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MAGNETISM IN NANOSCALE MATERIALS, EFFECT OF FINITE SIZE AND DIPOLAR INTERACTIONS By RITESH KUMAR DAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Ritesh Kumar Das I dedicate this to my parents and family for their active support. Without them it would have not been possible. ACKNOWLEDGMENTS I am truly indebted to ri: irn: individuals who have contributed to the success of my research work. Therefore, I express my sincerest regrets to any person not specifically mentioned here. First and foremost, I am thankful to my research advisor Prof. A. F. Hebard for giving me the opportunity to work with him. It has been a great experience to work under his supervision. His positive, openminded attitude toward research creates a unique laboratory environment full of encouragement. I have learned a lot from his unadulterated enthusiasm, willingness to learn and elegant but simple approach to understanding fundamental physics. I would like to thank all the present and former lab members for their helps and pleasant ( ".'r. ir:. I am grateful to John J. Kelly for teaching me many experimental techniques when I joined the group. Thanks to all the lab members Patrick, Rajiv, Sef, Siddhartha, Sanal, Xiaochang for their helps. I really enjoi, d working with you guys. I would also like to acknowledge the staffs of machine shop and electric shop. Specially cryogenic staffs, Greg and John, for their constant supply of liquid He and N2 all year around 24/7. Thanks to Jay (really a nice guy) for looking after all the pumps and chillers. I would like to thank all of my committee members. I will specially thanks Prof. Amlan Biswas. Though I did not have chance to collaborate with him, but his guidance and support towards my degree have been very helpful. I am also greatly thankful to Prof. D. Norton for the wonderful collaboration and for letting me use his lab facilities. I am thankful to my collaborators D. Kumar and A. Gupta from NCA&T. I am also very thankful to Matt, Patrick, KyeongWon from Prof. Norton's lab for their helps and being good friends. I am indebted to my parents for their support, encouragement and for alvi believing in me. I appreciate the warmth and affection of my sister Mridula. I could not have come this far without their blessings. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF TABLES ....................... ............. 8 LIST OF FIGURES .................................... 9 A BSTRA CT . . 11 CHAPTER 1 THEORY AND BACKGROUND ................... ..... 12 1.1 Single Domain and Multi Domain Particles ....... .......... 12 1.2 Hysteresis Loop of Single Domain Coherently Rotating Particles ...... 14 1.3 Hysteresis Loop of Multi Domain Particles ................. .. 24 1.4 Magnetization vs. Temperature .......... .. .. .. 29 1.4.1 Zero Field Cooled (ZFC) Magnetization 30 1.4.2 Field Cooled (FC) Magnetization ..... 31 2 DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL SINGLE DOMAIN GRAIN SIZE OF NI IN LAYERED Ni/A120O COMPOSITES 32 2.1 Abstract .. .. .. .. ... .. .. .. ... .. .. .. .. ...... .. 32 2.2 Introduction .................. ................ .. 32 2.3 Experimental Details .................. ........... 33 2.4 Data and Discussion .................. ........... .. 36 2.5 Conclusion .. ..... .. 40 2.6 M ethods .................. .................. .. 41 2.6.1 Mathematical Analysis .............. .. 41 2.6.2 Basic Physical Understanding ............... .. .41 3 EFFECT OF DIPOLAR INTERACTION ON THE COERCIVE FIELD OF MAGNETIC NANOPARTICLES: EVIDENCE FOR COLLECTIVE DYNAMICS 42 3.1 Abstract .................. .................. .. 42 3.2 Introduction .................. ................ .. 42 3.3 Results and Discussions .................. ......... .. 44 3.4 Conclusions .................. ................ .. 49 4 FINITE SIZE EFFECTS WITH VARIABLE RANGE EXCHANGE COUPLING IN THINFILM Pd/Fe/Pd TRILAYERS .................. ..... 51 4.1 Abstract .................. .................. .. 51 4.2 Introduction .................. ................ .. 51 4.3 Experimental Details .................. ........... .. 52 4.4 Results and Discussion .................. ........ .. .. 54 4.5 Conclusions .................. ................ .. 58 5 TEMPERATURE DEPENDENCE OF COERCIVITY IN MULTI DOMAIN NI NANOPARTICLES, EVIDENCE OF STRONG DOMAIN WALL PINNING 59 5.1 Abstract ...................... ............. 59 5.2 Introduction ...................... ........... 59 5.3 Results and Discussions ............................ 60 5.4 Relation Between Micromagnetic Parameter and Magnetic Parameters 65 5.5 Conclusions ...................... ........... 66 6 COERCIVE FIELD OF FE THIN FILMS AS THE FUNCTION OF TEMPERATURE AND FILM THICKNESS: EVIDENCE OF NEEL DISPERSE FIELD THEORY OF MAGNETIC DOMAINS ................... ....... 68 6.1 Abstract ...................... ............. 68 6.2 Introduction ...................... ........... 68 6.3 Experimental Details ................... ....... 69 6.4 Results and Discussion ................... ...... 70 6.5 Conclusion ...................... ........... 74 7 SCALING COLLAPSE OF THE IRREVERSIBLE MAGNETIZATION OF FERROMAGNETIC THIN FILMS ............. ........... 75 7.1 Abstract ....................... ............ 75 7.2 Introduction ...................... ........... 75 7.3 Experimental Results .................. ........... .. 76 7.4 Conclusions .................. ................ .. 84 7.5 Methods ................... ............... 85 7.5.1 Ni N i'" .q. iticle. .................. .......... .. 85 7.5.2 Gd Thin Film. .. ..... .. .......... 86 7.5.3 (Lai_yPry)0.67Ca,, M!O3 (LPC'1O) Thin Films. ... 86 7.5.4 Temperature Correction of Coercive Field. ..... 86 REFERENCES ................... ............. ...... 87 BIOGRAPHICAL SKETCH ........... ........ ... 96 LIST OF TABLES Table page 11 H, vs. T ...... ........... ............... .. .. 29 LIST OF FIGURES Figure page 11 SD and MD particle ............... ............. 13 12 Coherent and incoherent rotation ... ............ ..... .. 14 13 Single particle in magnetic field ............... ........ ..15 14 Two state energy ............... ............ .. .. 16 15 Hysteresis of SD particle ............... ........... ..18 16 Diagram of a particle .................... 20 17 Thermal average of magnetization .......... .. 21 18 Flow diagram .................. .................. .. 21 19 MH below TB .. ..................... .. ..24 110 M H below TB ....... ....... .... ..... .. 25 111 SD to MD transition and He .................. .......... .. 26 112 Magnetization loop for MD particle .................. ..... .. 27 113 Domain wall and He .................. .............. .. 28 114 M vs. T for 3 nm Ni nanoparticles .................. ..... .. 30 21 STEM image of Ni particle .................. ........... .. 34 22 H, vs. d, different T .................. ............ .. .. 35 23 d, vs. T ................................ .. .... 37 24 Hd and domain ............... ............... .. 41 31 Sample ..... ........... ... .............. 45 32 MH loop. .................... ....... ...... .. ... 46 33 He vs. d: dipolar interaction .................. .......... .. 47 34 Dipolar interaction .................. .. 49 41 Physical and magnetic view of sample .................. ...... 53 42 Saturation magnetization vs. x .................. ........ .. 54 43 Coercive field vs. x .................. ............... .. 55 44 Curie temperature vs. x 51 Three sets of sample . 52 MH loops of set 2 ..... 53 He vs. T2/3 set 1 samples 54 He vs. T2/3 set 2 samples 55 He vs. T2/3 set 3 samples 56 Hco and Eo of set 2 . 61 TEM image of Fe film . 62 MH loop of Fe film .. 63 He vs. T of Fe film . 64 He vs. K of Fe film .. 65 He vs. d of Fe film . 71 Irreversible Magnetization 72 Behavior of the AM(H, T) 73 The anstz .. ....... isotherms as the function of 74 Scaling collapse of variety of ferromagnetic materials .. ............ and scaling collapse Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Phil. .. hi, MAGNETISM IN NANOSCALE MATERIALS, EFFECT OF FINITE SIZE AND DIPOLAR INTERACTIONS By Ritesh Kumar Das August 2010 C'!I ir: A. F. Hebard Major: Physics Material physics is ahv, motivated by the materials with exotic properties. It was a common belief that exotic properties are only associated with exotic materials. Now it is clear that geometrical confinement at nanoscale dimensions can give rise to exotic properties even in simple materials. Ferromagnetic materials in restricted dimensions are extremely interesting because of their potential applications as well as the rich fundamental science involved. Magnetic nanoparticles are useful in high density magnetic data storage devices, sensors, contrast agents in MRI, drug delivery, treating hyperthemia and many more. All the applications of nanomagnets are very crucial in modern d4v life. But most of the applications are restricted due to the limitations in the fundamental properties arises in nanoscale and also due to the technical limitations of controlling things at nanoscale. For example particles become superparamagnetic as the size is reduced below a certain value and the magnetization direction fluctuates randomly due to the thermal energy which limits the density of data storage. The promises of nanomagnets are huge and to really achieve the grand challenges in nanomagnetism, it is necessary to understand the basic sciences involve at small scales. In this present work, the magnetic properties of systems in nanoscale (nanoparticles and thin films) have been investigated. The effect of dipolar interactions, particle size, particle size distribution, temperature, magnetic field etc. on the magnetic properties have been studied. CHAPTER 1 THEORY AND BACKGROUND Ferromagnetism is known for more than 2500 years to man. The first magnetic material discovered was magnetite (Fe2O3). The practical applications of ferromagnets was recognized from very ancient time. The first use of magnetic material was as a compass. According to the magnetic properties, materials can be divided into diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, spinglass etc. In this present work ferromagnetism will be the main topic. C'! lpters followed by this chapter will discuss the effect of finite size and dipolar interactions on the magnetic properties of some materials with nanoscale structures. In this chapter a general theoretical background will be given. 1.1 Single Domain and Multi Domain Particles When the size of the particle is very small, it will contain only one magnetic domain. 13 This is because the energy required to form a domain is larger than the magnetostatic self energy. The magnetostatic self energy for a spherical particle is given by1 Ema, ,1 2V/12 (11) where po is the free space permeability, i.. is the saturation magnetization and V is the volume of the particle. The energy required to form a Bloch domain wall is13 Ed, 4= AKR2 (12) where K is the anisotropy constant, A is the exchange stiffness and R is the radius of the particle. Note that Emag grows as R3 and Edw grows as R2. Domain formation is therefore favorable for larger particles as the magnetostatic energy will be large compared to the domain wall formation energy. The critical single domain radius (Rsd) where the transition from single domain (SD) to multidomain (M\1)) occurs is given by1 3 36V/A Rsd = (13) The above equation is determined by solving the equation Ema, = Ed.13 Thus particles having radius smaller than Rd are SD and particles having radius larger than Rd are MD (see Fig. 11). d < dc d >dc Figure 11. Smaller (larger) particles are SD (\!1)) as the magnetostatic self energy is smaller (higher) than the energy required to form domain. The critical size of the particle where the single domain to multidomain transition occurs is given by Eq. 13. 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. 12). The exchange length24 A (14) is a measure of the distance over which the atomic exchange interactions dominate and all the spins rotate coherently. Particles with size larger (smaller) than le rotate incoherently (coherently). The exchange length is usually larger than Rd for soft ferromagnets where the anisotropy energy is small. Thus in soft ferromagnets magnetization reversal occurs either by coherent rotation (small particles) or by domain wall motion (large particles). t tt t\ / ttttt Coherent rotation /ttit 1t t H H / t Incoherent rotation Figure 12. Coherent and incoherent rotation of the magnetization. In case of coherent rotation all the spins rotate together and the whole particle can be considered as a giant spin. Coherent rotation happens for SD particles with size smaller than the exchange length lex 1.2 Hysteresis Loop of Single Domain Coherently Rotating Particles The magnetization dynamics of the SD particles with R < lex will be coherent and the particle can be treated as a giant single spin of value M MsV. When a magnetic field (H) is applied along the easy axis of the magnetization (k) the energy of the particle is24 E(H) = KVsin20 i..VHcosO (15) where 0 is the angle between the applied magnetic field and the direction of magnetization as shown in the Fig. 13. The first term in Eq. 15 corresponds to the anisotropy energy and the second term corresponds to the Zeeman energy. The energy, E(H), is shown in Fig. 14 b) below as a function of 0 which shows two energy minima separated by a barrier. The energy minima occur at 0 = 0 (corresponding to the magnetization along the applied magnetic field or up direction) and 0 = 7 (corresponding to the magnetization opposite to the applied magnetic field or down direction). The maximum H Figure 13. A SD particle in an applied magnetic (H) field along the easy axis of magnetization (k). 0 is the angle between the magnetization M and the easy axis k. of the energy occurs at 0 = 7/2 separating the two energy minima. Figurel4 a) shows the energy diagram at zero magnetic field as a function of 0. In this case the particle will have magnetization parallel to the easy axis of magnetization since these correspond to minimum energy states (up or down). Any other directions will cost some anisotropy energy. The two states with minimum energy are separated by the anisotropy energy barrier equal to KV. In an applied magnetic field along the easy axis the two energy minima will be shifted due to the Zeeman energy (Fig. 14 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 (16) where E+(H) is the energy barrier seen by the up magnetized particles and E_(H) is the energy barrier seen by the down magnetized particles and Ho = 2K/M, .. Derivation of the Eq. 16 is given below. E Figure 14. At t Solu 0 7 0 7 Two state energy of a SD particle. Two energy minima correspond to the direction of the easy axis of magnetization. a) At zero magnetic field the particle will have magnetization along the easy axis of magnetization as those correspond to minimum energy states (up and down). Up and down states are separated by the energy barrier equal to KV. To reverse the magnetization direction from up to down or vice versa the system has to overcome the energy barrier.5 Brown proposed that this process requires a finite time given by Eq. 114.5 b) In an applied magnetic field along the easy axis, the two energy minima will be shifted due to the Zeeman energy. Now the up state which is along the applied magnetic field will be most stable and have the lowest energy. The state with opposite direction (down state) of magnetization will be metastable. The magnetic field dependent energy barrier for the spin down state is calculated by StonerWohlfarth (Eq. 16) 6 First order derivative of Eq. 15 with respect to 0 is JE(H) 2KVsin0cos0 + i1 ..VHsinO 60 he maxima and minima 2KVsin0cos0 + i1 ..VHsinO 0 tions of the above equation are sin0 0 cosO (17) (18) (19) (110) i 2K 2K Taking the second order derivative of Eq. 15, it can be shown that the Eq. 19 (Eq. 110) refers to minima (maximum) of the energy. Thus the energy minima are at 0 = 0 and 0 = 7 and maximum at when cosO=. [. 1/2K (see Fig. 14). Energies correspond to these extrema are Emi, = i [..VH (111) Emin, = .VH (112) Ema, = KV 1 + (f4[)] (113) 2K where Emin, and Emin, corresponds to 0 = 0 (spin up) and 0 = 7 (spin down) respectively. It is now easy to show that E+(H)= Emax Emin, and E_(H)= Emax Emin are given by Eq. 16. The energy barrier has to be overcome to reverse the magnetization direction from up to down or vice versa. Brown proposed that this process requires a finite time5 r oexp E ( ) (114) ( kBT ) where T is the temperature, 71 is the inverse attempt frequency of overcoming the energy barrier and kB is the Boltzmann constant. Figurel5 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. 16 (scenario 3). But still the temperature is not enough to overcome the energy barrier and the magnetization will still be trapped with a positive value. A further increase in magnetic field in the opposite direction will keep lowering the energy barrier until, at the coercive field, the energy barrier can be overcome by the thermal energy and magnetization reversal will occur (scenario 4). When H 4 3 Figure 15. He the energy 2 1 Hysteresis of a coherently rotating SD particle. Scenario 1) High positive magnetic field is applied and saturation magnetization is observed. Scenario 2) Magnetic field is reduced from positive value to zero. Magnetization is trapped in the positive direction as the thermal energy is not enough to overcome the energy barrier. Scenario 3) Magnetic field direction is reversed. Still the energy barrier is large compared to the thermal energy and magnetization is trapped in the positive direction. Scenario 4) Magnetic field equals to the coercive field. Now the energy barrier can be overcome by thermal energy and magnetization reversal occurs. barrier E_ (H) is such that the relaxation time 7 Tm at the temperature T. Where Tm is the experimental measurement time (around 100 sec for SQUID measurement). Thus when H= H, magnetization reversal occurs. Combining Eq. 16 and 114, the coercivity (He(T)) of the SD particle can be calculated as shown below. 7 = Tm= oexp E_(H)BT \kBT = E (H,) kBTln T To Now using the expression of E_(H) from Eq. 16 it is easy to show that6 H, Ho [ T 1) (115) where TB = KV/kBln(Tm/lo) is known as the blocking temperature. Below TB the anisotropic energy barrier is larger than the thermal energy and magnetization is blocked or trapped. Above TB the anisotropic energy barrier can be overcome easily by thermal energy and the particles are called superparamagnetic as will be discussed later. It is clear from the Eq. 115 that H, decreases with increasing temperature and above the blocking temperature (T > TB) the particles lose their coercive field. Note that the origin of H, in a SD particle is the finite time required to reverse the magnetization direction over the anisotropy energy barrier. The previous discussion is only true for an assembly of uniform size particles that have easy axis of magnetization oriented along the same direction with magnetic field applied along the easy axis. In real samples this is not the case since the easy axis of magnetization is usually randomly oriented and the particle size is not uniform. A more general case is shown in Fig. 16 below. Here a arbitrary angle between magnetic field and the easy axis of magnetization (Q) is considered. The energy of the particle in this case is7 (1 H The magnetization of the particle at an applied magnetic field is given by McosO,in, where O,i, is the angle corresponds to the minima of E(H). Note that here we have not considered the effect of temperature on the magnetization. At finite temperature other 0 values around the 0,i, will be occupied with a finite probability according to the Boltzmann factor as shown in Fig. 17 below. Thus the average over all the occupied direction with the occupation probability given by the Boltzmann factor will be the thermal average of the magnetization for a fixed value of H and Q. The procedure should be repeated for all values of H to get the MH loop for a particular value of Q. Then the Figure 16. Single particle in an applied magnetic field (H). k is the direction of the easy axis of magnetization. M and H are the magnetization and magnetic field vectors respectively. Without loosing any generality M and H can be considered in the same plane. The angle between M and H is 0. The angle between H and k is Q. The energy barrier for this general configuration is given by Eq. 116.7 MH loops for all possible i should be calculated. Averaging over all these MH loops will give a magnetization loop at temperature T for a sample of uniform particle size and a randomlyoriented easy axis of magnetization. All the above procedures should be done for all possible particle sizes as the real samples usually have some particle size distribution. The probability of a particular particle size can be modeled either as a lognormal or gaussian distribution function. In this way the magnetization loop of a real sample with nonuniform particle size and random orientation of the easy axis of magnetization can be determined. If the all the above procedures are repeated for different temperatures then the magnetization loop at different temperatures can be determined. Below we show a flow diagram for the above process.79 OT OT Omin Figure 17. At finite temperature other 0 values around min, will be occupied with a finite probability according to the Boltzmann factor and shown by the shaded region. The thermal average of the magnetization will be the average of the all magnetization directions over this shaded region.7 The probability of having some magnetization direction will be determined by the Boltzmann factor. 1 Start with the energy of the single particle. E(H) =2KV l(si 2 0) cose 2 Find the minima of E(H) S..VCosOi, will be the magnetization at T=0 for the given value of H, and V Figure 18. Flow diagram to show the process of calculating coercive field for real nanoparticle samples with particle size distribution and random orientation of the easy axis of magnetizations at finite temperature. kBT Thermal average (M(H, V))}T= 3 of the magnetization SAE f[2 Msvcosoexp 7BTde AE f02 exp ~WT d0 AE = Eo Eoin 01 and 02 are shown in Fig. 17 (M(H, p, V))T is the magnetization at temperature = T for the given value of H, and V Figure 18. continued 4 Step 1,2 and 3 should be repeated for different H. This will determine the MH loop for a given value of T, b and V 5 Step 1, 2, 3 and 4 should be repeated for all possible b and average of all those loops will determine the MH loop for a given value of T and V for an ensemble of particles with random orientation of the easy axis of magnetization. 6 Step 1, 2, 3, 4 and 5 should be repeated for different particle size to determine the MH 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 MH 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 Tablel1. Figure 18. continued Magnetization loops at different temperatures for a single l .r sample of Ni nanoparticles of average diameter around 18 nm are shown in Fig. 19. The coercive field is determined by the magnetic field where magnetization changes sign and passes through zero. It is clear from the Fig. 19 that coercive fields decreases with increasing temperature as discussed above. At temperatures high compared to the anisotropy energy KV, the magnetization directions can rotate freely over the barrier and the particles become superparamagnetic with H=0. In this case the system can be treated similar to the case of paramagnetism with each particle as a giant or super spin of value M1. V (thus called 4 2 E 2 4 6E5 1500 1000 500 0 500 1000 1500 H (Oe) Figure 19. Hysteresis loop of a Single 1~v. r Ni nanoparticles of 18 nm diameter embedded in an Al203 matrix at temperature, T < TB. The loops show well defined coercive field (where magnetization is zero) and decreases with increasing temperatures. superparamagnet). The magnetization for a collection of superparamagnetic particles is given by the Langevin equation MsVH kBT M(H, T) V N .V[coth M] (117) kBT 11.,VH where N is the number of particles. Note that M is a function of H/T in the above Eq. 117. Thus if M is plotted as the function of H/T for different T, all the MH loops will fall on top of each other as shown in Fig. 110 for a single l v.r sample of 12 nm Ni grains in an A1203 host matrix. 1.3 Hysteresis Loop of Multi Domain Particles In multidomain ferromagnetic system the origin of the hysteresis loop is dramatically different than the SD case. Usually in soft ferromagnets (Rsd < le) the SD and MD particles can be distinguished by the behavior of the coercive field as a function of particle size. Figure 111 below is a schematic showing the behavior of coercive field as a * 10 K . 50 K ==== 100 K **."u,,: * 150K U/n i U 200 K = , ,,,,"  / U/ U 200' K E 4.5E5 M275K 3 M300K E2 M325K 0 2 '3 At 4.5E5 3 2 1 0 1 2 3 H/T(Oe/K) Figure 110. Hysteresis loop of a SD coherently rotating particle at temperature (T > TB). Sample shows zero coercive field as expected for superparamagnetic particles. Note the H/T abscissa. Magnetization is plotted as a function of H/T for three different temperatures as indicated in the legend. Loops at all different temperatures fall on top of each other as predicted by the Langevin equation for superparamagnetic particles. function of particle size. For very small particles the coercive field is zero and particles are superparamagnetic (SP) with magnetization determined by the Langevin function. As the particle size is increased, the coercive field increases due to the fact that the energy barrier increases. Particles with size larger than the critical single domain radius are multidomain and the coercive field decreases with increasing particle size.2,3,9 This may be due to the fact that as particle size increases the number of domains increases and thus it is easier to have domain closure which decreases coercivity because there is less total magnetization. The size dependence of the coercivity in MD region is experimentally found to be2 HcMD = a + b/d" (118) where a, b are constants that depend on the real structure factor and materials, d is the diameter of the particles and x has value around 1.2 There is no theoretical model that Particle diameter Figure 111. 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. 118. Thus the the peak in the coercive field when plotted as the function of the particle size delineates the SD and MD behavior. In experiment we have found the same behavior for both rmultil vr and single l r samples of Ni particles in A1203 matrix as will be discussed in detail in chapter 2. Figurel12 shows the possible domain wall configuration for different points in the magnetization loop. Remember that compared to the SD case where the origin of the hysteresis was the hopping over a energy barrier, in case of MD the origin of hysteresis is irreversible domain wall motion. At very high magnetic field all the spins in the system will be aligned along the magnetic field and positive saturation (. [.) will be achieved (Fig. 112). As the magnetic field is reduced to zero a domain wall will be formed. Due to the imperfections in the sample, the domain wall will be stuck in a position such that the up domain is larger than the down domain and net magnetization or remanent magnetization (Mr) will be seen at zero magnetic field. Reversing the magnetic field will IH II Figure 112. I I Hysteresis loop of a MD system and possible domain wall configuration. At very large positive magnetic field all the spins are aligned along the magnetic field and saturation magnetization is achieved. When magnetic field is reduced to zero, a domain wall forms. Due to the imperfections in the sample, the domain wall will be stuck in a position such that the up domain is larger than the down domain and remanent magnetization is measured. If the direction of the magnetic field is reversed the domain wall will start to move to the right and the down domain will grow. At a magnetic field equal to the coercive field, the down and up domain will be equal in size and magnetization will be zero. For a large negative magnetic field the domain wall be moved to the right and all the spins will be in the direction of the magnetic field and negative saturation will be reached. move the domain wall to the right side and thus the down domain will start to grow and magnetization will be reduced. When the negative magnetic field is equals to the coercive field the up and down domain will have same size and magnetization will be zero. Further increase in magnetic field in the negative direction will force the domain wall to move all the way to right making all spins aligned along the magnetic field and negative saturation will be reached. * M M S/ To derive the coercive field in MD domain case consider a simple case, as shown in Fig. 113, 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 113. Single domain wall separating two magnetic domains. Right hand side is a spin up domain and left hand side is a spin down domain. In an applied magnetic field due to the Zeeman energy the domain wall will experience a pressure and some work need to be done to move the wall against this pressure. The origin of hysteresis in MD sample is the irreversible motion of the domain wall. along the spin up domain, the Zeeman energy of the up (down) domain will be 1 I[ (+M I) per unit volume. Thus the energy difference across the domain wall will be 2HMs per unit volume. This energy difference can be considered as a pressure on the wall and some work has to be done to move the domain wall against this pressure. The work done to move the wall a distance dx is2,11 dW = 21. [Sdx (119) where S the area of the domain wall. Thus the work done to move the wall by unit distance is2,11 dW/dx = 21. IS (120) where dw/dx can be thought of as the resistance of the domain wall motion. In real samples due to the impurities, imperfections, strains etc, dW/dx passes through maxima and minima. The wall motion over these maxima and minima is irreversible in magnetic field and that is the origin of the hysteresis. The coercive field, the measure of irreversibility, is usually given by2'11 1 H = (dW/dx)Tmax 211. 