Small hole polarons in rare-earth titanates

Rare-earth titanates (RTiO₃, where R is a trivalent rare-earth ion), in which the Ti³⁺ ion has a single 3d electron, are prototypical Mott insulators.^1,2 Doping of these materials, for instance, by substituting divalent Sr on the R-site (hole doping) can be used to induce insulator-to-metal transitions.^3,4 For an ideal Mott insulator, any incremental doping away from the half-filled insulating state should cause a transition to the metallic state, but in practice the insulating phase persists even for substantial doping concentrations. Several explanations have been proposed in the literature, including Anderson localization caused by the disorder introduced by the alloying, and the formation of small polarons.⁴ In the insulating state, transport is observed to be thermally activated.^2,5 A number of different mechanisms could give rise to such Arrhenius-type temperature dependence. Zhou and Goodenough² pointed out the discrepancy between the activation energies in DC transport and the band gap and proposed that the transport is caused by small polaron hopping.

Here, we investigate the role of self-localized carriers (small polarons) in rare-earth titanates. The formation of such polarons due to strong coupling with the lattice is well documented;^3,4,6 however, their role in optical conductivity has not been fully appreciated until now.

Optical conductivity spectra of rare-earth titanates typically display onsets in the range of 0.2–0.7 eV,^1,4,5,7 which have been attributed to excitations across the Mott-Hubbard gap. Recent photoluminescence measurements on GdTiO₃, combined with first-principles calculations, have shown that the gap is actually significantly larger, closer to 2 eV.⁸ This raises the question of the origin of the low-energy features in the optical conductivity spectra. Computational investigations of small hole polarons indicate they may give rise to excitations in this energy range below 1 eV.⁹ In the present work, we study the behavior of small hole polarons by measuring optical conductivity of high-quality epitaxial Gd_1−xSr_xTiO₃ films grown by molecular beam epitaxy (MBE). In parallel, we compute absorption spectra corresponding to excitations of small polarons, either from a localized site to an adjacent site^10–12 or from a localized to a delocalized configuration.¹³ We find an excellent match between the low-energy feature in the optical conductivity spectra and the calculated small polaron excitations.

20-nm thick Gd_1−xSr_xTiO₃ thin films with x = 0, 0.04, and 0.13 were grown on (001) surfaces of (La_0.3Sr_0.7) (Al_0.65Ta_0.35)O₃ (LSAT) substrates using MBE. LSAT is cubic with a lattice parameter of 7.72 Å. (001) LSAT is closely lattice matched to (110) GdTiO₃ (lattice parameters a = 5.4031 Å, b = 5.7009 Å, and c = 7.6739 Å). References for the crystal structures and details of the film growth and characterization are included elsewhere.^14,15

Optical properties were obtained from near-normal reflectance (∼10°) measurements over the frequency range from 0.05 to 3 eV using a Bruker 66v/S Fourier transform infrared (FTIR) spectrometer, referenced to Au films and a bare LSAT substrate.^16–18 The optical conductivity, σ(ω), is obtained from a Drude-Lorentz fit to the reflectance using a multilayer model containing the full measured dielectric response of the LSAT substrate.¹⁹ Due to the multiple orientation variants of the films on the LSAT,¹⁴ the optical conductivity is an (weighted) average of the conductivity along all three principal axes.

In the frequency range of interest, above 0.1 eV, we can estimate the uncertainty in epitaxial layer conductance by using an approximate relation between the reflectivity induced by the epitaxial layer and the optical conductivity of the film.¹⁶ The uncertainty in measured film conductivity then stems from inaccuracies in the reflectivity of the film/substrate composite. These experimental uncertainties grow in the high-frequency regime and are reflected in the error bars in Fig. 1. The error bars are shown for GdTiO₃ only, but are of the same magnitude for the Sr-doped films.

The first-principles calculations used density functional theory (DFT) with the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE),²⁰ as implemented in the Vienna Ab-initio Simulation Package (VASP).^21,22 The hybrid functional approach has been shown to give accurate electronic and structural parameters for a wide range of materials,^23,24 including Mott insulators.^9,25 In order to simulate a small hole polaron, a single electron was removed from a 160-atom supercell, which is constructed based on a 2 × 2 × 2 replication of the 20-atom GdTiO₃ unit cell. The crystal symmetry was broken by slightly displacing the O atoms around a given Ti atom, followed by the relaxation of the atomic positions in the supercell. A 400 eV cutoff for the plane-wave expansion was used, and the integrations over the Brillouin zone were performed with a (1/4, 1/4, 1/4) special k-point.

The mid- to near-IR conductivity, Re(σ(ω)), at 10 K is shown in Fig. 1. In all samples, a 67 meV phonon mode associated with Ti-O stretching is observed as a sharp peak. The broad feature close to 0.1 eV is not understood at this time, but we note that the substrate dielectric constant is changing very rapidly in this frequency region and may introduce artifacts. The temperature dependence of the conductivity was not investigated, but we expect no change in the features up to 300 K, as found in the previous work on undoped and doped GdTiO₃.^6,7 With increasing doping (x = 0.13), a broad feature, labeled as A, appears at energies around 1 eV, similar to what has been observed in bulk titanates.²⁶ This feature is interpreted in the following as being due to a small polaron. The difference in small hole polaron concentration between the x = 0 and x = 0.04 samples is not large enough to be able to detect a difference, given the magnitude of the error bar.

