1. Igor Lukyanchuk
Amiens University
Quantum Oscillations and Magnetic Properties of Graphene-related nanostructures
How to identify Dirac Fermions ???

2. Graphite …

3. Unusual electronic phenomena in Graphite
Y. Kopelevich et al. Phys. Rev. Lett. 87, (2001) 93, (2004)
Weak ferromagnetism
MIT transition
Linear magnetoresistance (old problem)
Quantum Hall
35K Superconductivity in Graphite-S

4. (2D graphite monolayer, Semimetal)
Graphene:
(2D graphite monolayer, Semimetal)
Brillouin
zone
Special points
of Brillouin zone
Linear Dirac
spectrum
4-component (Dirac) wave function

5. Graphite:
Band structure:
Slonczewski-McClure Model
Fitting parameters

6. holes
Is it real ????
electrons

7. Problems with band interpretation
Sh > Se
Problems with band interpretation 1)
Se > Sh 2)
H: point
Dirac Spectrum
Phase volume ~0
holes
no Dirac Fermions
should be seen in experiment
Normal Spectrum
electrons
Another possibility:
Independent layers ???

8. ρc/ ρa > 50000 (instead of 300 in Kish) ρa ~ 3 μΩ cm (300K)
ρ(T), HOPG
In best samples
ρc/ ρa > (instead of 300 in Kish)
ρa ~ 3 μΩ cm (300K)
n3D~3x1018 cm-3
n2D~1011 cm-2 ( in Graphene)
Mobility:
μ~107cm2/Vs (105 in Graphene)
Metals: 300μΩ cm, Ioffe-Regel μΩ cm

9. Dirac Fermions vs Normal Carriers ?
Graphite:
2D vs 3D ?
Dirac Fermions vs Normal Carriers ?
Tools: Quantum oscillations
I. Lukyanchuk, Y. Kopelevich et al.
- Phys. Rev. Lett. 93, (2004)
- Phys. Rev. Lett. 97, (2006)

10. Landau quantization: Normal vs Dirac
Normal electrons
‘’gap’’
Dirac electrons
no ‘’gap’’ !!!

11. F Magnetic Quantum oscillations
► Susceptibility (H) : de Haas–van Alphen (dHvA)
etc…
…due to Landau quantization of Density of States
Clean 2D
Quasi 2D
3D or dirty
t, 
hc F H

12. Transport properties: Shubnikov de Haas (SdH) oscillations
► Hall Resistance
Rxy(H) :
In-field resistivity
measurements…
In bulk sample
► Resistance
Rxx(H)
In 2D film:
Quantum Hall effect

13. and Phase ??? … difficult to extract
Quantum oscillations: What is usually studied ?
Profile: Information about
e-e interaction (in 2D)
Damping: Information about
e-scattering (Dingle factor G )
Period: Information about
Fermi surface cross section S(e)
and Phase ??? … difficult to extract
We propose the method.!!!

14. ► 2D + 3D case, normal spectrum
dHvA oscillations
► 3D case, arbitrary spectrum
► 2D case, normal spectrum
► 2D D case, normal spectrum
► 2D case, Dirac spectrum

15. g is extracted from phase !!!
More general…
3D spectrum, model
In terms of extremal cross sections
( arbitrary )
Bohr-Sommerfeld quantization
Topological index :
Mikitik, Sharlai, Phys. Rev. Lett. 82, 2147 (1999)
► for Normal electrons
g is extracted from phase !!!
► for Dirac electrons

16. Generalized formula: 2D, 3D, arbitrary spectrum
I. Luk’yanchuk and Y. Kopelevich
Phys. Rev. Lett. 93, (2004)
where
Normal:
Dirac:
Fermi surface cross section
Limit cases
3D :
=> Lifshitz-Kosevich
=> Shoenberg
2D :

17. { { SdH: Oscillations of xx (H) (1st harmonic)
Phase depends on :
Normal:  = 1/2
Dirac:  = 0
► Spectrum : {
2D:  = 0
3D:  = ± 1/8
► Dimensionality : {
dHvA: Oscillations of  (H) (1st harmonic)

18. Quantum oscillations in Graphite
Resistance Rxy (H)
Susceptibility c(H)
T. Berlincourt et al. Phys. Rev. 98, 956 (1955)
Hall effect Rxy(H) in 2D Graphene
Hall effect Rxy(H)
Y. Kopelevich et al. Phys. Rev. Lett. 90, (2003)
Novoselov, K. S. et al. Nature 438, 197 (2005);
Zhang, Y. et al. Nature 438, 201 (2005).

