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Journal of Computational Electronics

, Volume 13, Issue 3, pp 689–700 | Cite as

COOS: a wave-function based Schrödinger–Poisson solver for ballistic nanotube transistors

  • Martin Claus
  • Sven Mothes
  • Stefan Blawid
  • Michael Schröter
Article

Abstract

This paper gives an in depth overview on a wave-function based simulation framework (called coos) for modeling ballistic nanotube transistors by solving the effective-mass Schrödinger equation. The framework considers non-parabolic electronic band structure effects, band-to-band tunneling as well as a heterojunction-like model for extended contacts to describe the injection and reception of charge carriers into and from the channel. Special emphasis is put on an efficient and reliable numerical implementation. The applicability of the simulation framework and the necessity to include the aforementioned phenomena are shown by comparing simulation results with experimental data of a \(50\hbox { nm}\) long carbon nanotube transistor (cntfet). The intrinsic transit frequencies and the output characteristics for higher drain-source voltages are predicted and analyzed.

Keywords

Transistor Channel Charge Neutrality Level Inject Charge Carrier Tungsten Disulfide Transit Frequency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The authors acknowledge the financial support from the Cfaed, the dfg projects CL384/2 and SCHR695/6 as well as the namitec.

Supplementary material

References

  1. 1.
    Schroter, M., Claus, M., Sakalas, P., Wang, D., Haferlach, M.: An overview on the state-of-the-art of carbon-based radio-frequency electronics. In: IEEE Bipolar/BiCMOS Circuits and Technology Meeting (BCMT), pp. 112–119 (2012)Google Scholar
  2. 2.
    Tenne, R.: Inorganic nanotubes and fullerene-like nanoparticles. Nat. Nano. 1(2), 103–111 (2006)CrossRefGoogle Scholar
  3. 3.
    Remskar, M., Mrzel, A., Virsek, M., Godec, M., Krause, M., Kolitsch, A., Singh, A., Seabaugh, A.: The mos2 nanotubes with defect-controlled electric properties. Nanoscale Res. Lett. 6(1), 1–6 (2010)Google Scholar
  4. 4.
    Kim, F.S., Ren, G., Jenekhe, S.A.: One-dimensional nanostructures of \(\pi \)-conjugated molecular systems: assembly, properties, and applications from photovoltaics, sensors, and nanophotonics to nanoelectronics. Chem. Mater. 23(3), 682–732 (2011)CrossRefGoogle Scholar
  5. 5.
    Seidel, R.V., Graham, A.P., Kretz, J., Rajasekharan, B., Duesberg, G.S., Liebau, M., Unger, E., Kreupl, F., Hoenlein, W.: Sub-20 nm short channel carbon nanotube transistors. Nano Lett. 5(1), 147–150 (2005)CrossRefGoogle Scholar
  6. 6.
    Franklin, A.D., Chen, Z.: Length scaling of carbon nanotube transistors. Nat. Nano. 5(12), 858–862 (Dec. 2010)Google Scholar
  7. 7.
    Franklin, A.D., Luisier, M., Han, S.-J., Tulevski, G., Breslin, C.M., Gignac, L., Lundstrom, M.S., Haensch, W.: Sub-10 nm carbon nanotube transistor. Nano Lett. 12(2), 758–762 (2012)CrossRefGoogle Scholar
  8. 8.
    Lemay, S.G., Janssen, J.W., van den Hout, M., Mooij, M., Bronikowski, M.J., Willis, P.A., Smalley, R.E., Kouwenhoven, L.P., Dekker, C.: Two-dimensional imaging of electronic wavefunctions in carbon nanotubes. Nature 412(6847), 617–620 (2001)CrossRefGoogle Scholar
  9. 9.
    Knoch, J., Appenzeller, J.: Tunneling phenomena in carbon nanotube field-effect transistors. phys Status Solidi (a) 205(4), 679–694 (2008)CrossRefGoogle Scholar
  10. 10.
    Guo, J., Datta, S., Anantram, M., Lundstrom, M.: Atomistic simulation of carbon nanotube field-effect transistors using non-equilibrium green’s function formalism. In: Computational Electronics, 2004. IWCE-10 2004. Abstracts. 10th International Workshop on, pp. 71–72 (2004)Google Scholar
  11. 11.
    Klimeck, G.: Nemo 1-d: The first negf-based tcad tool and network for computational nanotechnology. Available: https://nanohub.org/resources/178 (2004)
  12. 12.
    Alam, K., Lake, R.K.: Leakage and performance of zero-Schottky-barrier carbon nanotube transistors. J. Appl. Phys. 98(6), 064307 (2005)CrossRefGoogle Scholar
  13. 13.
    Pourfath, M., Kosina, H., Selberherr, S.: Numerical study of quantum transpost in carbon nanotube transistors, Sci. Direct (2007)Google Scholar