9 There are different theoretical models to calculate (dW/dx),ax for different imperfections in the sample and the results for some of them are listed in Tablel1. Table 11. Table here lists some known models along with the variation of coercive field according to the model. Theory He System References StonerWohlfarth H = 2K [1 (kBg I /KV)1/2] SD, CR nanoparticle with uniaxial StonerWohlfarth Micromagnetic Inclusion Theory Inclusion Theory Inclusion Theory Inclusion Theory Inclusion Theory H, = [1 ( Ig H = (kB H, 2K Ms S/KV)3/4] r 2/3 2/3 Hc = M,/d H,  1/2(ln2L) H, ( )3/2, H 2 = [0.386 + log 2 ] H, = 3ylo/Md2 anisotropy along the applied magnetic field SD, CR nanoparticle with uniaxial anisotropy randomly oriented MD, 2 phase material, hard magnet, a3 is the micromagnetic parameter and depends on the K, Ms, A MD system, d < 6, free pole energy is ignored, coercivity is assumed to be equal to the maximum pining field, d is the diameter of the inclusion, 6 is the domain wall thickness, 7 is the domain wall energy per unit area, a is the volume fraction of the inclusion MD system, d > 6, free pole energy is ignored, L is the linear dimension of the sample MD system, d < 6, free pole energy is ig nored MD system, d < 6, free pole energy is con sidered MD system, closer domain, large inclusion, commonly seen in the case of Neel's spike, I is the equilibrium length of the spike 1.4 Magnetization vs. Temperature Until now we have been discussing the behavior of magnetization as a function of magnetic field at a fixed temperature. Now we will discuss how magnetization changes with the temperature at a fixed magnetic field. At small applied magnetic field, spins are trapped in metastable energy minima separated by energy barriers from the global (121) 7 12 11,13 11,14 11,14 15 16 minima. As the temperature is increased the spins can hop over the energy barrier to reach the global minima. Due to this trapping of spins in local minima, magnetization values depends strongly on the cooling protocol. There are mainly two different cooling protocols, field cooled (FC) and zero field cooled (ZFC). The behavior of magnetization as a function of temperature for the two protocols is shown in Fig. 114 below for the sample of Ni nanoparticles of 3 nm diameter at an applied field of 20 Oe. The temperature where the difference between FC and ZFC disappears is generally called the irreversible temperature (Ti,). For nanoparticles Ti, is same as the blocking temperature (TB).2 1.5x10 0 Im FC H = 20 Oe 1.0x10  i/4 5.0x10^  ZFC 0.0 0 50 100 150 200 250 300 T(K) Figure 114. Magnetization vs. temperature at an applied magnetic field of 20 Oe for the 3 nm diameter Ni nanoparticles. The red color is the field cooled (FC) magnetization and the black one is the zero field cooled (ZFC) magnetization. 1.4.1 Zero Field Cooled (ZFC) Magnetization Zero field cooled magnetization is measured by cooling the sample from high temperature (temperature above the irreversible temperature (Tir)) without any applied magnetic field. At low temperature a small magnetic field is applied and magnetization is measured as a function of temperature during the warm up while keeping the magnetic field on. Here we will discuss the shape of the ZFC magnetization in a qualitative manner. In general the magnetic system can be treated as a twostate problem as shown previously in Fig. 14 where spin up and down correspond to the energy minima separated by some energy barrier. The origin of the energy barrier in the SD case is the anisotropy whereas for the case of MD the origin is domain wall pinning at defects. At high temperature the energy barrier is easily overcome due to the thermal energy and the spin up and down states will be equally populated. Thus at high temperature above Ti,,, magnetization will be zero. Now if the sample is cooled to a low temperature without any applied magnetic field, then zero magnetization state will be blocked as the energy barrier is now large compared to the thermal energy.2,3,6 If a small magnetic field is applied the change in magnetization will occur only for the small energy barriers that can be overcome at that temperature and a small magnetization will be achieved. As temperature is increased, the probability of overcoming the larger barriers increases and magnetization increases. At temperature Tir the probabilities to overcome the barrier for spin up and down become nearly equal and the spin up and down mixing starts to happen and thus magnetization decreases with further increase in temperature. 1.4.2 Field Cooled (FC) Magnetization Field cooled magnetization is measured by cooling the sample from high temperature to the low temperature in an applied magnetic field and magnetization is measured during the warm up process.2 In this case at high temperature due to the applied magnetic field, the spin up states are more populated than the spin down states. Cooling the sample at a low temperature while keeping the field on will thus lock the system in magnetized state. An increase in temperature will increase the probability of spin up and down mixing and thus magnetization will gradually decrease. CHAPTER 2 DIPOLAR INTERACTIONS AND THEIR INFLUENCE ON THE CRITICAL SINGLE DOMAIN GRAIN SIZE OF NI IN LAYERED NI/AL20O COMPOSITES 2.1 Abstract Pulsed laser deposition has been used to fabricate Ni/Al2Os mruiltili, r composites in which Ni nanoparticles with diameters in the range of 360 nm are embedded as 1l. _i in an insulating A1203 host. At fixed temperatures, the coercive fields plotted as a function of particle size show welldefined peaks, which define a critical size that delineates a crossover from coherently rotating single domain to multiple domain behavior. We observe a shift in peak position to higher grain size as temperature increases and describe this shift with theory that takes into account the decreasing influence of dipolar magnetic interactions from thermally induced random orientations of neighboring grains. 2.2 Introduction The magnetic properties of nanoparticles have been the focus of many recent experimental and theoretical studies. Technological improvements have now made it possible to reproducibly fabricate nanomagnetic particles with precise particle size and interparticle distances. 1722 These controlled systems have enabled study of the fundamental properties of single as well as interacting particles. Most applications require that the particles be single domain with a uniform magnetization that remains stable with a sufficiently large anisotropy energy to overcome thermal fluctuations,23 which establishes a temperaturedependent lower bound to the particle size. These considerations must take into account the effect of interactions on magnetic properties as is evident for highdensity recording media24 where particles are very close to each other. Considerable insight has already been gained from experimental studies of the effect of dipolar interaction on superparamagnetic relaxation time2534 and blocking temperature.29 Less understood however is the effect of dipolar interactions on the establishment of an upper bound to particle size, which defines the crossover from single domain (SD) to multi domain (M!l)) behavior. In the following we show using coercivity measurements on Ni/A12Os composites that with increasing temperature this upper bound to particle size increases and then saturates due to attenuated dipolar interactions from thermally induced coherent motions of the magnetization of the neighboring randomly oriented particles. 2.3 Experimental Details The composite system studied in this paper comprises elongated and pi li, iv H11i.' Ni particles with diameters in the range of 360 nm embedded as lvir in an insulating A1203 host. The muiltiliv r samples were fabricated on Si(100) or sapphire (caxis) substrates using pulsed laser deposition from alumina and nickel targets. High purity targets of Ni (99.9' '. ) and A12O3 (99.9'" .) were alternately ablated for deposition. Before deposition, the substrates were ultrasonically degreased and cleaned in acetone and methanol each for 10 min and then etched in a I!' I'_ hydrofluoric acid (HF) solution to remove the surface silicon dioxide lv,_, thus forming hydrogen terminated surfaces.35 The base pressure for all the depositions was of the order of 10' Torr. After substrate l., lii. the pressure increased to the 10Torr range. The substrate temperature was kept at about 550" C during growth of the A1203 and Ni 1 .ir. The repetition rate of the laser beam was 10 Hz and energy density used was ~ 2 Jcm2 over a spot size 4 mm x 1.5 mm. A 40 nmthick buffer li;vr of A12O3 was deposited initially on the Si or sapphire substrate before the sequential growth of Ni and A1203. This procedure results in a very smooth starting surface for growth of Ni as verified by high resolution scanning transmission electron microscopy studies (Fig. 21). Multill , samples were prepared having 5 1 ,ris of Ni nanoparticles spaced from each other by 3 nmthick Al20s3 1I ,r. A 3 nmthick cap l ,r of A1203 was deposited to protect the topmost l1*ir of Ni nanoparticles. Shown in Fig. 21 is a crosssectional TEM image from a multi1 i,, 1 (5 l.. rs) NiA1203 sample grown on cplane 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 crosssectional view), which is comparable to the thickness of the A1203 spacer l iVt. For the purposes of this Figure 21. Cross sectional dark field STEM image of a 51i r NiAl203 sample grown on caxis 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 diskshaped grains shown in the figure. This calibration was obtained from crosssectional TEM micrographs of single l1.r  i1 !1. '' by comparing the average grain size with d. The TEM observation also shows that the Al203 spacer l.ri~ are partially crystallized. Due to the large surface energy difference between Ni and A1203 Ni forms welldefined, separated islands within the Al203 matrix.36 Previous studies on similarlyprepared samples using atomic number (Z) contrast imaging in TEM together with electron energy loss spectroscopy (EELS) have confirmed the absence of NiO at the Ni/A1203 interfaces.36 The Ni/A1203 interfaces were chemically abrupt without an intermixing between Ni, Al and oxygen. In addition we did not observe exchangebias induced .*,mmetric magnetization loops, thus lending support to the conclusions of previous studies36 that antiferromagnetic NiO is absent in our 1 ,, i, t Ni/A1203 system. Previous TEM studies on single lir samples have shown the particles to be p&I, *1 ,1 ilii,. For example, a three nm particle comprising three crystalline grains has been observed.3 P li. i lii, i11 1!: particles will therefore have crystalline grains oriented in different directions, thus tending to average any net crystalline anisotropy to zero. Accordingly, temperatureindependent shape anisotropy is dominant and temperaturedependent ( important to note that the exchange length l,, = 14.6 nm for Ni,37 which is the length scale below which atomic exchange interactions dominate over magnetostatic fields, determines the critical radii (Rcoh) for coherent rotation: Rcoh ~ 51,, for spherical particles and Rcoh ~ 3.51,, for nanowires.3 The particle sizes (1.530 nm in radius) that we have investigated are thus smaller than the critical radius below which coherent rotation of Ni prevails. 400 2001[ Figure 22. 0 20 40 60 Particle Size (nm) Coercivity for 51,ir Ni/Al20 muiltil ,r samples (5 repeated units) plotted as a function of particle size (diameter) at the temperatures indicated in the legend. The peak positions at d = d, for each isotherm, indicated by vertical arrows, delineate the crossover from single domain (SD) to multiple domain (Ml1)) behavior (d > de). Inset shows the behavior of H, as a function of 1/d for the particles with d > de at 10 K. The linear dependence up to 24 nm diameter particles with saturation at a constant value for large par i !. is consistent with the behavior expected for multidomain particles. Thus particles on the righthand side of the peak are multidomain. In Fig. 22 we show plots of H, as a function of particle size d at each of the temperatures indicated in the legend. Coercive fields were extracted from magnetization SD < > MD 45  H js. Id 10 K 24..r. 002 0.04 m0.6 0.00 lid(nm l S\100 K Sr 1o K '/ 4 15 K loops measured by a Quantum Design superconducting quantum interference device (SQUID) after subtracting out the diamagnetic contribution from the substrate. Magnetic field was applied along the plane of the films. To obtain the magnetization loops, the magnetic field was varied over the full range (5 T) while keeping temperature fixed. The high magnetic field data show linear magnetization with magnetic field, which is due to the diamagnetic contribution from the substrate (as signal from ferromagnetic Ni particles saturates at high magnetic fields) and can thus be subtracted from the data. The decrease of H, with increasing temperature for fixed d is clearly apparent and can be understood as the effect of thermal fluctuations.2 For the lowtemperature isotherms, there are pronounced peaks which define a temperaturedependent critical particle size dc delineating SD (d < dc ) behavior of coherently rotating particles from MD (d > d,) behavior. 2,8,3945 The reason why there is a peak in H,(d) is explained in the introduction chapter, page 29. In the inset of Fig. 22 we have plotted H, versus 1/d for the particles of size d > d, at 10 K. It is clear that H, behaves linearly with 1/d up to particle size of 24 nm and then saturates. This behavior is consistent with the dependence expected for multidomain particles. 3 Thus particles of size d > d, are multidomain and the peak defines the crossover from SD to MD behavior. The formation of domain structure is driven by the reduction of long range magnetostatic energy, which at equilibrium is balanced by shorter range exchange and anisotropy energy costs associated with the spin orientations within a domain wall. The purpose of this chapter is to show that this welldefined SD region of coherently rotating particles extends over a larger range of grain sizes at higher temperatures because of the diminishing influence of dipolar interactions from neighboring grains. 2.4 Data and Discussion The influence of dipolar interactions on the SD/\!I) crossover can be understood in a qualitative way by considering the three randomly oriented particles shown schematically 16 Simulation Experiment 14 * S12 S10 I I 8 6 I 0 50 100 150 200 250 T(K) Figure 23. Peak position, d,, plotted as a function of temperature (red circles). The black squares are the results derived from equation 25. The blue star represents the observed value of dc for a series of single lwr 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. 23. Particle 1 experiences dipolar fields from particles 2 and 3, which are not collinear for most orientations of a randomly oriented particle system. Because dipolar fields decrease rapidly with interparticle separation, the dipolar field due to particle 3 (2) will be stronger than particle 2 (3) on the left (right) side of the particle 1. The separate and unequal influence of the neighboring particles thus favors the formation of domains in particle 1. .To make these notions more quantitative, we modify the treatment of Dormann et a126 for interacting paramagnets to include the temperature region below the blocking temperature TB and find the temperaturedependent dipolar magnetic field Hd arising from temperature induced fluctuations in the magnetization of nearest neighbor nanometer size particles to be, Hd .a eC(1 e) 1 47r v/j(erfi(3) erfi(v  T )) where 'erfi' is the imaginary error function, i. ., is the saturation magnetization, / KV/kBT and a = V(3cos2 l)/s3 is a dimensionless parameter with and s corresponding respectively to an angle parameter and the separation between two .idi i:ent particles each with volume V The parameter Q is alv,v greater than one for T < TB where there is still coercivity; i.e., the magnetization is fluctuating but not going over barriers. Then Eq. 21 has the limiting value at T > 0 as given below. Hd T T 0 (22) 47 The derivation of Eq. 21 includes averaging over the accessible directions of magnetization weighted by a Boltzmann factor. Higher temperatures give smaller magnetizations since the particles fluctuate over larger angles. Specifically, spin up and down particles will be in energy minima separated by an anisotropy energy barrier. At absolute zero temperature only the direction corresponding to the minima of the energy will be occupied. At finite temperatures, according to the Boltzmann law, other energy states will be occupied around this minimum and will have different directions of magnetizations. Thus to obtain the actual magnetization, an average over all these accessible directions is calculated, constrained by the fact that the probability of those states to be occupied is given by the Boltzmann factor OT exp E(0) OTep[ ( ]'( ' fK exp[ f]dO where at zero magnetic field E(O) =KVsin20 Thus Omin = 0 and OT is temperature dependent, obeying the relation, sin20T = kBT/KV The parameter OT (see Fig. 17 on page 21 of chapter) will be higher at higher temperatures and thus the thermal average of the magnetization will diminish at higher temperatures. Using Eq. 23 one can determine the temperature dependence of the dipolar magnetic field Hd as shown in Eq. 21 for particles treated as simple dipoles. .In the absence of interactions (Hd = 0) the condition for the SD to MD transition is given for spherical particles with radius d/2 by, Ad = Bd2 where Ad' is the total magnetostatic energy and Ed, = Bdc is the domain wall energy.46 We have absorbed the factor of two, which relates diameter to radius, into the constants A and B In the presence of the dipolar magnetic field Hd the formation of domain walls will be assisted by a Zeeman term which is proportional to the volume of the affected particle. The condition determining the SD to MD transition now becomes, Ad = Bd 7ri.. .,1'/6 (24) When the dipolar interaction is a small perturbation, i.e., 3 [f,/A < 1, Eq. 21 and 24 can be combined to give the relation, (1 e() dc = dco ddw 3( (25) V/7P(erfi(3) erfi(vP 1 )) where dco = B/A is the temperatureindependent critical diameter in the absence of interactions (hightemperature limit) and dd, = poBM2Tr/(72A2) for a = r/3. The second term on the righthand side of Eq. 25 thus becomes a temperaturedependent correction to dc due to interactions from neighboring particles and decreases with increasing T . Since the magnetic field due to the dipoledipole interactions are weaker at higher temperatures Eq. 21, the nanoparticles remain in the SD state to a larger size, which by Eq. 25 results in a shift of dc towards higher values at higher temperatures. This is indeed evident in Fig. 23, which shows the temperature dependence of dc as determined from the data in Fig. 22. The black squares are the simulated data according to Eq. 25 using the two fitting parameters: dco and dd, Qualitatively, the data agree quite well with the prediction of the theoretical model without taking into account the topology and size distribution of the particles. We have found dco = 84 nm from our simulation (Fig. 23, black squares) to be close to the value for a particle with shape anisotropy constant Kshape = 3.1 x 104 J3 (do = 72AeK/,,,JV where Ae, is exchange stiffness, K is anisotropy constant).3 Values of A (oc ,,,, 1' ) and B (oc A,,K ) have been found to be 1.44 x 104Jm3 and 1.21 x 10lJm2 respectively. This value of A is very close to the theoretical predicted value3 and the value of B is again consistent with the value of the shape anisotropy. The value of the shape anisotropy can also be predicted from the zero temperature extrapolation H,, ~ K/M1. for randomly oriented particles.3 For Shape = 3.1 x 104Jm3, H,, ~ 620 Oe. This is in good agreement with the 500 Oe coercive field observed at 10 K for the 6 nm sample. For a separate series of single livr samples the coercivities at 10 K peak at de S14 nm as shown in Fig. 23 by the blue star. In the single lvr samples the peak position occurs at higher particle size (14 nm) than muiltiliv r samples (8 nm). This difference reinforces our interpretation and can be understood by realizing that the dipolar interactions of the single 1 ir samples are significantly reduced compared to the rniitil i.r samples because of the smaller number of nearest neighbors. 2.5 Conclusion In summary, we have fabricated magnetic nanoparticles in an insulating thin film matrix with tunable properties achieved by varying particle size and temperature. The peaks in the coercivity isotherms delineate a critical grain size de which identifies the crossover from SD to MD behavior. The presence of dipolar interactions and their diminishing influence with increasing temperature is responsible for the observed dependence of de on temperature and is in good qualitative agreement with our modification of present theory26 of interacting particles. The wellestablished influence of dipolar interactions on superparamagnetic relaxation time2534 together with the connection between relaxation time r and coercivity H, sil. 1 that there is a concomitant influence of dipolar interactions on the coercivity observed near the superparamagnetic limit where He = 0. The work reported here extends this connection to the upper limits on the size of SD particles by showing that dipolar interactions can facilitate the formation of multi domain particles especially at low temperatures. 2.6 Methods 2.6.1 Mathematical Analysis The Eq. 25 is self consistent (as the term / contains dc) and can not be solved analytically. The equation, d dco + dd w (ei(1 ), )) w3ir(erfi~jQ I I 3P1) 0, is solved by numerical approach and simultaneously the solution is fitted to the experimental data according to a nonlinear list square method. Mathematica, a commercial software, is used for this purpose. 2.6.2 Basic Physical Understanding A simplified physical understanding of the problem is shown in Fig. 24. 1 1 Hd \i\~s~ wy Hd' Figure 24. The net effect of dipolar magnetic field (Hd) is shown on the particle 1. As particles are randomly oriented, Hd from particle 3 will be in different direction than that from particle 2. As dipolar interaction decreases rapidly with distance, particle 1 will experience local dipolar magnetic fields in different directions from different neighboring particles and thus making it easy to form domains. CHAPTER 3 EFFECT OF DIPOLAR INTERACTION ON THE COERCIVE FIELD OF MAGNETIC NANOPARTICLES: EVIDENCE FOR COLLECTIVE DYNAMICS 3.1 Abstract The effect of dipolar interaction on the coercive field is discussed for the single domain and coherently rotating Ni nanoparticles embedded in A1203 matrix. Results for two sets of 5 livr samples with different interlayer spacing and a set of single l1ir samples of Ni nanoparticles are compared. The dipolar interactions are strongest in the samples with shorter interlayer distances and weakest for the single 1ivr samples. In this present study, the dipolar interaction is found to increase the coercive field. On the other hand the critical single domain radius decreases due to the dipolar interactions. These two behaviors together indicate that collective dynamics pl i', an important role in understanding the origin of the coercive field. 3.2 Introduction The origin of coercive field (H,) for coherently rotating ferromagnetic nanoparticles is remarkably different than that of the bulk,47 where irreversible domain wall motion is the dominant mechanism.4 In the case of nanoparticles, when the size of the particle is smaller than a critical size (dc), the most favorable energy state is to have single magnetic domain and particles are called single domain (SD) particles. When H, is plotted as a function of particle diameter (d), there is a well defined peak at de. Particles with d < d, (d > d,) are SD (multidomain (I\l))). 2,8,41,42,44,48 Kittel3,46 has shown that for a spherical particle, d, is given by the relation (see Eq. 13 on page 13 of chapter) dc 72 (31) S' ,, /.' where A is the exchange stiffness, K is the anisotropy constant, /o is the free space permeability and [.. is the saturation magnetization. In SD particles there is no domain wall. The origin of H, in this case is the finite time required to reverse the magnetization direction over the magnetic field dependent anisotropy energy.47 The time required to reverse the direction of the magnetization of a coherently rotating SD particle is given by the relation2'5'49'50 T = Toexp (32) kBT_ Here, To is the inverse of the attempt frequency to overcome the energy barrier, V is the volume of the particle, kB is the Boltzmann constant and T is the temperature. Stoner and Wohlfarth have calculated He for SD particles in the simple case when particles are coherently rotating and the applied magnetic field is along the easy axis of magnetization of the particles. The coercive field for a StonerWohlfarth particle is given by 2K I (ntr1n (2f Hesw = 1 n ( (33) where Tm is the time of measurement. From the simple StonerWohlfarth model it is clear that He for the nanoparticle can depend on many different factors. He increases with decreasing T,, increasing T and increasing K. In the presence of dipolar interactions the above equation will be modified. The widely accepted modification is achieved by treating the dipolar interactions to result in an effective anisotropy energy.5161. Thus if due to the dipolar interactions K increases (decreases) then ', according to Eq. 32, will also increase (decrease) and as a net result He will increase (decrease). A more familiar famous form of Eq. 33 is Hsw(T) = 2K/M..