TY - JOUR
AU - Murakami,, Yasukazu
AB - Abstract This review examines methods of magnetic flux density measurements from the narrow grain boundary (GB) regions, the thickness of which is of the order of nanometers, produced in Nd–Fe–B-based sintered magnets. Despite of the complex crystallographic microstructure and the significant stray magnetic field of the sintered magnet, recent progress in electron holography allowed for the determination of the intrinsic magnetic flux density due to the GB which is embedded in the polycrystalline thin-foil. The methods appear to be useful as well for intensive studies about interface magnetism in a variety of systems. electron holography, Nd–Fe–B magnet, magnetic induction, interface, phase shift Introduction For the last two decades, significant progress has been attained in the peripheral techniques of electron holography, which appear to be useful for the examination of magnetic systems. With reference to the progress triggered by new apparatus, the multiple biprisms enabled us to collect electron holograms which are free from undesired Fresnel fringes (i.e. a source of artificial phase shift) [1–3]. Sophisticated aberration–correction systems brought a dramatic improvement in the lateral resolution in the reconstructed phase images [4–7], which accelerated the atomic-scale observations of electromagnetic field within materials. Both the sensitivity and precision in electron holography have been improved by the application of a direct detection camera showing an excellent modulation transfer function [8]. A challenge for ultrafast electron holography, with the aid of femto-/nanosecond electron pulses, is another topic of significant interest [9,10]. Progress has been attained as well in the field of the data analysis. The achievements include the three-dimensional reconstruction of magnetic field using artificial and/or nanostructured model specimens [11–13], process to determine the magnetic moment in the unit of Bohr magneton [14], separation between magnetic and strain information using dark-field electron holography [15,16], precision improvement in the phase analysis by using automated data collections and the subsequent averaging [17,18], noise reduction in electron holograms based on the principles of information science [19,20], etc. With reference to the applications of electron holography to magnetic materials, understanding of the spin textures has been one of the most crucial issues in the previous decade, from the viewpoints of both fundamental physics and spintronics. Actually, magnetic flux mapping by electron holography played an essential role on the determination of the complex structures of magnetic skyrmions [21–24]. The flux mapping has been also a powerful tool for the self-assemble mechanisms of magnetic nanoparticles which are hopeful candidate systems applied to the future ultrahigh-density recording devices [25–28]. Electron holography observations from the artificial structures (i.e. multiple layered films, nanowires, core/shell particles, ion-irradiated systems, etc.) provided useful information for the research area of micromagnetics [29–32]. Another essential target of the electron holography study is of permanent magnets, as they are highly important materials for industries. Among the permanent magnets developed so far, the Nd–Fe–B system [33–36] is the strongest and accordingly allows for reducing the volume of electronic components such as traction motors. For example, both electric and hybrid vehicles use many traction motors made of the Nd–Fe–B magnet, which is in a polycrystalline state composed of the ferromagnetic Nd2Fe14B crystal grains and other non-ferromagnetic grains. However, the coercivity (i.e. a measure of the critical magnetic field to induce undesired magnetization reversal) of commercially available sintered magnets remains much smaller than the theoretical upper limit. A key to further improvement of the coercivity is tailoring of the magnetization within a narrow grain boundary (GB) region, the width of which is only 3 nm or less. Electron holography can provide a route for determination of the intrinsic magnetic flux density of the GB region [i.e. magnetic flux density B (=μ0M) due to the spontaneous magnetization M of the GB region, where μ0 is the permeability of vacuum]. The aim of this review article is to summarize the methods of magnetization measurements from the GB regions that were produced in the Nd–Fe–B-based permanent magnets [37–39]. For the state of the art of material science and engineering of Nd–Fe–B magnets, refer to the review paper authored by Hono and Sepehri-Amin [40]. Magnetic flux density measurement from grain boundary region Method Figure 1a schematizes a cross section of single-crystalline Nd2Fe14B foil magnetized in the –y direction: refer to the same figure for the x-y-z coordinate. For this specimen, phase shift ϕ observed in the R–S line (i.e. a line along x direction) can be expressed by [41] $$\begin{equation} \phi =\sigma \int{V}_0\mathrm{d}z-\frac{e}{\mathrm{\hslash}}\iint{B}_y\mathrm{d}x\mathrm{d}z. \end{equation}$$(1) Fig. 1 Open in new tabDownload slide Methods of revealing the magnetic flux density in the GB phase. (a) Schematic cross-sectional view of a thin-foil specimen composed of only one Nd2Fe14B grain, A. (b) Phase shift of an electron wave (schematic representation) observed along the line connecting points R and S in (a). (c) Schematic cross-sectional view of a thin-foil specimen containing two Nd2Fe14B grains, A and B, and a thin GB phase indicated by the red line. (d) Phase shift of an electron wave (schematic representation) observed along the line connecting points R and S in (c). When electron holograms were acquired, the specimen was inclined by (θ1=) 16.9° around the y-axis and by 6.8° around the x-axis, to suppress undesired diffraction contrast. Reprinted with permission from [37]. Fig. 1 Open in new tabDownload slide Methods of revealing the magnetic flux density in the GB phase. (a) Schematic cross-sectional view of a thin-foil specimen composed of only one Nd2Fe14B grain, A. (b) Phase shift of an electron wave (schematic representation) observed along the line connecting points R and S in (a). (c) Schematic cross-sectional view of a thin-foil specimen containing two Nd2Fe14B grains, A and B, and a thin GB phase indicated by the red line. (d) Phase shift of an electron wave (schematic representation) observed along the line connecting points R and S in (c). When electron holograms were acquired, the specimen was inclined by (θ1=) 16.9° around the y-axis and by 6.8° around the x-axis, to suppress undesired diffraction contrast. Reprinted with permission from [37]. The first term represents the phase shift due to the electric potential V0, where σ stands for the interaction constant. Refer to Ref. [41] [the chapter authored by Smith and McCartney] for greater detail about the interaction constant σ. The second term provides the phase shift due to the y component of the magnetic flux density By, where e and |$\hslash$| stand for elementary charge and Planck’s constant divided by 2π, respectively. Since the Nd–Fe–B specimen is metallic, undesired electric charging due to electron exposure could be negligible. Thus, we assume that V0 simply represents the mean inner potential which depends on the crystal phases. The phase shift due to V0 could be separated from that of By by using a method referred to as ‘time reversal operation using electron wave’ [42]. For this analysis, we acquired another electron hologram from the specimen flipped upside down with reference to the incident electrons. This operation leads to a simple relationship between ϕ and By $$\begin{equation} \phi =-\frac{e}{\hslash}\iint{B}_y\mathrm{d}x\mathrm{d}z \end{equation}$$(2) Note that when this operation is applied to the observations, the phase shift due to magnetic field is amplified by a factor 2, although the phase shift due to electric field can be negligible: i.e. Eq. (2) has been multiplied by 1/2 to restore the original value of phase. Because of the relationship in Eq. (2), for the single-crystalline Nd2Fe14B specimen with a constant thickness, the plot of ϕ [along the R–S line] can be approximated to be a linear function, as illustrated in Fig. 1b. Figure 1c presents another thin-foil specimen in which a grain boundary [GB: shown in a red color] is sandwiched by two Nd2Fe14B grains A and B [37]. To simplify the discussion, both of the grains are assumed to be magnetized in the –y direction. Since the GB is tilted away from the foil plane, it provides a wide range of projection WGB, as illustrated in Fig. 1c. Importantly, when the magnetization of GB region is smaller than that of Nd2Fe14B matrix, the slope of ϕ is reduced over the area of WGB (GB area, in what follows) as shown in Fig. 1d. We can measure the phase difference Δϕ between the observation [red line in Fig. 1d] and the reference plot which is the extrapolation of curve fitting for the grain A [blue dotted line in Fig. 1d]. Thus, measuring Δϕ can be a key for the magnetic flux density measurement from the GB region. Fig. 2 Open in new tabDownload slide Observations of commercial Nd–Fe–B magnet. (a) TEM image showing crystallographic grains, A–I. (b) Reconstructed