Topological Interface Modes in Graphene Multilayer Arrays

Transcription

1 Topological Interface Modes in Graphene Multilayer Arrays Feng Wang a,b, Shaolin Ke c, Chengzhi Qin a, Bing Wang a *, Hua Long a, Kai Wang a, Peiiang Lu a, c a School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan , China b Institute for Quantum Materials and School of Mathematics and Physics, Hubei Polytechnic University, Huangshi , China c Laboratory for Optical Information Technology, Wuhan Institute of Technology, Wuhan , China Keywords: Topological interface modes Graphene multilayer waveguide A B S T R A C T We investigate the topological interface modes of surface plasmon polaritons in a multilayer system composed of graphene waveguide arrays. The topological interface modes emerge when two topologically distinct graphene multilayer arrays are connected. In such multilayer system, the non-trivial topological interface modes and trivial modes coeist. By tuning the configuration of the graphene multilayer arrays, the associated non-trivial interface modes present robust against structural disorder. The total number of topological modes is related to that of graphene layers in a unit cell of the graphene multilayer array. The results provide a new paradigm for topologically protected plasmonics in the graphene multilayer arrays. The study suggests a promising approach to realize light transport and optical switching on a deep-subwavelength scale. 1. Introduction As many recent prominent discoveries in physics such as quantum Hall effect [1-3], topological superconductors [4], and topological insulators [5-7], the nontrivial topological properties of matter play the crucial role and have attracted intense attention. So far similar concepts have been analogized to the field of optics [8, 9] and acoustics [10]. The nontrivial topological effects have been demonstrated across a variety of optical systems including metallic-dielectric waveguides [11] and photonic crystals [12]. A paradigm of topological structure is the famous Su-Schrieffer-Heeger (SSH) model [13, 14], in which the topological interface modes emerge at the interface between topological trivial and non-trivial structures with distinct topological invariants [8]. Generally, the winding number can be regard as the topological invariant and is used to characterize the topological properties of the SSH model and the other one-dimensional structures [15]. According to the ratio between intra- and inter-cell couplings, the winding number is either zero or unity separated by the Dirac point. Such SSH model can be realized in many optical systems, including dimerized dielectric waveguides [16], nanoparticles [17], and metallic nanodisks [18]. Recently, the topological nontrivial modes associated with the SSH model have been investigated in graphene waveguide system [19, 20]. Graphene can support surface plasmon polaritons (SPPs) in a wide range of spectrum from terahertz to * Corresponding author. address: wangbing@hust.edu.cn (B. Wang). infrared frequencies [21-23]. The SPPs in graphene can be fleibly tuned by gate voltage and chemical doping [24-27], which also ehibit relatively low propagation loss and strong field confinement [28-30]. These properties enable graphene a promising platform to study topologically nontrivial SPP modes. In this work, we generalize the traditional SSH model from dimerized waveguide array to graphene multilayer arrays (GMAs), which can also constitute a hyperbolic metamaterial for application in negative refraction and transformation optics [31, 32]. Here the topologically nontrivial modes may emerge at the interface between two GMAs with distinct topological invariants, which is characterized by the binary winding number of the bulk bands. Meanwhile, the trivial interface modes coeist at the interface, leading to beat between the trivial and nontrivial modes. The topological interface modes of SPPs are also found to be robust against the perturbations of the structure. The dependence of the total number of topological modes on that of graphene layers in each unit cell of GMAs is also discussed in detail. 2. Concept and theory We start by investing the Bloch modes in the period GMAs. As shown in Fig. 1, each graphene sheet can support TM polarized SPP mode that propagates along z direction. By applying the tight-binding approimation [33], we can obtain the coupled-mode equation for the mode amplitude an () z igan ( z) icn 1, nan1 ( z) icn, n1an 1( z), (1) z 1

