Meir-WinGreen Formula Meir-WinGreen Formula

1

Quantum dot

U

LV RV

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

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

† †, , ,

, ,

†, , ,

, ,

( . .)

( . .)

d Lk Lk L Lkd dk k

Rk Rk R Rkk k

H d d Un n n V d c h c

n V d c h c

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)

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)

2

Quantum dot

U

LV RV

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 spinat the dot by

( )2( ) ( ) ( Im )

rL RL R d

L R

eI d f f G

This has been used for weak V also in the presence of strong U.3

2

2 2

Define ( ) ( ) ( ), where even in the presence of the bias. :

( ) Re ( ) , ( ) Re ( ) ,

We need the imaginary parts

Im( ( )),

kL R

k k

Lk RkL L L R R R

k kLk Rk

L L

V

i

V Vi i

i i

Im( ( )) R R

*

, ,,

Thisissomewhat similartoLandauerformula

L R mn n mmn

eJ d ff t t

44

General partition-free framework and rigorous Time-dependent current 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!

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

55

† †1 1

Current operator at site (Caroli et al.,1970)

, hopping integral

Poses a genuine many-body problem even without interactions,

because of Pauli principle depends on

hoppingm m m m m hopping

m

m

etJ c c c c t

i

J

population and Fermi level.

Time-dependent Quantum Transport System is in equilibrium until at time t=0 blue sites are shifted to V and J starts

device

J

6

, 1 , 1' 0

, 1

, 1 , 1

Current in discrete model

lim ,

of course depends on Fermi level;

but 1-body Schrodinger equation yields ,

hopping T Tm m m m m

t t

Tm m

advanced retardedm m m m

etJ G G

i

G

G G

†' '

'

( )' '

' 0 ''

For anyone-bodydensity ( ) ( ) ( ) ( '),

the expectation value is ( ) lim lim ( ) ( , ' ').T

t t x x

f x f x x x

f x i f x G xt x t

( ) †' '

† †' '

( , ' ') ( ) ( ' ')

( ') ( ) ( ' ') ( ' ) ( ' ') ( )

depends on Fermi level

TG xt x t i T xt x t

i t t xt x t i t t x t xt

( ) †' '( , ' ') ( ) ( ' ') ' is independent of Fermi level (if nointeractions)rG xt x t i xt x t t t

Use of Green’s functions

†1 ' '

0

( ) ( ) ( ) i.e. bias is on

eigenstates fo

at time0

r 0,

Fermi function f unbiaor syst ms eed .

q

q

kk k kk

Let H q q

H t V t a a t

V

f

Rigorous Time-dependent current formula

*, ,

( ') †

2( ) Im ( ) ( )

where m and n are connected by a bond .

and ( , ') ( ') vac ( ')

hopping retarded retardedmn q m q n q

q

hopping

retarded iH t tij j i

etJ t f G t G t

t

iG t t t t c e c t vac

7

The current at bond is rewritten with retarded GF : m n

Note:Occupation numbers refer to H before the time dependence sets in.

System remembers initial conditions!

derived by equation of motion or Keldysh method

8

† † †1 1

This formula can also be rewritten in terms of evolution operator,

ˆ ( ) ( ) ( ) with .

( ( ) for constant H but this is quite general ).

Also, onecan put in

hoppingm q m m m m m m

q

iHt

etJ t f q U t J U t q J c c c c

i

U t e

f

† ˆˆside, ( ) ( ) ( ) .

The states that were occupied initially are those that evolve and carry the current.

m mq

J t q U t J U t f q

0( )

In terms of Heisenberg operators,

1ˆ( ) [ ( )]1

.

mn mn HJ t Tr fJ t fe

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

9

2 2

0

22

2

half-filled 1 system left wire right wire

hopping 1

( ) 1 (1 )

( ) ( )8 2( ) in units( ( ) ( ))

2 4V

d

g

Vg g e

J V dV V

g g

1 2 3 4V

0 .1

0 .2

0 .3

0 .4

J

( ) , 0

quantum of conductivity

( ) 0, 4 bands mismatch

eVJ V V

e

J V V

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

10

Current in the bond from site 0 to -1

Transient current

asymptote

0.5*hopping integralV

In addition one can study transient phenomena

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

1

0

( ) ( ) ,

0, t<0 ( )

0.5, t>0

= =0

current at bond (-3, -4)

hopping

bias mm

a b

t

H t V t n

V t

Example:

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

1

0

( ) ( ) ,

0, t<0 ( )

0.5, t>0

= =0

current at a-b bond

hopping

bias mm

a b

t

H t V t n

V t

13

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!

