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Study of magnetic features of Nd₂Fe₁₄B through the spin reorientation transition by magnetic force microscopy Saleem, Muhammad 2017

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Study of Magnetic Features of Nd2Fe14B throughthe Spin Reorientation Transition by MagneticForce MicroscopybyMuhammad SaleemM.Sc., King Fahd University of Petroleum and Minerals, 2012a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)April 2017c© Muhammad Saleem, 2017AbstractNd2Fe14B is one of the high performance permanent magnets that has ap-peared as an appealing compound for commercial applications. The under-standing of its macroscopic magnetic properties through the study of itsmagnetic domain structures has received great attention. In this study, weuse magnetic force microscopy (MFM) to image the magnetic features as afunction of temperature through the spin-reorientation transition tempera-ture (TSR ∼ 135 K) of a Nd2Fe14B single crystal. We observe a pronouncedchange in the anisotropy of the magnetic features upon cooling from 170 K to100 K. Our autocorrelation analysis of the MFM images reveals an increasein the four-fold component of the anisotropy below TSR. The magnetic fea-ture size is estimated from the two-fold and four-fold components and foundto be between 4.5 µm and 6 µm below TSR. We observe an average mag-netic feature size around 5 µm above the spin reorientation transition. Thecomplexity in the geometry of magnetic features is studied from the fractaldimension (FD) analysis. Higher values of FD below TSR indicate that themagnetic features possess more rugged boundaries. Average values of FDincrease from 1.17 ± 0.05 for T > TSR to 1.29 ± 0.04 for T < TSR.iiPrefaceThe magnetic force microscopy (MFM) experiment was carried out in Hoff-man lab in Harvard University by M. Huefner. The sample used in the ex-periment was provided by R. Prozorov and P. C. Canfield from Iowa StateUniversity. Xiaoyu Liu contributed in developing MATLAB code for access-ing the MFM data. The data was analyzed and interpreted by the authorand Jason Hoffman. The main results of this thesis will be published onarXiv and in suitable peer-reviewed journals.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Brief description of chapters . . . . . . . . . . . . . . . . . . . 42 Fundamentals and Literature Review . . . . . . . . . . . . . 62.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . 62.1.2 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . 72.1.3 Spin reorientation transition . . . . . . . . . . . . . . . 102.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 123 Magnetic Force Microscopy . . . . . . . . . . . . . . . . . . . 183.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . 18iv3.2 Tip-sample interaction . . . . . . . . . . . . . . . . . . . . . . 193.3 MFM operational modes . . . . . . . . . . . . . . . . . . . . . 203.3.1 Static mode . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Dynamic mode . . . . . . . . . . . . . . . . . . . . . . 213.4 MFM lift-mode . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Experimental details . . . . . . . . . . . . . . . . . . . . . . . 244 Magnetic Feature Analysis . . . . . . . . . . . . . . . . . . . 264.1 Magnetic features . . . . . . . . . . . . . . . . . . . . . . . . . 264.1.1 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 284.1.2 Autocorrelation results . . . . . . . . . . . . . . . . . . 284.2 Analysis approach . . . . . . . . . . . . . . . . . . . . . . . . 294.2.1 Magnetic feature anisotropy . . . . . . . . . . . . . . . 294.3 Magnetic feature size . . . . . . . . . . . . . . . . . . . . . . . 304.3.1 T ≤ TSR . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.2 T > TSR . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Fractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1 Introduction to fractals . . . . . . . . . . . . . . . . . . . . . 345.1.1 Fundamental concept . . . . . . . . . . . . . . . . . . 345.1.2 FD computing methods . . . . . . . . . . . . . . . . . 385.2 Fractal analysis of magnetic features . . . . . . . . . . . . . . 415.2.1 FD analysis approach . . . . . . . . . . . . . . . . . . 415.2.2 Fractal results . . . . . . . . . . . . . . . . . . . . . . 436 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49vList of TablesTable 2.1 Summary of domain size observed by different measure-ment techniques. . . . . . . . . . . . . . . . . . . . . . . . . 17viList of FiguresFigure 1.1 Development in the energy density (BH)max of hard mag-netic materials in the 20th century (adapted from Ref.[8]).Nd-Fe-B type magnets dominate other type of magneticcompounds due to highest energy product. . . . . . . . . 2Figure 2.1 Tetragonal unit cell of Nd2Fe14B with lattice constants a= 8.80 A˚ and c = 12.20 A˚ . The c-axis is elongated toemphasize the puckering of hexagonal iron net. The tableindicates the atomic sites, occupancies, and coordinates(x,y,z ) of constituent atoms. This figure is adapted fromRef.[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.2 Magnetization along easy- and hard-axis in Nd2Fe14B sin-gle crystal measured at room temperature (adapted fromRef.[11]). Large magnetic field up to 7 MA/m is requiredto achieve the magnetization perpendicular to c-axis i.e.,along hard-axis, while magnetic field less than 1 MA/mis sufficient to magnetized Nd2Fe14B single crystal alongc-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.3 Temperature dependence of magnetic anisotropy constantsof Nd2Fe14B (adapted from Ref.[16]). The spin reorien-tation takes place at 135 K, where K1 becomes negativeand K2 and K3 increase abruptly. . . . . . . . . . . . . . 9viiFigure 2.4 Three different easy magnetization directions for tetrag-onal symmetry. M is the net magnetization aligned to apreferred direction. Easy-axis representation is when Mis aligned along the c-axis, easy-plane representation iswhen M is in the basal plane, and easy-cone representa-tion is when M is on the surface of cone. . . . . . . . . . 10Figure 2.5 Tilt angle as a function of temperature in single crystalof Nd2Fe14B (adapted from Ref.[14]). The canting angleis zero for T > TSR and increases for T ≤ TSR. Themaximum canting angle is about 30◦ at 4.2 K. . . . . . . 11Figure 2.6 MOKE images of single crystal of Nd2Fe14B (adaptedfrom Ref.[18]). Transformation of domain structures onbasal (a – f) and prismatic (g – i) planes as a function oftemperature. T = (a) 285, (b) 200, (c) 165, (d) 118, (e)113, (f) 20, (g) 285, (h) 135, and (i) 20 K. Different typesof domains are present on different planes of observation.Star-like domains exist on basal plane and transform intorectangular-like domains below TSR. Stripe-like domainsemerge on prismatic plane. . . . . . . . . . . . . . . . . . 13Figure 2.7 Both TEM and SEM images reveal maze-like pattern ofmagnetic domains of a (001) crystal of Nd2Fe14B. . . . . 14Figure 2.8 Scanning transmission X-ray microscopy image of 20 µm× 20 µm Nd-Fe-B sintered magnet with thickness 50 –150 nm. Maze-like domain patterns are clearly observed.This figure is adapted from Ref. [22]. . . . . . . . . . . . 15Figure 2.9 MFM image of Nd-Fe-B film. Small 1 µm – 2 µm grainsof Nd2Fe14B show magnetic domains with domain sizeabout 150 nm. This figure is adapted from Ref. [23]. . . 16viiiFigure 3.1 A schematic diagram showing the working principle ofMFM measurements. A cantilever with magnetic tip isscanned at constant height above the sample surface. Theoptical interferometer detects the shift in the resonancefrequency of the cantilever due to magnetic interaction(constant height mode). . . . . . . . . . . . . . . . . . . . 19Figure 3.2 Range of detection of different forces that contribute tothe tip-sample interaction. The electric and magneticforces are long range forces and can be sensed up to fewhundred of nanometers, which is beyond the detectionlimit of van der Waals forces. This figure is adapted fromRef.[51]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.3 Working principle of lift-mode MFM showing two passscanning. The first scan is done in tapping mode with-out magnetic information, while magnetic information isrecorded during the second pass in lift mode (adaptedfrom Ref.[53]). . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.4 The resonance frequency shift of cantilever detected inthree different ways as result of changes in magnetic forceon the tip (adapted from Ref.[54]). . . . . . . . . . . . . 23Figure 3.5 The home-built MFM setup situated in Hoffman lab atHarvard university. . . . . . . . . . . . . . . . . . . . . . 24Figure 3.6 Comparison of topography of the sample obtained fromtwo different tips. (a) Shows the topography of the sample(30 µm × 30 µm) using magnetic tip. (b) Shows thetopography of the sample (3 µm × 3 µm) using the non-magnetic Si-tip. . . . . . . . . . . . . . . . . . . . . . . . 25ixFigure 4.1 (a) and (c) show 59 µm × 59 µm MFM scans of the samearea of the sample at different temperatures below andabove TSR. We can see clearly the temperature depen-dent evolution in magnetic features. (b) and (d) show thenormalized 2D autocorrelation (59 µm × 59 µm) of theMFM-scans shown in part (a) and (c). Autocorrelationof the images is also distinct below and above TSR due topresence of different magnetic features. . . . . . . . . . . 27Figure 4.2 Zoomed in (20 µm × 20 µm) autocorrelated images evi-dently show the four-fold anisotropy below TSR. . . . . . 29Figure 4.3 (a) and (b) show the autocorrelation intensity as a func-tion of angle at 150 K and 115 K, respectively. Sym-bols show the intensity for different length scales fromFig. 4.1(b) and (d). The black line is the fitting curve byequation 4.3. (c) Variation in four-fold coefficient A4 withtemperature, computed through fitting as shown in (a)and (b) for all the temperatures. (d) Temperature depen-dence of four-fold component computed from FFT. Four-fold component and four-fold coefficient display identicaltemperature dependent behavior. . . . . . . . . . . . . . 31Figure 4.4 (a) and (b) show the length scale dependence of four-fold and two-fold components, respectively. These com-ponents are calculated from intensity of autocorrelationand show different response for temperatures below andabove TSR. . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.5 (a) Autocorrelation of MFM image at 150 K with two line-profiles through the center. (b) Line-profiles show peakcorrespond to the central part of autocorrelation. Peakswere fitted to obtain the FWHM. . . . . . . . . . . . . . 33Figure 5.1 A comparison between topological dimension and fractaldimension of a rugged line adapted from Ref. [58]. . . . 35xFigure 5.2 The notion of geometry for defining the dimension andscaling in Euclidean shapes. This figure is adapted fromRef. [59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 5.3 Sierpinski Triangle shows fractional dimension. It showsthe self-similarity where triangle looks exactly the samewhether it is viewed from close or far. This figure isadapted from Ref. [60]. . . . . . . . . . . . . . . . . . . . 37Figure 5.4 Binary images obtained as a result of cutoff values from20 % to 60 %. . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 5.5 Binary images obtained as a result of different range ofcutoff values for each image at 170 K, 150 K, and 135 K(T ≥ TSR). . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 5.6 Binary images as a result of chosen cutoff values for com-puting the fractal dimension. (a) MFM images. (b) Bi-nary images obtained for a uniform cutoff value of 50%.(c) Each image is binarized for subjectively-determinedoptimal cutoff values for T ≥ 135 K (TSR) which is largerthan 50 % cutoff value, while images from 130 K to 100K are binarized for 42 % cutoff values. . . . . . . . . . . 