Atomic insight to lattice distortions caused by carrier self-trapping in oxide materials

We gain hitherto missing access to the spatio-temporal evolution of lattice distortions caused by carrier self-trapping in the class of oxide materials - and beyond. The joint experimental/theoretical tool introduced combines femtosecond mid-infrared probe spectroscopy with potential landscape modeling and is based on the original approach that the vibration mode of a biatomic molecule is capable to probe strongly localized, short-lived lattice distortions in its neighborhood. Optically generated, small, strong-coupling polarons in lithium niobate, mediated by OH− ions present as ubiquitous impurities, serve as a prominent example. Polaron trapping is found to result in an experimentally determined redshift of the OH− stretching mode amounting to Δνvib = −3 cm−1, that is successfully modeled by a static Morse potential modified by Coulomb potential changes due to the displacements of the surrounding ions and the trapped charge carrier. The evolution of the trapping process can also be highlighted by monitoring the dynamics of the vibrational shift making the method an important tool for studying various systems and applications.

the presence of a short-range lattice displacement. With knowledge on the lattice position of both the molecular ligand and the small polaron site, the structural distortion can be reconstructed from the measurement of Δ ν vib . Thus, a variety of small polaron features like the associated displacements of atomic equilibrium positions or the electron-phonon coupling strength can be obtained. A particular strength of the approach is the possibility to study the temporal evolution of localized atomic displacements if dynamic measurements for the detection of Δ ν vib (t) are applied. Here we use a femtosecond visible pump, mid-infrared (MIR) probe absorption spectrometer (cf. e.g. ref. 17) with a remarkable shot-to-shot sensitivity below 10 −3 OD (Optical Density) at a spectral and temporal resolution of δν vib (t)/ν vib ≈ 9 ⋅ 10 −4 , and δt = 1 ms, respectively. It is shown that the transient absorption spectrum Δ A(ν, t) = A(ν, t) pumped − A(ν, t) reveals a frequency shift of Δ ν vib (OH − ) upon light exposure that can be related to the formation of short-lived, small O − hole polarons. Furthermore, the dynamic features from ms to seconds in the dark resemble the well-known stretched-exponential behavior that is characteristic for hopping-transport 18 . The main purpose of this article is (i) to verify the underlying experimental approach, i.e., to demonstrate and explain the small polaron access using changes in a vibrational fingerprint, (ii) to sketch the physical relation between experimentally determined frequency shifts and the structural properties of small, strong-coupling polarons, and (iii) to show-up the possibility for dynamic measurements with small polaron hopping-transport as an example.

Fundamentals
Carrier self trapping in lithium niobate. In what follows, we will focus on small, strong-coupling O − hole polarons (HP) 19 in direct vicinity to a V Li vacancy as well as on + Nb Li 4 electron polarons (GP) 20 . Structural details of the oxygen plane of LN are depicted in Fig. 1a. In both cases the trapped charge can be transferred by light to a neighboring equivalent or nearly equivalent site, thereby stripping the trapped charge of its polarization halo 18,19 . This results in broad absorption features playing a central role in the study and application of dynamic properties of small polarons in LN 9 . The absorption maxima are reported in the visible region (VIS, 2.5 eV; ≈ 20,000 cm −1 ) for HP and in the near infrared (NIR, 1.6 eV; ≈ 13,000 cm −1 ) for GP. In nominally undoped LN, pairs of HP and GP can be generated via two-photon excitation at 2.5 eV (cf. Fig. 1b, left part), i.e., with photon energies much below the band edge energy (E gap ≈ 4.1 eV 21 ). To raise signal changes in the MIR range (≈ 10 mOD, see below), it is necessary to generate a large number density N sp > 10 18 cm −3 of small polarons. This is realized by using intense laser pulses (I pump = (500 ± 100) TW/m 2 , extraordinary light polarization) with sub-ps pulse duration (τ pump = (120 ± 10) fs), as N sp depends quadratically on the intensity of the pump-pulse, = .⋅ N I const sp pump 2 22 , but also on the pulse duration τ pump 23 . Furthermore, repetitive fs-pulse exposure (f rep = 1 kHz) is applied, which leads, after a few seconds of illumination, to a saturation of the MIR effect. The average small polaron lifetimes τ GP ≈ τ HP ≈ (1.1 ± 0.3)s ≫ 1/f rep accord with the mutual HP-GP-recombination path, as shown in Fig. 1b 26 ). Due to the dipolar character of this defect complex it may be assumed that light-induced, small polarons are formed or get temporarily trapped in its direct vicinity. We consider an O − hole polaron localized at an oxygen ion next to the same V Li (see Fig. 1a). It is already established that the energy of the OH − -stretching vibration depends on the surroundings of the particular OH − -site resulting in several components of its MIR absorption feature. In as-grown, near stoichiometric LN up to five distinct bands have been observed 29,30 ; the most prominent bands appear at 3,466 cm −1 , 3,480 cm −1 , and 3,490 cm −1 (see Fig. 2a Fig. 2b using the color coding: red for

