2003/8/18ISSP Int. Summer School Interaction effects in a transport through a point contact...

2003/8/18 ISSP Int. Summer School

Interaction effects in a transport through a point contact

Collaborators A. Khaetskii (Univ. Basel) Y. Hirayama (NTT)

Contents1. Quantum Point Contact 　 (QPC)2. Conductance Anomaly3. Brief review of proposed Theories4. Scattering by spin fluctuation5. Open questions and Outlook

Yasuhiro Tokura (NTT Basic Research Labs.)


Two terminal conductance of quasi-1D system

n

nTh

eG )(

2

Landauer’s formula

Non-interacting, zero temperature

Quantum Point Contact (QPC)

Ballistic and adiabatic limit

)()( nn ET

occupiedNh

eG

22

B.J.van Wees, et al, Phys. Rev. B 43 (’91) 12431.

Conductance quantization (Zero field)


Field, Temperature, and Bias dependence

In-plane field B// dependence:

Finite temperature:

Bias dependence:

We restrict only to linear transport

)()(2

// occocc

NNh

eBG

nTkE

Tk

nn

Bn

B

eh

e

ef

Tf

dh

eTG

1

1

]1

1)([

)())(

()(

/)(

2

/)(

2

}2

,min{ RLBRL TkeV

gBB//


Conductance anomaly

Mesoscopic mystery:Anomalous conductance plate

au near 0.7 X 2e2/hIn-plane field drives the anomaly smoothly to 0.5Spin related phenomena ?

The structure is enhanced with temperatures Not a simple quantum interf

erence effect Ground state property seem

s not responsible K.J.Thomas, et al, Phys. Rev. Lett. 77 (’96) 135.


Temperature dependence

Quantum interference simply disappears for higher temperature

The structure persists after raster scan – imperfection is negligibleActivation behavior

Collective excitation on the contact?

A. Kristensen, et al, Physica B 249-251 (’98) 180.

gcgaTkE VVEeG Ba ,/


Interaction is more important for lower density (rs=Eee/EF ~1/nl)

Absence of polarized ground state in 1D Lieb-Mattice theory

Conduction band pinningExplains experiments amazingly well

Homogeneous 1D model is not relevant!

Spontaneous spin polarization?

H. Bruus, et al, Physica E 10 (’01) 97.

E.Lieb and D. Mattis, Phys. Rev. 125 (’62) 163.

)ln(/

/*

1*

2*

d

arrRy

rRy

ss

potential

skinetic

C.-K. Wang and K.-F. Berggren, Phys. Rev. B54(’96) 14257.


Inhomogeneous system

T. Rejec, et al, Phys. Rev. B 67 (’03) 75311.

Y. Meir, et al, Phys. Rev. Lett. 89 (’02) 196802.O.P.Sushkov, Phys. Rev.

B 67 (’03) 195318.

Singlet-triplet origin Naturally formed bulge Effective attractive potential

Ground state calculation by mean field theory Hartree-Fock (HF) Local spin density functional theory

　 (LSDF)Spontaneous local charge/spin formation?


Kondo effect ?

Kondo-like characteristics in dI/dVEffective Anderson modelHow robust is spin ½ state?

Other models Phonon scattering

Wigner crystal

S.M.Cronenwet, et al, Phys. Rev. Lett. 88 (’02) 226805.

Y. Meir, et al, Phys. Rev. Lett. 89 (’02) 196802.

G.Seeling and K. A. Matveev,Phys. Rev. Lett. 90 (’03) 176804.

B.Spivak and F. Zhou,Phys. Rev. B61 (’00) 16730.


Effective Hamiltonian

Adiabatic approximation

0);2

)((

);()(),(

),()(2

22

*

2

0

xxd

xyxyx

yxUm

H

n

nnEnE

yx

)(2

,

)(

12

*

2

1

,110

xUm

H

RL

HHHH

nxD

DD

1D+reservoirs modelA.Shimizu and T.Miyadera,Physica B249-251 (’98) 518.

A.Kawabata, J. Phys.Soc.Jpn. 67 (’98) 2430.


Interaction

: thickness of 2DEGEffective 1D model

Hartree-Fock approximation

)2

'(|)'(|

~

|)';'(||);(|)'(')',(

|'|)'(

21

211

22

2

xxwxxV

xyxyrrVdydyxxV

rr

errV

D

x

x’

L/2

L/2

-L/2

-L/2

V1D(x,x’)

)',()'())()(()',(

)'()',()('

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

1)()(

11

1

11

xxVxxxVxHxxH

xxxHxdxdx

xxxxVxxdxdxxHxdxH

FockHartreeD

HFD

HFD

DD


Scattering with Friedel oscillations

Correction to transmission amplitude

K.A.Matveev,D.Yue,and L.I.Glazman,Phys. Rev. Lett. 71 (’93) 3351.

