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Quantum transport beyond the independent-electron approximation Rex Godby

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

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Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July Experiments & modelling X. D. Cui et al. Science, Vol 294, (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 ≈ G 0 for Benzene-di-amin/Au [Quek, Nano Lett. (2007).]

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

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1. Ab Initio Quantum Conductance with e-e Interaction

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

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

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Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July The 4-point conductance ? P. Bokes, J. Jung and RWG, PRB 2007 For constrictions the correction term goes to 1 G 2P

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

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

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

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

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

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

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

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

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

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

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2. Stroboscopic Wavepacket Approach for Calculating and Interpreting Quantum Transport

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

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

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

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

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

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

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

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

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

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

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

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

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Acknowledgements and Summary

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

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

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

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

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

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