Dirac fermions in Graphite and Graphene Igor Lukyanchuk Amiens University I. Lukyanchuk, Y....
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Transcript of Dirac fermions in Graphite and Graphene Igor Lukyanchuk Amiens University I. Lukyanchuk, Y....
Dirac fermions in Graphite and Graphene
Igor Lukyanchuk Amiens University
I. Lukyanchuk, Y. Kopelevich et al.
- Phys. Rev. Lett. 93, 166402 (2004)- Phys. Rev. Lett. 97, 256801 (2006)
Graphene2005
Novoselov, et al. Nature 438, 197 (2005
Y. Zhang, et al., Nature 438, 201 (2005
Why graphene is interesting ?
- Fundamental physics
- Applications (carbon-based microelectronics )
3D 2D 1D 0D
(Nobel prize) (Nobel prize)
2 view of Graphene
Nanotube-grapheneGraphite-graphene
“ “:
November 2005
• Graphene active area covering an entire 8-inch wafer• Carrier mobility of the FET exceeding 15,000 cm2/V-s• Drain voltage of the FET smaller than 0.25 V• ft and fmax both larger than 500 GHz• W-band low noise amplifier with >15 dB of gain and <1dB of noise figure• Wafer yield of the low noise amplifiers is more than 90%
30 000 000 $
HP, Intel, IBM…
Wanted:
Linear Dirac spectrum
Graphene: (2D graphite monolayer, Semimetal)
Special points of Brillouin zone
Brillouin zone 4-component (Dirac ????) wave function
"Normal electrons" “Dirac fermions"
Schrödinger equation Dirac equation
Dirac spinor
Free Relativistic Electrons
Gap formation, excitonic insulator, weak ferromagnetism, … ???
Abrikosov Phys. Rev. B60, 4231 (1999) B61, 5928 (2000)
Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); 87, 246802 (2001)
González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996)
In magnetic field: 2 component equations
Schroedinger cond-mat physics
Dirac cond-mat physics !!!
Klein effect:
U(x)
U(x)
Ef
Ef
electron
electron
holehole
Metal (semiconductor)
Semimetal:
No electron localization !!!
Minimal conductivity
Band structure: Slonczewski-McClure Model
Graphite:F
ittin
g pa
ram
eter
s
holes
electrons
ρ(T), HOPG
In best samples
ρc/ ρa > 50000 (instead of 300 in Kish)
ρa ~ 3 μΩ cm (300K)
n3D~3x1018 cm-3
n2D~1011 cm-2 (1012-1013 in Graphene)
Mobility:
μ~106cm2/Vs (104 in Graphene)
Metals: 300μΩ cm, Ioffe-Regel 1000 μΩ cm
Novoselov, K. S. et al. Nature 438, 197 (2005); Zhang, Y. et al. Nature 438, 201 (2005).
2005: Discovery of Quantum Hall Effect in 2D Graphene Due to Dirac fermions …
From: - phase analysis - semi-integerr QHE
Quantum Hall Effect, different samples (2003)
0 1 2 3 4 50
1
2
3
4
5
Filling Factor
-
Gx
y /
G0
xy
Normal QHE
-8
-4
0
4
8
-
Rx
x
( m
)
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
9
-
Gxy
/G0
xy
B0/B
HOPG, Y. Kopelevich et al. PRL´2003
Few Layer Graphite (FLG)K.S.Novoselov et al., Science´2004
B0= 20 T, = > n ~ 2x1012cm-2
B0 = 4.68 T
Fig. 1
1
2
B0 = 4.68 T
Few Layer Graphite (FLG)K.S.Novoselov et al., Science´2004
B0= 20 T, = > n ~ 2x1012 cm-2
.
QHE: Graphite vs multi graphene
HOPG, Y. Kopelevich et al. PRL´2003
Do Dirac Fermions Exist in Graphite ?
Normal electrons
Dirac electrons
Landau quantization: Normal vs Dirac
‘’gap’’
no ‘’gap’’ !!!
SdH: Oscillations of xx (H) (1st harmonic)
Normal: = 1/2Dirac: = 0► Spectrum : {
2D: = 03D: = ± 1/8► Dimensionality :{
Phase depends on :
dHvA: Oscillations of (H) (1st harmonic)
Cyclotron mass(detection of e and h)
SdHdHvA
Experiment:
Electrons or Holes ?
Normal or Dirac ?
SdH dHvA
dHvASdH
Pass-band filtering
spectrum
Comparison of dHvA and SdH
electrons
holes
In-phase
Out-phase
Fan Diagram for SdH oscillations in Graphite
Dirac
Normal
Novoselov, 2005
graphene
Multilayer 5nm graphite
holes
electrons
Dirac Spectrum
Normal Spectrum
H: point
Phase volume ~0
no Dirac Fermionsshould be seen in experiment
Problems with band interpretation
Se > Sh1)
2)
Sh > Se
Independent layers ???
Another possibility:
2006 Confirmation: Angle Resolved Photoemission Spectroscopy
Dirac holes
Normalelectrons
(ARPES)
E. Andrei et al. 2007, Nature Phys.
Dirac+Normal fermions in HOPGTEM results:
Another confirmation of Dirac fermions:
Interlayer tunneling spectroscopy of Landau levels in graphiteYu. I. Latyshev1, A. P. Orlov1, V. A. Volkov1, A. V. Irzhak2, D. Vignolles3, J. Marcus4 and T. Fournier4
0 1 2 3 4 5 6-400
-300
-200
-100
0
100
200
300
400
Mag.trans 2x(01) STM 2x(01) STM 2x(02) Inter.tunn -11 Inter.tunn -22 Inter.tunn -33
V (
mV
)
B (T)
Graphite #1
OPTICAL PROPERTIES
- Visible- Infrared- Raman
Graphite
Graphene
C=
Reflectance and transmitance coefficients
Optical properties are defined by HF conductivity
πα ≈ 2.3%
INFRARED SPECTROSCOPY
nE)n(sign
nBe2v)n(signE
10
Fn
2006
Graphite, interpretation, ??? =>
RAMAN SPECTROSCOPY
RAMAN SPECTROSCOPY
« Graphene Fingerprint »
E =
2.3
3 e
V
0.4 0 0.20.400.2-2
-1
0
1
2
E (
eV
)
KK MM M
q’
q
q’’A
B C
double-resonant
0 1500 3000Raman shift (cm-1)
Inte
nsity
(a.
u) graphite 2.33 eV
D
G
D‘ G‘
Raman spectra of graphite
HOPG, Raman
model