(1 (T/TB)1/2), where TB = KV/25kB is the blocking temperature. The factor 25 comes from the fact that Tm, 100 s is a typical measurement time and rol ~ 109 sec1 is a typical attempt rate. The effect of dipolar interaction on the coercive field (He) has been investigated extensively. The first theoretical treatment by Neel62 showed that He decreases with the increase in the packing fraction (c) or the dipolar interaction as shown below in Eq. 34, where the interaction effect has been introduced as an "Interaction Field" and shown to lower the anisotropy energy. H, H,. (t e) (34) The "Interaction Field" is a function of the packing fraction (e). Later Wohlfarth51 showed that the effect of the interaction on the He can be increasing or decreasing depending on the particle orientation as the dipolar interaction is direction dependent. But all of those results have been constructed considering the fact that the anisotropy constant, K, either increases or decreases due to the interactions. Previous theoretical and experimental works have been reported either showing an increase or decrease in H, and explained in terms of a corresponding increase or decrease in the anisotropy energy.5161,63 In this present experiment we find that an increase in the dipolar interaction increases H, but decreases de. Equation 31 sil. 1 that the decrease in dc may be due to a decrease in the K. But a decrease in the K will also decrease r (Eq. 32) and thus will decrease H, (Eq. 33) which is contradictory to the present experimental result. Thus the change of K due to the dipolar interactions must not be applicable in the present case. As any change in K will give rise to change in H, and d, both in the same direction (both increase or decrease at the same time). Below, we show qualitatively that the increase in the He can be realized in terms of the collective dynamics of the magnetization of the particles and decrease in d, can be understood as discussed in reference48 3.3 Results and Discussions Samples were grown using pulsed laser deposition technique.48 Base pressure of the growth chamber was on the order of 10' Torr and the growth temperature was around 550" C. Multilayer structure of Al203 and Ni nanoparticle were grown without breaking the vacuum of the chamber. First a thick (40 nm) buffer livr of Al203 is grown on top of the substrate. The purpose of this buffer livr is to prevent any diffusion of the Ni into the substrate. Then Ni nanoparticles and A1203 are sequentially deposited on this buffer l?vr (see Fig. 1). The top 1v.r of A1203 acts as a capping l,r which prevents oxidation of the nanoparticles.36 Three different sets of samples are grown. Set 1 and set 2 samples consist of 5 1l ri of Ni nanoparticles separated by A12O3 1V, r. For set 1 (set 2) the A1203 separation is 3 nm (40 nm). Set 3 samples are single l1vr of Ni nanoparticles in A1203 matrix. Dipolar interactions are strongest in set 1, moderate in set 2 and weakest in set 3. The dipolar interactions are stronger in Set 1 compared to set 2 as the interlayer separation of the Ni particles is smaller in set 1 compared to set 2. Set 3 consists of only a single 1 v, of Ni particles and thus the dipolar interactions are weakest. All sets of samples consist of different samples with varying particle size from 3 nm to 60 nm. C) 1 41 1 1 4 4 4 Estate Multi Layer Figure 31. Fig la) shows the TEM image of a single 1i,r sample with average particle diameter of ~ 24 nm. Particles are well defined with inter particle distance of around 4 nm. Ib) shows a schematic of the single lv, r sample. A 40 nm thick buffer 1v,.r of A1203 is first grown on top of substrate. Then the Ni nanoparticles are grown on to of the buffer 1V.r. Finally a 3 nm thick capping lIv,r of A12O3 is grown to protect it from oxidation. ic) shows the schematic of 5 liv.i~ of Ni nanoparticle sample. Figure 3la) shows the TEM image of the single lv, r Ni particles with average particle diameter of ~ 24 nm (set 3). The simplified schematic of the single and rmiltil ivr samples are shown in 31 b) and c). Typical magnetization loops at three different temperatures are shown in Fig. 32a) for the sample with 3 nm A1203 spacer lv1 r (set 1) and 6 nm in diameter. The coercive field H,(T) is determined from the loop as shown by the arrow. This procedure to determine H, is repeated for all samples belonging to all three sets. At temperatures a) b) 4 44 statee Single Layer 15 M10K 10 M50K M100K, / 0 5 5 10 H, 15 1.5 1.0 0.5 0.0 0.5 1.0 1.5 H (KOe) b) 10 5 0 S5 S10 a) E L Figure 32. a) Magnetization loop of a sample from set 1 of average particle diameter of ~ 6nm. Coercive field (H,) is determined from the loop as shown by the arrow. H, decreases with increasing temperature and goes to zero above the blocking temperature. b) Magnetization loops above blocking temperatures. Magnetization is plotted as the function of H/T to show the superparamagnetic behavior as expected for the SD particles above the blocking temperature. above the blocking temperatures (TB) SD samples behave as superparamagnetic particles. Figure 32b) 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 33 shows H, plotted as a function of d for the set 1, set 2 and set 3 samples. The data that correspond to the different sample sets are indicated in the legends. The peak in the He separates SD and MD particles. 2,8,41,42,44,48 It is clear from the data that dc decreases with increasing dipolar interactions (de1 < d,2 < d). H, on the other hand increases with the increasing dipolar interactions (vertical dotted arrow) in the SD region. These two results can not be explained in terms of the commonly reported change in K due to the dipolar interactions.5161 The decrease in d, due to the dipolar interactions has been discussed elsewhere.48 In this present study, the collective dynamics of the particles 6 4 2 0 2 4 6 H/T(Oe/K) M250K o M300K M325K ' J , d at \ 10 K 3 nm separation (Set 1) 40 nm separation (Set 2) Single layer (Set 3)  30 40 Figure 33. Coercive field (He) as a function of particle diameter (d). The peak separates the single domain (SD) and multidomain (\ l)) particles. Particles with diameter higher (smaller) than the peak diameter (de) are MD (SD). Data for the 3 different sample sets are shown and indicated in the legends. The critical diameters d1,, dc2, d,3 are shown from the samples of set 1, set 2 and set 3 respectively. In the single domain region (below d,) the coercivity increases with increasing dipolar interactions as shown by the vertical dotted arrow. magnetization due to the dipolar interactions is found to be responsible for the increase in He. These observations are shown in Fig. 33 and summarized in Fig. 34. We first discuss the effect of dipolar interactions on He as presented in previous investigations. 5161 The treatment begin by including the change in anisotropy energy Edrp, due to dipolar interaction into the expression for r, as given by63 SKV + Ed ] T7 Toexp L j kL IT (35) 600 400 0 o 200 0 10 20 d (nm) d 0 Equation35 can be rewritten as shown in Eq. 36. Thus the effect of the dipolar interactions is treated as either an increase (+ Edip) or decrease ( Edip) of anisotropy energy. 7 0oexp K ) V (3 6) The effect of dipolar interactions on the He can be explained according to Eq. 36. In our case a + Edrp increases 7 and give rise to an increase in He with increasing dipolar interactions (Eq. 32 and 33). If this is to be true in our case then according to Eq. 31, de should also increase with increasing dipolar interactions. According to the previous approach both He and de should change in the same way, both increase or both decrease. In the present experiment we find however that He increases and de decreases due to dipolar interactions (see Fig. 34) and strongly ii. 1 an alternative approach to the problem. The effect of dipolar interactions on d, is discussed in reference48, where it has been shown that the local dipolar magnetic field from the nearby randomly oriented particles try to align the magnetization direction of the particle in different directions and thus favoring domain formation. The effect of dipolar interactions on He will be discussed below in terms of collective dynamics. It is well known that the magnetization dynamics can be collective in nature due to the interactions between the particles and the relaxation time (7*) in this case is given by 64,65 T T > T, (37) where T is the relaxation time of the single non interacting particle (Eq. 32), T 1 ,, f2/47kBr3 is the critical temperature and depends on the interparticle distance and particle magnetization and z is a critical exponent. The above equation clearly i I i that the relaxation time will be larger in the presence of dipolar interactions and thus according to Eq. 33 He will be larger, and thus agreeing with our experimental 14  Hc 4 Set 2 12 0 E * 2 o 10 t 10 Set 2 Set 1 8 Set 3 0 Dipolar interaction strength Figure 34. Coercive field (He) and critical diameter (de) as the function of the increasing dipolar interaction. H, (de) increases (decreases) with increasing dipolar interaction. The opposite behavior of He and d, , r. I that the collective dynamics and the critical slowdown is responsible for the increase in H, due to the dipolar interactions. The decrease in d, is discussed elsewhere.48 observations (Fig. 34). Note that in this case the anisotropy energy is unaffected by dipolar interactions and the increase in relaxation time is due to the fact that the reversal of magnetization is collective in nature.64,65 3.4 Conclusions A study of dipolar interactions is presented for the single and muiiltil i structure of Ni nanoparticles. The coercive field has been found to increase with increasing dipolar interactions and can be understood qualitatively in terms of collective dynamics. Three sets of samples are investigated. Each set consists of samples having particle size varying from 3 nm to 60 nm in diameter. Dipolar interactions are stronger in set 1 and decreases for set 2 and set 3. Behavior of coercive field and critical single domain radius are observed. Coercive field increases and critical single domain radius decreases Set 1 Set 3 *dc with increasing dipolar interactions. These two behaviors together si' I a collective dynamics of the magnetization reversal process in the SD region in the presence of dipolar interactions. To our knowledge, this is the first time that the effect of collective dynamics on a coercive field of the nanoparticle system has been observed. CHAPTER 4 FINITE SIZE EFFECTS WITH VARIABLE RANGE EXCHANGE COUPLING IN THINFILM PD/FE/PD TRILAYERS 4.1 Abstract The magnetic properties of thinfilm Pd/Fe/Pd trilayers in which an embedded ~1.5Athick ultra thin 1. r of Fe induces ferromagnetism in the surrounding Pd have been investigated. The thickness of the ferromagnetic trilayer is controlled by varying the thickness of the top Pd 1v,r over a range from 8 A to 56 A. As the thickness of the top Pd 1 r decreases, or equivalently as the embedded Fe 1, r moves closer to the top surface, the saturated magnetization normalized to area and the Curie temperature decrease whereas the coercivity increases. These thicknessdependent observations for proximitypolarized thinfilm Pd are qualitatively consistent with finite size effects that are well known for regular thinfilm ferromagnets. The functional forms for the thickness dependence, which are strongly modified by the nonuniform exchange interaction in the polarized Pd, provide important new insights to understanding nanomagnetism in twodimensions. 4.2 Introduction The presence of 3d magnetic transition metal ions in palladium (Pd) gives rise to giant moments thus significantly enhancing the net magnetization 6670. Pd is known to be in the verge of ferromagnetism because of its strong exchange enhancement with a Stoner enhancement factor of ~ 1071. The magnetic impurities induce small moments on nearby Pd host atoms thereby creating a cloud of polarization with an associated giant moment71'72. Neutron scattering experiments show that the cloud of induced moments can include ~200 host atoms with a spatial extent in the range 10 to 50 A72,73. Thus a thin 1I, of Fe encapsulated within Pd will be sandwiched between two ..11i i'.ent thin l,' is of ferromagnetic Pd with nonuniform magnetization and a total thickness in the range 20 to 100 A. We have investigated thinfilm Pd/Fe/Pd trial 'r;i in which the thickness dFe of the Fe is held constant near 1.5A and the thickness of the polarized ferromagnetic Pd is varied by changing the top Pd 1vr thickness x. The magnetic properties are studied as a function of x. Our experiments are motivated by the recognition that ferromagnetism in restricted dimensions has attracted significant research interest74 78. For example, the coercive field He increases as the thickness of the ferromagnetic film is decreased toward a thickness comparable to the width of a typical domain wall79'80. Moreover, the Curie temperature T, decreases as the thickness of the ferromagnetic film is decreased toward a thickness comparable to the spinspin correlation length8183. We will show below that similar phenomenology applies to ferromagnetically polarized Pd films, albeit with different functional dependence arising from the fact that exchange coupling, which decays with distance from the ferromagnetic impurity84, is not uniform throughout the film. 4.3 Experimental Details The samples were grown on glass substrate by RF magnetron sputtering. The base pressure of the growth chamber was of the order of 109 Torr. First a thick livr of Pd of thickness 200 A is grown on top of the substrate. The root mean square surface roughness of this Pd l1 r was measured by atomic force microscopy to be ~ 6 A. Then a very thin (1.5 A as recorded by a quartz i I I1 monitor) lIvr of Fe is deposited on top of the first Pd 1 i.r. Finally a top 1V.r of Pd with thickness x is grown to complete the trili vr structure shown schematically in Fig. 4la. We discuss six different samples with the top Pd 1~vr having a thickness x varying from 8 to 56 A. The total thickness y of the polarized Pd (see Fig. 41b) can range from 20 to 100 A72,73. Thus for x < y/2, changes in x will give rise to changes in y. Auger electron spectroscopy (AES) was used to verify the presence of a well defined Fe l. r. The AES measurements were performed in a 1010 Torr vacuum at sequential intervals following removal of sub angstrom amounts of Pd using an Argon etch. The depth profile of the high intensity Fe3 (703.0 eV) LMM Figure 41. 14.0k S12.0k  10.0k 3 8.0k 6.0k  4.0k S2.0k 0.0 a) Physical structure Paramagnetic Pd Poromngmnetic Pd Paramagnetic Pd b) Magnetic structure a) Multil]i r structure of a Pd/Fe/Pd trilayer. The bottom l ir of Pd is 200 A thick. The thickness of the Fe 1. r is 1.5 A as recorded by the quartz crystal monitor. The thickness x of the top 1I,r of Pd is varied from 8 to 56 A. b) Magnetic structure of the sample. The total thickness y of polarized Pd is in the range 20 to 100 A (shaded red area). Thus by varying x, it is possible to vary the thickness y of the polarized ferromagnetic Pd li. r. c) Intensity of Fe3 (703.0 eV) LMM Auger electron peak plotted as a function of material removed by argon sputtering. The data (solid black circles) are fit to a Gaussian distribution (red line). The full width half maximum value of 1.85 A is consistent with crystal monitor measurements Auger electron peak of Fig. 41c shows that the Fe is embedded in the Pd as a distinct 2D l ,r with a FWHM thickness of 1.8 A. All of these steps were performed without breaking vacuum. Measurements of the magnetization M (Fig. 42) were performed using a Quantum Design MPMS system. The magnetic field H was along the plane of the substrate. Since the magnetization measurements were ex situ, x was constrained to be greater than 8 A; otherwise the exposure of the sample to air caused unwanted oxidation of the Fe. The magnetic parameters H((x) (Fig. 43) and T((x) (Fig. 44) are calculated respectively from magnetization loops taken at 10 K (see inset of Fig. 43) and linear extrapolations of the temperaturedependent magnetization taken at H = 20 Oe (see inset of Fig. 44). The magnetic contribution from the bottom ferromagnetic Pd li, r is independent of x, since y/2 < 200A, the constant thickness of the bottom 1 VT. Data  Gaussian fit FWHM is i.asA 0 10 20 30 40 50 60 Thickness of the film from the top (A) c) Auger electron spectroscopy *4 o10  E 5) < 8 MsA (Fe)= 2.63x105 emu/cm I 6 I 0 10 20 30 40 50 60 x(A) Figure 42. The saturation magnetization normalized to the area of the sample if i shows a smooth increase with increasing thickness x. The experimental data are shown as solid black circles and the dashed black line is a guide to the e, Saturation to a constant value occurs near 30A (vertical arrow). 4.4 Results and Discussion For large values of x, the thickness y of the combined polarized ferromagnetic Pd 1~..i~ and the associated saturated magnetization M = I ., will reach a constant value. This expectation is borne out in Fig. 42 which shows the xdependence of saturated magnetization i. i normalized to sample area. We note that this normalized saturated magnetization i [. (x) increases with increasing x as the total amount of polarized Pd increases. The onset of saturation, near x = 30 A indicates that the polarization cloud including the embedded Fe lir is ~ 60 A thick. This value is consistent with previous observation73. The increase of if.. with x shown in Fig. 42 is thus straightforward to understand. As x increases the thickness of the top polarized ferromagnetic Pd lv1r increases with a concomitant increase of magnetic material in the system. Variation of x clearly controls the thickness of the polarized ferromagnetic Pd l1~Vr. When normalized to o 40 2 400 200 0 200 400 20 H(Oe) 10 20 30 40 50 60 x(A) Figure 43. The coercive field He shows a strong increase as the thickness x of the top V, r of the Pd decreases. The data are shown as solid black circles and the black solid line is a power law fit with exponent Tl 2.3(0.1). The inset shows magnetization loops at T 10 K for x 8A (solid black squares) and x 56A (solid red circles). the number of Fe atoms present, the saturated magnetization i = 1.1 x 104emu/cm2 corresponds to 9.2 fi per Fe atom, in close agreement with previous observations of the giant moment of Fe in Pd to be near 10 PB72. Modeling the x dependence of i (x) shown in Fig. 42 for our Pd/Fe/Pd trilhv'ir is not straightforward. For regular ferromagnets with if., uniform throughout the thickness, we would expect if. ,(x) to be linear in x; clearly it is not. A reasonable model will incorporate an exchange interaction J that decays radially with the distance from the point ferromagnetic impurity84. This complication requires modeling J as a function of distance x from the plane of impurity. A starting point would be to write the magnetization M is a function of J4, M(H, T, x) i= 1. 8, PBH + 2pMJ(x) (41) \kBT 200  / 3 180 % Mvs T(x = 56A)at200e 2 U E 0 T ............ 140 c i 100 200 300 T(K) 0 20 40 60 x (A) Figure 44. The Curie temperature T, rapidly increases with increasing x. Data are shown as solid black circles and the dashed black line is a guide to the ev. Saturation to a constant value occurs near 20A (vertical arrow) The inset with T, indicated by the vertical arrow shows the temperaturedependent magnetization taken in a field H = 20 Oe. where B8 is the Brillouin function and p is the number of the nearest neighbors beyond which J is zero. In principle the experimentally determined values of M(H, T, x) can be fit to Eq. 41 to find the best fit values of J(x) for different values of the parameter p. We have not performed such an ,in !1i Fig. 43 shows the behavior of the coercivity H,(x) as a function of x (solid black circles). The data are well described by a powerlaw dependence (solid black line), H,(x) oc x", where the exponent r = 2.3(0.1) is close to the ratio 7/3. Similar powerlaw behavior reveals itself in regular ferromagnetic thin films where q has a somewhat smaller value varying from 0.3 to 1.576. Because rl depends strongly on strain, roughness, impurity, and the nature of the domain wall (Bloch or Neel type)76, it is not surprising to see a wide variation in Tl. Neel predicted for example that for Bloch domain walls, He of a ferromagnetic thin film should vary as x4/3 when the thickness x of the film is comparable to the domain wall thickness w79. For the case of Neel walls, He depends only on the roughness of the film and does not depend on film thickness77. The variation of Hc(x) becomes particularly pronounced when the film thickness becomes comparable to w. A qualitative understanding of the steeper H((x) dependence becomes evident by recognizing that the formation of domain structure is driven by the reduction of long range magnetostatic energy which at equilibrium is balanced by shorter range exchange and anisotropy energy costs associated with the spin orientations within a Bloch or Neel domain wall. Domain wall thickness is given by w = /A/K3'82 where K is the ii 1 H!iii anisotropy constant and A is the exchange stiffness, proportional to the exchange energy, Jss. The domain wall size w increases for decreasing K and increasing J. If K, which depends on the relatively constant spinorbit interaction4 within the Pd component of the Pd/Fe/Pd trili ris, remains constant, then variations in w are dominated by variations in J. Thus as x decreases toward zero, the increase in J84 gives rise to an increase in w which in turn gives rise to a more rapid increase in He than would be seen in regular ferromagnets with constant J. As discussed above, this rapid variation with q ~ 7/3 is observed experimentally. The data in Fig. 44 show that T, increases as x increases and reaches a relatively constant value near x = 20 A. The dashed black line is a guide to the eye and is qualitatively similar to the behavior of [. !(x) shown in Fig. 42 which saturates at a larger value near 30 A. These observations are again qualitatively consistent with the finite size effect associated with critical phenomena in ferrei,, iii, i" ". Although the data are not of sufficient quality to distinguish the powerlaw behavior that is predicted for finite size eff. I i183, we expect that the dependence is further complicated by the previously discussed dependence of J on x in polarized ferromagnetic Pd. The behavior of Tc(x) Ii, i that Pd/Fe/Pd trili' r should be treated as a single livr with a well defined spinspin correlation length. If the Pd l zVi~ are treated separately, then the bottom 1 rl.r with fixed thickness y/2 would have a Tc equal to the highest T, of the top livr. In this case the overall measurement would not show a strong change in T, as a function of x, since the T, of the bottom l?vr would dominate for all x. We note that for our planar geometry, Tc decreases with decreasing thickness as has also been shown for thinfilm Nisl and epitaxial thinfilm structures based on Ni, Co and Fe82. On the other hand T, increases with decreasing size of ferrimagnetic MnFe204 nanoscale particles with diameters in the range 526 nm83. This increase of T, with decreasing size is attributed to finite size scaling in three dimensions where all three dimensions simultaneously collapse83. In our twodimensional planar thin films only one of the dimensions, the thickness, collapses and T, decreases rather than increases in accord with the observations of previous studies81'82 4.5 Conclusions In conclusion, we have characterized the magnetic properties of thinfilm Pd/Fe/Pd tril1vis and determined that critical size effects apply to I! iiii ;iwtic" Pd where the ferromagnetism is induced by proximity to an underlying ultra thin Fe film. The critical size, or equivalently the critical thickness, is controlled by varying the thickness x of the top Pd l~.vr. The dependence on film thickness of the coercive field He and the Curie temperature T, are in qualitative agreement with finite size effects seen in regular ferromagnetic films where the exchange coupling J is constant throughout the film. The results presented here increase our understanding of nanomagnetism in ultra thin systems by showing that the spatial variations of J in the proximity coupled Pd have a pronounced influence on the form of thicknessinduced variations, namely: a nonlinear dependence of I.[. (x), an unusually strong powerlaw dependence of H (x) and a dependence of T,(x) which indicates that the trilayer acts as a single l1v.r that necessarily includes the constant thickness Pd l?vr serving as a substrate for the Fe 1l.r. CHAPTER 5 TEMPERATURE DEPENDENCE OF COERCIVITY IN MULTI DOMAIN NI NANOPARTICLES, EVIDENCE OF STRONG DOMAIN WALL PINNING 5.1 Abstract The temperature dependence of the coercivity of the single and 5 liv r samples of Ni nanoparticles in A1203 matrix is studied. A linear T2/3 dependence of coercivity over a wide range of temperature (10 K to 350 K) is observed. All the samples consists of particles with multiple magnetic domains as the size of the particles are larger than the critical single domain size (see Eq. 13 on page 13 of chapter 1 and Fig. 33 on page 47 of chapter 3). The experimental results are understood in terms of strong domain wall pinning. 5.2 Introduction The temperature dependence of the extrinsic magnetic properties, for example coercive field (H,(T)), arise from two mechanisms. The first mechanism is, due to the temperature dependence of the intrinsic magnetic properties 11,15'86'7 such as saturation magnetization (. [.), magnetic anisotropy (K) and exchange stiffness (A) and will be discussed in chapter 6. The second mechanism is, due to the thermally activated hoping of the metastable states over some energy barrier.24,6 From the magnetization loops at different temperatures (Fig. 52) we have found that [., does not change with temperature. All the samples comprise pcl i, i I l 11iiw. par !I. and thus magne' i, 1 iii.w anisotropy can be neglected and temperature independent shape anisotropy is dominant.48 The experimental temperature range is 10 K to 300 K which is much smaller than the curie temperature of Ni (630 K)3 and A can be considered constant over this temperature range."8 In this chapter we will discuss the second mechanism as the origin of the temperature dependence of the coercive field (11.,, K and A are temperature independent). To understand the temperature dependence of the He due to the thermally activated hopping over metastable energy minima separated by some energy barrier, it is necessary to find out the magnetic field dependence of the energy barrier. A commonly used phenomenological energy barrier is2,3,47,89 AE =Eo[1 H/Ho] (51) where Eo is the energy barrier at zero magnetic field and energy barrier vanishes at H = H0 at T = 0. At H = H,, thermal energy, kBT, is sufficiently high to cause most of the moments to be thermally activated over the barrier. For example for the case of StonerWohlfarth particles m = 2, Eo = KV and Ho = 2K/M... For StonerWohlfarth particles the scenario is very simple and the Eq. 51 can be derived analytically (see Eq. 16 on page 15 of chapter 1). Remember that if AE(H) is known, it is possible to calculate H,(T). In this chapter we will discuss how to derive AE(H) (Eq. 51) for the MD nanoparticles and will compare H,(T) with the model. 5.3 Results and Discussions The sample preparation technique is discussed in chapter 2 and chapter 3. Three different sets of samples are investigated. Set 1 consists of single 1ir Ni particles in an A1203 matrix. Set 2 and Set 3 consists of 5 1. is, of Ni particles separated by A1203 Li,. 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. 51 below. In this chapter we will focus on the temperature dependence of H, for the MD Ni nanoparticles. A total of 15 samples are studied, 5 samples from each set. Magnetization loops are measured for every sample for seven (on average) different temperatures. This means a total of around 105 magnetization loops have been measured for the present study. Magnetization loops for the sample of average particle diameter of 12 nm of set 2 at different temperatures (indicated in the legends) are shown in Fig. 52. The arrow shows H, at 10 K. Note that H, decreases with increasing temperature. The temperature dependence of H, normalized to Ho for five different samples belonging to set 1 is shown in Fig. 53. The particle diameters are indicated in the legends. Note T2/3 in x axis. All the data follow a linear T2/3 dependence. To understand the above data, we will start Alumina Ni particles Substrate Set 1 Set 2 Set 3 Figure 51. Schematic of three sets of samples. Set 1 comprises a single lIr of Ni particles embedded in an A1203 matrix. Set 2 and Set 3 comprises of 5 1l.ri of Ni particles separated by different distances in an A1203 matrix. The interlayer distances in Set 2 and Set 3 are 3 nm and 40 nm respectively. with a general magnetic energy landscape of the system written as a polynomial expansion of the domain wall position (x) around a strong pinning center.24,6 E(x) = ao + aix + a2x2 + :3 boHx (52) 2 3 where ao, al, a2, a3 and bo are micromagnetic parameters that depend on the magnetic parameters K, ., and A. For the strong pinning center the x3 term is included as the effect of the pinning center is long distance compared to the weak pinning center where the x3 term is neglected.3 The relation between micromagnetic and magnetic parameter can be determined from the particular model used. Note that the micromagnetic parameters are temperature independent in our case as they only depend on the temperature independent magnetic parameters. First, we will derive the energy barrier separating the metastable minima from the global minima. The maxima or minima of E(x) are determined by setting the first order derivative to zero. E = at + a2x + a3X 2 boH = 0 (53) 6x 4w4w4w IM IMIM 40 4040 8.0xl104  4 6.0x104  4.0x104  2.0x104  0.0 2.0x104  4.0x104  6.0x104  8.0x104  e 500 1000 1500 H (Oe) Figure 52. Magnetization loops for the sample of average particle diameter of 12 nm of set 2 at different temperatures (indicated in the legends). The coercive field (He) at 10 K is indicated by the arrow. He decreases with increasing temperature. Saturation magnetization (3[.,) is constant at different temperatures. The two solutions for the above equations are a2 + 4/3(aj boH) 2a3 a2 V 4a3(a bo0H) (54) (55) Taking the second derivative of E(x) with respect to x it is easy to show that 62E/6X2 Ix,> 0 (62E/6X2 IX2< 0) and corresponds to the maximum (minimum). Thus the energy barrier is (a2 4ala3 + 4a3boH)3/2 AE(H) = E(xi) E(x2) a 6a3 o MsublOK  Msub50K MsublOOK  Msubl50K  Msub200K  Msub250K  Msub300K  Msub325K 24 nm Ni/AIl20 5 Layer l) 11111J11 1500 1000 500 __ (56) O 0.4 I 0.2 0.0 . 