a Stoner enhancement factor of The magnetic impurities induce small moments on nearby Pd host atoms thereby creating a cloud of polarization with an associated giant moment 71,72. 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 Å 72,73. Thus a thin layer of Fe encapsulated within Pd will be sandwiched between two adjacent thin layers of ferromagnetic Pd with nonuniform magnetization and a total thickness in the range 20 to 100 Å. 51

52 We have investigated thin-film Pd/Fe/Pd trilayers in which the thickness d F e of the Fe is held constant near 1.5Å and the thickness of the polarized ferromagnetic Pd is varied by changing the top Pd layer 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 interest For example, the coercive field H c increases as the thickness of the ferromagnetic film is decreased toward a thickness comparable to the width of a typical domain wall 79,80. Moreover, the Curie temperature T c decreases as the thickness of the ferromagnetic film is decreased toward a thickness comparable to the spin-spin correlation length We will show below that similar phenomenology applies to ferromagnetically polarized Pd films, albeit with different functional dependences arising from the fact that exchange coupling, which decays with distance from the ferromagnetic impurity 84, 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 layer of Pd of thickness 200 Å is grown on top of the substrate. The root mean square surface roughness of this Pd layer was measured by atomic force microscopy to be 6 Å. Then a very thin (1.5 Å as recorded by a quartz crystal monitor) layer of Fe is deposited on top of the first Pd layer. Finally a top layer of Pd with thickness x is grown to complete the trilayer structure shown schematically in Fig. 4-1a. We discuss six different samples with the top Pd layer having a thickness x varying from 8 to 56 Å. The total thickness y of the polarized Pd (see Fig. 4-1b) can range from 20 to 100 Å 72,73. 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 layer. The AES measurements were performed in a 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 52

53 Figure 4-1. a) Multilayer structure of a Pd/Fe/Pd trilayer. The bottom layer of Pd is 200 Å thick. The thickness of the Fe layer is 1.5 Å as recorded by the quartz crystal monitor. The thickness x of the top layer of Pd is varied from 8 to 56 Å. b) Magnetic structure of the sample. The total thickness y of polarized Pd is in the range 20 to 100 Å (shaded red area). Thus by varying x, it is possible to vary the thickness y of the polarized ferromagnetic Pd layer. 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 Å 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 layer with a FWHM thickness of 1.8 Å. 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 Å; otherwise the exposure of the sample to air caused unwanted oxidation of the Fe. The magnetic parameters H c (x) (Fig. 4-3) and T c (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 layer is independent of x, since y/2 < 200Å, the constant thickness of the bottom layer. 53

54 12 MsA(10-5 emu/cm 2 ) 10 8 M sa (Fe) = 2.63x10-5 emu/cm x(å) Figure 4-2. The saturation magnetization normalized to the area of the sample M sa 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 eye. Saturation to a constant value occurs near 30Å (vertical arrow). 4.4 Results and Discussion For large values of x, the thickness y of the combined polarized ferromagnetic Pd layers and the associated saturated magnetization M = M s will reach a constant value. This expectation is borne out in Fig. 4-2 which shows the x-dependence of saturated magnetization M sa normalized to sample area. We note that this normalized saturated magnetization M sa (x) increases with increasing x as the total amount of polarized Pd increases. The onset of saturation, near x = 30 Å indicates that the polarization cloud including the embedded Fe layer is 60 Å thick. This value is consistent with previous observation 73. The increase of M sa with x shown in Fig. 4-2 is thus straightforward to understand. As x increases the thickness of the top polarized ferromagnetic Pd layer increases with a concomitant increase of magnetic material in the system. Variation of x clearly controls the thickness of the polarized ferromagnetic Pd layer. When normalized to 54