14

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)

1515

ring L wire r wire L ring R ring biasH H H H H H H

Tight-binding model †

,

. .sidesN

ring mn m nm n

H h c c h c

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

Current excited by bias magnetic moment.

bias L L R RH U N U N

Quantum ring connected to leads in asymmetric way

current

1616

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

† †1 1

Semiclassical Recipe:

ˆ ˆinsert quantum current hoppingm m m m m m m

etJ J J c c c c

i

into classical formula

magnetic moment of ring current Classi (Jackscal on):

ring1sides

ring areaaring *

Nrea

latiN

mn averagem n

J

M J

1 2 3 4 5 6 7

7

average

J J J J J J JJ over oriented bonds

J1 J2

J3

J4

J5

J7

J6

this is arbitrary and physically unsound.

17

problems with the standard approach

Insert flux by Peierls Phases:

1

mn sidesN

ic

mn mn mnm n

h h eNN S

ring

ˆˆ ˆ ˆ, ring area*

HHellmann-Feynman H J M J

Isolated ring: vortex current excited by B magnetic moment

Connected ring: current excited by E magnetic moment.

Bias

current

h1 h2

h3

h4

h5

h7

h6

h5exp(i5)

h6exp(i6)

h1exp(i1) h2exp(i2)

h4exp(i4)

h3exp(i3)h7exp(i7)

ˆ ?

Hellmann-Feynman H

18

c

a b

Insert flux by Peierls Phases:

1

mn sidesN

ic

mn mn mnm n

h h e

NN S

ac ab bac b H J

00 0

0 bc

ab

0 ac

All real orbitals, all hoppings=

Gauges

Blue orbital picks phase , previous bond e i, following bond e-i

Physics does not change

ie ie

Probe flux, vanishes eventually

19

c

a b

Insert flux by Peierls Phases:

1

mn sidesN

ic

mn mn mnm n

h h e

NN S

00 0

abH Jac cb ab

bc ca 0

0 ab

0gauge 0 0

ca bc a bb cH J

counted counterclockwise

2020

Thought experiment: Local mechanical measurement of ring magnetic moment.

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.

Atomic force microscopeA commercial Atomic Force Microscope setup (Wikipedia)

Thus one can measure a torque, or a force.

System also performs self-measurement (induction effects)

2121

†

, S=ring area

Equivalently, the Magnetic Moment operator is:

ˆ

here .

ring

Gring G G

ring ring rin

n

g

ri gM S

HM

E

H H

w

 

,

= many-body state

What should we meas

of the biased wire

ure? The energy of the ring during the

d current-carrying system

ˆ ˆ ˆ grand-canonic

experim

al Ring Ha

ent is:

mGr

Gring

ing

H

H

ring ri

in

ng

r g

Ψ

= H

= Ψ

- N

Ψ

μ

E

iltonian.

Inserting an infinitesimal probe flux , one could measure a torque .

This can be interpreted as the interaction of with the probe field.

r

ring

ing

M

E

Quantum theory of Magnetic moments of ballistic Rings

2222

n ring

†

ring degrees of

( )2

= n ... n is taken over the

h=1-body Hamiltonian with Peierls phases,

freedom,

( ) ( )

ring

ring

di Tr h G

Tr

G t t t

ringE

0lim ( )

2

ring R A R R A A

ring

M d dh dh dhTr G G h G G G G

iS d d d

Re Re ,Re

1 1, , embeddingt Adv t Adv

t AdvG G

h h

, , ,, 2 ( ) Im R A R

L R L R L R L RG G G if U

Wires accounted for by embedding self-energy

Green’s function formalism

Explicit formula:

