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Zuo Feng, Wenxuan Wang, Yilong You, Yifei Chen, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Chang Liu, Kaihui Liu, Xiaobo Lu

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© 2025 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Rapid infrared imaging of rhombohedral graphene](https://mdr.nims.go.jp/datasets/80f88f2b-0ed6-4630-a1e4-772a95a2fa1d)

## Fulltext

Rapid infrared imaging for rhombohedral graphene  Zuo Feng1,2†, Wenxuan Wang2†, Yilong You1,2†, Yifei Chen3, Kenji Watanabe4, Takashi Taniguchi5, Chang Liu1,2,ǂ, Kaihui Liu1,2,§ and Xiaobo Lu2,6*  1State Key Laboratory for Mesoscopic Physics, Frontiers Science Centre for Nano-optoelectronics, School of Physics, Peking University, 100871, Beijing, China 2International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 3International School of Beijing, Beijing, 101318 4Research Center for Electronic and Optical Materials, National Institute of Material Sciences, 1-1 Namiki, Tsukuba 305-0044, Japan 5Research Center for Materials Nanoarchitectonics, National Institute of Material Sciences, 1-1 Namiki, Tsukuba 305-0044, Japan 6Collaborative Innovation Center of Quantum Matter, Beijing 100871, China  †Z.F., W.W., and Y.L. contributed equally to this work. ǂ Contact author : liu.chang@pku.edu.cn §Contact author : khliu@pku.edu.cn *Contact author : xiaobolu@pku.edu.cn  ABSTRACT. The extrinsic stacking sequence based on intrinsic crystal symmetry in multilayer two-dimensional materials plays a significant role in determining their electronic and optical properties. Compared with Bernal-stacked (ABA) multilayer graphene, rhombohedral (ABC) multilayer graphene hosts stronger electron-electron interaction due to its unique dispersion at low-energy excitations and has been utilized as a unique platform to explore strongly correlated physics. However, discerning the stacking sequence has always been a quite time-consuming process by scanning mapping methods. Here, we report a rapid recognition method for ABC-stacked graphene with high accuracy by infrared imaging based on the distinct optical responses at infrared range. The optical contrast of the image between ABC and ABA stacked graphene is strikingly clear, and the discernibility is comparable to traditional optical Raman microscopy but with higher consistency and throughput. We further demonstrate that the infrared imaging technique can be integrated with dry transfer techniques commonly used in the community. This rapid and convenient infrared imaging technique will significantly improve the sorting efficiency for differently stacked multilayer graphene, thereby accelerating the exploration of the novel emergent correlated phenomena in ABC stacked graphene.  I. INTRODUCTION.  Rhombohedral (ABC) stacked multilayer graphene has been a highly tunable platform to explore strongly correlated and topological phenomena. At low-energy excitation, the dispersion of ABC stacked multilayer graphene follows the relation of E=ApN for single-particle level, where E is the kinetic energy, A is a constant, p is the momentum and N is the number of layers. The unique band dispersion leads to strong correlation between electrons dominating over their kinetic energy at low Fermi energy, particularly for the layer number N≥3 [1-8]. In recent years, a plethora of novel correlated phenomena have come to light in ABC stacked graphene systems, including Mott insulators[5], layer-polarized antiferromagnetic insulators [9-11], superconductivity [12,13], Stoner ferromagnetism [14], orbital multiferroicity[15], integer Chern insulators [16-18] and fractional Chern insulators [19]based on a small number of devices reported by several laboratories. So far, the fabrication of standard ABC graphene devices compatible with transport measurement is still challenging due to two reasons: (i) the energetically metastable ABC stacked sequence will easily relax into Bernal stacked (ABA) sequence with lower energy during the fabrication processes; (ii) identifying ABC stacked region of graphene flakes is time-consuming and involves scanning techniques, such as scanning near-field optical microscopy [20-22], scanning Raman spectroscopy [23-25], scanning Kelvin probe microscopy [26], second harmonic generation [27,28] and angle-resolved photoemission spectroscopy [29]. Developing a rapid detection method for discerning the graphene stacking sequence, both before and after the assembly process of the heterostructures, would significantly facilitate device fabrication.   