0 5 10 15 20 25 30 35 40 45 50 T2/3 (K2/3) Figure 53. Coercive field (He) vs. T2/3 for five different samples of set 1. The linear behavior is observed for samples with particle size from 18 nm to 42 nm in diameter. For the reverse field, ie H  AE(H) H the above equation reduces to (a 4a3(al + boH))3/2 (a 4ala3)3/2 6aj 1 which is in the same form of Eq. 51, where (a 4ala3)3/2 Eo6a a 4ala3 4a3bo From Eq. 57 it is clear that AE(H) decreases with increasing H and when H energy barrier can be overcome by thermal energy (definition of the coercive field). Thus H a 481a3 4a3bo / (57) (58) (59) He the at H = H, the Eq. 57 can be rewritten as kBT (a 4aia3)3, 6a H )3/2 H) ao4ala3 4a360bo The above equation can be solved for He H Ho 1 \ ( 2/3 where Hco and Eo are given by Eq. 58 and 59. This temperature dependence of He is consistent with the experimental results shown in Fig. 53, 54 and 55. 1.0 0.9 Set 2 12 nm 0.8 18 nm A 24 nm 0.7 v 42 nm 4 60 nm 0 0.6 o 0.5 I 0.4 0.3 0.2 0 5 10 15 20 25 30 35 40 45 50 T2/3 (K2/3) Figure 54. Coercive field (He) vs. T2/3 for five different samples of set 2. The linear behavior is observed for samples with particle size from 12 nm to 60 nm in diameter. 2 (1 (510) (511) O 0 0.5 0.4 0.3 0.2 0.1 0.0 0 10 20 30 40 50 T2/3 (K2/3) Figure 55. Coercive field (He) vs. T2/3 for five different samples of set 3. The linear behavior is observed for samples with particle size from 16 nm to 44 nm in diameter. 5.4 Relation Between Micromagnetic Parameter and Magnetic Parameters Here we will outline a roadmap to relate the micromagnetic parameters ao, al, a2, a3 and bo to the magnetic parameters K, ., and A. To do that we will start with the magnetic energy expression,3 E(x) = A (v ( ))2 K(x)(k.M())2 oM(x).H M(x).Hd(M) dV (512) where the first term corresponds to the exchange energy cost due to the spin misalignment, the second term is the anisotropy energy, the third term is the Zeeman energy and the fourth term is the magnetostatic self energy. The position of the domain wall is given by x and k is the unit vector along the easy axis. The above equation should be solved for real samples while taking into account real structure and imperfections. The real structure and imperfections are responsible for the x dependence of the magnetic parameters (3 ..(x), A(x), K(x)). After solving Eq. 512 and by comparing the coefficients of the different power of x, it is possible to find out the micromagnetic parameters in terms of magnetic parameters. The behaviors of H o and Eo/kB are shown in the figure below. 500 450 0 O 400 350 350 10 20 30 40 50 60 d (nm) 700 600 500 c C) LU 400 300 Figure 56. The behaviors of H o and Eo on particle diameter are shown for set 2 samples. Ho decreases and Eo/kB increases with increasing particle size. The increasing behavior of Eo and decreasing behavior of Hco are consistent with the literature.3 The actual behavior can be very complicated as it depends on the real structure factors and imperfections in the material.3 5.5 Conclusions We have investigated the temperature dependence of the coercive field of MD Ni nanoparticles in A1203 matrix. He decreases linearly with the T2/3. This behavior can be understood according to the strong domain wall pinning. We show that the general energy Set 2  Hco e EO/kB barrier that arises due to strong domain wall pinning depends on the magnetic field with a power of 3/2 and is responsible for the temperature dependence of the H,. CHAPTER 6 COERCIVE FIELD OF FE THIN FILMS AS THE FUNCTION OF TEMPERATURE AND FILM THICKNESS: EVIDENCE OF NEEL DISPERSE FIELD THEORY OF MAGNETIC DOMAINS 6.1 Abstract The temperature dependence of the coercive field of Fe thin films has been investigated. Three different samples of different thickness are studied. The coercive field decreases with temperature and follows the same temperature dependence as the first order anisotropy constant. This behavior is consistent with the theoretical prediction made by Neel15 based on the disperse field theory of magnetic domain which takes in to account the effect of free poles on the coercive field that occurs at small inclusions. The value of coercive field increases with decreasing film thickness. This behavior is expected for multi domain ferromagnetic systems at nanoscale where the domain wall thickness is comparable to or larger than the film thickness. 6.2 Introduction The most interesting aspect of ferromagnetism is the hysteresis loop,90 which refers to the history dependent behavior of magnetization with applied magnetic field (Fig. 62). Hysteresis is a complex nonlinear, nonequilibrium and nonlocal phenomenon, reflecting the existence of anisotropyrelated metastable energy minima separated by fielddependent energy barriers.3 An extrinsic property of crucial importance in permanent magnetism is the coercive field, the magnetic field where magnetization changes sign as it passes through zero. The coercive field basically describes the stability of the remnant state and is a very important concept for most practical applications.9199 Coercivity in ferromagnets is known from very long time. 90 But, due to the complex nature, the origin of coercive field is still a subject of study. In this present work the behavior of coercive field of three different iron thin films with different thicknesses has been investigated. The temperature dependence of the coercive field agrees well with the theory of domain wall pinning arising from small inclusions (for example free pole is not negligible. 15 impurity or vacancy defects) where the energy of the 6.3 Experimental Details Figure 61. TEM image of Fe thin film of thickness 9 nm. Thin films of Fe were fabricated on Si(100) and sapphire (caxis) substrates using pulsed laser deposition from alumina and iron targets. High purity targets of Fe (99.9' .) and A1203 (99.9'.) were alternately ablated for deposition. Before deposition, the substrates were ultrasonically degreased and cleaned in acetone and methanol each for 10 min and then etched in a 49'. hydrofluoric acid (HF) solution to remove the surface silicon dioxide l ir (for the Si substrates only), thus forming hydrogen terminated surfaces.35 The base pressure for all the depositions was of the order of 107 Torr. After substrate 1i. lii. the pressure increased to the 10Torr range. The substrate temperature was kept at about 550" C during growth of the A1203 and Fe lv ir. The repetition rate of the laser beam was 10 Hz and energy density used was ~ 2 Jcm2 over a spot size 4 mm x 1.5 mm. A 40 nmthick buffer livr of A12O3 was deposited initially on the Si or sapphire substrate before the sequential growth of Fe and A1203. This procedure results in a very smooth substrate independent starting surface for the growth of Fe, as verified by high resolution scanning transmission electron microscopy studies (Fig. 21). A 40 nmthick cap 1. r of A1203 was deposited to protect the Fe 1vr from oxidation. Three different samples with thickness of 9 nm, 21 nm and 30 nm were prepared for the present study. Magnetization measurements are performed in a quantum design Magnetic Property Measurement System (\!PMS). The magnetic field was along the plane of the films. The dependence of temperature and film thickness on coercive field is investigated. 6.4 Results and Discussion 4.0x10 O M10K m M50K 2.0x10 M100K ? M150K / E 0M.0 5 2.0x10 M250K M300K 4.0x104 M325 6.0x10 0 8.0x1 0  600 400 200 0 200 400 600 H (Oe) Figure 62. MH loop of Fe film of thickness 21 nm at different temperatures as indicated in the legend. The coercive field defined at M=0, decreases with increasing temperature. Shown in Fig. 61 is the TEM image of the 9 nm thick Fe sample. Due to the low surface energy difference between Fe and A1203 (650 mJ/m2) and high adhesion energy (1205 mJ/m2) between Fe and Al203,100 Fe wets the surface of Al203 and thus a continuous thin film is grown instead of grains as is the case for the Ni/AO203 system discussed in chapter 2 and 3. Three different samples with thickness 9 nm, 21 nm and 30 nm were grown. The magnetic hysteresis loops at different temperatures from 10 K to 350 K are measured with the applied magnetic field along the plane of the thin film. MH loops for 21 nm sample are shown in Fig. 62. The magnetic field is swept from 5 T to 5 T and again from 5 T to 5 T to complete the loop. At high magnetic fields the magnetization from the Fe saturates and the magnetic field dependence arises due to the diamagnetic contribution from the substrate. The diamagnetic contribution from the Fe film is negligible as the thickness of the film is very small compared to the thickness of the substrate. Thus the high magnetic field data is linear in magnetic field with a negative slope. The slope of the line is determined from the linear fit of the high field data and subtracted from the raw data to extract the ferromagnetic signal due to the Fe thin film. The procedure is repeated for all samples and for all temperatures. The coercive field is found to decrease with increasing temperatures. The results for all three samples are presented in Fig. 63. Figure 64 shows the relation between K(T)s7 and H,(T) where T is the implicit variable."7 The linear behavior of H, and K i r, I that the coercivity mechanism is similar to the disperse field theory of magnetic domain which takes in to account the effect of free poles on the coercive field that occurs at small inclusions. 15 600 S 99 nm 550  500 450 8 400  350 21 nm 300 250 25 30 nm 200  . 0 50 100 150 200 250 300 T (K) Figure 63. H, vs. T of Fe films of thickness 9 nm, 21 nm and 30 nm. The temperature dependence of all three samples is similar, which ir . i that the origin of the coercive field depends on the intrinsic property of the iron, which in our case is the magnetocrystalline anisotropy. This is true as for the case of extrinsic origin the energy barrier depends on the size of the sample as discussed in chapter 5 56000 K Linear Fit of Data4 C 54000 52000 50000 48000 380 400 420 440 H (Oe) Figure 64. H, vs. K of Fe film of thickness 9 nm. The linear behavior ii I the validation of Neel's disperse field theory of magnetic domains which takes in to account the effect of free poles on the coercive field that occurs at small inclusions (Eq. 63). The similar behavior is observed for other two samples which is expected as they have similar temperature dependence. Theories concerned with the coercivity of the multi domain ferromagnetic materials begin with the consideration of the change in magnetic energy across a domain wall.2 Since the magnetization changes from [ ., to I .. across a 180 domain wall, the effective pressure on the wall will be the difference in the energy across the wall per unit volume. At an applied magnetic field H, the pressure will be 2H.1 .. Thus the work done to move a domain wall of area s to a distance dx will be2 dW 2H.1[.sdx (61) The above equation can be generalized for any angle (not only 180") domain wall.2 The term dW/dx, which is basically the energy required to move a domain wall to unit distance, is the measure of the resistance of domain wall motion.2 In real samples due to impurities, imperfections or irregularities in crystal, dW/dx passes through maximum and minimum. The origin of reversibility in the magnetization loop is the irreversible motion of domains through these maximum and minimum. 2 The coercive field is usually calculated using the equation below.2 2 1.  Different theoretical approaches concentrate on the derivation of (dW/dx)max. Substantial wall motion may require fields of tens or hundreds of Oersted. Evidently real materials contain crystal imperfections of one sort or another which hinder the easy motion of domain walls. These hindrances are generally of two kinds: inclusions and residual microstress.2 From a magnetic point of view, an "inclusion" in a domain is a region which has a different spontaneous magnetization from the surrounding material, or no magnetization at all. According to the result obtain by Neel in his disperse field theory (which basically calculates the term (dW/dx)max), coercivity is given by Eq. 63 below.15 1 A2a2 Kv2 H, = v + (63) 4 K, 1. 11. The first term is due to the residual microstress and vl is the volume fraction of the free pole at the residual microstress and A, a, K and i.., are the saturation magnetostriction, internal stress, magnet' il ii.1_,i. anisotropy constant and saturation magnetization respectively. The second term in Eq. 64 originates due to the poles at the inclusions of volume fraction v2. For iron, i1., = 1743 emu/cm3,2 K=8x106 ergs/cm3,101 A 10xt06102 and a=1010 dynes/cm2.103 Putting these values in the Eq. 63 we obtain, H, 0.18vl + 46002 Oe (64) Thus in our case the coercive field will be dominated by the 2nd term in Eq. 63, which is proportional to K and agrees well with the experimental results as shown in Fig. 63 and Fig. 64. Remember that i.., is constant. In Fig. 65 the variation of H,(d) is shown. The increase in H, with decreasing film thickness, d, could be due to the enhancement of K with decreasing d.101,104,105 The increase in coercive field with decreasing thickness is well known for the multidomain 600 \ H S Linear Fit of Data7_B 500  0) 400 0 300  200 U 5 10 15 20 25 30 d(nm) Figure 65. He vs. d of Fe films at 10 K. Coercive field increases with decreasing d. This behavior is due to fact that K increases with decreasing d which is very common in multidomain thin films when the domain wall thickness is comparable or larger than the film thickness. The domain wall thickness of iron is about 60 nm,3 which is larger than the thickness of the films studied here. magnetic thin films when the domain wall thickness is comparable to or higher than the film thickness.7 The domain wall thickness of iron is about 60 nm,3 which is larger than the thickness of the film studied here. Note that we have used the value of K in Eq. 63, which is larger than the bulk value. From Eq. 64, we find that v2 ~ 0.12. 6.5 Conclusion The coercive field of multidomain Fe thin films has been investigated. The temperature and thickness dependence of the coercive field agrees well with the prediction made by Neel, which takes in to account the effect of free poles on the coercive field that occurs at the small inclusions. The contribution of strain to the coercivity is very small for the case of Fe and the dominant contribution comes from the free poles at the inclusions. The volume fraction of inclusions has been estimated from the coercive field data to be around 0.12. CHAPTER 7 SCALING COLLAPSE OF THE IRREVERSIBLE MAGNETIZATION OF FERROMAGNETIC THIN FILMS 7.1 Abstract In ferromagnetic materials, hysteresis, or equivalently the history dependent behavior of the magnetization, reflects complex nonlinear and non equilibrium phenomenology that has been recognized for many years5'6'106108. Hysteresis depends strongly on materials properties such as structural length scales spanning the nanometer to micrometer range3 and manifests complex behaviors including magnetic relaxation with aging dependence109 dimensionally dependent dipoledipole interactions48'110, spinglass like memory effects 1 and supermagnetism112. Here we show that the irreversible magnetization, defined as the difference between fieldcooled and zerofield cooled magnetization, has a striking similarity for a wide variety of ferromagnetic materials. This similarity becomes apparent when the irreversible magnetization is normalized to its maximum value and plotted with respect to a temperature dependent renormalized field. The collapse of the irreversible magnetization onto a single curve for a given system implies an underlying symmetry to hysteresis that is not captured by previous analytical3,'5'6'48,106,107'109112 and computational treatments108s113 and thus provides a unifying theme that embraces a broad range of complex hysteretic behavior. 7.2 Introduction In general, hysteresis is a complex nonlinear non equilibrium phenomenon which reflects the presence of fielddependent energy barriers between anisotropydependent metastable minima. Accordingly, hysteresis is affected by a combination of intrin sic properties such as magnetocrystalline anisotropy which depends on crystal field energy and spinorbit coupling and extrinsic properties such as sample shape, grain boundaries, disorder and imperfections. For example, in bulk ferromagnets hysteresis is often described as a superimposition of domain wall motion and domain rotation with energy barriers related to magneto' i i 1iiw., anisotropy together with imperfections and/or impurities in the material107. For the simplest case of single domain coherently rotating nanoparticles, the origin of hysteresis is the finite time scale for magnetization reversal as the magnetization overcomes a magnetic field dependent energy barrier by thermal activation and rotates from one easy axis direction to another5'6'106. Hysteresis and associated magnetization reversals p1 i, an important role in applications such as magnetic data storage devices91 93, GMR94'95 or MRAM96 devices, magnetic , i i.'' and motors98, generators99 etc. 7.3 Experimental Results Measurements of irreversible magnetization are usually accomplished by one or both of two techniques illustrated respectively in panels a) and b) of Fig. 71. The sample under investigation in this figure (hereafter referred to a sample A) is a 20 Athick pulsed laser deposited thin film comprising five l~V iS of 3 nm elongated Ni particles embedded in an insulating Al203 host [see Methods]. The Ni particles are small enough to be in the single domain (SD) regime where all the spins are aligned in the same direction and rotate coherently together in response to a changing magnetic field. In the first technique (panel a), the sample is field cooled (FC) in a field H = 20 Oe (black squares) from 300 K to 5 K and then zero field cooled (ZFC). The irreversible magnetization (AM(H, T)), which is a function of magnetic field (H) and temperature (T), is shown by the twoheaded dashed vertical arrow. In the second technique, the sample is held at fixed temperature T and magnetization M loops obtained by repeatedly cycling the applied field H about H = 0 between two symmetric limits. The history dependent trajectories form closed loops shown schematically in the insets of panel (b). These loops, which can be acquired at different temperatures, are each characterized by a coercive field H,(T), a saturated moment .[.,(T) and a remnant magnetization Mr(T). The coercive field H,(T), plotted versus T1/2 in panel b for sample A discussed above, is shown as a blue line connecting the starred data points. The absence of a T1/2 dependence for coherently rotating SD particles with easy axes oriented along the field6 will be discussed below. While both AM(H, T) and H,(T) 0 50 100 150 200 250 300 T(K) b) 300 S200 U 100 0 4 5 6 7 T'2K) Figure 71. Irreversible magnetization AM(H, T) defined as the difference of the FC and ZFC magnetizations is a quantitative measure of hysteresis. a, Black squares and red circles represent respectively the FC and ZFC temperaturedependent magnetizations for sample A in a 20 Oe field applied parallel to the film surface. The irreversible magnetization AM(H, T) is shown by the twoheaded vertical dotted arrow. Inset: Large thermally blocked magnetic particles (1 and 3) with respective vertical and horizontal easy axes of magnetization indicated by the arrows. Particle 2 is small enough to be superparamagnetic (thermally unblocked). For magnetic fields applied along the vertical direction, particles 2 and 3 do not contribute to AM(H, T). b, The coercive field (He) for the same sample shown as a function of T1/2 (Blue stars) does not show the linear behavior expected for ideal StonerWohlfarth particles6 where all the particles are uniform size and aligned with the applied field (particle 1). The solid blue line is a fit to the data using a lognormal distribution of particle size together with a random orientation of the easy axis of the magnetizations. Inset: Schematic magnetization curves for the cases where (bl) only particles 1 and 2 and (bl) only particles 1 and 3 are present. The resultant magnetization loops (black curves) for the two cases show the pronounced effects of particle size and easy axis orientation on the determination of H,. are commonly accepted measures of hysteresis, the underlying phenomenology for each is considerably different. For example in nanoparticle magnetic systems such as shown in Fig. 71, AM(H, T) and Hc(T) are sensitive in different vo to both the presence of superparamagnetic particles and the random orientation of the easy axis of magnetization of each particle. The insets of Fig. 71 a) and b) schematically illustrate these differences using three particles: particles 1 (red) and 3 (blue) with easy axes of magnetization S5 0  mFC *ZFC I I I respectively along the vertical (H) and perpendicular directions and large enough so that at the temperature of measurement, changes in magnetization are blocked by energy barriers that cannot be overcome by thermal activation, and particle 2 (green) with easy axis along the vertical axis and small enough so that it is superparamagnetic with a nonhysteretic magnetization depending only on H and T (i.e., unblocked). Consider the case where only particles 1 and 2 are present. Since the superparamagnetic particle 2 will have zero AM(H, T), the total AM(H, T) value will be only due to the blocked particle 1. On the other hand, the MH loop will be the summed contributions of the hysteretic loop for particle 1 and the reversible loop for particle 2 as shown in inset bl of Fig. 7lb. Thus for this case AM(H, T) is not affected by superparamagnetic particles but He(T) is. Consider now the case where only particles 1 and 3 are present, i.e., both particles are blocked but with different (parallel and perpendicular) easy axis orientations with respect to the applied field. Simple theory for coherently rotating SD particles shows that upon reversing the field particle 1 must surmount a Hdependent energy barrier whereas particle 3 can line up without having to overcome an energy barrier3. In like manner to the previous case, The MH loop for particle 1 shows hysteresis but particle 3 does not (inset b2 of Fig. 71b). Hence H,(T) is affected by the presence of particles with perpendicular orientation but AM(H, T) is not. The above arguments can be generalized for SD magnetic nanoparticles with a broad size distribution and a random orientation of the easy axes of magnetization. Particles with an easy axis making an arbitrary angle with the applied magnetic field will alvwb contribute less to hysteresis than aligned particles. Because the perpendicular component does not contribute, the correction is a straightforward integration over angle. Accordingly. the measured value of AM(H, T) will be only due to blocked particles and dominated by particles with easy axis of the magnetization along the applied magnetic field. On the other hand, H,(T) will be strongly affected by the random orientation of the easy axis of blocked particles and the presence of the particles which are small enough to be superparamagnetic. For an ideal StonerWohlfarth particle system6 in which all the particles are aligned along the H direction, a linear square root temperature dependence of H,(T) is expected. For the considerably more complicated case of randomly oriented SD particles with a size distribution in which some of the particles are superparamagnetic (e.g., sample A), H,(T) can be described (solid line of Fig. 71b) using a model with a log normal size distribution and three fitting parameters9'10 (see Methods). The Hdependent behavior of AM(H, T) for sample A is shown in Fig. 72a for the temperatures indicated in the legend. The isotherms show peaks, AMma,(T), at magnetic fields, H,(T), indicated by the vertical arrows. These peaks are expected, since at H = 0 the FC and ZFC measurements are equivalent and the difference in magnetization should be zero, whereas at high H both FC and ZFC magnetizations saturate to the same value and again the difference should go to zero. The similarities in the AM(H, T) isotherms are ,i'. .1i'. and become manifest as an unexpected data collapse onto a single curve when the reduced irreversible magnetization, AM(H, T)/AMmax(T), is plotted as a function of reduced magnetic field, h(T) = H/H,(T), as done in Fig. 72b. The characteristic field H,(T) deviates significantly below Hc(T) for T < 25 K (inset of Fig. 72b) and identifies the Tdependent field where irreversibility is at a maximum. An important physical insight into the scaling collapse shown in Fig. 72b is gained by plotting AMmax(T) as a function of T1/2. The observed linear behavior shown in Fig. 73a is identical to the predicted temperaturedependent coercivity Hs W(T) of StonerWohlfarth (SW) particles6 mentioned above. Guided by this similarity, we make the ansatz: AMl a( T) oc H'w. This ansatz is physically reasonable since as shown above, AM(H, T) measurements are not effected by the presence of superparamagnetic particles, and in addition the contribution from particles with easy axis of magnetization along the magnetic field is dominant. As all real samples comprise particle size distributions and random easy axis orientations, the conventional MH loops from which coercivities are extracted are markedly different than would occur for idealized H(Oe) 0 2 4 H/Hm Figure 72. All of the data for AM(T, H) of sample A can, with proper normalization, be made to collapse onto a single curve, a, Isotherms of AM plotted as a function of H show selfsimilar behavior with maxima AMma,x(T) occurring at characteristic fields Hm(T) marked by the vertical arrows. With decreasing temperature AMmax(T) increases and H,(T) moves to higher fields. b, The scaling collapse occurs when AM(H, T)/AMma,(T) is plotted against the normalized field H/Hm(T). Inset: Plots of H, (squares) and H,(T) (stars) as a function of T. The data and fit for H, are the same as shown in Fig. Ib, but plotted with respect to T rather than T1/2. The solid line for Hm(T) is a guide to the cv SW behavior. The insensitivity of AM(H, T) measurements to superparamagnetic particles and perpendicular orientations of blocked particles together with our ansatz imply that more useful information about the magnetization reversal process is obtained from AM(H, T) measurements than H,(T) measurements. In 0 o b) 1.0 0.9 0.8 0 100 200 300 400 H,(T) (From MH) (Oe) Figure 73. For single domain particles the ansatz AMmx(T) oc H,w is verified, a, The linear dependence of AMmax(T) on T1/2 is in accordance with the StonerWohlfarth theoretical prediction that Hc(T) oc T1/2 for the coherently rotating single domain particles of sample A. The use of AMmax(T) rather than H,(T) as a measure of hysteresis removes the effects of nonuniform particle size and random orientation. b, The values of H,"I(T), computed for uniform size FePt particles (sample B) from the ansatzderived Eq. 2 at the indicated temperatures, scale linearly with HSW(T) which is determined from the coercive fields of M H loops after correction for the random orientation of the easy axis of magnetization (see Methods). The scaling collapse behavior plotted for sample A in Fig. 72b si. , that AM has the form, AM(H, T) AM,,(T)F (H/H,(T)) , (71) where F is an unknown function with the property F(1) 1. Taking the second partial derivative of the both sides of this equation and solving for AMmax(T) gives the result: (92F(h)F2 /h 2) 1 h 1H, T)a2 AM(H, T)/aH2 ,H(T) evaluated at the hl 1mT Slope 1.09(4) 10K 20 K 30 K 500 600 AMmC(T) maximum where h(T) = H/Hm(T) = 1. The first term, (2F(h)/0h2) 1 h must be a constant because of the scaling collapse. The remaining two terms have the same form as the expected value of an effective anisotropy field114. Thus according to our ansatz, AMmax(T) oc Hf we can write the relation Hw(T) CH, (T)a2AM(H, T)/aH2 = CHf'(T), (72) where C is a constant and Ha(T) = H2 (T)2 AM(H, T)/OH2 In (Tis computed from experimental data. To test the result expressed in Eq. 2, we use magnetization data on a system of a;,,' .rm size (6 nm diameter) FePt nanoparticles (sample B, see Methods) synthesized via thermal decomposition of Fe(CO)5 and reduction of Pt(acac)2115. The measured coercive field HKH (T) is obtained from MH loops at different temperatures. Assuming that the particles easy axes of magnetization are randomly oriented, a temperature dependent correction to HfH(T) must be made so that HSW(T) can