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Confinamiento dieléctrico 255DIELECTRIC MISMATCH EFFECTS IN TWO-ELECTRON…of the dielectric constant at the interface has been employedin order to bypass the non-integrability of Eq. 2 when asteplike dielectric function and a finite confinement barrieroccur simultaneously. A cosine-like profile within a layer atthe interface with a width of 0.3 nm the order of a latticeconstant is assumed. We have found that, as in Ref. 6, selfenergyeffects are nearly no sensitive to the smoothing modeland to small changes in the interface width.Finally, V I r j is the Coulomb potential including polarizationgenerated by a shallow donor impurity located at theoriginV I r j =− 1 i r j−1 0− 1 i 1 Rif r R,− 1 0 r jif r R,4where R stands for the interface radius, and i and 0 are thedielectric constants of the confined and surrounding media,respectively. Z in Eq. 2 is 1 0 when the impurity is includedexcluded.The spherical symmetry of the problem allows the angularcoordinates of the electron to be integrated analytically inEq. 2. The radial parts of the exact one-particle wave functions nlm r and the corresponding energies are obtained bymeans of numerical integration finite differences in a gridextended far beyond the interface radius R. The numericalnature of this integration requires the discretization of thecontinuous r function yielding a multistep profile withinthe interfacial layer, so that new, artificially introduced divergencesare encountered. Such numerical divergences havebeen overcome by means of a discretization scheme thatavoids calculating at the interfaces. 34Products of the basis functions nlm are then used to constructconfiguration-interaction CI expansions LS = j jof the symmetry- and spin-adapted two-electron configurations,where L and S are the total angular and total spinquantum numbers, respectively. The two-electron Hamiltoniancontaining Coulomb interaction and polarizationterms 1 is then diagonalized in the CI basis set. As a result,we get two-particle wave functions LS r 1 ,r 2 and energiesE 2S+1 L. We use as many single-particle basis functions nlmand as long a CI expansion as are needed to achieve convergenceand the required accuracy.Here we will not use the standard quantum-chemical definitionof correlation energy and correlation effects related todifferences between CI and Hartree-Fock variational procedures.Electronic correlation is understood in the presentpaper as the contribution of the excited configurations to theexact ground state 1 S g wave function in comparison to theground configuration 1s 2 . The quantification of this correlationcan be expressed, then, as c corr =1−c 1s 2 2 , where c 1s 2 isthe coefficient of the 1s 2 configuration in the CI expansion.From the wave functions we define the radial pair densityPr 1 ,r 2 ,PHYSICAL REVIEW B 74, 125322 2006Pr 1 ,r 2 =2 r 1 ,r 2 2 r 1 2 r 2 2 sin 1 sin 2 d 1 d 2 d 1 d 2 ,to study radial correlations, and the angular correlation densityZ,Z = N Z r max ,0,0,r max ,,0 2 ,with r max corresponding to the coordinates r 1 =r 2 of thePr 1 ,r 2 maximum and N Z to the appropriate normalizationfactor, to study angular correlations.The D − center will be characterized by its binding energyE b , which is defined asE b D − = E 0 + ED 0 − ED − .Here E 0 is the lowest energy of the Hamiltonian Eq. 2 withZ=0, i.e., the single-particle ground state energy of the undopedQD, and ED 0 and ED − are the single- and thetwo-particle ground state energies of the doped Z=1 QD,respectively.III. NUMERICAL RESULTSA. Quantum dot in air or a vacuumThis section is devoted to studying the effects of the polarizationof Coulomb interaction on the electronic correlationof a two-electron QD in the large dielectric mismatchregime i.e., in the presence of a large QD-surrounding dielectricmismatch leading to a transition from volume to surfacestates. To this end we consider a two-electron QD similarto the one studied in Ref. 29, defined by an R=5.35 nmradius, a V 0 =0.9 eV confining barrier and an m * =0.5 effectivemass. This QD is surrounded by air or a vacuum 0=m 0 * =1. The polarization of the Coulomb interaction is variedfrom zero QD =1 up to a maximum value achieved bya conductor QD, QD =. From a numerical point of view,no significant changes occur beyond QD =40–80, and welimit our study up to this range of QD permittivities.First of all, we should mention that, as expected, an increasein the dielectric mismatch is accompanied by an increasein radial localization. This is basically a singleparticleself-polarization effect. In our case, as the QDsurroundings consist of air or a vacuum 0 =1, the largerthe QD , the deeper the self-polarization well and, hence, thestronger the electron localization in this well. The transitionfrom volume to surface states takes place, in our case, at QD 4. This transition can be monitored very well by plottingthe radial density or the pair radial density Pr 1 ,r 2 vs QD . The corresponding figures, similar to Figs. 1 and 2 inRef. 29, have been omitted for the sake of conciseness.From now on, we should keep in mind that in our casethat is, QD 0 , the surface states are localized in a sphericalcrown beyond the QD border outside crown, i.e., theelectrons are mostly in a vacuum. Therefore, regardless ofthe QD permittivity, the electron-electron interaction containsan enhanced bare Coulomb term including a null567125322-3