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

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

3. 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

4. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 4 Experiments & modelling X. D. Cui et al. Science, Vol 294, 571-574 (2001) 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 G 0 (≈ 1/(13 kΩ))‏ [R.H.M.Smit et al. Nature (2002).] Theory – Density Functional Theory + NEGF: for G ≈ G 0 generally good for G << G 0 poor e.g. G ≈ 0.046 G 0 for Benzene-di-amin/Au [Quek, Nano Lett. (2007).]

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

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

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

8. 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

9. 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 For constrictions the correction term goes to 1 G 2P

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

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

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

13. 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:

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

15. 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

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

17. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 17 Electronic Excitations: G 1 and G 2 | N,0  | N+1,s  | N–1,s  | N,s 

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

19. Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 19 GW(ST) method Au (6s), FCC 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

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

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

22. 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 Example: free space in 1D 2. Occupying subset of the basis we recover some many-electron eigenstate 3. Basis functions generated by time propagation Density

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

24. 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  n =2  /  n guarantees orthogonality... and completeness

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

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

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

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

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

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

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

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

33. Acknowledgements and Summary

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

35. 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

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

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

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