Scanning Gate Microscopy of a Nanostructure inside which electrons interact

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Scanning Gate Microscopy of a Nanostructure
inside which electrons interact

• Axel Freyn, Ioannis Kleftogiannis
• and Jean-Louis Pichard
• CEA / IRAMIS
• Service de Physique de lEtat Condensé
• Phys. Rev. Lett. 100, 226802 (2008)

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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.

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The simplest spinless lattice model with a
single nearest neighbor interactionInteracting
nanosystem with six parameters

• 3 Hopping integrals ( td , tc, th 1)
• Nearest neighbor repulsion U n1no
• Gate potential VG
• Filling factor (Fermi energy EF)

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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)

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Effective nanosystem transmission ts2
(Hartree-Fock approximation)

• Large effect of the AB-flux upon the effective
  transmission ts2
• This effect occurs only if the electrons interact
  inside the nanosystem

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The effect of the AB-flux upon the nanosystem
effective transmission falls off with the
distance LC
Decay expected for Friedel oscillations
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2 nanosystems in seriesY.Asada, A. Freyn and
JLP Eur. Phys. J. B 53, 109 (2006)

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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
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Role of the temperature

• The effect disappears when

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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

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• 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
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Scanning gate microscope Topinka,LeRoy,Westervelt
,Shaw,Fleishmann,Heller,Maranowski,GossardLetters
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

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

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A lattice 2d model for SGM
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HF study of the nanosystemLandauer-Buttiker
conductance of the system (nanosystem tip)
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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 EltEF
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Self-consistent solution of coupled integral
equations
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Conductance of the combined system(nanosystem
tip)
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Nanosystem conductance without tip(g0lt1)
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Effect of the tip upon the nanosystem HF
self-energies
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The effect of the tip upon the Fock self-energy
falls off with rT as the Friedel oscillations
causing it.
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(Relative) Effect of the tip upon the
conductanceSGM images
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Without interaction, the effect of the tip upon g
falls off as 1/rT
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With interaction, there is an additional 1/rT2
decay(U1.7)
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Strength of the interaction effect upon the SGM
images as a function of the nanosystem parameters
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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

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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).

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Equivalent setup (orthogonal transformation)
Role of the internal hopping td
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Hartree-Fock Equations
1. Original basis
2. Transformed basis
(vAS 0 because of inversion symmetry)