# Response of the Verwey transition in magnetite to a controlled
point-like disorder induced by 2.5 MeV electron irradiation

Ruslan Prozorov<EMAIL_ADDRESS>Ames National Laboratory, Ames, Iowa
50011, USA  Department of Physics & Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Makariy A. Tanatar  Ames National Laboratory, Ames, Iowa
50011, USA  Department of Physics & Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Erik I. Timmons  Ames National Laboratory, Ames, Iowa 50011,
USA  Department of Physics & Astronomy, Iowa State University, Ames, Iowa
50011, USA  Marcin Kończykowski  Laboratoire des Solides Irradiés, École
Polytechnique, CNRS, CEA, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Tanya Prozorov  Ames National Laboratory, Ames, Iowa 50011, USA
Department of Chemical and Biological Engineering, Iowa State University,
Ames, Iowa 50011, USA

(14 May 2023)

###### Abstract

A controlled point-like disorder induced by low temperature 2.5 MeV electron
irradiation was used to probe the nature of the Verwey transition in
magnetite, $\textnormal{Fe}_{3}\textnormal{O}_{4}$. Two large single crystals,
one with optimal transition temperature, $T_{V}\approx 121$ K, and another
with $T_{V}\approx 109$ K, as well as biogenic nanocrystals, $T_{V}\approx
110$ K, were examined. Temperature-dependent resistivity is consistent with
the semiconductor-to-semiconductor sharp, step-like Verwey transition from a
state with a small bandgap of around 60 meV to a state with a larger bandgap
of about 300 meV. The irradiation causes an up-shift of the resistivity curves
above the transition without transition smearing or broadening. It also causes
an apparent down-shift of the resistivity maximum at high temperatures. In the
lower $T_{V}$ crystal, the electron irradiation drives the transition
temperature into a “forbidden” regime believed to separate the first order
from the second order phase transition. Contrary to this belief, the
transition itself remains sharp and hysteretic without a significant change in
the hysteresis width. We conclude that the sudden change of the bandgap
accompanied by the monoclinic distortion and the change of magnetic anisotropy
is the reason for the Verwey transition in magnetite and the effect of
additional disorder is mostly in the smearing of the sharp gap edges near the
Fermi level.

##  I Introduction

The original studies of magnetite, $\textnormal{Fe}_{3}\textnormal{O}_{4}$,
have initiated one of the most fascinating topics in condensed matter physics,
the metal-insulator transition (MIT) [1, 2, 3]. (Although now we know that,
specifically in magnetite, the transition is a semiconductor-to-semiconductor
type [4]). It all started in 1926 when an unexpected heat absorption at around
115 K - 117 K was found in heat capacity measurements by Parks and Kelley [1].
The authors write: “The heat absorption, found in the case of the magnetite
crystals at about 115 K, constitutes still another noteworthy feature. This
temperature would seem to be very low for a transformation in crystal
structure in a metallic oxide, the melting-point of which is above 1500 K. …
we think that the observed heat effect may possibly be connected with a change
in the magnetic properties of the substance. … Certainly this heat absorption
phenomenon should be investigated further.” No graphics/figures were shown,
though in Ref.[1]. Next, magnetic anomalies at similar temperatures were
reported [5, 6], seemingly agreeing with the assumption of the magnetic origin
of the observed behavior because no difference was found in x-ray diffraction
measured at room temperature and at liquid “air” (nitrogen) [6]. Finally,
thirteen years after the initial work, Verwey reported now famous two orders
of magnitude sharp increase of resistivity at around 120 K [2, 3]. Since that
time, it became known as the “Verwey transition,” and its microscopic origins
have been a subject of relentless experimental and theoretical research to
this day. Here we can only mention a very incomplete set of review articles
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Stoichiometric magnetite is a mix of iron oxides,
$\textnormal{Fe}_{3}\textnormal{O}_{4}=\textnormal{Fe}_{2}^{3+}\textnormal{O}_{3}^{2-}\cdot\textnormal{Fe}^{2+}\textnormal{O}^{2-}$,
which has an inverse spinel crystal structure with iron ions occupying two
distinct sites,
$\textrm{Fe}_{3}\textrm{O}_{4}=\left[\textrm{Fe}^{3+}\textrm{(A)}+\textrm{Fe}^{3+}\textrm{Fe}^{2+}\textrm{$\left(\textrm{B}\right)$}\right]\cdot\textrm{O}_{4}$
where A sites are coordinated in a tetrahedron and B sites are coordinated in
an octahedron. Ionic magnetic moments at the A and B sites are antiparallel,
resulting in ferrimagnetism with an excess magnetic moment of about $4\mu_{B}$
per formula units (f.u.). For each formula unit, there are two B sites with
spin $S=2.25\mu_{B}$ and one A site with $S=2.5\mu_{B}$. There are 8 f.u. in a
deformed cubic cell with lattice constant of $a=8.4\>$Å, so each cubic unit
cell contributes $32\mu_{B}$ of net moment. Magnetite’s Curie temperature is
about 800 K [21]. Above the Verwey transition temperature, $T_{V}\approx
121\>\textrm{K}$, the B sites are charge-frustrated leading to a significant
delocalization of electrons, which results in a moderately conducting state
with the resistivity of about $0.005-0.01\>\Omega\cdot\mathrm{cm}$ at room
temperature and about $0.02\>\Omega\cdot\mathrm{cm}$ at $T_{V}$. Verwey
discovered that below $T_{V}$, the resistivity increases sharply by about two
orders of magnitude and continues to increase significantly upon further
cooling. He has explained this behavior in terms of charge ordering in the B
subsystem: Fe2+ ions along the $\left[110\right]$ and Fe3+ along
$\left[\bar{110}\right]$ directions. This is called a “Verwey model” [9, 2,
3]. The resistive and thermodynamic signatures of the Verwey transition are
accompanied by the step-like change in sample magnetization and all other
thermodynamic, spectroscopic and transport properties [9]. Similar to the
resistive Verwey transition, the physics of the magnetic anomaly apparently
associated with it, is still debatable. It is known that magnetic easy axis
changes from the $\left[111\right]$ to $\left[100\right]$ (within $0.2\>$Å due
to small monoclinic distortion) direction below $T_{V}$. The $K_{1}$ term of
magnetic anisotropy energy increases by an order of magnitude, but the
amplitude of the total magnetic moment remains unchanged.

