Gyroid Optical Metamaterials: Calculating the Effective Permittivity of

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Gyroid Optical Metamaterials: Calculating the Effective Permittivity of Multi-Domain Samples James A. Dolan, Matthias Saba, Raphael Dehmel, Ilja Gunkel, Yibei Gu, Ulrich Wiesner, Ortwin Hess, Timothy D. Wilkinson, Jeremy J. Baumberg, Ullrich Steiner, and Bodo D. Wilts ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00400 • Publication Date (Web): 06 Sep 2016 Downloaded from http://pubs.acs.org on September 7, 2016

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Gyroid Optical Metamaterials: Calculating the Eective Permittivity of Multi-Domain Samples James A. Dolan,

Ulrich Wiesner,

†, ‡ k

Matthias Saba,

Ortwin Hess,

¶

¶

Raphael Dehmel,

Timothy D. Wilkinson,

Ullrich Steiner,

†Department

‡,⊥

§

Ilja Gunkel,

†

and Bodo D. Wilts

§

Yibei Gu,

Jeremy J. Baumberg,

k,# ‡

∗,§

of Engineering, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0FA, UK

‡Department

of Physics, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK

¶Department

of Physics, Imperial College, Prince Consort Road, London SW7 2BB, UK

§Adolphe kDepartment

Merkle Institute, Chemin des Verdiers 4, 1700 Fribourg, Switzerland of Materials Science and Engineering, Cornell University, 214 Bard Hall, Ithaca, NY 14853-1501, USA

⊥Current #Current

Address: Papierfabrik Louisenthal GmbH, 83701 Gmund a.T. Germany

Address: The Dow Chemical Company, 2301 N. Brazosport Blvd, Freeport, TX 77541, USA

E-mail: [email protected] Phone: +41 (0)26 300 89 69

1

ACS Paragon Plus Environment

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Abstract Gold gyroid optical metamaterials are known to possess a reduced plasma frequency and linear dichroism imparted by their intricate sub-wavelength single gyroid morphology. The anisotropic optical properties are however only evident when a large individual gyroid domain is investigated the long range order of the metamaterial exceeds a few microns. Multi-domain gyroid metamaterials, fabricated using a polyisoprene-b polystyrene-b -poly(ethylene oxide) triblock terpolymer and consisting of multiple small gyroid domains with random orientation and handedness, instead exhibit isotropic optical properties. Comparing three eective medium models, we here show that the specular reectance spectra of such multi-domain gyroid optical metamaterials can be accurately modelled over a broad range of incident angles by a Bruggeman eective medium consisting of a random wire array. This model accurately reproduces previously published results tracking the variation in normal incidence reectance spectra of gold gyroid optical metamaterials as a function of host refractive index and volume ll fraction of gold. The eective permittivity derived from this theory conrms the change in sign of the real part of the permittivity in the visible spectral region (i.e. that gold gyroid metamaterials exhibit both dielectric and metallic behaviour at optical wavelengths). That a Bruggeman eective medium can accurately model the experimental reectance spectra implies that small multi-domain gold gyroid optical metamaterials behave both qualitatively and quantitatively as an amorphous composite of gold and air (i.e. nanoporous gold), and that coherent electromagnetic contributions arising from the sub-wavelength gyroid symmetry are not dominant.

Keywords gyroid, optical metamaterial, eective medium model, block copolymer, polymer self-assembly

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Electromagnetic metamaterials are articially structured materials for which the electromagnetic response is a function of not only their chemical composition but also their

1

structure . When the length scales of the constituent structural units are signicantly smaller than the wavelength of interest, the response of the metamaterial may be described by the homogenised eective permittivity

εeff (ω)

and permeability

µeff (ω).