In this study, we have reported a real-time imaging method for rapidly distinguishing the stacking sequence of multilayer graphene in the infrared light range. The infrared imaging technique provides high-quality images of differently stacked domains (ABC or ABA stacking) with a resolution better than 1 μm, consistent with results obtained from Raman spectroscopy mapping. In multilayer graphene, more complex stacking orders can also be distinguished, such as ABAB, ABCA, and ABCB configurations in tetralayer graphene. Moreover, the stacking sequence can also be distinguished when graphene flakes are encapsulated with hBN flakes, allowing for convenient monitoring stacking sequence changes during assembly process.   II. METHODS  In ABC stacked graphene, the second layer atoms sit in the center of the hexagonal lattice of the first layer atoms, and the third layer atoms continue to shift in the same direction. In contrast, in ABA stacked graphene, the third layer aligns directly above the first layer (Fig. 1a and b). The different stacking sequences result in radically different band structures for multilayer graphene [30,31]. For tetralayer graphene, ABA stacked graphene has two hyperbolic bands near the K point and two split-off bands while ABC graphene has one hyperbolic band and three split-off bands (Fig. 1c and d). Details of the tight-binding calculation are shown in APPENDIX A. The optical conductivity of a material arises from two distinct types of absorption mechanisms: the intraband absorption and interband absorption. Interband transition describes the process in which an electron absorbs a photon and jumps from the valence band to the conduction band directly. Intraband absorption requires the scattering with phonons to satisfy the momentum conservations. In graphene, optical conductivity is primarily attributed to intraband transitions in the far-infrared region. In contrast, for visible and near-infrared wavelengths, optical conductivity is predominantly determined by interband transitions [32-36]. In Fig. 1e and f, we plot the interband transition distribution for photons with energy of 0.78 eV (corresponding to wavelength of 1600 nm) for ABC tetralayer graphene with two different stacking sequences. For ABC stacked graphene, the transition from the V1 band to the C1 band accounts for ~40% of the optical conductivity contribution. The transition from the V2 band to the C2 band accounts for ~25% of the optical conductivity contribution. For ABA stacked graphene, the transition from the V1 band to the C1 band and from the V2 band to the C2 band both contribute approximately 40% of the optical conductivity. Details of the tight-binding calculation are shown in APPENDIX B. Considering all the optical transitions, ABC stacked tetralayer graphene exhibits lower optical conductivity compared to ABA stacked graphene in the infrared range (Fig. 1g). The optical conductivity of ABC stacked graphene decreases with increasing wavelength, while the optical conductivity of ABA stacked graphene increases. Specifically, in the range of 1200 nm to 1600 nm, the conductivity of ABA stacked graphene is consistently higher than that of ABC stacked graphene.   FIG 1. The different optical reflectance in the infrared range of tetralayer ABC and ABA graphene (a) Atomic structures of rhombohedral ABC stacked graphene. (b) Atomic structures of Bernal ABA stacked graphene. (c) Band structure of ABC tetralayer graphene. (d) Band structure of ABA tetralayer graphene. (e) Contributions of different band transitions to optical conductivity of ABC tetralayer graphene at 1600nm. (f) Contributions of different band transitions to optical conductivity of ABA tetralayer graphene at 1600nm. (g) Calculation results of the optical conductivity of ABA and ABC tetralayer graphene.   