Exact analytical solutions describing quantum dot, ring and wire wavefunctions

Abstract

For the first time, exact analytical solutions of electronic wavefunctions in quantum ring-like structures are found using a symmetry-adapted coordinate system. In the case of an infinite potential barrier, it is related to a more general problem of finding resonance modes in a ring-like cavity. The solutions in the parabolic coordinate system are Bessel or confluent hypergeometric functions for quantum dot, ring and wire including objects with flat bases. The continuous evolution from the 0D quantum dot system to the perfect 1D quantum ring system is analysed. The electronic properties (energy levels, wavefunctions and density of states) are continuously followed during the dot to ring transformation until the limit of a perfect 1D quantum ring is reached. The creation of a ring structure by changing the material composition in a dot centre leads to a sharp variation in the energy levels of totally symmetric states. This is a consequence of the immediate build-up of a potential barrier energy in the centre of the ring. Finally the use of the structure of electronic levels to identify the signature of self-organized quantum rings when compared to quantum dots is proposed.