Precursor effects and field-induced short-range order above the Verwey transition in single Fe3O4 crystals

The internal fields in single crystals of magnetite (Fe3O4) have been previously studied through muon-spin rotation (μSR). By Maximum-Entropy (ME) μSR, we analyze Fe3O4 μSR data with external fields parallel to the , or axis. The ME peak-to-noise ratio is optimized by varying the filter time and time interval. Several μSR time series indicate a beat pattern. Using MEμSR, a second frequency signal is seen at non-zero fields in the temperature range above the Verwey transition (TV = ∼123 K). At zero field, MEμSR confirms with much-improved precision the existence of one frequency signal found earlier by curve fitting (CF) and Fourier transformation (FT). We compare our room temperature (RT) field-dependent MEμSR transforms for Fe3O4 with those found at 205 K to study a second order phase transition at the Wigner temperature (TW = ∼247 K). At RT and 205 K for fields below the demagnetization field and parallel to Fe3O4, a second MEμSR frequency is observed, missed by CF and FT. These extra magnetic fields fall on the extended magnetization curves below and above TW. At RT, a small field induces a short-range order similar to the precursor effects in the TV – TW interval. At 205 K within that precursor T-interval, we observe a comparable RT-disordered state. The existence of these additional internal fields is likely related to phonon-assisted 3d-electron(-spin) hopping and short-range order behaviors. Our MEμSR studies lead to a better understanding of the local magnetism in this Mott-Wigner glass.The internal fields in single crystals of magnetite (Fe3O4) have been previously studied through muon-spin rotation (μSR). By Maximum-Entropy (ME) μSR, we analyze Fe3O4 μSR data with external fields parallel to the , or axis. The ME peak-to-noise ratio is optimized by varying the filter time and time interval. Several μSR time series indicate a beat pattern. Using MEμSR, a second frequency signal is seen at non-zero fields in the temperature range above the Verwey transition (TV = ∼123 K). At zero field, MEμSR confirms with much-improved precision the existence of one frequency signal found earlier by curve fitting (CF) and Fourier transformation (FT). We compare our room temperature (RT) field-dependent MEμSR transforms for Fe3O4 with those found at 205 K to study a second order phase transition at the Wigner temperature (TW = ∼247 K). At RT and 205 K for fields below the demagnetization field and parallel to Fe3O4, a second MEμSR frequency is observed, missed by CF and FT. T...


I. INTRODUCTION
Magnetite (Fe 3 O 4 ) is a ferrimagnetic oxide. At the Verwey temperature (T V ≈ 123 K) Fe 3 O 4 shows a semimetal-to-insulator transition, which is related to the properties of the delocalized "extra 3d" (3d * ) electrons. The Verwey transition is a first order transition. 1,2 For the conduction of Fe 3 O 4 two models are being considered: either phonon-assisted electron hopping with the 3d * being the only spin-current carries or a broad energy band (about 1 eV) conduction mechanism. 1 The Wannier states of these 3d * conduction electrons in Fe 3 O 4 indicate a mixture of localized and delocalized electron & hole states. 3,4 Magnetic anomalies, observed between T V and the Wigner temperature (T W ≈ 247 K) show, Fe 3 O 4 can be considered a Wigner electron glass. 5 The resistivity is a minimum at T W suggesting glassy, precursor effects in the T V -T W region.
Fe 3 O 4 has a fully spin-polarized band, making an ideal compound to study basic spintronics. 6 The magneto-chemical formula is: 4 2-. The Fe ions have two different configurations: in the tetrahedral site (A) the Fe 3+ ion is surrounded by four O 2ions, while in the octahedral site (B) the Fe ion is surrounded by six O 2ions. 7,8 The electron configuration of (Fe 3+ ) A is 3d 5 and all 5 spins are parallel. These spins on the A sublattice are antiparallel to those on the B sublattice; the 3d * electron has a spin-down orientation. The net magnetization is 4 μ B per unit cell. 3,4 The top energy band is half filled by the fully spin-polarized 3d * electrons. Our studies support the phonon-assisted electron-spin hopping model and the Mott-Wigner glass description of Fe 3 O 4 . 5

ARTICLE
scitation.org/journal/adv II. μSR STUDIES 5,9 USING FOURIER TRANSFORMATION AND CURVE FITTING The behavior of the internal magnetic field in Fe 3 O 4 as a function of temperature and applied magnetic field has been reported in previous μSR studies. 5,9 At Tw, there appears to be a 2nd order phase transition. See Fig. 1. 5 These studies have used Fourier transformation (FT) and curve fitting (CF). The CF fits have only produced reasonable frequency (or local field (B loc ) values) with large error bars for its amplitudes and relaxation rates. B loc can be written as B loc = Bint + Bext -B dem for B > B dem and B loc = B loc (ZF) for B < B dem . B int is the internal field; B dem is the demagnetization field; ZF means zero field.
An external field (Bext) dependency for Fe 3 O 4 at RT for the B // <110> shows that the frequencies follow an expected linear trend with a slope of 13.55 MHz/kOe and B dem = ∼ 0.9 kOe. The broadening with increasing Bext was interpreted as two frequency signals. At Bext = 5 kOe, B loc (high) is somewhat larger than theoretically allowed. 5 The frequency behavior at 205 K, B // <110> differs from the one observed at RT, B // <110>. External field dependence of μSR frequencies for B // <110> at 205 K shows that two frequency signals follow the expected linear parallel trends with slopes of 13.55 MHz/kOe with both a B dem of about 0.5 kOe and a 5-MHz shift. The lower frequency at zero field and 100 Oe has not been seen by FT and CF analysis. At zero field, FT & CF studies 5,9 indicated only one frequency signal. Note, the relaxation rates at 205 K are much smaller; thus, smaller errors in frequency are seen at 205 K compared to those at RT.