be inferred. This correction is needed (see Methods) since the magnetization of randomly oriented uniform size particles shows a T3/4 dependence7 compared to the T1/2 dependence expected for an idealized sample in which all the particles are aligned along the applied magnetic field. The plot of Fig. 73 includes this correction and shows a linear dependence of Hcal( T) on HKH(T) as would be expected for a system of coherently rotating SD uniform size FePt nanoparticles. The linear behavior with slope of 1.09(4) confirms the validity of our model as expressed by Eq. 2 with C ~ 1. In Fig. 74 the reduced irreversible magnetization, AM(H, T)/AMm,a(T), is plotted as a function of reduced magnetic field, h(T) = H/Hm(T) for a wide variety of thinfilm magnetic materials labeled in the inset and described in Methods. Unexpectedly, when plotted in this manner, the magnetization data for each materials system collapse onto single curves which have similar shapes described by a unique function F(H/Hm(T)) (see Eq. 1) for each system. This data collapse is quite remarkable considering the variety of mechanisms responsible V VMT Sr .y FePt nanoparticles (6 nm) 0 2 4 6 H/Hm Figure 74. Scaling collapse describes irreversible magnetization (hysteresis) in a wide variety of ferromagnetic materials. a, Plots of the reduced irreversible magnetization AM(H, T)/AMlma(T) as a function of reduced magnetic field H/H,(T) for the six different magnetic materials labeling each curve. The implicit temperature variable increases from left to right and each color on a given curve represents a different AM(H, T) isotherm. For clarity, the ordinate values have been shifted and the solid lines have been added as guides to the eye. The bottommost curve for single crystal spinglass Cu:Mn (at. 1.5 .) material is taken from the literature6 for magnetic ordering together with a wide range of materials properties. The scaling collapse applies equally well: to single domain (SD) coherently rotating Ni nanoparticles with average particle diameter of 3nm embedded in an insulating matrix; to multidomain (l 1)) incoherently rotating Ni nanoparticles with average particle diameter of 12 nm also embedded in an insulating matrix; to coherently rotating uniform size FePt nanoparticles (6 nm diameter); to continuous metallic Gd thin films with magnetization derived from local moments supplemented by band structure exhibiting some itinerant character; and to mixedphase manganite (LPC'\ 10) where the Mn spins order by a double exchange mechanism in an environment where chargeordered and paramagnetic insulating phases compete with a ferromagnetic metallic phase. We have also included a spin glass material, single crystal Cu:Mn (1.5 at ), described in the literature1l6. The collapse occurs for nanoparticle systems which include SD, MD and superparamagnetic particles with both broad and uniform size distributions and magnetic moment reorientations arising from complex superimpositions of domain wall motion and domain rotation which can be very different in continuous films compared to nanoparticles where the extent of the domain wall is comparable to the size of the particle. Thus the materials measured here are very much different in terms of the origin of the irreversible magnetization. In conclusion, we have presented a heretofore unreported phenomenological result showing that the temperature and field dependent hysteresis of at least six distinctly different magnetic systems can be collapsed onto single curves (Fig. 74) using the particularly simple functional form expressed by Eq. 1. Our finding that such a scaling collapse applies to magnetic systems totally different than the spin glass for which similar scaling has been previously noted116'117 (the lowest curve in Fig. 74) t' I that explanations116 relying on spin glass phenomenology are too narrow. Thus a more general theory is needed to explain the scaling collapse. This collapse must imply an underlying symmetry that is not captured by previous analytical and computational treatments and may be a crucial clue to understand the complex history dependent magnetization process. The similarity with the spin glass material is interesting and may be because of the fact that all ZFC magnetization is a metastable state of the system and shows properties varying with time. Thus the dynamics of the spins may pl. i a very important role for the scaling collapse. The behavior of A/Mmax(T) is investigated for the 3 nm Ni particles. We it'' 1 a new approach to investigate the magnetization reversal process from the AAM,,x(T) measurement. We have successfully applied the method for the coherently rotating SD particles. 7.4 Conclusions In conclusion, we have presented a surprisingly general and unrecognized phenomenological result showing that the temperature and field dependent hysteresis of at least six distinctly different magnetic systems can be collapsed onto single curves (Fig. 74) using the particularly simple functional form expressed by Eq. 1. We have not yet found any exceptions. For the particularly simple case of coherently rotating single domain particles (samples A and B), our analysis of scaling collapse bypasses the complications of nonuniform size distributions and random easy axis orientation, unveiling an underlying StonerWohlfarth behavior6. Our finding that the same scaling collapse more generally applies to magnetic systems with a wide variety of mechanisms giving rise to hysteresis, ~i.' I that explanations116 relying on spin glass phenomenology are too narrow. Thus a more general theory is needed to explain the scaling collapse. This collapse must imply an underlying symmetry that is not captured by previous analytical and computational treatments and may be a crucial clue to understand complex history dependent magnetization processes. The similarity with the spin glass material116 is , ..,~ li' . and may be related to the fact that all ZFC magnetizations represent metastable states of the system, which if given sufficient time would relax toward the fieldcooled equilibrium state. Accordingly, the dynamics of the spins may pl iv a very important role in understanding the scaling collapse. 7.5 Methods 7.5.1 Ni Nanoparticle. Composite films comprising magnetic Ni nanoparticles embedded in an Al203 host matrix were synthesized by pulsed laser deposition (PLD)36. High purity targets of Ni (99.9' . ) and Al203 (99.99 .) were alternately ablated in the same deposition run. The base pressure of the deposition was on the order of 10' Torr. The substrate temperature was maintained near 550C during the growth. The repetition rate of the laser beam was 10 Hz and energy density used was 2 J cm2 over a spot size of 4 mm x 1.5 mm. A 40 nmthick buffer lvr of Al203 was deposited initially on the sapphire substrate before the sequential growth of Ni and A1203. This procedure results in a very smooth starting surface of growth of Ni as verified by the high resolution scanning transmission electron microscopy (STEM) studies. Samples consists of 5 l' ,irs of Ni and Al203. A cap livCr of A1203 was alvii used to protect the sample from oxidation. Zcontrast STEM image verifies the absence of the Ni oxide. Samples studied here consists of Ni particle size of 3 nm and 12 nm in diameter. 7.5.2 Gd Thin Film. Gd thin films were deposited on Si substrates by DC magnetron sputtering. The base pressure of the chamber was on the order of 5 x 107 Torr. The samples are continuous with thickness near 100 nm. 7.5.3 (LalyPry)o.67Cao.33MnO3 (LPCMO) Thin Films. Phase separated manganite (LalyPry)o.67Can, _,u03 (LPC'\O) films were grown using pulsed laser deposition (PLD) at a rate of 0.05 nm/s on NdGaO3 (NGO) (110) substrates kept at 820C in an oxygen atmosphere of 420 mTorr11 7.5.4 Temperature Correction of Coercive Field. For ideal StonerWohlfarth particles the coercive field is given by, Hw = Ho,(T)(1  (T/TB)1/2). For the case of randomly oriented coherently rotating particles all of which have the same size, the coercive field is given by, H,"(T) = 0.48H,,(T)(1 (T/TB)3/4). Since the FePt particles of sample B all have the same size, we can write H,"a(T) HNH(T) where HNH(T) is the coercive field extracted from the magnetization loops. 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Phys. Rev. B 75(9), 092404 (2007). 119. Sappey, R., Vincent, E., Hadacek, N., C'! .put, F., Boilot, J. P., and Zins, D. Nonmonotonic field dependence of the zerofield cooled magnetization peak in some systems of magnetic nanoparticles. Phys. Rev. B 56(22), 1455114559 (1997). BIOGRAPHICAL SKETCH Ritesh Kumar Das was born on 1981 in a very small village, Mohanbati, located east of India. At Haripal Guru Dayal high school, Ritesh was introduced to physics and fell in love with the subject. Ritesh got interested in science subjects during his high school di, where he found wonderful teachers, Mr. Robin C'i i 1. ijee and Mr. Uttam Saha. After completing the high school, Ritesh enrolled in the Ramakrishna Mission Vidyamandira (Belur) in August 1999 and graduated with a Bachelor of Science degree with honors in physics on August 2002. During this time Ritesh met his favorite teacher Dr. Deepak Ghsosh, who p1 i, d a very crucial role to make Ritesh interested in experimental physics. After completing the Bachelor of Science degree, Ritesh enrolled in Master of Science degree at Indian Institute of Technology (Kanpur) and completed the degree on August 2004. After this Ritesh joined University of Florida to pursue his Ph.D. degree. Ritesh was ahv, interested in the properties of materials at nanoscale and found that Prof. A. F. Hebard's lab to be the perfect place for the research in the field of thin films and nanoparticles. Ritesh graduated in August 2010 with a Doctor of Philosophy degree in physics. PAGE 1 MAGNETISMINNANOSCALEMATERIALS,EFFECTOFFINITESIZEANDDIPOLARINTERACTIONSByRITESHKUMARDASADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 1 PAGE 2 c2010RiteshKumarDas 2 PAGE 3 Idedicatethistomyparentsandfamilyfortheiractivesupport.Withoutthemitwouldhavenotbeenpossible. 3 PAGE 4 ACKNOWLEDGMENTS Iamtrulyindebtedtomanyindividualswhohavecontributedtothesuccessofmyresearchwork.Therefore,Iexpressmysincerestregretstoanypersonnotspecicallymentionedhere.Firstandforemost,IamthankfultomyresearchadvisorProf.A.F.Hebardforgivingmetheopportunitytoworkwithhim.Ithasbeenagreatexperiencetoworkunderhissupervision.Hispositive,openmindedattitudetowardresearchcreatesauniquelaboratoryenvironmentfullofencouragement.Ihavelearnedalotfromhisunadulteratedenthusiasm,willingnesstolearnandelegantbutsimpleapproachtounderstandingfundamentalphysics.Iwouldliketothankallthepresentandformerlabmembersfortheirhelpsandpleasantcompany.IamgratefultoJohnJ.KellyforteachingmemanyexperimentaltechniqueswhenIjoinedthegroup.ThankstoallthelabmembersPatrick,Rajiv,Sef,Siddhartha,Sanal,Xiaochangfortheirhelps.Ireallyenjoyedworkingwithyouguys.Iwouldalsoliketoacknowledgethestasofmachineshopandelectricshop.Speciallycryogenicstas,GregandJohn,fortheirconstantsupplyofliquidHeandN2allyeararound24/7.ThankstoJay(reallyaniceguy)forlookingafterallthepumpsandchillers.Iwouldliketothankallofmycommitteemembers.IwillspeciallythanksProf.AmlanBiswas.ThoughIdidnothavechancetocollaboratewithhim,buthisguidanceandsupporttowardsmydegreehavebeenveryhelpful.IamalsogreatlythankfultoProf.D.Nortonforthewonderfulcollaborationandforlettingmeusehislabfacilities.IamthankfultomycollaboratorsD.KumarandA.GuptafromNCA&T.IamalsoverythankfultoMatt,Patrick,KyeongWonfromProf.Norton'slabfortheirhelpsandbeinggoodfriends. 4 PAGE 5 Iamindebtedtomyparentsfortheirsupport,encouragementandforalwaysbelievinginme.IappreciatethewarmthandaectionofmysisterMridula.Icouldnothavecomethisfarwithouttheirblessings. 5 PAGE 6 TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1THEORYANDBACKGROUND .......................... 12 1.1SingleDomainandMultiDomainParticles .................. 12 1.2HysteresisLoopofSingleDomainCoherentlyRotatingParticles ...... 14 1.3HysteresisLoopofMultiDomainParticles .................. 24 1.4Magnetizationvs.Temperature ........................ 29 1.4.1ZeroFieldCooled(ZFC)Magnetization ................ 30 1.4.2FieldCooled(FC)Magnetization ................... 31 2DIPOLARINTERACTIONSANDTHEIRINFLUENCEONTHECRITICALSINGLEDOMAINGRAINSIZEOFNIINLAYEREDNi/Al2O3COMPOSITES 32 2.1Abstract ..................................... 32 2.2Introduction ................................... 32 2.3ExperimentalDetails .............................. 33 2.4DataandDiscussion .............................. 36 2.5Conclusion .................................... 40 2.6Methods ..................................... 41 2.6.1MathematicalAnalysis ......................... 41 2.6.2BasicPhysicalUnderstanding ..................... 41 3EFFECTOFDIPOLARINTERACTIONONTHECOERCIVEFIELDOFMAGNETICNANOPARTICLES:EVIDENCEFORCOLLECTIVEDYNAMICS 42 3.1Abstract ..................................... 42 3.2Introduction ................................... 42 3.3ResultsandDiscussions ............................ 44 3.4Conclusions ................................... 49 4FINITESIZEEFFECTSWITHVARIABLERANGEEXCHANGECOUPLINGINTHINFILMPd/Fe/PdTRILAYERS ...................... 51 4.1Abstract ..................................... 51 4.2Introduction ................................... 51 4.3ExperimentalDetails .............................. 52 6 PAGE 7 4.4ResultsandDiscussion ............................. 54 4.5Conclusions ................................... 58 5TEMPERATUREDEPENDENCEOFCOERCIVITYINMULTIDOMAINNINANOPARTICLES,EVIDENCEOFSTRONGDOMAINWALLPINNING 59 5.1Abstract ..................................... 59 5.2Introduction ................................... 59 5.3ResultsandDiscussions ............................ 60 5.4RelationBetweenMicromagneticParameterandMagneticParameters .. 65 5.5Conclusions ................................... 66 6COERCIVEFIELDOFFETHINFILMSASTHEFUNCTIONOFTEMPERATUREANDFILMTHICKNESS:EVIDENCEOFNEELDISPERSEFIELDTHEORYOFMAGNETICDOMAINS ............................. 68 6.1Abstract ..................................... 68 6.2Introduction ................................... 68 6.3ExperimentalDetails .............................. 69 6.4ResultsandDiscussion ............................. 70 6.5Conclusion .................................... 74 7SCALINGCOLLAPSEOFTHEIRREVERSIBLEMAGNETIZATIONOFFERROMAGNETICTHINFILMS ......................... 75 7.1Abstract ..................................... 75 7.2Introduction ................................... 75 7.3ExperimentalResults .............................. 76 7.4Conclusions ................................... 84 7.5Methods ..................................... 85 7.5.1NiNanoparticle. ............................. 85 7.5.2GdThinFilm. .............................. 86 7.5.3(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)ThinFilms. .......... 86 7.5.4TemperatureCorrectionofCoerciveField. .............. 86 REFERENCES ....................................... 87 BIOGRAPHICALSKETCH ................................ 96 7 PAGE 8 LISTOFTABLES Table page 11Hcvs.T ........................................ 29 8 PAGE 9 LISTOFFIGURES Figure page 11SDandMDparticle ................................. 13 12Coherentandincoherentrotation .......................... 14 13Singleparticleinmagneticeld ........................... 15 14Twostateenergy ................................... 16 15HysteresisofSDparticle ............................... 18 16Diagramofaparticle ................................. 20 17Thermalaverageofmagnetization .......................... 21 18Flowdiagram ..................................... 21 19MHbelowTB ..................................... 24 110MHbelowTB ..................................... 25 111SDtoMDtransitionandHc ............................. 26 112MagnetizationloopforMDparticle ......................... 27 113DomainwallandHc ................................. 28 114Mvs.Tfor3nmNinanoparticles ......................... 30 21STEMimageofNiparticle .............................. 34 22Hcvs.d,dierentT ................................. 35 23dcvs.T ........................................ 37 24Hdanddomain .................................... 41 31Sample ......................................... 45 32MHloop ........................................ 46 33Hcvs.d:dipolarinteraction ............................. 47 34Dipolarinteraction .................................. 49 41Physicalandmagneticviewofsample ....................... 53 42Saturationmagnetizationvs.x ........................... 54 43Coerciveeldvs.x .................................. 55 9 PAGE 10 44Curietemperaturevs.x ............................... 56 51Threesetsofsample ................................. 61 52MHloopsofset2 ................................... 62 53Hcvs.T2=3set1samples .............................. 63 54Hcvs.T2=3set2samples .............................. 64 55Hcvs.T2=3set3samples .............................. 65 56Hc0andE0ofset2 .................................. 66 61TEMimageofFelm ................................ 69 62MHloopofFelm .................................. 70 63Hcvs.TofFelm .................................. 71 64Hcvs.KofFelm .................................. 72 65Hcvs.dofFelm .................................. 74 71IrreversibleMagnetization .............................. 77 72BehavioroftheM(H;T)isothermsasthefunctionofHandscalingcollapse 80 73Theanstz ....................................... 81 74Scalingcollapseofvarietyofferromagneticmaterials ............... 83 10 PAGE 11 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMAGNETISMINNANOSCALEMATERIALS,EFFECTOFFINITESIZEANDDIPOLARINTERACTIONSByRiteshKumarDasAugust2010Chair:A.F.HebardMajor:Physics Materialphysicsisalwaysmotivatedbythematerialswithexoticproperties.Itwasacommonbeliefthatexoticpropertiesareonlyassociatedwithexoticmaterials.Nowitisclearthatgeometricalconnementatnanoscaledimensionscangiverisetoexoticpropertieseveninsimplematerials.Ferromagneticmaterialsinrestricteddimensionsareextremelyinterestingbecauseoftheirpotentialapplicationsaswellastherichfundamentalscienceinvolved.Magneticnanoparticlesareusefulinhighdensitymagneticdatastoragedevices,sensors,contrastagentsinMRI,drugdelivery,treatinghyperthemiaandmanymore.Alltheapplicationsofnanomagnetsareverycrucialinmoderndaylife.Butmostoftheapplicationsarerestrictedduetothelimitationsinthefundamentalpropertiesarisesinnanoscaleandalsoduetothetechnicallimitationsofcontrollingthingsatnanoscale.Forexampleparticlesbecomesuperparamagneticasthesizeisreducedbelowacertainvalueandthemagnetizationdirectionuctuatesrandomlyduetothethermalenergywhichlimitsthedensityofdatastorage.Thepromisesofnanomagnetsarehugeandtoreallyachievethegrandchallengesinnanomagnetism,itisnecessarytounderstandthebasicsciencesinvolveatsmallscales.Inthispresentwork,themagneticpropertiesofsystemsinnanoscale(nanoparticlesandthinlms)havebeeninvestigated.Theeectofdipolarinteractions,particlesize,particlesizedistribution,temperature,magneticeldetc.onthemagneticpropertieshavebeenstudied. 11 PAGE 12 CHAPTER1THEORYANDBACKGROUND Ferromagnetismisknownformorethan2500yearstoman.Therstmagneticmaterialdiscoveredwasmagnetite(Fe2O3).Thepracticalapplicationsofferromagnetswasrecognizedfromveryancienttime.Therstuseofmagneticmaterialwasasacompass.Accordingtothemagneticproperties,materialscanbedividedintodiamagnetic,paramagnetic,ferromagnetic,antiferromagnetic,ferrimagnetic,spinglassetc.Inthispresentworkferromagnetismwillbethemaintopic.Chaptersfollowedbythischapterwilldiscusstheeectofnitesizeanddipolarinteractionsonthemagneticpropertiesofsomematerialswithnanoscalestructures.Inthischapterageneraltheoreticalbackgroundwillbegiven. 1.1SingleDomainandMultiDomainParticles Whenthesizeoftheparticleisverysmall,itwillcontainonlyonemagneticdomain. 1 { 3 Thisisbecausetheenergyrequiredtoformadomainislargerthanthemagnetostaticselfenergy.Themagnetostaticselfenergyforasphericalparticleisgivenby 1 Emag=0M2sV=12(1{1) where0isthefreespacepermeability,MsisthesaturationmagnetizationandVisthevolumeoftheparticle.TheenergyrequiredtoformaBlochdomainwallis 1 { 3 Edw=4p AKR2(1{2) whereKistheanisotropyconstant,AistheexchangestinessandRistheradiusoftheparticle. NotethatEmaggrowsasR3andEdwgrowsasR2.Domainformationisthereforefavorableforlargerparticlesasthemagnetostaticenergywillbelargecomparedtothedomainwallformationenergy.Thecriticalsingledomainradius(Rsd)wherethe 12 PAGE 13 transitionfromsingledomain(SD)tomultidomain(MD)occursisgivenby 1 { 3 Rsd=36p AK 0M2s(1{3) TheaboveequationisdeterminedbysolvingtheequationEmag=Edw. 1 { 3 ThusparticleshavingradiussmallerthanRsdareSDandparticleshavingradiuslargerthanRsdareMD(seeFig. 11 ). Figure11. Smaller(larger)particlesareSD(MD)asthemagnetostaticselfenergyissmaller(higher)thantheenergyrequiredtoformdomain.ThecriticalsizeoftheparticlewherethesingledomaintomultidomaintransitionoccursisgivenbyEq. 1{3 ThemagnetizationdynamicsforSDandMDparticlesaredramaticallydierent.SDparticlesreversetheirmagnetizationbyrotationonly.MDparticlesreversetheirmagnetizationbydomainwallmotionandrotation.RotationofmagnetizationfortheSDparticlesismainlyoftwotypes:1)coherentand2)incoherent(Fig. 12 ).Theexchangelength 2 { 4 lex=s A 0M2s(1{4) isameasureofthedistanceoverwhichtheatomicexchangeinteractionsdominateandallthespinsrotatecoherently.Particleswithsizelarger(smaller)thanlexrotateincoherently(coherently).TheexchangelengthisusuallylargerthanRsdforsoftferromagnetswheretheanisotropyenergyissmall.Thusinsoftferromagnetsmagnetizationreversaloccurseitherbycoherentrotation(smallparticles)orbydomainwallmotion(largeparticles). 13 PAGE 14 Figure12. Coherentandincoherentrotationofthemagnetization.Incaseofcoherentrotationallthespinsrotatetogetherandthewholeparticlecanbeconsideredasagiantspin.CoherentrotationhappensforSDparticleswithsizesmallerthantheexchangelengthlex 1.2HysteresisLoopofSingleDomainCoherentlyRotatingParticles ThemagnetizationdynamicsoftheSDparticleswithR PAGE 15 Figure13. ASDparticleinanappliedmagnetic(H)eldalongtheeasyaxisofmagnetization(k).istheanglebetweenthemagnetizationMandtheeasyaxisk. oftheenergyoccursat==2separatingthetwoenergyminima.Figure 14 a)showstheenergydiagramatzeromagneticeldasafunctionof.Inthiscasetheparticlewillhavemagnetizationparalleltotheeasyaxisofmagnetizationsincethesecorrespondtominimumenergystates(upordown).Anyotherdirectionswillcostsomeanisotropyenergy.ThetwostateswithminimumenergyareseparatedbytheanisotropyenergybarrierequaltoKV.InanappliedmagneticeldalongtheeasyaxisthetwoenergyminimawillbeshiftedduetotheZeemanenergy(Fig. 14 b)).NowthestatealongthemagneticeldwillbemoststableastheenergyisloweredduetotheZeemanterm.Thestatewithoppositedirectionofmagnetizationwillbemetastable.Themagneticelddependentenergybarrierforthespinup(E+(H))anddown(E)]TJ /F1 11.955 Tf 7.08 1.79 Td[((H))stateiscalculatedbyStonerandWohlfarthtobe, 6 E(H)=KV1H Hco2(1{6) whereE+(H)istheenergybarrierseenbytheupmagnetizedparticlesandE)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)istheenergybarrierseenbythedownmagnetizedparticlesandHc0=2K=Ms.DerivationoftheEq. 1{6 isgivenbelow. 15 PAGE 16 Figure14. TwostateenergyofaSDparticle.Twoenergyminimacorrespondtothedirectionoftheeasyaxisofmagnetization.a)Atzeromagneticeldtheparticlewillhavemagnetizationalongtheeasyaxisofmagnetizationasthosecorrespondtominimumenergystates(upanddown).UpanddownstatesareseparatedbytheenergybarrierequaltoKV.Toreversethemagnetizationdirectionfromuptodownorviceversathesystemhastoovercometheenergybarrier. 5 BrownproposedthatthisprocessrequiresanitetimegivenbyEq. 1{14 5 b)Inanappliedmagneticeldalongtheeasyaxis,thetwoenergyminimawillbeshiftedduetotheZeemanenergy.Nowtheupstatewhichisalongtheappliedmagneticeldwillbemoststableandhavethelowestenergy.Thestatewithoppositedirection(downstate)ofmagnetizationwillbemetastable.ThemagneticelddependentenergybarrierforthespindownstateiscalculatedbyStonerWohlfarth(Eq. 1{6 ) 6 FirstorderderivativeofEq. 1{5 withrespecttois E(H) =2KVsincos+MsVHsin(1{7) Atthemaximaandminima 2KVsincos+MsVHsin=0(1{8) Solutionsoftheaboveequationare sin=0(1{9) cos=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(MsH 2K(1{10) 16 PAGE 17 TakingthesecondorderderivativeofEq. 1{5 ,itcanbeshownthattheEq. 1{9 (Eq. 1{10 )referstominima(maximum)oftheenergy.Thustheenergyminimaareat=0and=andmaximumatwhencos=MsH=2K(seeFig. 14 ).Energiescorrespondtotheseextremaare Emin+=)]TJ /F6 11.955 Tf 9.3 0 Td[(MsVH (1{11) Emin)]TJ /F1 11.955 Tf 16.71 2.87 Td[(=MsVH (1{12) Emax=KV"1+MsH 2K2# (1{13) whereEmin+andEmin)]TJ /F1 11.955 Tf 10.66 2.87 Td[(correspondsto=0(spinup)and=(spindown)respectively.ItisnoweasytoshowthatE+(H)=EmaxEmin+andE)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)=EmaxEmin)]TJ /F1 11.955 Tf 10.66 2.87 Td[(aregivenbyEq. 1{6 Theenergybarrierhastobeovercometoreversethemagnetizationdirectionfromuptodownorviceversa.Brownproposedthatthisprocessrequiresanitetime 5 =0expE(H) kBT(1{14) whereTisthetemperature,)]TJ /F3 7.97 Tf 6.58 0 Td[(10istheinverseattemptfrequencyofovercomingtheenergybarrierandkBistheBoltzmannconstant. Figure 15 showsthemagnetizationprocesswhenthemagneticeldissweptfromalargepositivevaluetoalargenegativevalueandagainfromanegativetopositivevaluetocompletethemagnetizationloop.Whenmagneticeldislarge(scenario1)allparticleswillbemagnetizedalongthemagneticeldandapositivesaturationmagnetizationisachieved.Asmagneticeldisreducedtozero(scenario2)themagnetizationdirectionwillbetrappedintheupdirectionasthetemperatureisnotenoughtoovercometheenergybarrier.Nowasthemagneticeldisreversedtheenergybarrier,E)]TJ /F1 11.955 Tf 7.09 1.8 Td[((H)willbereducedaccordingtoEq. 1{6 (scenario3).Butstillthetemperatureisnotenoughtoovercometheenergybarrierandthemagnetizationwillstillbetrappedwithapositivevalue.Afurtherincreaseinmagneticeldintheoppositedirectionwillkeeploweringtheenergy 17 PAGE 18 barrieruntil,atthecoerciveeld,theenergybarriercanbeovercomebythethermalenergyandmagnetizationreversalwilloccur(scenario4).WhenH=Hctheenergy Figure15. HysteresisofacoherentlyrotatingSDparticle.Scenario1)Highpositivemagneticeldisappliedandsaturationmagnetizationisobserved.Scenario2)Magneticeldisreducedfrompositivevaluetozero.Magnetizationistrappedinthepositivedirectionasthethermalenergyisnotenoughtoovercometheenergybarrier.Scenario3)Magneticelddirectionisreversed.Stilltheenergybarrierislargecomparedtothethermalenergyandmagnetizationistrappedinthepositivedirection.Scenario4)Magneticeldequalstothecoerciveeld.Nowtheenergybarriercanbeovercomebythermalenergyandmagnetizationreversaloccurs. barrierE)]TJ /F1 11.955 Tf 7.09 1.79 Td[((Hc)issuchthattherelaxationtime)]TJ /F1 11.955 Tf 10.99 1.79 Td[(=matthetemperatureT.Wheremistheexperimentalmeasurementtime(around100secforSQUIDmeasurement).ThuswhenH=Hc,magnetizationreversaloccurs.CombiningEq. 1{6 and 1{14 ,thecoercivity(Hc(T))oftheSDparticlecanbecalculatedasshownbelow. )]TJ /F1 11.955 Tf 17.05 1.79 Td[(=m=0expE)]TJ /F1 11.955 Tf 7.08 1.79 Td[((Hc) kBT)E)]TJ /F1 11.955 Tf 7.09 1.79 Td[((Hc)=kBTlnm 0 18 PAGE 19 NowusingtheexpressionofE)]TJ /F1 11.955 Tf 7.09 1.79 Td[((H)fromEq. 1{6 itiseasytoshowthat 6 Hc=Hc0"1)]TJ /F8 11.955 Tf 11.96 16.86 Td[(T TB1 2#(1{15) whereTB=KV=kBln(m=0)isknownastheblockingtemperature.BelowTBtheanisotropicenergybarrierislargerthanthethermalenergyandmagnetizationisblockedortrapped.AboveTBtheanisotropicenergybarriercanbeovercomeeasilybythermalenergyandtheparticlesarecalledsuperparamagneticaswillbediscussedlater.ItisclearfromtheEq. 1{15 thatHcdecreaseswithincreasingtemperatureandabovetheblockingtemperature(T>TB)theparticleslosetheircoerciveeld.NotethattheoriginofHcinaSDparticleisthenitetimerequiredtoreversethemagnetizationdirectionovertheanisotropyenergybarrier. Thepreviousdiscussionisonlytrueforanassemblyofuniformsizeparticlesthathaveeasyaxisofmagnetizationorientedalongthesamedirectionwithmagneticeldappliedalongtheeasyaxis.Inrealsamplesthisisnotthecasesincetheeasyaxisofmagnetizationisusuallyrandomlyorientedandtheparticlesizeisnotuniform.AmoregeneralcaseisshowninFig. 16 below.Hereaarbitraryanglebetweenmagneticeldandtheeasyaxisofmagnetization( )isconsidered.Theenergyoftheparticleinthiscaseis 7 E(H)=2KV1 2sin2( )]TJ /F6 11.955 Tf 11.95 0 Td[())]TJ /F6 11.955 Tf 16.89 8.09 Td[(H Hc0cos(1{16) ThemagnetizationoftheparticleatanappliedmagneticeldisgivenbyMcosmin,whereministheanglecorrespondstotheminimaofE(H).Notethatherewehavenotconsideredtheeectoftemperatureonthemagnetization.AtnitetemperatureothervaluesaroundtheminwillbeoccupiedwithaniteprobabilityaccordingtotheBoltzmannfactorasshowninFig. 17 below.ThustheaverageoveralltheoccupieddirectionwiththeoccupationprobabilitygivenbytheBoltzmannfactorwillbethethermalaverageofthemagnetizationforaxedvalueofHand .TheprocedureshouldberepeatedforallvaluesofHtogettheMHloopforaparticularvalueof .Thenthe 19 PAGE 20 Figure16. Singleparticleinanappliedmagneticeld(H).kisthedirectionoftheeasyaxisofmagnetization.MandHarethemagnetizationandmagneticeldvectorsrespectively.WithoutloosinganygeneralityMandHcanbeconsideredinthesameplane.TheanglebetweenMandHis.TheanglebetweenHandkis .TheenergybarrierforthisgeneralcongurationisgivenbyEq. 