Plasmon Reflections by Topological Electronic Boundaries in Bilayer

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Plasmon reflections by topological electronic boundaries in bilayer graphene Bor-Yuan Jiang, Guangxin Ni, Zachariah Addison, Jing K. Shi, Xiaomeng Liu, ShuYang Zhao, Philip Kim, Eugene J. Mele, Dimitri N. Basov, and Michael M. Fogler Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03816 • Publication Date (Web): 02 Oct 2017 Downloaded from http://pubs.acs.org on October 2, 2017

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Plasmon reflections by topological electronic boundaries in bilayer graphene Bor-Yuan Jiang,∗,†,k Guang-Xin Ni,†,k Zachariah Addison,‡ Jing K. Shi,¶ Xiaomeng Liu,¶ Shu Yang Frank Zhao,¶ Philip Kim,¶ Eugene J. Mele,‡ Dimitri N. Basov,†,§ and Michael M. Fogler∗,† †Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA ‡Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA ¶Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA §Department of Physics, Columbia University, New York, New York 10027, USA kThese authors contributed equally to this work E-mail: [email protected]; [email protected] Phone: +1 (858) 534-0852; +1 (858) 534-5978

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Abstract Domain walls separating regions of AB and BA interlayer stacking in bilayer graphene have attracted attention as novel examples of structural solitons, topological electronic boundaries, and nanoscale plasmonic scatterers. We show that strong coupling of domain walls to surface plasmons observed in infrared nanoimaging experiments is due to topological chiral modes confined to the walls. The optical transitions among these chiral modes and the band continua enhance the local conductivity, which leads to plasmon reflection by the domain walls. The imaging reveals two kinds of plasmonic standing-wave interference patterns, which we attribute to shear and tensile domain walls. We compute the electronic structure of both wall varieties and show that the tensile wall contains additional confined bands which produce a structure-specific contrast of the local conductivity, in agreement with the experiment. The coupling between the confined modes and the surface plasmon scattering unveiled in this work is expected to be common to other topological electronic boundaries found in van der Waals materials. This coupling provides a qualitatively new pathway toward controlling plasmons in nanostructures.

Keywords Nanoplasmonics, plasmon reflection, topological electronic boundaries, structural solitons, bilayer graphene, scanning near-field microscopy Topological band theory has become a valuable tool for interpreting ground-state properties and low frequency transport in electronic materials with nontrivial momentum-space geometry. 1 In this paper we demonstrate that topological states also play a significant role in the response of such materials at finite frequencies. Our objects of study are domain walls in bilayer graphene that separate regions of local AB and BA stacking order. Because of their prevalence in exfoliated samples 2–5 and their intriguing electronic 6–8 and optical properties, 9 these domain walls have been under intense investigation. 6,10–12 From the point of view of 2


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the crystal structure, the stacking wall is a line of partial dislocations. The magnitude of √ its Burgers vector |~b| = a/ 3 is equal to the bond length rather than the lattice constant a = 0.246 nm. 5 The domain wall can have an arbitrary angle α with respect to ~b, the two limiting varieties being the tensile wall [α = 0, Figure 1a] and the shear wall [α = π/2, Figure 1d]. The width l = 6–10 nm of a domain wall is determined by a competition between the stacking-dependent interlayer interaction and intralayer elastic strain. 3,13 The electronic structure of the domain walls and their topological properties are best elucidated using a long-wavelength effective theory 6,14 valid for states near the Brillouin zone corners. In this approach the two inequivalent corners are assigned different valley quantum numbers that are practically decoupled since l ≫ a. Far from the wall, the limit of perfect AB or BA stacking is approached. There, neglecting small “trigonal warping” effects, the dispersion of each valley consists of parabolic conduction and valence bands touching at a point. An electric field applied normal to the bilayer can be used to separate the bands by a tunable energy gap, Figure 1a and d. However, at the domain wall gapless electron states must remain since the valley-Chern number of the filled valence band (of a given valley) differs by two in the AB and BA stacked regions. Accordingly, the wall must host a minimum of two copropagating one-dimensional (1D) conducting channels per spin per valley (PSPV). 11,15,16 In particular, for the tensile wall which runs along a zigzag direction (henceforth, the xdirection), the two valleys project to widely separated conserved momenta parallel to the wall. These 1D channels are protected if one neglects intervalley scattering, and so they can support unidirectional valley currents along the wall. While the above arguments have been used to interpret dc transport experiments, 8 topological considerations alone do not delineate the response at infrared frequencies ω where optical transitions between many different electron states may contribute. Recent scatteringtype scanning near-field optical microscopy (s-SNOM) experiments have demonstrated that in this frequency range the AB-BA walls act as reflectors for surface plasmons. 9 Here we explicitly show that these reflections probe a structure-sensitive local conductivity of the wall.

