Charge localization in the Verwey structure of magnetite

The thermal evolution of electronic order in the complex Verwey ground state of magnetite $\left(\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4\right)$ has been determined through 22 high-accuracy synchrotron x-ray structure refinements using three $10–40\mu \mathrm{m}$ grains of stoichiometric magnetite. A robust fitting function is introduced to extract values of order parameterlike quantities at zero temperature and at the upper limit of the Verwey phase $T_{\mathrm{u}}=123.4\mathrm{K}$. The low-temperature structural distortion is found to be almost frozen below the Verwey transition but small changes in lattice and local mode amplitudes and Fe-Fe distances reveal an increase in electron localization on cooling. These distortions confirm that electron localization within trimerons is the driving force behind the Verwey transition. Electron localization is also revealed by anomalous decreases in the largest principal thermal displacement factors of Fe cations as electron-phonon decoupling occurs on cooling.

Figure 1

The low-temperature $Cc$ unit cell of magnetite with $B$ site $\mathrm{F}{\mathrm{e}}^{2+}/\mathrm{F}{\mathrm{e}}^{3+}$-like states drawn as blue/yellow spheres, and trimeron connections between $B$ sites shown. Each of the 16 unique $B$ sites is labeled once with minority spin density calculated in Ref. [15] shown.

Figure 2

(a) Variable temperature powder x-ray diffraction patterns for magnetite showing the cubic (440) peak at $2\theta =15.{49}^{\circ }$ and split components from the low temperature monoclinic phase. (b) Temperature evolution of the monoclinic phase fraction.

Figure 3

Thermal variation of the lattice parameters of magnetite showing fits of Eq. (3) to the monoclinic phase values.

Figure 4

(a) Plots of representative frozen phonon mode amplitudes against temperature, showing fits of Eq. (3). Points shown as triangles/circles/diamonds are from refinements of Crystal $8/13/17$ here and in Fig. 7. (b) Plot of $T=T_{\mathrm{u}}$ against $T=0$ values of the $168\mathrm{frozen}$ modes according to their symmetry class and the atom type $(A$- or $B$-site Fe, or O). The $q_{\mathrm{u}}=q_0$ frozen limit (broken line) and the best-fit slope are shown on each plot.

Figure 5

Root mean squared displacements (RMSDs) of $B$ site Fe ions in the low-temperature magnetite structure. (a) Temperature variations of the three RMSDs for representative B sites as labeled, showing fits of Eq. (3). (b) Plot of $T=T_{\mathrm{u}}$ against $T=0$ values of the largest RMSD for each of the $16B$ sites. (B31 has the largest $T=T_{\mathrm{u}}$ value and B32 the smallest.) The $\mathrm{RMS}{\mathrm{D}}_{\mathrm{u}}=\mathrm{RMS}{\mathrm{D}}_0$ straight line shown corresponds to the frozen limit for the low-temperature structure.

Figure 6

Plot of the local distortion mode amplitudes $Q_{\mathrm{JT}}$ against $Q_{\mathrm{rad}}$ showing the connected pair of extrapolated $T=0$ and $T=T_{\mathrm{u}}$ points at for each of the $16B$ sites in the $Cc$ magnetite structure. Domains of the $8\mathrm{F}{\mathrm{e}}^{2+}$-like and $8\mathrm{F}{\mathrm{e}}^{3+}$-like sites at $T=0$ and $T_{\mathrm{u}}$ are shown as rectangles.

Figure 7

(a) Representative plots of changes in nearest-neighbor $B\text{−}B$ distances $\mathrm{\Delta }{D}_{\mathit{\text{BB}}}$ against temperature showing fits of Eq. (3). (b) Plot of $T=T_{\mathrm{u}}$ against $T=0$ values of $\mathrm{\Delta }{D}_{\mathit{\text{BB}}}$ with different symbols for trimeron and nontrimeron distances. Averages for the pair of $B\text{−}B$ distances in each trimeron are also plotted. The $\mathrm{\Delta }{D}_{\mathit{\text{BB}}\mathrm{u}}=\mathrm{\Delta }{D}_{\mathit{\text{BB}}0}$ line shown corresponds to the frozen limit for the low-temperature structure.