Universal behavior of dispersive Dirac cone in plasmonic metamaterials

Matthias Maier,1, ∗ Marios Mattheakis,2, † Efthimios Kaxiras,2, 3 Mitchell Luskin,1 and Dionisios Margetis4

1School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA2School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA4Department of Mathematics, and Institute for Physical Science and Technology,

and Center for Scientific Computation and Mathematical Modeling,University of Maryland, College Park, Maryland 20742, USA.

(Dated: Draft of November 8, 2017)

We demonstrate analytically and numerically that the dispersive Dirac cone emulating an epsilon-near-zero (ENZ) behavior is a universal property of plasmonic crystals consisting of 2D metals. Ourstarting point is a periodic array of two-dimensional metallic sheets embedded in an inhomoge-neous and anisotropic dielectric host. By invoking a systematic bifurcation argument for arbitrarydielectric profiles, we show how transverse-magnetic Bloch waves experience an effective dielectricfunction that averages out microscopic details of the host medium. The corresponding effective dis-persion relation reduces to a Dirac cone when the conductivity of the metallic sheet and the periodof the array satisfy a critical condition for ENZ behavior. Our analytical findings are in excellentagreement with numerical simulations.

In the past few years the dream of manipulating thelaws of optics at will has evolved into a reality with theuse of metamaterials. These structures have made it pos-sible to observe aberrant behavior like no refraction, re-ferred to as epsilon-near-zero (ENZ) [1–3], and negativerefraction [4]. This level of control of the path and dis-persion of light is of fundamental interest and can lead toexciting applications. In particular, plasmonic metama-terials offer significant flexibility in tuning permittivity orpermeability values. This advance has opened the door tonovel devices and applications that include optical holog-raphy [5], tunable metamaterials [6, 7], optical cloaking[8, 9], and subwavelength focusing lenses [10, 11].

Plasmonic crystals, a class of particularly interestingmetamaterials, consist of stacked metallic layers arrangedperiodically with subwavelength distance, and embed-ded in a dielectric host. These metamaterials offer new‘knobs’ for controlling optical properties and can serveas negative-refraction or ENZ media [12–14]. The ad-vent of truly two-dimensional (2D) materials with a widerange of electronic and optical properties, comprisingmetals, semi-metals, semiconductors, and dielectrics [15],promise exceptional quantum efficiency for light-matterinteraction [16]. In this Letter, we characterize the ENZbehavior of a wide class of plasmonic crystals by using ageneral theory based on Bloch waves.

The ultra-subwavelength evanescent waves (plasmons)found in plasmonic crystals based on 2D metals, in addi-tion to providing extreme control over optical properties[17–20], also demonstrate low optical losses due to re-duced dimensionality [3, 4]. In particular, graphene isa rather special 2D plasmonic material exhibiting ultra-subwavelength plasmons, and a high density of free car-riers which is controllable by chemical doping or biasvoltage [19, 21–23]. An important finding is that theENZ behavior introduced by subwavelength plasmons is

characterized by the presence of dispersive Dirac conesin wavenumber space [2, 3]. This linear iso-frequencydispersion relation was shown for the special case ofplasmonic crystals containing 2D dielectrics with spatial-independent dielectric permittivity. This relation re-quires precise tuning of system features [3]. It is notclear from this earlier result to what extent the ENZ be-havior depends on the homogeneity of the 2D dielectric,or could be generalized to a wider class of materials.

In this Letter, we show that the occurrence of disper-sive Dirac cones in wavenumber space is a universal prop-erty of plasmonic crystals with dielectrics characterizedby any spatial-dependent dielectric permittivity within aclass of anisotropic materials. We provide an exact ex-pression for the critical structural period at which themultilayer system behaves as an ENZ medium. This dis-tance between adjacent sheets depends on the permittiv-ity profile of the dielectric host as well as on the surfaceconductivity of the 2D metallic sheets. In addition, wegive an analytical derivation and provide computationalevidence for our predictions. To demonstrate the applica-bility of our approach, we investigate numerically electro-magnetic wave propagation in finite multilayer plasmonicstructures, and verify the ENZ behavior at the predictedstructural period. These results suggest a systematic ap-proach to making general and accurate predictions aboutthe optical response of metamaterials based on 2D mul-tilayered systems. An implication of our method is theemergence of an effective dielectric function in the dis-persion relation, which can be interpreted as the resultof an averaging procedure (homogenization). This viewfurther supports the universal character of our theory.

