photonic band structures of periodic arrays of pores in a

Photonic band structures of periodicarrays of pores in a metallic host:

tight-binding beyond the quasistaticapproximation

Kwangmoo Kim1,2 and D. Stroud1,∗1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

2School of Physics, Korea Institute for Advanced Study, Seoul, 130-722, South Korea∗[email protected]

Abstract: We have calculated the photonic band structures of metallicinverse opals and of periodic linear chains of spherical pores in a metallichost, below a plasma frequency ωp. In both cases, we use a tight-bindingapproximation, assuming a Drude dielectric function for the metalliccomponent, but without making the quasistatic approximation. The tight-binding modes are linear combinations of the single-cavity transversemagnetic (TM) modes. For the inverse-opal structures, the lowest modes areanalogous to those constructed from the three degenerate atomic p-states infcc crystals. For the linear chains, in the limit of small spheres comparedto a wavelength, the results bear some qualitative resemblance to thedispersion relation for metal spheres in an insulating host, as calculatedby Brongersma et al. [Phys. Rev. B 62, R16356 (2000)]. Because theelectromagnetic fields of these modes decay exponentially in the metal,there are no radiative losses, in contrast to the case of arrays of metallicspheres in air. We suggest that this tight-binding approach to photonic bandstructures of such metallic inverse materials may be a useful approach forstudying photonic crystals containing metallic components, even beyondthe quasistatic approximation.

© 2013 Optical Society of America

OCIS codes: (160.5293) Photonic bandgap materials; (230.5298) Photonic crystals.

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1. Introduction

The photonic band structures of composite materials have been studied extensively. Such bandstructures are defined by the relation between frequency ω and Bloch vector k in media inwhich the dielectric constant is a periodic function of position. A major reason for such interestis the possibility of producing photonic band gaps, i.e., frequency regions, extending through allk-space, where electromagnetic waves cannot propagate through the medium. Such media havemany potentially valuable applications, including possible use as filters and in films with rejec-tion-wavelength tuning [1]. In systems with a complete photonic band gap, the spontaneousemission of atoms with level splitting within the gap can be strongly suppressed [2].

Since light cannot travel through the photonic band gap materials (Bragg diffracted back-wards), one of their applications can be a complete control over wasteful spontaneous emissionin unwanted directions when a device, such as a laser, is embedded inside a three-dimensional(3D) photonic crystal [3]. Two-dimensional (2D) photonic crystals can be used as optical mi-crocavities, microresonators [4], waveguides [5], lasers [6], or fibers [7] while one-dimensional(1D) photonic crystals can be used as Bragg gratings or optical switches [8].

The photonic band structure of a range of materials has been studied using a plane waveexpansion method. Typically, the method converges easily when the dielectric function is ev-erywhere real, but more slowly, or not at all, when the dielectric function has a negative realpart, as occurs when one component is metallic. For example, McGurn et al. [9] used thismethod to calculate the photonic band structure of a square lattice of metal cylinders in 2Dand of an fcc lattice of metal spheres embedded in vacuum in 3D. They found that that methodconverged well when the filling fraction f (i.e., volume fraction of metal spheres or cylinders)satisfied f ≤ 0.1%.

Kuzmiak et al. [10] used the same method to calculate the photonic band structures for 2Dmetal cylinders in a square or triangular lattice in vacuum. For low f and ω >ωp, the calculatedphotonic band structures are just slightly perturbed versions of the dispersion curves for elec-tromagnetic waves in vacuum. However, for ω < ωp and H-polarized waves (magnetic field Hparallel to the cylinders), they obtained many nearly flat bands for ω < ωp; these bands werefound to converge very slowly with increasing numbers of plane waves. They later extendedthis work to systems with dissipation [11]. To describe dispersive and absorptive materials,they used a complex, position-dependent form of dielectric function. They also introduced astandard linearization technique to solve the resulting nonlinear eigenvalue problem.

Zabel et al. [12] extended the plane wave method to treat periodic composites withanisotropic dielectric functions. In particular, they studied the photonic band structures of aperiodic array of anisotropic dielectric spheres embedded in air. They found that the anisotropysplit degenerate bands, and narrowed or even closed the band gaps. Much further work onanisotropic photonic materials has been carried out since this paper (see, e.g., [2]).

A different type of periodic metal-insulator composite is a periodic arrangement of metallicspheres in an insulating host. Brongersma et al. [13] studied the dispersion relation for coupledplasmon modes in such a linear chain of equally spaced metal nanoparticles, using a near-fieldelectromagnetic (EM) interaction between the particles in the dipole limit. They also studied thetransport of EM energy around the corners and through tee junctions of the nanoparticle chain-


array. Many other workers have carried extensive work, both theoretical and experimental, onplasmonic waves in 1D and occasionally 2D arrays of metallic nanoparticles in an insulatinghost [14–27]. Park and Stroud [28] also studied the surface-plasmon dispersion relations for achain of metallic nanoparticles in an isotropic medium. They used a generalized tight-bindingcalculations, including all multipoles, but still in the quasistatic approximation where it is as-sumed that ∇×E = 0, where E is the electric field.

