Dynamical screening of the Coulomb interaction for two confined electrons in a magnetic field

We show that a difference in time scales of vertical and lateral dynamics permits one to analyze the problem of interacting electrons confined in an axially symmetric three-dimensional potential with a lateral oscillator confinement by means of the effective two-dimensional Hamiltonian with a screened Coulomb interaction. Using an adiabatic approximation based on action-angle variables, we present solutions for the effective charge of the Coulomb interaction (screening) for a vertical confinement potential simulated by parabolic, square, and triangular wells. While for the parabolic potential the solution for the effective charge is given in a closed analytical form, for the other cases similar solutions can be easily calculated numerically.

Figure 1

Left-hand side: The localization of QD in the layer of the thickness $a$. Right-hand side: The schematic representation of the position of zero-point motion in the parabolic confinement relative to the layer thickness.

Figure 2

(Color online) The ratio $k_{\mathrm{eff}}∕k$ for the lowest states with different $m$ for the vertical confinement approximated by (a) the parabolic potential; (b) the hard wall (square or triangular well) potentials. (a) The results obtained in the adiabatic approximation and by means of the plain quantum-mechanical averaging procedure (see Sec. 3A) are connected by dashed (blue) and dotted-dashed (green) lines, respectively. (b) The results for the square and triangular well are connected by solid (pink) and dotted-dashed lines, respectively.

Figure 3

(Color online) Lowest energy levels (only the relative motion is considered) of the two-electron QD with fully parabolic potential. The results of the 2D, the 3D and the effective charge (adiabatic) approximations are connected by solid (orange), solid (with a corresponding quantum number m, red), and dashed (blue) lines, respectively. The result of the plain quantum-mechanical averaging for the $m=0$ level $(k_{\mathrm{eff}}=⟪\rho V_C⟫)$ is connected by the dotted-dashed (green) line.

Figure 4

(Color online) (a) The additional energy $\Delta \mu$ as a function of the magnetic field in the parabolic model for a two-electron QD. The results of calculations with a lateral confinement only (the 2D approach, $\hslash {\omega }_0=2.9\mathrm{meV}$, $\mid g^{*}\mid =0.3$) and full 3D approach [10] $(\hslash {\omega }_z=8\mathrm{meV})$ are connected by solid (orange) and (red) lines, respectively. The vertical grey lines indicate the position of the experimental crossings between different ground states [7]. Ground states are labeled by $(m,S)$, where $m$ and $S$ are the quantum numbers of the operators $l_z$ and the total spin, respectively. The results based upon the adiabatic approximation, Eq. (17), and the plain quantum-mechanical averaging procedure, Eq. (19), are connected by dashed (blue) and dotted-dashed (green) lines, respectively. (b) The ratio $k_{\mathrm{eff}}∕k$ as functions of the magnetic field based on Eq. (17) and the plain quantum-mechanical averaging, Eq. (19) are connected by dashed (blue) and dot-dashed (green) lines, respectively.

Figure 5

(Color online) (a) The magnetic dependence of the additional energy $\Delta \mu$ for a two-electron QD with the lateral confinement $\hslash {\omega }_0=2.9\mathrm{meV}$ and $\mid g^{*}\mid =0.3$. The results of the pure 2D approach are connected by the solid (orange) line with a label 2D. The results of the adiabatic approach (effective charge calculations) are connected by the dashed (blue) line for the parabolic model $(\hslash {\omega }_z=8\mathrm{meV})$, by the solid (pink) line for the square well model $(a=32.5\mathrm{nm})$, and by the dotted-dashed (black) line for the triangular well model $(F=220\mathrm{kV}∕\mathrm{m})$. See the text for details. The vertical grey lines indicate the position of the experimental crossings between different ground states labeled by $(m,S)$ [7]. (b) The ratio $k_{\mathrm{eff}}∕k$ for three different approximations of the vertical confinement as a function of the magnetic field.

Figure 6

(Color online) (a) The weight functions $w_{\mathrm{qm}}$ (solid line), $w_{\mathrm{ad}}$ (dashed line) and (b) the ratio $R=V_{\mathrm{int}}^{\mathrm{eff}\left(\mathrm{ad}\right)}∕V_{\mathrm{int}}^{\mathrm{eff}\left(\mathrm{qm}\right)}$ for the parabolic vertical confinement.