![]() Every homogeneous lineshape can be correctly called a dynamical lineshape, but dynamical lineshapes are not always homogeneous. The dynamical lineshape arises from the motions of electrons, holes, spins, ligands, and vibrations of the atomic lattice. Instead, a quantum dot sample has a static inhomogeneous distribution of bandgaps and an average single-dot lineshape for each bandgap called the dynamical lineshape ( 19). But even with narrow size distributions a few atomic layers wide ( 15– 18), colloidal nanocrystals have an astonishing variety of atomic structures ( 14), emission spectra of single quantum dots differ from each other ( 18), and there is no homogeneous spectrum that is the same for every quantum dot. For polydisperse samples, the resulting broadening of the absorption spectrum has successfully guided synthetic efforts to reduce the size dispersion. With particle-in-a-box quantum confinement, the size distribution from a synthesis generates a static bandgap distribution that is diagnostic for size dispersion. Nanocrystal applications benefit from narrow size and structure distributions ( 14). The rate constant measurements needed to determine the standard chemical potential are not easy ( 4), and the approach is not valid if spectra are broadened by static inhomogeneities ( 10), such as the conformational heterogeneity of photosynthetic proteins or the size distribution of quantum dots.Ī quantum dot is a semiconductor nanocrystal large enough to have bulk lattice properties but small enough that three-dimensional (3D) quantum confinement increases its bandgap above the bulk bandgap ( 13). In contrast, the relations have appeared to be inapplicable to molecular spectra ( 10) we suggest that distinct isomers ( 7, 12) or protonation states ( 7) in the ionic dye solutions used to test the relations may contribute to both this apparent inapplicability ( 10) and the apparent inhomogeneity of some dyes in photon echo experiments ( 12). These relations are widely applicable to impurity ions in crystals ( 11) and bulk crystalline semiconductors ( 3, 4). These also connect the rate constants for absorption and emission to the standard chemical potential that determines the maximum photovoltage. The homogeneous generalized Einstein relations connect the equilibrium Stokes’ shift to the linewidth. In condensed phases, thermalization shifts the emission spectrum to lower frequency than the absorption spectrum, this is known as Stokes’ shift. Determination uses a generalization of Einstein’s rate theory for absorption, stimulated emission, and spontaneous emission from gas-phase line spectra ( 5) to thermalized, homogeneously broadened condensed-phase spectra ( 6– 11). For homogeneous materials such as bulk crystalline semiconductors, the thermodynamic maximum photovoltage can be determined from the rate constants for photon absorption and emission as a function of frequency ( 4). Electronically excited molecules and materials can store energy ( 1, 2), generate power ( 3, 4), or emit light ( 1, 4).
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