LuttingerLuttinger LiquidLiquid atat thethe EdgeEdge ofof aa GrapheneGraphene VacuumVacuum H.A. Fertig, Indiana University Luis Brey, CSIC, Madrid I. Introduction: Graphene Edge States (Non-Interacting) II. Quantum Hall Ferromagnetism and a Domain Wall at the Edge III. Properties of the Domain Wall IV. Excitations from Filled (Graphene) Landau Levels (with Drew Iyengar and Jianhui Wang) V. Summary Funding: NSF I. Edge States for Graphene Honeycomb lattice, two atoms per unit cell B Lattice constant: 2.46Å Nearest neighbor distance: 1.42Å A Simple tight-binding model for pz orbitals: H = −t ∑ n1 n2 n1n2 =n.n. t ≈ 2.5-3 eV A and B sublattice sites in unit cell • For each k there are eigenvalues at ±|ε| ⇒ particle-hole symmetry • Fermi energy at ε=0 EF Wavefunctions in a magnetic field: ik x ⎛±φ (y − k 2 )⎞ ik x ⎛±φ (y − k 2 )⎞ Ψ(K,n) = e x ⎜ n−1 xl ⎟ Ψ(K′,n) = e x ⎜ n xl ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ φn (y − kxl ) ⎠ ⎝φn−1(y − kxl ) ⎠ 0 φ ikx x ⎛ ⎞ ikx x ⎛ 0 ⎞ Ψ(K,0) = e ⎜ ⎟ Ψ(K′,0) = e ⎜ ⎟ ⎝φ0 ⎠ ⎝ 0 ⎠ φn = harmonic oscillator state Energies: ε a Particle-hole ε (τ ,n) = ± 3 n t 0 l conjugates With valley and spin indices, each Landau level is 4-fold degenerate kx • Real samples in experiments are very narrow (.1-1µm) ⇒ edges can have a major impact on transport • Can get a full description of QHE within Dirac equation • Edge structure can be probed directly via STM at very small length scales. Nothing comparable is possible in standard 2DEG’s (GaAs samples, Si MOSFET’s) K´ K Ky K´y Tight-binding results, armchair edge II. Quantum Hall Ferromagnetism and the Graphene Edge EF DOS Energy EF Interactions DOS Energy • Exchange tends to force electrons into the same level even when bare splitting between levels is small • Renormalizes gap to much larger value than expected from non-interacting theory (even if bare gap is zero!) This does happen in graphene (Zhang et al., 2006). • Plateaus at ν=0?,±1. • System may be a quantum Hall ferromagnet. cf. Alicea and Fisher, 2006 Nomura and Macdonald, 2006 Fuchs and Lederer, 2006 “Vacuum” state (undoped graphene): n=2 n=1 K,K’ n=0 EF K,K’ n=-1 n=-2 Low-lying excitations: 2 (+2) spin (+valley) waves EF Spin stiffness ⇒ Analogy with Heisenberg ferromagnet. Consequences for edge states: Electron-like ∆(X ) 4 n=0 states edge state. 0 (Ez=0) Hole-like −∆(X ) edge state. 0 Include Zeeman coupling Domain wall Spin polarized Spin unpolarized Description of the domain wall: ⎡ θ(X ) θ(X ) ⎤ Ψ = cos 0 C+ +sin 0 eiϕ C+ C+ |0> ∏⎢ +,X0,↑ −,X0,↓⎥ −,X0,↑ X0<L⎣ 2 2 ⎦ X 0 → −∞ θ = 0; X 0 → L θ = π; ϕ = 0 2 ⎛ dθ ⎞ E =π 2 ρ ⎜ ⎟ + ()E − ∆(X ) cosθ (X ) l s ∑ ⎜ dX ⎟ ∑ z 0 0 X 0 <L ⎝ 0 ⎠ X 0 <L Pseudospin stiffness 0.5 0.4 Result of minimizing π /2 0.3 ) energy. Width of 0 X 0.2 ( domain wall set by θ 0.1 strength of confinement. 0.0 0 2 4 6 8 10 12 X0/ℓ III. Properties of the Domain Wall = +,↑ y = -,↓ x 1. φ=0: Broken U(1) symmetry ⇒ linearly dispersing collective mode φ ~ in-plane angle of “spins” m ~ position of domain wall 2. Spin-charge coupling ⇒ gapless charged excitations! Twist phase once 2 X =k l2 X0=kyl 0 y = weight in w/f Fermion operator: STM tip 3. Tunneling from STM tip: power law IV ⇒ not a Fermi liquid! tunneling y t Power law exponent a function x of confinement potential Domain wall Graphene sheet I ~ t 2 dE G adv E G ret E − eV − G ret E − eV G adv E ∫ []tip ( ) DW ( ) tip ( ) DW ( ) 1 G()τ ~ T ψ (y = 0;τ )ψ + (y = 0;0 )~ τ τ κ κ = (x +1/ x)/ 2; x = 4π ρ~ / Γ U(1) spin stiffness Γ ~ confinement potential ⇒ Exponent sensitive to edge confinement! 