Auxiliary Material
Supporting evidence for the assignment of different stacking orders of tetralayer
graphene samples
In the main text we indicated that the infrared conductivity spectra measured for 10
different samples could be divided into two well-defined groups. Representative spectra
of each type are shown in fig. 1a. The assignment of different stacking order to these two
classes of spectra is, as we have discussed, supported by several considerations. These
include theoretical predictions of the existence of precisely two low-energy stacking
orders and the excellent match of the experimental spectra for each class with the
calculated optical conductivity spectra for the corresponding crystal structure. To further
support the validity of the assignments, below we discuss the role of doping and other
environmental effects and describe an independent experiment based on Raman
spectroscopy.
Effect of unintentional doping and other environmental effects: Doping is known to
play an important role on the low-frequency optical conductivity for graphene
monolayers. The main effect of doping is to shift the sample Fermi level and reduce the
optical conductivity due to Pauli blocking. In few-layer graphene (FLG) samples,
however, the effect is significantly reduced both due to the increased sample thickness
and the increased densities of states as a result of the modification of the electronic
structure by the interlayer coupling (from linear dispersion in monolayers to parabolic or
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k4 dispersion in tetralayers). We estimated the Fermi level shift due to unintentional
doping in our samples to be < 20 meV. The influence of such a small Fermi energy shift
on the infrared optical conductivity spectrum of the samples will be very slight. Indeed,
the effect of doping has been investigated in bilayers [A1-A4]. Modest doping generally
changes only the detailed line shape, but not the position of the interband transitions.
Although the electronic structure of tetralayers is more complex, we expect similar
effects due to doping in tetralayers. Therefore, doping does not change the peak futures of
the observed optical conductivity spectrum (fig. 1a) and has no influence on the
assignment of the stacking order. We note that tetralayers (relatively thick FLG) have
been purposely chosen in this work because of the reduced doping effect.
To further support the validity of the statement, in figure A1 we show the infrared
conductivity spectra obtained from 4 different tetralayer samples. [The upper panels (A)
& (B) correspond to Bernal stacking and the lower panels (C) & (D) to rhombohedral
stacking.]
These spectra illustrate the reproducibility of the data for different samples,
which rules out the possibility that the two classes of spectra reflect unintentional
perturbations in the samples, such as inhomogeneous doping or strain.
Raman spectroscopy: We examined the tetralayer samples by Raman spectroscopy as
an independent experiment. It has been shown [A5] that the Raman spectrum can be used
as a finger print for monolayer and FLG. Similar to the infrared conductivity spectra, two
distinct groups of Raman response were observed and were correlated with the infrared
absorption spectra. A prominent feature that differentiates the two groups of Raman
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response is the width (and shape) of the 2D mode centered around 2700 cm-1 (a secondorder double resonant Raman mode). A linewidths of ~ 37 cm-1 and 43 cm-1 were
observed, respectively, from samples identified as ABAB and ABCA from the infrared
measurements. The unambiguous correlation between the two independent measurements
(Raman and infrared) provides a strong evidence of the existence of two distinct
polytypes of graphene.
Furthermore, in additional studies of a larger set of FLG samples, we occasionally
found tetralayer graphene samples with observable lateral spatial domains. In these
instances, the infrared conductivity spectra of the different domains matched either the
group 1 or group 2 response shown in fig. 1a. The boundaries between the domains were
observed to be abrupt, limited by the spot size of the probe laser (500 nm). The existence
of such abrupt boundaries can be readily understood as the consequence of crystal
domains of the tetralayer graphene with regions having the two different crystallographic
stacking orders. Such boundaries have, however, no natural explanation in terms of
impurities or externally induced perturbations.
In conclusion, the preparation and characterization of two distinct polytypes of
tetralayer graphene are supported by the following independent observations:
(1) Both rhombohedral and Bernal stacking order have been predicted theoretically and
are known to exist in our source graphite material.
(2) Two independent experimental techniques (infrared and Raman) have identified two
distinct pairs of response of tetralayer graphene samples.
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(3) An excellent match of the experimental optical conductivity spectra with the
calculated spectra for both rhombohedral and Bernal crystal structure has been
obtained.
(4) Domains with abrupt boundaries have been observed that can only be readily
understood as the consequence of crystal domains with regions of two different
crystallographic stacking orders.
(5) Both optical conductivity and Raman spectrum are highly reproducible from sample
to sample, thus excluding unintentional doping and other environmental effects.
Figures:
Figure A1. Optical sheet conductivity spectra, σ(ħω), for four of the ten tetralayer
graphene samples that were investigated in this study. Samples A and B show the spectral
response of group 1 materials and were assigned as Bernal tetralayers; Samples C and D
show the response of group 2 materials and were assigned as rhombohedral tetralayers.
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References:
A1. Z. Li et al., Phys. Rev. Lett. 102, 037403 (2009).
A2. K. F. Mak, C. H. Lui, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 102, 256405 (2009).
A3. Y. Zhang et al., Nature 459, 820 (2009).
A4. A. B. Kuzmenko et al., Phys. Rev. B 80, 165406 (2009).
A5. For instance, see Ferrari et al., Phys. Rev. Lett. 97, 187401 (2006).
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