  14. 14.
    Frensley, W.R.: Numerical evaluation of resonant states. Superlattices Microstruct. 11, 347–350 (1992)CrossRefGoogle Scholar
  15. 15.
    Claus, M., Schröter, M.: Design study of cnt transistor layouts for analog circuits. In: Proceedings of NSTI Workshop on Compact Modeling, Vol. 3, pp. 566–569 (2009)Google Scholar
  16. 16.
    Claus, M., Gross, D., Haferlach, M., Schröter, M.: Critical review of cntfet compact models. In: NSTI-Nanotech (Workshop on Compact modeling), Vol. 2, pp. 770–775 (2012)Google Scholar
  17. 17.
    Claus, M., Blawid, S., Schröter, M.: Impact of near-contact barriers on the subthreshold slope of short-channel cntfets. In: International Conference on Simulation of Semiconductor Devices and Processes (SISPAD), pp. 159–162 (2013)Google Scholar
  18. 18.
    Claus, M., Mothes, S., Schröter, M.: Modeling of NQS effects in carbon nanotube transistors. In: International Conference on Simulation of Semiconductor Devices and Processes (SISPAD), Bologna, Italy, pp. 203–206 (2010)Google Scholar
  19. 19.
    Claus, M., Blawid, S., Sakalas, P., Schröter, M.: Analysis of the frequency dependent gate capacitance in cntfets. In: International Conference on Simulation of Semiconductor Devices and Processes (SISPAD), pp. 336–339 (2012)Google Scholar
  20. 20.
    Claus, M., Blawid, S., Mothes, S., Schröter, M.: High-frequency ballistic transport phenomena in schottky-barrier cntfets. IEEE Trans. Electron. Devices 59(10), 2610–2618 (2012)CrossRefGoogle Scholar
  21. 21.
    Castro, L.C., John, D.L., Pulfrey, D.L., Pourfath, M., Gehring, A., Kosina, H.: Method for predicting fT for carbon nanotube FETs. IEEE Trans. Nanotechnol. 4(6), 699–704 (2005)CrossRefGoogle Scholar
  22. 22.
    Pourfath, M., Kosina, H., Selberherr, S.: A fast and stable Poisson–Schrödinger solver for the analysis of carbon nanotube transistors. J. Comput. Electron. 5, 155–159 (2006)CrossRefGoogle Scholar
  23. 23.
    Javey, A., Guo, J., Farmer, D.B., Wang, Q., Yenilmez, E., Gordon, R.G., Lundstrom, M., Dai, H.: Self-aligned ballistic molecular transistors and electrically parallel nanotube arrays. Nano Lett. 4, 1319–1322 (2004)CrossRefGoogle Scholar
  24. 24.
    Yang, M.H., Teo, K.B.K., Milne, W.I.: Carbon nanotube Schottky diode and directionally dependent field-effect transistor using asymmetrical contacts. Appl. Phys. Lett. 87, 253116 (2005)CrossRefGoogle Scholar
  25. 25.
    Chen, Z., Appenzeller, J., Knoch, J., Lin, Y.-M., Avouris, P.: The role of metal-nanotube contact in the performance of carbon nanotube field-effect transistors. Nano Lett. 5(7), 1497–1502 (2005)Google Scholar
  26. 26.
    Cummings, A.W., Leonard, F.: Enhanced performance of short-channel carbon nanotube field-effect transistors due to gate-modulated electrical contacts. ACS Nano 6(5), 4494–4499 (2012)CrossRefGoogle Scholar
  27. 27.
    Schroter, M., Claus, M., Sakalas, P., Haferlach, M., Wang, D.: Carbon Nanotube FET Technology for Radio-Frequency Electronics: State-of-the-Art Overview (invited). IEEE J. Electron Devices Soc. 1(1), 9–20 (2013)CrossRefGoogle Scholar
  28. 28.
    Nemec, N., Tomcanek, D., Cuniberti, G.: Modeling extended contacts for nanotube and graphene devices. Phys. Rev. B 77, 125 420–125 432 (2008)CrossRefGoogle Scholar
  29. 29.
    Claus, M., Fediai, A., Mothes, S., Knoch, J., Ryndyk, D., Blawid, S., Cuniberti, G., Schröter,M.: Towards a multiscale modeling framework for metal-cnt interfaces (accepted). In: International Workshop on Computional Electronics (IWCE), (2014)Google Scholar
  30. 30.
    Blawid, S., Claus, M., Schröter, M.: Phenomenological modeling of charge injection - beyond the schottky barrier paradigm, in 27th Symposium on Microlelectronics Technology and Devices (SBMicro), ECS Transactions, vol. 49, Brasília pp. 85–92 (2012)Google Scholar
  31. 31.
    Di Ventra, M.: Electrical Transport in Nanoscale Systems. Cambridge University Press, New York (2008)CrossRefGoogle Scholar
  32. 32.
    Cummings, A.W., Léonard, F.: Electrostatic effects on contacts to carbon nanotube transistors. Appl. Phys. Lett. 98(26), 263503 (2011)CrossRefGoogle Scholar
  33. 33.