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ventional ones do not cross the band gap, see below.) Incorporating the position-dependent σij (r) into a long-wavelength theory for the plasmon dispersion, we develop a quantitative description of the observed standing waveform and make predictions for future experiments. The presence of the extra 1D channels at the domain wall can be understood as arising from a smooth variation of the stacking order across the wall. If we envision this variation is very gradual (“adiabatic”), then each point is characterized by a local band structure of bilayer graphene with one layer shifted uniformly relative to the other layer by a certain ~δ = (0, δ). 6 In this limit the local band structure at the middle of the wall is approximated by SP (saddle point) stacking, while far away the band structure reverts to that of AB or BA, as shown in Figure 1a and d for the tensile and shear walls, respectively. These Figures can be understood as projections from the 2D momentum space to the 1D momentum axis parallel to the wall: kk = kx for the tensile wall and kk = ky for the shear wall. The 2D SP dispersion has two Dirac points shifted in both momentum and energy by the amount determined by the interlayer hopping amplitude γ1 = 0.40 eV, 17 the bias voltage Vi , and the valley index. For a tensile wall, these Dirac points project to different kx and remain distinct. For a shear wall, they occur at the same ky and overlap with other states. In both cases the dispersion at the middle of the wall is gapless. This is a crucial property which implies there is a range of energies E in the SP stacking that fall into the gap of the AB stacking. The states at such energies must be confined to the wall. These states can be thought of as electronic “waveguide modes.” For an infinitely wide waveguide there are infinitely many such states, and the band structure of the entire system is essentially an overlap of the SP and the AB bands. For the realistic situation of finite-width domain walls, the number of 1D bound states is finite because the quantum confinement produces a finite number of dispersing 1D branches. Figure 1b and c illustrate this for a positive and negative interlayer bias Vi , respectively. Among all the confined branches, only one pair PSPV crosses the gap. These are the topological chiral modes inferred from the valley-Chern number mismatch mentioned earlier. 11,16 The propagation direction of these states is determined by the sign

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of Vi and the valley index. The remaining confined branches are clearly inherited from the SP band structure, in agreement with our qualitative picture. These waveguide branches have mostly fixed direction regardless of the sign and magnitude of the bias. Actually, their existence is also a topological effect to some extent since it is facilitated by the gaplessness of the SP dispersion, which is ensured by the persistence of the Dirac points. The latter is a topological property protected by the spatial inversion and the time-reversal symmetries. 6 The electronic structure of the shear wall, Figure 1e and f, can be understood in a similar manner. The shear wall is narrower than the tensile wall, so that the quantum confinement effect is stronger. As a result, the dispersions of the waveguide modes are pushed so close to the boundaries of the conduction and valence bands that they are hardly visible in Figure 1e and f. For all practical purposes, the waveguide modes are absent and only the gap-crossing doublets of the chiral states survive. These differences in the band structures for the two types of walls result in different local optical responses, which we observed by imaging plasmonic reflections using s-SNOM. The principles of this experimental technique have been presented in prior works 18–21 and review articles. 22–24 In brief, the s-SNOM utilizes a sharp metallized tip of an atomic force microscope as a nano-antenna that couples incident infrared light to the surface plasmons in the bilayer graphene sheet (Figure 2e, inset). These plasmons propagate radially away from the tip and are subsequently reflected by inhomogeneities, in this case, the AB-BA walls. The intensity of the total electric field underneath the tip has a correction determined by the interference of the launched and reflected plasmon waves. The amplitude of this interference term oscillates as a function of the distance between the tip and the reflector, with the period equal to one-half of the plasmon wavelength. The detection of these interference fringes is made by measuring the light backscattered by the tip into the far field and isolating the genuine near-field signal s. For further details of the experimental procedures, see the end of this Letter.

