A Unified Model for Size-, Shape-, and Composition-Dependent Bandgap in Semiconductor Nanocrystals: Beyond the Effective Mass Approximation

Authors

Keywords:

Quantum confinement, Bandgap engineering, Semiconductor nanocrystals, Shape effects, Nanoalloys

Abstract

Quantum confinement in semiconductor nanocrystals enables precise bandgap engineering for optoelectronic applications. The Brus equation provides a foundational description for spherical particles but cannot account for shape anisotropy, surface states, or compositional inhomogeneity in alloyed systems. This work presents a unified analytical model that extends the Brus equation by incorporating three physically motivated elements: a shape-dependent confinement energy derived from an infinite-barrier box model and expressed through an anisotropy factor α_conf; a negative surface-state correction proportional to the surface-to-volume ratio that defines a critical radius for surface-dominated behaviour; and three mixing rules for binary semiconductor nanoalloys. The most advanced mixing rule, Rule S, couples a modified Butler isotherm to the confinement model to capture surface segregation. Validation against 28 experimental data points encompassing CdSe spheres, cubes, and rods, PbS quantum dots, and alloyed Cd1-xZnxS nanocrystals yields an overall root-mean-square error of 0.07 eV, representing up to a factor-of-five improvement over the original Brus equation for anisotropic particles (3.2× for CdSe rods, 4.7× for PbS spheres). The fully analytical model evaluates a single composition–size–shape configuration in 2 - 5 ms, enabling the screening of thousands of candidates per minute on standard hardware. This framework provides a computationally efficient and physically transparent tool for predictive bandgap engineering in semiconductor nanocrystals.

Dimensions

Alivisatos, A. P. (1996). Semiconductor clusters, nanocrystals, and quantum dots. Science, 271(5251), 933–937. https://doi.org/10.1126/science.271.5251.933.

Aubert, T., Golovatenko, A. A., Samoli, M., Lermusiaux, L., Zinn, T., Abécassis, B., Rodina, A. V., & Hens, Z. (2022). General expression for the size-dependent optical properties of quantum dots. Nano Letters, 22(4), 1778–1785. https://doi.org/10.1021/acs.nanolett.1c04199.

Boles, M. A., Ling, D., Hyeon, T., & Talapin, D. V. (2016). The surface science of nanocrystals. Nature Materials, 15(2), 141–153. https://doi.org/10.1038/nmat4526.

Brus, L. E. (1984). Electron–electron and electron–hole interactions in small semiconductor crystallites: The size dependence of the lowest excited electronic state. Journal of Chemical Physics, 80(9), 4403–4409. https://doi.org/10.1063/1.447218.

Efros, A. L., & Efros, A. L. (1982). Interband absorption of light in a semiconductor sphere. Soviet Physics—Semiconductors, 16(7), 772–775.

Ekimov, A. I., & Onushchenko, A. A. (1984). Size quantization of the electron energy spectrum in microscopic semiconductor crystals. JETP Letters, 40(8), 1136–1139.

García de Arquer, F. P., Talapin, D. V., Klimov, V. I., Arakawa, Y., Bayer, M., & Sargent, E. H. (2021). Semiconductor quantum dots: Technological progress and future challenges. Science, 373(6555), eaaz8541. https://doi.org/10.1126/science.aaz8541.

Hu, J., Li, L., Yang, W., Manna, L., Wang, L., & Alivisatos, A. P. (2001). Linearly polarized emission from colloidal semicouctor quantum rods. Science, 292(5524), 2060–2063. https://doi.org/10.1126/science.1060810.

Ithurria, S., & Dubertret, B. (2008). Quasi-2D colloidal CdSe platelets with thicknesses controlled at the atomic level. Journal of the American Chemical Society, 130(49), 16504–16505. https://doi.org/10.1021/ja807724e.

Kahmann, S., & Loi, M. A. (2020). Trap states in lead chalcogenide colloidal quantum dots—origin, impact, and remedies. Applied Physics Reviews, 7(4), 041305. https://doi.org/10.1063/5.0013166.

Kamat, P. V. (2008). Quantum dot solar cells. Semiconductor nanocrystals as light harvesters. Journal of Physical Chemistry C, 112(48), 18737–18753. https://doi.org/10.1021/jp806791s.