1{16 7 MHloopsforallpossible shouldbecalculated.AveragingoveralltheseMHloopswillgiveamagnetizationloopattemperatureTforasampleofuniformparticlesizeandarandomlyorientedeasyaxisofmagnetization.Alltheaboveproceduresshouldbedoneforallpossibleparticlesizesastherealsamplesusuallyhavesomeparticlesizedistribution.Theprobabilityofaparticularparticlesizecanbemodeledeitherasalognormalorgaussiandistributionfunction.Inthiswaythemagnetizationloopofarealsamplewithnonuniformparticlesizeandrandomorientationoftheeasyaxisofmagnetizationcanbedetermined.Ifthealltheaboveproceduresarerepeatedfordierenttemperaturesthenthemagnetizationloopatdierenttemperaturescanbedetermined.Belowweshowaowdiagramfortheaboveprocess. 7 { 9 20 PAGE 21 Figure17. AtnitetemperatureothervaluesaroundminwillbeoccupiedwithaniteprobabilityaccordingtotheBoltzmannfactorandshownbytheshadedregion.Thethermalaverageofthemagnetizationwillbetheaverageoftheallmagnetizationdirectionsoverthisshadedregion. 7 TheprobabilityofhavingsomemagnetizationdirectionwillbedeterminedbytheBoltzmannfactor. 1 Startwiththeenergyofthesingleparticle. E(H)=2KV1 2sin2( )]TJ /F6 11.955 Tf 11.96 0 Td[())]TJ /F5 7.97 Tf 16.59 4.7 Td[(H Hc0cos + 2 FindtheminimaofE(H) MsVCosminwillbethemagnetizationatT=0forthegivenvalueofH, andV Figure18. Flowdiagramtoshowtheprocessofcalculatingcoerciveeldforrealnanoparticlesampleswithparticlesizedistributionandrandomorientationoftheeasyaxisofmagnetizationsatnitetemperature. 21 PAGE 22 + 3 Thermalaverageofthemagnetization hM(H; ;V)iT=R21MsVcosexp)]TJ /F11 5.978 Tf 8.92 2.35 Td[(E kBTd R21exp)]TJ /F11 5.978 Tf 8.92 2.34 Td[(E kBTd E=E)]TJ /F6 11.955 Tf 11.96 0 Td[(Emin1and2areshowninFig. 17 hM(H; ;V)iTisthemagnetizationattemperature=TforthegivenvalueofH, andV + 4 Step1,2and3shouldberepeatedfordierentH.ThiswilldeterminetheMHloopforagivenvalueofT, andV + 5 Step1,2,3and4shouldberepeatedforallpossible andaverageofallthoseloopswilldeterminetheMHloopforagivenvalueofTandVforanensembleofparticleswithrandomorientationoftheeasyaxisofmagnetization. + Figure18. continued 22 PAGE 23 6 Step1,2,3,4and5shouldberepeatedfordierentparticlesizetodeterminetheMHloopforagivenvalueofTforasampleconsistingofnonuniformparticlesizeandrandomorientationoftheeasyaxisofmagnetization.Inrealsamplestheparticledistributionfunctionisusuallylognormalorgaussian. 9 10 + 7 Step1,2,3,4,5and6shouldberepeatedfordierentTtodeterminethetemperaturedependenceoftheMHloop. 7 + 8 Completionofstep7willprovideanopportunitytodeterminethetemperaturedependenceofthecoerciveeld,remanentmagnetizationetc.SomeofthetemperaturedependentofcoerciveeldsarelistedinTable 11 Figure18. continued MagnetizationloopsatdierenttemperaturesforasinglelayersampleofNinanoparticlesofaveragediameteraround18nmareshowninFig. 19 .Thecoerciveeldisdeterminedbythemagneticeldwheremagnetizationchangessignandpassesthroughzero.ItisclearfromtheFig. 19 thatcoerciveeldsdecreaseswithincreasingtemperatureasdiscussedabove.AttemperatureshighcomparedtotheanisotropyenergyKV,themagnetizationdirectionscanrotatefreelyoverthebarrierandtheparticlesbecomesuperparamagneticwithHc=0.InthiscasethesystemcanbetreatedsimilartothecaseofparamagnetismwitheachparticleasagiantorsuperspinofvalueMsV(thuscalled 23 PAGE 24 Figure19. HysteresisloopofaSinglelayerNinanoparticlesof18nmdiameterembeddedinanAl2O3matrixattemperature,T PAGE 25 Figure110. HysteresisloopofaSDcoherentlyrotatingparticleattemperature(T>TB).Sampleshowszerocoerciveeldasexpectedforsuperparamagneticparticles.NotetheH=Tabscissa.MagnetizationisplottedasafunctionofH=Tforthreedierenttemperaturesasindicatedinthelegend.LoopsatalldierenttemperaturesfallontopofeachotheraspredictedbytheLangevinequationforsuperparamagneticparticles. functionofparticlesize.Forverysmallparticlesthecoerciveeldiszeroandparticlesaresuperparamagnetic(SP)withmagnetizationdeterminedbytheLangevinfunction.Astheparticlesizeisincreased,thecoerciveeldincreasesduetothefactthattheenergybarrierincreases.Particleswithsizelargerthanthecriticalsingledomainradiusaremultidomainandthecoerciveelddecreaseswithincreasingparticlesize. 2 3 9 Thismaybeduetothefactthatasparticlesizeincreasesthenumberofdomainsincreasesandthusitiseasiertohavedomainclosurewhichdecreasescoercivitybecausethereislesstotalmagnetization. ThesizedependenceofthecoercivityinMDregionisexperimentallyfoundtobe 2 HcMD=a+b=dx(1{18) wherea,bareconstantsthatdependontherealstructurefactorandmaterials,disthediameteroftheparticlesandxhasvaluearound1. 2 Thereisnotheoreticalmodelthat 25 PAGE 26 Figure111. Coerciveeldplottedasafunctionofparticlediameter.Forverysmallparticlesthecoerciveeldiszeroandtheparticlesareknownassuperparamagneticparticles(SP).Astheparticlesizeisincreasedthecoerciveeldincreasesduetothefactthattheenergybarrierincreases.Particleswithsizelargerthanthecriticalsingledomainradiusaremultidomainandcoerciveelddecreaseswithincreasingparticlesize. explainsthebehaviorinEq. 1{18 .ThusthethepeakinthecoerciveeldwhenplottedasthefunctionoftheparticlesizedelineatestheSDandMDbehavior.InexperimentwehavefoundthesamebehaviorforbothmultilayerandsinglelayersamplesofNiparticlesinAl2O3matrixaswillbediscussedindetailinchapter 2 Figure 112 showsthepossibledomainwallcongurationfordierentpointsinthemagnetizationloop.RememberthatcomparedtotheSDcasewheretheoriginofthehysteresiswasthehoppingoveraenergybarrier,incaseofMDtheoriginofhysteresisisirreversibledomainwallmotion.Atveryhighmagneticeldallthespinsinthesystemwillbealignedalongthemagneticeldandpositivesaturation(Ms)willbeachieved(Fig. 112 ).Asthemagneticeldisreducedtozeroadomainwallwillbeformed.Duetotheimperfectionsinthesample,thedomainwallwillbestuckinapositionsuchthattheupdomainislargerthanthedowndomainandnetmagnetizationorremanent 26 PAGE 27 magnetization(Mr)willbeseenatzeromagneticeld.Reversingthemagneticeldwill Figure112. HysteresisloopofaMDsystemandpossibledomainwallconguration.Atverylargepositivemagneticeldallthespinsarealignedalongthemagneticeldandsaturationmagnetizationisachieved.Whenmagneticeldisreducedtozero,adomainwallforms.Duetotheimperfectionsinthesample,thedomainwallwillbestuckinapositionsuchthattheupdomainislargerthanthedowndomainandremanentmagnetizationismeasured.Ifthedirectionofthemagneticeldisreversedthedomainwallwillstarttomovetotherightandthedowndomainwillgrow.Atamagneticeldequaltothecoerciveeld,thedownandupdomainwillbeequalinsizeandmagnetizationwillbezero.Foralargenegativemagneticeldthedomainwallbemovedtotherightandallthespinswillbeinthedirectionofthemagneticeldandnegativesaturationwillbereached. movethedomainwalltotherightsideandthusthedowndomainwillstarttogrowandmagnetizationwillbereduced.Whenthenegativemagneticeldisequalstothecoerciveeldtheupanddowndomainwillhavesamesizeandmagnetizationwillbezero.Furtherincreaseinmagneticeldinthenegativedirectionwillforcethedomainwalltomoveallthewaytorightmakingallspinsalignedalongthemagneticeldandnegativesaturationwillbereached. 27 PAGE 28 ToderivethecoerciveeldinMDdomaincaseconsiderasimplecase,asshowninFig. 113 ,whereasingledomainwallseparatestwodomains.Therighthandsideisaspinupdomainandlefthandsideisaspindowndomain.Inanappliedmagneticeld,H, Figure113. Singledomainwallseparatingtwomagneticdomains.Righthandsideisaspinupdomainandlefthandsideisaspindowndomain.InanappliedmagneticeldduetotheZeemanenergythedomainwallwillexperienceapressureandsomeworkneedtobedonetomovethewallagainstthispressure.TheoriginofhysteresisinMDsampleistheirreversiblemotionofthedomainwall. alongthespinupdomain,theZeemanenergyoftheup(down)domainwillbe)]TJ /F6 11.955 Tf 9.3 0 Td[(MsH(+MsH)perunitvolume.Thustheenergydierenceacrossthedomainwallwillbe)]TJ /F1 11.955 Tf 9.3 0 Td[(2HMSperunitvolume.Thisenergydierencecanbeconsideredasapressureonthewallandsomeworkhastobedonetomovethedomainwallagainstthispressure.Theworkdonetomovethewalladistancedxis 2 11 dW=2MsHSdx(1{19) whereStheareaofthedomainwall.Thustheworkdonetomovethewallbyunitdistanceis 2 11 dW=dx=2MsHS(1{20) 28 PAGE 29 wheredw=dxcanbethoughtofastheresistanceofthedomainwallmotion.Inrealsamplesduetotheimpurities,imperfections,strainsetc,dW=dxpassesthroughmaximaandminima.Thewallmotionoverthesemaximaandminimaisirreversibleinmagneticeldandthatistheoriginofthehysteresis.Thecoerciveeld,themeasureofirreversibility,isusuallygivenby 2 11 Hc=1 2MsS(dW=dx)max(1{21) Therearedierenttheoreticalmodelstocalculate(dW=dx)maxfordierentimperfectionsinthesampleandtheresultsforsomeofthemarelistedinTable 11 Table11. Tableherelistssomeknownmodelsalongwiththevariationofcoerciveeldaccordingtothemodel. TheoryHc SystemReferences StonerWohlfarthHc=2K Ms[1)]TJ /F11 5.978 Tf 7.45 0 Td[((kBTlnm 0=KV)1=2] SD,CRnanoparticlewithuniaxialanisotropyalongtheappliedmagneticeld 2 6 StonerWohlfarthHc=0:96K Ms[1)]TJ /F11 5.978 Tf 7.45 0 Td[((kBTlnm 0=KV)3=4] SD,CRnanoparticlewithuniaxialanisotropyrandomlyoriented 7 MicromagneticHc=2K Ms)]TJ /F11 5.978 Tf 7.45 0 Td[([3p ja3j 4kBTlnm 0]2=3 MD,2phasematerial,hardmagnet,a3isthemicromagneticparameteranddependsontheK,Ms,A 12 InclusionTheoryHc=2=3 Msd MDsystem,d<,freepoleenergyisignored,coercivityisassumedtobeequaltothemaximumpiningeld,disthediameteroftheinclusion,isthedomainwallthickness,isthedomainwallenergyperunitarea,isthevolumefractionoftheinclusion 11 13 InclusionTheoryHc=1:751=2 MsL(ln2L d) MDsystem,d>,freepoleenergyisignored,Listhelineardimensionofthesample 11 14 InclusionTheoryHc=2:81=2 MsL(d )3=2(ln2L ) MDsystem,d<,freepoleenergyisignored 11 14 InclusionTheoryHc=2K Ms[0.386+logr 2M2s K] MDsystem,d<,freepoleenergyisconsidered 15 InclusionTheoryHc=3l=Msd2 MDsystem,closerdomain,largeinclusion,commonlyseeninthecaseofNeel'sspike,listheequilibriumlengthofthespike 16 1.4Magnetizationvs.Temperature Untilnowwehavebeendiscussingthebehaviorofmagnetizationasafunctionofmagneticeldataxedtemperature.Nowwewilldiscusshowmagnetizationchangeswiththetemperatureataxedmagneticeld.Atsmallappliedmagneticeld,spinsaretrappedinmetastableenergyminimaseparatedbyenergybarriersfromtheglobal 29 PAGE 30 minima.Asthetemperatureisincreasedthespinscanhopovertheenergybarriertoreachtheglobalminima.Duetothistrappingofspinsinlocalminima,magnetizationvaluesdependsstronglyonthecoolingprotocol.Therearemainlytwodierentcoolingprotocols,eldcooled(FC)andzeroeldcooled(ZFC).ThebehaviorofmagnetizationasafunctionoftemperatureforthetwoprotocolsisshowninFig. 114 belowforthesampleofNinanoparticlesof3nmdiameteratanappliedeldof20Oe.ThetemperaturewherethedierencebetweenFCandZFCdisappearsisgenerallycalledtheirreversibletemperature(Tirr).FornanoparticlesTirrissameastheblockingtemperature(TB). 2 Figure114. Magnetizationvs.temperatureatanappliedmagneticeldof20Oeforthe3nmdiameterNinanoparticles.Theredcoloristheeldcooled(FC)magnetizationandtheblackoneisthezeroeldcooled(ZFC)magnetization. 1.4.1ZeroFieldCooled(ZFC)Magnetization Zeroeldcooledmagnetizationismeasuredbycoolingthesamplefromhightemperature(temperatureabovetheirreversibletemperature(Tirr))withoutanyappliedmagneticeld.Atlowtemperatureasmallmagneticeldisappliedandmagnetizationismeasuredasafunctionoftemperatureduringthewarmupwhilekeepingthemagneticeldon.HerewewilldiscusstheshapeoftheZFCmagnetizationinaqualitativemanner.Ingeneralthemagneticsystemcanbetreatedasatwostateproblemasshownpreviously 30 PAGE 31 inFig. 14 wherespinupanddowncorrespondtotheenergyminimaseparatedbysomeenergybarrier.TheoriginoftheenergybarrierintheSDcaseistheanisotropywhereasforthecaseofMDtheoriginisdomainwallpinningatdefects.Athightemperaturetheenergybarrieriseasilyovercomeduetothethermalenergyandthespinupanddownstateswillbeequallypopulated.ThusathightemperatureaboveTirr,magnetizationwillbezero.Nowifthesampleiscooledtoalowtemperaturewithoutanyappliedmagneticeld,thenzeromagnetizationstatewillbeblockedastheenergybarrierisnowlargecomparedtothethermalenergy. 2 3 6 Ifasmallmagneticeldisappliedthechangeinmagnetizationwilloccuronlyforthesmallenergybarriersthatcanbeovercomeatthattemperatureandasmallmagnetizationwillbeachieved.Astemperatureisincreased,theprobabilityofovercomingthelargerbarriersincreasesandmagnetizationincreases.AttemperatureTirrtheprobabilitiestoovercomethebarrierforspinupanddownbecomenearlyequalandthespinupanddownmixingstartstohappenandthusmagnetizationdecreaseswithfurtherincreaseintemperature. 1.4.2FieldCooled(FC)Magnetization Fieldcooledmagnetizationismeasuredbycoolingthesamplefromhightemperaturetothelowtemperatureinanappliedmagneticeldandmagnetizationismeasuredduringthewarmupprocess. 2 Inthiscaseathightemperatureduetotheappliedmagneticeld,thespinupstatesaremorepopulatedthanthespindownstates.Coolingthesampleatalowtemperaturewhilekeepingtheeldonwillthuslockthesysteminmagnetizedstate.Anincreaseintemperaturewillincreasetheprobabilityofspinupanddownmixingandthusmagnetizationwillgraduallydecrease. 31 PAGE 32 CHAPTER2DIPOLARINTERACTIONSANDTHEIRINFLUENCEONTHECRITICALSINGLEDOMAINGRAINSIZEOFNIINLAYEREDNI/AL2O3COMPOSITES 2.1Abstract PulsedlaserdepositionhasbeenusedtofabricateNi/Al2O3multilayercompositesinwhichNinanoparticleswithdiametersintherangeof360nmareembeddedaslayersinaninsulatingAl2O3host.Atxedtemperatures,thecoerciveeldsplottedasafunctionofparticlesizeshowwelldenedpeaks,whichdeneacriticalsizethatdelineatesacrossoverfromcoherentlyrotatingsingledomaintomultipledomainbehavior.Weobserveashiftinpeakpositiontohighergrainsizeastemperatureincreasesanddescribethisshiftwiththeorythattakesintoaccountthedecreasinginuenceofdipolarmagneticinteractionsfromthermallyinducedrandomorientationsofneighboringgrains. 2.2Introduction Themagneticpropertiesofnanoparticleshavebeenthefocusofmanyrecentexperimentalandtheoreticalstudies.Technologicalimprovementshavenowmadeitpossibletoreproduciblyfabricatenanomagneticparticleswithpreciseparticlesizeandinterparticledistances. 17 { 22 Thesecontrolledsystemshaveenabledstudyofthefundamentalpropertiesofsingleaswellasinteractingparticles.Mostapplicationsrequirethattheparticlesbesingledomainwithauniformmagnetizationthatremainsstablewithasucientlylargeanisotropyenergytoovercomethermaluctuations, 23 whichestablishesatemperaturedependentlowerboundtotheparticlesize.Theseconsiderationsmusttakeintoaccounttheeectofinteractionsonmagneticpropertiesasisevidentforhighdensityrecordingmedia 24 whereparticlesareveryclosetoeachother.Considerableinsighthasalreadybeengainedfromexperimentalstudiesoftheeectofdipolarinteractiononsuperparamagneticrelaxationtime 25 { 34 andblockingtemperature. 29 Lessunderstoodhoweveristheeectofdipolarinteractionsontheestablishmentofanupperboundtoparticlesize,whichdenesthecrossoverfromsingledomain(SD)tomultidomain(MD)behavior.InthefollowingweshowusingcoercivitymeasurementsonNi/Al2O3composites 32 PAGE 33 thatwithincreasingtemperaturethisupperboundtoparticlesizeincreasesandthensaturatesduetoattenuateddipolarinteractionsfromthermallyinducedcoherentmotionsofthemagnetizationoftheneighboringrandomlyorientedparticles. 2.3ExperimentalDetails ThecompositesystemstudiedinthispapercompriseselongatedandpolycrystallineNiparticleswithdiametersintherangeof360nmembeddedaslayersinaninsulatingAl2O3host.ThemultilayersampleswerefabricatedonSi(100)orsapphire(caxis)substratesusingpulsedlaserdepositionfromaluminaandnickeltargets.HighpuritytargetsofNi(99.99%)andAl2O3(99.99%)werealternatelyablatedfordeposition.Beforedeposition,thesubstrateswereultrasonicallydegreasedandcleanedinacetoneandmethanoleachfor10minandthenetchedina49%hydrouoricacid(HF)solutiontoremovethesurfacesilicondioxidelayer,thusforminghydrogenterminatedsurfaces. 35 Thebasepressureforallthedepositionswasoftheorderof10)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Aftersubstrateheating,thepressureincreasedtothe10)]TJ /F3 7.97 Tf 6.59 0 Td[(6Torrrange.Thesubstratetemperaturewaskeptatabout550oCduringgrowthoftheAl2O3andNilayers.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.59 0 Td[(2overaspotsize4mm1.5mm.A40nmthickbuerlayerofAl2O3wasdepositedinitiallyontheSiorsapphiresubstratebeforethesequentialgrowthofNiandAl2O3.ThisprocedureresultsinaverysmoothstartingsurfaceforgrowthofNiasveriedbyhighresolutionscanningtransmissionelectronmicroscopystudies(Fig. 21 ).Multilayersampleswerepreparedhaving5layersofNinanoparticlesspacedfromeachotherby3nmthickAl2O3layers.A3nmthickcaplayerofAl2O3wasdepositedtoprotectthetopmostlayerofNinanoparticles. ShowninFig. 21 isacrosssectionalTEMimagefromamultilayered(5layers)NiAl2O3samplegrownoncplanesapphire.TheNiparticleshaveasizeof235nminwidthand9nminheight.Theseparationbetweenneighboringparticlesisontheorderof3nm(measuredasaprojecteddistanceincrosssectionalview),whichiscomparabletothethicknessoftheAl2O3spacerlayers.Forthepurposesofthis 33 PAGE 34 Figure21. CrosssectionaldarkeldSTEMimageofa5layerNiAl2O3samplegrownoncaxissapphire experimentthegrainsized,asmeasuredbytheamountofNidepositedreferencedtoacalibratedstandard,representstheaveragesizeofthediskshapedgrainsshowninthegure.ThiscalibrationwasobtainedfromcrosssectionalTEMmicrographsofsinglelayersamples 36 bycomparingtheaveragegrainsizewithd.TheTEMobservationalsoshowsthattheAl2O3spacerlayersarepartiallycrystallized.DuetothelargesurfaceenergydierencebetweenNiandAl2O3,Niformswelldened,separatedislandswithintheAl2O3matrix. 36 Previousstudiesonsimilarlypreparedsamplesusingatomicnumber(Z)contrastimaginginTEMtogetherwithelectronenergylossspectroscopy(EELS)haveconrmedtheabsenceofNiOattheNi/Al2O3interfaces. 36 TheNi/Al2O3interfaceswerechemicallyabruptwithoutanintermixingbetweenNi,Alandoxygen.Inadditionwedidnotobserveexchangebiasinducedasymmetricmagnetizationloops,thuslendingsupporttotheconclusionsofpreviousstudies 36 thatantiferromagneticNiOisabsentinourlayeredNi/Al2O3system. PreviousTEMstudiesonsinglelayersampleshaveshowntheparticlestobepolycrystalline.Forexample,athreenmparticlecomprisingthreecrystallinegrains 34 PAGE 35 hasbeenobserved. 36 Polycrystallineparticleswillthereforehavecrystallinegrainsorientedindierentdirections,thustendingtoaverageanynetcrystallineanisotropytozero.Accordingly,temperatureindependentshapeanisotropyisdominantandtemperaturedependentcrystallineanisotropycanbeneglected.Inaddition,itisalsoimportanttonotethattheexchangelengthlex=14.6nmforNi, 37 whichisthelengthscalebelowwhichatomicexchangeinteractionsdominateovermagnetostaticelds,determinesthecriticalradii(Rcoh)forcoherentrotation:Rcoh5lexforsphericalparticlesandRcoh3:5lexfornanowires. 3 Theparticlesizes(1.530nminradius)thatwehaveinvestigatedarethussmallerthanthecriticalradiusbelowwhichcoherentrotationofNiprevails. Figure22. Coercivityfor5layerNi/Al2O3multilayersamples(5repeatedunits)plottedasafunctionofparticlesize(diameter)atthetemperaturesindicatedinthelegend.Thepeakpositionsatd=dcforeachisotherm,indicatedbyverticalarrows,delineatethecrossoverfromsingledomain(SD)tomultipledomain(MD)behavior(d>dc).InsetshowsthebehaviorofHcasafunctionof1=dfortheparticleswithd>dcat10K.Thelineardependenceupto24nmdiameterparticleswithsaturationataconstantvalueforlargeparticles 38 isconsistentwiththebehaviorexpectedformultidomainparticles.Thusparticlesontherighthandsideofthepeakaremultidomain. InFig. 22 weshowplotsofHcasafunctionofparticlesizedateachofthetemperaturesindicatedinthelegend.Coerciveeldswereextractedfrommagnetization 35 PAGE 36 loopsmeasuredbyaQuantumDesignsuperconductingquantuminterferencedevice(SQUID)aftersubtractingoutthediamagneticcontributionfromthesubstrate.Magneticeldwasappliedalongtheplaneofthelms.Toobtainthemagnetizationloops,themagneticeldwasvariedoverthefullrange(5T)whilekeepingtemperaturexed.Thehighmagneticelddatashowlinearmagnetizationwithmagneticeld,whichisduetothediamagneticcontributionfromthesubstrate(assignalfromferromagneticNiparticlessaturatesathighmagneticelds)andcanthusbesubtractedfromthedata.ThedecreaseofHcwithincreasingtemperatureforxeddisclearlyapparentandcanbeunderstoodastheeectofthermaluctuations. 2 Forthelowtemperatureisotherms,therearepronouncedpeakswhichdeneatemperaturedependentcriticalparticlesizedcdelineatingSD(d PAGE 37 Figure23. Peakposition,dc,plottedasafunctionoftemperature(redcircles).Theblacksquaresaretheresultsderivedfromequation25.Thebluestarrepresentstheobservedvalueofdcforaseriesofsinglelayersamplesat10K.Theinset,aschematicofthreeneighboringparticlesorientedindierentdirections,illustrateshowthedipolareldsfromparticle2and3facilitatetheformationofdomainsinparticle1,asthedipolarmagneticeldsareindierentdirections. intheinsetofFig. 23 .Particle1experiencesdipolareldsfromparticles2and3,whicharenotcollinearformostorientationsofarandomlyorientedparticlesystem.Becausedipolareldsdecreaserapidlywithinterparticleseparation,thedipolareldduetoparticle3(2)willbestrongerthanparticle2(3)ontheleft(right)sideoftheparticle1.Theseparateandunequalinuenceoftheneighboringparticlesthusfavorstheformationofdomainsinparticle1. .Tomakethesenotionsmorequantitative,wemodifythetreatmentofDormannetal 26 forinteractingparamagnetstoincludethetemperatureregionbelowtheblockingtemperatureTBandndthetemperaturedependentdipolarmagneticeldHdarisingfromtemperatureinduceductuationsinthemagnetizationofnearestneighbornanometersizeparticlestobe, Hd=0Msa 4e(1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F3 7.97 Tf 6.59 0 Td[(1) p (er())]TJ /F1 11.955 Tf 11.96 0 Td[(er(p )]TJ /F1 11.955 Tf 11.96 0 Td[(1))(2{1) 37 PAGE 38 where'er'istheimaginaryerrorfunction,Msisthesaturationmagnetization,=KV=kBT,anda=V(3cos2)]TJ /F1 11.955 Tf 13.36 0 Td[(1)=s3isadimensionlessparameterwithandscorrespondingrespectivelytoanangleparameterandtheseparationbetweentwoadjacentparticleseachwithvolumeV.TheparameterisalwaysgreaterthanoneforT PAGE 39 .Intheabsenceofinteractions(Hd=0)theconditionfortheSDtoMDtransitionisgivenforsphericalparticleswithradiusd/2by,Ad3c=Bd2c,whereAd3cisthetotalmagnetostaticenergyandEdw=Bdcisthedomainwallenergy. 46 Wehaveabsorbedthefactoroftwo,whichrelatesdiametertoradius,intotheconstantsAandB.InthepresenceofthedipolarmagneticeldHd,theformationofdomainwallswillbeassistedbyaZeemantermwhichisproportionaltothevolumeoftheaectedparticle.TheconditiondeterminingtheSDtoMDtransitionnowbecomes, Ad3c=Bd2c)]TJ /F6 11.955 Tf 11.96 0 Td[(MsHdd3c=6(2{4) Whenthedipolarinteractionisasmallperturbation,i.e.,MsHd=A1,Eq. 2{1 and 2{4 canbecombinedtogivetherelation, dc=dc0)]TJ /F6 11.955 Tf 11.96 0 Td[(ddwe(1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F3 7.97 Tf 6.58 0 Td[(1) p (er())]TJ /F1 11.955 Tf 11.95 0 Td[(er(p )]TJ /F1 11.955 Tf 11.96 0 Td[(1))(2{5) wheredc0=B=Aisthetemperatureindependentcriticaldiameterintheabsenceofinteractions(hightemperaturelimit)andddw=0BM2s=(72A2)fora==3.ThesecondtermontherighthandsideofEq. 2{5 thusbecomesatemperaturedependentcorrectiontodcduetointeractionsfromneighboringparticlesanddecreaseswithincreasingT. SincethemagneticeldduetothedipoledipoleinteractionsareweakerathighertemperaturesEq. 2{1 ,thenanoparticlesremainintheSDstatetoalargersize,whichbyEq. 2{5 resultsinashiftofdctowardshighervaluesathighertemperatures.ThisisindeedevidentinFig. 23 ,whichshowsthetemperaturedependenceofdcasdeterminedfromthedatainFig. 22 .TheblacksquaresarethesimulateddataaccordingtoEq. 2{5 usingthetwottingparameters:dc0andddw.Qualitatively,thedataagreequitewellwiththepredictionofthetheoreticalmodelwithouttakingintoaccountthetopologyandsizedistributionoftheparticles.Wehavefounddc0=84nmfromoursimulation(Fig. 23 ,blacksquares)tobeclosetothevalueforaparticlewithshapeanisotropyconstantKshape=3:1104Jm3(dc0=72AexK=0M2s,whereAexisexchangestiness, 39 PAGE 40 Kisanisotropyconstant). 3 ValuesofA(/0M2s)andB(/AexK)havebeenfoundtobe1.44104Jm3and1.21103Jm2respectively.ThisvalueofAisveryclosetothetheoreticalpredictedvalue 3 andthevalueofBisagainconsistentwiththevalueoftheshapeanisotropy.ThevalueoftheshapeanisotropycanalsobepredictedfromthezerotemperatureextrapolationHcoK=Msforrandomlyorientedparticles. 3 ForKshape=3:1104Jm3;Hco620Oe.Thisisingoodagreementwiththe500Oecoerciveeldobservedat10Kforthe6nmsample. Foraseparateseriesofsinglelayersamplesthecoercivitiesat10Kpeakatdc=14nmasshowninFig. 23 bythebluestar.Inthesinglelayersamplesthepeakpositionoccursathigherparticlesize(14nm)thanmultilayersamples(8nm).Thisdierencereinforcesourinterpretationandcanbeunderstoodbyrealizingthatthedipolarinteractionsofthesinglelayersamplesaresignicantlyreducedcomparedtothemultilayersamplesbecauseofthesmallernumberofnearestneighbors. 2.5Conclusion Insummary,wehavefabricatedmagneticnanoparticlesinaninsulatingthinlmmatrixwithtunablepropertiesachievedbyvaryingparticlesizeandtemperature.ThepeaksinthecoercivityisothermsdelineateacriticalgrainsizedcwhichidentiesthecrossoverfromSDtoMDbehavior.Thepresenceofdipolarinteractionsandtheirdiminishinginuencewithincreasingtemperatureisresponsiblefortheobserveddependenceofdcontemperatureandisingoodqualitativeagreementwithourmodicationofpresenttheory 26 ofinteractingparticles.Thewellestablishedinuenceofdipolarinteractionsonsuperparamagneticrelaxationtime 25 { 34 togetherwiththeconnectionbetweenrelaxationtimeandcoercivityHcsuggeststhatthereisaconcomitantinuenceofdipolarinteractionsonthecoercivityobservednearthesuperparamagneticlimitwhereHc=0.TheworkreportedhereextendsthisconnectiontotheupperlimitsonthesizeofSDparticlesbyshowingthatdipolarinteractionscanfacilitatetheformationofmultidomainparticlesespeciallyatlowtemperatures. 40 PAGE 41 2.6Methods 2.6.1MathematicalAnalysis TheEq. 2{5 isselfconsistent(asthetermcontainsdc)andcannotbesolvedanalytically.Theequation,dc)]TJ /F6 11.955 Tf 12.13 0 Td[(dc0+ddwe(1)]TJ /F5 7.97 Tf 6.58 0 Td[(e)]TJ /F11 5.978 Tf 5.76 0 Td[(1) p (erfi())]TJ /F5 7.97 Tf 6.58 0 Td[(erfi(p )]TJ /F3 7.97 Tf 6.59 0 Td[(1))=0,issolvedbynumericalapproachandsimultaneouslythesolutionisttedtotheexperimentaldataaccordingtoanonlinearlistsquaremethod.Mathematica,acommercialsoftware,isusedforthispurpose. 2.6.2BasicPhysicalUnderstanding AsimpliedphysicalunderstandingoftheproblemisshowninFig. 24 Figure24. Theneteectofdipolarmagneticeld(Hd)isshownontheparticle1.Asparticlesarerandomlyoriented,Hdfromparticle3willbeindierentdirectionthanthatfromparticle2.Asdipolarinteractiondecreasesrapidlywithdistance,particle1willexperiencelocaldipolarmagneticeldsindierentdirectionsfromdierentneighboringparticlesandthusmakingiteasytoformdomains. 41 PAGE 42 CHAPTER3EFFECTOFDIPOLARINTERACTIONONTHECOERCIVEFIELDOFMAGNETICNANOPARTICLES:EVIDENCEFORCOLLECTIVEDYNAMICS 3.1Abstract TheeectofdipolarinteractiononthecoerciveeldisdiscussedforthesingledomainandcoherentlyrotatingNinanoparticlesembeddedinAl2O3matrix.Resultsfortwosetsof5layersampleswithdierentinterlayerspacingandasetofsinglelayersamplesofNinanoparticlesarecompared.Thedipolarinteractionsarestrongestinthesampleswithshorterinterlayerdistancesandweakestforthesinglelayersamples.Inthispresentstudy,thedipolarinteractionisfoundtoincreasethecoerciveeld.Ontheotherhandthecriticalsingledomainradiusdecreasesduetothedipolarinteractions.Thesetwobehaviorstogetherindicatethatcollectivedynamicsplaysanimportantroleinunderstandingtheoriginofthecoerciveeld. 3.2Introduction Theoriginofcoerciveeld(Hc)forcoherentlyrotatingferromagneticnanoparticlesisremarkablydierentthanthatofthebulk, 47 whereirreversibledomainwallmotionisthedominantmechanism. 4 Inthecaseofnanoparticles,whenthesizeoftheparticleissmallerthanacriticalsize(dc),themostfavorableenergystateistohavesinglemagneticdomainandparticlesarecalledsingledomain(SD)particles.WhenHcisplottedasafunctionofparticlediameter(d),thereisawelldenedpeakatdc.Particleswithd PAGE 43 reversethedirectionofthemagnetizationofacoherentlyrotatingSDparticleisgivenbytherelation 2 5 49 50 =0expKV kBT:(3{2) Here,0istheinverseoftheattemptfrequencytoovercometheenergybarrier,Visthevolumeoftheparticle,kBistheBoltzmannconstantandTisthetemperature.StonerandWohlfarthhavecalculatedHcforSDparticlesinthesimplecasewhenparticlesarecoherentlyrotatingandtheappliedmagneticeldisalongtheeasyaxisofmagnetizationoftheparticles.ThecoerciveeldforaStonerWohlfarthparticleisgivenby HcSW=2K Ms241)]TJ /F8 11.955 Tf 11.95 20.44 Td[( ln(m 0) ln( 0)!1 235;(3{3) wheremisthetimeofmeasurement.FromthesimpleStonerWohlfarthmodelitisclearthatHcforthenanoparticlecandependonmanydierentfactors.Hcincreaseswithdecreasingm,increasingandincreasingK. Inthepresenceofdipolarinteractionstheaboveequationwillbemodied.Thewidelyacceptedmodicationisachievedbytreatingthedipolarinteractionstoresultinaneectiveanisotropyenergy. 51 { 61 .ThusifduetothedipolarinteractionsKincreases(decreases)then,accordingtoEq. 3{2 ,willalsoincrease(decrease)andasanetresultHcwillincrease(decrease).AmorefamiliarfamousformofEq. 3{3 isHcSW(T)=2K=Ms(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)1=2),whereTB=KV=25kBistheblockingtemperature.Thefactor25comesfromthefactthatm100sisatypicalmeasurementtimeand)]TJ /F3 7.97 Tf 6.58 0 Td[(10109sec)]TJ /F3 7.97 Tf 6.58 0 Td[(1isatypicalattemptrate. Theeectofdipolarinteractiononthecoerciveeld(Hc)hasbeeninvestigatedextensively.ThersttheoreticaltreatmentbyNeel 62 showedthatHcdecreaseswiththeincreaseinthepackingfraction()orthedipolarinteractionasshownbelowinEq. 