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The observed plasmonic scattering and interference patterns are related to the spatial dependence of the optical conductivity σ⊥ of the bilayer graphene sheet. This parameter determines the momentum qp of the plasmons according to the formula qp =

iκω 2πσ⊥

where κ

is the effective permittivity of the environment. 24 Depending on the type of wall and their respective local electronic structure, the perturbation of local qp by the wall will vary, leading to distinct plasmonic signatures for each type of wall. Additionally, as the gate voltage Vg is tuned, the carrier density and chemical potential of the bilayer graphene sheet are varied. By tuning Vg and studying the corresponding s-SNOM signal, one effectively probes different parts of the electronic spectrum. For our quantitative analysis, we calculate the local optical conductivity of the domain wall following these steps. We start with a model 6 of 4 × 4 Dirac-type Hamiltonian H = H(kk , k⊥ ) of bilayer graphene with a uniform arbitrary stacking δ. We modify it by allowing smooth spatial variations of the stacking parameter δ = δ(x⊥ ). Note that x⊥ = y for the tensile wall and x⊥ = x for the shear wall where x is the zigzag direction of the graphene lattice, see Figure 1. The momentum kk remains a good quantum number. The momentum k⊥ perpendicular to the wall is replaced by the operator −i∂/∂x⊥ . The resulting eigenproblem is solved numerically on a 1D grid of x⊥ to obtain electronic band dispersions such as those shown in Figure 1. Next, these energy dispersions and the wave functions are used to evaluate the Kubo formula for the nonlocal conductivity Σ(x⊥ , x′⊥ ). We further deR fine an effective local conductivity σ⊥ (x⊥ ) ≡ Σ(x⊥ , x′⊥ )dx′⊥ , which is appropriate when the

total electric field on the sheet varies slowly on the scale of the wall width. 25 The presence

of the bound states is manifest in a strongly enhanced σ⊥ at the tensile domain wall. The profile of σ⊥ (x⊥ ) obtained from this nonlocal model can be compared with the results for an adiabatic model in which the Hamiltonian for the uniform stacking is used instead at every point x⊥ . As shown in Figure 3a, the real part of σ⊥ is much larger than that obtained from the adiabatic approximation. The difference occurs precisely because of the lack of the bound states in the latter. On the other hand, for shear walls, which host no bound states

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(a)

(b)

(c)

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Figure 3: (a) Local optical conductivity σ⊥ for the tensile wall at interlayer bias Vi = 0, calculated using the lattice approach and the adiabatic approach. Conventional bound states residing at the wall lead to the prominent peak in the real part of the conductivity, an effect missing in the adiabatic approach. (b) Comparison between the tensile and shear walls. At zero bias the shear wall hosts no confined modes and the conductivity is relatively flat. (c) Conductivity of the shear wall at Vi = ±0.2 V. The bias opens the band gap, induces the topological chiral modes, and gives rise to the peak in Re σ⊥ . Reversing the bias has no effect as it is equivalent to interchanging the valleys. (d) Conductivity of the tensile wall at Vi = ±0.1 V. Reversal of the sign of the bias alters the dispersion of the conventional bound states, changing the conductivities. All quantities were calculated at µ = 0.1 V, η = 0.1 and ω = 890 cm−1 . when Vi = 0, the conductivity is relatively flat at the wall, see Figure 3b. The large contrast of the local optical conductivity for the two types of walls leads directly to the distinct plasmonic profiles they produce. To simulate the measured s-SNOM profile, we converted σ⊥ into the plasmon momentum qp and used it as an input to the electromagnetic solver developed in previous work. 18,19,25–28 The chemical potential µ, interlayer bias Vi and the phenomenological damping rate η are tuned until the output matches the experimental profile. An example of a fit for the tensile wall is shown in Figure 4a-d and for the shear wall in Figure 4e-h, where the magnitude and phase of the simulated s9