phase image of the rectangular area shown in (a). Reprinted with permission from [37]. Fig. 2 Open in new tabDownload slide Observations of commercial Nd–Fe–B magnet. (a) TEM image showing crystallographic grains, A–I. (b) Reconstructed phase image of the rectangular area shown in (a). Reprinted with permission from [37]. Regarding the y component of magnetic flux density By, which is a function of x and z, we need to consider two contributions. One of the contributions is the intrinsic magnetic flux density B (=μ0M) associated with the magnetization M (depending on the crystal phases). Our goal is to determine the intrinsic magnetic flux density due to the magnetization of GB (BGB, in what follows). The other contribution is the stray magnetic field from the ferromagnetic crystal grains (BS, in what follows). BS can be calculated straightforwardly, both inside and outside the specimen, once the specimen shape, the crystal orientations of individual grains and other such parameters are known, as mentioned later in greater detail. With aid of those calculations, BGB [as the source of the phase difference Δϕ] can be determined by electron holography. Note that because of the large magnetocrystalline anisotropy of the Nd2Fe12B phase (~4.5 MJm−3), the magnetization vector should be parallel to the easy magnetization axis (i.e. c-axis) [40]. Regarding the nanometer-scaled GB region which is ferromagnetically coupled with the neighboring Nd2Fe12B grains, the direction of magnetization vectors was assumed to be the average of the direction cosines of those neighboring grains when the magnetic flux density was calculated [37]. Fig. 3 Open in new tabDownload slide Experimental results of the phase shift due to the ultrathin GB phase. (a) Phase shift observed along the R–S line shown in Fig. 2a. Labels A and B indicate the regions of grains A and B (i.e. areas A and B), respectively. In the middle (area GB), a thin GB phase (tilted away from the incident electrons) is sandwiched between portions of grains A and B; refer to Fig. 1c. The light-blue dots show the observed phase shift. The red line represents a curve fitting result for area A, although it is plotted for the extended regions including GB and B. (b) Difference between the observations and the fitting curve, which determined the phase shift due to the GB phase to be −0.34 rad at the border of area GB. Reprinted with permission from [37]. Fig. 3 Open in new tabDownload slide Experimental results of the phase shift due to the ultrathin GB phase. (a) Phase shift observed along the R–S line shown in Fig. 2a. Labels A and B indicate the regions of grains A and B (i.e. areas A and B), respectively. In the middle (area GB), a thin GB phase (tilted away from the incident electrons) is sandwiched between portions of grains A and B; refer to Fig. 1c. The light-blue dots show the observed phase shift. The red line represents a curve fitting result for area A, although it is plotted for the extended regions including GB and B. (b) Difference between the observations and the fitting curve, which determined the phase shift due to the GB phase to be −0.34 rad at the border of area GB. Reprinted with permission from [37]. Fig. 4 Open in new tabDownload slide Determination of magnetic flux density due to the GB phase. (a) Comparison of observations and calculations (simulations), specifically Δϕ; i.e. difference between the observed phase shift and the fitting curves for area A. The calculations of Δϕ were obtained for several values of BGB of 1.2 T, 1.1 T, 1.0 T, 0.9 T, 0.5 T and 0 T. (b) Relationship between Δϕ and magnetic flux density due to the GB phase, BGB. Reprinted with permission from [37]. Fig. 4 Open in new tabDownload slide Determination of magnetic flux density due to the GB phase. (a) Comparison of observations and calculations (simulations), specifically Δϕ; i.e. difference between the observed phase shift and the fitting curves for area A. The calculations of Δϕ were obtained for several values of BGB of 1.2 T, 1.1 T, 1.0 T, 0.9 T, 0.5 T and 0 T. (b) Relationship between Δϕ and magnetic flux density due to the GB phase, BGB. Reprinted with permission from [37]. Analysis of commercial Nd–Fe–B magnet As mentioned in Introduction, the coercivity of the commercial magnet (~1.2 T) is much smaller than the theoretical upper limit (~7 T). Engineering of the magnetism in a narrow GB region is a key for further improvement of the coercivity. Until recently, the GB region in the commercial magnet was believed to be non-ferromagnetic. However, Sepehri-Amin et al. [43] reported an unexpectedly high concentration of Fe (>60%) within