Critical Size and Doping Thresholds Governing Band Gap Evolution in Semiconductors
Abstract
Understanding how the band gap ( Eg) of semiconductors evolves with size, dimensionality, and doping concentration is crucial for optimizing modern electronic and optoelectronic devices. In this work, we perform a systematic analysis of critical sizes ( Lc) and doping thresholds (Nc ) governing significant band gap modification in Si, GaAs, InP, CdS, and GaN. Using effective mass theory with Coulomb corrections, Varshni temperature dependence, and numercical simulations, we identify that quantum confinement dominates when L ≲ 2a*B, yielding Lc = 10 nm for Si, 22 nm for GaAs, 6 nm for CdS, and 5–6 nm for GaN. The corresponding Mott-like critical doping thresholds satisfy Nc1⁄3 a*B≈0.25, giving Nc = 1.8·1018 cm-3 (Si), 5.6·1017 cm-3 (GaAs), and 2.9·1018cm-3(CdS). For quantum dots (0D) at r = 2nm, band gaps increase by ~0.9 eV for Si, ~1.3 eV for GaAs, and ~5.5 eV for GaN, while 1D nanowires exhibit 20–35% smaller shifts due to partial carrier delocalization along the wire axis. Temperature effects are minor (~0.01–0.03 eV from 50–500 K), confirming that dimensional confinement is the dominant factor. These results provide quantitative guidelines for engineering tunable band gaps in LEDs, lasers, Si tandem solar cells, UV optoelectronics, and photodetectors, offering a predictive framework for IV, III–V, and II–VI semiconductor nanostructures.
Downloads
References
C.Y. Ho, and C.H. Ho, “Phonon frequency and energy of electron-phonon interactions at the band gap of InAs, InP, GaP, and GaN semiconductors,” Discov. Mater. 6, 8 (2026). https://doi.org/10.1007/s43939-025-00474-6
M.T. Islam, and H. Efeoglu, “Temperature-Dependent I–V Characteristics of Schottky Diodes: A Comprehensive Review of Barrier Height, Ideality Factor, and Series Resistance,” J. Electron. Mater. 54, 10824–10857 (2025). https://doi.org/10.1007/s11664-025-12432-2
D.Q. Fang, A.L. Rosa, Th. Frauenheim, and R.Q. Zhang, “Band gap engineering of GaN nanowires by surface functionalization,” Applied Physics Letters, 94(7), 073116 (2009). https://doi.org/10.1063/1.3086316
S. Fan, Z.J. Yu, Y. Sun, W. Weigand, P. Dhingra, M. Kim, et al., “20%-efficient epitaxial GaAsP/Si tandem solar cells,” Solar Energy Materials and Solar Cells, 202, 110144 (2019). https://doi.org/10.1016/j.solmat.2019.110144
M. Verma, S. Routray, G.S. Sahoo, and G.P. Mishra, “Bandgap engineered GaAs₀.₉₅P₀.₀₅ solar cell with double BSF,” Advanced Natural Sciences: Nanoscience and Nanotechnology, 14(1), 015010 (2023). https://doi.org/10.1088/2043-6254/acb6e6
M. Verma, S. Routray, G.S. Sahoo, and G.P. Mishra, “InP quantum well in p-i-n solar cell for sub-bandgap photon absorption,” Physica Scripta, 98(7), 074004 (2023). https://doi.org/10.1088/1402-4896/ad00c6
J.Sh. Abdullayev, I.B. Sapaev, J.Sh. Abdullayev, D.A. Juraev, M.J. Jalalov, and E.E. Elsayed, “Mathematical modeling of incomplete ionization in radial p-Si/n-GaAs heterojunctions: temperature and doping effects,” Journal of Electronic Materials, 54, 1–9 (2025). https://doi.org/10.1007/s11664-025-12391-8
J.Sh. Abdullayev, U.S. Rakhmonov, N. Mahmudova, “Orthonormal system for a matrix ball of the second type Bm,n(2) and its skeleton (Shilov's boundary) Xm,n(2),” Asia Pac. J. Math. 10, 27 (2023). https://doi.org/10.28924/APJM/10-27
D. Li, C. Luo, H. Wang, F. Ling, and J. Yao, “Active control of plasmon-induced transparency based on a GaAs/Si heterojunction in the terahertz range,” Optical Materials, 114, 111609 (2021). https://doi.org/10.1016/j.optmat.2021.111609
