Correlations between Optical Properties and Voronoi-Cell Area of Quantum Dots

TL;DR

This study reveals strong correlations between optical properties of GaAs quantum dots and their growth-related capture zone areas, enabling better control and preselection for quantum communication applications.

Contribution

It demonstrates the applicability of the capture-zone model to low-density GaAs quantum dots and links optical properties to growth parameters.

Findings

Optical properties correlate with capture zone area.

Capture-zone model applies even at low QD densities.

Preselection of QDs based on optical properties is possible.

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 nano-holes, 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 $\mu$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.

Equations

$$E=E_0+\frac{h^2}{8\mu }\cdot {\left({\left(\alpha \cdot A_{\text{VC}}\right)}^{\beta /3}+d\right)}^{-2}$$

$$E\left(B\right)=E(B=0)+\gamma B^2\pm \frac{1}{2}g{\mu }_{B}B,$$

$$f_{\text{VC}}(x\mid \eta )=a_{\eta }\cdot x^{\eta }\cdot \exp \left(-b_{\eta }{x}^2\right).$$

$$P\left(\{x^{(i)}{\}}_{i\in \{\mathrm{1..}N\}}\mid \eta \right)=\prod _{i=1}^N{f}_{\text{VC}}(x^{(i)}\mid \eta ).$$

$$P\left(\eta \mid \{x^{(i)}{\}}_{i\in \{\mathrm{1..}N\}}\right)=\frac{P\left(\{x^{(i)}{\}}_{i\in \{\mathrm{1..}N\}}\mid \eta \right)\cdot P\left(\eta \right)}{P\left(\{x^{(i)}{\}}_{i\in \{\mathrm{1..}N\}}\right)}.$$

Full text

Semiconductor quantum dots (QDs) are excellent as a bright source of highly indistinguishable single photons Huber et al. (2017); Ding et al. (2016); Somaschi et al. (2016); Kiršanskė et al. (2017); Liu et al. (2018); He et al. (2017); Gschrey et al. (2015); Schöll et al. (2019) and entangled photon pairs Stevenson et al. (2006); Akopian et al. (2006); Hudson et al. (2007); Chen et al. (2018); Huber et al. (2018); Liu et al. (2019); Basset et al. (2019); Wang et al. (2019). A QD can host a single spin Atatüre et al. (2006); Bechtold et al. (2015); Javadi et al. (2018); Löbl et al. (2017) which, however, has a too short coherence time for applications in quantum communication Prechtel et al. (2016); Delteil et al. (2016); Éthier-Majcher et al. (2017); Ding et al. (2019). A hybrid-system of a QD and an atomic quantum memory is more promising in that respect Sangouard et al. (2007); Rakher et al. (2013). To connect a QD to an atomic memory based on rubidium, the QD should emit photons matched both in emission energy and bandwidth to the memory Wolters et al. (2017). The emission energy can be matched by using GaAs QDs embedded in AlGaAs Jahn et al. (2015); Keil et al. (2017); bandwidth matching can be achieved by using a Raman-scheme Béguin et al. (2018); Pursley et al. (2018); Lee et al. (2018); He et al. (2013).

The growth of QDs has been intensively studied employing scanning probe microscopy Rastelli et al. (2004a); Bruls et al. (2002); Atkinson et al. (2012); Fanfoni et al. (2012); Heyn et al. (2007); Rastelli et al. (2008); Nothern and Millunchick (2012); Miyamoto et al. (2009); Bietti et al. (2014). Aiming at entangled photon-pair generation, a connection between such an analysis and the optical properties Rastelli et al. (2004b); Surrente et al. (2017); Sapienza et al. (2017) has focused mostly on the fine-structure splitting of the QD-emission Huo et al. (2013); Plumhof et al. (2010); Abbarchi et al. (2010); Mereni et al. (2012); Huo et al. (2014); Liu et al. (2014); Skiba-Szymanska et al. (2017); Basso Basset et al. (2018). To tailor all the optical QD-properties, it is important to understand how they are connected to the QD-growth (Atkinson et al., 2012).

