Simulating Transport in Nanodevices Using the Usuki Method

  • Richard AkisEmail author
  • Matthew Gilbert
  • Gil Speyer
  • Aron Cummings
  • David Ferry


To calculate the conductance of mesoscopic structures such as quantum wires and dots at low temperature and bias, one typically employs the Landauer–Büttiker formalism, which relates quantum mechanical transmission probability to conductance. In this chapter, we discuss a numerically stable method to solve this transmission problem, the Usuki method, which is closely related to both the scattering matrix approach and recursive Green’s functions. It has a major advantage over the latter in that the electron density can be obtained far more efficiently. Various applications of this approach are presented: transport through open quantum dots, the study of spin filtering effects in quantum wire structures, computing the conductance of molecules and the application of the method to study MOSFETS. The extensions to the basic method required for each case are also discussed, the most extensive of which are required for the MOSFET problem, where inelastic scattering effects play a crucial role.


Nanostructures Quantum dots Quantum wires Molecular electronics MOSFETs Spin-Hall Effect 



We would like to thank the financial support from the Office of Naval Research, the Department of Energy and Intel Corporation. The experiments of Prof. Jonathan Bird and colleagues at ASU and at the University of Buffalo were the inspiration for much of the simulation work we have done over the years. The group of Prof. Yuichi Ochiai at Chiba University provided similar inspiration. At ASU, we have also had fruitful collaborations with the groups of Prof. Dragica Vasileska, Prof. Ying-Chen Lai, Prof. Otto Sankey, and Prof. Stephen Goodnick. The team of Roland Brunner and his adviser Prof. Friedemar Kuchar at the University of Leoben helped illuminate the correspondence between our quantum simulations and the classical behavior in quantum dots. Our thanks also go out to Jan Jacob and his advisors Prof. Meier and Prof. Matsuyama at the University of Hamburg for the collaborative work that they initiated on the spin Hall effect.


  1. 1.
    Akis R., Ferry D.K Quantum waveguide array generator for performing Fourier transforms: Alternate route to quantum computing. Appl. Phys. Lett. 79, 2823–2825 (2001).Google Scholar
  2. 2.
    Akis R., Ferry D.K.: Simulations of Spin Filtering Effects in a Quantum Point Contact. J. Phys. Condensed Matter (2008). doi: 10.1088/0953–8984/20/16/164201Google Scholar
  3. 3.
    Akis R., Bird J.P., Ferry D.K. Magnetotransport fluctuations in regular semiconductor ballistic quantum dots. Phys. Rev. B 54, 17705–17715 (1996).CrossRefGoogle Scholar
  4. 4.
    Akis R., Bird J.P., Ferry D.K.: The persistence of eigenstates in open quantum dots. Appl. Phys. Lett. (2002). doi: 10.1063/1.1490404Google Scholar
  5. 5.
    Akis R., Bird J.P., Vasileska D., Ferry D.K., deMoura A.P.S., Lai Y.-C.: On the Influence of Resonant States on Ballistic Transport in Open Quantum Dots: Spectroscopy and Tunneling in the Presence of Multiple Conducting Channels, In: Bird J.P. (ed.) Electron Transport in Quantum Dots, pp. 209–276. Kluwer Academic Publishers, Boston (2003)Google Scholar
  6. 6.
    Akis R., Gilbert M., Ferry D.K.: Fully quantum mechanical simulations of gated silicon quantum wire structures: investigating the effects of changing wire cross-section on transport. J. Phys. Conf. Series 36, 87–90 (2006).CrossRefGoogle Scholar
  7. 7.
    Ando T.: Quantum point contacts in magnetic fields. Phys. Rev. B 44, 8017–8027 (1991).CrossRefGoogle Scholar
  8. 8.
    Assad F., Ren Z., Vasileska D., Datta S., M. Lundstrom: On the performance limits for Si MOSFETs: a theoretical study. IEEE Trans. Elec. Dev. 47, 232–240 (2000)Google Scholar
  9. 9.
