Dirac fermions in Graphite:

Transcription

1 Igor Lukyanchuk Amiens University, France, Yakov Kopelevich University of Campinas, Brazil Dirac fermions in Graphite: I. Lukyanchuk, Y. Kopelevich et al. - Phys. Rev. Lett. 93, (2004) - Phys. Rev. Lett. 97, (2006) Graphene 2005 Novoselov, et al. Nature 438, 197 (2005 Y. Zhang, et al., Nature 438, 201 (2005

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

3 Do Dirac Fermion exist in Graphite Experiment: Yes!!! Lukyanchuk&Kopelevich 2004 Band Theory: No!!!

4 BAND THEORY

5 Graphite: holes Band structure: Slonczewski-McClure Model electrons

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

7 EXPERIMENT

8 ρ(t), HOPG In best samples ρ c / ρ a > (instead of 300 in Kish) ρ a ~ 3 μω cm (300K) n 3D ~3x10 18 cm -3 n 2D ~10 11 cm -2 ( in Graphene) Mobility: μ~σ/en ~ 10 6 cm 2 /Vs (10 4 in Graphene)

9 SdH oscillations and Quantum Hall Effect, (Kopelevich 2003) Kopelevich 2003 QHE Rxy Rxx SdH oscillations Linear!!!

10 Finding of the Quantum Hall Effect in Graphite Y. Kopelevich et al., Phys. Rev. Lett. 90, (2003); R. Ocaña et al., Phys. Rev. B 68, (2003).

11 Dirac fermions from quantum oscillations in Graphite Linear!!! SdH oscillations

12 Landau quantization: Normal vs Dirac Normal electrons gap Dirac electrons no gap!!!

13 Bohr-Sommerfeld quantization Quantum oscillations SdH, dhva. Normal: Dirac:

14 Comparison of dhva and SdH SdH dhva SdH dhva Pass-band filtering In-phase spectrum electrons Out-phase holes

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

16 Last LL Tanuma et al. 1981: CDW(?) in Graphite, Ultraquantum limit Rxy??

17 Dirac Fermions from QHE in Graphite

18 QHE effect : Normal vs Dirac Normal electrons, σxy σ xy Dirac- like electrons (expected for graphene) /H σxy 1 / H 1 / H Unconventional Integer Quantum Hall Effect in Graphene V. P. Gusynin and S. G. Sharapov Phys. Rev. Lett. 95, (2005)

19 HOPG, Y. Kopelevich et al. PRL 2003 B0 = 4.68 T QHE: Graphite vs multi graphene. 9 8 Fig. 1 Few Layer Graphite (FLG) K.S.Novoselov et al., Science 2004 B0= 20 T, = > n ~ 2x10 12 cm -2 - G xy /G 0xy HOPG, Y. Kopelevich et al. PRL 2003 B 0 = 4.68 T 2 Few Layer Graphite (FLG) K.S.Novoselov et al., Science B 0 = 20 T, = > n ~ 2x10 12 cm B 0 /B

20 Normal (Integer QHE) 5 8 GRAPHITE: Normal vs Dirac carriers separation - G xy / G 0xy Normal QHE Δ R xx ( mω ) Rxy Filling Factor ν B (T) Rxx Filtering - G xy / G 0xy Dirac (Semi-integer QHE) Δ R xx ( mω ) 1 Dirac QHE Filling Factor ν

21 5 4 - G xy /G 0xy e 2 /h HOPG-3 6e 2 /h 8e 2 /h HOPG-UC Dirac fermions 12e 2 /h 10e 2 /h Massive carriers HOPG-UC is "best quality" sample Massive carriers Dirac fermions /2 3/2 Filling Factor = nh/4eb, B 0 /B

22 Peculiarity of QHE in Graphite: Rxx > Rxy!!! Compare with TMTSF-PF S. Hannahs et al. Phys. Rev. Lett. 63, 1988 (1989) -Rxy Rxx

23 Other confirmation of Dirac Fermions in Graphite

24 2006 Confirmation: Angle Resolved Photoemission Spectroscopy (ARPES) Dirac holes Normal electrons

25 Another confirmation of Dirac fermions: Dirac+Normal fermions in HOPG TEM results: E. Andrei et al. 2007, Nature Phys.

26

27 RAMAN SPECTROSCOPY «Graphene Fingerprint»

28

29 HOPG, Raman

30 model

31 Conclusion: Both types of carriers (Normal and Dirac-like) exist in Graphite. They have the same nature as carriers recently identified in mono- and bi-layer Graphene Band theory vs independent layers?

