Quantum oscillations in graphene in the presence of disorder and interactions

Abstract

Quantum oscillations in graphene is discussed. The effect of interactions are addressed by Kohn's theorem regarding de Haas-van Alphen oscillations, which states that electron-electron interactions cannot affect the oscillation frequencies as long as disorder is neglected and the system is sufficiently screened, which should be valid for chemical potentials not very close to the Dirac point. We determine the positions of Landau levels in the presence of potential disorder from exact transfer matrix and finite size diagonalization calculations. The positions are shown to be unshifted even for moderate disorder; stronger disorder, can, however, lead to shifts, but this also appears minimal even for disorder width as large as one-half of the bare hopping matrix element on the graphene lattice. Shubnikov-de Haas oscillations of the conductivity are calculated analytically within a self-consistent Born approximation of impurity scattering. The oscillatory part of the conductivity follows the widely invoked Lifshitz-Kosevich form when certain mass and frequency parameters are properly interpreted.

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Quantum oscillations in graphene in the presence of disorder and interactions
Pallab Goswami, Xun Jia, and Sudip Chakravarty
Department of Physics, University of California Los Angeles, Los Angeles, CA 90095-1547
(Dated: September 23, 2008)
Quantum oscillations in graphene is discussed. The effect of interactions are addressed by Kohn’s
theorem regarding de Haas-van Alphen oscillations, which states that electron-electron interactions
cannot affect the oscillation frequencies as long as disorder is neglected and the system is sufficiently
screened, which should be valid for chemical potentials not very close to the Dirac point.
determine the positions of Landau levels in the presence of potential disorder from exact transfer
matrix and finite size diagonalization calculations. The positions are shown to be unshifted even
for moderate disorder; stronger disorder, can, however, lead to shifts, but this also appears minimal
even for disorder width as large as one-half of the bare hopping matrix element on the graphene
lattice. Shubnikov-de Haas oscillations of the conductivity are calculated analytically within a
self-consistent Born approximation of impurity scattering. The oscillatory part of the conductivity
follows the widely invoked Lifshitz-Kosevich form when certain mass and frequency parameters are
properly interpreted.
We
I. INTRODUCTION
Quantum oscillations of magnetization (de Haas-van
Alphen effect: dHvA), of conductivity (Shubnikov-de
Haas effect: SdH) and that of Hall coefficient (RH)
are excellent probes for topologies of Fermi surfaces and
properties of Fermi liquids.1Recently interesting experi-
ments have led to considerable insight into physical sys-
tems such as graphene2and high temperature supercon-
ductors.3,4These oscillations fundamentally arise from
Landau level quantization and hence their frequencies
should be robust with respect to electron-electron inter-
action, crystalline potential, as long as an effective con-
tinuum theory exists, and under modest broadening of
the Landau levels due to impurity scattering. On the
other hand the waveform (harmonic content) or ampli-
tudes are sensitive to many details.
An elegant theorem for a translationally invariant
continuum system in two dimensions and short range
electron-electron interactions were formulated by Kohn5
to reveal the robustness of the frequencies. This is in con-
trast to the more complex many-particle analysis of Lut-
tinger,6which, in principle, can also address the wave-
form or the amplitude of the oscillations, but appears to
fail in two dimensions.7The reason behind Kohn’s the-
orem is intuitively clear. The magnetic field is never a
small perturbation, especially in two dimensions, where
the spectrum converts from continuous to discrete in its
presence. Thus, the problem must be exactly diago-
nalized with the magnetic field before considering the
perturbative effects of short-ranged electron-electron in-
teraction, which can in principle be well behaved, mod-
ulo a quantum phase transition or a broken symmetry.
Moreover, when the Landau levels are completely filled,
the ground state is non-degenerate, resulting in a special
stability, similar to magic nuclei. As the magnetic field
is tuned through a sequence of filled Landau levels, the
macroscopic state repeats itself, hence the periodicity; of
course, this is strictly valid only if the relevant Landau
level index, n ? 1.