    Baranger H.U., Stone A. D.: Electrical linear-response theory in an arbitrary magnetic field: A new Fermi-surface formation. Phys. Rev. B 40, 8169–8193 (1989).CrossRefGoogle Scholar
  10. 10.
    Benisty H.: Reduced electron-phonon relaxation rates in quantum-box systems: Theoretical analysis. Phys. Rev. B 51, 13281–13292 (1995).CrossRefGoogle Scholar
  11. 11.
    Bird J. P., Olatona D. M., Newbury R., Taylor R. P., Ishibashi K., Stopa M., Aoyagi Y., Sugano T., Ochiai Y.: Lead-induced transition to chaos in ballistic mesoscopic billiards. Phys. Rev. B 52, R14336–R14339 (1995).CrossRefGoogle Scholar
  12. 12.
    Bird J.P., Ferry D.K. R., Ishibashi K., Aoyagi Y., Sugano T., Ochiai Y.: Periodic conductance fluctuations and stable orbits in mesoscopic semiconductor billiards. Europhys. Lett. 35, 529–534 (1996).CrossRefGoogle Scholar
  13. 13.
    Bird J.P., Akis R., Ferry D.K. Vasileska D., Cooper J., Aoyagi Y., Sugano T.: Lead-orientation-dependent wave function scarring in open quantum dots. Phys. Rev. Lett. 82, 4691–4694 (1999a).CrossRefGoogle Scholar
  14. 14.
    Bird J.P., Akis R., Ferry D.K.: Magnetoprobing of the discrete level spectrum of open quantum dots. Phys. Rev. B 60, 13676–13681 (1999b).CrossRefGoogle Scholar
  15. 15.
    Bird J.P., Akis R., Ferry D.K., de Moura A.P.S., Lai Y.-C., Indlekofer K.M.: Interference and interactions in open quantum dots. Rep. Prog. Phys. 66, 583–632 (2003).CrossRefGoogle Scholar
  16. 16.
    Blume-Kohout R., Zurek W.H.: Quantum Darwinism in Quantum Brownian Motion. Phys. Rev. Lett. (2008). doi: 10.1103/PhysRevLett.101.240405MathSciNetGoogle Scholar
  17. 17.
    Brunner R., Kuchar F., Meisels R., Akis R., Ferry D.K., Bird J.P.,: Draining of the Sea of Chaos: Role of Resonant Transmission and Reflection in an Array of Billiards. Phys. Rev. Lett. (2007). doi: 10.1103/PhysRevLett.98.204101Google Scholar
  18. 18.
    Brunner R., Akis R., Ferry D.K., Kuchar F., Meisels R.: Coupling-induced bipartite pointer states in arrays of electron billiards: Quantum Darwinism in action?. Phys. Rev. Lett. (2008). doi: 10.1103/PhysRevLett.101.024102Google Scholar
  19. 19.
    Burke, A. M., Akis, R., Day T. E., Speyer G., Ferry D.K., Bennett B.R.: Imaging scarred states in quantum dots. J. Phys. Condensed Matter (2009). doi: 10.1088/0953–8984/21/21/212201Google Scholar
  20. 20.
    Burke, A. M., Akis, R., Day T. E., Speyer G., Ferry D.K., Bennett B.R.: Periodic Scarred States in Open Quantum Dots as Evidence of Quantum Darwinism. Phys. Rev. Lett. (2010). doi: 10.1103/PhysRevLett.104.176801Google Scholar
  21. 21.
    Büttiker M.: Role of quantum coherence in series resistors. Phys. Rev. B 33, 3020–3026 (1986).CrossRefGoogle Scholar
  22. 22.
    Büttiker M., Imry Y., Landauer R., Pinhas S.: Generalized many-channel conductance formula with application to small rings. Phys. Rev. B 31, 6207–6215 (1985).CrossRefGoogle Scholar
  23. 23.