In this Letter, we establish a strong correlation between the optical properties of GaAs QDs, such as emission energy and diamagnetic shift, and a basic concept from nucleation theory, the capture-zone Mulheran and Blackman (1996); Blackman and Mulheran (1996); Miyamoto et al. (2009). This correlation is not obvious since the applicability of the capture zone model depends on conditions such as sudden nucleation Fanfoni et al. (2007) and spatially uniform diffusion Ratto et al. (2006). Both are not necessarily fulfilled. The correlations that we find here are absent for InGaAs QDs Spencer et al. (2019) and locally very weak for SiGe QDs Miyamoto et al. (2009); Ratto et al. (2006). The system we investigate consists of GaAs QDs grown by infilling of Al-droplet-etched nano-holes in an AlGaAs surfaceWang et al. (2007); Huo et al. (2013).

We apply the capture-zone model to the first phase of this process, the formation and growth of Al-droplets. The capture-zone model implies a correlation between Al-droplet volume (island size Pratontep et al. (2004); Miyamoto et al. (2009)) and capture-zone area. We show experimentally that this results in a strong correlation between the capture zone area and the optical QD-properties.

We use spatially-resolved photoluminescence imaging to determine simultaneously the position and the optical properties of individual QDs. We investigate samples of low QD-density where a QD is small relative to the distances between QDs (point-island model Blackman and Mulheran (1996)). The capture zone of each QD is determined as its Voronoi-cell (VC) Mulheran and Blackman (1996); Miyamoto et al. (2009); Fanfoni et al. (2007); Ratto et al. (2006). From the distribution of the VC-areas, we estimate a critical nucleus size of the Al-droplets (supplemental material). We find a strong negative correlation between the VC-area, $A_{\text{VC}}$, of a QD and its emission energy, in turn, a strong positive correlation between $A_{\text{VC}}$ and the diamagnetic shift of the emission. These results can be well explained with the capture-zone model describing the aggregation dynamics of the Al-droplets. Correlations are measured for QDs of particularly low density, $n_{\text{QD}}<1\ \mu\text{m}^{-2}$, implying a diffusion length on the $\mu$m-scale during the growth phase of Al-droplets Bietti et al. (2014). This is a striking result: The optical properties of a QD, which are directly related to its structure on an nm-scale, are strongly correlated with its surroundings on a $\mu$m-scale.

Sample growth – The samples are grown by molecular-beam epitaxy (MBE) on a (001)-substrate. We investigate two different samples (denoted here as A and B) that are grown in two different MBE-chambers. QDs are grown by GaAs-infilling of Al-droplet-etched nano-holes. A schematic depiction of the growth is shown in Fig. 1. First, aluminum is deposited on an AlGaAs-surface in the absence of an As-flux. The growth parameters are: 0.5 ML on Al0.4Ga0.6As, $T=600^{\circ}$C, flux $F=0.5$ ML/s for sample A; and 0.16 ML on Al0.33Ga0.67As, $T=630^{\circ}$C, flux $F=0.18$ ML/s for sample B. For both samples, the layer thicknesses correspond to an equivalent amount of AlAs. The Al-atoms nucleate (Volmer-Weber mode Heyn et al. (2015)) in the form of liquid nano-droplets on the sample surface (see Fig. 1(a)).

Underneath an Al-droplet the substrate material is unstable leading to a nano-etching process (Fig. 1(b)) Wang et al. (2007). Under a low As-flux, the etching proceeds until the whole Al-droplet is consumed and the material is recrystallized in the surrounding region. Another $2$ nm of GaAs are grown on top, filling up the nano-holes via diffusion during a $2$-minute annealing step (Fig. 1(c)). Overgrown with AlGaAs, the filled nano-holes become optically active QDs (Fig. 1(d)).