Experiment. The time dependent change of the absorption feature
In the time regime t < t on , i.e., without the presence of the pump-pulse, we detect a weak background signal that is applied for signal correction. At t on , the repetitive pump-pulse exposure is switched on and induces spectral changes ν ∆  A t ( , ) well above the noise limit. The temporal evolution resembles a mono-exponential behavior with a characteristic time constant in the order of 100 ms. The spectral evolution shows changes in the shape of all three MIR absorption bands with maximum values of about − 14 mOD at 3,466 cm −1 and about + 10 mOD at 3,463 cm −1 . Pump-induced changes at 3,480 cm −1 and 3,490 cm −1 are also observed but do not exceed 2 mOD; accordingly we will focus in the following on the strongest absorption band. Switching off the pump-pulse exposure (t > t off ), the absorption change decays to nearly zero and the process can be repeated from the beginning. Exemplarily, the spectrum ν ∆ =  A t t ( , ) off is extracted and plotted in Fig. 2c. The remarkable shape, in particular in the region of the strongest absorption band, can be understood as a shift of the OH − absorption frequency by − 3 cm −1 to 3,463 cm −1 .
The next step is to verify the relation of this shift to the presence of small, strong-coupling polarons. For this purpose, the temporal relaxation dynamics in the mid-infrared spectral range is highlighted in a semi-logarithmic plot in Fig. 3 (red data points) and compared with the temporal decay dynamics of the light-induced absorption detected in the near-infrared and visible spectral range (blue data points) close to the maximum of the GP and  HP absorption features. It is evident from Fig. 3 that the temporal behaviors of the absorption changes in all spectral ranges are strongly correlated, despite the differences in the sign of Δ A. As shown in the inserts of Fig. 3, the Δ A NIR/VIS (Δ A MIR ) plot is essentially linear. For a quantitative comparison, a stretched-exponential function with time constant τ and stretching exponent β is fitted to each data set. The values (τ,β) obtained from fitting the experimental data are listed in Table 1 and are comparable within the error margins. These findings are particularly remarkable since the absorption in the NIR and VIS region is related to changes in the electronic structure, whereas the absorption in the MIR is characteristic for the molecular vibration of OH − . The linear relation between the signals of different origin, thus, gives clear evidence for their common physical background which is the hopping transport of GP and HP leading to their recombination (cf. Fig. 1b). We note that the absorption dynamics at 3,463 cm −1 in Fig. 3b shows a reduced signal-to-noise ratio in comparison with data at 3,466 cm −1 , that is due to the lower signal level at this wavenumber. Concerning the dynamics at 20,492 cm −1 , a slight variation to long decay times in comparison to the stretched-exponential decay is observed, that may be attributed to the presence of A hole polaron is added at another oxygen neighbor of the lithium vacancy out of the three nearest ones. The potential change seen by the proton is accounted for by the Coulomb potential due to the added point charge on its host ion and by the changes of the Coulomb potentials due to the displacements of all neighboring ions with the exception of the oxygen in OH − . For simplicity the displacements are assumed to be radial (with respect to the polaron host) and scaled by the factor q i /d i where q i is the charge and d i the distance (to the host) of the i th neighbor (see Fig. 4). This scaling corresponds to considering only long range electron-phonon interaction characterized by the static dielectric constant ε 0 having a large value in comparison to the optical dielectric constant ε ∞ (ε 0,11 /ε ∞ = 57.6, ε 0,33 /ε ∞ = 19.0 37 , with ε ∞ ≈ 1.47 ⋅ ε vacuum 38 ) in LN. Choosing the displacement δd Nb of the nearest Nb neighbor ( = .
− + − d 0 188nm Nb O 5 2 ) as a numerical scaling parameter and neglecting possible changes of the OH − -direction, the resulting potential along the bond can be approximated by a new Morse potential of the same type.
The three dimensional Coulomb potential is calculated for a 5 × 5 × 2 hexagonal unit cell (1588 atoms), using the structural data of the LiNbO 3 lattice 10 with a lithium and an adjacent oxygen ion at the center of the cell replaced by an OH − -molecular-ion represented by a static Morse potential. The site of the proton substituting V Li , is assumed to be in the bisecting plane of the largest oxygen triangle (O-O distance 336 pm) 28  . We note that a OH − − V Li -complex causes only negligible distortions of the lattice as calculated by Lengyel et al. 25,27 . Cation charges in the boundary area of the cell are reduced to secure an overall neutral charge. The changes of the potential along the direction of the O-H bond resulting from the presence of a hole polaron are added to the Morse potential, neglecting the difference between the center of mass of OH − and the corresponding oxygen site. The hole polaron state is modeled  Fig. 3).
by nm) upon hole capture by Fe 2+ , thus, supporting our model assumptions. The estimates derived from our simplified model may already serve as useful input parameters for more complex model calculations and/or as a crosscheck of the validity of various small polaron model approaches. Appropriate calculations may be used straightforwardly for the determination of further small polaron parameters, like the electron phonon coupling strength.

