CONFINAMIENTO NANOSC´OPICO EN ESTRUCTURAS ... - It works!

Confinamiento dieléctrico 203148 J.L. Movilla, J. Planelles / Computer Physics Communications 170 (2005) 144–152⎧⎨ (l+1)(ǫ m −ǫ m+1 )+˜p m+1,l ((l+1)ǫ m +lǫ m+1 )( RmR ) 2l+1m+1˜p ml =if m4000, far beyond the achievementof convergence.Since the off-centered impurity system has axial symmetry, we use cylindrical coordinates in our calculations sothat we can obtain a differential equation only dependent on two electron coordinates (ρ and z). This equation hasbeen solved numerically using the finite-difference method on the two-dimensional grid (ρ,z). The kinetic energyoperator in Eq. (1) reads (in cylindrical coordinates),ˆT =− 1 ( )∂ ρ ∂(23)2ρ ∂ρ m ∗ − 1 ( )∂ 1 ∂(ρ,z) ∂ρ 2 ∂z m ∗ ,(ρ,z) ∂zwhere m ∗ (ρ,z) is the step-like variable effective mass. In order to avoid the source of inaccuracy arising from theδ-function nature of ∂m∗ ∂m∗∂ρand∂z, we discretize Eq. (23) by applying first central finite differences to the derivativesoutside the brackets and, in a second step, to the derivatives inside them. The resulting scheme is robust to largechanges in the effective mass across the interfaces [26].Summing up, the discretization of Eq. (1) yields eigenvalue problems of asymmetric, huge and sparse matrices(about 40000 × 40000) that have been solved by employing the iterative Arnoldi factorizations [27] implementedin the ARPACK package [28].(21)3. Illustrative calculations: dielectric effects and impurity off-centeringThe problem of an off-centered hydrogenic impurity in a confined spherical geometry has been widely treated inthe literature. However, as stated in the introduction, to a large extent, these studies rely on variational calculationswhich disregard the effects of the image charge induced at the dot boundary. This is a good approach if we dealwith systems where the materials involved have similar dielectric response, but breaks down for materials with alarge dielectric mismatch, as occurs when the system is immersed in air or a vacuum (ǫ o = 1) [29].Most calculations including these polarization effects when off-centered impurities are involved have been carriedout within the framework of perturbation theory. However, these calculations only incorporate first orderperturbation estimations of the coulombic terms. Therefore, the polarization correcting terms due to off-centering(φ C and φ E ) are not taken into account when, as usual, s-like unperturbed states are employed in the estimations(see Ref. [2] for more details).The procedure described in the previous section takes fully into account the coulombic polarization contributionsto the electronic energy. This tool will provide, then, a reliable understanding of the dielectric mismatch effects asa function of the impurity location inside the quantum dot.In order to broach this issue we proceed first to artificially isolate the dielectric confinement influence on theelectronic ground state of a QD with a donor impurity. To this end, we disregard, for the time being, spatial andmass confinements. We consider, then, a doped 3 nm radius, 0.2 effective mass QD in air or a vacuum (ǫ o = 1),and apply our algorithm to calculate the electronic ground state energy vs. ǫ i for different off-centering values z I