Dielectric effects, crystal field, and shape anisotropy tuning of the exciton fine structure of halide perovskite nanocrystals

We present a theoretical study of the excitonic band edge states applied to the more commonly studied inorganic perovskite nanocrystals: ${\mathrm{CsPbBr}}_3$. We highlight the key role played by the electron-hole exchange interaction, including long-range and short range terms, in the positioning and splitting of dark and bright exciton sublevels, and discuss the influence of dielectric confinement, crystal field and shape anisotropy effects on the excitonic fine structure. The electron-hole exchange interaction splits the four excitonic states into three bright excitonic states at higher energy and a singlet dark state at lower energy. We find that, whatever the crystal phase, the bright-dark splitting is weakly sensitive to shape anisotropy. However, a strong enhancement, of about 50%, is obtained at the maximum of the dielectric contrast. For two crystalline symmetries, we particularly analyze how the bright exciton triplet energies vary, taking into account the states polarizations and considering the directions along which shape elongation/contraction applies. The main effect of the dielectric contrast is to increase significantly the exciton energies, for all the configurations. For a cubic crystal—${\mathrm{O}}_h$ symmetry—the bright exciton triplet degeneracy can be partially or fully lifted with anisotropic shape along one or two directions, respectively. For a tetragonal crystal—${\mathrm{D}}_{4h}$ symmetry—the partial triplet degeneracy can be completely lifted by a single distortion from the cubic shape along a direction perpendicular to the tetragonal axis but a partial degeneracy is maintained for small distortions along the tetragonal axis. The amplitude of bright exciton splittings are basically driven by the shape anisotropy and is almost insensitive to the dielectric contrast. On the basis of this model, we discuss recent results on ${\mathrm{CsPbBr}}_3$ nanocrystals photoluminescence.

Figure 1

Exciton fine structures (EFSs) for nanocrystals in absence of shape anisotropy are shown schematically for the cases with cubic (${\mathrm{O}}_h$ as point group), tetragonal (${\mathrm{D}}_{4h}$ as point group), and orthorhombic crystal symmetry (${\mathrm{D}}_{2h}$ as point group). The three bright states are fully degenerate in cubic symmetry.

Figure 2

Case of isotropic NC: effect of dielectric contrast on the corrective factors, ${\mathcal{C}}_{\mathrm{sph}}$ and ${\mathcal{C}}_{\mathrm{cub}},$ of the long-range exchange interaction due to dielectric contrast for an isotropic NC. ${\mathcal{C}}_{\mathrm{sph}}$ [Eq. (13)] is for a spherical NC, and ${\mathcal{C}}_{\mathrm{cub}}$, is calculated within the image charges formalism when considering a cuboid shaped NC. The factors are plotted as a function of $\eta =({\epsilon }_1-{\epsilon }_2)/({\epsilon }_1+{\epsilon }_2).$

Figure 3

(a) Dependence of the exchange integrals with the anisotropy parameter $r$ while $s=1$. For $r>1 (r<1),$ the NC size along one direction is elongated (reduced). (b) Exchange integrals as a function of the anisotropy parameter $r$ while $rs=1$. Different dielectric contrasts are considered (blue curves, outside dielectric constant ${\epsilon }_2=7.3$; green, ${\epsilon }_2=2$; red, ${\epsilon }_2=1$).

Figure 4

Effect of the shape anisotropy and dielectric contrast on the normalization factor, $\mathcal{N}\left(a\right)$, of the trial function Eq. (5). Different outside dielectric constants are considered (blue curves for ${\epsilon }_2=7.3$; green for ${\epsilon }_2=2$; red for ${\epsilon }_2=1$). (a) $\mathcal{N}\left(a\right)$ as a function of $r$ when elongating or reducing the NC size along a single direction. (b) $\mathcal{N}\left(a\right)$ vs $r$ when assuming a shape distortion along two directions. The NC volume $\mathit{V}={\mathit{L}}^3$ is kept constant ($\mathit{L}=\mathit{8}$ nm).

Figure 5

Exciton fine structure of NCs with crystal phase ${\mathrm{O}}_h$. Different dielectric contrasts are considered (blue color for ${\epsilon }_2=7.3$; green for ${\epsilon }_2=2$; red for ${\epsilon }_2=1)$. (a) Bright states energies, (b) bright-dark splitting, ${\delta }_{\mathrm{BD}}$, and (c) bright-bright splitting, $\mathrm{\Delta }E,$ when elongating or reducing the NC size along a single direction. (d) Bright states energies, (e) bright-dark splitting ${\delta }_{\mathrm{BD}}$, and (f) bright-bright splitting $\mathrm{\Delta }E$, calculated when assuming a shape distortion along two directions.

Figure 6

Exciton fine structure of NCs with crystal phase ${\mathrm{D}}_{4h}$. Different outside dielectric constants are considered (blue curves for ${\epsilon }_2=7.3$; green for ${\epsilon }_2=2$; and red for ${\epsilon }_2=1$). (a) Bright states energies, (b) bright-dark splitting ${\delta }_{\mathrm{BD}}$, and (c) bright-bright splitting $\mathrm{\Delta }E,$ when elongating or reducing the NC size along the $Oz$ direction (or c axis).

Figure 7

Exciton fine structure of $D_{4h}$ crystal symmetry NCs. Different contrasts are considered (blue curves for ${\epsilon }_2=7.3$; green for ${\epsilon }_2=2$; red for ${\epsilon }_2=1$). (a) Bright states energies and (b) bright-dark splitting ${\delta }_{\mathrm{BD}}$, when elongating or reducing the NC size along $Oy$ direction. (c) Bright states energies and (d) bright-dark splitting, ${\delta }_{\mathrm{BD}}$, when assuming a shape distortion along two directions in the plane orthogonal to $O_x$.