Electric field induced tuning of electronic correlation in weakly confining quantum dots

We conduct a combined experimental and theoretical study of the quantum confined Stark effect in GaAs/AlGaAs quantum dots obtained with the local droplet etching method. In the experiment, we probe the permanent electric dipole and polarizability of neutral and positively charged excitons weakly confined in GaAs quantum dots by measuring their light emission under the influence of a variable electric field applied along the growth direction. Calculations based on the configuration-interaction method show excellent quantitative agreement with the experiment and allow us to elucidate the role of Coulomb interactions among the confined particles and—even more importantly—of electronic correlation effects on the Stark shifts. Moreover, we show how the electric field alters properties such as built-in dipole, binding energy, and heavy-light hole mixing of multiparticle complexes in weakly confining systems, underlining the deficiencies of commonly used models for the quantum confined Stark effect.

Figure 1

(a) AFM depth profile of a typical Al-droplet-etched NH. The solid and dashed lines were taken along the [110] and $[1-10]$ crystal direction of (Al)GaAs. The orange indicates the GaAs filling and the “wetting layer” (WL). (b) Sketch of the used $p\text{−}i\text{−}n$ ${\mathrm{Al}}_{0.4}{\mathrm{Ga}}_{0.6}\mathrm{As}$ diode with GaAs QDs in the intrinsic layer. The top and bottom of the diode membrane are protected by 10 nm of highly doped GaAs (with dimensions included within the thickness of the doped layers). The bottom Au layer is electrically grounded. (c) The $I\text{−}V$ characteristics of a diode at the PL measurement temperature of $7\mathrm{K}$. The built-in voltage ($\sim 2.3\mathrm{V}$) was estimated by the intersection of the dotted line marking the forward-bias region with the saturation current. In the inset, we show the schematic band profiles of the diode in the forward bias and near the zero field (flatband condition). For positive $F_d$ ($F_d$ directed along the growth direction, i.e., from the diode surface towards the gold layer), the $e^{-}$ ($h^{+}$) wave function is pulled towards the tip (base) of the QD. The solid and dotted arrows mark the positive direction of $F_d$ and $p_z$, respectively. (d) Color-coded $\mu \text{−}\mathrm{PL}$ spectra of QD1 embedded in a $p\text{−}i\text{−}n$ diode as a function of $F_d$ and corresponding applied voltage $V_d$. Inset: zoomed-in and intensity-enhanced part of the spectra, where we observe the crossing of $X^0$ and $X^{+}$.

Figure 2

Measured and calculated Stark shifts of $X^0$ and $X^{+}$ for different QDs. (a) Experimental data for two QDs and corresponding fits using Eq. (1) with and without setting $p_z=0$, respectively. (b) Data for five QDs (symbols) and simulation (curves) for $X^0$ (left) and $X^{+}$ (right), respectively. The simulated cone-shaped dots have a fixed base diameter of $40\mathrm{nm}$ and height varying from 4 to $9.5\mathrm{nm}$. Note that the QD1 data correspond to those in Fig. 1, while data of QD2–QD5 were taken from a series of polarization-resolved measurements for $F_d$ in the range of 100–$250\mathrm{kV}/\mathrm{cm}$. The $\mu \text{−}\mathrm{PL}$ spectra of $X^0$ were fitted using Gaussian curves on one single polarization component.

Figure 3

(a) Permanent electric dipole moments plotted as a function of the zero field energy $E_0$ of the corresponding complex $X^0$ or $X^{+}$. The parameter $p_z/e$ was obtained for experimental data (open symbols) by fitting the Stark shift data in Fig. 2 by Eq. (1). The theoretical data of $X^0$ ($X^{+}$) marked by full circles (full triangles) for cone- (lens-) shaped QD are given in dark blue and dark brown (light blue and pink) and were obtained using Eq. (2). Insets: Sketch of the cone- and lens-shaped dots used in the simulation, respectively, and the corresponding position of $e^{-}$ and $h^{+}$ wave functions for $p_z/e<0$. Note that the height and diameter of the dot are not shown in the same scale (the dots are actually rather flat). (b) Polarizability ($\beta$) as a function of $E_0$. For (a) and (b) the experimental data (discrete symbols) were extracted from the Stark shift of five measured QDs in Fig. 2 and presented in the corresponding color. (c) Cross-sectional view of the probability densities of $e^{-}$, $h^{+}$, $X^0$, and $X^{+}$ for several values of $F_d$.

Figure 4

(a) $E_{b,0}$ and (b) ${\beta }_{E_b}^{*}/e$ as a function of $E_0$ fitted by Eq. (4) for five QDs obtained from experiments (symbols, color marked in Fig. 2) and simulation (dark blue for cone-shaped QDs, light blue for lens-shaped QDs). The theory values of $E_{b,0}$ in (a) were obtained directly from CI calculations, i.e., without fitting, while ${\beta }_{E_b}^{*}/e$ in (b) were obtained by fitting theory values using Eq. (4). (c) Dependence of $E_b$ on the number of SP $e^{-}$ and SP $h^{+}$ states used in a CI basis calculated for QD with a height $h=9.5\mathrm{nm}$. Note that we used a symmetric basis, i.e., the number of SP $e^{-}$ states and SP $h^{+}$ states is equal. (d) Polarizabilities (full circles) of the Coulomb integrals $J_{eh}$ (red), $J_{hh}$ (green), $J_{eh}-J_{hh}$ (purple), and of $E_b$ computed by CI with a $12\times 12$ SP basis (blue). The corresponding fits by Eq. (4) are shown in Appendix pp5.

Figure 5

Contribution of $|\mathrm{HH}\rangle$, $|\mathrm{LH}\rangle$, and $|\mathrm{SO}\rangle$ states normalized to a total sum of contributions of these components, i.e., $\varkappa \left(\text{HH}\right)+\varkappa \left(\text{LH}\right)+\varkappa \left(\text{SO}\right)$, in $X^0$ (top row) and $X^{+}$ (bottom row) vs electric field $F_d$. The colors identify the heights of QDs in the same fashion as in Fig. 2, where blue corresponds to $h=4\mathrm{nm}$ and red to $h=9.5\mathrm{nm}$. The Bloch state contents for both $X^0$ and $X^{+}$ were calculated using the CI model with the basis consisting of 12 SP $e^{-}$ and 12 SP $h^{+}$ states, with the effects of the direct and the exchange Coulomb interaction, and the correlation effect being included; see also Appendix pp3.

Figure 6

(a) Permanent electric dipole moments ($p_z$) and (b) polarizability ($\beta$) plotted as a function of the zero field energy $E_0$ of the corresponding complex $X^0$ or $X^{+}$. The fits of the theoretical data (full symbols and curves) were obtained by fitting with Eq. (1) for different ranges of $F_d$ values as indicated by the inset of each panel.

Figure 7

Dependence of $J_{eh}$, $J_{hh}$, and $J_{eh}-J_{hh}$ on $F_d$ computed by CI with a $12\times 12$ SP basis. Blue data correspond to $h=4\mathrm{nm}$ and red to $h=9.5\mathrm{nm}$.