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Effects of screened Coulomb impurities on autoionizing two-electron resonances in spherical quantum dots

In a recent paper (Phys. Rev. B {\bf 78}, 075316 (2008)), Sajeev and Moiseyev demonstrated that the bound-to-resonant transitions and lifetimes of autoionizing states in spherical quantum dots can be controlled by varying the confinment strength. In the present paper, we report that such control can in some cases be compromised by the presence of Coulomb impurities. It is demonstrated that a screened Coulomb impurity placed in the vicinity of the dot center can lead to bound-to-resonant transitions and to avoided crossings-like behavior when the screening of the impurity charge is varied. It is argued that these properties also can have impact on electron transport through quantum dot arrays.


I. INTRODUCTION
in more detail in the following section II along with the computational procedure, and the results are presented in section III. A discussion and conclusions are given in section IV.

II. METHOD
The Hamiltonian of a spherical Gaussian two-electron quantum dot with a Coulomb impurity reads where q 1 , q 2 are the coordinates of the two electrons, m * the effective electron mass, ǫ the dielectric constant of the semiconductor, U 0 the depth of the confining potential, α a parameter describing the range of the latter and η the effective charge of the impurity. For convenience, we introduce scaled parameters as suggested in Ref. 8 : where m e is the electron mass, so that the Hamiltonian can be written as: From here on, all quantities will be given in effective atomic units, and the values for the corresponding semiconductor can be reobtained from Eqs. (2). For example, in the case of GaAs we have ǫ = 12.4 and m * /m e = 0.067, which gives the effective energy unit 1 Ha * ≈ 11.857 meV and the effective length unit a * 0 ≈ 9.794 nm. To determine the bound as well as the resonant states of the system, we diagonalize the complex scaled Hamiltonian within a B-Spline basis set. This method was earlier applied to describe autoionizing resonances in atomic 22,23 and exotic [24][25][26] systems. The Hamiltonian (Eq.(3)) is dilation-analytic, so that the uniform complex scaling [27][28][29][30] of the coordinates can be imposed: with a real parameter 0 < θ < π/4. The eigenvalues of the scaled Hamiltonian are complex.
However, the physical states are invariant with respect to θ while pseudo-continuum states are rotated into the complex plane by the angle 2θ. After diagonalizing the scaled Hamiltonian for different values of θ, one obtains the bound states of the system as the eigenvalues with vanishing imaginary parts and the resonant states as complex eigenvalues, the real part of which gives the position of the resonance and the negative imaginary part the halfwidth of the latter. It is connected to the lifetime τ as τ = /Γ. The diagonalization of the scaled Hamiltonian is carried out numerically using LAPACK-routines, and below we briefly describe the computational procedure. The radial part of the one-particle Hamiltonian h is given by and in the first step, the corresponding radial Schrödinger equation is solved in a sufficiently large box using piecewise polynomial functions B k (so called B-Splines 31,32 of a certain order k) defined on a given knot sequence. Thus, the radial one-particle eigenfunctions ϕ j are obtained in the form with expansion coefficients c kj . Subsequently, the matrix elements of the interaction potential m e ǫ 2 m * V 12 = e 2 4πǫ 0 |r 1 − r 2 | between the obtained states are computed using multipole expansion, where the radial integration can be carried out to machine accuracy using Gaussian quadrature, while the angular integration is performed analytically using Racah algebra 33 . The full Hamiltonian is then set up in the basis of coupled eigenstates to the one-particle Hamiltonians under consideration of the Pauli principle. Complex scaling is imposed by multiplying the kinetic terms by exp(−2iθ) and the Coulomb-terms by exp(−iθ). The real and imaginary part of the complex rotated Gaussian confining potential are given by and are scaled accordingly. After the setup of the Hamiltonian matrix is completed, the latter is diagonalized in the final step.

