Topological edge modes in multilayer graphene systems

Lixin Ge; Li Wang; Meng Xiao; Weijia Wen; C. T. Chan; Dezhuan Han

doi:10.1364/OE.23.021585

1. Introduction

Topological insulators (TIs) possess topological quantum states protected by time reversal symmetry [1, 2 ]. Inspired by the unique properties in TIs, the photonic analogs of TIs have attracted attentions recently [3]. For instance, one-way transport can be attained by means of topological edge states without breaking time reversal symmetry in many optical systems [3–7 ]. Topological edge states have been proposed theoretically or realized experimentally in photonic systems such as photonic crystals made by bi-anisotropic materials [4], chiral hyperbolic metamaterials [5], uniaxial metacrystals [6] and optical ring resonators [7]. The topological edge states have also been investigated in one-dimensional photonic crystals [8, 9 ] and other one-dimensional photonic systems [10–12 ].

Being only one atom thick, graphene becomes a promising platform for strong light-matter interactions on which plasmons can be supported [13, 14 ]. Graphene plasmons have many remarkable merits especially in Terahertz and far infrared frequencies, including deep sub-wavelength, low loss, and high tunability. The periodic graphene sheet arrays show many interesting phenomena due to the Bragg scattering such as band gaps [15, 16 ] and hyperbolic dispersion [17]. Moreover, the strong coupling of plasmon waves between adjacent graphene sheets lead to interesting negative coupling and additional waves [18, 19 ]. Graphene multilayer structures may have potential applications in new tunable optical devices [20] as the chemical potential can be adjusted by electric gating or chemical doping.

In this paper, we study the topological properties of the photonic bands in one-dimensional periodic graphene sheet arrays, analogous to the Su-Schrieffer-Heeger (SSH) model [21] for electrons. Firstly, we develop a self-consistent method to describe the properties of plasmons propagating on the monolayer and multilayer graphene sheets, including eigenstates, dispersions, and band structures. Due to the deep sub-wavelength nature of graphene plasmons, the fields decay very fast in the direction normal to the graphene sheets and the field confinement is strong. The coupling between the graphene sheets through evanescent waves can be truncated appropriately. The short range coupling can be well described using a tight-binding model for graphene plasmons. Based on this model, we then investigate the topological characteristics of plasmonic bands. Different values of Zak phases are found for different configurations similar to that in SSH model and topological edge plasmon modes can be realized by connecting two periodic graphene sheet arrays with different Zak phases. These topological edge modes stem from the topology of bulk bands so that they are indifferent to the detailed structure in a unit cell such as the separation distance between graphene sheets. Compared to the edge modes in the metallic spheres’ chain [10], the corresponding edge states in multilayer graphene systems have a dispersion which depends on the tangential wave vector. Furthermore, the edge-state dispersion can be tuned by adjusting the chemical potential of graphene sheets [14].

2. Tight-binding model for graphene plasmons

The plasmon dispersions for monolayer and multilayer graphene can be derived rigorously by a transfer matrix method [15, 16 ]. Though extensively used in electromagnetics, this method cannot directly manifest the characteristics of graphene plasmons. Here, we adopt a self-consistent method to calculate the plasmon dispersions of monolayer and multilayer graphene sheets. The key is to treat the surface current density J in the graphene sheet as a basic physical quantity, while other quantities, such as the electric field E and magnetic field H, are expressed through the surface current density J. This strategy is similar to that in coupled dipole equations [22] where dipole moments P are employed instead of E or H. In graphene systems, H can be related to J through boundary conditions, E can be calculated by H and then expressed by J, except when J = 0. J = 0 is a special case in which graphene sheets play no role in the optical responses. Finally, E and J are associated by Ohm’s law J = σE, where σ is the optical conductivity of graphene. In this way we can solve the equation for J self-consistently. The optical conductivity of graphene can be given, within a random phase approximation, by the following relation [23, 24 ]:

\tilde{σ} \equiv \frac{σ}{ε_{0} c} = 4 α \frac{i}{Ω} + π α [θ (Ω - 2) + \frac{i}{π} \ln | \frac{Ω - 2}{Ω + 2} |],

where

Ω \equiv ℏ ω / E_{F}

is the dimensionless frequency, ħ is the reduced Planck constant, c is the speed of light in vacuum, E_F is the Fermi energy,

α \approx 1 / 137

is the fine structure constant, and θ(x) is the Heaviside step function. The temperature T is assumed to be low enough and satisfies

k_{B} T ≪ E_{F}

, where k _B is the Boltzmann constant. The first and second terms on the right side arise from the intraband and interband contributions, respectively. The intraband conductivity is of the Drude-like form as that in metals. The dissipation in the Drude-term is neglected here. We focus on transverse magnetic (TM) plasmon modes in this study.

2.1. A monolayer graphene

Considering a monolayer graphene sheet set at z = 0 as schematically shown in the inset of Fig. 1(a) , the left (right) half space is dielectric medium with a dielectric constant ε₁ (ε₂). For the TM plasmon mode, we assume that the tangential component of the electric field is $E_{y}^{(i)} = E_{0} e^{i (k_{y} y + k_{z}^{(i)} | z |)}$ , where the superscript i = 1, 2 denote respectively the region z ≤ 0 and z > 0; E ₀ is the amplitude. This plasmon mode propagates along y axis with a wave vector k_y, and $k_{z}^{(i)} = \sqrt{ε_{i} k_{0}^{2} - k_{y}^{2}}$ is the normal component. $k_{z}^{(i)}$ is positive imaginary as $k_{y} > \sqrt{ε_{i}} k_{0}$ . The tangential component of the magnetic field can be calculated through $H_{x}^{(i)} = \mp (k_{0} / k_{z}^{(i)}) ε_{i} ε_{0} c E_{y}$ . By applying the boundary condition $H_{x}^{+} - H_{x}^{-} = J_{y}$ and Ohm’s law $J_{y} = σ E_{y}$ on the graphene sheet at z = 0, the dispersion equation reads:

\frac{ε_{1}}{\sqrt{ε_{1} k_{0}^{2} - k_{y}^{2}}} + \frac{ε_{2}}{\sqrt{ε_{2} k_{0}^{2} - k_{y}^{2}}} = - \frac{\tilde{σ}}{k_{0}},

which is the same as the rigorous one for graphene plasmons [15].

Fig. 1 Dispersions of graphene plasmons. (a) for monolayer graphene, and (b) for triple layer sheets. In (a), the dielectric constants of background media are taken to be ε ₁ = 1, ε ₂ = 2. In (b), the background dielectric constant and separation distance of two adjacent graphene sheets are set to be ε = 1, d = 0.1, respectively. The black and blue curves are calculated by the tight-binding method. Only the nearest-neighbor interactions are considered for the blue dashed curves, while the next-nearest neighbor interactions are also included for the black solid curves. The red dotted curves are the rigorous result calculated using the transfer matrix method. Here Ω = ħω/E _F, k_y = ħck_y _,SI/E _F and d = d _SI E _F/(ħc) are the dimensionless frequency, the parallel component of the wave vector, and the separation distance, respectively. E _F is the chemical potential, k_y _,SI and d _SI are the corresponding wave vector and separation distance in the SI units.