    Bychkov Y.A., Rashba E.I.: Oscillatory effects and the magnetic susceptibility of carriers in inversion layers. J. Phys. C. 17, 6039–6045 (1984).CrossRefGoogle Scholar
  24. 24.
    Cui Y., Lieber C. M.: Functional nanoscale electronic devices assembled using silicon nanowire building blocks. Science 291, 851–853 (2001).CrossRefGoogle Scholar
  25. 25.
    Cummings A.W., Akis R., Ferry D.K.: Electron spin filter based on Rashba spin-orbit coupling. Appl. Phys. Lett. (2006). doi: 10.1063/1.2364859Google Scholar
  26. 26.
    Cummings A.W., Akis R., Ferry D.K.: The Rashba Effect and Non-Abelian Phase in Quantum Wire Devices. J. Comp. Elect. 6, 101–104 (2007).CrossRefGoogle Scholar
  27. 27.
    Cummings A.W., Akis R., Ferry D.K., Jacob J., Matsuyama T., Merkt U., Meier G.: Cascade of Y-shaped spin filters in InGaAs/InAs/InGaAs quantum wells. J. Appl. Phys. (2008). doi: 10.1063/1.2980328Google Scholar
  28. 28.
    Datta S.: Nanoscale device modeling: the Green’s function method. Superlattices and Microstructures 28, 253–278 (2000).CrossRefGoogle Scholar
  29. 29.
    de Moura A.P.S., Lai Y.-C., Akis R., Bird J.P., Ferry D.K.: Tunneling and Nonhyperbolicity in Quantum Dots. Phys. Rev. Lett. (2002). doi: 10.1103/PhysRevLett.88.236804Google Scholar
  30. 30.
    Di Ventra M., Pantelides S. T., Lang, N. D.: First-Principles Calculation of Transport Properties of a Molecular Device. Phys. Rev. Lett. 84, 979–982 (2000).CrossRefGoogle Scholar
  31. 31.
    Dresselhaus G.: Spin-Orbit Coupling Effects in Zinc Blende Structures. Phys. Rev., 100, 580–586, (1955).zbMATHCrossRefGoogle Scholar
  32. 32.
    Ferry D. K.: Effective potentials and the onset of quantization in ultrasmall MOSFETs. Superlatt. Microstruct. 28, 419–423 (2000).CrossRefGoogle Scholar
  33. 33.
    Ferry D.K., Akis R., Bird J.P.: Einselection in action: Decoherence and pointer states in open quantum dots. Phys. Rev. Lett. (2004). doi: 10.1103/PhysRevLett.93.026803Google Scholar
  34. 34.
    Ferry D.K., Akis R., Bird J.P.: Einselection and the quantum to classical transition in quantum dots. J. Phys. Condensed Matter (2005a). doi: 10.1088/0953–8984/17/13/001Google Scholar
  35. 35.
    Ferry D.K., Akis R., Gilbert M.J., Ramey S.M.: Physics of Silicon Nanodevices. In: Oda S., Ferry D.K. (eds.) Silicon Nanoelectronics, pp. 200–210. Taylor & Francis, Boca Raton (2005b)CrossRefGoogle Scholar
  36. 36.
    Ferry D.K., Goodnick S.M., Bird J.P.: Transport in Nanostructures, Second Edition Cambridge, Cambridge (2009)Google Scholar
  37. 37.
    Fetter A. L., Walecka J. D.: Quantum Theory of Many-Particle Systems. McGraw-Hill, New York (1971)Google Scholar
  38. 38.
    Fischetti M.V.: Theory of electron transport in small semiconductor devices using the Pauli master equation. J. Appl. Phys., 83, 270–291 (1988).CrossRefGoogle Scholar
  39. 39.
    Gilbert M.J., Akis R., Ferry D.K: Magnetically and electrically tunable semiconductor quantum waveguide inverter. Appl. Phys. Lett. (2002). doi: 10.1063/1.1525073Google Scholar
  40. 40.
