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Keywords:

  • FORC;
  • domain wall;
  • low temperature;
  • pinning;
  • susceptibility;
  • titanomagnetite

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Domain wall pinning in titanomagnetite has been investigated at low temperatures using first-order reversal curve (FORC) diagrams, AC magnetic susceptibility, and Lorentz transmission electron microscopy. A discontinuous transition from a low-coercivity extrinsic pinning regime to a high-coercivity intrinsic pinning regime is evident in low-temperature FORC diagrams on cooling from 100 to 50 K. Intrinsic pinning is characterized by a “crescent moon” FORC distribution with narrow coercivity distribution centered on 10–20 mT. This crescent-shaped FORC distribution is reproduced using a modification of Néel's (1955) one-dimensional theory of domain wall pinning in a random field. The pinning transition coincides with a thermally activated relaxation process (activation energy 0.13 ± 0.01 eV), attributed to electron hopping. The relaxation and intrinsic pinning are explained as a magnetoelastic aftereffect caused by enhancement of magnetocrystalline anisotropy due to rearrangement and localization of Fe2+−Fe3+ cations within the domain walls. This study provides experimental verification that Néel's theory is an appropriate quantitative framework for the analysis of FORC diagrams in multidomain titanomagnetite and suggests a potential method for the quantitative unmixing of multidomain signals from FORC diagrams in rock and environmental magnetic studies.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

A key task in many rock and environmental magnetic studies is to quantify the type, concentration, and grain size distribution of magnetic minerals within a geological sample. While current techniques are highly successful at revealing qualitative trends, they do not lend themselves readily to obtaining an unambiguous quantitative unmixing of the superparamagnetic (SP), single-domain (SD), pseudo single domain (PSD), and multidomain (MD) fractions present. A recent study demonstrated the potential of first-order reversal curve (FORC) diagrams [Pike et al., 1999; Roberts et al., 2000] to provide a quantitative method of separating domain state–specific signals from magnetic mixtures [Egli et al., 2010]. Quantitative models for noninteracting and interacting SD particles have been developed by Newell [2005] and Egli [2006], respectively. A numerical framework for quantitative analysis of MD FORC signals was proposed by Pike et al. [2001a], based on Néel's [1955] one-dimensional theory of domain wall pinning in a random field. Here we provide the first experimental verification of Néel's theory applied to low-temperature titanomagnetite, which paves the way for a quantitative method of unmixing MD signals from FORC diagrams. Low-temperature magnetometry is becoming increasingly important as a method of characterizing the magnetic mineralogy of rocks, as it allows samples to be studied without fear of thermal alteration [e.g., Engelmann et al., 2010]. Here we reveal dramatic changes in the FORC diagram of MD titanomagnetite at low temperature, which provides a powerful method of identifying this phase in natural samples.

Previous studies of iron-rich MD titanomagnetite (Fe3O4)1−x(Fe2TiO4)x have noted dramatic changes in magnetic properties below 150 K. Radhakrishnamurty et al. [1990] and Radhakrishnamurty and Likhite [1993] first reported a sharp increase in alternating current (AC) magnetic susceptibility above 50 K that is dependent on the frequency of the applied field. Magnetic aftereffect (MAE) studies demonstrate relaxation over the same temperature range [Torres et al., 1997; Walz et al., 1997]. The increase in AC susceptibility was reproduced by Moskowitz et al. [1998] and Carter-Stiglitz et al. [2006], who attributed the phenomenon to thermally activated electron hopping between Fe2+ and Fe3+ ions. These authors also observed that low-temperature remanence decreases suddenly upon warming through the temperature range where relaxation is observed, which was attributed to either unpinning of magnetic domain walls at an isotropic point [Moskowitz et al., 1998; Özdemir and Dunlop, 2003] or to the loss of anisotropy caused by the onset of electron hopping [Carter-Stiglitz et al., 2006]. There is still no consensus on the origin of anomalous low-temperature magnetic properties in MD titanomagnetite. This problem is exacerbated by the lack of measurements of the temperature (T) and composition (x) dependence of anisotropy and magnetostriction for T < 77 K [Klerk et al., 1977; Kąkol et al., 1991a]. Here we apply a series of techniques to examine domain wall pinning and relaxation in the titanomagnetite solid solution at low temperatures, with the aim of gaining further insight into the physics behind these phenomena.

2. Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

2.1. Sample Synthesis

Samples synthesized for this study are single-phase titanomagnetites with 0 ≤ x ≤ 0.6. With few exceptions, all samples are polycrystalline sintered pellets synthesized in a vertical gas-mixing furnace and quenched into water. The oxygen fugacity was regulated by CO/CO2 gas mixtures at 1 atm [Deines et al., 1974]. Stoichiometric quantities of 99.999% Fe2O3 and 99.99% TiO2 powders (Sigma-Aldrich) were weighed and mixed in an agate mortar and pestle under acetone and pressed into pellets. Pellets were suspended in the furnace using a platinum wire cage to minimize physical contact and incorporation of iron into the platinum holder [Merrill and Wyllie, 1973]. Gas mixtures were chosen to avoid approaching the titanomagnetite/titanohematite phase boundary [Lattard et al., 2006], as titanomagnetites near this boundary have a high degree of nonstoichiometry [Hauptmann, 1974; Senderov et al., 1993]. For nominal compositions x < 0.3, the sintering temperature was 1300°C. Samples with x ≥ 0.3 were sintered at 1250°C. With the exception of samples of pure magnetite and x = 0.1, which were fired for one run of 48 h, all samples were sintered in at least 2 runs of 24 h each, with regrinding and repressing of the pellets between runs. Kąkol et al. [1991b] observed that titanium ions tend to aggregate at grain boundaries in polycrystalline samples, and Lattard et al. [2006] found in their samples that the pellet edges were enriched in titanium after the first sintering. Our magnetic susceptibility measurements (section 2.2) indicated that such effects were eliminated after the second sintering run.

