Skip to main content
SpringerLink
Log in

An improved 2D–3D model for charge transport based on the maximum entropy principle

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

To study the electron transport in a some tens of nanometers long channel of a metal oxide field effect transistor, in order to reduce the computational cost of simulations, it can be convenient to divide the electrons into a 2D and a 3D population. Near the silicon/oxide interface the two populations coexist, while in the remaining part of the device only the 3D component needs to be considered because quantum effects are negligible there. The major issue is the description of the scattering mechanisms between the 2D and the 3D electron populations, due to interactions of electrons with nonpolar optical phonons and interface modes. Here, we propose a rigorous treatment of these collisions based on an approach similar to that used in Fischetti and Laux (Phys Rev B 48:2244–2274, 1993), in the context of a Monte Carlo simulation. We also consider all the other main scatterings, which are those with acoustic phonons, surface roughness, and impurities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fischetti, M.V., Laux, S.E.: Monte Carlo study of electron transport in silicon inversion layers. Phys. Rev. B 48, 2244–2274 (1993)

    Article  ADS  Google Scholar 

  2. Camiola, V.D., Romano, V.: 2DEG–3DEG charge transport model for MOSFET based on the maximum entropy principle. SIAM J. Appl. Math. 73, 1439–1459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Camiola, V.D., Romano, V.: Mathematical structure of the transport equations for coupled 2D–3D electron gasses in a MOSFET. In: Idelsohn, S., Papadrakakis, M., Schrefler, B. (eds.) Computational Methods for Coupled Problems in Science and Engineering V, pp. 515–526. CIMNE (Int. Center for Num. Meth. in Engineering), Barcelona (2013)

    Google Scholar 

  4. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  6. Dreyer, W., Struchtrup, H.: Heat pulse experiment revisited. Contin. Mech. Therm. 5, 3 (1993)

    Article  MathSciNet  Google Scholar 

  7. Mascali, G., Romano, V.: Hydrodynamic subband model for semiconductors based on the maximum entropy principle. II Nuovo Cimento C 33, 155–163 (2010)

    MATH  Google Scholar 

  8. Mascali, G., Romano, V.: A non parabolic hydrodynamical subband model for semiconductors based on the maximum entropy principle. Math. Comput. Model. 55, 1003–1020 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Muscato, O., Di Stefano, V.: Hydrodynamic modeling of silicon quantum wires. J. Comput. Electron. 11, 45–55 (2012)

    Article  Google Scholar 

  10. Muscato, O., Di Stefano, V.: Hydrodynamic simulation of a \(n^+ - n - n^+\) silicon nanowire. Contin. Mech. Thermodyn. 26, 197–205 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Barletti, L.: Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle. J. Math. Phys. 55, 083303 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Morandi, O., Barletti, L.: Particle dynamics in graphene: collimated beam limit. J. Comput. Theor. Phys. 43, 1–15 (2014)

    MathSciNet  Google Scholar 

  13. Davies, J.H.: The Physics of Low-Dimensional Semiconductors. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  14. Lundstrom, M.: Fundamentals of Carrier Transport. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  15. Polizzi, E., Ben Abdallah, N.: Subband decomposition approach for the simulation of quantum electron transport in nanostructures. J. Comput. Phys. 202, 150–180 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Herring, C., Vogt, E.: Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering. Phys. Rev. 101, 944–961 (1956)

    Article  ADS  MATH  Google Scholar 

  17. Hess, K., Vogl, P.: Remote polar phonon scattering in silicon inversion layers. Solid State Commun. 30, 797–799 (1979)

    Article  ADS  Google Scholar 

  18. Moore, B.T., Ferry, D.K.: Remote polar phonon scattering in Si inversion layers. J. Appl. Phys. 51, 2603–2605 (1980)

    Article  ADS  Google Scholar 

  19. Moore, B.T., Ferry, D.K.: Scattering of inversion layer electrons by oxide polar mode generated interface phonons. J. Vac. Sci. Technol. 17, 1037 (1980)

    Article  ADS  Google Scholar 

  20. Dahl, D.A., Sham, I.J.: Electrodynamics of quasi-two-dimensional electrons. Phys. Rev. B 16, 651–661 (1977)

    Article  ADS  Google Scholar 

  21. Fetter, A.L.: Electrodynamics and thermodynamics of a classical electron surface layer. Phys. Rev. B 10, 3739–3745 (1974)

    Article  ADS  Google Scholar 

  22. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many Particles Systems. Academic Press, New York (1971)