One of the purposes of the present paper is to revisit
Kohn’s theorem in the context of graphene where SdH
oscillation frequencies have been shown to be remark-
ably close to what a pure noninteracting theory would
predict.2We show that this is a consequence of Kohn’s
theorem despite the broken translational symmetry due
to the crystalline potential, strengthening further the ev-
idence of the Weyl fermionic character of the excitation
spectra. In the process, we bring out some of the subtle
aspects of the theorem and its connection with the more
familiar Luttinger theorem regarding the volume of the
Fermi surface.8
The second purpose is to provide an exact calcula-
tion of the Landau levels in the presence of disorder and
an analytical self-consistent calculation of SdH within
self-consistent Born approximation (SCBA). A non-self
consistent calculation was previously reporrted,9as well
as a self-consistent calculation for unitary scatterers10—
perhaps such extreme strong scattering mechanism is not
applicable to realistic graphene samples. Not only do we
demonstrate that the oscillation frequencies are unshifted
to a good approximation for even moderate disorder, but
also obtain the amplitude and the waveform. The re-
sults, when suitably interpreted in terms of a mass pa-
rameter and a frequency scale, are similar to the widely
used Lifshitz-Kosevich formula,11which, in a strict sense,
cannot be applied in two dimensions. From our exact
numerical calculations, we also show that for very strong
disorder the Landau levels are shifted, but only mini-
mally, bearing some resemblance to the unitary scatterers
treated previously.10It is probably true that our results
also imply that, to lowest order, results would remain
unchanged when both electron-electron interaction and
impurity scattering are considered together, at least for
weak disorder relevant to experiments. The extension to
longer ranged impurity scattering is straightforward and
is not discussed.
The plan of the paper is as follows: in Sec. II we dis-
cuss Kohn’s theorem and apply it to graphene. In Sec. III
we provide exact transfer matrix and diagonalization cal-
arXiv:0809.3536v2 [cond-mat.mes-hall] 23 Sep 2008

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culations to determine the position of the Landau levels
and their shifts due to impurity scattering. In the second
part of this section we calculate SdH oscillations within
a self-consistent Born approximation (SCBA) developed
by Ando.12The brief Sec. IV contains our concluding re-
marks and there is an Appendix containing calculational
details.
II.KOHN’S THEOREM FOR QUANTUM
OSCILLATIONS IN TWO DIMENSIONS
A. Nonrelativistic fermions
There is a precise theorem by Kohn that in a two
dimensional continuum system electron-electron interac-
tion cannot shift the dHvA frequency in all orders in
perturbation theory.5To clarify its content, it is useful
to reconsider it. Let the unperturbed problem be defined
by the non-interacting Hamiltonian, H0, in a rectangular
box Lx×Ly. In the Landau gauge for the magnetic field
B (c being the velocity of light and m the mass of the
electrons),
?
The solution to this Landau level problem is of course
familiar. The energy eigenvalues and eigenfunctions of
an independent electron are
?
where the frequency ωc=eB
energy level is
H0=
1
2m
?
i
p2
x,i+
?
py,i+e
cBxi
?2?
(1)
?n,k= ?ωc
n +1
2
?
, ψn,k= eikyun
?
x +?ck
eB
?
,
(2)
mcand the degeneracy of each
g = 2Φ
Φ0,
(3)
where the total flux threading the system is Φ = BLxLy
and the flux quantum is Φ0= hc/e; the factor of 2 is for
spin.
Even though momenta are not good quantum num-
bers, it is still useful to depict the spectra on the two-
dimensional kx− ky-plane. The degenerate spectra, of
degeneracy g, lie on concentric circles in this plane, sep-
arated by ?ωc. Since momentum is no longer a good
quantum number, the states are not located on specific
points on the circle but can be viewed as rotating with
frequency ωc. The total number of available states in a
volume in the momentum space is unchanged in spite of
the Landau quantization. In particular, if we denote ∆A
to be the area between the concentric circles, then
LxLy
(2π)2∆A =g
2.
