Outline:

1
Magneto-transmission spectroscopy of graphene
Gérard Martinez Grenoble High Magnetic Field
Laboratory Centre National de la Recherche
Scientifique
Main collaborators M. Sadowski, M. Potemski
(GHMFL) C. Berger
and W. deHeer (Georgia Institute of Technology)
Y. Bychkov (Landau
Institute)

• Outline
• Introduction One electron transitions in
  graphene
• Experimental results
• Magnetoplasmon picture for tansitions involving
  n0 Landau levels
• Magnetoplasmon picture for interband transitions
• Discussion
• Conclusions

2
Introduction One electron transitions in graphene
The band structure of graphene is composed of two
cones located at two inequivalent corners K and
K of the Brillouin zone at which conduction and
valence band merge. For each valley
at B0
For finite value of B
Each LL ? 0 is four times degenerated due to
spin and valley degeneracy.
Depending on the filling factor, different
optical transitions are allowed either of
 cyclotron type  (A ) or of  interband
transitions type  (C and D) or mixed (B)
In experiments an effective value of the
velocity called larger than vF is
found for all transitions (Sadowski et al., PRL
, 97, 266405 (2006))
Here we take the value of vF 0.86 106 m/s
3
Experimental results Graphene samples
The SiC-4H sample is graphetized heated to1500
C in high vacuum ? sublimation of Si atoms
leaving behind C planes. A variety of
characterizations techniques lead to the
conclusion that the active part of this type of
structures consists of a few graphene layers.
We got likely a structure with broken sheets as
One expects structured samples such as
The dc conductivty data show that the carrier
concentration is in the range of 4 1012 cm-2
which is due to the built-in electric field at
the interface SiC-graphene and self-consistent
calculations indicate that it should reduced to
zero within the next five layers.
4
Experimental results Instrumentation
Magneto-transmission measurements are performed
in an absolute way using a rotating sample holder
working in situ in order to eliminate the
magnetic dependent response of the detector.
Sample graphitized SiC Reference pure SiC
The SiC substrate with a thikness of about 300 ?m
is opaque for energies between 85 and 200 meV.
5
Experimental results Example of transmission
spectrum in graphene
C
B
D
A
The relative intensity of transition A versus
transition B gives the electronic density of the
layer which depending on the sample is of the
order of 1010 cm-2. (Fermi energy less than 10
meV)
6
Transmission experiments main features
C
Two kinds of transitions were followed
constinuously for magnetic fields up to 4T
A first estimate of their energy position
demonstrate that they vary like B1/2, leading to
assume that they reflect the presence of
graphene layers in the sample.
These transitions correspond to an oscillator
strength which also clearly increases with the
magnetic field like B1/2 in contrast to
conventional 2DEG.
B
7
Transmission experiments global behavior
All transitions vary linearly with the square
root of the magnetic field with a slope,
independent of the transitions corresponding to
an effective velocity of
much larger than vF.
All these findings lead to the conclusion that
all observed transitions are originating from
graphene layers doped at a level much lower
than the one measured in transport measurements.
Sadowski et al., PRL , 97, 266405 (2006)
8
Transmission experiments exfoliated graphene
Z. Jiang et al., cond-mat/0703822
Measurements at fixed field ratio of
transmission at nu /-2 with respect to nu -10
Significantly larger than values found with SiC-G
samples Ratio of effective velocities ? 1.05
9
Magnetoplasmon plasmon approach Magnetoplasmon
excitations in conventional 2DEG
Transitions among Landau levels are not single
electron transitions
electron-electron
interactions CR is an excitonic transition ?
magneto-plasmon dispersion
MP model developped for any filling factor Phys.
Rev. B 66, 193312 (2002) also including the
corresponding optical conductivity. Phys. Rev. B
72, 195328 (2005)
Three one-electron transitions? Three MP curves
describing the dispersion of excitonic-like
transitions
10
Magnetoplasmon picture in graphene
In the magnetoplasmon picture, derived in the
Hartree-FockRPA approximations, all the spin and
valley dependent transitions could be a priori
mixed
Call the creation operator governing the
transition n?n, with spin ?, in a valley ?
Difference of exchange energies of the two
levels n and n
Electron-hole interaction Same spin, same valley
Simultaneous creation and destruction of an
exciton at different points of the Brillouin zone
for any value of the spin and in any valley.
The model assumes that the Coulomb energy
Ece2/?lB is smaller than the different energy
transitions.
11
Magnetoplasmon picture for transitions involving
the n0 Landau Levels
Results of the model are presented for filling
factors ? lt 2
Because the interaction is mainly important for
energy transitions which are of the same order
of magnitude, it is possible to treat the problem
independently for the different types of
transitions B, C and others
For transitions B there are five possible
transitions corresponding to the energy E10.
Comparaison of the tansition energy E10 and
Coulomb energy Ec e2/?lB
For a 2DEG (GaAs) ??c(meV) 1.7 B(T)
Ec (meV) 4.45 (B(T))1/2
??c/Ec 0.38 (B(T))1/2
For graphene E10(meV) 31.1 (B(T))1/2
Ec(meV) 11.2 (B(T))1/2 E1/Ec 2.78
independent on the field
The condition EcltE1 is better fulfilled for
Graphene than for GaAs.
One assumes that there is a splitting DS of the
valleys K and K larger than the spin splitting
in such a way the electrons remain in the same
valley (here K) for any value of nlt2.
12
Magnetoplasmon picture for transitions involving
the n0 Landau Levels
One has to solve the Hamiltonian for the exciton
energies E10 2.77 e2/?lB
Two degenerate solutions for non integer value
of ? and three degenerate solutions for ? 1
or 2.