Kayanuma, Y. (1988). Quantum-size effects of interacting electrons and holes in semiconductor microcrystals with spherical shape. Physical Review B, 38(14), 9797–9805. https://doi.org/10.1103/PhysRevB.38.9797.

Loss, D., & DiVincenzo, D. P. (1998). Quantum computation with quantum dots. Physical Review A, 57(1), 120–126. https://doi.org/10.1103/PhysRevA.57.120.

Michalet, X., Pinaud, F. F., Bentolila, L. A., Tsay, J. M., Doose, S., Li, J. J., Sundaresan, G., Wu, A. M., Gambhir, S. S., & Weiss, S. (2005). Quantum dots for live cells, in vivo imaging, and diagnostics. Science, 307(5709), 538–544. https://doi.org/10.1126/science.1104274.

Moreels, I., Lambert, K., Smeets, D., De Muynck, D., Nollet, T., Vanhaecke, F., Vantomme, A., Li, J., Allan, G., & Hens, Z. (2009). Size-dependent optical properties of colloidal PbS quantum dots. ACS Nano, 3(10), 3023–3030. https://doi.org/10.1021/nn900863a.

Munyebvu, N., Lane, E., Grisan, E., & Howes, P. D. (2022). Accelerating colloidal quantum dot innovation with algorithms and automation. Materials Advances, 3(18), 6950–6967. https://doi.org/10.1039/D2MA00139J.

Murray, C. B., Norris, D. J., & Bawendi, M. G. (1993). Synthesis and characterization of nearly monodisperse CdE (E = S, Se, Te) semiconductor nanocrystallites. Journal of the American Chemical Society, 115(19), 8706–8715. https://doi.org/10.1021/ja00072a025.

Nozik, A. J., Beard, M. C., Luther, J. M., Law, M., Ellingson, R. J., & Johnson, J. C. (2010). Semiconductor quantum dots and quantum dot arrays and applications of multiple exciton generation to third-generation photovoltaic solar cells. Chemical Reviews, 110(11), 6873–6890. https://doi.org/10.1021/cr900289f.

Peng, X., Manna, L., Yang, W., Wickham, J., Scher, E., Kadavanich, A., & Alivisatos, A. P. (2000). Shape control of CdSe nanocrystals. Nature, 404(6773), 59–61. https://doi.org/10.1038/35003535.

Resch-Genger, U., Grabolle, M., Cavaliere-Jaricot, S., Nitschke, R., & Nann, T. (2008). Quantum dots versus organic dyes as fluorescent labels. Nature Methods, 5(9), 763–775. https://doi.org/10.1038/nmeth.1248.

Sklénard, B., Mugny, G., Chehaibou, B., Delerue, C., Arnaud, A., & Li, J. (2022). Size and solvation effects on electronic and optical properties of PbS quantum dots. Journal of Physical Chemistry Letters, 13(39), 9044–9050. https://doi.org/10.1021/acs.jpclett.2c02197.

Uwaoma, C. J., Oriaku, C. I., Nwaokorongwu, E. C., & Okwara, E. I. (2025). Theoretical Investigation of Quantum Confinements in Spherical PbSrSe Semiconductor Quantum Dots. Nigerian Journal of Theoretical and Environmental Physics, 3(1), 29–36. https://doi.org/10.62292/njtep.v3i1.2025.60.

Uwaoma, C. J., Oriaku, C. I., Joseph, U., & Nnanna, L. A. (2024). Theoretical Study of Exciton Properties of Dilute GaInAsN Semiconductors. Nigerian Journal of Theoretical and Environmental Physics, 2(2), 194–202. https://doi.org/10.62292/njtep.v2i2.2024.52.

Zhong, X., Feng, Y., Knoll, W., & Han, M. (2003). Alloyed ZnₓCd₁₋ₓS nanocrystals with highly narrow luminescence spectral wiAlivisatos, A. P. (1996). Semiconductor clusters, nanocrystals, and quantum dots. Science, 271(5251), 933–937. https://doi.org/10.1126/science.271.5251.933. DOI: https://doi.org/10.1126/science.271.5251.933

Aubert, T., Golovatenko, A. A., Samoli, M., Lermusiaux, L., Zinn, T., Abécassis, B., Rodina, A. V., & Hens, Z. (2022). General expression for the size-dependent optical properties of quantum dots. Nano Letters, 22(4), 1778–1785. https://doi.org/10.1021/acs.nanolett.1c04199. DOI: https://doi.org/10.1021/acs.nanolett.2c00056

