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Topological Darkness in Self-Assembled Plasmonic Metamaterials Ludivine Malassis, Pascal Massé, Mona Tréguer-Delapierre, Stephane Mornet, Patrick Weisbecker, Philippe Barois, Constantin R. Simovski, Vasyl G. Kravets, and Alexander N. Grigorenko* Artificial nanostructured metamaterials can demonstrate new exciting properties not easily achieved in natural materials. With the help of nanostructuring, one can realize surface plasmon subwavelength optics,[1] efficient plasmonic nanotrapping,[2] negative index of refraction,[3] visible light magnetic permeability,[4,5] negative optical phase,[5,6] a plasmonic blackbody based on properties of Fresnel coefficients[7] and connected to coherent absorbers,[8] plasmonic waveguides,[9] quantum plasmon tunneling,[10] quantized plasmonic absorption,[11] optical cloaking,[12,13] and many other unusual phenomena important for applications. Recently, we have shown that nanostructured plasmonic materials can demonstrate socalled topological darkness[14] which could be important for phase-sensitive plasmonic detection[15,16] based on localized surface plasmons,[17–19] which have attracted a lot of attention recently.[20] Topological darkness was achieved with the help of collective plasmon resonances[21–26] exploiting light diffraction in nanoparticle arrays to attain the desired optical properties. Here we show that topological darkness can be realized in bulk plasmonic metamaterials produced by inexpensive self-assembly methods[27] which could make its application to phase-sensitive detection[14] practical. Figure 1 briefly illustrates the concept of topological darkness. Consider light reflection from a thin layer characterized by a refractive index nˆ = n + ik placed on a dielectric substrate. For a given angle of incidence and given film thickness there is a set of values (n, k) which guarantees the absence of Dr. L. Malassis, Prof. P. Barois CNRS, CRPP, UPR8641, F-33600, Pessac, France Dr. L. Malassis, Prof. P. Barois Univ. Bordeaux, CRPP, UPR8641, F-33600, Pessac, France Dr. P. Massé, Dr. M. Tréguer-Delapierre, Dr. S. Mornet CNRS, ICMCB, UPR9048, F-33600, Pessac, France Dr. P. Massé, Dr. M. Tréguer-Delapierre, Dr. S. Mornet Univ. Bordeaux, ICMCB, UPR9048, F-33600, Pessac, France Dr. P. Weisbecker CNRS, LCTS, UMR 5801, F -33600, Pessac, France Prof. C. R. Simovski Dept. Radio Sci. Eng., School ELEC Aalto University FI 00076, Aalto, Finland Dr. V. G. Kravets, Prof. A. N. Grigorenko School of Physics and Astronomy University of Manchester Manchester, M13 9PL, UK E-mail: [email protected]
DOI: 10.1002/adma.201303426
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reflection from the structure (obtained by tuning the light wavelength). This set can be found by analyzing the Fresnel coefficients and is usually represented by zero-reflection curves (the thicker the film, the more curves could provide zero reflection for a given spectral range). Figure 1 shows the p-polarized zeroreflection curve (i.e., for the light polarized parallel to the plane of incidence) computed for a 200 nm film on a glass substrate at an angle of incidence 63°. Normally, continuous metal films cannot provide the optical constants necessary for achieving zero reflection. For example, the curve A–B of Figure 1 shows the dispersion curve (n(λ), k(λ)) for silver and the top inset shows the measured reflection from a 200 nm silver film at an incident angle of 63° which is quite large and never reaches zero. The situation changes dramatically for nanostructured films. In this case, the effective dispersion curve (neff(λ), keff(λ)) shown in Figure 1 as the curve P–Q can have endpoints in two different areas separated by the zero-reflection curve. Hence the effective dispersion curve has to intersect the zero-reflection curve at some point due to Jordan curve theorem.[28] (The Jordan curve theorem states that the line connecting points inside and outside of an enclosed area intersects the boundary of this area at some point. For 3D, the Jordan curve theorem suggests that when two interlocking loops are deformed to remove their interlock they will intersect each other at some stage.) By changing the angle of incidence, one can adjust the intersection point in such a way that it will happen at the same wavelength for both curves (the zero-reflection curve and the curve P–Q), which results in zero reflection for a nanostructured material, see the bottom inset of Figure 1. We refer to this absence of reflection as topological darkness because the effect survives the imperfections of the internal nanostructure geometry due to the curve topology.[14] We have also experimentally showed that it can drastically improve the phase sensitivity of localized plasmon resonances.[14] For this reasons, we believe that materials which demonstrate topological darkness could find a wide scope of sensing applications. Before describing the main result of our work, it is worth putting the concept of topological darkness into a historical context. The absence of reflection was previously observed in various optical systems. For example, light reflection from a single interface at the Brewster angle should in theory provide zero reflection due to the nature of a dipole radiation pattern[29] (the absence of radiation in the direction of an oscillating dipole). Also, light reflection in the Kretschmann configuration (for a thin metal film deposited on a dielectric substrate) should in theory provide zero reflection.[30] The closest analogue of topological darkness can be found in the plasmonic