Metamaterials greatly

expand the range of electromagnetic responses otherwise found in natural and synthetic bulk materials. They therefore oer huge exibility to exploit exotic situations, combinations or spatial arrangements of eective permittivity and permeability in electromagnetic material design. Potential applications famously include cloaking devices and super lenses

2,3

. How-

ever, of perhaps more immediate technological interest is the access to particularly large and small, or strongly anisotropic (e.g. hyperbolic), material parameters and their associated

4

science and engineering . Whereas metamaterials which operate in the microwave regime have existed for decades, it is only relatively recently that advances in nanofabrication have allowed the manufacture of

5

optical metamaterials . Due to the requirement that the structural units of a metamaterial be deeply sub-wavelength, ideally by an order of magnitude or more, optical metamaterials demand feature sizes on the length scale of tens of nanometres. However, many early topdown lithographic techniques were either intrinsically two dimensional (e.g. a single material layer) or ventured into the third dimension only by tedious and non-scalable stacking of individual layers. These techniques are arguably still incapable of fabricating three dimensional nanostructures over macroscopically large areas with sucient speed and cost-eectiveness to allow optical metamaterials to attain true technological relevance. Therefore, until recently,

6

three dimensional optical metamaterials presented a considerable challenge . Polymer self-assembly presents an alternative and particularly intriguing means by which to fabricate intricate one, two and three dimensional morphologies on the appropriate length-

7

scales, from which optical metamaterials may be replicated .

Block copolymers are mac-

romolecules consisting of two or more covalently linked homopolymers (blocks) which can

3


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micro-phase separate into various well-dened morphologies due to the chemical dissimilarity between blocks. In block copolymers consisting of three distinct and sequential blocks (i.e. linear triblock terpolymers) one such morphology is the alternating gyroid: two single gyroid networks, each composed of one block, separated by a matrix of the third. The single gyroid is a continuous, chiral and triply periodic cubic network with a constant mean curvature

8

surface which is found in a range of natural and synthetic self-assembled systems . As the two single gyroid networks in the alternating gyroid are chemically distinct, it is possible to selectively remove just one such network and use the remaining voided terpolymer as a sacricial template for metal deposition

9,10

.

It was using this technique that Vignolini et

al. fabricated the rst truly three dimensional self-assembled optical metamaterial: a gold single gyroid with a

≈50 nm

unit cell size and 30% ll fraction

11

.

Gyroid metamaterials possess a range of optical properties not exhibited by the pure

8

metal, including an unambiguous optical anisotropy . Large individual domains exhibit a signicantly decreased plasma frequency and linear dichroism, evidenced by the shift of the extinction peak by up to 100 nm upon rotation under linearly polarised light

11

. Both the

plasma frequency and the linear dichroism are additionally a function of the unit cell size, ll fraction and dielectric environment

12,13

. Although much is now known about the optical

properties of gyroid metamaterials, suitable estimates of the eective material parameters remain elusive, thereby hindering a true appraisal of the metamaterial's potential utility. A tri-helical metamaterial (THM) model of the gyroid as periodic disconnected helices allows the analytical derivation of the permittivity, permeability and chiral coupling parameter

14,15

.

From these the plasma frequency of the metamaterial and its variation with unit cell size, ll fraction and dielectric environment can be successfully predicted

12

. However, the THM

model does not describe the linear dichroism of gyroid metamaterials and the model is unsuitable to average over an ensemble of small domains of random orientation and handedness. It is therefore of great interest to correctly identify the eective material parameters of gyroid metamaterials and to understand to what extent the ordered sub-wavelength structure con-

4


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tinues to aect the optical response when the macroscopic order is relatively poor. In this work, we investigate the optical properties of multi-domain gold gyroid optical metamaterials fabricated from the alternating gyroid phase of a polyisoprene- b -polystyreneb -poly(ethylene oxide) (ISO) triblock terpolymer.

The specular reection of the isotropic

metamaterial is measured under a range of angles of incidence of light and the resulting reectance spectra are compared to those predicted by three of the simplest (and therefore most common) analytical eective medium theories applicable to amorphous composites of spherical or ellipsoidal inclusions of one material in a host matrix of another.

