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Simulating Transport in Nanodevices Using the Usuki Method

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Nano-Electronic Devices

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Abstract

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.

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References

  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. Akis R., Ferry D.K.: Simulations of Spin Filtering Effects in a Quantum Point Contact. J. Phys. Condensed Matter (2008). doi: 10.1088/0953–8984/20/16/164201

    Google Scholar 

  3. Akis R., Bird J.P., Ferry D.K. Magnetotransport fluctuations in regular semiconductor ballistic quantum dots. Phys. Rev. B 54, 17705–17715 (1996).

    Article  Google Scholar 

  4. Akis R., Bird J.P., Ferry D.K.: The persistence of eigenstates in open quantum dots. Appl. Phys. Lett. (2002). doi: 10.1063/1.1490404

    Google Scholar 

  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. 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).

    Article  Google Scholar 

  7. Ando T.: Quantum point contacts in magnetic fields. Phys. Rev. B 44, 8017–8027 (1991).

    Article  Google Scholar 

  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. 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).

    Article  Google Scholar 

  10. Benisty H.: Reduced electron-phonon relaxation rates in quantum-box systems: Theoretical analysis. Phys. Rev. B 51, 13281–13292 (1995).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  16. Blume-Kohout R., Zurek W.H.: Quantum Darwinism in Quantum Brownian Motion. Phys. Rev. Lett. (2008). doi: 10.1103/PhysRevLett.101.240405

    MathSciNet  Google Scholar 

  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.98.204101

    Google Scholar 

  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.101.024102

    Google Scholar 

  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: 10.1088/0953–8984/21/21/212201

    Google Scholar 

  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.104.176801

    Google Scholar 

  21. Büttiker M.: Role of quantum coherence in series resistors. Phys. Rev. B 33, 3020–3026 (1986).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  24. Cui Y., Lieber C. M.: Functional nanoscale electronic devices assembled using silicon nanowire building blocks. Science 291, 851–853 (2001).

    Article  Google Scholar 

  25. Cummings A.W., Akis R., Ferry D.K.: Electron spin filter based on Rashba spin-orbit coupling. Appl. Phys. Lett. (2006). doi: 10.1063/1.2364859

    Google Scholar 

  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).

    Article  Google Scholar 

  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: 10.1063/1.2980328

    Google Scholar 

  28. Datta S.: Nanoscale device modeling: the Green’s function method. Superlattices and Microstructures 28, 253–278 (2000).

    Article  Google Scholar 

  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.88.236804

    Google Scholar 

  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).

    Article  Google Scholar 

  31. Dresselhaus G.: Spin-Orbit Coupling Effects in Zinc Blende Structures. Phys. Rev., 100, 580–586, (1955).

    Article  MATH  Google Scholar 

  32. Ferry D. K.: Effective potentials and the onset of quantization in ultrasmall MOSFETs. Superlatt. Microstruct. 28, 419–423 (2000).

    Article  Google Scholar 

  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.93.026803

    Google Scholar 

  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: 10.1088/0953–8984/17/13/001

    Google Scholar 

  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)

    Chapter  Google Scholar 

  36. Ferry D.K., Goodnick S.M., Bird J.P.: Transport in Nanostructures, Second Edition Cambridge, Cambridge (2009)

    Google Scholar 

  37. Fetter A. L., Walecka J. D.: Quantum Theory of Many-Particle Systems. McGraw-Hill, New York (1971)

    Google Scholar 

  38. Fischetti M.V.: Theory of electron transport in small semiconductor devices using the Pauli master equation. J. Appl. Phys., 83, 270–291 (1988).

    Article  Google Scholar 

  39. Gilbert M.J., Akis R., Ferry D.K: Magnetically and electrically tunable semiconductor quantum waveguide inverter. Appl. Phys. Lett. (2002). doi: 10.1063/1.1525073

    Google Scholar 

  40. Gilbert M.J., Akis R., Ferry D.K.: Dual computational basis qubit in semiconductor heterostructures. J. Appl. Phys. (2003). doi: 10.1063/1.1599633

    Google Scholar 

  41. Gilbert M.J., Akis R., Ferry D.K.: Phonon-assisted ballistic to diffusive crossover in silicon nanowire transistors. J. Appl. Phys. (2005). doi: 10.1063/1.2120890

    Google Scholar 

  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)

    Article  Google Scholar 

  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.70.241301

    Google Scholar 

  44. Harris J., Akis R., Ferry D.K.: Magnetically switched quantum waveguide qubit”, Appl. Phys. Lett. 79, 2214–2215 (2001).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  46. Heller E. J.:. Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits. Phys. Rev. Lett. 53, 1515–1518 (1984).