3{4 ,wheretheinteractioneecthasbeenintroducedasan"InteractionField"andshownto 43 PAGE 44 lowertheanisotropyenergy. Hc=Hc1(1)]TJ /F6 11.955 Tf 11.95 0 Td[()(3{4) The"InteractionField"isafunctionofthepackingfraction().LaterWohlfarth 51 showedthattheeectoftheinteractionontheHccanbeincreasingordecreasingdependingontheparticleorientationasthedipolarinteractionisdirectiondependent.Butallofthoseresultshavebeenconstructedconsideringthefactthattheanisotropyconstant,K,eitherincreasesordecreasesduetotheinteractions.PrevioustheoreticalandexperimentalworkshavebeenreportedeithershowinganincreaseordecreaseinHcandexplainedintermsofacorrespondingincreaseordecreaseintheanisotropyenergy. 51 { 61 63 InthispresentexperimentwendthatanincreaseinthedipolarinteractionincreasesHcbutdecreasesdc.Equation 3{1 suggeststhatthedecreaseindcmaybeduetoadecreaseintheK.ButadecreaseintheKwillalsodecrease(Eq. 3{2 )andthuswilldecreaseHc(Eq. 3{3 )whichiscontradictorytothepresentexperimentalresult.ThusthechangeofKduetothedipolarinteractionsmustnotbeapplicableinthepresentcase.AsanychangeinKwillgiverisetochangeinHcanddcbothinthesamedirection(bothincreaseordecreaseatthesametime).Below,weshowqualitativelythattheincreaseintheHccanberealizedintermsofthecollectivedynamicsofthemagnetizationoftheparticlesanddecreaseindccanbeunderstoodasdiscussedinreference 48 3.3ResultsandDiscussions Samplesweregrownusingpulsedlaserdepositiontechnique. 48 Basepressureofthegrowthchamberwasontheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(7Torrandthegrowthtemperaturewasaround550oC.MultilayerstructureofAl2O3andNinanoparticleweregrownwithoutbreakingthevacuumofthechamber.Firstathick(40nm)buerlayerofAl2O3isgrownontopofthesubstrate.ThepurposeofthisbuerlayeristopreventanydiusionoftheNiintothesubstrate.ThenNinanoparticlesandAl2O3aresequentiallydepositedonthisbuerlayer(seeFig.1).ThetoplayerofAl2O3actsasacappinglayerwhichpreventsoxidationofthenanoparticles. 36 Threedierentsetsofsamplesaregrown.Set1andset2samples 44 PAGE 45 consistof5layersofNinanoparticlesseparatedbyAl2O3layers.Forset1(set2)theAl2O3separationis3nm(40nm).Set3samplesaresinglelayerofNinanoparticlesinAl2O3matrix.Dipolarinteractionsarestrongestinset1,moderateinset2andweakestinset3.ThedipolarinteractionsarestrongerinSet1comparedtoset2astheinterlayerseparationoftheNiparticlesissmallerinset1comparedtoset2.Set3consistsofonlyasinglelayerofNiparticlesandthusthedipolarinteractionsareweakest.Allsetsofsamplesconsistofdierentsampleswithvaryingparticlesizefrom3nmto60nm. Figure31. Fig1a)showstheTEMimageofasinglelayersamplewithaverageparticlediameterof24nm.Particlesarewelldenedwithinterparticledistanceofaround4nm.1b)showsaschematicofthesinglelayersample.A40nmthickbuerlayerofAl2O3isrstgrownontopofsubstrate.ThentheNinanoparticlesaregrownontoofthebuerlayer.Finallya3nmthickcappinglayerofAl2O3isgrowntoprotectitfromoxidation.1c)showstheschematicof5layersofNinanoparticlesample. Figure 31 a)showstheTEMimageofthesinglelayerNiparticleswithaverageparticlediameterof24nm(set3).Thesimpliedschematicofthesingleandmultilayersamplesareshownin 31 b)andc). TypicalmagnetizationloopsatthreedierenttemperaturesareshowninFig. 32 a)forthesamplewith3nmAl2O3spacerlayer(set1)and6nmindiameter.ThecoerciveeldHc(T)isdeterminedfromtheloopasshownbythearrow.ThisproceduretodetermineHcisrepeatedforallsamplesbelongingtoallthreesets.Attemperatures 45 PAGE 46 Figure32. a)Magnetizationloopofasamplefromset1ofaverageparticlediameterof6nm.Coerciveeld(Hc)isdeterminedfromtheloopasshownbythearrow.Hcdecreaseswithincreasingtemperatureandgoestozeroabovetheblockingtemperature.b)Magnetizationloopsaboveblockingtemperatures.MagnetizationisplottedasthefunctionofH=TtoshowthesuperparamagneticbehaviorasexpectedfortheSDparticlesabovetheblockingtemperature. abovetheblockingtemperatures(TB)SDsamplesbehaveassuperparamagneticparticles.Figure 32 b)showsthesuperparamagneticbehavioroftheset1,6nmdiametersample.NotethemagnetizationdatafallontopofeachotherwhenplottedasafunctionofH=T.ThisbehaviorisadirectconsequenceofthesuperparamagneticbehaviorasexpectedfromthecoherentlyrotatingSDparticles. Figure 33 showsHcplottedasafunctionofdfortheset1,set2andset3samples.Thedatathatcorrespondtothedierentsamplesetsareindicatedinthelegends.ThepeakintheHcseparatesSDandMDparticles. 2 8 41 42 44 48 Itisclearfromthedatathatdcdecreaseswithincreasingdipolarinteractions(dc1 PAGE 47 Figure33. Coerciveeld(Hc)asafunctionofparticlediameter(d).Thepeakseparatesthesingledomain(SD)andmultidomain(MD)particles.Particleswithdiameterhigher(smaller)thanthepeakdiameter(dc)areMD(SD).Dataforthe3dierentsamplesetsareshownandindicatedinthelegends.Thecriticaldiametersdc1,dc2,dc3areshownfromthesamplesofset1,set2andset3respectively.Inthesingledomainregion(belowdc)thecoercivityincreaseswithincreasingdipolarinteractionsasshownbytheverticaldottedarrow. magnetizationduetothedipolarinteractionsisfoundtoberesponsiblefortheincreaseinHc.TheseobservationsareshowninFig. 33 andsummarizedinFig. 34 WerstdiscusstheeectofdipolarinteractionsonHcaspresentedinpreviousinvestigations. 51 { 61 ThetreatmentbeginbyincludingthechangeinanisotropyenergyEdip,duetodipolarinteractionintotheexpressionfor,asgivenby 63 =0expKVEdip kBT(3{5) 47 PAGE 48 Equation 3{5 canberewrittenasshowninEq. 3{6 .Thustheeectofthedipolarinteractionsistreatedaseitheranincrease(+Edip)ordecrease(Edip)ofanisotropyenergy. =0expK(eff)V kBT(3{6) TheeectofdipolarinteractionsontheHccanbeexplainedaccordingtoEq. 3{6 .Inourcasea+Edipincreases+andgiverisetoanincreaseinHcwithincreasingdipolarinteractions(Eq. 3{2 and 3{3 ).IfthisistobetrueinourcasethenaccordingtoEq. 3{1 ,dcshouldalsoincreasewithincreasingdipolarinteractions.AccordingtothepreviousapproachbothHcanddcshouldchangeinthesameway,bothincreaseorbothdecrease.InthepresentexperimentwendhoweverthatHcincreasesanddcdecreasesduetodipolarinteractions(seeFig. 34 )andstronglysuggestsanalternativeapproachtotheproblem.Theeectofdipolarinteractionsondcisdiscussedinreference 48 ,whereithasbeenshownthatthelocaldipolarmagneticeldfromthenearbyrandomlyorientedparticlestrytoalignthemagnetizationdirectionoftheparticleindierentdirectionsandthusfavoringdomainformation.TheeectofdipolarinteractionsonHcwillbediscussedbelowintermsofcollectivedynamics. Itiswellknownthatthemagnetizationdynamicscanbecollectiveinnatureduetotheinteractionsbetweentheparticlesandtherelaxationtime()inthiscaseisgivenby 64 65 =T Tg)]TJ /F1 11.955 Tf 11.95 0 Td[(1)]TJ /F5 7.97 Tf 6.58 0 Td[(z;T>Tg(3{7) whereistherelaxationtimeofthesinglenoninteractingparticle(Eq. 3{2 ),Tg=0M2=4kBr3isthecriticaltemperatureanddependsontheinterparticledistanceandparticlemagnetizationandzisacriticalexponent.TheaboveequationclearlysuggeststhattherelaxationtimewillbelargerinthepresenceofdipolarinteractionsandthusaccordingtoEq. 3{3 Hcwillbelarger,andthusagreeingwithourexperimental 48 PAGE 49 Figure34. Coerciveeld(Hc)andcriticaldiameter(dc)asthefunctionoftheincreasingdipolarinteraction.Hc(dc)increases(decreases)withincreasingdipolarinteraction.TheoppositebehaviorofHcanddcsuggeststhatthecollectivedynamicsandthecriticalslowdownisresponsiblefortheincreaseinHcduetothedipolarinteractions.Thedecreaseindcisdiscussedelsewhere. 48 observations(Fig. 34 ).Notethatinthiscasetheanisotropyenergyisunaectedbydipolarinteractionsandtheincreaseinrelaxationtimeisduetothefactthatthereversalofmagnetizationiscollectiveinnature. 64 65 3.4Conclusions AstudyofdipolarinteractionsispresentedforthesingleandmultilayerstructureofNinanoparticles.Thecoerciveeldhasbeenfoundtoincreasewithincreasingdipolarinteractionsandcanbeunderstoodqualitativelyintermsofcollectivedynamics.Threesetsofsamplesareinvestigated.Eachsetconsistsofsampleshavingparticlesizevaryingfrom3nmto60nmindiameter.Dipolarinteractionsarestrongerinset1anddecreasesforset2andset3.Behaviorofcoerciveeldandcriticalsingledomainradiusareobserved.Coerciveeldincreasesandcriticalsingledomainradiusdecreases 49 PAGE 50 withincreasingdipolarinteractions.ThesetwobehaviorstogethersuggestacollectivedynamicsofthemagnetizationreversalprocessintheSDregioninthepresenceofdipolarinteractions.Toourknowledge,thisisthersttimethattheeectofcollectivedynamicsonacoerciveeldofthenanoparticlesystemhasbeenobserved. 50 PAGE 51 CHAPTER4FINITESIZEEFFECTSWITHVARIABLERANGEEXCHANGECOUPLINGINTHINFILMPD/FE/PDTRILAYERS 4.1Abstract ThemagneticpropertiesofthinlmPd=Fe=Pdtrilayersinwhichanembedded1.5AthickultrathinlayerofFeinducesferromagnetisminthesurroundingPdhavebeeninvestigated.ThethicknessoftheferromagnetictrilayeriscontrolledbyvaryingthethicknessofthetopPdlayeroverarangefrom8Ato56A.AsthethicknessofthetopPdlayerdecreases,orequivalentlyastheembeddedFelayermovesclosertothetopsurface,thesaturatedmagnetizationnormalizedtoareaandtheCurietemperaturedecreasewhereasthecoercivityincreases.ThesethicknessdependentobservationsforproximitypolarizedthinlmPdarequalitativelyconsistentwithnitesizeeectsthatarewellknownforregularthinlmferromagnets.Thefunctionalformsforthethicknessdependences,whicharestronglymodiedbythenonuniformexchangeinteractioninthepolarizedPd,provideimportantnewinsightstounderstandingnanomagnetismintwodimensions. 4.2Introduction Thepresenceof3dmagnetictransitionmetalionsinpalladium(Pd)givesrisetogiantmomentsthussignicantlyenhancingthenetmagnetization 66 { 70 .PdisknowntobeinthevergeofferromagnetismbecauseofitsstrongexchangeenhancementwithaStonerenhancementfactorof10 71 .ThemagneticimpuritiesinducesmallmomentsonnearbyPdhostatomstherebycreatingacloudofpolarizationwithanassociatedgiantmoment 71 72 .Neutronscatteringexperimentsshowthatthecloudofinducedmomentscaninclude200hostatomswithaspatialextentintherange10to50A 72 73 .ThusathinlayerofFeencapsulatedwithinPdwillbesandwichedbetweentwoadjacentthinlayersofferromagneticPdwithnonuniformmagnetizationandatotalthicknessintherange20to100A. 51 PAGE 52 WehaveinvestigatedthinlmPd/Fe/PdtrilayersinwhichthethicknessdFeoftheFeisheldconstantnear1.5AandthethicknessofthepolarizedferromagneticPdisvariedbychangingthetopPdlayerthicknessx.Themagneticpropertiesarestudiedasafunctionofx.Ourexperimentsaremotivatedbytherecognitionthatferromagnetisminrestricteddimensionshasattractedsignicantresearchinterest 74 { 78 .Forexample,thecoerciveeldHcincreasesasthethicknessoftheferromagneticlmisdecreasedtowardathicknesscomparabletothewidthofatypicaldomainwall 79 80 .Moreover,theCurietemperatureTcdecreasesasthethicknessoftheferromagneticlmisdecreasedtowardathicknesscomparabletothespinspincorrelationlength 81 { 83 .WewillshowbelowthatsimilarphenomenologyappliestoferromagneticallypolarizedPdlms,albeitwithdierentfunctionaldependencesarisingfromthefactthatexchangecoupling,whichdecayswithdistancefromtheferromagneticimpurity 84 ,isnotuniformthroughoutthelm. 4.3ExperimentalDetails ThesamplesweregrownonglasssubstratebyRFmagnetronsputtering.Thebasepressureofthegrowthchamberwasoftheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(9Torr.FirstathicklayerofPdofthickness200Aisgrownontopofthesubstrate.TherootmeansquaresurfaceroughnessofthisPdlayerwasmeasuredbyatomicforcemicroscopytobe6A.Thenaverythin(1.5Aasrecordedbyaquartzcrystalmonitor)layerofFeisdepositedontopoftherstPdlayer.FinallyatoplayerofPdwiththicknessxisgrowntocompletethetrilayerstructureshownschematicallyinFig. 41 a.WediscusssixdierentsampleswiththetopPdlayerhavingathicknessxvaryingfrom8to56A.ThetotalthicknessyofthepolarizedPd(seeFig. 41 b)canrangefrom20to100A 72 73 .Thusforx PAGE 53 Figure41. a)MultilayerstructureofaPd/Fe/Pdtrilayer.ThebottomlayerofPdis200Athick.ThethicknessoftheFelayeris1.5Aasrecordedbythequartzcrystalmonitor.ThethicknessxofthetoplayerofPdisvariedfrom8to56A.b)Magneticstructureofthesample.ThetotalthicknessyofpolarizedPdisintherange20to100A(shadedredarea).Thusbyvaryingx,itispossibletovarythethicknessyofthepolarizedferromagneticPdlayer.c)IntensityofFe3(703.0eV)LMMAugerelectronpeakplottedasafunctionofmaterialremovedbyargonsputtering.Thedata(solidblackcircles)arettoaGaussiandistribution(redline).Thefullwidthhalfmaximumvalueof1.85Aisconsistentwithcrystalmonitormeasurements AugerelectronpeakofFig. 41 cshowsthattheFeisembeddedinthePdasadistinct2DlayerwithaFWHMthicknessof1.8A.Allofthesestepswereperformedwithoutbreakingvacuum.MeasurementsofthemagnetizationM(Fig. 42 )wereperformedusingaQuantumDesignMPMSsystem.ThemagneticeldHwasalongtheplaneofthesubstrate.Sincethemagnetizationmeasurementswereexsitu,xwasconstrainedtobegreaterthan8A;otherwisetheexposureofthesampletoaircausedunwantedoxidationoftheFe.ThemagneticparametersHc(x)(Fig. 43 )andTc(x)(Fig. 44 )arecalculatedrespectivelyfrommagnetizationloopstakenat10K(seeinsetofFig. 43 )andlinearextrapolationsofthetemperaturedependentmagnetizationtakenatH=20Oe(seeinsetofFig. 44 ).ThemagneticcontributionfromthebottomferromagneticPdlayerisindependentofx,sincey=2<200A,theconstantthicknessofthebottomlayer. 53 PAGE 54 Figure42. ThesaturationmagnetizationnormalizedtotheareaofthesampleMsAshowsasmoothincreasewithincreasingthicknessx.Theexperimentaldataareshownassolidblackcirclesandthedashedblacklineisaguidetotheeye.Saturationtoaconstantvalueoccursnear30A(verticalarrow). 4.4ResultsandDiscussion Forlargevaluesofx,thethicknessyofthecombinedpolarizedferromagneticPdlayersandtheassociatedsaturatedmagnetizationM=Mswillreachaconstantvalue.ThisexpectationisborneoutinFig. 42 whichshowsthexdependenceofsaturatedmagnetizationMsAnormalizedtosamplearea.WenotethatthisnormalizedsaturatedmagnetizationMsA(x)increaseswithincreasingxasthetotalamountofpolarizedPdincreases.Theonsetofsaturation,nearx=30AindicatesthatthepolarizationcloudincludingtheembeddedFelayeris60Athick.Thisvalueisconsistentwithpreviousobservation 73 .TheincreaseofMsAwithxshowninFig. 42 isthusstraightforwardtounderstand.AsxincreasesthethicknessofthetoppolarizedferromagneticPdlayerincreaseswithaconcomitantincreaseofmagneticmaterialinthesystem.VariationofxclearlycontrolsthethicknessofthepolarizedferromagneticPdlayer.Whennormalizedto 54 PAGE 55 Figure43. ThecoerciveeldHcshowsastrongincreaseasthethicknessxofthetoplayerofthePddecreases.Thedataareshownassolidblackcirclesandtheblacksolidlineisapowerlawtwithexponent=2:3(0:1).TheinsetshowsmagnetizationloopsatT=10Kforx=8A(solidblacksquares)andx=56A(solidredcircles). thenumberofFeatomspresent,thesaturatedmagnetizationMsA=1:110)]TJ /F3 7.97 Tf 6.59 0 Td[(4emu/cm2correspondsto9.2BperFeatom,incloseagreementwithpreviousobservationsofthegiantmomentofFeinPdtobenear10B 72 .ModelingthexdependenceofMsA(x)showninFig. 42 forourPd=Fe=Pdtrilayersisnotstraightforward.ForregularferromagnetswithMsuniformthroughoutthethickness,wewouldexpectMsA(x)tobelinearinx;clearlyitisnot.AreasonablemodelwillincorporateanexchangeinteractionJthatdecaysradiallywiththedistancefromthepointferromagneticimpurity 84 .ThiscomplicationrequiresmodelingJasafunctionofdistancexfromtheplaneofimpurity.AstartingpointwouldbetowritethemagnetizationMisafunctionofJ 4 M(H;T;x)=MsBs Ms kBT"gBH+2pMJ(x)#!;(4{1) 55 PAGE 56 Figure44. TheCurietemperatureTcrapidlyincreaseswithincreasingx.Dataareshownassolidblackcirclesandthedashedblacklineisaguidetotheeye.Saturationtoaconstantvalueoccursnear20A(verticalarrow)TheinsetwithTcindicatedbytheverticalarrowshowsthetemperaturedependentmagnetizationtakeninaeldH=20Oe. whereBsistheBrillouinfunctionandpisthenumberofthenearestneighborsbeyondwhichJiszero.InprincipletheexperimentallydeterminedvaluesofM(H;T;x)canbettoEq. 4{1 tondthebesttvaluesofJ(x)fordierentvaluesoftheparameterp.Wehavenotperformedsuchananalysis. Fig. 43 showsthebehaviorofthecoercivityHc(x)asafunctionofx(solidblackcircles).Thedataarewelldescribedbyapowerlawdependence(solidblackline),Hc(x)/x)]TJ /F5 7.97 Tf 6.58 0 Td[(,wheretheexponent=2:3(0:1)isclosetotheratio7/3.Similarpowerlawbehaviorrevealsitselfinregularferromagneticthinlmswherehasasomewhatsmallervaluevaryingfrom0.3to1.5 76 .Becausedependsstronglyonstrain,roughness,impurity,andthenatureofthedomainwall(BlochorNeeltype) 76 ,itisnotsurprisingtoseeawidevariationin.NeelpredictedforexamplethatforBlochdomainwalls,Hcofaferromagneticthinlmshouldvaryasx)]TJ /F3 7.97 Tf 6.59 0 Td[(4=3whenthethicknessxofthelm 56 PAGE 57 iscomparabletothedomainwallthicknessw 79 .ForthecaseofNeelwalls,Hcdependsonlyontheroughnessofthelmanddoesnotdependonlmthickness 77 .ThevariationofHc(x)becomesparticularlypronouncedwhenthelmthicknessbecomescomparabletow. AqualitativeunderstandingofthesteeperHc(x)dependencebecomesevidentbyrecognizingthattheformationofdomainstructureisdrivenbythereductionoflongrangemagnetostaticenergywhichatequilibriumisbalancedbyshorterrangeexchangeandanisotropyenergycostsassociatedwiththespinorientationswithinaBlochorNeeldomainwall.Domainwallthicknessisgivenbyw=p A=K 3 82 whereKisthecrystallineanisotropyconstantandAistheexchangestiness,proportionaltotheexchangeenergy,J 85 .ThedomainwallsizewincreasesfordecreasingKandincreasingJ.IfK,whichdependsontherelativelyconstantspinorbitinteraction 4 withinthePdcomponentofthePd=Fe=Pdtrilayers,remainsconstant,thenvariationsinwaredominatedbyvariationsinJ.Thusasxdecreasestowardzero,theincreaseinJ 84 givesrisetoanincreaseinwwhichinturngivesrisetoamorerapidincreaseinHcthanwouldbeseeninregularferromagnetswithconstantJ.Asdiscussedabove,thisrapidvariationwith7=3isobservedexperimentally. ThedatainFig. 44 showthatTcincreasesasxincreasesandreachesarelativelyconstantvaluenearx=20A.ThedashedblacklineisaguidetotheeyeandisqualitativelysimilartothebehaviorofMsA(x)showninFig. 42 whichsaturatesatalargervaluenear30A.Theseobservationsareagainqualitativelyconsistentwiththenitesizeeectassociatedwithcriticalphenomenainferromagnets 81 { 83 .Althoughthedataarenotofsucientqualitytodistinguishthepowerlawbehaviorthatispredictedfornitesizeeects 81 { 83 ,weexpectthatthedependenceisfurthercomplicatedbythepreviouslydiscusseddependenceofJonxinpolarizedferromagneticPd.ThebehaviorofTc(x)suggeststhatPd=Fe=Pdtrilayershouldbetreatedasasinglelayerwithawelldenedspinspincorrelationlength.IfthePdlayersaretreatedseparately,thenthebottomlayer 57 PAGE 58 withxedthicknessy=2wouldhaveaTcequaltothehighestTcofthetoplayer.InthiscasetheoverallmeasurementwouldnotshowastrongchangeinTcasafunctionofx,sincetheTcofthebottomlayerwoulddominateforallx. Wenotethatforourplanargeometry,TcdecreaseswithdecreasingthicknessashasalsobeenshownforthinlmNi 81 andepitaxialthinlmstructuresbasedonNi,CoandFe 82 .OntheotherhandTcincreaseswithdecreasingsizeofferrimagneticMnFe2O4nanoscaleparticleswithdiametersintherange526nm 83 .ThisincreaseofTcwithdecreasingsizeisattributedtonitesizescalinginthreedimensionswhereallthreedimensionssimultaneouslycollapse 83 .Inourtwodimensionalplanarthinlmsonlyoneofthedimensions,thethickness,collapsesandTcdecreasesratherthanincreasesinaccordwiththeobservationsofpreviousstudies 81 82 4.5Conclusions Inconclusion,wehavecharacterizedthemagneticpropertiesofthinlmPd=Fe=Pdtrilayersanddeterminedthatcriticalsizeeectsapplyto\ferromagnetic"PdwheretheferromagnetismisinducedbyproximitytoanunderlyingultrathinFelm.Thecriticalsize,orequivalentlythecriticalthickness,iscontrolledbyvaryingthethicknessxofthetopPdlayer.ThedependencesonlmthicknessofthecoerciveeldHcandtheCurietemperatureTcareinqualitativeagreementwithnitesizeeectsseeninregularferromagneticlmswheretheexchangecouplingJisconstantthroughoutthelm.TheresultspresentedhereincreaseourunderstandingofnanomagnetisminultrathinsystemsbyshowingthatthespatialvariationsofJintheproximitycoupledPdhaveapronouncedinuenceontheformofthicknessinducedvariations,namely:anonlineardependenceofMsA(x),anunusuallystrongpowerlawdependenceofHc(x)andadependenceofTc(x)whichindicatesthatthetrilayeractsasasinglelayerthatnecessarilyincludestheconstantthicknessPdlayerservingasasubstratefortheFelayer. 58 PAGE 59 CHAPTER5TEMPERATUREDEPENDENCEOFCOERCIVITYINMULTIDOMAINNINANOPARTICLES,EVIDENCEOFSTRONGDOMAINWALLPINNING 5.1Abstract Thetemperaturedependenceofthecoercivityofthesingleand5layersamplesofNinanoparticlesinAl2O3matrixisstudied.AlinearT2=3dependenceofcoercivityoverawiderangeoftemperature(10Kto350K)isobserved.Allthesamplesconsistsofparticleswithmultiplemagneticdomainsasthesizeoftheparticlesarelargerthanthecriticalsingledomainsize(seeEq. 1{3 onpage 13 ofchapter 1 andFig. 33 onpage 47 ofchapter 3 ).Theexperimentalresultsareunderstoodintermsofstrongdomainwallpinning. 5.2Introduction Thetemperaturedependenceoftheextrinsicmagneticproperties,forexamplecoerciveeld(Hc(T)),arisefromtwomechanisms.Therstmechanismis,duetothetemperaturedependenceoftheintrinsicmagneticproperties 11 15 86 87 suchassaturationmagnetization(Ms),magneticanisotropy(K)andexchangestiness(A)andwillbediscussedinchapter 6 .Thesecondmechanismis,duetothethermallyactivatedhopingofthemetastablestatesoversomeenergybarrier. 2 { 4 6 Fromthemagnetizationloopsatdierenttemperatures(Fig. 52 )wehavefoundthatMsdoesnotchangewithtemperature.Allthesamplescomprisepolycrystallineparticles 36 andthusmagnetocrystallineanisotropycanbeneglectedandtemperatureindependentshapeanisotropyisdominant. 48 Theexperimentaltemperaturerangeis10Kto300KwhichismuchsmallerthanthecurietemperatureofNi(630K) 3 andAcanbeconsideredconstantoverthistemperaturerange. 88 Inthischapterwewilldiscussthesecondmechanismastheoriginofthetemperaturedependenceofthecoerciveeld(Ms,KandAaretemperatureindependent).TounderstandthetemperaturedependenceoftheHcduetothethermallyactivatedhoppingovermetastableenergyminimaseparatedbysomeenergybarrier,itisnecessarytondoutthemagneticelddependenceoftheenergybarrier.Acommonly 59 PAGE 60 usedphenomenologicalenergybarrieris 2 3 47 89 E=E0[1)]TJ /F6 11.955 Tf 11.96 0 Td[(H=Hc0]m(5{1) whereE0istheenergybarrieratzeromagneticeldandenergybarriervanishesatH=Hc0atT=0.AtH=Hc,thermalenergy,kBT,issucientlyhightocausemostofthemomentstobethermallyactivatedoverthebarrier.ForexampleforthecaseofStonerWohlfarthparticlesm=2,E0=KVandHc0=2K=Ms.ForStonerWohlfarthparticlesthescenarioisverysimpleandtheEq. 5{1 canbederivedanalytically(seeEq. 1{6 onpage 15 ofchapter 1 ).RememberthatifE(H)isknown,itispossibletocalculateHc(T).InthischapterwewilldiscusshowtoderiveE(H)(Eq. 5{1 )fortheMDnanoparticlesandwillcompareHc(T)withthemodel. 5.3ResultsandDiscussions Thesamplepreparationtechniqueisdiscussedinchapter2andchapter3.Threedierentsetsofsamplesareinvestigated.Set1consistsofsinglelayerNiparticlesinanAl2O3matrix.Set2andSet3consistsof5layersofNiparticlesseparatedbyAl2O3layers.TheinterlayerseparationinSet2andSet3are3nmand40nmrespectively.TheschematicofallthreesetsofsamplesareshowninFig. 51 below. InthischapterwewillfocusonthetemperaturedependenceofHcfortheMDNinanoparticles.Atotalof15samplesarestudied,5samplesfromeachset.Magnetizationloopsaremeasuredforeverysampleforseven(onaverage)dierenttemperatures.Thismeansatotalofaround105magnetizationloopshavebeenmeasuredforthepresentstudy.Magnetizationloopsforthesampleofaverageparticlediameterof12nmofset2atdierenttemperatures(indicatedinthelegends)areshowninFig. 52 .ThearrowshowsHcat10K.NotethatHcdecreaseswithincreasingtemperature.ThetemperaturedependenceofHcnormalizedtoHcOforvedierentsamplesbelongingtoset1isshowninFig. 53 .Theparticlediametersareindicatedinthelegends.NoteT2=3inxaxis.AllthedatafollowalinearT2=3dependence.Tounderstandtheabovedata,wewillstart 60 PAGE 61 Figure51. Schematicofthreesetsofsamples.Set1comprisesasinglelayerofNiparticlesembeddedinanAl2O3matrix.Set2andSet3comprisesof5layersofNiparticlesseparatedbydierentdistancesinanAl2O3matrix.TheinterlayerdistancesinSet2andSet3are3nmand40nmrespectively. withageneralmagneticenergylandscapeofthesystemwrittenasapolynomialexpansionofthedomainwallposition(x)aroundastrongpinningcenter. 2 { 4 6 E(x)=a0+a1x+a2 2x2+a3 3x3)]TJ /F6 11.955 Tf 11.95 0 Td[(b0Hx(5{2) wherea0,a1,a2,a3andb0aremicromagneticparametersthatdependonthemagneticparametersK,MsandA.Forthestrongpinningcenterthex3termisincludedastheeectofthepinningcenterislongdistancecomparedtotheweakpinningcenterwherethex3termisneglected. 3 Therelationbetweenmicromagneticandmagneticparametercanbedeterminedfromtheparticularmodelused.Notethatthemicromagneticparametersaretemperatureindependentinourcaseastheyonlydependonthetemperatureindependentmagneticparameters.First,wewillderivetheenergybarrierseparatingthemetastableminimafromtheglobalminima.ThemaximaorminimaofE(x)aredeterminedbysettingtherstorderderivativetozero. E x=a1+a2x+a3x2)]TJ /F6 11.955 Tf 11.95 0 Td[(b0H=0(5{3) 61 PAGE 62 Figure52. Magnetizationloopsforthesampleofaverageparticlediameterof12nmofset2atdierenttemperatures(indicatedinthelegends).Thecoerciveeld(Hc)at10Kisindicatedbythearrow.Hcdecreaseswithincreasingtemperature.Saturationmagnetization(Ms)isconstantatdierenttemperatures. Thetwosolutionsfortheaboveequationsare x1=)]TJ /F6 11.955 Tf 9.3 0 Td[(a2+p a22)]TJ /F1 11.955 Tf 11.96 0 Td[(4a3(a1)]TJ /F6 11.955 Tf 11.96 0 Td[(boH) 2a3(5{4) x2=)]TJ /F6 11.955 Tf 9.3 0 Td[(a2)]TJ /F8 11.955 Tf 11.96 10.37 Td[(p a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a3(a1)]TJ /F6 11.955 Tf 11.96 0 Td[(boH) 2a3(5{5) TakingthesecondderivativeofE(x)withrespecttoxitiseasytoshowthat2E=x2jx1>0(2E=x2jx2<0)andcorrespondstothemaximum(minimum).Thustheenergybarrieris E(H)=E(x1))]TJ /F6 11.955 Tf 11.95 0 Td[(E(x2)=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3+4a3b0H)3=2 6a23(5{6) 62 PAGE 63 Figure53. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset1.Thelinearbehaviorisobservedforsampleswithparticlesizefrom18nmto42nmindiameter. Forthereverseeld,ieH=)]TJ /F6 11.955 Tf 9.3 0 Td[(Htheaboveequationreducesto E(H)=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a3(a1+b0H))3=2 6a23=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23 1)]TJ /F6 11.955 Tf 27.42 8.09 Td[(H a22)]TJ /F3 7.97 Tf 6.59 0 Td[(4a1a3 4a3b0!3=2 (5{7) whichisinthesameformofEq. 5{1 ,where E0=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23(5{8) Hc0=a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3 4a3b0(5{9) FromEq. 5{7 itisclearthatE(H)decreaseswithincreasingHandwhenH=Hctheenergybarriercanbeovercomebythermalenergy(denitionofthecoerciveeld).Thus 63 PAGE 64 atH=Hc,theEq. 5{7 canberewrittenas kBT=(a22)]TJ /F1 11.955 Tf 11.95 0 Td[(4a1a3)3=2 6a23 1)]TJ /F6 11.955 Tf 25.8 8.09 Td[(Hc a22)]TJ /F3 7.97 Tf 6.59 0 Td[(4a1a3 4a3b0!3=2(5{10) TheaboveequationcanbesolvedforHc Hc=Hc0"1)]TJ /F8 11.955 Tf 11.95 16.86 Td[(kBT E02=3#(5{11) whereHc0andE0aregivenbyEq. 5{8 and 5{9 .ThistemperaturedependenceofHcisconsistentwiththeexperimentalresultsshowninFig. 53 54 and 55 Figure54. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset2.Thelinearbehaviorisobservedforsampleswithparticlesizefrom12nmto60nmindiameter. 64 PAGE 65 Figure55. Coerciveeld(Hc)vs.T2=3forvedierentsamplesofset3.Thelinearbehaviorisobservedforsampleswithparticlesizefrom16nmto44nmindiameter. 5.4RelationBetweenMicromagneticParameterandMagneticParameters Herewewilloutlinearoadmaptorelatethemicromagneticparametersa0,a1,a2,a3andb0tothemagneticparametersK,MsandA.Todothatwewillstartwiththemagneticenergyexpression, 3 E(x)=Z"ArM(x) Ms2)]TJ /F6 11.955 Tf 11.96 0 Td[(K(x)(k:M(x))2 M2s)]TJ /F6 11.955 Tf 11.96 0 Td[(0M(x):H)]TJ /F6 11.955 Tf 13.15 8.09 Td[(0 2M(x):Hd(M)#dV(5{12) wherethersttermcorrespondstotheexchangeenergycostduetothespinmisalignment,thesecondtermistheanisotropyenergy,thethirdtermistheZeemanenergyandthefourthtermisthemagnetostaticselfenergy.Thepositionofthedomainwallisgivenbyxandkistheunitvectoralongtheeasyaxis.Theaboveequationshouldbesolvedforreal 65 PAGE 66 sampleswhiletakingintoaccountrealstructureandimperfections.Therealstructureandimperfectionsareresponsibleforthexdependenceofthemagneticparameters(Ms(x),A(x),K(x)).AftersolvingEq. 5{12 andbycomparingthecoecientsofthedierentpowerofx,itispossibletondoutthemicromagneticparametersintermsofmagneticparameters.ThebehaviorsofHc0andE0=kBareshowninthegurebelow. Figure56. ThebehaviorsofHc0andE0onparticlediameterareshownforset2samples.Hc0decreasesandE0=kBincreaseswithincreasingparticlesize. TheincreasingbehaviorofE0anddecreasingbehaviorofHc0areconsistentwiththeliterature. 3 Theactualbehaviorcanbeverycomplicatedasitdependsontherealstructurefactorsandimperfectionsinthematerial. 3 5.5Conclusions WehaveinvestigatedthetemperaturedependenceofthecoerciveeldofMDNinanoparticlesinAl2O3matrix.HcdecreaseslinearlywiththeT2=3.Thisbehaviorcanbeunderstoodaccordingtothestrongdomainwallpinning.Weshowthatthegeneralenergy 66 PAGE 67 barrierthatarisesduetostrongdomainwallpinningdependsonthemagneticeldwithapowerof3/2andisresponsibleforthetemperaturedependenceoftheHc. 67 PAGE 68 CHAPTER6COERCIVEFIELDOFFETHINFILMSASTHEFUNCTIONOFTEMPERATUREANDFILMTHICKNESS:EVIDENCEOFNEELDISPERSEFIELDTHEORYOFMAGNETICDOMAINS 6.1Abstract ThetemperaturedependenceofthecoerciveeldofFethinlmshasbeeninvestigated.Threedierentsamplesofdierentthicknessarestudied.Thecoerciveelddecreaseswithtemperatureandfollowsthesametemperaturedependenceastherstorderanisotropyconstant.ThisbehaviorisconsistentwiththetheoreticalpredictionmadebyNeel 15 basedonthedisperseeldtheoryofmagneticdomainwhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions.Thevalueofcoerciveeldincreaseswithdecreasinglmthickness.Thisbehaviorisexpectedformultidomainferromagneticsystemsatnanoscalewherethedomainwallthicknessiscomparabletoorlargerthanthelmthickness. 6.2Introduction Themostinterestingaspectofferromagnetismisthehysteresisloop, 90 whichreferstothehistorydependentbehaviorofmagnetizationwithappliedmagneticeld(Fig. 62 ).Hysteresisisacomplexnonlinear,nonequilibriumandnonlocalphenomenon,reectingtheexistenceofanisotropyrelatedmetastableenergyminimaseparatedbyelddependentenergybarriers. 