# Exciton in Halide Perovskite Nanoplatelets: Finite Confinement and Dielectric Effect in Effective Mass Approximation

Kaouther Tlili, Maria Chamarro, Kais Boujdaria, Christophe Testelin

► To cite this version:

Kaouther Tlili, Maria Chamarro, Kais Boujdaria, Christophe Testelin. Exciton in Halide Perovskite Nanoplatelets: Finite Confinement and Dielectric Effect in Effective Mass Approximation. Journal of Physical Chemistry C, 2025, 129 (2), pp.1270-1281. 10.1021/acs.jpcc.4c06006. hal-04917602

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Exciton in Halide Perovskite Nanoplatelets :
Finite Confinement and Dielectric Effect in
Effective Mass Approximation

Kaouther Tlili,†,‡ Maria Chamarro,‡ Kais Boujdaria,† and Christophe Testelin*,‡

†Université de Carthage, Faculté des Sciences de Bizerte, LR01ES15 Laboratoire de Physique des Matériaux : Structure et Propriétés, 7021 Zarzouna, Bizerte, Tunisia
‡Sorbonne Université, CNRS, Institut des NanoSciences de Paris, F-75005, Paris, France

E-mail: christophe.testelin@insp.upmc.fr

Abstract

In recent years, there has been a surge of interest in two-dimensional (2D) lead halide perovskites (LHPs) due to their high potential for optoelectronic devices. There are experimental evidences that carriers and excitons are not totally confined in these atomically thin 2D crystals, surrounded by organic ligands. Taking into account previous results on density functional theory calculations, we use variational method and **k.p** approach to solve the Hamiltonian of excitons in nanoplatelets (NPL) including quantum and dielectric confinements, surface stress, Coulomb interaction and finite barrier potential. Essential characteristics such as exciton energy, binding energy and reduced mass are discussed in two inorganic LHPs, *CsPbBr*₃ and *CsPbI*₃, as model systems. For NPL thickness comparable or smaller than the Bohr exciton radius, a good agreement between theory and experiment is obtained for the exciton energy. This agreement is achieved through consideration of the band offset between NPL and
surrounding ligands, coupled to a carrier mass thickness dependence, showing that pre-
vious studies which consider an infinite confinement largely overestimate the exciton
energy of thin NPLs. The dielectric environment of the NPL emerges as a pivotal
parameter influencing the exciton binding energy.

# 1 Introduction

Recently, lead halide perovskites (LHPs) have garnered significant attention, due to their impressive opto-electronic properties, making them promising candidates for multiple applications,¹ including but not limited to solar cells,² light emitting diodes,³ and photodetectors.⁴ To these outstanding properties, we must add the very easy and low-cost chemical manufacturing methods used to obtain these materials. In this context and to enlarge the applications of LHPs, the ability to synthesize LHP colloidal nanocrystals (NCs) has then attracted a large interest, particularly two-dimensional nanostructures, named nanoplatelets (NPL). They are very anisotropic NCs showing a strong carrier confinement along one direction. Indeed, their thickness can be as small as few monolayers (MLs) and is much smaller than their lateral dimensions. NPLs can then offer outstanding optical and electronic properties, as high absorption,⁵ narrow and fast emission lines,⁶ large two-photon absorption⁷ or a huge exciton binding energy⁸,⁹,¹⁰ and as a consequence, room-temperature stable excitons. These NPLs are then very good candidates for photonic and polaritonic applications.¹¹ Moreover, as they can have a large surface area, they can be easily incorporated in planar electro-optical devices.

Understanding the optical emission directly related to the band edge excitons, and modeling
their excitonic properties is then fundamental, particularly to predict or design new nanos-
tructures.

Literature is very rich on theoretical and experimental studies of II-VI semiconductor NCs
and, in particular, NPLs, since their first synthesis of II-VI NPLs dates from 2008.¹² How-
ever, fewer studies concern LHP NPLs that have been more recently synthesized. The unique
optical properties of 2D confined excitons in LHPs compounds start then to be explored in
experimental studies. Spectroscopic experiments on perovskite NPLs have measured the
exciton emission and binding energy as a function of thickness, mainly in CsPbBr₃ com-
pounds¹³¹⁴¹⁵,¹⁶ but also in CsPbI₃¹⁷¹⁸¹⁹ and hybrid LHPs MAPbX₃ (X=I, Br, Cl)²⁰²¹.²²
II-VI and LHPs colloidal NPLs share a main common characteristic : their close environment
is constituted by organic ligands, except those that are encapsulated by other semiconductor
materials. Ligands induce strain on colloidal objects²³,²⁴ create a dielectric environment with
a dielectric constant that differs from the dielectric constant of the colloidal object²⁵²⁶²⁷²⁸,²⁹
and impose a finite potential barrier²³,³⁰ as evidenced by transport and conductivity mea-
surements³¹.³²

From a theoretical point of view, we find early density functional theory (DFT) work on
the exciton energy in CsPbBr₃ NPLs that takes into account the dielectric contrast between
the NPL and the outside medium. However, in this study, the presence of organic ligands at
the boundary of the NPLs is not explicitly described.²⁸ A much more recent *ab-initio* work
investigated the exciton fine structure of 2D metal halide layered perovskites composed of
thin slabs of lead halide octahedra separated by large organic spacers.³³ *Ab-initio* methods
are well adapted to describe the electronic properties of a complex system of NPLs containing
a semiconductor NC core and ligands attached to the NC surface, but they are constrained
by nonlinear scaling with system size.

Recent theoretical studies in the framework of the effective mass approximation (EMA)
consider dielectric effects but infinite barriers<sup>26 27 34</sup>,<sup>35</sup> At small thicknesses, the theoretical
simulations predict an exciton energy blue shift, overestimated compared to the spectroscopic
measurements.<sup>15</sup> The exciton fine structure has also been explored<sup>15 26 35</sup> in this framework.
Considering a finite barrier at the boundary between a crystalline material and an amor-
phous material is, strictly speaking, the limit of application of the EMA and the envelope
function method. Nonetheless, DFT and pseudopotential calculations carried out in II-VI
semiconductors<sup>23 30 36</sup> showed that the EMA and envelope model accurately describe elec-
tronic properties such as electronic wavefunction penetration in ligands and exciton energies
by adjusting parameters that describe band offsets, electronic masses, and dielectric. In line
with this approach, J. A. Sichert et al. used EMA in an early publications to describe size
confinement effects in MAPbBr₃<sup>22</sup> and in CₙPbI₄ NPLs.<sup>37</sup>

The aim of this work is to investigate and model the effects of electronic and dielectric
confinement on exciton emission and binding energy in LHP NPLs using a **k.p** method
within the EMA. Key innovations of the present study include incorporating a finite barrier
model and introducing a thickness dependence in Bloch functions and carrier effective masses.
By accounting for Bloch function variations with NPL thickness, we capture the anisotropic
nature of the excitonic masses more precisely, and by introducing thickness-dependent carrier
masses we achieve a more accurate representation of the band-edge states in NPLs. EMA
allows us to address the dependence of electron and hole masses on thickness<sup>38,39</sup>

Unlike previous approaches that assumed an infinite potential barrier,²⁷ our model addresses finite potential offsets between the NPL and its surrounding ligands, as well as dielectric mismatches at the NPL-ligand interface. For the amorphous ligand medium, a mass can be defined for charge carriers.⁴⁰ To describe the NPL surrounding media, we use the free electron mass, as commonly used by other authors²²,²³,³⁶,³⁷ Dielectric effects are incorporated using the charge image formalism, enabling calculation of the electron and hole self-energies and the electron-hole Coulomb interaction. Additionally, we discuss the impact of ligand-induced strain on the exciton energy of CsPbBr₃ NPLs. This more realistic treatment of boundary conditions significantly enhances the predictive accuracy for excitonic energy, especially in thinner NPLs with high aspect ratios and reduced dimensionality.
# 2 THEORETICAL FRAMEWORK

## 2.1 Bulk Band Structure

In LHP, a reverse band ordering is observed for the band-edge states, compared to conventional III-V and II-VI semiconductors$^{41,42,43}$. As a direct band-gap semiconductors, LHPs have total angular momentum $j_v = 1/2$ for the states of the upper VB (which are anti-bonding combinations mainly of metal s but also of halide p orbitals, and denoted by $|S_V \uparrow (\downarrow)\rangle$ in Figure. 1). Strong spin-orbit coupling (SOC) mostly affects the conduction band (CB) in the cubic crystal structure's CB, splitting it into a lower split-off conduction band with $j_e = 1/2$ (which are anti-bonding combinations mainly of metal p but also halide s orbitals)$^{44}$ and an upper band with $j_e = 3/2$. We must also consider the effect of lattice distortion in the tetragonal phase in comparison to the cubic phase. The upper 4-fold degenerate CB (originating from $j_e = 3/2$ states of the cubic phase) splits into two separate 2-fold degenerate bands: $|j_e = 3/2, j_z^e = \pm 3/2\rangle_C$ and $|j_e = 3/2, j_z^e = \pm 1/2\rangle_C$ (refer to Figure. 1). The lower spin-orbit split-off band has ($j_e = 1/2, j_z^e = \pm 1/2$) and is denoted by $|j_e = 1/2, j_z^e = \pm 1/2\rangle_C$ in Figure. 1. We show in Figure. 1 an illustration of the 8-band k.p model focused on the zone center, obtained by eliminating the coupling between the 8 bands and the remaining bands. The Bloch states in p-conduction bands are the eigenstates at $k = 0$ of the effective Hamiltonian $H(0)$ which includes the contributions of the crystal field (CF) and SOC; for further information see Supporting Information section 1 which contains the Bloch functions for the p-like CB. The Bloch phase angle $\theta$, a key parameter in determining these Bloch states, is related to the parameters $T_0$ (bulk tetragonal CF parameter) and $\Delta_{SO}$ (spin-orbit split-off parameter) via the formula:

$$ \tan 2\theta = \frac{2\sqrt{2}\Delta_{SO}}{(\Delta_{SO} - 3T_0)} \quad (1) $$
<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic representation of the bulk tetragonal crystal conduction and valence bands at the Γ point when considering 8 bands.</figcaption></figure>
(with $0 < \theta < \pi/2$).