43Figure 5.7 Log-log plots of perimeter versus area of magnetic featuresfrom the binary images. Linear regression is perform tofit the data. The slope of fitted line is used to computethe fractal dimension using equation FD = 2×slope. . . 44Figure 5.8 Fractal dimension (FD) as a function of temperature. Theerror bars are obtained from slope uncertainty. (a) FDof the binary images obtained for uniform cutoff valueof 50 %. (b) FD of images binarized for subjectively-determined optimal cutoff values. Dashed lines show av-erage FD for T > TSR and T < TSR. (c) A comparisonof FD for all binary images produced for different cutoffvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xiAcknowledgmentsI would like to express my sincere gratitude to my research supervisor Prof.Jennifer Hoffman for giving me the opportunity to work in her group. I amvery grateful for the moral and financial support she provided to completemy master's degree. I would like to acknowledge with much appreciationher guidance, encouragement, and kindness during my studies.I am really thankful to my academic supervisor Prof. Robert Kiefl forreading my thesis and for his moral support.A special gratitude to Dr. Jason Hoffman for his availability and fruitfuldiscussion over analysis and results. I am thankful to him for helping me inMATLAB and reviewing my thesis draft. I really appreciate the motivationthat you gave me during my research.In addition, I would like to say “thank you” to all of the staff and facultymembers in the department of physics and astronomy. I am also thankfulto all my friends for being with me and having many useful discussions oncourse material and research related activities.Finally, I would like to thank my family for their best wishes, unlim-ited support and encouragement. I wish to dedicate this thesis to my bothbeloved sons Moaz S. Khan and Hassan S. Khan.xiiChapter 1Introduction1.1 BackgroundPermanent magnets are important due to applications in turbines, electron-ics, electric motors, automobile engines, hard drives, nanoscale devices, med-ical applications, research tools/equipments and many more [1–7]. Theseapplications can be optimized by developing materials with high coercivity,high remanence, large energy product (BH)max, and an almost rectangularhysteresis loop. Figure 1.1 shows the chronological development of energyproduct of magnets in the 20th century. Nd-Fe-B type magnets show thehighest energy products compared to other permanent magnets.Most of the commercially available permanent magnets are rare earthtransition metals compounds. In these materials, the rare earth elementsprovide most of the magneto-crystalline anisotropy responsible for the coer-civity, while magnetization emerges because of the transition metal sublat-tice [9]. Nd2Fe14B has become an effective candidate due to its low cost, itshigh energy product (50 MGOe) [10], and its high saturation magnetization(1.6 T) [11]. It is also suitable for light and small designs due to its lowspecific gravity. Thus, Nd2Fe14B has become the most widely used highperformance permanent magnet.Nd2Fe14B like other permanent magnets possess magnetic domains. Mag-netic domains describe small regions of uniform magnetization. Generally,1O GutfleischFigure 1. Development in the energy density (BH)max of hardmagnetic materials in the 20th century and presentation ofdifferent types of materials with comparable energy densities.Section 3 describes the complex task of transferring theintrinsic properties into extrinsic properties by appropriateprocessing. Microstructures with energy barriers preservingthe metastable, permanently magnetized state need to bedeveloped so that useful remanences and coercivities acrossa large temperature range arise. The different manufacturingroutes for high-energy-density RPMs are reviewed withparticular emphasis being placed on their limitations withrespect to the physical properties and thermal stabilities ofthe micro- and nanocrystalline materials. The first partis devoted to maximum energy density magnets based onNd–Fe–B (including a brief reference to their corrosionbehaviour) and to recent developments in the field of high-temperature magnets based on Sm–Co, which become moreand more relevant as RPMs are used in devices which operateat high temperatures. The different preparation methods toobtain the nanoscale structures such as non-equilibrium andhydrogen assisted processing routes as well as the importantgroup of interstitially modified compounds are described inthe following section. Finally, concepts of maximizing theenergy product in nanostructured magnets by either inducinga texture via HDDR (hydrogenation disproportionationdesorption and recombination) processing or hot deformationor enhancing the remanence via exchange coupling arereviewed. The paper ends with a short summary and outlook.2. R–T magnetic materialsThe search for new compounds with superior propertiesfocuses on materials with high values of the Curietemperature (TC > 500 K), high saturation magnetization(Ms > 1 T) and high anisotropy field, HA. These intrinsicproperties depend on the crystal structure and chemicalcomposition and a favourable combination of these valuesdoes not lead automatically to a good hard magnetic material,but can only be regarded as a prerequisite. The finalsuitability can only be assessed when the extrinsic propertiessuch as the coercive field HC , remanent magnetizationBr and maximum energy product (BH)max , derived fromthe intrinsic properties by the preparation of adequateFigure 2. Three prototypes of RPMs based on Nd2Fe14B withidealized microstructures. Type (I) is R rich and the individualcrystallites are separated by a thin paramagnetic layer (grey).Long-range dipolar interaction is dominant and each hardmagnetic grain behaves like a small permanent magnet, whichresults in high coercivities. The grains in type (II) are based onstoichiometric Nd2Fe14B and are exchange coupled as noadditional phase is present, leading to remanence enhancement.Type (III) is a nanocomposite magnet, where a Nd-deficientcomposition is used and the coupling occurs between theNd2Fe14B grains and soft magnetic Fe-rich grains (grey), resultingin a further increase in remanence.microstructures, fulfil certain criteria. The intrinsic magneticproperties and micromagnetic parameters of the mostimportant R–T compounds are summarized in table 1.Included are Nd2Fe14B- and SmCo-based compounds, whichare currently the most relevant for applications, and also theother recently discovered hard magnetic iron-rich compoundssuch as ThMn12-type compounds and the interstitial solidsolutions of N and C in R2Fe17-type compounds. The domainwall width δw in hard magnets is comparable in size to theexchange length lex (see section 3.2.3), the latter describingthe scale of the perturbed area when a spin is unfavourablyaligned; whereas in soft magnets, δw  lex is valid. Thecritical single-domain particle size dc describes the size ofthe largest possible crystallite in which the energy cost forthe formation of a domain wall is higher than the gain inmagnetostatic energy. Typical values for iron-based RPMsare of the order of 200–300 nm. There are two basic conceptsfor the prevention of nucleation and growth of reversedomains from the fully magnetized state (magnetizationreversal): (a) for microcrystalline sintered magnets and(b) for single-domain grains found in nanostructured magnetsobtained by melt spinning, mechanical alloying or HDDR.The former concept is realized firstly in the nucleation type,as found in sintered NdFeB-type magnets consisting of multi-R158Figure 1.1: Development in the energy density (BH)max of hard magneticmaterials in the 20th century (adapted from Ref.[8]). Nd-Fe-B type magnetsdominate other type of magnetic compounds due to highest energy product.the direction of magn tization varies from domain to domain. An exter-nal magnetic field is applied to align them, as a result, magnetic materialsgive non-zero bulk magnetization. The domains are separated by small re-gions are called domain walls. Domain walls are the transition region thatrepresent the continuous change in spin direction. The deep understand-ing of magnetic microstructures enable us to predict the magnetic behaviorof hard magnets such as Nd2Fe14B. Further, magnetic microstructure ofhard magnets is of interest for nanoscale devices. In micromagnetic modelsthe domain wall energy density ‘γ’ is a fundamental parameter, knowledgeof which is very important for the application of micromagnetics [12]. Inaddition, the study of magnetic domai s is useful for understanding anddetermining domain wall energy, magnetic behavior and thermal remagneti-zation [13]. Therefore, a detailed understanding and complete knowledge ofmagnetic domains and intrinsic magnetic properties of Nd2Fe14B magnetsare worth studying.Nd2Fe14B has a Curie temperature Tc = 565 K and shows a spin reori-entation transition at TSR = 135 K [9]. Above TSR, the easy magnetizationdirection is the c-axis. At ≤ TSR, the easy magnetization direction cantsaway from c-axis and changes into four equivalent magnetization axes in2{110} planes, forming an easy-cone-like magnetization [14, 15]. Magneticmicrostructures change significantly from temperature above TSR to belowTSR.Several different experimental techniques, including magneto-optic Kerreffect, transmission electron microscopy, scanning electron microscopy, andmagnetic force microscopy (MFM) have been used to investigate the mag-netic domain properties of Nd2Fe14B [16–25]. At room temperature, themagnetic domains form star-like patterns, which evolve into larger rect-angular shape domain features below TSR [16]. Smaller domains, with acharacteristic length scale of 0.02 µm – 0.5 µm are observed well-above TSR[24–29], while at 4 K, domains with size of 2 µm – 5 µm are observed [16].At 200 K, magnetic features at nano-meter length scale (from 6 nm to 40nm) in the bulk with domain wall thickness of 6 nm has been reported us-ing small-angle neutron scattering technique [26]. Furthermore, magneticdomain of size 20 nm and 25 nm have also been reported by Huang, et al.