2 where g is the propagation constant of SPPs in a single layer graphene. C n,n+1 is the coupling coefficient between the nth and (n+1)th graphene sheets. According to Bloch theorem [34, 35], the mode amplitude in the nth graphene at the position n is a ( z) u ep( ik )ep( i z), (2) n n n where and k are the propagation constant and Bloch wave vector. u n = u n+n is the period-in-cell part of the Bloch mode. The band structure (k ) can thus be obtained from the Eqs. (1) and (2). Since there eists N layers of graphene in each period, the GMAs can support N Bloch bands [25]. For each Bloch band m, the topological invariant is the winding number, which is defined as [36, 37] i / D umk, ( ) Wm [ u / m, k ( ) d] dk, D cell (3) k with the period function given by u m, k N ( ) sgn( n ) unep( n ). (4) n1 where = ( g dk 0 2 ) 1/2 and g = k 0[ d (2 d/ 0 g) 2 ] 1/2 are the decay constant and propagation constant of SPPs in a single layer graphene. σ g is the surface conductivity of graphene, which can be modeled by the Kubo formula [38, 39]. k 0 = 2π/λ is the wave number in air with 0 being the corresponding air impedance. The coupling coefficient can be derived as C n,n+1 = ( 0 )/4 with 0 and being the propagation constants of Bloch modes in monolayer graphene array as k = 0 and /D, respectively. It should be mentioned that the coupling coefficient C n,n+1 can be tuned by the intra-cell spacing of d 1 and inter-cell spacing of d 2, which will basically determine the topological property of the GMAs. Fig. 1. Schematic of the periodic GMAs. In each unit cell, there are N graphene sheets. The spatial period is D = (N1)d1+d2 with d1 and d2 being the spacing of the intra-cell within a unit and the inter-cell between adjacent unit, respectively. σg is the surface conductivity of graphene, and εd is the relative permittivity of the dielectric medium between graphene. Figures 2(a) and 2(b) show the band structures of the Bloch modes for different inter-cell spacing d 2 as the fied intra-cell spacing of d 1 = 40 nm. We choose N = 3, thus there are three Bloch bands. As d 2 increases from d 2 < d 1 to d 2 > d 1, the band gaps close and then reopen, indicating the emergence of a band inversion. Specially as d 2 = d 1 shown in Fig. 2(c), the band gaps vanish with the Dirac points emerging at Brillouin zone center between band 2 and 3 and Brillouin zone edge between band 1 and 2, respectively [40, 41]. The GMAs thus eperience a topological phase transition at the Dirac points. Fig. 2. Diffraction relation of SPPs and topological invariant in the GMAs with parameters given by N = 3, d1 = 40 nm, d = 2.13, = 10 m, c = 0.15 ev, and = 0.5 ps at room temperature. (a) Real and (b) imaginary parts of diffraction relation as the inter-cell spacing d2 is varying. (c) Band structures of Bloch modes as d2 = d1 = 40 nm. (d) Real part of winding numbers for band m = 3 as the inter-cell spacing d2 is varying. To understand the mechanisms for topological phase transition at the Dirac points, we investigate the topological invariants of the winding number W m for the bulk bands. The real parts of the winding number of band m = 3 for different inter-cell spacing d 2 are shown in Fig. 2(d). Due to low loss of graphene, the imaginary part of winding number is etremely tiny and can be ignored. As epected, the real part of winding number is a binary integer which equals to either zero or unity. It shows that W m = 1 for d 2 < d 1 and W m = 0 for d 2 > d 1, corresponding to topological nontrivial and trivial phases, respectively. So d 2 = d 1 is the topological phase transition point, which also manifests as the Dirac points in the band structure. As will be shown in the following, if we connect the two GMAs with distinct winding numbers together, there will eist topological nontrivial interface modes at the boundaries. 3. Topological interface modes of SPPs Now we truncate the periodic GMAs into finite graphene layers and join two GMAs with different topological phases together, which forms a hetero-structure array. We denote N as the total number of graphene sheets in the array. For the nth layer graphene, the amplitude of SPP mode satisfies a ( z) a ep( i z), (5) n n s where s is the collective propagation constant of the SPP supermode with s being the corresponding mode inde. Substituting Eq. (5) into the coupled-mode equation of Eq. (1), we can obtain the eigen equation 2

