Importance of nonlinear long-range electron-phonon interaction on the carrier mobility of anharmonic halide perovskites

Abstract

The interaction between the electrons and the lattice vibrations in a solid is responsible for various important effects, such as formation of polarons, temperature dependent bandgaps, phonon-limited carrier transport, and conventional superconductivity. Most works assume a linear electron-phonon interaction, where the electron only interacts with one phonon at a time. However, the validity of this assumption has not been verified in polar anharmonic materials, where large ionic displacements may invalidate the assumption of linear interaction. Here, we show that nonlinear electron-phonon interactions contribute significantly to the finite-temperature electron mobility of the inorganic lead halide perovskite CsPbI$_3$. We calculate the electron mobility from first principles using the self-energy relaxation time approximation and the long-range approximation. The effect of nonlinear interaction is taken into account using the recently derived expression for the long-range part of the one-electron-two-phonon matrix element. We show that due to the low phonon frequencies of CsPbI$_3$, the one-electron-two-phonon interaction changes the temperature scaling of the mobility and contributes about 10\% to the mobility at room temperature. The results underscore the importance of including nonlinear electron-phonon interaction in anharmonic halide perovskites.

Figures (4)

• Figure 1: (a)-(b) Feynman diagrams for the linear and lowest-order nonlinear electron-phonon interaction. In (a), the long-range approximation amounts to $\mathbf{q}\rightarrow \mathbf{0}$, and in (b) it amounts to $-\mathbf{q}_1 \rightarrow \mathbf{q}_2$. (c)-(d) Contributions to the imaginary part of the retarded self energy that are taken into account in this article, where (c) is the Fan-Migdal diagram, and (d) is its nonlinear analog. We do not include the Debye-Waller diagram since it is real.
• Figure 2: First principles results for the nonlinear electron-phonon properties of CsPbI$_3$ in the cubic phase. (a) High-symmetry Brillouin zone path for a simple cubic material, with an electric field breaking the symmetry in the $z$-direction. (b) Phonon frequencies $\omega_{\mathbf{q}\nu}$ of cubic CsPbI$_3$ at zero temperature. The overlaid circles have areas proportional to $\sum_{\nu'}|Y_{\nu\nu'z}(\mathbf{q})|^2$, i.e. the magnitude of the long-range interaction strengths $|Y_{\nu\nu'z}(\mathbf{q})|^2$ between the phonon $(\mathbf{q}\nu)$ and all other phonons of the form $(-\mathbf{q}\nu')$. There are imaginary phonon modes because the cubic phase is unstable at zero temperature: we set $Y_{\nu\nu',z}(\mathbf{q}) = 0$ for any branches with $\omega_{\mathbf{q},\nu} < 0.1~\text{THz}$. (c) One-electron-two-phonon spectral function $\mathcal{T}(\omega)$ calculated from Eq. (\ref{['TomegaDef']}) and the data from (b). $\mathcal{R}(\omega)$ is shown as a sequence of vertical lines representing the Dirac delta functions, with the areas under the delta functions indicated for reference. The effect of the one-electron-two-phonon interaction is significantly enhanced at finite temperatures: this is ultimately due to the very low phonon frequencies of CsPbI$_3$ (see main text).
• Figure 3: Inverse electron lifetimes in cubic CsPbI$_3$, close to the conduction band minimum at the R-point, calculated using Eqs. (\ref{['tauDef']})-(\ref{['SelfEnergy']}) and the first-principles data of Fig. \ref{['fig:Inputs']}. Full lines are calculated using only the linear electron-phonon interaction from Eq. (\ref{['RomegaDef']}), while dashed lines also include the nonlinear interaction from Eq. (\ref{['TomegaDef']}). The nonlinear interaction is negligible at low temperatures but contributes more at room temperature.
• Figure 4: SERTA mobility of CsPbI3 as a function of temperature, calculated using Eq. (\ref{['muSERTA']}) and the electron lifetimes of Fig. \ref{['fig:Lifetimes']}. The solid blue line is calculated using only the linear electron-phonon interaction from Eq. (\ref{['RomegaDef']}), and the solid red line is calculated including the one-electron-two-phonon interaction from Eq. (\ref{['TomegaDef']}). Both are calculated using a frequency cutoff of $0.1~\text{THz}$; the red shaded area shows the result when this cutoff is varied between $0.05~\text{THz}$ and $0.2~\text{THz}$, indicating that the nonlinear contribution remains significant regardless of this cutoff value. Dashed lines indicate power law fits in the region between $200-500~\text{K}$. The inset shows the relative change of the mobility due to the one-electron-two-phonon interaction, relative to the mobility when nonlinear interaction is neglected.