the GB region and accordingly predicted the presence of the ferromagnetic GB region, in contrast to the traditional understanding. Thus, their study triggered a direct magnetization measurement from the narrow GB region by electron holography [37]. Figure 2a shows a transmission electron microscopy (TEM) image of a thin-foil specimen, which contains five ferromagnetic Nd2Fe14B grains A–E. The other grains, F–I, were non-ferromagnetic crystal phases: i.e. NdOx grains F–G (x varies depending on the extent of oxidization), NdFe4B4 grain H and metallic Nd (α-Nd) grain I. We shall focus on the GB sandwiched by the Nd2Fe14B grains A and B: i.e. the GB highlighted in yellow. TEM observations confirmed the amorphous state in the GB region, the width (thickness) of which was 3 nm. This GB was tilted off from the foil plane by θ2 = 33.7°: refer to Fig. 1c for the definition of θ2. The value of θ2 was determined by the cross-sectional observation from the thin-foil specimen: i.e. after all the holography observations were collected, the thin-foil specimen was subject to cutting (by using focused-ion beam) to determine the specimen thickness and the tilting angle θ2 [37,38]. Electron diffraction revealed that the direction of c-axis (i.e. easy magnetization axis) was [0.077, −0.961, −0.267] for grain A and [0.103, −0.965, −0.242] for grain B, with reference to the x–y–z coordinate system shown in Fig. 2a. Deviation of these crystal orientations was only 2.1°. For the area indicated by the R–S line in Fig. 2a, the directions of the c-axis were approximately parallel to the plane of GB. For this geometry, the stray magnetic field (BS) within the narrow GB region was calculated to be 20 mT, which provides only negligible phase shift in electron holography observations. Thus, the area indicated by the R–S line was ideal for the determination of the intrinsic magnetic flux density of GB (BGS). Fig. 5 Open in new tabDownload slide Crystallographic microstructure of the thin-foiled specimen. (a) TEM image. (b) Schematic illustration of the thin-foiled specimen. Red arrows indicate the directions of the c-axis (projection in the x–y plane). (c) HAADF-STEM image of the GB phase. Reprinted with permission from [39]. Fig. 5 Open in new tabDownload slide Crystallographic microstructure of the thin-foiled specimen. (a) TEM image. (b) Schematic illustration of the thin-foiled specimen. Red arrows indicate the directions of the c-axis (projection in the x–y plane). (c) HAADF-STEM image of the GB phase. Reprinted with permission from [39]. When the electron holograms were collected, in order to suppress the undesired diffraction contrast, the specimen was tilted by (θ1=) 16.9° around the y-axis and by 6.8° around the x-axis. In this experimental setup, the length of WGB was about 110 nm: refer to Fig. 1c. Figure 2b provides a reconstructed phase image which was acquired from the rectangular area shown in Fig. 2a. The phase contour lines demonstrate that both grains A and B were magnetized in approximately one direction indicated by the red arrows. Figure 3a shows the phase shift ϕ which was observed along the R–S line. The plot of ϕ contains three regions: (i) area A, including only grain A; (ii) area GB, in which the GB is sandwiched between grains A and B; and (iii) area B, including only grain B. To obtain the phase information due to the GB, curve fitting with a quadric function was applied to area A. Using a quadratic function, in place of a linear function, was suited for the specimen showing a gradual change in thickness [37]. The result of fitting was extrapolated to the other areas, GB and B, as indicated by the red curve in Fig. 3a. For further convenience, the phase difference between observation and fitting, Δϕ, was plotted as a function of position along line R–S: see Fig. 3b. The value of Δϕ continued to decrease over the GB area, reaching a minimum of −0.34 rad at the border of this area. An increase of Δϕ, observed in the area B, was attributed to the deviation of the crystal orientations between grain A and grain B. It is noted that split-illumination electron holography [2] allowed for the hologram collections with sufficient beam intensity. As this technique improves the signal-to-noise ratio in holograms, the precision in the phase analysis ±0.08 rad (≈2π/79) could be attained [37]. Fig. 6 Open in new tabDownload slide Electron holography observations. (a) Schematic cross-sectional view of the specimen showing the tilt angle used to acquire the electron holograms. (b) TEM image obtained from the rectangular region shown in Fig. 5a. The yellow dots indicate the points from which