2.2. Sample Characterization

Curie temperatures were determined from susceptibility measurements made with an AGICO MFK1–FA Kappabridge, with low-temperature crysostat for measurements down to liquid nitrogen temperatures and a furnace for measurements up to 700°C. High-temperature measurements were performed under flowing argon gas to minimize oxidation during analysis. The derivative method described by Tauxe [1998] was used to determine the transition temperatures. Curie temperatures for all samples are listed in Table 1 and are in good agreement with the results of Bleil and Petersen [1982]. The measured Curie temperatures agree well with their nominal compositions. X-ray diffraction (XRD) using Bruker D8 Advanced powder diffractometer with CuKα1 X-ray source and an internal Si standard was applied as an additional check on stoichiometry and to highlight the presence of any impurities. All samples are single phase. A full list of lattice parameters measured by XRD is presented in Table 1. Results agree well with values from Wechsler et al. [1984] and Bosi et al. [2009].

Table 1. Curie Temperatures, Unit Cell Parameters, and Derived Compositions for the Studied Synthetic Titanomagnetites
Nominal Composition (x)Tc (±3°C)Composition From Tc (x)Unit Cell Length a (Å)Composition From Unit Cell Length (x)
0.15280.080(5)8.40674(39)0.120(4)
0.24680.180(5)8.41763(19)0.210(1)
0.34000.290(5)8.43162(12)0.310(1)
0.43170.410(5)8.44963(18)0.420(1)
0.52390.520(5)8.46339(16)0.500(1)
0.61520.630(5)8.48126(14)0.610(1)

2.3. FORC Diagrams

FORC measurements were carried out at the Institute for Rock Magnetism at the University of Minnesota on a Princeton Measurements Corporation (PMC) vibrating sample magnetometer (VSM) equipped with a helium cryostat, and were processed using the FORCinel software package [Harrison and Feinberg, 2008]. In addition to the single-phase, polycrystalline specimens described in section 2.1, single-crystal titanomagnetites [Bosi et al., 2009] were also measured. FORCs were obtained at room temperature, 150 K and 50 K for all samples. Additional measurements at 70 and 60 K were performed for a polycrystalline sample with x = 0.4. For a detailed description of the measurement and interpretation of FORC diagrams, the reader is referred to Pike et al. [1999], Roberts et al. [2000], Muxworthy and Roberts [2007], and Harrison et al. [2007].

2.4. Low-Temperature Hysteresis and Remanence Measurements

Hysteresis loops were measured for both polycrystalline and single-crystal specimens at 50 K, 150 K, and room temperature using the PMC VSM. A maximum field of 500 mT was sufficient to saturate all samples at all temperatures. Low-temperature saturation isothermal remanence (SIRM) measurements were carried out with a Quantum Design Magnetic Properties Measurement System (MPMS) at the Institute for Rock Magnetism. Field-cooled (FC) analyses were made by first cooling the sample from room temperature to 20 K in a 2.5 T field and then measuring remanence at 5 K steps during warming to 300 K in zero field. Zero-field-cooled (ZFC) analyses were made by first cooling the sample from 300 K to 20 K in zero field, applying a 2.5 T SIRM at 20 K, then measuring remanence at 5 K steps during warming to 300 K in zero field.

2.5. AC Magnetic Susceptibility

The frequency dependence of magnetic susceptibility was measured as a function of temperature using the MPMS. After cooling from room temperature to 10 K in zero field, each sample was given a 2.5 T SIRM. In-phase (χ′) and out-of-phase (χ″) magnetic susceptibility was then measured every 10°C during warming to room temperature using a 240 A/m AC field at five frequencies (1, 3, 10, 32, and 100 Hz). In some instances, a DC bias field was applied during warming to sweep away weakly pinned domain walls from the sample, thereby allowing us to examine the magnetic properties of strongly pinned domain walls.

2.6. Fresnel-Mode Lorentz Transmission Electron Microscopy (TEM)

Specimens for TEM were prepared from the polycrystalline samples using mechanical polishing and argon-ion beam thinning. Fresnel-mode Lorentz images were acquired on a FEI CM300 TEM. This instrument uses both liquid nitrogen and liquid helium sample holders for low-temperature observations. Magnetic domain walls are visible in Fresnel-mode images when the electron beam is slightly defocused [Chapman, 1984; Kasama et al., 2009, 2010]. The images presented were obtained using a low-magnification Lorentz lens and by underfocusing the beam.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

3.1. FORC Diagrams

The FORC diagram obtained for x = 0.4 and T = 100 K (Figure 1a) is representative of those obtained for all samples between room temperature and 100 K, and also of those obtained for 0 ≤ x ≤ 0.2 and T < 100 K. The FORC diagram is typical of annealed MD samples containing weakly pinned domain walls [Pike et al., 2001a]: the peak of the FORC distribution is at the origin, the coercivity is low (Hc ≤ 5 mT), and the intensity is spread vertically along the ±Hu axis. The vertical spread of the FORC function results mainly from the self-demagnetizing field of the sample [Pike et al., 2001a]. In most cases, however, an additional contribution to the signal along −Hu results from a time-dependent relaxation of the domain wall response, as modeled by Pike et al. [2001b], leading to an asymmetry of the FORC function about the Hc axis.

image

Figure 1. (a–d) FORC diagrams as a function of temperature in polycrystalline titanomagnetite with x = 0.4, evolving from (Figure 1a) classic multidomain behavior at 100 K to (Figure 1d) distinctive crescent-shaped distribution at 50 K. Diagrams are shown with a smoothing factor of 2.5 after Harrison and Feinberg [2008].

