Meir-WinGreen  Formula
This presentation is the property of its rightful owner.
Sponsored Links
1 / 33

Meir-WinGreen Formula PowerPoint PPT Presentation


  • 173 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Meir-WinGreen Formula

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Meir wingreen formula

Meir-WinGreen Formula

Quantum dot

U

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


Meir wingreen formula

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 wingreen formula

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


Meir wingreen formula

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)


Meir wingreen formula

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


Meir wingreen formula

Use of Green’s functions


Meir wingreen formula

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!


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

In addition one can study transient phenomena

Transient current

asymptote

Current in the bond from site 0 to -1


Meir wingreen formula

Example:

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


Meir wingreen formula

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


Meir wingreen formula

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!


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

S

Insert flux f by Peierls Phases:

c

a

b

counted counterclockwise

NN

19


Meir wingreen formula

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


Meir wingreen formula

Quantum theory of Magnetic moments of ballistic Rings

21

21


Meir wingreen formula

Green’s function formalism

Wires accounted for by embedding self-energy

This is easily worked out

Explicit formula:

22

22


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

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


Meir wingreen formula

A

B

A

B

A

B

A

B

A

B

A

B


Meir wingreen formula

A

B

A

B

A

B

A

B

A

B

A

B


Meir wingreen formula

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


  • Login