# Raman Spectroscopy of Twisted Bilayer Graphene

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## Abstract

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## 1. Introduction

## 2. Background Theory

## 3. Raman Signature in Twisted Graphene Layers

## 4. Giant G Band Enhancement by Resonances with the Moiré Pattern Singularities

## 5. Intralayer and Interlayer Electron–Phonon Processes

## 6. Electronic Transitions in TBG

**k**from the Dirac cone of one layer (dark green curve) is scattered by a phonon with momentum $\hslash {\mathbf{q}}_{\mathbf{M}}$ (red arrows) to a state with momentum $\hslash {\mathbf{k}}^{\prime}$ in the Dirac cone of the other layer (light green curve). The lowest possible value of ${E}_{\mathrm{e}}$ occurs when the equi-energies curves of the two layers tangentiate, as shown in Figure 6c. In this situation, the anti-crossing between states of the Dirac cones gives rise to van Hove singularities (vHs) in the density of states (DOS), shown in Figure 1d. For photons with energies below ${E}_{\mathrm{e}}$, the difference $|\mathbf{k}-{\mathbf{k}}^{\prime}|$ is always smaller than $|{\mathbf{q}}_{\mathbf{M}}|$ and the interlayer condition cannot be satisfied. Therefore, the minimum energy for the interlayer scattering process, ${E}_{\mathrm{e}}^{-}$, corresponds to the energy separation ${E}_{\mathrm{vHs}}$ between the vHs in the valence and conduction bands of a TBG.

**k**and ${\mathbf{k}}^{\prime}$ in the extended BZ in Figure 6c are folded to the same point in the reduced BZ, which are represented by the green dots in Figure 6e. They are located at the saddle point in the electronic structure of TBG, near the M point in the reduced BZ that gives rise to a van Hove singularity (vHs). The intralayer process can be also represented in the reduced BZ scheme. Figure 6f shows that the two excited states

**k**and ${\mathbf{k}}^{\prime}$ in the extended BZ in Figure 6d are folded to the blue dots within the reduced BZ. Differently from the case of the interlayer process that occurs near the M point of the reduced BZ, the intralayer process occur for states at general positions within the interior of the reduced BZ.

**k**-points grid and, for each twisted angle $\theta $, the number of joint electronic states that satisfy the restriction $\left|{E}^{\alpha}\left(\mathbf{k}\right)-{E}^{\alpha}\left({\mathbf{k}}^{\prime}\right)\right|\le \epsilon $ is stored, where the superscript $\alpha $ symbolizes the valence or the conduction bands and $\epsilon \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.02$ eV is an arbitrary tolerance. The electronic structures were obtained by folding the SLG calculation based on a fifth neighbors tight-binding approach, following the procedure given in Ref. [11], in which the parameters are fitted to reproduce density functional theory (DFT) calculations with many-body corrections. For the intralayer case, we have that ${\mathbf{k}}^{\prime}=\mathbf{k}+{\mathbf{q}}_{\mathrm{M}}$ because the two electronic states, $\mathbf{k}$ and ${\mathbf{k}}^{\prime}$, are connected by ${\mathbf{q}}_{\mathrm{M}}$ while, for the interlayer process, ${\mathbf{k}}^{\prime}=\mathrm{R}\left(\theta \right)\mathbf{k}+{\mathbf{q}}_{\mathrm{M}}$, being $\mathrm{R}\left(\theta \right)$ the rotation matrix that takes the Dirac cone of one layer into the Dirac cone of the other layer. Considering these two conditions in the model, momentum conservation is always achieved by a given pair of states $\mathbf{k}$ and ${\mathbf{k}}^{\prime}$, as long as the density of

**k**-points considered in the calculation is large enough that the energy difference between the states becomes less than the value of $\epsilon $, as shown in Figure 6c,d, where the small circles in the equi-energy edges represent the tolerance.

## 7. Raman Spectra of TBG Using Infrared and Ultraviolet Photons

## 8. Intralayer Process in Any Graphene Heterostructure

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Real and reciprocal spaces of TBG—adapted from Ref. [38]. (

**a**) Schematics of two rotated monolayer graphene with the blue layer sitting on the top of the golden layer. The layers are twisted by an angle $\theta $ (13.2${}^{\circ}$ in this case), generating a periodic Moiré pattern. Vectors ${\mathbf{T}}_{1}$ and ${\mathbf{T}}_{2}$ define the supercell, while ${\mathit{a}}_{1}$ and ${\mathit{a}}_{2}$ (${\mathit{a}}_{1}^{\prime}$ and ${\mathit{a}}_{2}^{\prime}$) are the untwisted (twisted) graphene lattice vectors. (