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(a)

(c)

(b)

(d)

(e)

(g)

(f)

(h)

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Figure 4: (a), (b) Simulated s-SNOM amplitude (¯ s3 ) and phase (φ) profiles for the tensile wall at Vg = −80 V. Experimental data is shown in gray in (a). (c), (d) The plasmon wavelength (λp ) and damping (γ) profile used for the fit. Peaks in these profiles arise from optical transitions involving the bound states and determine the plasmonic response around the domain walls. (e)-(h) Similar plots for the shear wall at Vg = −110 V. SNOM signal are shown. Here the plasmon momentum qp is parametrized by the plasmon wavelength λp and plasmonic damping rate γ, qp =

2π (1 λp

+ iγ). The interaction of the walls

with plasmons can be characterized by the plasmon reflection coefficient r. According to the first-order perturbation theory, 19 this reflection coefficient is proportional to the amount of excess conductivity at the wall with respect to the background value σ∞ far from the wall: R r = −iq∞ dx⊥ (σ⊥ (x⊥ ) − σ∞ )/σ∞ . The tensile wall has a larger |r|; therefore, it reflects plasmons more strongly than the shear wall, see Figure 2e. The phase θ of the reflection

coefficient r = |r|eiθ , (Figure 2f) dictates the position of the extrema of the interference pattern ∼ cos(2qp |x⊥ | + θ + θt ), where θt ≈ −π/2 is the tip-dependent phase shift. 19 At the tensile wall (x⊥ = 0, θ ≈ −π/2) the pattern has a minimum; at the shear wall (θ ≈ −π) it has a small local maximum, as observed in the experiments. Note that we are treating the interlayer bias Vi as a fitting parameter because in the experiment the value of Vi is determined primarily by (uncontrolled) dopants in the SiO2 substrate. 17 The calculated s-SNOM profiles are strongly Vi -dependent because the sign and value of Vi can alter the bound state dispersions (Figure 1b and c) and the conductivity σ⊥ dramatically (Figure 3c and d). This 10


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suggests that determination of Vi from the fits to these profiles is a robust procedure. More generally, the remarkable fidelity of our fits across a range of gate voltages indicates that s-SNOM can be a reliable method for probing the local band structure of electronic materials. Note that prior to the discovery of AB-BA walls, topological confinement of electrons at the “electric-field walls” in bilayer graphene was studied. 10,11,16,29 These are the regions where Vi and therefore, the valley Chern number changes sign. Hence, there is a minimum of N = 2 such chiral states PSPV, same as for the AB-BA walls. Based on our study, we expect that N can be system-dependent and larger than two. Recently 1D plasmons confined to “electric-field walls” were also studied theoretically. 30 Below we discuss similar collective modes for the AB-BA walls (Figure 2e, inset). The propagating 1D plasmons may exist if their Landau damping by the surrounding bulk is weak enough. This condition can be met if i) the chemical potential µ resides in the band gap so that the confined single-particle modes are the only low-energy degrees of freedom and ii) the frequency ω is small enough so that the optical transitions to bulk bands do not occur. Devices where these requirements can be achieved would likely need a thin top gate in addition to the back gate to control µ and also substrates/gate dielectrics other than SiO2 to reduce disorder and unintentional doping. The 1D plasmon mode may then be amenable for study by the s-SNOM nanoimaging, similar to plasmons in carbon nanotubes. 31 The dispersion of the domain-wall plasmon is determined by the divergences 32 of the loss function −Im ε−1 1D kk , ω . Here ε1D is the effective dielectric function ε1D