[25] and Al-Khafaji, et al. [24], respectively, using MFM measurements atroom temperature.1.2 MotivationAlthough most commercial applications of Nd2Fe14B magnets are well abovethe TSR, the microscopic study of its domain features as a function of tem-perature crossing through the TSR is important for understanding the mag-netic phase transition. In previous studies, not much attention was paid tothe study of magnetic features through the spin reorientation transition.In this work, we track the magnetic features of a Nd2Fe14B single crystalthrough TSR using magnetic force microscopy (MFM). MFM has establisheditself as a phenomenal tool to gain spatially resolved information about mag-netic structures with high resolution [30, 31]. In our analysis, we exploretwo main characteristics of the magnetic features of Nd2Fe14B in the tem-perature range 170 K to 100 K.1. We investigate the typical length scale of magnetic features in bothtemperature regimes i.e., above and below TSR.32. We explore the fractal dimension (FD) associated with the magneticfeatures below and above TSR.1.3 Brief description of chaptersChapter twoThis chapter summarizes the fundamental properties of Nd2Fe14B such ascrystal structure, magnetic anisotropy, and the spin reorientation transi-tion. These properties are discussed in light of previous studies. A reviewof the literature on Nd2Fe14B is also provided, with an emphasis on theexperimental studies of magnetic domain structure.Chapter threeIn this chapter, we discuss fundamentals of the MFM, including the workingprinciple and its different operational modes. The experimental approachcarried out to obtain the MFM images is also discussed.Chapter fourIn this chapter we describe the analysis of the MFM images. We estimatethe length scale associated with magnetic domain features. We calculatethe two-dimensional autocorrelation of MFM images. The autocorrelatedimages possess two-fold, as well as, four-fold anisotropies. We track the two-fold and four-fold components of the anisotropies as a function of differentradii. The radii represent the average length scale associated with magneticfeatures.Chapter fiveThis chapter is on the fractal dimension (FD) analysis of magnetic features.We introduce the fundamentals of fractals in general, and the various meth-ods that have been developed to study the fractals. A very brief descriptionof prevalent methods of fractal analysis is provided. The area-perimeter4method is discussed and used to obtain the FD as a function of tempera-ture.Chapter sixIn this last chapter of the thesis, conclusions of our analysis are presented.Future work is also proposed.5Chapter 2Fundamentals and LiteratureReview2.1 Fundamentals2.1.1 Crystal structureShortly after its discovery, the crystal structure of Nd2Fe14B was studied byHerbst, et al. [32], Givord, et al. [33] and Shoemaker, et al. [34] indepen-dently. The crystal structure is tetragonal, with lattice constants a = 8.80A˚ and c = 12.20 A˚ . Each Nd2Fe14B unit cell is composed of 68 atoms with56 Fe, 8 Nd and 4 B atoms arranged in an eight-layer structure, as shownin Fig. 2.1. Every Nd and B atom is bonded to 4 Fe atoms in the mirrorplanes at z = 0 and z = 1/2. In between these planes, the remaining Featoms construct a hexagonal net. Therefore, a unit cell of Nd2Fe14B showssheets of Fe, Nd and B atoms separated by hexagonal iron nets. The Fe-Feseparation is less than Fe-Nd separation, while all the pairs in the unit cellare separated by less than 4 A˚ [9]. There are six different sites for Fe atoms,two different sites for Nd atoms and only one site for B atoms as shown bythe table in Fig. 2.1.614Figure 12. Unit cell of the Nd2Fe14B compound according to Herbst. Theexperimental lattice parameters are a = 0.88 nm, c = 1.22 nm. The c-axis wasmanually elongated to emphasize the puckering of iron layers as in Fa¨hnle et al(1993).2c and 3d positions, which corresponds to the atomic composition Sm2Fe12Co5. As concerns theexchange parameters, a maximum value for JRT is obtained for the 2c–6f double substitutions,i.e. for Sm2Fe9Co8. This finding again supports the above statement that linear correlationof magnetic properties with chemical compositions is not generally valid and explicit DFTcomputation is better for analyzing and predicting magnetic properties.3.3. Magnetic properties of Nd–Fe–B magnets and the RE2TM14 B seriesSince the discovery of the hard ferromagnetic compound Nd2Fe14B, for its development to thestrongest available permanent magnet it has been a challenge to further improve its magneticproperties by compositional variations of the RE2Fe14B system. This may be achieved forinstance by substituting Nd with another RE element, replacing totally or in part Fe by otherTM elements like Co or Ni, or by exchanging B with other light IS (interstitial) atoms suchas H, C and N. Because the combinatorial variety of possible intermetallic RE2TM14IS phasesincreases very rapidly, it is essential to obtain guidelines from theory for a systematic search fornew materials.The exact stoichiometry and crystal structure of Nd2Fe14B were determined independentlyand simultaneously by three research groups in 1984 (Givord et al 1984a, 1984b, Herbst et al1984, Shoemaker et al 1984). Figure 12 displays the Nd2Fe14B unit cell. The crystal structurehas tetragonal symmetry (space group P4z/mnm), and each unit cell contains four formula units,i.e. 68 atoms. There are six crystallographically distinct TM sites (16k1, 16k2, 8 j1, 8 j2, 4c and4e), two different RE positions (4f and 4g) and one IS site (4g).New Journal of Physics 15 (2013) 125023 (http://www.njp.org/)Figure 2.1: Tetragonal unit cell of Nd2Fe14B with lattice co stants a =8.80 A˚ and c = 12.20 A˚ . The c-axis is elongated to emphasize the puckeringof hexagonal iron net. The table indicates the atomic sites, occupancies,and coordinates (x,y,z ) of constituent atoms. This figure is adapted fromRef.[35].2.1.2 Magnetic anisotropyMagnetic anisotropy is the characteristic of magnetic materials (ferromag-netic and ferrimagnetic) when magnetic moments are aligned to any en-ergetically favorable crystallographic axis to achieve the minimum energy.Magnetic anisotropy or magneto-crystalline anisotropy describes the direc-tional dependence of magnetic properties. The primarily origin of magneto-rystalline anisotropy is the spin-orbit interaction.For magnetic materials, there are two magnetization directions: the easy-magnetization direction and the hard-magnetization direction. The easymagnetization direction is obtained through a small applied field to achievethe saturation magnetization inside the crystal. On th oth r hand, the hardmagnetization axis represents that direction in space where large magnetic7field is required to achieve saturation magnetization. Thus, one would needa strong magnetic field to achieve the saturation magnetization along hard-axis compared to easy-axis. The magnetization curves of Nd2Fe14B alongeasy- and hard-axis are shown in Fig. 2.2.Magnetic properties of rare-earth-uron-boron permanent magnet materials M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, and S. Hirosawa SumitomoSpecial Metals Co., Ltd., Egawa, Shimamotocho, Mishimagun, Osaka 618. Japan Static magnetic measurements have been carried out on single crystals ofNd2 Fe'4 B, Sm2 Fe'4 B, and Y 2 Fe'4 B from 4.2 to 590 K. Values of K, estimated from high field measurements at room temperature are 4.5, - 12, and 1.1 MJ/m3 for Nd1Fe'4B, Sm2Fe'4B, and Y2Fe'4B, respectively. Anisotropic behavior of the magnetization versus magnetic field curves in the basal plane has been observed for Sm2Fe'4B, indicating large amplitude of the high order coefficients, K 2 and K 3' In Nd2 Fe14 B, the magnetization has been found to tilt from the c axis and simultaneously increase in magnitude. Average Fe moment is estimated to be 2.23 /-lBIFe at 4.2 K from the saturation magnetization of Y 2 Fe 14 B. I. INTRODUCTION Much attention has been focused on the R2Fe14 B inter-metallics since high energy permanent magnets having max-imum energy products larger than 280 kJ/m3 were devel-oped on the basis of Nd2Fe14B.1 Recent improvements involving addition of Co and Dy, which offer increased ener-gy products and thermal stability, have broaden the fields of potential applications for this new class of permanent mag-nets. 2 Because of this technological impact, there have been increasing quests for fundamental investigations of the basic magnetic properties of the R2 Fel4 B intermetallics which form in the tetragonal structure except for R = La. Especial-ly an accurate measurement of the temperature dependence of the magnetocrystalline anisotropy constants has been most desired in connection with a relatively rapid fall of co-ercive force ofthe Nd2 Fe14 B-based magnets at elevated tem-peratures. For this reason, we have carried out magnetic measurements on single crystals of Ndz Fe14 B, Sm2Fe14 B, and Y ZFe14 B. The yttrium compound was chosen because its magnetic properties arise only from the Fe sublattice and hence provide the Fe background of magnetic properties of the R 2Fe'4 B series. SmZFe14 B was examined as a typical example of Rz Fe 14 B with planar anisotropy. II. EXPERIMENTAL PROCEDURE Single crystals ofNdzFe14 B and Y 2Fe'4 B were grown in n infrar d imaging furnace by the floating-zone melting 1----------------------l.6i H ,c-axis 1.4 r ~ 1.2 t.. g 1.0-'.ii ~ 0,8-e ' 00 '" ~23456 891011 Magnetic Field(MA/m) FIG. 1. Magnetization vs magnetic field curves of a single crystal of Nd2 Fe,.B at room temperature. technique. Single crystals of Sm2 Fe14 B were prepared in a BN crucible in an Ar atmosphere by very slow cooling of the alloy from a temperature just above the melting point of Sm2 Fe'4B. The purity of starting materials was 99.5% for rare earths, 99.9% for Fe, and 99.5% for B. The single crys-tals thus obtained were qualified by means of x -ra y diffraction analysis and optical microscope observation. Magnetic mea-surements were performed on cleaved single crystals of the size of about O.5mm embedded in epoxy resin. Alignment of the crystals was achieved by the use of external magnetic fields at approximatel.y 370 K before the epoxy resin hard-ened. The temperature dependence of magnetization was measured in our laboratory using a vibrating sample magne-tometer(VSM), in an external magnetic field of 800 kA/m. High fiel.d magnetic measurements up to 11.5 MAim were carried ou t with a VSM and a water cooled Bi tter type magnet at the High Field Laboratory for Super Conducting Materi-als, Research Institute ofIron, Steel and other Metals of To-hoku University. III. RESULTS AND DISCUSSION Results of the high field measurements ofR2Fe14 B (R = Nd, Sm and Y) at room temperature are shown in Figs. 1-3. No appreciable coercivity is observed. The high field suscepti-bility at r om temperature is negligibly small. In the case of Nd2Fe14 B (Fig. 1 ) the magnetization versus magnetic field 1.6 1.4 i==' '; 1.0 o ~ 0,8 a; e l: 0.6 ~ 0,4 0.2 HI/a-axis Sm, Fe"B 2 3 4 5 6 7 8 9 10 11 Magnetic Field (MAim) FIG, 2. Magnetization vs magnetic field curves of a single crystal 01 Sm2Fe,.B at room temperature, 4094 J, Appl. Phys. 57 (1), 15 April 1985 0021-8979/85/014094·03$02,40 © 1985 American Institute of Physics 4094 Figure 2.2: Magnetization along easy- and hard-axis in Nd2Fe14B sin-gle crystal measured at room temperature (adapted from Ref.[11]). Largemagnetic field up to 7 MA/m is required to achieve the magnetization per-pendicular to c-axis i.e., along hard-axis, while magnetic field less than 1MA/m is sufficient to magnetized Nd2Fe14B single crystal along c-axis.Nd2Fe14B has uniaxial magnetic anisotropy i.e., easy-axis is alignedwith tetragonal c-axis above TSR. In tetragonal symmetry, the magneto-crystalline anisotropy energy can be approximated by:E = E0 +K1 sin2 θ +K2 sin4 θ +K3 sin4 θ cos 4φ, (2.1)where E0 is constant, θ is the angle between the magnetization vector andthe c-axis, and φ is the angle between the magnetization vector and thea-axis. These angles determine the orientation of the magnetization vectorwith respect to crystallographic axes. K1, K2 and K3 are anisotropy con-stants. The anisotropy constants have no direct correlation with the physicalorigin of magnetic anisotropy, rather they reflect only magnetic anisotropyphenomenologically [36]. However, these constants are strongly temperature8Figure 2.3: Temperature dependence of magnetic anisotropy constants ofNd2Fe14B (adapted from Ref.