Charge-Lattice Coupling in Hole-Doped ${\mathrm{LuFe}}_2{\mathrm{O}}_{4+\delta }$: The Origin of Second-Order Modulation

Understanding singularities in ordered structures, such as dislocations in lattice modulation and solitons in charge ordering, offers great opportunities to disentangle the interactions between the electronic degrees of freedom and the lattice. Specifically, a modulated structure has traditionally been expressed in the form of a discrete Fourier series with a constant phase and amplitude for each component. Here, we report atomic scale observation and analysis of a new modulation wave in hole-doped ${\mathrm{LuFe}}_2{\mathrm{O}}_{4+\delta }$ that requires significant modifications to the conventional modeling of ordered structures. This new modulation with an unusual quasiperiodic singularity can be accurately described only by introducing a well-defined secondary modulation vector in both the phase and amplitude parameter spaces. Correlated with density-functional-theory (DFT) calculations, our results reveal that those singularities originate from the discontinuity of lattice displacement induced by interstitial oxygen in the system. The approach of our work is applicable to a wide range of ordered systems, advancing our understanding of the nature of singularity and modulation.

Figure 1

Illustrations of different types of modulation. (a) Traditional modulation (single wave vector $\mathbf{q}$). Both phase (${\varphi }_0$) and amplitude ($A_0$) are constants. (b) A singularity in the modulation phase, i.e., a phase shift ($\mathrm{\Delta }\varphi$), generates phase variations with position vector ($\mathbf{r}$). (c) Second-order modulation. Phase and amplitude are modulated by a secondary wave, being a function of second-order modulation wave vector (${\mathbf{q}}_{\mathbf{s}}$) and position vector ($\mathbf{r}$).

Figure 2

Second-order incommensurate modulation. (a) The left-hand side of the upper panel demonstrates the three-dimensional rhombohedral structure of ${\mathrm{LuFe}}_2{\mathrm{O}}_4$, and its projection along the $\mathbf{a}$ axis is shown on the right-hand side. Lower panel illustrates charge frustration. (b) [100] zone axis EDP with a series of satellite reflections. (c) Close-up of region marked in (b). Two vectors, ${\mathbf{q}}_{\mathbf{p}}$ ($\sim 0.135{\mathbf{g}}_{\mathbf{1}}$) and ${\mathbf{q}}_{\mathbf{s}}$ ($\sim 0.110{\mathbf{g}}_{\mathbf{2}}$), are assigned for PM and SOM spots, respectively. (d) Simulated EDP considering both PM and SOM [Eq. (2)]. The fourth ($m_1$) and fifth ($m_2$) index correspond to ${\mathbf{q}}_{\mathbf{p}}$ and ${\mathbf{q}}_{\mathbf{s}}$, respectively. (e) HAADF-STEM image along the $\mathbf{a}$ axis. Spacings of the parallelogram correspond to real-space distances of ${\mathbf{q}}_{\mathbf{p}}$ and ${\mathbf{q}}_{\mathbf{s}}$. $d_1$ and $d_2$ are (027) and $(01\overline{7})$ planar distance, respectively. (f) PLD map along the [001] direction. Periodic phase shift occurs at $(01\overline{7})$ planes, indicated by the broken white lines. (g) Displacement vector map of Lu extracted from delimiters-framed area in (f). An enlargement of the marked section is shown. Arrows follow displacement directions. Arrow length and color-coded background represent the amplitude. (h) Displacement line profile along ${\mathbf{q}}_{\mathbf{p}}$ marked in (f), showing the phase shift ($\mathrm{\Delta }\varphi =2\pi d/\lambda$) and amplitude oscillation, outlined by envelope curves. All scale bars are 2 nm.

Figure 3

EELS characterizations showing charge discontinuity. (a) EELS spectra at sites A–C extracted from the atomically resolved spectrum image of $\mathrm{Fe}\text{−}{L}_{2,3}$ edges in the inset. (b) Color-coded $\mathrm{Fe}\text{−}{L}_3/L_2$ ratios extracted from each atomic column in spectrum image. Mesh spacings correspond to real-space distances of ${\mathbf{q}}_{\mathbf{p}}$ and ${\mathbf{q}}_{\mathbf{s}}$. (c) Integration of $L_3/L_2$ ratios at each (027) plane at the arrow position in (b). Experimental data can be fitted by two sinusoidal waves with a phase shift ($\mathrm{\Delta }\varphi$). Scale bars are 1 nm.

Figure 4

DFT calculations on the origin of SOM. (a) Five isolated interstitial sites considered in calculations, indexed from A to E in the $\mathbf{a}\text{−}\mathbf{b}$ plane (left-hand panel). Each site consists of a series of $z$ positions, indicated by vertical lines (right-hand panel). (b) The relative total energies ($\mathrm{eV}/{\mathrm{O}}_i$) as a function of the fractional coordinate. Polyhedron exhibits the lowest-energy site (site $A_0$) with $z/c=0.7816$. (c) A schematic illustrating the charge-lattice SOM. ${\mathrm{O}}_i$ resides at the lowest-energy site based on calculations. Lutetium is omitted for clarity. The discontinuity of displacement magnitude-damping curves (blue and yellow) induces the phase glide ($\mathrm{\Delta }\varphi$).