    Gilbert M.J., Akis R., Ferry D.K.: Dual computational basis qubit in semiconductor heterostructures. J. Appl. Phys. (2003). doi: 10.1063/1.1599633Google Scholar
  41. 41.
    Gilbert M.J., Akis R., Ferry D.K.: Phonon-assisted ballistic to diffusive crossover in silicon nanowire transistors. J. Appl. Phys. (2005). doi: 10.1063/1.2120890Google Scholar
  42. 42.
    Grubin H.L., Kreskovsky J.P., Govindan T.R., Ferry D.K.: Uses of the quantum potential in modeling hot-carrier semiconductor devices. Semicond. Sci. Technol. 9, 855–858 (1994)CrossRefGoogle Scholar
  43. 43.
    Hankiewicz E. M., Molenkamp L. W., Jungwirth T., and Sinova J.: Manifestation of the spin Hall effect through charge-transport in the mesoscopic regime. Phys. Rev. B, vol. 70, p., Dec. 2004. doi: 10.1103/PhysRevB.70.241301Google Scholar
  44. 44.
    Harris J., Akis R., Ferry D.K.: Magnetically switched quantum waveguide qubit”, Appl. Phys. Lett. 79, 2214–2215 (2001).CrossRefGoogle Scholar
  45. 45.
    He H., Zhu J., Tao N. J., Nagahara L. A., Amlani I., Tsui R.: A Conducting Polymer Nanojunction Switch. J. Am. Chem. Soc., 123, 7730–7731 (2001).CrossRefGoogle Scholar
  46. 46.
    Heller E. J.:. Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits. Phys. Rev. Lett. 53, 1515–1518 (1984).MathSciNetCrossRefGoogle Scholar
  47. 47.
    Huang L., Lai Y.-C., Ferry D. K., Goodnick S. M., Akis R.: Transmission and scarring in graphene quantum dots. Phys. Condensed Matter (2009). doi: 10.1088/0953–8984/21/34/ 344203Google Scholar
  48. 48.
    Huang L., Lai Y.-C., Ferry D. K., Goodnick S. M., Akis R.: Relativistic quantum scars. Phys.Rev. Lett. (2010). doi: 10.1103/PhysRevLett.103.054101Google Scholar
  49. 49.
    Jacob J., Meier G., Peters S., Matsuyama T., Merkt U., Cummings A.W., Akis R., Ferry D.K.: Generation of highly spin-polarized currents in cascaded In As spin filters. J. Appl. Phys. (2009). doi: 10.1063/1.3124359Google Scholar
  50. 50.
    Kadanoff L. P., Baym G.: Quantum Statistical Mechanics. Benjamin/Cummings, Reading (1962)zbMATHGoogle Scholar
  51. 51.
    Ke S.-H., Baranger H.U., Yang W.: Electron transport through molecules: Self-consistent and non-self-consistent approaches. Phys. Rev B (2004). doi: 10.1103/PhysRevB.70.085410Google Scholar
  52. 52.
    Kedzierski J., Bokor J., Anderson E.: Novel method for silicon quantum wire transistor fabrication. J. Vac. Sci. Tech. B 17, 3244–3247 (1999).CrossRefGoogle Scholar
  53. 53.
    Kluksdahl N.C., Kriman A.M., Ferry D.K., and Ringhofer C.: Self-consistent study of the resonant tunneling diode. Phys. Rev. B, 39, 7720–7735 (1989).CrossRefGoogle Scholar
  54. 54.
    Ko D.Y.K., Inkson J.C.: Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems. Phys. Rev. B, 38, 9945–9951 (1988).CrossRefGoogle Scholar
  55. 55.
    Kotlyar R., Obradovic B., Matagne P., Stettler M., Giles M.D.: Assessment of room-temperature phonon-limited mobility in gated silicon nanowires. Appl. Phys. Lett. 84, 5270–5272 (2004).CrossRefGoogle Scholar
  56. 56.
