Temperature- and doping-dependent optical absorption in the small-polaron system $\mathrm{P}{\mathrm{r}}_{1-x}\mathrm{C}{\mathrm{a}}_x\mathrm{Mn}{\mathrm{O}}_3$

Small polaron optical properties are studied comprehensively in thin film samples of the narrow bandwidth manganite $\mathrm{P}{\mathrm{r}}_{1-x}\mathrm{C}{\mathrm{a}}_x\mathrm{Mn}{\mathrm{O}}_3$ by optical absorption spectroscopy as a function of doping and temperature. A broad near infrared double-peak absorption band in the optical conductivity spectras is observed and interpreted in the framework of photon-assisted small polaron intersite hopping and on-site Jahn-Teller excitation. Application of quasiclassical small polaron theory to both transitions allows an approximate determination of polaron specific parameters like the polaron binding energy, the characteristic phonon energy, as well as the Jahn-Teller splitting energy as a function of temperature and doping. Based on electronic structure calculations, we consider the impact of the hybridization of $\mathrm{O}2p$ and Mn $3d$ electronic states on the Jahn-Teller splitting and the polaron properties. The interplay between hopping and Jahn-Teller excitations is discussed in the alternative pictures of mixed valence $\mathrm{M}{\mathrm{n}}^{3+}/\mathrm{M}{\mathrm{n}}^{4+}$ sites (Jahn-Teller polaron) and equivalent $\mathrm{M}{\mathrm{n}}^{\left(3+x\right)+}$ sites (Zener polaron). We give a careful evaluation of the estimated polaron parameters and discuss the limitations of small polaron quasiclassical theory for application to narrow bandwidth manganites.

Figure 1

Scheme of small polaron excitations presented in the framework of the two-site model in the adiabatic limit using parabolic potential surfaces to visualize a Franck-Condon-like excitation at $\hslash \omega \approx 2E_P$ and two relaxation channels, leading to either hopping transfer or on-site relaxation. At low temperatures, the broadening of the electronic levels $\mathrm{\Delta }$ is determined by zero-point phonon fluctuations. The potential energy of such two-site system is determined by the electronic energy gain $E_{\mathrm{el}}\propto -g{\sum }^{}{f}_i{q}_i$ (where $f_i$ denotes the occupation of site $i, \sum f_i=1)$ and the energy expended for the distortion of the lattice $E_{\mathrm{dist}}\propto +\hslash {\omega }_0\sum q_i^2$ [40]. Hence, the potential energy at each site as a function of $q_i$ is described by parabolic potential curves depending on the occupation $0\le f_i\le 1$ of the one-electron orbital. Note that $q$ denotes the relative distortion of both sites.

Figure 2

(a) Temperature-dependent resistivity of PCMO $0.2\le x\le 0.8$ shows insulating behavior in the whole temperature and doping range. The sample for $x=0.35$ and $x=0.5$ exhibit a pronounced CMR (inset) at $B=9\mathrm{T}$. (b) The apparent activation energies calculated from resistivity via Eq. (2) within finite temperature intervals of 3 K. The almost temperature-independent apparent energy at temperatures above 200 K gives the activation barrier in the paramagnetic charge-disordered state.

Figure 3

Optical conductivity of PCMO films for increasing doping level from $x=0$ (a) to $x=0.8$ (e) measured at $T=300\mathrm{K}$ (solid black line) and $T=80\mathrm{K}$ (dashed black line). The red hatched curves [peaks (C) and (D)] and blue hatched curves [peaks (A) and (B)] designate fit results from application of small polaron theory Eq. (4) to peaks (A) and (B) and Lorentzian functions to peaks (C) and (D), respectively. In the spectra (b)–(e) the sums of the fit curves are illustrated by the light-gray dashed lines.

Figure 4

Temperature dependence of the optical conductivity for increasing doping level from $x=0$ (a) to $x=0.8$ (e) after subtraction of UV absorption bands. Temperature variations from 80 to 300 K are indicated by arrows. The inset in (a) shows the whole spectral range of the NIR feature for $x=0$. The additional marginal absorption at $0.8\mathrm{eV}$ may originate from defect states. This feature will not be considered in the following discussion. Hatched Gaussian shaped areas represent the fit of peaks (A) and (B) at room temperature according to Eq. (4). The light-gray dashed line is the sum of the fit curves.

Figure 5

Temperature dependence of the excitation energies $\hslash {\omega }_{\max }^{\left(A\right)}\left(T\right)$ (solid symbols) and $\frac{1}{2}\hslash {\omega }_{\max }^{\left(B\right)}\left(T\right)$ (open symbols). The error bars representing the fit error are included only if they are larger than the symbol size.