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Dramatic changes occur in FORC diagrams on cooling from 70 to 50 K (Figures 1b1d). At 70 K (Figure 1b), a weak MD signal with a higher coercivity (Hc = 10–25 mT) appears in addition to the low-coercivity signal discussed above. At 60 K (Figure 1c), the high-coercivity signal grows in intensity and takes on a distinctive crescent shape, while the intensity of the low-coercivity signal decreases. At 50 K (Figure 1d), the FORC diagram is dominated by the high-coercivity crescent. The coexistence of low- and high-coercivity signals at intermediate temperatures suggests that the transition does not involve a gradual, homogeneous increase in coercivity, but rather a discontinuous transition. This is particularly apparent at 60 K (Figure 1c), where the low- and high-coercivity MD signals are both visible with a clear gap between them. At this temperature the sample contains regions with mobile walls and regions with strongly pinned walls.

Crescent-shaped FORC diagrams were observed at 50 K for all samples with 0.3 ≤ x ≤ 0.5. Typical results for both polycrystalline samples synthesized in this study and the single crystals synthesized by Bosi et al. [2009] are shown in Figure 2. Similarity between FORC diagrams for polycrystalline and single-crystal samples, each synthesized under different conditions, illustrates that the crescent-shaped FORC signal is an intrinsic low-temperature property of titanomagnetite. The minimum coercivity is around 10 mT for x = 0.3, 0.35 and 0.4, around 20 mT for x = 0.5, and around 60 mT for x = 0.6. The trend of increasing coercivity with increasing x reflects increasing magnetostriction with increasing x at low temperatures [Klerk et al., 1977]. Common features of the crescent FORC distribution include (1) a well-defined and narrow coercivity distribution for a given value of Hu; (2) a symmetrical distribution of intensity about the Hc axis; (3) a maximum intensity at Hu = 0, with the intensity dropping off toward the tips of the crescent; and (4) high noise levels within the crescent and in the background signal to the right of the crescent. The vertical extent of the crescent varies from sample to sample (especially for single crystals). If we compare the polycrystalline samples, there is a generally decreasing vertical spread with increasing x, which may reflect the decreasing saturation magnetization, and hence the decreasing magnitude of the demagnetizing field (HD = −NM), with increasing x.

image

Figure 2. FORC diagrams measured at 50 K for a series of compositions from the titanomagnetite solid solution. Samples are either polycrystalline (poly) or single crystals (SC), with the exception of the first sample, which consisted of several randomly oriented single crystals measured together. Diagrams are shown with a smoothing factor of (a–e) 2.5 and (f) 3.5.

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3.2. Hysteresis and Remanence Measurements

The crescent-shaped FORC signal indicates dramatically enhanced domain wall pinning at low temperatures for x > 0.2. Hysteresis loops measured at 50 and 150 K for a single crystal with x = 0.46 are compared in Figure 3. The remanence in Figure 3a is nearly 100% of the saturation magnetization, compared with 43% at 150 K (Figure 3b) and 2% at room temperature (not shown). The large remanence at 50 K indicates that the domain walls are strongly pinned. The pinning is so strong that large Barkhausen steps are observed in the low-temperature loop. Similar Barkhausen jumps in the measured FORCs are responsible for the high levels of noise in the crescent FORC diagrams at low temperatures.

image

Figure 3. Hysteresis loops for a single crystal with x = 0.46. Horizontal and vertical steps in the 50 K loop correspond to individual Barkhausen jumps, which indicates that the domain walls are strongly pinned.

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The decrease in remanence can be observed in more detail in zero-field-cooled (ZFC) and field-cooled (FC) low-temperature SIRM warming curves (Figure 4). With the exception of x = 0.2 (included for reference) all samples presented in Figure 4 yielded crescent-shaped FORC diagrams at 50 K and have a sharp drop in both FC and ZFC remanence between 50 and 70 K, followed by a slow decline above 70 K. All samples except a single crystal with x = 0.46 (Figure 4c) have a higher ZFC remanence than FC. Higher ZFC remanence has been reported in similar samples by Schmidbauer and Readman [1982] and Carter-Stiglitz et al. [2006], and was attributed by the latter group to the role of randomly oriented magnetocrystalline easy axes in ZFC samples (by analogy with the explanation for similar behavior in the low-temperature monoclinic phase of magnetite by Kosterov [2003] (section 3.4)). The exceptional behavior of the x = 0.46 sample, where FC remanence was higher than ZFC, may be the result of incomplete saturation of this platy single crystal specimen, but it is worth noting that Schmidbauer and Readman [1982] reported a similar inversion in a single crystal with x = 0.4.

image

Figure 4. (a–d) FC and ZFC remanence of titanomagnetites with 0.2 ≤ x ≤ 0.5. All samples are polycrystalline except for the third sample (Figure 4c). Isotropic points (Ti) are interpolated from Kąkol et al. [1991a], although the complete loss of remanence at the upper decay temperature provides an alternative estimation of these values.

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3.3. AC Susceptibility Measurements

The frequency-dependent AC susceptibility of two iron-rich, polycrystalline titanomagnetites was measured in a range of DC bias fields. Samples with x = 0.2 and 0.4 were chosen for comparison, as both have been observed to undergo relaxation below 150 K yet only the x = 0.4 sample has a crescent-shaped FORC diagram. The in-phase and out-of-phase components of susceptibility, with and without bias fields, are summarized in Figure 5. Relaxational phenomena (i.e., magnetic responses that develop over a characteristic relaxation time) give rise to susceptibility signals that shift to higher temperatures with increasing measurement frequency. Both samples have a prominent relaxation centered around 70 K, coinciding with the remanence and pinning transitions discussed in sections 3.1 and 3.2. This relaxation is seen as a rapid rise in the in-phase component of susceptibility on warming and a corresponding peak in the out-of-phase component. A second relaxation, centered around 250 K, is seen for x = 0.2 (Figures 5a and 5b). This relaxation is most obvious in the out-of-phase signal, shown in detail in Figure 6a. An estimate of the activation energy for this high-temperature relaxation can be made by noting the temperature where each curve crosses a reference value of susceptibility (an arbitrary value of 6 × 10−3 m3/kg was chosen here, indicated by the dashed line in Figure 6a). For a thermally activated process, the relaxation time, τ = (2πf)−1, obeys the Arrhenius equation:

