## Abstract

We introduce a new class of plasmonic crystals possessing graphene-like internal symmetries and Dirac-type spectrum in *k*-space. We study dynamics of surface plasmon polaritons supported in the plasmonic crystals by employing the formalism of Dirac dynamics for relativistic quantum particles. Through an analogy with graphene, we introduce a concept of pseudo-spin and chirality to indicate built-in symmetry of the plasmonic crystals near Dirac point. The surface plasmon polaritons with different pseudo-spin states are shown to split in the crystals into two beams, analogous to spin Hall effect.

©2010 Optical Society of America

## 1. Introduction

Often advances in one field of science inspire others and lead to important discoveries in unexpected ways. Through analogies between seemingly different systems, novel physical phenomena have been discovered. One representative example in optics is photonic crystals. Influenced by solid state crystals and electronic band gap, a number of innovative discoveries for manipulating photons have been made.

Graphene, a two dimensional monolayer of carbon atoms, is one of the subjects under active research due to its unique electronic properties and promising applications in various fields [1]. The uniqueness of graphene stems from its crystal symmetry of honeycomb lattice, leading to Dirac-like Hamiltonian and Dirac-type conical dispersion for low energy quasiparticles [2]. Interesting physics associated with the honeycomb crystal symmetry has been recently reinterpreted in optics regime. By directly employing the hexagonal structure in photonic crystals, novel optical phenomena corresponding to the electronic counter parts in graphene have been demonstrated. Such structures can indeed reproduce the Dirac-type linear dispersion near the Dirac point for photons. By utilizing the unique dispersion, edge states, zitterbewegung, and Klein tunneling in photonic honeycomb structures have been investigated [3–7].

Recently, we have theoretically shown that Dirac-type spectrum in k-space can be produced in plasmonic nanostructures as well [8]. Conical-like diffraction was demonstrated as a consequence of the linear dispersion in periodic one-dimensional (1-D) metal-dielectric structures. Here we present theoretical description of surface plasmon polariton (SPP) dynamics in graphene-like plasmonic crystals in the context of 1-D quantum electrodynamics. The presented theoretical approach not only reproduces the previous numerical results for SPP diffraction anomalies but also reveals built-in symmetries such as pseudo-spin and chirality for SPPs in the crystal. Because of the resemblance in the lattice structure and dispersion property, graphene and the proposed plasmonic crystals are described by similar Hamiltonians. The presented theoretical frame and description for the plasmonic crystals in analogy with graphene physics enables new understanding and interpretation, and could lead to designing a new class of metamaterials and discovering exotic phenomena as we have experienced with graphene.

## 2. Lattice symmetry, pseudo-spin, and chirality

Two-dimensional honeycomb lattice in graphene is composed of two interpenetrating equivalent sublattices A and B distinguished by geometrical topology as shown in the inset of Fig. 1(a) . As previously mentioned, unique properties of graphene is originated from this crystal symmetry. In a similar manner we can construct analogous 1-D plasmonic crystal with equivalent two sublattices A and B as shown in Fig. 1(a). SPP mode is supported in each waveguide schematically represented by blue bars. In the lattice each SPP mode is evanescently coupled to the neighbors forming a coupled array. The couplings between adjacent waveguides are not all identical, rather two sets of alternating coupling coefficients, ${C}_{+}$ and ${C}_{-}$. With${C}_{+}\ne {C}_{-}$, the two sublattices A and B are distinguishable and the array becomes binary, analogous to the graphene’s bipartite lattice. Especially, we are interested in ${C}_{+}>0$and ${C}_{-}<0$cases. Such a lattice can be realized by stacks of a unit cell with metal/low/high/low index dielectric layers [8]. From the crystal symmetry we expect that Dirac equation formalism used for graphene can be applied in a similar manner.

The governing Hamiltonian of the plasmonic crystal is found as Eq. (1) from the coupled differential equations based on coupled mode theory (CMT) [8–10].

*β*and ${\beta}_{0}$ being the propagation constant of the array and single isolated waveguide, respectively.

*A*and

*B*are the amplitudes of the Floquet-Bloch eigenmodes in A and B sublattice, respectively.

