#  Superparamagnetic dynamics and blocking transition in Fe3O4 nanoparticles
probed by vibrating sample magnetometry and muon spin relaxation

Benjamin A. Frandsen<EMAIL_ADDRESS>Department of Physics and Astronomy,
Brigham Young University, Provo, Utah 84602, USA.  Charlotte Read  Department
of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA.
Jade Stevens  Department of Physics and Astronomy, Brigham Young University,
Provo, Utah 84602, USA.  Colby Walker  Department of Physics and Astronomy,
Brigham Young University, Provo, Utah 84602, USA.  Mason Christiansen
Department of Physics and Astronomy, Brigham Young University, Provo, Utah
84602, USA.  Roger G. Harrison  Department of Chemistry and Biochemistry,
Brigham Young University, Provo, Utah 84602, USA.  Karine Chesnel
<EMAIL_ADDRESS>Department of Physics and Astronomy, Brigham Young
University, Provo, Utah 84602, USA.

###### Abstract

The magnetic properties of Fe3O4 nanoparticle assemblies have been
investigated in detail through a combination of vibrating sample magnetometry
and muon spin relaxation ($\mu$SR) techniques. Two samples with average
particle sizes of 5 nm and 20 nm, respectively, were studied. For both
samples, the magnetometry and $\mu$SR results exhibit clear signatures of the
superparmagnetic state at high temperature and the magnetically blocked state
at low temperature. The $\mu$SR data demonstrate that the transition from the
superparamagnetic to the blocked state occurs gradually throughout the sample
volume over a broad temperature range due to the finite particle size
distribution of each sample. The transition occurs between approximately 3 K
and 45 K for the 5 nm sample and 150 K and 300 K for the 20 nm sample. The
magnetometry and $\mu$SR data are further analyzed to yield estimates of
microscopic magnetic parameters including the nanoparticle spin-flip
activation energy $E_{A}$, magnetic anisotropy $K$, and intrinsic nanoparticle
spin reversal attempt time $\tau_{0}$. These results highlight the
complementary information about magnetic nanoparticles that can be obtained by
bulk magnetic probes such as magnetometry and local magnetic probes such as
$\mu$SR.

##  I Introduction

Magnetic nanoparticles have stimulated significant interest over the past
several decades both as a platform for fundamental studies of magnetism on the
nanoscale as well as a route to novel technologies in sectors ranging from
information technology to medicine Pankhurst  _et al._ (2003); De  _et al._
(2008); Frey  _et al._ (2009); Majetich  _et al._ (2011). One of the
interesting phenomena associated with magnetic nanoparticles is that of
superparamagnetism, whereby ferro- or ferrimagnetic nanoparticles small enough
to be in the mono-domain limit exhibit strong thermal fluctuations of the
magnetization. Referring to the net magnetic moment on each nanoparticle as a
nanospin, we can describe superparamagnetism as the dynamical flipping of
nanospins under the influence of temperature, similar to the random flipping
of electronic magnetic moments in a conventional paramagnet. When cooled to
sufficiently low temperatures, the nanospins freeze, thereby entering the
blocked state. This blocking transition is a complex process that depends on
the characteristics of the individual nanoparticles in question and can also
be strongly influenced by any inter-particle interactions present in the
nanoparticle assembly Goya  _et al._ (2003); Lima  _et al._ (2006); Varón  _et
al._ (2013).

Although much experimental and theoretical work has been performed to
understand superparamagnetism and the blocking transition Majetich and Sachan
(2006); Roca  _et al._ (2006); Yang  _et al._ (2005), many details relating to
these behaviors remain unclear. Among others, these behaviors include the
precise manner in which the the blocking transition proceeds throughout the
sample, and how this affects experimental data Livesey  _et al._ (2018); the
influence of the structure and morphology across varying length scales Varón
_et al._ (2013); Klomp  _et al._ (2020); and even the estimation of
microscopic parameters such as the nanoparticle spin flip activation energy
and fluctuation time Périgo  _et al._ (2015). There is increasing recognition
that a more complete understanding is most likely to be obtained by attacking
the problem with a variety of complementary techniques, such as magnetometry
Roca  _et al._ (2006), Mössbauer spectroscopy Gabbasov  _et al._ (2015),
electronic microscopy Yamamuro  _et al._ (2002) and holography Yamamoto  _et
al._ (2008), neutron scattering Mishra  _et al._ (2015), x-ray magnetic
spectroscopy Cai  _et al._ (2014), or x-ray magnetic scattering Chesnel  _et
al._ (2018). In this context, probes of local magnetism—i.e. methods that are
sensitive to magnetism on length scales of several interatomic
spacings—provide important information not easily obtained from bulk-averaged
probes such as conventional magnetization measurements. Muon spin
relaxation/rotation ($\mu$SR) is one such local magnetic probe that has
heretofore seldom been applied to the topic of superparamagnetism. This
technique can provide a wealth of information, such as the volume fraction of
competing magnetic states and the nature of magnetic fluctuations on time
scales of 10-12 to 10-4 s (highly complementary to slower probes such as
nuclear magnetic resonance or ac/dc magnetic susceptibility measurements and
faster probes such as neutron scattering) Uemura (1999); Blundell (2010);
Yaouanc and de Réotier (2011).

In this work, we combine $\mu$SR experiments with vibrating sample
magnetometry (VSM) to investigate the superparamagnetic behavior and blocking
transition in Fe3O4 nanoparticle assemblies. Two samples with average particle
sizes of approximately 5 nm and 20 nm were studied. As established in previous
studies Chesnel  _et al._ (2014); Klomp  _et al._ (2020), we find that the
blocking temperature $T_{B}$ increases significantly as the particle size is
increased from 5 nm to 20 nm. The magnetometry results provide insight into
the drastic dependence of the blocking temperature on particle size and on how
the particle size distribution affects the extent of the blocking transition,
but quantitative information about the size distribution dependence of the
blocking transition requires complex modeling Livesey  _et al._ (2018). Even
without such modeling, however, we are still able to obtain average
information from the magnetometry data, including various estimates of the
average blocking temperature. We then use the $\mu$SR data for more
quantitative analysis of the blocking transition. We establish that the
blocking transition occurs gradually throughout the sample volume over a broad
temperature interval due to the finite size distribution of the particles.
More importantly, we extract crucial microscopic magnetic parameters directly
from the temperature-dependent $\mu$SR relaxation rate, including the
nanoparticle spin flip activation energy, the magnetic anisotropy, and the
nanoparticle spin fluctuation time (i.e., the magnetic reversal attempt time).
Our study shows that, complementary to magnetometry, the $\mu$SR technique
provides an accurate and independent means of probing crucial properties of
magnetic NP assemblies.

##  II Experimental Methods

Fe3O4 NPs with average diameters of approximately 5 nm were synthesized using
an organic solution method detailed in Ref. Altavilla  _et al._ , 2005, and
made of Fe3O4 particles surrounded by a ligand shell of oleic acid molecules.
To purify the NPs, they were dissolved in toluene, precipitated from ethanol,
and extracted after centrifugation at 5000 revolutions per minute for
approximately 15 minutes. The end material was a black paste containing a
dense assembly of the NPs. Fe3O4 NPs with average diameters of approximately
20 nm were purchased from Cytodiagnostics Inc in Ontario, Canada. These NPs
came dissolved in toluene and were similarly encased in an oleic acid ligand
shell and were shipped in solution with toluene. The toluene was evaporated by
bubbling nitrogen gas through the sample, leaving a thick residue consisting
of the NPs. Bulk Fe3O4 powder for comparison with the NPs was purchased from
Alfa Aesar.