8e2 kk , ω = κ − ln h

A kk l

X N j=1

kk2 |vj |

ω 2 − kk2 vj2

,

(1)

κ is the dielectric constant of the environment, N is the total number PSPV of the 1D electron states, vj is the velocity of jth state, and A ∼ 1 is a numerical coefficient. The dispersion curves calculated using Eq. (1) (cf. Supplementary Material for details) are approximately linear in the experimentally accessible range of momenta kk , see Figure 5a. The

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(b)

Figure 5: (a) Dispersion of 1D plasmons at µ = 0 for i. the shear wall at Vi = 0.1 V (green, N = 2), ii. the tensile wall at Vi = 0.1 V (orange, N = 2) and iii. the tensile wall at Vi = −0.1 V (blue, N = 6). The difference in the √ plasmon velocity vp = ∂ω/∂kk is due to the different number of plasmonic channels N , vp ∝ N . (b) (Left panel) Plasmon wavelength for the three cases in (a) at ω = 100 cm−1 . For case iii the plasmon wavelength changes sharply at particular µ’s due to a change in N . (Right panel) Corresponding band structure for case iii, where N varies between 4 (cyan) and 6 (magenta) at different chemical potentials µ. slope of each curve, i.e., the plasmon group velocity scales as

√

N , similar to the case of

carbon nanotubes. 31 For shear walls N = 2 (Figure 1e and f), and so the plasmon wavelength λp = 2π/kk as a function of µ at given fixed ω and Vi is approximately constant, see the red and green curves in Figure 5b. For tensile walls, we can have N = 2, 4, or 6, depending on the chemical potential and interlayer bias (Figure 5b, right panel). Sharp changes in λp should therefore occur at some µ where N changes in steps of two, see the blue curve in Figure 5b. These properties of plasmons at AB-BA boundaries may be interesting for exploring fundamental physics of interacting 1D systems (so-called Luttinger liquids 32 ) or for implementing ultrasmall plasmonic circuits. Note that theoretical calculations for “electric-field walls” predict macroscopically large electron mean-free paths of chiral electronic 29 (or plasmonic 30 ) states if the intervalley scattering is dominated by long-range disorder. This prediction should equally apply to the AB-BA boundaries but it is yet to be verified experimentally. In summary, the local electrodynamic impedance of bilayer graphene is strongly sensitive to the atomic-scale stacking order, leading to plasmonic reflection that depends on the

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strength and orientation of the strain at the domain wall. Topological arguments give an important initial insight into the physics of this reflection. However, further microscopic analysis of the electronic structure proves to be necessary to account quantitatively for the plasmonic interference fringes observed in near-field nanoimaging. We anticipate many useful applications of these theoretical concepts and experimental approaches to other types of electronic boundaries found in a wide family of van der Waals heterostructures. Experimental procedures are as follows. Graphene flakes were exfoliated onto a 285 nmthick SiO2 layer on top of a highly doped Si substrate. Regions of bilayer graphene were identified by their contrast under optical microscopy. Metal contacts were defined on graphene using shadow masks. The infrared nanoimaging experiments were performed at ambient conditions using an s-SNOM based on an atomic force microscope operating in the tapping mode. Infrared light (λ = 11.2 µm) was focused on the tip of the microscope. A pseudoheterodyne interferometric detection was used to extract the scattering amplitude s and phase φ of the near-field signal. To remove the background, the signal was demodulated at the third harmonic of the tapping frequency 270 kHz.

