Topological Edge Plasmon Modes between Diatomic Chains of Nanoparticles

We study the topological edge plasmon modes between two"diatomic"chains of identical plasmonic nanoparticles. Zak phase for longitudinal plasmon modes in each chain is calculated analytically by solutions of macroscopic Maxwell's equations for particles in quasi-static dipole approximation. This approximation provides a direct analogy with the Su-Schrieffer-Heeger model such that the eigenvalue is mapped to the frequency dependent inverse-polarizability of the nanoparticles. The edge state frequency is found to be the same as the single-particle resonance frequency, which is insensitive to the separation distances within a unit cell. Finally, full electrodynamic simulations with realistic parameters suggest that the edge plasmon mode can be realized through near-field optical spectroscopy.

In analogy to the electronic topological edge states at the interface between two domains of gapped crystals, we demonstrate that plasmonic topological edge states also exist between two gapped plasmonic domains made of plasmonic nanoparticle dimers. Zak phases for arrays of plasmonic nanoparticle dimers (diatomic chain) are calculated in the quasistatic limit. It is found that the edge state frequency is independent of several variable separation distances without changing the topological invariants. The quasi-static theory is found to be consistent with full electrodynamic three-dimensional simulations of photon emission within the system.
Topology band theory has explained many important electronic phenomena in condensed matter physics, like quantum hall effect, topological insulators, and edge states of 2D graphenes [1][2][3][4][5]. Apart from electronic systems, the theory has enriched the physics of some novel photonic systems [6][7][8]. It turns out that this type of photonic states can also be supported in plasmonic systems in the nanoscale [9]. Recently, it is found that topological Majorana-like states [10] can be supported in a single chain of plasmonic nanoparticles even though the freespace environment is a pass band for photons [11].
One-dimensional chains of plasmonic nanoparticles have been intensively studied in recent years because of their abilities of guiding and confining light in the nanoscale [12][13][14][15]. The guided modes in chains of plasmonic nanoparticles are due to the coupling among localized plasmons in plasmonic nanoparticles. In that sense, each plasmonic nanoparticle plays a role of an "atom" supporting nearly localized "orbitals" for photons. Considering a dimer made of two plasmonic nanoparticles as a unit cell and an array of these dimers as a "crystal" domain (simply called a "diatomic" chain), here we study the plasmonic topological edge states formed between the two domains. The Zak phase and the edge state frequency in "diatomic" plasmon chains are studied analytically. We show that those plasmonic edge states can also be topologically protected and the mode frequency is independent of several spacing parameters. The edge state properties are confirmed by numerical simulations of the plasmon enhanced photon emission rate.
We start by calculating the band dispersion of an infinite diatomic chain system (see Figure 1). As long as the nanoparticles are not too close together, we can regard excited plasmons on spheres by point dipole moments p n [16]. To give a close analogy with known electronic models, we first ignore the long-range couplings and retardation effect and only consider the nearest-neighbor coupling. There are transverse (xy component) and longitudinal mode (z component) in this one-dimensional chain system. It has been shown that the nearest neighbor approximation is pretty good in the longitudinal case [16] because it is orthogonal to free transverse photon mode propagating in the same direction. For the longitudinal modes, the z components of the dipole moments on spheres satisfy the following coupled dipole equations: where ǫ 0 is the permitivity of free space, t and d are geometrical parameters defined in Fig. 1. Here, all nanoparticles have the same electric dipole polarizability α(ω), which depends on plasmon frequency ω.  shows an infinite diatomic chain with spherical metal nanoparticles aligned along z-direction. There are two spherical metal nanoparticles in a unit cell, and are denoted by sphere A and B. Unit cell length, separation between spheres(within a unit cell), and the radii of the spheres are denoted by d, t, and a with respectively. The chain is embedded in a dielectic host. We use half unit cell length d0 = d/2 as length scale. This figure is drawn in scale such that a = 0.33d0, and t = 0.8d0. (b) shows the longitudinal mode dispersion relation of the diatomic chain(solid lines). There are two non-degenerated longitudinal bands(solid lines) as there are two atoms in a unit cell. The dispersion of a monatomic chain is also plotted (dashed line) for comparison. In that case, t = d0, which means the spheres are equally separated. moments can be written in the form where k is the wave number of the guided plasmon mode. Substituting the above into the coupled dipole equations [Eq. (1)], we obtain a matrix equation, which is equivalent to a two level problem: where Note that the above matrix equation is quite similar to the Su-Schrieffer-Heeger (SSH) model, except that in this case we have considered the actual near-field plasmonic interaction that is proportional to 1/t 3 . Solving Eq. 3, we obtained the dispersion relation: The polarizability of the sphere is given by α = 4πǫ 0 a 3 (ǫ − ǫ 0 )/(ǫ + 2ǫ 0 ), and we adopt simple plasma model (i.e., lossess Drude model) such that dielectric constant ǫ/ǫ 0 = 1 − ω 2 p /ω 2 [16,17], in which ω p is plasma frequency. Denoting V as sphere's volume, the above dispersion relation becomes ω 2 /ω p 2 = 1/3 ∓ V ǫ 0 a 12 (k)a 21 (k), and is plotted in Fig. 1 (b) for real k. The − and + signs refer to the lower and the upper bands in the figure, respectively. We see there are two plasmon bands separated by a gap at 0.606ω p > ω > 0.547ω p , which is caused by the two types of coupling between adjacent spheres [18].