(4)
Imagine that the Fermi level, ?F, at T = 0 is situated
on one such concentric level such that all states with
energy E ≤ ?F are completely filled and all the levels for
E > ?F are completely empty. Then the total number of
occupied states is eaxctly the same as the system without
the magnetic field and the total energy per electron is also
exactly the same. The area enclosed by the Fermi level,
A(?F), follows trivially from Eq. (4) and is
2A(?F)
(2π)2=
N
LxLy,
(5)
where N is the total number of particles. That is none
other than the Luttinger sum rule. It is important to
note that even though a magnetic field is never a small
perturbation, the Luttinger sum rule is unchanged.
The magnetic field corresponding to the ground state
of a system with an integer number, n(?F), of Landau
levels completely filled and all the rest completely empty
will satisfy
1
Bn
Note that this ground state is an isolated nondegenerate
state separated by a gap ?ωcfrom the excited state. As
we increase B, the quantized orbits are drawn out of
the Fermi level, and sequentially pass through essentially
identical set of nondgenerate isolated ground states. This
of course results in the periodicity in the properties of the
electron gas; the correction in the limit that n(?F) ? 1 is
negligible. Periodicity of course does not imply sinusoidal
wave form and can contain higher harmonics.
Now, fix Bn to a completely filled Landau level and
turn on the electron-electron interaction. To all orders
in perturbation theory, a nondegenerate isolated ground
state will remain nondegenerate and therefore the se-
quence of states corresponding to fully filled Landau lev-
els as a function of the magnetic field will be the same,
as in the noninteracting case. The periodicity is there-
fore unchanged and is determined by the enclosed area
A(?F), which in turn is fixed by the Luttinger sum rule.
The argument will clearly break down if electron-
electron interaction drives a quantum phase transition
as a function of the ineteraction strength. A greater like-
lihood for this happening is when the Landau level is
partially filled and degenerate. Indeed, even for higher
Landau levels, we know that a zoo of density wave states
is a possibility,13but Kohn’s theorem should be robust
for fully filled Landau levels. Note that Kohn’s theo-
rem makes no statement about the amplitude of oscilla-
tions and is also silent about the waveform of the oscilla-
tions. In general the periodicity, when Fourier analyzed
will contain harmonics, and the harmonic content of the
nonrelativistic case will be different from the relativistic
case.
= n(?F)2πe
?c
1
A(?F).
(6)
B.Crystalline system: Dirac fermions in graphene
We now extend Kohn’s theorem to graphene.
known from experiments that SdH oscillation frequen-
It is

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cies are in excellent agreement with the noninteracting
system, as though electron-electron interaction plays no
role. This is as it should be if the theorem holds. There
are two basic issues that need to be dealt with: the crys-
talline potential that breaks translational invariance and
disorder in graphene samples. In this subsection we will
discuss the former and will leave the latter to the follow-
ing section.
As before, first consider the non-interacting problem.
The low energy spectrum for which the quasiparticles are
well described by continuum Lorentz invariant Hamilto-
nians based on two inequivalent nodes K = (2π
and K?= (2π
of the graphene lattice, with a the lattice spacing (cf.
below).14Near the Dirac point at K the energy eigen-
functions are given by
3a,
4π
3√3a)
3a,−
4π
3√3a) in the tight binding description
− i?vFσ · ∇ψ(r) = Eψ(r) (7)
where σ are the Pauli matrices and vF is the Fermi ve-
locity. In the plane wave basis, the k-space Hamiltonian
HK= ?vFσ·k. Similarly, for K?, HK? = ?vFσ∗·k. The
spectra for each energy is two-fold degenerate, ignoring
spin.