[16]). The spin reorientation takes place at135 K, where K1 becomes negative and K2 and K3 increase abruptly.dependent and have been estimated experimentally as a function of temper-ature [37]. The temperature dependent behavior of anisotropy constantsfor Nd2Fe14B are shown in Fig. 2.3. With increasing temperature from 4.2K, K1 increases from negative to positive. On the other hand, at 4.2 K,both K2 and K3 are positive and decrease with increasing temperature. K2remains positive, while K3 becomes negligibly small above TSR.The anisotropy constants K1 and K2 are dominant and basal planeanisotropy constant K3 is small. The energy minimization can be derivedfrom the conditions ∂E∂θ = 0 and∂2E∂θ2> 0. Depending on values of K1and K2, three cases namely, easy-axis, easy-plane, and easy-cone can beconsidered. Figure 2.4 illustrates the easy-axis, easy-cone, and easy-planemagnetization in tetragonal symmetry.i. For K1 ≥ 0 and K1 + K2 > 0 , then θ = 0◦ or 180◦. This shows thepreferred magnetization direction along c-axis.9ii. For K1 < 0 and K1 + 2K2 < 0 or K1 > 0 and K1 + K2 < 0, thenθ = 90◦. This is an easy-plane where the preferred magnetizationdirection is perpendicular to c-axis.iii. For K1 < 0 and K1 + 2K2 > 0, then sin2 θ = − K12K2 . This indicatesthat the preferred magnetization direction will be some where betweeneasy-axis and easy-plane in the space. This is the situation when therewill be four equivalent preferred axes form an easy-cone.c-axisM8.5 cmc-axisMφM a-axiseasy-axis easy-plane easy-coneFigure 2.4: Three different easy magnetization directions for tetragonalsymmetry. M is the net magnetization aligned to a preferred direction.Easy-axis representation is when M is aligned along the c-axis, easy-planerepresentation is when M is in the basal plane, and easy-cone representationis when M is on the surface of cone.2.1.3 Spin reorientation transitionGenerally the application of three factors (temperature, magnetic field, andexternal pressure) change the direction of easy-axis of magnetization fromone crystal axis to any other axis. The phenomena of net magnetizationdeflection away from one crystallographic axis at high temperature to an-other at low temperature is called the spin reorientation transition (SRT).The temperature at which the SRT takes place is known as the spin re-orientation transition temperature (TSR). The occurrence of SRT destroys10the uniaxial anisotropy and worsen the magnetic performance of permanentmagnets for technological applications [9].The SRT can proceed either through first-order (discontinuous) transi-tion or through second-order (continuous) transition. In the first-order tran-sition i.e., easy-axis to easy-plane, the easy-axis of magnetization changesrapidly from axial to planar (in basal plane) with decreasing temperature.This transition is not mediated by conical arrangement. Among R2Fe14Bcompounds, Er2Fe14B, Tm2Fe14B, and Yb2Fe14B have shown the first-orderSRT [9]. In these compounds, Fe and R (= Er, Tm, Yb) sublattices havetemperature induced competition of magnetic moments alignment. Inter-metallic compounds such as TbFe11Ti, DyFe11Ti, and NdCo5 have shownboth types i.e., first- and second-order transitions.Nd2Fe14B has second-order SRT, where easy-axis magnetization changesto easy-cone magnetization. Above TSR (T = 135 K), Fe and Nd magneticmoments are ferrimagnetically coupled and have collinear alignment with336 MEASUREMENTS ON Nd2Fe|4B SINGLE CRYSTALS Vol. 56, No. 4 . .30  g, "0  v 20 U.I .. J  Z I-- I ! I I \ \ \ 0 0 i I 50 100 TENIPERTURE NdzFe14B lq~o--o-o--o-o--o--e-o-- I I 150 200 T (K)  Fig.7 The tilt angle 8 of the direction of easy magnetization from the [001] axis to the [ii0] axis determined by the torque measurement for the (ii0) plane of Nd2FeI4B at temperatures below and above the spin reorientatlon temper- ature. those obtained from magnetization measurements. WRen we determine the spin reorientatlon temper- ature from the kink point of the magnetization vs. temperature curve, it is 135K. When we determine the temperature, however, as the start point where the direction of easy magnetization begins to tilt from the [001] axis in torque measurements, the temperature is found to be 133K. A rather good agreement exists between these two temperature values taking into consideration the fact that we cannot bring the thermocouple into contact with the specimen in torque measurements. References [i] M. Sagawa, S. Fujimura, N. Togawa, N. Yamamoto and Y. Matsuura; J. Appl. Phys., 5--5 2083 (1984). [2] J.F. Herbst, J.J. Croat, F.E. Pinkerton and W.B. Telon; Phys. Rev. B 294176 (1984). [3] D. Givord, H.S. Li and J.M. Moreau; Solid State Commun., 50 497 (1984). [4] D. Givord, H.S. Li, R. Perrier de la B~thle; Solid State Commun., 51 857 (1984). [5] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura and S. Hirosawa; Conf. Magn. Magn. Mat. (San Diego, 1984). Figure 2.5: Tilt angle as a function of temperature in single crystal ofNd2Fe14B (adapted from Ref.[14]). The canting angle is zero for T > TSRand increases for T ≤ TSR. The maximum canting angle is about 30◦ at 4.2K.11c-axis. Therefore, easy-axis magnetization remains along [001] direction.At TSR, easy-axis magnetization changes into four easy magnetization di-rections (in {110} planes) on the surface of a cone with some canting angle.The canting angle is temperature sensitive, and increases from 0◦ (uniaxial)to 30◦ at 4.2 K [14, 38]. Figure 2.5 shows the increase in canting angle withtemperature.There are several studies carried out to understand the origin of SRT inNd2Fe14B. It is considered that the strong interplay between crystal field andFe-Nd exchange interaction causes the SRT [15, 39]. Generally, crystal fieldand exchange interaction are temperature dependent. It was revealed thatthe reorientation of the magnetization towards the c-axis at higher temper-ature is due to the relative decrease of crystal field interaction with respectto exchange interaction, which favors a collinear arrangement of Nd and Femoments [15, 40]. Some experimental findings have shown the existence ofnon-collinear arrangement of both Fe and Nd magnetic moments throughoutthe spin reorientation transition [41–43]. These findings indicated that Ndand Fe canting angles reach up to 58◦ and 27◦, respectively, at 4.2 K.2.2 Literature reviewA wide-variety of experimental techniques have been used to investigate themagnetic domains of Nd2Fe14B. The majority of studies were conducted atroom temperature (RT) or well above the TSR. Some studies have also beenconducted well below TSR. The domains of different types, geometry, anddifferent orientations depend on several factors. For instance, the domainsobserved in prismatic plane [100] (plane parallel to c-axis) are completelydifferent than domains observed in the basal plane [001] (plane perpendicularto c-axis). Domain features are distinct and vary significantly from T > TSRto T < TSR.Room temperature magneto-optical Kerr effect (MOKE) measurementsrevealed star-like domains in the basal plane, while stripe-like domains wereobserved in the prismatic plane of Nd2Fe14B single crystals [16]. Figure 2.6shows the temperature dependent domain evolution in basal and prismatic12Figure 2.6: MOKE images of single crystal of Nd2Fe14B (adapted fromRef.[18]). Transformation of domain structures on basal (a – f) and prismatic(g – i) planes as a function of temperature. T = (a) 285, (b) 200, (c) 165,(d) 118, (e) 113, (f) 20, (g) 285, (h) 135, and (i) 20 K. Different types ofdomains are present on different planes of observation. Star-like domainsexist on basal plane and transform into rectangular-like domains below TSR.Stripe-like domains emerge on prismatic plane.planes of Nd2Fe14B single crystal. In the prismatic plane, stripe domains,with width 63 µm were observed above TSR. Similar type of domain struc-ture (53 µm) with immediate appearance of closure domains of width 2 µm– 5 µm were observed below TSR [16]. On the other hand, star-like domainsin the basal plane change into a new type of rectangular domain upon de-creasing the sample temperature below TSR. A domain size of about 5 µmin the basal plane was reported below TSR [17]. The emergence of new typeof domains is apparent on all planes of observation and are the consequence13of change in the preferred orientation of domain walls [16–18].Scanning electron microscopy (SEM) [19–21] and transmission electronmicroscopy (TEM) [20, 22] revealed magnetic domains in maze-like patternwith spike reversed domains having opposite polarity on the basal plane ofNd2Fe14B single crystals. Maze-like pattern observed by SEM and TEMare shown in Fig. 2.7. The spike domains are the surface domains, while(a) TEM image adapted from Ref.[20]. (b) SEM image adapted from Ref.[21].Figure 2.7: Both TEM and SEM images reveal maze-like pattern of mag-netic domains of a (001) crystal of Nd2Fe14B.the maze domains extend into the bulk. The stray field at the surface wasalso viewed three-dimensionally forming the hillocks separated by troughs[19, 21], which revealed that the origin of stray field at the sample surface ismaze domain pattern. Apart from the Nd2Fe14B phase, magnetic domainsof Nd-Fe-B alloys have also been investigated. Scanning transmission X-ray microscopy study of 50 nm – 150 nm thin c-plane sample of Nd-Fe-Bsintered magnet showed maze-like pattern domains configuration of width150 nm – 200 nm [22]. Similar domains of width about 150 nm with domainwall of thickness 3 nm in Nd-Fe-B alloy were reported by Mishra, et al.,[44]. A Bitter pattern technique reported the same type of domains [28] i.e.,maze pattern and spike reversed domains on the basal plane of thermallydemagnetized sample. However, observed domains were found to be 3 µm –6 µm wide. Interestingly, the very same Nd-Fe-B sample revealed domainsof width 1 µm – 3 µm from SEM study and fine magnetic domains withwide range of width 20 nm – 250 nm from MFM measurements [28]. Thesediscrepancies in the domain size of same surface clearly evidence the limited14resolution of Bitter and SEM compared to MFM.Figure 2.8: Scanning transmission X-ray microscopy image of 20 µm × 20µm Nd-Fe-B sintered magnet with thickness 50 – 150 nm. Maze-like domainpatterns are clearly observed. This figure is adapted from Ref. [22].MFM study of thin Nd-Fe-B films composed of small rectangular grainsof 1 µm – 2 µm showed the stripe domains of width of 100 nm – 300 nm[23]. Al-Khafaji, et al. [24] reported the magnetic features on the order of25 nm in a rick-rack domain pattern in the basal plane of Nd2Fe14B singlecrystal. It was also noticed that the small tip-sample separation enhancesthe image resolution and fine scale magnetic features can be seen. Recently,another MFM study showed star-like magnetic domains at RT as a result ofcomplex network of elongated domains [25]. Magnetic domains of width 20nm with domain wall of thickness about 2 nm were reported. The featuresize (domain wall thickness) of 2 nm is much less than MFM resolution,and suggests the possibility of a tip artifact. In MFM studies, tip-inducedartifacts should not be ruled out, as they can cause a misinterpretation ofthe domain structure [45].The domain structure in the basal plane of a Nd2Fe14B single crystal wasalso studied by small angle neutron scattering [26]. The magnetic domains15Figure 2.9: MFM image of Nd-Fe-B film. Small 1 µm – 2 µm grains ofNd2Fe14B show magnetic domains with domain size about 150 nm. Thisfigure is adapted from Ref. [23].of length scales 6 nm – 40 nm were found and are in comparable rangeto length scale observed by MFM. Furthermore, this study also showedthat high temperature (T = 200 K) magnetic features are smoother thanmagnetic features below TSR at 20 K.Table 2.1 summarizes the magnetic feature sizes observed by differentexperimental techniques. Domains with characteristic length scale of 0.02µm – 0.5 µm appear above TSR. Domain with larger length scale (2 µm – 5µm) are observed below TSR (4.2 K). These experimental results indicate thediscrepancies in domain size and observed domain structure. For instance,star-like magnetic features were not observed in TEM and SEM imagesabove TSR, unlike MOKE and MFM images. The difference in domainfeatures might have the following reasons:i. different surface sensitivity of the measurement technique.ii. different spatial resolution of the measurement technique.iii. plane of observation of the sample.16iv. temperature and thickness of the sample.v. presence or absence of applied magnetic field.Table 2.1: Summary of domain size observed by different measurementtechniques.T(K) MeasurementTechniqueTechniqueResolutionDomain Size ReferencesRT MFM 25 nm 25 nm Al-Khafaji, etal. [24]RT MFM – 100 – 300 nm Neu, et al. [23]RT MFM 15 nm 20 nm Huang, et al.