III. RESULTS
In the present calculations, we considered both donor and acceptor impurities, and therefore we varied the effective charge in the regions η > 0 and η < 0. This is a quite simple picture of the screening mechanism, but it should be sufficient to illustrate the effect of impurities on autoionizing states. In a more advanced approach, one could also include a possible spatial dependence of the screening. For example, Kwon 34 recently presented a model where the screening is modelled by an exponentially decreasing potential where q is the true impurity charge and r s the screening length which depends on the doping concentration in the semiconductor and the temperature (see Eqs.(9)- (15) in Ref. 34 ). We simplify the treatment by varying η independently of r i , as it is often done in atomic manyelectron systems to model the screening effects by core electrons. Also, since we are mostly interested in autoionizing states which are situated not very close to the dot center, this approximation seems reasonable.
The parameters of the confining potential are chosen in the same region as suggested in Ref. 8 : throughout the calculations, the potential depth is kept fixed at V 0 = 3 Ha * and for the range parameter β we take certain values which are well suited to illustrate the physical behavior we aim to demostrate. To represent the basis states, a sequence of 48 knot points with a box size of R = 24 a * 0 is used to generate the B-Spline set, the order of which is k = 7 throughout the paper. We restrict our treatment to singlett resonances and include all configurations of s − s, p − p and d − d type for the 1 S symmetry and all configurations of s−p and p−d type for the 1 P symmetry, which is enough to reach sufficient convergence. The numerical stability of the method was confirmed by successfully reproducing the positions and widths of the resonances as given in Ref. 8 for the case η = 0 obtained with a Gaussian basis set. Figures 1 and 2 show the positions and halfwidths of the three lowest states and the 1s-threshold for donor and acceptor impurities with different screening strengths. We observe that a donor impurity can turn an autoionizing state into a bound state, while for an acceptor impurity the opposite behavior is seen. Somewhat similar situations are known to occur in atomic systems, when a certain electronic configuration gives or does not give rise to autoionizing resonances depending on the nuclear charge. For example, the (1s 2 2p 2 ) 1 S state is autoionizing in beryllium 35 , while in the case of any other beryllium-like ion (e.g. beryllium-like carbon 36 ) this state is bound. Of course, this analogy does not fully hold in the case studied here, since the confining potential remains unchanged, but to some extent it allows a qualitative insight in the observed behavior. Another interesting aspect in the case of donor impurities is that the width of the state which remains autoionizing (blue curve in Fig. 1) is affected differently for different total angular momenta: in the case of  Fig. 1 for an acceptor impurity (η < 0). Here, we observe that two states (blue dotted and red solid lines) cross the threshold. However, in the case of 1 S symmetry, the width of the second state (red solid line) is orders of magnitude smaller than the one of the third state and is not shown in the plot since it is not distinguishable from zero on the given scale. The values for the range parameter were taken as β = 0.10 (a * 0 ) −2 for the 1 S symmetry and β = 0.08 (a * 0 ) −2 for the 1 P symmetry.
1 S symmetry it is slightly increasing while for the 1 P symmetry it is slightly decreasing as the effective impurity charge grows. Furthermore, we would like to point out that, for an acceptor impurity, the positions of the second and third states become very close to each other after crossing the threshold (blue and red curves in the upper left panel in Fig. 2).
By "zooming in" into the relevant region, we see that these curves show an avoided-crossing like behavior. It is illustrated in Fig. 3, where we also plot the energy difference ∆E of the states vs. the effective impurity charge. To summarize, we observe that the dot spectra are quite sensitive with respect to impurities, both concerning the positions and, in case of autoionizing states, also the lifetimes, in particular because impurities can cause thresholdcrossings so that bound states become resonant or vice versa. In the following section IV, we discuss the physical implications of the observed behavior which allows us to draw some In conclusion, we studied autoionizing resonances in the presence of Coulomb impurities in spherical Gaussian-shaped two-electron quantum dots using the complex scaled direct diagonalization method. We found that donor impurities can turn resonant states with a finite lifetime into bound states, while acceptor impurities have the opposite effect. Implications of these features were discussed in the context of photoionization and transport processes in quantum dots, underlining the importance of the semiconductor purity in these particular applications.