This paper is concerned with the solution of the nanoscale structures consisting of the with an effective mass envelope function theory, the electronic states of the quantum ring are studied. In calculations, the effects due to the different effective masses of electrons in and out the rings are included. The energy levels of the electron are calculated in the different shapes of rings, i.e., that the inner radius of rings sensitively change the electronic states. The energy levels of the electron are not sensitively dependent on the outer radius for large rings. The structures of quantum rings are studied by the one electronic band Hamiltonian effective mass approximation, the energy- and position-dependent on electron effective mass approximation, and the spin-dependent on the Ben Daniel-Duke boundary conditions. In the description of the Hamiltonian matrix elements, the Finite elements method with different base piecewise linear function is adopted. The non-linear energy confinement problem is solved approximately by using the Finite elements method with piecewise linear function, to calculate the energy of the one electron states for the quantum ring. The results of numerical example are compared for accuracy and efficiency with the finite element method of linear triangular element. This comparison shows that good results of numerical example.