Figure 6

The vibrational energies $E_{\mathrm{vib}}^{\left(A\right)}\left(T\right)$ (solid symbols) and $E_{\mathrm{vib}}^{\left(B\right)}\left(T\right)$ (open symbols) as a function of temperature for $x=0.2–0.8$ calculated via Eq. (5). The significance of the fitting results is reduced for 0.8 because a substantial part of the absorption band (A) is outside the measured spectral range. Especially the fitting of the low temperature spectra of $x=0.8$ yields large error bars because peaks (A) and (B) shift together hampering their separation in the fitting procedure.

Figure 7

Doping dependence of (a) $\hslash {\omega }_{\max }^{\left(A\right)}$ and $\frac{1}{2}\hslash {\omega }_{\max }^{\left(B\right)}$ at room temperature and (b) $E_{\mathrm{vib}}^{\left(A\right)}$ and $E_{\mathrm{vib}}^{\left(B\right)}$ obtained via fitting feature (A) (solid symbols) and feature (B) (open symbols). The value of $\frac{1}{2}\hslash {\omega }_{\max }^{\left(B\right)}$ for $x=0$ is deduced from the spectral center of gravity of the NIR band and the parameters for $x=0.8$ are put in parentheses (see explanation in the text). (c) Doping dependence of the spectral weights of peak (A) (solid symbols) and (B) (open symbols) at room temperature. The black line and the blue lines in (c) represent theoretical doping dependencies according to $\sim x(1-x)$ and $\sim (1-x)$, respectively. These relations will be considered in Sec. 5. The errors of the spectral weights are estimated by the spectral weight of the respective peak fit which is outside the measured spectral range, i.e., the fit area below the measurement threshold of $0.38\mathrm{eV}$. In (a), (b), and (c) the error bars are only shown if they exceed the symbol size.

Figure 8

(a) Determination of the onset energies $E_{\mathrm{onset}}^{\left(A\right)}$ and $E_{\mathrm{onset}}^{\left(B\right)}$ via linear extrapolation of the low energy shoulder of fit curves. (b) $E_{\mathrm{onset}}^{\left(A\right)}$ as a function of temperature for $x\ge 0.2$. (c) The doping dependence of $E_{\mathrm{onset}}^{\left(B\right)}$ at $T=80\mathrm{K}$. Error bars are smaller than the symbol size.

Figure 9

Band structure PCMO for $x=0$ (top), $x=0.5$ (middle), and $x=1$ (bottom) presented as staple diagram. It reveals oxygen states (red areas) and manganese states $(t_{2g}$ green and $e_g$ yellow) near the Fermi energy. The total DOS including $\mathrm{Pr}/\mathrm{Ca}$ states is illustrated in black. The blue arrows in the band structure for $x=0$ (top) indicate the optical transitions associated with peaks (B), (C), and (D) (cf. Fig. 3). We attribute the absorption peak (A) (see text) to intraband polaron hopping which cannot be illustrated in the band structure scheme.

Figure 10

Doping dependence of (a) $\sigma (0,T)$ gained from fitting the optical conductivity (black spheres) and for comparison the dc conductivity obtained via $\rho \left(T\right)$ measurement (open stars) and (b) the transfer integral $J$ determined via Eq. (7). Error bars are included if they exceed the symbol size. For $x=0.8$ the reliability of $J$ (parenthesized) is questionable.

Figure 11

Two alternative mechanisms leading to the double peak shape of ${\sigma }_{\mathrm{NIR}}$ and the relation $2E_P=\frac{1}{2}\hslash {\omega }_{\max }^{\left(B\right)}$ for hole doped PCMO in the limiting pictures of mixed Mn valence states (left) and equivalent Mn valence states (right): (a) Electronic scheme of adiabatic JT on-site excitation representing a $d-d$ transition in the JT polaron picture (left). In the Zener polaron picture (right) the JT transition may have $p-d$ charge transfer character. Adiabatic hopping between $\mathrm{M}{\mathrm{n}}^{3+}/\mathrm{M}{\mathrm{n}}^{4+}$ sites in the JT polaron picture and nonadiabatic hopping in the Zener polaron picture between equal $\mathrm{M}{\mathrm{n}}^{\left(3+x\right)+}$ sites involving a $\mathrm{M}{\mathrm{n}}^{4+}$ transition state. (b) Schematic lattice configuration and local electronic structure of the initial and final states for JT polaron hopping (left) and Zener polaron hopping (right). Notably, the JT energy in case of a Zener polaron (right) is reduced compared to the JT polaron (left).