Machine Learning Models for Evaluating Efficient Carrier Mobility in Silicon
DOI:
https://doi.org/10.15407/ujpe71.8.682Keywords:
carrier mobility, silicon, Klaassen model, symbolic regression, machine learningAbstract
The application of machine learning (ML) models – specifically Random Forest (RF), Gradient Boosting (GB), Support Vector Regression (SVR), and Deep Neural Networks (DNNs) – alongside Symbolic Regression (SR) for predicting carrier mobility in monocrystalline silicon over wide temperature (200–500 K) and doping concentration (1013–1019 cm–3) ranges has been evaluated. The analytical expressions derived via SR are notably more parsimonious, requiring 50% fewer parameters than the Klaassen model. Furthermore, these expressions bypass the need for preliminary impurity ionization calculations while maintaining a relative error below 0.1% in most cases. A comparative analysis demonstrated that the SR and SVR models significantly outperform the Arora approximation in terms of accuracy, particularly for minority carriers. The proposed approach offers a robust framework for TCAD systems, substantially reducing computational overhead while ensuring high modeling accuracy.
References
1. C. Hamaguchi. Basic Semiconductor Physics. 4th edition (Springer Cham, 2023) [ISBN: 978-3-031-25510-6].
2. J. Leveillee, X. Zhang, E. Kioupakis, F. Giustino. Ab initio calculation of carrier mobility in semiconductors including ionized-impurity scattering. Phys. Rev. B 107, 125207 (2023)
https://doi.org/10.1103/PhysRevB.107.125207
3. K. Seeger. Semiconductor Physics. An Introduction. 9th edition (Springer Berlin, 2004) [ISBN: 978-3-540-21957-6].
4. K.W. Böer, U.W. Pohl. Semiconductor Physics 2nd edition (Springer Cham, 2023) [ISBN: 978-3-031-18285-3].
https://doi.org/10.1007/978-3-031-18286-0
5. E. Bourbaba, S. Kadri, M. Bensafi, S. Larguech, Y. Menni, B.M. Alshammari, L. Kolsi. Empirical model integration for accurate charge carrier mobility simulation in silicon mosfets. Open Physics 23, 20250248 (2025).
https://doi.org/10.1515/phys-2025-0248
6. N.H. Fletcher. The high current limit for semiconductor junction devices. Proc. IRE 45, 862 (1957).
https://doi.org/10.1109/JRPROC.1957.278485
7. S. Rudra, D. Rao, S. Poncé, B. Saha. Dominant scattering mechanisms in limiting the electron mobility of scandium nitride. Nano Lett. 24, 11529 (2024).
https://doi.org/10.1021/acs.nanolett.4c02920
8. D. Caughey, R. Thomas. Carrier mobilities in silicon empirically related to doping and field. Proc. IEEE 55, 2192 (1967).
https://doi.org/10.1109/PROC.1967.6123
9. G. Masetti, M. Severi, S. Solmi. Modeling of carrier mobility against carrier concentration in arsenic-, phosphorus-, and boron-doped silicon. IEEE Trans. Electron Devices 30, 764 (1983).
https://doi.org/10.1109/T-ED.1983.21207
10. S. Berrada, H. Carrillo-Nunez, J. Lee, C. Medina-Bailon, T. Dutta, O. Badami, F. Adamu-Lema, V. Thirunavukkarasu, V. Georgiev, A. Asenov. Nano-electronic simulation software (ness): A flexible nano-device simulation platform. J. Comput. Electron. 19, 1031 (2020).
https://doi.org/10.1007/s10825-020-01519-0
11. N. Arora, J. Hauser, D. Roulston. Electron and hole mobilities in silicon as a function of concentration and temperature. IEEE Trans. Electron Devices 29, 292 (1982).
https://doi.org/10.1109/T-ED.1982.20698
12. D. Klaassen. A unified mobility model for device simulation - I. Model equations and concentration dependence. Solid-State Electron. 35, 953 (1992).
