Outline:

Introduction: One electron transitions in graphene

Experimental results

Magnetoplasmon picture for tansitions involving n=0 Landau levels

Magnetoplasmon picture for interband transitions

Discussion

Conclusions

Magneto-transmission spectroscopy of graphene

Gérard MartinezGrenoble 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)

IntroductionOne electron transitions in graphene

pv)p(E F

nE)n(sign

nBe2v)n(signE

10

Fn

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 B=0 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))

c~

A

B

E10

DC

n = 0

n = 1

n = -1

n = -2

n = 2

n = -3

n = 3

Here we take the value of vF= 0.86 106 m/s

Experimental results :Graphene samples

The SiC-4H sample is graphetized: heated to1500 °Cin 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.

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.

One expects structured samples such as: We got likely a structure with broken sheets as:

Experimental results : Instrumentation

The SiC substrate with a thikness of about 300 m is opaque for energies between 85 and 200 meV.

FTS spectrometer Brucker IFS-66v

Reference

Sample

Bolometer (Si)

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

Experimental resultsExample of transmission spectrum in graphene

A

B

E10

DC

n = 0

n = 1

n = -1

n = -2

n = 2

n = -3

n = 3

A

BC

D

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)

20 30 40 50 60 70 800.85

0.90

0.95

1.00

20 30 40 50 60 70 80

0.99

1.00

4T

2T

1T

0.4T

Rel

ativ

e tr

ansm

issi

on

Energy (meV)

0.7T0.5T

0.3T

0.2T

Transmission experiments: main features

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 conventional2DEG.

B

C

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:

s/m10)01.003.1(c~ 6

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.

much larger than vF.

Sadowski et al., PRL , 97, 266405 (2006)

Transmission experiments: exfoliated grapheneZ. Jiang et al., cond-mat/0703822

Measurements at fixed field: ratio of transmission at nu= +/-2 with respect to nu= -10

s/m10)03.018.1(c~

s/m10)02.012.1(c~

6C

6B

Significantly larger than values found with SiC-G samplesRatio of effective velocities 1.05

N0 + 1

ab

c

N0

N0 + 2

Transitions among Landau levels are not single electron transitions

+electron-electron interactionsCR is an excitonic transition magneto-plasmon dispersion

Magnetoplasmon plasmon approachMagnetoplasmon excitations in conventional 2DEG

Three one-electron transitions Three MP curves describing the dispersion of excitonic-like transitions

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)

Magnetoplasmon picture in graphene

In the magnetoplasmon picture, derived in the Hartree-Fock+RPA 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 : ,,'n,nA

Difference of exchange energies of the two levels n and n’

Simultaneous creation and destruction of an exciton at different points of the Brillouin zone for any value of the spin and in any valley.

Electron-hole interactionSame spin, same valley

The model assumes that the Coulomb energy Ec=e2/lB is smaller than the different energy transitions.

2

32

232

2

32

42

4242

222

2

22

,n,n

,,n,n'nnyxn,n,n,'n

n,n,,n,n'nnxyn,n,n,'n

,,'n,nnn,n,n,nn

n,'n,n,'n,'n,n

,,'n,nex

0A)ff)(k,k(V~

0A)ff)(k,k(E

0Af))0(E)0(E(

0A)k(E

Magnetoplasmon picture for transitions involving the n=0 Landau Levels

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

Results of the model are presented for filling factors < 2

For transitions B there are five possible transitions corresponding to the energy E10.

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

Comparaison of the tansition energy E10 and Coulomb energy Ec= e2/lB :

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 Ec<E1 is better fulfilled for Graphene than for GaAs.

B

E10

n = 0

n = 1

n = -1

Valley K Valley K’

One assumes that there is a splitting S 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 <2.

Magnetoplasmon picture for transitions involving the n=0 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 V corrections, all dispersion curves converge to a single value for klB 0. One single line (red curve) in infrared absorption

Only one dispersion curve (red curve) will give rise to singularities in the density of states which could possibly be seen in Raman experiments.

0 1 2 3 40.4

0.6

0.8

1.0

1.2

Exci

ton

ener

gies

(e

2 /l B

uni

ts)

k lB

= 0.5

X 2

0 1 2 3 4

X 3

= 1.0

k lB

Graphene: transitions n= -1,0 to n= 0,1

0 1 2 3 4

X 2

= 1.5

k lB

0 1 2 3 4 5

X 3

= 2.0

k lB

Curves are displayed with respect to the one-electron energy E10.