The Hamiltonian ${H}_{0}$ can be transformed into the familiar form of Dirac Hamiltonian through proper transformation. A direct analogy between Hamiltonian ${H}_{0}$ for SPPs in the graphene-like plasmonic crystals and Dirac Hamiltonian for relativistic quantum particles is described in the Appendix. Here we use the original Hamiltonian form ${H}_{0}$ with bases of sublattices A and B for further discussions since it allows easier interpretation of the results.

The eigenvalues and the corresponding eigenvectors of Eq. (1) are given in Eq. (2) with diagonalized Hamiltonian${H}_{0}{}^{\text{'}}$.

The two eigenvalues $\pm K(\eta ,\kappa )$indicate two-sheeted isofrequency surfaces in momentum space. With the asymmetry constant$\eta =-1$, the band structure in momentum space for low *K* exhibits linear Dirac-type spectrum near Dirac point (or diabolical point) as shown in Fig. 1(b). Along the each linear dispersion curves (blue and red lines), the eigenvectors take approximately the same sublattice symmetry, which can be easily confirmed from Eq. (2b) since $\left|\Delta \right|/\Delta \approx \left|\kappa \right|/(-i\kappa )$when $\eta =-1$ and $\left|\kappa \right|<<1$. The blue and red lines emphasize the origin of the linear spectrum with same characteristic symmetry of sublattice. This property allows us to introduce the concept of pseudo-spin for the two component eigenstates in Eq. (2b). The pseudo-spin states have a form of ${\left(\begin{array}{cc}i{e}^{i\theta},& 1\end{array}\right)}^{T}$and θ takes only two values θ = 0 (‘up’ spin) and θ = π (‘down’ spin) corresponding to the blue and the red lines, respectively, since the dimensionality is one. This two-component description is analogous to the graphene’s quasiparticle wave functions and the spinor wave functions in QED. However, the pseudo-spin states for binary plasmonic crystals and graphene indicate the sublattices, not the real spin of electrons [11]. The SPP mode with positive dimensionless propagation constant *K* and positive momentum *κ* belongs to the same branch of the spectrum (blue) for the SPP mode with the opposite *K* and *κ*.

We further introduce chirality *C* as Eq. (3), which is a projection of pseudo-spin on the direction of momentum *κ*.

*γ*is the eigenvalue of

*C.*

The eigenvalue *γ* of the chirality operator *C* takes + 1 and −1 for positive *K* and negative *K* SPPs, respectively. For$\gamma =1$ (positive *K*), the direction of pseudo-spin (θ = 0 for blue line and θ = π for red line) is parallel to the momentum *κ* and antiparallel for $\gamma =-1$ (negative *K*). This property indicates the SPP states near Dirac point have well defined chirality. The chirality is used to refer to the additional built-in symmetry between the SPP modes with positive and negative *K* analogous to graphene [11]. The chirality is conserved since$\left[{H}_{0},C\right]=0$. The conservation of chirality and pseudo-spin could be utilized exploring novel SPP properties such as transmission or tunneling in the plasmonic crystals as for Klein tunneling and backscattering immune property in graphene [12,13]. We will address this in detail elsewhere. We also note that the bands are deformed with appearance of a bandgap for unbalanced coupling ($\eta \ne -1$). This is also analogous to the energy spectrum of graphene with deformed crystal lattice by strong lateral strain [14,15].

## 3. Dirac dynamics of SPP

In the graphene-like plasmonic crystals, a normally incident Gaussian beam splits into two wave packets as shown in Fig. 1(c), which was previously explained in the context of classical conical diffraction [8]. With the proposed concept of pseudo-spin, we gain a new understanding of the phenomenon: The diffraction anomaly is analogous to spin Hall effect, spin accumulation at each side of a sample due to coupling of spin state and motion of the electrons. The SPPs with opposite pseudo-spin states are spatially separated and we can interpret this as an analogous effect that effective magnetic field proportional to the momentum *κ* acts on the pseudo-spin${\sigma}_{y}$, as seen in the Hamiltonian form in Eq. (3a). Since ${\sigma}_{y}$is purely off-diagonal, the effective field *κ* accounts for SPP coupling between A and B sublattices [16]. This interpretation also reminds us spin Hall effect of light [17–19]. However, circular polarization state of the input beam and the gradient of the dielectric constant are not involved here.

