# Modeling of the Electronic Structure of Semiconductor Nanoparticles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Numerical Method

## 4. Results and Comparison with the Experimental Data

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Nicolais, L.; Carotenuto, G. Metal/Polymer Nanocomposites; John Wiley and Sons: New York, NY, USA, 2005. [Google Scholar]
- Trakhtenberg, L.I.; Lin, S.H.; Ilegbusi, O.J. Physico-Chemical Phenomena in Thin Films and at Solid Surfaces; Elsevier Inc.: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Achermann, M. Exciton-Plasmon Interactions in Metal-Semiconductor Nanostructures. J. Phys. Chem. Lett.
**2010**, 1, 2837–2843. [Google Scholar] [CrossRef] - Garcia, G.; Buonsanti, R.; Runnerstrom, E.L.; Mendelsberg, R.J.; Llordes, A.; Anders, A.; Richardson, T.J.; Milliron, D.J. Dynamically Modulating the Surface Plasmon Resonance of Doped Semiconductor Nanocrystals. Nano Lett.
**2011**, 11, 4415–4420. [Google Scholar] [CrossRef] [PubMed] - Boulais, E.; Lachaine, R.; Hatef, A.; Meunier, M. Plasmonics for Pulsed-Laser Cell Nanosurgery: Fundamentals and applications. J. Photochem. Photobiol. C
**2013**, 17, 26–49. [Google Scholar] [CrossRef] - Faucheaux, J.A.; Stanton, A.L.D.; Jain, P.K. Plasmon Resonances of Semiconductor Nanocrystals: Physical Principles and New Opportunities. J. Phys. Chem. Lett.
**2014**, 5, 976–985. [Google Scholar] [CrossRef] [PubMed] - Schröfel, A.; Kratošová, G.; Safark, I.; Šafaříková, M.; Raška, I.; Shor, L.M. Applications of Biosynthesized Metallic Nanoparticles—A Review. Acta Biomater.
**2014**, 10, 4023–4042. [Google Scholar] [CrossRef] [PubMed] - Granitzer, P.; Rumpf, K. Nanostructured Semiconductors—From Basic Research to Applications; Pan Stanford Publishing Ltd.: Singapore, 2014. [Google Scholar]
- Chen, T.; Liu, Y. Semiconductor Nanocrystals and Metal Nanoparticles: Physical Properties and Device Applications; CRC Press Taylor and Francis Group: Boca Raton, FL, USA, 2016. [Google Scholar]
- Jimenez, L.C.; Mendez, H.A.; Paez, B.A.; Ramırez, M.E.; Rodrıguez, H. Production and Characterization of Indium Oxide and Indium Nitride. Braz. J. Phys.
**2006**, 36, 1017–1020. [Google Scholar] [CrossRef] - Prathap, P.; Gowri, D.G.; Subbaiah, Y.P.V.; Ramakrishna, R.K.T.; Ganesan, V. Growth and Characterization of Indium Oxide Films. Curr. Appl. Phys.
**2008**, 8, 120–127. [Google Scholar] [CrossRef] - Barsan, N.; Weimar, U. Conduction Model of Metal Oxide Gas Sensors. J. Electroceram.
**2001**, 7, 143–167. [Google Scholar] [CrossRef] - Yamazoe, N.; Shimanoe, K. Theoretical Approach to the Gas Response of Oxide Semiconductor Film Devices under Control of Gas Diffusion and Reaction Effects. Sens. Actuators B
**2011**, 154, 277–282. [Google Scholar] [CrossRef] - Lewis, B.G.; Paine, D.C. Applications and Processing of Transparent Conducting Oxides. MRS Bull.
**2000**, 25, 22–27. [Google Scholar] [CrossRef] - Bodneva, V.L.; Ilegbusy, O.J.; Kozhushner, M.A.; Kurmangaleev, K.S.; Posvyanskii, V.S.; Trakhtenberg, L.I. Modeling of Sensor Properties for Reducing Gases and Charge Distribution in Nanostructured Oxides: A Comparison of Theory with Experimental Data. Sens. Actuators B Chem.
**2019**, 287, 218–224. [Google Scholar] [CrossRef] - Kurmangaleev, K.S.; Ikim, M.I.; Kozhushner, M.A.; Trakhtenberg, L.I. Electron Distribution and Electrical Resistance in Nanostructured Mixed Oxides CeO
_{2}-In_{2}O_{3}. Appl. Surf. Sci.**2021**, 546, 149011. [Google Scholar] [CrossRef] - Gerasimov, G.N.; Gromov, V.F.; Ilegbusi, O.J.; Trakhtenberg, L.I. The Mechanisms of Sensory Phenomena in Binary Metal-Oxide Nanocomposites. Sens. Actuators B
