# Dielectric Confinement and Exciton Fine Structure in Lead Halide Perovskite Nanoplatelets

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## Abstract

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## 1. Introduction

**k.p**method [15,16] or tight binding theory [17] and more recently in halide inorganic perovskite NPL [18]. Neglecting the e-h EI, these authors have shown that the self-interaction with image charges is important and increases the exciton binding energy.

**k.p**approach including dielectric effects under the image charge formalism and a variational excitonic wave function. Including both spatial and dielectric confinements, we deduce the eigen-energies of the band-edge excitonic states, and discuss the self and binding energies. The e-h EI, including short-range (SR) and long-range (LR) contributions, is then considered to calculate the FSS. In the following, by including both spatial and dielectric confinements, we first deduce the eigen-energies of the band-edge excitonic states and discuss their self and binding energies. Then, in a second part, the e-h EI, including short-range (SR) and long-range (LR) contributions, is considered to calculate the FSS and discussed. The theoretical methods are presented independently in each part.

## 2. Electronic and Dielectric Confinement Effects on the Exciton Energy

#### 2.1. Theoretical Methods

#### 2.2. Results and Discussion on Exciton Energy

## 3. Electronic and Dielectric Confinement Effects on the Exciton Fine Structure

#### 3.1. Theoretical Methods

#### 3.2. Results and Discussion on Exciton FSS

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Free e-h pair energy E${}_{eh}$ (black line) and excitonic energy E${}_{X}$ with different dielectric mismatches (the outside dielectric constant ${\epsilon}_{2}$ varying from 7.3 to 1), for CsPbBr${}_{3}$. The dashed line correspond to the bulk gap energy E${}_{g}$. The black and red symbols are experimental data are from the works in [28,29], respectively. (

**b**) Energy difference $\Delta ={E}_{X}-{E}_{eh}$ for different dielectric mismatches.

**Figure 2.**Different energy contributions induced by the Coulomb interaction and the dielectric effects in CsPbBr${}_{3}$ NPLs with different dielectric mismatches (the outside dielectric constant ${\epsilon}_{2}$ varying from 7.3 to 1). (

**a**) Summation of the electron and hole self energies ${\Delta}_{S}$. (

**b**) Direct e-h Coulomb interaction ${\Delta}_{C}$. (

**c**) Change in the kinetic energy ${\Delta}_{T}$ in presence of e-h coupling. (

**d**) Binding energy ${E}_{b}$ of the e-h pair including the Coulomb interaction and the dielectric effects in CsPbBr${}_{3}$ NPLs with different dielectric mismatches. The black symbols are experimental values of the binding energy from in [29]. The dashed line corresponds to the bulk binding energy R = −32 meV.

**Figure 3.**Energy labeling of the fine structure exciton states with cubic (O${}_{h}$), tetragonal (D${}_{4h}$), and orthorhombic (D${}_{2h}$) symmetry, for (

**top**) a cube-shaped NC and (

**bottom**) a quasi-2D NPL. In O${}_{h}$ symmetry, the three bright states are totally degenerate in a cube-shaped NC, while the degeneracy is partially lifted in the NPL. In tetragonal and orthorhombic symmetry, the FSSs have comparable ordering.

**Figure 4.**(

**a**) Bright–bright splitting $\Delta E$ and (

**b**) bright–dark splitting ${\delta}_{BD}$, in CsPbBr${}_{3}$ NPLs with a cubic symmetry and different dielectric mismatches (the outside dielectric constant ${\epsilon}_{2}$ varying from 7.3 to 1). The vertical line indicates a size comparable to the exciton Bohr radius.

**Figure 5.**(

**a**) Bright–bright splitting $\Delta E$ and (

**b**) bright–dark splitting ${\delta}_{BD}$, in CsPbBr${}_{3}$ NPLs with a tetragonal symmetry and different dielectric mismatches (the outside dielectric constant ${\epsilon}_{2}$ varying from 7.3 to 1). The vertical line indicates a size comparable to the exciton Bohr radius.

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**MDPI and ACS Style**

Ghribi, A.; Ben Aich, R.; Boujdaria, K.; Barisien, T.; Legrand, L.; Chamarro, M.; Testelin, C. Dielectric Confinement and Exciton Fine Structure in Lead Halide Perovskite Nanoplatelets. *Nanomaterials* **2021**, *11*, 3054.
https://doi.org/10.3390/nano11113054

**AMA Style**

Ghribi A, Ben Aich R, Boujdaria K, Barisien T, Legrand L, Chamarro M, Testelin C. Dielectric Confinement and Exciton Fine Structure in Lead Halide Perovskite Nanoplatelets. *Nanomaterials*. 2021; 11(11):3054.
https://doi.org/10.3390/nano11113054

**Chicago/Turabian Style**

Ghribi, Amal, Rim Ben Aich, Kaïs Boujdaria, Thierry Barisien, Laurent Legrand, Maria Chamarro, and Christophe Testelin. 2021. "Dielectric Confinement and Exciton Fine Structure in Lead Halide Perovskite Nanoplatelets" *Nanomaterials* 11, no. 11: 3054.
https://doi.org/10.3390/nano11113054