# Strong Shape Dependence of the Morin Transition

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Strong Shape Dependence of the Morin Transition

Nano Research Nano Res DOI 10.1007/s12274-014-0699-1 Strong Shape Dependence of the Morin Transition in α-Fe2O3 Single-Crystalline Nanostructures Jun Wang1,a(), Victor Aguilar2, Le Li1, Fa-gen Li1, Wen-zhong Wang3, Guo-meng Zhao1,2,b() Nano Res., Just Accepted Manuscript • DOI: 10.1007/s12274-014-0699-1 http://www.thenanoresearch.com on December 16 2014 © Tsinghua University Press 2014 Just Accepted This is a “Just Accepted” manuscript, which has been examined by the peer-review process and has been accepted for publication. A “Just Accepted” manuscript is published online shortly after its acceptance, which is prior to technical editing and formatting and author proofing. Tsinghua University Press (TUP) provides “Just Accepted” as an optional and free service which allows authors to make their results available to the research community as soon as possible after acceptance. 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The font is ArialMT 16 (automatically inserted by the publisher) The Moring transition in -Fe2O3 nanostructures depends strongly on the morphology: There is a sharp Morin transition at about 210 K for nanorings while it disappears for nanotubes. Provide the authors’ website if possible. Author 1, website 1 Author 2, website 2 1 Nano Res DOI (automatically inserted by the publisher) Research Article Strong Shape Dependence of the Morin Transition in α-Fe2O3 Single-Crystalline Nanostructures Jun Wang1,a(), Victor Aguilar2, Le Li1, Fa-gen Li1, Wen-zhong Wang3, Guo-meng Zhao1,2,b() 1Department of Physics, Faculty of Science, Ningbo University, Ningbo, P. R. China of Physics and Astronomy, California State University, Los Angeles, CA 90032, USA 3School of Science, Minzu University of China, Beijing 100081, P. R. China 2Department Received: day month year / Revised: day month year / Accepted: day month year (automatically inserted by the publisher) © Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2011 ABSTRACT Single-crystalline α-Fe2O3 nanorings (short nanotubes) and nanotubes were synthesized by a hydrothermal method. High-resolution transmission electron microscope and selected-area electron diffraction confirm that the axial directions of both nanorings and nanotubes are parallel to the crystalline c-axis. What is intriguing is that the Morin transition occurs at about 210 K in the short nanotubes with a mean tube length of about 115 nm and a mean outer diameter of about 169 nm while it disappears in the nanotubes with a mean tube length of about 317 nm and a mean outer diameter of about 148 nm. Detailed analyses of magnetization data, x-ray diffraction spectra, and room-temperature Mössbauer spectra demonstrate that this very strong shape dependence of the Morin transition is intrinsic to hematite. We can quantitatively explain this intriguing shape dependence in terms of opposite signs of the surface magnetic anisotropy constants in the surface planes parallel and perpendicular to the c-axis. KEYWORDS Hematite nanostructures, Morin transition, shape dependence, surface magnetic anisotropy 1. Introduction Hematite (α-Fe2O3) has a corundum crystal structure and orders antiferromagnetically below its Néel temperature of about 950 K. Bulk hematite exhibits a Morin transition [1] at about 260 K, below which it is in an antiferromagnetic (AF) phase, where the two antiparallel sublattice spins are aligned along the rhombohedral [111] axis. Above the Morin transition temperature TM, α-Fe2O3 is in the weak ferromagnetic (WF) phase, where the antiparallel spins are slightly canted and lie in the basal (111) plane rather than along [111] axis. The Morin transition is companied by the ———————————— Address correspondence to Guo-meng Zhao, [email protected]; Jun Wang, [email protected] 2 change of the total magnetic anisotropic constant from a negative value at T > TM to a positive value at T < TM. Interestingly, this AF-WF transition was found to depend on magnetic field. An applied magnetic field parallel to the rhombohedral [111] axis below TM was shown [2-4] to induce the spin-flip transition in the entire temperature range below TM. The AF-WF transition can also be induced by an applied magnetic field perpendicular to the [111] direction [5]. The magnetic structure, the Morin transition, and the field dependence of TM were explained [5, 6] in terms of phenomenological thermodynamic potential of Dzyaloshinsky. Figure 1 Transmission electron microscopic (TEM) images of the -Fe2O3 nanostructures prepared with different phosphate concentrations. (a) TEM image for the sample prepared with the phosphate concentration of 0.05 