  • equation image

where f is the measurement frequency, τ0 is the characteristic attempt time, Ea is the activation energy and kB is the Boltzmann constant. Hence a plot of ln(τ) versus 1/T yields a straight line with slope Ea/kB and intercept ln(τ0) (Figure 6b). The fitted value of Ea = 0.45(2) eV agrees, within error, with the activation energy for a prominent relaxation process observed in MAE studies between 200 and 250 K (Ea = 0.53 ± 0.12 eV [Walz et al., 1997]). This relaxation is attributed to the MAE caused by diffusional reorientation of interstitial Fe2+ defects. The MAE study of Walz et al. [1997] also demonstrated that the interstitial relaxation mechanism is most prominent for x = 0.2, and is of only minor importance for x = 0.4, which explains the lack of a similar feature in Figures 5c and 5d.

image

Figure 5. Frequency dependence of magnetic susceptibility for compositions (a and b) x = 0.2 and (c and d) x = 0.4 at low temperatures and with bias fields from 0 to 150 mT. Graphs are offset on the y axis, but the scale is preserved.

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image

Figure 6. Relaxation behavior around 250 K in samples with composition x = 0.2, caused by diffusion of interstitial Fe2+ defects. The (b) Arrhenius plot is obtained from the temperature shift of the curves measured at different frequencies, estimated by (a) the temperature at which each curve crosses the dashed line.

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The low-temperature relaxation around 70 K shares many of the characteristics of relaxations observed in ferroelastic materials related to the thermally activated motion of twin walls driven by an alternating stress [e.g., Harrison and Redfern, 2002; Harrison et al., 2003; Harrison et al., 2004a, 2004b; Carpenter et al., 2010] and in ferroelectric materials related to the motion of domain walls driven by an alternating electric field [e.g., Huang et al., 1997]. These characteristics include (1) a sharp increase in the in-phase component of susceptibility on warming, (2) a pronounced peak in the out-of-phase susceptibility, and (3) an out-of-phase susceptibility that tends to zero for temperatures far below the peak but adopts a temperature-independent plateau above the peak. Both x = 0.2 and 0.4 samples have a shoulder on the low-temperature side of the peak in out-of-phase susceptibility (Figures 5c and 5d). The x = 0.4 sample has a second peak in the out-of-phase susceptibility at 118 K in zero and 50 mT bias fields (Figure 5d, upper two data sets), which is accompanied by a distinct cusp in the in-phase component of susceptibility (Figure 5c, upper two data sets). The cusp does not shift to higher temperatures with increasing measurement frequency, which indicates that it is associated with a fixed transition temperature rather than a relaxation process. It also coincides with the upper remanence decay temperature for x = 0.4 (Figure 4b).

Arrhenius plots for x = 0.2, 0.35 and 0.4 in zero and low bias fields (Figure 7) were determined using a Lorentzian fit to the peak in the out-of-phase susceptibility to obtain an accurate measurement of the peak position for each measurement frequency. The average activation energy for the relaxation process is 0.13 ± 0.01 eV with an attempt time of 3.7 × 10−11 s. The activation energy and attempt time values agree well with previous studies, which attribute the modulation of the domain wall motion to electron hopping effects [Walz and Kronmüller, 1994]. Carter-Stiglitz et al. [2006] reported an activation energy for polycrystalline samples with x = 0.16 and 0.35 of ∼ 0.1 eV, although they did not report the attempt time. Torres et al. [1997] and Walz et al. [1997] reported relaxation behavior in MAE studies in this temperature range, which they attributed to long-range electron hopping between Fe2+ and Fe3+ ions. First principles calculations of electron hopping in bulk magnetite yield attempt times of 1–4.2 × 10−11 s and activation energies of 0.17–0.22 eV [Skomurski et al., 2010]. Hence, we are confident in assigning the relaxation processes to an electron hopping process.

image

Figure 7. Arrhenius plots derived from the peak in out-of-phase susceptibility at around 70 K (Figure 5), as well as those of a single crystal with composition x = 0.35. This latter sample plots in a different part of the graph because it was measured using higher-frequency AC fields but yields similar values for the attempt time and activation energy of the relaxation process.

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The effect of measuring AC susceptibility in a DC bias field up to 50 mT is negligible for x = 0.2 (Figures 5a and 5b). Larger bias fields, up to 150 mT, were used for the x = 0.4 sample. The high-temperature plateau in the out-of-phase susceptibility is suppressed for bias fields of 100 and 150 mT. The frequency-independent cusp in the in-phase susceptibility and the accompanying peak in the out-of-phase susceptibility, both associated with the isotropic point at 118 K, are similarly suppressed in high bias fields. Most notable, however, is the significant change in nature of the relaxation process that occurs for bias fields ≥ 100 mT: rather than the simple increase in susceptibility with increasing temperature seen for x = 0.2, the in-phase susceptibility now decreases to a distinct minimum just below the relaxation temperature and rises to a distinct peak just above the relaxation temperature, before falling to a temperature-independent plateau on further warming. This behavior (shown in detail in Figure 8) becomes more pronounced with increasing bias field and increasing measurement frequency.

image

Figure 8. Detail of the in-phase susceptibility of a polycrystalline sample with x = 0.4 and a bias field of 150 mT (Figure 5c). The distinctive peak immediately above, and the trough immediately below, the relaxation temperature is observed in analyses with high bias fields, which have the effect of eliminating weakly pinned domain walls. These features are attributed to domain wall resonance in strongly pinned walls.

