Crystal orientation effects on the optical properties of wurtzite GaN/AlN quantum dots on semi-polar substrates

The study of the optical properties of epitaxial quantum dots is important in the development of high-performance opto-electronic devices and room-temperature quantum light sources. This study focuses on the effects of crystal orientation on the spontaneous emission peak of wurtzite GaN/AlN quantum dots grown on semi-polar substrates. Theoretical analysis shows that a tenfold increase in the spontaneous emission peak can occur as the crystal orientation changes from θ=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =0^\circ$$\end{document} to θ=90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =90^\circ$$\end{document}. This significant shift is attributed to the change in the optical dipole matrix elements, which is caused by the change in the screening of the built-in potential of the wurtzite quantum dots with the variation in crystal orientation. The spontaneous emission peak is seen to increase rapidly when the crystal angle exceeds θ=50∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =50^\circ$$\end{document}.


Introduction
Epitaxial Quantum Dots (QDs) are of increasing interest due to their potential as quantum light sources for room temperature applications in optoelectronic devices (Reith Maier et al. 2015;Lodahl et al. 2015;Schliwa et al. 2009;Castelleto et al. 2014;Mizuochi et al. 2012;Michler et al. 2000;Ahn et al. 2020;Adelmann et al. 2004;Tamariz et al. 2019). Specifically, GaN QDs embedded in AlN have been shown to exhibit large exciton binding energies of over 150 meV (Honig et al. 2014), making them ideal for use as quantum light emitters at room temperature and higher. The growth of Self assembled GaN/AlN quantum dots by means of the Stranski-Krastanov (SK) method is being achieved from about two decades (Adelmann et al. 2004), followed by the recent the growth of GaN QDs on bulk AlN single crystal substrates (Tamariz et al. 2019), despite the previously believed difficulty due to the small lattice mismatch of 2.5% between GaN and AlN. For the development of successful quantum light sources, it is critical to consider the feasibility of enhancing spontaneous emission from these III-nitride QDs (Dovzhenko et al. 2020;Bimberg et al. 1999). Wurtzite (WZ) III-nitride structures possess both a large spontaneous electric dipole along the [0001] (c-axis) direction of the lattice, as well as a piezoelectric effect (Bernardini et al. 1997;Martin et al. 2541). This built-in field leads to significant deterioration of optical recombination rates.
Studies have previously been conducted on the crystal orientation effect on the optical characteristics of WZ GaN-based Quantum Wells (QWs) (Park and Chuang 1999;Mireles and Ulloa 2000;Takeuchi et al. 2000). Similarly, research has been conducted on the electronic and optical properties of non-polar QDs (Mireles and Ulloa 2000;Takeuchi et al. 2000;Marquardt et al. 2009;Schulz et al. 2009;Schulz and O'Reilly 2010;Caro et al. 2011;Young et al. 2296;Budagosky et al. 2016). Our recent work has shown that the light emission intensity of the non-polar 1120 -GaN/AlN QD structure is expected to be greater than that of the c-plane (0001) GaN/AlN QD structure, due to a reduced internal potential, despite the presence of facets along the [0001] direction even when grown on a non-polar substrate . However, limited research has been performed on the general crystal orientation effect on the optical properties of WZ IIInitride QDs grown on semi-polar substrates (Park et al. 2015;Patra et al. 2016;Patra and Schulz 2019).
In this study, we examine the general crystal orientation effect on the spontaneous emission characteristics of WZ GaN/AlN QDs grown on semi-polar substrates for various crystal orientation angles . In this work, we focused on crystal orientation effects on the optical properties of WZ GaN/AlN quantum dots by considering the simplified case of cubic-shaped QDs.

Theoretical model
The Hamiltonian for the valence-band structure has been derived by using the k ⋅ p method The full Hamiltonian for the (0001)-oriented WZ crystal can be written as (Park and Chuang 1999) where (1)

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Here, the A i 's are the valence-band effective-mass parameters, the D i 's are the deformation potentials for WZ crystals, k i is the wave vector, ij is the strain tensor, Δ 1 is the crystal-field split energy, and Δ 2 and Δ 3 account for spin-orbit interactions. The bases for the Hamiltonian are defined as, The Hamiltonian for an arbitrary crystal orientation can be obtained using a rotation matrix.