3 g C a1 a1 C12 g C23 0 a2 a2 0 C23 g 0 a 3 s a 3, (6) g a N a N Here s is the eigen value and Φ s = (a 1, a 2,,a N) T is the eigen vector for the sth supermode. Since there are total N layers of graphene in the array, the number of supermodes is also N, that is s = 1, 2,, N [42]. So we can obtain the transverse mode profile of the magnetic field distribution for the sth supermode H N ys, n an n n1 ( ) sgn( ) ep( ). (7) topological nontrivial interface modes at the boundaries of the two GMAs. Apart from the topological nontrivial interface modes, there also eist two topological trivial interface modes of s = 1 and s = 48 marked by the green dots in Fig. 3(b). To validate the theoretical analysis, we also perform numerical calculation of SPP supermodes by using transfer matri method [42]. The results are shown in Fig. 3(c). The analytical and numerical results agree fairly with each other. Figure 4 illustrates the mode profiles (H y) of the topological nontrivial and trivial interface modes corresponding to Fig. (3). Two non-trivial topological modes are well confined around the interface, displayed as blue lines in Figs. 4(a) and 4(b). In addition, there are also two trivial interface modes which are also located around the interface of the structure as shown in Figs. 4(c) and 4(d). The trivial defect modes are formed as the four equidistant graphene sheets around the interface constitute a region of higher refractive inde than the rest of the structure. Fig. 3. (a) Schematic of the connection with two different GMAs. The blue arrows denote the unit cell of left and right spacing. (b) The analytical eigenvalues of the supermodes corresponding to (a). (c) The numerical results of the supermodes. The red dots represent the topological interface modes, and the green dots represent the trivial interface modes. Figure 3(a) shows the structure of the two connected GMAs. There is N = 3 layers of graphene in each unit cell of the left and the right arrays, respectively. The spatial periods of the left and right arrays are DL = 2d L 1 + d L 2 and D R = 2d R 1 + d R 2 with d L 1 (d R 1 ) and d L 2 (d R 2 ) being the respective intra-cell and inter-cell spacing. We choose d L 1 = 60 nm, d L 2 = 40 nm, d R 1 = 40 nm, and d R 2 = 60 nm, respectively. Thus the left array is topological nontrivial and the right one is topological trivial. Therefore, there should eist topological nontrivial modes at the interface of two arrays. The propagation constants of all SPP supermodes can be obtained by solving Eq. (6) as shown in Fig. 3(b). The number of graphene layers in the left and right arrays are both set as n' = 24 with the total layer number being N = 48. The supermodes are sorted in an ascending order of propagation constants from s = 1 to s = 48. It shows that the supermodes of s = 16 and s = 33 locate in the gap of the sub-bands denoted by the red dots, which corresponds to the Fig. 4. (a) and (b) Transverse magnetic field (Hy) distributions of the topological interface modes corresponding to Fig. 3. (c) and (d) Transverse magnetic field (Hy) distributions of the trivial interface modes corresponding to Fig. 3. The positions of graphene sheets are denoted by the red dashed lines. The topological and trivial interface modes can also greatly influence the beam propagation of SPPs. We perform numerical simulations of the SPP beam propagation by using the finite element method (FEM). Graphene is modeled by employing the surface current boundary condition [22]. As shown in Fig. 5(a), the nontrivial interface mode of s = 16 is injected from the bottom edge of the structures shown in Fig. 3(a). The interface mode can propagate steadily in the structure with the mode profile remaining almost constant ecept for the propagation loss. The field is concentrated at the interface since only one topological interface mode is ecited. The results are similar for other interface modes of s = 33, s = 1 and s = 48 as shown in Figs. 5(b)-5(d). In Fig. 5(e), the waveguides are ecited from the single graphene sheet at the interface of the structures shown in Fig. 3(a). The field is also confined around the interface. Here both the trivial and nontrivial interface modes are ecited simultaneously, leading 3

