Author’s Accepted Manuscript Investigation of nanostructural, optical and magnetic properties of Cerium-substituted yttrium iron garnet films prepared by a sol gel method N.B. Ibrahim, A.Z. Arsad www.elsevier.com/locate/jmmm
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S0304-8853(15)30712-5 http://dx.doi.org/10.1016/j.jmmm.2015.10.078 MAGMA60771
To appear in: Journal of Magnetism and Magnetic Materials Received date: 17 August 2015 Revised date: 8 October 2015 Accepted date: 21 October 2015 Cite this article as: N.B. Ibrahim and A.Z. Arsad, Investigation of nanostructural, optical and magnetic properties of Cerium-substituted yttrium iron garnet films prepared by a sol gel method, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.10.078 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigation of nanostructural, optical and magnetic properties of Cerium-substituted yttrium iron garnet films prepared by a sol gel method N. B. Ibrahim and A.Z. Arsad School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor *Corresponding author:
[email protected]
Abstract Cerium substituted yttrium iron garnet films with a chemical formula Y3-xCexFe5O12 ( x = 0.00.3) have been successfully prepared by a sol-gel method. The microstructure analysis showed that all films exist in the cubic garnet structure. The lattice parameter and grain size increased with the increment of Ce concentrations up to 0.25, indicating the complete Ce substitution in yttrium site. For a film with x = 0.3, the lattice parameter remained unchanged and grain size decreased. The film thickness increased and surface roughness varied with the increment of Ce content. All of the films have high optical transparency (above 80%). The Ce content reduced the saturation magnetization of the film up to a certain limit where above this limit the value increased. Overall, the findings showed that the films with x ≤ 0.25 exhibited very excellent properties, hence they are promising materials for magneto-optical devices. Keywords Cerium- Yttrium iron garnet; Sol-gel method; Solubility limit; Surface morphology 1. Introduction Recently, yttrium iron garnet (YIG) films have been extensively used in magneto-optical applications such as optical isolators and magnetic field sensors [1-3] due to their good characteristic in Faraday rotation. Since the demands in the applications keep on increasing,
therefore there is a need of films with a large magneto-optical effect. Strong magneto-optic property can be achieved if the transition of 4f-5d intraorbital electronic transition of light rare earth ions occurs at the lower photon energy side. Ce3+ ion has the lowest 5d energy level among trivalent rare-earth ions, thus the lowest 4f-5d transition energy, hence substituting cerium in yttrium iron garnet (YIG) films is expected to increase the magneto-optic property of YIG films. This prediction has been proven experimentally by Gomi et al. [4] who have prepared Ce:YIG films by radio-frequency (RF) diode sputtering. The magneto-optic properties has been reported to increase linearly with the Ce content. However, the substitution of Ce3+ ion for the Y3+ ion is hard to control [1] since the tendency of cerium to form CeO2 due to the direct oxidation of Ce3+ [5,6]. This produces a certain limitation of Ce ion solubility into YIG structure, creating a serious problem because CeO2 impurity degrade the magnetic and magneto-optical properties in Ce:YIG materials [1, 2]. There are a variety of techniques that can be used to synthesize Ce:YIG materials such as a pulsed-laser deposition (PLD) [3, 7], a radio-frequency (RF) sputtering [8, 9], a glycothermal process [10], a solid-state sintering method [1] and a sol-gel method [11]. In this study, a sol-gel method is chosen due to its low cost, simple preparation and easy to control the material composition during the synthesized process. Furthermore, this technique provides more advantages for the film deposition such as high homogeneity, high purity [12], providing a good uniform layer over a substrate surface [13] and resulting material in nanocrystalline formed. Xu et al. [11] recently reported, the solubility limit of Ce3+ in Y3-xCexFe5O12 nanoparticles powder annealed in air that prepared by a sol-gel method was around 0.1. The highest Ce3+ solubility limit in Y3-xCexFe5O12 nanocrystals has been reported by Rongjin et al. [10]. The nanocrystals were prepared by a glycothermal process. To date, there are no reports yet on the solubility limit of Ce doped YIG films. Hence, this research reports on the influence of cerium substitution on the nanostructure, surface roughness, optical and magnetic properties of yttrium iron garnet films. The findings show the optimum substitution of Ce3+ into YIG film is 0.25. This finding is important because increasing Ce3+ substitution in Y3+ site can increase the Faraday rotation [2, 8]. 2. Experimental details
