Short-Range Correlations in Magnetite above the Verwey Temperature

Magnetite, Fe$_3$O$_4$, is the first magnetic material discovered and utilized by mankind in Ancient Greece, yet it still attracts attention due to its puzzling properties. This is largely due to the quest for a full and coherent understanding of the Verwey transition that occurs at $T_V=124$ K and is associated with a drop of electric conductivity and a complex structural phase transition. A recent detailed analysis of the structure, based on single crystal diffraction, suggests that the electron localization pattern contains linear three-Fe-site units, the so-called trimerons. Here we show that whatever the electron localization pattern is, it partially survives up to room temperature as short-range correlations in the high-temperature cubic phase, easily discernible by diffuse scattering. Additionally, {\it ab initio} electronic structure calculations reveal that characteristic features in these diffuse scattering patterns can be correlated with the Fermi surface topology.


I. INTRODUCTION
Discovered in the first half of the 20th century, the Verwey transition in magnetite [1] remains one of the most intriguing phenomena in solid state physics. Magnetite is a ferrimagnetic spinel with anomalously high Curie temperature T C = 850 K. Hence, it is viewed as an ideal candidate for room temperature spintronic applications. It crystalizes in the inverse spinel cubic structure, with two types of Fe sites: the tetrahedral A sites and the octahedral B ones [2][3][4][5][6][7][8]. At T V = 124 K, a first order phase transition occurs as the electric conductivity drops by two orders of magnitude [1] with the simultaneous change of the crystal structure from the cubic to monoclinic symmetry [2] and with spectacular anomalies in practically all physical characteristics [3][4][5][6][7][8]. The low-temperature structure of magnetite, as deduced from recent studies, is identified to be of monoclinic Cc space group symmetry, with complex displacement pattern [9].
In recent years, the main research effort was focused on the low-temperature phase in order to elucidate the character of charge ordering (CO) first proposed by Verwey as the primary mechanism of the phase transition. From these investigations a complex picture of a low-symmetry state arises involving charge, orbital, and lattice degrees of freedom [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Diffraction studies performed below T V have revealed a fractional charge disproportionation [6,7], with a CO and associated orbital ordering on B sites, which can be explained by strong Coulomb interactions [10,11] which amplify the coupling between 3d electrons and lattice deformation [12][13][14]. This charge and orbital order was recently suggested to exist in the form of so-called trimerons, distributed over three octahedral Fe sites, and coupled to the lattice distortion [9].
This short-range order above the Verwey transition, its subtleties and its connection to the low-temperature phase is still not completely understood. Thus, to gain further insights into the transition, the observation of this short range order and its coupling to the electronic properties of the t 2g minority-spin states of the Fe octahedral (B) atoms (which contribute to the metallic state above the Verwey transition) is crucial. We will show below that Fermi surface nesting features may be responsible for the observed short range order.
Typical spot-like diffuse scattering was reported a few Kelvin above T V at positions q = (h, 0, l + 1/2), which become Bragg reflections below the phase transition [29]. Another type of diffuse scattering with a disc-like shape was revealed close to the Γ and X points over a wide range of temperatures [30][31][32]. In order to clarify the exact shape and behavior of diffuse scattering, as well as to unambiguously demonstrate its relation to the lowsymmetry structure, we have studied the evolution of x-ray diffuse scattering in magnetite as a function of temperature down to the Verwey transition and below. Thanks to the use of a state-of-the-art large area detector (PILATUS 6M), the detailed three-dimensional (3D) reciprocal space mapping could be performed revealing an extremely rich diffuse scattering pattern, inherited from the complex low-temperature structure below the Verwey transition.
The remaining of the paper is organized as follows. The experimental part is described in Sec. II. In Sec. III we give the details of the performed ab initio electronic structure calculations which serve to determine the Fermi surface in the metallic state. In Sec. IV we present and discuss the obtained results of diffuse scattering experiments (Sec. IV A), and present the consequences of charge ordering (Sec. IV B). The main results for the Fermi surface of magnetite are given in Sec. IV C. There we also report a remarkable agreement between the experimentally obtained nesting signatures in the reciprocal space of magnetite, and the calculated Fermi surface nesting. The paper is summarized in Sec. V.

