General boundary conditions for envelope wave functions at semiconductor nanocrystals surface

Theoretical calculations of electron energy spectrum and wave functions in spherical semiconductor nanocrystals (NC) surrounded by a dielectric media are presented. The case of high, but finite potential barrier at the NC surface, i. e. at the boundary between semiconductor and dielectric, is considered with account taken for a large difference between electron effective mass inside and outside of NC. We argue that within effective mass method such NC surface can be described as impenetrable for electron with nonvanishing envelope wave functions at the boundary. General boundary conditions that provide a consistent description of quantum size energy levels of localized electron states are suggested and the conditions of their applicability are determined. General boundary conditions are characterized by a single surface parameter that depends only on the height $U$ of the potential barrier and electron effective mass $m_B$ outside NC. We show that the energies of electron levels decrease while the probability of finding electron at the NC surface increases with increasing $m_B$. The analytical asymptotic expressions for the dependence of the electron ground state energy on $U$ and $m_B$ are obtained.