Boles, M. A., Ling, D., Hyeon, T., & Talapin, D. V. (2016). The surface science of nanocrystals. Nature Materials, 15(2), 141–153. https://doi.org/10.1038/nmat4526. DOI: https://doi.org/10.1038/nmat4526

Brus, L. E. (1984). Electron–electron and electron–hole interactions in small semiconductor crystallites: The size dependence of the lowest excited electronic state. Journal of Chemical Physics, 80(9), 4403–4409. https://doi.org/10.1063/1.447218. DOI: https://doi.org/10.1063/1.447218

Efros, A. L., & Efros, A. L. (1982). Interband absorption of light in a semiconductor sphere. Soviet Physics—Semiconductors, 16(7), 772–775.

Ekimov, A. I., & Onushchenko, A. A. (1984). Size quantization of the electron energy spectrum in microscopic semiconductor crystals. JETP Letters, 40(8), 1136–1139.

García de Arquer, F. P., Talapin, D. V., Klimov, V. I., Arakawa, Y., Bayer, M., & Sargent, E. H. (2021). Semiconductor quantum dots: Technological progress and future challenges. Science, 373(6555), eaaz8541. https://doi.org/10.1126/science.aaz8541. DOI: https://doi.org/10.1126/science.aaz8541

Hu, J., Li, L., Yang, W., Manna, L., Wang, L., & Alivisatos, A. P. (2001). Linearly polarized emission from colloidal semicouctor quantum rods. Science, 292(5524), 2060–2063. https://doi.org/10.1126/science.1060810. DOI: https://doi.org/10.1126/science.1060810

Ithurria, S., & Dubertret, B. (2008). Quasi-2D colloidal CdSe platelets with thicknesses controlled at the atomic level. Journal of the American Chemical Society, 130(49), 16504–16505. https://doi.org/10.1021/ja807724e. DOI: https://doi.org/10.1021/ja807724e

Kahmann, S., & Loi, M. A. (2020). Trap states in lead chalcogenide colloidal quantum dots—origin, impact, and remedies. Applied Physics Reviews, 7(4), 041305. https://doi.org/10.1063/5.0013166. DOI: https://doi.org/10.1063/5.0019800

Kamat, P. V. (2008). Quantum dot solar cells. Semiconductor nanocrystals as light harvesters. Journal of Physical Chemistry C, 112(48), 18737–18753. https://doi.org/10.1021/jp806791s. DOI: https://doi.org/10.1021/jp806791s

Kayanuma, Y. (1988). Quantum-size effects of interacting electrons and holes in semiconductor microcrystals with spherical shape. Physical Review B, 38(14), 9797–9805. https://doi.org/10.1103/PhysRevB.38.9797. DOI: https://doi.org/10.1103/PhysRevB.38.9797

Loss, D., & DiVincenzo, D. P. (1998). Quantum computation with quantum dots. Physical Review A, 57(1), 120–126. https://doi.org/10.1103/PhysRevA.57.120. DOI: https://doi.org/10.1103/PhysRevA.57.120

Michalet, X., Pinaud, F. F., Bentolila, L. A., Tsay, J. M., Doose, S., Li, J. J., Sundaresan, G., Wu, A. M., Gambhir, S. S., & Weiss, S. (2005). Quantum dots for live cells, in vivo imaging, and diagnostics. Science, 307(5709), 538–544. https://doi.org/10.1126/science.1104274. DOI: https://doi.org/10.1126/science.1104274

Moreels, I., Lambert, K., Smeets, D., De Muynck, D., Nollet, T., Vanhaecke, F., Vantomme, A., Li, J., Allan, G., & Hens, Z. (2009). Size-dependent optical properties of colloidal PbS quantum dots. ACS Nano, 3(10), 3023–3030. https://doi.org/10.1021/nn900863a. DOI: https://doi.org/10.1021/nn900863a

Munyebvu, N., Lane, E., Grisan, E., & Howes, P. D. (2022). Accelerating colloidal quantum dot innovation with algorithms and automation. Materials Advances, 3(18), 6950–6967. https://doi.org/10.1039/D2MA00139J. DOI: https://doi.org/10.1039/D2MA00468B

