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Xingzhou Chen, Zheng Sun, Min Zhang, Ming Li, Zhigao Hu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), David Snoke, Zhe-Yu Shi, Jian Wu

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[Broadband enhancement of absorption by two-dimensional atomic crystals modeled as non-Hermitian photonic scattering](https://mdr.nims.go.jp/datasets/c0e0b375-5443-4f6a-9e28-23e7d1081378)

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Broadband enhancement of absorption by two-dimensional atomic crystals modeled as non-Hermitian photonic scatteringAppl. Phys. Lett. 122, 041105 (2023); https://doi.org/10.1063/5.0134789 122, 041105© 2023 Author(s).Broadband enhancement of absorption bytwo-dimensional atomic crystals modeled asnon-Hermitian photonic scatteringCite as: Appl. Phys. Lett. 122, 041105 (2023); https://doi.org/10.1063/5.0134789Submitted: 14 November 2022 • Accepted: 12 January 2023 • Published Online: 24 January 2023Xingzhou Chen,  Zheng Sun, Min Zhang, et al.ARTICLES YOU MAY BE INTERESTED INNoncontact evaluation of the interface potential in VO2/Si heterojunctions across metal–insulator phase transitionApplied Physics Letters 122, 041601 (2023); https://doi.org/10.1063/5.0136610Tuned band offset in homogenous TMDs via asymmetric ferroelectric semiconductor gatestoward simultaneous rectification and memoryApplied Physics Letters 122, 042103 (2023); https://doi.org/10.1063/5.0130587Graphene-assisted low temperature growth of nearly single-crystalline GaN thin films viaplasma-enhanced atomic layer depositionApplied Physics Letters 122, 041602 (2023); https://doi.org/10.1063/5.0128372https://images.scitation.org/redirect.spark?MID=176720&plid=1992743&setID=378288&channelID=0&CID=730412&banID=520904761&PID=0&textadID=0&tc=1&type=tclick&mt=1&hc=7436a5fec7817cf0d9cfb2168f6c07a5e1904203&location=https://doi.org/10.1063/5.0134789https://doi.org/10.1063/5.0134789https://aip.scitation.org/author/Chen%2C+Xingzhouhttps://orcid.org/0000-0002-5209-2563https://aip.scitation.org/author/Sun%2C+Zhenghttps://aip.scitation.org/author/Zhang%2C+Minhttps://doi.org/10.1063/5.0134789https://aip.scitation.org/action/showCitFormats?type=show&doi=10.1063/5.0134789http://crossmark.crossref.org/dialog/?doi=10.1063%2F5.0134789&domain=aip.scitation.org&date_stamp=2023-01-24https://aip.scitation.org/doi/10.1063/5.0136610https://aip.scitation.org/doi/10.1063/5.0136610https://doi.org/10.1063/5.0136610https://aip.scitation.org/doi/10.1063/5.0130587https://aip.scitation.org/doi/10.1063/5.0130587https://doi.org/10.1063/5.0130587https://aip.scitation.org/doi/10.1063/5.0128372https://aip.scitation.org/doi/10.1063/5.0128372https://doi.org/10.1063/5.0128372Broadband enhancement of absorptionby two-dimensional atomic crystals modeledas non-Hermitian photonic scatteringCite as: Appl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789Submitted: 14 November 2022 . Accepted: 12 January 2023 .Published Online: 24 January 2023Xingzhou Chen,1 Zheng Sun,1,2,a) Min Zhang,1 Ming Li,3 Zhigao Hu,2,3,4 Kenji Watanabe,5Takashi Taniguchi,6 David Snoke,7 Zhe-Yu Shi,1,a) and Jian Wu1,2,8,a)AFFILIATIONS1State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200241, China2NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China3Technical Center for Multifunctional Magneto-Optical Spectroscopy (Shanghai), Engineering Research Center of Nanophotonicsand Advanced Instrument (Ministry of Education), Department of Materials, School of Physics and Electronic Science, East ChinaNormal University, Shanghai 200241, China4Shanghai Institute of Intelligent Electronics and Systems, Fudan University, Shanghai 200433, China5Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan6International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044,Japan7Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA8Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, Chinaa)Authors to whom correspondence should be addressed: zsun@lps.ecnu.edu.cn; zyshi@lps.ecnu.edu.cn;and jwu@phy.ecnu.edu.cnABSTRACTWe report the design and fabrication of a vertical structure using a distributed Bragg reflector and dielectric material layer to achieveoptimized optical absorption enhancement for a stack of monolayer WS2 and MoS2, namely, a tenfold increase in absorption over a 100 nmspectral range. Our research indicates that we can approach over 50% absorption by finely tuning the thickness of the spacer layer. Ourtheoretical model shows that the dependence of the absorption coefficient