Experiment – Optical measurements are performed in a helium bath-cryostat. Photoluminescence (PL) is measured under above-band excitation ($\lambda=633\ \text{nm}$). An aspheric objective lens ($\text{NA}=0.68$) collects the PL. The PL of the QD-ensemble is centered at wavelength $787.4\ \text{nm}$ for sample A, $798.0\ \text{nm}$ for sample B (values referring to the neutral exciton, $X^{0}$), with ensemble standard-deviations $3.4\ \text{nm}$ and $1.4\ \text{nm}$, respectively. A typical spectrum of a single QD is shown in Fig. 2(a). The neutral exciton has the highest emission energy Jahn et al. (2015); the positively charged exciton is redshifted by on average $2.7$ meV (sample B: $2.1$ meV). Additional exciton complexes appear at even lower energy Wang et al. (2009) but are not the focus of our analysis.

Spatially resolved micro-PL is performed by scanning the sample with a low-temperature piezoelectric xy-scanner (attocube ANSxy100/lr). A spatially resolved PL-measurement is shown in Fig. 1(e). QDs can be identified as bright regions on this PL-map. The lateral positions of the QDs are obtained by fitting two-dimensional Gaussians. A slight non-linearity of the piezo-scanner is corrected by using the widths of the fitted Gaussians as a reference (supplemental material). We determine the capture zone around a QD by its Voronoi-cell (VC) – the area that is closer to this particular QD than to any other one. A Voronoi-diagram together with the corresponding QD-positions is shown in Fig. 1(e). We find an average VC-area of $\langle A_{\text{VC}}\rangle=3.04\pm 0.08\ \mu\text{m}^{2}$ (sample A) and $\langle A_{\text{VC}}\rangle=6.87\pm 0.44\ \mu\text{m}^{2}$ (sample B), corresponding to a quantum dot density of $n_{\text{QD}}=0.329\pm 0.009\ \mu\text{m}^{-2}$ and $n_{\text{QD}}=0.146\pm 0.009\ \mu\text{m}^{-2}$, respectively.

Shown in Fig. 1(f) is the emission energy of the neutral exciton, $E_{X^{0}}$, for many QDs as a function of the VC-area, $A_{\text{VC}}$. The plot is obtained by combining three independent PL-maps from sample A. We find a strong negative (Pearson) correlation coefficient of $\rho=-0.812$ (sample B: $\rho=-0.809$) between emission energy and $A_{\text{VC}}$ ($\rho=\pm 1$ maximum correlation; $\rho=0$ no correlation). We explain this correlation by applying the capture-zone model to the growth phase of the Al-droplets (Fig. 1(a)). An Al-atom, impinging at a random position on the sample, moves on the surface via diffusion and is most likely to attach to the closest Al-droplet. In the capture-zone model, the growth rate of an Al-droplet is thus assumed to be proportional to the VC-area. If all Al-droplets form at about the same time (sudden nucleation Fanfoni et al. (2007)), it leads to a correlation between Al-droplet volume and VC-area. For an Al-droplet with a small VC-area, much material nucleates at its nearest neighbors reducing its own accumulation rate. In turn, an Al-droplet with a larger VC-area accumulates more atoms and the droplet-volume $V_{d}$ becomes larger. For a larger Al-droplet, the nano-hole etched underneath it becomes deeper (Atkinson et al., 2012). The QD subsequently formed from this nano-hole has a weaker confinement in the growth direction lowering its emission energy.

