A fractional-differential approach to numerical simulation of electron-induced charging of ferroelectrics

L. I. Moroz ; A. G. Maslovskaya

Sibirskij žurnal industrialʹnoj matematiki, Tome 27 (2024) no. 1, pp. 55-71

Résumé

The paper proposes a fractional-differential modification of the mathematical model of the process of nonstationary charging of polar dielectric materials under conditions of irradiation with medium-energy electron beams. The mathematical formalization is based on a spherically symmetric diffusion-drift equation with a fractional time derivative. An implicit finite-difference scheme is constructed using the Caputo derivative approximation. An application program has been developed in Matlab software that implements the designed computational algorithm. Verification of an approximate solution of the problem is demonstrated using a test example. The results of computational experiments to evaluate the characteristics of field effects of injected charges in ferroelectrics when varying the order of fractional differentiation in subdiffusion regimes are presented.

Détail

Keywords: electron-induced charging, fractional partial differential equation, Caputo derivative, implicit finite-difference scheme, computational experiment.
Mots-clés : ferroelectric

Bibliographie

[1] Otten D., Mathematical models of reaction diffusion systems, their numerical solutions and the freezing method with Comsol Multiphysics, Bielefeld University, Bielefeld, 2000

[2] Patankar S. V., Numerical heat transfer and fluid flow, Hemisphere Publ. Corp., Washington, 1980 | Zbl

[3] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems, LIBROKOM, M., 2015 (in Russian)

[4] Kan T., Suzuki M., “Uniform estimates and uniqueness of stationary solutions to the drift-diffusion model for semiconductors”, Appl. Anal, 5:10 (2019), 1799–1810 | DOI | MR

[5] Rekhviashvili S. S., Alikhanov A. A., “Simulation of drift-diffusion transport of charge carriers in semiconductor layers with a fractal structure in an alternating electric field”, Semiconductors, 51 (2017), 755–759 | DOI

[6] He J., Tang S. H., Qin Y. Q., Dong P., Zhang H. Z., Kang C. H., Sun W. X., Shen Z. X., “Two-dimensional structures of ferroelectric domain inversion in $\mathrm{LiNbO_3}$ by direct electron beam lithography”, J. Appl. Phys, 93 (2003), 9943–9947 | DOI

[7] L. S. Kokhanchik and D. V. Irzhak, “Formation of regular domain structures and peculiarities of switching of the spontaneous polarization in lithium tantalate crystals during discrete electron irradiation”, Phys. Solid State, 52 (2010), 306–310 | DOI

[8] E. I. Rau, E. N. Evstaf'eva, and M. V. Andrianov, “Mechanisms of charging of insulators under irradiation with medium-energy electron beams”, Phys. Solid State, 50 (2008), 621–630 | DOI

[9] Chezganov D. S., Kuznetsov D. K., Shur V. Ya., “Simulation of spatial distribution of electric field after electron beam irradiation of $\mathrm{MgO}$-doped $\mathrm{LiNbO_3}$ covered by resist layer”, Ferroelectrics, 496 (2016), 70–78 | DOI

[10] Chan D. S. H., Sim K. S., Phang J. C. H., “A simulation model for electron irradiation induced specimen charging in a scanning electron microscope”, Scan. Microsc, 7:3 (1993), 847–859 https://digitalcommons.usu.edu/microscopy/vol7/iss3/9

[11] Melchiger A., Hofmann S., “Dynamic double layer model. Description of time dependent charging phenomena in insulators under electron beam irradiation”, J. Appl. Phys, 78:10 (1995), 6224–6232 | DOI

[12] Cazaux J., “About the mechanisms of charging in EPMA, SEM, and ESEM with their time evolution”, Microsc. Microanal, 10:6 (2004), 670–680 | DOI

[13] Kotera M., Yamaguchi K., Suga H., “Dynamic simulation of electron-beam-induced charging up of insulators”, Jpn. J. Appl. Phys, 38:12S (1999), 7176–7179 | DOI

[14] Ohya K., Inai K., Kuwada H., Hauashi T., Saito M., “Dynamic simulation of secondary electron emission and charging up of an insulting material”, Surf. Coat. Technol, 202 (2008), 5310–5313 | DOI

[15] Maslovskaya A., Pavelchuk A., “Simulation of dynamic charging processes in ferroelectrics irradiated with SEM”, Ferroelectrics, 476:2 (2015), 157–167 | DOI

[16] Kalmanovich V. V., Seregina E. V., Stepovich M. A., “Comparison of analytical and numerical modeling of distributions of nonequilibrium minority charge carriers generated by a wide beam of medium-energy electrons in a two-layer semiconductor structure”, J. Phys. Conf. Ser, 1479 (2020), 012116 | DOI

