Soft tilt and rotational modes in the hybrid improper ferroelectric ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$

Raman spectroscopy is employed to probe directly the soft rotation and tilting modes, which are two primary order parameters predicted in the hybrid improper ferroelectric material ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$. We observe a giant softening of the $107-{\mathrm{cm}}^{-1}$ octahedron tilting mode by $26{\mathrm{cm}}^{-1}$ on heating through the structural transition from a ferroelectric to paraelectric orthorhombic phase. This is contrasted by a small softening of the $150-{\mathrm{cm}}^{-1}$ rotational mode by $6{\mathrm{cm}}^{-1}$. In the intermediate phase, the competing soft modes with different symmetries coexist, bringing about many-faceted anomalies in spin excitations and lattice vibrations. Our work demonstrates that the soft rotation and tilt patterns, relying on a phase-transition path, are a key factor in determining ferroelectric, magnetic, and lattice properties of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$.

Figure 1

(a) Low-temperature crystal structure of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$ in its distorted ferroelectric state. Mn ions (gray) are surrounded by oxygen octahedra (red O ions) with Ca ions (green) interspersed. The arrows depict the polar displacements of Ca ions. (b) Octahedral rotation ($X_2^{+}$) and octahedral tilt ($X_3^{-}$) distortions.

Figure 2

Raman spectra of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$ at an excitation wavelength of $\lambda =532$ nm in ($xx$), ($xy$), ($x^{\prime }{x}^{\prime }$), ($x^{\prime }{y}^{\prime }$), and ($zz$) polarizations measured at $T=8$ K. The most intense peaks are marked with indexes.

Figure 3

Schematic representation of calculated eigenvectors for the $A{2}_{1}am$ symmetry. The relative amplitude of the vibrations is represented by the arrow length. The blue, red, and green balls are Mn, O, and Ca atoms, respectively. The numbers are the calculated frequencies using shell-model lattice-dynamical calculations.

Figure 4

Raman spectra of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$ at an excitation wavelength of $\lambda =532$ nm measured in the ($xx$) and ($zz$) polarizations at $T=290$ K. The intense peaks are marked with indexes.

Figure 5

Schematic representation of the calculated modes for the $Acaa$ symmetry. The numbers are the calculated frequencies. The related amplitude of the vibrations is represented by the arrow length. The blue, red, and green balls are Mn, O, and Ca atoms, respectively.

Figure 6

(a) Raman spectra of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$ measured at $T=8$ K in the ($xx$), ($xy$), and ($zz$) polarizations. The spectra were recorded with an excitation wavelength of $\lambda =532$ nm. The pink and purple shaded areas denote a two-magnon excitation and a background continuum, respectively. (b) Temperature dependence of the Raman spectra in the ($xx$) polarization. The spectra are vertically shifted by a constant amount. (c) Temperature dependence of the frequency and the linewidth of the one-phonon mode at 623 ${\mathrm{cm}}^{-1}$ and hybrid excitation at 1542 ${\mathrm{cm}}^{-1}$. The solid lines are fits to the anharmonic model. (d) Normalized scattering intensity of the phonons and background continuum as well as their intensity ratios as a function of temperature and polarization. The solid lines are fits of the intensity using the exponential form $I\left(T\right)\propto exp(-k_{B}T/\mathrm{\Delta })$.

Figure 7

(a) Resonant Raman spectra of ${\mathrm{Ca}}_3{\mathrm{Mn}}_2{\mathrm{O}}_7$ measured at $T=8$ K in the ($xx$) polarization. The excitation energy is varied from 1.92 to 2.54 eV. The spectra are vertically shifted for clarity. (b) Normalized Raman scattering intensity of the first-order phonon modes as a function of the incident phonon energy. The dashed lines represent guides to the eye. (c) Normalized Raman scattering intensity of the higher-order peaks as a function of the energy excitation. (d) Wavelength dependence of the normalized intensity of the magnetic excitation (yellow triangles) and the background continuum (green squares). (e) Second moment of the background continuum versus the laser excitation energy.

Figure 8

(a) and (b) Evolution of two-magnon excitation (gray shading) in the ($xx$) and ($zz$) polarizations. The purple solid line is a Fano resonance. (c) Temperature dependence of the frequency, linewidth, and intensity of two-magnon scattering. The yellow shading marks $T_{\mathrm{N}}$. (d) Cartoons of the two-magnon Raman scattering process. (e) Schematic representation of the eigenvector of the 338 ${\mathrm{cm}}^{-1} A_1$ symmetry mode. (f) Temperature dependence of the frequency, linewidth, Fano asymmetry, and normalized intensity. The solid lines are fits using an anharmonic model.

Figure 9

(a) Temperature dependence of the low-frequency Raman spectra ranging from 60 to 160 ${\mathrm{cm}}^{-1}$ in the ($zz$) polarization. T and ${\mathrm{T}}^{\prime }$ (R and ${\mathrm{R}}^{\prime }$) denote the soft tilt (rotational) modes. (b) Color contour plot of the soft-mode intensity in the temperature-Raman-shift plane. (c) and (d) Temperature dependence of the frequencies of the soft modes. The solid lines are fittings using the mean-field model described in the text. The inset shows a representative fit of the low-energy spectrum to four Lorentzian profiles at $T=350$ K, representing the T, ${\mathrm{T}}^{\prime }$, R and ${\mathrm{R}}^{\prime }$ soft modes. (e) Schematic representation of eigenvectors of the soft modes in the $Acaa$ and $A{2}_{1}am$ phases.