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blackbody[7] (which realizes maximal light absorption from a thin layer of an optical material placed on a dielectric substrate) also known as a coherent absorber.[8] However, all these examples of systems with zero reflection can be easily affected by sample disorder, inhomogeneity, or irregularities. As a result, light reflected under Brewster conditions is not exactly zero and is elliptically polarized due to a transitional layer;[31] light reflection at the level of 1–5% is still detected in the Kretschmann configuration for surface plasmon resonance in gold films due to surface corrugation[16]; nonzero reflection can be observed for a plasmonic blackbody due to, e.g., sample disorder. Topological darkness suggested by Kravets et al.[14] develops the concept of zero reflection further and allows one to elevate problems of sample disorder, inhomogeneity, or irregularities. This concept suggests that a wide class of optical systems with a mild restriction on the effective dispersion curve (neff(λ), keff(λ)) demonstrates exactly zero reflection at certain angles of incidence and light wavelength. This mild restriction on the effective dispersion curve of a structure only implies that the dispersion curve should start and end in two different areas separated by a zero-reflection curve. Then, exactly zero reflection can be achieved due to the Jordan curve theorem. More importantly, this zero reflection is topologically stable and cannot disappear in the presence of disorder. (A simple analogy of such topological stability can be found in the spectrum of massless Dirac fermions in graphene.[32] This spectrum is topologically stable and cannot be affected by the presence of disorder as an
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Figure 1. Topological darkness in plasmonic metamaterials. The graph plots: p-polarized zero-reflection curve for a thin layer (thickness 200 nm) placed on a glass substrate (for an angle of incidence 63°), the dispersion of the bulk silver (the curve A–B), the dispersion curve of a hypothetic metamaterial (the curve P–Q). The color of the points corresponds to the light wavelength. The top inset shows the p-polarized reflection for a silver 200-nm film (angle of incidence 63°). The bottom inset shows the reflection for a metamaterial with the dispersion curve P–Q. Zeroreflection point near 420 nm corresponds to the topological darkness.
intersection point between two curves cannot disappear after small deformations of the curves.) It is worth stressing that topological darkness provides “inevitable” exactly zero reflection for a large class of optical systems. The important application for “dark” metamaterials is phase-sensitive optical transduction. There are other techniques which enjoys the benefits of enhanced phase sensitivity: interferometry,[33] whisperinggallery modes,[34] polarimetry[35] (e.g., based on Pancharatnam phase[36]), etc. However, these methods are either not compatible with high-throughput analysis, or do not enjoy plasmonic enhancement of the sensitivity, or are much more complicated. In the work by Kravets et al.,[14] the topological darkness was observed in a variety of regular plasmonic nanoarrays made by electron-beam lithography. Here, we report the topological darkness in bulk plasmonic metamaterials made by self-assembly which are more attractive for applications due to relatively low fabrication costs. Three-dimensional samples made of N layers of core-shell Ag@SiO2 nanoparticles were prepared by repeating N times a single-layer procedure based on the inverse Langmuir–Shaefer method developed by Lee et al.[37] This method enabled the fabrication of dense assemblies of nanoparticles with a uniform coverage of the whole substrate.[27] Figure 2a and b show a monolayer and a three-layer structure of fabricated metamaterials, respectively, while Figure 2c shows schematics of the studied samples. Core–shell nanoparticles Ag@SiO2 were prepared in two steps. First, silver nanoparticles were made and centrifuged to achieve a monodisperse solution. Then, monodisperse silver particles have been covered with a silica layer using the well-established Stöber method.