Summary and Conclusions
Summarizing the results, a pump-induced − 3 cm −1 shift of the frequency of the OH − -stretching vibration is experimentally verified and attributed to the presence of small, strong-coupling O − hole polarons next to the OH − ion. Sign and magnitude of the shift can only be explained by taking into account, in addition to the polaronic charge, also the distortion of the lattice in the imminent neighborhood -a very striking result that supports the microscopic approach for the bulk photovoltaic effect based on small, strong-coupling polarons in LN 16 and will foster its advances. The measured shift can be reconstructed by modeling the change of the three-dimensional Coulomb potential caused by the polaron, leading to reasonable values of the ligand displacements. The measured frequency shift may also provide a stringent test for ab initio/DFT calculations desirable for a deeper understanding of the properties of polarons interacting with molecular defects, and vice versa. It should be stressed that the presented experimental approach for determining local lattice deformation, although introduced with small, strong-coupling polarons, can also be applied for the detection of small, weak-coupling and large polarons 1 . It is furthermore neither limited to the chosen molecular hydoxide group nor to lithium niobate as a host material, and may be adopted, e.g., to further oxides like TiO 2 , ZnO or MgO, but also to probe localized atomic distortions in DNA 40 . As its most important aspect, however, the approach enables dynamic studies at room temperature and, thus, general physical questions about polaron formation, hopping-transport and recombination phenomena, encompassing time scales from sub-ps to seconds, become accessible. If necessary, the time resolution can be increased to the sub-ps time regime by using up-conversion MIR-spectroscopy; this enables a larger signal-to-noise ratio accompanied with a better time resolution.