• Without introducing DV corrections,
• all dispersion curves converge
• to a single value for klB ? 0.
• One single line (red curve) in
• infrared absorption

Curves are displayed with respect to the
one-electron energy E10.
Only one dispersion curve (red curve) will give
rise to singularities in the density of states
which could possibly be seen in Raman experiments.
13
Magnetoplasmon picture for transitions involving
the n0 Landau Levels transition B
For k values of the exciton ? 0, the Hamiltonian
is essentially diagonal with terms involving
the difference of exchange contributons and
.The MP
energy is
Independent of the filling factor!
With that formulation C1 diverges and the
summation has to be truncated but will remain
much larger than 3/4 a0.
Therefore the evolution of the energy of the
transition B with will display a slope
larger than vF.
On the other hand the intensity of the transition
remains proportional to (vF)2!
14
Magnetoplasmon picture for transitions C
One has to treat now 8 transitions four
corresponding to n-1? n2 (labelled I ) and
four corresponding to n-2? n1 (labelled J ).
In the one electron picture all these transitions
correspond to the same energy
In Coulomb units E21 6.693
The resulting matrix to be diagonalized has a
very high degree of degeneragy.
15
Magnetoplasmon picture for transitions involving
the LL n-2,-1 to n 1,2 (transition C)
The one electron picture gives an energy E21
6.693 e2 /klB
Results are identical for ? 1 or 2 and very
sligthly dependent on ? for non integer values
Two single solutions (left part of the figure)
and two groups of three times degenerate
solutions .
At klB ? 0 there are two dictinct solutions for
integer values of n.
The only optical active transitions are those
corresponding to the non degenrate solutions
16
Magnetoplasmon picture for interband transitions
at klB ?0
Transition C
For k lB ? 0, the Hamiltonian is essentially
diagonal and for integer values of n there is a
splitting . (
very
small)
The mean variation of the transition
is given by
Also divergent but
DC2 converges
Therefore one can write
All the divergence remains in C1.
17
Magnetoplasmon picture for transitions involving
the LL n-3,-2 to n 2,3 (transition D)
The one electron picture gives an energy E32
8.722 e2 /klB
Results are identical for ? 1 or 2 and very
sligthly dependent on ? for non integer values
Two single solutions (left part of the figure)
and two groups of three degenerate solutions .
At klB ? 0 there are two dictinct solutions for
integer values of n.
The only active optical transitions are those
corresponding to the non degenrate solutions.
18
Magnetoplasmon picture for interband transitions
at klB ?0
Transition D
For k lB ? 0, the Hamiltonian is essentially
diagonal and for integer values of n there is a
splitting . (
very
small)
The mean variation of the transition
is given by
Again divergent but
DC3 converges
Therefore one can write
All the divergence remains in C1.
19
Magnetoplasmon picture for interband transitions
at klB?0
Model to treat the divergence of C1
The most reliable experimental results, because
obtained on a large scale of magnetic field are
those relative to the transitions C and D and E.
For these transitions, the effective velocity is
We use this experimentall value to determine the
upper index of LL, Nmax beyond which the
