Magnetic structure and phase transition at the surface region of Fe3O4(100)

The magnetic structure and phase transition of the near-surface region of Fe3O4(100) was investigated by 57Fe conversion electron Mössbauer spectroscopy (CEMS) and theoretical calculations. It is revealed that at 300 K the magnetization is in-plane in the surface region and cants from the in-plane to the 〈111〉 direction in a deeper region suggesting the presence of a noncollinear magnetic structure. The critical exponents for the tetrahedral and octahedral sites are estimated to be 0.24 ± 0.01 and 0.28 ± 0.01, respectively. Near the critical temperature, furthermore, the magnetization direction in the surface region was found to deviate from the in-plane direction.


Introduction
Lowering of symmetry at surfaces exerts significant effects on the magnetic properties at surfaces such as spin orientation and phase transition, because the exchange interaction and magnetic anisotropy at surfaces are possibly modified as compared to the bulk. In the trend that the size of various devices is reduced to nanoscale, the magnetic properties at surfaces are increasingly important for device operation. Whereas surface magnetic anisotropy due to the broken symmetry could induce a noncollinear magnetic structure near surfaces [1,2], modification of the surface exchange interaction might bring about critical phenomena characteristic to surfaces. It has been shown that the critical temperature and critical exponent can be different than those of bulk when the exchange interaction at the surface is different [3,4].
Magnetite (Fe 3 O 4 ) has attracted much attention due to its fascinating physical properties. It has an inversespinel structure consisting of two iron sites of tetrahedral A and octahedral B, where the A site is Fe 3+ and the B site is a mixture of Fe 3+ and Fe 2+ at room temperature. Magnetite is a ferrimagnet, where spin 5/2 at the A site is antiferromagnetically coupled with spin 5/2 and 3/2 at the B site with a curie temperature of 858 K. The internal magnetic fields of the A and B sites are 48.9 and 45.2 T as measured by 57 Fe Mössbauer spectroscopy. The critical exponents of the magnetic phase transition are consistent with the value obtained by the three-dimensional Heisenberg model, although the values for the A and B sites are slightly different [5,6].
Magnetite thin films with various thicknesses have been grown on substrates and their magnetic properties have been studied in detail. The magnetization direction is determined by the competition between the bulk magnetocrystalline anisotropy, surface/interface magnetocrystalline anisotropy, exchange interaction, and the shape anisotropy [7][8][9]. Due to the large shape anisotropy energy, the magnetization of thin films tend to be inplane [10][11][12]. When the film thickness increases, it is reported that the magnetization direction changes from in-plane towards the bulk easy magnetization direction [13]. This might point to a non-collinear magnetic structure, although the magnetic structure was tacitly assumed to be collinear in the entire film. It is also reported that the magnetic anisotropy of the Fe 3 O 4 films are controllable by the lattice distortion and electric coupling at the interface [13][14][15][16]. Although the magnetic phase transition at the surface of Fe 3 O 4 is reported [17], Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the difference of the magnetic critical behavior between the surface region and bulk of Fe 3 O 4 is yet to be clarified. The surface magnetic anisotropy plays an important role for the magnetic properties of nanoparticles. Due to the surface effect, the shell part of nanoparticles could have a spin-canting structure as compared to the core part [18][19][20], which also corresponds to a noncollinear magnetic structure.
To clarify the surface magnetic properties, depth-resolved investigation of the magnetic properties is strongly required. The conversion electron Mössbauer spectroscopy (CEMS) probes the 57 Fe nuclear levels by detecting the internal conversion electrons from the sample. Since electrons suffer from energy loss when passing through solids, energy-resolved CEMS allows for depth-resolved analysis of the magnetism of materials [21]. Furthermore, by doping 57 Fe in thin films, 57 Fe CEMS can selectively probe the 57 Fe doping layer, which allows depth-selective investigation of magnetic properties [22,23].
We recently revealed the presence of noncollinear magnetic structures at the surfaces of iron and magnetite [24,25]. We also suggested that the magnetic phase transition at the interface is different from that in the film for an iron film epitaxially grown on MgO(100) [9]. The magnetic critical behavior at the surface region of magnetite may also be different from that of bulk, which is expected to give important information for the interpretation of the surface magnetism of the magnetite. In the present study, we investigated the magnetic structure and phase transition at the surface region of Fe 3 O 4 (100) by means of CEMS and theoretical calculations. It is revealed that the magnetization is in-plane in the surface region and cants from the in-plane to the 〈111〉 direction in a deeper region. It is also shown that the critical exponent at the surface region is significantly smaller than those for bulk magnetite suggesting that the surface region is described by the two-dimensional XY model rather than the three-dimensional Heisenberg model. As the temperature rises above 700 K, furthermore, the magnetization is observed to deviate from the in-plane direction.