Weber and Ford [29] have shown that all calculations within the quasistatic approximationomit important interactions between transverse plasmon waves and free photon modes, even ifthe interparticle separation is small compared to the wavelength of light. Thus, most quasistaticcalculations need to have certain corrections included at particular values of the wave vector.

Later, Gaillot et al. [30] have studied the photonic band structures of another type of structure,a so-called inverse opal structure. This structure is an fcc lattice of void spheres in a host ofanother material. Such a structure can be prepared, e.g., starting from an opal structure madeof spheres of a convenient substance, infiltrating it with another material, then dissolving awaythe spheres. In the work of [30], the photonic band structure of Si inverse opal was calculatedas a function of the infiltrated volume fraction f of air voids using 3D finite difference timedomain (FDTD) method. It was found that for certain values of f , a complete band gap opensup between the eighth and ninth bands.

In the present work, first we study the photonic band structure of an inverse opal structure,such as that investigated in [30], but instead of dielectric materials such as Si, we consider met-als as the infiltrated materials. Thus, the material we study is also the inverse of the fcc array ofmetal spheres studied by McGurn et al. [9] Such metallic inverse opal structures have recentlybecome of great interest, because it has been found that Pb inverse opals exhibit superconduc-tivity [31]. These workers have studied the response of these materials to an applied magneticfield, and have found a highly non-monotonic fractional flux penetration into the Pb spheres asa function of the applied field.

As a second example, we study the photonic band structure of a linear chain of nanoporesin a metallic medium. This is an inverse structure of a linear chain of metallic nanospheres, ofwhich the dispersion relation is given in [13].

For both types of structures, our primary method for studying the photonic band structuresbelow the plasma frequency ωp is a tight-binding approximation which is valid even in the non-quasistatic regime. Because the analogs of the tight-binding atomic states decay exponentiallyin the metallic host medium, the resulting tight-binding waves do not lose energy radiatively,as do the corresponding waves along 1D chains of metallic nanoparticles in air. The absenceof radiative decay has been previously noted in a multiple-scattering calculation of the bandstructure of a periodic array of pores in a host metal [32]. Furthermore, because the modes areexpanded in “atomic” states rather than plane waves, there is no convergence problem as therecan be in the plane wave case.

Wave propagation through void networks is of interest, in part, because of the special for-malism needed to treat it. First, it is straightforward to go beyond the quasistatic regime (inwhich the electric field is assumed to be curl-free), as we discuss below. Thus, calculations canreadily be carried out even for voids which are not much smaller than the wavelength of light.Furthermore, in contrast to waves propagating along chains or other periodic arrays of metalparticles, there are no radiative losses, because the waves in the host region (i.e., in this case,the metallic region) are exponentially decaying.

The remainder of this paper is organized as follows. In Section 2, we first present the formal-ism for calculating the transverse magnetic (TM) and transverse electric (TE) modes of a singlespherical cavity in a metallic host. We then describe the method for calculating the photonicband structures of metallic inverse opals and of linear chains of nanopores in a metallic host,


using a simple tight-binding approach for ω < ωp. In Section 3, we give the numerical resultsfor the TM and TE modes of a single cavity and those of the tight-binding method for the metalinverse opals and the linear chain of nanopores. Section 4 presents a summary and discussion.

2. Formalism

In this section, we present a summary of the equations determining the band structure of aphotonic crystal containing a metallic component with Drude dielectric function ε(ω) = 1−ω2

p/ω2 and an insulating component of dielectric constant unity. The insulating component isassumed to be present in the form of identical spherical cavities of radius R. We first writedown the equations for the TM and TE modes of a spherical cavity in a Drude metal. Then, wepresent a tight-binding method for ω < ωp.

2.1. Spherical Cavity

As a preliminary to calculating the photonic band structure, we first discuss the modes of asingle spherical cavity in a Drude metal host. We begin with the TM modes of the cavity, thenthe TE modes.