4 to the beating pattern between them. For comparison, we also simulate the dynamics of beam propagation for the graphene waveguide array with equal spacing d L 1 = d L 2 = d R 1 = d R 2 =40 nm. The result is shown in Fig. 5(f) in which we ecite the SPP mode at the center graphene sheet. The beam will spread during the propagation, which ehibits the discrete diffraction pattern in waveguide array. Fig. 6. Eigenvalues of supermode corresponding to Fig. 3(a). In each unit cell, there are N graphene sheets. (a)-(d) N = 5, 6, 8, and 9, respectively. The red and green dots represent the topological and trivial interface modes, respectively. Fig. 5. (a)-(d) Beam evolutions for the eigen supermodes of s = 16, s = 33, s = 1, and s = 48, respectively ecitation corresponding to the structures shown in Fig. 3(a). (e) Beam evolutions for a single-waveguide ecitation corresponding to the structures shown in Fig. 3(a). (f) Beam evolutions for a single-waveguide ecitation corresponding to the arrays with equidistance. The number of graphene layers in each unit cell of GMAs also greatly influence the topological interface modes. Figures 6(a)-6(d) show the propagation constants of supermodes for different number of layers in one unit cell as N = 5, 6, 8, and 9, respectively. The spatial periods for the left and right multilayer arrays are D L = (N 1)d L 1 + d L 2 with d L 1 = 60 nm and d L 2 = 40 nm and D R = (N 1)d R 1 + d R 2 with d R 1 = 40 nm and d R 2 = 60 nm. Thus the structure has an interface with left and right arrays ehibiting different topological phases. As the layer number N increases in each unit cell, the band of the supermodes will be divided into more sub-bands. As a result, there are more topological nontrivial interface modes emerging in the bandgaps. However, the number of the topological trivial interface mode keeps unchanged, which remains as s = 1 and s = N' for different N. Although both the topological trivial and nontrivial interface modes are located at the boundaries of the two GMAs, they ehibit different responses to the structure perturbations. The topological nontrivial modes are robust against structural disorder while the topological trivial interface modes will penetrate into the bulk bands under sufficient perturbations. Figure 7(a) shows the propagation constants of the supermodes for different spacing d L 1 (d R 2 = d L 1 ) as d L 2 = d R 1 = 40 nm. It shows that the band gap becomes narrow when d L 1 approaches d L 2. At the same time, the topological interface modes will eperience a weaker confinement. Interestingly, as the waveguide spacing changes, the eigenvalues of the topological interface modes remain stable and undergo little change provided that d 1 and d 2 don t change their relative magnitudes. However, the topological trivial interface modes eperience greater variation and can penetrate into the bulk bands. Figure 7(b) shows the calculated eigenvalues for different spacing d L 2 (d R 1 = d L 2 ) as d L 1 = d R 2 = 60 nm, which ehibits similar results as Fig. 6(a). Fig. 7. (a) Real part of eigenvalues of supermodes as different waveguide spacing d L 1 for the structures shown in Fig. 3(a). The spacing d L 2 is fied at 40 nm. (b) Real part of eigenvalues of supermodes as different waveguide spacing d L 2 for the structures shown in Fig. 3(a). The spacing d L 1 is fied at 60 nm. Here d R 2 equals to d L 1, and d R 1 equals to d L 2. 4