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2.2. Multilayer graphene

Equation (2) can be regarded as the contribution of the on-site energy of the plasmon mode. We can further generalize this method to multilayer graphene sheets with mutual interactions taken into account. For simplicity, all graphene sheets are embedded in a homogenous medium with a dielectric constant ε. We assume that the oscillating surface current density $J_{n} = J_{n} e^{i (k_{y} y - ω t)} {\hat{e}}_{y}$ exists in the n-th graphene sheet. This current will generate magnetic fields $H_{n, x}^{\pm} (z) = \pm \frac{1}{2} J_{n} e^{i k_{y} y + i k_{z} | z - z_{n} |}$ , where the wave vector component $k_{z} = \sqrt{ε k_{0}^{2} - k_{y}^{2}}$ ; z_n is the location of the n-th graphene sheet; and the superscripts ± stand for the right and left side of the graphene. The tangential component of the electric field is $E_{n, y}^{\pm} (z) = - (k_{z} J_{n} / 2 ω ε ε_{0}) e^{i k_{y} y + i k_{z} | z - z_{n} |}$ which is continuous crossing the graphene sheet. In the coupled dipole equation, the electric field induced by a point dipole is divergent at the source point [25]. However, for the graphene sheet, the induced electric field will not diverge since we now have a plane source instead of a point source. In the absence of external electric fields, the total electric field is the summation of the electric fields induced by all the graphene sheets. The electric field at the n-th graphene sheet should be consistent with the induced surface current density according to Ohm’s law, namely,

J_{n} = σ \sum_{m} E_{m, y} (z_{n}) = - \frac{k_{z} \tilde{σ}}{2 k_{0} ε} \sum_{m} J_{m} e^{i k_{z} | z_{m} - z_{n} |} .

This is a self-consistent equation for multilayer graphene sheets. Analogous to the tight-binding method for electrons, the interaction between graphene sheets caused by evanescent waves can even be truncated to the nearest-neighbor ones as the scale of confinement is close to the separation distance. Thus we only consider m = n−1, n, n + 1 in Eq. (3). With this truncation, the underlying physics can be interpreted directly.

Based on the tight-binding method, the dispersion of plasmons in the multilayer graphene system can be solved by the self-consistent equation as shown above. In Fig. 1(b), we plot the dispersion for triple layer graphene sheets. The separation distance between the adjacent graphene sheets is d = 0.1. Here, we adopt a dimensionless wave vector and length with k = ħck_SI/E _F and d = d_SIE _F/(ħc), where k_SI and d_SI are respectively the wave vector and separation distance in SI units. For instance, the distance d = 0.1 is about 98.9 nm for E _F = 0.2 eV. Due to the interactions among the graphene sheets, the plasmon dispersions for triple layer graphene sheets split to three branches compared with the monolayer one. From Fig. 1(b), the tight binding result (blue) agrees quite well with that from transfer matrix method (red) for large wave vectors and relatively high frequencies.

2.3. Truncation in the tight binding model

In the above sub-section, the mutual interactions between graphene sheets have been truncated to the nearest neighbors. This truncation is a good approximation if $| k_{z} d | ≫ 1$ , i.e., the evanescent waves decay very fast within the separation distance d. As the frequency decreases, the parallel component of the wave vector k_y, along with the normal component $| k_{z} |$ , also decreases. The decay length L is about $1 / | k_{z} |$ , therefore the electric field from the m-th graphene sheets in the region that satisfies $| z_{n} - z_{m} | \leq L$ should all be included in Eq. (3).

For the triple layer graphene sheet configuration, only the interaction between the next-nearest neighbor needs to be included. In Fig. 1(b), the blue dashed curves correspond to the dispersion with only the nearest-neighbor interaction considered. At zero frequency ω = 0, the lower branch of dispersions approaches a finite wave vector. This behavior is an artifact of the truncation in the small k_y regime and can be eliminated if the interactions between the next-nearest neighbors are incorporated. The result with next-nearest neighbor interactions included is shown by the black solid curves in Fig. 1(b), coinciding with the exact one calculated by the transfer matrix method (red dotted) in the complete range of k_y. While the nearest-neighbor result (blue dashed) agrees with the exact result in the range of large k_y.

