Charge-Stripe Order in the Electronic Ferroelectric ${\mathrm{LuFe}}_2{\mathrm{O}}_4$

The structural features of the charge ordering states in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ are characterized by in situ cooling transmission electron microscopy observations from 300 K down to 20 K. Two distinctive structural modulations, a major ${\mathbf{q}}_{\mathbf{1}}=(1/3,1/3,2)$ and a weak ${\mathbf{q}}_{\mathbf{2}}={\mathbf{q}}_{\mathbf{1}}/10+(0,0,3/2)$, have been well determined at the temperature of $\sim 20\mathrm{K}$. Systematic analysis demonstrates that the charges at low temperatures are well crystallized in a charge-stripe phase, in which the charge-density wave behavior in a nonsinusoidal fashion results in elemental electric dipoles for ferroelectricity. It is also noted that the charge ordering and ferroelectric domains often change markedly with lowering temperatures and yield a rich variety of structural phenomena.

Figure 1

Superstructure modulations in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ observed at about 20 K. Electron-diffraction patterns taken (a) along the $[1\overline{1}0]$ zone-axis direction and (b) slightly off the $\mathbf{c}$ axis direction, showing clear superstructure spots. (c) Schematic representation of Fig. 1a illustrating the ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$ modulations. (d) A structural model for the ionic order of ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$. (e) A theoretical diffraction pattern showing the satellite spots corresponding with the ${\mathbf{q}}_{\mathbf{1}}$ modulation.

Figure 2

Charge ordering and spontaneous polarization in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$. (a), (b) Structural models schematically illustrating the ordered charge stripes in the ground state. (c) The CDW in nonsinusoidal fashion along $⟨116⟩$ direction, corresponding with ${\mathbf{q}}_{\mathbf{1}}$ modulation. (d) A schematic illustration for electric polarizations in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ crystal; the spontaneous polarization is along the $c$ direction.

Figure 3

Evolution of charge ordering with lowering temperature. (a) A series of electron-diffraction patterns and schematic illustrations in the right column directly exhibiting the superstructure features at different temperatures. (b) Diffraction patterns and (c) schematic illustration demonstrating the twinning structure of the charge stripes.

Figure 4

Relationship between ferroelectric domains and CO twinning. (a) A structural model for twins in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$. (b) A schematic illustration for the 180° ferroelectric domains corresponding with the CO twinning. (c) Dark-field TEM images at the temperature of 100 K; the stripelike twinning lamellae (180° ferroelectric domains) are shown. (d) High-resolution TEM image using the spots from ${\mathbf{q}}_{\mathbf{1}}$ modulation clearly exhibiting the short-range ordered state for the CO state at 300 K.