4. Tunneling from a bulk lead: possibility of a quantum phase transition (into 3D metal). Lead Model lead as non-interacting electrons in a magnetic field y ⇒ Tunneling t x with Perturbative dt 2 = −(κ − 2)t 2 Shrinking t ⇒ DW a Luttinger liquid RG: dl Growing t ⇒ DW + lead = Fermi liquid? IV. Inter-Landau Level Excitations (Magnetoplasmons) Low-lying excited states = particle-hole pairs Electron in empty band cf. Kallin and Standard 2DEG: 2 ql q Halperin, 1984 Hole in filled band • Measurable in cyclotron resonance, inelastic light scattering. • This picture is largely the same for graphene, just need to be careful about spinor structure of particle and hole states. Two-Body Problem Electron Hole (ħ=ωc=l=1) To diagonalize (A= -Byx): 1. Adopt center and relative coordinate R=(r1+r2)/2, r= r1-r2 + 2. Apply unitary transformation H´0 = U H0U with r r r U = eip⋅(zˆ×P)e−ixY P = center of mass momentum with z=x+iy Wavefunctions constructed from: with Wavefunctions are 4-vectors |n+,n-> constructed from with energies Electron Hole 3. Apply unitary transformation to interaction H1: 4. Compute eigenvalues of ⇒ two-body eigenenergies with fixed P Results: γ0=4 γ1=4 4, 0, 2 2, 0 1,1 5, 1 3, 0 4, 1 2 2, 0 F F v v 1 3, 1 1, 0 Energy Energy 2, 1 1 Lowest filled LL = 0 Lowest filled LL = 1 2 4 6 8 10 12 2 4 6 8 10 12 P P 2 Interaction scale: β=(e /εl)/(ħvF/l ) ≈ (c/vFε)/137 = 0.73 γ0<4 γ1<4 2 0, 2 , 1, 0, 2, 0 4, 1 0, 1 , 1 3, 1 1, 0 1 F F v 2, 1 v Energy Energy 0 0, 0 0 1, 1 2 4 6 8 10 12 2 4 6 8 10 12 P P Note negative energy Comments: 1. Negative energies because we have not included loss of exchange self-energy ⇒ many-body approach needed 2. Landau level mixing relatively small 0.2 0.1 Probability 2 4 6 8 10 12 P Note however for β≈1, LL mixing becomes much more pronounced ⇒ system on cusp between weakly and strongly interacting Many-Body Particle-Hole Approach • A generalization of spin-wave calculation Almost, but not quite. Must watch out for degeneracies n=2 n=1 1 n=0 EF 2 n=-1 n=-2 • Excitations characterized K,K’ by ∆Sz and ∆τz Also: Exchange energy with “infinite” number of filled hole levels leads to (logarithmically) divergent self-energy. Fix this with an explicit cutoff in number of filled Landau levels. µ (1 ,−1) (1, 1) (−1,−1) (−1, 1) (0 , 0) (0 ,−1) (0 , 1) (−1 , 0) (1 , 0) γ=4 ↑,↓=spin, double arrows=pseudospin (m,n)=(sz,tz) Energy generically involves four terms: Noninteracting Direct (Ladders) Exchange (Bubbles - RPA) Exchange self-energy Results: N=0 1 E0 Ε 2 Two-body result (up to constant) γ <4 E 0 Intra-Landau level 0 0 2 4 6 8 10 12 P Comments: 1. Change in kinetic energy and γ0<4 Zeeman energy must be added in E3 E4 2 2. Gapless excitations for ν=-1,1 1 E γ ≤4 3. Excitation spectra identical for Ε 0 2 ν=-1,1: particle-hole symmetry E E2 ∆n=1 0 2 4 6 8 10 12 P Dashed lines equivalent to 1 two-body result. E γ1=4 Ε 2 1 E Very large many-body E2 correction! 0 2 4 6 8 10 12 P 2 E1 γ =4 2 Ε 2 E 1 E2 0 2 4 6 8 10 12 P • Minima/maxima may be visible in inelastic light scattering or microwave absorption. Summary • Graphene: a new and interesting material both for fundamental and applications reasons. • Clean system is likely a quantum Hall ferromagnet. •Armchair edges: oppositely dispersing spin up and down bands ⇒ domain wall • Domain wall supports gapless collective excitations, and gapless charged excitations through pseudospin texture. • Domain wall supports power law IV (Luttinger liquid). • Domain wall may undergo quantum phase transition when coupled to a bulk lead. • Collective inter-Landau level excitations = excitons • Many-body corrections split and distort dispersions found in two-body problem.