[25]RT MFM – 20 – 250nm Szmaja, et al.[28]RT Bitter pattern 0.5 µm 3 – 6 µm Szmaja, et al.[28]RT SEM 0.1 µm 1 – 2 µm Szmaja, et al.[28]4.2 MOKE – 2 – 5 µm Pastushenkov,et al. [16]200 SANS – 40 nm Kreyssig, et al.[26]Furthermore, very little attention has been paid to the magnetic featuresnear the spin reorientation transition. Therefore, it is worthwhile to studythe magnetic features and associated characteristic length scale of magneticfeatures through TSR.17Chapter 3Magnetic Force Microscopy3.1 Working principleMagnetic force microscopy (MFM) is a scanning probe techniques that isused to study the surface properties of magnetic samples with submicronresolution. MFM was first introduced in 1987 [46, 47], soon after the in-vention of the atomic force microscope (AFM) in 1986 by Binnig, et al.[48]. MFM has received considerable attention in academic, as well as, inindustrial research due to its high resolution and minimal sample prepara-tion requirements. Further, this technique provides the direct observationof magnetic features.The simple schematic shown in Fig. 3.1 illustrates the working principleof the MFM. It consists of a cantilever, which has a small needle-shapedmagnetized tip on its free end. When the cantilever is brought close to thesurface of the sample, generally few hundred of nanometers, the tip-sampleinteraction leads to a change in cantilever resonance frequency. The tip-sample interaction is either attractive or repulsive depending on the directionof tip magnetization and stray field. The amount of deflection is recordedby an optical detector. The cantilever resonance and phase deflections areobtained while scanning the tip in a raster-like pattern over the samplesurface and are processed by a computer program to construct the MFMimage in real time.18The cantilever and tip are usually made of Si or Si3N4. In order to makethe Si tip magnetic, the tip is coated with a thin layer of magnetic materiale.g., CoPt, NiFe, or CoCr. The cantilever is typically a few hundred microns(200 µm – 300 µm) in length, while a tip up to a few microns (5 µm – 20µm) in length is generally used. The radius of the end of the tip plays avital role in the image resolution. Therefore, tips with small apex radii arepreferred for high resolution images. Coated tips of apex radii 15 nm – 50nm are used.xyzSampleMagnetic TipCantileverOptical fiber interferometer8.5 cmFigure 3.1: A schematic diagram showing the working principle of MFMmeasurements. A cantilever with magnetic tip is scanned at constant heightabove the sample surface. The optical interferometer detects the shift in theresonance frequency of the cantilever due to magnetic interaction (constantheight mode).3.2 Tip-sample interactionThe tip-sample surface interaction depends on several parameters that in-cludes tip shape, radius of tip apex, thickness of tip coating, and tip-sampleseparation. Tip-sample separation is chosen with care during the MFMmeasurements, because interaction becomes stronger close to the surface.The tip can be remagnetized if the surface stray field is stronger than the19tip coercivity [28, 49], which might cause an artifact in the MFM results. Ithas also been reported that the type of coated material on the tip (soft orhard tip) and radius of the apex of the tip may produce different magneticstructures [45, 49, 50].Tip of the MFM acts as a tiny magnet that interacts with the stray fieldof the sample above the surface. The magnetic interaction i.e., magneticforce on the tip can be calculated by first calculating the magnetic potentialenergy E and magnetic force−→F acting on the MFM tip [51]:E = −µ0∫ −→M tip · −→H sampledVtip (Joule) (3.1)−→F = −−→∇E = µ0∫ −→∇(−→M tip · −→H sample)dVtip (Newton) (3.2)where Mtip is tip magnetization and Hsample is the stray field from thesample. The integration is carried out over the tip volume.3.3 MFM operational modesMFM has two modes of operations, the static and the dynamic mode.3.3.1 Static modeStatic mode is also sometimes called constant or DC mode. In this mode ofdetection, the force of interaction (repulsive or attractive) between tip andsurface causes the cantilever to bend. The force of interaction is measuredthrough the detection of vertical deflection in the cantilever's equilibriumposition. According to Hook's law, the magnetic force |−→F | detected in thismode of operation is [51]:|−→F | = −c∆z, (Newton) or ∆z = −|−→F |c(3.3)where ∆z is the vertical displacement of cantilever as a result of the forceand c is the cantilever constant.203.3.2 Dynamic modeIn the dynamic or AC mode of operation, the cantilever oscillates close toits free resonance frequency f0 i.e., the resonance frequency when there isno tip-sample interaction. Here, the cantilever is treated as a harmonicoscillator such that the force gradient ∂F∂z changes the resonance frequencyto [51, 52]f = f0√1−∂F∂zc(Hz). (3.4)For ∂F∂z  c, using a Taylor expansion, the change in resonance frequency isapproximated by∆f = f − f0 ≈ − f2c∂F∂z(Hz). (3.5)If the tip-sample interaction is attractive, then ∂F∂z > 0. This leads toa decrease in the resonance frequency. In the case of repulsive interaction(∂F∂z < 0), the resonance frequency increases and leads to a positive frequencyshift.3.4 MFM lift-modeThe cantilever deflection may emerge due to the presence of electrostatic,van der Waals, and quantum mechanical forces other than the magneticforce. These forces are distance dependent and are listed with their rangeof detection in Fig. 3.2. MFM scans may therefore, contain both topo-graphic and magnetic information. If the tip is in close proximity of thesurface, where the AFM tapping mode operates, then topographic featureswill dominate the magnetic contrast. Thus, magnetic features will not beobvious. The topographic features generally result from van der Waals in-teractions. On the other hand, magnetic forces are long-range forces, whichcan be sensed at distances much longer than van der Waals. Therefore, thetip is lifted from the surface and scan proceeds at constant height in orderto avoid (minimize) the influence of van der Waals forces.21mechanic and magnetic dissipation [38]. The last two can provide some informationon the sample. An example of a technique making use of the magnetic dissipationwhen scanning for example across domain walls is the Magnetic dissipation forcemicroscopy. We may describe the damping in terms of the so called quality factor:𝑄 = 𝜔𝐸mech𝑃loss. (3.1)This is just ratio of mechanical energy stored in the cantilever and the powerdissipated during one period of oscillation T=2𝜋/𝜔, with 𝜔 being angular frequencyof oscillation. The higher Q, the less damping - enhanced sensitivity.AFM electronics is controlled via computer and many task can be automated.The system also involves a feedback loop. When the feedback loop is turned on,it keeps constant deflection or oscillation amplitude of the cantilever by adjustingtip-sample distance.Dominant interaction of the probe with the sample depends on the tip-sampleseparation as illustrated in Figure 3.4. Magnetic forces are long-range, thus inorder to sense mainly the magnetic contribution, the tip-sample distance shouldbe at least 10 nm. In practise, for the separation of topography and long-rangedmagnetic contribution, so called lift mode is employed. The tapping/lift mode willbe described in section 3.3.Fig. 3.4: Forces acting on a magnetic tip and tip-sample distances where they prevail.Adapted from [38].To conclude, the most common scheme involves sensing the force or its gradientswith flexural deflection in the contact mode or change in resonance of the cantileverin the dynamic mode. The probe can sense also lateral force acting on the lever.In addition to the flexural resonance, torsional resonance of the cantilever can beexploited for lateral forces imaging. The torsion is excited by two piezo-elementswhich are excited out-of-phase.More about AFM can be found in a very nice book by Eaton and West [39].28igure 3.2: Range f detection of different forces tha contribute to the tip-sample interaction. The electric and magnetic forces are long range forcesand can be sensed up to few hundred of nanometers, which is beyond thedetection limit of van der Waals forces. This figure is adapted from Ref.[51].In constant height mode, dual (two pass) scanning is carried out. Thefirst scan is the topographic scan, where tapping mode with a feedback loopon is op rated. The feedback loop adjusts the tip-sample distance and main-tains the constant amplitude of cantilever oscillation. In the second scan,the tip is lifted to a constant height and retraces the first scanned profilewithout feedback loop. The resulting interaction between the MFM tip andsurface stray field produces a magnetic field gradient with no dependenceon surface topography. A cartoon of lift-mode is shown in Fig. 3.3.The cantilever frequency shift is detected in three ways: frequency mod-ulation, phase modulation, and amplitude modulation, as shown in Fig.3.4. In frequency modulation, the shift in cantilever resonance frequency isdetected. Phase modulation detects the variations in the phase of the can-tilever oscillation relative to the piezo drive frequency, whereas amplitudemodulation measures changes in the cantilever oscillation amplitude. Fre-quency and phase modulation generally produce better results compared toamplitude modulation, with greater ease of use, higher signal-to-noise ratio,and fewer artifacts [55].22   Fig. 2 Modeled tip-sample interaction, where k denotes cantilever constant and the force derivative F∂ ⁄ z∂  is schematically depicted here as an additive force interaction constant.  There are two ways to deal with resonance frequency measurement. The amplitude detection is based on the cantilever oscillated at given frequency (its value is greater then free resonant frequency), that means the changes in resonant frequency cause deflections of the cantilever. The frequency detection can be realized as follows: the cantilever is vibrated accurately at its resonant frequency f where the amplitude is controlled by the feedback loop. Resulting detection is assured by FM demodulator. With respect to the fact that MFM can be operated in constant frequency shift mode, two-pass (tapping-lift) mode or constant height mode, the crucial issue is to minimize surface topography features on the image of the magnetic forces distribution. To solve this problem, the major part of the MFM measurements are performed in terms of two-pass mode.            ©FORMATEX 2007Modern Research and Educational Topics in Microscopy.                                   A. Méndez-Vilas and J. Díaz (Eds.)808      _______________________________________________________________________________________________Figure 3.3: Working principle of lift-mode MFM showing two pass scan-ning. The first scan is done in tapping mode without magnetic information,while magnetic information is recorded during the second pass in lift mode(adapted from Ref.[53]).Figure 3.4: The resonance frequency shift of cantilever detected in threedifferent ways as result of changes in magnetic force on the tip (adaptedfrom Ref.