Magnetoplasmon picture for transitions involving the n=0 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:)ff)(0(E 'nn'n,n,n,'n

10100010010MP

10 C4

3Ef

2

3C)1f2(

4

3E

B

On the other hand the intensity of the transition remains proportional to (vF)2!

c~

0m 0

21m

x1 )x(Ledx

2

1C

2

Independent of the filling factor!

With that formulation C1 diverges and the summation has to be truncated but will remain much larger than3/4 0.

Therefore the evolution of the energy of the transition B with will display a slope larger than vF.

Magnetoplasmon picture for transitions C

n = 0

n = 1

n = -1

Valley K Valley K’

n = 2

n = -2

J

I

One has to treat now 8 transitions four corresponding to n=-1 n=2 (labelled I ) and four corresponding to n=-2 n=1 (labelled J ).

The resulting matrix to be diagonalized has a very high degree of degeneragy.

In the one electron picture all these transitions correspond to the same energy : )12(EE 1021

In Coulomb units E21= 6.693

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 /lB

Two single solutions (left part of the figure) and two groups of three times degenerate solutions .

Results are identical for =1 or 2 and very sligthly dependent on for noninteger values

At klB 0 there are two dictinct solutions for integer values of

The only optical active transitions are those corresponding to the non degenrate solutions

0 1 2 3 40.6

0.8

1.0

1.2

1.4

1.6

1.8

= 1 or 2

k lB

0 1 2 3 4 5

X 3

X 3

= 1 or 2

Exc

iton

en

erg

ies

(e2 /

l B u

nits

)

k lB

Graphene: transitons n= -2,-1 to n=1,2

Magnetoplasmon picture for interband transitions at klB 0

For k lB 0, the Hamiltonian is essentially diagonal and for integer values of

there is a splitting . ( very small))T(B874.0)meV(8/0

Transition C

8/0MP21

The mean variation of the transition is given by:MP21m 2021

MP21 C

32

33Em

Also divergent but :

0m 0

211

21mx

22

)x(L1

1m

)x(Ledx

2

1C

2

0m 0

21m

4x

2

212

)x(L1m

xedx

2

1C

CC)12(C

2

C2 converges

Therefore one can write: 20110MP21 C

32

33)CE)(12(m

All the divergence remains in C1.

1021 E)12(E

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 /lB

Two single solutions (left part of the figure) and two groups of three degenerate solutions .

Results are identical for =1 or 2 and very sligthly dependent on for noninteger values

At klB 0 there are two dictinct solutions for integer values of

The only active optical transitions are those corresponding to the non degenrate solutions.

0 1 2 3 41.8

1.9

2.0

2.1

2.2

2.3

2.4

= 1 or 2

k lB

0 1 2 3 4 5

X 3

X 3

= 1 or 2Exc

iton

en

erg

ies

(e2 /

l B u

nits

)

k lB

Graphene: transitons n=-3,-2 to n=2,3

Magnetoplasmon picture for interband transitions at klB 0

For k lB 0, the Hamiltonian is essentially diagonal and for integer values of

there is a splitting . ( very small))T(B437.0)meV(16/0

Transition D

16/0MP21

The mean variation of the transition is given by:MP32m 3032

MP32 C

256

233Em

Again divergent but :

3

)x(L

2

)x(L

1m

)x(Ledx

2

1C

212

0m 0

211

21mx

32

0m 0

4221

m

2x

3

313

32

xx)

2

13()x(L

1m

xedx

2

1C

CC)32(C

2 C3 converges

Therefore one can write: 30110MP32 C

256

233)CE)(23(m

All the divergence remains in C1.

1032 E)23(E

Magnetoplasmon picture for interband transitions at klB0

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 : s/m1003.1c~ 6

We use this experimentall value to determine the upper index of LL, Nmax beyond which the summation for C1 is truncated.

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

In both cases: s/m1003.1c~ 6

s/m1099.0c~ 6

The scaling is performed with the transition C

for the transition D

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

s/m10025.1c~ 6 for the transition E

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.

Consequences of the introduction of the electron-phonon interaction

What is expected if we introcude a valley splitting V ?

Valley K'

Valley K

LL diagram for B >0 with valley splitting

2F

2v )kv()2/()k(E

B’B

C C

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.

Consequences: In such a case one expects a splitting of the transition B which should providea direct measurement of V .

Transitions C and the following ones should not be splitted but their variation with the field should acquire a component linear in the field.

Splitting observed in transportmeasurements in high fields:Y.Zhang et al., PRL, 96,136806 (2006)

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 n=0 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 V which should be splited by V .

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 V.

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.

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 V .

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 !!