Strain-induced control of a pillar cavity-GaAs single quantum dot photon source

Herein, we present the calculated strain-induced control of single GaAs/AlGaAs quantum dots (QDs) integrated into semiconductor micropillar cavities. We show precise energy control of individual single GaAs QD excitons under multi-modal stress fields of tailored micropillar optomechanical resonators. Further, using a three-dimensional envelope-function model, we evaluated the quantum mechanical correction in the QD band structures depending on their geometrical shape asymmetries and, more interestingly, on the practical degree of Al interdiffusion. Our theoretical calculations provide the practical quantum error margins, obtained by evaluating Al-interdiffused QDs that were engineered through a front-edge droplet epitaxy technique, for tuning engineered QD single-photon sources, facilitating a scalable on-chip integration of QD entangled photons.


Results and Discussions
Adjusting the nano-or micro-scale distributed Bragg reflector (DBR) cavity of a singly clamped pillar offers several degrees of freedom with completely different natures. Figure 1 shows how the frequencies of the four characteristic oscillation eigenmodes in a standard micropillar cavity depend on the pillar aspect ratio g (defined as R/h, where R and h are the radius and height of the pillar, respectively). These results were obtained via a finite element simulation. As the pillar widened relative to its height (g < 0. 3), the eigenfrequencies f n of the first two flexural modes increased such that β π ρ = f h YI A /(2 ) / n n 2 2 41 . Here, the first two eigenvalues of the vibration modes correspond to β 1 = 1.875 and β 2 = 4.694 42 . For Al x Ga 1−x As epitaxial structures, the alloy-composition-dependent Young's modulus Y and mass density ρ correspond to (85.3 − 1.8x) [MPa] and (5320 − 1560x) [kg/m 3 ], respectively 43 . In addition, I = πR 4 /4 is the area moment of inertia and A is the cross-sectional area. While the longitudinal eigenfrequencies were almost independent of g, the hybrid mode frequencies degenerated at two points within a relative frequency splitting (Δf/f) of 10 −3 , which are labeled D1 and D2 (with g D1 ~ 0.1 and g D2 ~ 0.2, respectively) in Fig. 1. The hybrid resonances with higher-order modes can be set in a strong-coupling regime 44,45 , controlling the resonant dynamics via the coherent exchange of phonons [46][47][48] . In our DBR-based SPS prototype, the degenerate eigenfrequencies of the two hybrid modes were tunable from several tens of MHz to ~650 MHz.
Controlling the pillar ellipticity adds additional degrees of freedom, leading to asymmetric mechanical polarization distributions. The semi-major and semi-minor radii of the elliptic pillar resonators are denoted as R x and R y , respectively. As the ellipse's axial ratio R y /R x reduced toward 0.5, the fundamental oscillation frequency of the major axis increased with a relative frequency splitting of 0.4. Elliptical VCSEL resonators with low axial ratios developed torsional breathing modes (green open circles in Fig. 1(b)). The longitudinal breathing characteristics were independent of the pillar ellipticity. In the second flexural mode, two separate elliptic modes appeared within 2% of the resonant frequencies.
These distinct multimode mechanical vibrations were accompanied by non-uniform strain fields in the pillar geometries. Figure 2(a) shows typical in-plane principal-stress components of the various modes applied on the QD embedded in the middle of the AlGaAs cavity layer inside the pillar. The stress components linearly varied by several tens of MPa. Owing to the quadratic distributions of stress in the flexural modes and the non-uniformity of the longitudinal stress, our investigation of the stress-field gradients Δ σ x z m is restricted in the ranges |r/R| < 0.4. The Δ σ x z m along the x-axis were determined as 4 and 17 MPa/μm for the first two flexural modes, respectively. We adjusted the aspect ratio of the VCSEL pillar from 0.2 to 0.9 at a fixed radius of 1.5 μm such that a high Purcell factor could be obtained 49 . Uniaxial stresses, σ z L , of up to 40 MPa can be generated in the QD plane by longitudinally displacing the free-end of the cylindrical pillar by 1 nm (≈10 6 x zpa ) at g ~ 0.4 (see Fig. 2(b)). Here, x zpa denotes the zero-point amplitude of the first flexural mode. In addition, we calculated the uniform in-plane mean stress profiles at 0 MPa and 22 MPa for the radial and longitudinal breathing modes, respectively. As Fig. 2(c) demonstrates, increasing the ellipticity decreased the axial stress gradient. The gradient of a highly elliptic cavity resonator (axial ratio 0.5) was almost 50% that of the symmetric flexural resonances. The (ellipticity-independent) longitudinal resonance simply increased the relative principal stress by 15%. Here, the sign of the applied stress depends on the vibrational phase.
