Correlations between optical properties and Voronoi-cell area of quantum dots

Matthias C. Löbl, Liang Zhai, Jan-Philipp Jahn, Julian Ritzmann, Yongheng Huo, Andreas D. Wieck, Oliver G. Schmidt, Arne Ludwig, Armando Rastelli, and Richard J. Warburton

Phys. Rev. B 100, 155402 – Published 3 October 2019

Abstract

A semiconductor quantum dot (QD) can generate highly indistinguishable single photons at a high rate. For application in quantum communication and integration in hybrid systems, control of the QD optical properties is essential. Understanding the connection between the optical properties of a QD and the growth process is therefore important. Here, we show for GaAs QDs, grown by infilling droplet-etched nanoholes, that the emission wavelength, the neutral-to-charged exciton splitting, and the diamagnetic shift are strongly correlated with the capture-zone area, an important concept from nucleation theory. We show that the capture-zone model applies to the growth of this system even in the limit of a low QD density in which atoms diffuse over $μ m$ distances. The strong correlations between the various QD parameters facilitate preselection of QDs for applications with specific requirements on the QD properties; they also suggest that a spectrally narrowed QD distribution will result if QD growth on a regular lattice can be achieved.

Received 23 May 2019
Revised 11 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.155402

Physics Subject Headings (PhySH)

Excitons Growth Nucleation on surfaces

Quantum dots

Photoluminescence

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Matthias C. Löbl ^1,*, Liang Zhai¹, Jan-Philipp Jahn¹, Julian Ritzmann², Yongheng Huo^3,4, Andreas D. Wieck², Oliver G. Schmidt³, Arne Ludwig², Armando Rastelli⁵, and Richard J. Warburton¹

¹Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
²Lehrstuhl fur Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
³Institute for Integrative Nanosciences, Leibniz IFW Dresden, Helmholtzstrasse 20, D-01069 Dresden, Germany
⁴Hefei National Laboratory for Physical Sciences at the Microscale (HFNL) & Shanghai Branch of the University of Science and Technology of China (USTC), No.99, Xiupu Road, Pudong New District 201315, Shanghai, China
⁵Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria

^*matthias.loebl@unibas.ch

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Vol. 100, Iss. 15 — 15 October 2019