The transmission electron microscopy (TEM) images were collected on a
ThermoFisher Scientific Tecnai F20 UT operating at 200 kV. For TEM imaging,
the NPs were deposited on very thin carbon membranes by dropping solutions of
the NPs on the membranes and letting the solvents, either toluene or
chloroform, evaporate. For each NP sample, the concentration of the solution
was finely adjusted so as to achieve a monolayer or less of NPs on the
membrane, thus allowing individual NPs to be observed.

The vibrating sample magnetometry (VSM) data were collected on a Quantum
Design Physical Properties Measurement System (PPMS) that includes a
superconducting magnet, generating a ramping field up to 9 T, and a cryogenic
sample holder using liquid helium. For the VSM measurements, a small amount of
the dried NP material was inserted into capsules of approximate volume 1 mm3.
The material was tightly compacted inside each capsule, which was then placed
inside a cylindrical brass sample holder to be inserted in the VSM coilset.
Field cooling (FC) and zero-field cooling (ZFC) measurements were carried out
under a field of 100 Oe (10 mT) at a warming speed of 1 K/min over a
temperature range of 10 K – 400 K. Data points were collected in intervals of
1 s.

The $\mu$SR experiments were performed at TRIUMF in Vancouver, Canada on the
LAMPF (Los Alamos Meson Physics Facility) spectrometer. The $\mu$SR technique
involves implanting spin-polarized positive muons one at a time in the sample.
The local magnetic field at the muon stopping site, i.e. the vector sum of the
internal field intrinsic to the sample and any externally applied field,
results in Larmor precession of the muon spin. The muon decays into a positron
and two neutrinos after a mean lifetime of 2.2 $\mu$s, with the positron being
emitted preferentially in the direction of the muon spin at the moment of
decay. The positron events are recorded by detectors near the sample position.
The main experimental quantity of interest is the asymmetry, defined as the
normalized difference in positron events between a pair of detectors placed on
opposite sides of the sample. Physically, the asymmetry is proportional to the
projection of spin polarization of the muon ensemble along the axis defined by
the detector pair. Thus, the internal magnetic field distribution in the
sample can be probed via analysis of the time-dependent $\mu$SR asymmetry
Yaouanc and de Réotier (2011). A helium gas flow cryostat was used to control
the temperature between 2 K and 300 K. Asymmetry spectra were collected to a
maximum time window of 9.7 $\mu$s. Fits to the $\mu$SR spectra and other
analyses were done with MUSRFIT Suter and Wojek (2012) and home-built python
software. The $\mu$SR parameter $\alpha$ was calibrated using the spectra
showing the fastest relaxation, allowing us to measure the true zero of the
asymmetry spectrum directly. The usual calibration method using a weak
transverse field in the paramagnetic state was not possible due to the high
magnetic ordering temperature of Fe3O4 ($\sim$858 K).

##  III Results

###  III.1 5 nm nanoparticles: TEM and Magnetometry

We first present the magnetometry and TEM characterization of the 5 nm sample.
After deposition onto a thin membrane, the NPs tend to self-assemble and form
a close-packed hexagonal lattice, as indicated by the TEM image in Fig. 1(a).

Figure 1:  (a) TEM image of a self assembly of 5 nm NPs deposited on a thin
carbon membrane. Inset: Histogram of particle sizes yielding an average size
of 5.16 $\pm$ 0.96 nm. (b) FC/ZFC curves collected under a field of 100 Oe (10
mT) over the temperature range 10 – 400 K. (c) Magnetization difference
$\Delta M=M_{\mathrm{FC}}-M_{\mathrm{ZFC}}$ and (d) its derivative with
respect to temperature, $\mathrm{d}\Delta M/\mathrm{d}T$, plotted over the
temperature range 10 – 50 K.

When the initial NP concentration is optimal, one can achieve a uniform
monolayer of NPs, such as the one seen in Fig. 1(a). A fine analysis of the
particle size carried out on multiple TEM images containing about 465 NPs
yields an average particle diameter of 5.16 $\pm$ 0.96 nm. Each particle is
surrounded by an oleic acid ligand shell of approximately 1 nm in thickness,
so that the average inter-particle distance is in fact closer to 6 nm or
larger, as x-ray magnetic scattering measurements on identically prepared
samples have revealed Chesnel  _et al._ (2018). The formation of a vast,
uniform, close-packed lattice is enabled by the narrow size distribution of
the sample.

The FC/ZFC measurements shown in Fig. 1(b) suggest a superparamagnetic (SPM)
behavior where the associated blocking transition is indicated by the presence
of a peak in the ZFC curve accompanied by the merging of the ZFC and FC
curves. The narrowness of the peak exhibited by the ZFC curve reflects the
narrowness of the particle size distribution. This result is consistent with
FC/ZFC measurements on previous batches of 5 nm NPs prepared using the same
method Chesnel  _et al._ (2014); Klomp  _et al._ (2020). The proper way to
estimate the blocking temperature $T_{B}$ from magnetometry data has been
subject to debate in recent years Bruvera  _et al._ (2015); Livesey  _et al._
(2018). We provide here two estimates of $T_{B}$ using the following two
different criteria: 1) the location of the peak in the ZFC curve, as
illustrated in Fig. 1(b); and 2) the merging point of the FC/ZFC curves, where
the difference $\Delta M(T)=M_{\mathrm{FC}}-M_{\mathrm{ZFC}}$ and its
derivatives, $\mathrm{d}\Delta M/\mathrm{d}T$, $\mathrm{d}^{2}\Delta
M/\mathrm{d}T^{2}$, etc., reach zero Livesey  _et al._ (2018). As illustrated
in Fig. 1(c), $\Delta M(T)$ is the largest at the lowest temperature and
decreases as $T$ increases until it reaches a merging point where $\Delta
M=0$, above which it plateaus at zero (marking the SPM phase). Likewise, the
magnitude of the derivative $|\mathrm{d}\Delta M/\mathrm{d}T|$ decreases when
$T$ increases until plateauing at $\mathrm{d}\Delta M/\mathrm{d}T=0$, as seen
in Fig. 1(d). For the 5 nm NPs, the estimated blocking temperature using the
ZFC peak is $T_{B}\approx 24$ K, whereas the merging criterion yields an
estimated value of $T_{B}\approx 28$ K, somewhat higher but not significantly
different.

The VSM data and the estimate of $T_{B}$ may be used to obtain further
information about the dynamics of the nanospins in the SPM state. Based on the
Néel-Arrhenius law Néel (1949), the mean time between two nanoparticle spin
flips, called the Néel relaxation time $\tau_{N}$, is given by

$\displaystyle\tau_{N}=\tau_{0}\exp\left(\frac{E_{A}}{k_{B}T}\right),$ (1)

where $\tau_{0}$ is the intrinsic nanoparticle spin fluctuation time (i.e.,
the magnetic reversal attempt time, which is also the value of $\tau_{N}$ at
infinite temperature), $E_{A}$ is the activation energy or energy barrier for
a nanospin flip, $k_{B}$ is the Boltzmann constant, and $T$ is the temperature
of the material. For measurements where the sampling time $\tau_{m}$ is kept
constant, the blocking temperature $T_{B}$, delineating the SPM state from the
blocked state, is the temperature for which $\tau_{m}=\tau_{N}$ such that

$\displaystyle\tau_{m}=\tau_{0}\exp\left(\frac{E_{A}}{k_{B}T_{B}}\right).$ (2)

If $\tau_{m}$, $T_{B}$, and $E_{A}$ are measured experimentally, an estimate
of $\tau_{0}$ may then be obtained.