Since the original proposal, different mechanisms of the Verwey transition
considering new experimental results have been put forward. Resonant X-ray
studies found no evidence of charge ordering and strong strong electron-phonon
interaction was suggested as the cause of the Verwey transition [22]. Anderson
argued that in spinel’s octahedral sites, nearest-neighbor Coulomb
interactions can never lead to the long-range order, and even the long-range
Coulomb interaction are practically ineffective at least at higher
temperatures [23]. Based on the large thermopower and low conductivity
observed in magnetite both below and above the Verwey transition, it was
suggested that Anderson localization due to randomness of electrons and
impurities is involved [24, 25, 26]. In general, some form of electron
localization (e.g., trimerons [27, 28, 29, 30, 31, 32]) is considered as one
of the primary mechanisms of the transition. It was also demonstrated that the
collective Jahn-Teller distortion due to interacting local degeneracies may
play an important role [33, 34, 35, 28]. It supports some other ideas, such as
the cooperative ordering of molecular polarons below $T_{V}$ [36]. From
another angle, Mott suggested that the Verwey transition is a transition from
Wigner glass to Wigner crystal in an electronic subsystem, but there was not
much of experimental followup [37, 38]. Considering non-metallic behavior, it
was also suggested that the condensation of active optical phonon mode could
be the reason for the transition [39] and it seems that the oxygen-18 isotope
effect study supported it [40].

On the opposite side from electron localization mechanisms, there is
overwhelming experimental and theoretical evidence for the fairly large ($\sim
200$ meV) bandgap opening below the Verwey temperature [41, 42, 4, 16, 18].
This gap could originate from the charge-orbital ordering due to the on-site
Coulomb interactions [43]. Yet, some details are not quite clear. Some studies
assert that the energy gap only opens below $T_{V}$, while above the
transition the electronic bandstructure is gapless and involves molecular
polarons hopping conductivity [42, 16, 18]. Scanning tunneling spectroscopy
(STS) shows directly a clear smaller gap, of the order of 70 meV, above
$T_{V}$ transition and a larger gap below [41, 4]. Diffusive x-ray scattering
study finds short-range charge ordering above $T_{V}$ related to trimeron
ordering below the transition. This short-range ordering persist at least up
to room temperature and is, apparently, correlated with the Fermi surface
nesting signatures [32], which again would imply opening the small energy gaps
at least on parts of the Fermi surface.

Another experimental puzzle in magnetite is the discontinuity of the Verwey
transition separating first order phase transition in relatively clean
compositions from the (alleged) second order transition in less stoichiometric
compounds. For example, in Fe3(1-δ)O4, the critical value of
$\delta_{c}=0.0039$ [44, 45, 46, 10, 47, 48]. Ironically, the temperature-
dependent resistivity in the latter regime is highly hysteretic, pointing to a
first-order nature of the transition. No satisfactory explanation for this
disparity exists [46, 10, 49]. Similar bifurcation was observed in chemically
substituted magnetite,
$\text{Fe}_{3\left(1-x\right)}\text{M}_{3x}\text{O}_{4}$, with M = Ni, Co, Mg,
Al, Ti, Mn, Li (sorted from the lowest to the highest rate of $T_{V}$
suppression) [50, 51, 52, 53]. In this work, we show that the Verwey
transition continues to remain first order when driven to the “forbidden”
range of $T_{V}$ values by disorder. It shows that off-stoichiometry and
disorder are not equivalent.