    Lake R., Klimeck G., Bowen R.C., Jovanovic D: Single and multiband modeling of quantum electron transport through layered semiconductor devices. J. App. Phys. (1997). doi: 10.1063/1.365394Google Scholar
  57. 57.
    Landauer R.: Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Develop. 1, 223–231 (1957).MathSciNetCrossRefGoogle Scholar
  58. 58.
    Landauer R.: Electrical resistance of disordered one-dimensional lattices. Phil. Mag. 21, 863–867 (1970).CrossRefGoogle Scholar
  59. 59.
    Laux S.E., Kumar A., Fischetti M.V.: Ballistic FET modeling using QDAME: quantum device analysis by modal expansion. IEEE Trans. Nano. 1, 255–259 (2002).CrossRefGoogle Scholar
  60. 60.
    Lee M.H., Speyer G., Sankey O.F.: Theory of electron transport through single molecules of polyaniline. J. Phys. Condensed Matter (2007). doi: 10.1088/0953–8984/19/21/215204Google Scholar
  61. 61.
    Lundstrom M.: Elementary scattering theory of the Si MOSFET. IEEE Elect. Dev. Lett. 18, 361–363 (1997).CrossRefGoogle Scholar
  62. 62.
    Lundstrom M.: Fundamentals of Carrier Transport. Cambridge, Cambridge (2000).CrossRefGoogle Scholar
  63. 63.
    Marinescu D.C., Marinescu G.M.: Approaching Quantum Computation. Pearson Prentice Hall, Upper Saddle River (2005).Google Scholar
  64. 64.
    MacDiarmid A.G., Chiang J.-C., Richter A.F., Epstein A.J.: Polyaniline: A New Concept in Conducting Polymers. Synth. Met., 18, 285 (1987)CrossRefGoogle Scholar
  65. 65.
    Moore G.E.: Cramming more components onto integrated circuits. Electronics 38 (1965).Google Scholar
  66. 66.
    Namatsu H., Kurihara K., Nagase M., Makino T.: Fabrication of 2 nm wide silicon quantum wires through a combination of a partially-shifted resist pattern and orientation-dependent etching. Appl. Phys. Lett. 70, 619–621 (1997).CrossRefGoogle Scholar
  67. 67.
    Natori K.: Ballistic metal-oxide semiconductor field effect transistor. J. Appl. Phys. 76, 4879–4890 (1994).CrossRefGoogle Scholar
  68. 68.
    Neofotistos G., Lake R., Datta S.: Inelastic-scattering effects on single-barrier tunneling. Phys. Rev. B 43, 2442–2445 (1991).CrossRefGoogle Scholar
  69. 69.
    Ollivier H., Poulin D., Zurek W.H.: Objective Properties from Subjective Quantum States: Environment as a Witness. Phys. Rev. Lett. (2004). doi: 10.1103/PhysRevLett.93.220401Google Scholar
  70. 70.
    Pala M.G., Iannaccone G.: Effect of dephasing on the current statistics of mesoscopic devices. Phys. Rev. Lett. 93, 256803 (2004).CrossRefGoogle Scholar
  71. 71.
    Pikus F.G., Likharev K.K.: Nanoscale field effect transistors: an ultimate size analysis. Appl. Phys. Lett. 71, 3661–3663 (1997).CrossRefGoogle Scholar
  72. 72.
    Ramamoorthy A., Akis, R., Bird J.P: Influence of Realistic Potential Profile of Coupled Electron Waveguide on Electron Switching Characteristics. IEEE Trans. Nanotechnology (2006). doi: 10.1109/TNANO.2006.883478Google Scholar
  73. 73.
    Reed M. A., Zhou C., Muller C. J., Burgin T. P., Tour J. M.: Conductance of a Molecular Junction. Science, 278, 252–254 (1997).Google Scholar
  74. 74.
    Schliemann J., Loss D., Westervelt R.M: Zitterbewegung of ElectronicWave Packets in III-V Zinc-Blende Semiconductor Quantum Wells. Phys. Rev. Lett. (2005). doi: 10.1103/PhysRevLett.94.206801Google Scholar
  75. 75.
