Nonparabolic dispersion of charge carriers in CsPbI$_3$ in the orthorhombic phase

Abstract

The dispersion curves for the electrons and holes in CsPbI$_3$ in the orthorhombic phase are calculated using the density functional theory (DFT), with the spin-orbit coupling taken into account. The effective masses of the charge carriers are obtained using the parabolic approximation of the dispersion curves in different directions in the $k$-space. It is found that the dispersion curves demonstrate strong nonparabolicity at energies above 0.2 eV for electrons and above 0.1 eV for holes, available for experimental study by the means of optical spectroscopy. We propose a model that describes the dispersion dependences of charge carriers at those energies, where the effective masses of the quasiparticles depend quadratically on the wave vector. An expression is obtained according to the model, which can accurately approximate the dispersion curves for the electron and the hole in all symmetric directions from the center to the boundary of the Brillouin zone.

Figures (8)

• Figure 1: (a) The unit cell of the orthorhombic CsPbI$_3$ crystal lattice jain_commentary_2013horton_accelerated_2025. The image of the unit cell is taken from the Materials Project for CsPbI$_3$ (mp-1120768) from database version v2025.09.25. The lattice constants obtained from the DFT calculation are $a_x = 0.907$ nm, $a_y = 1.254$ nm, and $a_z = 0.801$ nm. The values of the constants agree well with those obtained via X-ray diffraction in Sutton2018_CsPbI3_effective_mass. (b) The first BZ with the primitive reciprocal lattice vectors, $b_{\alpha} = 1/a_{\alpha}$, where $\alpha=x,y,z$. The labels $\Gamma$, $\mathrm{X}$, $\mathrm{Y}$, etc. show the high-symmetry points used in the DFT calculations.
• Figure 2: The dispersion dependences for the valence and the conduction bands in $\gamma$-CsPbI$_3$ is calculated via the DFT method, with the spin-orbit coupling taken into account.
• Figure 3: The dispersion curves in the $\Gamma$-X, $\Gamma$-Z, and $\Gamma$-U directions for electrons (a) and holes (b). The dispersion curves for the electrons (c) and the holes (d) in the $\Gamma$-X direction are provided as an example. The points show the dispersion curves obtained from the DFT calculation; the dashed line shows the effective mass approximation (see Eq. (\ref{['eq:kp']})).
• Figure 4: Examples of the dispersion curves for electrons (a,b) and holes (c,d) in directions $\Gamma$-X (a,c) and $\Gamma$-U (b,d) in the Brillouin zone, fitted using the nonparabolic model (see Eqs. (\ref{['eq:Non-Parabolic']}) and (\ref{['eq:monk']})). The dots represent the DFT data, and the solid curves correspond to the least-squares fit of the data to the model.
• Figure 5: The cross-sections for energy isosurfaces for electron (top) and hole (bottom) in the $\Gamma$XZ (a,c) and the $\Gamma$XY (b,d) planes. Symbols are the DFT calculations. Solid lines are the modeling using Eqs. (\ref{['eq:Non-Parabolic']}), (\ref{['eq:monk']}), and values of parameters listed in Tab. \ref{['tab:coefficients']}. The cross-section energies are $\Delta E_{e,1}=0.03$ eV; $\Delta E_{e,2}=0.10$ eV; $\Delta E_{e,3}=0.31$ eV; $\Delta E_{e,4}=0.43$ eV; $\Delta E_{e,5}=0.53$ eV for electrons, and $\Delta E_{h,1}= -0.04$ eV; $\Delta E_{h,2}= -0.10$ eV; $\Delta E_{h,3}= -0.25$ eV; $\Delta E_{h,4}= -0.35$ eV; $\Delta E_{h,5}= -0.45$ eV for holes.