volume | PIER Journals

Abstract

We apply the third-order susceptibility tensor generated by the Simplified Bond Hyperpolarizability Model (SBHM) to address the long standing challenges in distinguishing the bulk and surface quadrupolar second-harmonic-generation (SHG) contributions in diamond lattices, such as silicon, which exhibit bulk inversion symmetry. Assuming that the quadrupolar contribution originates from the interface gradient of the excited electric field, we demonstrate through symmetry considerations and numerical calculations for Si(001) and Si(111) facet orientations that it is not possible to separate the different quadrupolar contributions when the incoming light is incident normally. However, we show that such separation is achievable with oblique incidence. Furthermore, we propose a novel experimental design to measure the bulk and surface quadrupolar SHG contributions separately by introducing a semi-vicinal surface. Using numerical SBHM simulations, we show for the first time that this semi-vicinal setup can prove the existence of spatial dispersion, a nonlinear dipolar bulk effect recently proposed. This approach may lead to a better understanding of various nonlinear contributions in silicon and enable precise nonlinear surface monitoring.

1. Franken, P. A., Alan E. Hill, C. W. Peters, and Gabriel Weinreich, "Generation of optical harmonics," Physical Review Letters, Vol. 7, No. 4, 118, 1961.

2. Maiman, T. H., "Stimulated optical radiation in ruby," Nature, Vol. 187, 493-494, 1960.

3. Bloembergen, Nicolaas and P. S. Pershan, "Light waves at the boundary of nonlinear media," Physical Review, Vol. 128, No. 2, 606, 1962.

4. Armstrong, J. A., N. Bloembergen, J. Ducuing, and Peter S. Pershan, "Interactions between light waves in a nonlinear dielectric," Physical Review, Vol. 127, No. 6, 1918, 1962.

5. Maker, P. D. and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Physical Review, Vol. 137, No. 3A, A801, 1965.

6. Ward, J. F. and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Physical Review, Vol. 185, No. 1, 57, 1969.

7. Mendoza, Bernardo S. and W. Luis Mochán, "Polarizable-bond model for second-harmonic generation," Physical Review B, Vol. 55, No. 4, 2489, 1997.

8. Arzate, N. and Bernardo S. Mendoza, "Polarizable bond model for optical spectra of Si (100) reconstructed surfaces," Physical Review B, Vol. 63, No. 11, 113303, 2001.

9. Aspnes, D. E., "Bond models in linear and nonlinear optics," Ultrafast Nonlinear Imaging and Spectroscopy III, Vol. 9584, 21-31, 2015.

10. Powell, G. D., J.-F. Wang, and D. E. Aspnes, "Simplified bond-hyperpolarizability model of second harmonic generation," Physical Review B, Vol. 65, No. 20, 205320, 2002.

11. Wang, J.-F. T., G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, "Simplified bond-hyperpolarizability model of second harmonic generation: Application to Si-dielectric interfaces," Journal of Vacuum Science & Technology B, Vol. 20, No. 4, 1699-1705, 2002.

12. McGilp, J. F., "Using steps at the Si–SiO2 interface to test simple bond models of the optical second-harmonic response," Journal of Physics: Condensed Matter, Vol. 19, No. 1, 016006, 2007.

13. Hardhienata, Hendradi, Andrii Prylepa, David Stifter, and Kurt Hingerl, "Simplified bond-hyperpolarizability model of second-harmonic-generation in Si (111): theory and experiment," Journal of Physics: Conference Series, Vol. 423, No. 1, 012046, 2013.

14. Alejo-Molina, Adalberto, Hendradi Hardhienata, and Kurt Hingerl, "Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann's principle," Journal of the Optical Society of America B, Vol. 31, No. 3, 526-533, 2014.

15. Alejo-Molina, Adalberto, Kurt Hingerl, and Hendradi Hardhienata, "Model of third harmonic generation and electric field induced optical second harmonic using simplified bond-hyperpolarizability model," Journal of the Optical Society of America B, Vol. 32, No. 4, 562-570, 2015.

16. Hardhienata, Hendradi, Adalberto Alejo-Molina, Cornelia Reitböck, Andrii Prylepa, David Stifter, and Kurt Hingerl, "Bulk dipolar contribution to second-harmonic generation in zincblende," Journal of the Optical Society of America B, Vol. 33, No. 2, 195-201, 2016.