Since the dynamics of SPP in the plasmonic graphene is governed by Dirac-like Hamiltonian, we can directly apply the formalism of Dirac dynamics for relativistic 1-D quantum systems [20]. For the positive and negative eigenvalues, we can determine plane wave solutions for a fixed value of transverse momentum *κ* as in Eq. (4):

*K*states are analogous to positive and negative energy states of relativistic particles in QED. If a Gaussian beam is normally incident on the plasmonic crystal and the beam width ${w}_{0}$is much wider than the lattice period

*Λ*, the two sublattices are approximately equally excited. Thus the initial spinor state in position space may be expressed as Eq. (5a). By performing Fourier transform, we also obtain an expression in momentum space as Eq. (5b):

*A*is a normalization factor of the initial state given as $A=1/\sqrt{\sqrt{2\pi}{\omega}_{0}}$with the normalization condition $\int {\left|\varphi (x)\right|}^{2}dx=1$.

The initial state can be decomposed into the positive and negative eigenstates ${u}_{\pm}{}_{K}(\kappa )$as Eq. (6):

*z*is finally given as Eq. (8) in the momentum and position space, respectively.

Figure 2
shows the results for $\eta =-0.99$ and the dimensionless beam width${w}_{0}=8$. The left figures in Fig. 2(a) and 2(b) are the intensity of input beam (probability density of initial spinor state) in momentum, ${\left|\widehat{\varphi}(\kappa )\right|}^{2}$and position space, ${\left|\varphi (x)\right|}^{2}$respectively. The beam width is chosen to ensure that the distribution of the momentum remains in a narrow region centered at $\kappa =0$where the dispersion is linear as seen in the upper right in Fig. 2(c). The middle and the right figures in Fig. 2(a) are projections of ${\left|\widehat{\varphi}(\kappa )\right|}^{2}$onto the positive and negative *K* states in the momentum space, ${\left|{\widehat{\varphi}}^{+}(\kappa )\right|}^{2}$ and ${\left|{\widehat{\varphi}}^{-}(\kappa )\right|}^{2}$, respectively. The input Gaussian beam couples more to the positive *K* state than the negative one ($\int {\left|{\widehat{\varphi}}^{+}(\kappa )\right|}^{2}}d\kappa >{\displaystyle \int {\left|{\widehat{\varphi}}^{-}(\kappa )\right|}^{2}}d\kappa $) and at $\kappa =0$, the input beam only couples to the positive eigenstate. This is due to the mode symmetry at the band edges at $\kappa =0$, where the eigenmodes of degenerate positive and negative *K* state are symmetric and antisymmetric, respectively. Thus the input Gaussian wave packet preferentially couples to the positive state at $\kappa =0$. We observe the same trends in position space as seen in the middle and the right in Fig. 2(b) displaying ${\left|{\varphi}^{+}(x)\right|}^{2}$ and ${\left|{\varphi}^{-}(x)\right|}^{2}$, respectively. Interestingly, the width of ${\left|\varphi (x)\right|}^{2}$ is slightly narrower than ${\left|{\varphi}^{+}(x)\right|}^{2}$and${\left|{\varphi}^{-}(x)\right|}^{2}$, which is attributed to destructive interference between ${\varphi}^{+}(x)$and${\varphi}^{-}(x)$.

The evolution of the initial Gaussian beam along the z direction ${\left|\psi (x,z)\right|}^{2}$ is calculated by Eq. (8-)b) and shown in the left of Fig. 2(c). The input beam is split into two branches with an angle due to the linear Dirac-type spectrum. As we previously stated, we view this as spin Hall effect of SPPs with two pseudo-spin states. For comparison purpose, the diffraction pattern by numerical simulation for specific metal-dielectric nanostructures with balanced coupling in Ref. [8]. is also shown in the lower right. The unit cell parameters for this particular plasmonic crystal are metal (Au, 8nm)/low index dielectric (n = 1.34, 50nm)/high index dielectric (n = 3.48, 200nm)/low index dielectric (n = 1.34, 50nm) at the wavelength of 1550nm. As we observe, the pure analytical approach based on Dirac dynamics reproduces the diffraction pattern by numerical simulation, which proves the validity of our effective Hamiltonian model and Dirac dynamics approach.

