Optical properties of epitaxial quantum dots are of considerable interest for high efficiency opto-electronic devices and room-temperature quantum light sources. In this work, the general crystal orientation effects on the spontaneous emission peak of wurtzite (WZ) GaN/AlN quantum dots (QDs) grown on semi-polar substrates is studied theoretically. As crystal orientation changes from Θ=0° to Θ=90°, the spontaneous emission peak is predicted to increase almost ten-fold. This drastic change of the spontaneous emission peak is caused by the rapid increase of the optical dipole matrix elements with crystal orientation angle. The change of the optical matrix element is due to the change of the screening of built-in potential of the WZ QDs with the crystal orientation change. In particular, the spontaneous emission peak rapidly increases when the crystal angle exceeds Θ=50°.
Research Article
Crystal orientation effects on the optical properties of wurtzite GaN/AlN quantum dots on semi-polar substrates
https://doi.org/10.21203/rs.3.rs-1860375/v1
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Epitaxial quantum dots (QDs) are of growing interest for not only for optoelectronic devices but also for potential quantum light sources for room-temperature applications [1–8]. For example, GaN QDs embedded in AlN exhibits large exciton binding energies in excess of 150 meV [9], which is comparable to I-VII semiconductors such as CuCl [7], making them ideal candidates for quantum light emitters at room temperature and above. Recently, by relying on bulk AlN single crystal substrate, the growth GaN QDs using Stranski-Krastanov (SK) growth mode was demonstrated [8]. Until now, it was believed that the SK growth mode transition would be difficult because of the small lattice mismatch of 2.5% between GaN and AlN.
For successful quantum light sources, feasibility of enhancing spontaneous emission of above mentioned III-nitride QDs is of considerable importance [10, 11]. Wurtzite (WZ) III-nitride structures have a large spontaneous (SP) electric dipoles along the [0001] direction (c-axis) of the lattice in addition to piezoelectric (PZ) effect [12, 13]. Built-in field due to both SP and PZ The built-in field leads to the substantial deterioration of the optical recombination rates.
In the case of quantum wells (QWs), the crystal orientation effect on optical characteristics of WZ GaN-based QWs has been studied [14–16]. Similarly, in the case of QDs, electronic and optical properties on non-polar QDs have been studied [15–22].
Recently, we have reported that the light emission intensity of the non-polar \(\left(11\overline{2}0\right)\)GaN/AlN QD structure is expected to be larger than that of the c-plane (0001) GaN/AlN QD structure due to a reduced internal potential, although facets along the [0001] direction still remain even when grown on a non-polar substrate [23]. On the other hand, not much work has been done on the general crystal orientation effect on the optical properties of WZ III-nitride QDs grown on semi-polar substrates. Here, we study the general crystal orientation effect on the spontaneous emission characteristics of WZ GaN/AlN QDs grown on semi-polar substrate for various crystal orientation angle \(\theta\).
The spontaneous emission coefficient for the quantum dot is given by [23]
where \({m}_{o}\) is the free-electron mass, \(\omega\)is the angular frequency, \({\mu }_{o}\) is the vacuum permeability,
\({\tau }_{in}\) is the intraband relaxation time, which is assumed to be \(1\times {10}^{-13}s\), \({{M}}_{m}^{n}\) is the optical dipole matrix element between the quantum state \(n\) of the conduction band and the state \(m\) of the valence band, \({f}_{c}^{n}\) and \({f}_{v}^{m}\) are the distribution functions for the conduction band and valence band, respectively, and \({E}_{hm}^{en}\) is the excitation energy. Here, we consider a cubic QD structure (GaN) grown on GaN of a length \(d\), which is embedded in AlN cladding material with a size of \(200\times 200\times 200\left({Å }^{3}\right)\). The electronic properties of QDs on the semi-polar substrates is calculated from the generalized Luttinger-Kohn \(6\times 6\) Hamiltonian for WZ crystals [24].
