1. Axel Freyn, Ioannis Kleftogiannis and Jean-Louis Pichard
Scanning Gate Microscopy of a Nanostructure inside which electrons interact
Axel Freyn, Ioannis Kleftogiannis
and Jean-Louis Pichard
CEA / IRAMIS
Service de Physique de l’Etat Condensé
Phys. Rev. Lett. 100, (2008)

2. 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.

3. 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)

4. 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, (2007)

5. 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

6. Decay expected for Friedel oscillations
The effect of the AB-flux upon the nanosystem effective transmission falls off with the distance LC
Decay expected for Friedel oscillations

7. 2 nanosystems in series Y. Asada, A. Freyn and JLP; Eur. Phys. J
2 nanosystems in series Y.Asada, A. Freyn and JLP; Eur. Phys. J. B 53, 109 (2006)

8. 1/Lc correction with even-odd oscillations at half filling.
The results can be simplified at half-filling (particle-hole symmetry) Hartree corrections are compensated.
Renormalization of the internal hopping term td because of exchange
1/Lc correction with even-odd oscillations at half filling.
Friedel Oscillations; RKKY interaction

9. Role of the temperature
The effect disappears when

10. 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

11. Scanning gate microscopy
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

12. Scanning gate microscope Topinka,LeRoy,Westervelt,Shaw,Fleishmann,Heller,Maranowski,Gossard Letters to Nature, 410,183 (2001)
2DEG , QPC
AFM cantilever
The charged tip creates a depletion region inside the 2deg which can be scanned around the nanostructure (qpc)
Conductance without the tip

13. Conductance of the QPC as a function of the tip position
SGM images
Conductance of the QPC as a function of the tip position
g(without tip)=2e²/h
Dg falls off with distance r from the QPC, exhibiting fringes spaced by lF/2

14. 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

15. A lattice 2d model for SGM

16. HF study of the nanosystem Landauer-Buttiker conductance of the system (nanosystem + tip)

17. This self-energy has to be calculated
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

18. Self-consistent solution of coupled integral equations

19. Conductance of the combined system (nanosystem + tip)

20. Nanosystem conductance without tip (g0<1)

21. Effect of the tip upon the nanosystem HF self-energies

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

23. (Relative) Effect of the tip upon the conductance SGM images

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

25. With interaction, there is an additional 1/rT2 decay (U=1.7)

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

27. 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…

28. 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, (2007).
A. Freyn and JLP, Eur. Phys. J. B 58, 279 (2007).
A. Freyn, I. Kleftogiannis and JLP, Phys. Rev. Lett. 100, (2008).
D. Weinmann, R. Jalabert, A. Freyn, G.-L. Ingold and JLP, arXiv: (2008).

34. Equivalent setup (orthogonal transformation)
Role of the internal hopping td
Equivalent setup (orthogonal transformation)

35. Hartree-Fock Equations
1. Original basis
2. Transformed basis
(vAS = 0 because of inversion symmetry)