Ce-substituted Yttrium iron garnet (Ce-YIG) films with (x = 0.0, 0.05, 0.01, 0.15, 0.2, 0.25, 0.3) were prepared by a sol-gel method. The iron (III) nitrate nanohydrate (Fe(NO3)3.9H2O) with purity 99.99% was dissolved in 2 ml ethanol and 1 ml ethylene glycol, Yttrium (III) nitrate hexahydrate (Y(NO3)3.6H2O) with purity 99.8% was dissolved in 1 ml ethanol and Cerium (III) nitrate hexahydrate (Ce(NO3) 3.6H2O) with high purity 99.999% was dissolved in 1 ml ethanol. These solutions were stirred separately with a magnetic stirrer at the room temperature (29 oC) to obtain a homogeneous gel. After 5h, the solutions were mixed together, filtered and aged for 2 days at room temperature (29 oC). To obtain Ce-YIG films, 200 μl of the sol was dropped on a clean quartz substrate which was mounted on a spin coater. Then the substrate was spun at 500 rpm for 15 seconds and 3500 rpm for 30 seconds. The gel film was dried at 70ºC for 20 minutes in ambient atmosphere. Then the films were annealed at 900ºC for 2 hours in pure oxygen ambient with the gas flow rates of 400 sccm. This process is to crystallize the films. The crystalline phases of films were characterized by an X-ray diffractometer (Bruker Discover 8 XRD) with Cu-Kα radiation (λ = 0.15406 nm). The count scanning rate is 2θ step/1s and the scanning angle is from 20° to 90°. The surface morphology was studied using 2 different field emission scanning electron microscopes (FE-SEM). FE-SEM model ZEISS Supra 55VP operated at 15 KV was used for backscattered electron imaging and FESEM model Zeiss Merlin Compact operated at 3 KV was used for secondary electron imaging. The surface roughness of the films were determined by an atomic force microscope (AFM, NIEGRA Prima, NT-MDT, Russia). The optical properties were investigated using a UV-Vis spectrophotometer (PerkinElmer double beam Lambda 900). The magnetic properties of films at 298K were studied by a vibrating sample magnetometer (Lake shore 7404 VSM) with a magnetic field applied parallel to the film plane. The film was cut into (4x4) mm dimension. Figure 1 shows the position of film in the applied magnetic field during the measurement. The maximum applied magnetic field was 5 KOe. 3. Results and Discussion 3.1 Structural characterization using X-ray diffractometer The XRD patterns of CexY3-xFe5O12 (0 ≤ x ≤ 0.3) films deposited on quartz substrates are depicted in Figure 2. The XRD analysis reveals that all films consists of eight strong diffraction garnet peaks of (400), (420), (422), (521), (532), (444), (640) and (642) with the nonexistence of
impurity peak. The peaks can be indexed with JCPDS card no. 43-0507. These diffractograms show all films exist in the cubic garnet structure. The increase in Ce concentrations in films shift the garnet peaks towards the lower diffraction angle, indicating the change of lattice parameter of films [14]. Lattice parameter, a can be calculated from the combination of Bragg’s law equation, 2d sin θ = nλ and d-spacing expression of cubic lattice structure, a = √
(
) where λ =
1.5406Å is the wavelength of Cu Kα radiation, θ is the angle between the incident beam and the reflecting lattice planes, d is the lattice constant, (hkl) is the Miller indices for cubic lattice and n = 1 is the interference order [14-16]. Following Bradley and Jay’s extrapolation method [15], the lattice parameter was obtained from each diffraction peak, then a graph of a versus cos2θ was plotted. The measured values of lattice parameter with different Ce concentrations are shown in Figure 3. The lattice parameter increases with the increment of Ce concentrations up to 0.25. Sekijima et al. [6] have reported that the continuous increasing in lattice parameter of Ce:YIG single crystal grown by a floating zone method gives a good sign that all of Ce3+ ions completely substitute in Y3+ sites, due to the fact that the ionic radius of Ce3+ ion (1.14 Å) is larger than ionic radius of Y3+ ion (1.02 Å) [1]. Hence the increasing trend in this study suggests that the Ce3+ ion is being completely substituted in the Y3+ site. However, for film x ≥ 0.3, the lattice parameter remains constant indicating that the solubility limit of Ce in YIG has been reached. The obtained trend of lattice parameter and Ce contents of Ce: YIG film result is similar to that reported by Mao et al. [1, 6] for their powder Ce-YIG sample. 3.2 Morphology and thickness analysis Figure 4-5 (a) show the FESEM backscattered electron (BSE) images of film surface at different magnification. The micrographs illustrate that the surface morphology of films varies with the Ce concentrations. As shown in Figure 4, the morphology of films with (x = 0.0-0.25) show a very homogeneous structure with single contrast images which means that, Ce completely substituted into YIG structure and formed in a single phase structure. However, further Ce substitution (x = 0.3), film show inhomogeneous structure due to the existence of some undissolved spots appear on the film surface (bright contrast in Figure 5 (a)). Using FESEM-EDX analysis, the spot was confirmed as CeO2. Hence, the present result reveals that
