# Stability of equidimensional pseudo–single-domain magnetite over billion-year timescales

^{a}Earth and Planetary Science, School of Geosciences, University of Edinburgh, Edinburgh EH9 3FE, United Kingdom;^{b}Department of Earth Science and Engineering, Royal School of Mines, Imperial College London, London SW7 2AZ, United Kingdom;^{c}Marine Geology, Norges Geologiske Undersøkelse, 7040 Trondheim, Norway;^{d}Materials and Condensed Matter Physics, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom;^{e}Geophysical Observatory of Russian Academy of Sciences, Borok, Yaroslvaskaya Oblast 152742, Russia;^{f}Institute of Geology and Petroleum Technologies, Kazan Federal University, Kazan 420008, Russia

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Edited by Dennis V. Kent, Rutgers University, New Brunswick, NJ, and approved August 8, 2017 (received for review May 22, 2017)

## Significance

When magnetic crystals form in rocks and meteorites, they can record the ancient magnetic field and retain this information over geological timescales. Scientists use these magnetic recordings to study the evolution of the Earth and the Solar System. Previous theories for the recording mechanisms of the magnetic crystals in rocks and meteorites are based on the idea that the magnetic structures within crystals are near uniform. However, from numerous studies we know this not to be true. The crystals are too large in size and display complex nonuniform “vortex” structures. In this study we have shown that vortex structures are capable of recording and retaining magnetic signals over billions of years.

## Abstract

Interpretations of paleomagnetic observations assume that naturally occurring magnetic particles can retain their primary magnetic recording over billions of years. The ability to retain a magnetic recording is inferred from laboratory measurements, where heating causes demagnetization on the order of seconds. The theoretical basis for this inference comes from previous models that assume only the existence of small, uniformly magnetized particles, whereas the carriers of paleomagnetic signals in rocks are usually larger, nonuniformly magnetized particles, for which there is no empirically complete, thermally activated model. This study has developed a thermally activated numerical micromagnetic model that can quantitatively determine the energy barriers between stable states in nonuniform magnetic particles on geological timescales. We examine in detail the thermal stability characteristics of equidimensional cuboctahedral magnetite and find that, contrary to previously published theories, such nonuniformly magnetized particles provide greater magnetic stability than their uniformly magnetized counterparts. Hence, nonuniformly magnetized grains, which are commonly the main remanence carrier in meteorites and rocks, can record and retain high-fidelity magnetic recordings over billions of years.

Since the 1900s magnetic recordings observed in rocks and meteorites have been studied to understand the evolution of the Earth and the Solar System. The validity of the findings from these studies depends on a theoretical understanding of rock-magnetic recordings provided by Néel (1, 2) and numerous experimental studies, for example, Strangway et al. (3) and Evans and Wayman (4). The overwhelming evidence from these authors was that stable natural magnetic remanence in rocks resides within ultrafine, uniformly magnetized particles, called single-domain (SD) particles. Néel’s theory (1, 2) for the behavior of thermally activated SD particles describes a unique relationship between thermal and temporal stability and gave confidence that paleomagnetic recordings that become unstable (unblocked) only at high temperatures retain magnetic recordings from the time of their mineral crystallization, possibly as far back in time as 4 billion y ago.

However, in the 1970s and 1980s the widespread use of hysteresis parameters to characterize magnetic mineralogy (5) found that the majority of magnetic particles in rocks are not in uniform magnetic states, but are larger in size (80–1,000 nm) and contain complex magnetic states that are not described by either SD theory or the multidomain (MD) theory of micrometer-sized particles (2, 6). The term pseudo-single domain (PSD) was coined for such particles and much effort was spent in determining the origin of their magnetic fidelity (7, 8). Due to the complexity of the problem it has not been possible to determine the temporal stability of magnetization in PSD grains on geological timescales from their observed thermal stability in the laboratory, although this stability is usually assumed (9, 10). It was therefore difficult to claim with any degree of certainty that the measured paleomagnetic signal of most rocks and meteorites containing PSD grains represents the magnetic field recorded when the mineral formed.

We consider here only the simplest form of PSD grains, which are single-vortex (SV) magnetic domain states that form in equidimensional magnetite grains from 80 nm to 1,000 nm. Beyond this grain size more complex nonuniform PSD states will exist that may consist of multiple vortices or MD regions with broad domain wall-like structures.