Experimental and theoretical methods
The sample used in the present study is the (100) surface of a single crystal Fe 3 O 4 . To enhance the surface sensitivity of Mössbauer spectroscopy, we fabricated a 57 Fe 3 O 4 layer by depositing 57 Fe (95% enriched) on the Fe 3 O 4 (100) surface with a thickness of 20 nm at a rate of 0.01 nm/s in oxygen of 3×10 −4 Pa at 550 K after cleaning the surface in an ultra-high vacuum chamber (base pressure: 1×10 −8 Pa) [25]. The 57 Fe 3 O 4 was epitaxially grown as confirmed by reflection high energy electron diffraction.
The magnetic property of the sample was investigated by CEMS using a conventional 57 Co radioisotope with a γ-ray energy of 14.413 keV. The electron emitted via the internal conversion process associated with the deexcitation of 57 Fe nuclei was detected with a sealed-off proportional counter filled with rare gases. In the present study, a mixture of Ar (10%) and He (90%) was used, which improved the energy resolution to ΔE/ E=30% compared to ΔE/E=80% of the pure-He-filled proportional counter. The internal magnetic field and magnetization direction were analyzed by the Zeeman-split sextet of the CEMS spectrum.
The CEMS measurement was performed in a depth-selective manner. Figure S1 available online at stacks. iop.org/JPCO/4/115001/mmedia shows an output pulse height distribution of the proportional counter, which corresponds to the energy spectrum of the electrons emitted from the surface [26]. The main peak denoted by b corresponds to the K-shell conversion electron with an energy of 7.3 keV. The energy spectrum also reveals a low-energy tail below 5.4 keV, which corresponds to the electrons coming from a deeper region that suffer from energy loss in the sample. In previous studies, the depth range of the inelastically scattered conversion electrons was investigated in detail by measuring CEMS for Fe films with various thicknesses [27,28]. The depth ranges for the energy ranges of 4.9-7.3 and 3.3-5.7 keV are estimated to be 0-260 and 200-350 nm, respectively. On the other hand, the absorption length of the incident γ-ray is determined by the 57 Fe nuclear excitation and the photoelectron excitation. The cross section of the former process is 2.56 × 10 −18 cm 2 , which is about 400 times larger than the latter [29,30]. We can then estimate the absorption fraction of the incident γray in a specific depth region. In the case of the Fe 3 O 4 sample with a 57 Fe 3 O 4 layer (20 nm), 20% of the incident γ ray is absorbed within a depth of 0-20 nm. In depth regions of 20-260 nm and 200-350 nm, on the other hand, we estimate 4.8 and 3% of the incident γ ray to be absorbed. By combining the above two effects, i.e. the electron escape depth and the γ-ray absorption depth, it follows that the predominant probing depth of the energyresolved CEMS is 0-20 and 200-350 nm when the electrons at energies of 4.9-7.3 and 3.3-5.7 keV are detected, respectively.
The magneto-crystalline anisotropy (MCA) energy was evaluated by first-principles density-functional theory (DFT) calculations using the Vienna ab initio simulation package (VASP) [31][32][33]. In the calculations, wavefunctions were expanded in a plane-wave basis set, and the behaviors of core electrons were described by the projector augmented wave potential [34,35]. For the exchange and correlation energy, the spin-polarized generalized gradient approximation proposed by Perdew, Becke, and Ernzerhof was adopted [36].

Results and discussion
3.1. Magnetic structure 3.1.1. Experimental results Figure 1 shows the energy-resolved CEMS spectra at room temperature, where electrons with energies above 4.9 keV and of 3.5-5.7 keV are acquired. Two sets of six lines representing the Fe A and B sites are clearly observed. The origin of the horizontal axis in Mössbauer spectra is shown with respect to the center of the spectrum for α-Fe. The solid curves are fits with fit parameters of the internal magnetic fields (B hf ), the intensity ratio of the transitions of Δm=±1 (I Δm=±1 ) and Δm=0 (I Δm=0 ), where m is the magnetic quantum number of the 57 Fe nuclear spin, the isomer shift (δ) and the line widths of the peaks at the Fe sites. Because the quadrupole splitting is negligibly small at 300 K, the quadrupole splitting is excluded from the fit parameters in figure 1.
