Semi-Deterministic Quantum Dot Placement in Heteroepitaxy

TL;DR

The work addresses deterministic quantum dot placement during epitaxial growth by using engineered boundary geometry to bias diffusion and nucleation. It develops an overdamped Langevin framework where the adatom density $n$ and current $\mathbf{J}$ follow a Boltzmann-weighted field, with a quasi-stationary balance $\partial_t n + \nabla \cdot \mathbf{J} = 0$. It demonstrates that boundary-anchored QDs create a repulsive field that reshapes the chemical potential landscape, enabling secondary QD nucleation within the bulk, i.e., semi-deterministic placement. This approach enables engineered QD patterns for scalable quantum photonics and cQED systems.

Abstract

Achieving deterministic placement of self-assembled quantum dots (QDs) during epitaxial growth is essential for the reliable and efficient fabrication of high-quality single-photon sources and solid-state cavity quantum electrodynamics (cQED) systems, yet it remains a significant challenge due to the inherent stochasticity of QD nucleation processes. In this work, we theoretically and numerically demonstrate that deterministic QD nucleation within a pristine growth region, e.g., InAs on a (001)-oriented GaAs substrate, can be achieved by engineering the boundary geometry of that region. During epitaxial growth, adatoms initially move toward the boundary and promote the formation of primary QDs along the boundary, driven by curvature and diffusion anisotropy. The resulting primary QDs distribution will generate many-body interactions that dynamically reshape the chemical potential landscape for subsequently deposited adatoms, enabling the formation of secondary QDs within the pristine growth region. These findings provide a theoretical foundation for reliable patterning of high optical-quality QDs, with potential applications in next-generation quantum photonic devices.

Figures (9)

• Figure 1: Illustration of patterned geometries used to generate engineered boundary fields (e.g., strain fields). Red and blue lines indicate the boundary contours of patterned regions that induce localized field gradients. For illustrative purposes, solid black circles represent QD nucleating directly at these boundaries, while open circles denote QDs formed under the collective influence of both the boundary fields and QDs already formed on the boundary. Various pattern geometries demonstrate the flexibility of geometric design in controlling the spatial distribution of boundary-induced QD nucleation. The dark gray regions represent areas that adatoms cannot physically access, such as hard masks (forbidden regions), while light gray regions denote areas where adatom access is energetically unfavorable (partially forbidden). We refer to the exposed pristine surface (white regions) as the bulk in this work.
• Figure 2: The free energy landscapes with four regimes (shaded with different colors) of quantum dot formation, detailed in the main text. The formation self-limited QDs exhibits two local minima representing a dissolved state and self-limited QD state, separated by a free energy barrier, peaked (local maximum) at the density $\hat{n}_\mathrm{LaMer}$. As discussed in Eq. \ref{['LaMer eq']}, the LaMer free energy $\Delta G_\mathrm{LaMer}$ (purple curve) represents, as a function of cluster density $\hat{n}$, the balance between the free energy contributions from interfacial elasticity (cyan curve) and the formation of the cluster surface (red curve). Four regimes (A-D) are separated by three characteristic densities, subcritical density $\hat{n}_{\mathrm{seed}}$, supercritical density $\hat{n}_{\mathrm{supercritical}}$, and self-limited density $\hat{n}_{\mathrm{self-limited}}$, associated with the formation of temporary clusters, clusters lead to coherent QD growth, and self-limited (stable) QDs. In regime D (shaded in blue), the net free energy $\Delta G$ given in Eq. \ref{['total E eq']} may exhibit a convex behavior around the self-limited (stable) QDs. $\hat{n}_{\mathrm{supercritical}}$ and $\hat{n}_{\mathrm{self-limited}}$ are referred as a single critical density $\hat{n}_{\mathrm{crit}}$.
• Figure 3: Detailed comparison of empirical adatom density field (Monte Carlo sampled), QD nucleation, and mean-field analytical adatom density field solutions under a homogeneous circular boundary field, for both isotropic diffusion ($D_x = D_y$) and anisotropic diffusion ($D_x = 3D_y$) along the boundary. A: Atomic flux is uniformly deposited from an epitaxy source onto the center of a circular patterned substrate with radius $R_0=10\, \ell_y$, where $\ell_y$ is the diffusion length in $\mathbf{\hat{y}}$ direction. B--C: The first (leftmost) column shows the empirical adatom density field at a given radius $R_0=10\, \ell_y$. Due to diffusion, adatoms accumulate at the boundary, and the resulting spatial distribution depends on the diffusion anisotropy. For isotropic diffusion, the empirical density exhibits radial symmetry, while for anisotropic diffusion, the distribution becomes directionally biased. The second column shows radial statistics extracted from various radii with respect to the diffusion length $5\, \ell_y <R< 20 \, \ell_y$, further illustrating the breakdown of axial symmetry under anisotropy. In both columns, pairwise interactions are turned off ($w = 0$) and the nucleation potential is absent ($V_{\mathrm{nc}} = 0$). The third column incorporates a nonzero nucleation potential ($V_{\mathrm{nc}} \neq 0$) and displays the resulting QD nucleation sites for various radii $5\, \ell_y <R< 20 \, \ell_y$. Nucleation events are strongly correlated with regions of high adatom density. The fourth column presents mean-field analytical solutions obtained from Eq. \ref{['path-integral']}. D: Cross-section comparison between mean-field analytical solutions and Langevin-based Monte Carlo simulations at a given radius $R_0=10\, \ell_y$, evaluated with respect to angular parametrization under both isotropic and anisotropic conditions.
• Figure 4: Detailed comparison between empirical adatom density fields (Monte Carlo) and mean-field analytical adatom density field solutions under a homogeneous elliptical boundary field for isotropic diffusion ($D_x = D_y$) and anisotropic diffusion ($D_x = 3D_y$) along the boundary. A: Schematic of elliptical boundaries with varying aspect ratios and the corresponding empirical adatom density fields. B--C: Radial statistics extracted from ellipses with different aspect ratios under isotropic diffusion ($D_x = D_y$) and anisotropic diffusion ($D_x = 3D_y$), respectively. The semi-major axis $a$ is oriented parallel ($a > b$), at a $45^{\circ}$ tilt ($a < b$), and orthogonal ($a < b$) to one of the diffusion axes $\mathbf{\hat{x}}$. By incorporating local curvature contributions, the resulting mean-field analytical solutions reproduce key features observed in the Monte Carlo–sampled empirical density fields.
• Figure 5: Geometric tuning of empirical adatom density fields under isotropic diffusion ($D_x = D_y$). A: By incorporating a nonzero repulsive pairwise interaction potential $w$ between adatoms, the resulting empirical adatom density field exhibits a smeared-out asymptotic density profile along the boundary contour. B: Starting from a lozenge boundary, elongation of the right-side flat edge with an aspect ratio $a/R$ induces local anisotropic confinement, leading to enhanced adatom accumulation near the elongated region. This demonstrates that local curvature can modulate the spatial profile of the adatom density field along the boundary. C: Simulations with various boundary geometries under initially uniform atomic deposition illustrate the flexibility and adaptability of the modeling framework in capturing geometry-tunable adatom density profiles and QD nucleation behavior along the boundary contour.