19. Quantum oscillations in Graphite:
Fermi surface
Y. Kopelevich
high fields
SdH
spectrum
dHvA
low fields
minority
majority

20. Pass-band filtering Comparison of dHvA and SdH SdH dHvA dHvA SdH
high fields
spectrum
electrons
holes
low fields
holes (?)

21. Fan Diagram for SdH oscillations in Graphite
Dirac
Normal
Novoselov, 2005
Multilayer 5nm graphite
graphene

22. QHE and DF

24. 2005: Discovery of Quantum Hall Effect in 2D Graphene
Due to Dirac fermions …
From: - phase analysis
- semi-integerr QHE
Novoselov, K. S. et al. Nature 438, 197 (2005);
Zhang, Y. et al. Nature 438, 201 (2005).

25. sxy sxy QHE effect : Normal vs Dirac Normal electrons,
Dirac- like electrons
(expected for graphene)
sxy
1 / H

26. « Dirac » QHE Hall effect in Graphene monolayer n
Novoselov, K. S. et al. Nature 438, 197 (2005);
Zhang, Y. et al. Nature 438, 201 (2005).
« Dirac » QHE n

27. QHE effect in 2-layer graphite film
(K.Novoselov et al, cond-mat 2005, Nature-Physics)
-sxy
12T
1 / H
« Normal » QHE
due to normal electrons

28. Graphite, Quantum Hall Effect, different samples Kopelevich, (2003)

29. QHE: Graphite vs multi graphene
HOPG, Y. Kopelevich et al. PRL´2003
QHE: Graphite vs multi graphene
B0 = 4.68 T .
Few Layer Graphite (FLG)
K.S.Novoselov et al., Science´2004
B0= 20 T, = > n ~ 2x1012 cm-2

30. Filtering Normal (Integer QHE) GRAPHITE: Normal vs Dirac
carriers separation
Lukyanchuk, Kopelevich
- Phys. Rev. Lett. 97, (2006)
Rxy
Dirac (Semi-integer QHE)
Rxx
Filtering
B (T)

31. FQHE in Graphite

32. Rxy
Rxx
Rxy
Rxy
Rxx

33. Another manifestations
of DF in Graphite

34. 2006 Confirmation: Angle Resolved Photoemission Spectroscopy
(ARPES)
Dirac
holes
Normal
electrons

35. Another confirmation of Dirac fermions:
Dirac+Normal fermions in HOPG
TEM results:
C. Li and E. Andrei. 2007, Nature Phys.

36. Interlayer tunneling spectroscopy of Landau levels in graphite
Yu. I. Latyshev1, A. P. Orlov1, V. A. Volkov1, A. V. Irzhak2, D. Vignolles3, J. Marcus4 and T. Fournier4

37. INFRARED SPECTROSCOPY

38. 2006
Graphite, interpretation, ??? =>

40. RAMAN SPECTROSCOPY

41. Raman spectra of graphite
double-resonant 
graphite
2.33 eV D G D‘ G‘

42. RAMAN SPECTROSCOPY
« Graphene Fingerprint »

43. HOPG-Graphite, Raman, Peak 2G
I. Lukyanchuk, M. El Marssi, Y. Kopelevich
Physica B, 2009
Trace of graphene

45. model

46. B
As in bilayer ?

47. Nernst-Ettingshausen (NE) Quantum Oscillations
Graphite, Graphene, Bismuth…
Perpendicular
magnetic field
NE - effect
Temperature
gradient
V, Potential
difference

48. NE coefficient:
Advantages:
- oscillations have the thermodynamic origin,
related with and
- can be measured in films

49. NE oscillations, calculated from thermodynamics
for:

50. Results: 2D 3D
Giant !!!
(even in 3D)
Phase
shift

51. As function of gate voltage …
Normal
Dirac …

52. EXPERIMENT
Graphene, DF

53. Graphite, ??
Transport
Nernst
Giant (!), 3D (?), Phase ???

54. Bismuth
Giant (!), 3D (?), Phase ???