II. EXPERIMENT
The single crystalline magnetite samples were grown at Purdue University by the skull melter, crucibleless technique [36]. This allowed controlling of the oxygen partial pressure during growth, thereby ensuring that the melt remains within the stability range of the material. After preparation, the crystals were subjected to subsolidus annealing under CO/CO 2 gas mixtures to establish the appropriate metal/oxygen ratio [37]. Due to rapid quenching from high temperatures, this procedure generates octahedral defects and introduces stress [38]. However, the stoichiometry is maintained and most of the low temperature electronic processes are not affected, as is evidenced by the sharp Verwey transition with high T V ∼ 124 K, see Figure 1 in the Supplemental Material [39].
Similar low-temperature (low-T ) processes were revealed by microscopic probes [40,41] for samples prepared in a different way. Thus, electronic processes that cause diffuse scattering result from the intrinsic properties of magnetite, and not from particular preparation conditions.
Prior to the experiment, the nearly stoichiometric crystals (nominal δ = 0.00003 and δ = −0.0001 for Fe 3(1−δ) O 4 ) were mechanically put to the shape of a needle and etched down to 50 µm diameter with HCl in order to remove the damaged surface layer. Both samples gave identical results; the data for δ = −0.0001 are presented and referred to. It is noteworthy, that the diffuse patterns are stable with respect to the nonstoichiometry above T V , while below the same pattern can coexist with the superstructure, see Figure 2 in the Supplemental Material [39].
The scattering data were collected at a room temperature, T = T V + 2.5 K and T = T V − 2.5 K, and a number of intermediate temperatures (155 K, 195 K, 245 K) in shutter-less mode with the PILATUS 6M detector [42]. The preliminary measurements were performed at beamline X06SA at the SLS, the follow-up data recording took place at beamline ID29 at the ESRF; in both cases a wavelength of 0.7Å was employed. The crystal was mounted on a horizontal rotation stage, and the diffuse scattering patterns were recorded with an increment of 0.1 • over an angular range of 360 • with 0.25 s exposure per frame, i.e., 15 min per full dataset.
The experimental geometry was refined with the help of the CrysAlis software [43] that was also used for the preliminary data evaluation. The reconstruction of the selected reciprocal space layers was performed with a locally developed software. The reconstructed volume was averaged with its symmetrically equivalent orientations employing the Laue symmetry of the average structure, thus improving the signal-to-noise ratio and removing the gaps between individual detector elements. The lowtemperature data were collected utilizing an attenuator in order to avoid intensity saturation effects of the superstructure reflections.

III. ELECTRONIC STRUCTURE CALCULATIONS
Ab initio calculations of the Fermi surface (FS) were performed in the cubic inverse-spinel unit cell within the generalized-gradient approximation of the density functional theory (GGA/DFT) using the all-electron WIEN2k code [44]. The linearized augmented plane wave basis with local orbitals was expanded to k max given by r · k max = 7 outside the atomic spheres with radius r = 1.63 a.u. for oxygen and r = 1.83 a.u. for iron.
A symmetry reduced grid of 21 × 21 × 21 points in momentum space was used for convergence of the total energy. The lattice parameter was relaxed to 15.871 a.u. which agrees very well with the experimental value 15.862 a.u. A well-converged ferrimagnetic arrangement was obtained, with opposite orientations of magnetic moments in the A and B sites, as observed in magnetite below the Curie temperature T C = 850 K.

