Outline: Introduction: One electron transitions in graphene Experimental results
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Transcript of Outline: Introduction: One electron transitions in graphene Experimental results
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 !!