Murray, C. B., Norris, D. J., & Bawendi, M. G. (1993). Synthesis and characterization of nearly monodisperse CdE (E = S, Se, Te) semiconductor nanocrystallites. Journal of the American Chemical Society, 115(19), 8706–8715. https://doi.org/10.1021/ja00072a025. DOI: https://doi.org/10.1021/ja00072a025

Nozik, A. J., Beard, M. C., Luther, J. M., Law, M., Ellingson, R. J., & Johnson, J. C. (2010). Semiconductor quantum dots and quantum dot arrays and applications of multiple exciton generation to third-generation photovoltaic solar cells. Chemical Reviews, 110(11), 6873–6890. https://doi.org/10.1021/cr900289f. DOI: https://doi.org/10.1021/cr900289f

Peng, X., Manna, L., Yang, W., Wickham, J., Scher, E., Kadavanich, A., & Alivisatos, A. P. (2000). Shape control of CdSe nanocrystals. Nature, 404(6773), 59–61. https://doi.org/10.1038/35003535. DOI: https://doi.org/10.1038/35003535

Resch-Genger, U., Grabolle, M., Cavaliere-Jaricot, S., Nitschke, R., & Nann, T. (2008). Quantum dots versus organic dyes as fluorescent labels. Nature Methods, 5(9), 763–775. https://doi.org/10.1038/nmeth.1248. DOI: https://doi.org/10.1038/nmeth.1248

Sklénard, B., Mugny, G., Chehaibou, B., Delerue, C., Arnaud, A., & Li, J. (2022). Size and solvation effects on electronic and optical properties of PbS quantum dots. Journal of Physical Chemistry Letters, 13(39), 9044–9050. https://doi.org/10.1021/acs.jpclett.2c02197. DOI: https://doi.org/10.1021/acs.jpclett.2c02247

Uwaoma, C. J., Oriaku, C. I., Nwaokorongwu, E. C., & Okwara, E. I. (2025). Theoretical Investigation of Quantum Confinements in Spherical PbSrSe Semiconductor Quantum Dots. Nigerian Journal of Theoretical and Environmental Physics, 3(1), 29–36. https://doi.org/10.62292/njtep.v3i1.2025.60. DOI: https://doi.org/10.62292/njtep.v3i1.2025.60

Uwaoma, C. J., Oriaku, C. I., Joseph, U., & Nnanna, L. A. (2024). Theoretical Study of Exciton Properties of Dilute GaInAsN Semiconductors. Nigerian Journal of Theoretical and Environmental Physics, 2(2), 194–202. https://doi.org/10.62292/njtep.v2i2.2024.52. DOI: https://doi.org/10.62292/njtep.v2i2.2024.52

Zhong, X., Feng, Y., Knoll, W., & Han, M. (2003). Alloyed ZnₓCd₁₋ₓS nanocrystals with highly narrow luminescence spectral width. Journal of the American Chemical Society, 125(44), 13559–13563. https://doi.org/10.1021/ja0353598. DOI: https://doi.org/10.1021/ja036683a

dth. Journal of the American Chemical Society, 125(44), 13559–13563. https://doi.org/10.1021/ja0353598.

Published

2026-07-04

How to Cite

Ogunji, A. B., Stephen, A., Almaghamsi, H., & Atilade, A. O. (2026). A Unified Model for Size-, Shape-, and Composition-Dependent Bandgap in Semiconductor Nanocrystals: Beyond the Effective Mass Approximation. Nigerian Journal of Theoretical and Environmental Physics, 4(1), 106-118. https://doi.org/10.62292/njtep.v4i1.2026.141

Issue

Vol. 4 No. 1 (2026): Nigerian Journal of Theoretical and Environmental Physics - Vol. 4 No. 1

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Articles
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Copyright (c) 2026 Abdulrahaman Babajide Ogunji, Abayomi Stephen, Hanaa Almaghamsi, Adesanya Oluwafemi Atilade

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite

Ogunji, A. B., Stephen, A., Almaghamsi, H., & Atilade, A. O. (2026). A Unified Model for Size-, Shape-, and Composition-Dependent Bandgap in Semiconductor Nanocrystals: Beyond the Effective Mass Approximation. Nigerian Journal of Theoretical and Environmental Physics, 4(1), 106-118. https://doi.org/10.62292/njtep.v4i1.2026.141

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