Calculation of two-photon absorption by nanocrystals of CsPbBr3

We present calculations of the two-photon absorption cross section ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})$ for nanocrystals (NCs) of the inorganic perovskite ${\mathrm{CsPbBr}}₃$ for photon energies $\ensuremath{\omega}$ ranging from the absorption threshold at $2\ensuremath{\omega}\ensuremath{\approx}2.3\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ up to $2\ensuremath{\omega}=3.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$. The calculations employ a $4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ envelope-function model, with final-state excitons described in a self-consistent Hartree-Fock approximation. The $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ corrections to ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})$ are found to be rather large, giving a reduction of about 30% in the cross section at the largest energies considered. The cross section is shown to be independent of polarization in the effective-mass approximation (EMA), but including $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ corrections leads to a small difference in ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})$ between circular and linear polarization, which rises to about 16% at $2\ensuremath{\omega}=3.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$. The theoretical cross section follows closely a power-law dependence on NC size, ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})\ensuremath{\propto}{L}^{\ensuremath{\alpha}}$, with theoretical exponents $\ensuremath{\alpha}=3.4$ (EMA) or $\ensuremath{\alpha}=3.2$ ($4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{0.16em}{0ex}}\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model), in excellent agreement with experiment. The dominant contribution to the exponent $\ensuremath{\alpha}$ is shown to be the number of final-state excitons per unit energy. Measured values of the absolute (normalized) cross section ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})$ show a large spread of values, differing by as much as a factor of 25 for some NC sizes. Our calculations strongly favor a group of measurements at the lower end of the reported range of ${\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})$.