Electronic interactions in graphene sheets

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Madeira, feb. 08
Electronic instabilities and disorder in
graphene
María A. H. Vozmediano, ICMM, CSIC

• Outline
• Electronic inhomogeneities in graphene.
  Observations.
• Estability of the Fermi points.
• Midgap states due to ripples.
• Summary and future.

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Collaborators Paco Guinea, Juan Mañes, Mijail
Katsnelson José González , Pilar López-Sancho,
Tobias Stauber, Belén Valenzuela, Fernando de
Juan, Alberto Cortijo
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Charge inhomogeneities
Scanning single electron transistor, J. Martin
et al , cond-mat/0705.2180.
Courtesy of Amir Yacoby
The position of the maximum fixes the Dirac
point. It changes from point to point in the
sample. Local compressibility transmuted into
electronic density.
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Instabilities in graphene
Opening of a gap Fermi points protected by
topology A gap can not open if time
reversalinversion symmetry are preserved. (J.
Mañes, F. Guinea and MAHV, PRB 75, 155424
(2007))
Charge segregation favored by a high density of
states at the FL. The DOS of neutral graphene
is zero at eF
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Observation of ripples in graphene

(A. Geim)
(Old!)

• The free standing graphene samples show
  corrugations.
• Can be modeled by effective magnetic fields.
• Invoked to suppress weak antilocalization
• Morozov et al PRL 97, 016801 (2006).
• Morpurgo-Guinea, PRL 97, 196804 (2006).

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TEM
Ripples of size 0.5nm and 5nm of lateral size
adjust best the experimental data
Meyer et. al. Nature 446, 60-63 (2007)
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STM-1
1nm roughness variation in STM figures of
exfoliated graphene samples placed on on top of
SiO2.
High-Resolution Scanning Tunneling Microscopy
Imaging of Mesoscopic Graphene Sheets on an
Insulating Surface E. Stolyarova et al. PNAS
104, 9209 (2007).
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STM-2
Boundary graphene-substrate
Masa Ishigami et al, Nanoletters7, 1643 (07)
The structure of graphene seems to follow the
morphology of the substrate.
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Modeling curvature
Tight binding Stretching of the nn bonds seen as
effective gauge fields
If the ripples are randomly distributed the
system can be studied as Dirac fermions in
random magnetic fields.
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Dirac in curved space
We can include curvature effects by coupling the
Dirac equation to a curved space
Need a metric and a tetrad.
Generate r-dependent Dirac matrices and an
effective gauge field.
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Effects of the curvature (1)
1. The curved gamma matrices
Can be seen as a position-dependent Fermi velocity
Examples
(no real r dependence)
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Effects of the curvature (2)
The spin connection

• It can be seen as an effective gauge field
  with
• a matrix structure. Geometrical s3.
• It has different signs in the two Fermi points
• (unlike a real magnetic field).

Extra gauge fields
In the case of topological defects (pentagons,
heptagons) other gauge fields arise due to
topology and lattice effects.
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Modeling the ripples
F. de Juan, A. Cortijo, MAHV PRB76, 165409
(2007).
For future purposes we define

• measures the departure
• from flat space (h/L)2

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Space-dependent Fermi velocity
2D surface in polar coordinates with radial
symmetry
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The spin connection
After a simple calculation..
Adding dimensionful constants get O(1T)
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The electron Greens function
Equation for the Dirac propagator in curved
space.
Expand around flat space and rewrite at first
order in a as flat in potential
From the gauge field
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The local density of states
The gauge field gives a zero contribution to the
correction due to traces of matrices
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Effect on the local DOS
A smooth ripple
Correction to the LDOS
LDOS vs. curvature and shape
Energy dependence
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Midgap states induced by ripples
F. Guinea, M. Katsnelson , and MAHV PRB 77 (08) .
Effective B
Axial 2D ripple
1D ripples
Ripples generate eff magnetic fields giving rise
to midgap states. Interaction effects
induce instabilities.
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Conclusions and future
Ripples induce charge modulations in undoped
graphene. Affect transport properties. The
effective gauge fields associated to the ripples
give rise to midgap states which can induce
electronic instabilities.

• Experimental test .
• Interplay with elasticity theory.
• Influence on transport properties.