In tetralayer graphene, three inequivalent layer stackings
should
exist; however, only rhombohedral (ABCA) and Bernal (ABAB) stacking
have so far been observed. The three stacking sequences differ in
their electronic structure, with the elusive third stacking (ABCB)
being unique as it is predicted to exhibit an intrinsic bandgap as
well as locally flat bands around the K points. Here, we use scattering-type
scanning near-field optical microscopy and confocal Raman microscopy
to identify and characterize domains of ABCB stacked tetralayer graphene.
We differentiate between the three stacking sequences by addressing
characteristic interband contributions in the optical conductivity
between 0.28 and 0.56 eV with amplitude and phase-resolved near-field
nanospectroscopy. By normalizing adjacent flakes to each other, we
achieve good agreement between theory and experiment, allowing for
the unambiguous assignment of ABCB domains in tetralayer graphene.
These results establish near-field spectroscopy at the interband transitions
as a semiquantitative tool, enabling the recognition of ABCB domains
in tetralayer graphene flakes and, therefore, providing a basis to
study correlation physics of this exciting phase.
The functional renormalization group (FRG), an established computational method for quantum many-body phenomena, has been subject to a diversification in topical applications, analytic approximations and numerical implementations. Despite significant efforts to accomplish a coherent standard through benchmarks and the reproduction of previous results, no systematic and comprehensive comparison has been provided until now. While this has not prevented the publication of relevant scientific results we argue that established mutual agreement across realizations will strengthen confidence in the method. To this end, we report explicit implementational details and numerical data reproduced thrice independently up to machine accuracy. To substantiate the reproducibility of our calculations, we scrutinize pillar FRG results reported in the literature, and discuss our calculations of these reference systems. We mean to entice other groups to reproduce and establish this set of benchmark FRG results thus propagating the joint effort of the FRG community to engage in a shared knowledge repository as a reference standard for FRG implementations
Graphical abstract
The electronic and structural properties of atomically thin materials can be controllably tuned by assembling them with an interlayer twist. During this process, constituent layers spontaneously rearrange themselves in search of a lowest energy configuration. Such relaxation phenomena can lead to unexpected and novel material properties. Here, we study twisted double trilayer graphene (TDTG) using nano-optical and tunneling spectroscopy tools. We reveal a surprising optical and electronic contrast, as well as a stacking energy imbalance emerging between the moiré domains. We attribute this contrast to an unconventional form of lattice relaxation in which an entire graphene layer spontaneously shifts position during assembly, resulting in domains of ABABAB and BCBACA stacking. We analyze the energetics of this transition and demonstrate that it is the result of a non-local relaxation process, in which an energy gain in one domain of the moiré lattice is paid for by a relaxation that occurs in the other.
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