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3.4. TEM

Fresnel-mode Lorentz TEM images of the magnetic domain structure in a polycrystalline sample with x = 0.5 are shown in Figure 9. In Lorentz mode, magnetic domain walls appear as either thin bright lines or broader dark lines. The magnetic contrast is achieved by simply taking the image out of focus. Additional contrast in the images is caused by a combination of diffraction effects (broad areas of dark and light contrast caused by bending of the TEM foil) and thickness effects (the vein-like web of thin dark lines, most obvious in Figure 9c, is the result of a scalloped sample surface imparted by the sample preparation method).

image

Figure 9. Lorentz microscopy images of a polycrystalline sample with composition x = 0.5 at (a) room temperature, (b) 100 K, and (c) 27 K. Domain walls appear as light and dark lines; two are indicated by white arrows (Figure 9a). At low temperatures (Figures 9b and 9c), domain walls are crystallographically constrained because of increased magnetocrystalline anisotropy and magnetostriction at low temperatures.

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At room temperature (Figure 9a), the sample is above the isotropic point (easy axes <111>). There are few domain walls and they are often curved. At 100 K (Figure 9b), the sample is below the isotropic point but above the low-temperature remanence transition (easy axes <100>). The domains are smaller, the walls are straighter, and their orientations are crystallographically constrained. Images of the domain structures are evocative of ferroelastic twins in low-temperature (monoclinic) magnetite, which have been shown to influence magnetic behavior and create pinning sites [Kasama et al., 2010]. These observations are consistent with a dramatic increase in the magnitude of magnetocrystalline anisotropy at low temperature. A set of faint domain walls oriented NW-SE in Figure 9b appear to flow parallel to the surface edge, and may be stress induced, consistent with the high values of magnetostriction expected for x = 0.5 at 100 K [Klerk et al., 1977]. Strong contrast domain walls appear in two dominant orientations perpendicular to each other. Each set of domain walls appears to exist in its own discrete region, with the formation of needle domains whenever two regions containing perpendicular domain orientations intersect (Figure 9c). Such structures are reminiscent of the twinning microstructures observed in ferroelastic minerals [Salje, 1993] and provide a strong hint that significant magnetoelastic effects control the magnetic domain structures.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

4.1. Mechanism of the Remanence Transition

Various mechanisms have been proposed to explain sharp remanence transitions like those in Figure 4 between 50 and 70 K: (1) the presence of an isotropic point, Ti, where the anisotropy constant, K1, changes from negative to positive and the magnetocrystalline easy axis switches from <111> to <100> [Moskowitz et al., 1998; Özdemir and Dunlop, 2003]; (2) the presence of a structural phase transition, such as the Verwey transition in magnetite [Özdemir and Dunlop, 1999]; (3) the presence of a spin glass transition (e.g., as proposed for titanohematite by Burton et al. [2008]); and (4) sudden unpinning of domain walls caused by rapid changes in anisotropy as a function of temperature [Carter-Stiglitz et al., 2006]. The role of the isotropic point in causing the remanence transition depends critically on the sample composition. Estimates of Ti were obtained from the data of Kąkol et al. [1991a] and are indicated on Figure 4. The isotropic point occurs at 130 K in magnetite, falls to below liquid nitrogen temperatures in the compositional range 0.05 < x < 0.35, then rises again to 230 K at x = 0.5. In more titanium-rich compositions, K1 is always positive and there is no low-temperature isotropic point. Hence, for x = 0.2 and 0.3 (Figure 4a) the isotropic point has the potential to play a role in the remanence transition. However, for x = 0.4, 0.46 and 0.5 (Figures 4b4d) the remanence transition temperature is well below Ti, and remains constant as Ti changes from 110 to 230 K. There is, instead, a good agreement between Ti and the temperature at which the slow remanence decay above 70 K reaches completion. We propose, therefore, that this upper remanence decay temperature is coincident with the isotropic point, giving revised estimates of Ti = 112, 188, and 195 K for samples with x = 0.4, 0.46, and 0.5, respectively. The isotropic point in these samples is responsible for the frequency-independent cusp in the in-phase component of susceptibility (Figure 5c) noted in section 3.3.

Given the evidence for a dramatic transition in domain wall pinning between 50 and 70 K presented in section 3.1, we favor a pinning-related model for the remanence transition [e.g., Carter-Stiglitz et al., 2006]: elevated remanences are obtained when a saturating field is applied below 50 K due to strong domain wall pinning (Figure 3a); this remanence is lost suddenly during the pinning transition between 50 and 70 K as domain walls respond to the demagnetizing field by moving into the sample (Figure 3b). For x = 0.4, 0.46, and 0.5 the remanence that remains above 70 K is due to weaker domain wall pinning by extrinsic defects, such as dislocations. This remanence is lost gradually as domain walls broaden and unpin on approaching the isotropic point.

4.2. FORC Diagrams

An abrupt pinning transition on cooling from 100 to 50 K produces a characteristic crescent-shaped FORC diagram for samples with 0.3 ≤ x ≤ 0.5. The form of the FORC diagram is similar to that predicted by Pike et al. [2001a] using a one-dimensional model of domain wall pinning in a random field [Néel, 1955]. Here we adapt the Pike et al. [2001a] method and demonstrate that the key features of the crescent-shaped FORC diagram can be explained by taking account of how the parameters of the one-dimensional model are modified by the three-dimensional nature of the magnetization process close to saturation.

The sample is considered as a one-dimensional grain with cross-sectional area A and length L. A domain wall with position x (0 ≤ xL) is able to move through the grain under the influence of an applied field, H (dynamic effects, such as those discussed in section 4.3, are ignored). The magnetization of the grain is M(x) = AMS (L−2x) and the demagnetizing field, HD, is approximated by a uniform field HD(x) = −NM(x), where N is the demagnetizing factor. The total energy of the grain is the sum of pinning, magnetostatic and demagnetizing energies:

  • equation image

where EW(x) denotes the variation in domain wall energy as a function of position due to the presence of pinning potentials. A dimensionless version of equation (2) can be written

  • equation image

where et = ET/μ0A2NMs2, ew = EW/μ0A2NMs2, and h = H/NAMs. The domain wall sits at a local minimum in total energy:

  • equation image

where hp = −1/2 dew/dx is the pinning field. The domain wall is stationary when h = hp + (L−2x), i.e., when the applied field equals the sum of the pinning field and the demagnetizing field. If the pinning field is known, then the position of the wall can be calculated (see Pike et al. [2001a] for details). The partial hysteresis loops that make up FORC diagrams can be modeled starting at positive saturation and calculating the domain wall position reached for a given reversal field, Ha. The progress of the wall is then tracked as the field, Hb, is ramped back to positive saturation.