https://doi.org/10.1016/0038-1101(92)90325-7
13. S. Swirhun, Y.-H. Kwark, R. Swanson. Measurement of electron lifetime, electron mobility and band-gap narrowing in heavily doped p-type silicon. In: Proc of the 1986 International Electron Devices Meeting, Los Angeles, CA, USA, December 07-10, 1986, p. 24.
https://doi.org/10.1109/IEDM.1986.191101
14. J. del Alamo, S. Swirhun, R. Swanson. Simultaneous measurement of hole lifetime, hole mobility and bandgap narrowing in heavily doped n-type silicon. In: Proc of the 1985 International Electron Devices Meeting, Washington, DC, USA, December 01-04, 1985, p. 290.
https://doi.org/10.1109/IEDM.1985.190954
15. D.L. Rode, J.S. Cetnar. Electron mobility of heavily doped semiconductors including multiple scattering by ionized impurities. J. Appl. Phys. 134, 075701 (2023).
https://doi.org/10.1063/5.0165201
16. T. Sadi, C. Medina-Bailon, M. Nedjalkov, J. Lee, O. Badami, S. Berrada, H. Carrillo-Nunez, V. Georgiev, S. Selberherr, A. Asenov. Simulation of the impact of ionized impurity scattering on the total mobility in si nanowire transistors. Materials 12, 124 (2019).
https://doi.org/10.3390/ma12010124
17. Z. Wang, Z. Tang, A. Guo, X. Luo, C. Cao, Y. Yuan, X. Zhang, L. Liu, J. Li, Y. Cao, O. Shao, S. Hu, S. Chen, Y. Zhao, X. Kou. Temperature-driven gate geometry effects in nanoscale cryogenic mosfets. IEEE Electron Device Lett. 41, 661 (2020).
https://doi.org/10.1109/LED.2020.2984280
18. L. Abenante. Incomplete activation and ionization of dopants in si at room temperature. AIP Adv. 13, 015109 (2023).
https://doi.org/10.1063/5.0117615
19. D.J. Cummings, A.F. Witulski, H. Park, R.D. Schrimpf, S.E. Scott, M.E. Low. Mobility modeling considerations for radiation effects simulations in silicon. IEEE Trans. Nucl. Sci. 57, 2318 (2010).
https://doi.org/10.1109/TNS.2010.2052831
20. Q. Wu, A. Sojka, B.D. Price, N.I. Agladze, A. Yadav, S.L. Pain, J.D. Murphy, T. Niewelt, M.S. Sherwin. Two-fluid mobility model from coupled hydrodynamic equations for simulating laser-driven semiconductor switches. Phys. Rev. Appl. 24, 014007 (2025).
https://doi.org/10.1103/47kj-1g91
21. S. Noor Mohammad, A.V. Bemis, R.L. Carter, R.B. Renbeck. Temperature, electric field, and doping dependent mobilities of electrons and holes in semiconductors. Solid-State Electron. 36, 1677 (1993).
https://doi.org/10.1016/0038-1101(93)90213-A
22. Y.-N. Wu, X.-G. Zhang, S.T. Pantelides. First-principles calculations reveal controlling principles for carrier mobilities in semiconductors. Semicond. Sci. Technol. 31, 115016 (2016).
https://doi.org/10.1088/0268-1242/31/11/115016
23. D.-Y. Liu, L.-M. Xu, X.-M. Lin, X. Wei, W.-J. Yu, Y. Wang, Z.-M. Wei. Machine learning for semiconductors. Chip 1, 100033 (2022).
https://doi.org/10.1016/j.chip.2022.100033
24. R. Li, E. Lee, T. Luo. A unified deep neural network potential capable of predicting thermal conductivity of silicon in different phases. Mater. Today Phys. 12, 100181 (2020).