Supporting Information Available Calculation of the band structure and optical conductivity of the domain walls. Details of the fitting procedure for the measured s-SNOM profiles. Derivation of the effective 1D dielectric function and plasmonic dispersion. This material is available free of charge via the Internet at http://pubs.acs.org/. AUTHOR INFORMATION Corresponding Author ∗

E-mail: [email protected]; [email protected]

Author Contributions

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D.N.B. and E.J.M. conceived the project. B.-Y.J., Z.A., E.J.M. and M.M.F. developed the theoretical model. G.N. performed the experiments and analysed the data. J.K.S., X.L., F.Z. and P.K. prepared the samples. All authors contributed to the manuscript. Notes The authors declare that they have no competing financial interests.

Acknowledgement This work is supported by the DOE under Grant DE-SC0012592, by the ONR under Grant N00014-15-1-2671 and N00014-15-1-2761, by the NSF under Grant ECCS-1640173, and by the SRC. D.N.B. and P.K. are investigators in Quantum Materials funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No. GBMF4533 and GBMF4543. Work by Z.A. and E.J.M. was supported by the DOE under grant DE FG02 84ER45118.

References (1) Bansil, A.; Lin, H.; Das, T. Rev. Mod. Phys. 2016, 88, 021004. (2) Brown, L.; Hovden, R.; Huang, P.; Wojcik, M.; Muller, D. A.; Park, J. Nano Lett. 2012, 12, 1609–1615. (3) Alden, J. S.; Tsen, A. W.; Huang, P. Y.; Hovden, R.; Brown, L.; Park, J.; Muller, D. A.; McEuen, P. L. Proc. Natl Acad. Sci. USA 2013, 110, 11256–11260. (4) Lin, J.; Fang, W.; Zhou, W.; Lupini, A. R.; Idrobo, J. C.; Kong, J.; Pennycook, S. J.; Pantelides, S. T. Nano Lett. 2013, 13, 3262–3268. (5) Butz, B.; Dolle, C.; Niekiel, F.; Weber, K.; Waldmann, D.; Weber, H. B.; Meyer, B.; Spiecker, E. Nature 2013, 505, 533–537.

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(6) Koshino, M. Phys. Rev. B 2013, 88, 115409. (7) San-Jose, P.; Gorbachev, R.; Geim, A.; Novoselov, K.; Guinea, F. Nano Lett. 2014, 14, 2052–2057. (8) Ju, L.; Shi, Z.; Nair, N.; Lv, Y.; Jin, C.; Velasco, J.; Ojeda-Aristizabal, C.; Bechtel, H. A.; Martin, M. C.; Zettl, A.; Analytis, J.; Wang, F. Nature 2015, 520, 650–655. (9) Jiang, L.; Shi, Z.; Zeng, B.; Wang, S.; Kang, J.-H.; Joshi, T.; Jin, C.; Ju, L.; Kim, J.; Lyu, T.; Shen, Y.-R.; Crommie, M.; Gao, H.-J.; Wang, F. Nature Mater. 2016, 15, 840–844. (10) Yin, L.-J.; Jiang, H.; Qiao, J.-B.; He, L. Nature Commun. 2016, 7, 11760. (11) Zhang, F.; MacDonald, A. H.; Mele, E. J. Proc. Natl Acad. Sci. USA 2013, 110, 10546– 10551. (12) Vaezi, A.; Liang, Y.; Ngai, D. H.; Yang, L.; Kim, E.-A. Phys. Rev. X 2013, 3, 021018. (13) Lebedeva, I. V.; Lebedev, A. V.; Popov, A. M.; Knizhnik, A. A. Phys. Rev. B 2016, 93, 235414. (14) McCann, E.; Koshino, M. Rep. Prog. Phys. 2013, 76, 056503. (15) Li, J.; Wang, K.; McFaul, K. J.; Zern, Z.; Ren, Y.; Watanabe, K.; Taniguchi, T.; Qiao, Z.; Zhu, J. Nature Nanotech. 2016, 11, 1060–1065. (16) Martin, I.; Blanter, Y. M.; Morpurgo, A. F. Phys. Rev. Lett. 2008, 100, 036804. (17) Zhang, L. M.; Li, Z. Q.; Basov, D. N.; Fogler, M. M.; Hao, Z.; Martin, M. C. Phys. Rev. B 2008, 78, 235408. (18) Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; Neto, A. H. C.; Lau, C. N.; Keilmann, F.; Basov, D. N. Nature 2012, 487, 82–85. 15