55 80 2 x = 8 Å x = 56 Å 10 K H C (Oe) M(10-5 emu) 0-2 H c H(Oe) x(å) Figure 4-3. The coercive field H c shows a strong increase as the thickness x of the top layer of the Pd decreases. The data are shown as solid black circles and the black solid line is a power law fit with exponent η = 2.3(±0.1). The inset shows magnetization loops at T = 10 K for x = 8Å (solid black squares) and x = 56Å (solid red circles). the number of Fe atoms present, the saturated magnetization M sa = emu/cm 2 corresponds to 9.2 µ B per Fe atom, in close agreement with previous observations of the giant moment of Fe in Pd to be near 10 µ 72 B. Modeling the x dependence of M sa (x) shown in Fig. 4-2 for our Pd/Fe/Pd trilayers is not straightforward. For regular ferromagnets with M s uniform throughout the thickness, we would expect M sa (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 impurity 84. 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 J 4, ( [ ]) M s M(H, T, x) = M s B s gµ B H + 2pMJ(x), (4 1) k B T 55

56 T C (K) M(10-5 emu) M vs T (x = 56 Å) at 20 Oe T C T(K) 0 20 x (Å) Figure 4-4. The Curie temperature T c rapidly increases with increasing x. Data are shown as solid black circles and the dashed black line is a guide to the eye. Saturation to a constant value occurs near 20Å (vertical arrow) The inset with T c indicated by the vertical arrow shows the temperature-dependent magnetization taken in a field H = 20 Oe. where B s 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 analysis. Fig. 4-3 shows the behavior of the coercivity H c (x) as a function of x (solid black circles). The data are well described by a power-law dependence (solid black line), H c (x) x η, where the exponent η = 2.3(±0.1) is close to the ratio 7/3. Similar power-law behavior reveals itself in regular ferromagnetic thin films where η has a somewhat smaller value varying from 0.3 to Because η 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 η. Neel predicted for example that for Bloch domain walls, H c of a ferromagnetic thin film should vary as x 4/3 when the thickness x of the film 56

57 is comparable to the domain wall thickness w 79. For the case of Neel walls, H c depends only on the roughness of the film and does not depend on film thickness 77. The variation of H c (x) becomes particularly pronounced when the film thickness becomes comparable to w. A qualitative understanding of the steeper H c (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/K 3,82 where K is the crystalline anisotropy constant and A is the exchange stiffness, proportional to the exchange energy, J 85. The domain wall size w increases for decreasing K and increasing J. If K, which depends on the relatively constant spin-orbit interaction 4 within the Pd component of the Pd/Fe/Pd trilayers, remains constant, then variations in w are dominated by variations in J. Thus as x decreases toward zero, the increase in J 84 gives rise to an increase in w which in turn gives rise to a more rapid increase in H c than would be seen in regular ferromagnets with constant J. As discussed above, this rapid variation with η 7/3 is observed experimentally. The data in Fig. 4-4 show that T c increases as x increases and reaches a relatively constant value near x = 20 Å. The dashed black line is a guide to the eye and is qualitatively similar to the behavior of M sa (x) shown in Fig. 4-2 which saturates at a larger value near 30 Å. These observations are again qualitatively consistent with the finite size effect associated with critical phenomena in ferromagnets Although the data are not of sufficient quality to distinguish the power-law behavior that is predicted for finite size effects 81 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 T c (x) suggests that Pd/Fe/Pd trilayer should be treated as a single layer with a well defined spin-spin correlation length. If the Pd layers are treated separately, then the bottom layer 57