3. Band structures for periodic graphene sheet arrays

We now consider graphene sheets arranged in a periodic lattice as shown in Fig. 2(a) . The separation distance of two adjacent graphene sheets in a unit cell is t and the period is d. The permittivities of the two dielectric media are respectively ε ₁ and ε ₂. The normal components of the wave vectors in the two different media are $k_{i, z} = \sqrt{ε_{i} k_{0}^{2} - k_{y}^{2}}$ , i = 1, 2. k_i,z are purely imaginary as k_y lies outside the light cone. We note that k _1, _z≈k _2, _z≡k_z≈ik_y under the approximation of large wave vector (or deep sub-wavelength), i.e., $k_{y} ≫ k_{0}$ , $\sqrt{ε_{1}} k_{0}$ , and $\sqrt{ε_{2}} k_{0}$ . By applying the tight-binding method presented in the previous section, the interactions between the surface current sources are given by:

α (Ω) J_{n} = {\begin{matrix} J_{n - 1} e^{i k_{n - 1, z} (d - t)} + J_{n + 1} e^{i k_{n + 1, z} t}, for n is even \\ J_{n - 1} e^{i k_{n - 1, z} t} + J_{n + 1} e^{i k_{n + 1, z} (d - t)}, for n is odd \end{matrix}

where

α (Ω) = - 1 - k_{0} (ε_{1} + ε_{2}) / (\tilde{σ} (Ω) k_{z})

which corresponds to the polarizability in the coupled dipole equations [10, 22 ]. For the frequency region Ω< 2, σ(Ω) is purely imaginary if the dissipation is neglected. k_z is also purely imaginary outside the light cone, therefore α(Ω) is real. If Ω > 2, the real part in σ(Ω) due to the interband transition becomes dominating. In this regime, graphene plasmon will be strongly damped by electron hole excitations and the field will not be confined near the graphene sheets, and the tight-binding model will breakdown at this frequency region. To calculate the band structures, the following Bloch’s condition is utilized:

J_{n} (q) = {\begin{matrix} J_{A} (q) e^{i q n d / 2}, for n is even \\ J_{B} (q) e^{i q (n - 1) d / 2}, for n is odd \end{matrix}

where q is the Bloch wave vector, J_A and J_B are surface current densities in the graphene sheets for n = 0 and n = 1 respectively. With the Bloch’s condition, Eq. (4) can be represented by a two by two matrix form with its eigenvector {J_A, J_B}:

[\begin{matrix} 0 & a_{12} (q) \\ a_{21} (q) & 0 \end{matrix}] [\begin{matrix} J_{A} \\ J_{B} \end{matrix}] = α (Ω) [\begin{matrix} J_{A} \\ J_{B} \end{matrix}]

The matrix elements are given by a ₁₂(q) = e^ikzt + e^ikz ⁽ ^d-t ⁾ e^-iqd and a ₂₁(q) = e^ikzt + e^ikz(d-t)e^iqd. The eigenvalue problem is similar to that in the SSH model. α(Ω) corresponds to the energy eigenvalue and can be directly found as follows:

α (Ω) = \pm \sqrt{a_{12} (q) a_{21} (q)} .

This dispersion relation is also similar to the case of diatomic plasmonic nanoparticles [10]. In Fig. 2(b), we show the band structures calculated by the above equation. We choose the parallel component of the wave vector k_y = 30, and the period d = 0.2 without loss of generality. It is found that for t = 0.5d (i.e., simple lattice) there is no band gap, whereas for the “dimerized” lattice with t = 0.4d and t = 0.3d, large band gaps are opened. The band projections are shown in Figs. 2(c) and 2(d) for t = 0.4d, calculated with both the tight-binding method (red) and transfer matrix method (grey). In Fig. 2(c), as expected, the truncation of the interaction to the nearest neighbor works very well for large k_y which almost coincides with the result from the transfer matrix method. However, this truncation to the nearest neighbor eventually breakdown as k_y decreases since the decay length of plasmons becomes large and more mutual interactions need to be taken into account. In Fig. 2(d), it is shown that the tight binding result agrees with the rigorous one better as the next-nearest neighbor interactions are incorporated. On the other hand, for the large k_y, the field confinement of plasmons is so strong that the interactions between the nearest-neighbors are very small, leading to extremely narrow bands and gap.

