Electronic properties of graphene. Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay)

Transcription

1 Electronic properties of graphene Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay) Cargèse, September 2012

2 3 one-hour lectures in 2 x 1,5h on electronic properties of graphene 1. Band structure & massless Dirac electrons 2. Graphene's quantum Hall effect 3. Topological stability and merging of Dirac points

3 Graphene: a new 2D electron gas Discovered in 2004 by Novoselov and Geim: scotch tape trick A truly 2D crystal Honeycomb lattice of carbon Conducting Carrier density electrically tunable Carriers are unusual: massless Dirac electrons Therefore a new 2DEG as shown by its peculiar QHE Nobel prize in physics 2010

4 Band structure & massless Dirac electrons Outline 1. Crystal structure 2. Band structure: tight-binding model 3. Massless Dirac equation 4. Berry phase 5. Relativistic quantum mechanics 6. Interactions and screening 7. Transport

5 1. Crystal structure

6 Crystal structure Graphene = 2D honeycomb crystal of carbon Carbon atom: 6 electrons 1s2 (core) 2s2 2p2 (valence) hybridization: 1 2s orbital and 2 2p orbitals 3 sp2 orbitals 1 2pz orbital left - 3 coplanar σ bonds, with 120 angle: honeycomb structure - 1 conduction electron per C atom, 2pz orbital, perpendicular to the plane, giving π bands: electronic properties

7 Crystal structure Honeycomb crystal = triangular (2D) Bravais lattice + 2 atoms basis (important for Bloch s theorem) Direct space: y B A A A B B B B B A x O A B B A A A B

8 Crystal structure Reciprocal space: Reciprocal lattice of the triangular lattice = triangular lattice (whatever the atomic basis) ky g1 g2 K K' K' Γ K K g1 K' 1st Brillouin zone (1Bz) kx

9 Crystal structure Hexagonal Brilloun zone is a torus

10 Summary Graphene is a truly 2D crystal: thickness of a single carbon atom. It can be seen as a 2D membrane in a 3D world. Carbon atoms have a sp2 character as in graphite, carbon nanotubes, fullerenes, etc. But unlike diamond Honeycomb lattice is not a Bravais lattice. It is a triangular Bravais lattice with a two atom basis: two atoms per unit cell Two inequivalent corners of the hexagonal Brillouin zone: two valleys

11 2. Band structure: tight-binding model

12 Goal To understand the motion of conduction electrons in graphene (band structure of the so-called π bands coming from the 2pz carbon orbitals).

13 Band structure Nearest neighbor tight-binding model built on the 2pz orbitals Wallace 1947 Following the notations/conventions of Bena and Montambaux, New J. Phys. 11, (2009). Lattice spacing = a 3=2.46 Å and C-C distance = a = 1.42 Å. Real space Reciprocal space Atomic basis Hexagonal (first) Brillouin zone

14 Band structure

15 Band structure

16 Band structure

17 Band structure +3t CB(α=+1) 0-3t K' K VB(α=-1)

18 Band structure

19 Band filling CB 1Bz K VB DoS εf=0 CB VB ε -3t -t t 0 εf 3t K

20 Exfoliated graphene on SiO2 Apply an electric tension to the heavily n-doped Si backgate. contacts (Au) graphene (~1 to 100 μm2) n-si 300nm of SiO2 Novoselov et al., Science 2004 and PNAS 2005

21 Graphene capacitor: electric field effect A gate tension Vg allows to control the filling of electrons in the graphene sheet (electrical doping) Capacitor = plate(graphene)/dielectric(sio2)/plate(n-doped Si) Novoselov et al., Science 2004 and PNAS 2005 typical mobility μ ~ 1 m2/v.s = cm2/v.s typical carrier density nc ~ 1012 cm-2 [varies between 1011 and 1013 cm-2] Zhang et al., Nature 2005 geometrical capacitor law: nc=α x Vg with α=7x1010 cm-2/v

22 Density of states near Dirac point DoS εf=0 VB CB g1 g1-3t -t 0 VB ε t εf 3t

23 Electron and hole puddles Record with suspended graphene: ncmin = 108 cm-2 EFmin = 10 K Bolotin et al Martin et al., Nature Phys. 2008

24 3. Massless Dirac equation

25 Low energy effective theory CB CB VB diabolo Linear: photon-like but fermion VB

26 Low energy effective theory

27 Low energy effective theory

28 The Dirac equation (redux) g1 Dirac equation 1928

29 Dirac 3+1 versus Weyl 2+1 g1 electron electron 2mc 2 εf=0 positron Dirac sea hole Fermi sea Dirac equation 1928 Weyl equation for neutrino 1929 Positron (Dirac 1930) Hole in semiconductors (Peierls 1929)

30 Effective theory: summary g1

31 4. Berry phase

32 Geometric phase & Berry phase Parallel transport of a vector along a closed path in a curved space gives rise to a geometrical phase. Solid state context: Vector = bispinor (Bloch's u(k)>) Parameter space = reciprocal space (Brillouin zone is a torus) Closed path = isoenergy trajectory (cyclotron orbit) e.g. In quantum mechanics, geometrical phase is called Berry phase (M. Berry 1984): For review see e.g. Resta, EPJB 2011 or Q. Niu et al., RMP 2010

33 Berry phase See for example Fuchs, Piechon, Goerbig and Montambaux, EPJB 2010.

34 Dirac point seen as a vortex in reciprocal space carrying a topological charge ±1 (Berry phase ±π) They appear in vortex-antivortex pairs (fermion doubling) There is more to the Hamiltonian than its spectrum: eigenenergies & eigenvectors. Graphene is a coupled band system.