This is easily worked out

2323

2-2 0

1Density of States of wires

Example:

chemical potential 1 2-2 0

1

U=0 (no bias)

no current

Left wire DOS

Right wire DOS

U=1

current no current

U=2

2424

U1.00.0 0.5 1.5

0.04

0.0

-0.02

ringI

3

ab bc caJ J J

1

2-2 0

1

Slope=0 for U=0

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

2525

1.0

0.04

-0.04

0.0

0.0 0.5 1.0 1.5 2.0U

abJ

i

ca

r ngand I

J

2-2 0

1Ring conductance vanishes by quantum interference(no laminar current at small U)

Slope=0 for U=0

2626

0.0 0.5 1.0 1.5 2.0U

-0.04

0.04

0.0

ringI

3

ab bc caJ J J

0.5

2-2 0

1

Slope=0 for U=0

2727

1.5

0.0 0.5 1.0 1.5 2.0U

-0.1

0.1

0.0

ringI

3

ab bc caJ J J

2-2 0

1

Slope=0 for U=0

28

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

ringI conductivity 0 in linear response. Big effect! What is the Message?

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.

Quantized adiabatic particle transport

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

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

2

( ) V(x+a)=V(x) 2

Block functions of band n ( ) ( )

( ) ( ) ( ) ( )

iqxnq nq

iqx iqx iqx iqxnq nq nq nq nq nq nq

pH V x

m

x e u x

He u x e u e He u x H q u x u

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

2( , ) ( , ) ( , ) H(q+ , ) ( , ).

The parameter space is a torus.

H H q t H q t T H q t t H q ta

Bloch states e ( , ) Parameters q,t. FL in gap Adiabatic theorem holds

2 parameters Berry A =-i( , )

iqxn

n nn n n

u q t

u uvector potential u u

q t

A B A B A B A B A B A B

The gauge invariant Berry is a velocity.

Thouless by adiabatic perturbation theory shows that indeed the velocity of band electrons

1becomes v ( )

n n n n nqt

n

u u u ucurvature i

q t t q

q

bands n

0

( ) ( )1, usual band term, .

( )1The average of over the filled band vanishes.

Pumped current: j=- .2

Charge in cycle c =- is a Berry2

n nn nqt qt

n

nqt

BZ

T nn qt

BZ

q qdrift

q q

q

q

dq

dqdt

phase (flux of magnetic field on Torus surface).

A B A B A B A B A B A B

2 flux of 'magnetic field'=Berry phase

1c =- ( ,0) (1, ) ( ,1) (0, )

2

n

B C D A

n x y x yA B C D

c

dxA x dyA y dxA x dxA y

1 1

0 0

1

0( ,0) ( ,0) (

( ,1) and ( ,0) are physically equivalent can only differ by a phase: ( ,1) ( ,0)exp( ( )) :

[ ( ,0) ( ,1)] [ ( ,0) ( ,0) ( ,0) ( ( ,0) ) ]

[ ,0)

x

x

x

x

i ix x x x

ix

u x u x u x u x i x

dx A x A x dx u x i u x u x e i u x e

d u x i u x u x ex

1

0( ,0) ( ( ,0( ( ) ],0 () )) 1 (0)xx xi i

x x x x xi

x u x e u x i e xi u x e d

1Thererfore, Charge in cycle c = (1) (0) (1) (0) .

2n x x y y

Pumped charge for band n over a cycle:

Charge in cycle first Chern number=integer nc

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

1Thererfore, c = (1) (0) (1) (0) .

2n x x y y

(1)(0)

(1)(0)

(1)

Matching relations at the corners A,B,C,D

AD: u(0,0)=e (0,1) but since along DC: (0,1) e (1,1),

u(0,0)=e e (1,1).

However along DB: (1,1) e (1,0),

u(0,0)=e

yx

yx

x

x

ii

ii

i

i

u u u

u

u u

(1) (0)(0) (1)

(1) (0)(0) (1)

e e (1,0) but since along AB: (1,0) e (0,0),

u(0,0)=e e e e (0,0) (1) (0) (0) (1) 2

y yx

y yx x

i ii

i ii ix y x y

u u u

u n