The parameters $\Delta_{SO}$ and $T_0$ can be deduced from the bulk parameters given in TABLE 1.

The exciton states, at the band-gap edge, are then bound states, pairing an upper VB hole $|j_h = \frac{1}{2}, j_z^h = \pm \frac{1}{2}\rangle$ (deduced from the VB electron states $j_v = \frac{1}{2}$ via time-reversal operator) with an electron from the lowest split-off CB $|j_e = \frac{1}{2}, j_z^e = \pm \frac{1}{2}\rangle_C$.

## 2.2 Quantum Well Structure

In the context of our research, it is imperative to note that the 2D LHP NPLs display a quantum well electronic structure. Our model system is designed to emulate a quasi-2D structure, in which the thickness of the NPLs ($L_z$) can be close to the bulk Bohr radius of the respective material and is significantly smaller than their lateral dimensions ($L_x$ and $L_y$) which are infinite in our model. The schematic structure of our model system is shown in Figure. 2. This configuration includes a single quantum well surrounded by barrier layers having a lower dielectric constant ($\epsilon_2$) than the well ($\epsilon_1$). This dielectric mismatch induces a strong dielectric confinement which has to be added to the spatial confinement created by the conduction and valence band offsets. These two contributions have to be simultaneously considered to determine the electronic states.

## 2.3 Hamiltonian of a Bound electron-hole Pair

The Hamiltonian for the electron, hole, and electron-hole pair (exciton) systems are presented as follows:

$$ 
\begin{aligned}
H^{(e)} &= T_{kin}^{(e)} + V_{conf}^{(e)} + V_{self}^{(e)} \\
H^{(h)} &= T_{kin}^{(h)} + V_{conf}^{(h)} + V_{self}^{(h)}
\end{aligned}
\qquad (2) $$

$$ H_{system} = H^{(e)} + H^{(h)} + V_C + E_g $$

Here the suffixes e and h refer to the electron and hole, where
<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic representation of the dielectric quantum-well structure, where $\epsilon_2 < \epsilon_1$.</figcaption></figure>

* $T_{kin}^{(e)}$ and $T_{kin}^{(h)}$ are the kinetic energies of the electron and hole respectively.

* $V_{conf}^{(e)}$ and $V_{conf}^{(h)}$ are the confinement potentials of the electron and hole respectively.

* $V_{self}^{(e)}$ and $V_{self}^{(h)}$ are the self potentials of the electron and hole respectively.

* $V_C$ is the electron-hole Coulomb potential.

* $E_g$ is the band-gap energy of the bulk.

The impact of the dielectric contrast between the well and barrier layers is considered in the Coulomb potential ($V_C$), as well as in the self-potential ($V_{self}$), by incorporating the effect of image charges.²⁹ Recently, such an electrostatic procedure has been used to compute the exciton fine structure in NPLs²⁶ and nanocrystals⁴⁵,⁴⁶ with LHP as constituent material. This well-established image charge method effectively replicates the electric field generated by polarization charges at the interface between materials possessing different dielectric constants. The Coulomb interaction is a potential that describes interactions between two particles, the electron and the hole. In presence of a dielectric mismatch, one has to include the interaction of the electron with the hole images, as shown in the equation 3. The dielectric contrast will increase the attractive e-h interaction, leading to a stronger binding between
electrons and holes, which in turn results in a reduction of the optical gap. The self-energy potential is a one-particle potential. It describes the interaction of charge carriers (electron or hole) with their own charge images. It is given in the equations 4 and 5 respectively for electron and hole. Self-energy potentials play a crucial role, especially to precise the excitonic energy<sup>47</sup> or the interactions of excitons across the organic barrier at the boundary of the inorganic layers.<sup>48</sup>

$$V_C = -\frac{e^2}{4\pi\epsilon_0\epsilon_1} \times \sum_{n=0}^{+\infty} \frac{q_n}{\left[ (\rho_e - \rho_h)^2 + [z_e - (-1)^n z_h + nL_z]^2 \right]^{\frac{1}{2}}} \quad (3)$$

where $\rho_l = (x_l, y_l)$ and $z_l$ are in-plane and out-of-plane components of the position vector $r_l$ of carrier l, $q_n = \eta^{|n|}$, with $\eta$ the dielectric contrast, defined as $\eta = \frac{\epsilon_1 - \epsilon_2}{\epsilon_1 + \epsilon_2}$.

$$V_{self}^{(e)} = \frac{e^2}{4\pi\epsilon_0\epsilon_1} \sum_{n=\pm 1, \pm 2} \frac{q_n}{2 \left| z_e - (-1)^n z_e + nL_z \right|} \quad (4)$$

$$V_{self}^{(h)} = \frac{e^2}{4\pi\epsilon_0\epsilon_1} \sum_{n=\pm 1, \pm 2} \frac{q_n}{2 \left| z_h - (-1)^n z_h + nL_z \right|} \quad (5)$$

Using the variational calculus method, the exciton energy, $E_X$, and the exciton ground state will be determined. Our normalized trial wave function (WF) for the ground state is a two-parameters-dependent WF constructed by combining the lowest one-particle envelope WF of an electron and a hole multiplied by an anisotropic hydrogenoid function. It takes the following format:

$$\psi_{e,h}(\mathbf{r}_e, \mathbf{r}_h) = N(a, \alpha) \frac{e^{i\mathbf{K}_\perp \cdot \mathbf{R}_\perp}}{\sqrt{L_x L_y}} \chi_e(z_e) \chi_h(z_h) \times \exp\left(-\frac{\sqrt{(x_e - x_h)^2 + (y_e - y_h)^2 + \alpha^2(z_e - z_h)^2}}{a}\right) \quad (6)$$

in which $N(a, \alpha)$ is a normalization factor, a and $\alpha$ are the variational parameters. $R_\perp$ is the in-plane coordinate of the e-h pair center of mass and $K_\perp$ is its kinetic momentum
(K⊥ = 0 in the ground state). $\chi_{\ell}(z_{\ell})$ is the single particle envelope WF along the z axis ($\ell = e, h$), eigenstate of the Hamiltonian $\bar{H}^{(\ell)} = T_{kin}^{(\ell)} + V_{conf}^{(\ell)}$.

The type of spatial confinement that will be taken into account determines the expression of $\chi_{\ell}(z_{\ell})$.

In the following, based on the Hamiltonian established for the bound electron-hole pair, we proceed to investigate the effects of spatial confinement in LHP NPLs under both infinite and finite barrier models.

## 2.4 Confinement by Infinite Barrier

As shown in the Figure. 2, the LHPs materials occupy the region within the boundaries of $-\frac{L_z}{2} \le z \le \frac{L_z}{2}$. We will first assume an infinite confinement potential outside this designated zone so that $\chi_{\ell}(z_{\ell}) = \sqrt{\frac{2}{L_z}} \cos(\frac{\pi z_{\ell}}{L_z})$. Let us focus on the variational calculation for the exciton ground state. Its trial state should have an average energy as close as possible from the true ground state energy of the Hamiltonian $H_{system}$. The best trial WF is such that the average of the Hamiltonian of our system (equation 2) in a state $|\psi\rangle$ (equation 6) is minimum with respect to the variational parameters a and $\alpha$. This minimization process allows us to determine the exciton energy as shown in the equation 7.

$$E_X = \langle \psi_{e,h} | H_{system} | \psi_{e,h} \rangle = \langle E_{conf} \rangle + \langle V_{eh}^{self} \rangle + \langle V_C \rangle + E_g \quad (7)$$

in which $\langle E_{conf} \rangle$ is the quantum confinement energy, $\langle V_{eh}^{self} \rangle$ the sum of electron and hole repulsion self-energy and $\langle V_C \rangle$ the Coulomb attraction energy. The binding energy ($E_b$) is then defined by isolating the exciton energy ($E_X$) from all single-particle contributions, as described in the equation 8:

$$E_b = \left( \langle V_{eh}^{self} \rangle + E_g + \frac{\hbar^2 \pi^2}{2\mu_{\parallel} L_z^2} \right) - E_X \quad (8)$$
where $\frac{\hbar^2\pi^2}{2\mu_{\parallel}L_z^2}$ is the confinement energy associated to both the electron and the hole, and $\mu_{\parallel}$ is the effective exciton mass along the z-axis : $\frac{1}{\mu_{\parallel}} = \frac{1}{m_{\parallel}^e} + \frac{1}{m_{\parallel}^h}$. $m_{\parallel}^\ell$ is the effective mass along the z-axis for carrier $\ell$.

