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
= ⊥ = 3.0 eVtV 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
)( +−Δ= λαkkkE 42
α k 2 −= 2πn min,σ 2λ σ
α k 2 += 2πn max,σ 2λ σ Coulomb interaction
1 + + Hint = )( +qk ,σ − ′ pqp ,, σσ′ccccqV k ,σ ∑∑ 2A ,,,qpk σσ′
2πe2 qV )( = a) Self-energy corrections to the dispersion ε 0q
b) Exchange energy k a = min kmax Self-energy corrections
n
12 -2 nc =1.5x10 cm -->“exactly” filled Mexican hat Ferromagnetism
Density of states:
1 1 ερ)( = 4 −εελπmin
Stoner criterion:
+= n λπεε)2( 2 F min σ ρ ε F U >1)( Ginzburg-Landau functional
= + nnn −+ = − nnm −+
2 )()( mnbmnaF 4 ++= … 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 = nm 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 −= kkkE 0 )|(|)( High energy regime (b → 0)
Asymptotic behaviour: −5 Im P +0.
3
1 2.5 qP ω),(Im → 2 ω 2 ω k ω 0 1.5 Quasiparticle lifetime: 1 −∝Γ εεε||)( 0.5 F 0 0 0.5 1 1.5 2 2.5 3 kq 0 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: br)2cos( δ rn )( → 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 →2 0 +qkq 1 1 0 rkbr )2cos()2cos( δ rn )( → 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: ε ω = − qPqVq ω),()(1),(
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.