A. MEμSR Fe 3 O 4 investigations
Our ME μSR study analyzes original Fe 3 O 4 data to investigate these T V & T W transitions and potential precursor effects in the T V -T W region. For more detail on our ME-Burg μSR technique, we refer to Refs. 10 and 11.  The field-dependent μSR data of B // <111> Fe 3 O 4 at RT have been analyzed using MEμSR. An optimized T f of 0.6 μs is found to be about twice the relaxation time of the frequency signal. The MEμSR <111> Fe 3 O 4 transform observations are consistent and yet are more precise than those previously made by CF. 5,9 In Fig. 2, we show the <111> ME transform at 5 kOe, RT fitted with two Lorentzians (Lor) that describes the asymmetric broad peak best. A fit with two Gaussians (Gau) or a Gau/Lor combination gave a higher χ 2 . In Table I, our fit parameters are given. The fact that the Lor fits are better implicates exponential μ-spin relaxation, caused by the muons moving among the μ-O sites within the empty O octahedrons at RT. 9,11 Note, for a perfect alignment B / / <111> the six muon-stop sites within an empty O-octahedron 9,11 are magnetically and electrically equivalent, due to rotation symmetry around the <111> axis. Assuming one μ-site and one B dem , the highest 111-MHz frequency is about the maximum allowable. 5,9 A slight misalignment of the B // <111> alignment causes the μ-O sites to be magnetically different, resulting in an asymmetric ME distribution.  With an optimized T f of 0.5 μs, two peaks become overlapping. See Fig. 3. The lower, less intense, frequency signal corresponds with those reported in the previous studies. The higher frequency signal (not found previously by CF an FT) is found below B dem . This strong alternative interpretation suggests that a small field induces a shortrange order, similar to what is seen in the T V -T W interval.  signal at 60 MHz. This sharp signal corresponds to the signal seen by CF. The 55 & 57 -MHz signals were not seen in FT and CF analysis.
In Fig. 5, an MEμSR transform (T f = 0.5 μs) is shown for B (720 Oe) // <110> at T = 205 K. Besides the peak at 68 MHz, a second signal is seen at 60 MHz. The frequency difference between the two signals is ∼8 MHz which is in the order of the frequency shifts seen at T V and at T W in zero field. 5,9 This suggests that in the T V -T W region, two magnetically different subregions in the B sublattice exists: one following the normal magnetization curve, and one for which a comparable RT-disordered state has been induced by the small applied field. This may well be glassy, precursor effects.
A summary of our MEμSR Fe 3 O 4 <110> analysis is shown in Table II.

D. MEμSR Fe 3 O 4 B // <100> field dependence at RT
We have evaluated the MEμSR transforms (T f = 1 μs) in Fe 3 O 4 for small B // <100> at RT. These ME transforms for B // <100> indicate no substantial change up to 1 kOe. Only one frequency signal at zero field, 50 Oe and 1 kOe is seen. The fitted frequencies of about 55

III. CONCLUSIVE REMARKS
Using MEμSR, we find with improved precision and sensitivity the local magnetic fields in Fe 3 O 4 single crystals and find frequency signals not seen by FT nor CV. Two signals close in frequency is consistent with the beat patterns seen in the μSR time series. We have found two frequencies for the <111> orientation at 5 kOe and RT; the smaller signal indicates a slight misalignment of the <111> Fe 3 O 4 crystal. For the <100> orientation and fields less than 1 kOe, one peak with no change in frequency is found. Thus, the <100> B dem is larger than 1 kOe; effectively zero field exists for B < 1 kOe.
For the <110> orientation: at RT and 500 Oe, two frequency signals are observed. These two broad signals suggest two separate magnetizations at RT. These B-dependent MEμSR peaks reveal a much different behavior at RT than at 205 K. The sharp MEμSR signals at 205 K suggest also two separate magnetizations. The difference between the two frequencies and the expected values is below 3 %. See Table II. In both cases, these effects are plausibly caused by glassy, precursor effects above T V and induced by small magnetic fields below B dem .
Our new MEμSR outcomes are consistent with the single crystal diffuse scattering findings of short-range correlations in Fe 3 O 4 above T V . 12 Further, analysis of interatomic pair distribution derived from x-ray scattering data has revealed that short-range order can be observed even up to the ferrimagnetic Neél temperature of ∼850 K. 13 These short-range correlations may also explain the spontaneous magnetization reversals seen in a low field and in the T V -T W region. 14,15 In sum, Fe 3 O 4 is more likely a narrow-band (degenerate) semiconductor than a semimetal at RT. The 3d * electrons are important ingredients of the conduction mechanism in Fe 3 O 4 , supporting the phonon-assisted electron(-spin) hopping model. 5,9 Through MEμSR analysis, a clearer picture of the magnetic environments in Fe 3 O 4 is found. This new interpretation indicates two magnetization trends, reflecting different short-range orders in the ZF phase diagram.