We have introduced the nudged–elastic-band (NEB) method, which minimizes the action along the energy path (11⇓⇓–14) into a finite-element method (FEM) micromagnetic model to determine the energy barriers between possible magnetic states as a function of temperature. We used the NEB algorithm to determine the thermal behavior, the temperature of magnetic stability (blocking temperature), and relaxation times for equidimensional SD and SV grains of magnetite. This study finds that magnetically nonuniform SV magnetite particles that are ubiquitous in nature have high blocking temperatures close to their Curie temperature and long relaxation times and as such offer a much higher magnetic recording fidelity than equidimensional SD particles.

To determine the thermally activated stability of magnetite particles, all possible local-energy minimum (LEM) states for particles with a given volume at a given temperature need to be found. This was achieved by initializing the geometry with random magnetization directions and minimizing the free energy with respect to temperature-dependent values for the saturation magnetization, magnetocrystalline anisotropy, and exchange constant (15⇓–17) (*Materials and Methods*). Although no computational method can guarantee that all possible LEM states have been found, after executing several thousand simulations, the only domain states seen to nucleate are the ones described below. For example, in a cuboctahedral particle equivalent to a spherical volume diameter (ESVD) of 100 nm, at 20 °C, we identified 16 LEM states corresponding to SV structures with vortex cores directed along one of four cubic diagonals, which are the magnetocrystalline easy axes in magnetite. Fig. 1 identifies eight such structures for a right-handed vortex, and there are a further eight for a left-handed vortex with equal preference for both left- and right-handed structures.

The NEB method (11⇓⇓–14) was used to calculate the energy barriers between two LEM states by determining the most energetically favorable path between those two states. We found it unnecessary to compute energy barriers across all possible (120) pairs of LEM states, because many energy paths and barriers are equivalent (Fig. 1). For example, equivalent paths were identified for the 100-nm (ESVD) cuboctahedron at 20 °C by calculating and directly comparing energy paths for all possible permutations of start/end pairs; the summary of these results is shown in Fig. 2 and indicates two fundamental classes of transition: same-sense reorientations, where the handedness of the vortex core does not change; and vortex-core unwindings, where the vortex core switches from clockwise to anticlockwise or vice versa.

We identified two types of same-sense vortex-core reorientations: (*i*) small-angle reorientations and (*ii*) large-angle reorientations. Small-angle reorientations of the vortex core represent a rotation of ∼71°, corresponding to the angle between two of the nearest-neighbor cubic diagonals. In this transition, the vortex core remains right or left handed, and the energy graph illustrates a single smooth peak (Fig. 2*A*) corresponding to a single energy barrier between start and end states. Large-angle reorientations are vortex-core transitions through angles >71°. The energy graph for a large-angle transition illustrates a smooth initial path, corresponding to a 71° reorientation, and a second, noisier section (Fig. 2*B*). This noisy section reflects the fact that the NEB method we used guarantees only the resolution of a single energy barrier between any two LEM states; the second transition is therefore not effectively resolved. However, the full transition can be determined by breaking the path into several shorter sections and individually determining the barriers along each section. Consequently, large-angle transitions are simply resolved as chains of small-angle transitions. When we consider transition paths with a change of vortex-core orientation from a left-handed sense to a right-handed sense (Fig. 2*C*), an initial large energy peak is seen, followed by three much smaller peaks each of the same maximum energy as a small-angle reorientation, as seen in Fig. 2*A*. This behavior corresponds to an initial high-energy unwinding of the vortex core propagating from one end of the vortex to the other, followed by a vortex-core rotation through 180° via three successive small-angle rotations (Movies S1–S3).

Through exploration of the energy paths of each transition pair we reduced the number of NEB calculations for each cuboctahedral geometry from potentially 120 pairs down to 1, greatly simplifying the problem of mapping out relaxation times for a range of temperatures and grain sizes. This was possible because all of the unwinding–class transitions were discounted as being energetically unlikely, and transitions between same-handed structures are energetically equivalent to a single small-angle rotation.

To determine thermal stability, we then calculated energy barriers as a function of temperature for equidimensional truncated cuboctahedral magnetite particles (Fig. 1), with sizes from 40 nm to 170 nm (ESVD) increasing in steps of at most 10 nm. The temperature range for NEB calculations was 20 °C to 550 °C, increasing in steps of 10 °C.

In Fig. 3 we summarize this study’s results for the blocking temperatures of equidimensional magnetite particles from the SD size up to well above the SD/SV transition size (

For SV magnetite (with easy-aligned vortex cores), the blocking temperature increases with particle size from less than 100 °C for particles at

It is interesting to compare the SD and SV calculated blocking temperatures shown in Fig. 3. For cuboctahedral magnetite, the calculations predict that SV particles have higher blocking temperatures than SD particles, in contrast to previous models (2, 6, 21) for nonuniform magnetization structures, which predict the opposite trend. As the effective volumes associated with SCR are larger than in coherent SD switching, this increases the blocking temperatures to close to the Curie temperature and explains the experimental observation of high paleomagnetic stability (Fig. 3).