The form of the pinning field used in this study was the same as that used by Pike et al. [2001a]. A pinning function was generated by dividing the particle into 100,000 discrete points along its length. The pinning function was then generated using

  • equation image

where the index i refers to each of the discrete points in the pinning function (0 ≤ i < 100,000), ξ is a correlation parameter that links the pinning field at each point to the value of the preceding point, R is randomly selected from a normal distribution with mean zero and variance 1, and Ω defines the amplitude of the pinning field. The pinning field for i = 0 was arbitrarily fixed at zero. After generation of a pinning function for the particle, 100 individual FORCs were calculated. To minimize the effect of Barkhausen noise on the simulated FORCs, the pinning field was regenerated 200 times and the resulting FORCs averaged.

The FORC diagram obtained for L = 1, Ω = 0.1 and ξ = 0.01 is shown in Figure 10a, which simply reproduces the original result of Pike et al. [2001a]. The FORC distribution has a well-defined and narrow coercivity distribution that is spread symmetrically along ±Hu due to the demagnetizing field. Two key features of the experimental FORC diagrams (Figures 1 and 2) are not reproduced by the Pike et al. [2001a] model, however. The model distribution has a “cigar” shape rather than a crescent shape, and the intensity of the FORC distribution is evenly spread throughout the cigar, rather than being maximum at Hu = 0 and tailing off toward the tips. The curvature of the FORC distribution can be obtained by modifying the nature of the pinning function. To obtain larger values of Hc at the tips of the distribution (i.e., when ∣Hu∣ is large), it is necessary to increase the amplitude of the pinning field as the sample approaches saturation (i.e., when the domain wall approaches the edges of the particle). An example pinning function and the resulting FORC diagram obtained after multiplying equation (5) by an amplification factor of [1 + 3(x − 0.5)2] is shown in Figure 10b. The cigar is now transformed to a “banana”, but still lacks the appropriate variation in intensity from the center to the tips. This intensity variation can be obtained by relaxing the assumption that the demagnetizing factor N is constant. The effect of increasing the demagnetizing field slightly as the sample approaches saturation, by adding an additional contribution of −5(x − 0.5)3 to the original demagnetizing field of (1 − 2x) (keeping the amplitude of the pinning field constant), is shown in Figure 10b. The distribution of intensity within the cigar now corresponds more closely to the experimental results. The effects of simultaneously increasing both the amplitude of the pinning and demagnetizing fields as the sample approaches saturation is shown in Figure 10d. The simulated and experimental FORC diagrams are compared in more detail in Figure 11 and agree well.

image

Figure 10. Simulated FORC diagrams based on the Néel [1955] one-dimensional domain wall pinning model. (a) Model corresponding to that of Pike et al. [2001a] with constant amplitude of pinning and demagnetizing fields. (b) Effect of increasing amplitude of pinning field near sample saturation. (c) Effect of increasing demagnetizing field near saturation. (d) Effect of increasing the amplitude of both pinning and demagnetizing fields near saturation.

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image

Figure 11. Comparison of (a) a measured FORC diagram for a single crystal with composition x = 0.35 at 50 K and (b) a modeled diagram using modifications of the method described by Pike et al. [2001a]. The scaling used in Figure 11b is arbitrary, but the shape and profile of the FORC distribution is similar to that of the experimental diagram.

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Physical justification for the increase in pinning and demagnetizing fields as the sample approaches saturation can be found in the low-temperature domain images (Figure 9). As argued by Kosterov [2003] for the case of low-temperature magnetite, the microcoercivity due to domain wall displacement varies as sec(θ), where θ is the angle between the applied field and the magnetization within a domain. In Figure 9, discrete regions in low-temperature titanomagnetite have uniaxial easy axes oriented 90° to each other. If the field is applied parallel to the easy axis of one region, the domains within this region will move more easily than those in the perpendicular region. Close to saturation, the sample will be dominated by the perpendicular regions, and hence the average effective pinning field will increase. A similar argument can be used to explain the increase in demagnetizing field. It is well known that the demagnetizing field of a soft ferromagnet increases close to saturation because of the elimination of closure domains [Dunlop, 1983]. Although closure domains are unlikely to form in low-temperature titanomagnetite, due to the high uniaxial anisotropy, the sample can reduce its demagnetizing field by forming surface regions with easy axis parallel to the surface. As before, domains within these regions require a higher critical field before they move, and are more likely to survive until the sample approaches saturation. As these surface domains are removed, the demagnetizing factor will increase, just as it does for a soft ferromagnet.

4.3. AC Susceptibility

In larger bias fields, the in-phase susceptibility for x = 0.4 develops a peak immediately above, and a trough immediately below, the relaxation temperature (Figure 8). Here we demonstrate that these features can be explained by a resonant component to the domain wall relaxation.