The measurements under applied isostatic or directional (uniaxial strain)
pressure support this assertion. Isostatic pressure suppresses the Verwey
temperature and, at around a critical value of $P_{c}\approx 8$ GPa, recovers
the higher-conductance “metallic” state, although the details of $P_{c}$ and
other features vary between different studies [54, 47, 55, 16, 56].
Directional stress leads to the enhancement of $T_{V}$, most likely due to
favorable structural domain selectivity [21, 17, 57]. In either case,
pressure/strain change $T_{V}$ continuously (but at $P_{c}$ it disappears
abruptly) and the significant hysteresis in the measured quantity is observed,
supporting our statement above.

Finally, another line of study was inspired by the reports of a significant
smearing and shifting of the Verwey transition in magnetite nanoparticles of
different size with initial reports showing the effect already at about 100
nm, which is quite large for the local-moment magnetism [58, 59, 60, 61, 62,
63, 18]. This sub-field of magnetite nanoparticles grew very rapidly and,
unfortunately, it is very difficult or mostly impossible to know whether those
numerous reports have actually had nanocrystalline magnetite of a mix of iron
oxides or just maghemite, $\gamma-$Fe2O3. Many studies rely on the
conventional x-ray diffraction, but at room temperature it is hard to
distinguish between different oxides because both have the same cubic
structure and their lattice parameters are almost identical [64]. Moreover, it
is likely that nanoparticles powder-like samples actually contain different
compositions in comparable proportions [64]. The only reliable feature,
present only in magnetite, the Verwey transition is smeared and broadened in
nanoparticles assemblies due to size and shape distributions, stress, strain
and the variation of stoichiometry, hence is difficult to identify. The
biological magnetite is the exception, as it comes as well-shaped particles of
a similar size [65, 66] and is proven to be Fe3O4 by the magnetic induction
mapping using electron holography [67, 68] and NV-centers in diamond [69]).
One of the most detailed systematic investigations where a great deal of
attention was devoted to the identification of magnetite as the primary phase,
expectantly showed that the Verwey transition remains practically unchanged
down to a nanoparticle size of 8 nm, below which $T_{V}$ drops very rapidly
[70]. Similar conclusions were drawn by considering biological magnetite of
magneto-tactic bacteria where the chemical composition is preserved by the
lipid membranes preventing oxidation [66]. Of course, an additional technical
problem is the size-dependent superparamagnetic nature of magnetite
nanoparticles [61, 71, 66, 72]. The magnetic signature of the Verwey
transition measured in nanoparticle assemblies is smeared simply because of
the size-dependent blocking temperature, $T_{B}$, which quickly becomes lower
than bulk $T_{V}$ rendering magnetic signature of the Verwey transition ill-
defined, while direct transport measurements on single nanoparticles are not
generally possible in most labs. In case of fairly large (50 nm) biological
nanoparticles, the Verwey transition is somewhat smeared, yet, it is quite
well-defined because blocking temperature is close to room temperature [66]
(also, see the data below).

With so much experimental and theoretical effort devoted to magnetite and the
Verwey transition over almost 100 years, what new can one possibly do and
find? In order to shed some more light on this fascinating topic, we used a
controlled point-like disorder as a probe. We stress that chemical doping or
intentional variation of Fe/O ratio conducted in many works in the past, is
not equivalent to the artificial uncorrelated disorder, because such disorder
does not “dope” the system and does not change the Fermi energy. (This was
checked using Hall effect measurements on pristine and disordered samples,
albeit in a different system [72].) If Verwey’s charge ordering, Anderson
localization, or the formation of Wigner crystal scenarios were realized, we
would expect to observe a substantial smearing of the transition upon the
introduction of random disorder. In the case of polarons, it is unclear what
to expect with the added disorder. If, on the other hand, the disorder only
affects the bandwidth and/or introduces the extra “impurity” band, the
transition temperature, $T_{V}$, would shift to the lower temperature, because
the clean band-gap will become smaller but the transition itself would remain
sharp and hysteretic. Unfortunately, we did not find systematic theoretical
studies of the possible effects of point-like disorder on the Verwey
transition. It was suggested that impurities may lead to closing up the gap
above $T_{V}$ [73, 52]. However, this is not supported by the experiment and
the Authors considered an actual ionic substitution rather than dilute
scattering centers. Perhaps our work will serve as the motivation for further
studies.

Figure 1: Partial cross-sections of knock-out defect creation (both, vacancies
and interstitials) for iron (blue line) and oxygen (green line) ions in
stoichiometric magnetite, $\textrm{Fe}_{3}\textrm{O}_{4}$, as a function of
electron energy assuming the typical displacement energy threshold,
$E_{d}=25\>\mathrm{eV}$. At the operating energy of 2.5 MeV, the total cross-
section is $\sigma=56.7$ barn, which leads to the estimate of $2.48\times
10^{-3}$ defects per 1 C/cm2 per formula unit ($dpf$),
$\textnormal{Fe}_{3}\textnormal{O}_{4}$.

##  II Experimental

###  II.1 Controlled artificial point-like disorder

The point-like disorder was introduced at the SIRIUS facility in the
Laboratoire des Solides Irradiés at École Polytechnique, Palaiseau, France.
Electrons were accelerated using a pelletron-type linear accelerator to the
energy of 2.5 MeV, reaching 0.985 of the speed of light [74]. Such electrons
are capable of knocking out ions from crystal lattice creating the so-called
Frenkel pairs of vacancy-interstitial [75, 76, 77, 78, 79].

