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Numerical Algorithms for Time-Resolved Quantum Transport

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Numerical Methods for Time-Resolved Quantum Nanoelectronics

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In Sect. 2.3.3 we saw that wavefunction-based methods are the current state of the art for the types of systems we want to study (a large number of degrees of freedom and long times). In this chapter we shall explain the algorithms used to compute time-dependent observables using our wavefunction-based “source-sink” algorithm.

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References

  1. B. Gaury et al., Numerical simulations of time-resolved quantum electronics. Phys. Rep. 534(1), 1–37 (2014)

    Google Scholar 

  2. R. Li et al., A corrected NEGF+DFT approach for calculating electronic transport through molecular devices: filling bound states and patching the non-equilibrium integration. Chem. Phys. 336(2–3), 127–135 (2007)

    Article  ADS  Google Scholar 

  3. A. Dhar, D. Sen, Nonequilibrium Green’s function formalism and the problem of bound states. Phys. Rev. B 73(8), 085119 (2006)

    Article  ADS  Google Scholar 

  4. G. Stefanucci, Bound states in ab initio approaches to quantum transport: a time-dependent formulation. Phys. Rev. B 75(19), 195115 (2007)

    Article  ADS  Google Scholar 

  5. E. Khosravi et al., Bound states in time-dependent quantum transport: oscillations and memory effects in current and density. Phys. Chem. Chem. Phys. 11(22), 4535–4538 (2009)

    Article  Google Scholar 

  6. E. Khosravi et al., The role of bound states in time-dependent quantum transport. Appl. Phys. A 93(2), 355–364 (2008)

    Article  ADS  Google Scholar 

  7. R.E.V. Profumo et al., Quantum Monte Carlo for correlated out-of-equilibrium nanoelectronic devices. Phys. Rev. B 91(24), 245154 (2015)

    Article  ADS  Google Scholar 

  8. C.W. Groth et al., New J. Phys. 16(6), 063065 (2014)

    Google Scholar 

  9. R.P. Feynman, Forces in molecules. Phys. Rev. 56(4), 340–343 (1939)

    Article  ADS  MATH  Google Scholar 

  10. M. Wimmer, Quantum transport in nanostructures: from computational concepts to spintronics in graphene and magnetic tunnel junctions. 1. Aufl. Dissertationsreihe der Fakultät für Physik der Universität Regensburg 5 (Univ.-Verl. Regensburg, Regensburg, 2009)

    Google Scholar 

  11. I. Rungger, S. Sanvito, Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition. Phys. Rev. B 78(3), 035407 (2008)

    Article  ADS  Google Scholar 

  12. P. Amestoy et al., A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM. J. Matrix Anal. Appl. 23(1), 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Weston, X. Waintal, A linear-scaling source-sink algorithm for simulating timeresolved quantum transport and superconductivity (2015). arXiv:1510.05967 [cond-mat]

  14. X. Antoine et al., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4(4), 729–796 (2008)

    MathSciNet  MATH  Google Scholar 

  15. J.G. Muga et al., Complex absorbing potentials. Phys. Rep. 395(6), 357–426 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  16. O. Shemer, D. Brisker, N. Moiseyev, Optimal reflection-free complex absorbing potentials for quantum propagation of wave packets. Phys. Rev. A 71(3), 032716 (2005)

    Article  ADS  Google Scholar 

  17. U.V. Riss, H.-D. Meyer, The transformative complex absorbing potential method: a bridge between complex absorbing potentials and smooth exterior scaling. J. Phys. B: At. Mol. Opt. Phys. 31(10), 2279 (1998)

    Article  ADS  Google Scholar 

  18. U.V. Riss, H.-D. Meyer, Reflection-free complex absorbing potentials. J. Phys. B: At. Mol. Opt. Phys. 28(8), 1475 (1995)

    Article  ADS  Google Scholar 

  19. D.J. Kalita, K. Ashish, J. Chem. Phys. 134(9), 094301 (2011)

    Article  ADS  Google Scholar 

  20. Z.H. Zhang, Use of negative complex potential as absorbing potential. J. Chem. Phys. 108(4), 1429–1433 (1998)

    Google Scholar 

  21. R. Baer et al., Ab initio study of the alternating current impedance of a molecular junction. J. Chem. Phys. 120(7), 3387–3396 (2004)

    Article  ADS  Google Scholar 

  22. L. Zhang, J. Chen, J. Wang, First-principles investigation of transient current in molecular devices by using complex absorbing potentials. Phys. Rev. B 87(20), 205401 (2013)

    Article  ADS  Google Scholar 

  23. S. Kurth et al., Time-dependent quantum transport: a practical scheme using density functional theory. Phys. Rev. B 72(3), 035308 (2005)

    Article  ADS  Google Scholar 

  24. D.E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten- Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6(1–2), 61–71 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  25. W.H. Press (ed.), Numerical Recipes: The Art of Scientific Computing, 3rd edn (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  26. S.O. Fatunla, Numerical integrators for stiff and highly oscillatory differential equations. Math. Comput. 34(150), 373–390 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Madroñero, B. Piraux, Explicit time-propagation method to treat the dynamics of driven complex systems. Phys. Rev. A 80(3), 033409 (2009)

    Article  ADS  Google Scholar 

  28. A.L. Frapiccini et al., Explicit schemes for time propagating many-body wave functions. Phys. Rev. A 89(2), 023418 (2014)

    Google Scholar 

  29. J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Math. Proc. Cambridge Philos. Soc. 43(01), 50–67 (1947)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. G. Stefanucci, E. Perfetto, M. Cini, Time-dependent quantum transport with superconducting leads: a discrete-basis Kohn-Sham formulation and propagation scheme. Phys. Rev. B 81(11), 115446 (2010)

    Article  ADS  Google Scholar 

  31. S. Blanes, P.C. Moan, Splitting methods for the time-dependent Schrödinger equation. Phys. Lett. A 265(1–2), 35–42 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. B.A. Shadwick, W.F. Buell, Unitary integration with operator splitting for weakly dissipative systems. J. Phys. A: Math. Gen. 34(22), 4771 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. W. Magnus, On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7(4), 649–673 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Hochbruck, C. Lubich, On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 41(3), 945–963 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Blanes et al., The Magnus expansion and some of its applications. Phys. Rep. 470(5–6), 151–238 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  37. C. Groth et al., Algorithms for Quantum Transport (To be published) (2016)

    Google Scholar 

  38. R. Piessens et al., Quadpack. Springer Series in Computational Mathematics, vol. 1 (Springer, Berlin, Heidelberg, 1983)

    Google Scholar 

  39. R.P. Brent, Algorithms for Minimization Without Derivatives. Prentice-Hall series in Automatic Computation (Prentice-Hall, Englewood Cliffs, 1972)

    Google Scholar 

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Weston, J. (2017). Numerical Algorithms for Time-Resolved Quantum Transport. In: Numerical Methods for Time-Resolved Quantum Nanoelectronics. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-63691-7_3

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