For our VSM measurements, $\tau_{m}\approx 1$ s, and the blocking temperature
for the 5 nm NPs is $T_{B}\approx 24$ K. As explained below, the $\mu$SR data
provides an estimate of the activation energy of the 5 nm NPs to be
$E_{A}\approx 1.4\times 10^{-21}$ kJ. From these values, we obtain the ratio
$E_{A}/(k_{B}T_{B})\approx 4.2$ and $\tau_{0}\approx 1.4\times 10^{-2}$ s (14
ms). However, this estimated value for $\tau_{0}$ is several orders of
magnitude higher than values found in the literature, which typically range
from $10^{-9}-10^{-10}$ s Majetich and Sachan (2006) or even as short as
$10^{-13}$ s Périgo  _et al._ (2015). This large discrepancy may be explained
by missing terms in the expression for the energy barrier, such as magnetic
interaction terms as described by W.F. Brown Brown (1963).

Here we attempt a simple correction to the energy barrier magnetic interaction
between the nanospins and the applied magnetic field (Zeeman effect):
$E_{M}=\Delta\vec{M}\cdot\vec{H}$ where $\Delta\vec{M}$ is the change in
magnetic moment carried by each NP caused by a flip and $\vec{H}$ is the
applied magnetic field. Assuming a full up-down flip occurs along the
direction of the applied field, $E_{M}=2M_{s}H$ where $M_{s}$ is the
magnetization at saturation for an individual NP. The adjusted energy barrier
is, to a first approximation, $\Delta E=E_{A}+E_{M}$ and the adjusted
fluctuation time $\tau_{0}^{*}$ is then given by

$\displaystyle\tau_{0}^{*}=\tau_{m}\exp\left(-\frac{E_{A}+E_{M}}{k_{B}T_{B}}\right).$
(3)

The magnitude of $M_{s}$ may be estimated by using the volume of each NP and
the density of magnetic ions in the material. Based on previous
crystallographic studies of these samples Klomp  _et al._ (2020), the cubic
lattice parameter is 8.35 Å and the atomic density is about 13.7 Fe3O4 units
per nm3. Accounting for combined spins carried by the Fe2+ and Fe3+ ions in
the spinel structure Cai  _et al._ (2014), the saturated magnetization density
is about 55 $\mu_{B}$/nm3. Assuming a spherical NP shape with a radius of 2.6
nm, we have $M_{s}=4000$ $\mu_{B}$. The applied field during the VSM
measurement was $H=100$ Oe (10 mT). The resulting value for the Zeeman
magnetic interaction then comes to $E_{M}=7.4\times 10^{-22}$ J. The adjusted
energy ratio $\Delta E/k_{B}T_{B}\approx 6.46$ leads to an adjusted attempt
time $\tau_{0}^{*}\approx 1.5\times 10^{-3}$ (1.5 ms), only a factor of 10
smaller than the initial estimate of $\tau_{0}$.

Evidently, this simple model does not appear to reflect accurately the actual
energy barrier $\Delta E$ experienced by the NPs during the VSM measurements,
but may be refined by including additional energy terms, such as the
ferromagnetic exchange energy, the magnetostatic energy, the crystalline and
shape anisotropy energy, and interparticle magnetic interactions. The
contributions from these factors could be derived from various models, such as
the Stoner-Wohlfarth model, the Gilbert equations, and the Fokker-Planck
equations, as suggested theoretically by Brown Brown (1963) and experimentally
by more recent works Majetich and Sachan (2006); Torres  _et al._ (2015). In
the scope of our present paper, instead of attempting to refine our estimate
of the energy barrier experienced by the NPs, we show in the following how
$\mu$SR provides a unique and reliable way to probe spin fluctuations in
magnetic NP systems, making the technique highly complementary to bulk
magnetometry.

###  III.2 5 nm nanoparticles: $\mu$SR

To obtain additional insight into the behavior of this system, we now turn to
the $\mu$SR characterization of the 5 nm NPs. Fig. 2 displays $\mu$SR
asymmetry spectra collected at representative temperatures and field
conditions.

Figure 2:  (a) $\mu$SR asymmetry spectra for 5 nm Fe3O4 NPs collected at 150 K
in various applied longitudinal fields (LF). (b) Same as (a) but for data
collected at 3 K. (c) Asymmetry spectra collected in a weak longitudinal field
of 0.01 T at 100 K (superparamagnetic), 35 K (mixed superparamagnetic and
blocked states), and 3 K (blocked). The solid black curves are fits described
in the main text.

In panel (a), we display spectra collected at 150 K in various values of an
externally applied longitudinal field (LF), i.e. an applied field directed
along the initial muon spin polarization direction. Significant relaxation of
the asymmetry is observed even in the strongest applied field LF = 0.3 T,
indicating that the intrinsic magnetic field at the muon site is dynamically
fluctuating. This is expected, since the sample is in the SPM state at this
temperature. We note that the spectrum collected with zero applied field
appears to consist of two asymmetry components, one that relaxes rapidly
within the first $\sim$2 $\mu$s, and the other which displays a more gentle
relaxation. The slowly relaxing asymmetry from the latter component is fully
recovered when a modest field of LF = 0.01 T is applied, as seen by the blue
symbols in Fig. 2(a), indicating that the muons contributing to this asymmetry
component experience a very weak internal field. We attribute this slowly
relaxing asymmetry component to muons stopping in the ligand shells around the
NPs, where the intrinsic field is much weaker. The faster relaxing asymmetry
component arises from muons stopping within the NPs, where the large intrinsic
field dominates the externally applied field. Since the asymmetry component
corresponding to muons stopping within the NPs is of primary interest here, we
calibrated the zero-asymmetry level (i.e. the $\alpha$ parameter) using the
rapidly relaxing spectra in LF = 0.01 T. Hence, the negative asymmetry
observed for LF = 0 in Fig. 2(a) is artificial.

The persistent relaxation of the spectra shown in Fig. 2(a) demonstrates the
presence of SPM dynamics at 150 K, as expected. Similar measurements performed
at 3 K show strongly contrasting behavior, as seen in Fig. 2(b). In this case,
the spectra for LF = 0.01 T and 0.3 T both show very little relaxation with
time. Furthermore, the overall asymmetry level for LF = 0.3 T is much higher
than for 0.01 T. This “decoupling” of the asymmetry with increasing LF,
together with the lack of strong relaxation, demonstrate that the local
internal field for the vast majority of the muons is static, consistent with
the blocked state expected for this behavior. Interestingly, the slight
relaxation still observed at 3 K as seen in Fig. 2(b) indicates that a very
small fraction of the muons experience dynamically fluctuating fields even at
3 K, suggesting that some small portion of the sample remains in the SPM state
at this low temperature. Nevertheless, the data shown in Fig. 2(a) and (b)
clearly reveal that the SPM behavior at high temperature gives way to a
blocked state at low temperature, fully in accord with the expectations for
this sample.