35 5. Relativistic quantum mechanics

36 Velocity operator, current & Zitterbewegung Schrödinger 1930 Zitterbewegung is an inter-band effect. It disappears for a wavepacket with all its weight on a single band

37 Chiralities Several (different) meanings: 1) chirality in the sense of helicity: projection of the (sublattice) pseudo-spin on the direction of motion σ p/ p 2) chirality in the sense of sublattice symmetry S: A A and B-> -B. In the context of graphene: S = σz anticommutes with the Hamiltonian. Consequence: E->-E symmetry of the spectrum. Remark: bipartite lattices, γ5 Dirac matrix. 3) chirality in the sense of direction of rotation, such has The chirality of quantum Hall edge states.

38 Chirality (helicity) and sublattice spin g1 K' - K + CB + - VB

39 Absence of backscattering g1 p' θ p Ando, Nakanishi, Saito, J. Phys. Soc. Jpn

40 Klein tunneling g1 V0 V0 energy ε 0 V0 V0 ε εf K O. Klein 1929 K p <v> α=1 1D proof: x α=-1 p' <v'>=<v> Review: Allain and Fuchs, Eur. Phys. J. B 2011 Proposal for an experimental observation with graphene: Katsnelson, Novoselov, Geim, Nature Physics 2006.

41 Summary Graphene is a 2D gapless semiconductor (or 2D semimetal with zero band overlap) Its low energy carriers are massless Dirac fermions, which exist in 4 flavors (2 for the valley K/K' degeneracy and 2 for the spin ½ degeneracy) These carriers are chiral (helicity) Dirac points in the Brillouin zone can be seen as vortices carrying a topological charge +/-1 which is related to a quantized Berry phase +/- π. They come in pairs (K/K', fermion doubling) Graphene is a coupled bands system

42 6. Interactions and screening

43 Electron-electron interactions

44 Screening

45 7. Electronic transport

46 Understanding the electronic transport properties of standard graphene Three experimental facts (the V-shaped curve): - σ(t)= almost constant - σ(nc) linear or slightly sublinear - σmin ~ 4 e2/h>0 at the neutrality point Novoselov et al., Nature 2005 Diffusive and incoherent transport ( classical transport ) in a degenerate (semi-)metal of non-interacting electrons.

47 Characteristic length scales and transport regimes

48 T Graphene's phase diagram Thermally Generated Electron-hole pairs T [K] Thermal Electron-hole pairs plasma TF Deg. hole metal Deg. electron metal Sheehy and Schmalian, PRL 2007

49 Including inhomogeneities (puddles) T [K] Thermal Electron-hole pairs plasma TF TFmin Deg. metal

50 Drude conductivity

51 Carrier mobility

52 Disorder strength 1/kFl Disorder strength grows towards the neutrality point. System becomes more and more diffusive (less ballistic). Geim and Novoselov, Nature Mat. 2007

53 What are the most relevant impurities?

54 Minimal conductivity: puddles?

55 Minimal conductivity: disorder broadening? Self-consistent Born approximation (SCBA: Ando and Shon 1998): DoS is never really vanishing as planewaves now have a finite energy width ~ ћ/τ

56 Minimal conductivity: ballistic graphene Evanescent modes transport? Tworzydlo et al.; Katsnelson 2006

57 Summary on transport - σ(t)=constant: because TFmin > T (graphene's phase diagram) - σ(nc) linear or slightly sublinear: probably resonant (ln2, not unitary) scatterers (for doped graphene) AND/OR charged impurities (depends on sample quality) - σmin ~ e2/h>0 at the neutrality point: classical transport can give a rough explanation (puddles due to charged impurities induce a non-zero local carrier density) but out of its validity domain. Otherwise SCBA or evanescent wave transport in ballistic regime.

58 Conclusion: graphene versus usual 2DEG

59 References - Popular scientific reviews: M. Wilson, Physics Today (January 2006), page 21; J.N. Fuchs, M.O. Goerbig, Pour la Science (mai 2008) et Images de la physique (CNRS 2007) [both in french]. - General reviews: A. Geim, K. Novoselov, Nature Materials 6, 183 (2007); A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81, 109 (2009) - Two important experiments: Novoselov et al., Nature 438, 197 (2005) [Geim s group] Zhang et al., Nature 438, 201 (2005) [Kim s group]