on the spacer thickness can be understood as a solution of a non-Hermitian Schr€odinger equation. These results advance the development of broadband optical devices, including solar energy conversionand sensitive optical sensors, by using two-dimensional excitonic materials.VC 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0134789Modifying light absorption while diminishing its reflection is oneof the most fundamental processes in light–matter interactions and iscritical for designing modern optoelectronic devices, for example, sig-nal conversion and energy harvesting. Controlled absorption of lighthas been used in optical sensors,1 photodetectors,2 photodiodes,3 solarcells,4–6 photocatalysis,7,8 and luminous energy storage.9,10 Usingmonolayers aids the general requirement for on-chip and integrateddevices. One method is to use the destructive interference of light,11,12which often involves complicated designs and complex structures.13,14In contrast, in this paper, we consider a dissipative optical system thatis able to localize a photonic state to enhance the intrinsic opticalabsorption of two-dimensional materials. By customizing the opticalresonances to match the absorption energy bands in the materials, theoptical absorption of the devices can thus be strongly modified. Thisinvolves taking into account the real and imaginary terms of the com-plex susceptibility. As shown below, this theoretical model maps to thecase of a Schr€odinger equation with a non-Hermitian Hamiltonian.Two-dimensional transition metal dichalcogenides (2D TMDCs)are promising candidates as building blocks for next-generationoptoelectronic devices not only for their thinness but also for theirultra-large exciton binding energy15 and generally excellent opticalproperties.16,17 The layers can be stacked together by interlayer vanAppl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789 122, 041105-1VC Author(s) 2023Applied Physics Letters ARTICLE scitation.org/journal/aplhttps://doi.org/10.1063/5.0134789https://doi.org/10.1063/5.0134789https://www.scitation.org/action/showCitFormats?type=show&doi=10.1063/5.0134789http://crossmark.crossref.org/dialog/?doi=10.1063/5.0134789&domain=pdf&date_stamp=2023-01-24https://orcid.org/0000-0002-5209-2563https://orcid.org/0000-0003-0575-2191https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-5203-9335https://orcid.org/0000-0002-1318-2291mailto:zsun@lps.ecnu.edu.cnmailto:zyshi@lps.ecnu.edu.cnmailto:jwu@phy.ecnu.edu.cnhttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://doi.org/10.1063/5.0134789https://scitation.org/journal/aplder Waals force in many ways, like building with “Lego” blocks.18,19When two TMDC monolayers are used to make a bilayer, exotic opti-cal phenomena can appear that depend on the relative angle betweenthe crystal axes of the two layers.20,21Although these monolayers have a relatively strong light absorp-tion in a visible light range, as high as 5%–10%,22 this absorption effi-ciency is not sufficient for practical application in optoelectronicdevices. Therefore, it is a design goal to extract the greatest possibleenergy using these atomic layer materials.23,24 In recent years, studieshave been performed on enhancing the absorption rate of TMDCs,especially using metal layers to enhance absorption.25–28 Although thepreparation of structures with metal layers is convenient, the reflectiv-ity of the thin metal layer is not high, and the reflectivity cannot beflexibly adjusted to different wavelengths, resulting in a poor absorp-tion enhancement effect. Some other groups reported methods thatexploit grating and photonic crystal structures to increase opticalabsorptions.29,30 However, those structures are often more complexdesigned and need more sophisticated fabrications.Here, we propose to approach the absorption of the monolayersusing a distributed Bragg reflector (DBR). The absorption can betuned by the variation of the thickness of the dielectric material layer.In addition to offering a strongly increased absorption in a monolayer,we find that our devices offer an attractive route toward the develop-ment of broadband light absorption enhancement by taking advantageof the easy stacking methods to form van der Waals heterostructures.This strategy presented in this work offers high universality and, inparticular, can be required in solar energy conversion and sensitiveoptical sensors applications.Our system structure is designed based on the theory of non-Hermitian optics.31 Mathematically, the reflection of