The exact energy eigenvalues for this two-component
Weyl fermion problem are of course trivial, and the corre-
sponding eigenvalues and eigenfunctions in the presence
of a perpendicular magnetic field are well known. While
the energy eigenvalues differ from the corresponding non-
relativistic Landau level problem, the spinor eigenfunc-
tions are easily constructed from the corresponding non-
relativistic problem. However, we will eschew the ex-
act solutions and only take advantage of the validity of
the continuum Hamiltonian to formulate the semiclassi-
cal dynamics for large Landau levels that leads to the
correct quantum oscillation frequencies. This is exactly
what is necessary to demonstrate Kohn’s theorem for the
general case of electrons in a crystalline environment. We
want to show that the Onsager quantization condition
follows from the semiclassical quantization of the Landau
orbits as the long wavelength low energy Hamiltonian is
a valid description, regardless of whether not the descrip-
tion is in terms of the relativistic Weyl Hamiltonian, as
for graphene.
In general, for the relativistic quasiparticles the semi-
classical equations of motion of a Bloch electron are dif-
ferent due to the presence of a nontrivial Berry curva-
ture.15These are
1
?∇kEα(k) −˙k × Ωα(k),
?˙k = −eE −e
where E and B are the electric and magnetic fields re-
spectively. The Berry curvature is defined as
˙ r =
(8)
c˙ r × B,
(9)
Ωα(k) = ∇k× Aα(k),
(10)
where
Aα(k) = i?uα(k)|∇kuα(k)?
(11)
is the Berry vector potential. The periodic part of the
Bloch wavefunction is denoted by uα(k), where α is the
band index.
When E = 0 and B = Bˆ z, following Chang and Niu,15
it can be shown that these set of equations of motion
lead to the the following semiclassical quantization rule
for the area of the orbit in momentum space
?
and the Maslov index
γ =1
Ak=1
2
(k × dk) · ˆ z =2πeB
?c
(n + γ).
(12)
2−Γ
2π,
(13)
with
Γ =
?
Aα(k) · dk.
(14)
the Berry phase for the orbit.
This phase easy is to compute for graphene because
the eigenspinors for K and K?can be chosen to be
?
where ϕk= tan−1(ky/kx). A simple computation shows
that
u±,K=
1
√2
1
±eiϕk
?
, u±,K? =
1
√2
?
1
±e−iϕk
?
(15)
AK(k) = i?u±,K(k)|∇ku±,K(k)? = −
ˆθ
2k,
ˆθ
2k.
(16)
AK?(k) = i?u±,K?(k)|∇ku±,K?(k)? =
whereˆθ is the azimuthal angle in the kx− kyplane and
k =
k2
The semiclassical quantization formula is then
(17)
?
x+ k2
y. The Berry phase Γ is therefore ±π.
Ak=2πeB
?c
n.
(18)
This implies once again that the magnetic field corre-
sponding to a fully filled Landau level at the Fermi energy
is given exactly by Eq. (6). The only remaining subtlety
now is to note that the area of the Fermi pocket is
πk2
F=2πeB
?c
n(?F),
(19)
from which it immediately follows that the Fermi energy
for the relativistic Dirac spectrum is given by
?F= ?vFkF=vF
?B
?
2n(?F).
(20)
The magnetic length
?B=
?
?c/(eB).
(21)
The appearance of this Berry phase Γ = ±π for massless
nodal quasiparticles with linear spectrum is a well known
topological property of the Hamiltonian.16,17

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From here on the argument proceeds identically to the
non-relativistic case, the only difference being the spac-
ing between the Landau level, which is not constant as
a function of n. As we turn on the electron-electron in-
teraction, the energy levels are drawn out of the Fermi
level in exactly the same sequence as the non-interacting
case. Therefore the frequency of dHvA is unchanged. To
the extent that the low energy continuum theory ade-
quately describes graphene, there should be no effect of
electron-electron interaction on the oscillation frequency.