[17] Raftari B., “Self-consistence drift-diffusion-reaction model for the electron beam interaction with dielectric samples”, J. Appl. Phys, 118 (2015), 204101 | DOI

[18] Pavelchuk A. V., Maslovskaya A. G., “Simulation of delay reaction-drift-diffusion system applied to charging effects in electron-irradiated dielectrics”, J. Phys. Conf. Ser, 1163 (2019), 012009 | DOI

[19] R. V. Brizitskii, N. N. Maksimova, and A. G. Maslovskaya, “Theoretical analysis and numerical implementation of a stationary diffusion–drift model of polar dielectric charging”, Comput. Math. Math. Phys., 62:10 (2022), 1680–1690 | DOI | DOI | MR | Zbl

[20] S. S. Borisov, E. A. Grachev, and S. I. Zaitsev, “Modeling the polarization of a dielectric during irradiation with an electron beam”, Prikl. Fiz., 2004, no. 1, 118–123 (in Russian)

[21] Uchaikin M., Sibatov R., Fractional kinetics in solids. Anomalous charge transport in semiconductors, dielectrics and nanosystems, World Scientific, Singapore, 2012 | MR

[22] Holm S., Natural occurrence of fractional derivatives in physics, arXiv: 2305.07074 | DOI

[23] Deng W., Hou R., Wang W., Xu P., Modeling Anomalous Diffusion. From Statistics to Mathematics, World Scientific, Singapore, 2020 | MR | Zbl

[24] Evangelista L. R., Lenzi E. K., Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, Cambridge, 2018 | MR | Zbl

[25] R. P. Meilanov and S. A. Sadykov, “Fractal model of polarization switching kinetics in ferroelectrics”, Zh. Tekh. Fiz., 69:5 (1999), 128–129 (in Russian)

[26] Galiyarova N. M., “Fractal dielectric response of multidomain ferroelectrics from the irreversible thermodynamics standpoint”, Ferroelectrics, 222:1 (1999), 381–387 | DOI

[27] Ducharne B., Sebald G., Guyomar D., “Time fractional derivative for frequency effect in ferroelectrics”, Proc. 18th IEEE Internat. Sympos. Appl. Ferroelectr, 2009, 1–4 | DOI

[28] Wang X., “Analytical solitary wave solutions of a time-fractional thin-film ferroelectric material equation involving beta-derivative using modified auxiliary equation method”, Results Phys, 48 (2023), 106411 | DOI | MR

[29] Moroz L. I., Masslovskaya A. G., “Numerical simulation of an anomalous diffusion process based on the higher-order accurate scheme”, Math. Models Comput. Simul, 13:3 (2021), 492–501 | DOI | MR

[30] Maslovskaya A. G., Moroz L. I., “Time-fractional Landau—Khalatnikov model applied to numerical simulation of polarization switching in ferroelectrics”, Nonlinear Dyn, 111 (2023), 4543–4557 | DOI

[31] Moroz L. I., Maslovskaya A. G., “Simulation of nonlinear pyroelectric response of ferroelectrics near phase transition: fractional differential approach”, Mater. Sci. Forum, 992 (2022), 843–848 | DOI

[32] P. T. Oreshkin, Physics of Semiconductors and Dielectrics, Vyssh. Shkola, M., 1977 (in Russian)

[33] Joy D. C., Monte Carlo modeling for electron microscopy and microanalysis, Oxford University Press, N. Y., 1995

[34] Meng R., “Application of fractional calculus to modeling the non-linear behaviors of ferroelectric polymer composites: viscoelasticity and dielectricity”, Membranes, 11:6 (2021), 409 | DOI

[35] Amadou Y., Justin M., Hubert M. B., Betchewe G., Doka S. Y., Crepin K. T., “Fractional effects on solitons in a 1D array of rectangular ferroelectric nanoparticles”, Waves Random Complex Media, 30 (2020), 581–592 | DOI | MR | Zbl

[36] Guzelturk B., Yang T., Liu Y., Wei Ch.-Ch., Orenstein G., Trigo M., Zhou T., Diroll B. T., Holt M. V., Wen H., Chen L.-Q., Yang J.-Ch., Lindenberg A. M., “Sub-nanosecond reconfiguration of ferroelectric domains in bismuth ferrite”, Adv. Mater, 35 (2023), e2306029 | DOI

[37] Liu F., Meerschaert M. M., McGough R. J., Zhuang P., Liu Q., “Numerical methods for solving the multi-term time-fractional wave-diffusion equation”, Fract. Calc. Appl. Anal, 16 (2013), 9–25 | DOI | MR | Zbl

[38] Buckova Z., Ehrhardt M., Gunther M., “Alternating direction explicit methods for convection diffusion equations”, Acta Math. Univ. Comenianae, 84:2 (2015), 309–325 | MR | Zbl