## 2.5 Confinement by Finite Barrier

In real system, the barrier potential is finite, fixed by the nature of the NPL and the surrounding medium which is constituted by organic ligands. There are experimental evidences for penetration of charge carrier on organic ligands$^{23,30,36}$. Then the WF $\chi_{\ell}(z_{\ell})$ has to be calculated for finite band offset $V_o^{\ell}$, leading to an evanescent function in the barrier. The carriers being delocalized in the NPL and the barriers, the self-energies and the electron-hole Coulomb interaction have to be fully defined in the whole space, using the charge image formalism. $V_C$ and $V_{self}^{\ell}$ are defined in the Supporting Information section 2. Within the context of dielectric interfaces, the potential exhibits non-integrable divergences, due to $\chi_{\ell}$ non-zero values at the interface, which constitutes a significant challenge. To address this issue, a viable approach known as "the method of the shifted mirror faces" has been employed.$^{29}$ This method involves the introduction of a phenomenological parameter denoted as $\delta$, which is associated to the center of mass of the electron density close to the interface.$^{49}$ Mathematically, this technique entails the replacement of the $n = \pm 1$ terms in equation 4 and equation 5 with the following expression:

$$ \frac{\eta}{2|2z_{\ell} + L_z + 2\delta|} + \frac{\eta}{2|2z_{\ell} - L_z - 2\delta|} \qquad (9) $$

where $\ell$ refers to the charge carrier corresponding to the electron or hole. Non-zero values $\delta$, smaller than the lattice parameter solve the interface divergence.

As previously indicated, in the final step of our approach and within the context of k.p theory, we consider a square quantum well along z direction with thickness $L_z$ and finite barrier, but with an energy-dependent effective mass.
In the quantum well with finite potential barriers, we consider a system characterized by a carrier mass in the quantum well ($m_l$, where $l$ denotes either an electron or a hole), a barrier mass ($m_B$), a potential barrier height of electron or hole ($V_0^l$), and a NPL width ($L_z$). The wave vector ($k_l$) and the decay constant of the charge carrier ($\kappa_l$) are defined as:

$$k_l = \sqrt{\frac{2m_l E_{conf}^{(l)}}{\hbar^2}}, \quad \kappa_l = \sqrt{\frac{2m_B}{\hbar^2}(V_0^l - E_{conf}^{(l)})} \qquad (10)$$

with $E_{conf}^{(l)}$, the energy level of the carrier. The corresponding WFs in different regions are given by:

$$\chi_l(z_l) = \begin{cases} A_l \cos(k_l z_l) & \text{for } |z_l| \le \frac{L_z}{2} \\ B_l e^{\kappa_l (z_l + \frac{L_z}{2})} & \text{for } z_l \le -\frac{L_z}{2} \\ B_l e^{-\kappa_l (z_l - \frac{L_z}{2})} & \text{for } z_l \ge \frac{L_z}{2} \end{cases} \qquad (11)$$

To determine the energy levels ($E_{conf}^{(l)}$) corresponding to electron or hole confinement energy, we solve the following transcendental equation:

$$\kappa_l - \frac{m_B}{m_l} k_l \tan \left( \frac{k_l L_z}{2} \right) = 0 \qquad (12)$$

After finding $E_{conf}^{(p)}$, we can compute $k_l$ and $\kappa_l$. The normalization constants ($A_l$ and $B_l$) are then derived as:

$$A_l = \left[ \frac{L_z}{2} + \frac{1}{\kappa_l} \left( \cos^2 \frac{k_l L_z}{2} + \frac{m_B}{m_l} \sin^2 \frac{k_l L_z}{2} \right) \right]^{-1/2} \qquad (13)$$

$$B_l = A_l \cos \frac{k_l L_z}{2} \qquad (14)$$

Finally, a normalization factor ($F_e$) for the electron envelope function is determined as follows:
$$F_e = \left\langle \chi_e \left| \frac{-d^2}{dz^2} \right| \chi_e \right\rangle = \frac{A_e^2 k_e^2}{2} \left[ L_z + \frac{\sin k_e L_z}{k_e} \right] - B_e^2 \kappa_e \quad (15)$$

This detailed description includes the essential equations and boundary conditions required to solve the Schrödinger equation for a quantum well with a potential barrier, accounting for the carrier mass ($m_l$) in the quantum well, which could be an electron or a hole, and the electron mass ($m_B$) in the barrier$^{36,23,50}$

## 2.6 Carrier Mass and Bloch function Dependence with Nanoplatelet Thickness

<table><caption>Table 1: Bulk and **k.p** model parameters of CsPbBr$_3$ in tetragonal phase.</caption><thead><tr><th>Basic parameters</th><th>Bulk</th><th>4-band</th><th>8-band</th></tr></thead><tbody><tr><td>E<sub>g</sub>(eV)</td><td>2.342<sup>a</sup>(2K)</td><td>-</td><td>-</td></tr><tr><td></td><td>2.403<sup>a</sup>(270K)</td><td>-</td><td>-</td></tr><tr><td>E<sub>b</sub>(meV)</td><td>33<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>ε<sub>1</sub></td><td>7.3<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>a<sub>B</sub>(nm)</td><td>3.1<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>μ(m<sub>0</sub>)</td><td>0.126<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>θ(°)</td><td>40.1<sup>d</sup></td><td>-</td><td>-</td></tr><tr><td>Δ<sub>SO</sub>(eV)</td><td>1.4<sup>d</sup></td><td>-</td><td>-</td></tr><tr><td>T<sub>0</sub>(eV)</td><td>0.24<sup>d</sup></td><td>-</td><td>-</td></tr><tr><td>E<sub>2</sub>(eV)</td><td>3.68<sup>c,b</sup></td><td>-</td><td>-</td></tr><tr><td>E<sub>3</sub>(eV)</td><td>3.83<sup>c,b</sup></td><td>-</td><td>-</td></tr><tr><td>E<sub>P<sub>S,Z</sub></sub>(eV)</td><td>15.65<sup>b</sup></td><td>22.55<sup>d</sup></td><td>15.55<sup>d</sup></td></tr><tr><td>E<sub>P<sub>S,p</sub></sub>(eV)</td><td>16.8<sup>b</sup></td><td>32.27<sup>d</sup></td><td>18.32<sup>d</sup></td></tr></tbody></table>

<sup>a</sup>, <sup>51</sup> <sup>b</sup>, <sup>26</sup> <sup>c</sup>, <sup>52</sup> <sup>d</sup> This work.

The **k.p** theory enables the incorporation of the dependence of carrier masses on band-gap energy, while taking into account the confinement effect. We next estimated the effective carrier masses as a function of thickness using the bulk band parameters, which are determined in previous studies using the multiband **k.p** theory$^{26,53,54}$ or experimentally measured$^{51}$ and shown in TABLE 1. As observed by Yang et al.$^{51}$ the energy of the exciton ground state
(1S) transition exhibits a continuous blue-shift with increasing temperature, indicating that $CsPbX_3$ (X = Br, I) compounds do not undergo a structural phase transition within the temperature range of 4K to 270K. They conclude that both $CsPbBr_3$ and $CsPbI_3$ remain in a single phase across this temperature range. Furthermore, Yang et al. report a constant diamagnetic coefficient from 4K to 180K, which, being directly related to the dielectric constant and the effective exciton mass, suggests that these parameters remain largely unaffected by temperature. Therefore, we have adopted temperature-independent **k.p** parameters. In order to compare experimented results both at room and low temperature, we take into account only the temperature-dependent shift in the bandgap energy.

The band diagram of these compounds near the high-symmetry point that corresponds to the direct band-gap can be obtained using the multiband **k.p** description. It can also be used to describe the carriers effective masses at the band edge.

The dispersion relations for electrons and holes near the $\Gamma$ point is given as following:

$$ \left\{ \begin{aligned} E_e(k) &= E_g + \frac{\hbar^2 k_z^2}{2m_{e\parallel}} + \frac{\hbar^2 k_\rho^2}{2m_{e\perp}} \\ E_h(k) &= \frac{\hbar^2 k_z^2}{2m_{h\parallel}} + \frac{\hbar^2 k_\rho^2}{2m_{h\perp}} \end{aligned} \right. \qquad (16) $$

Here $k_\rho^2 = k_x^2 + k_y^2$, $m_{e\parallel}$ and $m_{e\perp}$ represent the effective masses of the lowest CB along and perpendicular to the tetragonal-axis, respectively (the symmetry axis of the tetragonal structure). Similarly, $m_{h\parallel}$ and $m_{h\perp}$ denote the hole effective masses in the upper VB along and perpendicular to the c-axis.