mM/L. A ring-like (short nanotube) morphology is seen in the sample. (b) TEM image for the sample prepared with the phosphate concentration of 0.50 mM/L. A tube-like morphology is observed in the sample. In recent years, magnetic nanostructures have attracted much attention, not only because of their interesting physical properties but also because of their broad technological applications. Of particular interest is a finite-size effect on ferromagnetic or ferromagnetic transition temperature. Finite-size effects have been studied in quasi-two-dimensional ultra-thin ferromagnetic films [7-10] and in quasi-zero-dimensional ultra-fine ferromagnetic or ferromagnetic nanoparticles [11-15]. The studies on thin films [7-10] and recent studies on nanoparticles [13-15] have consistently confirmed the finite-size scaling relationship predicted earlier [16]. Similarly, a finite-size effect on the Morin transition temperature was observed in nanosized α-Fe2O3 spherical particles [17-21]. The data show that TM decreases with decreasing particle size [17, 19], similar to the case of ferromagnetic or ferrimagnetic nanoparticles [13-15]. The reduction in the Morin transition temperature was interpreted as due to inherent lattice strain (lattice expansion) of nanocrystals [17, 20]. It was also shown [19] that the TM suppression is caused by both strain and the finite-size effect, commonly observed in ferromagnetic/ferromagnetic materials. More recently, Mitra et al. [22] have found that TM shifts from 251 K for ellipsoidal to 221 K for rhombohedra nanostructure, which suggests observable shape dependence of the Morin transition. Here we show that the Morin transition temperature depends very strongly on the shape of nanocrystals: TM shifts from 210 K for the nanorings (short nanotubes with a mean tube length of about 115 nm) to <10 K for the nanotubes with a mean tube length of about 317 nm. The very strong shape dependence of the Morin transition is quite intriguing considering the fact that the lattice strains of both nanoring and nanotube crystals are negligibly small, and that the sizes of nanotubes are too large to explain their complete suppression of TM by a finite-size effect. Instead, we can quantitatively explain this intriguing shape dependence in terms of opposite signs of the surface magnetic anisotropy constants in the surface planes parallel and perpendicular to the c-axis 2 2. Experimental α-Fe2O3 nanorings treatment at 220 ℃ for 48 h. After the autoclave were a was allowed to cool down to room temperature, hydrothermal method, which is similar to that the precipitate was separated by centrifugation, reported in [23]. In the typical process, FeCl 3, washed with deionized water and ethanol several NH4H2PO4 times, and finally dried in air at 80℃. While (phosphate), prepared and Na2SO4 by were dissolved in deionized water with concentrations keeping of 0.002, 0.05 and 0.55 mM/L, respectively. After unchanged, vigorous stirring for 15 min, 80 mL of solution concentration from 0.05 to 0.50 mM/L to produce was transferred into a 100 mL Teflon-lined nanotubes. stainless steel autoclave for all other experimental increasing the parameters phosphate hydrothermal Figure 2 Length (L) and wall-thickness (t) histograms of the -Fe2O3 nanostructures. (a), (b) Length and wall-thickness histograms of the -Fe2O3 nanorings. (c), (d) Length and wall-thickness histograms of the -Fe2O3 nanotubes. The solid lines are the best fitted curves by log-normal distribution functions. performed on a field-emission transmission The morphology of the samples was electron microscope (TEM, JEOL, JEM 2000, checked by transmission electron microscopy accelerating voltage 200 KV). X-ray diffraction (JEOL-2010, operated at 200 kV). High-resolution (XRD) spectra were taken by Rigaku Rint transmission electron microscopy (HRTEM) was D/Max-2400 X-ray diffractometer. Magnetic 3 moment was measured using a Quantum Design phosphate vibrating sample magnetometer (VSM) with a morphology is seen in the sample prepared with resolution better than 1×10-6 emu. The Mössbauer the phosphate concentration of 0.05 mM/L (Fig. spectra (MS) were taken by a WISSEL MB-500 1(a)) and a tube-like morphology is observed in (Germany) using a standard absorption method the with a source of concentration of 0.50 mM/L (Fig. 1(b)). In fact, Co under room temperature 57 and fitted using the Gauss-Newton method. sample concentrations. prepared with A the ring-like phosphate these nanorings can be described as short nanotubes with tube lengths shorter than outer diameters. 