Figure 1 shows the potential along \(z\)-axe as a function of crystal angle (\(\theta\)) for GaN/AlN QD structures grown on GaN substrate. The length \(d\) of cubic QD is set to be 40 Å
The (0001)-oriented GaN/AlN QD structure with \(\theta ={0}^{o}\) shows that there exists a large potential along \(z\)-axis which is caused by the piezoelectric polarization arising from the strain. On the other hand, QDs grown on semi-polar substrate, this potential is gradually decreasing with increasing crystal angle. As a result, in the case of the GaN/AlN QD structure with \(\theta ={90}^{o}\), the potential is shown to almost vanish.
Figure 2 shows quasi-Fermi levels in conduction and valence bands \(z\)-axe for various crystal angle (\(\theta\)) for GaN/AlN QD structures grown on semi-polar GaN substrate. Here, the quasi-Fermi-level separation \(\varDelta {E}_{fc}\) (\(\varDelta {E}_{fv}\)) is defined as the energy difference between the quasi-Fermi level and the ground-state energy in the conduction band (the valence band). The quasi-Fermi-level separation are obtained at the sheet carrier density of \({N}_{3D}=10\times {10}^{24}c{m}^{-3}\). In the case of the conduction band, the quasi-Fermi-level separation is independent of the crystal angle. On the other hand, the quasi-Fermi-level separation in the valence band is shown to be affected by the crystal angle. That is, the absolute value of the quasi-Fermi-level separation gradually decreases with increasing crystal angle, shows a minimum value at about \(\theta ={50}^{o}\), and begins to increase when the crystal angle is further increased.
Figure 3 shows (a) optical matrix elements for the TE-polarization with \(\theta ={0}^{o}\) and 90\({}^{o}\) and (b) averaged optical matrix elements as a function of the crystal angle for GaN/AlN QD structures. The \(x\) or \(y\)coordinate indicates the subband index in the conduction and valence bands.
The optical matrix elements were obtained for the first two subbands in the conduction band and the first four subbands. This is attributed by the fact that light intensities for the transitions between the
higher subbands are negligible because the quasi-Fermi levels in the conduction and valence bands is below the first subbands in both bands. All optical matrix elements for a given crystal orientation were summed and averaged over total transition number between the conduction and valence bands for the TE-polarization. The averaged optical matrix element for the TE-polarization, as shown in
Figure 3(b) is enhanced with increasing crystal angle. In particular, it rapidly increases when the crystal angle exceeds \(\theta ={50}^{o}\).
In Fig. 4, we plotted the spontaneous emission peak for the TE-polarization as a function of the crystal angle for GaN/AlN QD structures. The spontaneous emission peak for the TE-polarization gradually increases with increasing crystal angle. However, we observe that it rapidly increases when the crystal angle exceeds \(\theta ={50}^{o}\). This is mainly attributed to the rapid increase in the averaged optical matrix element for the TE-polarization because the absolute value of the quasi-Fermi levels in the valence band increases when the crystal angle exceeds \(\theta ={50}^{o}\).
These results suggest that the successful QDs growth on non-polar substrates with \(\theta >{50}^{o}\) would be interesting for high efficiency optoelectronic devices as well as room-temperature quantum light source applications.
In summary, the spontaneous emission peak of wurtzite GaN/AlN quantum dots grown on semi-polar substrates is investigated as a function of the crystal orientation. The quasi-Fermi-level separation in the valence band is affected by the crystal angle while that in the conduction band is independent of the crystal angle. The spontaneous emission peak for the TE-polarization rapidly increases with increasing crystal angle. This is mainly attributed to the rapid increase in the averaged optical matrix element for the TE-polarization. Thus, we expect that semipolar or nonpolar GaN/AlN quantum dot structures with \(\theta >{50}^{o}\) would be of considerable interest.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(2021R1F1A1048588). D. A. was also supported by Korea National Research Foundation (NRF) grant No. NRF-2020M3E4A1080031, NRF grant No. NRF-2020M3H3A1105796, ICT R&D program of MSIT/IITP 2021-0-01810 and AFOSR grant FA2386-21-1-0089.
