Scanning Gate Microscopy of a Nanostructure

inside which electrons interact

Axel Freyn, Ioannis Kleftogiannis and Jean-Louis Pichard

CEA / IRAMISService de Physique de l’Etat Condensé

Phys. Rev. Lett. 100, 226802 (2008)

Outline

• Part I : The quantum transmission of a nanosystem inside which the electrons interact becomes non local.

• Part II : Method for probing electron-electron interactions inside a nanostructure using a scanning gate microscope.

The simplest spinless lattice model with a single nearest neighbor interaction

Interacting nanosystem with six parameters

• 3 Hopping integrals: ( td , tc, th =1)• Nearest neighbor repulsion: U n1no

• Gate potential: VG

• Filling factor (Fermi energy: EF)

Interacting nanosystem in series with a one body scatterer(attached ring pierced by an Aharonov-Bohm flux)

A. Freyn and JLP, Phys.Rev. Lett. 98, 186401 (2007)

Effective nanosystem transmission |ts|2 (Hartree-Fock approximation)

Large effect of the AB-flux upon the effective transmission |ts|2

This effect occurs only if the electrons interact inside the nanosystem

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C

F

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The effect of the AB-flux upon the nanosystem effective transmission falls off with the distance LC

line) (solid )2cos(

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c

CF

CR

F

G

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Decay expected for Friedel oscillations

2 nanosystems in seriesY.Asada, A. Freyn and JLP; Eur. Phys. J. B 53, 109 (2006)

CL

The results can be simplified at half-filling (particle-hole symmetry)

Hartree corrections are compensated.

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

2 ; 1 ; 0 ;

2

FdG ktUUV

Friedel Oscillations; RKKY interaction

Renormalization of the internal hopping term td because of exchange

Role of the temperature

• The effect disappears when

).2

.(Tk

hvL

LL

BFT

Tc

Origin of the non local transmission(Hartree-Fock theory)

• The external scatterer induces Friedel oscillations of the electron density inside the interacting nanosystem

• This modifies the Hartree potentials and the Fock corrections inside the nanosystem.

• The nanosystem effective transmission can be partly controlled by external scatterers when the electrons interact inside the nanosystem

1

ii ccii cc

• To neglect electron-electron interactions outside the nanosystem is not realistic when 1d wires are attached to it.

• This assumption becomes more realistic if one attaches 2d strips of large enough electron density

Scanning gate microscopy

Scanning gate microscope Topinka,LeRoy,Westervelt,Shaw,Fleishmann,Heller,Maranowski,Gossard

Letters to Nature, 410,183 (2001)

Conductance without the tip

2DEG , QPC AFM cantilever

The charged tip creates a depletion region inside the 2deg which can be scanned around the nanostructure

(qpc)

g falls off with distance r from the

QPC, exhibiting fringes spaced by F/2

SGM images Conductance of the QPC as a function of the tip position

g(without tip)=2e²/h

)pwithout ti() tipwith( ggg

The electron-electron interactions inside the QPC can be probed by SGM images

• By lateral gates (or additional top gate), one reduces the electron density inside the QPC. This makes the interactions non negligible inside the QPC, [0.7 (2 e2 /h) anomaly]. The density remains important and the interactions negligible outside the QPC.

• The Friedel oscillations created by the charged tip can modify the effective QPC transmission, if the electrons interact inside the QPC

A lattice 2d model for SGM

HF study of the nanosystemLandauer-Buttiker conductance of the system

(nanosystem + tip)

Hartree-Fock theory for the interacting nanosystem coupled to 2d non interacting strips

This self-energy has to be calculated using a recursive method for different positions of the tip and energies E<EF

Self-consistent solution of coupled integral equations

Conductance of the combined system(nanosystem + tip)

Nanosystem conductance without tip(g0<1)

Effect of the tip upon the nanosystem HF self-energies

The effect of the tip upon the Fock self-energy falls off with rT as the Friedel oscillations causing it.

(Relative) Effect of the tip upon the conductanceSGM images

Without interaction, the effect of the tip upon g falls off as 1/rT

With interaction, there is an additional 1/rT2 decay

(U=1.7)

Strength of the interaction effect upon the SGM images as a function of the nanosystem parameters

Summary• The effective transmission can be modified by external scatterers when the electrons

interact inside the nanosystem.

• This non local effect can be probed using a scanning gate microscope (enhanced fringes near the nanostructure + phase shift of the fringes).

• In the HF approximation, the effect is induced by the Friedel (Hartree) or related (exchange) oscillations created by the external scatterers inside the nanosystem.

• One can make the effect very large by a suitable choice of the nanosystem parameters. Reducing td enhances the effect. But an orbital Kondo effect (yielded by inversion symmetry) occurs when td goes to 0.

• Comparison between HF, DMRG, NRG results…

References

• R. Molina, D. Weinmann and JLP, Eur. Phys. J. B 48, 243, (2005).• Y. Asada, A. Freyn and JLP, Eur. Phys. J. B 53, 109 (2006).• A. Freyn and JLP, Phys. Rev. Lett. 98, 186401 (2007).• A. Freyn and JLP, Eur. Phys. J. B 58, 279 (2007).• A. Freyn, I. Kleftogiannis and JLP, Phys. Rev. Lett. 100, 226802

(2008).• D. Weinmann, R. Jalabert, A. Freyn, G.-L. Ingold and JLP, arXiv:

0803.2780 (2008).

Equivalent setup (orthogonal transformation)

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dGA

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tVV

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Role of the internal hopping td

Hartree-Fock Equations

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ASSAASS

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vVVnUVV

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1. Original basis

2. Transformed basis

(vAS = 0 because of inversion symmetry)