5 The chemical potential of graphene also has effect on the topological interface modes. Figure 8(a) depicts the propagation constant of the supermodes as c = 0.18 ev for the structure shown in Fig. 3(a). It shows that the topological interface modes remain stable, the propagation constants of the supermodes reduce significantly compared to c = 0.15 ev shown in Fig. 3(b). Figure 8(b) displays the propagation constant of the supermodes as different chemical potential of graphene. As the chemical potential increases, the propagation constants of the topological interface modes decrease. It s worth noting that the topological interface modes may penetrate into the bulk bands as chemical potential increases to around c = 0.36 ev. Fig. 8. (a) Real part of eigenvalues of supermodes as c = 0.18eV of graphene for the structures shown in Fig. 3(a). (b) Real part of eigenvalues of supermodes as different chemical potential of graphene. Here the spacing are d R 2 = d L 1 = 60 nm and d L 2 = d R 1 = 40 nm. 4. Conclusions In conclusion, we have investigated the topological nontrivial interface modes of SPPs in hetero-structure GMAs. The GMAs ehibits a topological phase transition as the intra-cell and inter-cell couplings equal to each other. By combining two GMAs with distinct topological phases, we realize both trivial and nontrivial interfaced modes at the boundaries. The nontrivial interface modes are robust against structural disorder while the trivial modes will penetrate into the bulk bands under perturbations. The number of topological interface modes increases as that of graphene layers in each unit cell increases. The results provide a promising approach to control light propagation on the deep-subwavelength scale and may find application in SPP waveguides and optical switches. Funding This work is supported by the 973 Program (No. 2014CB921301), the National Natural Science Foundation of China (Nos , ), Natural Science Foundation of Hubei Province (2015CFA040). References [1] D. Thouless, M. Kohmoto, M. Nightingale, and M. Den Nijs, Phys. Rev. Lett. 49(6), (1982). [2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95(14), (2005). [3] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, (2006). [4] M. Z. Hasan, S.Y. Xu, and G. Bian, Phys. Scr. 2015, (2015). [5] L. H. Wu and X. Hu, Phys. Rev. Lett. 114(22), (2015). [6] H. Wang, L. Xu, H. Chen, and J. Jiang, Phys. Rev. B 93(23), (2016). [7] L. Xu, H. Wang, Y. Xu, H. Chem, and J. Jiang, Opt. Ep. 24(16), (2016). [8] L. Lu, J. D. Joannopoulos, and M. Soljačić, Nat. Photon. 8(11), (2014). [9] S. Ke, B. Wang, H. Long, K. Wang and P. Lu, Opt Quant Electron 49, 224 (2017). [10] Y. Peng, C. Qin, D. Zhao, Y. Shen, X. Xu, M. Bao, H. Jia and X. Zhu, Nat. Commun. 7, (2016). [11] H. Deng, X. Chen, N. C. Panoiu, and F. Yei, Opt. Lett. 41(18), (2016). [12] T. J. Atherton, Phys. Rev. B 93(12), (2016). [13] W. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42(25), (1979). [14] A. Blanco-Redondo, I. Andrea, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, Phys. Rev. Lett. 116(16), (2016). [15] J. Zak, Phys. Rev. Lett. 62(23), (1989). [16] H. Schomerus, Opt. Lett. 38(11), (2013). [17] A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 114(12), (2015). [18] I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, Nanoscale 7, (2015). [19] L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, Opt. Ep. 23(17), (2015). [20] S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, Opt. Ep. 25(10), (2013). [21] Z. Wang, B. Wang, H. Long, K. Wang and P. Lu, Journal of Lightwave Technology, 35(14), (2017). [22] A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, Phys. Rev. B 84(16), (2011). [23] Q. Bao and K. P. Loh, ACS Nano 6(5), (2012). [24] H. Da and C. W. Qiu, Appl. Phys. Lett. 100(24), (2012). [25] F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, IEEE J. Sel. Top. Quantum Electron. 23(1), (2017). [26] D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, Opt. Quantum Electron. 49(4), 163 (2017). [27] H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, J. Lightwave Technol. 35(2), (2017). [28] A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photon. 6(11), (2012). [29] C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, J. Lightwave Technol. 34(16), (2016). [30] C. H. Gan, Appl. Phys. Lett. 101(11), (2012). [31] A. Vakil and N. Engheta, Science 332, (2011). [32] M. A. K. Othman, C. Guclu, and F. Capolino, Opt. Ep. 21(6), (2013). [33] Y. Fan, B. Wang, H. Huang, K. Wang, H. Long, and P. Lu, Opt. Lett. 39(24), (2014). 5

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