Consider a plasmonic crystal that is periodic in the xdirection and consists of 2D metallic sheets of isotropicconductivity σ. Each sheet is parallel to the yz-planeand positioned at x = nd for integer n. The material

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filling the space between any two consecutive sheets isdescribed by the anisotropic relative permittivity ten-sor diag (εx, εy, εz), where εx = constant, and εy(x) =εz(x) depends on the spatial coordinate x with periodd. Here, we set the vaccuum permittivity equal to unity,ε0 = 1. We seek solutions of time-harmonic Maxwell’sequations with transverse-magnetic (TM) polarization,that is, with electric and magnetic field componentsE = (Ex, 0, Ez) and H = (0, Hy, 0). The assumed TM-polarization and the symmetry of the physical systemsuggest that

Ez(x, z) = E(x) eikzz,

which effectively reduces the system to a 2D problem.Substituting the ansatz into time-harmonic Maxwell’sequations and eliminating Ex and Hy leads to the fol-lowing ordinary differential equation for E(x):

−∂2xE + κ(kz)εz(x)E = 0, κ(kz) =

k2z − ω2µεx

εx, (1)

where µ denotes the permeability of the ambient materialand ω is the angular frequency. The metallic sheets giverise to transmission conditions at x = nd, viz.,E

+ = E−,

−i(ω/σ)[(∂xE

)+ − (∂xE)−] = κ(kz) E+,

where (.)± indicates the limit from the right (+) or theleft (−) of the metallic interface. In order to close thesystem, we make a Bloch-wave ansatz in the x-direction,with kx denoting the real Bloch wavenumber:

E(x) = eikxdE(x− d), ∂xE(x) = eikxd∂xE(x− d).

The combination of the transmission condition and theperiodicity assumption leads to a closed system consist-ing of Eq. (1) and the following boundary conditions:[

E(d−)E ′(d−)

]= eikxd

[1 0

−i(σ/ω)κ(kz) 1

] [E(0+)E ′(0+)

],

with E ′(x) = ∂xE(x). We next identify the dispersionrelation between kx and kz. In the following analy-sis, we work in the 2D wavenumber space with k =(kx, kz). To render Eqs. (1) with above boundary condi-tions amenable to analytical and numerical investigation,we perform an additional algebraic manipulation: LetE(1)(x) and E(2)(x) be solutions of Eq. (1) with initialconditions

E(1)(0) = 1, E ′(1)(0) = 0, E(2)(0) = 0, E ′(2)(0) = 1. (2)

These solutions are linearly independent and thereforethe general solution of the system is given by E(x) =

c1E(1)(x) + c2E(2)(x). The existence of a non-trivial solu-tion implies the condition

D[k] = det

(eikxd

[1 0

−i(σ/ω)κ(kz) 1

]

−[E(1)(d) E(2)(d)E ′(1)(d) E ′(2)(d)

])= 0. (3)

Equation (3) expresses an implicit dispersion relation,namely, to find the locus of points k such that D[k] = 0.We will present a number of analytical and computa-tional results for different examples of permittivity pro-files εz(x). We also address the problem of arbitraryεz(x). Note that the case of constant permittivity,εz(x) = const., was discussed in [3].

For certain permittivity profiles εz(x) with period d,the system of Eqs. (1) and (2) admits exact, closed-formsolutions. Hence, Eq. (3) is made explicit. For example,consider the parabolic profile

εz(x) = εz,0

[1 + 6α

x

d

(1− x

d

)], (4)

which is well known in optics [24, 25]. Here, α > 0 isa scaling parameter with background dielectric permit-tivity εz,0 > 0. In this case, E(1) and E(2) can be writ-ten in terms of closed-form special functions (see supple-mentary material). Relation (3) can be further simpli-fied in the vicinity of the center of the Brillouin zone,where |kxd| � 1, by choosing the branch of the disper-sion relation containing k∗ = (k∗x, k

∗z) = (0,±ω√µεx).

As a result of this simplification, the Bloch wave seesa homogeneous medium with effective permittivity ε =diag

(εx, ε

effz , ε

effz

). The dispersion relation is

k2x

εeffz

+k2z

εx= k2

0,εeffz

εz,0= 1 + α− ξ0

d. (5)

In the above, ξ0 denotes the plasmonic thickness, ξ0 =−iσ/(ωεz,0) [3, 4, 18]. Here, we assume for the sake ofargument that σ is a purely imaginary number so thatξ0 is real valued. Below, we will provide a derivation of(5) for general profiles εz(x). Dispersion relation (5) isvalid in a neighborhood of k∗. For εeff

z ≷ 0, this relationdescribes an elliptic, or hyperbolic band, respectively.