**2017**, 240, 613–624. [Google Scholar] [CrossRef] - Zandi, O.; Agrawal, A.; Shearer, A.; Reimnitz, L.C.; Dahlman, C.J.; Staller, C.M.; Milliron, D. Impacts of Surface Depletion on the Plasmonic Properties of Doped Semiconductor Nanocrystals. Nat. Mater.
**2017**, 17, 710–717. [Google Scholar] [CrossRef] - Ghini, M.; Curreli, N.; Lodi, M.B.; Petrini, N.; Wang, M.; Prato, M.; Fanti, A.; Manna, L.; Kriegel, I. Control of Electronic Band Profiles through Depletion Layer Engineering in Core–Shell Nanocrystals. Nat. Commun.
**2022**, 13, 537. [Google Scholar] [CrossRef] - Kurmangaleev, K.S.; Bodneva, V.L.; Posvyanskii, V.S.; Trakhtenberg, L.I. Sensoty Effect toward Hydrogen in a Nanostructured CeO
_{2}-In_{2}O_{3}System. Russ. J. Phys. Chem. A**2022**, 96, 2056–2058. [Google Scholar] [CrossRef] - Madon, M.J.; Morrison, S.R. Chemical Sensing with Solid State Devices; Academic Press: San Diego, CA, USA, 1989. [Google Scholar]
- Williams, F.A. Combustion Theory; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Wolkenstein, F.F. Electronic Processes on Semiconductor Surfaces during Chemisorption; Consultants Bureau, Cop.: London, UK, 1990. [Google Scholar]
- Morrison, S.R. The Chemical Physics of Surfaces, 2nd ed.; Springer: Dordrecht, The Netherlands, 1990; 456p. [Google Scholar]
- Vol’pert, A.I.; Khudyaev, S.I. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, 1st ed.; Springer: Dordrecht, The Netherlands, 1985; pp. 626–640. [Google Scholar]
- Gelfand, I.M.; Fomin, S.V. Calculus of Variations; Prentice-Hall: Hoboken, NJ, USA, 1963. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Statistical Physics; Elsevier Science: Amsterdam, The Netherlands, 1980; Volume 5. [Google Scholar]
- Kozhushner, M.A.; Lidskii, B.V.; Oleynik, I.I.; Posvyanskii, V.S.; Trakhtenberg, L.I. Inhomogeneous Charge Distribution in Semiconductor Nanoparticles. J. Phys. Chem. C
**2015**, 119, 16286–16292. [Google Scholar] [CrossRef] - Pines, D. Elementary Excitations in Solids; W.A. Benjamin: New York, NY, USA, 1963. [Google Scholar]
- Kittel, C. Introduction to Solid State Physics, 8th ed.; John Wiley and Sons: New York, NY, USA, 2005. [Google Scholar]
- Godunov, S.K.; Ryaben’kii, V.S. Difference Schemes; North Holland: Amsterdam, The Nederlands, 1987. [Google Scholar]
- Kurmangaleev, K.S.; Trakhtenberg, L.I.; Mikhailova, T.Y. Oxygen Chemisorption on the Surface of an In
_{2}O_{3}(011) nanocrystal. Inorg. Mater.**2020**, 56, 1138–1146. [Google Scholar] [CrossRef] - Dixit, A.; Sudakar, C.; Naik, R.; Naik, V.M.; Lawes, G. Undoped Vacuum Annealed In
_{2}O_{3}Thin Films as a Transparent Conducting. Appl. Phys. Lett.**2009**, 95, 192105. [Google Scholar] [CrossRef] - Zuev, D.A.; Lotin, A.A.; Novodvorsky, O.A.; Lebedev, F.V.; Khramova, O.D.; Petuhov, I.A.; Putilin, P.N.; Shatohin, A.N.; Rumyanzeva, M.N.; Gaskov, A.M. Pulsed Laser Deposition of ITO Thin Films and Their Characteristics. Semiconductors
**2012**, 46, 410–413. [Google Scholar] [CrossRef] - Dumesic, J.A.; Milligan, B.A.; Greppi, L.A.; Balse, V.R.; Sarnowski, K.T.; Beall, C.E.; Kataoka, T.; Rudd, D.F.; Trevino, A.A. A Kinetic Modeling Approach to the Design of Catalysts—Formulation of a Catalyst Design Advisory Program. Ind. Eng. Chem. Res.
**1987**, 26, 1399–1407. [Google Scholar] [CrossRef] - Berhanu, T.; Hoffnagle, J.; Rella, C.; Kimhak, D.; Nyfeler, P.; Leuenberger, M. High-Precision Atmospheric Oxygen Measurement Comparisons Between a Newly Built CRDS Analyzer and Existing Measurement Techniques. Atmos. Meas. Tech.
**2019**, 12, 6803–6826. [Google Scholar] [CrossRef] - Varley, J.; Schleife, A. Bethe-Salpeter Calculation of Optical-Absorption Spectra of In
_{2}O_{3}and Ga_{2}O_{3}. Semicond. Sci. Technol.**2015**, 30, 024010. [Google Scholar] [CrossRef] - Trakhtenberg, L.I.; Gerasimov, G.N.; Gromov, V.F.; Belysheva, T.V.; Ilegbusy, O.J. Effect of Composition on Sensing Properties of SnO
_{2}+ In_{2}O_{3}Mixed Nanostructured Films. Sens. Actuators B**2012**, 169, 32–38. [Google Scholar] [CrossRef] - Gerasimov, G.N.; Gromov, V.F.; Ikim, M.I.; Ilegbusi, O.J.; Trakhtenberg, L.I. Effect of Interaction between Components of In
_{2}O_{3}-CeO_{2}and SnO_{2}-CeO_{2}Nanocomposites on Structure and Sensing Properties. Sens. Actuators B**2019**, 279, 22–30. [Google Scholar] [CrossRef]