3. Results From TEM images we obtain histograms of their lengths and wall thicknesses, which are displayed in Fig. 2. Both length (Figs. 2(a), (Fig. 2(c)) and thickness 2(b), 2(c)) distributions are well described by a log-normal distribution function [24]: P(x) ln 2 x x 0 exp 2 2 x 2 1 (1) where σ is the standard deviation and lnx0 is the mean value of lnx. This distribution function was used to fit the diameter histogram of -Fe2O3 nanoparticles well [24]. The best fit of Eq. 1 to the data yields L0 =113.8±0.9 nm and t0 = 47.8±1.3 nm for the nanorings; L0 = 309.5±3.7 nm and t0 = 25.6±0.9 nm for the nanotubes. Figure 3 TEM images (left panels) and SAED patterns (right panels) for a single nanoring. The results consistently demonstrate the single-crystalline nature of the nanoring with its axis parallel to the [001] direction. Figure 4 TEM image (left panel) and SAED pattern (right panel) for a single nanotube. The result demonstrates the single-crystalline nature of the nanotube with its axis parallel Figure 1 shows transmission electron to the [001] direction. microscopic (TEM) images of the two α-Fe2O3 nanostructures prepared with different 4 Since or direction or the crystalline c-axis. From the SAED magnetic moment of a particle is proportional to pattern, we can evaluate the c-axis lattice constant. its volume, the mean value of length or thickness The obtained c = 13.77(2) Å is close to that should be length- or thickness-weighted [21], that determined from the XRD data (see below). is, x a x-ray diffraction intensity For the nanotube sample, it is very 0 2 x P(x)dx 0 (2) xP(x) Based on Eq. 2 and the fitting parameters for the histograms, the mean length and thickness are calculated to be 115 and 50 nm for the nanorings, respectively, and 317 and 30 nm for the nanotubes, respectively. The mean outer diameter of the short nanotubes is 169 nm, much larger than the mean tube length (115 nm), which is consistent with the ring morphology. The mean outer diameter of the nanotubes is 148 nm, much smaller than the mean tube length (317 nm), which is consistent with the tube morphology. It is remarkable that the mean thicknesses (50 and unlikely to get a top-view TEM image since the axes of the tubes tend to be parallel to the surface of the sample substrate. So we can only take TEM images of a single nanotube from the side-wall view. The left panel of Fig. 4 displays a side-wall-view TEM image of a nanotube. The image indicates that the length of the tube is about 200 nm. The single-crystalline nature of the nanotube is clearly confirmed by the SAED pattern (see the right panel of Fig. 4). The red arrow marks the [001] direction, which is determined by the SAED pattern. It is striking that the tube axis is also parallel to the [001] direction. 30 nm) of the nanorings and nanotubes inferred from the TEM images are very close to those (58 and 32 nm) deduced from x-ray diffraction spectra (see below). In the left panel of Figure 3(a), we show the TEM image of a single nanoring (the top view). This ring has a wall thickness of about 50 nm, which is very close to the mean value deduced from the histogram above and slightly smaller than the mean value of 58 nm deduced from the XRD peak widths (see below). The selected-area electron diffraction (SAED) pattern (right panel of Fig. 3(a)) with a clear hexagonal symmetry indicates that the nanoring is a single crystal with a ring axis parallel to the crystalline c-axis. In order to further prove the single-crystalline nature of the nanoring, we show the side-wall view of the ring (left panel of Fig. 3(b)) and the corresponding SAED pattern (right panel of Fig. 3(b)). The red arrow indicates the [001] direction, which is determined by the SAED pattern. It is apparent that the ring axis is parallel to the [001] 5 Figure 5 X-ray diffraction (XRD) spectra of the two -Fe2O3 nanostructures. (a), B) XRD spectra for the nanorings and Figure 5 shows x-ray diffraction (XRD) nanotubes, respectively. Rietveld refinement of the XRD data spectra of two α-Fe2O3 nanostructures prepared (solid blue lines) with a space group of R3c (trigonal hematite with the NH4H2PO4 concentrations of 0.05 and lattice) was carried out to obtain the lattice parameters and 0.50 mM/L, respectively. These samples are phase fractional coordinates of the atoms. The red lines represent the pure, as the spectra do not show any traces differences between the data and the refined curves. Figure 6 Expanded views of some X-ray diffraction peaks of the -Fe2O3 nanostructures. (a), (b), (c) The (110), (300), (220) peaks of the nanorings, respectively. (d), (e), (f) The (110), (300), (220) peaks of the nanotubes, respectively. The peaks are best fitted by two Lorentzians (solid lines), which are contributed from the Cu K1 and K2 radiations. The intrinsic peak width (after correcting for the instrumental broadening) is indicated in each figure. of other phases. Rietveld refinement of the XRD