    Mintmire, J.W., White, C.T.: Universal density of states for carbon nanotubes. Phys. Rev. Lett. 81(12), 2506–2509 (1998)CrossRefGoogle Scholar
  34. 34.
    Léonard, F.M.C., Tersoff, J.: Role of fermi-level pinning in nanotube schottky diodes. Phys. Rev. Lett. 84(20), 4693–4696 (2000)CrossRefGoogle Scholar
  35. 35.
    Kane, E.: Zener tunneling in semiconductors. J. Phys. Chem. Solids 12(2), 181–188 (1960)MathSciNetGoogle Scholar
  36. 36.
    Lundstrom, M.: Fundamentals of Carrier Transport. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  37. 37.
    Jüngel, A.: Transport equations for semiconductors, ser. Lecture notes in physics. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  38. 38.
    Claus, M.: Modeling of ballistic carbon nanotube transistors for analog high-frequency applications, Ph.D. Dissertation, Technische Universität Dresden (2011)Google Scholar
  39. 39.
    Einspruch, N. G., Frensley, W. R. (eds): Heterostructures and Quantum Devices. Academic Press, San Diego (1994)Google Scholar
  40. 40.
    Fernando, C.L., Frensley, W.R.: An efficient method for the numerical evaluation of resonant states. J. Appl. Phys. 76(5), 2881–2886 (1994)CrossRefGoogle Scholar
  41. 41.
    Arnold, A.: Mathematical concepts of open quantum boundary conditions. Transp. Theory Stat. Phys. 30, 561–584 (2001)CrossRefzbMATHGoogle Scholar
  42. 42.
    Arnold, A., Ehrhardt, M.: Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics. J. Comput. Phys. 145(2), 611–638 (1998) Google Scholar
  43. 43.
    Deuflhard, P.: Newton Methods for Nonlinear Problems : Affine Invariance and Adaptive Algorithms. Springer series in computational mathematics. Springer, Heidelberg (2006)Google Scholar
  44. 44.
    Espelid, T. O.: Doubly adaptive quadrature routines based on newton-cotes rules. In: Reports in Informatics 229, Department of Informatics, Univerity of Bergen, (2002)Google Scholar
  45. 45.
    Pinaud, O.: Transient simulations of a resonant tunneling diode. J. Appl. Phys. 92(4), 1987–1994 (2002)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Cheng, C., Lee, J.-H., Lim, K.H., Massoud, H.Z., Liu, Q.H.: 3d quantum transport solver based on the perfectly matched layer and spectral element methods for the simulation of semiconductor nanodevices. J. Comput. Phys. 227(1), 455–471 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Claus, M., Mothes, S., Schröter, M.: A numerical device simulator for nanoscale carbon nanotube transistors. In: Proceedings of the Semiconductor Conference Dresden, vol. A3–3, (2009)Google Scholar
  48. 48.
    Gander, W., Gautschi, W.: Adaptive quadrature—revisited. BIT Numer. Math. 40(1), 84–101 (2000)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Reich, S., Thomsen, C., Maultzsch, C.: Carbon Nanotubes: Basic Concepts and Physical Properties. Wiley, Weinheim (2004)Google Scholar
  50. 50.
    Koswatta, S.O., Neophytou, N., Kienle, D., Fiori, G., Lundstrom, M.S.: Dependence of DC characteristics of CNT MOSFETs on bandstructure models. IEEE Trans. Nanotechnol. 5(4), 368–372 (2006)CrossRefGoogle Scholar
  51. 51.
    Johnson, S. G.: Notes on Perfectly Matched Layers (PMLs), lecture Notes at Massachusetts Institute of Technology (MIT), USA (2008)Google Scholar
  52. 52.
    Odermatt, S., Luisier, M., Witzigmann, B.: Bandstructure calculation using the k \(\cdot \) p method for arbitrary potentials with open boundary conditions. J. Appl. Phys. 97(4), 046104 (2005)CrossRefGoogle Scholar
  53. 53.
    Karner, M., Gehring, A., Kosina, H., Selberherr, S.: Efficient calculation of quasi-bound state tunneling in CMOS devices, In: Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices SISPAD 2005(01–03), 35–38 (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Martin Claus
    • 1
  • Sven Mothes
    • 1
  • Stefan Blawid
    • 2
  • Michael Schröter
    • 1
  1. 1.Chair for Electron Devices and Integrated Circuits, Department of Electrical and Computer Engineering, Center for Advancing Electronics Dresden (Cfaed)Technische Universität DresdenDresdenGermany
  2. 2.Laboratory for Devices and Integrated Circuits, Department of Electrical EngineeringUniversidade de BrasíliaBrasíliaBrazil

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