J. S. Abdullayev, M. S. Ibragimova, J. S. Abdullayev, and I. B. Sapaev, “Cryogenic material and electrophysical changes in Si and GaAs,” East European Journal of Physics, (1), 343–350 (2026). https://doi.org/10.26565/2312-4334-2026-1-40
J.S. Abdullayev, M.S. Ibragimova, J.S. Abdullayev, and I.B. Sapaev, “Thermal expansion characteristics of planar and radial Si/GaAs p–n heterojunctions,” East European Journal of Physics, (1), 388–395 (2026). https://doi.org/10.26565/2312-4334-2026-1-46
M.N. Hasan, Y. Zheng, J. Lai, E. Swinnich, O.G. Licata, M.A. Baboli, B. Mazumder, et al., “Influences of native oxide on the properties of ultrathin Al₂O₃-interfaced Si/GaAs heterojunctions,” Advanced Materials Interfaces, 9(13), 2101531 (2022). https://doi.org/10.1002/admi.202101531
M. Jurisch, F. Börner, T. Bünger, S. Eichler, T. Flade, U. Kretzer, A. Köhler, et al., “LEC- and VGF-growth of SI GaAs single crystals—Recent developments and current issues,” Journal of Crystal Growth, 275(1–2), 283–291 (2005). https://doi.org/10.1016/j.jcrysgro.2004.10.092
J.Sh. Abdullayev, and I.B. Sapaev, “Factors influencing the ideality factor of semiconductor p-n and p-i-n junction structures at cryogenic temperatures,” East European Journal of Physics, (4), 329–333 (2024). https://doi.org/10.26565/2312-4334-2024-4-37
C.D. Thurmond, “The standard thermodynamic functions for the formation of electrons and holes in Ge, Si, GaAs, and GaP,” Journal of The Electrochemical Society, 122(8), 1133 (1975). https://doi.org/10.1149/1.2134410
J. Herfort, H.-P. Schönherr, and K.H. Ploog, “Epitaxial growth of hybrid structures,” Applied Physics Letters, 83(18), 3912 3914 (2003). https://doi.org/10.1063/1.1625426
J.Sh. Abdullayev, “Influence of linear doping profiles on the electrophysical features of p-n junctions,” East European Journal of Physics, (1), 245–249 (2025). https://doi.org/10.26565/2312-4334-2025-1-26
S. Heun, M. Sugiyama, S. Maeyama, Y. Watanabe, K. Wada, and M. Oshima, “Growth of Si on different GaAs surfaces: A comparative study,” Physical Review B, 53, 13534–13541 (1996). https://doi.org/10.1103/PhysRevB.53.13534
P.K. Saxena, P. Srivastava, and A. Srivastava, “Defect analysis of MBE reactor-grown HgCdTe on Si, GaAs, GaSb, and CZT substrates through the TNL-Epigrow simulator,” Journal of Electronic Materials, 53, 5803–5812 (2024). https://doi.org/10.1007/s11664-024-11082-0
A.B. Roy, N.H.R. Valiji, R. Mohammad, P. Giridhar, and P. Mondal, “Performance enhancement of Si/GaAs based heterojunction solar cells by opto-electronics modeling and optimization,” in: 2024 International Conference on Recent Advances in Electrical, Electronics, Ubiquitous Communication, and Computational Intelligence (RAEEUCCI) (pp. 1–6). IEEE. https://doi.org/10.1109/RAEEUCCI61380.2024.10547792
J.Sh. Abdullayev, and I.B. Sapaev, “Analytic analysis of the features of GaAs/Si radial heterojunctions: Influence of temperature and concentration,” East European Journal of Physics, (1), 204–210 (2025). https://doi.org/10.26565/2312-4334-2025-1-21
M. Piriyev, G. Loget, Y. Léger, L. Chen, A. Létoublon, T. Rohel, C. Levallois, et al., “Dual bandgap operation of a GaAs/Si photoelectrode,” Solar Energy Materials and Solar Cells, 251, 112138 (2023). https://doi.org/10.1016/j.solmat.2022.112138