We show

that a Bruggeman model with ellipsoidal inclusions is the only model considered which can reproduce the features present in the measured reectance spectra of the investigated multi-domain gold gyroid optical metamaterials, and that such optical metamaterials behave substantially similar to nanoporous gold.

Eective Medium Theories For a material which is homogenous at a particular length scale, its physical properties may be described by a reduced set of eective material parameters

16

.

Eective medium

theories are approximate analytical means to ascribe a homogeneous (although potentially anisotropic) material parameter, in this case

εeff (ω),

to a heterogeneous mixture of two or

1

more components . Such theories are anticipated to hold when the inhomogeneity length scale of the material and the local variation in the electric eld are both signicantly smaller than the wavelength of interest

17

.

Although numerous approaches of varying complexity

exist, the two considered in this work are arguably the most prevalent theories: the MaxwellGarnett theory for spherical inclusions and the Bruggeman theory

5


1,17

.

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a)

b)

c)

d)

ε1(ω)

ε2(ω)

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εeff(ω)

Sketches of the mathematical representations of various eective medium theories. a) Maxwell-Garnett, Equation (1), where the eective permittivity ( εeff ) is Figure 1:

derived by considering spherical inclusions ( A

= 1/3) of one material (ε1 ) in a host matrix of another (ε2 ). b) Bruggeman ( A = 1/3), Equation (2), where the eective permittivity ( εeff ) is derived by averaging over single inclusions ( ε1 and ε2 ) individually embedded in an eective medium consisting of all other inclusions ( εeff ). c) Bruggeman ( Ak ), Equation (3), where the eective permittivity ( εeff ) is derived analogously to the case where A = 1/3 except that all individual spherical inclusions are replaced by randomly oriented ellipsoids. d) The single gyroid morphology for which an eective medium theory is sought.

Maxwell-Garnett Theory: Case of Spherical Inclusions The simplest form of the Maxwell-Garnett eective medium theory considers spherical inclusions of one material in a host matrix of another material (Figure 1a).

The eective

1

permittivity is found by solving the following implicit equation :

ε2 (ω) − εeff (ω) (ε1 (ω) − ε2 (ω))f1 + = 0, ε2 (ω) + (ε1 (ω) − ε2 (ω))A ε2 (ω) + (εeff (ω) − ε2 (ω))A

6


(1)

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where

εeff (ω) is the (isotropic) eective permittivity of the composite, ε1 (ω) is the permittiv-

ity of the rst component (the inclusion), (the host),

f1

ε2 (ω)

is the permittivity of the second component

is the volume fraction of the rst component,

(1/3 for spherical inclusions), and

ω

is frequency.

A

is the depolarisation factor

*

The above theory is strictly valid only

when the volume ll fraction of the inclusions is small. As a result, the above equation is not invariant under the transformation

ε1 ↔ ε2

and

f1 → f2 .

Asymmetric formulations such

as this are only valid for low volume fractions of the polar phase (see e.g.

Table 1 in

18

).

Nonetheless, the Maxwell-Garnett eective medium theory should be considered due to its widespread usage and having previously been employed to justify the diering behaviour of gold gyroid metamaterials from an amorphous composite

11

.

Bruggeman Theory: Case of Spherical Inclusions We adopt here a version of the Bruggeman eective medium theory, in which all constituents of the composite are treated symmetrically. If the inclusions are again assumed to be spherical (Figure 1b), the eective permittivity is found by solving the following implicit equation (see Supplementary Information for more information)

where

18

:

(ε2 (ω) − εeff (ω)) f2 (ε1 (ω) − εeff (ω)) f1 + = 0, εeff (ω) + (ε1 (ω) − εeff (ω)) A εeff (ω) + (ε2 (ω) − εeff (ω)) A

(2)

f2 = 1 − f1

is the volume

εeff (ω), ε1 (ω), ε2 (ω), f1 , A

and

ω

are dened as above, and

ll fraction of the second component. Note that the above equation is now invariant under the transformation

ε1 ↔ ε2

and

f1 ↔ f2 ,

indicating that the Bruggeman eective medium

theory is applicable for any given volume ll fraction. It is therefore anticipated that this simplest form of the Bruggeman theory may more accurately model the behaviour of 30% ll fraction gyroid metamaterials, such as those considered here.

