Quantum dot defined in a two-dimensional electron gas at a $n\text{−}\mathrm{Al}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ heterojunction: Simulation of electrostatic potential and charging properties

We present a self-consistent Schrödinger–Poisson scheme for simulation of electrostatic quantum dots defined in gated two-dimensional electron gas formed at $n\text{−}\mathrm{Al}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ heterojunction. The computational method is applied to a quantitative description of transport properties studied experimentally by Elzermann et al. [Appl. Phys. Lett. 84, 4617 (2004)]. Our three-dimensional model describes the electrostatics of the entire device with a quantum dot that changes shape and floats inside a gated region when the applied voltages are varied. Our approach accounts for the metal electrodes of arbitrary geometry, includes magnetic field applied perpendicular to the growth direction, electron-electron correlation in the confined electron system, and its interaction with the electron reservoir surrounding the quantum dot. We calculate the electric field, the space charge distribution, and energies as well as wave functions of confined electrons to describe opening of two transport channels between the reservoir and the confined charge puddle. We determine the voltages for charging the dot with up to four electrons. The results are in qualitative and quantitative agreement with the experimental data.

Figure 1

(Color online) Structure of layers used for formation of a gated quantum dot in the two-dimensional electron gas (according to Ref. 9).

Figure 2

Geometry, size, and position of the metal gates deposited on top of the semiconductor heterostructure (according to Ref. 9).

Figure 3

(Color online) Electron potential energy (solid line) and the charge density of the electron gas (dashed curve) at the $\mathrm{Al}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ heterojunction (GaAs is at the positive side of $z$). Thin horizontal dashed line shows the Fermi energy pinned at the donor impurity level in AlGaAs. $E^D$ is the donor binding energy that enters formula (8).

Figure 4

(Color online) Electron potential energy $U(x,y)=-\mid e\mid {\varphi }_{\mathrm{elst}}(x,y,z_o)$ calculated at a distance of $z_o=12\mathrm{nm}$ of the AlGaAs barrier for voltages $V_P=0$, $V_T=1.5\mathrm{V}$, $V_M=-1.07\mathrm{V}$, and $V_R=-0.96\mathrm{V}$. Assumed donor impurity concentration is $n_D=20\times {10}^{16}{\mathrm{cm}}^{-3}$ in (a), $n_D=25\times {10}^{16}{\mathrm{cm}}^{-3}$, in (b) and $n_D=21.641\times {10}^{16}{\mathrm{cm}}^{-3}$ in (c). Blue line shows the $U=0$ contour.

Figure 5

(Color online) Electron potential energy calculated for voltages $V_P=0$, $V_T=1.5\mathrm{V}$, $V_M=-1.07\mathrm{V}$, $V_R=-0.96\mathrm{V}$, and the donor concentration $n_D=21.641\times {10}^{16}{\mathrm{cm}}^{-3}$ calculated along the lines that are parallel to one of the axes (solid line to $z$, dashed line to $y$, and dotted line to the $z$ axis) and pass through the potential minimum $(x_{\min },y_{\min },z_{\min })$. The solid line is therefore $V\left(x\right)=U(x,y_{\min },z_{\min })$, the dashed curve is $V\left(y\right)=U(x_{\min },y,z_{\min })$, and the dotted line is $V\left(z\right)=U(x_{\min },y_{\min },z)$. Dashed-dotted line shows the Fermi energy $[U=0]$.

Figure 6

(Color online) Transmission lines as calculated (solid lines) and measured (Ref. 9) (thick dashed lines). $N$ stands for the number of electrons that is fixed between the transmission lines. The dots marked by letters (a)–(f) correspond to voltages for which the potential distribution is presented in Fig. 7.

Figure 7

(Color online) Equipotential contours and regions of negative electron potential energy (plotted in blue) for $n_D=21.641\times {10}^{16}{\mathrm{cm}}^{-3}$ at $12\mathrm{nm}$ of the barrier. The voltages corresponding to plots (a)–(f) are marked in Fig. 6 by black dots (for the exact numbers, see text).

Figure 8

Contour plot of the density of the electron gas at $12\mathrm{nm}$ of the barrier for voltages marked by point a in Fig. 6; the darker shade of gray the larger density.