32 QHE effect in Graphite: from bulk to monolayer: Spectrum of SdH oscillations in bulk 1.2 In bulk: Normal electrons are dominated in QHE Rxy Rxx 0.6 In monolayer: Dirac (holes) are dominated in QHE Normal electrons Dirac holes x10 5

33 2 frequency fit experiment

34 Spectrum, 2D model Brillouin zone Fermi surface Double Ir. Repr. Wave function ~ Dirac spinor Hamiltonian : Dirac spectrum and Landau quantization

35 dhva oscillations 3D case, arbitrary spectrum 2D case, normal spectrum 2D + 3D case, normal spectrum 2D case, Dirac spectrum

36 Generalized CM formula: 2D, 3D, arbitrary spectrum where Normal: Dirac: Limit cases 3D : 2D : => Shoenberg => Lifshitz-Kosevich

37 Normal vs Dirac spectrum

38 In bulk Graphite: 3D Fermi surface Extremal cross sections S(ε) different groups of carriers Normal or Dirac???

39 Strongly correlated electronic phenomena (Experiment) Weak ferromagnetism MIT transition 35K SC in Graphite-S Linear R(H) (old problem) Y. Kopelevich et al. Phys. Rev. Lett. 87, (2001) 93, (2004)

40 1) What are the Dirac Fermions??? 2) Why in Graphite??? 3) Why it is interesting???

41 3b) Why it is interesting??? (Theory) Dirac-like equation Linear Dirac spectrum Model of strongly coupled relativistic Dirac fermions Gap formation, excitonic insulator, weak ferromagnetism,??? In magnetic field: 2 component equations Abrikosov Phys. Rev. B60, 4231 (1999) B61, 5928 (2000) González, Guinea, Vozmediano, Phys. Rev. Lett. 77, 3589 (1996) Khveshchenko, Phys. Rev. Lett. 87, (2001); 87, (2001)

42 3c) Why it is interesting??? (Experiment) Strongly correlated electronic phenomena in Graphite Weak ferromagnetism Y. Kopelevich et al. Phys. Rev. Lett. 87, (2001) 93, (2004) MIT transition Linear magnetoresistance (old problem) Quantum Hall 35K Superconductivity in Graphite-S

43 Objective Do Dirac Fermions exist in Graphite? ( yes!! )* How to test? I. Luk yanchuk and Y. Kopelevich Phys. Rev. Lett. 93, (2004) Appropriate tool: quantum oscillations!!! 2D Phase-frequency analysis filtering

44 Magnetic Quantum oscillations Susceptibility χ(h) : de Haas van Alphen (dhva) etc due to Landau quantization of Density of States Clean 2D Quasi 2D 3D or dirty H εf hωc t, Γ

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

46 Quantum oscillations in Graphite Resistance Rxy (H) Susceptibility χ(h) Y. Kopelevich et al. Phys. Rev. Lett. 90, (2003) Hall effect Rxy(H) T. Berlincourt et al. Phys. Rev. 98, 956 (1955)

47 dhva oscillations 3D case, arbitrary spectrum 2D case, normal spectrum 2D + 3D case, normal spectrum 2D case, Dirac spectrum

48 More general 3D spectrum, model In terms of extremal cross sections Bohr-Sommerfeld quantization ( arbitrary ) Berry phase Topological index : for Normal electrons Mikitik, Sharlai, Phys. Rev. Lett. 82, 2147 (1999) for Dirac electrons

49 Decoupling of φ onto μ, γ and δ Electrons or holes? (determination of sign of μ) By comparison of dhva χ ( H ) and SdH σ ( H ) oscillations: Akhieser 39, Kosevich, Andreev 60 Oscillations: - In phase (μ = + 1): electrons - Out of phase (μ = - 1): holes By comparison of SdH σ ( H ) and Hall σ ( H ) oscillations: -

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

51 QHE effect : 0 vs 2π Berry Phase Normal electrons, 0 Berry Phase σxy σ xy /H 1 / H Normal electrons, 2π Berry phase proposed for bi-layer graphene * McCanne, Falko, Phys. Rev. Lett. 96, (2006) σxy 1 / H

52 QED: Dirac Fermions, Lorentz Group Graphene: 4dim IR of Hexagonal Crystal Group

53

54 H-point: E n ~n 1/2 (Dirac) K-point: E n ~ n (normal)

55 2 view of Graphene Nanotube-graphene Graphite-graphene

56 MIT vs Compensated metal

57 GRAPHITE: EXPERIMENTAL BACKGROUND: Statement: = stack of graphene monolayers

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59 Why QHE in graphite has not been reported before?

60 Graphite is strongly diamagnetic material χ ~ χ superconductor Room temperature levitating HOPG

61 MIT? Graphite, Magneto-resistance R(T,H)

62 Experimental Findings of (quasi) 2D Dirac Fermions in Graphite and Graphene Bulk Graphite: I. A. Luk yanchuk and Y.K., Phys. Rev. Lett. 93, (2004) Graphene: K. S. Novoselov et al., Nature 438, 197 (2005) Y. Zhang et al., Nature 438, 201 (2005)

63 Electrons or Holes? Normal or Dirac? Experiment: SdH dhva