uncertainties of these fitting parameters are even data (see solid blue lines) with a space group of much larger than themselves. A large reliability R3c (trigonal hematite lattice) was carried out to factor (Rwp~11%) of the refinement makes it obtain fractional impossible to yield reliable fitting parameters for coordinates of the atoms. The atomic occupancy lattice strains (which are negligibly small) and was assumed to be 1.0 and not included in the particle sizes (which are quite large). In contrast, refinement. We tried to include the lattice strains the and particle sizes in the refinement but the refinements are quite accurate: a = b = 5.0311(12) the cell parameters and lattice parameters obtained from the 6 Å , c = 13.7760(33) Å for the nanotube sample; a = b by Eq. 3. The fitting parameters are indicated in each figure. = 5.0340(6) Å , c = 13.7635(17) Å for the nanoring The error bars in (a) are inside the symbols and not visible. sample. These parameters are slightly different from those for a bulk hematite [25]: a = b = 5.0351 Å , c = 13.7581 Å . It is known that the x-ray diffraction peaks are broadened by strain, lattice defects, and small Since the axes of both nanorings and particle size. When the density of lattice defects is nanotubes are parallel to the crystalline c-axis, the negligibly small, the broadening is contributed mean wall thickness of the nanorings and from both strain ε and particle size taυ. In this case, nanotubes can be quantitatively determined by there is a simple expression [26]: cos 0.89 2 sin (3) t a where the first term is the same as Scherrer's the peak widths of the x-ray diffraction peaks that are associated with the diffraction from the planes perpendicular to the c-axis. Figure 6 shows x-ray diffraction spectra of the (110), (300), and (220) peaks for the nanoring and nanotube samples. The peaks are best fitted by two Lorentzians (solid lines), which are contributed from the Cu Kα1 and Kα2 radiations. The fit has a constraint that the ratio of the Kα1 and Kα2 intensities is always equal to 2.0, the same as that used in Rietveld refinement. equation that is related to the particle size taυ, the second term is due to strain broadening, and ξ was found to be close to 2ε [27]. In Fig. 7, we plot βcosθ/λ versus 2ξsinθ/λ for the nanorings (Fig. 7(a)) and nanotubes (Fig. 7(b)). According to Eq. 3, a linear fit to the data gives information about the mean wall thickness taυ and strain εa along a and b axes. The strain is small and negative for both samples (see the numbers indicated in the figures). It is interesting that the magnitudes of the strain inferred from the XRD peak widths are very close to those calculated directly from the measured lattice parameters. For example, the strain is calculated to be 0.023(13)% from the lattice parameters for the nanorings and for the bulk, in excellent agreement with that (0.017(2)%) inferred from the XRD peak widths. Moreover, the mean wall thicknesses inferred from the XRD peak widths are very close to those determined from TEM images. This further justifies the validity of our Williamson-Hall analysis of the XRD peak widths. Figure 8 shows temperature and field dependences of the normalized magnetizations M(T)/M(350K) for the α-Fe2O3 singe-crystalline nanorings and nanotubes. The samples were initially cooled to 10 K in zero field and a field of 100 Oe was set at 10 K, and then the moment was Figure 7 Dependence of βcosθ/λ on 2sinθ/λ for the nanotings and nanotubes. The linear lines are the fitted curves taken upon warming up to 350 K and cooling down from 350 K to 10 K. At 10 K, other higher 7 fields (1 kOe, 10 kOe, and 50 kOe) were set and state) and it enhances significantly above TM (WF the moment was taken upon warming up to 350 state). It is interesting that TM for heating K and cooling down from 350 K to 10 K. It is measurements is significantly higher than that for remarkable that the magnetic behaviors of the cooling measurements (the arrows in the figure two nanostructures are very different. For the indicate the directions of the measurements). This nanorings, rapid difference is far larger than a difference (about 6 increase around 200 K upon warming, which is K) due to extrinsic thermal lag. This thermal associated with the Morin transition (see Fig. hysteresis was also observed in spherical α-Fe2O3 8(a)). It is worth noting that there seem to be two nanoparticles [19]. The observed intrinsic thermal transitions Morin hysteresis shows that the nature of the Morin transition temperatures. The reason for this is transition is of first-order. The result in Fig. 8(a) unclear. also the magnetization with slightly shows different suggests that the Morin transition temperature decreases with the increase