Limiting our analysis to the close band-edge bands, we obtain readily the effective masses of the different carriers in the vicinity of the $\Gamma$ point by employing the second-order Löwdin perturbation theory.⁵³ Within the 4-band and 8-band **k.p** models, the carriers effective masses can be expressed as:
**4-band Scheme:**

$$ \left\{ \begin{aligned} \frac{m_0}{m_{e\parallel}} &= 1 + \sin^2 \theta \frac{E_{P_{S,Z}}}{E_g} \\ \frac{m_0}{m_{h\parallel}} &= -1 + \sin^2 \theta \frac{E_{P_{S,Z}}}{E_g} \end{aligned} \right. \qquad (17) $$

$$ \left\{ \begin{aligned} \frac{m_0}{m_{e\perp}} &= 1 + \frac{1}{2} \cos^2 \theta \frac{E_{P_{S,\rho}}}{E_g} \\ \frac{m_0}{m_{h\perp}} &= -1 + \frac{1}{2} \cos^2 \theta \frac{E_{P_{S,\rho}}}{E_g} \end{aligned} \right. \qquad (18) $$

**8-band Scheme:**

$$ \left\{ \begin{aligned} \frac{m_0}{m_{e\parallel}} &= 1 + \sin^2 \theta \frac{E_{P_{S,Z}}}{E_g} \\ \frac{m_0}{m_{h\parallel}} &= -1 + E_{P_{S,Z}} \left[ \frac{\sin^2 \theta}{E_g} + \frac{\cos^2 \theta}{E_g} \right] \end{aligned} \right. \qquad (19) $$

$$ \left\{ \begin{aligned} \frac{m_0}{m_{e\perp}} &= 1 + \frac{1}{2} \cos^2 \theta \frac{E_{P_{S,\rho}}}{E_g} \\ \frac{m_0}{m_{h\perp}} &= -1 + \frac{1}{2} E_{P_{S,\rho}} \left[ \frac{1}{E_3} + \frac{\sin^2 \theta}{E_2} + \frac{\cos^2 \theta}{E_g} \right] \end{aligned} \right. \qquad (20) $$

<figure><img src="image_4.png" /><figcaption>Figure 3: Thickness-dependence of the Bloch phase angle $\theta$ in the finite barrier model.</figcaption></figure>

The band-gap energy $E_g$ and the inter-band transition energies ($E_2, E_3$) are defined from the VB maximum to the conduction states $|\frac{1}{2}, \pm \frac{1}{2}\rangle_C$, $|\frac{3}{2}, \pm \frac{1}{2}\rangle_C$, and $|\frac{3}{2}, \pm \frac{3}{2}\rangle_C$. In the bulk materials, they have been measured by optical absorption⁵² and are given in TABLE 1. The
Kane energies $E_{P_S,Z}$ and $E_{P_S,\rho}$ are derived from these equations by using the experimental values of the effective mass and the band-gap energy as provided in TABLE 1.

In NPLs, the Bloch phase angle $\theta$ as proposed recently by Gramlich and coworkers³⁵ and mentioned previously in section 2.1, is confinement dependent. Specifically, spatial confinement within the NPL structure induces modifications in the Bloch functions, which subsequently impact the parameter $\theta$. To account for this, we have to consider an extra effective crystal field, denoted $T_{eff}$ and defined as follows:³⁵

$$T_{eff} = -6\gamma_2 \frac{\hbar^2}{2m_0} F_e \qquad (21)$$

where $\gamma_2 = \frac{E_{P_S,Z}}{6E_g}$ is the second Luttinger parameter. $\theta$ is then given by equation 1 with a total crystal field $T = T_0 + T_{eff}$. Using a self-consistent numerical resolution, we can then deduce the thickness dependence of the carrier masses and angle $\theta$, by including the confinement contribution to the conduction states energies, and replacing $E_g$, $E_2$ and $E_3$ by:

$$\left\{ \begin{array}{l} E'_{g} = E_{g} + E_{conf}^{(e)} + E_{conf}^{(h)} \\ E'_{2} = E'_{g} + \Delta_{SO} - \frac{T}{3} \\ E'_{3} = E'_{g} + \Delta_{SO} + \frac{T}{3} \end{array} \right. \qquad (22)$$

The Figure 3 illustrates the variation of the angle $\theta$ in function of the NPLs thickness $L_z$. This figure demonstrates that while $\theta$ varies slightly with NPL thickness, it stays in a range close to the bulk value of $\theta = 40.1^{\circ}$, with no strong variation as claimed in Ref³⁵ for infinite barriers.

Equations (17–18), which are derived from a 4-band scheme, and (19–20), which are obtained from an 8-band scheme, can be used to straightforward determine the components of the reduced exciton mass tensor along $(\mu_\parallel)$ and perpendicular $(\mu_\perp)$ to the c-axis. We get:
**4-band Scheme:**

$$ \left\{ \begin{aligned} \frac{m_0}{\mu_{\parallel}} &= 2 \sin^2 \theta \frac{E_{P_{S,Z}}}{E'_{g}} \\ \frac{m_0}{\mu_{\perp}} &= \cos^2 \theta \frac{E_{P_{S,\rho}}}{E'_{g}} \end{aligned} \right. \qquad (23) $$

**8-band Scheme:**

$$ \left\{ \begin{aligned} \frac{m_0}{\mu_{\parallel}} &= 2 \sin^2 \theta \frac{E_{P_{S,Z}}}{E'_{g}} + \cos^2 \theta \frac{E_{P_{S,Z}}}{E'_{2}} \\ \frac{m_0}{\mu_{\perp}} &= \cos^2 \theta \frac{E_{P_{S,\rho}}}{E'_{g}} + \frac{1}{2} \sin^2 \theta \frac{E_{P_{S,\rho}}}{E'_{2}} + \frac{1}{2} \frac{E_{P_{S,\rho}}}{E'_{3}} \end{aligned} \right. \qquad (24) $$

Based on the 8-band **k.p** model, the effective masses variation versus the well width of CsPbBr₃ NPLs, Lz, is displayed in Figure 4 (a) for both carriers, along and perpendicular to the tetragonal c-axis. One observes an increase of the electron and hole masses as NPL thickness decreases. In the range of explored thicknesses, $m_{e\perp}$ is always the largest mass, down to 2 ML (the ML thickness is 0.58 nm¹⁴). The effective electron masses, $m_{e\perp}$ and $m_{e\parallel}$, are larger than the hole effective masses, $m_{h\perp}$ and $m_{h\parallel}$, except for 2-3 ML thicknesses where $m_{h\parallel}$ becomes comparable and even larger than $m_{e\parallel}$.
<figure><img src="image_5.png" /><figcaption>Figure 4: (a): Thickness-dependence of the effective mass for electrons ($m_{e\perp}$, $m_{e\perp}$), holes ($m_{h\parallel}$, $m_{h\perp}$) obtained from 8-band *k.p* calculations; dashed lines are defined as the bulk effective mass for different carriers. (b): Thickness-dependence of the reduced effective mass, parallel ($\mu_\parallel$) and perpendicular ($\mu_\perp$) to the c-axis, obtained from 4-band and 8-band *k.p* models, the black dashed line refers to the bulk reduced effective mass.</figcaption></figure>
Figure 4 (b) illustrates that the reduced effective masses decrease as the NPL thickness increases, converging towards the bulk experimental reduced mass ($\mu_{exp} = 0.126 m_0$). It can be observed that the reduced effective masses for both 4-band and 8-band models overlap for larger thicknesses (> 10 nm) and show slight differences for smaller thicknesses (< 10 nm), indicating no significant dependency on the specific number of bands considered. For this study, we will adopt the 8-band model due to its higher precision compared to the 4-band model.

# 3 Results and Discussions

## 3.1 Confinement by Infinite Barrier

In this part, we will focus on CsPbBr₃ NPLs, following the infinite barrier modelization described in subsection 2.4. The calculated exciton energy versus the NPL thickness is shown on Figure. 5 (a). We have taken into account the presence of a dielectric mismatch and fixed carrier masses. For the outside dielectric constant, we consider three configurations: (i) $\epsilon_2 = \epsilon_1$, no dielectric contrast ($\eta = 0$); (ii) $\epsilon_2 = 1$ (maximum dielectric contrast); (iii) $\epsilon_2 = 2.5$. The selection of this outside dielectric constant value ($\epsilon_2 = 2.5$) is based on the consideration that the external environment contains organic ligands such as oleic acid, oleylamine, and a solvent mixture of hexane and acetone⁵⁵,⁵⁶.⁵⁷ For comparison, among the available published results, we have only considered absorption measurements, best suited to extract the free exciton energy, either at low temperature (5-10K)¹⁴,¹⁶ or room temperature¹³,¹⁵ (for Ref,¹⁴ the effective NPL thickness has been shifted by 2 MLs, the previous analysis overestimating $L_z$). We notice a good agreement between our results and the experimental results for large thicknesses (> 4nm) of CsPbBr₃ NPLs. However, a strong discrepancy between theory and experiments is observed in the strong confinement regime ($a_B < L_z$), while a weak discrepancy is visible in the intermediate regime ($a_B \sim L_z$), where $a_B = 3.1$ nm.⁴⁶
Note that the dielectric mismatch slightly reduces the exciton energy for a given size. Nonetheless, it has a very moderate influence on the overall behavior of the exciton energy versus NPL thickness. That can be explained by the dielectric mismatch generating two extra contributions, with similar amplitudes but opposite signs : the self-energy and the Coulomb interaction.

From Figure. 5(a), we deduce that the infinite barrier model is not suitable for very thin NPL, due to an overestimation of the confinement energy (more than 700 meV for $L_z$ = 1.74 nm (3ML)) and a non realistic wavefunction fully localized in the NPL.

