Hexagonal lattice model of theAA-stacked bilayer graphene
A. A. Nikolaev1∗, M. V. Ulybyshev21Far Eastern Federal
University; 2Regensburg University, ITEP and Moscow State
University
∗[email protected]
Abstract
Tight-binding model of the AA-stacked bilayer graphenewith
screened electron-electron interactions has been stud-ied using the
Hybrid Monte Carlo simulations on the origi-nal double-layer
hexagonal lattice. Instantaneous screenedCoulomb potential is taken
into account using Hubbard-Stratonovich transformation. G-type
antiferromagnetic or-dering has been studied and the phase
transition with spon-taneous generation of the mass gap has been
observed.Dependence of the antiferromagnetic condensate on
theon-site electron-electron interaction is examined.
Monolayer graphene and bilayer graphene
Monolayer graphene is represented by a single sheet ofcarbon
atoms, which form hexagonal lattice. Each carbonatom has 4 valence
electrons, 3 of them form σ-bonds andthe last remains on
π-orbital.
Figure 1: Atomic structure of monolayer graphene. Elec-trons on
π-orbitals are oriented perpendicular to thegraphene plane.
Bilayer graphene is represented by two graphene sheets,stacked
upon each other.
Figure 2: Two common types of bilayer graphene. Energyspectrum
of electrons and transport properties dependsheavily on the way of
stacking.
We concentrate on the AA-stacked bilayer graphene (AA-BLG).
as
ls
Figure 3: Atomic structure of the elementary cell of AA-BLG (as
= 1.42 A, ls = 3.3 A).
Tight-binding Hamiltonian for electrons in the AA-BLG maybe
written as follows:
Ĥtb = −t2∑i=1
∑
∑σ=↑,↓
â+XiσâYiσ −
−t0∑X
∑σ=↑,↓
â+X1σâX2σ + h.c.,
where â+Xiσ and âXiσ are creation and annihilation opera-tors
for the electron with spin σ at the site X on the layer
irespectively, t = 2.57eV represents nearest-neighbour hop-ping
inside one layer and t0 = 0.36eV — between layers(values of
hoppings are taken from [1]).
Energy spectrum of such system has the following form:
s
Figure 4: Energy spectrum of AA-BLG from
tight-bindingHamiltonian.
Near the Dirac points:
� = vF |k|,
where vF = 32ta ≈1
315 ⇒• coupling constant α = e
2
vF≈ 2.3, in AA-BLG we have elec-
trodynamics with a strong interaction•magnetic and retardation
effects can be neglected, only
Coulomb interaction may be consideredInteraction potentials:
screened Coulomb inside thegraphene layer, usual Coulomb between
the layers.
1
10
0 0.2 0.4 0.6 0.8 1
V(r
), e
V
r, nm
Non-compact gauge field
Screened potentials
Coulomb
Figure 5: Electron-electron interaction potential inside
onegraphene layer. For detailed dicsussion about screening
byσ-electrons see [2].
Figure 6: There are arguments from mean-field theory,
thatCoulomb interaction in the AA-BLG may open the energygap and
lead to formation of G-type AFM condensate [3].
Lattice formulation
We start from the Hamiltonian formulation of partitionfunction.
Hamiltinian is formulated in terms of electrons andholes:
Ĥ = Ĥtb + Ĥstag. + Ĥint.
Ĥtb = −t2∑i=1
∑
(â+XiâYi + b̂+Xib̂Yi)−
− t0∑X
(â+X1âX2 + b̂+X1b̂X2) + h.c.
Ĥst. =2∑i=1
∑X,Y
±mâ+XiâYi ±mb̂+Xib̂Yi,
where the sign before m in the Ĥst. are chosen accordingto the
left part of Fig. 6. Electron-electron interaction areadded by
Ĥint. =1
2
2∑i,j=1
∑X,Y
q̂XiVijXY q̂Yj,
where q̂Xi = â+XiâXi − b̂
+Xib̂Xi and V
ijXY is the matrix of po-
tentials.Exponentials in partition function are splitted as
follows:
Z = Tr(e−βĤ
)= Tr
(e−∆τ (Ĥtb+Ĥstag.+Ĥint.)
)Nt=
= Tr(e−∆τ (Ĥtb+Ĥst.)e−∆τĤint.e−∆τ (Ĥtb+Ĥst.) . . .
)+ O(∆τ2)
Important feature: there are now 2Nt time layers due tosuch
splitting, only even time layers are physical.To deal with
e−∆τĤint. we perform Hubbard-Stratonovichtransformation [4]:
e−∆τ2
∑X,Y
q̂XVXY q̂Y=
∫Dϕe
− 12∆τ∑X,Y
ϕXV−1XYϕY−i
∑X
ϕX q̂X
Finally we arrive at the following expression:
Z =
∫DϕDηDηDχDχe−ηMη−χM
+χ− 12∆τϕT V̂ −1ϕ
=
∫Dϕdet(M+M)e−
12∆τϕ
T V̂ −1ϕ
Fermionic determinant is positive!The observable: AFM
condensate. Electron density opera-tors:
n̂iA↑ =1
Nsubl.
∑X∈A
â+XA,i↑âXA,i↑
n̂iB↓ =1
Nsubl.
∑X∈B
â+XB,i↓âXB,i↓
∆n =〈n̂1A↑
〉−〈n̂2A↑
〉=〈n̂1B↓
〉−〈n̂2B↓
〉In terms of inverse Dirac operator:
〈∆n〉 = 1NτNsubl.
∑τ
〈∑X∈A
(M̂−1X2X2 − M̂
−1X1X1
)〉
=1
NτNsubl.
∑τ
〈∑X∈B
(M̂−1X1X1 − M̂
−1X2X2
)〉
Numerical results
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
9 10 11 12 13 14
<∆
n>
Vxx, eV
T=0.19 eV, 122x35 lattice, ∆τ=0.15 eV
-1
m = 0.0, extrapolated
Figure 7: The dependence of the AFM condensate on theon-site
interaction potential Vxx at fixed temperature. AFMcondensate
vanishes at the value (8.89 ± 0.33) eV.
Here we have disagreement with MF result: ∆n ≈ 0.5 atVxx = 8.9
eV [3].
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
<∆
n>
T, eV
ε=3.0, ∆τ=0.15 eV-1
m = 0.0, extrapolated
Figure 8: The dependence of the AFM condensate on tem-perature,
measured on lattices with different temporal sizes.All potentials,
except Vxx, were rescaled by the factor of� = 3.0: V ijXY → V
ijXY /3.0.
Conclusions
•Original hexagonal lattice model for AA-bilayer graphenewith
long-range Coulumb interaction was studied.• Formation of the AFM
condensate was observed and its
dependence on the on-site electron-electron interactionwas
examined.
References
[1] J.-C. Charlier, J.-P. Michenaud, and X. Gonze, Phys.Rev. B
46, 4531 (1992).
[2] M. V. Ulybyshev, P. V. Buividovich, M. I. Katsnelson,M. I.
Polikarpov, Phys. Rev. Lett. 111, 056801 (2013).
[3] A. L. Rakhmanov, A. V. Rozhkov, A. O. Sboychakov,and Franco
Nori, Phys. Rev. Lett. 109, 206801 (2012).
[4] R. C. Brower, C. Rebbi, and D. Schaich,
PoS(Lattice2011)056.
The Helmholtz International Summer School ”Lattice QCD, Hadron
Structure and Hadronic Matter”, 25.08.2014 - 06.09.2014, Dubna,
Russia