normallyincident light by layered dissipative media is equivalent to the one-dimensional scattering problem of a non-Hermitian Schr€odingerequation with complex potential Vc. Assuming the input light is prop-agating along the x-direction, we have Vc xð Þ ¼ k20½1� n xð Þ� withk0 ¼ 2p=k the wavenumber in vacuum and n xð Þ is the local (com-plex) refractive index. Varying the thickness of different layers, such asthe spacer and the hexagonal boron nitride (hBN), is equivalent tochanging the width of the potential wells in Vc(x), which can be usedto adjust the optical reflectivity of the system. Via fine-tuning of thethickness of specific layers, the optical absorption of the light in thisstructure can be enhanced well beyond 50%, which is known to be theupper bound of single-pass light by a thin absorbing layer,32,33 andeven achieves perfect absorption if a non-Hermitian bound state istuned to be resonant to the incoming light.34 Furthermore, since boththe complex potential Vc and the effective energy of the input light arewavelength-dependent, the optical reflectivity for different wave-lengths can be changed separately, which aids the design of deviceswith broadband absorption.The multi-layer structure was obtained by alternately growingtwo dielectric materials for multiple periods (here, the samples haveeight periods) to get high light reflectivity in a particular wavelengthband called the stop band. By changing the thickness of the grownmaterials, we can modify the stop band. First, a DBR with a stop bandwidth of �100nm and a center wavelength of �630nm was grown.Then, monolayers of WS2 (exA� 610nm) and MoS2 (exA� 660nm)were used as the absorbing materials since the two materials possesssimilar band structures, with optical transitions within this stop band.Finally, between the WS2 and DBR, and between the MoS2 and theWS2, a spacer layer of silicon dioxide (SiO2) and a thin flake of hBNwere inserted, respectively.Figure 1(a) illustrates the schematic of the full structure designon the order of DBR/spacer/WS2/hBN/MoS2 from the bottom up.Figure 1(b) shows the transfer-matrix simulation results of the reflec-tion of the full structure. The red line is the reflectance of the bareDBR. It indicates that the stop band is from about 560 to 690nm, Theblue line is the calculated reflectance of the full structure. We can seetwo deep absorption valleys where the monolayer absorption occurs,at 610 and 660nm, within the stop band of DBR. When thermalbroadening of these lines at room temperature is taken into account,they will merge, resulting in broadband absorption.To tune the thickness of the spacer, we transferred the monolayerWS2 onto the top of the DBRs with different spacer thicknesses. TheDBR consisted of alternating layers of SiO2 and silicon nitride and wasgrown on a silicon substrate by the Plasma Enhanced Chemical VaporDeposition (PECVD). With the same system, the different spacerlayers were deposited on each DBR, with bare silicon exposedFIG. 1. (a) Schematic diagram of the designed structure with the order of DBR/Spacer/WS2/hBN/MoS2. (b) The simulated reflectance by transfer matrix method. The red lineis the reflectance of a bare eight periods DBR. The blue dotted line is the reflectivity of the entire structure with a 100 nm SiO2 spacer layer and 10 nm hBN flake accordingly.Applied Physics Letters ARTICLE scitation.org/journal/aplAppl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789 122, 041105-2VC Author(s) 2023https://scitation.org/journal/aplalongside as a control. The thickness of the spacer layer was deter-mined by a Filmmetrics system. Isolated monolayers of WS2 with thetypical size of 10� 10lm2, as shown in Fig. 2(a), were mechanicallyexfoliated from bulk crystals. The whole samples were fabricated bythe standard dry transfer method to the surface of the DBR substrateswith the different spacer layers of SiO2 coating. An Argon laser(532 nm) with a spot size of 1lm at normal incidence, focusedthrough a 100� microscope objective, was employed as the pumpsource. The PL spectra were collected by the same microscope objec-tive and directed toward a charged couple device (CCD) equippedspectrometer to help prescreen the monolayers. The differential reflec-tance spectra were carried out in the same configurations but with amirror in the middle of the light path that could be flipped to blockthe laser while letting the white light go through. The white light sizewas cut down to 2lm by an aperture and thus can precisely illuminatethe area of interest. Six