2.1.1. TM Modes

It is convenient to describe the modes of the embedded cavity in terms of the B field. To thatend, we combine the two homogeneous Maxwell equations ∇×E = iω

c B and ∇×B =− iωc εE

to obtain a single equation for B. We express the position- and frequency-dependent dielectricfunction ε(x,ω) as 1/ε(x,ω) = θ(x)/(1−ω2

p/ω2)+1−θ(x), where the step function θ(x) =1 inside the metallic region and θ(x) = 0 elsewhere. Then, after a little algebra, we obtain

[ω2 −ω2p (1−θ(x))]∇× (∇×B) =

ω2

c2 (ω2 −ω2p )B. (1)

This expression gives rise to different equations inside and outside the void.For a spherical void within a metallic host, these equations are conveniently solved in spher-

ical coordinates. For the TM modes, the solutions for B and E are given in the standard litera-ture [33]. The coefficients of the solutions inside and outside the cavity can then be determinedfrom the boundary conditions at r = R. The allowed frequencies for ω < ωp are then found tosatisfy

k′2[

j�(kR)+ kR j′�(kR)]=−k2 j�(kR)

k�(k′R)[k�(k

′R)+ k′Rk′�(k′R)

], (2)

where k = ω/c and k′ = (ω2p −ω2)1/2/c are the wave vectors inside and outside the void, j� is

the spherical Bessel function, and k� is the modified spherical Bessel function (note that this k�is different from the wave vectors k and k′). In the limit kR � 1 and k′R � 1, we can readilyobtain the asymptotic forms of the solutions to Eq. (2). The result is k′2(�+1) = k2�, or, for aDrude metal,

ω2 =�+1

2�+1ω2

p . (3)

The largest value, ω =√

2/3ωp, occurs at � = 1 and the limiting value for large � is ω =

ωp/√

2.

2.1.2. TE Modes

A similar procedure for the electric field of the TE modes leads to the self-consistency condition

j�(kR)+ kR j′�(kR) =j�(kR)

k�(k′R)[k�(k

′R)+ k′Rk′�(k′R)

]. (4)


It is readily found that, in the asymptotic regime when kR� 1 and k′R� 1, there are no allowedeigenvalues for the TE modes.

2.2. Tight-Binding Approach to Modes for ω < ωp

We now turn from describing the single-cavity modes to a discussion of the band structure for aperiodic array of such cavities. In conventional periodic solids, the tight-binding method is veryuseful in treating narrow bands. In what follows, we try to suggest an analogous tight-bindingapproach for the lowest set of TM modes in a periodic lattice of spherical cavities in a metallichost, in the frequency range ω < ωp. We apply the resulting method, first, to an fcc lattice ofpores, and then to a linear chain of spherical pores in a metallic host.

Even though these are TM modes, it is convenient to describe them now in terms of theirelectric fields. We denote the electric field of the λ th mode by Eλ (x). This field satisfies

∇× (∇×Eλ (x))+ω2

p θ(x)c2 Eλ (x)≡ OEλ (x) =

ω2λ

c2 Eλ (x), (5)

where O = ∇× (∇×)+ (ω2p/c2)θ(x) is the “Hamiltonian” of this system. Since O is a Her-

mitian operator, the eigenstates corresponding to unequal eigenvalues ω2λ/c2 and ω2

μ/c2 areorthogonal and may be chosen to be orthonormal. (The orthogonality may also be proved di-rectly by integration by parts.) The orthonormality relation is

∫E∗

λ (x) ·Eμ(x)dx = δλ ,μ . (6)

Since Eλ (x) is real for ω < ωp, the complex conjugation is, in fact, unnecessary.In Sec. 2.1.1, our paper already gives the equations determining the electric and magnetic

fields of isolated TM modes for a spherical cavity. The lowest set corresponds to �= 1, and thereshould be three of these. For a spherical cavity, all three are degenerate, i.e., all three have thesame eigenfrequencies. Even though the three modes have equal frequencies, one can alwayschoose an orthonormal set, with electric fields E1, E2, and E3 satisfying the orthonormalityrelation in Eq. (6).

In order to obtain the tight-binding band structure built from these three modes, we need tocalculate matrix elements of the form

Mα,β (R) =∫

E∗α(x) ·OEβ (x−R)dx, (7)

corresponding to two single-cavity modes associated with different cavities centered at theorigin and at R. Here, O is the “Hamiltonian” of the system as defined implicitly in Eq. (5).

Next, we introduce normalized Bloch states associated with the three � = 1 single-cavitymodes. In order to do this, we first make the standard tight-binding assumption that the “atomic”states corresponding to different cavities are orthogonal:

∫E∗

λ (x−R) ·Eμ(x−R′)dx = δλ ,μ δR,R′ . (8)

This orthogonality of states on different cavities is reasonable since the fields fall off exponen-tially with separation.

The orthonormal Bloch states then take the form

Ek,λ (x) = N−1/2 ∑R

eik·REλ (x−R), (9)


where k is a Bloch vector, and the R’s are the Bravais lattice vectors. In writing Eq. (9), wehave assumed that there are N identical spherical cavities, and that the Bloch states satisfy theusual periodic boundary conditions of Born-von Karman type. We also introduce the elementsof the “Hamiltonian” matrix

Mλ ,μ(k) = ∑R

eik·RMλ ,μ(R). (10)

We can then obtain the frequencies ω(k) by diagonalizing a 3×3 matrix as follows:

det

∣∣∣∣Mλ ,μ(k)−

(ω2(k)

c2 − ω2at

c2

)δλ ,μ

∣∣∣∣= 0, (11)

where ωat is the eigenvalue of a single-cavity mode. The solutions to these equations give thethree p-bands for a periodic lattice of cavities in a metallic host. This procedure is analogous tothat used in the well-known procedure for obtaining tight-binding bands from three degeneratep-bands in the electronic structure of conventional solids (see, e.g., [34]).

