quantum transport beyond the independent-electron approximation

Quantum transport beyond the

independent-electron approximation

Rex Godby

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

+

Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 3

Bothersome aspects of quantum transport

Ab-initio modelNon-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

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 G0 (≈ 1/(13 kΩ)) [R.H.M.Smit et al. Nature (2002).]

Theory – Density Functional Theory + NEGF:

for G ≈ G0 generally goodfor G << G0 poor e.g. G ≈ 0.046 G0 for Benzene-di-amin/Au [Quek, Nano Lett. (2007).]

Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 5

The standard modelMads Brandbyge et al. PRB (2002).

1. Ab Initio Quantum Conductance with e-e

Interaction

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

F

2

ETheG

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

VIG P /4 /2 IG P

For constrictions the correction term goes

to 1

G2P

Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 10

Integrals - real space formulation

irreducible polarization:

Correction factor:

“2P” conductance:

irr z , z ' n zV to t z '

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- behaviour limited by system size

Order of limit essential: First L then 0

Lv /2 Fmin

Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 13

Relation to electronic structure

Model nanojunction: wire with a gapNumber 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 /G relation, exact closed equations for G, 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

')';',()';',(2

);',( '

deGWi ixxxxxx

where ),( rx is space+spin

In the time domain

)';',()';',()';',( ttGttiWtt xxxxxx

W is the dynamically screened Coulomb interaction between electrons

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

2. Stroboscopic Wavepacket Approach

for Calculating and Interpreting Quantum

Transport

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)

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 eigenstate3. Basis functions generated by time propagation

Density Hie ˆ

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Stroboscopic Wavepacket Basis II.

Reference Hamiltonian: infinite systemcontinuous spectrum translational symmetry (locality)

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 n=2/n guarantees orthogonality

... and completeness

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

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Stroboscopic propagation

Quantum transport beyond the indep.-electron approx. - Towler Institute, 30 July 2008 27

Compactness of basisIllustrated 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

td-NEGF from G. Stefanucci and C. O. Almbladh PRB (2004).

Wavepackets give physical picture for the period of the non-linear oscillations

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.

Acknowledgements and Summary

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

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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