Short-Range Correlations in Magnetite above the Verwey Temperature

Magnetite, ${\mathrm{Fe}}_3{\mathrm{O}}_4$, 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 $T_V=124\mathrm{K}$ 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, ab initio electronic structure calculations reveal that characteristic features in these diffuse scattering patterns can be correlated with the Fermi surface topology.

Popular Summary

Magnetite, ${\mathrm{Fe}}_3{\mathrm{O}}_4$, was the first magnetic material discovered and utilized by man in Ancient Greece. Yet, some of its physical behavior continues to challenge scientists. Upon cooling to the temperature of 124 K, its electrical resistivity jumps by 2 orders of magnitude and its structure changes from a simple cubic lattice to a more complex monoclinic one. Although seven decades ago Verwey, after whom this transition is named, proposed charge ordering as the primary mechanism for the transition, its nature has not been unambiguously resolved. Rather, a complex picture involving interactions between electrons, the characteristics of their orbitals in the Fe atoms, and lattice vibrations has emerged. In this combined experimental and theoretical paper, we identify a few major pieces in this puzzle that link the two apparently different material states across the Verwey transition.

The remarkable 2012 publication of Senn, Wright, and Attfield provides an important clue into the low-temperature state. It suggests that electron localization, responsible for the dramatically increased resistivity, appears in basic three-Fe-site units, called “trimerons.” Our measurements, based on state-of-the-art x-ray diffuse scattering, clearly show that behind the remarkable apparent sharpness of the Verwey transition is the electronic correlation seen in the low-temperature state persisting in the high-temperature state above the transition. Indeed, even at room temperature, which is considerably higher than the 124-K transition temperature, electrons appear to be correlated over length scales larger than the trimeron size. Moreover, by combining ab initio electronic-structure calculations with our experimental diffuse-scattering data, we could associate some characteristic experimental features with the topology of the Fermi surface of magnetite. While the structure of the low-temperature phase was frequently considered as a key to the understanding of the Verwey transition, our work indicates that the diffuse scattering pattern above the transition actually encodes some (if not all) key information.

Observing the building blocks of the displacement pattern of the low-temperature phase at temperatures more than twice the transition is remarkable in a highly correlated electron material. This observation warrants the extension of the search for such persistent structural fingerprints to other materials.

Figure 1

Isosurface representation of diffuse scattering in magnetite slightly above $T_V$. 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 visualization. White circles mark strong Bragg reflections in the H0L plane, and arrows denote HK0 and HK4 cuts perpendicular to the image plane.

Figure 2

Diffuse scattering in magnetite at variable temperature. Magnetite reciprocal space cuts HK0 (a) and HK4 (b) are shown at room temperature (RT), slightly above ($T_V+\delta$) and below ($T_V-\delta$) the Verwey transition, with $\delta \sim 2.5\mathrm{K}$. Left-hand bottom panels: Sections perpendicular to the cell doubling direction. Right-hand bottom panels: Sections parallel to the cell doubling direction. Selected regions of interest (see text) are shown in (c) together with the unit cell vectors $a*$ and $b*$. The room-temperature cut is scaled taking into account the thermal expansion for visualization purposes; the labels in (c) denote the central Bragg spot. Cubic $m3m$ (above $T_V$) and tetragonal $4/mmm$ (below $T_V$) Laue symmetries are applied.

Figure 3

Top: Similarity of scattered intensity distribution in magnetite above and below the Verwey transition. Magnetite reciprocal space cuts, (a) HK0 and (b) HK4, are compared for temperatures above $T_V$ ($T_V+\delta$, upper part) and below $T_V$ ($T_V-\delta$, lower part), with $\delta \sim 2.5\mathrm{K}$. Gaussian blurring is applied to the superstructure reflections; main reflections are replaced by constant intensity peaks to avoid masking the superstructure-related features. Middle and bottom panels: Isosurface representation of diffuse scattering in magnetite as given in the proximity of the (c) (440) and (d) (008) node. Indices and vector lengths are given in r.l.u. of the cubic structure. Cubic $m3m$ Laue symmetry is applied. Artificial twinning of the low-temperature data recovers the cubic symmetry.

Figure 4

Temperature dependence of characteristic diffuse intensity in magnetite: (a) linear scans across the arclike diffuse feature of magnetite for temperatures between $T_V$ and a room temperature, and (b) the temperature decay of the derived $1/w^3$ value (symbols), together with its exponential fit $A\exp (-T/\tau )$ (solid line). Here, $w$ is the width of the linear diffuse scattering scans and $a_0$ denotes the lattice parameter of the cubic lattice.

Figure 5

Fermi surface of magnetite in the cubic phase. (a) Total (multiband) Fermi surface of magnetite as derived from the spin-polarized electronic band structure using the Wien2k code, (b) isolated minority-spin $t_{2g}$ band of interest.

Figure 6

Nesting signatures in the reciprocal space of magnetite. Selected reciprocal space patterns of magnetite at $T_V+\delta$ [left-hand panels of (a) and (b)] in the (a) HK0 plane and (b) HK4 plane are compared to the Fermi surface nesting construction [right-hand panels of (a) and (b)]. Dark features in the nesting construction correspond to more efficient nesting. Common features are underlined or encircled by dashed lines. The contours of the Brillouin zone are superposed on the right-hand subpanels, $X$ points are indicated. Note that the $X$ point is avoided, in both the experiment and the calculation.