We briefly comment on the connection between this approach and that used by earlier work-ers [13,28]. In this work, the authors treat wave propagation along a chain of metallic nanopar-ticles. They use the tight-binding approximation, as we do, but in the quasistatic approximationin which one assumes that ∇×E = 0. This approximation is reasonable when both the particleradii and the interparticle separations are small compared to a wavelength, but is not accurate inother circumstances. Furthermore, even in the small-particle and small-separation regime, thisapproximation still fails to account for the radiation which occurs at certain wave numbers andfrequencies. The present approach would generalize this tight-binding method to (a) 3D as wellas 1D; (b) pore modes instead of small particle modes; and most importantly (c) larger poresand larger interparticle separations, via extension beyond the quasistatic approximation.

Next, we discuss the numerical evaluation of the required matrix elements, Eq. (7). Therelevant electric fields are given in this paper, but in spherical coordinates. It is not difficult toconvert these into Cartesian coordinates. The operator O is just a little trickier. We first notethat O = OR +O ′, where OR is the single-cavity operator: OR = ∇× (∇×) if x is inside theRth cavity and OR = ∇× (∇×)+ω2

p/c2 otherwise. Now we also have

OREβ (x−R′) =ω2

at

c2 Eβ (x−R′), (12)

since Eβ is an eigenstate of OR with an eigenvalue ω2at/c2.

But since we are assuming that the overlap integral between “atomic” electric field statescentered on different sites vanishes, the term involving OR does not contribute to the matrixelement Mα ,β , which is therefore just given by

Mα ,β (R) =∫

Eα(x) ·O ′Eβ (x−R)dx. (13)

We can also write

O ′ =ω2

p

c2 ∑R′

θR′(x), (14)

whereθR′(x) = θ(x−R′), (15)

is a step function which is unity inside the cavity centered at R′ and is zero otherwise.


A reasonable approximation to Eq. (14) might be to include just R′ = 0. In this case, wefinally will get

Mα ,β (R)∼ ω2p

c2

∫Eα(x) ·Eβ (x−R)dx, (16)

where the integral runs just over the cavity centered at the origin. As a further approximation,we can just replace Eβ (x−R) by the value of this function at the origin, i.e., Eβ (−R). Thenthis field can be taken outside the integral and we just have

Mα ,β (R)∼ ω2p

c2 Eβ (−R) ·∫

Eα(x)dx, (17)

where once again the integral runs over the cavity centered at the origin.The calculation of this matrix element from this integral expression is straightforward. The

Cartesian components of the normalized eigenfunctions Eα(x) are readily calculated from thesolutions discussed in Sec. 2.1.1 and the integral in Eq. (17). In fact, it turns out that all the in-tegrals entering the matrix element can be obtained analytically. Given this matrix element, thecomputation of the full tight-binding band structure of the three p-bands is also straightforward.

3. Numerical Results

Before showing our numerical results, we first point out that, for a metal with a Drude di-electric function, the photonic band structures can be entirely expressed in terms of suitabledimensionless parameters. Specifically, for the inverse opal structure, the two relevant dimen-sionless parameters are R/d, the ratio of the void radius to the fcc lattice constant, and ωpd/c.Of course, other combinations of these parameters would serve equally well. Given these pa-rameters, the scaled frequencies ω/ωp are functions of the scaled wave vector kd. In whatfollows, we consider only those scaled units.

For the inverse opals we arbitrarily assume that the ratio of void sphere radius R to the nearestneighbor distance d/

√2 is 3/10, or R/d = 3/(10

√2), corresponding to a sphere filling fraction

f = 0.160. In our calculations, we also arbitrarily use the value ωpd/c = 1. For typical metallicvalues of ωp, this would correspond to d of order 20 nm. For the linear chain of nanopores (seebelow), we use d to denote the separation between the centers of two adjacent nanopores andR to denote the radius of a nanopore as in Fig. 1(b); we take the ratio R/d = 1/3 at first, thenchange it later.

Our band structures for the inverse opals are expressed in terms of the standard nota-tion for k values at symmetry points in the Brillouin zone. These are Γ = (0,0,0), X =(2π/d)(0,0,1), U = (2π/d)(1/4,1/4,1), L = (2π/d)(1/2,1/2,1/2), W = (2π/d)(1/2,0,1),and K = (2π/d)(3/4,0,3/4).