the magnetic flux density in the GB phase was determined. (c) Reconstructed phase image showing the in-plane (x–y component) magnetic flux density. (d) Phase shift measured along the P–Q line shown in (c). (e) Plot of the phase difference Δϕ, between the observation and the fitting shown in (d). Reprinted with permission from [39]. Fig. 6 Open in new tabDownload slide Electron holography observations. (a) Schematic cross-sectional view of the specimen showing the tilt angle used to acquire the electron holograms. (b) TEM image obtained from the rectangular region shown in Fig. 5a. The yellow dots indicate the points from which the magnetic flux density in the GB phase was determined. (c) Reconstructed phase image showing the in-plane (x–y component) magnetic flux density. (d) Phase shift measured along the P–Q line shown in (c). (e) Plot of the phase difference Δϕ, between the observation and the fitting shown in (d). Reprinted with permission from [39]. To determine the intrinsic magnetic flux density of the narrow GB phase, BGB, the observation of Δϕ was compared with simulations. The calculations of Δϕ needed the input data about the specimen shape, specimen size, crystal orientations of individual grains, thickness of GB, tilting angle of GB, tilting angle of the specimen when the hologram was collected and other crystallographic/geometric information. All these parameters were determined by TEM and electron diffraction. The other input parameters were of the magnetization of the ferromagnetic crystal phases. The intrinsic magnetic flux density of the Nd2Fe14B phase was 1.6 T [40]. As the intrinsic magnetic flux density of GB (BGB) was unknown, the curves of Δϕ were calculated for several values of BGB: i.e. BGB = 0, 0.5, 0.9, 1.0, 1.1, and 1.2 T. Note that for the calculations of Δϕ, the effect of the stray magnetic field (i.e. phase modulation in the reference wave) was taken into consideration following the method proposed by Matteucci et al. [44]. In the presence of the gradual change in the specimen thickness [i.e. 101 nm at R, 109 nm at S, determined by the cross-sectional TEM observations], the phase shift due to the mean inner potential was considered as well [37]. The results of the calculations of Δϕ are summarized in Fig. 4a. The calculations reproduce the characteristic features of the observation: i.e. the decrease in Δϕ over the GB area, the minimum appearing at the GB border and the gradual increase in Δϕ in area B. Figure 4b shows the relationship between BGB and Δϕ calculated at the GB border. The value of BGB that explains the observation (Δϕ = −0.34 rad) was 1.0 T. Thus, we concluded that the GB region produced in the commercial Nd–Fe–B magnetic is ferromagnetic, in contrast to the traditional understating as non-ferromagnetic. Analysis of 0.1% Ga-doped Nd–Fe–B magnet Doping a small amount of Ga is an efficient way for controlling the magnetic properties of Nd–Fe–B permanent magnets [45]. The 0.1% Ga-doped Nd–Fe–B system has attracted attentions from researchers because this magnet exhibits a large coercivity (~2.0 T) and excellent squareness in the magnetic hysteresis loop [46]. Understanding of the GB magnetism is essential as well for the Ga-doped system [38]. However, since this specimen shows only a narrow GB region (1.6 nm), which is a half of that in the commercial Nd–Fe–B magnet, the electron holography needed further improvement in the precision. Three methods were applied to the precision improvement [39]. First, as it was employed for the observations of the commercial Nd–Fe–B magnet free from Ga doping, the specimen was tilted away from the crystallographic zone axis to suppress undesired phase shift due to electron diffraction. Secondly, to separate the phase shift due to magnetic flux density from that of the mean inner potential, another set of electron hologram was collected by making the specimen upside down with respect to the incident electrons: i.e. application of the time reversal operation using the electron wave. Thirdly, over 100 reconstructed phase images were averaged to reduce the statistical error in phase analysis. Because of the applications of those methods, the precision of ±0.03 rad (≈2π/209) could be attained for the phase analysis, which was determined by the standard deviation with reference to the curve fitting. As shown in Figs. 5a and b, a thin-foiled specimen of 0.1% Ga-doped Nd–Fe–B contained seven Nd2Fe14B grains A–G [39]. The other grains H and I were in the non-ferromagnetic |$Ia\overline{3}$| phase [45]. The red arrows in Fig. 5b indicate the in-plane (x–y) component of the c-axis of Nd2Fe14B grains, determined