At each resonance, an applied stress can change the exciton energy levels of the strain-free GaAs QDs embedded in a cylindrical VCSEL pillar resonator. A single crystal under uniaxial stress undergoes various deformation processes. First, a hydrostatic stress produces the isotropic crystal lattice distortion, shifting the QD conduction/ valence band extrema to higher/lower energies, relative to the mechanical phase. Second, a shear stress lowers the symmetry of zinc-blend crystals, splitting their heavy-and light-hole valence states. Consequently, the bandgap  (5), 17 (15), 27 (25), and 37 (35). energy between the heavy hole and conduction bands also shifts to a lower energy. Under an external (tensile) uniaxial stress, the reduced QD height modifies the quantum potential. Such a change in the confinement potential not only increases the electron-hole sub-band confinement energies, but also alters the strength of the electron-hole Coulomb interaction. Even small intrinsic strains in GaAs/AlGaAs QDs can significantly alter the quantum confinement parameters 15 . Figure 3(a) depicts the calculated shift in the QD excitonic bandgap energy  δE G under characteristic multimode strain fields (e.g., Fig. 2(a)). To exploit our system as an optomechanical oscillator for QD SPSs, we examined the magnitudes of the rates of change of  δ Δ E x G in the core region |r/R| < 0.4. We tuned the cross-sectional rates of change to 0.2, 1.4, and 0.7 meV/μm, obtaining the first flexural mode, the second flexural mode, and the radial breathing mode at g ~ 0.9, respectively ( Fig. 3(b)). In the longitudinal mode, we observed an almost uniform energy shift of the GaAs QDs embedded in the micropillar cavity. Notably,  δ Δ E x G was characterized by higher degree polynomials undergoing the nonlinear stress fields in the flexural and longitudinal modes outside the core region (see Figs. 2(a) and 3(a)). The combined effects of the hydrostatic and shear stresses experienced by a direct bandgap semiconductor QD can be described by the Bir-Pikus Hamiltonian 50 where the hydrostatic and shear deformation energies, E H and Q ε , R ε , and S ε , are defined, respectively, as Here, ε ij denotes the components of the strain tensor. The coefficient a denotes the hydrostatic deformation potential, whereas the coefficients b and d denote the valence-band shear deformation potentials, corresponding to the strain tensors with symmetries Γ 1 , Γ 3 , and Γ 4 50 . For the bulk case of GaAs, the shear deformation energy is simply given by Q ε based on an external uniaxial stress applied along the [001] growth direction. The simple case of strain correspondingly induces the bandgap energy shift δ Fig. 3(a) and symbols). In practice, under the stress applied by the fundamental flexural oscillation (cf. Fig. 2), various types of shear deformation potentials produce a two-fold decrease in the rate of bandgap change  δ Δ E x G (red lines and closed red circles, in Fig. 3(a,b)). Shear deformation effects in pillar cavities with three different aspect ratios induce non-zero rates of change (between 0.3 and 0.7 meV/μm; Fig. 3(b)) in the radial breathing mode. The size of the bump in the center (|r/R| < 0.02) of the energy response curve of the flexural modes increases with the increase in axial ratio of the elliptic pillar cavity (Fig. 3(c)). At an axial ratio of 0.5, the slope of the least-squares-fitted line  δ Δ E x G for the first flexural mode was twice that of the symmetric cavity resonator. As the quadratic behaviors of the shear deformations of the bump, was approximately 70 μeV under a uniaxial hydrostatic stress of 1 MPa (Fig. 3(d)). The substantial correction to the quantum confinement surpasses the stress-dependent bandgap change (60 μeV/ MPa; Figs. 2 and 3). The geometrical parameters were obtained from the cross-sectional transmission electron microscope (TEM) images 15,52 . Here, the QD emission energy (1.653 eV) was engineered to couple to the cavity's resonant mode. Figure 3(d) demonstrates the considerable quantum confinement effects by precisely controlling the ellipticity R R / y x QD QD of pure GaAs QDs. The stress-induced confinement characteristics,  Δ E S Q , were determined by the slope of a quantum confinement energy-stress curve obtained using a 3D envelope-function formalism. We evaluated the 3D QD confining potentials relative to structural