could be identified by selecting for products in
each of the three sets with added masses corresponding to that of the additional reagent. (45–66)
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the yield of 16aa was observed after an additional 10 hours
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described in (40) produced (E)-(1,2-diphenylvinyl)triethylsilane
as a major product as well as trace quantities of isomers of
43. The reaction of 2a with diethyl (1-phenylallyl) phosphate
using the conditions described in (39) resulted in formation of 16aa in 34% yield as well as formation of two
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by GC-MS). The same conditions afforded 18% of 16aa
and a larger quantity of its isomer (detected by GC-MS)
when cinnamyl diethyl phosphate was used as an allyl
44. Our method is not limited to analysis by GC-MS and
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on products that are less volatile than those in the
This work was supported by the Director, Office of Science,
U.S. Department of Energy, under contract no. DE-AC02- 05CH11231
and the Deutsche Forschungsgemeinschaft (Forschungsstipendium
TR 1239/1-1 to K. T.). We are grateful to S. D. Dreher at Merck
Research Laboratories for advice on conducting experiments in
multiwell formats and to S. Herzon for initial discussions on the use
of MS to identify products in mixtures. All data are provided in
the supplementary materials.
Materials and Methods
Figs. S1 to S22
Tables S1 to S28
Data 1 to 3
13 March 2017; accepted 30 May 2017
oscillations caused by recurring
Bloch states in graphene superlattices
R. Krishna Kumar,1,2,3 X. Chen,2 G. H. Auton,2 A. Mishchenko,1 D. A. Bandurin,1
S. V. Morozov,4,5 Y. Cao,2 E. Khestanova,1 M. Ben Shalom,1 A. V. Kretinin,2,6
K. S. Novoselov,2 L. Eaves,2,7 I. V. Grigorieva,1 L. A. Ponomarenko,3
V. I. Fal’ko,1,2 A. K. Geim1,2*
Cyclotron motion of charge carriers in metals and semiconductors leads to Landau
quantization and magneto-oscillatory behavior in their properties. Cryogenic
temperatures are usually required to observe these oscillations. We show that graphene
superlattices support a different type of quantum oscillation that does not rely on Landau
quantization. The oscillations are extremely robust and persist well above room
temperature in magnetic fields of only a few tesla. We attribute this phenomenon to
repetitive changes in the electronic structure of superlattices such that charge carriers
experience effectively no magnetic field at simple fractions of the flux quantum per
superlattice unit cell. Our work hints at unexplored physics in Hofstadter butterfly systems
at high temperatures.
Oscillations of physical properties of mate- rials with magnetic field are a well known and important phenomenon in condensed matter physics. Despite having a variety of experimental manifestations, there are only
a few basic types of oscillations: those of either
quantum or semiclassical origin (1–5). Semiclas-
sical size effects, such as Gantmakher and Weiss
oscillations, appear owing to commensurability
between the cyclotron orbit and a certain length
in an experimental system (1–4). Quantum magneto-
oscillations are different in that they arise from
periodic changes in the interference along closed
electron trajectories (1–5). Most commonly, quan-
tum oscillations involve cyclotron trajectories. This
leads to Landau quantization and, consequently,
Shubnikov–de Haas (SdH) oscillations in mag-
netoresistance and the associated oscillatory be-
havior in many other properties (1–3). In addition,
quantum oscillations may arise from interference
SCIENCE sciencemag.org 14 JULY 2017 • VOL 357 ISSUE 6347 181
1School of Physics and Astronomy, University of Manchester,
Manchester M13 9PL, UK. 2National Graphene Institute,
University of Manchester, Manchester M13 9PL, UK.
3Department of Physics, University of Lancaster, Lancaster
LA1 4YW, UK. 4Institute of Microelectronics Technology and
High Purity Materials, Russian Academy of Sciences,
Chernogolovka 142432, Russia. 5National University of
Science and Technology (MISiS), Moscow 119049, Russia.
6School of Materials, University of Manchester, Manchester
M13 9PL, UK. 7School of Physics and Astronomy, University
of Nottingham, Nottingham NG7 2RD, UK.
*Corresponding author. Email: firstname.lastname@example.org
(V.I.F.); email@example.com (A.K.G.)