The metallic dielectric functions we assume for the inverse opals and linear chain ofnanopores are of the usual Drude form,

ε(ω) = 1− ω2p

ω2 , (18)

where ωp is the plasma frequency of the conduction electrons. ε(ω) < 0 when ω < ωp, whileε(ω) > 0 when ω > ωp. Our calculations are thus carried out assuming that the Drude relax-ation time τ → ∞. For a metal in its normal state, ω2

p = 4πne2/m, where n is the conductionelectron density and m is the electron mass. Note that with this choice of dielectric function,the entire band structure can be expressed in scaled form of ω/ωp.

Since we are considering void spheres in inverse opals and linear chains of nanopores, it is ofinterest to consider electromagnetic wave modes in a single cavity, which could be considered


Fig. 1. Schematic diagram for (a) an inverse opal structure with a lattice constant d anda void sphere radius R; (b) a linear chain of nanopores with a pore separation d and ananopore radius R.

a single “atom” of the void lattice. We show only results for ω < ωp, since these are the resultsmost relevant to possible narrow-band photonic states in the inverse opal structure. Our resultsfor ω <ωp for an isolated spherical cavity in an infinite medium, and when kR� 1 and k′R� 1are given in Table 1. These two inequalities are reasonable for the choice of “inverse opal”system parameters R/d = 3/(10

√2) and ωpd/c = 1, because

kR =ωc

R <ωp

cR =

ωpd

cRd=

3

10√

2= 0.2121,

k′R =

√ω2

p −ω2

cR =

√(ωpR

c

)2

−(

ωRc

)2

=

√(ωpd

cRd

)2

− (kR)2

=

√(3

10√

2

)2

− (kR)2 =√

0.045− (kR)2 <√

0.045 = 0.2121. (19)

The (modified) spherical Bessel functions in Eq. (2) are extremely close to the ω axis for � > 5,so that it is difficult to get eigenfrequencies for � > 5 in the isolated spherical cavity. Howeverthe eigenfrequencies continue to exist even for � > 5 when kR � 1 and k′R � 1.

The solutions to Eq. (4) do not exist for ω < ωp with ωpd/c = 1. This fact is consistent withthat the eigenvalues for ω < ωp do not exist for TE modes when kR � 1 and k′R � 1.


Table 1. TM mode frequencies ω ′ = ωd/(2πc), where ω < ωp and ωpd/c = 1, calculatedfor an isolated spherical cavity (“Infinite medium”) and those when both kR� 1 and k′R�1. The (modified) spherical Bessel functions are extremely close to the ω ′ axis for � > 5, sothat it is difficult to get eigenfrequencies for � > 5 in the isolated spherical cavity. Howeverthis does not happen when kR � 1 and k′R � 1.

Infinite medium kR � 1, k′R � 1�= 1 0.1296 0.1299�= 2 0.1232 0.1233�= 3 0.1203 0.1203�= 4 0.1186 0.1186�= 5 0.1178 0.1175

For our fcc calculations, we calculate the band structure including only the 12 nearest-neighbors of the cavity at the origin. Thus R = (d/2)(±1,±1,0), (d/2)(±1,∓1,0),(d/2)(±1,0,±1), (d/2)(±1,0,∓1), (d/2)(0,±1,±1), and (d/2)(0,±1,∓1). Assumingωpd/c = 1.0 and using ωatd/(2πc) = 0.1296 (ωat = 0.8143ωp) for �= 1 in an infinite medium,we get the tight-binding results in Fig. 2. This Figure shows three separate bands in the X-U-Lregion and X-W -K region, which behave as expected for the p-bands. The bandwidth is rela-tively small as Mα ,β (R)d2 ∼ 0.001, which proves the general relation between the bandwidthand the overlap integral [34]. All three bands are degenerate at k = 0 (the Γ point). In addition,there is a double degeneracy when k is directed along either a cube axis (Γ-X) or a cube bodydiagonal (Γ-L), the higher (concave upward) bands being degenerate in both cases. The lowertwo bands have a band gap at the U point, and these bands cross at the W point.

0.74

0.76

0.78

0.80

0.82

0.84

0.86

Χ U L Γ Χ W K

ω/ω

p

Fig. 2. Tight-binding inverse opal band structure for ω < ωp with R/d = 3/(10√

2) andωpd/c= 1.0, using ωatd/(2πc)= 0.1296 (ωat = 0.8143ωp) for �= 1 in an infinite medium.The horizontal dotted line represents the “atomic” level.

In Fig. 3, we plot the frequency of the triply degenerate Γ point as a function of the numberof nearest neighbor shells included, up to seven shells. It can be seen that the frequency changessignificantly over this range of shells, though we find, of course, that the triple degeneracy is re-tained (since it is required by symmetry). The band structure will certainly converge extremelywell for a sufficient number of neighbor shells, because the hopping integral will eventually fall


off exponentially with separation, with an inverse decay length (ω2p −ω2)1/2/c. For the present

case, ωpd/c = 1.0, so this exponential decay does not fully set in until a fairly large number ofshells is included. For a larger value of ωpd/c, the eigenfrequencies will converge much morequickly with number of shells. This problem is worse in 3D than in 1D (see below), because themagnitude of the hopping integral, as a function of separation r, varies as 1/r3 for small ωpr/c,while the number of terms in each shell increases as r2. Convergence is assured, however, witha sufficient number of shells, because of the exponential decay which sets in at large r. As isseen below, the convergence is much faster in 1D.