**b**) The grey and black hexagons correspond to the Brillouin zones (BZ) of two graphene layers, denoted by A and B, twisted by the angle $\theta $ = 13.2${}^{\circ}$. The small red hexagon represents the reduced BZ and the vectors ${\mathbf{q}}_{1}$, ${\mathbf{q}}_{2}$ and ${\mathbf{q}}_{3}$ correspond to the unit vectors of the Moiré reciprocal lattice; (

**c**) energy versus momentum diagram calculated for $\theta $ = 13.2${}^{\circ}$. The grey (black) curves represent the Dirac cone of layer A (layer B). The vertical blue arrow represents the optical transition for the intralayer process, and the vertical green arrow represents the transition for the interlayer case. The horizontal red arrows represent the wave vector ${\mathbf{q}}_{1}$ of the phonon; (

**d**) density of electronic states (DOS) of the TBG and optical transition between vHs in the valence and conduction bands.

**Figure 2.**Experimental and simulated Raman spectra—adapted from Ref. [14]. (

**a**) Raman spectra obtained in four different regions of a graphene sample that contains regions with single layer and bilayer regions. The spectrum on the top was recorded inside the Cu enclosure, and the other three were obtained in different regions outside of the Cu enclosure, with bilayer graphene (BLG) domains. The peak marked by an asterisk is an artifact that comes from the substrate; (

**b**) Raman spectra taken in the same spot of the sample, showing the extra peak at 1383 cm${}^{-1}$ recorded with four different laser energies 3.40, 3.53, 3.69 and 4.13 eV; (

**c**) simulated Raman spectra using Equation (1) from Ref. [14] with sets of ${\mathbf{q}}_{\mathrm{M}}$ vectors for several possible twist angles between graphene layers. The G band is not shown in the figure.

**Figure 3.**Raman excitation profile of the extra peaks in TBG—adapted from Ref. [19]. (

**a**) Raman spectra in the 1200–1700 cm${}^{-1}$ range, obtained with different laser lines, showing the resonance behavior of the sharp extra peak around 1380 cm${}^{-1}$. The peaks marked with asterisks are broadened since the laser lines of the diode laser were very broad; (

**b**,

**c**) simulated umklapp double-resonance Raman spectra, $I\left(\omega \right)$, for two possible twisting angles and three different laser energies, indicated in the figures; (

**d**) laser energy dependence of the relative intensity of the extra M band with respect to the G-band intensity. The black squares correspond to the experimental results presented in (

**a**) and the open and black circles correspond to the simulated results for TBG with 7.34${}^{\circ}$ and 13.17${}^{\circ}$, respectively.

**Figure 4.**G band intensity and width in TBG—adapted from Ref. [23]. (

**a**) ${I}_{\mathrm{TBG}}$/${I}_{\mathrm{SLG}}$ for more than 100 TBG samples with many different twisting angles between ${0}^{\circ}$ and ${30}^{\circ}$. Blue, green, and red dots are data taken at 488, 532, and 633 nm. Solid lines are the Gaussian fits to the observed angle dependence ${I}_{\mathrm{TBG}}$/${I}_{\mathrm{SLG}}$. The insets show a zoom of ${I}_{\mathrm{TBG}}$/${I}_{\mathrm{SLG}}$ for low ($<{9}^{\circ}$) and high twisting angles ($>{17}^{\circ}$); (

**b**) angular dependence of the ratio between the FWHM of TBG and SLG. The insets show the fitting of the G Band with two and one Lorentzians for $\theta ={1}^{\circ}$ and $\theta ={28}^{\circ}$, respectively.

**Figure 5.**Raman results of two different samples of TBG in the visible range—adapted from Ref. [38]. (

**a**,

**b**) Raman spectra in two samples of TBG with $\theta $ = 6${}^{\circ}$ and 13${}^{\circ}$ recorded with the 2.18 eV and 2.41 eV laser lines, respectively. The vertical scale, ${I}_{\mathrm{TBG}}/{I}_{\mathrm{SLG}}$, corresponds to the ratio of the peak intensities of the Raman spectra in TBG and single-layer graphene (SLG). The peak around 1620 cm${}^{-1}$ in (

**a**) is called L${}_{\mathrm{a}}$ since it comes from the LO phonon branch and is activated by the intralayer electron–phonon scattering process, whereas the peak at 1480 cm${}^{-1}$ in part (

**b**) is called T${}_{\mathrm{e}}$ since it comes from the in-plane TO phonon branch and is activated by the interlayer process; (