    Article  MathSciNet  Google Scholar 

  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/ 344203

    Google Scholar 

  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.103.054101

    Google Scholar 

  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: 10.1063/1.3124359

    Google Scholar 

  50. Kadanoff L. P., Baym G.: Quantum Statistical Mechanics. Benjamin/Cummings, Reading (1962)

    MATH  Google Scholar 

  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.70.085410

    Google Scholar 

  52. Kedzierski J., Bokor J., Anderson E.: Novel method for silicon quantum wire transistor fabrication. J. Vac. Sci. Tech. B 17, 3244–3247 (1999).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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: 10.1063/1.365394

    Google Scholar 

  57. Landauer R.: Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Develop. 1, 223–231 (1957).

    Article  MathSciNet  Google Scholar 

  58. Landauer R.: Electrical resistance of disordered one-dimensional lattices. Phil. Mag. 21, 863–867 (1970).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  60. Lee M.H., Speyer G., Sankey O.F.: Theory of electron transport through single molecules of polyaniline. J. Phys. Condensed Matter (2007). doi: 10.1088/0953–8984/19/21/215204

    Google Scholar 

  61. Lundstrom M.: Elementary scattering theory of the Si MOSFET. IEEE Elect. Dev. Lett. 18, 361–363 (1997).

    Article  Google Scholar 

  62. Lundstrom M.: Fundamentals of Carrier Transport. Cambridge, Cambridge (2000).

    Book  Google Scholar 

  63. Marinescu D.C., Marinescu G.M.: Approaching Quantum Computation. Pearson Prentice Hall, Upper Saddle River (2005).

    Google Scholar 

  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)

    Article  Google Scholar 

  65. Moore G.E.: Cramming more components onto integrated circuits. Electronics 38 (1965).

    Google Scholar 

  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).

    Article  Google Scholar 

  67. Natori K.: Ballistic metal-oxide semiconductor field effect transistor. J. Appl. Phys. 76, 4879–4890 (1994).

    Article  Google Scholar 

  68. Neofotistos G., Lake R., Datta S.: Inelastic-scattering effects on single-barrier tunneling. Phys. Rev. B 43, 2442–2445 (1991).

    Article  Google Scholar 

  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.93.220401

    Google Scholar 

  70. Pala M.G., Iannaccone G.: Effect of dephasing on the current statistics of mesoscopic devices. Phys. Rev. Lett. 93, 256803 (2004).

    Article  Google Scholar 

  71. Pikus F.G., Likharev K.K.: Nanoscale field effect transistors: an ultimate size analysis. Appl. Phys. Lett. 71, 3661–3663 (1997).

    Article  Google Scholar 

  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.2006.883478

    Google Scholar 

  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. 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.94.206801

    Google Scholar 

  75. Svishenko A. and Anantram M.P.: Role of scattering in nanotransistors. IEEE Trans. Elect. Dev. 50, 1459–1466 (2003).

    Article  Google Scholar 

  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.92.126603

    Google Scholar 

  77. Speyer G., Akis R., Ferry D.K.: Rapid molecular conductance calculations using transfer matrix method. Physica E, 19, 145–148 (2003).

    Article  Google Scholar 

  78. Speyer G., Akis R., Ferry D.K.: Conductance investigations of stretched molecules. IEEE Trans. Nanotechnology (2005). doi: 10.1109/TNANO.2005.851287

    Google Scholar 

  79. Speyer G., Akis R., Ferry D.K.: Using local orbitals in DFT to examine oligothiophene conductance anomalies. J. Phys. Conf. Ser. (2006). doi: 10.1088/1742–6596/38/1/007

    Google Scholar 

  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. 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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  86. Winkler R.: Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer, Berlin (2003).

    Google Scholar 

  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. 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.72.115321

    Google Scholar 

  89. Zutic I., Fabian J., Das Sarma S.: Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004).

    Google Scholar 

  90. Zurek W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  91. Zurek W.H.: Quantum Darwinism. Nature Physics (2009). doi: 10.1038/nphys1202

    Google Scholar 

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Acknowledgements

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.

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  1. Department of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ, USA

    Richard Akis

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  1. Richard Akis
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  2. Matthew Gilbert
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  3. Gil Speyer
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  4. Aron Cummings
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Correspondence to Richard Akis .

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  1. Dept. Electrical Engineering &, Center for Solid State, Arizona State University, Tempe, 85287-5706, Arizona, USA

    Dragica Vasileska

  2. , Dept. of Electrical Engineering, Arizona State University, Tempe, 85287-5706, Arizona, USA

    Stephen M. Goodnick

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Akis, R., Gilbert, M., Speyer, G., Cummings, A., Ferry, D. (2011). Simulating Transport in Nanodevices Using the Usuki Method. In: Vasileska, D., Goodnick, S. (eds) Nano-Electronic Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8840-9_6

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