0.794

0.796

0.798

0.800

0.802

0.804

0.806

0.808

0.810

1 2 3 4 5 6 7

ω/ω

p

nth n.n.

Fig. 3. Dependence of the triply degenerate frequency at Γ on the number of nearest neigh-bor shells included in the tight-binding calculation, for the inverse-opal calculation shownin Fig. 2. Up to seven shells are included.

Next, we turn to the band structure of a periodic linear chain of spherical nanopores in aDrude metal host. For this linear chain, the Bravais lattice vectors are R = d(0,0,±n), where±n is the nth nearest-neighbor, d is the separation between two nanopores and we assumethat the chain is directed along the z axis. We can calculate the tight-binding band structureincluding as many sets of neighbors ±n as we wish. We use the ratio R/d = 1/3 and thedimensionless parameter ωpd/c = 0.35. These are arbitrarily chosen to be the same as used in[13]. The “atomic” frequency is found by solving Eq. (2) and gives ωatd/(2πc) = 0.0454 (ωat =0.8150ωp) for � = 1 in an infinite medium. Our resulting tight-binding dispersion relationsare shown by open triangles in Fig. 4 with only nearest-neighbors included. The transverse(T) branches are twofold degenerate, while the longitudinal (L) branch is non-degenerate. Aswe increase the number of nearest-neighbors (nn’s) included, the separation between the Land T branches increases at the zone center but decreases at the zone boundary, as shownin Fig. 4. The sum also converges quickly, so there is only a slight difference between thedispersion relation including through the next-nearest-neighbors and that including throughthe 5th nearest-neighbors. However, if one includes more than nearest-neighbor overlap, the Land T branches no longer cross exactly at k = ±π/(2d). [32] considered the long-wavelengthtight-binding limit of their multiple-scattering dispersion relation for a periodic linear chain ofdielectric cavities in a metallic host, and also found that the L and T branches of the � = 1dispersion relations crossed at k =±π/(2d) if only nearest-neighbor hopping is included.

To check the convergence of the number of nn’s included, we have calculated the ω at k = 0since the difference between different numbers of nn’s is the most evident there. For ωpd/c =1.0 and ωatd/(2πc) = 0.1291 (ωat = 0.8112ωp), we find that both the L and T frequencies


0.78

0.79

0.80

0.81

0.82

0.83

0.84

0.85

-π/d -π/2d 0 π/2d π/d

ω/ω

p

k

L

T

nn’snnn’s5nn’s

Fig. 4. Tight-binding results of a periodic chain of nanopores in a Drude metal host, for ω <ωp. We take R/d = 1/3 and ωpd/c = 0.35, using ωatd/(2πc) = 0.0454 (ωat = 0.8150ωp)for � = 1 in an infinite medium. The horizontal dotted line represents the “atomic” level.Three different numbers of neighbors are included: nearest-neighbors (nn’s), next-nearest-neighbors (nnn’s), and fifth-nearest-neighbors (5nn’s). In this and the following two plots,“L” and “T” denote the longitudinal and transverse branches, respectively.

change by less than 0.3% in going to the 5th nn shell, and are unchanged to within 0.05%thereafter, up to 10 nn shells. This convergence is quicker than in 3D and can be readily seen inthe ω(k) plots.

We have carried out similar calculations using other values of the parameter ωpd/c, namely1.0, 2.0, and 5.0. Such calculations are possible here because our calculations are non-quasistatic, so that the overlap integral between neighboring spheres falls off exponentiallywith separation. The results, and the corresponding results including more overlap integrals,for a typical example, ωpd/c = 5.0, are shown in Fig. 5, since the results for ωpd/c = 1.0 and2.0 are similar to Fig. 4 except for the increase of ω along the y-axis and ωat. It is also strikingthat, as ωpd/c increases in going from Fig. 4 to 5, the ratio rLT of the width of the L band tothat of the T band decreases. In Fig. 4, rLT > 1, while in Fig. 5, rLT < 1.

One could also say that, except for an overall scale factor, Fig. 5 looks like an inverted imageof Fig. 4 about the horizontal line of ωat. For the nn case, the T and L bands cross at ±π/(2d),while they cross at smaller values than |π/(2d)| when further neighbors are included, as can beseen in Figs. 4 and 5, but the crossing points get closer to ±π/(2d) as ωpd/c increases. Also,the effects of including further neighbors become smaller as ωpd/c increases; they are smallestat ωpd/c = 5.0, as can be seen in Fig. 5.