of the applied magnetic field. The zero-field TM in the nanorings is about 210 K. What is striking is that the Morin transition is almost completely suppressed in the nanotubes (see Fig. 8(b)). Figure 8 Temperature and field dependences of the normalized magnetizations M(T)/M(350K) for the nanorings (a) and nanotubes (b). For the nanorings, the mean Morin transition temperatures for heating and cooling measurements are 210.5, 208.6, 199.5, and 175.2 K in the fields of 0.1, 1.0, 10, and 50 kOe, respectively. For the nanotubes, the Morin transition disappears down to 10 K. Figure 9 Magnetic hysteresis loops at 300 K (a) and 10 K (b) Furthermore, the magnetization below the for the nanorings and nanotubes. The inset in (a) is the Morin transition temperature TM is small (AF expanded view of the loop. The room-temperature saturation 8 magnetization Ms, as inferred from a linear fit to the magnetization data above 15 kOe, is the same (0.303±0.001 4. Discussion emu/g) for both samples. The saturation magnetizations for both samples are nearly the same as that (0.29±0.02 emu/g) [21] The completely different magnetic behaviors for a polycrystalline sample with a mean grain size of about 3 observed in the nanoring and nanotube samples m. are intriguing considering the fact that the two samples have the same saturation magnetization In Figure 9(a), we compare magnetic at 300 K and nearly the same lattice parameters. It hysteresis loops at 300 K for the nanorings and is known that the lattice strain can suppress TM nanotubes. There is a subtle difference in the according to an empirical relation deduced for magnetic hysteresis loops of the two samples. spherical nanoparticles [19]: ∆TM = −600ε K, The remanent magnetization Mr for the nanotube where ε is isotropic lattice strain in %. For a sample is about 30% higher than that for the uniaxial strain, the formula may be modified as nanoring sample, which is related to a higher ∆TM = −200εi K, where εi is the strain along coercive field in the former sample (see the inset certain crystalline axis. For the nanoring sample, of saturation a = 5.0340(6) Å , which is slightly smaller than magnetization Ms, as inferred from a linear fit to (5.0351 Å ) for a bulk hematite [25]. This implies the magnetization data above 15 kOe, is the same that εa = −0.023(13)% for the nanoring sample, in (0.303±0.001 emu/g) for both samples. The excellent saturation magnetizations for both samples are inferred from the XRD peak widths. For the also the same as that (0.29±0.02 emu/g) [21] for a nanotube sample, a = 5.0311(12) Å , so εa = polycrystalline sample with a mean grain size of −0.080(24)%, in good agreement about 3 μm. Since the saturation magnetization is (-0.048(6)%) inferred from the XRD peak widths. very sensitive to the occupancy of the Fe site, The negative strain would imply an increase in the fact that the saturation magnetizations of both TM according to the argument presented in Ref.17. nanaoring and nanotube samples are nearly the Therefore the suppression of TM cannot arise same as the bulk value suggests that the from the lattice strains along the a and b occupancies of the Fe site in the nanostructural directions. On the other hand, the lattice strain samples are very close to 1.0, which justifies our along the c direction is positive. Comparing the XRD Rietveld refinement. The fact that the measured c-axis lattice parameters of the two saturation magnetizations of the two nanostructures are close to the bulk value argues against the possibility of α-Fe2O3 to Fe3O4 phase transformation. Figure 9(b) shows magnetic samples with that for a bulk hematite [24], we can hysteresis loops at 10 K for the two samples. It is by 8(2) K and 26(5) K for the nanoring and clear that the nanotube sample remains weak nanotube ferromagnetic at 10 K (the absence of the Morin negative strains along a and b directions are transition down to 10 K) while the nanoring compensated by the positive strain along c sample is antiferromagnetic with zero saturation direction (also see Table 2 below) so that the magnetization. volume of unit cell remains nearly unchanged. Fig. 9(a)). However, the 3+ 3+ agreement with that (−0.017(2)%) with that readily calculate that εc = 0.040(12)% for the nanoring sample and 0.130(24)% for the nanotube sample. This would lead to the suppression of TM samples, respectively. The small This implies that the TM suppression due to lattice strains should be negligibly small. 