17. Reitböck, Cornelia, David Stifter, Adalberto Alejo-Molina, Kurt Hingerl, and Hendradi Hardhienata, "Bulk quadrupole and interface dipole contribution for second harmonic generation in Si (111)," Journal of Optics, Vol. 18, No. 3, 035501, 2016.

18. Alejo-Molina, A., Hendradi Hardhienata, P. A. Márquez-Aguilar, and Kurt Hingerl, "Facet-dependent electric-field-induced second harmonic generation in silicon and zincblende," Journal of the Optical Society of America B, Vol. 34, No. 6, 1107-1114, 2017.

19. Reitböck, Cornelia, David Stifter, Adalberto Alejo-Molina, Hendradi Hardhienata, and Kurt Hingerl, "Nonlinear ellipsometry of Si (111) by second harmonic generation," Applied Surface Science, Vol. 421, 761-765, 2017.

20. Hardhienata, Hendradi, Tony Ibnu Sumaryada, Benedikt Pesendorfer, and Adalberto Alejo-Molina, "Bond model of second-and third-harmonic generation in body-and face-centered crystal structures," Advances in Materials Science and Engineering, Vol. 2018, No. 1, 7153247, 2018.

21. Hardhienata, Hendradi, Adalberto Alejo-Molina, Muhammad Danang Birowosuto, Amirreza Baghbanpourasl, and Husin Alatas, "Spatial dispersion contribution to second harmonic generation in inversion-symmetric materials," Physical Review B, Vol. 103, No. 12, 125410, 2021.

22. Hardhienata, Hendradi, Salim Faci, Adalberto Alejo-Molina, Mohammad Ryan Priatama, Husin Alatas, and Muhammad Danang Birowosuto, "Quo vadis nonlinear optics? An alternative and simple approach to third rank tensors in semiconductors," Symmetry, Vol. 14, No. 1, 127, 2022.

23. Chen, Wei-Ting, Ting-Yu Yen, Yang-Hao Hung, and Kuang-Yao Lo, "Structure of an in situ phosphorus-doped silicon ultrathin film analyzed using second harmonic generation and simplified bond-hyperpolarizability model," Nanomaterials, Vol. 12, No. 23, 4307, 2022.

24. Chen, Wei-Ting, Ting-Yu Yen, Yang-Hao Hung, Yu-Hsiang Huang, Shang-Jui Chiu, and Kuang-Yao Lo, "Second harmonic generation and simplified bond hyperpolarizability model analyses on the intermixing of Si/SiGe stacked multilayers for gate-all-around structure," Nanotechnology, Vol. 34, No. 14, 145702, 2023.

25. Bloembergen, Nicolaas, Richard K. Chang, S. S. Jha, and C. H. Lee, "Optical second-harmonic generation in reflection from media with inversion symmetry," Physical Review, Vol. 174, No. 3, 813, 1968.

26. Bauer, Klaus-Dieter, Martin Panholzer, and Kurt Hingerl, "Bulk quadrupole contribution to second harmonic generation from a microscopic response function," Physica Status Solidi (B), Vol. 253, No. 2, 234-240, 2016.

27. Bauer, Klaus-Dieter and Kurt Hingerl, "Bulk quadrupole contribution to second harmonic generation from classical oscillator model in silicon," Optics Express, Vol. 25, No. 22, 26567-26580, 2017.

28. Hardhienata, Hendradi, Ignu Priyadi, Husin Alatas, Muhammad Danang Birowosuto, and Philippe Coquet, "Bond model of second-harmonic generation in wurtzite ZnO (0002) structures with twin boundaries," Journal of the Optical Society of America B, Vol. 36, No. 4, 1127-1137, 2019.

29. Boyd, R. W., Nonlinear Optics, Academic Press, 2003.

30. Aspnes, David E. and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Physical Review B, Vol. 27, No. 2, 985, 1983.

31. Hardhienata, Hendradi, Hasan Al Kharfan, Salim Faci, Muhammad Danang Birowosuto, and Husin Alatas, "Bond model of second harmonic generation in tetragonal and orthorhombic perovskite structures," Journal of the Optical Society of America B, Vol. 40, No. 11, 2773-2781, 2023.

Dr. Damián Zúiga-Avelar

Dr. Omar Palillero-Sandoval

Dr. Rosibel Carrada-Legaria

Dr. Muhammad Ahyad

Dr. Hendradi Hardhienata

Dr. Adalberto Alejo-Molina