58 with fixed thickness y/2 would have a T c equal to the highest T c of the top layer. In this case the overall measurement would not show a strong change in T c as a function of x, since the T c of the bottom layer would dominate for all x. We note that for our planar geometry, T c decreases with decreasing thickness as has also been shown for thin-film Ni 81 and epitaxial thin-film structures based on Ni, Co and Fe 82. On the other hand T c increases with decreasing size of ferrimagnetic MnFe 2 O 4 nanoscale particles with diameters in the range 5-26 nm 83. This increase of T c with decreasing size is attributed to finite size scaling in three dimensions where all three dimensions simultaneously collapse 83. In our two-dimensional planar thin films only one of the dimensions, the thickness, collapses and T c decreases rather than increases in accord with the observations of previous studies 81, Conclusions In conclusion, we have characterized the magnetic properties of thin-film Pd/Fe/Pd trilayers and determined that critical size effects apply to ferromagnetic 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 layer. The dependences on film thickness of the coercive field H c and the Curie temperature T c 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 M sa (x), an unusually strong power-law dependence of H c (x) and a dependence of T c (x) which indicates that the trilayer acts as a single layer that necessarily includes the constant thickness Pd layer serving as a substrate for the Fe layer. 58

59 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 layer samples of Ni nanoparticles in Al 2 O 3 matrix is studied. A linear T 2/3 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 c (T )), arise from two mechanisms. The first mechanism is, due to the temperature dependence of the intrinsic magnetic properties 11,15,86,87 such as saturation magnetization (M s ), 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 M s does not change with temperature. All the samples comprise polycrystalline particles 36 and thus magnetocrystalline 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. 88 In this chapter we will discuss the second mechanism as the origin of the temperature dependence of the coercive field (M s, K and A are temperature independent). To understand the temperature dependence of the H c 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 59

60 used phenomenological energy barrier is 2,3,47,89 E = E 0 [1 H/H c0 ] m (5 1) where E 0 is the energy barrier at zero magnetic field and energy barrier vanishes at H = H c0 at T = 0. At H = H c, thermal energy, k B T, 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, E 0 = KV and H c0 = 2K/M s. 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 E(H) is known, it is possible to calculate H c (T ). In this chapter we will discuss how to derive E(H) (Eq. 5 1) for the MD nanoparticles and will compare H c (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 layer Ni particles in an Al 2 O 3 matrix. Set 2 and Set 3 consists of 5 layers of Ni particles separated by Al 2 O 3 layers. 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 c 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 The arrow shows H c at 10 K. Note that H c decreases with increasing temperature. The temperature dependence of H c normalized to H co for five different samples belonging to set 1 is shown in Fig The particle diameters are indicated in the legends. Note T 2/3 in x axis. All the data follow a linear T 2/3 dependence. To understand the above data, we will start 60

61 Alumina Ni particles Substrate Set 1 Set 2 Set 3 Figure 5-1. Schematic of three sets of samples. Set 1 comprises a single layer of Ni particles embedded in an Al 2 O 3 matrix. Set 2 and Set 3 comprises of 5 layers of Ni particles separated by different distances in an Al 2 O 3 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) = a 0 + a 1 x + a 2 2 x2 + a 3 3 x3 b 0 Hx (5 2) where a 0, a 1, a 2, a 3 and b 0 are micromagnetic parameters that depend on the magnetic parameters K, M s and A. For the strong pinning center the x 3 term is included as the effect of the pinning center is long distance compared to the weak pinning center where the x 3 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. δe δx = a 1 + a 2 x + a 3 x 2 b 0 H = 0 (5 3) 61

62 M (emu) 8.0x10-4 Msub10K 6.0x10-4 Msub50K Msub100K 4.0x10-4 Msub150K Msub200K 2.0x10-4 Msub250K Msub300K Msub325K x x x nm Ni/Al 2 O 3 5 Layer H c -8.0x 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 (H c ) at 10 K is indicated by the arrow. H c decreases with increasing temperature. Saturation magnetization (M s ) is constant at different temperatures. The two solutions for the above equations are x 1 = a 2 + a 2 2 4a 3 (a 1 b o H) 2a 3 (5 4) x 2 = a 2 a 2 2 4a 3 (a 1 b o H) 2a 3 (5 5) Taking the second derivative of E(x) with respect to x it is easy to show that δ 2 E/δx 2 x1 > 0 (δ 2 E/δx 2 x2 < 0) and corresponds to the maximum (minimum). Thus the energy barrier is E(H) = E(x 1 ) E(x 2 ) = (a2 2 4a 1 a 3 + 4a 3 b 0 H) 3/2 6a 2 3 (5 6) 62