The ENZ behavior is characterized by εeffz ≈ 0 in dis-

persion relation (5) [3]. In the case of the parabolic profilethis condition is achieved if ξ0/d = 1+α. This motivatesthe definition of a critical ENZ structural period,

d0 = ξ0/(1 + α

). (6)

A breakdown of Eq. (5) due to εeffz = 0 is a necessary

condition to observe linear dispersion and thus dispersiveDirac cones [3]. Even though Eq. (5) is an approximateformula describing the dispersion relation in the neigh-borhood of k∗, the ENZ condition d = d0 is exact for the

3

(a) (b) (c)

FIG. 1: Parameter study of numerically computed dispersion curves for (a) the parabolic profile (4) with the scalingparameter α = 20/3, (b) double-well profile fdw and (c) non-symmetric profile fns. d/d0 is chosen in the range from0.8 to 1.2. The red solid lines indicate the Dirac cones dispersion at d = d0. The inset shows the dispersion relation

at the center of the Brillouin zone k∗ as function of d (blue and green cutlines in the major image).

existence of a Dirac cone for this example of a parabolicprofile.

Next, we verify the effective theory given by Eqs. (5)and (6) numerically. In order to compute all real-valueddispersion bands located near k∗, we solve the system ofEqs. (1), (2), and (3) (for details see supplementary ma-terial). We carry out a parameter study with the scalingparameter α = 20/3, background permittivity compo-nents εz,0 = 2 (in-plane) and εx = 1 (out-of-plane), andd/d0 in the range from 0.8 to 1.2. The numerically com-puted dispersion bands are shown in Fig. 1a. A band gapappears for values of d different than d0.

We now claim that effective dispersion relation (5) andENZ condition (6) are in fact universal. This means thatthey are valid for any tensor permittivity diag (εx, εz, εz)with arbitrary, spatial-dependent εz(x). To develop ageneral argument, we set

εz(x) = εz,0f(x/d), f(x) > 0, (7)

where f(x) is an arbitrarily chosen, continuous and pe-riodic positive function. Guided by our results for theparabolic profile, we now make the conjecture that dis-persion relation (5) still holds with the definitions

d0 = ξ0

[∫ 1

0

f(x)dx

]−1

,εeffz

εz,0= ξ0

(1

d0− 1

d

). (8)

In the following analysis, we give a formal bifurcationargument justifying definition (8). We start by expand-ing Eq. (3) in the neighborhood of k∗ in powers of thecomponents of δk = (δkx, δkz) = k∗ − k. First, it canbe readily shown that at k = k∗ Eq. (1) reduces to−∂2

xE = 0. Thus, the system of fundamental solutionsis given by E(1)(x) = 1, E(2)(x) = x. This implies thatD[k∗] = 0. The expansion of D[k] up to second order inδk leads to an expression of the form

D[k∗ + δk] =

bxδkx + bzδkz + bxx(δkx)2 + bzz(δkz)2 + bxzδkxδkz.

The occurrence of a Dirac point is identified with the ap-pearance of a critical point D[k], when bx = bz = 0. Inorder to express bx and bz in terms of physical param-eters, we notice that only the term

[ieikxd(σ/ω)κ(kz) +

E ′(1)(d)]E(2)(d) of D[(kx, kz)] can contribute to first order

in δk. Accordingly we find

D[k∗ + δk] =

− d(−iσ

ωεx2kzδkz − δE ′(1)(d)[δkz]

)+O((δk)2).

Here, δE [δkz] denotes the total variation of E with respectto kz in the direction δkz. It can be shown (see supple-mentary material) that δE(1)[δkz] solves the differentialequation−∂2

xδE(1) = −εz(x)/εx 2kzδkz. The solution hasthe derivative

δE ′(1)(x) = 2kzδkz

[εx

∫ x

0

εz(ξ) dξ

]−1

,

which enters D[k∗ + δk]. Thus, we obtain that bx = 0and

bz =

[ξ0 − d

∫ 1

0

f(x)dx

]2dkzεz,0εx

.

At the critical point, the expression in the bracket mustvanish, which produces Eq. (8).

A refined computation for the critical case of d = d0

gives bxx = −d2, bxz = 0, and bzz > 0. Thus, the effectivedispersion relation at d/d0 = 1 up to second order termsis bxxδk

2x+bzzδk

2z = 0 with bxxbzz < 0, which corresponds

to a Dirac cone. Moreover, for εeffz /εx ∼ 1 it can be shown

that

D[k∗ + δk] ≈ −d2[δk2

x +εeffz

εx(k∗z + δkz)2 − εeff

z

εx(k∗z)2

].