[54]).233.5 Experimental detailsThe MFM setup used in this experiment is shown in Fig. 3.5. The homebuilt scanning probe setup is located in a vacuum chamber within a Janisflow cryostat. Magnetic features at the sample surface are mapped usinga sharp, high-resolution magnetic tip (Team Nanotec MFM) with cobaltalloy coating. The magnetic coating has a coercivity of 950 Oe, with amagnetic moment of approximately 3.75× 10−14 emu. The tip has in-planeand perpendicular magnetic remanences of 540 emu-cm−3 and 160 emu-cm−3, respectively. The tip radius is less than 25 nm, while the length ofthe tip was greater than 9 µm. The tip has a conical shape with resonancefrequency of 75 kHz and force constant c 3.0 N/m.RSIScanning force microscope with both horizontal and vertical cantilever systemsJeehoon Kim, Martin Zech, Sang Chu, T. Williams, and J.E. HoffmanDepartment of Physics,Harvard University, Cambridge, MA 02138(Dated: January 11, 2010)put abstractPACS numbers:I. INTRODUCTIONII. INSTRUMENT DESIGNA. CryostatThe cryostat is a custom-designed, Janis-built 4He flowcryostat. A 4.5-inch four-way cross with two 2.75-inchconflate flanges is attached on the top of the cryostat. Aturbo molecular pump is directly attached on one portof the four-way cross through a 4.5-inch angle valve inorder to increase a vacuum conductance. The eddy cur-rent damping system located in the middle of the fridgeeffectively isolates mechanical noises of a turbo molecu-lar pump, so that the microscope can be operated underthe vacuum as low as 10−8 Torr at room temperature.This capability is good for Kelvin probe force microscopywhere the sample surface is easily contaminated with wa-ter moisture related absorbents in air that contribute un-wanted potential features. A 50-pin and 32-pin electricalfeedthrough are fixed on a 4.5-inch and 2.75-inch conflateflange of the cross, respectively. An optical fiber is epox-ied on a 0.01-inch stainless tubing that is silver-solderedto a 2.75 conflate flange. The long enough length of anoptical fiber is stored in a fiber spool sitting around themicroscope head for many fiber cleavages. The custom-spec 4.5-inch conflate flange with eight SMA connectorsis attached on the cross for the capacitance detectionof the walker motion, a piezo driving signal, and a tipbias. To make a wiring compact and neat, and to easilydetach the microscope head from the cryostat, the minia-tured 25-pin D-connectors (Oxford instruments, A8-401and A8-402) were used. The cryostat rests on the topof the 1-inch thick hexagon aluminum plate that sits onthe triangular wooden table filled with 2000 pounds oflead bricks to lower the system resonant frequency. Thewooden table supported by three air springs has a pul-ley system and a hoist to lift a 5-tesla superconductingdewar. The whole room is floated with 6 heavy duty airsprings.B. Fiber optic interferometer detection systemThe laser light from a 1550 nm diode laser (ThorlabsS3FC 1550) is fed into a standard 90/10 directional cou-FIG. 1: The whole MFM systempler (GouldFiber Optics Inc.). The 10 percent light trav-els a custom spec single mode fiber, and exits the fiberthrough a flat cleaved surface, then hits the cantileverand returns to fiber. The interference happens betweenthe reflected light at the end of the cleaved flat fiber andthe reflected light at the cantilever. This interferencesignal is measured by a balanced photodiode detector(New Focus 2117). The 90 percent light coming out ofthe other end of a fiber coupler as a reference signal isfed into a variable optical attenuator (Thorlabs VOA50-APC) to attenuate intensities of the reference light. Thisreference signal goes into one of the differential inputs ofthe balanced photodiode detector then it is subtractedfrom the interference signal to remove noises from thelaser fluctuation. The photo detector output voltage Voutfrom interference as a function of distance d between acleaved fiber end and the cantilever isVout =Vmax + Vmin2(1− νcos4pid/λ), (1)where Vmax and Vmin are maximum and minimum volt-age from the photodetector, and ν is the fringe visibility,and λ is the wave length of a laser. The interference isoperated at the most sensitive region d = λ(n+1)/8 withn integer. By differentiating Equation (1), the interfer-ometer calibration (the interference sensitivity) is givenby∆d∆V= ± λ2pi(Vmax − Vmin) , (+ : 2n,− : 2n + 1). (2)Figure 3.5: The home-built MFM setup situated in Hoffman lab at Harvarduniversity.A single crystal of Nd2Fe14B is used. The sam le is several millimeterin lateral ex ent in each direction. The desired surface of study is the asalplan i.e., surface perpendicular to crystallographic c-axis. All the MFMscans were recorded at a base pressure low r than 10−7 Torr in the absence24of any external applied magnetic field. Topography of the sample surface isobtained through scanning in contact mode i.e., with very small tip sampleseparation. Figure 3.6(a) shows a topographic scan measured at room tem-perature. The topography of the sample is flat with a roughness of severalnanometers. However, some particles rising up to several tens of nanometersin height have accumulated on the surface. Another topographic image istaken using a non-magnetic Si-tip, as shown in Fig. 3.6(b). We observe thesame kind of topographic structure, indicating no correlation between thetopography and the observed magnetic structure. The magnetic featuresare mapped using lift-up mode at constant height. The measurements aretaken with a step size of 5 K while the sample is cooled from 170 K to 100K.(a) (b)Figure 3.6: Comparison of topography of the sample obtained from twodifferent tips. (a) Shows the topography of the sample (30 µm × 30 µm)using magnetic tip. (b) Shows the topography of the sample (3 µm × 3 µm)using the non-magnetic Si-tip.25Chapter 4Magnetic Feature Analysis4.1 Magnetic featuresFigures 4.1(a) and (c) illustrate a series of 59 µm × 59 µm MFM imagesof the Nd2Fe14B sample taken while cooling from 170 K to 100 K. AboveTSR, magnetic features of different length scales over band-like regions areobserved. Below TSR, a new type of distorted rectangular magnetic fea-ture appears. There is an onset of magnetic feature rearrangement at TSR,which persists at lower temperatures. We believe that the emergence ofdistorted rectangular magnetic features below TSR are directly correlatedto the change of magnetic phase due to spin reorientation transition. Ithas been argued that the spin reorientation transition leads the formationof new type of magnetic features [16], because easy axis preferred directionchanges from an easy c-axis to four easy magnetization directions in {110}planes.During the temperature dependent evolution from T > TSR to T < TSR,magnetic features show some correlation, which remains in the new magneticphase. For instance, it can be noticed that some of the magnetic featurespreserve their location and orientation through the reorientation transition.Continuous modifications in the magnetic features take place due to thetemperature dependent change of existing domains and domain walls. Thewalls of the main magnetic features at high temperature (T > TSR) become26elements of the magnetic features of magnetic phase at low temperature(T < TSR), while retaining the parallel alignment to the tetragonal c-axis[18]. As a result, we believe that the formation of new type of magneticfeatures below TSR might be due to mingling of small magnetic features ofhigh temperature (T > TSR). Furthermore, the non-vanishing correlationbetween magnetic features above and below TSR also support this argument.Figure 4.1: (a) and (c) show 59 µm × 59 µm MFM scans of the samearea of the sample at different temperatures below and above TSR. We cansee clearly the temperature dependent evolution in magnetic features. (b)and (d) show the normalized 2D autocorrelation (59 µm × 59 µm) of theMFM-scans shown in part (a) and (c). Autocorrelation of the images is alsodistinct below and above TSR due to presence of different magnetic features.274.1.1 AutocorrelationCorrelation is a mathematical method that uses two signals to produce anew signal. If the two input signals are different, then the output signal iscalled cross-correlation. If the signal is correlated with itself, the resultingsignal is called autocorrelation. Correlation is a simple and useful operationsometimes is used to extract the information from images. The autocorre-lation of an image helps us in finding the periodic pattern (features) in theimage. Autocorrelation is a linear operation because every pixels in auto-correlated result is the linear combination of its neighbors. Furthermore,autocorrelation is also shift-invariant i.e., same operation is performed atevery pixel in the image.For an M×N image, the formula for autocorrelation function Gii(a,b) is[56]:Gii(a, b) =M∑xN∑yi(x, y) ∗ i(x− a, y − b), (4.1)where i(x,y) is the image intensity at position (x,y), and a and b representthe distance from the corresponding x and y positions. The formula inequation 4.1 gives theoretical result and are not suitable for limited areacalculation. There is an alternative suitable and efficient practical methodthat uses the fast Fourier transforms using the Weiner-Khinchin theorem[56]:F−1[Gii(a, b)] = S(i) = |F [i(x, y)]|2, (4.2)where S (i) is the power spectrum of the image and F is the Fourier transformof i(x,y).4.1.2 Autocorrelation resultsThe computed two-dimensional autocorrelation of each MFM image is shownin Figs. 4.1(b) and (d). Highly correlated features exist in the center of theautocorrelated images, and the autocorrelation signal diminishes as we go28away from the center of the autocorrelated images. The distinct magneticfeatures observed in the MFM images are reproduced in the autocorrelatedimages. A distorted squared type features are observed in the autocorrelatedimages below TSR, which are absent in the autocorrelated images above TSR,as shown in Fig. 4.2. The features of autocorrelated images seem to possesseither two-fold or four-fold anisotropy.10 µm170 K 150 K 135 K130 K 115 K 100 KFigure 4.2: Zoomed in (20 µm × 20 µm) autocorrelated images evidentlyshow the four-fold anisotropy below TSR.4.2 Analysis approach4.2.1 Magnetic feature anisotropyIn order to determine the magnetic feature size, we explore the two-foldand four-fold anisotropies in the features of the autocorrelated images. Toquantify the anisotropy of magnetic features, we plot the intensity of theautocorrelated images as a function of angle at fixed radii from the center.The radii represent the length scale of magnetic features. Therefore, from29now on, we shall use length scale instead of radius. Figures 4.3(a) and(b) show the autocorrelation amplitude for different length scales at 150 Kand 115 K, respectively. We see peaks corresponding to the two-fold andfour-fold anisotropies. At 150 K, we only see the two-fold peaks, whichbecome more prominent at larger length scale value. On the other hand, at115 K, the four-fold peaks emerge with two-fold peaks as the length scaleis increased from 2 µm to 6 µm. The appearance of additional four-foldpeaks only for low temperatures scans (T < TSR) as shown in Fig. 4.2, isattributed to the distorted rectangular magnetic features. To quantify thecontribution of the two-fold and four-fold anisotropy components, we fit theautocorrelation intensity ∆ using:∆ = A0 +A2 sin (2θ + φ2) +A4 sin (4θ + φ4) , (4.3)where A2 (φ2) and A4 (φ4) are the amplitudes (phases) corresponding to thetwo-fold and four-fold anisotropies, respectively. The variation in four-foldcoefficient (A4) with temperature is plotted in Fig. 4.3(c). Temperature de-pendent behavior of four-fold component displays an abrupt increase aroundTSR, which persists at lower temperatures.We use an alternative approach to quantify the changes in the magneticfeatures. We perform 1D fast Fourier transform (FFT) of the autocorrelationintensity as a function of angle curves exemplified in Figs. 4.3(a) and (b).Figure 4.3(d) shows the four-fold component of the FFT as a function oftemperature at several length scales. We find no four-fold contribution aboveTSR, but there is an abrupt increase in the magnitude of four-fold componentbelow TSR. This temperature dependent behavior of four-fold component isin agreement with analysis in Fig. 4.3(c).4.3 Magnetic feature size4.3.1 T ≤ TSRFigures 4.4(a) and (b) show the two-fold and four-fold components of theFFT as a function of length scale at different temperatures. Below TSR we30100 110 120 130 140 150 160 17004812160 60 120 180 240 300 3600.00.20.40.60.80 60 120 180 240 300 3600.00.20.40.60.8100 110 120 130 140 150 160 1700.000.020.040.060.08 Four Fold ComponentTemperature (K) 1 µm 2 µm 4 µm 6 µm 2 µm 4 µm 6 µmT=115K∆ (arb.units) θ (degree)  (a) T=150K  ∆ (arb.units)θ (degree) 2 µm 4 µm 6 µm(b)(c)  2 µm 4 µm 6 µm  A 4 (arb.units)Temperature (K)(d)Figure 4.3: (a) and (b) show the autocorrelation intensity as a function ofangle at 150 K and 115 K, respectively. Symbols show the intensity for differ-ent length scales from Fig. 4.1(b) and (d). The black line is the fitting curveby equation 4.3. (c) Variation in four-fold coefficient A4 with temperature,computed through fitting as shown in (a) and (b) for all the temperatures.