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We study the electronic structure of alternating-twist tetralayer graphene, especially near its magic angle $\theta = 1.75^\circ$, for different AA, AB, and SP sliding geometries at their middle interface that divides two twisted bilayer graphenes. This sliding dependence is shown for the bandwidths, band gaps, and $K$-valley Chern numbers of the lowest-energy valence and conduction bands as a function of twist angle and interlayer potential difference. Our analysis reveals that the AA sliding is most favorable for narrow bands and gaps, and the AB sliding is most prone to developing finite valley Chern numbers. We further analyze the linear longitudinal optical absorptions as a function of photon energy and the absorption map in the moir\'{e} Brillouin zone for specific transition energies. A self-consistent Hartree calculation reveals that the AA system's electronic structure is the most sensitive to variations in carrier density., Comment: 15 pages, 11 figures

We theoretically study the energy and optical absorption spectra of alternating twist multilayer graphene (ATMG) under a perpendicular electric field. We obtain analytically the low-energy effective Hamiltonian of ATMG up to pentalayer in the presence of the interlayer bias by means of first-order degenerate-state perturbation theory, and present general rules for constructing the effective Hamiltonian for an arbitrary number of layers. Our analytical results agree to an excellent degree of accuracy with the numerical calculations for twist angles $\theta \gtrsim 2.2^{\circ}$ that are larger than the typical range of magic angles. We also calculate the optical conductivity of ATMG and determine its characteristic optical spectrum, which is tunable by the interlayer bias. When the interlayer potential difference is applied between consecutive layers of ATMG, the Dirac cones at the two moir\'{e} Brillouin zone corners $\bar{K}$ and $\bar{K}'$ acquire different Fermi velocities, generally smaller than that of monolayer graphene, and the cones split proportionally in energy resulting in a step-like feature in the optical conductivity., Comment: 11 pages, 11 figures, 2 tables

Mechanical exfoliation of graphene and its identification by optical inspection is one of the milestones in condensed matter physics that sparked the field of 2D materials. Finding regions of interest from the entire sample space and identification of layer number is a routine task potentially amenable to automatization. We propose supervised pixel-wise classification methods showing a high performance even with a small number of training image datasets that require short computational time without GPU. We introduce four different tree-based machine learning algorithms -- decision tree, random forest, extreme gradient boost, and light gradient boosting machine. We train them with five optical microscopy images of graphene, and evaluate their performances with multiple metrics and indices. We also discuss combinatorial machine learning models between the three single classifiers and assess their performances in identification and reliability. The code developed in this paper is open to the public and will be released at github.com/gjung-group/Graphene_segmentation., Comment: 12 pages, 6 figures

We investigate the electronic structure of alternating-twist triple Bernal-stacked bilayer graphene (t3BG) as a function of interlayer coupling $\omega$, twist angle $\theta$, interlayer potential difference $\Delta$, and top-bottom bilayers sliding vector $\boldsymbol{\tau}$ for three possible configurations AB/AB/AB, AB/BA/AB, and AB/AB/BA. The parabolic low-energy band dispersions in a Bernal-stacked bilayer and gap-opening through a finite interlayer potential difference $\Delta$ allows the flattening of bands in t3BG down to $\sim 20$~meV for twist angles $\theta \lesssim 2^{\circ}$ regardless of the stacking types. The easier isolation of the flat bands and associated reduction of Coulomb screening thanks to the intrinsic gaps of bilayer graphene for finite $\Delta$ facilitate the formation of correlation-driven gaps when it is compared to the metallic phases of twisted trilayer graphene under electric fields. We obtain the stacking dependent Coulomb energy versus bandwidth $U/W \gtrsim 1$ ratios in the $\theta$ and $\Delta$ parameter space. We also present the expected $K$-valley Chern numbers for the lowest-energy nearly flat bands., Comment: 15 pages, 10 figures

Trilayer graphene with a twisted middle layer has recently emerged as a new platform exhibiting correlated phases and superconductivity near its magic angle. A detailed characterization of its electronic structure in the parameter space of twist angle $\theta$, interlayer potential difference $\Delta$, and top-bottom layer stacking $\tau$ reveals that flat bands with large Coulomb energy vs bandwidth $U/W > 1$ are expected within a range of $\pm 0.2^{\circ}$ near $\theta \simeq1.5^{\circ}$ and $\theta \simeq1.2^{\circ}$ for $\tau_{\rm AA}$ top-bottom layer stacking, between a wider $1^{\circ} \sim 1.7^{\circ}$ range for $\tau_{\rm AB}$ stacking, whose bands often have finite valley Chern numbers thanks to the opening of primary and secondary band gaps in the presence of a finite $\Delta$, and below $\theta \lesssim 0.6^{\circ}$ for all $\tau$ considered. The largest $U/W$ ratios are expected at the magic angle $\sim 1.5^{\circ}$ when $|\Delta| \sim 0$~meV for AA, and slightly below near $\sim 1.4^{\circ}$ for finite $|\Delta| \sim 25$~meV for AB stackings, and near $\theta \sim 0.4^{\circ}$ for both stackings. When ${\tau}$ is the saddle point stacking vector between AB and BA we observe pronounced anisotropic local density of states (LDOS) strip patterns with broken triangular rotational symmetry. We present optical conductivity calculations that reflect the changes in the electronic structure introduced by the stacking and gate tunable system parameters., Comment: 13 pages 8 figures

Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) on nearly aligned hexagonal boron nitride (BN) substrate builds on top of the electronic structure resulting from combined G/G and G/BN double moire interfaces. Here we show that tBG/BN commensurate double moire patterns can be classified into two types, each favoring the narrowing of either the conduction or valence bands on average, and obtain the evolution of the bands as a function of the interlayer sliding vectors and electric fields. Finite valley Chern numbers $\pm 1$ are found in a wide range of parameter space when the moire bands are isolated through gaps, while the local density of states associated to the flat bands are weakly affected by the BN substrate invariably concentrating around the AA-stacked regions of tBG. We illustrate the impact of the BN substrate for a particularly pronounced electron-hole asymmetric band structure by calculating the optical conductivities of twisted bilayer graphene near the magic angle as a function of carrier density. The band structures corresponding to other $N$-multiple commensurate moire period ratios indicate it is possible to achieve narrow width $W \lesssim 30$ meV isolated folded band bundles for tBG angles $\theta \lesssim 1^{\circ}$., Comment: 17 pages, 9 figures