Evolution of Raman G and G'(2D) Modes in Folded Graphene Layers

Bernal- and non-Bernal-stacked graphene layers have been systematically studied by Raman imaging and spectroscopy. Two dominant Raman modes, G and G' (or 2D) of folded graphene layers exhibit three types of spectral features when interlayer lattice mismatches, defined by a rotational angle varies. Among these folded graphene layers, the most interesting one is the folded graphene layers that present an extremely strong G mode enhanced by a twist-induced Van Hove singularity. The evolution of Raman G and G' modes of such folded graphene layers are probed by changing the excitation photon energies. For the first time, doublet splitting of the G' mode in folded double-layer (1 + 1) and of the G mode in folded tetra-layer (2 + 2) graphene are clearly observed and discussed. The G' mode splitting in folded double-layer graphene is attributed to the coexistence of inner and outer scattering processes and the trigonal warping effect as well as further downwards bending of the inner dispersion branch at visible excitation energy. While the two peaks of the G mode in folded tetra-layer graphene are assigned to Raman-active mode (E2g) and lattice mismatch activated infrared-active mode (E1u), which is further verified by the temperature-dependent Raman measurements. Our study provides a summary and thorough understanding of Raman spectra of Bernal- and non-Bernal-stacked graphene layers and further demonstrates the versatility of Raman spectroscopy for exploiting electronic band structures of graphene layers.Comment: 29 pages, 10 figure

Stacking sequence determines Raman intensities of observed interlayer shear modes in 2D layered materials - A general bond polarizability model

2D layered materials have recently attracted tremendous interest due to their fascinating properties and potential applications. The interlayer interactions are much weaker than the intralayer bonds, allowing the as-synthesized materials to exhibit different stacking sequences (e.g. ABAB, ABCABC), leading to different physical properties. Here, we show that regardless of the space group of the 2D material, the Raman frequencies of the interlayer shear modes observed under the typical configuration blue shift for AB stacked materials, and red shift for ABC stacked materials, as the number of layers increases. Our predictions are made using an intuitive bond polarizability model which shows that stacking sequence plays a key role in determining which interlayer shear modes lead to the largest change in polarizability (Raman intensity); the modes with the largest Raman intensity determining the frequency trends. We present direct evidence for these conclusions by studying the Raman modes in few layer graphene, MoS2, MoSe2, WSe2 and Bi2Se3, using both first principles calculations and Raman spectroscopy. This study sheds light on the influence of stacking sequence on the Raman intensities of intrinsic interlayer modes in 2D layered materials in general, and leads to a practical way of identifying the stacking sequence in these materials.Comment: 30 pages, 8 figure

Magnetic Oscillation of Optical Phonon in ABA- and ABC-Stacked Trilayer Graphene

We present a comparative measurement of the G-peak oscillations of phonon frequency, Raman intensity and linewidth in the Magneto-Raman scattering of optical E2g phonons in mechanically exfoliated ABA- and ABC-stacked trilayer graphene (TLG). Whereas in ABA-stacked TLG, we observe magnetophonon oscillations consistent with single-bilayer chiral band doublets, the features are flat for ABC-stacked TLG up to magnetic fields of 9 T. This suppression can be attributed to the enhancement of band chirality that compactifies the spectrum of Landau levels and modifies the magnetophonon resonance properties. The drastically different coupling behaviour between the electronic excitations and the E2g phonons in ABA- and ABC-stacked TLG reflects their different electronic band structures and the electronic Landau level transitions and thus can be another way to determine the stacking orders and to probe the stacking-order-dependent electronic structures. In addition, the sensitivity of the magneto-Raman scattering to the particular stacking order in few layers graphene highlights the important role of interlayer coupling in modifying the optical response properties in van der Waals layered materials.Comment: 25 pages, 6 figure

On the Stochastic Gradient Descent and Inverse Variance-flatness Relation in Artificial Neural Networks

Stochastic gradient descent (SGD), a widely used algorithm in deep-learning neural networks has attracted continuing studies for the theoretical principles behind its success. A recent work uncovered a generic inverse variance-flatness (IVF) relation between the variance of neural weights and the landscape flatness of loss function near solutions under SGD [Feng & Tu, PNAS 118,0027 (2021)]. To investigate this seemly violation of statistical principle, we deploy a stochastic decomposition to analyze the dynamical properties of SGD. The method constructs the true "energy" function which can be used by Boltzmann distribution. The new energy differs from the usual cost function and explains the IVF relation under SGD. We further verify the scaling relation identified in Feng's work. Our approach may bridge the gap between the classical statistical mechanics and the emerging discipline of artificial intelligence, with potential for better algorithm to the latter

Topology, Vorticity and Limit Cycle in a Stabilized Kuramoto-Sivashinsky Equation

A noisy stabilized Kuramoto-Sivashinsky equation is analyzed by stochastic decomposition. For values of control parameter for which periodic stationary patterns exist, the dynamics can be decomposed into diffusive and transverse parts which act on a stochastic potential. The relative positions of stationary states in the stochastic global potential landscape can be obtained from the topology spanned by the low-lying eigenmodes which inter-connect them. Numerical simulations confirm the predicted landscape. The transverse component also predicts a universal class of vortex like circulations around fixed points. These drive nonlinear drifting and limit cycle motion of the underlying periodic structure in certain regions of parameter space. Our findings might be relevant in studies of other nonlinear systems such as deep learning neural networks.Comment: Main body: 16 pages, 3 figures; Supplementary: 14 pages, 6 figure

Stacking Dependent Optical Conductivity of Bilayer Graphene

The optical conductivities of graphene layers are strongly dependent on their stacking orders. Our first-principle calculations show that while the optical conductivities of single layer graphene (SLG) and bilayer graphene (BLG) with Bernal stacking are almost frequency independent in the visible region, the optical conductivity of twisted bilayer graphene (TBG) is frequency dependent, giving rise to additional absorption features due to the band folding effect. Experimentally, we obtain from contrast spectra the optical conductivity profiles of BLG with different stacking geometries. Some TBG samples show additional features in their conductivity spectra in full agreement with our calculation results, while a few samples give universal conductivity values similar to that of SLG. We propose those variations of optical conductivity spectra of TBG samples originate from the difference between the commensurate and incommensurate stackings. Our results reveal that the optical conductivity measurements of graphene layers indeed provide an efficient way to select graphene films with desirable electronic and optical properties, which would great help the future application of those large scale misoriented graphene films in photonic devices.Comment: 20 pages, 5 figures, accepted by ACS Nan