by electron diffraction. We shall focus on the GB lying between grain A and grain B. The GB region was in the amorphous state, as revealed by the high-angle annular dark-field scanning TEM (HAADF-STEM) image of Fig. 5c. Figure 6a schematizes the experimental setup for the collection of electron holography, in which the foil plane was tilted away from the x-axis by 29.6°. For this condition, a TEM image provided a wide range of the GB area (i.e. projection of GB), in which the tilted GB was sandwiched by the Nd2Fe14B grains A and B: refer to the area indicated by the dotted lines in Fig. 6b. With reference to the reconstructed phase image of Fig. 6c, we obtained the plot of the phase vs position in the P–Q line [crossing the position 3 in (b)] as shown in Fig. 6d. As demonstrated in the study of commercial Nd–Fe–B magnet, application of the curve fitting to area A resulted in the phase difference (Δϕ) between the observation and fitting: see Fig. 6e. The value of Δϕ continued to decrease over the GB area, which indicates a reduction of the spontaneous magnetization in the GB phase. It was difficult to accurately determine the position of the minimum Δϕ point because of the insufficient lateral resolution [39]. Therefore, another curve fitting (simply using a linear function) was applied to the GB area: see the red dotted line in Fig. 6e. The value of Δϕ at the GB border was determined to be −0.19 rad for the P–Q line. Following the method explained by using Fig. 4 (i.e. comparison between the observation of Δϕ and the simulations), the intrinsic magnetic flux density of the GB region (BGB) was determined to be 0.8 T. In addition to position 3 which crossed the P–Q line, for the GB region of 0.1% Ga-doped magnet, BGB was determined from the other three positions (1, 2 and 4) in Fig. 6b. The results were plotted in Fig. 7, in which the error bars were determined by the standard deviations in the plots of Δϕ. The value of BGB shows only a negligible change over the range of observation (~700 nm). The average of BGB was determined to be 0.8 ± 0.1 T, which is smaller than that for a commercial Nd–Fe–B magnet subjected to the optimal heat treatment (~1.0 T) [37]. Following the calculations by Sakuma et al. [47], magnetization can be reduced by a depression of the Fe content. The Fe content in the GB phase produced in the 0.1% Ga-doped magnet was ~56 at% [46], which was smaller than that of commercial magnet, ~63 at% [37]. Thus, the electron holography observations on the magnetic flux density are consistent with the chemical composition analysis. Fig. 7 Open in new tabDownload slide Values of BGB determined from points 1–4 shown in Fig. 6b. Reprinted with permission from [39]. Fig. 7 Open in new tabDownload slide Values of BGB determined from points 1–4 shown in Fig. 6b. Reprinted with permission from [39]. Concluding remarks As expressed by Eqs. (1) and (2) given in the main text, for electron holography, the phase shift ϕ can be related to the in-plane component of magnetic flux density of a specimen. Because of this relationship, electron holography can be a powerful tool for collecting magnetic information from a nanometer-sized area. However, electron holography is only sensitive to the magnetic flux density B, in place of the magnetization M. This point has hampered the application to the study of permanent magnet, in which the signal (phase shift) due to M may be obscured by the stray magnetic field produced outside/inside of the specimen. For a solution to this problem, this article described the method which employed thorough simulations incorporating the undesired phase shift due to the stray magnetic field etc. The method can be applied to studies of various magnetic compounds showing complex microstructures. Acknowledgements The authors thank K. Hono, T. Ohkubo, T. T. Sasaki (NIMS), K. Niitus (Kyoto Univ.), K. Harada, K. Shimada, Y. Iwasaki, D. Shindo, H. S. Park (RIKEN), T. Tanigaki, Y. Takahashi, T. Akashi, T. Matsuda, H. Shinada (Hitachi Ltd.,), Y. Takeno (Tohoku Univ.), Y. Takada, T. Sato, Y. Kaneko (Toyota Central R&D Labs.), A. Kato (Toyota Motor Co.), R. Sawada, H. Nakajima, T. Tamaoka and A. Sato (Kyushu Univ.) for their collaborations about the studies of permanent magnets. 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TI - Magnetic flux density measurements from narrow grain boundaries produced in sintered permanent magnets
JF - Microscopy
DO - 10.1093/jmicro/dfaa032
DA - 2021-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/magnetic-flux-density-measurements-from-narrow-grain-boundaries-5CyRslLprm
SP - 17
EP - 23
VL - 70
IS - 1
DP - DeepDyve
ER -