Finally, the exciton binding energy $E_b$ is shown in Figure. 5(b). One observes a strong increase of the binding energy with the dielectric contrast by a factor 2-3 when $\epsilon_1$ varies from 7.3 to 2.5, and more for the maximum dielectric contrast. $E_b$ increases when decreasing the NPL thickness and reaches a value of 270 meV for 2 MLs when $\epsilon_2$ = 2.5 (to be compared to the 90 meV value in absence of dielectric effect), which closely aligns with the experimental results reported by Bohn et al$^{15}$ and Wang et al.$^{16}$ Even for thick NPLs, a significant $E_b$ increase is seen. Nonetheless, a discrepancy is observed between experiment and theory in the intermediate regime.
<figure><img src="image_6.png" /><figcaption>Figure 5: (a): CsPbBr₃ excitonic energy *E*<sub>X</sub> with different dielectric mismatches (ε₂ varying from 7.3 to 1) for an infinite barrier model with fixed masses. The symbols are experimental data: black, <sup>16</sup>red, <sup>14</sup>blue, <sup>13</sup>orange. <sup>15</sup> (b): CsPbBr₃ binding energy *E*<sub>b</sub> with different dielectric mismatches (ε₂ varying from 7.3 to 1) for an infinite barrier model with fixed masses. The orange <sup>15</sup> and black <sup>16</sup> symbols are experimental data.</figcaption></figure>
## 3.2 Confinement by Finite Barrier

We are now considering finite barriers for the NPL. Given that NPLs are enveloped by organic ligands,⁵⁸ we can define the heights of the electron and hole barriers in terms of band offsets: $V_0^e$ for the electron and $V_0^h$ for the hole. The band offsets can be defined by comparing the conduction (valence) edge to the LUMO (HOMO) levels in the ligand molecules (such as oleic acid). We have then compared conduction and valence band edges, determined experimentally⁵⁹,⁶⁰,⁶¹ with the HOMO-LUMO levels of oleic acid, obtained by DFT calculations⁶²,⁶³,⁶⁴ For our modelization, we have then considered for the band offset, the values $V_0^e = 3.1$ eV and $V_0^h = 1.1$ eV. As discussed in the previous section, we have to include a mirror shift parameter $\delta$.

<figure><img src="image_7.png" /><figcaption>Figure 6: CsPbBr₃ excitonic energy $E_X$ with different mirror shift $\delta$ varying from 0.06 nm to 0.015 nm for $\epsilon_2$ = 2.5. The symbols are experimental data as mentioned previously for exciton energy in Figure. 5(a).</figcaption></figure>

We first discuss the influence of the mirror shift parameter on the exciton energy. As explained in⁴⁹ the mirror shift $\delta$ depends on the electronic charge density $n$ and is related to the parameter $r_s = (3/(4\pi n^3))^{1/3}$. To calculate $n$, we consider that each CsPbBr₃ unit cell contains one Cs atom, one Pb atom, and three Br atoms, which contribute one, four, and
seven electrons, respectively. This results in $r_s = 2.3$ and a mirror shift $\delta = 0.08$ nm. The calculated mirror shift $\delta$ dependence of the exciton energy $E_X$ is shown on Figure. 6 (a small non-zero mirror shift alloying to solve the non-integrable divergence). We notice that the dependence of the exciton energy on $\delta$ is relatively weak, even in the case of very thin layers. We have then fixed the mirror shift at $\delta = 0.08$ nm and have incorporated this value in our computations further with the finite barrier model.

For finite barrier model with fixed masses, the calculated exciton energy is shown on Figure. 7(a), versus the NPL thickness, in presence of a dielectric mismatch. We observe a strong improvement between theory and experiment for small thicknesses, particularly in the intermediate regime ($E_X$ for infinite barriers is also shown for comparison). We notice a persistent disagreement between theory and experiment in the strong confinement regime for $\epsilon_2 = 2.5$ but much smaller when considering finite barrier offsets (for $L_z = 1.74$ nm, the theory-experience gap is around 150 meV). In order to improve the model, variable masses were taken into account, a topic explained in the next section.

The theoretical exciton binding energy $E_b$ shown in Figure. 7(b), illustrates a comparable agreement compared with the binding energy obtained in the previous model (infinite barrier) and the experimental results, especially in the high confinement regime.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7: (a): CsPbBr₃ excitonic energy *E*<sub>X</sub> with different dielectric mismatches (ε<sub>2</sub> varying from 7.3 to 1) for a finite barrier model with fixed masses. The symbols are experimental data as mentioned previously for exciton energy in Figure. 5(a). (b): CsPbBr₃ binding energy *E*<sub>b</sub> with different dielectric mismatches (ε<sub>2</sub> varying from 7.3 to 1) for a finite barrier model with fixed masses. The symbols are experimental data as mentioned previously for exciton energy in Figure. 5(b).</figcaption></figure>
### 3.3 Confinement by Finite Barrier and Variable Mass

For a realistic outside dielectric constant $ε_2 = 2.5$, the finite barrier model has improved our simulations down to 4 MLs. However, to more accurately address the impact of confinement and reduce the disparities observed for very narrow NPLs, we will incorporate effective masses that vary with confinement, as outlined in the section 2.6.

<figure><img src="image_10.png" /><figcaption>Figure 8: (a): CsPbBr₃ excitonic energy $E_X$ with different dielectric mismatches ($ε_2$ varying from 7.3 to 1) for a finite barrier model with variable masses. The symbols are experimental data as mentioned previously for exciton energy in Figure 5(a). (b): CsPbBr₃ binding energy $E_b$ with different dielectric mismatches ($ε_2$ varying from 7.3 to 1) for a finite barrier model with variable masses. The symbols are experimental data as mentioned previously for exciton energy in Figure 5(b).</figcaption></figure>
In the finite barrier model with variable masses, the computed exciton energy is given in Figure. 8(a) relative to the NPL thickness, considering the presence of a dielectric mismatch. For comparison, the fixed masses calculation is shown (red dashed lines); a strong improvement is observed while taking into account the variable masses in the *k.p* model. This model yields a favorable agreement with various experimental results, particularly for $\epsilon_2 = 2.5$, as illustrated in Figure. 8(a).

The ML thickness was fixed at $a_0 = 0.58$ nm, which is the c-axis lattice parameter measured by X-ray diffraction (XRD) on CsPbBr$_3$ bulk at low temperature.⁶⁵ Recent studies on thin CsPbBr$_3$ nanoplatelets (NPLs) with $L_z = 2$ MLs at room temperature have shown an increase in both in-plane and out-of-plane lattice parameters compared to bulk materials. This study explores the potential effects on energy and data analysis. We have then explored the effect of the unit cell expansion on the exciton energy for the low-dimensional system as NPLs.

The first effect is an underestimation of the c-axis lattice parameter and of the experimental thickness $L_z = na_0$ used to plot the spectroscopic data. For very thin NPLs, the out-of-plane axis (c-axis) expands by +0.75%,⁶⁶ with a smaller change expected for thicker NPLs. In absence of data on the thickness dependence of the out-of-plane lattice parameter, a constant $a_0$ value was assumed. This approximation results in a negligible uncertainty in the real NPL thickness, particularly when experimental data are plotted (less than 0.04 nm for $L_z = 5$ nm) and compared to theoretical models.

Additionally, unit cell expansion affects the conduction and valence band edges via hydrostatic pressure, altering the band gap energy by $\delta E_g = (a_c - a_v)(\varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz})$, where $\varepsilon_{ij}$ is the strain tensor (i,j=x,y,z), and $a_c$ and $a_v$ are the conduction and valence deformation potentials for CsPbBr$_3$. The deformation potentials are estimated as $a_c = -0.393$ eV and $a_v = -3.760$ eV.⁶⁷ From volume variation,⁶⁶ $\varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz} = 1.20\%$, resulting in a gap increase $\delta E_g = 40$ meV, which adds to the exciton energy for $L_z = 2$ ML. This $\delta E_g$ decreases
as the thickness increases, potentially explaining the small discrepancies between experimen-
tal results and theoretical predictions for small thicknesses.

The exciton binding energy $E_b$, given in Figure. 8(b) for $\epsilon_2$ = 2.5, shows similar results when compared to the binding energy derived from the previous models (finite barrier and fixed mass). The variation in effective exciton mass, unlike the impact on exciton energy, exhibits minimal influence on $E_b$. The dielectric contrast is the main factor to fix the exciton binding energy. For reasonable dielectric constants and small thicknesses, in agreement with NPL synthesis, $E_b$ can be increased by a factor 3 compared to NPLs without dielectric effect (as shown on Figure. 8(b)) and by a factor 9 with respect to the $CsPbBr_3$ bulk exciton binding energy. It is noteworthy that the finite-barrier model successfully replicated the binding energy for very thin NPLs, demonstrating a remarkable agreement with the results obtained by Wang et al.,¹⁶ with an accuracy within a 15% margin. It also reproduced the experimental findings of Bohn et al.¹⁵ for NPL thicknesses of 2 and 3 MLs. However, a discrepancy is present in the intermediate confinement; a reduction of the dielectric contrast, in case of NPL stacking, can not be excluded (the effective dielectric constant $\epsilon_2$ being increased by the proximity of other NPLs, thus reducing the exciton binding energy).