Data availability
The data that support the plots in this paper are available from the corresponding author upon reasonable request.
Disclosure
The authors declare no competing interests.
- G. Reith Maier, M. Kniberg, F. Flossing, S. Lichtmanecker, K. Muller, A. Andrejew, J. Vuckovic, R. Cross, J. J. Finley, Nano Lett. 15, 5208 (2015)
- P. Lodahl, S. Mahmoodian, S. Stobbe, Rev. Mod. Phys. 87, 347 (2015).
- A. Schliwa, M. Winkelnkemper, A. Lochmann, E. Stock, D. Bimberg, Phys. Rev. Lett. B 80, 161307 (2009).
- S. Castelleto, B. C. Johnson, V. Ivady, N. Stavrias, T. Umeda, A. Gali, T. Ohshima, Nat. Mater. 13, 151 (2014).
- N. Mizuochi, T. Makino, H. Kato, D. Takeuchi, M. Ogura, H. Okushi, M. Nothaft, P. Nuumann, A. Gali, E. Jelezko, J. Wrachtrup, S. Yamaski, Nat. Photonics 6, 299 (2012).
- P. Michler, A. Imamoglu, M. D. Mason, G. F. Strouse, S. K. Buratto, Nature 406, 968 (2000).
- D. Ahn, J. D. Song, S. S. Kang, J. Y. Lim, S. H. Yang, S. Ko, S. H. Park, S. J. Park, D. S. Kim, H. J. Chang, J. Chang, Sci. Reports 10, 3995 (2020).
- S. Tamariz, G. Callsen, N. Grandjean, Appl. Phys. Lett. 114, 082101 (2019).
- G. Honig, G. Callsen, A. Schliwa, S. Kalinoski, C. Kindel, S. Kato, Y. Arakawa, D. Bimberg, A. Hoffmann, Nat. Commun. 5, 5721 (2014).
- D. Dovzhenko et al., Opt. Express. 28, 22705 (2020).
- D.Bimberg, M.Grundmann and N.N.Ledentsov: Quantum Dot Heterostructure(Wiley, New York, 1999), Chap. 1.
- F. Bernardini, V. Fiorentini and D. Vanderbilt, Phys. Rev. B56, 10024 (1997).
- G. Martin, A. Botchkarev, A. Rockettand H. Morkoc, Appl. Phys. Lett. 68, 2541 (1996).
- S.-H.Park and S.L.Chuang: Phys. Rev.B 59, 4725 (1999).
- F.Mireles and S.E.Ulloa: Phys. Rev.B 62, 2562 (2000).
- T.Takeuchi, H.Amano and I.Akasaki: Jpn. J. Appl. Phys. 39, 413 (2000).
- O. Marquardt, T. Hickel, and J. Neugebauer, J. Appl. Phys. 106, 083707 (2009).
- S. Schulz, A. Berube, and E. P. OâReilly, Phys. Rev. B 79, 081401 (R) (2009).
- S. Schulz and E. P. O'Reilly, Phys. Status Solidi C 7, 80 (2010).
- M. A. Caro, S. Schulz, S. B. Healy, and E. P. OâReilly, J. Appl. Phys.109, 084110 (2011).
- T. D. Young, G. Jurczak, A. Lotsari, G. P. Dimitrakopulos, Ph. Komninou, and P. DÅuewski, Phys. Status Solidi B 252, 2296 (2015).
- J. A. Budagosky, N. Garro, A. Cros, and A. Garca-Cristbal, Materials Science in Semiconductor Processing 49, 76 (2016).
- S.-H. Park and D. Ahn, MDPI Electronics9, 1256 (2020).
- S.-H. Park and S. L. Chuang, Phys. Rev. B 59, 4725 (1999).
No competing interests reported.