# Electron Polarons in Lithium Niobate: Charge Localization, Lattice Deformation, and Optical Response

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

## 3. Results and Discussion

#### 3.1. Structure Optimization

#### 3.2. Electronic Properties

#### 3.3. Optical Properties

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

B(Nb) | bipolaron at a Nb${}_{\mathrm{Li}}$ antisite defect |

B(Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$) | bipolaron at a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair |

BSE | Bethe–Salpeter equation |

DFT | density-functional theory |

DOS | density of states |

EPR | electron paramagnetic resonance |

FP | free polaron |

IPA | independent-particle approximation |

IQA | independent-quasiparticle approximation |

LN | lithium niobate |

P(Nb${}_{\mathrm{Li}}$) | bound polaron at a Nb${}_{\mathrm{Li}}$ antisite defect |

P(Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$) | bound polaron at a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair |

PBEsol | Perdew-Burke-Ernzerhof for solids |

SLN | stoichiometric lithium niobate |

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**Figure 1.**Stacking order of lithium niobate: Stoichiometric crystal (

**a**), free polaron (

**b**), bound polaron at a Nb${}_{\mathrm{Li}}$ antisite (

**c**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**d**), bipolaron at a Nb${}_{\mathrm{Li}}$ antisite (

**e**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**f**). Black squares represent oxygen octahedra, while red squares indicate the formation of a (bi)polaron at this site. Gray fillings highlight deviations from the stoichiometric solid.

**Figure 2.**Atomic displacements in the vicinity of a bipolaron at the Nb${}_{\mathrm{Li}}$ antisite defect (

**a**) or the Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**b**) with respect to the positions in stoichiometric LN, plotted as a function of the distance from the defect niobium atom hosting the bipolaron. Green circles and red boxes refer to niobium and oxygen atoms, respectively. The numbering corresponds to the pictured section of the crystal structure, the label “0” marks the defect Nb${}_{\mathrm{Li}}$ or Nb${}_{\mathrm{V}}$ atom hosting the bipolaron. Large green and small red balls represent niobium and oxygen atoms, respectively. Oxygen atoms outside the central pillar of oxygen octahedra are indicated in a lighter color shade.

**Figure 3.**Crystal structure in the vicinity of the free polaron (

**a**,

**b**), the bound polaron at a Nb${}_{\mathrm{Li}}$ antisite defect (

**c**,

**d**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**e**,

**f**), and the bipolaron at Nb${}_{\mathrm{Li}}$ (

**g**) or Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ (

**h**). Subfigures (

**b**,

**d**,

**f**) refer to tilted configurations with broken trigonal symmetry, all other structures exhibit axial symmetry. Black balls indicate atomic positions in stoichiometric LN without polarons, while blue, green, and red balls denote lithium, niobium, and oxygen atoms in the relaxed structures, respectively. Note that (

**e**,

**f**,

**h**) show a different segment of the crystal than the other subfigures.

**Figure 4.**Charge densities of the free polaron (

**a**,

**b**), the bound polaron at a Nb${}_{\mathrm{Li}}$ antisite defect (

**c**,

**d**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**e**,

**f**), and the bipolaron at Nb${}_{\mathrm{Li}}$ (

**g**) or Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ (

**h**). Subfigures (

**b**,

**d**,

**f**) refer to tilted configurations, all other structures exhibit axial symmetry. Blue, green, and red balls represent lithium, niobium, and oxygen atoms, respectively. Reproduced from [12].

**Figure 5.**Electron densities of states for the free polaron (

**a**,

**b**), the bound polaron at a Nb${}_{\mathrm{Li}}$ antisite defect (

**c**,

**d**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**e**,

**f**), and the bipolaron at Nb${}_{\mathrm{Li}}$ (

**g**) or Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ (

**h**). Subfigures (

**b**,

**d**,

**f**) refer to tilted configurations, all other structures exhibit axial symmetry. The thick black line shows the imaginary part of ${\epsilon}_{zz}(\omega )$ within the independent-particle approximation (IPA), whose low-energy resonances correspond to transitions from the defect state into the conduction band.

**Figure 6.**Imaginary (

**top**) and real (

**bottom**) parts of the ${\epsilon}_{zz}$ component of the dielectric function for bipolarons localized at a Nb${}_{\mathrm{Li}}$ antisite defect (

**left**) or a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair (

**right**). We compare results from IPA (black), IQA (blue), and BSE (red). The arrows indicate the positions of the dominant defect-related peaks in the absorption spectra. The dashed horizontal line for $\mathrm{Re}{\epsilon}_{zz}$ marks zero.

**Figure 7.**Imaginary (solid lines) and real (dashed lines) parts of the dielectric function $\epsilon $ calculated within the BSE for z-polarized (red) and x-polarized (blue) light. The left and right panel correspond to bipolarons at a Nb${}_{\mathrm{Li}}$ antisite defect and a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair, respectively.

**Figure 8.**Optical properties derived from the dielectric function within the BSE: reflectivity R (

**top**), absorption coefficient $\alpha $ (

**middle**), and electron-energy-loss function L (

**bottom**) for z-polarized (red) and x-polarized (blue) light. The left panels correspond to bipolarons at a Nb${}_{\mathrm{Li}}$ antisite defect, the right panels to bipolarons at a Nb${}_{\mathrm{V}}$–V${}_{\mathrm{Li}}$ defect pair. In the plots of the absorption coefficients, vertical lines indicate the position of the absorption band assigned to bipolarons in different experiments [12,13].

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**MDPI and ACS Style**

Schmidt, F.; Kozub, A.L.; Gerstmann, U.; Schmidt, W.G.; Schindlmayr, A. Electron Polarons in Lithium Niobate: Charge Localization, Lattice Deformation, and Optical Response. *Crystals* **2021**, *11*, 542.
https://doi.org/10.3390/cryst11050542

**AMA Style**

Schmidt F, Kozub AL, Gerstmann U, Schmidt WG, Schindlmayr A. Electron Polarons in Lithium Niobate: Charge Localization, Lattice Deformation, and Optical Response. *Crystals*. 2021; 11(5):542.
https://doi.org/10.3390/cryst11050542

**Chicago/Turabian Style**

Schmidt, Falko, Agnieszka L. Kozub, Uwe Gerstmann, Wolf Gero Schmidt, and Arno Schindlmayr. 2021. "Electron Polarons in Lithium Niobate: Charge Localization, Lattice Deformation, and Optical Response" *Crystals* 11, no. 5: 542.
https://doi.org/10.3390/cryst11050542