Quantum conductance and indirect exchange interaction (RKKY interaction)

Conductance of nano-systems with interactions coupled via conduction electrons: Effect of indirect exchange interactions

cond-mat/0605756 to appear in Eur. Phys. J. B

Yoichi Asada (Tokyo Institute of Technology) Axel Freyn (SPEC), JLP (SPEC).

Interacting electron systems between Fermi leads:Effective one-body transmission and correlation clouds

Rafael Molina, Dietmar Weinmann, JLP Eur. Phys. J. B 48, 243 (2005)

Scattering approach to quantum transport

22

,2

UEth

e

V

IG Feff

UEt Feff ,

SContact (Fermi) Contact (Fermi)

S(U)Fermi Fermi

1. Nano-system inside which the electrons do not interact

One body scatterer

Many body scatterer

effective one body scatterer

Value of ?Size of the effective one body scatterer?

Relation with Kondo problem

Carbon nanotubeMolecule,Break junctionQuantum dot of high rs Quantum point contact g<1YBaCuO…

2. Nano-system inside which the electrons do interact

222

FEth

e

V

IG

How can we obtain the effective transmission coefficient?

The embedding method

How can we obtain ? Density Matrix Renormalization Group

Embedding + DMRG = exact numerical method.

Difficulty: Extension outside d=1

Permanent current of a ring embedding the nanosystem + limit of infinite ring size

2,UEtI Feff

I

How can we obtain the size of the effective one body scatterer?

2 scatterers in series

• Are there corrections to the combination law of one body scatterers in series? Yes

• This phenomenon is reminiscent of the RKKY interaction between magnetic moments.

Combination law for 2 one body scatterers in series

tt CF

T

SLST

Lcik

Lcik

L

S

Lk

tt

MMMM

e

eM

tt

rt

r

tM

F

F

42

42

*

*

*

22cos112

..

0

0

1

1

Half-filling: Even-odd oscillations + correction

The correction disappears when the length of the coupling lead increases with a power law

C

SC

L

LUALg

,

Correction:

Magnitude of the correction

U=2 (Luttinger liquid – Mott insulator)

RKKY interaction(S=spin of a magnetic ion or nuclear spin)

1S2S

).(1

'',

)'(kiikk

i kk

rkkieS

eSSe

SCaaeV

H

HHHH

i

4

)2sin()2cos(

.

ij

ijFijFijij

jii ij

ijRKKY

R

RkRkRJ

SSJH

R

RkJ

d

F )2cos(

1

Zener (1947)Frohlich-Nabarro (1940)Kasuya(1956)Yosida(1957)Ruderman-Kittel(1954)Van Vleck(1962)Friedel-Blandin(1956)

The two problems are related: Electon-electron interactions (many body effects) are necessary.

The spins are not

SPINS:

Nano-systems with many body effects:

Spinless fermions in an infinite chain with repulsion between two central sites.

L

iiiii cccctH

111

SL

iii VnVnU

21

)(v 01 vccUt

Mean field theory: Hartree-Fock approximation

1t2

1V (if half-filling)

Reminder of Hartree-Fock approximation

The effect of the positive compensating potential cancels

the Hartree term. Only the exchange term remains

0''.'

'''.')(2

2

22

rrrUdrrUrU

rErrrrrUdrrrUrm

h

jj

ion

iijijj

ii

Hartree-Fock approximation for a 1d tight binding model

0,'1,1,'0,',

*',',

'',

'

.'.1.1.

pppppp

EfEpp

HFpp

p

HFpp

UU

ppUt

pEptptpt

Uttt

ccUUUt

HF

EEfE

EHF

1,01,0

01*

1,0 .01.

1 nanosystem inside the chain

Hartree-Fock describes rather well a very short nanosystem

DMRG

Hartree-Fock

2 nanosystems in series

CL

The results can be simplified at half-filling in the limit

1/Lc correction with even-odd oscillations characteristic of half filling.

0U

Conductance of 2 nanosystems in series

Conductance for 2 scatterers

16

32

F

F

k

k

4.0U

Hartree-Fock reproduces the exact results (embedding method, DMRG + extrapolation)

when U<t

2

Fk

DMRG

Hartree-Fock

Correction ),( FkUA

Role of the temperature

• The effect disappears when

).2

.(Tk

hvL

LL

BFT

Tc

How to detect the interaction enhanced non locality of the conductance ?

(Remember Wasburn et al)

U

Ring-Dot system with tunable coupling (K. Ensslin et al, cond-mat/0602246)