Due to the relativistic spectrum of massless quasiparti-
cles in (2+1)-dimensions the density of states vanish lin-
early at the nodal points, and the Coulomb interaction is
not screened. This leads to strong long range interaction
between the quasiparticles in graphene. In the presence
of a moderate nonzero chemical potential the interaction
will be screened, however. For small chemical potentials
long range interactions can lead to broken symmetries
and hence to a failure of Kohn’s theorem.
The nodal relativistic spectra can also arise from a
condensate. An example is nodal fermionic quasiparti-
cles of a particle-hole condensate in l = 2 angular mo-
mentum channel, as in a singlet d-density wave (DDW),
staggered flux phase, or an orbital antiferromagnet.18
This linearized Dirac fermion theory is valid for momen-
tum small compared to inverse lattice spacing and en-
ergy small compared to the bandwidth. For a wide range
of the chemical potential, small compared to the band-
width, one can still use the linearized continuum theory
and results identical to those above hold.
III. EFFECTS OF DISORDER
A.Landau levels in graphene
FIG. 1: (Color online) Graphene lattice. Lattice sites belong-
ing to the same shaded horizontal or vertical stripes share
the same indices i or j. The solid circles correspond to the
A sublattice and the open circles to the B sublattice. The
three nearest neighbor vectors joining the two sublattices are
δk, k = 1,2,3.
Consider the tight-binding Hamiltonian, H, on a hon-
eycomb lattice of dimension Lx× Lyin a perpendicular
magnetic field, as shown in Fig. 1, which is
?
?
where the summation n ranges over all unit cells, and
cn,A, cn,B are the fermionic annihilation operators in
the unit cell n for the sublattices A and B, respec-
tively. The spin degrees of freedom are omitted, as we
assume that the magnetic field is sufficiently strong to
completely polarize them. In principle, disorder can take
various forms,19but here we shall consider the on-site
energy ?n,Aand ?n,B to be random variables uniformly
distributed in the range [−gV/2,gV/2] corresponding to
potential disorder. We choose the hopping matrix ele-
ment tn,k = t = 1.0, providing a natural energy scale.
The phases a? n,kare chosen such that the magnetic flux
per hexagonal plaquette, φ, is 1/Q, in units of the
flux quanta φ0 = hc/e. We choose a gauge such that
an,1= πi/Q for the vertical bonds in slice i as in Fig. 1,
and an,2= an,3= 0.
The Hall conductance σxycan be computed by impos-
ing periodic boundary conditions in both directions of the
system and diagonalizing the Hamiltonian (22) to obtain
a set of energy eigenvalues Eα and eigenvectors |α? for
α = 1,...,Lx× Ly. Then from the Kubo formula:20
4πie2
LxLyh
Eα<E<Eβ
H =
n
(?n,Ac†
n,Acn,A+ ?n,Bc†
n,Bcn,B)
−
n
3
?
k=1
(tn,keian,kc†
n,Acn+δk,B+ h.c.),
(22)
σxy(E) =
?
?α|vy|β??β|vx|α? − (x ↔ y)
(Eα− Eβ)2
,
(23)
where vx= [H,x]/i? is the velocity operator along the
x direction and similarly for vy. Note that the bonds in
Fig. 1 that are not parallel to y direction contribute to
both vxand vy. The summation corresponds to all states
below and above the energy E. Finally, the expression is
disorder averaged.
The longitudinal conductance σxxis studied using the
well developed transfer matrix method. Consider a quasi-
1D system, Lx ? Ly ≡ 2M with a periodic boundary
condition only along the y direction, where M denotes
the number of unit cells in a slice along y direction. Let
Ψi = (ψi,1,ψi,2,...,ψi,2M)Tbe the amplitudes on the
slice i for an eigenstate with a given energy E; then the
amplitudes on the successive slices are related by the ma-
trix multiplication:
?Ψi+1
where
Vi
isadiagonal
(ti,1,ti,2,...,ti,2M) representing the hopping matrix
elements connecting the slices i and i + 1, and Hi is
the Hamiltonian within the slice. All postive Lyapunov
exponents of the transfer matrix,21γ1> γ2> ... > γ2M,
are computed by iterating Eq. (24) and performing
Ψi
?