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(19) Fei, Z.; Rodin, A. S.; Gannett, W.; Dai, S.; Regan, W.; Wagner, M.; Liu, M. K.; McLeod, A. S.; Dominguez, G.; Thiemens, M.; Neto, A. H. C.; Keilmann, F.; Zettl, A.; Hillenbrand, R.; Fogler, M. M.; Basov, D. N. Nature Nanotech. 2013, 8, 821–825. (20) McLeod, A. S.; Kelly, P.; Goldflam, M. D.; Gainsforth, Z.; Westphal, A. J.; Dominguez, G.; Thiemens, M. H.; Fogler, M. M.; Basov, D. N. Phys. Rev. B 2014, 90, 085136. (21) Jiang, B.-Y.; Zhang, L.; Castro Neto, A.; Basov, D.; Fogler, M. J. Appl. Phys. 2016, 119, 054305. (22) Keilmann, F.; Hillenbrand, R. Phil. Trans. Roy. Soc. London, Ser. A 2004, 362, 787– 805. (23) Atkin, J. M.; Berweger, S.; Jones, A. C.; Raschke, M. B. Adv. Phys. 2012, 61, 745–842. (24) Basov, D. N.; Fogler, M. M.; Lanzara, A.; Wang, F.; Zhang, Y. Rev. Mod. Phys. 2014, 86, 959–994. (25) Jiang, B.-Y.; Ni, G.; Pan, C.; Fei, Z.; Cheng, B.; Lau, C.; Bockrath, M.; Basov, D.; Fogler, M. Phys. Rev. Lett. 2016, 117, 086801. ¨ (26) Ni, G. X.; Wang, H.; Wu, J. S.; Fei, Z.; Goldflam, M. D.; Keilmann, F.; Ozyilmaz, B.; Neto, A. H. C.; Xie, X. M.; Fogler, M. M.; Basov, D. N. Nature Mater. 2015, 14, 1217–1222. (27) Ni, G. X.; Wang, L.; Goldflam, M. D.; Wagner, M.; Fei, Z.; McLeod, A. S.; Liu, M. K.; ¨ Keilmann, F.; Ozyilmaz, B.; Neto, A. H. C.; Hone, J.; Fogler, M. M.; Basov, D. N. Nature Photon. 2016, 10, 244–247. (28) Goldflam, M. D.; Ni, G.-X.; Post, K. W.; Fei, Z.; Yeo, Y.; Tan, J. Y.; Rodin, A. S.; Chapler, B. C.; Ozyilmaz, B.; Castro Neto, A. H.; Fogler, M. M.; Basov, D. N. Nano Lett. 2015, 15, 4859–4864. 16


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(29) Qiao, Z.; Jung, J.; Niu, Q.; MacDonald, A. H. Nano Lett. 2011, 11, 3453–3459. (30) Hasdeo, E. H.; Song, J. C. W. arXiv:1706.00790 2017, (31) Shi, Z.; Hong, X.; Bechtel, H. A.; Zeng, B.; Martin, M. C.; Watanabe, K.; Taniguchi, T.; Shen, Y.-R.; Wang, F. Nature Photon. 2015, 9, 515–519. (32) Kane, C.; Balents, L.; Fisher, M. P. A. Phys. Rev. Lett. 1997, 79, 5086–5089.

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Graphical TOC Entry

for TOC only

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Plasmon Reflections by Topological Electronic Boundaries in Bilayer

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