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

212 PublicacionesJ. L. MOVILLA AND J. PLANELLES PHYSICAL REVIEW B 71, 075319 2005FIG. 1. a Ground state binding energy vs impurity offcenteringz I corresponding to a R=3 nm spherical SiO 2 nanocrystalin air or vacuum m o * = o =1. Calculations have been performedwithout S0, dotted line and with the inclusion of the selfpolarizationcontributions S1, full line. SiO 2 parameters are specifiedin the text. b Same as a but with V 0 =10 eV and the selfpolarizationpotential that is now calculated with i =4 and 0 =1.B. Strong confinement regime: exact calculations vsperturbational approachSmall crystallites built of materials with large dielectricconstants and light effective masses are in the strong confinementregime Ra 0 * . For calculating binding energies inthese nanocrystals, including full dielectric effects, Ferreyraet al. 17,32 have developed the so-called strong confinementapproach 55 and have carried out calculations employing twomodels of spatial confining potential, namely parabolic andinfinite hard wall, and for both on- and off-centered impurities.Their results show that the strong confinement approachyields meaningful results in all the cases studied, from whicha general trend seems to emerge: E b is a monotonous decreasingfunction of z I the same result as the one found inthe variational calculations discussed in the previous subsection.In order to check whether the strong confinement approachcan be generalized to the more realistic finite steplikespatial confining potential, we carry out the same calculationas in the previous subsection SiO 2 in vacuum but,this time, we have artificially reduced the effective mass tom * =0.05, in order to move into the strong confinement regime.The results are shown in Fig. 2a, where we can seethat if E b is calculated numerically excluding self-energyS0 it is almost insensitive to the off-centering, while this isnot the case when self-energy is included S1. This is sobecause for z I 0.7 the attractive self-polarization potentialFIG. 2. a Exact and first-order perturbational estimations ofbinding energy E b vs off-centering z I for a R=3 nm spherical QD inair or vacuum m i * =0.05, m 0 * =1, i =4, o =1 and V 0 =0.9 eV. S1full lines include while S0 dotted lines exclude self-polarizationcontributions. The zeroth-order wave functions employed in theperturbational calculations are those of the impurity-free QD in thepresence and absence of the self-polarization potential, respectively.b Same as a but with V 0 =10 eV and i =4. c Same as a butwith V 0 =10 eV and i =8.well squeezes part of the electronic density into the well,yielding a relevant increase in E b . The differences we findout between S0 and S1 and the fact that exact E b increaseswith z I lead us to suspect that the first-order perturbationapproach would not be appropriate in this case. On the onehand, the self-polarization potential s has not first-ordercontribution to E b and, on the other hand, the Coulomb cterm always predicts a reduction in E b vs z I Refs. 17 and31. This steady prediction can be explained as follows: inthe strong confinement approach E b is calculated as the oppositesign expectation value of the c potential, Eqs.3–9, in the impurity free QD ground state. The angularpart of the corresponding wave function is just a constantY 0 0 ,=1/ 4. Thus, when integrating over the angularcoordinates, only the terms P l cos of c contain the angular, variables. Sincem=lP l cos =4 Y m2l +1 l 0 , 0 * Y m l ,,m=−l11where 0 , 0 are the fixed coordinates of the impurity and075319-4