9 As mentioned above, there is also an independent finite-size effect on TM unrelated to Another possibility is that the nanotubes the strain. For spherical nanoparticles, TM is may contain more lattice defects than the suppressed according to ∆TM = −1300/d K (Ref. 19), nanorings. If this were true, the line width of the where d is the mean diameter of spherical Mössbauer spectrum for the nanotube sample particles in nm. For the nanoring and nanotube would be broader than that for the nanoring samples, the smallest dimension is the wall sample because the line width is sensitive to thickness, which should play a similar role as the disorder, inhomogeneity, and lattice defects. In diameter of spherical particles [14]. With taυ = 58 contrast, the observed line width for nm and 32 nm for the nanoring and nanotube nanotube sample is smaller than that for the samples, respectively, the suppression of TM is nanoring sample by 33% and very close to that calculated to be 22 K and 41 K, respectively. for the bulk sample (see Fig. 10 and Table 1). In Therefore, due to the finite-size effect, TM would fact, all the Mössbauer parameters for the be reduced from the bulk value of 258 K (Ref. 17) nanotube sample are the same as those of the to 236 K and 217 K for the nanoring and bulk nanotube samples, respectively. For the nanoring uncertainties (see Table 1). If there would exist sample, the zero-field TM is about 211 K, which is substantial lattice defects, they would mostly be 25 K lower than the expected value from the present finite-size effect only. This additional sample in within surface the layers. the experimental The narrower TM Mössbauer line width observed in the nanotube suppression of 25 K should be caused by other sample is consistent with the fact that the mechanism discussed below. For the nanotube nanotubes have a smaller fraction of surface sample, the Morin transition is almost completely layers. suppressed, which cannot be explained by the Mössbauer spectra of both samples show only strain and/or finite-size effect. one Moreover, set of superparamagnetic the sextet, room-temperature suggesting relaxation at no room temperature. This is consistent with the observed magnetic hysteresis loops (see Fig. 9(a)). Table 1 The fitting parameters for the Mössbauer spectra of the nanorings and nanotubes. The parameters for the bulk samples are taken from Fig. 3 of Ref. [28]. Width (mm/s) Hyperfine field (kOe) Isomer shifts (mm/s) Nanoring 0.48±0.01 513.3±0.2 0.44±0.01 shifts (mm/s) -0.22±0.01 Nanotube 0.36±0.01 511.9±0.2 0.44±0.01 -0.22±0.01 Bulk 0.38±0.05 510±4 0.37±0.02 -0.20±0.02 Quadrupole Figure 10 Room-temperature Mössbauer spectra for the nanorings and nanotubes. The spectra are fitted by a single sextet (solid lines) with the fitting parameters displayed in Finally, we can quantitatively explain the strong shape dependence of the Morin transition Table 1. 10 temperature if we assume that the surface magnetic anisotropy constant KS is negative in the surface planes parallel to the c-axis and positive Figure 11 Numerically calculated TM as a function of in the surface planes perpendicular to the c-axis. KMD/KMD(bulk), where KMD (bulk) is the bulk anisotropy Indeed a negative value of KS was found in Ni constant. The calculation is based on a simple model presented (111) surface [29] while Ks is positive in Co(0001) in Ref. [31] and on the assumption that KFS remains surface [30]. For the nanorings, the surface area unchanged. for the planes parallel to the c-axis is similar to that for the planes perpendicular to the c-axis. Following this simple model, we can Therefore, the total KS will have a small positive numerically calculate TM as a function of or negative value due to a partial cancellation of KMD/KMD(bulk) on the the KS values (with opposite signs) in different remains unchanged, where KMD(bulk) is the bulk surface planes. In contrast, the surface area of a anisotropy constant. The calculated result is nanotube for the surface planes parallel to the shown in Fig. 11. It is apparent that TM is c-axis is much larger than that for the planes suppressed to zero when the magnitude of KMD perpendicular to the c-axis. This implies that the increases by 2.2%. Near this critical point, TM total KS in the nanotubes should have a large decreases rapidly with increasing the magnitude negative value. of KMD. For nanotubes, contribution of the surface assumption that KFS For bulk hematite, the Morin transition anisotropy is substantial and should be added to temperature is uniquely determined by the total the total anisotropy constant. Following the bulk anisotropy constant K at zero temperature expressions used in Refs.