If the balance in coupling breaks, the evolution of the wave packet substantially changes. Figure 3
shows the SPP dynamics for$\eta =-0.85$. With breaking of the balance, the band gap opens up and the Dirac point disappears, analogous to massive particles. The term plays a role of mass (see the Appendix). The coupling to the negative *K* state is significantly reduced as seen in Fig. 3(a) and 3(b). This is due to the band deformation near $\kappa =0$and the mode symmetry as previously mentioned. The deformed band leads to more complicated diffraction pattern as seen in the left of Fig. 3(c). Because of the finite distribution of the dispersion slope centered at zero and the momentum distribution in the positive and negative state, the wave packet spatially overlaps and exhibits interference patterns with notable two side lobes at each side. The lower right figure in Fig. 3(c) shows the diffraction pattern for the same metal-dielectric structure except for the increased metal thickness to 10nm from 8nm in Fig. 2(c). It should be noted that properties of real metal-dielectric structures are not fully captured by the Hamiltonian ${H}_{0}$since the nearest neighbor coupling assumption does not hold accurately. For example, the positive and negative *K* dispersion curves are not symmetric due to strong coupling in the real structure. Nevertheless, they are in good agreement with the overall pattern.

With significant unbalance in coupling ($\eta =-0.5$), the input beam couples only to the positive *K* state [Fig. 4(a) and (b)
]. The wave packet evolution shows conventional diffraction broadening in Fig. 4(c) and no anomaly occurs, which agrees with the numerical simulation for the structure with increased metal thickness to 15nm in the lattice.

## 4. Conclusion

We have studied Dirac dynamics of SPP modes in one-dimensional plasmonic crystal with two identical sublattices A and B. The unique symmetry in the plasmonic crystal produces Dirac-type spectrum in k-space, two intersecting linear bands near Dirac point. In analogy with graphene, we have introduced the concept of pseudo-spin and chirality for the SPP mode in the graphene-like plasmonic crystals. The pseudo-spin induced spatial splitting of a normally incident Gaussian beam was interpreted as analogous to spin Hall effect. The developed analytical approach in the frame of 1-D quantum electrodynamics describes well the SPP dynamics in such a unique medium. This work could pave the way for discovering novel phenomena in plasmonic crystals or Dirac metamaterials.

## Appendix

Below we derive Dirac-like Hamiltonian form from the eigenvalue *K* in a similar manner the original Dirac Hamiltonian Eq. (A-2) was derived by linearizing the energy-momentum relation (A-1):

For massless quasiparticles in 1-D, ${H}_{D}={\sigma}_{1}cp$.

From the square of the eigenvalue for ${H}_{0}$in Eq. (A-3), we obtain Hamiltonian${H}_{p}$in Eq. (A-4) near the Dirac point, with which ${H}_{p}{}^{2}$produces the eigenvalue of${K}^{2}$.

By direct comparison with Eq. (A-2) and Eq. (A-4), both Hamiltonians take identical forms with substitution of parameters, $\eta +1\equiv \delta \to m{c}^{2}$ and$\sqrt{-\eta}\equiv {\upsilon}_{D}\to c$. By quick inspection, we notice $(\eta +1)$ plays a role of mass, thus we obtain Dirac-type linear dispersion corresponding to massless Dirac fermions when $\delta =0$ and the band gap, $2\left|\eta +1\right|$ shows up when $\delta \ne 0$.

Also note that the bases of the matrix operator ${H}_{p}$ are different from those of ${H}_{0}$ in Eq. (3a), which has bases of sublattices $|A\u3009$ and $|B\u3009$. If we retain the expansion terms in ${H}_{0}$ up to second order of *κ* as in Eq. (A-5), both Hamiltonians have the same eigenvalues $K=\pm \sqrt{{(\eta +1)}^{2}-\eta {\kappa}^{2}}\text{\hspace{0.17em}}(\kappa <<1)$ and we can obtain ${H}_{p}$ through proper unitary transformation of ${H}_{0}$ as in Eq. (A-6), or vice versa.

where $U=\left(\begin{array}{cc}|{u}_{1}\u3009,& |{u}_{2}\u3009\end{array}\right)$, $T=\left(\begin{array}{cc}|{t}_{1}\u3009,& |{t}_{2}\u3009\end{array}\right)$, and $|{u}_{1,2}\u3009$, $|{t}_{1,2}\u3009$are eigenvectors of ${H}_{0}$,${H}_{p}$, respectively.

## Acknowledgements

This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-6NA25396.

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