J. Alanis, S.J. Gutiérrez-Ojeda, R. Méndez-Camacho, and E. Cruz-Hernández, “Theoretical investigation of the growth of GaAs on Si(001), Si(110), Si(111), Si(113), and Si(331),” Surfaces and Interfaces, 44, 103792 (2024). https://doi.org/10.1016/j.surfin.2023.103792
G.K. Khudayberganov, J.S. Abdullayev, and U.S. Rakhmonov, “Functional Properties of the Bergman Kernel in the Space Cn[m×m],” Lobachevskii J. Math. 46, 1322–1335 (2025). https://doi.org/10.1134/S1995080225605247
J. Sadullayev, M. Akhmedov, M. Vapayev, I. Davletov, and G. Boltaev, “Modeling of Thermal Effects in a Polyimide Target Under Pulsed Laser Irradiation,” East European Journal of Physics, (1), 274-280 (2026). https://doi.org/10.26565/2312-4334-2026-1-31
R. Huang, Q. Wang, Y. Guo, and Z. Wang, “Comparative study on GaAs/Si heterojunction fabricated by nitrogen and oxygen plasma activated bonding,” Vacuum, 208, 111735 (2023). https://doi.org/10.1016/j.vacuum.2022.111735
M. Yamaguchi, T. Takamoto, H. Juso, K. Nakamura, R. Ozaki, and N. Kojima, “33.7% efficiency Si tandem solar cell modules,” in: Proceedings of the 2024 IEEE 52nd Photovoltaic Specialist Conference (PVSC), (IEEE, 2024). https://doi.org/10.1109/PVSC57443.2024.10749502
A. Šagátová, A. Novák, E. Kováčová, O. Riabukhin, S. Kotorová, and B. Zaťko, “Radiation-degraded Si GaAs detectors and their metallization,” AIP Conference Proceedings, 2778, 060009 (2023). https://doi.org/10.1063/5.0137383
J. Liang, L. Chai, S. Nishida, M. Morimoto, and N. Shigekawa, “Investigation on the interface resistance of Si/GaAs heterojunctions fabricated by surface-activated bonding,” Japanese Journal of Applied Physics, 54(3), 030211 (2015). https://doi.org/10.7567/JJAP.54.030211
J.Sh. Abdullayev, I.B. Sapaev, and S.R. Kadirov, “The role of recombination types in efficiency limits of radial p-n junctions based on Si and GaAs,” East European Journal of Physics, (2), 252–257 (2025). https://doi.org/10.26565/2312-4334-2025-2-30
M. Haris, S.A. Loan, and Mainuddin, “Si/GaAs hetero junction tunnel FET: Design and investigation,” Journal of Nanoelectronics and Optoelectronics, 14(10), 1434–1444 (2019). https://doi.org/10.1166/jno.2019.2575
J.S. Abdullayev, D.A. Qalandarova, M.S. Ibragimova, et al. “Experimental and Simulation-Based Investigation of p-Si/n-CdS Heterojunctions: From Cryogenic Freeze-Out to Room Temperature Operation,” J. Electron. Mater. 55, 2229–2239 (2026). https://doi.org/10.1007/s11664-025-12642-8
J. Sh. Abdullayev, D. A. Qalandarova, and M. Sh. Ibragimova, “Impact of incomplete ionization on the critical electric field of p–n junction structures based on Si and GaAs,” Low Temperature Physics, 52(2), 164–169 (2026). https://doi.org/10.1063/10.0042291
D. A. Qalandarova, M. S. Ibragimova, J. S. Abdullayev, and I. B. Sapaev, “Mathematical modeling of electrostatic potential in radial and planar p–n junctions: A comparative study,” East European Journal of Physics, (1), 333–342 (2026). https://doi.org/10.26565/2312-4334-2026-1-39
U.S. Rakhmonov, J.Sh. Abdullayev, “On properties of the second type matrix ball Bm,n(2) from space Cn[m×m],” J. Sib. Fed. Univ. Math. Phys. 15(3), 329–342 (2022). https://doi.org/10.17516/1997-1397-2022-15-3-329-342
J. Sh. Abdullayev, “Estimates the Bergman kernel for classical domains É. Cartan's,” Chebyshevskii Sb. 22(3), 20–31 (2021). https://doi.org/10.22405/2226-8383-2018-22-3-20-31
Copyright (c) 2026 J.Sh. Abdullayev, D.A. Qalandarova, M.Sh. Ibragimova, U.A. Akberadjiyeva, D.I. Yunusova, Zevarjon Jumaboyeva, Sh.A. Shoyusupov, I.O. Jumaniyozov
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
