New variational solution for the lowest subband level of the two-dimensional electron gas

An analytical formula for the first-subband energy level of the two-dimensional electron gas was derived using the variational method. Two different trial envelope wave functions were used: an Airy function with one variational parameter and an exponential-power function with two variational parameters. Solutions were obtained in the spherical effective-mass approximation with the envelope wave function assumed to vanish at the surface (or interface). In both cases we obtained quantitative agreement with the numerical self-consistent results within accuracies of 1% to 2% not only for the values of the first-subband energy but also for the average characteristics of the wave function. The accuracy of these solutions is several times better than the well-known variational solution proposed by F. F. Fang and W. E. Howard [Phys. Rev. Lett. 16, 797 (1966)]. Exchange-correlation energy is calculated in the local-density-functional approximation. Analytical form of this energy is obtained to first order of the perturbation theory using exponential-power variational wave function.