We obtain a quantitative relation between $A_{\text{VC}}$ and the emission energy by the following considerations. In the capture-zone model, the volume of each Al-droplet is proportional to $A_{\text{VC}}$. We assume that all Al-droplets have a similar aspect ratio Huo et al. (2014); Jo et al. (2013). Then the droplet-height $H_{d}$ is connected to the droplet-volume $V_{d}$ and to the VC-area by $H_{d}\propto V_{d}^{1/3}\propto A_{\text{VC}}^{1/3}$. For the relation between QD-height $H$ (nano-hole depth, respectively) and the droplet-height $H_{d}$ Atkinson et al. (2012), we assume a phenomenological relation $H\propto H_{d}^{\beta}$. Since $H$ is much smaller than the lateral extent of a QD Atkinson et al. (2012), it is this parameter which mainly determines the emission energy of the QD. In the case of a hard-wall confinement and without considering Coulomb interaction terms, the emission energy of a QD is given by the bandgap plus electron and hole confinement energy: $E=E_{0}+\frac{h^{2}}{8\mu H^{2}}$, where $E_{0}=1.519\ \text{eV}$ is the bandgap of the QD-material (GaAs) and $\mu=\left(\frac{1}{m_{e}^{*}}+\frac{1}{m_{h}^{*}}\right)^{-1}$ the reduced electron-hole effective mass ($m_{e}^{*}=0.067m_{e}$, $m_{h}^{*}=0.51m_{e}$). Including the $d=2\ \text{nm}$ thick quantum well above the QDs leads to $E=E_{0}+\frac{h^{2}}{8\mu(H+d)^{2}}$. Using the above relations one obtains an equation connecting $A_{\text{VC}}$ and the emission energy $E$ of the QD:

[TABLE]

A fit of Eq. 1 to the data is shown in Fig. 1(f). With the fit parameters $\alpha=1.268\cdot 10^{-32}\ \text{m}^{3/\beta-2}$ and $\beta=0.556$, we find a very good agreement with the data. The average height of a QD resulting from this fit is $H=8.7\ \text{nm}$ which agrees well with AFM-measurements (supplemental material). A direct correlation between the measured emission energy $E_{X^{0}}$ and the term on the right-hand side of Eq. 1 shows an even higher correlation of $\rho=0.879$ (supplemental material) than that between emission energy and $A_{\text{VC}}$. This strong correlation supports our model for the connection between QD-properties and capture zone.

We consider further QD-properties and their connection to the VC-area, $A_{\text{VC}}$: the diamagnetic shift of the QD-emission; and the splitting between neutral and positively-charged exciton, $E_{X^{0}}-E_{X^{+}}$. Both of these QD-properties are connected mainly to the lateral rather than the vertical confinement of the QD.

We measure the energy of the PL-emission ($X^{0}$) as a function of a magnetic field applied in the growth direction (inset to Fig. 2(a)). The magnetic field splits the emission lines by the Zeeman energy and leads to a diamagnetic shift. For every QD we fit the relation Van Hattem et al. (2013); Tholen et al. (2016)

[TABLE]

where $g$ is the exciton $g$-factor and $\mu_{B}$ the Bohr magneton. We approximate the diamagnetic shift with a parabola with prefactor $\gamma$ Van Hattem et al. (2013); Tholen et al. (2016). The fine structure splitting of the studied QDs Huo et al. (2014) is negligibly small in this context. For the diamagnetic shift, a probe of the lateral area of the exciton, we find values in the range $\gamma=15-20\ \mu\text{eV}/\text{T}^{2}$, in good agreement with Ref. Ulhaq et al., 2016. The dependence of $\gamma$ on the VC-area ($A_{\text{VC}}$) is shown in Fig. 2(b). We find a correlation of $\rho=0.805$ between $A_{\text{VC}}$ and $\gamma$ which reveals a connection between the capture-zone area and the lateral size of a QD. An Al-droplet with a larger VC-area has a larger lateral extent leading to a QD with a weaker lateral confinement potential. This finding is also fully compatible with the capture-zone model. For the direct dependence between the emission energy, $E_{X^{0}}$, and the diamagnetic shift, $\gamma$, we find an approximately linear relation (Fig. 2(c)) associated with a correlation of $\rho=-0.922$. This connection between vertical and lateral confinement is consistent with a reported correlation between emission energy and $s$-to-$p$-shell–splitting Wang et al. (2009).

