We theoretically study the temperature dependence of the J-band width in disordered linear molecular aggregates, caused by dephasing of the exciton states due to scattering on vibrations of the host matrix. In particular, we consider inelastic one- and two-phonon scattering between different exciton states (energy-relaxation-induced dephasing), as well as elastic two-phonon scattering of the excitons (pure dephasing). The exciton states follow from numerical diagonalization of a Frenkel Hamiltonian with diagonal disorder; the scattering rates between them are obtained using the Fermi Golden Rule. A Debye-like model for the one- and two-phonon spectral densities is used in the calculations. We find that, owing to the disorder, the dephasing rates of the individual exciton states are distributed over a wide range of values. We also demonstrate that the dominant channel of two-phonon scattering is not the elastic one, as is often tacitly assumed, but rather comes from a similar two-phonon inelastic scattering process. In order to study the temperature dependence of the J-band width, we simulate the absorption spectrum, accounting for the dephasing induced broadening of the exciton states. We find a power-law (T^p) temperature scaling of the effective homogeneous width, with an exponent p that depends on the shape of the spectral density of host vibrations. In particular, for a Debye model of vibrations, we find p ~ 4, which is in good agreement with experimental data on J-aggregates of pseudoisocyanine [J. Phys. Chem. A 101, 7977 (1997)].
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