    Google Scholar 

  23. Fried, B.D., Conte, S.D.: The Plasma Dispersion Function. Academic Press, New York (1961)

    Google Scholar 

  24. Ando, T., Fowler, A.B., Stern, F.: Electronic properties of two-dimensional systems. Rev. Mod. Phys. 58, 437–672 (1982)

    Article  ADS  Google Scholar 

  25. Fischetti, M.V., Laux, S.E.: DAMOCLES Theoretical Manual. IBM Corporation, Yorktown Heights (1995)

    Google Scholar 

  26. Ben Abdallah, N., Caceres, M.J., Carrillo, J.A., Vecil, F.: A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs. J. Comput. Phys. 17, 6553–6571 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Vecil, F., Mantas, J.M., Caceres, M.J., Sampedro, C., Godoy, A., Gamiz, F.: A parallel deterministic solver for the Schroedinger–Poisson–Boltzmann system in ultra-short DG-MOSFETs: comparison with Monte Carlo. Comput. Math. Appl. 67, 1703–1721 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Romano, V., Majorana, A., Coco, M.: DSMC method consistent with the Pauli exclusion principle and comparison with deterministic solutions for charge transport in graphene. J. Comput. Phys. 302, 267–284 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Polizzi, E., Abdallah, N.: Self-consistent three dimensional models for quantum ballistic transport in open system. Phys. Rev. B 66, 245301 (2002)

    Article  ADS  Google Scholar 

  30. Romano, V.: 2D numerical simulation of the MEP energy-transport model with a finite difference scheme. J. Comput. Phys. 221, 439–468 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Camiola, V.D., Mascali, G., Romano, V.: Numerical simulation of a double-gate MOSFET with a subband model for semiconductors based on the maximum entropy principle. Contin. Mech. Thermodyn. 14, 417–436 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Camiola, V.D., Mascali, G., Romano, V.: Simulation of a double-gate MOSFET by a non-parabolic energy-transport model for semiconductors based on the maximum entropy principle. Math. Comput. Model. 58, 321–343 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mascali, G., Romano, V.: Exploitation of the maximum entropy principle in mathematical modeling of charge transport in semiconductors. Entropy 19(1), 36 (2017). https://doi.org/10.3390/e19010036. (open access article)

    Article  ADS  Google Scholar 

  34. Anile, A.M., Romano, V.: Non parabolic band transport in semiconductors: closure of the moment equations. Contin. Mech. Thermodyn. 11, 307–325 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Romano, V.: Non parabolic band transport in semiconductors: closure of the production terms in the moment equations. Contin. Mech. Thermodyn. 12, 31–51 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Alì, G., Mascali, G., Romano, V., Torcasio, R.C.: A hydrodynamic model for covalent semiconductors with applications to GaN and SiC. Acta Appl. Math. 122, 335–348 (2012)

    MATH  Google Scholar 

  37. Mascali, G.: A hydrodynamic model for silicon semiconductors including crystal heating. Eur. J. Appl. Math. 26, 477–496 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Junk, M., Romano, V.: Maximum entropy moment system of the semiconductor Boltzmann equation using Kane’s dispersion relation. Math. Comput. Model. 17, 247–267 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Mascali, G., Romano, V.: Si and GaAs mobility derived from a hydrodynamical model for semiconductors based on the maximum entropy principle. Phys. A 352, 459–476 (2005)

    Article  Google Scholar 

  40. Romano, V.: Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices. Math. Methods Appl. Sci. 24, 439–471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mascali, G., Romano, V.: A hydrodynamical model for holes in silicon semiconductors: the case of non-parabolic warped bands. Math. Comput. Model. 53, 213–229 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jacoboni, C.: Theory of Electron Transport in Semiconductors. Springer, Berlin (2010)

    Book  Google Scholar 

Download references

Acknowledgements

G.M. acknowledges the financial support from the P.R.A. of the University of Calabria. V.R. acknowledges the financial support by progetto FIR 2016-2018 Modellistica, simulazione e ottimizzazione del trasporto di cariche in strutture a bassa dimensionalità, University of Catania. The authors G.M and V. R. are members of the INdAM research group GNFM.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    Vito Dario Camiola & Vittorio Romano

  2. Dipartimento di Matematica e Informatica, Università della Calabria and INFN-Gruppo c. Cosenza, 87036, Cosenza, Italy

    Giovanni Mascali

Authors
  1. Vito Dario Camiola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Mascali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vittorio Romano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vittorio Romano.

Additional information

Communicated by Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Camiola, V.D., Mascali, G. & Romano, V. An improved 2D–3D model for charge transport based on the maximum entropy principle. Continuum Mech. Thermodyn. 31, 751–773 (2019). https://doi.org/10.1007/s00161-018-0735-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-018-0735-6

Keywords

Search

Navigation