To gain a better understanding of the blocking transition, we collected
$\mu$SR data at a series of temperatures between 3 K and 150 K with LF = 0.01
T. The application of this weak field allows us to decouple the slowly
relaxing asymmetry arising from muons stopping in the ligand shell so that any
remaining relaxation can be related directly to the behavior of the NPs. Fig.
2(c) displays the spectra at the representative temperatures 100 K, 35 K, and
3 K. At 100 K, the asymmetry relaxes steadily and approaches zero due to the
SPM fluctuations. On the other hand, the spectrum collected at 3 K [the same
one as shown in Fig. 2(b)], remains relatively constant in time, as expected
for NPs in the blocked state. It is also noteworthy that at $t=0$, the 100 K
spectrum has a higher value of the initial observed asymmetry than does the 3
K spectrum. This is due to the well-known phenomenon by which the $\mu$SR
spectra for samples with randomly oriented, static magnetic domains
(individual NPs in this case) contain two components: a “two-thirds” asymmetry
component with oscillations and/or strong damping due to internal field
components that are perpendicular to the initial muon spin direction, and a
“one-third” tail showing greatly reduced relaxation arising from the internal
field components that are parallel to the muon spin and therefore cause no
Larmor precession. In the present case, only the one-third tail is visible for
the spectrum collected at 3 K; the two-thirds component apparently cannot be
resolved within the time resolution of the detectors due to very large
internal fields and/or a broad field distribution at the muon site. At 100 K,
the NPs undergo SPM fluctuations that do not result in a loss of the two-
thirds component, explaining the higher initial ($t=0$) asymmetry value for
100 K.

The spectrum collected at 35 K displays more rapid relaxation than either of
the other two spectra displayed in Fig. 2(c). Considering that this is close
to $T_{B}$ = 24 K observed in the magnetometry data, this enhanced relaxation
rate can be attributed to slow dynamics of the SPM fluctuations near the
blocking transition, very similar to the enhanced $\mu$SR relaxation due to
critical dynamics observed in the vicinity of any conventional magnetic
ordering transition in bulk materials Frandsen  _et al._ (2020); Yaouanc and
de Réotier (2011); Uemura (1999). In addition, close inspection of the 35 K
spectrum reveals that the initial observable asymmetry lies between those of
the 100 K and 3 K spectra, suggesting that the sample is in an intermediate
state at 35 K, with part of the sample blocked and the rest in the SPM state.

We now proceed with a more quantitative analysis of the $\mu$SR data to probe
the blocking transition. First, we display in Fig. 3(a) the results of
integrating the experimental asymmetry spectra over the first 8 $\mu$s for
each temperature point at which data were collected.

Figure 3:  (a) Integrated $\mu$SR asymmetry as a function of temperature for
the 5 nm NPs. (b) Temperature dependence of the superparamagnetic relaxation
rate $\sigma$ as determined by least-squares fits. (c) Volume fraction of the
sample exhibiting superparamagnetic behavior as a function of temperature,
obtained from the least-squares fits. The schematic diagrams illustrate (from
right to left) the fully superparamagnetic state, mixed state, and fully
blocked state. (d) Logarithm of the superparamagnetic relaxation rate $\sigma$
plotted against $1/T$. The solid line represents a fit using the activation
model described in the text.

The most striking feature is a sharp rise in the integrated asymmetry
beginning below 30 K, which results from an increasing portion of the sample
entering the blocked state and consequently experiencing greatly reduced
relaxation. This simple quantification of the $\mu$SR asymmetry spectra is an
effective way to gain a rough estimate of $T_{B}$.

More detailed understanding can be gained by fitting a mathematical model to
the asymmetry spectra using least-squares optimization. Since individual muons
act as local probes of their immediate surroundings, regions of the sample in
distinct magnetic states (such as SPM and blocked states) will make distinct
contributions to the total observed asymmetry. Accordingly, we use the
following two-component model to describe the asymmetry spectra collected in
LF = 0.01 T:

$\displaystyle a(t)=a_{0}\left[f_{\mathrm{SPM}}\times e^{-(\sigma
t)^{\beta_{1}}}+\frac{1-f_{\mathrm{SPM}}}{3}\times e^{-(\lambda
t)^{\beta_{2}}}\right],$ (4)

where $a_{0}$ is the total initial asymmetry observed in the SPM state,
$f_{\mathrm{SPM}}$ is the volume fraction of the sample in the SPM state,
$\sigma$ is the SPM relaxation rate, $\lambda=1/T_{1}$ is the conventional
electronic relaxation rate observed in the blocked state due to intra-particle
spin flip processes, and $\beta_{1}$ and $\beta_{2}$ are exponential
“stretching” parameters for the SPM and blocked states, respectively. The 3 in
the denominator of the coefficient on the second term reflects the fact that
we only observe the one-third tail of the asymmetry from the blocked state.
This model was successfully used in a $\mu$SR study of $\gamma$-Fe2O3
nanoparticles Rebbouh  _et al._ (2007) and is justified by an earlier
theoretical work Lord (2005). For ideally monodisperse nanoparticles with a
single fluctuation rate, $\beta_{1}=0.5$ and $\beta_{2}=1$. If these
assumptions are not met, $\beta_{1}$ and $\beta_{2}$ will typically deviate to
become lower than these ideal values Lord (2005).

We performed least-squares fits to the temperature-dependent $\mu$SR spectra
using this model with $a_{0}$, $\beta_{1}$, and $\beta_{2}$ set as global
parameters common for all spectra, while $f_{\mathrm{SPM}}$, $\sigma$, and
$\lambda$ were allowed to vary with temperature. The resulting fits provided
excellent matches to the data, as seen by the black curves in Fig. 2(c). The
value of $a_{0}$ came to 0.13, which is less than the full asymmetry of
$\sim$0.2 - 0.25 expected at TRIUMF due to the decoupled component from muons
landing in the ligand shells described previously. $\beta_{1}$ refined to
0.25, which is consistent with the finite size distribution of the sample.
$\beta_{2}$ tended to remain close to the ideal value of 1, so we fixed it to
1 for simplicity. The temperature-dependent values of $\lambda$ all remained
fairly small ($\lesssim 0.3~{}\mu$s).

The most important parameters for understanding the blocking transition in the
sample are $\sigma$ and $f_{\mathrm{SPM}}$, which are displayed versus
temperature in Fig. 3(b) and (c). The SPM relaxation rate $\sigma$ in panel
(b) displays a dramatic peak centered around 30 K indicative of the blocking
transition, consistent with the change of slope in the integrated asymmetry
seen in Fig. 3(a) and the divergence of the FC and ZFC magnetization curves
shown in Fig. 1(b). Inspecting $f_{\mathrm{SPM}}$ in panel (c), we see that
the SPM fraction is below 0.1 at the lowest measured temperature of 3 K,
indicating that at least 90% of the sample is fully blocked at this
temperature. With increasing temperature, $f_{\mathrm{SPM}}$ rises steadily,
passing the midpoint (i.e. $f_{\mathrm{SPM}}$ = 0.5) around 17 K and finally
reaching 1 for 50 K and above, indicating that the full sample volume is SPM
at these temperatures. Thus, for the entire temperature region between
approximately 3 K and 50 K, the data suggest sample is in a mixed state. In
this mixed state, some regions of the sample exhibit SPM behavior while other
regions are in the blocked state, as illustrated by the schematic diagrams in
Fig. 3(c). Based on these results, we conclude that the blocking transition
occurs gradually over a finite temperature window between approximately 3 K
and 50 K. From this, we can define three convenient temperatures related to
the blocking transition: the onset blocking temperature, which is the highest
temperature at which a nonzero volume of the sample is in the blocked state;
the midpoint blocking temperature, at which 50% of the sample volume is in the
blocked state; and a completion blocking temperature, at which 100% of the
sample is in the blocked state. For the current sample, the onset, midpoint,
and completion blocking temperatures are $\gtrsim 45$ K, $\sim 16.6$ K, and
$\lesssim 3$ K, respectively. We stress that the nature of $\mu$SR as a local
probe imbues this technique with a rather unique ability to quantify the
segregation or coexistence of distinct magnetic states as seen here, providing
complementary information to bulk probes such as magnetometry.