IV. RESULTS AND DISCUSSION
A. Diffuse scattering experiments Figure 1 shows the representative diffuse features of magnetite slightly above T V in form of isosurfaces. The strong main lattice Bragg spots are not visible here because their intensity is much higher than the value of the isosurfaces. The isolated diffuse clouds centered on weak Bragg reflections are removed, thus only keeping extended and/or interconnected fragments. Already this overview shows that the diffuse scattering cannot be reduced to simple objects, such as spots and discs, but in FIG. 1. Isosurface representation of diffuse scattering in magnetite slightly above TV . Color represents the distance to the (000) node; diffuse clouds in the proximity of weak Bragg spots are removed. The half-space above the H0L plane is removed. |Q|-dependent intensity scaling is applied for the purpose of better visualisation. White circles mark strong Bragg reflections in the HOL plan; arrows denote HKO and HK4 cuts perpendicular to the image plane.
contrast, reveals a rich structure which is discussed in detail below.
The results of the diffuse scattering measurements in magnetite for two representative reciprocal space cuts of the HK0 plane (left panel) and HK4 plane (right panel) are presented in Fig. 2. Besides the contributions from thermal diffuse scattering (TDS), which arises predominantly from acoustic phonons and is centered around Bragg reflections, we observe other distinct diffuse features already at room temperature. On cooling, these features gradually become stronger and sharper (see Fig.  2, upper panels).
Detailed mapping allows recovering the actual shape of these features, previously reported as spot-like and disk-like objects [30][31][32]. Among the most remarkable features, we can list squares centered on the strongest Bragg reflections of the spinel (400, 800, 440 and 448) structure and arcs (nearly symmetric pairs are visible around 400 and 804 reflections). Some distinct objects are shown in Fig. 2(c). It is worth noting that the local maxima of diffuse intensity never appear at X points, but are rather shifted aside, to incommensurate positions.
Below the Verwey temperature the diffuse intensity collapses into the superstructure Bragg reflections. While the real symmetry of the low-temperature phase is monoclinic, the diffraction pattern appears similar to that of a tetragonal structure. The modulation vector corresponding to the doubling of the unit cell in the c direction appears in only one direction of the three <100>-type equivalents; thus, not more than 8 out of 24 twins allowed by the symmetry are apparent. This is illustrated by the lower panels of Fig. 2, where the structure is considered to be pseudotetragonal with corresponding Laue symmetry operations applied.

B. Consequences of charge ordering
We will now compare the measured diffuse intensities with the scattering pattern of the ordered phase below T V (intensities taken from Ref. [9]). In the same linear greyscale, the superstructure reflections would be highly saturated. Thus, for the graphical representation we performed the following data transformations: (i) symmetrization by the operations of the m3m point group, (ii) attenuation of cubic spinel Bragg reflections to a constant level, and (iii) convolution of all the reflections with a Gaussian profile. In this way the intensity of superstructure reflections is visualized via both the size and the intensity of spots; the result of this procedure is shown in Figs. 3(a) and 3(b) (lower panels) and compared to the diffuse dataset taken slightly above T V . Comparision of the 3D intensity distribution above and below the transition is shown in Figs. 3c) and 3(d).
The qualitative similarity is apparent: all diffuse features have their counterparts with proportional inten- commensurability (note that the X points are avoided).
The characteristic length estimated from the width of diffuse features varies from ∼ 2 unit cells (u.c.) of the prototype cubic structure at T V + 2.5 K to a value slightly larger than ∼ 1 u.c. at room temperature (see Fig. 4). Thus, the ordering pattern cannot be reduced to the trimeron features [9], but rather to complexes of trimerons. Therefore our study supports the polaron picture, and we can state that its structure is in reality much more complex than ever expected previously.