Figure 1 shows the ion-type-resolved cross-sections of the defects creation
calculated using SECTE (“Sections Efficaces Calcul Transport d’Électrons”)
software, developed at École Polytechnique (Palaiseau, France) by members of
the “Laboratoire des Solides Irradiés”, specifically for the interpretation of
MeV-range electron irradiation using their pelletron-type linear accelerator,
SIRIUS [80]. Using atomic weights averages, the SECTE software interpolates
the ion knock-out cross-sections tabulated by O. S. Oen [81]. The input
parameters are the chemical composition of the substance of interest, the
direct head-on knockout energy, $E_{d}=25$ eV, and the energy of the
projectiles (electrons), 2.5 MeV in our case. We used the commonly assumed
value of $E_{d}=25$ eV [75, 76] since the exact numbers are unimportant
because we are only interested in the order-of-magnitude estimates. We
obtained the partial cross-sections, 40.6 barn for oxygen and 78.2 barn for
iron. Therefore, the total cross-section is 56.7 barn. With the total beam
current, measured behind the sample using a Faraday cage, $\mathrm{2.7\>\mu
A}$ through a 5 mm in diameter circular diaphragm, the electron beam flux is
$8.6\times 10^{13}$ electrons/($\mathrm{s}\cdot\mathrm{cm^{2}}$). The acquired
irradiation dose is conveniently measured in C/cm2, where 1 C/cm2 = 6.24
$\times$ 1018 electrons/cm2. This gives $3.54\times 10^{-4}$ defects per 1
C/cm2 per atom (dpa), or $2.48\times 10^{-3}$ defects per 1 C/cm2 per formula
unit ($dpf$), $\textnormal{Fe}_{3}\textnormal{O}_{4}$ (not conventional unit
cell with eight formula units!). An interesting feature of Fig.1 is that while
at 2.5 MeV iron defects dominate, if the energy is reduced to around 0.5 MeV,
the only defects produced will be oxygen interstitials and vacancies.
Therefore, it is possible to conduct ion-specific study to determine what kind
of defects affect the properties the most.

For the doses used in this work of 2, 3.8, 4.1, and 6.3 C/cm2 we estimate the
concentration of defects to be 5, 9, 10 and 16 defects per thousand of formula
units, or 40, 75, 81, 125 defects per thousand of conventional deformed cubic
unit cell, or 0.5, 0.9, 1.0, 1.5 $\times$ 106 defects/cm3. These estimates are
made under the assumption that all collisions are head-on and there is no
recombination of the Frenkel pairs. During electron irradiation, the sample is
immersed in liquid hydrogen at around 20 K to remove significant heat of
collisions and prevent immediate recombination and clusterization of the
defects. On warming to room temperature, the induced defects partially anneal
with some Frenkel pairs recombine and ions in interstitial positions migrate
to various sinks (dislocations, surfaces etc.) leaving behind a quasi-
equilibrium distribution of vacancies which have much higher barrier for
diffusion. The resultant level of the induced disorder is reflected in the
change of sample resistivity at room temperature where carrier density is
roughly constant, and the only change comes from the differences in the
residual resistivity before and after irradiation. Typically about 30%-50% of
the defects are lost as judged by the in-situ resistivity measurements [77,
78]. For magnetite, such measurement is difficult due to its gapped
semiconducting nature.

For comparison, 1% $\left(\delta=0.01\right)$ of iron vacancies,
$\textnormal{Fe}_{3\left(1-\delta\right)}\textnormal{O}_{4}$, found in
biogenic magnetite [12] and synthetic magnetite with different
synthesis/annealing protocols [44, 46, 45], or ionic substitutions
$\text{Fe}_{3\left(1-x\right)}\text{M}_{3x}\text{O}_{4}$, substitutions with
M=Ni, Co, Mg, Al, Ti, Mn, Li (sorted from lowest to highest rate of $T_{V}$
suppression) [50, 51, 52, 53] corresponds to $dpf=3\delta/7=4.3$ defects per
1000 formula units, comparable with the concentration of defects induced by
electron irradiation used in this paper. An important advantage of controlled
irradiation is that the same physical sample is studied before and after
irradiation without taking off the contacts for transport measurements. This
removes any ambiguity that exists when comparing different samples that were
subject to different annealing protocols or chemical substitutions.

Figure 2:  (a) Temperature-dependent resistivity on a $\log{}_{10}$ scale for
the samples used in this study, solid red line - optimal, solid green line -
oxygen deficient. For reference we plot the data after Honig [46] for oxygen-
deficient samples. (b) Arrhenius plots of conductivity, $\ln{\sigma}$ vs.
$1000/T$, for the samples before and after irradiation.