))arg(||2sin(||2

||)(

)()(),(),(

)(),(2)(

0

*

1

rxkx

rnxn

fxyyxVyxV

xnyxdxVxV

F

qqqq

Fock

DHartree

Friedel oscillation at absolute zero

),(),()()(1

),()()()(2

~

xyVxynyxdydxvi

t

yxVyydyxndxvi

t

tttt

LkRkk

Fk

LkRkk

Hk

Fk

Hkkk

|)(

2|ln||

|

2

|||,|2

F

yxL

Fk

Hk

kkLrt

tt

For sufficiently short-range potential, there is region of dG(T)/dT<0, but…

The HF contribution in the reservoirs:


Beyond Hartree-Fock approximation

In real 2D system,

FDFD

D

FF

FFDHartree

F

TVkV

fkV

rkr

rn

))0()2(2(

)()()2(2

1

)2sin(2

)(

22

2

2

G. Zala, et al., Phys. Rev. B64 (’01) 214204.

Only linear correction:in the context of “metal-insulator transition” in 2D

|||,|2

|yx

LFk

Hk tt

The HF contribution on the contact:

may show resonance at zero T.

Assuming featureless HF potential, we search for collective mode effective for electron scattering.


Collective mode - paramagnon

Homogeneous system with short range interaction, I:

)1(),(

),(1

),(),(

20

0

0

qiCAqq

qI

qq

R

R

RRRPA

Stoner mean-field condition is determined at q,~0

11

)0,0( 0

III

RRPA

Paramagnon excitation for I0<1, q,~0

A

Iqq

C

A

qq

q

q

qRRPA

022

2222

1),(

1),(

RPA:random phase approximation


Localized paramagnon

Y.Tokura and A. Khaetskii,Physica E12 (’02) 711.

)()()'()'()()',(

)',()(),()',()',(

'

'*''

*

',0

00

i

ffxxxxxx

xyywyxdyIxxxx

qq

qqqqq

qqq

R

RRPA

RRRRPA

Characteristic frequency:

01 IL

vFpm

To couple spin and charge, we need finite scattering:

))(2exp(1

1)(

2

1)( 2

1

Um

T

xUxU


Conductance by Kubo formula

D.L.Maslov and M. Stone,Phys. Rev. B52 (’95) R5539.A.Kawabata, J.Phys. Soc.Jpn. 65 (’96) 30.A. Shimizu, ibid, 65 (’96) 1162.

Neglect interaction in reservoirs(large density, 2D)

-Kubo formula is safely used.

',,,321)2('

23' 14

2

0

3241|],,,,[

2],',[

,)]'(),,([)(),',(

)),0,',(Im),',((Im1

limlimlim

xxxxxx

R

RR

xx

uxxxxG

xxxxmi

euxxK

xjtxjti

txxK

xxKxxKG


Lowest RPA correction

Ta vanishes at absolute zero.Both corrections vanishes when

|t|2=0 or 1.Y.Tokura, Proc. ICPS-26 (’03)Ed. A. R. Long and J. H.Davis.

)'()(),',(

)],,'(),'(Im)(

),,'(Im),'(1

1[)()'(')(

))()(Im(2

)(

)',|()',|(),',(Im')](1

1[

1)(

)]()(|)([|2

*

*

2

22

xxxxg

yyyyGf

yyyyGe

dyydydyF

FtT

yygyygyydydyfe

dT

TTtf

dh

eG

RR

RRRL

b

RLRa

ba


Numerical results

Model static potential U1(x)=U0cosh-2(2x/L)Using susceptibility function near |t|2=1

Energy and length in unit of U0 and kv=(2mU0)1/2/h


Equivalent semiclassical model

Y. Levinson and P. Wolfle,Phys. Rev. Lett. 83 (99) 1399.

)',',()','(),(

),(0

ttxxWtxUtxU

txUHH

Time-dependent scattering theory ..),(),(

2),(

,),,(),,(),,( 10

cctxtxm

ietxJ

txEtxEtxE

)',(),'()()'(')(

))()(Re(2

)(

)',|()',|()',('1

)(

)]()(|)([|2

*

2

22

yyWyyGd

yydydyF

FtT

yygyygyyWdydyd

T

TTtf

dh

eG

RRL

b

RLa

ba

Almost equivalent to Kubo formula result with replacement:

),',(Im1

2)',( yy

eyyW R


Adiabatic limit

O. Entin-Wohlman, et al.,Phys. Rev. B65 (’02) 195411.

)(''4

)()0,cos(

)0),(())(,(

,))(,(2

2

2

2

Ta

TtaT

tETtUT

tUTCd

h

eG

If low frequency fluctuation is dominant,

Therefore, temperature-dependent (classical) correction is proportional to the second derivative of T().


Why 0.7 ?

“Free” conductanceCorrection increase with temperature –classical correctionZero-temperature mass correction G enhancement

))(2exp(1

1)(

2

1)( 2

1

Um

T

xUxU

Total:


Outlook

Bias dependence – relevance to Kondo-like behavior ?Magnetic field dependence –suppress spin fluctuationsShot noise characteristics – suppression near 0.7 structure ? R. C. Liu, et al.,

Nature 391 (’98) 263.

Localized ½ spin is essential ?


Summary

The conductance anomaly found in a quantum point contact is critically reviewed.Electron interaction and spin effect are essential to understand the phenomena.Using an effective inhomogeneous one-dimensional model, conductance is derived in Kubo formula within random phase approximation.Scattering by paramagnon fluctuation can explain the anomaly and its temperature dependence.

2003/8/18ISSP Int. Summer School Interaction effects in a transport through a point contact...

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