Electronic properties of graphene Jean-Noël Fuchs Laboratoire de Physique des Solides Université Paris-Sud (Orsay) Cargèse, September 2012
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
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
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
1. Crystal structure
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
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
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
Crystal structure Hexagonal Brilloun zone is a torus
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
2. Band structure: tight-binding model
Goal To understand the motion of conduction electrons in graphene (band structure of the so-called π bands coming from the 2pz carbon orbitals).
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, 095003 (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
Band structure
Band structure
Band structure
Band structure +3t CB(α=+1) 0-3t K' K VB(α=-1)
Band structure
Band filling CB 1Bz K VB DoS εf=0 CB VB ε -3t -t t 0 εf 3t K
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
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 =10 000 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
Density of states near Dirac point DoS εf=0 VB CB g1 g1-3t -t 0 VB ε t εf 3t
Electron and hole puddles Record with suspended graphene: ncmin = 108 cm-2 EFmin = 10 K Bolotin et al. 2008 Martin et al., Nature Phys. 2008
3. Massless Dirac equation
Low energy effective theory CB CB VB diabolo Linear: photon-like but fermion VB
Low energy effective theory
Low energy effective theory
The Dirac equation (redux) g1 Dirac equation 1928
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)
Effective theory: summary g1
4. Berry phase
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
Berry phase See for example Fuchs, Piechon, Goerbig and Montambaux, EPJB 2010.
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.
5. Relativistic quantum mechanics
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
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.
Chirality (helicity) and sublattice spin g1 K' - K + CB + - VB
Absence of backscattering g1 p' 0.6 0.4 0.2 θ p 0.2-0.2-0.4 Ando, Nakanishi, Saito, J. Phys. Soc. Jpn 1998-0.6 0.4 0.6 0.8 1
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.
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
6. Interactions and screening
Electron-electron interactions
Screening
7. Electronic transport
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.
Characteristic length scales and transport regimes
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
Including inhomogeneities (puddles) T [K] Thermal Electron-hole pairs plasma TF TFmin Deg. metal
Drude conductivity
Carrier mobility
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
What are the most relevant impurities?
Minimal conductivity: puddles?
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
Minimal conductivity: ballistic graphene Evanescent modes transport? Tworzydlo et al.; Katsnelson 2006
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.
Conclusion: graphene versus usual 2DEG
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]