Effect of the shape on the spin state and exchange in quantum dots. Feynman path integral analysis

Abstract

The ab initio computer simulation of the mixed quantum states of 1-5 nm model ellipsoid quan tum dots with "soft" walls containing two and three quantum indistinguishable nonrelativistic electrons has been performed by the path integral method. The calculation has been carried out beyond the single electron and mean field approximations with the fundamentally exact inclusion of Coulomb and exchange correla tions of all orders and the spin variable. Distributions over the eigenfunctions of the spin squared operator, as well as the equilibrium spin numbers, have been obtained depending on the shape of a quantum dot and the temperature. The complete set of basis functions symmetrized in permutations according to the spin of the system has been obtained by application of the Young symmetry operators. The dependence of the energy on the shape of the quantum dot corresponds to the negative sign of the surface tension at its boundary. The cal culation indicates that the spin magnetic susceptibility in the system of two electrons decreases strongly for spherical quantum dots ("pairing" of spins) and the temperature dependences have a pronounced maximum whose position depends on the shape of the quantum dot. For three electrons in an oblate quantum dot, the inversion of the energy levels of spin states is observed and affects the spin magnetic susceptibility. The results indicate a strong dependence of the energy of collective spin states of electrons on the detailed inclusion of exchange and Coulomb spatial correlations.