^{1}, L. Covaci

^{2}, F. M. Peeters

^{1,2}and G. A. Farias

^{2}

### Abstract

The interaction of an injected electron towards a quantum dot(QD) containing a single confined electron is investigated using a flexible time-dependent quantum mechanics formalism, which allows both electrons to move and undergo quantum transitions. Different scenarios combining quantum dot dimensions, dielectric constant, injected wave packet energy, and width were explored, and our main results are: (i) due to the large characteristic transitions times between the confined state in the quantum dot and the delocalized state in the continuum, it is relatively difficult to ionize the occupied QD by Coulomb interaction solely and (ii) the charging state of the quantum dot can be sensed by direct injection of charges.

This work was financially supported by the Brazilian National Research Council (CNPq), under Contract No. NanoBioEstruturas 555183/2005-0, Fundao Cearense de Apoio ao Desenvolvimento Cientfico e Tecnolgico (Funcap), CAPES, Pronex/CNPq/Funcap, the Bilateral program between Flanders and Brazil, and the Flemish Science Foundation (FWO).

I. INTRODUCTION

II. MODELING

A. Construction of the initial state

B. Calculation of

III. COMPUTATIONAL APPROACH

IV. RESULTS

A. Two-particle wave function

B. Transmission and reflection coefficients

C. Width of the incident wave packet and effect of the dielectric constant

D. Intersubband transitions and QDionization

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Quantum dots
- 78.0
- Wave functions
- 21.0
- Quantum wells
- 15.0
- Dielectric constant
- 12.0
- Transmission coefficient
- 10.0

## Figures

Schematics of an injected electron state in the form of a gaussian wave packet (half-width of ) with a kinetic energy moving towards a QD (of width *W*) containing one electron in one of the confined states. The depth of the confining potential is (represented by the solid red line). The effective potential for the injected electron is represented by the dashed red line. The green arrows represent the possible quantum transitions that the confined electron may undergo due to the interaction with the injected electron. There are two types of transitions: (i) intersubband transitions for the case that the electron is excited from the ground to the first excited QD state, and (ii) ionization, for the case where the confined electron is excited to a delocalized continuum state. The whole system is embedded in a quantum wire of diameter *D* (see upper inset). In this work, we have adopted , and . The barrier sizes are of the order of 30 nm, and the size of the absorbing layers is 5 nm which are introduced in order to eliminate spurious reflections.

Schematics of an injected electron state in the form of a gaussian wave packet (half-width of ) with a kinetic energy moving towards a QD (of width *W*) containing one electron in one of the confined states. The depth of the confining potential is (represented by the solid red line). The effective potential for the injected electron is represented by the dashed red line. The green arrows represent the possible quantum transitions that the confined electron may undergo due to the interaction with the injected electron. There are two types of transitions: (i) intersubband transitions for the case that the electron is excited from the ground to the first excited QD state, and (ii) ionization, for the case where the confined electron is excited to a delocalized continuum state. The whole system is embedded in a quantum wire of diameter *D* (see upper inset). In this work, we have adopted , and . The barrier sizes are of the order of 30 nm, and the size of the absorbing layers is 5 nm which are introduced in order to eliminate spurious reflections.

Effective Coulomb potential for and different values of the dielectric constant of the semiconductor, and different diameters *D* of the quantum dot.

Effective Coulomb potential for and different values of the dielectric constant of the semiconductor, and different diameters *D* of the quantum dot.

Matrix form of the system of time dependent equations given by Eq. (3) where .

Matrix form of the system of time dependent equations given by Eq. (3) where .

(left)Time evolution of for different energies of the injected electron (0.1 eV and 0.5 eV) at intervals of 8 fs. The initial state has a single component (), such that no transitions between the angular modes are allowed. The QD dimensions are *W* = 3 nm, *D* = 10 nm, and the dielectric constant is . The width of the injected particle is . (right-top) Position-dependent probability to find either particle in the z direction within the time window up to 40 fs. The vertical dashed lines represent the QD region. The horizontal dotted lines are reference lines. (right-bottom) Momentum-dependent probability to find either particle with a momentum *k*.