To determine the magnetic stability of magnetic remanence at room temperature, we use Eq. **5** (*Materials and Methods*) and a temperature of 20 °C to determine relaxation times as a function of particle size (Fig. 3). For small (SD) particles, the stability peaks at

For single vortex-core cuboctahedrons just above the SD–SV transition size, the relaxation time is less than 60 s, but increases sharply to greater than the age of the universe by 100 nm and continues to increase for all of the particle sizes up to the maximum size of 170 nm considered in this study.

In comparison with SD particles, equidimensional magnetite with SV magnetization structures has longer relaxation times and hence greater stability than equidimensional SD grains. It is important to note, of course, that shape elongation of small SD particles will greatly enhance their magnetic stability and so SD grains will remain an ideal domain state for paleomagnetic recording. However, here we have shown that grains containing SV domain states, which in many rock samples may constitute the majority, are still able to hold a paleomagnetic recording with a high thermal stability and have relaxation times at room temperature that exceed the age of the universe. Although SV domain states have been identified theoretically (22) and experimentally (23) for some years, we believe that the results of the present study are a unique indication that these domain states have sufficient thermal and temporal stability to make them excellent carriers of paleomagnetic information. Indeed the high blocking temperatures of SV particles predicted by this model are supported by recent high-temperature nanometer-scale magnetic imaging (20, 24⇓–26) of SV magnetite particles.

By examining switching mechanisms in idealized magnetic grains, we can finally explain an observation central to paleomagnetism: the apparent contradiction between the recording stability and fidelity of natural magnetic systems that also display magnetic-hysteresis characteristics of particles with nonuniform magnetic structures, which until now were assumed to indicate poor magnetic recording fidelity. By demonstrating that SV magnetite particles are of high magnetic stability, we have shown that there is no inconsistency. While it is undoubtedly true that PSD grains with more complex domain states will contribute to the magnetic remanence, SV states will be some of the highest PSD remanence carriers while also accounting for a large grain-size range [

## Materials and Methods

Magnetization structures in particles are classified into three basic types: (*i*) SD, where all magnetization vectors are oriented in the same direction; (*ii*) PSD, where vectors are arranged into inhomogeneous structures, but are not able to form well-defined domains; and (*iii*) MD, which contain distinct uniformly magnetized regions separated by domain walls. In this study, calculation of SV states (being the simplest of PSD states) were made using FEM micromagnetic algorithms (18, 30). This technique allows accurate modeling of geologically realistic magnetic particles by taking a geometric description of a magnetic particle and dividing it into many small tetrahedral segments. The magnetic free energy is then calculated for each segment and the total free energy is the discrete sum over all tetrahedra. The FEM discretizes the total free energy *m* in a magnetic grain, given by ref. 31,

The temperature-dependent material parameters for Eq. **1** are given by experimentally derived expressions (16, 17, 32) and have the forms

To determine the thermal stability of a particle, it is necessary to find the energy barriers that exist between all possible stable domain states at all possible temperatures within a sample; energetically stable configurations correspond to magnetization states where the free energy,

Once two suitable start and end points are selected, the NEB method is used to compute an energy barrier. Fig. 4 illustrates the NEB concept in terms of the well-documented case of a uniformly magnetized ellipsoid (19). Path start and end points correspond to magnetizations aligned along the

To calculate the blocking temperature, **5**. A polynomial fit was then made of the calculated relaxation times to determine the blocking temperature—a reference time of 100 s was chosen since it is on the order of the time used in laboratory experiments (33). This time point is indicated by the dashed red line in Fig. 5.

All models shown in this paper were produced using the Micromagnetic Earth Related Robust Interpreter Language Laboratory (MERRILL), the source code of which is publicly available at https://bitbucket.org/wynwilliams/merrill. MERRILL scripts and file data formats used in this study are detailed in Dataset S1.

## Acknowledgments

We thank the reviewers for their helpful suggestions. This work is supported by the Natural Environment Research Council (Grant NE/J020966/1) and the European Research Council (Grant EC320832 Imaging Magnetism in Nanostructures Using Electron Holography).

## Footnotes

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^{1}To whom correspondence should be addressed. Email: l.nagy{at}ed.ac.uk.

Author contributions: L.N. and W.W. designed research; L.N. performed research; L.N., W.W., and A.R.M. analyzed data; and L.N., W.W., A.R.M., K.F., T.P.A., P.Ó.C., and V.P.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1708344114/-/DCSupplemental.

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