Furthermore, our modelization gives access to the carrier penetration depth in the organic
barrier. Recent DFT calculations in NC solids (made of ordered NCs or NPLs separated
by organic ligands) have studied the electron transport through organic barriers. For reg-
ular ligands, Khabibullin et al.³⁰ have predicted typical penetration length in the 1 to 1.5
Å, in agreement with experimental measurements⁶⁸.⁶⁹ From our calculation, one can derive
the electron and hole penetration lengths defined as $\ell_l = \kappa_l^{-1}(p = e, h)$. As shown in the
Supporting Information section 3, $\ell_l$ depends very little on the NPL thickness. Such quasi-
constant behavior has also been predicted by DFT, with a small change in $\ell_l$ when $L_z$ varies
from 2 to 4 MLs.³⁰ Finally, the values we obtain with a **k.p** theory, are typically 1.2 and
1.9 Å, for electron and hole respectively, very similar to DFT theory, making our approach
pertinent to study excitons in NPL but also carriers in organic barriers.
It is important to note that most of the parameters are deduced independently from experiments, either directly ($E_g, \epsilon_1, \epsilon_2$, exciton mass) or associated to well-established **k.p** models ($E_{P_{S,Z}}, E_{P_{S,\rho}}$, electron and hole masses).

## 3.4 CsPbI$_3$ Nanoplatelets

To validate our model, we have also considered CsPbI$_3$ NPLs. The dielectric and **k.p** parameters are given in Table 2 and are mainly deduced from experiments. We are presenting on Figure. 9, the exciton energy and the binding energy for CsPbI$_3$ NPLs, in the framework of finite barrier offsets and variable masses, for $\epsilon_2 = 2.5$, assuming the same environment as for CsPbBr$_3$ NPLs. The conduction and valence band offsets are fixed at $V_0^e = 3.86$ eV and $V_0^h = 1.2$ eV$^{61,70,71}$.

The theoretical simulations are compared with experimental data, obtained at low temperature$^{18,19}$ or room temperature.$^{17}$ The studies on CsPbI$_3$ NPLs being less extended than on CsPbBr$_3$ NPLs, we have included exciton emission energies obtained by photoluminescence measurements.$^{17}$
<figure><img src="image_11.png" /><figcaption>Figure 9: (a): CsPbI$_3$ excitonic energy $E_X$ for $\epsilon_2$=2.5 and a finite barrier model with variable masses. The symbols are experimental data: Black: $^{18}$Blue: $^{17}$Green: $^{19}$ (b): CsPbI$_3$ binding energy $E_b$ for $\epsilon_2$=2.5 and a finite barrier model with variable masses.</figcaption></figure>
On Figure. 9(a), we show our result for the exciton energy versus NPL thickness. We obtain again a very good agreement between simulations and experiments. The dispersion in the data is due to a Stoke Shift between photoluminescence¹⁷ and absorption measurements¹⁸.¹⁹ On Figure 9(b), the binding energy $E_b$ is given and can be compared with the result obtained for very thin NPLs and measured by absorption. The strong increase of $E_b$, compared to the bulk value is again a signature of the importance of the dielectric effects, coupled to quantum confinement. Once again, a very good agreement is observed in the strong coupling regime, confirming the relevance of our model for quasi-bidimensional systems.

<table><caption>Table 2: Bulk and k.p model parameters of CsPbI₃ in tetragonal phase.</caption><thead><tr><th>Basic parameters</th><th>Bulk</th><th>4-band</th><th>8-band</th></tr></thead><tbody><tr><td>E<sub>g</sub>(eV)</td><td>1.738<sup>a</sup>(2K)<br>1.818<sup>a</sup>(270K)</td><td>-</td><td>-</td></tr><tr><td>E<sub>b</sub>(meV)</td><td>15<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>ε<sub>1</sub></td><td>10<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>a<sub>B</sub>(nm)</td><td>4.6<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>μ(m<sub>0</sub>)</td><td>0.114<sup>a</sup></td><td>-</td><td>-</td></tr><tr><td>θ(°)</td><td>38.3<sup>c</sup></td><td>-</td><td>-</td></tr><tr><td>Δ<sub>SO</sub>(eV)</td><td>1.43<sup>b</sup></td><td>-</td><td>-</td></tr><tr><td>T<sub>0</sub>(eV)</td><td>0.155<sup>b</sup></td><td>-</td><td>-</td></tr><tr><td>E<sub>2</sub>(eV)</td><td>3.126</td><td>-</td><td>-</td></tr><tr><td>E<sub>3</sub>(eV)</td><td>3.220</td><td>-</td><td>-</td></tr><tr><td>E<sub>P<sub>S,Z</sub></sub>(eV)</td><td>17.03<sup>b</sup></td><td>20.42<sup>c</sup></td><td>14.54<sup>c</sup></td></tr><tr><td>E<sub>P<sub>S,p</sub></sub>(eV)</td><td>19.81<sup>b</sup></td><td>27.8<sup>c</sup></td><td>16.99<sup>c</sup></td></tr></tbody></table>

a51, b26 c This work.

# 4 Conclusion

In conclusion, we have developed a model describing the excitonic state within inorganic lead halide NPLs. In an effective mass approach, we incorporated considerations for quantum confinement, finite barrier potential, Coulomb interaction and dielectric effects. Employing this model, we investigated the thickness-dependent behavior of excitonic energy and bind-
ing energy. Strain effect has also been discussed.

The results underscore the significance of finite barriers, characterized by the band offset between the NPL and the surrounding ligands. For thicknesses smaller than or comparable to the Bohr radius, a good agreement, between theory and experiments requires the presence of finite barriers and a variable effective mass. This last consideration is enabled via k.p theory and precise parameters from bulk materials.

In this framework, we successfully simulated the excitonic energy for CsPbBr₃ and CsPbI₃ NPLs, demonstrating excellent agreement with experimental data. Additionally, a robust modeling of the exciton binding energy was conducted, showing the importance of the dielectric effects to strongly enhance the electron-hole interaction. This result evidences the influence of the surrounding environment on NPL emission.

Although other materials and NPL systems must still need to be tested, simulated, and reproduced, our model paves the way for accurately simulating and predicting NPL emission wavelength of NPLs, taking into account size, composition, environment, and key interactions such as confinement, dielectric effects, and Coulomb interactions.

## ACKNOWLEDGEMENTS

This work was financially supported by Tunisian Ministry of Higher Education and Scientific Research, the French Ministry of Foreign Affairs through the project PHC Utique (CMCU No.22G1305), and the French National Research Agency (ANR IPER-Nano2, ANR-18-CE30-0023).

## Supporting Information Available

The Supplementary Information (SI) includes:

* Bloch Functions for the Conduction Band.

* Coulomb Interactions and Self-Energy Potentials.
• Carrier Penetration Lengths in the Organic Barrier.

Consult the SI for an additional data and extended discussions that complement the findings
of this study.

References

(1) Li, H.; Luo, T.; Zhang, S.; Sun, Z.; He, X.; Zhang, W.; Chang, H. "Two-dimensional metal-halide perovskite-based optoelectronics: synthesis, structure, properties and applications". *Energy Environ. Mater.* **2021**, *4*, 46–64.

(2) Gan, X.; Wang, O.; Liu, K.; Du, X.; Guo, L.; Liu, H. "2D homologous organic-inorganic hybrids as light-absorbers for planer and nanorod-based perovskite solar cells". *Sol. Energy Mater. Sol. Cells* **2017**, *162*, 93–102.

(3) Cui, J.; Liu, Y.; Deng, Y.; Lin, C.; Fang, Z.; Xiang, C.; Bai, P.; Du, K.; Zuo, X.; Wen, K., et al. "Efficient light-emitting diodes based on oriented perovskite nanoplatelets". *Sci. Adv.* **2021**, *7*, eabg8458.

(4) Liu, X.; Yu, D.; Cao, F.; Li, X.; Ji, J.; Chen, J.; Song, X.; Zeng, H. "Low-voltage photodetectors with high responsivity based on solution-processed micrometer-scale all-inorganic perovskite nanoplatelets". *Small* **2017**, *13*, 1700364.

(5) Achtstein, A. W.; Antanovich, A.; Prudnikau, A.; Scott, R.; Woggon, U.; Artemyev, M. "Linear absorption in CdSe nanoplates: Thickness and lateral size dependency of the intrinsic absorption". *J. Phys. Chem. C* **2015**, *119*, 20156–20161.

(6) Ithurria, S.; Tessier, M.; Mahler, B.; Lobo, R.; Dubertret, B.; Efros, A. L. "Colloidal nanoplatelets with two-dimensional electronic structure". *Nat. Mater.* **2011**, *10*, 936–941.
(7) Scott, R.; Achtstein, A. W.; Prudnikau, A.; Antanovich, A.; Christodoulou, S.; Moreels, I.; Artemyev, M.; Woggon, U. "Two photon absorption in II-VI semiconductors: the influence of dimensionality and size". *Nano Lett.* **2015**, *15*, 4985-4992.

(8) Naeem, A.; Masia, F.; Christodoulou, S.; Moreels, I.; Borri, P.; Langbein, W. "Giant exciton oscillator strength and radiatively limited dephasing in two-dimensional platelets". *Phys. Rev. B* **2015**, *91*, 121302.

(9) Palummo, M.; Bernardi, M.; Grossman, J. C. "Exciton radiative lifetimes in two-dimensional transition metal dichalcogenides". *Nano Lett.* **2015**, *15*, 2794-2800.