The dynamic response of a domain wall to an applied field can be compared to that of a damped harmonic oscillator [Rado et al., 1950]:

  • equation image

where x is displacement, t is time, m is the mass of the domain wall, β is a viscous damping coefficient, α is a spring constant, Ms is the saturation magnetization, and H is the applied field. The first term in equation (6) describes the acceleration of the wall in response to the applied field. The mass of the domain wall is given by [Döring, 1948; Rado et al., 1950]

  • equation image

where γ is the gyromagnetic factor and δ = πequation image is the domain wall width (A is the exchange stiffness constant and K is the anisotropy constant). The second term in equation (6) describes the viscous drag force experienced by a moving domain wall. In conducting ferromagnets this drag force can arise from eddy currents created as the wall moves. In ferrites, however, the drag force is dominated by diffusion-induced MAE (disaccommodation) [Castro and Rivas, 1993; Hubert and Schäfer, 1998]. Rado et al. [1950] defined β as

  • equation image

where η is a viscosity coefficient. The third term in equation (6) describes the restoring force acting on a domain wall that is displaced from the origin. This restoring force may arise from the self-demagnetizing field of the sample or from the presence of a harmonic pinning potential. The solution to equation (6) for an AC field with frequency ω is given by [Rado et al., 1950]

  • equation image

for the complex susceptibility or

  • equation image

and

  • equation image

for the in-phase and out-of-phase components, where χ0 is the static susceptibility χ0 = Ms2/αd (d is the average domain size), and ω0 and ωc are characteristic resonance and relaxation frequencies of the system, respectively:

  • equation image

and

  • equation image

Equations (10) and (11) are plotted as two-dimensional functions of ln(ω0) (horizontal axis) and ln(ωc) (vertical axis) in Figure 12 for a fixed applied frequency ω = 1. A horizontal cross section through Figure 12 yields a purely resonant response, while a vertical cross section yields a purely relaxational response (see insets). The measured susceptibility depends on the exact path taken through ln(ω0)−ln(ωc) space. In our experiments, ω is kept constant and we measure the in-phase and out-of-phase response of the system as function of temperature. T is not an explicit variable in equations (10) and (11), therefore we must consider the temperature effect on ω0 and ωc separately. Numerous MAE studies [e.g., Walz et al., 1997], as well as our own AC susceptibility measurements, indicate that domain wall dynamics are controlled by thermally activated electron hopping in the vicinity of the low-temperature relaxation. The viscous term in equation (6) is, therefore, dominated by the MAE (i.e., local enhancement of anisotropy and pinning caused by thermally activated rearrangement of Fe2+ and Fe3+ ions within the wall) and the viscosity parameter (η in equation (8)) is directly proportional to the relaxation time for electron hopping (equation (1) [Castro and Rivas, 1993]). This relation causes ωc to vary as ln ωc = ln ωc0Ea/kBT. Assuming that ω0 is constant and that ω < ω0 (i.e., we are performing a subresonant experiment), the response of the system is sampled along a vertical path from positive ln(ωc) (high temperatures) to negative ln(ωc) (low temperatures). The result is a conventional Debye relaxation, as shown in the insets to Figure 12, which provides an explanation for the behavior observed for x = 0.2 and x = 0.4 in low bias fields.

image

Figure 12. (a) In-phase and (b) out-of-phase susceptibility as a function of lnω0 and lnωc for an applied frequency ω = 1 Hz, derived from equations (10) and (11), respectively. Insets represent the purely resonant response obtained by measuring a horizontal cross section and purely relaxational response obtained from a vertical cross section. Insets are not to the same scale.

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The behavior observed for x = 0.4 in high bias fields (Figure 8) can be explained by taking into consideration the T dependence of magnetocrystalline anisotropy. As K1 increases with decreasing temperature, the width of domain walls will decrease and their Döring mass will increase (equation (7)). This increase in mass decreases ω0 (equation (12)) and moves the system closer to the resonance peak. A nonlinear path through ln(ω0)−ln(ωc) space (Figure 13) gives a qualitative explanation for the data acquired at 1 Hz with a 150 mT bias field. For comparison with the data, values of ln(ωc) were converted to temperature values using the activation energy and attempt times measured in Figure 7. The path taken assumes that the rate of increase in domain wall mass accelerates significantly near the relaxation, causing the system to pass through the tip of the resonance peak. The modeled susceptibility has a sharper transition than the data because we assume a single characteristic relaxation time rather than a broad distribution. Nevertheless, the position and magnitudes of the minor peaks and troughs agree well with the measured values, and the area under the out-of-phase curve equals the area under the data.

image

Figure 13. (a) Nonlinear path that approaches both the resonance and relaxation regions in lnω0−lnωc space, yielding susceptibility curves that qualitatively explain the distinctive behavior highlighted in Figure 8. (b) Comparison of the calculated in-phase and out-of-phase susceptibility-temperature curves obtained from the path shown in Figure 13a with measurements of a sample with composition x = 0.4, measured with a 1 Hz AC field and a 150 mT bias field.

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This modeling suggests that the reduced rate of electron hopping precipitates a rapid increase in anisotropy with decreasing temperature below 77 K, which moves the system from pure relaxation toward a more resonant behavior. The observed behavior, however, depends on the proximity of the system to the resonant peak above 77 K. If ω << ω0, then simple relaxational behavior is expected. This is the case for weakly pinned (broad) walls with low mass and high ω0. If ω0 decreases significantly (or if we increase ω significantly) then resonant behavior becomes more likely. Enhancement of resonant behavior at higher measurement frequencies was noted in Figure 5. The enhancement of resonant behavior in higher bias fields might also be explained if there is a broad distribution of ω0 values in the system: the bias field eliminates weakly pinned walls with low mass and high ω0, leaving the strongly pinned walls with high mass and low ω0. Hence, increasing the bias field sweeps away the weakly pinned domain walls that otherwise dominate the susceptibility signal, effectively moving the path taken during analysis closer to the resonance peak.

Interpretation of Figure 8 in terms of a resonance effect might also provide an alternative explanation for the appearance of a “negative” Debye peak at ∼65 K in the MAE studies of Walz et al. [2003]. In MAE studies it is normal to identify positive Debye signals with a relaxation-induced, time-dependent decrease of the initial susceptibility due to development of enhanced anisotropy within domain walls. The negative signal identified by Walz et al. [2003], at a temperature corresponding to the low-temperature resonance seen here, is highly unusual, as it implies a relaxation-induced, time-dependent increase of the initial susceptibility due to the development of reduced anisotropy within the wall. This was explained by Walz et al. [2003] in terms of an electron-supported mechanism of local stress release due to rearrangement of differently sized Fe2+ and Fe3+ ions in the vicinity of Ti4+ ions. Alternatively, the resonance effect can be used to explain the increase in susceptibility, while retaining the usual interpretation of enhanced anisotropy at the domain wall: enhancement of anisotropy causes the system to move closer to the resonance, which enhances the susceptibility ahead of the main relaxation.