Fig. 2 (a) Schematic view of a periodic graphene sheet array. In each unit cell, there are two graphene sheets, denoted by A and B respectively. The separation distance of the two adjacent graphene sheets within a unit cell and the period are denoted respectively by t and d. (b) Band structures with k_y = 30, d = 0.2 fixed. The black, blue and red curves represent t = 0.5d, t = 0.4d and t = 0.3d, respectively. The solid and dashed curves are calculated respectively by the tight-binding method and transfer matrix method. (c) Projected band structures as t = 0.4d and d = 0.2. The red and grey parts stand for the results calculated by the tight-binding method and transfer matrix method, respectively. In (c), only the nearest- neighbor interactions are considered, while both the nearest and next-nearest neighbor interactions are taken into account in (d). We choose ε ₁ = ε ₂ = 1 here.

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4. Zak phases

To classify the geometric properties of the plasmonic bands, one can calculate the Zak phase which is the integration of the Berry connection over the whole reduced Brillouin zone. The nontrivial solution for Eq. (6) can be written as:

[\begin{matrix} J_{A} \\ J_{B} \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} \pm e^{i ϕ (q)} \\ 1 \end{matrix}],

where

ϕ (q) = a r g [1 + e^{i k_{z} (d - 2 t)} e^{- i q d}]

. Since the eigenvector is not unique up to an arbitrary phase factor e^iθ ⁽ ^k ⁾, there exists a symmetry group U(1) (one-dimensional unitary group) in the reciprocal space. The above solution satisfies the periodic gauge J_A,B(q) = J_A,B(q + 2π/d) automatically. The Zak phases can be calculated by the following formula [26]:

γ = i \int_{- π / d}^{π / d} (J_{A}^{*} \frac{\partial J_{A}}{\partial q} + J_{B}^{*} \frac{\partial J_{B}}{\partial q}) d q = - \frac{ϕ (π / d) - ϕ (- π / d)}{2} .

Since the periodic gauge is used, ϕ (−π/d) should be equal to either ϕ (π/d), or ϕ (π/d) + 2nπ. ϕ (q) is the argument of the function 1 + e^ikz ⁽ ^d- ² ^t ⁾e^-iqd in the complex plane. [ϕ (−π/d)−ϕ (π/d)]/2π = n is exactly the integer representing the number of cycles that the curve travels around the origin, namely, the winding number. It is easy to find that for t < d/2,

ϕ (π / d) = ϕ (- π / d)

and we have γ = 0; while for t > d/2,

ϕ (π / d) = ϕ (- π / d) + 2 π

, we have γ = π. In the latter case, both the lower and upper band have nontrivial topology.

5. Topological edge states

If two semi-infinite periodic arrays of graphene sheets with respectively zero and non-zero Zak phases are connected, as show in Fig. 3(a) , there should exist topological edge states at the interface. We assume that the graphene sheet arrays in the left and right half-space have the same period d, whereas the separation distances between the two adjacent graphene sheets in the unit cell are different, denoted respectively by t_L and t_R. The periodic structures in the left and right half-space are the same if t_L + t_{R =} d, however, the Zak phases are completely different since the choices of the unit cell are different. For instance, when t_L>d/2 and t_{R =} d-t_L, the Zak phases in the left half-space and right half-space are respectively π and 0. This is similar to the SSH model for electrons, wherein the different choices of the unit cell can give zero or non-zero Zak phases.

Fig. 3 Edge states between “dimerized” graphene sheet arrays. (a) Schematic view of the geometric structure of connected graphene sheet arrays. The separation distances for left and right half-space are respectively t = t_L and t = t_R. Both the left and right half-space have the same period d. (b) The eigen-frequencies for a finite system with 61 graphene sheets, where t_L = 0.6d, t_R = 0.4d and d = 0.2. The eigenfunctions of surface current density J_n for three different states, one lying in the gap, the other two in the pass bands, are shown in the right panel. (c) Eigen-frequencies as a function of t_R, where t_L is set to be d-t_R. The dielectric constants are ε ₁ = ε ₂ = 1 here.

