Simulations of Trions and Biexcitons in Layered Hybrid Organic-Inorganic Lead Halide Perovskites

Behaving like atomically precise two-dimensional quantum wells with non-negligible dielectric contrast, the layered hybrid organic-inorganic lead halide perovskites (HOIPs) have strong electronic interactions leading to tightly bound excitons with binding energies on the order of 500 meV. These strong interactions suggest the possibility of larger excitonic complexes like trions and biexcitons, which are hard to study numerically due to the complexity of the layered HOIPs. Here, we propose and parametrize a model Hamiltonian for excitonic complexes in layered HOIPs and we study the correlated eigenfunctions of trions and biexcitons using a combination of diffusion Monte Carlo and very large variational calculations with explicitly correlated Gaussian basis functions. Binding energies and spatial structures of these complexes are presented as a function of the layer thickness. The trion and biexciton of the thinnest layered HOIP have binding energies of 35 and 44 meV, respectively, whereas a single exfoliated layer is predicted to have trions and biexcitons with equal binding energies of 48 meV. We compare our findings to available experimental data and to that of other quasi-two-dimensional materials.

Figure 1

Illustration of our model of a layered HOIP, with $n=2$ [11] shown as an example. The electrons and holes move continuously in the $xy$ plane and hop discretely between inorganic sublayers in the $z$ direction. The electrostatics of the inorganic and organic layers are modeled with slabs of dielectric constant ${\epsilon }_1$ and ${\epsilon }_2$, respectively.

Figure 2

Exciton ($X$) binding energy of layered HOIPs obtained by DMC and SVM compared with previously reported atomistic tight-binding GW Bethe-Salpeter equation (BSE) calculations [40] (a). Experimentally measured values are from Ref. [14]. Trion ($X^{-}$) and biexciton ($XX$) binding energy calculated via DMC, SVM, and the three-parameter wave function (b). The error bar for the $n=1$ biexciton shows the range of experimentally measured binding energies [19, 21, 23]. Solid lines indicate $n\to \infty$ bulk binding energies inferred from accurate 3D calculations [63, 64]. Dashed lines show the scaling law for each binding energy as described in the text.

Figure 3

Effective radii of the exciton and trion as a function of number of sublayers (a). The black dashed line indicates the total thickness of the quantum well, $n\times d$. Radial probability distributions of electron-hole (solid line) and electron-electron (dashed line) separations of the exciton, trion, and biexciton for $n=1$ (b) and of the trion for $n=1–5$ (c), as calculated by the SVM.

Figure 4

Total electron probability distribution of the trion for (a) $n=3$ and $n=5$ with the hole fixed in the center sublayer, and (b) $n=5$ with the hole fixed at the center and the edge sublayers.