We can obtain microscopic details about the SPM properties of the sample
through further analysis of the relaxation rate $\sigma$. The monotonic
decrease of $\sigma$ as the temperature is raised from 30 K (roughly where the
peak in $\sigma$ occurs) to 150 K results from thermal activation of
nanoparticle spin flip processes and can be modeled by Rebbouh  _et al._
(2007)

$\displaystyle\sigma(T)=\sigma_{0}\exp\left({E_{A}/k_{\mathrm{B}}T}\right),$
(5)

where $T$ is the temperature in kelvin, $\sigma_{0}$ is a type of intrinsic
relaxation rate related to the asymptotic dynamics at high temperature,
$E_{A}$ is the activation energy for flipping the net magnetic moment of an
individual nanoparticle, and $k_{\mathrm{B}}$ is the Boltzmann constant. This
is most conveniently displayed as a logarithmic plot of $\sigma$ versus
inverse temperature, as shown in Fig. 3(d). Eq. 5 appears as a line in this
type of plot, with the slope equal to $E_{A}/k_{\mathrm{B}}$. A fit to the
data is shown by the black line in Fig. 3(d). The best-fit value of $E_{A}$ is
$1.4(1)\times 10^{-21}$ J, corresponding to a magnetic anisotropy constant
$K=E_{A}/V$ (with $V$ the NP volume) of $19\pm 11$ kJ/m3. While experimental
limitations and the finite size distribution of our sample preclude a highly
precise calculation of $K$ from our data, our estimate is nevertheless
generally consistent with other values reported in the literature Rebbouh  _et
al._ (2007); Tackett  _et al._ (2008); Périgo  _et al._ (2015); Dobosz  _et
al._ (2017), lending confidence to this result.

The best-fit value of $\sigma_{0}$ can further be used to estimate the
previously mentioned magnetic reversal attempt time $\tau_{0}$ in the limit of
high temperature using the relationship Lord (2005); Rebbouh  _et al._ (2007)

$\displaystyle\sigma_{0}=4(\gamma_{\mu}\Delta B)^{2}\tau_{0},$ (6)

where $\gamma_{\mu}=851.6$ $\mu$s-1T-1 is the muon’s gyromagnetic ratio and
$\Delta B$ is the half width at half maximum of the typical distribution of
internal fields at the muon stopping sites. Since the instantaneous internal
field is due primarily to the ferrimagnetic order of magnetite within each
nanoparticle, we can estimate $\Delta B$ from the transverse damping rate of
the spontaneous asymmetry oscillations observed in bulk Fe3O4 (data not shown)
Dunsiger  _et al._ (1996); Frandsen  _et al._ (2020). This yields an estimate
of $\tau_{0}$ of $2.8(5)\times 10^{-10}$ s, likewise in line with published
estimates for magnetite and related nanoparticle systems Mørup and Tronc
(1994); Tronc  _et al._ (2000); Rebbouh  _et al._ (2007); Tackett  _et al._
(2008); Périgo  _et al._ (2015).

As a final check for self-consistency, we insert the experimentally determined
values of $E_{A}$ and $\tau_{0}$, together with $\tau_{m}\approx 10^{-6}$ s
for $\mu$SR, into Eq. 2 to calculate $T_{B}$. The result is $T_{B}\sim 12$ K,
which falls comfortably within the range of temperatures over which the sample
is transitioning between the SPM and blocked states. This consistency supports
the reliability of the estimates of the microscopic parameters obtained from
the $\mu$SR analysis. It also highlights the value of using $\mu$SR to
estimate $\tau_{0}$ directly from the temperature-dependent relaxation rate,
providing an independent calculation in addition to the more frequently used
relationship in Eq. 2.

###  III.3 20 nm nanoparticles: TEM and Magnetometry

Equivalent characterizations using TEM, magnetometry, and $\mu$SR were carried
out on the sample of 20 nm Fe3O4 NPs. When deposited on a thin membrane for
TEM imaging, the 20 nm NPs tend to self-assemble in a hexagonal lattice,
similar to the 5 nm NPs. The TEM image in Fig. 4(a) indicates the formation of
monolayers of closely packed 20 nm NPs distributed in sparse islands
throughout the film.

Figure 4:  (a) TEM image of a 20 nm NP self-assembly deposited on a thin
carbon membrane. Inset: Histogram of particle sizes yielding an average size
of $19.22\pm 2.37$ nm. (b) FC/ZFC curves collected under a field of 100 Oe (10
mT). Inset: Zoomed-in view of the blocking transition region.

With optimal NP concentration, one may achieve a uniform monolayer coverage,
with a few particles stacking on top of each other. A fine analysis of the
particle size carried out on multiple TEM images including about 265 NPs
yields an average particle size of 19.23 $\pm$ 2.37 nm.

The FC/ZFC curves in Fig. 4(b) indicate that the 20 nm NPs also exhibit
superparamagnetism, with a blocking transition occurring at $T_{B}\approx 295$
K using the ZFC peak criterion and $T_{B}\approx 290$ K using the merging
point criterion, again not significantly different. It has been suggested Li
_et al._ (2017) that 20 nm may be the upper NP size limit for SPM to occur,
but in any case, the commercially supplied 20 nm NPs studied here
unambiguously display SPM above room temperature. Another remarkable feature
observed in both the FC and ZFC curves is a kink at about 120 K, which we
interpret as a signature of the Verwey transition, a magneto-crystalline
transition normally occurring in bulk Fe3O4 at $T_{\mathrm{V}}=125$ K Verwey
and Haayman (1941). This transition has been previously observed in FC/ZFC
curves collected on larger Fe3O4 NPs. One study on 50-100 nm NPs shows the
occurrence of a clear peak at 125 K Klomp  _et al._ (2020). Another study
shows a kink in the FC/ZFC curves located at lower temperatures, 16 K for 50
nm NPs and 98 K for 150 nm NPs Goya  _et al._ (2003). For NPs smaller than 20
nm, FC/ZFC studies have established the complete vanishing of the Verwey
transition Klomp  _et al._ (2020). It is therefore interesting to observe a
sign of the Verwey transition in the present 20 nm NP material.