###  II.2 Samples and experimental techniques

Single crystals of $\textnormal{Fe}_{3}\textnormal{O}_{4}$ were obtained from
J. M. Honig, which provided an excellent opportunity to directly compare our
measurements before electron irradiation with his published data that were the
first published systematic electrical resistivity study of
$\textnormal{Fe}_{3-\delta}\textnormal{O}_{4}$ on a set of samples with
different $\delta$. The crystals grown by using the so-called skull melting
technique described elsewhere [82, 46, 10]. Samples for transport measurements
with typical sample sizes of (1-2)$\times$0.3$\times$0.1 mm3
(length$\times$width$\times$thickness) were cut and polished from large single
crystals. The short side corresponded to [011] direction.

Electrical transport measurements were performed in a Quantum Design physical
property measurement system (PPMS) using a standard four-probe method with
electrical current flowing along the longer side. The contacts were made by
soldering silver wires with indium solder and the follow-up mechanical
strengthening with conductive Dupon 4929N silver paste [83]. The contact
resistance of these contacts was below 100 $\mu\Omega$, and they were
mechanically stable to withstand electron irradiation [84] . Importantly, the
contacts remained intact between measurements and irradiation runs, thus
enabling quantitative characterization of the resistance change without
invoking changes in the sample geometry.

Magnetite nanocrystals, approximately $20\times 20\times 50$ nm in size, where
extracted from the lysed cells of magnetotactic bacteria, MV-1 strain [15, 85,
66]. Details of the extraction process and magnetic characterization of
different biogenic magnetite strains and single crystals are published
elsewhere [66]. The retrieved nanoparticles were smeared on a standard carbon
grid for transmission electron microscopy (TEM) studies. For irradiation, the
grids were enclosed in thin aluminum pouches for protection. Magnetization
measurements were carried out using a _Quantum Design_ magnetic property
measurement system (MPMS). Magnetite nanocrystals were measured directly
before and after irradiation in their aluminum enclosures.

##  III Results

We first look at the stoichiometric single crystal and then compare the
results with thouse for off-stoichiometric crystal. The samples for electron
irradiation and measurements were cut and shaped from these large crystals as
described in the Experimental section. The close to optimal composition
crystal had $T_{V}\approx 121$ K [10, 49], while another had substantially
lower $T_{V}\approx 109$ K, it showed equally sharp transition and thermal
hysteresis in $\rho\left(T\right)$. For completeness, we also study the effect
of disorder on biological magnetite extracted from magnetotactic bacteria [86,
87, 85, 68, 66, 12].

Figure 3: Electrical conductivity in $(\Omega\cdot\textrm{cm})^{-1}$ of our
pristine magnetite crystal (solid red curve) and irradiated with the dose of
3.76 C/cm2 (solid blue curve), compared to the literature data up to almost
500 K in Ref. [88] (solid green curve), and extended to much higher
temperatures, well above the Curie temperature of 858 K, from Ref. [89], shown
by black open circles connected by a gray spline line. The inset focuses on a
region of conductivity maximum occurring at around 300 - 350 K.

Figure 2(a) shows temperature dependent resistivity on a $\log_{10}$ scale
versus temperature for two magnetite single crystals used in this study, with
$T_{V}\approx 121$ K and $T_{V}\approx 109$ K. For comparison, we re-plot
Honig’s data from his Fig.13 [46], which was the first systematic reported
study of this kind. Remarkably, both below and above $T_{V}$ , the slopes
$d\lg\rho/dT$ are practically identical between all these samples despite the
fact that Honig has grouped his samples into groups A (1st order transition)
and B (2nd order transition.). In order to examine the thermally activated
population of carriers in the conduction band, $n_{c}$, Fig. 2(b) presents the
natural logarithm of electrical conductivity, $\sigma\sim
n_{c}\sim$exp$\left(-U/T\right)$ plotted versus $1000/T$. The linear fits
around the transition are shown by solid lines in Fig. 2(b). The activation
energy, $U$, is estimated to be about 60 meV above $T_{V}$, and about 200 meV
just below, regardless of the $T_{V}$ value, and in a fair agreement with the
scanning tunneling spectroscopy (STS) data [41]. In temperature equivalent,
these gaps correspond to about 700 K above the transition and 2700 K just
below $T_{V}$. Above the transition this activation energy corresponds to an
energy gap in band-gap model [90], polaron activation energy in the polaron
transport model [91], and to energy distance to mobility edge in weak
localization model [25].

In Fig. 3 we compare temperature-dependent conductivity $\sigma$ of high
$T_{V}$ sample (red curve) with previous measurements over extended
temperature range, by DeGiorgi [88] and by Miles et al. [89]. Similar to these
studies, conductivity tends to saturate before reaching a peak at $\sim$350 K.
A metallic-like decrease of conductivity on heating above the peak may suggest
a cross-over to an intrinsic conductivity regime, supporting band transport
model [90]. Irradiation with 3.76 C/cm2 (blue curve), leads to a more
pronounced flattening of $\sigma(T)$ curve, suggesting a shift of the maximum
to lower temperatures. This may be natural for a semiconductor with higher
carrier density, as expected for crystals with higher defect density after
irradiation.

