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Meir-WinGreen Formula

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Meir-WinGreen Formula

Quantum dot

U

Consider a quantum dot ( a nano conductor, modeled for example by an Anderson model) connected with quantum wires

Quantum dot

U

Consider a quantum dot ( a nano conductor, modeled for example by an Anderson model) connected with wires

where L,R refers to the left and right electrodes. Due to small size, charging energy U is important. If one electron jumps into it, the arrival of a second electron is hindered (Coulomb blockade)

Meir and WinGreen have shown, using the Keldysh formalism, that the current through the quantum dot is given in terms of the local retarded Green’s function for electrons of spin s at the dot by

This has been used for weak V also in the presence of strong U.

3

General partition-free framework

and rigorous Time-dependent current formula

Partitioned approach has drawbacks: it is different from what is done experimentally, and L and R subsystems not physical, due to specian boundary conditions. It is best to include time-dependence!

4

Interactions can be included by Keldysh formalism, (now also by time-dependent density functional)

Time-dependent Quantum Transport

device

J

System is in equilibrium until at time t=0 blue sites are shifted to V and J starts

5

Use of Green’s functions

Rigorous Time-dependent current formula

derived by equation of motion or Keldysh method

Note:

Occupation numbers refer to H before the time dependence sets in. System remembers initial conditions!

Current-Voltage characteristics

In the 1980 paper I have shown how one can obtain the current-voltage characteristics by a long-time asyptotic development. Recently Stefanucci and Almbladh have shown that the characteristics for non-interacting systems agree with Landauer

Long-Time asymptotics and current-voltage characteristics are the same as in the earlier partitioned approach

In addition one can study transient phenomena

Transient current

asymptote

Current in the bond from site 0 to -1

Example:

M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

G. Stefanucci and C.O. Almbladh (Phys. Rev 2004) extended to TDDFT LDA scheme

TDDFT LDA scheme not enough for hard correlation effects: Josephson effect would not arise

Keldysh diagrams should allow extension to interacting systems, but this is largely unexplored.

Retardation + relativistic effects totally to be invented!

Magnetic effects in quantum transport

Michele Cini, Enrico Perfetto and Gianluca Stefanucci

Dipartimento di Fisica, Universita’ di Roma Tor Vergata

and LNF, INFN, Roma, Italy

,PHYSICAL REVIEW B 81, 165202 (2010)

14

Quantum ring connected to leads in asymmetric way

current

Tight-binding model

Current excited by bias magnetic moment.

How to compute ring magnetic moment and copuling to magnetic field? (important e.g. for induction effects)

15

15

J1

J2

J3

J7

J4

J6

J5

State-of-the-art calculation of connected ring magnetic moment

this is arbitrary and physically unsound.

16

16

problems with the standard approach

h1exp(ia1)

h2exp(ia2)

h7exp(ia7)

h3exp(ia3)

h4exp(ia4)

h6exp(ia6)

h5exp(ia5)

h1

h2

h3

h7

h4

h6

h5

S

Isolated ring: vortex current excited by B magnetic moment

Insert flux f by Peierls Phases:

current

Bias

NN

Connected ring: current excited by E magnetic moment.

17

S

Insert flux f by Peierls Phases:

c

a

b

Probe flux, vanishes eventually

Gauges

NN

All real orbitals, all hoppings= t

Blue orbital picks phase a , previous bond t e ia, following bond t e-ia

Physics does not change

18

S

Insert flux f by Peierls Phases:

c

a

b

counted counterclockwise

NN

19

Thought experiment: Local mechanical measurement of ring magnetic moment.

Atomic force microscope

A commercial AtomicForce Microscope setup(Wikipedia)

The information is gathered by "feeling" the surface with a mechanical probe. Piezoelectric elements that facilitate tiny but accurate and precise movements on (electronic) command enable the very precise scanning.

The atom at the apex of the "senses" individual atoms on the underlying surface when it forms incipient chemical bonds.

Thus one can measure a torque, or a force.

System also performs self-measurement (induction effects)

20

20

Quantum theory of Magnetic moments of ballistic Rings

21

21

Green’s function formalism

Wires accounted for by embedding self-energy

This is easily worked out

Explicit formula:

22

22

Density of States of wires

1

1

U=0 (no bias)

U=1

U=2

Left wire

DOS

-2

-2

2

2

0

0

Right wire

DOS

no current

current

no current

23

23

Slope=0 for U=0

0.04

1

0.0

-2

2

0

-0.02

0.0

0.5

1.0

1.5

U

Cini Michele, Enrico Perfetto and Gianluca Stefanucci, Phys.Rev. B 81, 165202-1 (2010)

24

24

Ring conductance vanishes by quantum interference

(no laminar current at small U)

Slope=0 for U=0

0.04

0.0

1

-2

2

0

-0.04

1.0

1.5

2.0

U

0.0

0.5

25

25

0.04

Slope=0 for U=0

1

0.0

-2

2

0

-0.04

1.0

1.5

2.0

0.0

0.5

U

26

26

Slope=0 for U=0

0.1

1

0.0

-2

2

0

-0.1

1.0

1.5

2.0

0.0

0.5

U

27

27

Law: the linear response current in the ring is always laminar and produces no magnetic moment

The circulating current which produces the magnetic moment is localized and does not shift charge from one lead to the other, contrary to semiclassical formula.

28

Quantized adiabatic particle transport

(Thouless Phys. Rev. B 27,6083 (1983) )

Consider a 1d insulator with lattice parameter a; electronic Hamiltonian

Consider a slow perturbation with the same spatial periodicity as H whici ia also periodic in time with period T , such that the Fermi level remains in the gap. This allows adiabaticity. The perturbed H has two parameters

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

Niu and Thouless have shown that weak perturbations, interactions and disorder cannot change the integer.