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

Graphite …

Unusual electronic phenomena in Graphite Y. Kopelevich et al. Phys. Rev. Lett. 87, 147001 (2001) 93, 166402 (2004) Weak ferromagnetism MIT transition Linear magnetoresistance (old problem) Quantum Hall 35K Superconductivity in Graphite-S

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

Graphite: Band structure: Slonczewski-McClure Model Fitting parameters

holes Is it real ???? electrons

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 ???

ρc/ ρa > 50000 (instead of 300 in Kish) ρa ~ 3 μΩ cm (300K) 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~107cm2/Vs (105 in Graphene) Metals: 300μΩ cm, Ioffe-Regel 1000 μΩ cm

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, 166402 (2004) - Phys. Rev. Lett. 97, 256801 (2006)

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

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

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

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.!!!

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

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

Generalized formula: 2D, 3D, arbitrary spectrum I. Luk’yanchuk and Y. Kopelevich Phys. Rev. Lett. 93, 166402 (2004) where Normal: Dirac: Fermi surface cross section Limit cases 3D : => Lifshitz-Kosevich => Shoenberg 2D :

{ { 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)

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, 156402 (2003) Novoselov, K. S. et al. Nature 438, 197 (2005); Zhang, Y. et al. Nature 438, 201 (2005).

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

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

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

QHE and DF

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).

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

« 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

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

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

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

Filtering Normal (Integer QHE) GRAPHITE: Normal vs Dirac carriers separation Lukyanchuk, Kopelevich - Phys. Rev. Lett. 97, 256801 (2006) Rxy Dirac (Semi-integer QHE) Rxx Filtering B (T)

FQHE in Graphite

Rxy Rxx Rxy Rxy Rxx

Another manifestations of DF in Graphite

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

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

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

INFRARED SPECTROSCOPY

2006 Graphite, interpretation, ??? =>

RAMAN SPECTROSCOPY

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

RAMAN SPECTROSCOPY « Graphene Fingerprint »

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

model

B As in bilayer ?

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

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

NE oscillations, calculated from thermodynamics for:

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

As function of gate voltage … Normal Dirac …

EXPERIMENT Graphene, DF

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

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