To explain the behavior of feature A, we consider possible excitation mechanisms. One of these is the excitation of the small polaron out of its self-trapping potential well via a hopping process, with the structure subsequently relaxing by creating a lattice distortion at a neighboring site [Fig. 2(a)]. In this process, the optical absorption is expected to peak near twice the polaron self-trapping energy, E_ST [defined in Fig. 2(b)].¹² Considering the probability of hopping to an adjacent site within perturbative solutions to the Holstein model, the optical absorption peak is expected to be four times as large as the DC conductivity activation energy.^27–29 Therefore, we also expect the self-trapping energy to be twice as large as the activation energy.

We compare the incoherent absorption below the Mott-Hubbard gap to an approximate expression for the optical conductivity of a small polaron^10–12 

where n_p is the polaron density and $t=ℏ22ma2$ is the electronic bandwidth parameter with a being the lattice constant and m the effective mass. At low temperatures, $kT≪ℏω0$⁠, the width is determined by the zero-point phonon motion, $Δ=(4ESTℏω0)1/2$⁠, where ω₀ is the frequency of the relevant phonon mode. The factor t/Δ represents a transition probability and should be replaced by unity in the adiabatic limit, t > Δ.¹² 

To separate the polaron contribution from the Mott-Hubbard feature at higher energy, the optical conductivity of the undoped GdTiO₃ film was subtracted from that of the Gd_0.87Sr_0.13TiO₃ film at 10 K and the difference was fitted to Eq. (1). The fit is shown as the dashed line in Fig. 1. The resulting self-trapping energy is E_ST = 0.58 eV, about twice as large as the 0.24 eV activation energy that has been reported for bulk samples.² The broadening factor, Δ = 0.51 eV, corresponds to a phonon energy $ℏω0=110 meV$⁠, which is larger than the highest-frequency optical phonon mode (67 meV) reported for GdTiO₃.³⁰ That there is some difference is expected, since Eq. (1) is an approximate expression involving multiple fitting parameters. The presence of Sr_Gd substitutional impurities will also lead to an increased mode energy, as will be discussed further below. Taking the polaron density as n_p = x = 0.13, we obtain a bandwidth t = 0.18 eV and mass m = 1.4 m_e, where m_e is the free-electron mass.

The atomic structure of a small hole polaron, as obtained from our first-principles calculations, is shown in Fig. 3. We consider two types of excitation processes: (a) the hopping process described above and (b) the excitation of a small hole polaron to a delocalized-hole configuration. The calculated configuration coordinate diagrams associated with these two mechanisms are shown in Fig. 2. The generalized coordinate Q describes the displacement of the atoms with respect to the initial state (Q = 0) weighted by the mass of each atomic species

Here, m_α are the atomic masses of the atoms, labeled with index α. The atomic positions of the intermediate configurations were obtained by interpolation between the initial and final configurations. A parabola was fitted to the data points.

Based on the configuration coordinate diagrams, the broadening of the transition energies due to lattice vibrations was calculated using the formalism developed by Huang and Rhys.^31,32 Each configuration corresponds to a harmonic oscillator, with quantized levels corresponding to different vibronic states. The vibrational problem is therefore approximated by a single effective phonon frequency. The calculated vibrational modes are 75 meV for the harmonic oscillators in transition (a), and 65 meV for those of transition (b), which are both close to the highest-frequency optical phonon mode in bulk GdTiO₃ (67 meV).³⁰ 

As seen in the configuration coordinate diagram in Fig. 2(b), we find a polaron self-trapping energy of 0.55 eV (E_ST), and in Fig. 2(a), we find the energy barrier for polaron migration to be 0.29 eV (E_m). This is close to half of the polaron self-trapping energy, consistent with the Holstein model. The calculated peak energy for the excitation of an electron from the lower Hubbard band to the polaron state in the gap is 1.12 eV (E_T), and for the excitation of the polaron out of its self-trapping potential well via a hopping process it is 1.09 eV (⁠$ETh$⁠). Both of these values are in good agreement with the observed 1 eV peak (Fig. 1), and once again consistent with the Holstein model, being about twice as large as the polaron self-trapping energy.

The calculated absorption curves for the two mechanisms are shown in Figs. 2(c) and 2(d). Since the experiments were performed at 10 K, we only take transitions from the first vibronic state of the initial configuration into account. The onsets are at a slightly higher energy than in experiment. We attribute this to the presence of the Sr_Gd substitutional impurities in the samples, which is likely to lead to a broadening of the absorption peak that is not included in the calculations. This is consistent with the phonon mode found by fitting the experimental data to Eq. (1) being larger (110 meV) than the calculated modes [75 meV for transition (a) and 65 meV for transition (b)].

In summary, we have identified small hole polarons in Gd_1−xSr_xTiO₃ thin films. The polaron self-trapping energy is found to be 0.6 eV, both from optical conductivity measurements as well as from first-principles calculations. Calculations for excitation of a small hole polaron to a delocalized-hole configuration and for excitation of the small polaron out of its self-trapping potential well via a hopping process both yield an optical excitation peak at 1.1 eV, in good agreement with the experimental peak at 1 eV. We conclude that this feature in the optical conductivity spectra is caused by small hole polarons and not by excitations across the Mott-Hubbard gap.

This work was supported by the NSF MRSEC program (No. DMR-1121053). P.M. was supported by NSF Grant No. DMR-1006640. Experimental work by T.A.C. was supported through the Center for Energy Efficient Materials, an Energy Frontier Research Center funded by the DOE (Award No. DE-SC0001009). B.H. was supported by ONR (N00014-12-1-0976). Computational resources were provided by the Center for Scientific Computing at the CNSI and MRL (an NSF MRSEC, DMR-1121053) (NSF CNS-0960316), and by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF (ACI-1053575).