sets of samples were made with the thicknessof the spacer layers ranging from 60 to 160nm. A maximum of 50.6%absorbance was obtained in the sample with the spacer layer thicknessof 100nm. To quantify whether or not this absorbance is significantlysensitive to the spacer thickness, finite-element modeling usingCOMSOL Multiphysics 6.0 was performed. We set the centerwavelength of the stop band at 630nm and varied the thickness of thespacer layers. A plane wave was employed in the model. The simula-tion results and the experimental data were calculated using the fol-lowing equation:DRR¼ Isub � IsampIsub: (1)Here, Isub and Isamp stand for the reflected light intensity when theincident light sheds on the substrate and the sample, respectively. Weplot the differential reflectivity as a function of the spacer thickness forwavelength �610nm (monolayer WS2 exA) in Fig. 2(d). Both showthe same behavior: the absorbance increases when the spacer layer isthin, reaches a maximum when the thickness is about 100nm, andthen decreases at greater thickness. A control sample was made of amonolayer WS2 on a piece of bare silicon. As shown in Fig. 2(b), theabsorbance of the WS2 monolayer on the Si substrate is only 6.6%,which agrees with that of TMDCmonolayers given in the other litera-ture.22 We have achieved enhancement of the absorbance by a factorof 8 using the DBR. (The absorbance spectra of the WS2 monolayerfor the different spacer layer thicknesses are shown in supplementaryFig. S2.)FIG. 2. (a) An optical microscope image of a WS2 monolayer on a DBR with a SiO2 spacer layer. (b) The differential reflectance spectrum of the monolayer WS2 transferredon top of a DBR with a 100 nm SiO2 spacer (in blue) compared to the same type of monolayer on a piece of bare silicon (in red). (c) Left: schematic of the sample, with the dis-tances shown on the vertical axis of the plot on the right. Right: calculated electric field magnitude (assuming the jEj2 of incident field is 1) as a function of distance. An anti-node of the field occurs at the location of the monolayer. (d) Spacer layer thickness-dependent differential reflectivity for wavelength �610 nm. The blue dots are theexperimental results for six different thicknesses. The red dots are from our simulation. Dotted lines connect the data points.Applied Physics Letters ARTICLE scitation.org/journal/aplAppl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789 122, 041105-3VC Author(s) 2023https://scitation.org/journal/aplFigure 2(c) shows the simulated distribution of the electric fieldalong in sample, which clearly shows that the maximum E-field inten-sity is at the position of the WS2 monolayer. The light passing throughthe WS2 monolayer is affected by the reflection of the DBR and thespacer layer. Theoretically, this structure can lead to photonic boundstates located near the position of the WS2 monolayer.As a next step, a thin flake hBN was selected and used as anotherspacer layer, and then a monolayer MoS2 was deposited. The hBN hereplays two roles, both as a protector for the monolayer WS2 beneath andas a dielectric medium to compensate for the spacer thickness mis-match. An atomic force microscope was used to determine the thick-nesses of the hBN, giving the step profile in the inset of Fig. 3(a).According to our calculations, about 10nm hBN was the best for realiz-ing the broadband absorption enhancement. In Fig. 3(a), the mono-layers of WS2 and MoS2 are highlighted with the blue and the reddotted lines, respectively, while the hBN is shown by the region belowthe black dotted line. The effective area (green highlighted) where thethree materials overlap is about 5� 10lm. Thus, the interesting site cancompletely cover the white light spot with a diameter�2lm.Figure 3(b) exhibits the differential reflectivity spectra of the fullstack on the DBR compared to monolayer WS2 and monolayer MoS2control samples on a bare Si substrate. As expected, the overlayed areapossesses two dominant peaks, and they are the characteristic absorp-tion peaks of the WS2 and the MoS2. Compared to the monolayers onthe Si substrate, the absorptances increased from 6.6% to 32.2% and2.2% to 26.9%, respectively. However, as seen by comparison to Fig.2(b), the enhanced absorption of a single monolayer of WS2 wasreduced from 50.6% down to 32%, presumably because the monolayeris vulnerable to being affected by its environment. As shown in Fig.3(b), the absorption peak of the WS2 monolayer was at 616 nm, whichis about 6 nm red-shifted compared to the WS2 