https://doi.org/10.1016/j.mtphys.2020.100181
25. R. Li, E. Lee, T. Luo. Physics-informed deep learning for solving coupled electron and phonon boltzmann transport equations. Phys. Rev. Appl. 19, 064049 (2023).
https://doi.org/10.1103/PhysRevApplied.19.064049
26. R. Li, J. Zhou, J.-X. Wang, T. Luo. Physics-informed bayesian neural networks for solving phonon boltzmann transport equation in forward and inverse problems with sparse and noisy data. ASME J. Heat and Mass Transfer 147, 032501 (2024).
https://doi.org/10.1115/1.4067163
27. C. Fai, A.J. Ladd, C.J. Hages. Machine learning for enhanced semiconductor characterization from time-resolved photoluminescence. Joule 6, 2585 (2022).
https://doi.org/10.1016/j.joule.2022.09.002
28. Y. Xu, A.V. Lalwani, K. Arora, Z. Zheng, A. Renteria, D.G. Senesky, P. Wang. Hall-effect sensor design with physics-informed gaussian process modeling. IEEE Sensors J. 22, 22519 (2022).
https://doi.org/10.1109/JSEN.2022.3216499
29. X. Chen, S. Lu, Q. Chen, Q. Zhou, J. Wang. From bulk effective mass to 2d carrier mobility accurate prediction via adversarial transfer learning. Nat. Commun. 15, 5391 (2024).
https://doi.org/10.1038/s41467-024-49686-z
30. J.H. Hong, C.H. Lee, H.W. Kim, D.W. Jeon, J.H. Jun, Y.-S. Kim, J.-S. Bae, S. Heo, S. Lee, J. Na, J. You, S. Hong, H. Cho, J.-S. Park, S.B. Cho. Ai-driven quantitative review of mobility-stability trade-off in oxide semiconductors. Nano Convergence 13, 4 (2026).
https://doi.org/10.1186/s40580-026-00535-3
31. T. Nematiaram, Z. Lamprou, Y. Moshfeghi. Accelerating the discovery of high-mobility molecular semiconductors: a machine learning approach. Chem. Commun. 61, 3676 (2025).
https://doi.org/10.1039/D4CC04200J
32. P. Reiser, M. Konrad, A. Fediai, S. Léon, W. Wenzel, P. Friederich. Analyzing dynamical disorder for charge transport in organic semiconductors via machine learning. J. Chem. Theory Comput. 17, 3750 (2021).
https://doi.org/10.1021/acs.jctc.1c00191
33. A.K. Pimachev, S. Neogi. First-principles prediction of electronic transport in fabricated semiconductor heterostructures via physics-aware machine learning. npj Comput. Mater. 7, 93 (2021).
https://doi.org/10.1038/s41524-021-00562-0
34. V. Bhat, B. Ganapathysubramanian, C. Risko. Rapid estimation of the intermolecular electronic couplings and charge-carrier mobilities of crystalline molecular organic semiconductors through a machine learning pipeline. J. Phys. Chem. Lett. 15, 7206 (2024).
https://doi.org/10.1021/acs.jpclett.4c01309
35. T. Tan, D. Wang. Machine learning based charge mobility prediction for organic semiconductors. J. Chem. Phys. 158, 094102 (2023).
https://doi.org/10.1063/5.0134379
36. K.M. Mamun, N. Pala, M.S.A. Shawkat. A comprehensive review of machine learning approaches for semiconductor device modeling and simulation. IEEE Access 13, 162969 (2025).
https://doi.org/10.1109/ACCESS.2025.3605856
37. P. Raut, D.K. Panda, A.K. Goyal. A comprehensive review on next-generation modeling and optimization for semiconductor devices. IEEE Access 13, 123724 (2025).
https://doi.org/10.1109/ACCESS.2025.3587721
38. R. Khatua, B. Das, A. Mondal. Physics-informed machine learning with data-driven equations for predicting organic solar cell performance. ACS Appl. Mater. Interfaces 16, 57467 (2024).