=
?
V−1
i
(E − Hi) −V−1
1
i
Vi−1
0
??
Ψi
Ψi−1
?
,
(24)
matrixwithelements

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orthonormalization regularly.
algorithm is guaranteed by the well known Osledec
theorem.22The conductance per square, σxx, is given by
the Landauer formula23,24,25,26(note the factor of
the argument of cosh due to the honeycomb lattice):
The convergence of this
√3 in
σxx=e2
h
2M
?
i=1
1
cosh2(2√3Mγi).
(25)
The localization length in the quasi-1D system with Ly=
2M is given by λM= 1/γ2M.
FIG. 2: (Color online) The Hall conductance σxy (plateaus)
and the longitudinal conductance σxx as a function of en-
ergy E. The magnetic flux through the hexagonal plaquette,
φ = 1/2000 in units of the flux quantum φ0 = hc/e and
the potential disorder gV = 0.5,0.05,0.01,0.001 from top to
bottom. For σxx, M was chosen to be 48, and, for σxy, the
system size was chosen to be 4000 × 10 for the exact diago-
nalization. In physical units the magnetic field corresponds
to B ≈ 40T. The vertical dashed lines indicate the locations
of the Landau levels at En = sgn(n)(√3π|n|/1000)1/2for the
parameters considered here. There are virtually no shifts of
the positions of the Landau levels except for gV = 0.5 for
which it is minimal.
The results are shown in Fig. 2, where the Hall
conductance σxy and the longitudinal conductance σxx
are computed as a function of energy E with param-
eters φ = 1/2000 and for potential disorder gV
0.5,0.05,0.01,0.001. The longitudinal conductance σxx
peaks almost exactly at the Landau levels
?√2vF
where
=
En= sgn(n)?
?B
??
|n|,
(26)
vF= 3ta/2?
(27)
is the fermi velocity, a being the lattice spacing. The
only exception is the case gV = 0.5, as shown in Fig. 3,
for which there is a minimal shift. Recall that this is
very large disorder as the energy scale is in terms of the
hopping parameter, t, which is set to unity. Each level is
broadened due to disorder and finite size effects. Because
the spacings between successive Landau bands become
smaller as |n| increases, the overlap between the neigh-
boring bands increases, resulting in the overall parabolic
shape of the background σxx. The Hall conducivity σxy
jumps by 2e2/h every time the energy passes through a
Landau level, corresponding to the two fold valley degen-
eracy in this system.
n1/2
gV= 0.005
gV= 0.01
gV= 0.05
gV= 0.5
FIG. 3: (Color online) The location of the peaks in Fig. 2.
Note that the slope is essentially the same as the noninter-
acting system except for gV = 0.5 for which it is slightly
different.
B. SdH Oscillations: analytical results in the
self-consistent Born approximation
Shubnikov-de Haas oscillations in graphene can be cal-
culated numerically by the method described in the previ-
ous subsection. However, for realistic fields, it is difficult
to control the accuracy because of the essential singu-
larity corresponding to the magnetic field corresponding
to all quantum oscillation phenomena.
dependence on the physical parameters are not particu-
larly transparent. For this reason we adopt an analytical
SCBA developed by Ando.12
In the presence of disorder, the imaginary part of the
self energy at the Fermi level is non-zero. Hence, Lut-
tinger’s theorem and Kohn’s argument are not immedi-
ately applicable. However, for weak disorder, the quasi-
particle lifetime at the fermi energy, τ(?F) ≡ τ (hence-
forth by τ we will mean τ(?F)), can be long and the
Fermi surface can be reasonably well defined within the
uncertainty ?/τ. If the Fermi energy ?F is very large,
we can use ?/?Fτ as a small parameter. Due to disor-
der the Landau levels are broadened into bands and in
the limit of overlapping Landau levels (ωcτ ? 1) explicit
Moreover, the