III. RESULTS  We have further experimentally investigated the infrared properties of ABA and ABC stacked graphene (including trilayer, tetralayer and decalayer graphene) with differential reflectance spectroscopy covering wavelengths from 1200 nm to 1600 nm. In our experiment, the multilayer graphene was exfoliated onto a highly doped silicon substrate with 285 nm oxide layer which is widely used in the community. The stacking sequence was initially confirmed with Raman spectrum. The differential reflectance ∆R/R is defined as ∆R/R=(Rgr-Rsub)/Rsub, where Rgr is the reflectance of graphene on substrate, Rsub is the reflectance of bare substrate. Fig. 2a shows the calculated differential reflectance ∆R/R considering the multi-reflection process for a monolayer graphene. We use a four-layer model composed of air (n0), graphene (ngr), silicon oxide (n2) and silicon substrate (n3). More details are shown in APPENDIX C. Theoretically, Rgr=| (r1+r2e-2iβ1+ r3e-2i(β1+β2)+r1r2r3e-2iβ2)/(1+r1r2e-2iβ1+r1r3e-2i(β1+β2)+r2r3e-2iβ2)|2, where r1=(n0-ngr)/(n0+ngr), r2=(ngr-n2)/(ngr+n2), r3=(n2-n3)/(n2+n3) are the reflection coefficients between the interface, respectively.  β1=2πngrd1/λ, β2=2πn2d2/λ are the phase changes due to the different optical path. The refractive index of graphene (ngr) sensitively depends on the optical conductivity by the relation σ = iω(ngr2− ε0) ,where ω is the frequency of incident light, ε0 is the dielectric constant of vacuum. For multilayer graphene, different values of optical conductivity σ originating from different stacking sequence (Fig. 1g) can lead to distinct different values of ∆R/R as shown in Fig. 2b-d.      FIG.2 The differential reflectance spectroscopy of ABA and ABC graphene. (a) Calculated contrast of monolayer graphene considering multi-reflection. Inset: Schematic of multi-reflection model. (b), (c), (d) The differential reflectance spectroscopy of ABA and ABC trilayer graphene (b), tetralayer graphene (c) and decalayer graphene (d).   Based on the distinct differential reflectance results, we have further designed a real-time infrared imaging system to distinguish different stacking sequences of multilayer graphene. As shown in Fig.3a, the infrared light with effective wavelength l>900nm achieved by a long-wave-pass filter is used for illumination. The reflected signal is collected through an objective with a magnification value of 100 and a numerical aperture value of 0.85. The signal is further recorded with an InGaAs CCD with a detection range from 900 nm to 1700 nm. Representative graphene samples, ranging from trilayer to heptalayer graphene, were characterized with the infrared imaging system. Traditional optical images, shown in Fig. 3b, reveal only uniform optical contrast. In contrast, the infrared imaging system provides sharp differentiation between ABC- and ABA-stacked graphene due to their distinct optical reflectance in the infrared range (Fig.3c). As illustrated in Fig. 3c, the brighter regions correspond to ABC-stacked areas, while the darker regions correspond to ABA-stacked areas. The stacking order were confirmed by the Raman spectra[23]. ABC-stacked graphene shows a more asymmetric feature with a pronounced shoulder compared to ABA-stacked graphene around 2680 cm-1. Full width at half maximum (FWHM) mapping of the 2D Raman mode, shown in Fig. 3d was also used to independently verify regions with different stacking sequences. Notably, the infrared imaging results are highly consistent with Raman mapping. To demonstrate the universality of this imaging technique, different graphene samples with layer numbers ranging from trilayer to heptalayer were imaged. Contrast differences between different stacking sequences vary from 15% to 40%, enabling a straightforward identification (Fig. 3e).    FIG 3. Rapid infrared imaging system. (a) Schematic of infrared imaging system. (b) Optical image of graphene from trilayer to heptalayer. The scale bar is 6 μm. (c) Infrared image of the graphene flakes shown in (b). (d) Raman FWHM 2D mode mappings and Raman spectra of the same graphene flakes shown in (b) and (c). Raman spectra for ABA-stacked (black) and ABC-stacked (red) graphene were extracted from Raman mapping with the positions indicated with little red circles. (e) Contrast difference in the infrared images between ABA and ABC graphene for different layers.   