Following the earlier analysis of the VSM data to obtain an estimate of the
attempt time $\tau_{0}$, we calculate the ratio $E_{A}/k_{B}T_{B}$ to be
$\sim$5.7, using $T_{B}=290$ K and $E_{A}=2.3\times 10^{-20}$ J as obtained
from the $\mu$SR data (see below). This yields an attempt time of
$\tau_{0}\approx 3.2\times 10^{-3}$ s (3.2 ms), once again far too large a
value to be realistic. Using the energy correction method described
previously, we find that the saturated magnetization for the 20 nm NPs is
$M_{s}\approx 204000$ $\mu_{B}$, leading to a Zeeman energy of $E_{M}\approx
3.8\times 10^{-20}$J. The adjusted energy ratio $\Delta E/k_{B}T_{B}\approx
15.2$ corresponds to an adjusted attempt time $\tau_{0}^{*}\approx 2.5\times
10^{-7}$ s (250 ns). This is about 4 orders of magnitude smaller than the
initial estimate of $\tau_{0}$, but still too large with respect to the
expected value, meaning not all of the relevant interactions have been
properly modeled.

###  III.4 20 nm nanoparticles: $\mu$SR

The $\mu$SR data collected on the 20 nm NPs are qualitatively similar to those
of the 5 nm NPs, confirming a broad transition from superparamagnetism at high
temperature to the blocked state at low temperature. Spectra collected at 300
K in zero field (ZF) and LF = 0.3 T are shown in Fig. 5(a), where the strong
relaxation in the LF spectrum confirms the prevalence of SPM dynamics.

Figure 5:  (a) $\mu$SR asymmetry spectra for 20 nm Fe3O4 NPs collected at 300
K in zero field (ZF) and LF = 0.3 T. (b) ZF asymmetry spectra at 300 K (mostly
superparamagnetic), 220 K (mixed superparamagnetic and blocked states), and 3
K (blocked). The solid black curves are fits described in the main text.

Interestingly, the ZF spectrum does not show the slowly relaxing component
observed in the 5 nm data. This is likely because the much greater volume of
the NPs in this sample compared to the 5 nm sample results in a much larger
fraction of muons landing inside the NPs rather than in the ligand shell, so
the weak relaxation from muons stopping in the ligand shell is negligible. ZF
spectra at representative temperatures are shown in Fig. 5, where we once
again observe fast relaxation at high temperature (300 K) in the SPM state,
nearly negligible relaxation in the blocked state at low temperature (3 K),
and intermediate behavior at 220 K suggesting a mixed state with both SPM and
blocked regions of the sample.

We carried out the same quantitative analysis for the 20 nm data as we did for
the 5 nm data. The integrated asymmetry is plotted as a function of
temperature in Fig. 6(a).

Figure 6:  (a) Integrated $\mu$SR asymmetry as a function of temperature for
the 20 nm NPs. (b) Temperature dependence of the superparamagnetic relaxation
rate $\sigma$ as determined by least-squares fits. (c) Volume fraction of the
sample exhibiting superparamagnetic behavior as a function of temperature. (d)
Logarithm of the superparamagnetic relaxation rate $\sigma$ plotted against
$1/T$. The solid line represents a fit using the activation model described in
the text.

A minimum is observed around 220 K, with the integrated asymmetry increasing
monotonically as the temperature is lowered below 220 K. As before, this
indicates that a significant fraction of the sample has entered the blocked
state as the temperature is reduced beyond this point.

Eq. 4 was once again used to model the ZF spectra, with representative fits
shown as the black curves in Fig. 5. The globally refined value of $a_{0}$
came to 0.21, $\beta_{1}$ refined to 0.25 as in the 5 nm case, and $\beta_{2}$
refined to 0.6. The temperature dependence of $\sigma$ and $f_{\mathrm{SPM}}$
are displayed in Fig. 6(b) and (c), where we again observe a prominent peak in
$\sigma$ corresponding to the blocking transition and a gradual evolution of
$f_{\mathrm{SPM}}$, revealing a wide temperature range over which the mixed
state with coexisting SPM and blocked regions exists in the sample.
Interestingly, even at the highest temperature measured (300 K), the fits
indicate that approximately 5-10% of the sample is already in the blocked
state. This fraction could plausibly be even higher, but without a reference
measurement fully in the SPM state, there is some difficulty in unambiguously
separating the two components in Eq. 4 at high temperature. Regardless, the
$\mu$SR data indicate that the onset of the blocking transition in the $\mu$SR
time scale is above room temperature and therefore higher than the
$T_{B}\approx 295$ K estimated from the magnetometry data in Fig. 4. From the
$f_{\mathrm{SPM}}$, the onset, midpoint, and completion blocking temperatures
for the 20 nm sample are $>300$ K, $\sim 200$ K, and $\sim 150$ K,
respectively.

Fig. 6(d) shows the result of the thermal activation analysis carried out on
the best-fit values of $\sigma$ using Eq. 5. With only four data points
comprising the high-temperature side of the peak in $\sigma$ [see Fig. 6(b)],
our ability to determine the activation energy, anisotropy, and intrinsic
nanoparticle spin fluctuation time reliably is quite limited compared to the 5
nm results. Pressing forward nonetheless, we estimate the activation energy to
be $E_{A}=2.3(6)\times 10^{-20}$ J, the anisotropy constant to be $K=6.1\pm
2.9$ kJ/m3, and the intrinsic nanoparticle spin fluctuation time to be
$\tau_{0}\sim 10^{-11}$ s. We note that the estimated uncertainty for
$\tau_{0}$ was larger than the calculated value itself, so this result should
be considered cautiously. Nevertheless, inserting these values into Eq. 2
yields an estimated $T_{B}$ of 143 K, which is no more than a factor of
$\sim$2 different from the observed blocking temperature.

##  IV Discussion and Conclusion

The $\mu$SR and magnetometry results presented here provide valuable
information about the magnetic properties of Fe3O4 nanoparticles. Key
quantities related to the magnetism in both samples are listed in Table 1.

Table 1: Selected magnetic properties of 5 and 20 nm Fe3O4 nanoparticles. The various estimates of the blocking temperature $T_{B}$ and the magnetic reversal attempt time $\tau_{0}$ are explained in the main text. | Particle size
---|---
| 5 nm | 20 nm
VSM results |  |
$T_{B}$ ZFC peak (K) | 24 | 295
$T_{B}$ merging (K) | 28 | 290
$E_{M}$ at 10 mT ($10^{-21}$ J) | 0.74 | 37
$\tau_{0}^{*}$ (s) | $1.5\times 10^{-3}$ | $2.5\times 10^{-7}$
$\mu$SR results |  |
$T_{B}$ onset (K) | 45 |  $>$300
$T_{B}$ midpoint (K) | 16.6 | 200
$T_{B}$ completion (K) |  $<$3 | 150
$E_{A}$ ($10^{-21}$ J) | 1.4 $\pm$ 0.1 | 23 $\pm$ 6
$K$ (kJ/m3) | 19 $\pm$ 11 | 6.1 $\pm$ 2.9
$\tau_{0}$ (s) | $(0.28\pm 0.05)\times 10^{-10}$ | $10^{-11\dagger}$
†Calculated uncertainty is larger than the value itself;
upper limit is $10^{-9}$ s.

