Band Collapse and the Quantum Hall Effect in Graphene Page: 2 of 10
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2
Semenoff4:
H= -tZ c (A)c(A+&,)+ct(A+&)c(A)
+I3gZct(A)c(A) - ct(z+&)c(+&) (1)
where c(A), c(A + ') are the annihilation operators for
sites on sublattice A and B, and ) is an energy difference
for electrons localized on the A and B sublattices. We
will call this term the Semenoff mass. Graphene is effec-
tively massless which is approximated by taking ) -> 0.
In this limit the band structure is gapless at two inequiv-
alent points K - -4 , ), K' -K, where a is the
nearest-neighbor lattice constant. Around these points,
the Hamiltonian is described by (in the ideal case mass-
less) Dirac fermions with4,5:HK - =xkx+yky; HK/
-axkx + aoky (2)
which act on a two-spinor wavefunction describing the
sublattices A and B, see Fig[1]. There is also an overall 2-
fold spin degeneracy which we neglect for the remainder
of the paper. Note that parity switches A = B and
K - K' while time reversal switches K - K'. The
Semenoff term opens a gap of value m = 2)/3/3ta at
K and -m at K' -K so time reversal symmetry is
preserved.where 'bn(x) are harmonic oscillator eigenstates and u
are the eigenstates of HK with energies E . Notice that
all the energy levels are paired except the ni 0 level.
There is a common misconception that unpaired "zero-
modes" occur only for a massless fermion but observe
that for m > 0 we have uo 0 while for m < 0 we
have uto= 0, so such levels are unpaired even for non-
zero mass. In the field theory formalism, the current is
defined to be J = - zery [vv/] and is odd w.r.t.
charge conjugation symmetry. We find that(01J010) = p = (N-
2N+) weB I
(4)
where N+ and N_ are the numbers of filled positive and
negative energy Landau levels (LL). Hence the Hall con-
ductance is1
N+)
(5)
in units of c2/h. Due to the unpaired level, this will
be half-integer and the position of the unpaired level de-
pends on the sign of eB and m as in Fig[2].21
2j-2
2+1- 2A B
A A i AT)I c
c I3 Ii.
a4 ctl F
-~ .1nvl
FIG. 2: Zero mode in the Dirac Equation.
dimensional lattice
Now consider one Dirac fermion at the K-point with
mass m in magnetic field B. The Hamiltonian is H -
uxkx + ry(k - eBx)+ m z. For eB > 0, the eigenstates
are_ky
k,n 47ranwith
an
E0( an n( -
+ an m Vbn_1(xo (k))
-4 (k))This analysis is correct for the fermion located around
the K-point, but as mentioned before the graphene band-
structure contains two such fermions. For the purpose of
being well defined, we consider a small positive Semenoff
mass m at K which means a small negative mass at K'.
Consider the case of eB > 0. The Hall conductance gets
a contribution from both fermions and is zero when the
Fermi level is in the gap -m < p < m and odd inte-
ger otherwise. This is then an odd integer quantum Hall
effect as in Fig[3]. When the gap is vanishingly small,
m - 0 the region of zero Hall conductance becomes in-
finitely narrow.
II. HARPER EQUATION FOR GRAPHENE
We now present a different argument that reproduces
the experimental results and is valid for both high andFIG. 1: Graphene Lattice, BZ and One
on which the Harper equation is defined.2eB + Inm2
1
-(k an)
eB
cln
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Bernevig, B.Andrei; Hughes, Taylor L.; Zhang, Shou-Cheng; /Stanford U., Phys. Dept.; Chen, Han-Dong; /Illinois U., Urbana et al. Band Collapse and the Quantum Hall Effect in Graphene, article, March 16, 2010; United States. (https://digital.library.unt.edu/ark:/67531/metadc928231/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.