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Quantum transport beyond the independent-electron approximation

Quantum transport beyond the independent-electron approximation. Rex Godby. Outline. Introduction to the quantum transport problem Ab initio quantum conductance in the presence of e-e interaction TDDFT MBPT Stroboscopic wavepacket approach for calculating and interpreting quantum transport.

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Quantum transport beyond the independent-electron approximation

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  1. Quantum transport beyond the independent-electron approximation Rex Godby

  2. Outline • Introduction to the quantum transport problem • Ab initio quantum conductance in the presence of e-e interaction • TDDFT • MBPT • Stroboscopic wavepacket approach for calculating and interpreting quantum transport Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 +

  3. Bothersome aspects of quantum transport Ab-initio model Non-equilibrium Quantum Mechanics Many-body problem Conductance G=I/U Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  4. Experiments & modelling Experiments: Octanedithiol/Au:R ≈ 900 MΩ [X. D. Cui et al., Science (2001).] Benzene-di-amin/Au: R≈ 2 MΩ [Quek, Nano Lett. (2007).] Benzene-di-thiol/Au: R ≈ 18±12 MΩ [M. A. Reed et al. Science (1997).] H2/Pt: G ≈ 0.95 G0 (≈ 1/(13 kΩ))‏ [R.H.M.Smit et al. Nature (2002).] Theory – Density Functional Theory + NEGF: for G ≈ G0 generally good for G << G0 poor e.g. G ≈ 0.046 G0 for Benzene-di-amin/Au [Quek, Nano Lett. (2007).] X. D. Cui et al. Science, Vol 294, 571-574 (2001) Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  5. The standard model Mads Brandbyge et al. PRB (2002). Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  6. 1. Ab Initio Quantum Conductance with e-e Interaction

  7. Quantum Transport Theories • Conductance in 1-electron or mean-field theory given by Landauer formula • Drawbacks of usual approach: • Can be orders of magnitude wrong • Difficult to generalise to many-body case • Calculation of T not readily compatible with periodic bcs Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  8. Our Approach • “Beyond ground-state DFT” description of quantum transport still troublesome • Formulate the linear-response theory of conductance for rigorous ab-initio modelling within a supercell technique: • well defined conductance • converged basis set • realistic e-e interaction 4-point conductance Plane-wave basis GW method P. Bokes, J. Jung and RWG, PRB 2007 Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  9. The 4-point conductance P. Bokes, J. Jung and RWG, PRB 2007 ? G2P For constrictions the correction term goes to 1 Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  10. Integrals - real space formulation “2P” conductance: irreducible polarization: Correction factor: Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  11. Real-space integrals for G Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  12. Long time limit - periodic box Low-a behaviour limited by system size Order of limit essential: First L then a0 Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  13. Relation to electronic structure Model nanojunction: wire with a gap Number of atoms in the lead: Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  14. Gold contacts • Perpendicular k-point sampling improves extrapolation Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  15. Calculating  including interactions • Time-dependent density-functional theory • e.g. J. Jung, P. Bokes, and RWG, PRL 2007 • but the form of the XC kernel is delicate • Na Sai, PRL(2005), Koentopp PRB(2006), Toher PRL(2005), Burke PRL(2005)‏ • or, better: Many-body perturbation theory Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  16. Hedin’s Equations • With S/G relation, exact closed equations for G, S etc. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  17. Electronic Excitations: G1 and G2 | N,0  | N+1,s | N–1,s | N,s Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  18. The GW Approximation Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  19. GW(ST) method • Corrects band-gaps and band alignment • Introduces finite lifetime • GW and quantum transport: • Thygesen JCP (2007), Darancet PRB (2007), Neaton PRL (2007), .... • Our implementation: real-space / imaginary-time: • Rojas, RWG, Needs PRL (1995)‏ • Finite temperatures (metals): Verstraete (2008)‏ • Effect on conductance… in progress Au (6s), FCC Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  20. 2. Stroboscopic Wavepacket Approach for Calculating and Interpreting Quantum Transport

  21. Motivation • Non-linear transport? • Time-dependent transport? • How about physical insight? P. Bokes, F. Corsetti and RWG, PRL (July 2008) ‏ Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  22. Stroboscopic Wavepacket Basis I. 1. Each basis function to be localised in space 2. Occupying subset of the basis we recover some many-electron eigenstate 3. Basis functions generated by time propagation Density Example: free space in 1D Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  23. Stroboscopic Wavepacket Basis II. • infinite system • continuous spectrum • translational symmetry (locality)‏ Reference Hamiltonian: Choice of normalisation: The initial set: n – energy-band index Arbitrary unitary rotation Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  24. Stroboscopic Wavepacket Basis III. Propagation of the initial set The time step tn=2p/Den guarantees orthogonality ... and completeness Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  25. Steady-state transport, Landauer Scattering WP Left-going WP basis Right-going WP basis Reference H Different H', non-translationally-invariant Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  26. Stroboscopic propagation Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  27. Compactness of basis Illustrated for continuous-time propagation Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  28. Propagation through barrier Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  29. Time-dependence: switching-on Wavepackets give physical picture for the period of the non-linear oscillations td-NEGF from G. Stefanucci and C. O. Almbladh PRB (2004). Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  30. Edge-induced spin Hall effect I. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  31. Edge-induced spin Hall effect II. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  32. Edge-induced spin Hall effect III. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  33. Acknowledgements and Summary

  34. Collaborators • Peter Bokes • Matthieu Verstraete • Jeil Jung • Fabiano Corsetti Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  35. Summary • 4-point conductance PRB 2007 • well defined for interacting systems • numerically feasible in supercell geometry • GW calculations in progress • Stroboscopic wavepacket basis PRL 2008 • particularly suitable for transport problems • applications for TD transport and spin Hall effect Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  36. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  37. Numerical convergence of wavepacket basis Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

  38. Jellium wire vs. Na wire Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008

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