As an analogy of the Berry phase [5], the Zak phase [19] has been used to classify band topology for studying their edge states [20,21]. We now evaluate the Zak phase for our plasmonic diatomic chain by solving the nontrivial solutions of Eq. (3). Providing t = d/2, the solution is The above solution is not unique in general (up to a phase factor), which is known as gauge freedom. Since the matrix equation Eq. (3) is periodic in k, a natural choice is requiring that p n (k) = p n (k + G), where G = 2π/d is the reciprocal lattice vector, and is known as a choice of periodic gauge [22]. Eq. (5) satisfies periodic gauge already, and this gauge leads to a Z 2 invariant in calculating the Zak phase [23]: Here, we have the same γ for both the lower and the upper bands because the ± sign in Eq. (5) for different bands are canceled when multiplying by its conjugate. The Zak phase is also related to the winding number, see Fig. 2. From this we can classify the system into two classes, one with γ = π and one with γ = 0, see table I. It should be noted that choice of unit cell would also lead to different Zak phase. For example, if we shift the unit cell by half the period, the separation between atoms within the unit cell will change from t to d − t, and γ will change by π. The choice of unit cell is arbitrary for an infinite chain but not a truncated chain where the boundary conditions limit the choice.

Imaginary part
Real part , winding number is non-zero, which leads to non-zero Zak phase.
The Zak phase gives a simple topological classification of the coupled plasmon modes. If two chains with different Zak phases are connected together to form a new chain, it is expected that there exists an edge state localized at the interface between the chains. To verify the existence of such an edge state, we now consider two semiinfinite diatomic chains connected together [see Fig. 3 (a)]. The left and the right chains have the same lattice constant d, and the separations between the two atoms in the unit cell are t = t L and t = t R respectively.
Here, we first consider a special case with t L = 0.8d 0 and t R = 1.2d 0 . By adopting the lossless Drude model, we can rewrite Eq. 1 into an eigenvalue problem, with ω 2 /ω p 2 being the eigenvalue. For a finite connected chain with the left chain located between n = −61 to n = −1 and the right chain located between n = 0 and n = 61, the plasmon frequencies are shown in Fig. 3(b). The results confirm that there is a band gap in the region 0.606ω p > ω > 0.547ω p . The figure also indicates a state within the band gap, with ω = 0.577ω p . We show the dipole moments p n of the system for three states in Figs. 3 (c) to (e). Figs. 3 (c) and (e) show the states just above and below the band gap while Fig. 3 (d) shows the edge state at the interface of chains. It should be noted that the magnitudes of dipole moments in Fig. 3 (d) decays away from the interface. It should be noted that the edge state at the interface is neither symmetric nor antisymmetric because the system has a broken inversion (and reflection) symmetry. For the left and right chains with the same Zak phase, we have also verified that there is no edge state at the interface by solving the eigenvalue problem. Considering the same eigenvalue problem, we can show the robustness of the edge state. The plasmon frequencies of the system (123 spheres) with different t L and t R are shown in Fig. 4. We first set t R = d − t L [See Fig. 4 (a)]. In this case, we see a single edge state at the interface when t L = d 0 . It is very interesting to note that the state frequency is independent of t L . In Fig. 4 (b), we fix t R = 0.9d 0 but varies t L . Similar to the case in Fig. 4 (a), a single edge state exists when t L = d 0 and the edge state frequency is independent of the variable parameters (although the band frequencies change a lot).
In the following, we analytically show that there is always an edge state solution the two semi-infinite chains have different Zak phases, that is t R < d/2 < t L or t R > d/2 > t L . The state is localized at the boundary between the two chains and always has frequency ω/ω p = 1/ √ 3 (Fig. 4). Let's consider an infinite con- nected chain with t R < d/2 < t L as shown in Fig. 3(a) from now on. Dividing the connected chain into two semi-infinite regions, the right and the left chain, the plasmon wave vector k in different regions takes the form k = π/d+iµ R and k = −π/d−iµ L , where µ R and µ L are real positive constants and are related to decay length in different regions [24]. Substituting into a 12 and a 21 , and using superscript R and L to denote the expressions for the right and the left chain, we have In the above, a 12 corresponds to upper sign while a 21 corresponds to lower sign. It can be inferred from Eq. (3) that non-trivial solution exists when For ω/ω p = 1/ √ 3, Eq. (4) implies that α −1 = 0 (i.e., at least one of a R 12 and a R 21 is zero and at least one of a L 12 and a L 21 is zero). When t L < d/2, the factor will never equal zero as µ R varies. Therefore, in this case, a R 21 = 0 and a R 12 = 0, which implies With p n R and p n L denoting, respectively, the solution for the right and the left region, the edge state solution is written in the form p n = Cp n R , for n 0 (right region) Dp n L , for n < 0 (left region) , where C and D are constants to be determined by matching with the interface equations: in which t 0 = (t L + t R )/2. With Eq. (2) and (9), we have dipole moments near the interface (p −2 , p −1 , p 0 , p 1 ) = (0, D, 0, C). Putting into Eq. (10), we have Thus, the edge state solution is for n is odd and n < 0 (−1) for n is odd and n > 0 (11) This shows that the edge state exist at a frequency of ω/ω p = 1/ √ 3 = 0.577. The presence of exponential factors in the edge state indicate dipole moments decay away from the interface.