* The

depolarisation factor relates the dipole moment of a dielectric ellipsoid to the applied external electric eld and varies from 0 (disks) to 1 (wires).

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Bruggeman Theory: Case of Ellipsoidal Inclusions Although the Bruggeman eective medium theory is insensitive to the absolute size of the sub-wavelength inclusions of a composite, it is possible to extend the theory to take anisotropy of the inclusions into account. This is done by modelling such inclusions as aligned ellipsoids with semi-axes

ak .

replaced by

a1 a2 a3 Ak = 2

Z∞

Ak

du

in Equation (2) is

,

r 0

where

A

In such a case the depolarisation factor

(a2k + u) (a21 + u)(a22 + u)(a23 + u)

is the depolarisation factor for light linearly polarised along the

k th

axis of an

aligned ellipsoidal inclusion. Note that the derived eective permittivity is hence only valid for that same relative orientation of linearly polarised light and the eective permittivity of the material is anisotropic (i.e. when all ellipsoidal inclusions are assumed to be aligned). However, it is also possible to dene a macroscopically isotropic eective permittivity whilst allowing for microscopic anisotropy if the ellipsoidal inclusions are assumed to be randomly oriented in the composite (Figure 1c). In such a case, the eective permittivity is derived by solving the following implicit equation (see Supplementary Information)

3 X k=1

where

(ε1 (ω) − εeff (ω)) f1 (ε2 (ω) − εeff (ω)) f2 + =0 εeff (ω) + (ε1 (ω) − εeff (ω)) Ak εeff (ω) + (ε2 (ω) − εeff (ω)) Ak

εeff (ω), ε1 (ω), ε2 (ω), f1 , f2

and

ω

are as dened above, and

Ak

(3)

are the three depolar-

isation factors corresponding to the geometry of the ellipsoidal inclusions ( A1 +A2 +A3

= 1).

Note that this equation is identical for both a polycrystalline sample consisting of numerous randomly oriented anisotropic domains, and a composite sample consisting of equal amounts of three isotropic constituents of diering permittivities or depolarisation factors

16

. Note also

that the Maxwell-Garnett theory may similarly be extended to take into account microscopic anisotropy (vide infra ).

8


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Results An optical and electron micrograph of the multi-domain gyroid metamaterial, the geometry of the experiment and the resulting reectance spectra for transverse magnetic (TM) and transverse electric (TE) polarisations of incident light are shown in Figure 2. The optical micrograph captured under linearly polarised light (Figure 2a) highlights the light orange colour of the metamaterial and the the electron micrograph of an individual domain (Figure 2b) is consistent with the gyroid nanostructure previously observed for this terpolymer

1113

. The

small size and multiplicity of the domains can be more clearly seen in Figure S1. Both TM (Figure 2d) and TE (Figure 2e) reectance spectra are qualitatively similar to those of bulk gold (Figure S2), insofar as they exhibit low reectance at shorter wavelengths and increased reectance at longer wavelengths across all angles of incidence. (Note that in Figs. 2-5, reectance spectra are plotted on a log scale as they exhibit spectral features that span multiple orders of magnitude.) In bulk gold, these features are attributable to interband transitions and the plasma frequency respectively. However, the measured spectra dier quantitatively from those of bulk gold due both to a lowered eective plasma frequency and their variation with angle of incidence.

Otherwise, the spectra exhibit the general decrease (increase) in

reectance for TM (TE) polarised light with increasing angle of incidence, with an eective Brewster's angle of

≈50◦ .

A comparison between the measured and modelled reectance spectra across all wavelengths and angles of incidence for TM polarised light is shown in Figure 3; the same for TE polarised light is shown in Figure S3. The eective permittivity

εeff (ω)

used to generate the

modelled spectra in Figure 3b was calculated using the Maxwell-Garnett theory by setting

ε1 (ω) = εAu (ω), ε2 (ω) = 1, f1 = 0.3

and

A = 1/3

(i.e. spherical inclusions) in Equation (1).

