Mechanism of the Verwey Transition in Magnetite

By combining ab initio results for the electronic structure and phonon spectrum with the group theory, we establish the origin of the Verwey transition in ${\mathrm{Fe}}_3{\mathrm{O}}_4$. Two primary order parameters with $X_3$ and ${\Delta }_5$ symmetries are identified. They induce the phase transformation from the high-temperature cubic to the low-temperature monoclinic structure. The on-site Coulomb interaction $U$ between $3d$ electrons at Fe ions plays a crucial role in this transition—it amplifies the coupling of phonons to conduction electrons and thus opens a gap at the Fermi energy.

Figure 1

Low-frequency phonon dispersion relations for cubic phase of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ calculated with $U=4\mathrm{eV}$ and $J=0.8\mathrm{eV}$. The squares show the results of neutron scattering experiment [29]. Two primary OPs are related to ${\Delta }_5$ and $X_3$ phonons marked by circles. The high symmetry points from left to right in units of $\frac{2\pi }{a}$ are $L=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, $\Gamma =(0,0,0)$, $X=(0,0,1)$, $K=(\frac{1}{4},\frac{1}{4},1)$, $\Gamma =(1,1,1)$.

Figure 2

Total energies $E_{\mathrm{tot}}$ per supercell of 56 atoms for the structures: cubic $Fd\overline{3}m$, cubic with superimposed phonon modes of ${\Delta }_5$ at ${\mathbf{k}}_{\Delta }$ ($Fd\overline{3}m+{\Delta }_5$), and $X_3$ at ${\mathbf{k}}_X$ ($Fd\overline{3}m+X_3$) for increasing mode amplitude $Q$, and for the ground state monoclinic $P2/c$. Parameters as in Fig. 1.

Figure 3

Electron minority density of states for the structures of Fig. 2: cubic $Fd\overline{3}m$, cubic with superimposed phonon modes of ${\Delta }_5$ at ${\mathbf{k}}_{\Delta }$ ($Fd\overline{3}m+{\Delta }_5$), and $X_3$ at ${\mathbf{k}}_X$ ($Fd\overline{3}m+X_3$) symmetry with normal coordinate $Q=3$, and monoclinic $P2/c$ structures. Solid (dashed) lines correspond to $U=4\mathrm{eV}$ and $J=0.8\mathrm{eV}$ ($U=J=0$), respectively.