# 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 |

## References

- Imlau, M.; Badorreck, H.; Merschjann, C. Optical nonlinearities of small polarons in lithium niobate. Appl. Phys. Rev.
**2015**, 2, 040606. [Google Scholar] [CrossRef] [Green Version] - Günter, P.; Huignard, J.P. Photorefractive Materials and Their Applications 1; 0342-4111; Springer: New York, NY, USA, 2006; Volume 113. [Google Scholar]
- Bazzan, M.; Sada, C. Optical waveguides in lithium niobate: Recent developments and applications. Appl. Phys. Rev.
**2015**, 2, 040603. [Google Scholar] [CrossRef] - He, J.; Franchini, C.; Rondinelli, J.M. Lithium Niobate-Type Oxides as Visible Light Photovoltaic Materials. Chem. Mater.
**2016**, 28, 25–29. [Google Scholar] [CrossRef] - Tisdale, W.A.; Williams, K.J.; Timp, B.A.; Norris, D.J.; Aydil, E.S.; Zhu, X.Y. Hot-electron transfer from semiconductor nanocrystals. Science
**2010**, 328, 1543–1547. [Google Scholar] [CrossRef] - Pelaez, M.; Nolan, N.T.; Pillai, S.C.; Seery, M.K.; Falaras, P.; Kontos, A.G.; Dunlop, P.S.; Hamilton, J.W.; Byrne, J.A.; O’shea, K.; et al. A review on the visible light active titanium dioxide photocatalysts for environmental applications. Appl. Catal. B Environ.
**2012**, 125, 331–349. [Google Scholar] [CrossRef] [Green Version] - Migani, A.; Blancafort, L. Excitonic interfacial proton-coupled electron transfer mechanism in the photocatalytic oxidation of methanol to formaldehyde on TiO2 (110). J. Am. Chem. Soc.
**2016**, 138, 16165–16173. [Google Scholar] [CrossRef] - Zhong, Y.; Trinh, M.T.; Chen, R.; Purdum, G.E.; Khlyabich, P.P.; Sezen, M.; Oh, S.; Zhu, H.; Fowler, B.; Zhang, B.; et al. Molecular helices as electron acceptors in high-performance bulk heterojunction solar cells. Nat. Commun.
**2015**, 6, 1–8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schirmer, O.F.; von der Linde, D. Two-photon- and x-ray-induced Nb
^{4+}and O^{-}small polarons in LiNbO_{3}. Appl. Phys. Lett.**1978**, 33, 35–38. [Google Scholar] [CrossRef] - Schirmer, O.; Juppe, S.; Koppitz, J. Electron-spin-resonance, optical and photovoltaic studies of reduced undoped Fe:LiNbO
_{3}. Cryst. Lattice Defects Amorph. Mater.**1987**, 16, 353–357. [Google Scholar] - Schirmer, O.; Thiemann, O.; Wöhlecke, M. Defects in LiNbO
_{3}. experimental aspects. J. Phys. Chem. Solids**1991**, 52, 185–200. [Google Scholar] [CrossRef] - Schirmer, O.F. O
^{-}bound small polarons in oxide materials. J. Phys. Condens. Matter**2006**, 18, R667. [Google Scholar] [CrossRef] - Schirmer, O.F.; Imlau, M.; Merschjann, C.; Schoke, B. Electron small polarons and bipolarons in LiNbO
_{3}. J. Phys. Condens. Matter**2009**, 21, 123201. [Google Scholar] [CrossRef] - Schirmer, O.F.; Imlau, M.; Merschjann, C. Bulk photovoltaic effect of LiNbO
_{3}:Fe and its small-polaron-based microscopic interpretation. Phys. Rev. B**2011**, 83, 165106. [Google Scholar] [CrossRef] - Emin, D. Polarons; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Tatyana Volk, M.W. Lithium Niobate; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Schmidt, F.; Kozub, A.L.; Biktagirov, T.; Eigner, C.; Silberhorn, C.; Schindlmayr, A.; Schmidt, W.G.; Gerstmann, U. Free and defect-bound (bi)polarons in LiNbO
_{3}: Atomic structure and spectroscopic signatures from ab initio calculations. Phys. Rev. Res.**2020**, 2, 043002. [Google Scholar] [CrossRef] - Guilbert, L.; Vittadello, L.; Bazzan, M.; Mhaouech, I.; Messerschmidt, S.; Imlau, M. The elusive role of NbLi bound polaron energy in hopping charge transport in Fe: LiNbO3. J. Phys. Condens. Matter
**2018**, 30, 125701. [Google Scholar] [CrossRef] [PubMed] - Merschjann, C. Optically Generated Small Polarons: Time-Resolved Pump-Multiprobe Experiments in Lithium Niobate vs. Random-Walk Charge-Transport Simulations in Oxide Crystals. Ph.D. Thesis, Fachbereich Physik, Univesitat Osnabrueck, Osnabrück, Germany, 2007. [Google Scholar]
- Merschjann, C.; Schoke, B.; Conradi, D.; Imlau, M.; Corradi, G.; Polgár, K. Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO
_{3}. J. Phys. Condens. Matter**2009**, 21, 015906. [Google Scholar] [CrossRef] [PubMed] - Holstein, T. Studies of polaron motion. Ann. Phys.
**1959**, 8, 343–389. [Google Scholar] [CrossRef] - Marcus, R.A. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. I. J. Chem. Phys.
**1956**, 24, 966–978. [Google Scholar] [CrossRef] [Green Version] - Zylbersztejn, A. Thermally activated trapping in Fe-doped LiNbO3. Appl. Phys. Lett.
**1976**, 29, 778–780. [Google Scholar] [CrossRef] - Mhaouech, I.; Guilbert, L. Temperature dependence of small polaron population decays in iron-doped lithium niobate by Monte Carlo simulations. Solid State Sci.
**2016**, 60, 28–36. [Google Scholar] [CrossRef] - Austin, I.G.; Mott, N.F. Polarons in crystalline and non-crystalline materials. Adv. Phys.
**2001**, 50, 757–812. [Google Scholar] [CrossRef] - Vittadello, L.; Bazzan, M.; Messerschmidt, S.; Imlau, M. Small polaron hopping in Fe: LiNbO3 as a function of temperature and composition. Crystals
**2018**, 8, 294. [Google Scholar] [CrossRef] [Green Version] - Beyer, O.; Maxein, D.; Woike, T.; Buse, K. Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale. Appl. Phys. B
**2006**, 83, 527–530. [Google Scholar] [CrossRef] - Berben, D.; Buse, K.; Wevering, S.; Herth, P.; Imlau, M.; Woike, T. Lifetime of small polarons in iron-doped lithium niobate crystals. J. Appl. Phys.
**2000**, 87, 1034–1041. [Google Scholar] [CrossRef] - Herth, P.; Granzow, T.; Schaniel, D.; Woike, T.; Imlau, M.; Krätzig, E. Evidence for Light-Induced Hole Polarons in LiNbO
_{3}. Phys. Rev. Lett.**2005**, 95, 067404. [Google Scholar] [CrossRef] [PubMed] - Shklovskii, B.I.; Efros, A.L. Electronic Properties of Doped Semiconductors; Springer Science & Business Media: New York, NY, USA, 2013; Volume 45. [Google Scholar]

**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