Clearly the Maxwell-Garnett eective medium theory is a poor model for the eective permittivity of the gyroid metamaterial as it predicts a striking dip in reectance at an eective Brewster's angle of

≈60◦

which is entirely absent in the measured data. Despite also assum-

ing spherical inclusions, the eective permittivity calculated using the Bruggeman theory,

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θ

d) Angle of Incidence

a)

b)

15°

TM

25°

35° 45° 55° 65° 450 500 550 600 650 700 750 800

-0.1

-0.1

-1.8

log10 Reflectance TE

c)

-1.4

Angle of Incidence

e) log10 Reflectance TM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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15°

TE

25° 35° 45° 55° 65° 450 500 550 600 650 700 750 800 λ (nm)

Gyroid optical metamaterials optical and electron micrographs, experiment geometry and measured reectance spectra. a) Optical micrograph of Figure 2:

multi-domain gold gyroid metamaterial under linearly polarised light (scale bar 100

µm).

b) Electron micrograph of an individual domain of the sample (scale bar 200 nm). c) Specular reectance spectra of the gyroid metamaterial are measured as a function of angle of incid◦ ◦ ence between 15 and 65 and 450 and 800 nm. Although a single gyroid domain is shown for clarity, the measurement spot size in fact encompasses a large number of randomly oriented gyroid domains. d) The reectance spectra as a function of angle of incidence for transverse magnetic (TM) polarised light (log colour scale extends from

−1.8 to −0.1). e) The same for −1.8 to −0.1).

transverse electric (TE) polarised light (log colour scale extends from

10


25°

0

Maxwell-Garnett (A = 1/3) TM

-1.0 Maxwell-Garnett (A = 1/3)

15°

Experiment

25°

b)

Experiment TM

35°

45°

45°

55°

55°

65° 450 500 550 600 650 700 750 800

-1.8 65° 450 500 550 600 650 700 750 800

-6.0

15°

-0.6

15° 25°

d)

Bruggeman (A = 1/3) TM

-0.8

Bruggeman (Ak) TM

25°

35°

35°

45°

45°

55°

55°

65° 450 500 550 600 650 700 750 800 λ (nm)

-2.0 65° 450 500 550 600 650 700 750 800 λ (nm)

Bruggeman (Ak)

35°

Bruggeman (A = 1/3)

c)

15°

log10 Reflectance

Angle of Incidence

a)

Angle of Incidence

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log10 Reflectance

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-1.9

Comparison of measured and modelled TM reectance spectra as a function of angle of incidence using three eective medium theories. a) The measured Figure 3:

reectance spectra for transverse magnetic (TM) polarised light repeated from Figure 2d (log colour scale extends from

−1.8

to 0).

b) The reectance spectra generated using the

Maxwell-Garnett theory (log colour scale extends from using the Bruggeman theory and

−2.0 to −0.8). d) The A2 = 0.34 and A3 = 0, i.e. −0.6). from

A = 1/3,

−6.0 to −1.0).

c) The same generated

i.e. spherical inclusions (log colour scale extends

A1 = 0.66, from −1.9 to

same generated using the Bruggeman theory and ellipsoidal inclusions (log colour scales extends

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

by again setting

ε1 (ω), ε2 (ω), f1

and

A

Page 12 of 29

as above in Equation (2), results in a far better

qualitative match between the measured and modelled reectance spectra (Figure 3c). The global reectance minimum at

≈50◦

between 450 and 500 nm is reproduced well. However,

the spectra still exhibit a decrease in reectance at long wavelengths and high angles of incidence which is not present in the experimental data. This decrease in reectance is absent in the reectance spectra of Figure 3d which uses our microscopically anisotropic Bruggeman theory. These spectra were modelled by setting

A2 = 0.34

and

A3 = 0

ε1 (ω), ε2 (ω), f1

as above and

A1 = 0.66,

in Equation (3). These depolarisation factors were found by perform-

ing a constrained optimisation to t the TM reectance spectra generated by the model to the measured data (see Experimental Methods). Table 1:

Table of cost functions

f (A)

(see Experimental Methods) associated with the

various eective medium theories (Model), equations (Eqn.), polarisations (Pol.) and vector of depolarisation factors ( A). A lower value of cost function indicates a better t.