monolayer withoutcapping; this can be attributed to the stress-induced deformation ofthe lattice, which can shift bandgap of the material due to the deforma-tion potential effect.35We, thus, see that the absorption peaks of the two materials canbe superimposed, with at least an absorption rate of 10% throughout abroadband spectral region.To further illustrate the physical mechanism of the opticalabsorbing process in our system, especially the dependence of theabsorption coefficient on the system structure, we have developed atheoretical model that relates our optical system to the non-Hermitianphysics of a one-dimensional Schr€odinger equation. As we know,non-Hermitian quantum mechanical systems have recently attractedgreat attention in a variety of subfields in physics including condensedmatter, atomic gases, and optics as well.36 The non-Hermitian opticsdescribed by our theoretical approach, thus, not only helps in under-standing the absorption process in the systems from a different per-spective but also provides a novel approach for designing andsimulating various non-Hermitian quantum physics in two-dimensional monolayer systems.For generality, we consider systems with one-dimensional spa-tial-dependent (absolute) permittivity � xð Þ and assume a constant per-meability l � l0; where l0 is the permeability of vacuum, as allmaterials involved in this work are non-magnetic. It is then straight-forward to show that the electric field E satisfies@2E@t2� 1l�r2E �r r��1 � Eð Þð Þ ¼ 0: (2)For an incoming plane wave propagating along the x-direction, wemay write its electric field as E ¼ E xð Þêze�ixt with x being its fre-quency and êz being the unit vector along the z-direction. Equation(2) can then be written in the form of a time-independent Schr€odingerequation� @2@x2E þ Vc xð ÞE ¼ k20E; (3)where k0 ¼ x=c ¼ xffiffiffiffiffiffiffiffiffil0�0pis the wavenumber in vacuum and theeffective complex potential Vc is related to the local refractive indexthrough Vc xð Þ ¼ l0 �0 � � xð Þð Þx2 ¼ k20 1� n xð Þð Þ.The problem of calculating the absorbance in the experimentsetup can then be mapped to a non-Hermitian scattering problem asillustrated in Fig. 4(a) (see the supplementary material for moredetails). One can see that the effective potential is complex because ofFIG. 3. (a) Image of the structure with the layers labeled in order of deposition. Inset is a clear step profile showing the thickness of 10 nm of the selected hBN measured byAFM. (b) Differential reflectance spectra for the entire structure and the control samples.Applied Physics Letters ARTICLE scitation.org/journal/aplAppl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789 122, 041105-4VC Author(s) 2023https://www.scitation.org/doi/suppl/10.1063/5.0134789https://scitation.org/journal/aplthe complex refractive index of monolayer WS2. For further illustra-tion, we theoretically calculate the absorbance of light with 610nmwavelength in the WS2-spacer-DBR structure and plot it as a functionof the spacer thickness. The result clearly shows that the absorbancehas a peak when the spacer thickness is around 100nm, confirmingthe experimental measurements. It is worth mentioning that the theo-retical curve presented in Fig. 4(b) does not exactly match the mea-sured results in Fig. 2(d). The main reason that caused this deviation isthat we have simplified the non-Hermitian scattering process byassuming that the DBR can be treated as a perfect mirror (an infinitehard wall in the language of the quantum scattering process). While inthe experiment, the real DBR is never perfect, which necessarily causessome difference between the measured results and the simplifiedmodel. As a matter of fact, it would be quite easy to get rid of this devi-ation by choosing a “best fitted” reflection coefficient r in our theoreti-cal calculation. The comparison of the experimental data andthe theoretical calculation with various reflection coefficients can befound in the supplementary material. Nevertheless, as shown in Fig.4(b), the simplified theoretical model does not affect the optimalspacer thickness but only slightly changes the absorbance peak value.Furthermore, the calculation of the corresponding non-Hermitianscattering process through the DBR/spacer/WS2/hBN/MoS2 also helpsverify the optimal experimental setup of the hBN thickness for achiev-ing broadband absorbance.The theoretical approach we used here is equivalent to the tradi-tional method used in optics, such as the transfer matrix method.However, mapping the problem