Figure 4: (a) Temperature-dependent resistivity of optimal single crystal
before (black line) and after (red line) electron irradiation with a
substantial dose of 3.76 C/cm2 = 2.3 $\times$ 1019 electrons/cm2 producing a
maximum of 9 defects per 1000 formula units. There is a clear signature of the
Verwey transition at around 121 K where the resistivity changes by two orders
of magnitude from $0.02\>\mathrm{\Omega}\cdot\mathrm{cm}$ above $T_{V}$ to
$2\>\mathrm{\Omega}\cdot\mathrm{cm}$ below. (b) Resistivity plotted on a
logarithmic scale zooming in on the Verwey transition region revealing
substantial hysteresis signifying a strong 1st order phase transition. In
addition to the curves shown on the left panel, here green dashed line shows
the evolution of the Vervey transition after five months of passive room-
temperature annealing on the shelf. Importantly, all curves remain parallel at
the transition showing no broadening. Also note that irradiation causes an
increase of $\rho$ avove $T_{V},$ and a decrease below as indicated by violet
arrows.

Figure 4 presents electrical transport measurements in magnetite single
crystal of optimal composition. The black curve is the pristine sample and red
is the measurement of a sample after a substantial dose of electron
irradiation, 3.76 C/cm2 = 2.3 $\times$ 1019 electrons/cm2 producing a maximum
of 9 defects per 1000 formula units. Panel (a) shows $\rho\left(T\right)$ on
the linear scales. There is a clear signature of the Verwey transition at
around 121 K and, as it was first observed by Verwey [2], the resistivity
changes by the two orders of magnitude from
$0.02\>\mathrm{\Omega}\cdot\mathrm{cm}$ above $T_{V}$ to
$2\>\mathrm{\Omega}\cdot\mathrm{cm}$ below. At 90 K, resistivity grows
exponentially to $198\>\mathrm{\Omega}\cdot\mathrm{cm}$. Analysis shown in
Fig.2, gives the energy gap for for carriers in the conduction band, $U=171$
meV below $T_{V}$ and $57$ meV above. Panel (b) shows the resistivity plotted
on a logarithmic scale zooming in on the Verwey transition region, revealing
substantial hysteresis between cooling with its transition temperature,
$T_{V}^{c}$, and warming with, $T_{V}^{w}>T_{V}^{c}$, signaling a strong 1st
order phase transition. (Throughout the paper we use higher value,
$T_{V}^{w}$, speaking of the generic transition, $T_{V}.$) Remarkably, the
transition temperature in the irradiated sample shifts by a significant 2.2 K
($\sim 2\%$), but does not smear. All curves remain practically parallel at
the transition showing no broadening or smearing. If anything, the hysteresis
becomes a little bit smaller. In addition to the curves shown in panel(a), the
green dashed line in panel (b) shows the evolution of the Verwey transition
after five months of passive room-temperature annealing on the shelf.

Figure 5: Measurements similar to Fig.4 performed on an off-steocheometric
magnetite crystal with $T_{V}\approx 109$ K. Two large doses of electron
irradiation were applied in this case, 4.1 C/cm2 (red curve - cooling only) of
10 defects per 1000 formula units and 6.3 C/cm2 (green curve, cooling and
warming) corresponding to 16 defects per 1000 formula units. In addition,
zoomed in 6.3 C/cm2 curve in panel (b) shows two “minor” hysteresis loops when
the sample was cooled to the middle of the transition and then warmed up
showing switching between the two branches. Quite similar observations may be
drawn from this figure as compared to the optimal crystal data of Fig.4.

Let us switch to an off-stoichiometric magnetite crystal with $T_{V}\approx
109.1$ K. Figure 5 shows measurements similar to Fig.4. Two large doses of
electron irradiation were applied in this case, 4.1 C/cm2 (red curve - cooling
only) and 6.3 C/cm2 (green curve, cooling and warming). In addition, zoomed in
6.3 C/cm2 curve in Fig.5(b) shows two “minor” hysteresis loops when the sample
was cooled to the middle of the transition and then warmed up showing
switching between the two branches. This is very characteristic of an
irreversible behavior. The question is - what is the origin of the hysteretic
resistivity? Structural domains would contribute randomly and it is unclear
what kind of domains is formed in a slightly distorted monoclinic phase. It
seems that a substantial increase of magnetic anisotropy is responsible for
such behavior and the hysteresis is connected to under-cooling and super-
heating associated with the 1st order transition. Introduced point-like
disorder makes the system less perfect and the hysteresis even shrinks
somewhat.

Figure 6: Biogenic magnetite nanocrystals extracted from MV-1 strain of
magnetotactic bacteria. (a) TEM image of the whole bacteria with magnetosome
containing a chain of [111] oriented head-to-tail magnetite particles inside.
(b) Zoom in on a magnetosome chain. (c) bacteria lised using French press (d)
free magnetite nanoparticles with [111] direction along the longer side.