Both techniques confirm the trend that with increasing particle size, the
blocking temperature increases and the magnetic anisotropy decreases. This can
be understood as a consequence of the smaller surface-to-volume ratio in the
larger particles. The quantitative differences in the blocking temperature as
determined by the two different techniques are attributable to the differing
sensitivities of the techniques. Specifically, the $T_{B}$ values determined
by magnetometry are lower in temperature than the onset $T_{B}$ as detected by
$\mu$SR, which is consistent with the fact that $\mu$SR is a faster probe.
Fluctuations longer than $\sim 10^{-6}$ s begin to appear static to $\mu$SR,
so for a finite temperature range above $T_{B}$ as measured by magnetometry,
the nanoparticle spins fluctuate faster than the measurement time scale for
magnetometry but slower than that of $\mu$SR. On the other hand, the midpoint
of the blocking transition as measured by $\mu$SR (corresponding to
$f_{\mathrm{SPM}}=0.5$) occurs at lower temperatures than $T_{B}$ as measured
by magnetometry, suggesting that $T_{B}$ as identified from the VSM data
occurs when more than half of the sample volume is in the SPM state.

The greatest discrepancies between the magnetometry and $\mu$SR results are
for the magnetic reversal attempt time $\tau_{0}$. The values obtained from
the VSM data using Eq. 3 are orders of magnitude larger than expected. The
inclusion of the Zeeman energy term in the analysis of the VSM data improves
the estimate of $\tau_{0}$, but is still not sufficient to produce a
physically reasonable value. This demonstrates the inherent challenges of
extracting a reliable attempt time from bulk magnetometry data. On the other
hand, the $\mu$SR estimates of $\tau_{0}$ fall into the expected range of
values and produce a self-consistent estimate of the blocking temperature,
underscoring the accuracy of this approach. The favorable time scale of
$\mu$SR measurements, together with its sensitivity to the local magnetic
environment, make this technique a powerful approach to probing SPM dynamics
that can complement more commonly used experimental methods such as bulk
magnetometry.

Finally, we now discuss in more detail the observation of a gradual transition
from the blocked to the SPM state as revealed by the $\mu$SR data. This
indicates the presence of an intermediate mixed state containing coexisting
SPM and blocked regions within the sample. The finite size distribution of
each sample will be at least partially responsible for this gradual
transition, since particles with slightly different sizes will enter the
blocked state at slightly different temperatures. However, it is conceivable
that other factors such as interparticle interactions could also contribute to
the broadening of the blocking transition. To investigate this, we calculated
the expected SPM fraction as a function of temperature for both samples
assuming that $T_{B}$ depends linearly on particle size. This linear
dependence was estimated by considering the midpoint blocking temperatures
(where 50% of the volume is blocked) for the two samples in the present work,
and determining the equation of the line connecting those two points as a
function of particle size. This is shown by the black circles and dashed line
in Fig. 7(a). The equation of the dashed line is $T_{B}(D)=AD+B$, where $D$ is
the particle size in nm, $A=13~{}\mathrm{K/nm},$ and $B=-50.8~{}\mathrm{K}$.
For comparison, we additionally plot $T_{B}$ as determined from the ZFC peak
in the magnetometry data as brown triangles and other previously published
magnetometry results Klomp  _et al._ (2020) as open squares. Other
experimental stuides of magnetic NPs have likewise suggested a linear
dependence of $T_{B}$ on particle size as a first order approximation in the 5
-25 nm range Mohapatra  _et al._ (2013); Shim  _et al._ (2008); Aslani  _et
al._ (2018). The particle size distributions $p(D)$ for the two samples in the
present study are also shown by the blue and orange curves, approximating the
measured size distributions for the 5 and 20 nm NPs displayed in Fig. 1(a) and
Fig. 4(a), respectively, as normalized Gaussian distributions with the mean
and standard deviation determined from the TEM characterization.

Figure 7: (a) Blocking temperature $T_{B}$ versus particle size as determined
by various techniques. In the case of the $\mu$SR results (black circles), the
value of $T_{B}$ shown here corresponds to the temperature at which 50% of the
sample is in the blocked state. The dashed line represents the equation of the
line connecting the two $\mu$SR data points. The solid blue and orange curves
represent the particle size distributions for the two samples studied in this
work. (b) Calculated temperature dependence of the superparamagnetic fraction
$f_{\mathrm{SPM}}$ (dashed curves) compared to the experimentally determined
results (solid circles) for the two samples in the present work. Details about
the calculations are given in the main text.

The expected SPM fraction $f_{\mathrm{SPM}}$ can be calculated for a given
temperature $T$ by inverting the linear function $T_{B}(D)$ to determine the
cutoff particle size $D_{c}(T)=(T-B)/A$ that undergoes the blocking transition
at $T$, and integrating the normalized particle size distribution $p(D)$ up to
$D_{c}$. Mathematically, we have

$\displaystyle f_{\mathrm{SPM}}(T)=\int_{0}^{D_{c}(T)}p(D)\mathrm{d}D.$ (7)

The results of this calculation for both samples are shown as the dashed
curves in Fig. 7(b), together with the experimentally determined values shown
as the colored circles. The agreement between the experimental and calculated
results is relatively good, especially considering the simplicity of our model
with an assumed linear dependence of $T_{B}$ on particle size. This indicates
that the broad temperature range of coexisting blocked and superparmagnetic
regions revealed by $\mu$SR is primarily caused by the finite particle size
distribution, although interparticle interactions or other factors may play a
role as well Bruvera  _et al._ (2015); Majetich and Sachan (2006).

In summary, we have employed magnetometry and $\mu$SR in a complementary
fashion to gain detailed information about the SPM behavior and the blocking
transition in assemblies of Fe3O4 nanoparticles with average particle
diameters of 5 nm and 20 nm. We established that the transition between the
SPM and blocked states extends gradually throughout the sample volume over a
wide temperature interval due to the finite size distribution of each sample.
We provided estimates of the blocking temperature for each sample using
various methods, and we extracted microscopic information including the
activation energy, anisotropy constant, and magnetic reversal attempt time.
This allowed us to investigate the dependence of these properties on particle
size. Finally, we have also discussed some of the difficulties of estimating
microscopic parameters such as the attempt time $\tau_{0}$ using bulk
magnetometry alone, and instead have shown that local probes such as $\mu$SR
can be highly useful for reliably estimating $\tau_{0}$, determining the
volume fraction of competing states, and otherwise complementing data obtained
from bulk probes.

Acknowledgements

We thank Gerald Morris, Bassam Hitti, and Donald Arseneau for their support at
the $\mu$SR beamline at TRIUMF. This work was made possible by funds provided
by the College of Physical and Mathematical Sciences at Brigham Young
University.

