A new type of quantum structure is discussed where the probability distributions of the charge carriers are concentrated on the shell of a cone. These GaAs cone‐shell quantum structures (CSQSs) are filled into nanoholes in AlGaAs that are fabricated in a self‐assembled fashion using local droplet etching during molecular beam epitaxy. The structural properties of the CSQSs are studied with atomic force microscopy (AFM) and the optical emission with single‐dot photoluminescence (PL). Numerical simulations of the influence of a vertical electric field predict a strong field‐dependent displacement of either the electron or the hole away from the tip of the cone shell. This displacement has several consequences. First, the Coulomb interaction is strongly reduced. Accordingly, simulations as well as PL measurements indicate a non‐parabolic Stark‐shift for CSQSs with a regime of approximately constant emission energy. Second, the calculated exciton‐recombination lifetimes establish a variability from nanoseconds up to milliseconds. Third, regarding the shape of the electron or hole probability distributions, we predict a gate‐voltage controlled transformation from a dot into a ring shape. The respective other charge carrier remains as a dot.