(10) Schué, L.; Sponza, L.; Plaud, A.; Bensalah, H.; Watanabe, K.; Taniguchi, T.; Ducastelle, F.; Loiseau, A.; Barjon, J. "Bright luminescence from indirect and strongly bound excitons in h-BN". *Phys. Rev. Lett.* **2019**, *122*, 067401.

(11) Wang, J.; Su, R.; Xing, J.; Bao, D.; Diederichs, C.; Liu, S.; Liew, T. C.; Chen, Z.; Xiong, Q. "Room temperature coherently coupled exciton-polaritons in two-dimensional organic-inorganic perovskite". *ACS Nano* **2018**, *12*, 8382-8389.

(12) Ithurria, S.; Dubernet, C. "Colloidal Nanoplatelets with Two-Dimensional Electronic Structure". *J. Am. Chem. Soc.* **2008**, *130*, 16504-16505.

(13) Gramlich, M.; Bohn, B. J.; Tong, Y.; Polavarapu, L.; Feldmann, J.; Urban, A. S. "Thickness-dependence of exciton-exciton annihilation in halide perovskite nanoplatelets". *J. Phys. Chem. Lett.* **2020**, *11*, 5361-5366.

(14) Steinmetz, V.; Ramade, J.; Legrand, L.; Barisien, T.; Bernardot, F.; Lhuillier, E.; Bernard, M.; Vabre, M.; Saïdi, I.; Ghribi, A., et al. "Anisotropic shape of CsPbBr₃ colloidal nanocrystals: from 1D to 2D confinement effects". *Nanoscale* **2020**, *12*, 18978-18986.
(15) Bohn, B. J.; Tong, Y.; Gramlich, M.; Lai, M. L.; Doblinger, M.; Wang, K.; Hoye, R. L.; Muller-Buschbaum, P.; Stranks, S. D.; Urban, A. S., et al. "Boosting tunable blue luminescence of halide perovskite nanoplatelets through postsynthetic surface trap repair". *Nano Lett.* **2018**, *18*, 5231–5238.

(16) Wang, S.; Dyksik, M.; Lampe, C.; Gramlich, M.; Maude, D. K.; Baranowski, M.; Urban, A. S.; Plochocka, P.; Surrente, A. "Thickness-dependent dark-bright exciton splitting and phonon bottleneck in CsPbBr₃-based nanoplatelets revealed via magneto-optical spectroscopy". *Nano Lett.* **2022**, *22*, 7011–7019.

(17) Tong, Y.; Bladt, E.; Aygüler, M. F.; Manzi, A.; Milowska, K. Z.; Hintermayr, V. A.; Docampo, P.; Bals, S.; Urban, A. S.; Polavarapu, L., et al. "Highly luminescent cesium lead halide perovskite nanocrystals with tunable composition and thickness by ultrasonication". *Angew. Chem. Int. Ed.* **2016**, *55*, 13887–13892.

(18) Liu, A.; Nagamine, G.; Bonato, L. G.; Almeida, D. B.; Zagonel, L. F.; Nogueira, A. F.; Padilha, L. A.; Cundiff, S. T. "Toward engineering intrinsic line widths and line broadening in perovskite nanoplatelets". *ACS Nano* **2021**, *15*, 6499–6506.

(19) Ye, J. et al. "Direct Linearly-Polarised Electroluminescence from Perovskite Nanoplatelet Superlattices". *Nat. Photon.* **2024**, *18*, 586.

(20) Klein, E.; Black, A.; Tokmak, O.; Strelow, C.; Lesyuk, R.; Klinke, C. "Micron-size two-dimensional methylammonium lead halide perovskites". *ACS Nano* **2019**, *13*, 6955–6962.

(21) Cho, J.; Choi, Y.-H.; O'Loughlin, T. E.; De Jesus, L.; Banerjee, S. "Ligand-mediated modulation of layer thicknesses of perovskite methylammonium lead bromide nanoplatelets". *Chem. Mater.* **2016**, *28*, 6909–6916.

(22) Sichert, J. A.; Tong, Y.; Mutz, N.; Vollmer, M.; Fischer, S.; Milowska, K. Z.; García Cortadella, R.; Nickel, B.; Cardenas-Daw, C.; Stolarczyk, J. K.; Urban, A. S.; Feldmann, J.
“Size-Dependent Optical Properties of Perovskite Nanoplatelets”. *Nano Lett.* **2015**, *15*, 6521–6527.

(23) Grenwood, A.; Mazzotti, S.; Norris, D.; Galli, G. “Electronic Properties of Colloidal Nanocrystals: Applications to Lead Halide Perovskite Nanoplatelets”. *J. Phys. Chem. C* **2021**, *125*, 4820–4827.

(24) Antanovich, A.; Achtstein, A.; Matsukovich, A.; Prudnikau, A.; Bhaskar, P.; Gurin, V.; Molinari, M.; Artemyev, M. “Anisotropic Optical Properties of Perovskite Nanoplatelets”. *Nanoscale* **2017**, *9*, 18042–18053.

(25) Katan, C.; Mercier, N.; Even, J. “Quantum and Dielectric Confinement Effects in Halide Perovskite Nanoplatelets”. *Chem. Rev.* **2019**, *119*, 3140–3192.

(26) 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.

(27) Rajadell, F.; Climente, J. I.; Planelles, J. “Excitons in core-only, core-shell and core-crown CdSe nanoplatelets: Interplay between in-plane electron-hole correlation, spatial confinement, and dielectric confinement”. *Phys. Rev. B* **2017**, *96*, 035307.

(28) Sapori, D.; Kepenekian, M.; Pedesseau, L.; Katan, C.; Even, J. “Quantum confinement and dielectric profiles of colloidal nanoplatelets of halide inorganic and hybrid organic–inorganic perovskites”. *Nanoscale* **2016**, *8*, 6369–6378.

(29) Kumagai, M.; Takagahara, T. “Excitonic and nonlinear-optical properties of dielectric quantum-well structures”. *Phys. Rev. B* **1989**, *40*, 12359.

(30) Khabibulli, A.; Efros, A.; Erwin, S. “Dielectric Effects on the Electronic Properties of Colloidal Quantum Dots”. *Nanoscale* **2020**, *12*, 23028–23035.
(31) Mentzel, T.; Wanger, D.; Ray, N.; Walker, B.; Strasfeld, D.; Bawendi, M.; Kastner, M. "Transport and Conductivity in Colloidal Quantum Dot Solids". *Nano Lett.* **2012**, *12*, 4404–4408.

(32) Dreyer, A.; Rempel, T.; Gottschalk, M.; Zierold, R.; Weimer, A.; Feld, A.; Schneider, G.; Weller, H.; Hütten, A. "Spin- and Stress-Depending Electrical Transport in Nanoparticle Supercrystals: Sensing Elastic Properties of Organic Tunnel Barriers via Tunneling Magnetoresistance". *Adv. Electron. Mater.* **2022**, *8*, 2200082.

(33) Quarti, C.; Giorgi, G.; Katan, C.; Even, J.; Palummo, M. "Exciton Fine Structure in Layered Halide Perovskites: The Role of Organic Spacers". *Adv. Opt. Mater.* **2024**, *17*, 2202801.

(34) Movilla, J. L.; Planelles, J.; Climente, J. I. "Excitons in metal halide perovskite nanoplatelets: an effective mass description of polaronic, dielectric and quantum confinement effects". *Nano. Adv.* **2023**, *5*, 6093–6101.

(35) Gramlich, M.; Swift, M. W.; Lampe, C.; Lyons, J. L.; Döblinger, M.; Efros, A. L.; Sercel, P. C.; Urban, A. S. "Dark and bright excitons in halide perovskite nanoplatelets". *Adv. Sci.* **2022**, *9*, 2103013.

(36) Pellegrini, G.; Mattei, G.; Mazzoldi, P. "Finite depth square well model: Applicability and limitations". *J. Appl. Phys.* **2005**, *97*.

(37) Sichert, J. A.; Hemmerling, A.; Cardenas-Daw, C.; Urban, A. S.; Feldmann, J. "Effects of Confinement on the Optical Properties of Perovskite Nanoplatelets". *APL Mater.* **2019**, *7*, 041116.

(38) Wetzel, C.; Winkler, R.; Drechsler, M.; Meyer, B.; Rössler, U.; Scriba, J.; Kotthaus, J.; Härle, V.; Scholz, F. "Electron effective mass and nonparabolicity in Ga 0.47 In 0.53 As/InP quantum wells". *Phys. Rev. B* **1996**, *53*, 1038.
(39) Warburton, R.; Michels, J.; Nicholas, R.; Harris, J.; Foxon, C. "Optically detected cyclotron resonance of GaAs quantum wells: Effective-mass measurements and offset effects". *Phys. Rev. B* **1992**, *46*, 13394.

(40) Singh, J. "Effective Mass of Charge Carriers in Amorphous Semiconductors and its Applications". *J. Non-Cryst. Solids* **2002**, *299-302*, 444-448.

(41) Even, J.; Pedesseau, L.; Jancu, J.-M.; Katan, C. "Importance of spin-orbit coupling in hybrid organic/inorganic perovskites for photovoltaic applications". *J. Phys. Chem. Lett.* **2013**, *4*, 2999-3005.