By (k∗z)2/εx = k20, the above relation recovers the elliptic

profile of Eq. (5).

4

0.5

1

1.5

2

−1 −0.5 0 0.5 1

kz/k0

kx/k0

k∗

FIG. 2: Numerically computed dispersion curves (solidlines) and the effective dispersion relation (dashed lines)

for d/d0 = 1.1 computed for the parabolic (blue),double-well fdw(x) (orange), and nonsymmetric profilefns(x) (green). The curvatures agree at the critical

point k∗.

In order to support this bifurcation argument with nu-merical evidence, we test Eq. (8) for two additional di-electric profiles. In the spirit of [26], we study distinctlydifferent profiles εz(x). Specifically, we use the symmet-ric double-well profile fdw(x) = 1 − 3.2x + 13.2x2 −20x3 + 10x4 and the non-symmetric profile fns(x) =1 + 0.5 (e5x − 1)(1 − x). The computational results forthe dispersion relation are given in Fig. 1b-c. Further-more, for k in the neighborhood of k∗ and d/d0 = 1.1 wenotice excellent agreement of effective dispersion relation(5) with the numerically computed curve kz(kx) (Fig. 2).

To test the results of our model against more prac-tical configurations, we carry out direct numerical sim-ulations for a system with a finite number of metal-lic sheets. We choose graphene as the material forthe 2D conducting sheets, since it has been used ex-tensively in plasmonic and optoelectronic applications[16, 21]. In the THz frequency regime, doped graphenebehaves like a Drude metal because intraband transi-tions are dominant [21, 22]. Hence, the conductivity ofthe metallic sheets is approximated by the Drude for-mula, σ = ie2µc/[π~2(ω + i/τ)]. The doping amount isµc = 0.5 eV and the transport scattering time of electronsis τ = 0.5 ps to account for optical losses [3, 4]. Hence, inthis frequency regime doped graphene supports plasmons[21].

In Fig. 3, we present the spatial distribution ofHy(x, z) propagating through a structure of 100 graphenelayers embedded periodically in a lossless dielectric hostwith anisotropic and spatial-dependent permittivity. Thecomputation is carried out for parabolic profile (4) withα = 20/3, εz,0 = 2, εx = 1, as well as the double-wellprofile, with εz,0 = 2 and εx = 4. By setting the struc-tural period to d = d0, we observe the expected signatureof ENZ behavior, namely, wave propagation without dis-persion and with no phase delay through the periodic

structure [1–3].

The notion of an effective permittivity εeffz that arises

in Eqs. (5) and (8) bears a striking similarity to homog-enization results for Maxwell’s equations [27]. In fact,it can be shown that Eq. (8) can also be derived by ap-plying an asymptotic analysis procedure to the full sys-tem of time-harmonic Maxwell’s equations. For a generaltensor-valued permittivity ε(x/d) and sheet conductiv-ity σ(x/d), the effective permittivity of the metamaterialtakes the form

εeff = 〈ε χ〉host +i

ω〈σ χ〉sheet .

Here, 〈 . 〉R denotes the arithmetic average over region Rand χ is a weight function that solves a closed boundaryvalue problem in the individual layer [28]. In the specialcase of ε = diag (εx, εz, εz), the weight function reducesto the unit tensor, χ = I. Understanding the ENZ be-havior on the basis of this effective permittivity is thesubject of work in progress.

In conclusion, we have shown that dispersive Diraccones are universal for a wide class of plasmonic mul-tilayer systems consisting of 2D metals. We also deriveda general, exact condition on the structural period d toobtain a corresponding dispersion relation with epsilon-near-zero behavior. The universality of our approach iskey for the investigation of wave coupling effects in dis-crete periodic systems and the design of effective ENZmedia. Our results pave the way to a systematic studyof homogenization and effective parameters in the con-text of multilayer plasmonic systems.

We acknowledge support by ARO MURI Award No.W911NF-14-0247 (MMai, MMat, EK, ML, DM); EFRI2-DARE NSF Grant No. 1542807 (MMat); and NSFDMS-1412769 (DM). We used computational resourceson the Odyssey cluster of the FAS Research ComputingGroup at Harvard University.

∗ msmaier@umn.edu; http://www.math.umn.edu/~msmaier† mariosmat@g.harvard.edu; http://scholar.harvard.

edu/marios_matthaiakis

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FIG. 3: Spatial distribution of Hy in an anisotropic dielectric host with 100 layers of doped graphene with structuralperiod d = d0 (black rectangle). A magnetic dipole source is located below the multilayer structures (white dots)

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