film with x = 0.3 exceed the solubility limit. However, the CeO2 phase could not be detected using the XRD analysis due to its small amount quantity. Similar observation has been reported by Mao et al. [1] who reported that the CeO2 impurity phase of YIG ceramic could not be detected in the XRD pattern due to its very weak peak. The work done by Mitra et al. [17] also showed the XRD analysis could not trace the CeO2 impurity phase in La1-xCexMnO3 manganites due to its small quantity which was less than 5%. However, to date, the solubility limit of Cesubstituted YIG films in this study was approximately ~ x = 0.25 which is higher than the solubility limit of Ce:YIG powder (x = 0.1) prepared by a sol-gel method [11]. The morphology characteristic study of films also dependence with the Ce concentrations. Figure 4 shows the pure YIG film has very fine grains and free-agglomeration structure. When x = 0.05, the grains start to agglomerate and interconnect to each other. Further increasing in x concentrations, (see the circle in Figure 4) the grains become more agglomerate and bound together. The structure has also become densely packed and it creates voids in random formation. Figure 5(a) shows the film with x = 0.3 has a high void formation. It is predicted that the existence of CeO2 phase possible cause more defect in the film structure. The grain size of films has been calculated by averaging of 10 individual grains. The average values of grain size of all films are shown in Figure 6. The average grain size of films increases with increasing of x up to 0.25, but it decreases when x = 0.3. The large grain size of film with x = 0.25 is due to the optimization of Ce substitution in the Y3+ ions sites. Figure 5(b) shows the typical cross section FESEM image of film with x = 0.25 and Figure 6 shows the thickness of films strongly increased with the increment of Ce concentrations (x). Overall, the estimated thickness is recorded around 93 to 214 nanometers. Figure 7 shows a three dimensional AFM image of (5 µm x 5 µm) of films deposited on quartz substrates. The root mean square roughness of the pure YIG film was 4.11 nm. With increasing of Ce concentrations, it varied significantly. For x = 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3, the root mean square roughness were 2.98, 3.17, 3.19, 3.46, 4.45 and 7.30 nm, respectively. The high surface roughness for film with x = 0.3 is due to its high void formation. 3.3 Optical properties The effect of Ce-substitution on optical properties were also studied. Figure 8(a) shows the optical transmittance spectra of CexY3-xFe5O12 films with (x = 0.0-0.3) as a function of wavelength (200-800 nm). It is found that the thinnest film (pure YIG film) shows a high optical
transmittance above 95% in the visible and near infrared region. To further investigate the optical properties of the film, optical absorption coefficient spectra was plotted. Figure 8(b) shows the optical absorption coefficient spectra of CexY3-xFe5O12 films with (x = 0.0-0.3) as a function of wavelength. The absorption coefficient (α) of the film was calculated using the Beer – Lambert’s law equation: α = ln(1/T)/d, where T is transmittance and d is film thickness. It is clearly demonstrated that the pure YIG film exhibits a larger absorption coefficient compared to other films. With increasing of x concentrations, the absorption coefficient decrease with unsystematic patterns. However, the obtained results are different from the previous reported for Ce-substituted YIG material [6, 9]. Sekijima et al. [6] have reported that the optical absorption coefficient of Ce:YIG crystals (grown by a floating zone (FZ) method) at λ = 1550 nm decreased with increasing of Ce concentrations (x = 0, 0.1, 0.2, 0.3, 0.4, 0.5) due to the decreasing of Fe 2+ ions in the crystals. However, Gomi et al. [9] found that the optical absorption of Ce:YIG films (prepared by sputtering method) at λ = 829 and 592 nm increased with the increasing of Ce concentrations (x = 1.0, 2.0, 2.5). But, it should be noted, the preparation technique of Ce:YIG films reported are different compared to this study. The various techniques can attribute the difference in surface roughness [18]. This factor can provide various possibilities of surface scattering light [19] which possibly lead to the different optical properties of the films. However, our results show the absorption coefficient of Ce:YIG films which occur at wavelength (< 500 nm) is quite similar to that Ftema et al. [15] for Al doped Tb-YIG films prepared by the sol- gel method. 