Terahertz spectroscopy of low-energy excitations in charge-ordered ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$

The low-energy electrodynamic response of a series of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ samples of different morphology is systematically studied in the frequency range from 4 to $700{\text{cm}}^{-1}$ (0.1–20 THz, quantum energies 0.4–90 meV) with particular attention focused at frequencies below $40{\text{cm}}^{-1}$. In the antiferromagnetically ordered phase below 140 K, a resonancelike absorption band appears in the spectra around $20–40{\text{cm}}^{-1}$ that is assigned to acoustical phonons which acquire optical activity due to a fourfold superstructure appearance in the crystallographic $a$ direction and correspondent Brillouin-zone folding. Nanosize effects in charge and magnetic subsystems are investigated by measuring the spectra of samples with grain diameters reduced from several micrometers down to 40 nm. A strong relaxation in the lowest-frequency spectra is observed above 100 K in all samples, indicating that the charge-order transition shows characteristics typical for order-disorder transitions.

Figure 1

(Color online) (a) Real part of dielectric permittivity of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ sample made of grains with 1000 nm diameter, as a function of frequency at temperatures $T=300$, 140, and 5 K. The symbols correspond to data directly obtained by a Mach-Zehnder interferometer; the solid and dashed lines result from reflectivity measurements via Kramers-Kronig analysis. (b) Optical conductivity plotted as a function of frequency for various temperatures as indicated. Again, the symbols correspond to data directly obtained by a Mach-Zehnder interferometer, solid and dashed lines are calculated from reflectivity measurements; the lines below $40{\text{cm}}^{-1}$ are guides to the eye. The thin dotted line represents the ${\nu }^2$ behavior of the conductivity indicating the low-frequency tail of the lowest infrared phonon. In the inset of panel (a) the transmission through a 0.076 mm thick plane-parallel pellet is shown (dots), measured at $T=5\text{K}$. The solid line is obtained by a fit employing the Fresnel’s expressions for a plane-parallel slab. The inset in panel (b) demonstrates in an Arrhenius plot the activated behavior of the low-frequency $\left(\nu =4.7{\text{cm}}^{-1}\right)$ conductivity with an activation energy 0.13 eV (straight blue line).

Figure 2

Dependence of the charge-ordering temperature $T_{\text{CO}}$, the energy gap $2\Delta$, and of the ratio $2\Delta /T_{\text{CO}}$ of ${\text{La}}_{1-x}{\text{Ca}}_x{\text{MnO}}_3$ on calcium concentration $x$. The data given by open circles are obtained from infrared conductivity spectra from Ref. 42. The solid dots correspond to the values obtained in the present work based on the activated temperature behavior of the low-frequency $\left(\nu =4.7{\text{cm}}^{-1}\right)$ conductivity of polycrystalline ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ with grains of $1\mu \text{m}$ diameter.

Figure 3

Temperature dependence of the parameters of the absorption band observed in ${\text{La}}_{1-x}{\text{Ca}}_x{\text{MnO}}_3$ in the spectral range from 20 to $40{\text{cm}}^{-1}$: resonance frequency ${\nu }_0$, damping constant $\gamma$, and dielectric strength $\Delta ϵ$.

Figure 4

(Color online) Spectra of the dielectric permittivity ${ϵ}^{\prime }\left(\nu \right)$ and conductivity $\sigma \left(\nu \right)$ of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ sample with various grain sizes, measured at different temperatures $T=300$, 100, and 5 K, as indicated. Upper left frame: 200 nm, upper right frame 150 nm, lower left frame 85 nm, and lower right frame 40 nm. The dashed straight lines represent the ${\nu }^2$ behavior of the conductivity, which would correspond to the low-frequency tail of the lowest-frequency infrared phonon.

Figure 5

(Color online) Comparison of the dielectric permittivity ${ϵ}^{\prime }\left(\nu \right)$ and conductivity $\sigma \left(\nu \right)$ of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ sample consisting of different grain sizes measured at liquid-helium temperature. The curves are derived from reflectivity measurements, the dots are directly measured using the Mach-Zehnder interferometer. Lines below $40{\text{cm}}^{-1}$ are guides to the eye. Dashed straight lines represent the ${\nu }^2$ behavior of the conductivity that would correspond to the low-frequency tail of the lowest-frequency infrared phonon.

Figure 6

Grain-size dependences of various characteristic parameters of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$. (a) Dielectric strength $\Delta {ϵ}_D$ of the relaxation seen at the lowest frequencies in the permittivity and conductivity spectra at $T=100\text{K}$. (b) Dielectric strength $\Delta ϵ$ of the absorption band in the range $20–40{\text{cm}}^{-1}$ at $T=5\text{K}$. (c) Activation energy $E_{\text{act}}$, describing the exponential temperature behavior of the low-frequency $\left(5–10{\text{cm}}^{-1}\right)$ conductivity $\sigma \left(T\right)\propto \text{exp}\left\{-E_{\text{act}}/k_{B}T\right\}$ in the range $140<T<300\text{K}$. (d) Charge-ordering temperature $T_{\text{CO}}$ (left axis), defined from measurements of the magnetization and lattice parameters (Ref. 13), and “strength” of charge-ordering interaction $E_{\text{act}}/k_B{T}_{\text{CO}}$ (right axis). (e) Low-frequency $\left(10{\text{cm}}^{-1}\right)$ permittivity for $T=5$ and 300 K, and (f) optical conductivity at fixed frequency $\nu =10{\text{cm}}^{-1}$. The lines are guides to the eye.

Figure 7

(Color online) Temperature dependence of the low-frequency $\left(\nu =12{\text{cm}}^{-1}\right)$ permittivity and conductivity of ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ samples with different grain sizes as indicated. For the 1000 nm sample, data are available only below $T=140\text{K}$. The solid lines are guides to the eye. The dashed line shows the temperature dependence of the dc conductivity of polycrystalline ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ taken from Ref. 56.

Figure 8

(Color online) (a) Unit cell of $\left(\text{La},\text{Ca}\right){\text{MnO}}_3$ in the orthorhombic $Pnma$ phase. (b) Transformation of the Brillouin zone of a cubic perovskitelike unit cell at the $Pm3m\leftrightarrow Pnma\leftrightarrow P{2}_1/m$ phase transitions, leading to supercell of $4a$. (c) Qualitative scheme demonstrating the folding of the modes at the $Pnma\leftrightarrow P{2}_1/m$ phase transition. The acoustic phonons with $q_x=\pi /\left(4a\right)$ and $q_x=\pi /\left(2a\right)$ (gray dots) in $Pnma$ structure are folded to the center of the monoclinic zone and become Raman-active (open dots) and infrared-active (filled dots) optic modes. The band in the frequency range around $20–40{\text{cm}}^{-1}$ is formed by modes with wave vector $q_x=\pi /\left(4a\right)$ and the band around $100{\text{cm}}^{-1}$ is formed by modes with wave vector $q_x=\pi /\left(2a\right)$ from the Brillouin-zone boundary. Certain contributions to the low-frequency part of the band around $100{\text{cm}}^{-1}$ can come from the phonons with the wave vector $q_x=\pi /\left(3a\right)$ (phase with supercell $3a$) (Ref. 29).

Figure 9

(Color online) Low-frequency spectra of (a) real and (b) imaginary parts of dielectric permittivity, ${ϵ}^{\prime }$ and ${ϵ}^{″}$, of polycrystalline sample ${\text{La}}_{0.25}{\text{Ca}}_{0.75}{\text{MnO}}_3$ (grain size of 1000 nm) measured in Mach-Zehnder geometry at different temperatures as indicated. The lines are guides to the eye.