Tetralayer graphene exhibits three types of stacking orders: ABAB, ABCA, and ABCB. The optical image (Fig. 4a) shows uniform contrast across the tetralayer graphene, with no visible differences. However, these stacking configurations display distinct contrasts under infrared imaging (Fig. 4b), allowing clear differentiation between the ABAB, ABCA, and ABCB stacking orders. The ABAB stacking exhibits the lowest contrast and occupies the largest area, while the ABCA stacking displays the highest contrast. The ABCB stacking shows an intermediate contrast. We extracted the gray values for the three stacking orders from the infrared image (Fig. 4d), where a gray value of 1 corresponds to pure black and a gray value of 255 represents pure white. The average gray value for ABAB stacking is 42, the lowest among the three. The average gray value for ABCB stacking is 72, while ABCA stacking has an average gray value of 104. By comparing the grayscale contrasts of different regions in the infrared image, the various stacking configurations can be directly distinguished. The stacking orders were further confirmed through Raman spectroscopy, specifically by examining the G and 2D modes [24]. In ABCA stacked graphene, the G mode shows a blue shift. The G mode positions of ABCB and ABAB stacked tetralayer graphene are similar, but the G mode peak of ABCB stacking is narrower than that of ABAB stacking. Notably, the 2D mode of these three stacking types in tetralayer graphene exhibits distinct characteristics: ABCA stacking shows the most asymmetric feature with a pronounced shoulder, ABAB stacking shows a symmetric 2D band with two peaks of nearly equal intensity, and ABCB stacking presents a profile between those of ABAB and ABCA stacking. Furthermore, our imaging technique is capable of distinguishing even more complex stacking orders in multilayer graphene (APPENDIX D). Four stacking orders can be resolved in heptalayer graphene. The average gray values in the 1, 2, 3, and 4 regions are 21, 41, 56, and 82, respectively. By comparing infrared image with Raman mapping, we observe that the brightest regions exhibit a more asymmetric 2D mode with an enhanced shoulder around 2690 cm-1 while the darkest regions show an enhanced shoulder around 2720 cm-1. However, as the heptalayer graphene contains more than four stacking configurations, we are unable to precisely identify the specific stacking types in the regions with intermediate contrast. We further demonstrate the infrared imaging system as a reliable and versatile technique for distinguishing graphene stacking sequences when encapsulated with hBN flakes (APPENDIX E). The ABC stacking region also shows a bright contrast. Utilizing the rapid infrared imaging technique, pure ABC stacking regions were accurately selected for further device fabrication. With the help of rapid infrared imaging technique, several devices with thickness ranging from three to six layers, were fabricated (APPENDIX F). All the devices exhibit the unique electrical properties of ABC stacked graphene with previous studies[5,13,14,19].   FIG 4. Different stacking orders in tetralayer graphene. (a) Optical images of tetralayer graphene. Scale bar is 5 μm. (b) Infrared image of tetralayer graphene. (c) Raman mapping of the FWHM 2D mode of tetralayer graphene. (d) Grey value of tetralayer graphene with three stacking orders. (e) Raman spectra (G mode) of tetralayer graphene with three stacking orders. f, Raman spectra (2D mode) of tetralayer graphene with three stacking orders.  IV. CONCLUSIONS  In summary, we have introduced a rapid imaging technique for identifying ABC-stacked graphene. The ABC stacked regions exhibit a brighter contrast under infrared light compared to ABA stacked regions on SiO2/Si substrate. For thicker graphene layers, distinct stacking orders exhibit unique contrasts. This technique also enables real-time identification of stacking sequences in graphene encapsulated by hBN throughout the fabrication process. However, we also note that there are still some inherent limitations of this rapid imaging technique. Firstly, as the number of layers increases, the stacking configurations beyond pure ABC and ABA become increasingly intricate, making it challenging to definitively identify each unique stacking sequence. Secondly, this imaging technique is limited by the diffraction limit, resulting in a lower imaging resolution than that of scanning probe techniques.We anticipate that this technique will significantly advance research on stacking order control and deepen exploration of the strongly correlated physics emerging in ABC-stacked multilayer graphene.   ACKNOWLEDGMENTS Acknowledges support from the National Key R&D Program (Grant nos. 2022YFA1403500/02) and the National Natural Science Foundation of China (12141401, 52025023, 11888101, T2188101).  X.L. and K.L. supervised the projects. Z.F. and W.W. fabricated the devices and performed the measurement. Y.Y, Y.C and C.L. performed the theoretical modeling; K.W., T.T contributed materials; Z.F. and X.L. wrote the paper with input from all authors. APPENDIX APPENDIX A: Calculation of Band Structure We use the tight-binding model to calculate the band structure of multilayer graphene. Here we take the basis as |𝐴!