FermiFermi liquidliquid theorytheory ofof aa FermiFermi ringring T. Stauber, N. Peres, F. Guinea, and A. Castro Neto cond-mat/0611468 OverviewOverview Biased bilayer graphene One band approximation: Mexican hat dispersion Coulomb interaction: Self-energy correction Ferromagnetism: Ginzburg-Landau functional Polarizability A) High energies (Non-Fermi liquid behaviour) B) Low energies (Fermi liquid with modified Friedel oscillations) Plasmon dispersion Biased bilayer graphene A2 B2 V t⊥ A1 B1 t V= t⊥ = 3.0 eV Biased bilayer graphene Mexican hat dispersion Mexican hat dispersion a) b) Ek 0.4 , Ek 0.04 k 0.02 0.05 0.05 0 0.044 ky 0.02 0.02 0.4 0 kx 0.02002 0.04 0.04 Fermi ring: k k0 min b kmax Mexican hat dispersion E()= k Δα −2 k λ + 4 k α k 2 = −2πn min,σ 2λ σ α k 2 = +2πn max,σ 2λ σ Coulomb interaction 1 + + Hint = V q() ck+ q,σp c− qσ,,′ p σ′ ck ,σ c ∑∑ 2Ak,, p, σ q σ′ 2πe2 V() q= a) Self-energy orrectionsc to the dispersion ε 0q b) Exchange energy k a = min kmax Self-energy corrections n<nc n>nc 12 -2 nc =1.5x10 cm -->“exactly” filled Mexican hat Ferromagnetism Density fo states: 1 1 ρ() ε = π4 λ ε− min ε Stoner criterion: ε= ε( + 2 πn 2 λ ) F min σ (ρ ε F )U > 1 Ginzburg-Landau functional n= n++ − n m= + n− − n F a=()() n m2 + b4 n… + m Ginzburg-Landau functional 0.01 b(n) a(n) 0 0 A A -0.0001 -0.01 0.0002 -0.02 - -0.03 -0.0003 0 1 2 3 4 5 6 1 1.5 2 2.5 3 3.5 4 11 −2 11 −2 n [in 10 cm ] n [in 10 cm ] with self-energy correction m= n Ginzburg-Landau functional 0 F(m) A -2 -4 n =1 -6 n =5 n =2.5 -8 -10 0 2 4 6 11 −2 m [in 10 cm ] Polarizability “Mexican hat dispersion” 2 E( k )= (| k | −0 k ) High energy regime (b → 0) Asymptotic behaviour: −5 Im P +0. 3 1 2.5 ImP ( qω ,→ ) 2 ω 2 ω k ω 0 1.5 asiparticleQu lifetime: 1 Γ(ε ) ∝ ε | − ε | 0.5 F 0 0 0.5 1 1.5 2 2.5 3 q0 k Low-energy regime (ω ¡ b2) A) Forward scattering −0.5 Im P +0. Asymptotic behaviour 3 2.5 2 b2 ω 1.5 1 0.5 Perfect nesting at q=2b: 0 0 0.5 1 1.5 2 2.5 3 q b Friedel oscillations: cos(br 2 ) δn() r→ r 2 2 B) Backward scattering Low-energy regime (ω ¡ b ) −0.5 Im P +0. −0.5 Im P +0. 3 3 Friedel oscillations: 2.5 2.5 2 2 ω b2 ω b2 1.5 1.5 q→2 k0 + q 1 cos(1 2br ) cos( k0 r 2 ) δn() r → 2 0.5 0.5 r 0 0 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 q b q b Asymptotic behaviour: Perfect nesting: q=2k0 q=2k0-2b q=2k0+2b Example Numerical Numerical Analytical*1.5 Analytical/1.3 30 1,5 q=3b/2 q=2k0-b/2 20 1 -Im P -Im P 10 high-energy0,5 regime 0 0 024 024 2 2 ω/b ω/b 2 2 low-energyω/b regime ω/b 15 q=3b 1 q=2k0-3b 10 -Im P-Im P -Im P-Im P 0,5 5 0 0 0 5 10 15 5 0 5 10 15 20 2 2 ω/b ω/b Plasmon dispersion Random-phase approximation: (q ,ε )ω = 1 V− ( q ) Pω ( q , ) Plasmon dispersion: (ε q ,Ωq )= 0 2 DEG: Bilayer graphene: Summary 9 Biased bilayer graphene at low densities leads to a topologically non-trivial Fermi-surface. 9 At intermediate electronic densities, the one-band approximation becomes unstable. 9 Thegroundstateisunstabletowardferromagnetic ordering. 9 Including self-energy effects leads to a saturation in the magnetisation. 9 At low energies, the system is a Fermi-liquid, albeit with peculiar Friedel oscillations. 9 At high energies or low densities, the system shows non- Fermi liquid behavior. 9 The plasmons show the familiar two-dimensional square- root disperion, but with a larger energy scale..