We now examine biogenic magnetite extracted from magneto-tactic bacteria.
Figure 6 shows transmission electron microscopy (TEM) images of magnetotactic
bacteria MV-1 that produce some of the most perfect nanocrystals of magnetite
[67, 92, 66]. The extracted magnetite nanocrystals were smeared over the TEM
grid and the grid was encapsulated in a thin aluminum foil. This “pack” was
measured in _Quantum Design_ MPMS before and after irradiation.

Figure 7: Magnetic measurements of biogenic magnetite. Main panel shows zero-
field cooled (ZFC) measurements, followed by field-cooled (FC) data, both in a
$H=500$ Oe applied magnetic field. Black line is for pristine sample and red
after 2 C/cm2 electron-irradiated sample. The inset shows low-temperature
$M(H)$ loops of the same sample at $T=5$ K before (dashed black line) and
after (solid red line) irradiation.

Figure 7 shows temperature and field dependent magnetization of magnetite
nanocrystals extracted from the MV-1 magnetotactic bacteria. Main panel shows
magnetization measured on warming after the sample was cooled to 5 K in zero
field, then $H=500$ Oe magnetic field was applied and the data were collected
on warming (this protocol is abbreviated ZFC-W). After reaching maximum
temperature, the measurements continued on cooling without turning magnetic
field off (FC-C protocol). Pristine sample curve is shown by black filled
circles. The curve after electron irradiation of 2 C/cm2 is shown by open red
squares. The inset in Fig.7 shows $T=5$ K $M(H)$ loops of the same sample,
before (dashed black line) and after (solid red line) irradiation. Notice the
significant hysteresis between ZFC and FC measurements at all temperatures
closing only above 250 K. This means that blocking (collective
irreversibility) temperature is somewhere around room temperature [66, 61].
Therefore, the direction of the magnetic moments in mono-domain nanocrystals
is fixed ($k_{B}T_{V}\ll k_{B}T_{B}$), hence the sharp changes around 110 K
are due to the Verwey transition. Similar to our observations on crystals
described above, Verwey temperature shifts to the lower values and hysteresis
becomes somewhat smaller after irradiation.

Figure 8: Verwey transition temperature versus off-stoichiometric parameter,
$\delta$, in Fe3(1-δ)O4. The solid lines are the fits to the multiple
literature data collected on off-stoichiometric [44, 45, 46, 10, 47, 48] and
ion-substituted magnetite adjusted to the parameter $\delta$ [50, 51, 52, 53].
For $\delta\lesssim 0.0039$ the transition is of the first order and is well
approximated by $T_{V}=121.3-3.2\times 10^{-3}\delta$, whereas for
$\delta\gtrsim 0.0039$ the transition was thought to be of the 2nd order and
is described by $T_{V}=111.4-2.5\times 10^{-3}\delta$. Our measurements are
shown by symbols. Importantly, driven by irradiation, the off-stoichiometric
sample enters continuously the forbidden range of $T_{V}$ and shows the same
highly hysteretic behavior, against the second order expectations.

Finally, we examine the evolution of the Verwey transition temperature with
added point-like disorder induced by 2.5 MeV electron irradiation. Figure 8
shows Verwey transition temperature versus off-stoichiometric parameter
$\delta$ in Fe3(1-δ)O4. The solid lines are the fits to the multiple
literature data collected on off-stoichiometric [44, 45, 46, 10, 47, 48] and
ion-substituted magnetite adjusted to the parameter $\delta$ [50, 51, 52, 53].
Until now, it is a general believe that for $\delta\lesssim 0.0039$ the
transition is of the first order, and can be well approximated by
$T_{V}=121.3-3.2\times 10^{-3}\delta$, whereas for $\delta\gtrsim 0.0039$ the
transition is of the second order and is described by $T_{V}=111.4-2.5\times
10^{-3}\delta$. In Fig. 8 our measurements are shown by symbols. Clearly,
driven by disorder induced by the electron irradiation, the off-stoichiometric
sample enters continuously the “forbidden” range of $T_{V}$ and shows the same
highly hysteretic behavior, against the second order expectations. The
hysteresis has also been observed in all previous measurements with no
explanation provided of how is this compatible with the 2nd order transition.

##  IV Discussion

Our temperature-dependent resistivity measurements in pristine samples are
fully consistent with the literature data, so we have a reliable starting
point to investigate the effects of electron irradiation. There are several
important features observed in magnetite crystals and nanocrystals before and
after electron irradiation that introduces artificial controlled point-like
disorder.

(1) The increase of the resistivity above $T_{V}$ is counter-intuitive,
considering activated character of transport. This suggests that carrier
mobility is affected stronger by disorder than carrier density.

(2) The sharpness of the step-like $\rho\left(T\right)$ transition remains
practically unchanged (contrary to the substitutional disorder, which
significantly smears and broadens the transition [52]). All curves remain
parallel at the transition showing no broadening after the irradiation, they
just shift.

(3) The under-cooling / super-heating hysteresis becomes a little bit smaller
after irradiation.