3 Anextrinsicpropertyofcrucialimportanceinpermanentmagnetismisthecoerciveeld,themagneticeldwheremagnetizationchangessignasitpassesthroughzero.Thecoerciveeldbasicallydescribesthestabilityoftheremnantstateandisaveryimportantconceptformostpracticalapplications. 91 { 99 Coercivityinferromagnetsisknownfromverylongtime. 90 But,duetothecomplexnature,theoriginofcoerciveeldisstillasubjectofstudy.Inthispresentworkthebehaviorofcoerciveeldofthreedierentironthinlmswithdierentthicknesseshasbeeninvestigated.Thetemperaturedependenceofthecoerciveeldagreeswellwiththetheoryofdomainwallpinningarising 68 PAGE 69 fromsmallinclusions(forexampleimpurityorvacancydefects)wheretheenergyofthefreepoleisnotnegligible. 15 6.3ExperimentalDetails Figure61. TEMimageofFethinlmofthickness9nm. ThinlmsofFewerefabricatedonSi(100)andsapphire(caxis)substratesusingpulsedlaserdepositionfromaluminaandirontargets.HighpuritytargetsofFe(99.99%)andAl2O3(99.99%)werealternatelyablatedfordeposition.Beforedeposition,thesubstrateswereultrasonicallydegreasedandcleanedinacetoneandmethanoleachfor10minandthenetchedina49%hydrouoricacid(HF)solutiontoremovethesurfacesilicondioxidelayer(fortheSisubstratesonly),thusforminghydrogenterminatedsurfaces. 35 Thebasepressureforallthedepositionswasoftheorderof10)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Aftersubstrateheating,thepressureincreasedtothe10)]TJ /F3 7.97 Tf 6.59 0 Td[(6Torrrange.Thesubstratetemperaturewaskeptatabout550oCduringgrowthoftheAl2O3andFelayers.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.58 0 Td[(2overaspotsize4mm1.5mm.A40nmthickbuerlayerofAl2O3wasdepositedinitiallyontheSiorsapphiresubstratebeforethesequentialgrowthofFeandAl2O3.Thisprocedureresultsinavery 69 PAGE 70 smoothsubstrateindependentstartingsurfaceforthegrowthofFe,asveriedbyhighresolutionscanningtransmissionelectronmicroscopystudies(Fig. 21 ).A40nmthickcaplayerofAl2O3wasdepositedtoprotecttheFelayerfromoxidation.Threedierentsampleswiththicknessof9nm,21nmand30nmwerepreparedforthepresentstudy.MagnetizationmeasurementsareperformedinaquantumdesignMagneticPropertyMeasurementSystem(MPMS).Themagneticeldwasalongtheplaneofthelms.Thedependenceoftemperatureandlmthicknessoncoerciveeldisinvestigated. 6.4ResultsandDiscussion Figure62. MHloopofFelmofthickness21nmatdierenttemperaturesasindicatedinthelegend.ThecoerciveelddenedatM=0,decreaseswithincreasingtemperature. ShowninFig. 61 istheTEMimageofthe9nmthickFesample.DuetothelowsurfaceenergydierencebetweenFeandAl2O3(650mJ/m2)andhighadhesionenergy(1205mJ/m2)betweenFeandAl2O3, 100 FewetsthesurfaceofAl2O3andthusacontinuousthinlmisgrowninsteadofgrainsasisthecasefortheNi/Al2O3systemdiscussedinchapter 2 and 3 .Threedierentsampleswiththickness9nm,21nmand30nmweregrown.Themagnetichysteresisloopsatdierenttemperaturesfrom10Kto350Karemeasuredwiththeappliedmagneticeldalongtheplaneofthethinlm.MHloopsfor21nmsampleareshowninFig. 62 .Themagneticeldissweptfrom5 70 PAGE 71 Tto5Tandagainfrom5Tto5Ttocompletetheloop.AthighmagneticeldsthemagnetizationfromtheFesaturatesandthemagneticelddependencearisesduetothediamagneticcontributionfromthesubstrate.ThediamagneticcontributionfromtheFelmisnegligibleasthethicknessofthelmisverysmallcomparedtothethicknessofthesubstrate.Thusthehighmagneticelddataislinearinmagneticeldwithanegativeslope.TheslopeofthelineisdeterminedfromthelineartofthehighelddataandsubtractedfromtherawdatatoextracttheferromagneticsignalduetotheFethinlm.Theprocedureisrepeatedforallsamplesandforalltemperatures.Thecoerciveeldisfoundtodecreasewithincreasingtemperatures.TheresultsforallthreesamplesarepresentedinFig. 63 .Figure 64 showstherelationbetweenK(T) 87 andHc(T)whereTistheimplicitvariable. 87 ThelinearbehaviorofHcandKsuggestthatthecoercivitymechanismissimilartothedisperseeldtheoryofmagneticdomainwhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions. 15 Figure63. Hcvs.TofFelmsofthickness9nm,21nmand30nm.Thetemperaturedependenceofallthreesamplesissimilar,whichsuggeststhattheoriginofthecoerciveelddependsontheintrinsicpropertyoftheiron,whichinourcaseisthemagnetocrystallineanisotropy.Thisistrueasforthecaseofextrinsicorigintheenergybarrierdependsonthesizeofthesampleasdiscussedinchapter 5 71 PAGE 72 Figure64. Hcvs.KofFelmofthickness9nm.ThelinearbehaviorsuggeststhevalidationofNeel'sdisperseeldtheoryofmagneticdomainswhichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatsmallinclusions(Eq. 6{3 ).Thesimilarbehaviorisobservedforothertwosampleswhichisexpectedastheyhavesimilartemperaturedependence. Theoriesconcernedwiththecoercivityofthemultidomainferromagneticmaterialsbeginwiththeconsiderationofthechangeinmagneticenergyacrossadomainwall. 2 SincethemagnetizationchangesfromMsto)]TJ /F6 11.955 Tf 9.3 0 Td[(Msacrossa180odomainwall,theeectivepressureonthewallwillbethedierenceintheenergyacrossthewallperunitvolume.AtanappliedmagneticeldH,thepressurewillbe2HMs.Thustheworkdonetomoveadomainwallofareastoadistancedxwillbe 2 dW=2HMssdx(6{1) Theaboveequationcanbegeneralizedforanyangle(notonly180o)domainwall. 2 ThetermdW=dx,whichisbasicallytheenergyrequiredtomoveadomainwalltounitdistance,isthemeasureoftheresistanceofdomainwallmotion. 2 Inrealsamplesduetoimpurities,imperfectionsorirregularitiesincrystal,dW=dxpassesthroughmaximumandminimum.Theoriginofreversibilityinthemagnetizationloopistheirreversiblemotionofdomainsthroughthesemaximumandminimum. 2 Thecoerciveeldisusuallycalculatedusingtheequationbelow. 2 72 PAGE 73 Hc=1 2Mss(dW=dx)max(6{2) Dierenttheoreticalapproachesconcentrateonthederivationof(dW=dx)max.SubstantialwallmotionmayrequireeldsoftensorhundredsofOersted.Evidentlyrealmaterialscontaincrystalimperfectionsofonesortoranotherwhichhindertheeasymotionofdomainwalls.Thesehindrancesaregenerallyoftwokinds:inclusionsandresidualmicrostress. 2 Fromamagneticpointofview,an"inclusion"inadomainisaregionwhichhasadierentspontaneousmagnetizationfromthesurroundingmaterial,ornomagnetizationatall.AccordingtotheresultobtainbyNeelinhisdisperseeldtheory(whichbasicallycalculatestheterm(dW=dx)max),coercivityisgivenbyEq. 6{3 below. 15 Hc=1 4v122 KMs+Kv2 Ms(6{3) Thersttermisduetotheresidualmicrostressandv1isthevolumefractionofthefreepoleattheresidualmicrostressand;;KandMsarethesaturationmagnetostriction,internalstress,magnetocrystallineanisotropyconstantandsaturationmagnetizationrespectively.ThesecondterminEq. 6{4 originatesduetothepolesattheinclusionsofvolumefractionv2.Foriron,Ms=1743emu/cm3, 2 K=8x106ergs/cm3, 101 =10x10)]TJ /F3 7.97 Tf 6.58 0 Td[(6 102 and=1010dynes/cm2. 103 PuttingthesevaluesintheEq. 6{3 weobtain, Hc'0:18v1+4600v2Oe(6{4) Thusinourcasethecoerciveeldwillbedominatedbythe2ndterminEq. 6{3 ,whichisproportionaltoKandagreeswellwiththeexperimentalresultsasshowninFig. 63 andFig. 64 .RememberthatMsisconstant. InFig. 65 thevariationofHc(d)isshown.TheincreaseinHcwithdecreasinglmthickness,d,couldbeduetotheenhancementofKwithdecreasingd. 101 104 105 Theincreaseincoerciveeldwithdecreasingthicknessiswellknownforthemultidomain 73 PAGE 74 Figure65. Hcvs.dofFelmsat10K.Coerciveeldincreaseswithdecreasingd.ThisbehaviorisduetofactthatKincreaseswithdecreasingdwhichisverycommoninmultidomainthinlmswhenthedomainwallthicknessiscomparableorlargerthanthelmthickness.Thedomainwallthicknessofironisabout60nm, 3 whichislargerthanthethicknessofthelmsstudiedhere. magneticthinlmswhenthedomainwallthicknessiscomparabletoorhigherthanthelmthickness. 76 Thedomainwallthicknessofironisabout60nm, 3 whichislargerthanthethicknessofthelmstudiedhere.NotethatwehaveusedthevalueofKinEq. 6{3 ,whichislargerthanthebulkvalue.FromEq. 6{4 ,wendthatv20:12. 6.5Conclusion ThecoerciveeldofmultidomainFethinlmshasbeeninvestigated.ThetemperatureandthicknessdependenceofthecoerciveeldagreeswellwiththepredictionmadebyNeel,whichtakesintoaccounttheeectoffreepolesonthecoerciveeldthatoccursatthesmallinclusions.ThecontributionofstraintothecoercivityisverysmallforthecaseofFeandthedominantcontributioncomesfromthefreepolesattheinclusions.Thevolumefractionofinclusionshasbeenestimatedfromthecoerciveelddatatobearound0.12. 74 PAGE 75 CHAPTER7SCALINGCOLLAPSEOFTHEIRREVERSIBLEMAGNETIZATIONOFFERROMAGNETICTHINFILMS 7.1Abstract Inferromagneticmaterials,hysteresis,orequivalentlythehistorydependentbehaviorofthemagnetization,reectscomplexnonlinearandnonequilibriumphenomenologythathasbeenrecognizedformanyyears 5 6 106 { 108 .Hysteresisdependsstronglyonmaterialspropertiessuchasstructurallengthscalesspanningthenanometertomicrometerrange 3 andmanifestscomplexbehaviorsincludingmagneticrelaxationwithagingdependence 109 ,dimensionallydependentdipoledipoleinteractions 48 110 ,spinglasslikememoryeects 111 andsupermagnetism 112 .Hereweshowthattheirreversiblemagnetization,denedasthedierencebetweeneldcooledandzeroeldcooledmagnetization,hasastrikingsimilarityforawidevarietyofferromagneticmaterials.Thissimilaritybecomesapparentwhentheirreversiblemagnetizationisnormalizedtoitsmaximumvalueandplottedwithrespecttoatemperaturedependentrenormalizedeld.Thecollapseoftheirreversiblemagnetizationontoasinglecurveforagivensystemimpliesanunderlyingsymmetrytohysteresisthatisnotcapturedbypreviousanalytical 3 5 6 48 106 107 109 { 112 andcomputationaltreatments 108 113 andthusprovidesaunifyingthemethatembracesabroadrangeofcomplexhystereticbehavior. 7.2Introduction Ingeneral,hysteresisisacomplexnonlinearnonequilibriumphenomenonwhichreectsthepresenceofelddependentenergybarriersbetweenanisotropydependentmetastableminima.Accordingly,hysteresisisaectedbyacombinationofintrinsicpropertiessuchasmagnetocrystallineanisotropywhichdependsoncrystaleldenergyandspinorbitcouplingandextrinsicpropertiessuchassampleshape,grainboundaries,disorderandimperfections.Forexample,inbulkferromagnetshysteresisisoftendescribedasasuperimpositionofdomainwallmotionanddomainrotationwithenergybarriersrelatedtomagnetocrystallineanisotropytogetherwithimperfections 75 PAGE 76 and/orimpuritiesinthematerial 107 .Forthesimplestcaseofsingledomaincoherentlyrotatingnanoparticles,theoriginofhysteresisisthenitetimescaleformagnetizationreversalasthemagnetizationovercomesamagneticelddependentenergybarrierbythermalactivationandrotatesfromoneeasyaxisdirectiontoanother 5 6 106 .Hysteresisandassociatedmagnetizationreversalsplayanimportantroleinapplicationssuchasmagneticdatastoragedevices 91 { 93 ,GMR 94 95 orMRAM 96 devices,magneticsensors 97 andmotors 98 ,generators 99 etc. 7.3ExperimentalResults Measurementsofirreversiblemagnetizationareusuallyaccomplishedbyoneorbothoftwotechniquesillustratedrespectivelyinpanelsa)andb)ofFig. 71 .Thesampleunderinvestigationinthisgure(hereafterreferredtoasampleA)isa20Athickpulsedlaserdepositedthinlmcomprisingvelayersof3nmelongatedNiparticlesembeddedinaninsulatingAl2O3host[seeMethods].TheNiparticlesaresmallenoughtobeinthesingledomain(SD)regimewhereallthespinsarealignedinthesamedirectionandrotatecoherentlytogetherinresponsetoachangingmagneticeld.Inthersttechnique(panela),thesampleiseldcooled(FC)inaeldH=20Oe(blacksquares)from300Kto5Kandthenzeroeldcooled(ZFC).Theirreversiblemagnetization(M(H;T)),whichisafunctionofmagneticeld(H)andtemperature(T),isshownbythetwoheadeddashedverticalarrow.Inthesecondtechnique,thesampleisheldatxedtemperatureTandmagnetizationMloopsobtainedbyrepeatedlycyclingtheappliedeldHaboutH=0betweentwosymmetriclimits.Thehistorydependenttrajectoriesformclosedloopsshownschematicallyintheinsetsofpanel(b).Theseloops,whichcanbeacquiredatdierenttemperatures,areeachcharacterizedbyacoerciveeldHc(T),asaturatedmomentMs(T)andaremnantmagnetizationMr(T).ThecoerciveeldHc(T),plottedversusT1=2inpanelbforsampleAdiscussedabove,isshownasabluelineconnectingthestarreddatapoints.TheabsenceofaT1=2dependenceforcoherentlyrotatingSDparticleswitheasyaxesorientedalongtheeld 6 willbediscussedbelow.WhilebothM(H;T)andHc(T) 76 PAGE 77 Figure71. IrreversiblemagnetizationM(H;T)denedasthedierenceoftheFCandZFCmagnetizationsisaquantitativemeasureofhysteresis.a,BlacksquaresandredcirclesrepresentrespectivelytheFCandZFCtemperaturedependentmagnetizationsforsampleAina20Oeeldappliedparalleltothelmsurface.TheirreversiblemagnetizationM(H;T)isshownbythetwoheadedverticaldottedarrow.Inset:Largethermallyblockedmagneticparticles(1and3)withrespectiveverticalandhorizontaleasyaxesofmagnetizationindicatedbythearrows.Particle2issmallenoughtobesuperparamagnetic(thermallyunblocked).Formagneticeldsappliedalongtheverticaldirection,particles2and3donotcontributetoM(H;T).b,Thecoerciveeld(Hc)forthesamesampleshownasafunctionofT1=2(Bluestars)doesnotshowthelinearbehaviorexpectedforidealStonerWohlfarthparticles 6 wherealltheparticlesareuniformsizeandalignedwiththeappliedeld(particle1).Thesolidbluelineisattothedatausingalognormaldistributionofparticlesizetogetherwitharandomorientationoftheeasyaxisofthemagnetizations.Inset:Schematicmagnetizationcurvesforthecaseswhere(b1)onlyparticles1and2and(b1)onlyparticles1and3arepresent.Theresultantmagnetizationloops(blackcurves)forthetwocasesshowthepronouncedeectsofparticlesizeandeasyaxisorientationonthedeterminationofHc. arecommonlyacceptedmeasuresofhysteresis,theunderlyingphenomenologyforeachisconsiderablydierent.ForexampleinnanoparticlemagneticsystemssuchasshowninFig. 71 ,M(H;T)andHc(T)aresensitiveindierentwaystoboththepresenceofsuperparamagneticparticlesandtherandomorientationoftheeasyaxisofmagnetizationofeachparticle.TheinsetsofFig. 71 a)andb)schematicallyillustratethesedierencesusingthreeparticles:particles1(red)and3(blue)witheasyaxesofmagnetization 77 PAGE 78 respectivelyalongthevertical(H)andperpendiculardirectionsandlargeenoughsothatatthetemperatureofmeasurement,changesinmagnetizationareblockedbyenergybarriersthatcannotbeovercomebythermalactivation,andparticle2(green)witheasyaxisalongtheverticalaxisandsmallenoughsothatitissuperparamagneticwithanonhystereticmagnetizationdependingonlyonHandT(i.e.,unblocked).Considerthecasewhereonlyparticles1and2arepresent.Sincethesuperparamagneticparticle2willhavezeroM(H;T),thetotalM(H;T)valuewillbeonlyduetotheblockedparticle1.Ontheotherhand,theMHloopwillbethesummedcontributionsofthehystereticloopforparticle1andthereversibleloopforparticle2asshownininsetb1ofFig. 71 b.ThusforthiscaseM(H;T)isnotaectedbysuperparamagneticparticlesbutHc(T)is.Considernowthecasewhereonlyparticles1and3arepresent,i.e.,bothparticlesareblockedbutwithdierent(parallelandperpendicular)easyaxisorientationswithrespecttotheappliedeld.SimpletheoryforcoherentlyrotatingSDparticlesshowsthatuponreversingtheeldparticle1mustsurmountaHdependentenergybarrierwhereasparticle3canlineupwithouthavingtoovercomeanenergybarrier 3 .Inlikemannertothepreviouscase,TheMHloopforparticle1showshysteresisbutparticle3doesnot(insetb2ofFig. 71 b).HenceHc(T)isaectedbythepresenceofparticleswithperpendicularorientationbutM(H;T)isnot. TheaboveargumentscanbegeneralizedforSDmagneticnanoparticleswithabroadsizedistributionandarandomorientationoftheeasyaxesofmagnetization.Particleswithaneasyaxismakinganarbitraryanglewiththeappliedmagneticeldwillalwayscontributelesstohysteresisthanalignedparticles.Becausetheperpendicularcomponentdoesnotcontribute,thecorrectionisastraightforwardintegrationoverangle 7 .Accordingly.themeasuredvalueofM(H;T)willbeonlyduetoblockedparticlesanddominatedbyparticleswitheasyaxisofthemagnetizationalongtheappliedmagneticeld.Ontheotherhand,Hc(T)willbestronglyaectedbytherandomorientationoftheeasyaxisofblockedparticlesandthepresenceoftheparticleswhicharesmallenoughto 78 PAGE 79 besuperparamagnetic.ForanidealStonerWohlfarthparticlesystem 6 inwhichalltheparticlesarealignedalongtheHdirection,alinearsquareroottemperaturedependenceofHc(T)isexpected.FortheconsiderablymorecomplicatedcaseofrandomlyorientedSDparticleswithasizedistributioninwhichsomeoftheparticlesaresuperparamagnetic(e.g.,sampleA),Hc(T)canbedescribed(solidlineofFig. 71 b)usingamodelwithalognormalsizedistributionandthreettingparameters 9 10 (seeMethods).TheHdependentbehaviorofM(H;T)forsampleAisshowninFig. 72 aforthetemperaturesindicatedinthelegend.Theisothermsshowpeaks,Mmax(T),atmagneticelds,Hm(T),indicatedbytheverticalarrows.Thesepeaksareexpected,sinceatH=0theFCandZFCmeasurementsareequivalentandthedierenceinmagnetizationshouldbezero,whereasathighHbothFCandZFCmagnetizationssaturatetothesamevalueandagainthedierenceshouldgotozero.ThesimilaritiesintheM(H;T)isothermsaresuggestiveandbecomemanifestasanunexpecteddatacollapseontoasinglecurvewhenthereducedirreversiblemagnetization,M(H;T)=Mmax(T),isplottedasafunctionofreducedmagneticeld,h(T)=H=Hm(T),asdoneinFig. 72 b.ThecharacteristiceldHm(T)deviatessignicantlybelowHc(T)forT<25K(insetofFig. 72 b)andidentiestheTdependenteldwhereirreversibilityisatamaximum. AnimportantphysicalinsightintothescalingcollapseshowninFig. 72 bisgainedbyplottingMmax(T)asafunctionofT1=2.TheobservedlinearbehaviorshowninFig. 73 aisidenticaltothepredictedtemperaturedependentcoercivityHSWc(T)ofStonerWohlfarth(SW)particles 6 mentionedabove.Guidedbythissimilarity,wemaketheansatz:Mmax(T)/HSWc.Thisansatzisphysicallyreasonablesinceasshownabove,M(H;T)measurementsarenoteectedbythepresenceofsuperparamagneticparticles,andinadditionthecontributionfromparticleswitheasyaxisofmagnetizationalongthemagneticeldisdominant.Asallrealsamplescompriseparticlesizedistributionsandrandomeasyaxisorientations,theconventionalMHloopsfromwhichcoercivitiesareextractedaremarkedlydierentthanwouldoccurforidealized 79 PAGE 80 Figure72. AllofthedataforM(T;H)ofsampleAcan,withpropernormalization,bemadetocollapseontoasinglecurve.a,IsothermsofMplottedasafunctionofHshowselfsimilarbehaviorwithmaximaMmax(T)occurringatcharacteristiceldsHm(T)markedbytheverticalarrows.WithdecreasingtemperatureMmax(T)increasesandHm(T)movestohigherelds.b,ThescalingcollapseoccurswhenM(H;T)=Mmax(T)isplottedagainstthenormalizedeldH=Hm(T).Inset:PlotsofHm(squares)andHc(T)(stars)asafunctionofT.ThedataandtforHcarethesameasshowninFig.1b,butplottedwithrespecttoTratherthanT1=2.ThesolidlineforHm(T)isaguidetotheeye. SWbehavior.TheinsensitivityofM(H;T)measurementstosuperparamagneticparticlesandperpendicularorientationsofblockedparticlestogetherwithouransatzimplythatmoreusefulinformationaboutthemagnetizationreversalprocessisobtainedfromM(H;T)measurementsthanHc(T)measurements. 80 PAGE 81 Figure73. ForsingledomainparticlestheansatzMmax(T)/HSWcisveried.a,ThelineardependenceofMmax(T)onT1=2isinaccordancewiththeStonerWohlfarththeoreticalpredictionthatHc(T)/T1=2forthecoherentlyrotatingsingledomainparticlesofsampleA.TheuseofMmax(T)ratherthanHc(T)asameasureofhysteresisremovestheeectsofnonuniformparticlesizeandrandomorientation.b,ThevaluesofHcalc(T),computedforuniformsizeFePtparticles(sampleB)fromtheansatzderivedEq.2attheindicatedtemperatures,scalelinearlywithHSWc(T)whichisdeterminedfromthecoerciveeldsofM)]TJ /F6 11.955 Tf 11.95 0 Td[(Hloopsaftercorrectionfortherandomorientationoftheeasyaxisofmagnetization(seeMethods). ThescalingcollapsebehaviorplottedforsampleAinFig. 72 bsuggeststhatMhastheform, M(H;T)=Mmax(T)F(H=Hm(T));(7{1) whereFisanunknownfunctionwiththepropertyF(1)=1.TakingthesecondpartialderivativeofthebothsidesofthisequationandsolvingforMmax(T)givestheresult:Mmax(T)=(@2F(h)=@h2))]TJ /F3 7.97 Tf 6.59 0 Td[(1h=1H2m(T)@2M(H;T)=@H2Hm(T),evaluatedatthe 81 PAGE 82 maximumwhereh(T)=H=Hm(T)=1.Therstterm,(@2F(h)=@h2))]TJ /F3 7.97 Tf 6.58 0 Td[(1h=1,mustbeaconstantbecauseofthescalingcollapse.Theremainingtwotermshavethesameformastheexpectedvalueofaneectiveanisotropyeld 114 .Thusaccordingtoouransatz,Mmax(T)/HSWc,wecanwritetherelation HSWc(T)=CH2m(T)@2M(H;T)=@H2Hm(T)=CHcalc(T);(7{2) whereCisaconstantandHcalc(T)=H2m(T)@2M(H;T)=@H2Hm(T)iscomputedfromexperimentaldata. TotesttheresultexpressedinEq.2,weusemagnetizationdataonasystemofuniformsize(6nmdiameter)FePtnanoparticles(sampleB,seeMethods)synthesizedviathermaldecompositionofFe(CO)5andreductionofPt(acac)2 115 .ThemeasuredcoerciveeldHMHc(T)isobtainedfromMHloopsatdierenttemperatures.Assumingthattheparticleseasyaxesofmagnetizationarerandomlyoriented,atemperaturedependentcorrectiontoHMHc(T)mustbemadesothatHSWc(T)canbeinferred.Thiscorrectionisneeded(seeMethods)sincethemagnetizationofrandomlyorienteduniformsizeparticlesshowsaT3=4dependence 7 comparedtotheT1=2dependenceexpectedforanidealizedsampleinwhichalltheparticlesarealignedalongtheappliedmagneticeld.TheplotofFig. 73 includesthiscorrectionandshowsalineardependenceofHcalc(T)onHMHc(T)aswouldbeexpectedforasystemofcoherentlyrotatingSDuniformsizeFePtnanoparticles.Thelinearbehaviorwithslopeof1.09(4)conrmsthevalidityofourmodelasexpressedbyEq.2withC1.InFig. 74 thereducedirreversiblemagnetization,M(H;T)=Mmax(T),isplottedasafunctionofreducedmagneticeld,h(T)=H=Hm(T)forawidevarietyofthinlmmagneticmaterialslabeledintheinsetanddescribedinMethods.Unexpectedly,whenplottedinthismanner,themagnetizationdataforeachmaterialssystemcollapseontosinglecurveswhichhavesimilarshapesdescribedbyauniquefunctionF(H=Hm(T))(seeEq.1)foreachsystem.Thisdatacollapseisquiteremarkableconsideringthevarietyofmechanismsresponsible 82 PAGE 83 Figure74. Scalingcollapsedescribesirreversiblemagnetization(hysteresis)inawidevarietyofferromagneticmaterials.a,PlotsofthereducedirreversiblemagnetizationM(H;T)=Mmax(T)asafunctionofreducedmagneticeldH=Hm(T)forthesixdierentmagneticmaterialslabelingeachcurve.TheimplicittemperaturevariableincreasesfromlefttorightandeachcoloronagivencurverepresentsadierentM(H;T)isotherm.Forclarity,theordinatevalueshavebeenshiftedandthesolidlineshavebeenaddedasguidestotheeye.ThebottommostcurveforsinglecrystalspinglassCu:Mn(at.1.5%)materialistakenfromtheliterature 116 formagneticorderingtogetherwithawiderangeofmaterialsproperties.Thescalingcollapseappliesequallywell:tosingledomain(SD)coherentlyrotatingNinanoparticleswithaverageparticlediameterof3nmembeddedinaninsulatingmatrix;tomultidomain(MD)incoherentlyrotatingNinanoparticleswithaverageparticlediameterof12nmalsoembeddedinaninsulatingmatrix;tocoherentlyrotatinguniformsizeFePtnanoparticles(6nmdiameter);tocontinuousmetallicGdthinlmswithmagnetizationderivedfromlocalmomentssupplementedbybandstructureexhibitingsomeitinerantcharacter;andtomixedphasemanganite(LPCMO)wheretheMnspinsorderbyadoubleexchangemechanisminanenvironmentwherechargeorderedandparamagneticinsulatingphasescompetewithaferromagneticmetallicphase.Wehavealsoincludedaspinglassmaterial,singlecrystalCu:Mn(1.5at%),describedintheliterature 116 .ThecollapseoccursfornanoparticlesystemswhichincludeSD,MDandsuperparamagneticparticleswithboth 83 PAGE 84 broadanduniformsizedistributionsandmagneticmomentreorientationsarisingfromcomplexsuperimpositionsofdomainwallmotionanddomainrotationwhichcanbeverydierentincontinuouslmscomparedtonanoparticleswheretheextentofthedomainwalliscomparabletothesizeoftheparticle.Thusthematerialsmeasuredhereareverymuchdierentintermsoftheoriginoftheirreversiblemagnetization. Inconclusion,wehavepresentedaheretoforeunreportedphenomenologicalresultshowingthatthetemperatureandelddependenthysteresisofatleastsixdistinctlydierentmagneticsystemscanbecollapsedontosinglecurves(Fig. 74 )usingtheparticularlysimplefunctionalformexpressedbyEq.1.Ourndingthatsuchascalingcollapseappliestomagneticsystemstotallydierentthanthespinglassforwhichsimilarscalinghasbeenpreviouslynoted 116 117 (thelowestcurveinFig. 74 )suggeststhatexplanations 116 relyingonspinglassphenomenologyaretoonarrow.Thusamoregeneraltheoryisneededtoexplainthescalingcollapse.Thiscollapsemustimplyanunderlyingsymmetrythatisnotcapturedbypreviousanalyticalandcomputationaltreatmentsandmaybeacrucialcluetounderstandthecomplexhistorydependentmagnetizationprocess.ThesimilaritywiththespinglassmaterialisinterestingandmaybebecauseofthefactthatallZFCmagnetizationisametastablestateofthesystemandshowspropertiesvaryingwithtime.Thusthedynamicsofthespinsmayplayaveryimportantroleforthescalingcollapse.ThebehaviorofMmax(T)isinvestigatedforthe3nmNiparticles.WesuggestanewapproachtoinvestigatethemagnetizationreversalprocessfromtheMmax(T)measurement.WehavesuccessfullyappliedthemethodforthecoherentlyrotatingSDparticles. 7.4Conclusions Inconclusion,wehavepresentedasurprisinglygeneralandunrecognizedphenomenologicalresultshowingthatthetemperatureandelddependenthysteresisofatleastsixdistinctlydierentmagneticsystemscanbecollapsedontosinglecurves(Fig. 74 )usingtheparticularlysimplefunctionalformexpressedbyEq.1.Wehavenotyetfound 84 PAGE 85 anyexceptions.Fortheparticularlysimplecaseofcoherentlyrotatingsingledomainparticles(samplesAandB),ouranalysisofscalingcollapsebypassesthecomplicationsofnonuniformsizedistributionsandrandomeasyaxisorientation,unveilinganunderlyingStonerWohlfarthbehavior 6 .Ourndingthatthesamescalingcollapsemoregenerallyappliestomagneticsystemswithawidevarietyofmechanismsgivingrisetohysteresis,suggeststhatexplanations 116 relyingonspinglassphenomenologyaretoonarrow.Thusamoregeneraltheoryisneededtoexplainthescalingcollapse.Thiscollapsemustimplyanunderlyingsymmetrythatisnotcapturedbypreviousanalyticalandcomputationaltreatmentsandmaybeacrucialcluetounderstandcomplexhistorydependentmagnetizationprocesses.Thesimilaritywiththespinglassmaterial 116 issuggestiveandmayberelatedtothefactthatallZFCmagnetizationsrepresentmetastablestatesofthesystem,whichifgivensucienttimewouldrelaxtowardtheeldcooledequilibriumstate.Accordingly,thedynamicsofthespinsmayplayaveryimportantroleinunderstandingthescalingcollapse. 7.5Methods 7.5.1NiNanoparticle. CompositelmscomprisingmagneticNinanoparticlesembeddedinanAl2O3hostmatrixweresynthesizedbypulsedlaserdeposition(PLD) 36 .HighpuritytargetsofNi(99.99%)andAl2O3(99.99%)werealternatelyablatedinthesamedepositionrun.Thebasepressureofthedepositionwasontheorderof10)]TJ /F3 7.97 Tf 6.58 0 Td[(7Torr.Thesubstratetemperaturewasmaintainednear550Cduringthegrowth.Therepetitionrateofthelaserbeamwas10Hzandenergydensityusedwas2Jcm)]TJ /F3 7.97 Tf 6.58 0 Td[(2overaspotsizeof4mm1.5mm.A40nmthickbuerlayerofAl2O3wasdepositedinitiallyonthesapphiresubstratebeforethesequentialgrowthofNiandAl2O3.ThisprocedureresultsinaverysmoothstartingsurfaceofgrowthofNiasveriedbythehighresolutionscanningtransmissionelectronmicroscopy(STEM)studies.Samplesconsistsof5layersofNiandAl2O3.AcaplayerofAl2O3wasalwaysusedtoprotectthesamplefromoxidation.ZcontrastSTEMimage 85 PAGE 86 veriestheabsenceoftheNioxide.SamplesstudiedhereconsistsofNiparticlesizeof3nmand12nmindiameter. 7.5.2GdThinFilm. GdthinlmsweredepositedonSisubstratesbyDCmagnetronsputtering.Thebasepressureofthechamberwasontheorderof510)]TJ /F3 7.97 Tf 6.59 0 Td[(7Torr.Thesamplesarecontinuouswiththicknessnear100nm. 7.5.3(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)ThinFilms. Phaseseparatedmanganite(La1)]TJ /F5 7.97 Tf 6.59 0 Td[(yPry)0:67Ca0:33MnO3(LPCMO)lmsweregrownusingpulsedlaserdeposition(PLD)atarateof0.05nm/sonNdGaO3(NGO)(110)substrateskeptat820Cinanoxygenatmosphereof420mTorr 118 7.5.4TemperatureCorrectionofCoerciveField. ForidealStonerWohlfarthparticlesthecoerciveeldisgivenby,HSWc=Hco(T)(1)]TJ /F1 11.955 Tf 453.9 23.9 Td[((T=TB)1=2).Forthecaseofrandomlyorientedcoherentlyrotatingparticlesallofwhichhavethesamesize,thecoerciveeldisgivenby,Hranc(T)=0:48Hco(T)(1)]TJ /F1 11.955 Tf 12.36 0 Td[((T=TB)3=4).SincetheFePtparticlesofsampleBallhavethesamesize,wecanwriteHranc(T)=HMHc(T)whereHMHc(T)isthecoerciveeldextractedfromthemagnetizationloops.Itisthenstraightforwardtoshowthat,HSWc(T)=2HMHc(T)(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)1=2)=(1)]TJ /F1 11.955 Tf 12.13 0 Td[((T=TB)3=4).Usingthefactthatatloweldsthezeroeldcooledpeaktemperaturecanbeidentiedastheblockingtemperature 119 ,wendthatTB=60KfortheFePtnanoparticles.AccordinglyHSWc(T),theabscissainFig. 73 b,canbecalculatedinastraightforwardmannerfromthemeasuredvaluesofTBandHMHc(T). 86 PAGE 87 REFERENCES 1. Neel,L.Sometheoreticalaspectsofrockmagnetism.AdvancesinPhysics4(14),191{243(1955). 2. 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Sappey,R.,Vincent,E.,Hadacek,N.,Chaput,F.,Boilot,J.P.,andZins,D.Nonmonotonicelddependenceofthezeroeldcooledmagnetizationpeakinsomesystemsofmagneticnanoparticles.Phys.Rev.B56(22),14551{14559(1997). 95 PAGE 96 BIOGRAPHICALSKETCH RiteshKumarDaswasbornon1981inaverysmallvillage,Mohanbati,locatedeastofIndia.AtHaripalGuruDayalhighschool,Riteshwasintroducedtophysicsandfellinlovewiththesubject.Riteshgotinterestedinsciencesubjectsduringhishighschooldays,wherehefoundwonderfulteachers,Mr.RobinChatterjeeandMr.UttamSaha.Aftercompletingthehighschool,RiteshenrolledintheRamakrishnaMissionVidyamandira(Belur)inAugust1999andgraduatedwithaBachelorofSciencedegreewithhonorsinphysicsonAugust2002.DuringthistimeRiteshmethisfavoriteteacherDr.DeepakGhsosh,whoplayedaverycrucialroletomakeRiteshinterestedinexperimentalphysics.AftercompletingtheBachelorofSciencedegree,RiteshenrolledinMasterofSciencedegreeatIndianInstituteofTechnology(Kanpur)andcompletedthedegreeonAugust2004.AfterthisRiteshjoinedUniversityofFloridatopursuehisPh.D.degree.RiteshwasalwaysinterestedinthepropertiesofmaterialsatnanoscaleandfoundthatProf.A.F.Hebard'slabtobetheperfectplacefortheresearchintheeldofthinlmsandnanoparticles.RiteshgraduatedinAugust2010withaDoctorofPhilosophydegreeinphysics. 96 