Calculation of two-photon absorption by nanocrystals of CsPbBr 3

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 ${\\ensuremath{\\sigma}}^{(2)}(\\ensuremath{\\omega})$.