A plot of the exciton $g$-factor versus $A_{\text{VC}}$ is shown in Fig. 2(d). The $g$-factor shows a slight dependence on $A_{\text{VC}}$ with $\rho=-0.562$. For these QDs, the electron $g$-factor is very small such that the exciton $g$-factor is determined largely by the hole $g$-factor Ulhaq et al. (2016); Béguin et al. (2018). The hole states are predominantly heavy hole in character. However, even a small admixture of light-hole states reduces the $g$-factor from the heavy hole limit by a large amount Ares et al. (2013); Watzinger et al. (2016). This light-hole admixture is size-dependent which can explain the dependence of the $g$-factor on $A_{\text{VC}}$. However, the $g$-factor is more weakly correlated with $A_{\text{VC}}$ than the emission energy. We speculate that the hole $g$-factor is sensitive to the shape and not just the volume of the QD leading to a weaker correlation with $A_{\text{VC}}$.

Shown in Fig. 2(e) is the splitting between neutral and positively-charged exciton ($E_{X^{0}}-E_{X^{+}}$) as a function of $A_{\text{VC}}$. Using single-particle wavefunctions, $E_{X^{0}}-E_{X^{+}}$ can be associated with the term $E_{eh}-E_{hh}$, where $E_{hh}$, $E_{eh}$ are the direct Coulomb integrals between two holes, and an electron and a hole, respectively Warburton et al. (1998); Cheng et al. (2003). Both terms decrease with increasing lateral size of the QD and hence with increasing size of the VC-area, $A_{\text{VC}}$. Experimentally, we observe a monotonic relation between $A_{\text{VC}}$ and $E_{X^{0}}-E_{X^{+}}$ with a negative correlation ($\rho=-0.819$). For the direct relation between $E_{X^{0}}-E_{X^{+}}$ and the emission energy $E_{X^{0}}$, we find a linear dependence corresponding to a pronounced correlation of $\rho=0.942$ Dalgarno et al. (2008) (see Fig. 2(f)). This dependence also indicates a connection between the lateral and vertical confinements.

Conclusions – We show how the optical properties of QDs grown by GaAs-infilling of Al-droplet-etched nano-holes are connected to the capture-zone model, a concept from nucleation theory. The QD-positions and the optical QD-properties are obtained simultaneously by spatially resolved photoluminescence. The capture zone of QDs is determined by the Voronoi-diagram of the QD-positions. We find a strong negative correlation between the VC-area and the emission energy of QDs. This result can be explained with the capture-zone model applied to the growth-phase of the Al-droplets. A relation between VC-area and further optical properties (diamagnetic shift and neutral-to-charged exciton splitting) shows that not only the vertical but also the lateral QD-size is correlated with the area of the capture zone. We measure these correlations on samples with low QD-densities ($n_{\text{QD}}<1\ \mu\text{m}^{-2}$). Properties of a QD on a nm-scale, which determine its optical emission, are therefore connected to its surroundings on a $\mu$m-scale. This result might be transferable to other nanostructures when strong material diffusion is present during the growth. The correlations between different QD-parameters facilitate preselection of QDs for applications which place stringent requirements on the QD-properties. The correlation between emission energy and capture-zone area has a powerful implication: If all capture-zone areas are identical – e.g. by forcing the nucleation of the Al-droplets on a lattice – then a spectrally narrow wavelength distribution of the QD-ensemble can potentially be engineered. This idea has been successfully applied to stacked QD-layers and QDs in pyramidal nanostructures Teichert et al. (1996); Springholz et al. (1998); Pinczolits et al. (1999); Gogneau et al. (2004); Kiravittaya et al. (2006); Biasiol et al. (2011); Mohan et al. (2010). For the system investigated here, Fig. 1(f) indicates that the QD-ensemble would narrow by a factor of two if all Voronoi-cell areas were in a range of $3-4\ \mu\text{m}^{2}$.