to a non-Hermitian scattering prob-lem allows us to inspect the optical problem from a new perspective.For example, suppose we assume that all the mediums are non-dissipative. In that case, i.e., if the refractive index n xð Þ is realeverywhere, the Schr€odinger equation (3) becomes Hermitian, and theoptical theorem from Hermitian scattering theory38 naturally impliesthat the absorbance must vanish. For non-Hermitian scatterings, itcan be shown that a system with complex potential Vc can supportbound states of energies with positive real parts.39 Thus, through aproper design of the optical structure, one can fine-tune the energy ofa non-Hermitian bound state to be precisely in resonance with theeffective scattering energy [i.e., k20 as shown in Eq. (3)]. The“wavefunction” of the bound state then has the asymptotic formE xð Þ ’ eik0x for x! �1. Optically, this corresponds to a pureincoming plane wave without any reflection, and the system reachescritical coupling at this point.34In conclusion, we have explored the effect of the thickness ofeach spacer layer on the absorption of WS2 and MoS2 on a high-quality reflector. We have demonstrated that tuning the thickness ofthe spacer layer to form photonic bound states at the WS2 monolayerand MoS2 monolayer positions can induce strong absorption in two-dimensional materials, which may be useful for various applications,such as photon energy harvesting and sensitive sensing. Theoretically,we have shown that the optical system is equivalent to a non-Hermitian Schr€odinger equation. By calculating the one-dimensionalscattering problem of this Schr€odinger equation, we obtain the absorp-tion coefficient as a function of spacer thickness. The results agreewith our experimental measurements. Our research shows thatTMDC materials can have high optical absorption efficiency, atomi-cally thin thickness, and scalability in applications, making thempromising in studying modern optoelectronic devices.See the supplementary material for the complete study of thespacer layer thickness-dependent differential reflectivity spectra andthe theoretical details on non-Hermitian scattering.The work was supported by the National Key Research andDevelopment Program of China (Nos. 2021YFA1200803,2019YFB2203403, and 2018YFA0306303); National Natural ScienceFoundation of China (Nos. 12174111, 11834004, 12004115,62090013, 62074058, 61974043, and 61974044); Shanghai PujiangProgram (Nos. 21PJ1403000 and 20PJ1403600); Shanghai SailingProgram (20YF1411600); and Science and Technology Commissionof Shanghai Municipality (Nos. 21JC1402100 and 19511120100).D.S. acknowledges support from U.S. Army Research Office underMURI (W911NF-17-1-0312). K.W. and T.T. acknowledge supportFIG. 4. (a) A schematic plot of the effective potential Vc xð Þ for the non-Hermitian scattering problem for the WS2-spacer-DBR structure. The potential has a nonvanishing imag-inary part at the monolayer WS2 because it is a dissipative material. (b) The calculated absorbance for 610 nm light as a function of the spacer thickness for the WS2-spacer-DBR structure. The refractive index of WS2 is obtained from Ref. 37.Applied Physics Letters ARTICLE scitation.org/journal/aplAppl. Phys. Lett. 122, 041105 (2023); doi: 10.1063/5.0134789 122, 041105-5VC Author(s) 2023https://www.scitation.org/doi/suppl/10.1063/5.0134789https://www.scitation.org/doi/suppl/10.1063/5.0134789https://scitation.org/journal/aplfrom JSPS KAKENHI (Nos. 19H05790, 20H00354, and 21H05233)and A3 Foresight by JSPS. The authors acknowledge discussionswith Professor Vinod Menon at CUNY.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.Author ContributionsXingzhou Chen: Data curation (equal); Formal analysis (equal);Writing – original draft (equal). Jian Wu: Resources (supporting);Writing – review & editing (supporting). Zheng Sun:Conceptualization (lead); Project administration (lead); Supervision(lead); Writing – review & editing (lead). Min Zhang: Data curation(supporting); Formal analysis (supporting). Ming Li: Methodology(supporting). Zhigao Hu: Resources (supporting). Kenji Watanabe:Resources (supporting). Takashi Taniguchi: Resources (supporting).DavidW. Snoke:Writing – review & editing (equal). Zheyu Shi:Datacuration (equal); Formal analysis (equal); Writing – review & editing(equal).DATA AVAILABILITYThe data that support the findings of this study are availablewithin the article and its supplementary material.REFERENCES1N. Liu, M. Mesch, T. Weiss, M. 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