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 ε Particle-hole (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 domainθ wall:
⎡ (X ) θ(X ) ⎤ Ψ = cos 0 C+ +sin 0 eiϕ C+ C+ |0> ∏⎢ +,X0,↑ −,X0,↓⎥ −,X0,↑ X X 0 → −∞ θ = 0; X 0 → L θ = π; ϕ = 0 2 ⎛ d ⎞ E = 2 ⎜ ⎟ + ()E − ∆(X ) cosθ (X ) l s ∑ ⎜ dX ⎟ ∑ z 0 0 X 0 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 τ I ~ t 2 dE[]G adv (E)G ret (E − eV )− G ret (E − eV )G adv (E) ∫τ tip DW tip DW κ ψ 1 G()~ T y = (0;τ ψ + y )= 0;0 (~ ) τ 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