Model (Eqn.) Pol.

A

Max.-Garn. (1) Brugg. (2)

TM

Brugg. (3)

2.02

1/3

0.28 0.12

1/3

0.34

1/3

0.08

[0.66, 0.34, 0]

0.04

TE

Brugg. (3)

1/3 [0.66, 0.34, 0]

Max.-Garn. (1) Brugg. (2)

f (A)

A quantitative comparison of the goodness of t of each model is presented in Table 1. Cost functions

f (A)

(see Experimental Methods) are tabulated not only for TM but also

TE polarisation using the depolarisation factors derived from the TM measurements. Note that the cost function evaluated for TE polarisation using the depolarisation factors derived from the TM measurements ( A1

= 0.66, A2 = 0.34

and

A3 = 0)

is only 0.66% larger than

the same cost function evaluated for TE polarisation using the depolarisation factors derived from the TE measurements ( A1

= 0.70, A2 = 0.30

and

A3 = 0).

be the case (e.g. for an anisotropic sample). Both ts yield

This would not in general

A3 = 0, which thus selects a wire

(i.e. a uniaxially innitely extended ellipsoid) with an elliptical cross-section. It is therefore clear that the microscopically anisotropic Bruggeman model consisting of randomly oriented

12


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wire inclusions is the best of the three eective medium models considered and that only a single set of depolarisation factors

A1 = 0.66, A2 = 0.34

and

A3 = 0

are appropriate to

model both TM and TE measurements. A good model for the eective permittivity of the gyroid metamaterial should be capable not only of reproducing the measured reectance spectra presented in this work, but also the previously reported behaviours as a function of refractive index of the host and volume ll fraction of gold. Figure 4 compares measurements performed by Salvatore et al. on 30% ll fraction gold gyroids inltrated with various refractive index media with the optimised Bruggeman model

12

. Following the convention in that work, each normal incidence spectrum

is characterised by its plasma edge, the wavelength of the point of inection of the reectance spectrum. In the simplest Drude model of the permittivity of metals, the plasma wavelength is the wavelength at which the permittivity changes from positive to negative and is characteristic of the density and eective mass of the free electrons in the particular metal. The plasma edge is not the plasma wavelength but behaves similarly and may be more readily and systematically identied from experimental data than the plasma wavelength (see for example Figure S6 of

12

). Figure 4a reproduces the reectance spectra measured by Salvatore

et al. upon inltration of the gyroid metamaterial with media of refractive indices

nfill = 1.0,

1.33, 1.5 and 1.7, and indicates the plasma edge in each case (the labelled vertical lines). Note how the plasma edge is red-shifted with increasing host refractive index.

Figure 4b

plots equivalent normal incidence reectance spectra derived using the optimised Bruggeman model with

ε2 (ω)

of Equation (3) set to

n2fill

respectively.

Again the plasma edge is

red-shifted with increasing host permittivity. Although the measured and modelled plasma edges do not perfectly coincide, it is clear from Figure 4c that the modelled plasma edges are within experimental error for each refractive index. Therefore, despite the dissimilarity between the measured and modelled normal incidence reectance spectra, the model successfully predicts the plasma edge of the gold gyroid metamaterial. Furthermore, the optimised Bruggeman model performs better in this respect than the tri-helical metamaterial (THM)

13


0.1

Expt.

796

0.01

0.001 450 1 b)

500

550

600

650

700

750

800

Brugg.

