Optical Ceiling on Polaron Mobility in Disordered Molecular Semiconductors: Lessons from Eumelanin

Polaron hopping in disordered molecular semiconductors such as eumelanin is set by energetic disorder $σ$ and reorganization energy $λ$. We show that the reduced polaron absorption band encodes this transport problem. Its value at zero energy fixes the mean hop rate, which bounds the mobility in any dimension. For Gaussian disorder and classical vibrations its centroid is $E_\mathrm{op}=λ+(1-c)σ^2/k_BT$, its variance obeys the Hush relation, and the bound is the Marcus rate at $λ\to E_\mathrm{op}$; a 1D lower bound vanishes at $k_BT^*=(1-c)σ^2/λ$, read from optics.