superlattice-broadened LL, such that its energy
Þ disappears in the limit of vanishing superlattice modulation (12). If the Fermi
energy eF lies within these superlattice-broadened
LLs, the system should exhibit a metallic behavior
(25). The Hofstadter spectrum can then be understood as Landau quantization of BZ minibands
in the effective field Beff ¼ B ; f0 p=qS (20, 29).
With this concept in mind, let us take a closer
look at the experimental behavior of sxx and
the Hall conductivity sxy at high T and small
Beff—that is, in the absence of Landau quantization in BZ minibands (Fig. 2, D to F, and fig. S6).
One can see that every time BZ minibands are
formed, sxx exhibits a local maximum and sxy
shows a Beff/(1 + Beff2)–like feature on top of a
smoothly varying background. This local behavior
resembles changes in sxx and sxy expected for any
metallic system near zero B and approximated
by the functional forms 1/(1 + B2) and B/(1 + B2),
respectively (1–3). The latter are sketched in the
insets of Fig. 2D and match well the shape of local
changes in sxx and sxy near fractional fluxes
f ¼ f0p=q, which correspond to Beff = 0 (Figs.
2D and 3 and fig. S6).
The described analogy between magnetotransport in normal metals and in BZ minibands
can be elaborated using the approximation of a
constant scattering time t (1–3). We assume t to
be the same for all minibands and magnetic fields.
In this approximation, sxx ºv2t and is determined
by the group velocity of charge carriers, v (26).
Each BZ miniband effectively represents a different two-dimensional system with a different
k-dependent velocity. If T is larger than the cyclotron gaps, as in our case, the Fermi step becomes smeared over several minibands, which all
contribute to sxx. In this regime, one can define
hv2i averaged over an interval of ±T around eF.
We calculate hv2i using a representative miniband spectrum for a graphene-on-hBN superlattice, which was computed with the model
developed in (29). The resulting conductivity is
evaluated as (26)
sxx ¼ 4e2 h peFt h v2i v2 F
where e is the electron charge, vF is the Fermi-Dirac velocity, and h is Planck’s constant (26).
The only fitting parameter is t, which we choose
so that sxx fits the experimental values for f ¼
f0=2 (Fig. 3). For other p/q, the calculated sxx
are shown by black dots. Furthermore, according to the classical magnetotransport theory
(1–3), sxx near zero Beff should vary as sxx (Beff) =
sxx (0)/(1 + aBeff2), where a is a p- and q-dependent
coefficient. It can be evaluated (26) without
extra fitting parameters (narrow black parabolas in Fig. 3). One can see that the theory provides qualitative agreement for the observed
experimental peaks. The derived values of t yield
sxx(B = 0) ≈ 20 mS, again in qualitative agreement with experiment. It would be unreasonable
to expect any better agreement because of the
limited knowledge about the graphene-on-hBN
superlattice potential (20, 29) and the used t
approximation. The observed exponential T dependence of BZ oscillations (detailed in fig. S4)
can also be understood qualitatively as arising
from scattering on acoustic phonons such that the
scattering length ðº1=TÞ becomes shorter than
the characteristic size, aq, of supercells responsible for the q-peak in conductivity (26).
To conclude, graphene superlattices exhibit a
distinct quantum oscillatory phenomenon that
can be understood as repetitive formation of dif-
ferent metallic systems, the BZ minibands. At
simple fractions of f0, charge carriers effectively
experience zero magnetic field, which results in
straight rather than curved (cyclotron) trajecto-
ries. Straighter trajectories lead to weaker Hall
effect and higher conductivity. The smooth back-
ground (varying over many q) is attributed to
trajectories that involve transitions between dif-
ferent minibands and effectively become curved.
The reported oscillations do not require mono-
chromaticity, which allows them to persist up to
exceptionally high T, beyond the existence of
LLs. The extrapolation of the observed T depen-
dences (fig. S4B) suggests that the quantum oscil-
lations may be observable even at 1000 K. Further
theory is required to understand details of tem-
perature, field, and concentration dependences
of BZ oscillations; the origin of the electron-hole
asymmetry of phonon scattering; the behavior
of higher-order fractions; and the effect of inter-
miniband scattering, which is responsible for the
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This work was supported by the European Research Council,
Lloyd’s Register Foundation, the Graphene Flagship, and the
Royal Society. R.K.K. and E.K. acknowledge support from the
Engineering and Physical Sciences Research Council, D.A.B. and
I.V.G. from the Marie Curie program SPINOGRAPH, and S.V.M.
from the Russian Science Foundation and National University of
Science and Technology (MISiS).
Figs. S1 to S10
3 November 2016; accepted 9 June 2017
Fig. 3. BZ oscillations
as recurring Bloch
states in small effective
fields. Solid curves: sxx at
100 K for electron and
hole doping (n/n0 = ±1.6)
(top and bottom panels,
respectively) in a superlattice device with a ≈
13.6 nm. Black dots and
curves: sxx calculated in
the constant-t approximation for different p and
q. Inset image: BZ minibands eðk
Þ inside the first
Brillouin zones indicated
by the gray hexagons
(their size decreases with
increasing q). The minibands were calculated for
a generic graphene-on-hBN superlattice (29)
and correspond to
broadened LLs (for
example, LLs are 2 and 3
for q = 2 and range from 3 to 8 for q = 5). Only those minibands at energies relevant to the doping
level on the experimental curves are shown.
B ( T )
φ/φ0 = 1/5 1/4 1/3 1/2