**c**) Raman excitation profile (REP) of the G band (black squares) and the L${}_{\mathrm{a}}$ peak (blue circles) of the sample with a low twisting angle ($\theta $ = 6${}^{\circ}$); (

**d**) Raman excitation profile (REP) of the G band (black squares) and of the T${}_{\mathrm{e}}$ peak (green circles) of the sample with intermediate twisting angle ($\theta $ = 13${}^{\circ}$). The T${}_{\mathrm{e}}$ peak intensity was multiplied by ≈ 100 times for comparison with the G band REP and, in both (

**c**,

**d**) panels, the error bars represent the standard deviation.

**Figure 6.**Intralayer and interlayer el-ph scattering processes—adapted from Ref. [38]. (

**a**,

**b**) are representation of the low (−) and high (+) energy for the intralayer ${E}_{\mathrm{a}}$ and interlayer ${E}_{\mathrm{e}}$ processes, respectively; (

**c**) an interlayer el-ph process where a phonon with momentum $\hslash {\mathbf{q}}_{1}$ connects the states $\mathbf{k}$ and ${\mathbf{k}}^{\prime}$. The light and dark green curves correspond to the equi-energies ${E}_{\mathrm{e}}$ around K${}_{\mathrm{A}}$ and K${}_{\mathrm{B}}$; (

**d**) intralayer el-ph process where both states $\mathbf{k}$ and ${\mathbf{k}}^{\prime}$ are in the equi-energies ${E}_{\mathrm{a}}$ of the same layer (light and dark blue curves around K${}_{\mathrm{A}}$ and K${}_{\mathrm{B}}$, respectively); (

**e**) interlayer (green dots) and (

**f**) intralayer (blue dots) electronic states $\mathbf{k}$ and ${\mathbf{k}}^{\prime}$ represented in a reduced BZ scheme.

**Figure 7.**Resonance energies and phonon frequencies for the intralayer and interlayer el-ph processes—adapted from Ref. [38]. (

**a**,

**b**) joint density of states that satisfy the intralayer and interlayer processes, respectively, for some twisting angles $\theta $. (

**c**) The dashed blue, green, pink, and red curves represent the calculated values of ${E}_{\mathrm{a}}^{-}$, ${E}_{\mathrm{e}}^{-}$, ${E}_{\mathrm{a}}^{+}$, and ${E}_{\mathrm{e}}^{+}$ as a function of the twisting angle $\theta $. The blue and black open circles correspond to the resonance energies of the extra peaks observed in the visible Raman spectra in samples with, respectively, small and intermediate angles. The light blue and red open squares correspond to the laser energies where the extra peaks observed in the Raman spectra in samples with $\theta $ in the ranges 3${}^{\circ}$–5${}^{\circ}$ and 6${}^{\circ}$–10${}^{\circ}$ have the maximum intensity. The black open squares correspond to the resonance energies of the extra peaks observed in the UV Raman spectra for samples with large twisting angles (22${}^{\circ}$–28${}^{\circ}$). The black symbols

**×**correspond to the energies of the peaks in the optical absorption spectra of TBG reported in Ref. [26]. (

**d**) The red and black curves represent the dependence of the TO and LO phonon frequencies as a function of $\theta $. The blue and black circles, and the green, red, and black squares correspond to the frequencies of the extra peaks observed in the visible, IR and UV Raman spectra as described in part (

**c**). The black triangles correspond to the results reported by Wang et al. [21]. The error bars in the panels (

**c**,

**d**) represent the standard deviation.

**Figure 8.**Raman results for gr/h-BN—adapted from Ref. [38]. (

**a**–

**c**) Raman spectra in three different samples of graphene on the top of h-BN, with twisting angles $\theta $ = 2${}^{\circ}$, 5${}^{\circ}$, and 6${}^{\circ}$, recorded with the 1.96 eV excitation energy.

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**MDPI and ACS Style**

Moutinho, M.V.O.; Venezuela, P.; Pimenta, M.A.
Raman Spectroscopy of Twisted Bilayer Graphene. *C* **2021**, *7*, 10.
https://doi.org/10.3390/c7010010

**AMA Style**

Moutinho MVO, Venezuela P, Pimenta MA.
Raman Spectroscopy of Twisted Bilayer Graphene. *C*. 2021; 7(1):10.
https://doi.org/10.3390/c7010010

**Chicago/Turabian Style**

Moutinho, Marcus V. O., Pedro Venezuela, and Marcos A. Pimenta.
2021. "Raman Spectroscopy of Twisted Bilayer Graphene" *C* 7, no. 1: 10.
https://doi.org/10.3390/c7010010