## References

  * Pankhurst  _et al._ (2003) Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson, J. Phys. D: Appl. Phys. 36, R167 (2003).
  * De  _et al._ (2008) M. De, P. S. Ghosh, and V. M. Rotello, Adv. Mater. 20, 4225 (2008).
  * Frey  _et al._ (2009) N. A. Frey, S. Peng, K. Cheng, and S. Sun, Chem. Soc. Rev. 38, 2532 (2009).
  * Majetich  _et al._ (2011) S. A. Majetich, T. Wen, and R. A. Booth, ACS Nano 5, 6081 (2011).
  * Goya  _et al._ (2003) G. F. Goya, T. S. Berquó, F. C. Fonseca, and M. P. Morales, J. Appl. Phys. 94, 3520 (2003).
  * Lima  _et al._ (2006) E. Lima, A. L. Brandl, A. D. Arelaro, and G. F. Goya, J. Appl. Phys. 99, 083908 (2006).
  * Varón  _et al._ (2013) M. Varón, M. Beleggia, T. Kasama, R. J. Harrison, R. E. Dunin-Borkowski, V. F. Puntes, and C. Frandsen, Sci. Rep. 3, 1234 (2013).
  * Majetich and Sachan (2006) S. A. Majetich and M. Sachan, J. Phys. D: Appl. Phys. 39, R407 (2006).
  * Roca  _et al._ (2006) A. G. Roca, M. P. Morales, K. O’Grady, and C. J. Serna, Nanotechnology 17, 2783 (2006).
  * Yang  _et al._ (2005) T. Yang, C. Shen, Z. Li, H. Zhang, C. Xiao, S. Chen, Z. Xu, D. Shi, J. Li, and H. Gao, J. Phys. Chem. B 109, 23233 (2005).
  * Livesey  _et al._ (2018) K. L. Livesey, S. Ruta, N. R. Anderson, D. Baldomir, R. W. Chantrell, and D. Serantes, Sci. Rep. 8, 11166 (2018).
  * Klomp  _et al._ (2020) S. Klomp, C. Walker, M. Christiansen, B. Newbold, D. Griner, Y. Cai, P. Minson, J. Farrer, S. Smith, B. J. Campbell, R. G. Harrison, and K. Chesnel, IEEE T. Magn. 56, 2300109 (2020).
  * Périgo  _et al._ (2015) E. A. Périgo, G. Hemery, O. Sandre, D. Ortega, E. Garaio, F. Plazaola, and F. J. Teran, Appl. Phys. Rev. 2, 041302 (2015).
  * Gabbasov  _et al._ (2015) R. Gabbasov, M. Polikarpov, V. Cherepanov, M. Chuev, I. Mischenko, A. Lomov, A. Wang, and V. Panchenko, J. Magn. Magn. Mater. 380, 111 (2015).
  * Yamamuro  _et al._ (2002) S. Yamamuro, D. F. Farrell, and S. A. Majetich, Phys. Rev. B 65, 224431 (2002).
  * Yamamoto  _et al._ (2008) K. Yamamoto, S. A. Majetich, M. R. McCartney, M. Sachan, S. Yamamuro, and T. Hirayama, Appl. Phys. Lett. 93, 082502 (2008).
  * Mishra  _et al._ (2015) D. Mishra, O. Petracic, A. Devishvili, K. Theis-Bröhl, B. P. Toperverg, and H. Zabel, J. Phys: Condens. Mat. 27, 136001 (2015).
  * Cai  _et al._ (2014) Y. P. Cai, K. Chesnel, M. Trevino, A. Westover, R. G. Harrison, J. M. Hancock, S. Turley, A. Scherz, A. Reid, B. Wu, C. Graves, T. Wang, T. Liu, and H. Dürr, J. Appl. Phys. 115, 17B537 (2014).
  * Chesnel  _et al._ (2018) K. Chesnel, D. Griner, D. Smith, Y. Cai, M. Trevino, B. Newbold, T. Wang, T. Liu, E. Jal, A. H. Reid, and R. G. Harrison, Magnetochemistry 4, 42 (2018).
  * Uemura (1999) Y. Uemura, in  _Muon Science: Muons in Physics, Chemistry and Materials_ , edited by S. Lee, R. Cywinski, and S. Kilcoyne (Taylor & Francis, New York, 1999).
  * Blundell (2010) S. J. Blundell, Contemporary Physics 40, 175 (2010).
  * Yaouanc and de Réotier (2011) A. Yaouanc and P. D. de Réotier,  _Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter_ , 1st ed. (Oxford University Press, Oxford, 2011).
  * Chesnel  _et al._ (2014) K. Chesnel, M. Trevino, Y. Cai, J. M. Hancock, S. J. Smith, and R. G. Harrison, J. Phys.: Conf. Ser. 521, 012004 (2014).
  * Altavilla  _et al._ (2005) C. Altavilla, E. Ciliberto, D. Gatteschi, and C. Sangregorio, Adv. Mater. 17, 1084 (2005).
  * Suter and Wojek (2012) A. Suter and B. Wojek, Physics Procedia 30, 69 (2012), 12th International Conference on Muon Spin Rotation, Relaxation and Resonance ($\mu$SR2011).
  * Bruvera  _et al._ (2015) I. J. Bruvera, P. Mendoza Zélis, M. Pilar Calatayud, G. F. Goya, and F. H. Sánchez, J. Appl. Phys. 118, 184304 (2015).
  * Néel (1949) L. Néel, Annales de Géophysique 5, 99 (1949).
  * Brown (1963) W. F. Brown, Phys. Rev. 130, 1677 (1963).
  * Torres  _et al._ (2015) T. E. Torres, E. Lima, A. Mayoral, A. Ibarra, C. Marquina, M. R. Ibarra, and G. F. Goya, J. Appl. Phys. 118, 183902 (2015).
  * Frandsen  _et al._ (2020) B. A. Frandsen, K. A. Petersen, N. A. Ducharme, A. G. Shaw, E. J. Gibson, B. Winn, J. Yan, J. Zhang, M. E. Manley, and R. P. Hermann, Phys. Rev. Materials 4, 074405 (2020).
  * Rebbouh  _et al._ (2007) L. Rebbouh, R. P. Hermann, F. Grandjean, T. Hyeon, K. An, A. Amato, and G. J. Long, Phys. Rev. B 76, 174422 (2007).
  * Lord (2005) J. S. Lord, J. Phys.: Conf. Ser. 17, 81 (2005).
  * Tackett  _et al._ (2008) R. Tackett, C. Sudakar, R. Naik, G. Lawes, C. Rablau, and P. Vaishnava, J. Magn. Magn. Mater. 320, 2755 (2008).
  * Dobosz  _et al._ (2017) B. Dobosz, R. Krzyminiewski, J. Kurczewska, and G. Schroeder, J. Magn. Magn. Mater. 429, 203 (2017).
  * Dunsiger  _et al._ (1996) S. R. Dunsiger, R. F. Kiefl, K. H. Chow, B. D. Gaulin, M. J. P. Gingras, J. E. Greedan, A. Keren, K. Kojima, G. M. Luke, W. A. MacFarlane, N. P. Raju, J. E. Sonier, Y. J. Uemura, and W. D. Wu, Phys. Rev. B 54, 9019 (1996).
  * Mørup and Tronc (1994) S. Mørup and E. Tronc, Phys. Rev. Lett. 72, 3278 (1994).
  * Tronc  _et al._ (2000) E. Tronc, A. Ezzir, R. Cherkaoui, C. Chanéac, M. Noguès, H. Kachkachi, D. Fiorani, A. M. Testa, J. M. Grenèche, and J. P. Jolivet, J. Magn. Magn. Mater. 221, 63 (2000).
  * Li  _et al._ (2017) Q. Li, C. A. Kartikowati, T. Ogi, T. Iwaki, and K. Okuyama, Sci. Rep. 7, 9894 (2017).
  * Verwey and Haayman (1941) E. J. W. Verwey and P. W. Haayman, Physica 8, 979 (1941).
  * Mohapatra  _et al._ (2013) J. Mohapatra, A. Mitra, D. Bahadur, and M. Aslam, CrystEngComm 15, 524 (2013).
  * Shim  _et al._ (2008) H. Shim, P. Dutta, M. S. Seehra, and J. Bonevich, Solid State Commun. 145, 192 (2008).
  * Aslani  _et al._ (2018) A. Aslani, M. Ghahremani, M. Zhang, L. H. Bennett, and E. Della Torre, IEEE T. Magn. 54, 1 (2018).