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Images

Figure 1
(a) Schematic of the growth process of the quantum dots (QDs). In a first step, aluminum is deposited on an epitaxially grown AlGaAs surface. The aluminum atoms nucleate in the form of liquid nanodroplets. An atom is most likely to attach to the closest Al droplet. This is the Al droplet into whose capture zone (red area) the atom falls. (b) Underneath the Al droplet, the substrate material is unstable and nanohole etching takes place upon exposure to an arsenic flux. (c) After formation of nanoholes, GaAs is deposited. Diffusion leads to an infilling of the droplet-etched nanoholes with GaAs. (d) Finally, the sample is capped with AlGaAs. The GaAs within the nanohole is now embedded in higher band-gap AlGaAs and forms a QD. (e) Spatially resolved photoluminescence (PL) on a $24 \times 24 μ m^{2}$ -large region of sample A. Positions of individual QDs are obtained by Gaussian fitting (black dots). The red lines are the Voronoi cells (VCs) corresponding to the QD positions. (f) Relation between the VC area ( $A_{VC}$ ) and the emission energy of the neutral exciton, $X^{0}$ . The light-blue ellipse is a guide to the eye indicating the correlation in a linear approximation (correlation coefficient $ρ = - 0.812$ ). The red line is a fit of Eq. (1).
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Figure 2
(a) An exemplary PL spectrum of a QD. The emission line at the highest energy is the neutral exciton ( $X^{0}$ ). At lower energy, emission of a singly charged exciton ( $X^{+}$ ) and a broad emission from further excitons appear. The inset shows the $X^{0}$ emission energy as a function of the magnetic field. A diamagnetic shift and a Zeeman splitting are observed. The following subfigures refer to emission from $X^{0}$ on sample A. (b) Diamagnetic shift as a function of the Voronoi cell area, $A_{VC}$ (correlation coefficient $ρ = 0.805$ ). The light blue ellipse has the same slope as a linear fit to the data points; its widths indicate a $1.5 σ$ interval parallel and perpendicular to the slope. (c) Diamagnetic shift as a function of the emission energy of the neutral exciton. These parameters are negatively correlated ( $ρ = - 0.922$ ). (d) A weak correlation between the exciton $g$ -factor and the Voronoi cell area, $A_{VC}$ . (e) Splitting between neutral and positively charged excitons ( $E_{X 0} - E_{X +}$ ) as a function of $A_{VC}$ ( $ρ = - 0.819$ ). (f) $E_{X 0} - E_{X +}$ as a function of the $X^{0}$ emission energy ( $ρ = 0.942$ ).
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Figure 3
Frequency distribution $f_{VC}$ of the normalized Voronoi cell area, $A_{VC} / 〈 A_{VC} 〉 \equiv x$ . For better visibility, the normalized Voronoi cell areas are divided in finite intervals. The red line is a fit of Eq. (A1) obtained by likelihood optimization.
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Figure 4
Correction of the distortion due to the nonlinearity of the piezoscanners. The QD position is set by the voltages $V_{x / y}$ applied to the piezoscanners. The distortion correction is carried out independently for the (a) $x$ and (b) $y$ directions ( $x$ direction in the following explanation). Initially, the spot size $σ_{x}$ is measured for every QD in units of the voltage applied to the piezoscanner. It is obtained by fitting a two-dimensional Gaussian to the PL intensity which is spectrally filtered to select individual QDs. For a larger derivative $\frac{d x}{d V_{x}}$ , the spot size appears smaller than for a smaller value of $\frac{d x}{d V_{x}}$ . The spot size $σ_{x}$ is therefore inversely proportional to the derivative $\frac{d x}{d V_{x}}$ . The red curves in (a), (b) are fits to a phenomenological parabolic dependence between applied piezovoltage $V_{x}$ and $1 / σ_{x}$ : $\frac{d x}{d V_{x}} = \tilde{c} / σ_{x} = \tilde{c} (a_{0} + a_{1} V_{x} + a_{2} V_{x}^{2})$ . We use the fit results to map $V_{x}$ to the position $x (V_{x}) = \tilde{c} \int_{0}^{V_{x}} (a_{0} + a_{1} V + a_{2} V^{2}) dV$ . The prefactor $\tilde{c}$ is obtained by the constraint that the highest voltage corresponds to the full scan range. (c) PL intensity as a function of the $x y$ position. The PL is integrated over the full QD ensemble. The PL map is shown without the distortion correction assuming that position and applied piezovoltage are related linearly. Due to the described nonlinearity of the piezoscanners, the QD spots at low voltages appear slightly larger in comparison to the QD spots at high voltages — the PL map is distorted. (d) The same PL map with a distortion correction using the above method. The QD spot sizes are homogeneous, indicating a successful correction of the distortion.
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Figure 5
(a) An atomic force microscopy (AFM) image on a sample similar to sample A. The growth is stopped after etching the nanoholes. The size of the AFM scan is $5 \times 5 μ m^{2}$ with $512 \times 512$ pixels. The AFM image indicates that the more separated nanoholes (with a larger VC area) are deeper than those with a close-by neighbor. This finding is in good agreement with the results shown in the main text. A zoom-in of two nanoholes is shown in (b, c) to illustrate this observation. The first nanohole has several close-by neighbors and is shallow. In contrast, the second nanohole is more isolated and is particularly deep. The image size and resolution are too low to allow for a quantitative statistical analysis comparable to the main text. Note that an AFM image comparable in size to the presented PL measurements, simultaneously imaging individual nanoholes with $\sim 2 nm$ resolution, would have to be $\sim 10^{4} \times 10^{4}$ pixels large — a very time-consuming measurement.
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Figure 6
Emission energy ( $X^{0}$ ) as a function of Voronoi cell area ( $A_{VC}$ ). The red line is a fit to Eq. (1) for QDs on sample A (blue points). Inset: Emission energy of QDs on sample A as a function of the expression on the right-hand side of Eq. (1). The relation between both quantities is more linear (correlation coefficient $| ρ | = 0.879$ ) than the direct relation of emission energy and $A_{VC}$ ( $| ρ | = 0.812$ ) which supports Eq. (1). QDs on sample B (black data points) are red-shifted compared to QDs on sample A.
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