Next we consider values of R/d other than 1/3, but still keeping the same value of ωpd/c =0.35 and including up to the fifth nearest-neighbors. For a smaller R/d = 0.25, the variationof the band energies with k becomes smaller, as seen by open triangles in Fig. 6, than it is inFig. 4, but the crossing points between the L and T branches still occur at values of |k| slightlyless than |π/(2d)|. This behavior becomes clearer when the results for more values of R/d areplotted together as in Fig. 6. As R/d increases, the variation of the band energies with k, and theseparation between the L and T branches at both the zone center and zone boundary, increase,but the L and T branches still cross at values of |k| slightly less than |π/(2d)|. Furthermore,the separation between the L and T bands increases slightly at k = 0, but decreases slightly atk = ±π/d compared to the results with only nn’s included. We show only R/d up to 0.4 in


0.700

0.705

0.710

0.715

0.720

0.725

0.730

-π/d -π/2d 0 π/2d π/d

ω/ω

p

k

L

T

nn’snnn’s5nn’s

Fig. 5. Same as Fig. 4, except ωpd/c = 5.0 and ωatd/(2πc) = 0.5691 (ωat = 0.7152ωp).

this Figure because, in the quasistatic limit, there is evidence that for larger values of R/d thedispersion relations are significantly modified by higher values of � [28].

0.76

0.78

0.80

0.82

0.84

0.86

0.88

-π/d -π/2d 0 π/2d π/d

ω/ω

p

k

L

T

R/d = 0.25R/d = 0.33R/d = 0.40

Fig. 6. Plotting together three different results for ω < ωp, all with ωpd/c = 0.35, butwith different (R/d)’s: R/d = 0.25 and ωatd/(2πc) = 0.04546 (ωat = 0.8161ωp); R/d =0.33 and ωatd/(2πc) = 0.04544 (ωat = 0.8157ωp); R/d = 0.40 and ωatd/(2πc) = 0.04543(ωat = 0.8156ωp), with inclusion of up to the fifth nearest-neighbors. For the larger valuesof R/d, it may be necessary to include more than just �= 1.

4. Discussion

In this work we have calculated the photonic band structures of metal inverse opals and of alinear chain of spherical voids in a metallic host for frequencies below ωp, when � = 1 usinga tight-binding approximation. In both cases, we include only the � = 1 “atomic” states of thevoids. As a possible point of comparison, we have also computed the same band structuresusing the asymptotic forms of the spherical and modified spherical Bessel functions for smallvoid radius. In this asymptotic region, there are only TM modes. The results for the linear chain


of voids somewhat resemble those of [13] for a chain of metallic spheres in an insulating host,except that the L branch lies above the doubly degenerate T branch.

It is of interest to compare our work to that of other workers. In particular, Lidorikis etal. [35] have previously developed a method based on a linear combination of “atomic” orbitalsto treat propagation of photonic waves in a periodic composite. Their method gives an excellentaccount of the photonic band structure of a periodic array of parallel dielectric cylinders in ahost of a smaller dielectric constant, and agrees very well with that computed by expanding thefields in plane waves. However, our work deals with a very different system from [35]. We treata metal dielectric composite, in which the dielectric is a network of voids and the metal hasa frequency-dependent dielectric function which is negative at the frequencies of interest. Bycontrast, [35] deals with a composite in which both components have frequency-independent,positive dielectric constants. As a result, our tight-binding bands are linear combinations of theindividual plasmons associated with the voids, whereas those of [35] come from Mie resonancesof the 2D dielectric cylinders. We believe that these can never be very narrow if the cylindershave a positive dielectric constant, and therefore, their bands are not plasmonic in character asours are.

A somewhat different method, based on multiple scattering theory, has been used to treat 1Dchains of nanocavities in a metallic host in [32], and a 3D extension of this multiple-scatteringmethod has been used to treat an fcc crystal of silicon spheres in a metallic host in [36]. Al-though the approach in these two papers is quite different from the tight-binding method used inour calculations, it is still of interest to compare results from the two methods where possible.For example, we can compare our tight-binding results for the inverse opal structure [Fig. 2]along the Γ−X direction in the Brillouin zone with those presented in [36] in their Fig. 1(a). Aquantitative comparison is not possible, because our calculations and those of [36] are carriedout for a quite different fcc lattice constant, pore volume fraction, and dielectric constant of thematerial in the pore space (ε = 11.9 in [36], ε = 1 in our calculations). Nevertheless, the doublydegenerate bands in our calculation appear to have the same general shape as the correspond-ing bands shown in their Fig. 1(a), that is, an increase with increasing kz starting from the Γpoint followed by a flattening of these bands as the point X is approached. The non-degenerateband along Γ−X in our calculations falls monotonically starting from the Γ point (lower bandalong the line Γ−X in our Fig. 2), whereas the corresponding band shown in their Fig. 1(a)seems to be nearly flat. We tentatively attribute this difference primarily to the difference in theparameters of the two calculations.