Rotations of the Euler angles and transform the physical quantities from
The z-axis corresponds to the c-axis [0001], and the growth axis (defined as the z ′ -axis) is normal to the QW plane (hkil) . The relation between the coordinate systems for vectors and tensors is expressed as where summation over repeated indices is indicated.
The optical momentum matrix elements in (x � , y � , z � ) coordinates for a general crystal orientation are given by where Ψ � c and Ψ � v are the wave functions for the conduction and the valence bands, respectively, and = ↑ and ↓ denote electron spins. The polarization-dependent interband momentum-matrix elements can be written as follows: (i) TE-polarization ( ê � = cos �x� + sin �ŷ� ): (3) Also, The spontaneous emission coefficient for the QD is then given by Park and Ahn (2020): where m o is the free-electron mass, is the angular frequency, o is the vacuum permeability, in is the intraband relaxation time (assumed to be 1 × 10 −13 s ), M n m is the optical dipole matrix element between the quantum state n of the conduction band and state m of the valence band, f n c and f m v are the distribution functions for the conduction and valence bands, respectively, and E en hm is the excitation energy. In this study, we consider a cubic QD structure (GaN) grown on GaN of length d , embedded in AlN cladding material with size 200 × 200 × 200 (Å 3 ). The electronic properties of the QDs on semi-polar substrates are calculated using the generalized Luttinger-Kohn 6 × 6 Hamiltonian for WZ crystals (Park et al. 2015). Figure 1 displays the potential along the z-axis as a function of the crystal angle ( ) for GaN/AlN quantum dot (QD) structures grown on a GaN substrate. The length of the cubic QD is fixed at 40 Å. The potential along the z-axis of the (0001)-oriented GaN/AlN QD structure with = 0 • exhibits a large value due to the piezoelectric polarization caused by strain. However, for QDs grown on semi-polar substrates, this potential decreases as the crystal angle increases, reaching close to zero for a GaN/AlN QD structure with = 90 • . Figure 2 presents the quasi-Fermi levels in the conduction and valence bands along the z-axis for various crystal angles (θ) for GaN/AlN QD structures grown on a semi-polar GaN substrate. The quasi-Fermi-level separation ΔE fc ( ΔE fv ) defined as the energy difference between the quasi-Fermi level and the ground-state energy in the conduction or valence band, is calculated at a sheet carrier density ofN 3D = 10 × 10 24 cm −3 . The separation in the conduction band remains constant regardless of the crystal angle, while the separation in the valence band is influenced by the crystal angle, reaching a minimum value at approximately = 50 • and increasing again as the crystal angle continues to increase. Figure 3 shows (a) the optical matrix elements for the TE-polarization with = 0 • and 90° and (b) the averaged optical matrix elements as a function of the crystal angle for GaN/AlN QD structures. The optical matrix elements were obtained for the first two subbands in the conduction band and the first four subbands in the valence band, as transitions between higher subbands have negligible intensities. The average optical matrix elements for the TE-polarization, as presented in Fig. 3b, increase with increasing crystal angle, particularly rapidly exceeding = 50 • .

Results and discussion
In Fig. 4, the spontaneous emission peak for the TE-polarization is plotted as a function of the crystal angle for GaN/AlN QD structures. The peak gradually increases with increasing crystal angle, with a rapid increase observed after exceeds 50°. This is primarily due to the rapid increase in the average optical matrix element for the TE-polarization, which results from the increase in the absolute value of the quasi-Fermi levels in the valence band.
These results indicate that the successful growth of QDs on non-polar substrates with > 50 • would be of interest for high-efficiency optoelectronic devices and roomtemperature quantum light source applications. The findings suggest that the successful Fig. 1 Potential along z-axe as a function of crystal angle ( ) for GaN/AlN QD structures grown on GaN substrate. The length d of cubic QD is set to be 40 Å growth of quantum dots (QDs) on non-polar substrates with thickness greater than 50 Å could be particularly exciting for both high-efficiency optoelectronic devices and roomtemperature quantum light source applications. This is because non-polar substrates are advantageous in optoelectronic device applications, as they possess unique optical and electrical properties that allow for higher efficiencies. Additionally, QDs grown on these substrates could be used as room-temperature quantum light sources due to their high emission stability and efficiency.
Such high-efficiency optoelectronic devices and room-temperature quantum light sources could have a wide range of practical applications, including in optoelectronic data storage, laser systems, and in the development of quantum technologies for information Fig. 2 Quasi-Fermi levels in conduction and valence bands z-axe for various crystal angle ( ) for GaN/AlN QD structures grown on semi-polar GaN substrate Fig. 3 a Optical matrix elements for the TE-polarization with = 0° and 90° and b optical matrix elements as a function of the crystal angle for GaN/AlN QD structures processing and communication. Therefore, further research in this direction is highly justified and could lead to significant technological advancements.
In this study, we focused on the impact of crystal orientation on the optical properties of WZ GaN/AlN cubic-shaped quantum dots (QDs) for the sake of simplicity. It is widely acknowledged that these QDs are island-like in nature and can take various shapes, such as hexagonal-based pyramids or trapezoidal-based, based on the substrate's crystallographic orientation (Adelmann et al. 2004). The intricate relationship between the QD shape and the intrinsic electric field is crucial, as it is closely tied to the material's crystal orientation. Further research is needed to better understand the effects of crystal orientation in selfassembled quantum dots grown using the SK method, and to model more realistic structures that are more in line with experimentally achievable cases.

Summary
The study focuses on analyzing the impact of crystal orientation on the spontaneous emission peak of wurtzite GaN/AlN quantum dots grown on semi-polar substrates. It is found that the quasi-Fermi-level separation in the valence band varies with the crystal angle, while the separation in the conduction band remains constant. The emission peak for the TE-polarization significantly rises as the crystal angle increases, due to the significant increase in the average optical matrix element for the TE-polarization. As a result, it is believed that GaN/AlN quantum dot structures with a crystal angle greater than 50° on semi-polar or nonpolar substrates would be of significant interest.