https://doi.org/10.1021/acsami.4c10868
39. R. Couderc, M. Amara, M. Lemiti. Reassessment of the intrinsic carrier density temperature dependence in crystalline silicon. J. Appl. Phys. 115, 093705 (2014).
https://doi.org/10.1063/1.4867776
40. R. Pässler. Dispersion-related description of temperature dependencies of band gaps in semiconductors. Phys. Rev. B 66, 085201 (2002).
https://doi.org/10.1103/PhysRevB.66.085201
41. A. AlQurashi, C. Selvakumar. A new approximation of Fermi-Dirac integrals of order 1/2 for degenerate semiconductor devices. Superlattices Microstruct. 118, 308 (2018).
https://doi.org/10.1016/j.spmi.2018.03.072
42. A. Tonda. Review of pysr: high-performance symbolic regression in python and julia. Genet. Program. Evol. M. 26, 7 (2024).
https://doi.org/10.1007/s10710-024-09503-4
43. D. Angelis, F. Sofos, T.E. Karakasidis. Artificial intelligence in physical sciences: Symbolic regression trends and perspectives. Arch. Comput. Methods Eng. 30, 3845 (2023).
https://doi.org/10.1007/s11831-023-09922-z
44. V. Hruska, A. Furmanova, M. Bednarik. Analytical formulae for design of one-dimensional sonic crystals with smooth geometry based on symbolic regression. J. Sound Vibration 597, 118821 (2025).
https://doi.org/10.1016/j.jsv.2024.118821
45. B.L. Davis, Z. Jin. Discovery of a planar black hole mass scaling relation for spiral galaxies. Astrophys. J. Lett. 956, L22 (2023).
https://doi.org/10.3847/2041-8213/acfa98
46. V.V. Kuryliuk, O.Y. Olikh. Thermal conductivity of nanoporous silicon: Molecular dynamics simulations and machine learning prediction. Low Temp. Phys. 52, 50 (2026).
https://doi.org/10.1063/10.0042162
47. T. Mengel, P. Steffanic, C. Hughes, A.C.O. da Silva, C. Nattrass. Interpretable machine learning methods applied to jet background subtraction in heavy-ion collisions. Phys. Rev. C 108, L021901 (2023).
https://doi.org/10.1103/PhysRevC.108.L021901
48. Y. Li, H. Wang, Y. Li, H. Ye, Y. Zhang, R. Yin, H. Jia, B. Hou, C. Wang, H. Ding, X. Bai, A. Lu. Electron transfer rules of minerals under pressure informed by machine learning. Nat. Commun. 14, 1815 (2023).
https://doi.org/10.1038/s41467-023-37384-1
49. M. Cranmer. Interpretable machine learning for science with pysr and symbolicregression.jl. arXiv:2305.01582v3 [astro-ph.IM]
50. M.F. Hanif, M.U. Siddique, J. Si, M.S. Naveed, X. Liu, J. Mi. Enhancing solar forecasting accuracy with sequential deep artificial neural network and hybrid random forest and gradient boosting models across varied terrains. Advanced Theory and Simulations 7, 2301289 (2024).
https://doi.org/10.1002/adts.202301289
51. T. Akiba, S. Sano, T. Yanase, T. Ohta, M. Koyama. Optuna: A next-generation hyperparameter optimization framework. In: Proc of the 25th ACM SIGKDD Int. Conf. Knowl. Discov. Data Mining, Anchorage, AK, USA, August 04-08, 2019, p. 2623.
https://doi.org/10.1145/3292500.3330701
52. S.C. Choo. Theory of a forward-biased diffused-junction pl-n rectifier-part i: Exact numerical solutions. IEEE Trans. Electron Devices 19, 954 (1972).
https://doi.org/10.1109/T-ED.1972.17525
53. J. Dorkel, P. Leturcq. Carrier mobilities in silicon semiempirically related to temperature, doping and injection level. Solid-State Electron. 24, 821 (1981).
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