(42) Heidrich, K.; Schäfer, W.; Schreiber, M.; Söchtig, J.; Trendel, G.; Treusch, J.; Grandke, T.; Stolz, H. "Electronic structure, photoemission spectra, and vacuum-ultraviolet optical spectra of CsPbCl₃ and CsPbBr₃". *Phys. Rev. B* **1981**, *24*, 5642.

(43) Yu, Z. "Effective-mass model and magneto-optical properties in hybrid perovskites". *Sci. Rep.* **2016**, *6*, 28576.

(44) Tao, S.; Schmidt, I.; Brocks, G.; Jiang, J.; Tranca, I.; Meerholz, K.; Olthof, S. "Absolute energy level positions in tin-and lead-based halide perovskites". *Nat. Commun.* **2019**, *10*, 2560.

(45) Ghribi, A.; Ben Radhia, S.; Boujdaria, K.; Legrand, L.; Barisien, T.; Chamarro, M.; Testelin, C. "Dielectric effects, crystal field, and shape anisotropy tuning of the exciton fine structure of halide perovskite nanocrystals". *Phys. Rev. Mater.* **2022**, *6*, 106001.

(46) Guilloux, V.; Ghribi, A.; Majrab, S.; Margaillan, F.; Bernard, M.; Bernardot, F.; Legrand, L.; Lhuillier, E.; Boujdaria, K.; Chamarro, M., et al. "Exciton Fine Structure of CsPbCl₃ Nanocrystals: An Interplay of Electron-Hole Exchange Interaction, Crystal Structure, Shape Anisotropy, and Dielectric Mismatch". *ACS Nano* **2023**, *17*, 12266-12277.
(47) Polovitsyn, A.; Dang, Z.; Movilla, J. L.; Martín-García, B.; Khan, A. H.; Bertrand, G. H.; Brescia, R.; Moreels, I. "Synthesis of air-stable CdSe/ZnS core–shell nanoplatelets with tunable emission wavelength". *Chem. Mater.* **2017**, *29*, 5671–5680.

(48) Movilla, J. L.; Planelles, J.; Climente, J. I. "Dielectric confinement enables molecular coupling in stacked colloidal nanoplatelets". *J. Phys. Chem. Lett.* **2020**, *11*, 3294–3300.

(49) Lang, N.; Kohn, W. "Theory of Metal Surfaces: Induced Surface Charge and Image Potential". *Phys. Rev. B* **1973**, *7*, 3541.

(50) Salomon, A.; Cahen, D.; Lindsay, S.; Tomfohr, J.; Engelkes, V. B.; Frisbie, C. D. "Comparison of electronic transport measurements on organic molecules". *Adv. Mater.* **2003**, *15*, 1881–1890.

(51) Yang, Z.; Surrente, A.; Galkowski, K.; Miyata, A.; Portugall, O.; Sutton, R.; Haghighirad, A.; Snaith, H.; Maude, D.; Plochocka, P., et al. "Impact of the halide cage on the electronic properties of fully inorganic cesium lead halide perovskites". *ACS Energy Lett.* **2017**, *2*, 1621–1627.

(52) Ramade, J.; Andriambariarijaona, L. M.; Steinmetz, V.; Goubet, N.; Legrand, L.; Barisien, T.; Bernardot, F.; Testelin, C.; Lhuillier, E.; Bramati, A., et al. "Fine structure of excitons and electron–hole exchange energy in polymorphic $CsPbBr_3$ single nanocrystals". *Nanoscale* **2018**, *10*, 6393–6401.

(53) Ben Aich, R.; Ben Radhia, S.; Boujdaria, K.; Chamarro, M.; Testelin, C. "Multiband **k.p** model for tetragonal crystals: application to hybrid halide perovskite nanocrystals". *J. Phys. Chem. Lett.* **2020**, *11*, 808–817.

(54) Garcia-Arellano, G.; Boujdaria, K.; Chamarro, M.; Testelin, C. "Landé g factors in tetragonal halide perovskite: A multiband **k.p** model". *Phys. Rev. B* **2022**, *106*, 165201.
(55) Haynes, W. M.; Lide, D. R.; Bruno, T. J. "CRC Handbook of Chemistry and Physics". 2014; https://edisciplinas.usp.br/pluginfile.php/4557662/mod_resource/content/1/CRC%20Handbook%20of%20Chemistry%20and%20Physics%2095th%20Edition.pdf.

(56) Maryott, A. A.; Smith, E. R. "Table of Dielectric Constants of Pure Liquids". 1951; https://nvlpubs.nist.gov/nistpubs/Legacy/circ/nbscircular514.pdf.

(57) Marton, K.; Tomčo, L.; Cimbala, R.; Király, J.; Rajňák, I.; Timko, M.; Kopčanský, P.; Kolcunová, I.; Kurimský, J.; German-Sobek, M. "Magnetic fluid in ionizing electric field". *J. Electrostat.* **2013**, *71*, 467–470.

(58) Sichert, J. A.; Tong, Y.; Mutz, N.; Vollmer, M.; Fischer, S.; Milowska, K. Z.; García Cortadella, R.; Nickel, B.; Cardenas-Daw, C.; Stolarczyk, J. K., et al. "Quantum size effect in organometal halide perovskite nanoplatelets". *Nano Lett.* **2015**, *15*, 6521–6527.

(59) Ng, C. H.; Ripolles, T.; Hamada, K.; Teo, S. H.; Lim, H. N.; Bisquert, J.; Hayase, S. "Tunable open circuit voltage by engineering inorganic cesium lead bromide/iodide perovskite solar cells". *Sci. Rep.* **2018**, *8*, 2482.

(60) Zhou, S. "Rapid separation and purification of lead halide perovskite quantum dots through differential centrifugation in nonpolar solven". *RSC Adv.* **2021**, *11*, 28410.

(61) Ravi, V. K.; Markad, G. B.; Nag, A. "Band edge energies and excitonic transition probabilities of colloidal CsPbX₃ (X = Cl, Br, I) perovskite nanocrystals". *ACS Energy Lett.* **2016**, *1*, 665.

(62) Valencia, D.; Garcia-Cruz, I.; Uc, V. H.; Ramirez-Verduzco, L. F.; Amezcua-Allieri, M. A.; Aburto, J. "Unravelling the chemical reactions of fatty acids and triacylglycerides under hydrodeoxygenation conditions based on a comprehensive thermodynamic analysis". *Biomass Bioenergy* **2018**, *112*, 37.
(63) Liu, X.; Zhao, H.; Wei, L.; Ren, X.; Zhang, X.; Li, F.; Zeng, P.; Liu, M. "Ligand-modulated electron transfer rates from $CsPbBr_3$ nanocrystals to titanium dioxide". *Nanophotonics* **2021**, *10*, 1967–1975.

(64) Ituen, E. B.; Essien, E. A.; Udo, U. E.; Oluwaseyi, O. R. "Experimental and theoretical study of corrosion inhibition effect of Cucumeropsis mannii N. seed oil metallic soap of zinc on mild steel surface in sulphuric acid". *Adv. Appl. Sci. Res.* **2014**, *5*, 26.

(65) López, C. A.; Abia, C.; Alvarez-Galván, M. C.; Hong, B.-K.; Martínez-Huerta, M. V.; Serrano-Sanchez, F.; Carrascoso, F.; Castellanos-Gomez, A.; Fernandez-Díaz, M. T.; Alonso, J. A. "Crystal structure features of $CsPbBr_3$ perovskite prepared by mechanochemical synthesis". *ACS Omega* **2020**, *5*, 5931–5938.

(66) Toso, S.; Baranov, D.; Giannini, C.; Manna, L. "Structure and Surface Passivation of Ultrathin Cesium Lead Halide Nanoplatelets Revealed by Multilayer Diffraction". *ACS Nano* **2021**, *15*, 20341–20352.

(67) Wang, S.; Huang, M.; Wu, Y.-N.; Chen, S. "Absolute volume deformation potentials of inorganic ABX3 halide perovskites: the chemical trends". *Adv. Theory Simul.* **2021**, *4*, 2100060.

(68) Wuelfing, W. P.; Green, S. J.; Pietron, J. J.; Cliffel, D. E.; Murray, R. W. "Electronic conductivity of solid-state, mixed-valent, monolayer-protected Au clusters". *J. Am. Chem. Soc.* **2000**, *122*, 11465–11472.

(69) Samadi Khoshkhoo, M.; Lox, J. F.; Koitzsch, A.; Lesny, H.; Joseph, Y.; Lesnyak, V.; Eychmuller, A. "Highly conductive copper selenide nanocrystal thin films for advanced electronics". *ACS Appl. Electron. Mater.* **2019**, *1*, 1560–1569.

(70) Zhao, Q.; Hazarika, A.; Chen, X.; Harvey, S. P.; Larson, B. W.; Teeter, G. R.; Liu, J.; Song, T.; Xiao, C.; Shaw, L., et al. "High efficiency perovskite quantum dot solar cells with charge separating heterostructure". *Nat. Commun.* **2019**, *10*, 2842.
(71) Gaulding, E. A.; Hao, J.; Kang, H. S.; Miller, E. M.; Habisreutinger, S. N.; Zhao, Q.; Hazarika, A.; Sercel, P. C.; Luther, J. M.; Blackburn, J. L. "Conductivity tuning via doping with electron donating and withdrawing molecules in perovskite CsPbI3 nanocrystal films". *Adv. Mater.* **2019**, *31*, 1902250.