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To observe these edge modes, numerical calculations are performed for the finite layers of graphene sheets. We arrange 61 layers of graphene sheets periodically with the period d = 0.2, t_L = 0.6d and t_R = 0.4d as shown in Fig. 3(a). The left side is from n = −31 to n = −1 and the right side is from n = 0 to n = 29. By numerically solving the eigenvalue problem in Eq. (3), the eigen-frequencies for k_y = 30 are shown in Fig. 3(b). Obviously, there is an edge mode in the mid of the band gap at frequency Ω = 0.627. We also show the distributions of the surface currents for this mode, and for other two modes in the pass bands for comparison, which confirms field confinement near the edge for the edge state. The eigen-frequencies as a function of t_R with k_y = 30, d = 0.2 fixed are also shown in Fig. 3(c). As t_R increases, the band gap narrows, and finally closes as t_R = t_L = 0.5d, however, the eigen-frequencies of edge states are almost constant and robust to these geometrical parameters.

If two identical semi-infinite graphene sheet arrays with different choices of the unit cell (or different edge-cuts) are connected, the dispersion for edge states can be derived analytically. Since we can map Eq. (6) to the case of the metallic nano-spheres’ chain [10], a similar condition for the existence of topological edge states α(Ω) = 0 is found. Therefore we have $i k_{y} \tilde{σ} (Ω) + k_{0} (ε_{1} + ε_{2}) = 0$ , which is exactly the same dispersion as the plasmon dispersion for a monolayer graphene sheet under the deep sub-wavelength approximation. Since the dispersion relation of plasmons on a monolayer graphene sheet can be treated as the contribution from the on-site energy of plasmon modes as mentioned previously, the topological edge modes correspond to the zero-energy edge modes in the SSH model. The existence of these plasmonic edge modes are protected topologically by the inversion symmetry of the periodic systems. It should be mentioned that if one forms an interface or a truncated surface away from the inversion centers, the Zak phase can take up continuous and arbitrary values. In that case, interface states may still be found depending on the details of the system, however, their existence cannot be predicted by merely knowing the geometrical phases of the bulk bands. We note that in the metallic nano-spheres’ chain, the edge state frequency is pinned at the dipole resonance frequency $ω_{p} / \sqrt{3}$ , where $ω_{p}$ is the plasma frequency. However, in the graphene sheet arrays studied here, the topological edge states now have dispersion instead of being fixed at the resonance frequency. We also note that similar dispersions of interface states derived from the discontinuous jump of Zak phases across a boundary have been investigated in certain two-dimensional photonic crystals [27]. In the one-dimensional plasmonic chain, all the modes inside the light cone are radiative. The corresponding topological edge plasmon modes will have radiation loss even though the dissipation is neglected. For multilayer graphene structures, this is a “real” one-dimensional system and all the modes are eigenmodes unlike the above mentioned quasi-eigenmodes inside the light cone. Furthermore, the edge state dispersion can be tuned by adjusting the chemical potential through electrical gating or chemical doping [14].

To verify our conclusion, the dispersions of edge modes are shown by the red curves in Figs. 4(a) and 4(b) . In Fig. 4(a), the grey background stands for the projected band structure for the infinite periodic graphene sheet array with t = 0.4 d, d = 0.2, and ε ₁ = ε ₂ = 1.5. The blue curves are calculated by the rigorous transfer matrix method for a finite structure with 6 periods. The separation distances are t_L = 0.6d and t_R = 0.4d in the left and right half of this finite system, respectively. It can be observed that the dispersion solved analytically by α(Ω) = 0 (red) fits well with the rigorous result (blue) under large k_y assumption. The edge state dispersion α(Ω) = 0 eventually enters the projected bulk band as k_y decreases, which also implies more and more interactions beside the nearest-neighbor ones should be included as corrections. However, this tight-binding model provides us a simple way to extract the underlying physics.