**Figure 1.**The temperature dependence of the sensor effect for $I{n}_{2}{O}_{3}$ system at 2% hydrogen in the air. The dots represent the experimental data [38]. The curve is the calculation result (fitting).

**Figure 2.**The temperature dependence of the sensor effect for the mixed system $Ce{O}_{2}$-$I{n}_{2}{O}_{3}$ at 2% hydrogen in the air. The dots represent the experimental data [39]. The curve is the calculation result (fitting).

Mass unit in ASU | ${m}_{e}$ | $9.109\times \phantom{\rule{0.166667em}{0ex}}{10}^{-31}$ kg |

Length unit in ASU | ${a}_{0}$ | $5.292\times \phantom{\rule{0.166667em}{0ex}}{10}^{-11}$ m |

Time unit in ASU | ${t}_{h}$ | $2.419\times \phantom{\rule{0.166667em}{0ex}}{10}^{-17}$ s |

Energy unit in ASU | ${E}_{h}$ | $4.360\times \phantom{\rule{0.166667em}{0ex}}{10}^{-18}$ J |

Parameter | Meaning | Value |
---|---|---|

${K}_{dis}\phantom{\rule{0.166667em}{0ex}}\left[{t}_{h}^{-1}\right]$ | dissociation constant of adsorbed molecules ${O}_{2}$ | ${\nu}_{O-O}exp(-\frac{{\displaystyle {\epsilon}_{dis}}}{kT})$ |

${K}_{rec}\phantom{\rule{0.166667em}{0ex}}\left[{a}_{0}^{2}{t}_{h}^{-1}\right]$ | recombination constant of adsorbed atoms O | ${a}_{O}^{2}{\nu}_{O}exp(-\frac{{\displaystyle {\epsilon}_{a}^{O}}}{kT})$ |

${K}_{ad}\phantom{\rule{0.166667em}{0ex}}\left[{a}_{0}^{-2}{t}_{h}^{-1}\right]$ | adsorption constant of molecules ${O}_{2}$ | $\frac{{\displaystyle 1}}{{\displaystyle 4}}{\tilde{n}}_{{O}_{2}}\sqrt{\frac{3kT}{{\displaystyle {m}_{{O}_{2}}}}}{\alpha}_{{O}_{2}}\left(T\right)$ |

${K}_{des}\phantom{\rule{0.166667em}{0ex}}\left[{t}_{h}^{-1}\right]$ | desorption constant of molecules ${O}_{2}$ | ${\nu}_{{O}_{2}}exp(-\frac{{\displaystyle {\epsilon}_{des}}}{kT})$ |

${K}_{O}^{des}\phantom{\rule{0.166667em}{0ex}}\left[{t}_{h}^{-1}\right]$ | desorption constant of atoms O | ${\nu}_{O}^{des}exp(-\frac{{\displaystyle {\epsilon}_{des}^{O}}}{kT})$ |

${K}_{ret}\phantom{\rule{0.166667em}{0ex}}\left[{t}_{h}^{-1}\right]$ | electron return constant into nanoparticle volume | ${\nu}_{ret}exp(-\frac{{\displaystyle {\epsilon}_{ret}}}{kT})$ |