C. Fermi surface
Cooling down to the Verwey temperature provokes condensation of these dynamic objects to the monoclinic structure with static charge ordering. The entropy change at T V should be further reduced compared to the trimeron-based estimates [9]. The local symmetry of the complexes can be lower than cubic, but the average cubic symmetry is recovered by nanotwinning.
Such a complex CO, which cannot be reduced to a few frozen phonon modes, can, at least partially, be explained by the Fermi surface topology. In fact, it has been shown that the short range order part of the diffuse scattering can reveal details about the electronic structure [45,46]. This is caused by an anomaly in the static susceptibility of the conduction electrons at scattering vectors, k = 2k F (k F denotes the Fermi wave vector) and, therefore, in the Fourier transform of the pair-interaction potential. The effect can become particularly pronounced if different portions of the Fermi surface are connected by a single scattering vector (Fermi surface nesting).
In order to validate our hypothesis, a number of nesting constructions based on our ab initio calculations were evaluated for intra-and interband transitions. For the nesting construction we used |∇χ q |, where is the real part of the bare spin susceptibility (Lindhard function) [47] at frequency ω → 0; here stands for the Fermi-Dirac distribution function (k B is the Boltzmann constant), and ε denotes the kinetic energy of the electrons (the chemical potential is set to zero). The total Fermi surface has a quite complex shape, traced in Fig. 5(a) as the isosurface of n exp(−ε 2 k,n /α), where n refers to the band number, and α is chosen to smear out the discretization artifacts. We have found that the scattering within the minority-spin t 2g band shown in Fig. 5(b) can account for the observed diffuse scattering pattern. Figure 6 confronts the reciprocal space patterns of the HK0 and HK4 planes with the Fermi surface nesting construction. It can be appreciated that most of the non-TDS diffuse features can be associated to efficient nesting vectors (dark grey in the right panels). The parts of the squares and arcs observed in the diffuse scattering, and indicated by the dashed lines, can be found in the nesting constraction. Also the strong arcs inside the squares are fully reproduced. The spots with high intensity close to the X points are indicated by circles. Not all features obtained from calculations can be observed in the diffuse scattering patter as the nesting construction does not take into account the modulation related to the x-ray structure factor of the spinel structure, which should attenuate/suppress a number of features compared to others. The real Q dependence would take into account the structure factor arising from the average crystallographic positions and polaron-related displacements.
The absence of an efficient nesting vector directing to the X point is reflected in the displacement of the maximum of diffuse scattering away from this point. The origin of this displacement is not clear but it points towards an incommensurate character of short-range fluctuations and is consistent with the recent inelastic x-ray scattering studies [35].
We emphasize that the agreement between the diffuse scattering data and the calculated Fermi surface nesting, shown in Fig. 6, is remarkable. The matching between the observed data and calculated patterns appears to be very good indeed and this provides clear evidence that scattering within the minority spin d-band accounts for the diffuse scattering, and hence Fermi surface nesting may in part explain the nature of the Verwey transition.

V. SUMMARY
In summary, our diffuse scattering study of magnetite allowed us linking the nature of short-range charge ordering above the Verwey temperature with the long-range structure of the low-T phase. The short-range correlations generate a very rich pattern in large areas of reciprocal space with the highest intensities shifted from commensurate wavelengths. The complexes of trimerons are inherited from the low-T structure with minor modifications. Their characteristic correlation length can be in the order of ∼ 1.5 nm just above T V , and a local symmetry lower than cubic can be assumed.
In this context the study of external stimuli on the diffuse scattering pattern (i.e., magnetic field) might be interesting, as it could reduce the symmetry of diffuse scattering without major changes in the average cubic structure. The coupling between charge fluctuations and lattice distortions (phonons) leads ultimately to the structural phase transition with a complicated charge distribution. The underlying mechanism for the formation of nontrivial charge ordering can be related to nesting features of the Fermi surface and thus to the Fermi surface shape. We have previously demonstrated an example of successful use of Fermi surface reconstruction via 3D reciprocal space mapping associated with the electronphonon coupling [48], and now we show that this approach can be extended well beyond the "good" metals.
The most straightforward methods of Fermi surface measurements would fail in the case of magnetitefor example, the de Haas-van Alphen and Shubnikov-De Haas effects, as well as positron annihilation cannot be used due to the required low temperatures at which Fe 3 O 4 is insulating. The use of photoemission spectroscopy is far from being obvious due to the highly polar surface terminations that distort the electronic structure within the shallow probing depth, and potentially due to the surface instability under UHV conditions. Thus, providing constraints on the Fermi surface of magnetite from the diffuse scattering appears to be particularly attractive.