The edge state in Fig. 3(d) can be confirmed by putting t R = d − t L in Eq. (11). In this case, we further show that there is no edge state solution within band gap when ω/ω p = 1/ √ 3. By expanding expressions in Eq. (7), one have µ R = µ L ≡ µ and α −1 = 0, so the non-trivial state for the two level problem is Normalizing factor in the above is absorbed to the tuning constants C and D in the next step. Substituting the above into Eq. (11), we have dipole moments near the interface (p −2 , p −1 ) = D a L 12 /α −1 , 1 and (p 0 , p 1 ) = C a R 12 /α −1 , 1 . Putting p −2 , p −1 , p 0 , and p 1 into Eq. (10), one can verify that C = D = 0, which means there is no connected chain state whose frequency lies inside the band gap but ω/ω p = 1/ √ 3. So far, our calculations are based on quasi-static approximations. Here, we verify the existence of the edge state by doing full-wave simulation using the finitedifference time-domain method. To do this, we put a dipole emitter inside one of the nanoparticles. Using mesh size = 1 nm, we simulated the connected chain with parameters a = 25 nm, d = 150 nm, t L = 90 nm, and t R = 60 nm (so t L = 1.2d 0 and t R = 0.8d 0 ). Material permittivity is a Drude model with plasma frequency ω p = 1 × 10 16 rads −1 and electron collision frequency 3×10 14 rads −1 . The left chain runs from sphere n = −13 to n = −1, while the right chain runs from sphere n = 0 to n = 11. The dipole source is put inside the sphere n = −1, as it will attain maximum dipole moment in the edge state, and so we can have prominent results. Also, we etched a hole with radius 6 nm in the sphere n = −1 to avoid contact between the dipole source and the metallic structures, which eventually leads to diverging numerical errors. We define an emission rate, which has a meaning similar to the local density of states (LDOS), as the total power flowing out of the metal nanoparticle divided by the source power: emission rate(ω) = power flowing out(ω) source power(ω) , where ω is the dipole source angular frequency. Here, the source power means the dipole emission power in free space. The presence of a plasmon mode would result in larger emission rate, which can help us to locate the response and edge states. The rate is found by boxing the whole sphere containing the dipole source with power monitors. Let us first focusing on the emission rate for the connected chain (contains 25 spheres) as shown in Fig. 5 (a). We see there are two "prominent hills" peaked at 8.3 and 9.4 × 10 14 Hz, and one "small hill" peaked at 7.6 × 10 14 Hz. The middle hill within the band gap is due to the presence of the edge state, and the right and the left hills are the responses of the upper and the lower band. A comparison with a pure diatomic chain (where we set both t L = t R = 60 nm, contains 24 spheres) is also shown in Fig. 5. In this case, there are two prominent hills peaked at about 7.6 and 9.4 × 10 14 Hz, which are corresponding to the upper and the lower band. The valley in between, is noticed as a band gap. Response of band actually varies with source position. For the case of the connected chain in Fig. 5 (a), the small hill at 7.6 × 10 14 Hz will become prominent if we put the dipole source and measure the emission rate of sphere n = −2 instead. The edge state peak coincides with the the single sphere resonance frequency 8.3 × 10 14 Hz, which is consistent to the result ω/ω p = 1/ √ 3. In fact there is one more peak at a higher frequency for the single sphere, as there are two metal-dielectric surfaces [25], but it is outside the frequency range in the figure and so it is not shown here.
In Fig. 5 (b), we showed the emission rate for connected chain with different t L = 1.05d 0 , 1.10d 0 , 1.15d 0 , and 1.20d 0 , where d 0 = 75 nm. Each curve contains en edge state peak around 8.3 × 10 14 Hz, which shows the invariance of the edge state frequency. For the case t L = 1.05d 0 , as it is close to a monatomic plasmon chain, the band gap is no longer visible.
To conclude, we studied the plasmonic topological edge states between diatomic chains of plasmonic nanoparticles. When two diatomic chains with different Zak phases are connected, it is found that a new localized plasmon mode appears at the domain boundary. Even though our analytical results are calculated using quasistatic point dipole approximation with simple plasma model, they have correctly predicted the existence of the plasmonic edge modes in full electrodynamic simulation with Drude metal nanoparticles. This work was supported by the Hong Kong Research Grant Council through the Early Career Scheme (grant no. 509813) and the Area of Excellence Scheme (grant no. AoE/P-02/12). We thank C. T. Chan, Z. Q. Zhang, K. T. Law, C. H. Lam, and C. T. Yip for useful discussions. * khfung@polyu.edu.hk