The authors thank Immo Söllner for fruitful discussions on the experiments and Christoph Klöffel for fruitful discussions on the QD hole states. MCL, LZ, JPJ, and RJW acknowledge financial support from SNF Project No. 200020_156637 and from NCCR QSIT. LZ received funding from the European Union Horizon 2020 Research and Innovation programme under the Marie Skłodowska-Curie grant agreement No. 721394 (4PHOTON). AL, JR, and ADW gratefully acknowledge financial support from grants DFH/UFA CDFA05-06, DFG TRR160, and BMBF Q.Link.X 16KIS0867. AL acknowledges funding from DFG via project No. 383065199, LU2051/1-1. YH was supported by NSFC under grant No. 11774326, Shanghai Pujiang R&D Program, and SHSF under grant No. 17ZR1443900. AR acknowledges support from the FWF P29603.

MCL, LZ, and JPJ carried out the experiments. MCL, LZ, AR, and RJW analysed the data. YH and AR performed the AFM-measurement. AR, YH, and OGS grew sample A; JR, ADW, and AL grew sample B. MCL initiated the project. MCL and RJW wrote the paper with input from all authors.

II Supplemental Material

II.1 Frequency Distribution of the Voronoi Cell Areas

The frequency distribution of the VC-areas gives further information about the Al-droplet formation. The probability density distribution, $f_{\text{VC}}$, of the normalized Voronoi cell areas, $A_{\text{VC}}/\langle A_{\text{VC}}\rangle\equiv x$, is often modeled by a generalized Wigner distribution Pimpinelli and Einstein (2007); Groce et al. (2012):

[TABLE]

In our notation, $f(x\mid\eta)$ corresponds to a probability density distribution for $x$ under the condition $\eta$. The parameters $b_{\eta}$ and $a_{\eta}$ are defined by the constraint that $f_{\text{VC}}$ is a normalized probability density distribution with a mean of 1. Explicit expressions for $b_{\eta}$, $a_{\eta}$ are given in Refs. González et al., 2011; Einstein et al., 2014. The parameter $\eta$ can be connected to the critical nucleus size, $i$, via \eta\approx i+2\Groce et al. (2012); Einstein et al. (2014); Pimpinelli et al. (2014).

We fit Eq. S1 to the distribution of normalized VC-areas via likelihood optimization, without making approximations such as Poissonian or Gaussian error estimation Rossi (2018). Let $\{A_{\text{VC}}^{(i)}\}_{i\in\{1..N\}}$ be the set of the areas corresponding to the $N$ different Voronoi cells. We define $\{x^{(i)}\}_{i\in\{1..N\}}$ as the set of all normalized Voronoi cell areas, $x^{(i)}=A_{\text{VC}}^{(i)}/\langle A_{\text{VC}}\rangle$. Eq. S1 assigns a probability density to each value $x^{(i)}$. We assume that every value $x^{(i)}$ corresponds to an independent random variable, $X^{(i)}$. Then, under the condition of a fixed value for $\eta$, the probability density of measuring a set of normalized Voronoi cells, $\{x^{(i)}\}_{i\in\{1..N\}}$, is given by:

[TABLE]

This likelihood distribution, $P\left(\{x^{(i)}\}_{i\in\{1..N\}}\mid\eta\right)$, is maximum for $\eta=\eta_{opt}=3.66$. The mean of the likelihood distribution is $\eta_{m}=3.68$. The difference between mean and maximum of the likelihood distribution is small because the distribution is close-to symmetric and only slightly biased. The found value of the parameter $\eta$ corresponds to a critical nucleus size of $i\approx 2$.