3.4 Magnetic properties The magnetic measurement of CexY3-xFe5O12 films with (x = 0.0-0.3) was carried out using a VSM at room temperature (27oC). Figure 9(a) shows the in-plane hysteresis loops for all films. The magnetization loops were obtained by subtracting the diamagnetic contribution from the substrate from the obtained results. The low coercivity values indicate that the films are soft magnetic film. Figure 9(b) shows the saturation magnetization (Ms) of films as a function of Ce concentrations. The estimated error of Ms is recorded around ~ 0.1%. This error is attributed from the uncertainty of the substrate volume. It is clear that the pure YIG film shows the highest Ms with the value of 112 emu/cm3. The value is lower than the value of bulk YIG (~ 139.3 emu/cm3
[20]), but higher than the YIG films prepared by the sol-gel method ~ 82.34 emu/cm3 [21]. However, with the increasing of Ce concentrations (x ≤ 0.15), Ms decreases progressively then increases again when 0.15
x ≤ 0.25. After x = 0.3, Ms decreases to the lowest value (see
Figure 9(a)). In a magnetic YIG system, the magnetic properties of YIG are contributed by Fe3+ ions and their relationship to the surrounding oxygen ions [1]. The total net magnetic moment of YIG structure can be expressed as
|
| [10]. The sub lattices of (c) site contain of
non-magnetic 24Y3+ ions, while [a] and (d) sites are occupied by 16 Fe3+ and 24 Fe3+ ions, respectively. Based on the formula, Rongjin et al. [10] explained that the saturation magnetization of Ce-YIG structure is predicted to have lower than the pure YIG structure. As Ce3+ ions (1µB) enters to the Y site (0 µB), its moment would align anti-parallel to that moment of Fe3+ ions in the d-site [7]. Hence, the magnetic moment of Fe3+ ions in the (d) site can be reduced due to the decreasing in a-d exchange interaction. This explains the decrease of Ms when the sample was substituted with Ce up to 0.15 (see Fig 9b). However, above the limit (x > 0.15 to 0.25), Ms slightly increases due to the magnetic moment of the system has contrarily different magnetic behavior. Rongjin et al. [10] have also found the magnetic behavior of heavily Ce-doped YIG nanocrystal which prepared by a glycothermal process, different from the expected theory. They reported the Ms of Ce:YIG nanocrystal increases as the Ce increases due to the enhancement of the super-exchange interaction between Fe3+ through oxygen ions due to presence of Ce3+ ions. For sample x = 0.3, the Ms suddenly decreases to the lowest value (see Figure 9(a)) because the film exists in a mixed phase. This is due to the solubility in cerium has achieved the limits. The excess of cerium strongly favors to form as CeO2 phase (impurity) in the Ce:YIG structure as depicted in Figure 5(a). Thus, this result indicates the existence of CeO2 phase has no beneficial effect to enhance the magnetic properties in Ce:YIG film. The similar evidence has been reported by Mao et al. [1] who reported the saturation magnetization of Ce:YIG ceramic decreased due to the presence of CeO2. Therefore, we can state the variation Ms of films in the series of increasing Ce contents is due to the intrinsic property (compositional substitution) and the presence of CeO2 (impurity) phase. Figure 9(b) displays the coercivity values of all films. The coercivity values of films varied unsystematically with the increment of Ce concentrations. The lowest coercivity is given by the
pure YIG film. The measured coercivity values of all films are in the range of 38-68 Oe. The coercivity values are also affected by the grain size as shown in Figure 9(c) (the value of film with x = 0.3 is not included because the film contains mixed phases). The Hc value increases as the grain size increases up to ~ 12.7 nm then decreases as the grain further increases. These results show that there are 2 types of films i.e single domain film and multi domain film. The Hc of films with single domain increases as the grain size increases up to a critical size, in this study the critical grain size is ~ 12.7 nm. The reduction of Hc with grain size is due to thermal effects, according to the relation:
where D is the grain diameter, g and h are constants.