(d) Temperature dependence of four-fold component computed from FFT.Four-fold component and four-fold coefficient display identical temperaturedependent behavior.observe a contribution from the four-fold component, while no pronouncedcontribution until ∼ 7 µm appears above TSR. Below TSR, the four-foldcomponent peaks at length scale around ∼ 6 µm and then decreases after-ward. Furthermore, we observe a small variation in the length scale wherethe four-fold component peaks from 5.5 µm to 6 µm between 130 K and100 K. The two-fold component also appears below TSR and peaks at lengthscale around ∼ 4.5 µm. The two-fold component for T > TSR increases310 2 4 6 8 10 12051015200 2 4 6 8 10 12010203040 Four Fold ComponentLength Scale (µm) 100 K  115 K 130 K  135 K 150 K  170 K(a) Two Fold ComponentLength Scale (µm) 100 K  115 K 130 K  135 K 150 K  170 K(b)Figure 4.4: (a) and (b) show the length scale dependence of four-fold andtwo-fold components, respectively. These components are calculated fromintensity of autocorrelation and show different response for temperaturesbelow and above TSR.up to larger length scale which peaks at ∼ 12 µm. The length scales cor-responding to two-fold and four-fold components represent the average sizeof magnetic features below TSR. Thus, magnetic features of size ∼ 4.5 µm– 6 µm are present in MFM images for T < TSR. Our observed magneticfeature size is comparable to previously reported values of 5 µm below TSRusing magneto-optical Kerr effect [17]. The magnetic features are the re-gions between two stripe-like (bluish background) features as shown in Fig.4.1(c). The stripe-like features also evolve with temperature as their widthincreases with increase in temperature. For T > TSR, the stripe-like featuresbecome band-like features. The average width of band-like feature is ∼ 12µm which is displayed by two-fold component in Fig. 4.4(b).4.3.2 T > TSRThe magnetic features of various sizes do not possess any regular shape.Therefore, the size of magnetic features above TSR can not be estimatedthrough either two-fold or four-fold components. However, we estimate thesize of magnetic features from the central circular part of autocorrelatedimages. The average of full width at half maximum (FWHM) is calculated32from different directions as presented in Fig. 4.5 (b), reflects the averagesize magnetic features. We find feature size of ∼ 5 µm for all temperaturesabove TSR.#2 #1150 K(a) (b)Figure 4.5: (a) Autocorrelation of MFM image at 150 K with two line-profiles through the center. (b) Line-profiles show peak correspond to thecentral part of autocorrelation. Peaks were fitted to obtain the FWHM.33Chapter 5Fractal Analysis5.1 Introduction to fractals5.1.1 Fundamental conceptObjects of various shapes and sizes can easily be understood and interpretedmathematically through Euclidean geometry, where the characteristic fea-tures of regular shapes (i.e., length, area or volume) can easily be defined.For example, in Euclidean geometry, 0, 1, 2 and 3-dimensional space is usedto describe dots, lines, areas, and volumes, respectively. The dimension ofsuch objects is expressed in whole integer form and is independent of thesize of the measuring scale.In nature, objects such as trees, mountains, rivers and their banks,clouds, and the human body possess very complex shapes and morpholo-gies. In fact, the shapes of nature are so varied as to deserve being called“geometrically chaotic” [57]. The description of such objects is beyond thescope of Euclidean geometry because approximating such objects in Eu-clidean geometry produces an inaccurate description. The complexity ofsuch complicated objects was described by Mandelbrot, where he used word“fractal”.A comparison between topological dimension (Dt) and fractal dimension(FD) is described in Fig. 5.1. Both straight and rugged lines have Dt34Figure 5.1: A comparison between topological dimension and fractal di-mension of a rugged line adapted from Ref. [58].equal to 1. However, ruggedness of lines is not fully described by the Dt.Mandelbrot proposed that the ruggedness of a line is described by addingfractional number. Therefore, we notice FD exceeds the Dt (i.e., FD > Dt).This suggests the dimension of fractals is not equal to the space it residesin. Moreover, one can also notice the complexity in rugged line from FDvalues.There are few characteristics that distinguish fractals from Euclidean35shapes. First, FD is relatively new concept where dimension is a fractionalquantity unlike the topological dimension. Second, the characteristic prop-erties (length, size, etc...) describing the Euclidean shapes are independentof length of measuring scale. However, FD does change with the size ofmeasuring scale. Third, the Euclidean shape is usually described by math-ematical (algebraic) formula, whereas, FD is computed through the slopeof log-log plot of measuring scale versus measurement, which results in anon-integer values. Fourth, fractals are self-similar, which means that theobject is exactly or approximately a copy of the whole at reduced scale.Figure 5.2: The notion of geometry for defining the dimension and scalingin Euclidean shapes. This figure is adapted from Ref. [59].Self-similarity or scaling factor is a key concept of fractal geometry and isclosely connected to the notion of dimension. An example of self-similarityin Euclidean shapes is shown in Fig. 5.2. A one unit long line can be dividedinto self-similar parts, each of which is scaled down by scaling factor r. Thesecond row of Fig. 5.2 shows the case where r is equal to 2. Similarly, forthree self-similar parts, r is 3 and so on. A square (cube) will be divided into4 (8) and 9 (27) self-similar parts for scaling factors 2 and 3, respectively.36Thus, scaling factor r and number of copies N (r) are related as:N(r) = rD (5.1)D =log(N(r))log(r)(5.2)Thus, from the above equation the dimension D is equal to 1, 2, and 3 forline, square, and cube, respectively.Figure 5.3: Sierpinski Triangle shows fractional dimension. It shows theself-similarity where triangle looks exactly the same whether it is viewedfrom close or far. This figure is adapted from Ref. [60].The same method can be implemented to geometric fractal. For instance,the Sierpinski Triangle shown in Fig. 5.3 [60]. The 0 order triangle is scaleddown by factor of 2 and three self-similar copies are produced. For scalingfactor of 4, nine copies are produced and so on. So in this case:D =log(9)log(4)= 1.58 (5.3)The dimension D of the Sierpinski Triangle is fractional. These examplesdescribe mathematical fractals with true self-similarity, where the objectappears identical whether it is viewed from near or far. Each small portionof the structures reproduce the large portion. Such true self-similarity isnot present in objects in nature, instead they show statistical self-similarity,which means that shapes have some likeness [61]. It should be noted that37not all self-similar objects are fractal: for example a straight line is self-similar, but not fractal, because its fractal and topological dimensions areequal to 1 [62].5.1.2 FD computing methodsFD analysis has established itself as a powerful tool to analyze the irregularand complex shapes of different objects. So far, FD analysis has been imple-mented in almost all the disciplines of sciences and engineering.A multitudeof methods have been developed to analyze the fractal features. Since thesemethods are based on different theoretical bases, one should therefore ex-pect different results for the same feature. There are three main steps thatare common in these FD analysis methods:i. A relationship between measured quantities as a function of variousstep size (measuring scale) is developed.ii. A log-log plot of measured quantities versus step sizes is produced.iii. A linear regression slope of log-log plot is evaluated, which is used todetermine the FD.We briefly discuss several common methods for computing the fractaldimension below.Divider methodThe basic implementation of this method is to walk the divider (yardstick)along the fractal curve and record the number of steps required to cover thefractal curve with a fixed step length [63, 64]. The mathematical formula is:L(G) = FG(1−D), (5.4)where F is a positive constant, L(G) is the measured length of the line,which depends on step length G. In practice, to get the L(G), there are Nvalues of G. D is the fractal dimension, which is computed from the slope(1-D) of the log-log plot of above equation.38Box-counting methodThis is the most popular method for FD analysis introduced by Russel, etal. [65]. In this method a binary image is covered with a grid of N (r) boxesof size r. As the size of the box is reduced, the number of boxes increases.The minimum size of the box can become equal to the image resolution.The FD is computed using the formula:FD = − limr→0log(N(r))log(r)(5.5)The slope of best linear fitting curve of log-log plot gives the FD.Differential Box-counting methodN. Sarkar and Chaudhuri [66] proposed the differential box counting methodwhere binarization of image is avoided. In this method, an image is griddedinto various box size r and N (r) is computed in differential form. A image(M × M) is scaled down to the different box size (s × s). For example,an image of 3D space with (x,y) denoting 2D space, are partitioned intodifferent grids (s × s) and third coordinate (z ) denotes the gray levels. Themaximum and minimum gray level of image in the (i,j)th grid fall in lth andkth box number, respectively. Thennr(i, j) = l− k + 1, (5.6)is the contribution of N (r) in (i,j)th grid. The contribution from all the gridcell isNr =∑ijnr(i, j). (5.7)The FD is calculated using the equation 5.5.Variogram methodThis method is based on Gaussian modeling of the image. In this method,the variogram function, which describes how variance in surface height varieswith distance, is used for estimating the fractal dimension. Large number39of pairs of points of different spacing along the profile are considered anddifference in vertical (z ) values is computed. The FD is determined fromthe log-log plot between the square of the expected difference as a functionof distance between the point pair [63], according to〈[Zp − Zq]2〉 ∝ (dpq)4−2D,where D is fractal dimension, Zp and Zq are the elevations of points