(left)Time evolution of for different energies of the injected electron (0.1 eV and 0.5 eV) at intervals of 8 fs. The initial state has a single component (), such that no transitions between the angular modes are allowed. The QD dimensions are *W* = 3 nm, *D* = 10 nm, and the dielectric constant is . The width of the injected particle is . (right-top) Position-dependent probability to find either particle in the z direction within the time window up to 40 fs. The vertical dashed lines represent the QD region. The horizontal dotted lines are reference lines. (right-bottom) Momentum-dependent probability to find either particle with a momentum *k*.

(top) Time dependence of the average position of the particle confined in the QD for different kinetic energies of the injected particle. (middle) Average position of the position-dependent probability . (bottom) Average momentum of the incident (solid curves) and transmitted (dashed curves) total wave function for different kinetic energies of the injected particle. The QD parameters are the same as for Fig. 4.

(top) Time dependence of the average position of the particle confined in the QD for different kinetic energies of the injected particle. (middle) Average position of the position-dependent probability . (bottom) Average momentum of the incident (solid curves) and transmitted (dashed curves) total wave function for different kinetic energies of the injected particle. The QD parameters are the same as for Fig. 4.

Region-dependent probability to find one electron in the left (), center () and right () sides with respect to the QD position. The sum of these probabilities is also shown (solid circles). The parameters are the same as for Fig. 4. The injected kinetic energies are: 0.1 eV (black), 0.5 eV (blue), and 1.0 eV (red). The vertical lines mark the maximum traveling time of the total wave function before is drained by the absorbing layers. is also used to determine the transmission and reflection coefficients as and , respectively. represents the peak of .

Region-dependent probability to find one electron in the left (), center () and right () sides with respect to the QD position. The sum of these probabilities is also shown (solid circles). The parameters are the same as for Fig. 4. The injected kinetic energies are: 0.1 eV (black), 0.5 eV (blue), and 1.0 eV (red). The vertical lines mark the maximum traveling time of the total wave function before is drained by the absorbing layers. is also used to determine the transmission and reflection coefficients as and , respectively. represents the peak of .

(top) Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) coefficients on the kinetic energy of the injected particle. Results are shown for different QD widths: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). The QD diameters are *D* = 5 nm (so symbol) and *D* = 10 nm (circle). (bottom) Energy dependence of the peak probability of (see Fig. 6). The diameter dependence is negligible for . Here we used .

(top) Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) coefficients on the kinetic energy of the injected particle. Results are shown for different QD widths: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). The QD diameters are *D* = 5 nm (so symbol) and *D* = 10 nm (circle). (bottom) Energy dependence of the peak probability of (see Fig. 6). The diameter dependence is negligible for . Here we used .

Difference of *P* (solid), *T* (dashed) and *R* (dash-dotted) calculated with and without the Coulomb interaction for *D* = 10 nm. Different QD widths are shown: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). Here we used .

Difference of *P* (solid), *T* (dashed) and *R* (dash-dotted) calculated with and without the Coulomb interaction for *D* = 10 nm. Different QD widths are shown: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). Here we used .

Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) on the half-width of the incident wave packet for kinetic energy . Different QD widths are shown: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). The QD diameter is *D* = 10 nm. Here we used .

Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) on the half-width of the incident wave packet for kinetic energy . Different QD widths are shown: *W* = 3 nm (black), *W* = 5 nm (blue), and *W* = 7 nm (red). The QD diameter is *D* = 10 nm. Here we used .

Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) on the dielectric constant for a kinetic energy of . Different values of are shown: (black), (blue), and (red). The QD dimensions are *D* = 10 nm and *W* = 3 nm.

Dependence of *P* (solid), *T* (dashed), and *R* (dash-dotted) on the dielectric constant for a kinetic energy of . Different values of are shown: (black), (blue), and (red). The QD dimensions are *D* = 10 nm and *W* = 3 nm.

Characteristic transition time of a Coulomb mediated intersubband transition between the two lowest confined states in the QD. The QD dimensions are *W* = 3 nm and *D* = 10 nm.

Characteristic transition time of a Coulomb mediated intersubband transition between the two lowest confined states in the QD. The QD dimensions are *W* = 3 nm and *D* = 10 nm.

## Tables

Indexing scheme of the lowest two-electron radial states (*i,j*) in terms of the single-particle radial quantum numbers.

Indexing scheme of the lowest two-electron radial states (*i,j*) in terms of the single-particle radial quantum numbers.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content