[29, 30], we have [31]. Contributions to K are mainly dipolar K MD K MD (bulk) 2K S // t a 2K S La anisotropy constant KMD, arising from magnetic dipolar interaction, and fine structure anisotropy (magneto-crystalline anisotropy) KFS, arising from spin-orbit coupling [31]. With KMD = −9.2×10−6 (4) where KS// and KS is the surface anisotropy constants for the planes parallel and erg/cm in the bulk perpendicular to the c-axis, respectively, and Laυ hematite, the Morin transition temperature was is the average tube length. Here we have predicted to be 0.281TN = 270 K (Ref. 31), very assumed that the surface areas of the inner and close to the measured bulk value of 258 K (Ref. outer walls are the same for simplicity. For the 17). nanotubes, TM is nearly suppressed to zero. erg/cm and KFS= 9.4×10 3 −6 3 According to Fig. 11, KMD/KMD(bulk) should be close to 1.022 for the nanotubes with Laυ = 317 nm and taυ = 32 nm. For the nanorings (short nanotubes), the zero-field TM is about 211 K. This implies that TM is totally suppressed by 47 K compared with the bulk value of 258 K. Since the finite-size effect can suppress TM by 22 K (see discussion above), the additional suppression of TM by 25 K should be due to an increase in KMD by about 0.58% according to Fig. 11, that is, KMD/KMD(bulk) = 1.0058 for the short nanotubes 11 with Laυ = 115 nm and taυ = 58 nm. Substituting lowest 220.8 K for rhombohedra structure, with these KMD/KMD(bulk), Laυ, and taυ values of both intermediate values of TM for the other two nanoring and nanotube samples into Eq. 4, we structures. In Table 2, we compare some obtain parameters for four nanostructures reported in two equations with two unknown variables, KS// and KS. Solving the two equations for the unknown KS// and KS yields KS// = −0.37 erg/cm2 and KS = 0.42 erg/cm2. The deduced Ref. [22] and two nanostructures reported here. The total lattice strain ε is calculated using ε = 2εa + εc, where εi is the percentage difference in the lattice constant of a nanostructure and the bulk. It is apparent that TM does not correlate with any of these parameters. For example, the Fe occupancy magnitudes of the surface anisotropy constants (0.96) in the ellipsoidal structure is significantly are in the same order of the experimental values lower than 1.0, but TM is the highest and close to found for Ni and Co [KS = -0.22 erg/cm2 for Ni(111) the bulk value. This implies that the Fe vacancies and 0.5 erg/cm for Co(0001) [29,30]. Therefore, should have little effect on the Morin transition. the observed intriguing experimental results can We thus believe that the weak shape dependence be naturally explained by a negative and a of the Morin transition observed in the previous positive surface anisotropy constant in the work [22] should also arise from the opposite surface planes parallel and perpendicular to the signs crystalline c-axis, respectively. constants. 2 of the surface The much magnetic lower anisotropic TM in the rhombohedra structure can be explained as due Table 2 Some parameters for six nanostructures. The to a much larger surface area parallel to the c axis parameters for ellipsoidal, spindle, fattened, and rhombohedra in the structure, in agreement with the observed structures are calculated from the data reported in Ref. [22]. HRTEM image [22]. The Fe occupancies for the nanoring and nanotube structures are inferred from the measured saturation magnetizations (see 5. Conclusions discussion in the text). In summary, we have prepared single-crystalline (%) Ellipsoida TM (K) 251.4 Spindle c/a 0.028(1) Fe occupancy 0.9600(3) 245.4 0.063(1) 0.9921(10) 2.7329(1) Fattened 231.5 -0.132(2) 0.9787(3) 2.7327(1) Rhombohedra 220.8 0.053(2) 0.9934(16) 2.7340(1) Nanoring 211 -0.007(25) 1.0 2.7341(5) Nanotube <10 -0.028(48) 1.0 2.7381(9) hematite nanorings and nanotubes using a hydrothermal 2.7337(1) transmission method. electron High-resolution microscope and selected-area electron diffraction confirm that the axial directions of both nanorings and nanotubes are parallel to the crystalline c-axis. Magnetic measurements show that there exists a first-order Morin transition at about 210 K in the nanoring crystals while this transition disappears in nanotube crystals. The current results suggest that the Morin transition depends very strongly Now we discuss the shape dependence of the Morin transition temperature observed in other nanostructures [22]. It was shown that TM shifted from highest 251.4 K for ellipsoidal to on the shape of nanostructures. This strong shape dependence of the Morin transition can be well explained by a negative and a positive surface anisotropy constant in the surface planes parallel 12 and perpendicular to the crystalline c-axis, respectively. Acknowledgements This work was supported by the National Natural Science Foundation of China (11174165), the Natural Science Foundation of Ningbo (2012A610051), and the K. C. Wong Magna Foundation. VA acknowledges financial support from NIH and NIGMS under MBRS-RISE M.S.