    Svishenko A. and Anantram M.P.: Role of scattering in nanotransistors. IEEE Trans. Elect. Dev. 50, 1459–1466 (2003).CrossRefGoogle Scholar
  76. 76.
    Sinova J., Culcer D., Niu Q., Sinitsyn N. A., Jungwirth T., MacDonald A.H: Universal Intrinsic Spin Hall Effect. Phys. Rev. Lett. (2004). doi: 10.1103/PhysRevLett.92.126603Google Scholar
  77. 77.
    Speyer G., Akis R., Ferry D.K.: Rapid molecular conductance calculations using transfer matrix method. Physica E, 19, 145–148 (2003).CrossRefGoogle Scholar
  78. 78.
    Speyer G., Akis R., Ferry D.K.: Conductance investigations of stretched molecules. IEEE Trans. Nanotechnology (2005). doi: 10.1109/TNANO.2005.851287Google Scholar
  79. 79.
    Speyer G., Akis R., Ferry D.K.: Using local orbitals in DFT to examine oligothiophene conductance anomalies. J. Phys. Conf. Ser. (2006). doi: 10.1088/1742–6596/38/1/007Google Scholar
  80. 80.
    Speyer G., Akis R., Ferry D.K.: Complexities of the Molecular Conductance Problem. In: Lyshevski S.E. (ed.) Nano and Molecular Electronics Handbook, pp 21–1–68. CRC Press, Boca Raton (2007)Google Scholar
  81. 81.
    Thornton T.J., Pepper M., Ahmed H., Andrews D., Davies G.J.: One-Dimensional Conduction in the 2D Electron Gas of a GaAs-AlGaAs Heterojunction. Phys. Rev. Lett. 56, 1198–1201 (1986).CrossRefGoogle Scholar
  82. 82.
    Usuki T., Saito M., Takatsu M., Kiehl R.A., Yokoyama N.: Numerical analysis of electron wave detection by a wedge shaped point contact. Phys. Rev. B 520, 7615–7625 (1994).CrossRefGoogle Scholar
  83. 83.
    Usuki T., Saito M., Takatsu M., Kiehl R.A., Yokoyama N.: Numerical analysis of ballistic electron transport in magnetic-fields by using a quantum point contact and a quantum wire. Phys. Rev. B 52, 8244–8255 (1995).CrossRefGoogle Scholar
  84. 84.
    van der Vorst H. A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. J. SIAM J. Sci. Stat. Comp. 13, 631–644 (1992).zbMATHCrossRefGoogle Scholar
  85. 85.
    Vignolo P., Farchioni R., Grosso G.: Tight-Binding Effective Hamiltonians for the Electronic States of Polyaniline Chains. Phys. Stat. Sol. B 223, 853–866 (2001).CrossRefGoogle Scholar
  86. 86.
    Winkler R.: Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer, Berlin (2003).Google Scholar
  87. 87.
    Wong H.S., Taur Y.: Three-dimensional “atomistic” simulation of discrete random dopant distribution effects in sub-0.1 μm MOSFETs. IEDM Tech. Dig., 705–708 (1993).Google Scholar
  88. 88.
    Yamamoto M., Ohtsuki T., Kramer B: Spin polarization in a T-shaped conductor induced by strong Rashba spin-orbit coupling. Phys. Rev. B (2005). doi: 10.1103/PhysRevB.72.115321Google Scholar
  89. 89.
    Zutic I., Fabian J., Das Sarma S.: Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004).Google Scholar
  90. 90.
    Zurek W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Zurek W.H.: Quantum Darwinism. Nature Physics (2009). doi: 10.1038/nphys1202Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Richard Akis
    • 1
    Email author
  • Matthew Gilbert
  • Gil Speyer
  • Aron Cummings
  • David Ferry
  1. 1.Department of Electrical, Computer, and Energy EngineeringArizona State UniversityTempeUSA

Personalised recommendations