In tight-binding calculations, it is important to ascertain how sensitive the results are to thenumber n of nearest neighbor shells included in the calculations. In order to answer this ques-tion, we show in Fig. 3 the calculated energies at Γ as functions of n, for the inverse opal struc-ture. As mentioned earlier, the frequency will certainly converge extremely well for a sufficientnumber of neighbor shells, because the hopping integral will eventually fall off exponentiallywith separation. This convergence is faster in 1D than in 3D because the magnitude of the hop-ping integral varies as 1/r3 for small ωpr/c, while the number of terms in each shell increasesas r2.

In the case of 1D bands, [32] shows that their multiple-scattering formalism, in the limitka � 1 (where a is the cavity radius), leads to a dispersion relation of the tight-binding formω = ω0 +ω1 cos(kd), where d is distance between the sphere centers. This result is obtainedprovided that one neglects interactions other than between nearest neighbor cavities, and alsothat one disregards interactions between the lowest (� = 1) plasmon band and all the higherbands. In our present tight-binding calculation, we do not need to assume ka � 1, and we arealso able to go beyond nearest neighbor hopping, though we do include only the � = 1 bands.Thus, our approach is somewhat different from that of [32], though it gives similar results in


certain parameter regimes. While we do not include modes with � > 1, our approach couldreadily be extended to do so. Finally, our bands do not cross exactly at |k|d = π/2 when weinclude more than one shell of neighbors in the tight-binding calculation; so our dispersionrelations are not exactly of the form ω0 +ω1 cos(kd) found in [32]. Thus, in short, our 1Dresults bear some similarities to those of [32], but are obtained in a different way and arecalculated in somewhat different regimes.

Next, we briefly discuss the non-orthogonality of our basis functions for the TM “atomic”modes corresponding to different spherical voids. The electric fields of these modes are givenin [33]. This non-orthogonality also arises in the usual tight-binding method as applied to elec-tronic states in solids (see [34]), when one uses atomic states as basis functions. In that case,the common procedure is to neglect the overlap integral between states centered on differentatoms. In our case, we have neglected the overlap integral

Sλ μ(R) =∫

E∗λ (x) ·Eμ(x−R)dx (20)

for modes centered on voids separated by a nonzero lattice vector R. This neglect is reasonable,for our problem, because the basis functions decay exponentially into the metal, and thus theoverlap should be small.

If one wishes to correct for the non-orthogonality, there is a well-established procedurefor doing so. It has been discussed in the electronic case, e.g., in [37] and [38], based on atransformation originally developed by Lowdin [39]. For the present problem, one would ob-tain a Hamiltonian H with an orthonormal basis from the the original Hamiltonian H with anonorthogonal basis by means of the transformation

H = (I+S)−1/2H (I+S)−1/2, (21)

where S is a matrix whose elements are given by Eq. (20), and I is the identity matrix. To carryout this procedure, one would simply need to calculate the overlap matrix elements Sλ μ(R) andexecute the transformation in Eq. (21). We believe that the resulting corrections to the bandstructure would be small, because, as in the analogous electronic problem for narrow tight-binding bands, the off-diagonal elements of the overlap matrix are expected to be small forexponentially decaying basis functions, compared to those of the original Hamiltonian H .

In the quasistatic case, for metal grains in air, when R/d is greater than about 0.4, it becomesimportant to include more than just �= 1, as in [28]. Inclusion of such higher �’s might be ratherdifficult in the present dynamical case, though it would be straightforward in the quasistaticlimit for 1D chains of spherical nanopores.

In summary, we have described a tight-binding method for calculating the photonic bandstructure of a periodic composite of spherical pores in a metallic host, and have applied it toboth 1D and 3D systems. The method is fully dynamical, and is not limited to very small pores.The method does not have the convergence problems found when the magnetic or electric fieldis expanded in plane waves. Furthermore, there are no radiation losses to consider, unlike thecomplementary case of small metal particles in an insulating host, because the fields associatedwith these modes outside the pores are exponentially decaying. Thus, this method may be usefulfor a variety of periodic metal-insulator composites. It would be of interest to compare thesecalculations to experiments on such materials.

Acknowledgments

This work was supported by the National Science Foundation through the Materials ResearchScience and Engineering Center at The Ohio State University (DMR-0820414), and by De-partment of Energy Grant No. DE-FG02-07ER46424. All of the calculations using plane wave


expansions were carried out on the P4 Cluster at the Ohio Supercomputer Center, with the helpof a grant of time. We also thank Korea Institute for Advanced Study for providing computingresources (KIAS Center for Advanced Computation Abacus) for this work.


photonic band structures of periodic arrays of pores in a

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