63 1.0 Hc/Hc Set 1 18 nm 24 nm 30 nm 36 nm 42 nm T 2/3 (K 2/3 ) Figure 5-3. Coercive field (H c ) vs. T 2/3 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 = H the above equation reduces to E(H) = (a2 2 4a 3 (a 1 + b 0 H)) 3/2 6a 2 3 = (a2 2 4a 1 a 3 ) 3/2 6a 2 3 which is in the same form of Eq. 5 1, where ( 1 H a 2 2 4a 1a 3 4a 3 b 0 ) 3/2 (5 7) E 0 = (a2 2 4a 1 a 3 ) 3/2 6a 2 3 (5 8) H c0 = a2 2 4a 1 a 3 4a 3 b 0 (5 9) From Eq. 5 7 it is clear that E(H) decreases with increasing H and when H = H c the energy barrier can be overcome by thermal energy (definition of the coercive field). Thus 63

64 at H = H c, the Eq. 5 7 can be rewritten as k B T = (a2 2 4a 1 a 3 ) 3/2 6a 2 3 ( 1 H c a 2 2 4a 1a 3 4a 3 b 0 ) 3/2 (5 10) The above equation can be solved for H c ( ) ] 2/3 kb T H c = H c0 [1 E 0 (5 11) where H c0 and E 0 are given by Eq. 5 8 and 5 9. This temperature dependence of H c is consistent with the experimental results shown in Fig. 5-3, 5-4 and Hc/Hc Set 2 12 nm 18 nm 24 nm 42 nm 60 nm T 2/3 (K 2/3 ) Figure 5-4. Coercive field (H c ) vs. T 2/3 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. 64

65 Set 3 16 nm 24 nm 30 nm 36 nm 44 nm Hc/Hc T 2/3 (K 2/3 ) Figure 5-5. Coercive field (H c ) vs. T 2/3 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 a 0, a 1, a 2, a 3 and b 0 to the magnetic parameters K, M s and A. To do that we will start with the magnetic energy expression, 3 [ ( ( )) 2 M(x) E(x) = A K(x) (k.m(x))2 M s M 2 s µ 0 M(x).H µ 0 2 M(x).H d(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 65

66 samples while taking into account real structure and imperfections. The real structure and imperfections are responsible for the x dependence of the magnetic parameters (M s (x), A(x), K(x)). After solving Eq 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 c0 and E 0 /k B are shown in the figure below. Hc0 (Oe) Set 2 Hco E0/kB d (nm) E0/kB (K) Figure 5-6. The behaviors of H c0 and E 0 on particle diameter are shown for set 2 samples. H c0 decreases and E 0 /k B increases with increasing particle size. The increasing behavior of E 0 and decreasing behavior of H c0 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 Conclusions We have investigated the temperature dependence of the coercive field of MD Ni nanoparticles in Al 2 O 3 matrix. H c decreases linearly with the T 2/3. This behavior can be understood according to the strong domain wall pinning. We show that the general energy 66

67 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 c. 67

68 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 Neel 15 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 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 68

69 from small inclusions (for example impurity or vacancy defects) where the energy of the free pole is not negligible 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 (99.99%) and Al 2 O 3 (99.99%) 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 layer (for the Si substrates only), thus forming hydrogen- terminated surfaces. 35 The base pressure for all the depositions was of the order of 10 7 Torr. After substrate heating, the pressure increased to the 10 6 Torr range. The substrate temperature was kept at about 550 o C during growth of the Al 2 O 3 and Fe layers. The repetition rate of the laser beam was 10 Hz and energy density used was 2 Jcm 2 over a spot size 4 mm 1.5 mm. A 40 nm-thick buffer layer of Al 2 O 3 was deposited initially on the Si or sapphire substrate before the sequential growth of Fe and Al 2 O 3. This procedure results in a very 69