Magnetite often suffers from Fe deficiency described as Fe 3−x O 4 , where x is the deviation from stoichiometric magnetite Fe 3 O 4 . In a previous study, the relation between the deviation x and the intensity ratio of the tetrahedral site (P A ) to the octahedral site (P B ) is reported to be described as P B /P A =(2−6x)/(1+5x) [ 37]. The ratio P B /P A is clearly different in figure 1(a) and (b), where x is estimated at 0.03 and 0.02, respectively. This indicates that Fe is slightly more deficient near the surface. The values of δ for the A and B sites are estimated at 0.277±0.003 and 0.659±0.003mm s −1 , respectively, for both figure 1(a) and (b), which are similar to the bulk values. The temperature dependences of δ are also found to be similar to those of bulk [5]. This indicates that the chemical state of Fe atoms at the surface is similar to that of bulk.
The linewidth of the A (B) site spectrum is estimated at 0.24-0.28 (0.34-0.39) mm s −1 below 700 K, which is slightly larger than that of bulk [5,38]. With increasing temperature, the width is broadened to 0.50±0.07 (0.68±0.07)mm s −1 . Although the line broadening suggests enhanced diffusion of iron atoms [39], it is difficult to extract the diffusion coefficient from the line width because of other factors such as homogeneous and inhomogeneous broadening.
The magnetization direction of the sample can be evaluated from the intensity ratio of I Δm=±1 and I Δm=0 . The magnetization direction θ m as measured from the γ-ray direction is related to the intensity ratio as follows [40] In figure 1(a), the I Δm=±1 /I Δm=0 values at the A and B sites are evaluated to be 1.01 and 1.07, respectively. This means that the magnetization direction of Fe 3 O 4 (100) is almost in-plane in the surface region within a depth range of 20 nm. On the other hand, the magnetization directions of the A and B sites of figure 1(b)  The origin of the in-plane MCA of Fe 3 O 4 (100) is analyzed by the second-order perturbation of the spin-orbit interaction using the wavefunctions obtained by DFT calculations on the basis of the formulation presented in [44]. With the spin-orbit coupling constant of 54 meV for Fe and 24 meV for O, the MCA energy in the secondorder perturbation calculations is found to be consistent with the MCA energy from the first-principles calculations. Figure 2 The Fe A at the sub-surface (Fe 2 ) also shows a small contribution to the in-plane MCA.
Clarifying the surface MCA energy, the stability of the noncollinear magnetic structure shown in figure 3(a) was evaluated by the following equation for the total energy (E NC ) [9]; where A = 2.64×10 −6 erg/cm [45] is the exchange stiffness constant and 2πM 2 is the shape magnetic anisotropy energy. Since the present experimental result shows the magnetization is in-plane within 20 nm from the surface, we assume the surface region as z sur =20 nm and θ sur = 0. We also assume that the angle of the spin direction (θ) with respect to the surface in-plane (  figure 3(a). When 2πM 2 =1.58×10 6 erg cm −3 as obtained by DFT calculations, E NC is found to be positive irrespective of W indicating that the noncollinear magnetic structure is not stable. A previous study, on the other hand, has shown that the closure domain in magnetite is formed to reduce the magnetostatic energy [46] and the magnetostatic energy is reduced to 1/16 compared to a single magnetic domain [47]. Substituting 1.0×10 5 erg cm −3 for 2πM 2 , E NC is plotted in figure 3(b) as a function of W. It is found that the noncollinear magnetic structure is stabilized for W200 nm.
From these results, it is concluded that the surface region of Fe 3 O 4 (100) has a noncollinear magnetic structure, where the magnetization direction changes from in-plane to 〈111〉 towards a deeper region with a depth scale of ca. 200 nm. The in-plane magnetization observed on the (100) surface is similar to a previous study on the (111) surface. As discussed in previous papers, the noncollinear magnetic structure originates from the competition between the bulk magneto-crystalline anisotropy, surface magneto-crystalline anisotropy and magneto-static energy [9]. Figure 4 shows the CEMS spectra at various temperatures, where all electrons are acquired without energy resolution. Note that the CEMS spectra predominantly reflect the surface region of 0-20 nm because a 57 Fe 3 O 4 (20 nm) layer was formed on the sample surface. The Zeeman splitting is clearly decreased as the temperature increases from 578 to 862 K indicating a magnetic phase transition to a paramagnetic phase. An additional  paramagnetic component with a quadrupole splitting is observed at temperatures higher than 752 K. This additional component might be assigned to Fe 1−x O resulting from formation of oxygen vacancies at high temperature [48].

