# Polaron Trapping and Migration in Iron-Doped Lithium Niobate

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## Abstract

**:**

## 1. Introduction

## 2. Small Polarons in Fe:LiNbO_{3}

## 3. Monte Carlo Simulation of Polaron Hopping

## 4. Direct Trapping versus Migration-Accelerated Trapping

## 5. Estimation of the Escape Time

## 6. Pure Trapping Regimes

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

LN | Lithium Niobate |

KWW | Kohlrausch Williams and Watts stretched exponential function |

F | ${\mathrm{Nb}}_{\mathrm{Nb}}$ free polarons |

P | ${\mathrm{Nb}}_{\mathrm{Li}}$ bound polarons |

LIA | Light Induced Absorption spectroscopy |

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**Figure 1.**Electronic energy of the different polaron centers in Fe:LiNbO${}_{3}$ with respect to the ${\mathrm{Nb}}_{\mathrm{Nb}}$ level in a rigid lattice (i.e., not considering the lowering in energy due to the local lattice distortion associated to the polaronic effect).

**Figure 2.**(

**Top**) Survival probability of P polarons as measured by a typical LIA experiment. The black dashed line shows the decay shape when polaron migration is prohibited. The green dashed line is placed at ${\tau}_{0}=8\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$, as calculated from Equation (12). (

**Bottom**) Number of PP and FF hopping processes performed by each polaron during its walk.

**Figure 3.**(

**a**) Survival probability of P polarons at 298 K, at 248 K and at 198 K in a congruent sample with the same composition as the one in Figure 2. The dashed lines indicate the value of ${\tau}_{0}$ as approximately determined by (12). (

**b**) Survival probability for P polarons at room temperature in a sub-congruent (

**blue**), congruent (

**green**) and near-stoichiometric sample (

**yellow**). The blackdashed line shows for comparison a decay of a population of F polarons in a fully stoichiometric sample. The numbers in the legend indicate the ${\mathrm{Nb}}_{\mathrm{Li}}$ concentration in $\times {10}^{25}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$. (

**c**) Survival probability for P polarons at room temperature in a congruent sample for increasing Fe concentrations. The dashed line indicates the hopping characteristic time ${\tau}_{0}$. The numbers in the legend indicate the ${\mathrm{Fe}}^{3+}$ concentration in $\times {10}^{25}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$.

**Figure 4.**(

**a**) Fraction of hops as a free polaron ${\mathrm{N}}_{\mathrm{FF}}/{\mathrm{N}}_{\mathrm{tot}}$ as a function of temperature and $\left[{\mathrm{Nb}}_{\mathrm{Li}}\right]$ concentration for an undoped sample ([Fe] = 0). (

**b**) Fraction of hops as a free polaron ${\mathrm{N}}_{\mathrm{FF}}/{\mathrm{N}}_{\mathrm{tot}}$ as a function of the temperature and [Fe] concentration for a stoichiometric sample $(\left[{\mathrm{Nb}}_{\mathrm{Li}}\right]=0)$. The solid lines represent the theoretical boundary between the pure trapping regime calculated according to Equations (15) and (16). (

**c**) ${\mathrm{N}}_{\mathrm{PP}}/{\mathrm{N}}_{\mathrm{tot}}$ as a function of temperature and $\left[{\mathrm{Nb}}_{\mathrm{Li}}\right]$ concentration for an undoped sample ([Fe] = 0). (

**d**) ${\mathrm{N}}_{\mathrm{PP}}/{\mathrm{N}}_{\mathrm{tot}}$ as a function of the temperature and [Fe] concentration for a congruent sample ($\left[{\mathrm{Nb}}_{\mathrm{Li}}\right]=19\times {10}^{25}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$). Note the different temperature range in the two plots.

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**MDPI and ACS Style**

Vittadello, L.; Guilbert, L.; Fedorenko, S.; Bazzan, M. Polaron Trapping and Migration in Iron-Doped Lithium Niobate. *Crystals* **2021**, *11*, 302.
https://doi.org/10.3390/cryst11030302

**AMA Style**

Vittadello L, Guilbert L, Fedorenko S, Bazzan M. Polaron Trapping and Migration in Iron-Doped Lithium Niobate. *Crystals*. 2021; 11(3):302.
https://doi.org/10.3390/cryst11030302

**Chicago/Turabian Style**

Vittadello, Laura, Laurent Guilbert, Stanislav Fedorenko, and Marco Bazzan. 2021. "Polaron Trapping and Migration in Iron-Doped Lithium Niobate" *Crystals* 11, no. 3: 302.
https://doi.org/10.3390/cryst11030302