-to-Ph.D. Program (R25GM061331). References [1] Morin, F. J. Magnetic susceptibility of α-Fe2O3 and α-Fe2O3 with added titanium. Phys. Rev. 1950, 78, 819-820. [2] Besser, P.; Morrish, A. Spin flopping in synthetic hematite crystals. Phys. Lett. 1964, 13, 289-290. [3] Foner, S.; Williamson, S. Low-temperature antiferromagnetic resonance in α-Fe2O3. J. Appl. Phys. 1965, 36, 1154-1156. [4] Hirone, T. Magnetic studies at the research institute for iron, steel and other metals. J. Appl. Phys. 1965, 36, 988-992. [5] Flanders, P. J.; Shtrikman, S. Magnetic field induced antiferromagnetic to weak ferromagnetic transitions in hematite. Solid State Commun. 1965, 3, 285-288. [6] Dzyaloshinsky, I. A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 1958, 4, 241-255. [7] Huang, F.; Mankey, G.; Kief, M.; Willis, R. Finite-size scaling behavior of ferromagnetic thin films. J. Appl. Phys. 1993, 73, 6760-6762. [8] Li, Y.; Baberschke, K. Dimensional crossover in ultrathin Ni (111) films on W (110). Phys. Rev. Lett. 1992, 68, 1208-1211. [9] Elmers, H.; Hauschild, J.; Höche, H.; Gradmann, U.; Bethge, H.; Heuer, D.; Köhler, U. Submonolayer magnetism of Fe (110) on W (110): finite width scaling of stripes and percolation between islands. Phys. Rev. Lett. 1994, 73, 898-901. [10] Schneider, C.; Bressler, P.; Schuster, P.; Kirschner, J.; De Miguel, J.; Miranda, R. Curie temperature of ultrathin films of fcc-cobalt epitaxially grown on atomically flat Cu(100) surfaces. Phys. Rev. Lett. 1990, 64, 1059-1062. [11] Tang, Z.; Sorensen, C.; Klabunde, K.; Hadjipanayis, G. Size-dependent Curie temperature in nanoscale MnFe2O4 particles. Phys. Rev. Lett. 1991, 67, 3602-2605. [12] Du, Y. W.; Xu, M. X.; Wu, J.; Shi, Y. B.; Lu, H. X.; Xue, R. H. Magnetic properties of ultrafine nickel particles. J. Appl. Phys. 1991, 70, 5903-5905. [13] Wang, J.; Wu, W.; Zhao, F.; Zhao, G. M. Curie temperature reduction in SiO2-coated ultrafine Fe3O4 nanoparticles: Quantitative agreement with a finite-size scaling law. Appl. Phys. Lett. 2011, 98, 083107-083109. [14] Wang, J.; Wu, W.; Zhao, F.; Zhao, G. M. Finite-size scaling behavior and intrinsic critical exponents of nickel: Comparison with the three-dimensional Heisenberg model. Phys. Rev. B 2011, 84, 174440-174444. [15] Wang, J.; Zhao, F.; Wu, W.; Zhao, G. M. Finite-size scaling relation of the Curie temperature in barium hexaferrite platelets. J. Appl. Phys. 2011, 110, 123909-123913. [16] Fisher, M. E.; Barber, M. N. Scaling theory for finite-size effects in the critical region. Phys. Rev. Lett. 1972, 28, 1516-1519. [17] Schroeer, D.; Nininger Jr, R. Morin transition in α-Fe2O3 microcyrstals. Phys. Rev. Lett. 1967, 19, 632-635. [18] Gallagher, P.; Gyorgy, E. Morin transition and lattice spacing of hematite as a function of particle size. Phys. Rev. 1969, 180, 622-623. [19] Muench, G.; Arajs, S.; Matijević, E. The Morin transition in small α-Fe2O3 particles. Phys. Status Solidi (a) 1985, 92, 187-192. [20] Morrish, A. H. Canted antiferromagnetism: Hematite; World Scientific: Singapore, 1994. [21] Bødker, F.; Hansen, M. F.; Koch, C. B.; Lefmann, K.; Mørup, S. Magnetic properties of hematite nanoparticles. Phys. Rev. B 2000, 61, 6826-6838. [22] Mitra, S.; Das, S.; Basu, S.; Sahu, P.; Mandal, K. Shape-and field-dependent Morin transitions in structured α-Fe2O3. J. Mag. Mag. Mater. 2009, 321, 2925-2931. [23] Jia, C.-J.; Sun, L.-D.; Luo, F.; Han, X.-D.; Heyderman, L. J.; Yan, Z.-G.; Yan, C.-H.; Zheng, K.; Zhang, Z.; Takano, M. Large-scale synthesis of single-crystalline iron oxide magnetic nanorings. J. Am. Chem. Soc. 2008, 130, 16968-16977. [24] Bacri, J. C.; Perzyenski, R.; Salin, D.; Cabuil, D.; Massart, R. Magnetic Colloidal Properties of Ionic Ferrofluids. J. Mag. Mag. Mater. 1986, 62, 36-46. [25] Hill, A.; Jiao, F.; Bruce, P.; Harrison, A.; Kockelmann, W.; Ritter, C. Neutron diffraction study of mesoporous and bulk hematite, α-Fe2O3. Chem. Mater. 2008, 20, 4891-4899. [26] Williamson, G.; Hall, W. X-ray line broadening from filed aluminium and wolfram. Acta Metallurgica 1953, 1, 22-31. [27] Smith, C. S.; Stickley, E. The width of x-ray diffraction lines from cold-worked tungsten and α-brass. Phys. Rev. 1943, 64, 191-198. [28] Fleischer, I.; Agresti, D. G.; Klingelhöfer, G.; Morris, R. 13 V. Distinct hematite populations from simultaneous fitting of Mössbauer spectra from Meridiani Planum, Mars. J. Geo. Res. in antiferromagnetic corundum-type sesquioxides. Phys. Rev. 1965, 138, A912-917. 2010, 115, E00F06. [29] Gradmann, U.; Bergholz, R.; Bergter, E. Magnetic surface anisotropies of clean Ni. Magnetics, IEEE Transactions on 1984, 20, 1840-1845. [30] Chappert, C.; Le Dang, K.; Beauvillain, P.; Hurdequint, H.; Renard, D. Ferromagnetic resonance studies of very thin cobalt films on a gold substrate. Phys. Rev. B 1986, 34, 3192-3197. [31] Artman, J.; Murphy, J.; Foner, S. Magnetic anisotropy 14