Above critical diameter, the films become multidomain, hence the Hc decreases as the grain diameter increases according to relation
where a and b are constants.
4. Conclusion The effect of Ce-substituted YIG films on the nanostructure, surface roughness, optical and magnetic properties were investigated. Based on the microstructure analysis, the maximum substitutions of Ce3+ ions for Y3+ ions in Ce:YIG film was obtained at x = 0.25. It had been observed that the films with Ce concentrations (x = 0-0.25) showed very well homogeneous structure. Contrarily, film with x = 0.3 is inhomogeneous due to the existence of CeO2 impurity in the film structure. The measured average grain size of films was estimated around (11.1-14.4) nm. The films became thicker, however the surface roughness varied with the increasing of Ce concentrations. The optical studies showed all films have high optical transparency (above 80%) in the UV region. Based on the magnetic analysis, the Ce content reduced the saturation magnetization of the film at a certain x value then the saturation magnetization of films increased. In the presented result, films with x ≤ 0.25 attribute the excellent properties such as high crystallization, homogenous structure, small grain size and high optical transparency (above 80%). As regards to their best properties, the films are much more promising to be applied in magneto-optical devices. Acknowledgement
The authors would like to gratefully acknowledge for the fund grant by the Malaysian Ministry of Science, Technology and Innovation (MOSTI) via grant No. 02-01-02-SF0742 and Universiti Kebangsaan Malaysia (DPP-2014-040 and DLP-2014-006). References [1] T.-C. Mao, J.-C. Chen, Influence of the addition of CeO 2 on the microstructure and the magnetic properties of yttrium iron garnet ceramic, J. Magn. Magn. Mater. 302 (2006) 74-81. [2] M. Huang, S.-Y. Zhang, Growth and characterization of cerium-substituted yttrium iron garnet single crystals for magneto-optical applications, Appl. Phys. A. 74 (2002) 177-180. [3] S. Higuchi, K. Ueda, F. Yahiro, Y. Nakata, H. Uetsuhara, T. Okada, M. Maeda, Fabrications of cerium-substituted YIG thin films for magnetic field sensor by pulsed-laser deposition, IEEE Trans. Magn. 37 (2001) 2451-2453. [4] M. Gomi, K. Satoh, M. Abe, Giant Faraday rotation of Ce-substituted YIG films epitaxially grown by RF sputtering, Jpn. J. Appl. Phys. 27 (1988) L1536. [5] G. Anandha babu, G. Ravi, T. Mahalingam, M. Navaneethan, M. Arivanandhan, Y. Hayakawa, Size and Surface Effects of Ce-Doped NiO and Co3O4 Nanostructures on Ferromagnetism Behavior Prepared by the Microwave Route, J. Phys. Chem. C. 118 (2014) 23335-23348. [6] T. Sekijima, H. Itoh, T. Fujii, K. Wakino, M. Okada, Influence of growth atmosphere on solubility limit of Ce 3+ ions in Ce-substituted fibrous yttrium iron garnet single crystals, J. Cryst. Growth. 229 (2001) 409-414. [7] X. Zhou, F. Lin, X. Ma, W. Shi, Fabrication and magnetic properties of Ce 1 Y 2 Fe 5 O 12 thin films on GGG and SiO 2/Si substrates, J. Magn. Magn. Mater. 