Parameter | Meaning | Value |
---|---|---|

${K}_{cap}\phantom{\rule{0.166667em}{0ex}}\left[{t}_{h}^{-1}\right]$ | electron capture constant by oxygen atom | $1.25\times {10}^{-17}+3.07\times \phantom{\rule{0.166667em}{0ex}}{10}^{-19}{e}^{\frac{T-383.94}{96.07}}$ |

${K}_{{H}_{2}O}\phantom{\rule{0.166667em}{0ex}}\left[{a}_{0}^{3}{t}_{h}^{-1}\right]$ | reaction rate constant of hydrogen-oxygen reaction | $7.92\times {10}^{-11}T{e}^{-\frac{486.34}{T}}$ |

Parameter | Value | Dimension | References |
---|---|---|---|

${\epsilon}_{dis}$ | $4.32\times {10}^{-2}$ | ${E}_{h}$ | |

${\epsilon}_{a}^{O}$ | $6.6\times {10}^{-2}$ | ${E}_{h}$ | |

${\epsilon}_{des}$ | $2.8\times {10}^{-2}$ | ${E}_{h}$ | [32] |

${\epsilon}_{des}^{O}$ | $2.9\times {10}^{-2}$ | ${E}_{h}$ | |

${\epsilon}_{ret}$ | $2.3\times {10}^{-2}$ | ${E}_{h}$ | |

${\epsilon}_{v}$ | $7.35\times {10}^{-3}$ | ${E}_{h}$ | [33,34] |

${\epsilon}_{O}$ | ${10}^{-3}$ | ${E}_{h}$ | |

${\nu}_{O-O}$ | $2.4\times {10}^{-4}$ | ${t}_{h}^{-1}$ | [35] |

${\nu}_{{O}_{2}}$ | $2.4\times {10}^{-5}$ | ${t}_{h}^{-1}$ | [35] |

${\nu}_{O}^{des}$ | $7.5\times {10}^{-5}$ | ${t}_{h}^{-1}$ | [35] |

${\nu}_{O}$ | $2.4\times {10}^{-4}$ | ${t}_{h}^{-1}$ | [35] |

${\nu}_{ret}$ | $2.36\times {10}^{-13}$ | ${t}_{h}^{-1}$ | |

${n}_{v}$ | $2.16\times {10}^{-6}$ | ${a}_{0}^{-3}$ | [11] |

${\tilde{n}}_{{O}_{2}}$ | $1.15\times {10}^{-6}$ | ${a}_{0}^{-3}$ | [36] |

${\tilde{n}}_{{H}_{2}}$ | $4\times {10}^{-9}$ | ${a}_{0}^{-3}$ | |

${n}_{{O}_{2}}^{lim}$ | $1.12\times {10}^{-2}$ | ${a}_{0}^{-2}$ | |

${a}_{O}$ | 2 | ${a}_{0}^{2}$ | |

${\epsilon}_{r}$ | 5 | dimensionless | [37] |

${m}_{{O}_{2}}$ | $5.9\times {10}^{4}$ | ${m}_{e}$ | |

${m}^{*}$ | 2 | ${m}_{e}$ | |

R | 700 | ${a}_{0}$ | |

${\alpha}_{{O}_{2}}\left(T\right)$ | $3.8\times {T}^{1.69}\phantom{\rule{0.166667em}{0ex}}{e}^{-\frac{11664.3}{T}}$ | dimensionless |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Novozhilov, V.B.; Bodneva, V.L.; Kurmangaleev, K.S.; Lidskii, B.V.; Posvyanskii, V.S.; Trakhtenberg, L.I.
Modeling of the Electronic Structure of Semiconductor Nanoparticles. *Mathematics* **2023**, *11*, 2214.
https://doi.org/10.3390/math11092214

**AMA Style**

Novozhilov VB, Bodneva VL, Kurmangaleev KS, Lidskii BV, Posvyanskii VS, Trakhtenberg LI.
Modeling of the Electronic Structure of Semiconductor Nanoparticles. *Mathematics*. 2023; 11(9):2214.
https://doi.org/10.3390/math11092214

**Chicago/Turabian Style**

Novozhilov, Vasily B., Valeria L. Bodneva, Kairat S. Kurmangaleev, Boris V. Lidskii, Vladimir S. Posvyanskii, and Leonid I. Trakhtenberg.
2023. "Modeling of the Electronic Structure of Semiconductor Nanoparticles" *Mathematics* 11, no. 9: 2214.
https://doi.org/10.3390/math11092214