The likelihood distribution, $P\left(\{x^{(i)}\}_{i\in\{1..N\}}\mid\eta\right)$, has a standard deviation of $\sigma_{\eta}=0.33$ when varying $\eta$ for a fixed measurement, $\{x^{(i)}\}_{i\in\{1..N\}}$. Bayes’ theorem connects the probability density for measuring $\{x^{(i)}\}_{i\in\{1..N\}}$ under the condition of a fixed parameter $\eta$ to the probability density for $\eta$ under the condition of a measurement, $\{x^{(i)}\}_{i\in\{1..N\}}$:

[TABLE]

The distribution $P\left(\{x^{(i)}\}_{i\in\{1..N\}}\right)$ does not depend on $\eta$. Furthermore, we assume a uniform prior distribution, $P\left(\eta\right)$. In this case, both probability density distributions $P\left(\eta\mid\{x^{(i)}\}_{i\in\{1..N\}}\right)$ and $P\left(\{x^{(i)}\}_{i\in\{1..N\}}\mid\eta\right)$ are equal up to a prefactor which does not depend on $\eta$. Therefore, the width of both distributions is identical. The error on the parameter $\eta$ is given by $\sigma_{\eta}=0.33$, the standard deviation calculated for the likelihood distribution, $P\left(\{x^{(i)}\}_{i\in\{1..N\}}\mid\eta\right)$.

Shown in Fig. S1 is the distribution of the relative frequency, $f_{\text{VC}}$, of the normalized VC-area, $A_{\text{VC}}/\langle A_{\text{VC}}\rangle\equiv x$. In this figure, the normalized Voronoi cell areas, $\{x^{(i)}\}_{i\in\{1..N\}}$, are divided in finite intervals. The solid red line is the fit ($\eta=\eta_{opt}=3.66$) of Eq. S1 to the data, $\{x^{(i)}\}_{i\in\{1..N\}}$.

II.2 Correction of the Non-Linearity of the Piezo-Scanners

The determination of Voronoi-cell sizes is based on measuring the PL as a function of the position. Such a PL-map is carried out by scanning the sample position with piezoelectric xy-scanners. The scanners have a hysteresis which, however, does not affect our measurements as we perform all measurements while scanning in the forward direction. Besides, the piezo-scanner position depends non-linearly on the applied voltage. This non-linearity could potentially be corrected by calibrating the piezo-scanners with a well-defined reference-structure. Here, we use a different approach: The emission spot of a QD appears differently in size depending on the absolute position of the scanner. The QD spot size is a direct measure of the non-linear dependence between the applied piezo-voltage and the position. The lateral size of a QD itself ($<40\ \text{nm}$) is negligible in comparison to the spot size. We can, therefore, use the widths of the QD spots as a reference to compensate for the distortion of the PL-map. The corresponding procedure is explained in the caption of Fig. S2. The distortion correction works well due to the large number of QDs on each PL-map. A comparison of a PL-map with and without distortion correction is shown in Fig. S2(c, d).

II.3 AFM-Measurement

We perform an AFM-measurement on a reference sample for which the growth is stopped after etching of the nano-holes. An AFM-image of a $5\times 5\ \mu m^{2}$ large region is shown in Fig. S3. The measurement suggests that nano-holes with other nano-holes in their proximity are shallower in comparison to more separated ones. This finding is consistent with the capture zone model and the results obtained by spatially resolved photoluminescence. The size of the AFM-image does not allow for a quantitative analysis comparable to which is done based on the spatially resolved PL-measurements.

II.4 Emission Energy vs. Voronoi Cell Area

Shown in Fig. S4 is the dependence between the emission energy and the Voronoi cell (VC) area, $A_{\text{VC}}$, for the two different samples (A, B). For both samples, we observe that the emission energy decreases with increasing Voronoi-cell area, $A_{\text{VC}}$.

Eq. 1 of the main text suggests that the relation between $A_{\text{VC}}$ and emission energy is non-linear. In Fig. S4 we plot the emission energy as a function of the expression on the right-hand side of Eq. 1. The correlation of this dependence is higher ($\lvert\rho\rvert=0.879$) than the correlation between emission energy and $A_{\text{VC}}$ ($\lvert\rho\rvert=0.812$). This result suggests that Eq. 1 of the main text is a better description of the data than a linear approximation.