4.4. Mechanism of the Pinning Transition

Our observations provide general constraints on the mechanism of the pinning transition for the studied range of compositions and temperatures where the crescent-shaped FORC is observed. The pinning transition is only observed at 50 K for samples with x > 0.2, which according to O'Reilly and Banerjee [1965] corresponds to the composition where Fe2+ ions first enter the tetrahedral sites of the spinel lattice. The presence of tetrahedral Fe2+ is important due to its role in generating a tetragonal Jahn-Teller distortion [Goodenough, 1964]. The appearance of enhanced pinning only for x > 0.2 hints that magnetoelastic contributions to the magnetic anisotropy, originating from the presence of Fe2+ Jahn-Teller ions on the tetrahedral sublattice, are an essential component of the pinning transition. Sudden loss of pinning on heating indicates, however, that the presence of tetrahedral Fe2+ alone is insufficient to cause the effect, and that localization of Fe2+ ions on the octahedral sublattice due to cessation of electron hopping is also necessary for enhanced pinning. This observation is consistent with the analysis of Kataoka [1974], who explained the origin of giant magnetostriction in ulvöspinel (Fe2TiO4) in terms of an interaction between the Jahn-Teller distortion of Fe2+ on tetrahedral sites and the spin-orbit coupling (i.e., magnetocrystalline anisotropy) of octahedral Fe2+. We suggest that a similar mechanism operates here and that enhanced pinning below the low-temperature relaxation is predominantly magnetoelastic in origin.

Pinning of domain walls in minerals is normally considered to be due to the presence of extrinsic defects, such as dislocations, grain boundaries, impurities, etc. However, close association of the pinning transition and the MAE-driven domain wall relaxation suggests that pinning at low temperatures may be an intrinsic effect, directly related to enhancement of anisotropy within the walls. In this scenario, Fe2+ and Fe3+ ions within the wall rearrange in such as way as to enhance the tetragonal magnetoelastic distortion associated with the interaction between tetrahedral and octahedral Fe2+ [Kataoka, 1974]. Below the relaxation temperature, when electrons are no longer able to diffuse on the time scale of the experiment, this magnetoelastic MAE is no longer responsible for a viscous drag force on the domain walls (i.e., the β term in equation (6)), but instead causes a static, high-amplitude intrinsic pinning field (i.e., the hp term in equation (4)). Above the relaxation temperature, the MAE once again plays the role of a viscous drag force and pinning is controlled by extrinsic defects until the isotropic point is reached.

Evidence in support of the existence of an intrinsic pinning regime at low temperatures may be obtained from the large difference between ZFC and FC remanence below the pinning transition, which implies that application of a magnetic field during cooling can modify the nature of the pinning function (this would not be the case if pinning were entirely caused by extrinsic defects). Modification of the pinning function may be caused by the elimination of regions with differently oriented magnetocrystalline anisotropy axes [e.g., Carter-Stiglitz et al., 2006], or simply through elimination of domain walls themselves (thereby eliminating wall-related pinning sites) when cooling in a strong field. Another prediction of the intrinsic pinning regime is that domain walls will be rigidly pinned along their entire length, whereas in the extrinsic pinning regime walls are only pinned at discrete points, permitting sections of wall in between pinning sites to bow in response to an applied field. These two regimes might be distinguishable using hysteresis loops in the Barkhausen regime. Intrinsic pinning would not permit any change in magnetization between Barkhausen jumps, leading to a series of horizontal steps in the magnetization curve, as observed (Figure 3). Bowing would lead to a more pronounced gradient in the magnetization curve between Barkhausen jumps.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

We have demonstrated the existence of an intrinsic pinning regime in titanomagnetite at temperatures below 50 K and compositions 0.3 ≤ x ≤ 0.5. In this regime, magnetic domain walls are pinned along their entire length due to local enhancement of magnetocrystalline anisotropy within the wall. This enhancement is precipitated by the cessation of electron hopping at low temperatures, which leads to the freezing-in of favorable Fe2+−Fe3+ arrangements within the wall. The enhanced magnetocrystalline anisotropy leads to a narrowing of the domain wall width at low temperatures and an increase in their Döring mass. For the more strongly pinned walls this increase in mass results in a temperature-driven resonant response in AC susceptibility measurements. The dramatic changes in FORC diagram that occur at low temperatures provide a fingerprint for the presence of MD titanomagnetite in geological samples. FORC diagrams in the intrinsic pinning regime are accurately reproduced using a simple modification to the Néel [1955] one-dimensional model of pinning in a random field. Having verified the suitability of numerical simulations for modeling MD FORC diagrams, and the physical significance of the modifications required to obtain an accurate fit to the data, it is now possible to apply the method to the problem of quantitatively unmixing domain state–specific signals from FORC diagrams of geological samples. Given that only subtle modifications of Néel's theory are required to arrive at a good match between experiment and theory, low-temperature titanomagnetite provides an ideal system for studying the physics of domain wall pinning, and will be a fruitful area of future research.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

This work was funded by the European Science Foundation (ESF) under the EUROCORES program EuroMinScI, through contract ERAS-CT-2003-980409 of the European Commission, DG Research, FP6, NERC grant NE/D522203/1, and an IRM visiting fellowship. The authors thank Takeshi Kasama for his help in obtaining the Lorentz TEM image, Ferdinando Bosi and Ulf Hålenius for providing the single-crystal titanomagnetite samples, and Andrew Roberts, Adrian Muxworthy, and Özden Özdemir for their reviews of the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
ggge1961-sup-0001-tab01.txtplain text document0KTab-delimited Table 1.

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