Size-Dependent Band-Gap and Molar Absorption Coefficients of Colloidal CuInS₂ Quantum Dots

The knowledge of the quantum dot (QD) concentration in a colloidal suspension and the quantitative understanding of the size-dependence of the band gap of QDs are of crucial importance from both applied and fundamental viewpoints. In this work, we investigate the size-dependence of the optical properties of nearly spherical wurtzite (wz) CuInS₂ (CIS) QDs in the 2.7 to 6.1 nm diameter range (polydispersity ≤10%). The QDs are synthesized by partial Cu⁺ for In³⁺ cation exchange in template Cu_{2- x}S nanocrystals, which yields CIS QDs with very small composition variations (In/Cu = 0.91 ± 0.11), regardless of their sizes. These well-defined QDs are used to investigate the size-dependence of the band gap of wz CIS QDs. A sizing curve is also constructed for chalcopyrite CIS QDs by collecting and reanalyzing literature data. We observe that both sizing curves follow primarily a 1/ d dependence. Moreover, the molar absorption coefficients and the absorption cross-section per CIS formula unit, both at 3.1 eV and at the band gap, are analyzed. The results demonstrate that the molar absorption coefficients of CIS QDs follow a power law at the first exciton transition energy (ε _{E₁} = 5208 d^2.45) and scale with the QD volume at 3.1 eV. This latter observation implies that the absorption cross-section per unit cell at 3.1 eV is size-independent and therefore can be estimated from bulk optical constants. These results also demonstrate that the molar absorption coefficients at 3.1 eV are more reliable for analytical purposes, since they are less sensitive to size and shape dispersion.