(4) Resistivity is not metallic at all temperatures exhibiting semiconductor-
like Arrhenius activated behavior with characteristic energy barriers around
$U=200$ meV below $T_{V}$ and $60$ meV above. These values are similar to the
reported in literature [41].

(5) These barrier values remain practically unchanged even after substantial
increase of the disorder.

(6) The Verwey transition temperature in the irradiated sample shifts by a
significant amount of $\Delta T_{V}/T_{V}^{0}\approx 0.48\%$ per 1 C/cm2 (6.24
$\times$ 1018 electrons/cm2) in the optimal sample, and $0.45\%$ per 1 C/cm2
in the lower $T_{V}$ sample. These are quite similar values despite the large
difference in $T_{V}$ and the actual maximum doses acquired, 3.76 C/cm2 and
6.3 C/cm2, respectively.

(7) The Verwey transition of the lower-$T_{V}$ sample in pristine state is at
the border of an alleged cross-over from the 1st order to the 2nd order
transition. The irradiation, however, pushes Verwey temperature continuously
to lower values without broadening or smearing and with a substantial
hysteresis. This, combined with the original Honig’s observations of such
hysteresis means that the discontinuity in the $T_{V}\left(\delta\right)$ or
$T_{V}\left(\mathrm{doping}\right)$ is most likely due to sample inhomogeneity
and phase separation rather than the intrinsic behavior.

(8) Irradiated samples can be annealed toward the pristine state showing the
behavior expected for the lower-dose irradiation.

(9) Magnetite nanocrystals extracted from MV-1 strain of magnetotactic
bacteria show behavior qualitatively similar to the bulk crystals. The Verwey
temperature is suppressed and hysteresis becomes smaller. Of course, it was
only possible to measure magnetization, not electric transport.

Is it possible to understand all those features from a single point of view?
If the Verwey transition was driven by Fe2+/Fe3+ ordering and/or
electrons/polarons localization, it would be significantly smeared by the
additional randomly distributed defects. Also, resistivity would increase at
all temperatures. The same arguments can be used against the Wigner glass to
crystal transition. In our view, the opening of the band-gap the most
plausible explanation of the Verwey transition. Additional disorder, at our
low concentrations, does not affect the energy gap itself, but could form an
impurity (disorder) band close to the conduction band. Additionally, it would
certainly affect/increase the bandwidth making the band edge “fuzzy”.
According to the theory, this would lower the effective activation barrier
[93], thus leading to the reduction of $T_{V}$. Furthermore, the increase of
$\rho$ above $T_{V},$ and the decrease below is most likely due to the
magnetic component of the problem. At least some irradiation-induced defects
may become “magnetic” scatterers when both the defect’s and scattered
electron’s spins flip. This would be effective in the magnetically mildly
anisotropic cubic phase above $T_{V}$ but then become less significant in the
higher anisotropy monoclinic phase. However, the reason that resistivity
decreases in the low-temperature phase after irradiation is the same
broadening of the bandwidth lowering the effective barrier, hence leading to
more carriers in the conduction band. Apparently this effect dominates the
opposed trend of increased density of scattering centers.

##  V Conclusions

To summarize, a 2.5 MeV low-temperature (20 K) electron irradiation was used
to induce point-like defects in magnetite crystals and biogenic nanocrystals.
The Verwey transition shifts to the lower temperatures regardless of the
$T_{V}$ itself, approximately at the rate of $\Delta T_{V}/T_{V}^{0}\approx
0.5\%$ per 1 C/cm2 (6.24 $\times$ 1018 electrons/cm2). The transition itself
remains sharp without any sign of smearing or broadening. In $T_{V}=109.1$ K
sample, in the pristine state, it is right at the “border” of the alleged 1st-
to-2nd order transition. Upon irradiation, the resistivity curve parallel-
shifts to what so far was believed to be a ”forbidden” range for the Vervey’s
$T_{V}$. Notably, the transition itself remains sharp and hysteretic arguing
against the intrinsic mechanism of the _apparent_ 2nd order transition. The
wealth of obtained results can be explained within the bandwidth bandgap
theory with a smaller gap, around 60 meV above $T_{V}$, and a larger gap
around 200 meV below $T_{V}$. We hope that the obtained data will motivate
further work on the mechanisms of the fascinating Verwey transition in
magnetite.

###### Acknowledgements.

We are grateful to Jurgen M. Honig (passed away on November 4, 2022) and to
Robert J. McQueeney for illuminating discussions and for providing high-
quality single crystals of magnetite; and to Dennis A. Bazylinski for valuable
discussions and providing biogenic nanocrystals form the MV-1 magnetotactic
bacteria. This research was supported by the U.S. Department of Energy (DOE),
Office of Science, Basic Energy Sciences, Materials Science and Engineering
Division. Ames Laboratory is operated for the U.S. DOE by Iowa State
University under contract DE-AC02-07CH11358. We thank the SIRIUS team for
running electron irradiation at École Polytechnique supported by the EMIR&A
(the French Federation of Accelerators for Irradiation and Analysis of
Molecules and Materials) user proposals 15-2463, 16-0640 and 17-8487.

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