Phase transition and phonon
The temperature (T) dependences of the internal magnetic fields at the A and B sites are shown in figure 5 with fits by the following formula: where B 0 , β and T c are fit parameters corresponding to the internal magnetic field at T=0, the critical exponent and the critical temperature, respectively. The critical temperature is estimated at 861 K, which is consistent with a typical bulk value of ∼858 K within the data scatter reported in literatures. Therefore, it is considered that T c at the surface is the same as the bulk value. The critical exponents of the A and B sites are estimated at 0.24±0.01 and 0.28±0.01, respectively, which are significantly smaller than the values of 0.34±0.01 at the A site and 0.32±0.01 at the B site in the bulk [5]. The critical behavior near the Curie temperature is clearly different from those of bulk [5] as shown in figure 5. It is noted that the critical exponent of the B site is larger than that of the A site, which is also observed for bulk magnetite [5].
As theoretically investigated in previous studies [4], the surface critical exponent is lowered when the exchange interaction at the surface is enhanced as compared to the bulk exchange interaction. Although the inplane exchange interaction is not evaluated in the present calculations, the interlayer exchange interaction at the surface is found to be slightly smaller than that of bulk. Therefore, lowering of the critical exponent observed in the present study seems not to be caused by the difference of the surface exchange interaction. The critical exponents of the three-dimensional Heisenberg and two-dimensional XY systems are theoretically estimated at 0.365 and 0.231, respectively [49]. Actually, the critical exponent for bulk magnetite is theoretically estimated at 0.35±0.01 [50]. Since the values obtained in the present study are close to the theoretical value of the twodimensional XY model, the near-surface region (∼20 nm) of Fe 3 O 4 (100) seems to behave magnetically as a twodimensional XY system. In a previous study for FeAs with a noncollinear magnetic structure, on the other hand, the critical exponent is reported to be 0.16±0.02, which is significantly smaller than the values for the threedimensional Heisenberg and three-dimensional Ising models [51]. We therefore speculate that the noncollinear magnetic structure due to the competing interaction observed in the surface region of Fe 3 O 4 affects the critical behavior lowering the critical exponent.
We next discuss the magnetization direction near the critical temperature. Figure 6 shows the I Δm=±1 /I Δm=0 value measured at various temperatures. It is almost 1.0 up to 725 K indicating that the magnetization directions of the A and B sites are in-plane. Above 725 K, on the other hand, the values seem to deviate from 1.0 and become larger. The larger I Δm=±1 /I Δm=0 value indicates either a magnetization direction of near-[111] or random orientation. Note that the I Δm=±1 /I Δm=0 value is confirmed to return to 1.0 when the sample temperature is lowered to room temperature from T C . It has been shown that the magnetic anisotropy constant tends to get smaller with increasing temperature. We therefore argue here that the magnetization direction in the in-plane direction at room temperature becomes randomly oriented near the critical temperature.
The recoilless fraction ( f ) at a temperature of T is expressed with the Debye temperature (Θ) as [52] Since f is proportional to the total intensity of the CEMS spectrum, the Debye temperature can be estimated from the temperature dependence of CEMS. Because the total intensity I of the six lines with respect to the background intensity (I BG ) is proportional to f, the temperature dependence of I /I BG is plotted as shown in figure 7. To avoid the effect of suspected oxygen vacancies above 750 K, equation (4) is fitted to the data in the range of 375-750 K. The Debye temperature (Θ A(B) ) of the A (B) site is estimated at 191±7 (194±5) K, which The error bars in the figure are evaluated by the statistical uncertainty of the CEMS spectra. The dash-dotted curves are the data taken from a previous study [5], which is scaled so that the Curie temperature matches the present data.
is lower than the bulk value of 334 (314) K [52]. It is considered that the phonon in the surface region is softer than that in the bulk.

Conclusion
The magnetic structure and critical behavior in the surface region of Fe 3 O 4 (100) were investigated by CEMS and theoretical calculations. It is experimentally revealed that the surface region of Fe 3 O 4 (100) has a noncollinear magnetic structure in the depth direction, where the magnetization direction changes from in-plane to 〈111〉. The magneto-crystalline anisotropy energy at the Fe 3 O 4 (100) surface was estimated to be −1.75 erg cm −2 by DFT calculations indicating preference of the in-plane magnetization. By evaluating the total magnetic energy taking account of the effects of the closure domain structure, the non-collinear magnetic structure was found to be stable when the noncollinear region is larger than 200 nm, which is in good agreement with the present  experiment. While the critical temperature was similar to that of the bulk, the critical exponents were evaluated to be 0.24±0.01 and 0.28±0.01 for the A and B sites, respectively. Near the critical temperature, the magnetization direction at the surface seems to change from the in-plane direction to random orientation.