summation for C1 is truncated.
The scaling is performed with the transition C
This requires the imput of a value for vF
Taking vF 0.86 106 m/s ? Nmax 28, C1 0.880
and ?C2 -0.157
Taking vF 0.88 106 m/s ? Nmax 17, C1 0.805
and ?C2 -0.158
for the transition D
In both cases
for the transition E
for the transition B
The effective velocity of the transition B is
found to be lower than that of the two next
interband transitions by about 4. Not observed
in SiC-G
20
Electron-phonon interaction in Graphene
Models with different types of electron-phonon
interactions predict a splitting of the valley
which is as big as the spin splitting and vary
linearly with the magnetic field.
J. Yan et al., March meeting Denver (2007)
Fuchs and Lederer, PRL, 98, 016803 (2007)
All the optical branches are expected to give a
strong electron-phonon interaction which also
renormalizes the Fermi velocity by decreasing it.
21
Consequences of the introduction of the
electron-phonon interaction
What is expected if we introcude a valley
splitting DV ?
Models with different types of electron-phonon
interactions predict a splitting of the valley
which is as big as the spin splitting and vary
linearly with the magnetic field.
B
Consequences In such a case one expects a
splitting of the transition B which should
provide a direct measurement of DV . Transitions
C and the following ones should not be splitted
but their variation with the field should
acquire a component linear in the field.
B
C
C
Splitting observed in transport measurements in
high fields Y.Zhang et al., PRL, 96,136806 (2006)
22
Conclusions of the magnetoplasmon model

• There exist characteristic dispersion relations
  for graphene which depends on the optical
  transition. They are different for transitions
  implying the n0 LL (B) and those related to
  interband transitions (C, D,E ..). The results
  are not depending on the existence of a splitting
  between valleys K and K.
• For the transition B, near klB ? 0, one finds a
  single MP tansition in the absence of the valley
  spiltting DV which should be splited by DV .
• For interband transitions, near klB ? 0, the MP
  transitions are splitted due to the exchange
  terms but no extra splitting is expected due to
  the introduction of DV.
• The variation of the optical energies of the
  transitions, near klB ? 0, with the magnetic
  field corresponds to an effective velocity higher
  than the Fermi velocity vF.
• This effective velocity is found to be lower for
  the transition B than for interband transitions
  by about 4.
• The oscillator strength of the transitions
  remains proportional to (vF)2.

23
Problems which remain to be solved

• There are on the experimental side divergences
  between experimental results obtained on SiC-G
  and exfoliated Graphene. Why?
• We do not have yet any direct measurement of the
  electron-phonon interaction in Graphene or of the
  splitting DV .
• If the electron-electron interactions open the
  gap (increase of the renormalized Fermi velocity)
  the strong electron-phonon interaction in C-based
  compound, including Graphene will tend to
  decrease it. What is their relative weight? .
• In experiments we measure a combination of
  both and it is even not very clear on theoritical
  grounds that this combination should be the same
  with and without magnetic field !!