320 (2008) 1817-1821. [8] T. Uno, S. Noge, Growth of magneto-optic Ce: YIG thin films on amorphous silica substrates, J. Eur. Ceram. Soc. 21 (2001) 1957-1960. [9] M. Gomi, H. Furuyama, M. Abe, Strong magneto‐optical enhancement in highly Ce‐ substituted iron garnet films prepared by sputtering, J. Appl. Phys. 70 (1991) 7065-7067. [10] J. Rongjin, Y. Wenhui, F. Caixiang, Z. Yanwei, Glycothermal synthesis of heavily Cedoped YIG nanocrystals and their microstructures and magnetic properties, J. Mater. Chem. C. 1 (2013) 1763-1770. [11] H. Xu, H. Yang, Magnetic properties of YIG doped with cerium and gadolinium ions, J. Mater. Sci. - Mater. Electron. 19 (2008) 589-593. [12] D. Uhlmann, G. Teowee, Sol-gel science and technology: Current state and future prospects, J. Sol-Gel Sci. Technol. 13 (1998) 153-162. [13] H. Lee, T. Kim, S. Kim, Y. Yoon, S. Kim, A. Babajanyan, T. Ishibashi, B. Friedman, K. Lee, Magneto-optical imaging using a garnet indicator film prepared on glass substrates, J. Magn. Magn. Mater. 322 (2010) 2722-2727. [14] Y.-P. Fu, Electrical conductivity and magnetic properties of Li 0.5 Fe 2.5− x Cr x O 4 ferrite, Mater. Chem. Phys. 115 (2009) 334-338. [15] F.W. Aldbea, N. Ibrahim, M. Yahya, Effect of adding aluminum ion on the structural, optical, electrical and magnetic properties of terbium doped yttrium iron garnet nanoparticles films prepared by sol–gel method, Appl. Surf. Sci. 321 (2014) 150-157.
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Highlights
The maximum substitution of Ce3+ ions in the CexY3-xFe5O12 sol-gel film was x = 0.25.
The measured average grain size of films was estimated around (11.1-14.4) nm
All films produced high optical transparency above 80% in the UV region.
The Ce content reduced the saturation magnetization (Ms) of the film at a certain x value then the Ms increased
The coercivity values of films indicate there are multi domain and single domain film
The films with x ≤ 0.25 are suitable to be applied in magneto-optical devices.
LIST TO FIGURE Figure 1. The position of film in the applied magnetic field during the VSM measurement. H = applied field; N = north pole; S = south pole. Figure 2. X-ray diffraction pattern of CexY3-xFe5O12 films with (x = 0.0-0.3). Figure 3. Lattice parameter of CexY3-xFe5O12 films with (x = 0.0-0.3). Figure 4. BSE image shows the morphology of CexY3-xFe5O12 films at (A) x = 0.0, (B) x = 0.05, (C) x = 0.1, (D) x = 0.15, (E) x = 0.2 and (F) x = 0.25. The circle area shows the agglomeration of the grains and the arrow sign shows the void formation. Figure 5. (a) The BSE image of CexY3-xFe5O12 film with x = 0.3 and the undissolved spots of CeO2 phase appeares in the bright contrast in the film structure. (b) The typical cross-sectional image of CexY3-xFe5O12 film with x = 0.25. Figure 6. The average grain size and thickness of CexY3-xFe5O12 films as a function of Ce concentrations (x). Figure 7. Three dimensional AFM image of CexY3-xFe5O12 films with (x = 0.0-0.3). Figure 8. (a) Optical transmittance and (b) Optical absorption coefficient spectra of CexY3xFe5O12 films
with (x = 0.0-0.3).
Figure 9. (a) The variation of magnetization of CexY3-xFe5O12 films with (x = 0.0-0.3). (b) The variation of saturation magnetization and coercivity of CexY3-xFe5O12 films with (x = 0.0-0.25). (c) The coercivity values of CexY3-xFe5O12 films with (x = 0.0-0.25) versus grain size.
Figure 1. The position of film in the applied magnetic field during the VSM measurement. H = applied field; N = north pole; S = south pole.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
x =0
x = 0.05
x = 0.1
x = 0.15
x = 0.02
x = 0.25
x = 0.3
Figure 7
(a)
(b)
Figure 8
(a)
(b)
(c)
Figure 9