Fig. 4 Dispersion of topological edge modes. A finite structure (totally 13 graphene sheets) is considered with 3 periods of “dimerized” graphene sheet structure in both the left and right side. Plasmon dispersions calculated using the transfer matrix method are shown by the blue curves. The red curves represent the edge mode dispersion calculated by α(Ω) = 0 analytically. (a) Dispersions for the structure with different choices of the unit cell, t_L = 0.6d and t_R = 0.4d. The period and dielectric constants are respectively d = 0.2 and ε ₁ = ε ₂ = 1.5. (b) Dispersions for the graphene sheets embedded into the interfaces of a one-dimensional photonic crystal with dielectric constants ε ₁ = 1.5 and ε ₂ = 4. The separation distances t_L = t_R = 0.5 d have the same value here, and the period is chosen to be d = 10. The grey background stand for the projected band structures of the corresponding infinite periodic structures.

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As t_L = t_R = 0.5d, the gap between the lower and upper bands will be closed as shown in Fig. 3(c). The separation distance t gives us a degree of freedom to adjust the hopping energy in the tight-binding model. Yet there is another degree of freedom in our system since we can vary the dielectric constant, e.g., ε ₂. If the dielectric constants ε ₂≠ε ₁, in other words, the graphene sheets are embedded into the interfaces of a one-dimensional photonic crystal, the hopping energy from the nearest neighbors in the left and right side are different. Figure 4(b) gives an example for ε ₁ = 1.5, ε ₂ = 4 and d = 10. Topological edge modes can also be observed inside the gap of the projected band structure. Note that the frequencies of edge modes here are much lower than that in Fig. 4(a) since the period d = 10 is much larger than d = 0.2 used previously. As the separation distance between graphene sheets is large, the nearest-neighbor coupling dominates, which guarantees the validity of tight-binding approximation.

6. Conclusion

In summary, a self-consistent method has been developed to study the plasmon modes in the monolayer and multilayer graphene sheets, including plasmon dispersions and band structures. A tight-binding method has been employed to obtain analytic results. When the momentum of the plasmons are large enough, the analytical results calculated from the tight-binding method agree very well with that from the rigorous transfer matrix method. Based on the tight-binding model, the Zak phases can be easily found for periodic graphene sheet arrays, depending on the choice of the unit cell. Analogous to the SSH model, topological edge modes have been revealed as two “dimerized” graphene arrays with different Zak phases are connected. The edge modes are robust to the variation of separation distance of two adjacent graphene sheets in a unit cell. These edge states in periodic graphene sheet arrays are non-radiative and have dispersion dependent on the wave vector component parallel to graphene sheets. The edge state frequencies of these modes can be further tuned by electrical gating or chemical doping. The unique properties of topological edge modes in multilayer graphene systems may have potential applications in new optical devices such as structure-insensitive plasmonic waveguides, optical cavities, etc. See the Appendix for more information.

Appendix

The decay length of graphene plasmons is $L \sim 1 / | k_{z} |$ , therefore the electric fields from the m-th graphene sheets in the region $| z_{m} - z_{n} | \leq L$ should be incorporated in Eq. (6) to calculate the band structure. Equation (6) includes only the nearest-neighbor interaction with the truncation $N \equiv | n - m | = 1$ . If we take account of the next-nearest neighbor interaction, namely, N≤2, we have:

{\begin{matrix} α (Ω) J_{0, A} = J_{- 1, B} e^{i k_{z} (d - t)} + J_{1, B} e^{i k_{z} t} + J_{- 2, A} e^{i k_{z} d} + J_{2, A} e^{i k_{z} d} \\ α (Ω) J_{1, B} = J_{0, A} e^{i k_{z} t} + J_{2, A} e^{i k_{z} (d - t)} + J_{- 1, B} e^{i k_{z} d} + J_{3, B} e^{i k_{z} d} \end{matrix} .