450

500

550

c)

794

674

727

0.1 587

Reflectance

710

1.0 1.33 1.5 1.7

684

1

Page 14 of 29

609

a)

Plasma Edge (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Reflectance

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600 650 λ (nm)

700

750

1.3 1.4 nfill

1.5

1.6

800

900 Expt. THM Brugg.

800 700 600 1.0

1.1

1.2

1.7

Variation in normal incidence reectance spectra and plasma edge as a function of host refractive index. a) Measured reectance spectra at normal incidence Figure 4:

for 30% ll fraction gold gyroid metamaterials inltrated with media of refractive indices 1.0 (solid line), 1.33 (dashed line), 1.5 (dotted-dashed line) and 1.7 (dotted line), and the associated plasma edge of each spectrum in nanometres (labelled vertical lines of the same style).

Data reproduced from Salvatore et al

12

. b) Modelled normal incidence reectance 2 spectra using the optimised Bruggeman model where ε2 (ω) of Equation (3) is set to 1.0 2 2 2 (solid line), 1.33 (dashed line), 1.5 (dotted-dashed line) and 1.7 (dotted line), and the associated plasma edge of each spectrum in nanometres (labelled vertical lines of the same style). c) Plasma edge against refractive index of host for measured (Expt.) and modelled (Brugg.)

spectra, and the relationship predicted by the tri-helical metamaterial (THM)

model, reproduced from Salvatore et al

12

.

14


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ACS Photonics

analytical model, also reproduced from Salvatore et al. and shown in Figure 4c

12

.

The results of the optimised Bruggeman model may also be compared to previously measured data where the volume ll fraction of gold is varied.

Figure 5a shows normal

incidence reectance spectra from Salvatore et al. as the volume ll fraction of the gold gyroid is increased from 30% to 90%. Only four of the ve plasma edges are shown, since those for the 75% and 90% ll fraction gyroids are within 2 nm of one another.

Whereas

the plasma edge is red-shifted with increasing refractive index of the host, it is blue-shifted as the ll fraction increases and the reectance of the gyroid metamaterial tends towards that of bulk gold. Figure 5b plots equivalent normal incidence reectance spectra calculated from the optimised Bruggeman model. Here only two of the ve plasma edges are shown as all of those corresponding to ll fractions above 30% are extremely close (within 2 nm). Again the plasma edge is blue-shifted with increased ll fraction as it was in the measured spectra.

However, as can be seen in Figure 5c, there is signicant deviation between the

measured and predicted plasma edges for intermediate ll fractions between about 40% and 70%.

Outside of this region (i.e. at low and high ll fractions) the optimised Bruggeman

model performs well and is comparable to the THM model.

Discussion The failure of the Maxwell-Garnett theory to describe multi-domain gyroid samples is perhaps not surprising, since its validity is limited to a low volume fraction of spherical dielectric inclusions that are spatially separated. The gyroid samples considered here consist however of an interconnected, percolating network of gold struts. The better description of the experimental system by the Bruggeman model with spherical inclusions arises primarily from the fact that this model retains it validity for a high density of scatterers, expressed in the model by the ll fraction.

The results of Figure 3 indicate

that a model based on an eective medium approach is able to approximate the random

15


500

30% 45% 60% 75% 90%

600

554 569

513

0.1

650

700

750

600 650 λ (nm)

700

750

550

600

Brugg.

0.01 450

500

587

0.1 512

b)

Page 16 of 29

1 Expt.

0.01 450 1 Reflectance

a)

550

c) Plasma Edge (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Reflectance

ACS Photonics

700 650 600 550 500 450

Expt. THM Brugg.

20

30

40 50 60 70 Fill Fraction (%)

80

90

Variation in normal incidence reectance spectra and plasma edge as a function of gold volume ll fraction. a) Measured normal incidence reectance spectra Figure 5:

of gold gyroid metamaterials with volume ll fractions of 30% (solid line), 45% (dashed line), 60% (dotted-dashed line), 75% (dotted line) and 90% (solid line with circles), and the associated plasma edge of each spectrum in nanometres (labelled vertical lines of the same style).