⟩, |𝐵!⟩; |𝐴"⟩, |𝐵"⟩; … ; |𝐴#⟩, |𝐵#⟩. In order to capture the main features of our experiment data, we only consider the nearest intra-layer hopping parameter 𝛾$  and inter-layer hopping parameter 𝛾! . Then the Hamiltonian around the K point becomes ℋ = -𝐻$ 𝑉      𝑉% 𝐻$ 𝑉%      𝑉 𝐻$ 𝑉      ⋱ ⋱ ⋱1, (1)  with  𝐻$ = 2 0 𝑣𝑝&𝑣𝑝' 0 6 , 𝑉 = 2 0 0𝛾! 06,  (2) where 𝑝± = 𝑝) ± 𝑖𝑝* , with 𝒑 = −𝑖ħ𝛁, and the velocity of monolayer graphene is 𝑣. According to former researches,  𝑣 is related to the band parameter through the equation 𝑣 = √3𝑎𝛾$/2ħ.  𝐻$ describes the intra-layer hopping, while 𝑉 depicites the inter-layer hopping. Here we use 𝑎 = 0.246𝑛𝑚,  𝛾$ = 3.16𝑒𝑉 and 𝛾! =0.37𝑒𝑉 . Especially in 4-layer graphene as is mentioned in the article, the Hamiltonian for AB stacked graphene is  ℋ =⎝⎛𝐻$ 𝑉    𝑉% 𝐻$ 𝑉%    𝑉 𝐻$ 𝑉    𝑉% 𝐻$⎠⎞, (3) and the Hamiltonian for ABC-stacked graphene is ℋ =⎝⎜⎛𝐻$ 𝑉    𝑉% 𝐻$ 𝑉    𝑉% 𝐻$ 𝑉    𝑉% 𝐻$⎠⎟⎞, (4) APPENDIX B: Calculation of optical conductivity We employ the Kubo formula to calculate the optical conductivity:  𝜎(𝜔)+,- =.!ħ+0∑  1",1!,𝒌3#4.$"𝒌 5&3#4.$!𝒌 5.$"𝒌 &.$!𝒌6.$"𝒌 |𝐯&|.$!𝒌 96.$!𝒌 :𝐯':.$"𝒌 9ħ;'.$"𝒌 &.$!𝒌 '+<, (5) where  𝑒=𝒌  are the eigenenergies of the Hamiltonian in band b at momentum point k and V𝑒1>W  are the corresponding eigenvectors. 𝑛? refers to the Fermi–Dirac distribution.  In our calculation, we only consider the x-direction optical conductivity 𝑣) = −(𝑖/ħ)[𝑥,ℋ] = ∂ℋ/ ∂𝑝). While obviously ∂𝑉/ ∂𝑝) = 0, which means 𝑣) is   𝑣) = -∂𝐻$/ ∂𝑝)        ∂𝐻$/ ∂𝑝)        ∂𝐻$/ ∂𝑝)        ∂𝐻$/ ∂𝑝)1, (6) Here we take 𝜂 = 20𝑚𝑒𝑉 and 𝑇  =  300𝐾 into our calculation.  APPENDIX C: Calculation of multi-refection process To explain the contrast change of graphene on SiO2/Si substrate. We use the Fresnel’s equations. The multireflection model is composed of graphene, SiO2 and Silicon layers. The light is incident from air onto the three layers, the reflected light intensity can be calculated using the following equation:                  R@A = ` B"'B!.(!&)"'B*.(!&()",)!)'B"B!B*.(!&)!!'B"B!.(!&)"'B"B*.(!&()",)!)'B!B*.(!&)!`"                                        (7)                                                         RCD= = a(B!'B*.(!&)!)!'B!B*.(!&)!a"                                                    (8) Δ𝑅𝑅 =R@A − RCD=RCD= Where 𝑟" = 3./&3*3./'3*,𝑟" = 3./&3*3./'3* ,𝑟G = 3!&3*3!'3*  𝛽! = 2𝜋𝑛HB𝑑!/𝜆, 𝛽" = 2𝜋𝑛"𝑑"/𝜆.  Considering the refractive index can be described by the following equation:  σ = iω(𝑛HB" − ε$),  where σ is the optical conductivity of graphene, ω is the frequency of the incident light, ε$ is the dielectric constant of vacuum.  The reflected light intensity depended on optical conductivity can be described by the following equation:  R@A(σ) = nn01(23&4,5101,23&4,51'2 3&4,51(0!2 3&4,51,0!.(!&)"'0!(0*0!,0*.(!&()",)!)'01(23&4,5101,23&4,51∙2 3&4,51(0!2 3&4,6,0!∙0!(0*0!,0*.(!&)!!'01(23&4,5101,23&4,51∙2 3&4,51(0!2 3&4,51,0!.(!&)"'01(23&4,5101,23&4,51∙0!(0*0!,0*.(!&()",)!)'2 3&4,51(0!2 3&4,51,0!∙0!(0*0!,0*.(!&)!nn"      (9)                 RCD= = nn2 3&4,51(0!2 3&4,51,0!'0!(0*0!,0*.(!&)!!'01(23&4,5101,23&4,51∙0!(0*0!,0*.(!&)!nn"           (10)    APPENDIX D: Different stacking orders in heptalayer graphene  FIG 5. Different stacking orders in heptalayer graphene. (a) Optical images of heptalayer graphene. Scale bar is 5 μm. (b) Infrared image of heptalayer graphene. (c) Raman mapping of the FWHM 2D mode of heptalayer graphene. (d) Grey value of heptalayer graphene with four stacking orders. (e) Raman spectra (G mode) of heptalayer graphene with four stacking orders. (f) Raman spectra (2D mode) of heptalayer graphene with four stacking orders.   APPENDIX E: Graphene stacking sequences when encapsulated with hBN flakes  FIG 6.Graphene stacking sequences when encapsulated with hBN flakes. (a) Optical image of tetralayer graphene encapsulated with hBN flakes and the schematic of the sample. Scale bar is 3 μm. (b) Infrared image of tetralayer graphene encapsulated with hBN flakes. (c) Raman mapping of tetralayer graphene encapsulated with hBN flakes.    APPENDIX F: Transport measurement of four different layer ABC stacked graphene  FIG 7. Transport measurement of four different layer ABC stacked graphene. (a) Schematic of the dual-gated device. 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