p andq, respectively, while d is distance between p and q.Power spectrum methodIn this method, the power spectrum of the Fourier transform of each imageline is evaluated and then all the power spectra are averaged [67]. The FDis determined from the slope. This method is found to be very slow andrequires gridded data.Area-perimeter methodIn this method, the FD is computed from the relationship between theperimeter and the area of the features. This method determines the FDof linear features that form the closed loop [68]. The FD of lakes, islands,contour loops, grain boundaries (quartz), and magnetic domains have beencomputed using the area-perimeter method [69–73]. For fractal grain bound-aries, the perimeter P is related to the diameter d or the area A as [63, 70]:P ∝ AD2 ∝ dD,where D is FD, which is determined from the slope of linear regression oflog-log plot of A and P. The range of fractal dimension is 1 ≤ D ≤ 2 becausethe measurement is in 2D Euclidean space [71]. The FD obtained representsthe collective property of set of features of various sizes. The area-perimetermethod is most stable and simple method as compared to variogram andspectral methods [74].In the next section, we use area-perimeter method to compute the FD ofmagnetic features. The area-perimeter algorithm is developed in MATLABand is implemented on binary images.405.2 Fractal analysis of magnetic features5.2.1 FD analysis approachTo implement the area-perimeter algorithm, the MFM images were firstbinarized. A binary images is a digital image that has a series of 0s and1s. A cutoff value based on pixels intensity is used, where every pixel abovethe cutoff turns into 1, while pixel values below the cutoff turn into 0.Figure 5.4 shows the binary images as a result of several different cutoffvalues. It can be seen that for larger cutoffs, there is a greater loss of imageinformation. Therefore, one must be careful while choosing the right cutoffvalues. Initially, 30 % – 60 % cutoffs with step of 10 % were implemented.It is worth noting that the the chosen cutoff, for example 30 % cutoff meansthat 30 % pixels of lowest pixel values are discarded. The choice of 50 %cutoff is selected to be a single reasonable cutoff value for all images toobtain the binary image with magnetic features identical to MFM imagesas shown in Fig 5.6(b).(a)(b)20 % Cut-off 30 % Cut-off 40 % Cut-off 50 % Cut-off 60 % Cut-offFigure 5.4: Binary images obtained as a result of cutoff values from 20 %to 60 %.For T < TSR, the binary images obtained using initially chosen 50 %cutoff value accurately reproduce the main magnetic features of the MFMimages. However, one can notice that the 50 % cutoff values do not look ap-propriate for T > TSR, where binary images contains sizable features. Such41(b)72 % Cut-off 74 % Cut-off 76 % Cut-off 78 % Cut-off 80 % Cut-off78 % Cut-off 80 % Cut-off 82 % Cut-off 84 % Cut-off 86 % Cut-off(c)(a)58 % Cut-off 60 % Cut-off 62 % Cut-off 64 % Cut-off 66 % Cut-offFigure 5.5: Binary images obtained as a result of different range of cutoffvalues for each image at 170 K, 150 K, and 135 K (T ≥ TSR).sizable features do not display the disperse small magnetic features in MFMimages. Therefore, these MFM images (T > TSR) are binarized for highercutoff values. Figure 5.5 shows the binary images obtained for higher cutoffvalues. For 155 K – 170 K, 140 K – 150 K, 135 K, and 100 K – 130 K differentrange of cutoff values 70 % – 90 % , 60 % – 80 %, 50 % – 70 %, and 40 % – 50% with step of 1 % were implemented, respectively. Our approach of takingdifferent cutoff values for each image indicates that there is no single cutoffvalue for the images 170 K – 135 K. Figure 5.6(c) shows binary images forsubjectively-determined optimal cutoffs for selected images below and aboveTSR. For the higher cutoffs, the small magnetic features vanish, however,the main features persist. So in this case, subjectively-determined optimalcutoff value is found to be higher than 50 %, which decreases as temperaturedecreases. This makes sense because magnetic features are temperature sen-sitive and modify themselves as temperature decreases. However, magneticfeatures below TSR do not change much (apparently) as a function of tem-perature, a single cutoff value about 42 % produces all the binary images42identical to the MFM images.(b)(c)50 % 50 % 50 % 50 % 50 % 50 %82 % 80 % 78 % 60 % 42 % 42 %Figure 5.6: Binary images as a result of chosen cutoff values for computingthe fractal dimension. (a) MFM images. (b) Binary images obtained for auniform cutoff value of 50%. (c) Each image is binarized for subjectively-determined optimal cutoff values for T ≥ 135 K (TSR) which is larger than50 % cutoff value, while images from 130 K to 100 K are binarized for 42 %cutoff values.After obtaining the binary images, we implement the area-perimeter(AP) algorithm, which generates several areas and corresponding perimetervalues. Then log-log plot of area versus perimeter is produced. The datapoints are fitted through linear regression method and FD is computed fromthe slope of the fitted line.5.2.2 Fractal resultsFigure 5.7 shows the log-log plots of area versus perimeter of magnetic fea-tures at different temperatures. The fractal dimension (FD) is related tothe slope of fitted line by FD = 2×slope. We expect that the FD shouldvary with temperature, since magnetic features change with temperature.The computed FD as a function of temperature are shown in Fig. 5.8.Since fractal nature is related to the nonlinear, irregular, and complex struc-43101 102 103 104 105 106Area (arb.units)101102103104P er i me te r ( ar b .u ni t s)dataFit: log(y) = 0.66378log(x) + 0.7814101 102 103 104 105 106Area (arb.units)101102103104P er i me te r ( ar b .u ni t s)dataFit: log(y) = 0.6164log(x) + 0.94145101 102 103 104 105 106Area (arb.units)101102103104P er i me te r ( ar b .u ni t s)dataFit: log(y) = 0.60555log(x) + 0.9993101 102 103 104 105 106Area (arb.units)101102103104P er i me te r ( ar b .u ni t s)dataFit: log(y) = 0.61385log(x) + 0.92549170 K 150 K135 K 115 KFigure 5.7: Log-log plots of perimeter versus area of magnetic featuresfrom the binary images. Linear regression is perform to fit the data. Theslope of fitted line is used to compute the fractal dimension using equationFD = 2×slope.tures, therefore, FD is a useful parameter to describe them quantitatively.The computed FD describes the degree of complexity in boundary structureof the magnetic features. The growing behavior of the magnetic featureswith decrease in temperature can be understood from the structure of theboundaries using the FD information.The computed FD are found to be greater than 1 for all the images,which means that the boundaries of magnetic features are not smooth. Thecomputed values of FD indicate that the boundaries possess complexity dueto ruggedness in their structure. We also notice an increase in the FD belowspin reorientation transition (T = 135 K) as shown in Fig. 5.8(a). An44abrupt increase in FD proves the role of spin reorientation transition on themagnetic feature geometry. The higher values of FD below TSR indicatesthat magnetic features in this new magnetic phase not only produce newtype of magnetic features (discussed in Chapter 4), but also possess highercomplexity. The temperature dependent behavior also indicates the lackof consistency in FD values from one temperature to another temperaturewhich is quite obvious for temperatures T > TSR (135 K). The FD forimages at 170 K, 155 K and 150 K are higher than rest of the images in 170K – 140 K region. This inconsistency might be related to the experimentalfactor during the MFM scan. This is evident in images taken at 150 K and160 K as shown in Fig. 5.6(a), where image at 160 K looks slightly blurredcompared to 170 K and 150 K images.Figure 5.8(b) shows very similar FD behavior as a function of temper-ature. Although, the choice of higher cutoffs cause the small magnetic fea-tures to disappear and left the large magnetic feature. However, individualcutoffs for each image does not influence the overall FD of the magnetic fea-tures. Here, we also notice that FD for 150 K and 155 K are also higher thanimages at other temperatures above TSR similar to Fig. 5.6(a). This rulesout the possible effect of chosen cutoff values on the computed FD values. Itshould be noted that the computed FD do not represent the rugged natureof the boundaries of a single magnetic feature, rather gives us an average ofall the features in the image.Figure 5.8 (c) shows the effect of subjectively-determined optimal cutoffvalues on the computed FD. We can see that no matter what reasonablecutoff values we choose, the temperature dependent behavior of computedFD stays the same. Moreover, fractal values of binary images obtained atdifferent cutoffs stay very close to each other. The average FD changes from1.18 ± 0.04 to 1.28 ± 0.04 for the magnetic features observed above andbelow spin reorientation transition, respectively for 50 % cutoff values (Fig.5.8(a)). Similarly, binary images produced from subjectively-determinedoptimal cutoffs, FD (average values) changes from 1.17 ± 0.05 to 1.29 ±0.04 for magnetic features observed above and below TSR.45Figure 5.8: Fractal dimension (FD) as a function of temperature. Theerror bars are obtained from slope uncertainty. (a) FD of the binary imagesobtained for uniform cutoff value of 50 %. (b) FD of images binarized forsubjectively-determined optimal cutoff values. Dashed lines show averageFD for T > TSR and T < TSR. (c) A comparison of FD for all binary imagesproduced for different cutoff values.46Chapter 6ConclusionsWe studied the magnetic features of Nd2Fe14B single crystal from MFMimages obtained at various temperatures (170 K – 100 K) including thespin reorientation transition temperature (TSR = 135 K). A new type ofmagnetic features emerge below TSR, where four magnetization directionsform an easy-cone of magnetization. The magnetic features of irregularshapes observed above TSR, which transformed into distorted rectangularmagnetic features below TSR.The size of magnetic features below TSR is determined from the au-tocorrelation of MFM images. We noticed the presence of two-fold andfour-fold anisotropies in the autocorrelated images. Both two-fold and four-fold anisotropy components are found to be length scale dependent. Thetwo-fold and four-fold anisotropy components peak around length scale ∼6 µm and ∼ 4.5 µm, respectively, and are related to the size of magneticfeatures (4.5 µm – 6 µm) below TSR. Above TSR, no contribution fromthe four-fold component of anisotropy is seen in the autocorrelation images,while the two-fold component was attributed to the band-like structure inMFM images. Thus, anisotropy components did not incorporate the size ofmagnetic features above TSR. The average magnetic feature size ∼ 5 µmis estimated above spin reorientation temperature from the FWHM of thecentral part of the autocorrelated images.The geometry of magnetic features is studied from the fractal dimension47(FD). The complexity of magnetic features is quantified from the FD values.The area-perimeter method is used to compute the FD. To implement thearea-perimeter algorithm, MFM images were first binarized. FD shows atemperature dependent behavior and increased from 1.17 ± 0.05 to 1.29 ±0.04. The higher value of FD revealed the more complex nature of magneticfeatures below TSR compared to magnetic features observed above TSR.Furthermore, different cutoffs for binarization did not alter the FD valuesignificantly.Further investigation can be carried out to study the magnetic featuresdeeply and their correlation below and above TSR. 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