Data reproduced from Salvatore et al

12

.

b) Modelled normal incidence reectance

spectra using the optimised Bruggeman model where

f1

of Equation (3) is set to 0.30 (solid

line), 0.45 (dashed line), 0.60 (dotted-dashed line), 0.75 (dotted line) and 0.90 (solid line with circles), and the associated plasma edge of each spectrum in nanometres (labelled vertical lines of the same style). (Expt.)

c) Plasma edge against volume ll fraction of gold for measured

and modelled (Brugg.)

spectra, and the relationship predicted by the tri-helical

metamaterial (THM) model, reproduced from Salvatore et al

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12

.

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ACS Photonics

multi-domain gyroid network, despite the morphological dissimilarity. While a model with a larger number of free parameters (i.e. the three depolarisation factors) is expected to improve the level of agreement with the data, it is revealing that ts of both reected TE and TM polarised light result always in zero values for one of the depolarisation factors divergence for

Ak .

Since

Ak < 1/3

corresponds to a prolate ellipse, with a

Ak → 0, the optimisation of the Bruggeman model with free Ak

values results

in an eective medium consisting of randomly oriented wires of elliptical cross section (see Figure 6 inset). While a random wire assembly is a better description than isolated individual scatterers for an interconnected gyroid network, the good agreement between t and data is nevertheless surprising given that the Bruggeman theory does not model well percolating networks

19

. Although allowing for microscopic anisotropy within the Maxwell-Garnett model

(i.e. ellipsoidal inclusions) might similarly improve upon the results of the equivalent model with spherical inclusions, this variant of the Maxwell-Garnett model would still suer from a limited validity because of the high volume ll fractions of our samples and was therefore not considered. Like the classical wire-grid metamaterial, the gold gyroid metamaterial is shown to exhibit a strongly red-shifted plasma edge

20

.

The eective permittivity

εeff (ω)

derived from the

Bruggeman theory with optimised depolarisation factors, shown in Figure 6a, has a crossover of the real part of the permittivity just below 650 nm (arrow). In contrast, the real part of the permittivity of the constituent gold does not cross the zero axis at any wavelength in the visible (inset).

Notwithstanding the eect of losses associated with the imaginary

part of the permittivity, the gold gyroid metamaterial therefore behaves as both a dielectric (λ

. 650 nm)

and metal ( λ

& 650 nm)

in the visible.

In the simple Drude model of the

permittivity of a metal, this crossover is associated with the plasma frequency (plasma wavelength), an association which is hard to isolate in more complicated models (e.g. DrudeLorentz) or experimentally measured permittivities. The previously introduced plasma edge is thus useful to characterise the optical response of the gyroid metamaterial, despite the lack

17


ACS Photonics

2

a)

Permittivity

1

0 0

-1

-20 -2 450

450

800

500

550

600

650

Real (Au) Imag (Au)

700

750

800

Real (Bruggeman) Imag (Bruggeman)

1.6

2

1.5

nfill

1.4

1

1.3

1.2

0

1.1

c)

1.0 450 500 550 600 650 700 750 800 90

-1 0

80 70

-4

60

-8

50 -12 40 -16

30

Real Part of Permittivity

1.7

Real Part of Permittivity

b)

Fill Fraction (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 29

20 450 500 550 600 650 700 750 800 λ (nm)

Eective permittivity of small domain gyroid metamaterials derived from the Bruggeman eective medium theory. a) Real (solid line) and imaginary Figure 6:

εeff (ω) calculated using the Bruggeman theory ε1 (ω) = εAu (ω), ε2 (ω) = 1, f1 = 0.3 and A1 = 0.66, A2 = 0.34 and A3 = 0 in

(dashed line) parts of the eective permittivity by setting

Equation (3).

The insets are a sketch of the equivalent structure corresponding to the

optimised Bruggeman model, and the real (solid line) and imaginary (dashed lines) part of the permittivity of bulk gold

Gyroid Optical Metamaterials: Calculating the Effective Permittivity of

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