parameters (R x y / QD and h) and the uniaxial stress-induced change in the confinement geometry. At a minor-major axis ratio of ~0.9, the stress-induced rate  Δ E S Q was changed by 10% (7 μeV/MPa) relative to the symmetric QD. The quantum-confinement effect  δE Q has a standard deviation of 3% over an axial ratio of 0.5-1. Furthermore, we investigated the effect of Al interdiffusion on the excitonic band gap in the GaAs DE QDs. To construct a realistic confinement potential profile, we studied the empirical in-depth Al profile for an optimized type of GaAs DE QD 15,52 . In a previous two-dimensional study, we measured the position-dependent mole fraction x(z) of Al along the QD's growth axis using TEM-energy dispersive X-ray spectroscopy. Then, we applied the polynomials obtained via least-squares-fitting to the well-known bandgap formula 2 53 . According to this analysis wherein we used 3D envelope-function modeling, the DE QD recombination energy red-shifted by ~1.7% and the quantum confinement correction of ~16%. In addition, the stress-induced shift rate  Δ E S Q changed by 5% in its realistic confining potential profile of Al-interdiffused QDs (cf. Fig. 3(e)). Our 3D calculation showed that the individual rate of change  Δ E S Q , which depended on the shape anisotropies of the confinement potential in the GaAs QDs, was also shifted by approximately 5%. These results provide the practical margins for GaAs QD tuning and for improving the source performances of QD SPS applications (such as nondestructive position mapping and imaging methods). Our methodology presented in this study can be directly implemented in versatile www.nature.com/scientificreports www.nature.com/scientificreports/ micropillar-microcavity embedding quantum emitters. For quantum telecommunication on optical fiber networks, such precise control can be utilized in hybrid two-level systems such as InGaAsP/InP [54][55][56] and GaSb/GaAs QDs 57,58 , clad by lattice-matched DBR pillars. In general, our methods can be applied to improve the source performance of different types of solid-state quantum-emitters embedded in cylindrical structures. Because the quantum error in the energy shift was estimated down to E E 5% / Q Al X , the subwavelength imaging/positioning of QDs theoretically outperforms other optical nanoscopy such as the stimulated emission depletion, in which 5-nm resolution of nitrogen-vacancy centers was achieved 59 . Meanwhile, the practical quantum error is set by employing the quantum-mechanical corrections in the individual confinement geometries of solid-state emitters. Destructive composition depth-profile analyses of individual emitters are mainly needed for their high-precision control at the quantum level.

conclusion
We have theoretically demonstrated full frequency control of strain-free GaAs QDs embedded in micropillar cavity SPSs. By harnessing strain coupling, we fine-tuned the QD excitonic energy with cross-sectional shift rates of few meV/μm in tailored micropillar optomechanical resonators. Using an envelope-function model of 3D GaAs QDs, we demonstrated that the tuning/positioning precision of the quantum confinement energy is limited to 5% by varying shapes and compositions of the QDs. Our approach, which exploits the nano-optomechanics of QD-cavity systems, is fully compatible with integrated photonic circuits, providing an intriguing avenue for developing hybrid quantum and classical computers.

Methods
We employ a 3D envelope-function method developed within k ⋅ p perturbation theory to calculate the quantum mechanical characteristics of lens-shaped DE QDs of GaAs/Al 0.3 Ga 0.7 As. The geometrical parameters such as height and base lengths were extracted from the cross-sectional TEM images 15,52 . The QD energy of a single electron-hole pair comprises the GaAs bulk bandgap E G , the sub-band energies E S of the carriers, and the direct Coulomb interaction energy J. Here, the quantity = − | | E E J Q S characterizes the quantum confinement of an electron-hole system in the QD. To determine the sub-band confinement energies E S and the electron-hole Coulomb interaction energy J in the stress-dependent QD confinement potential V(r), we numerically solved the following Schrödinger equation: Using a finite-difference method, we evaluated the 3D Coulomb integral of J projected onto the exciton state |ψ e ψ h 〉 defined as 15,51