Equation (6) should be re-written by:

[\begin{matrix} a_{11} (q) & a_{12} (q) \\ a_{21} (q) & a_{22} (q) \end{matrix}] [\begin{matrix} J_{A} \\ J_{B} \end{matrix}] = α (Ω) [\begin{matrix} J_{A} \\ J_{B} \end{matrix}],

where

a_{11} (q) = a_{22} (q) = 2 \cos (q d) e^{i k_{z}}^{d}

,

a_{12} (q) = e^{i k_{z}}^{(d - t)} e^{- i q d} + e^{i k_{z} t}

, and

a_{21} (q) = e^{i k_{z} t} + e^{i k_{z}}^{(d - t)} e^{i q d}

. The corresponding eigen-values are:

α (Ω) = \pm \sqrt{a_{12} (q) a_{21} (q)} + a_{11} = \pm \sqrt{s + f (q)} + f (q),

where

s = e^{2 i k_{z} t} + e^{2 i k_{z} (d - t)}, f (q) = 2 \cos (q d) e^{i k_{z} d}

.

As the decay length further increases, more correction terms should be included in Eq. (6). The results can be summarized as follows.

If N is an odd number, the matrix elements are

\begin{array}{l} a_{11} (q) = a_{22} (q) = 2 \sum_{l = 1}^{(m - 1) / 2} \cos (q l d) e^{i k_{z} l}^{d}, \\ a_{12} (q) = e^{i k_{z}}^{(d - t)} e^{- i q d} (1 + \sum_{l = 1}^{(m - 1) / 2} e^{i k_{z} l d} e^{- i q l d}) + e^{i k_{z} t} (1 + \sum_{l = 1}^{(m - 1) / 2} e^{i k_{z}}^{l d} e^{i q l d}), \\ a_{21} (q) = e^{i k_{z} t} (1 + \sum_{l = 1}^{(m - 1) / 2} e^{i k_{z}}^{l d} e^{- i q l d}) + e^{i k_{z}}^{(d - t)} e^{i q d} (1 + \sum_{l = 1}^{(m - 1) / 2} e^{i k_{z}}^{l d} e^{i q l d}) . \end{array}

If N is an even number, the corresponding matrix elements are

\begin{array}{l} a_{11} (q) = a_{22} (q) = 2 \sum_{l = 1}^{m / 2} \cos (q l d) e^{i k_{z}}^{l d}, \\ a_{12} (q) = e^{i k_{z}}^{(d - t)} e^{- i q d} (1 + \sum_{l = 1}^{m / 2 - 1} e^{i k_{z}}^{l d} e^{- i q l d}) + e^{i k_{z} t} (1 + \sum_{l = 1}^{m / 2 - 1} e^{i k_{z}}^{l d} e^{i q l d}), \\ a_{21} (q) = e^{i k_{z} t} (1 + \sum_{l = 1}^{m / 2 - 1} e^{i k_{z}}^{l d} e^{- i q l d}) + e^{i k_{z}}^{(d - t)} e^{i q d} (1 + \sum_{l = 1}^{m / 2 - 1} e^{i k_{z}}^{l d} e^{i q l d}) . \end{array}

Acknowledgments

This work is supported the National Natural Science Foundation of China (Grant No. 11304038) and the Fundamental Research Funds for the Central Universities (Grant No. CQDXWL-2014-Z005). Work in Hong Kong is supported by Hong Kong Research Grant Council AOE/P-02/12.

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Topological edge modes in multilayer graphene systems

Abstract

1. Introduction

2. Tight-binding model for graphene plasmons

2.1. A monolayer graphene

2.2. Multilayer graphene

2.3. Truncation in the tight binding model

3. Band structures for periodic graphene sheet arrays

4. Zak phases

5. Topological edge states

6. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (4)

Equations (14)

Optics Express