[38] The size distribution of the nanoparticles was characterized by using a Hitachi H600 transmission electron microscope. The diameter of the Ag core was 27 ± 2 nm. The nanoparticles have a pseudospherical shape with an average size of 82 ± 3 nm. Figure 2d shows the TEM image of the Ag@SiO2 nanostructures with N = 6 layers. The absorption spectrum of the colloidal suspension in ethylene glycol exhibits a sharp band at 425 nm that is characteristic of the dipole resonance of the silica-coated Ag nanospheres. The details of the sample fabrication and characterization can be found in our previous work.[27] The optical measurements were performed in the 240– 1000 nm wavelength range with a variable-angle spectroscopic Woollam ellipsometer M2000F. The spot size was down to 30 μm which was small enough to ensure sensing applications and large enough to have small angular variation of the angle of incidence. In addition to the intensity reflection coefficients Rp and Rs, we also measured the ellipsometric parameters Ψ and Δ defined as rp/rs = tan(Ψ)exp(iΔ) where rp = Ep/Ei and rs = Es/Ei are the amplitude reflection coefficients for light polarization parallel (p), and perpendicular (s) to the plane of incidence, and Ei denotes the electrical field of the incident light. From the definition it follows that Ψ goes to zero whenever the p-polarized reflection goes to zero and Ψ is equal to 90° whenever the s-polarized reflection is zero. Figure 3 shows the main result of our work – the topological darkness observed in self-assembled core–shell plasmonic metamaterials. For the self-assembled metamaterial with N > 3 we found zero reflection for p-polarized light. Figure 3a shows the ellipsometric reflection Ψ for four, five, and six-layer metamaterials in the conditions of topological darkness observed
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Figure 2. Self-assembled core-shell metamaterials. a) Schematics of one layer. b) Three-layer core–shell metamaterial. c) Cross-section of the structure. d) TEM image of six-layer metamaterial shown at two magnifications ×21333 and ×42667 in inset (TEM operated at 10 kV).
near the wavelength of 400 nm at angles of incidence of 48°, 54°, and 63°, respectively. The ellipsometric reflection is just a convenient way to represent the p-reflection coefficient when the reflection becomes very small: Ψ = 0 if and only if Rp = 0. In addition, the ellipsometric data are measured with a better accuracy than normal reflection as they are stable under intensity variation of the light source and perturbations in ambient media. From Figure 3a–d, we see that minimal measured ellipsometric reflection was at the level Ψmin = 0.5° which gives the minimum light intensity reflected from our sample for p-polarization as low as Rpmin = tan2(Ψmin) Rsmin = 5 × 10−5% (where we used Rsmin = 0.7% from Figure 3d). In our installation, the spectral detection was performed by using a photodiode-array spectrophotometer and hence we measured digitized spectra with the wavelength step of about 1 nm. This procedure means that the zero-reflection point could be missed during digitization of the spectrum. The inset to Figure 3a displays the zoomed region of the spectra near 400 nm which shows that data points (corresponding to measured spectral data) can miss the zero-reflection point. In this case, the linear extrapolation (see the straight lines of the inset to Figure 3a) can be used to estimate the position of the theoretically expected topological darkness. As one would expect,[14,16,30,39] the phase Δ shows fast changes in the conditions close to topological darkness which can be used for plasmonic biosensing, see Figure 3b. For completeness, we provide the spectral dependence of intensity reflection Rp and Rs for the four-layer metamaterial under the conditions close to the topological darkness at angles of incidence of 45°, 48°, and 50°. It is worth noting that despite
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the fact that experimental structures (Figure 2d) are not as regular as ideal 3D structures shown in Figure 2a–c, the zero reflection for our “imperfect” samples survives due to topological nature of the suggested phenomenon, see the discussion above. We found that the optical properties of the self-assembled metamaterials with N > 3 are described well within the Maxwell–Garnett effective medium approximation (EMA).[40,41] When the incident light wavelength is much larger than the size of the particle, the core–shell sphere can be approximated by a dipole with polarizability as shown in Equation 1:[42]
"eff = 4B r 23
(g d − 1)(g m + 2g d ) + D (g m − g d )(1 + 2g d ) (g d + 2)(g m + 2g d ) + D (2g d − 2)(g m − g d )
(1)
where r2 is the outer radius of core-shell particle, ρ = (r1/r2)3 is the fraction of the total particle volume occupied by the metalcore, and εm and εd are the permittivity of the metal core and the dielectric shell, respectively. Consequently, a core–shell metamaterial can be described by an effective dielectric constant given by the Clausius–Mossotti formula:[40] (g eff − 1) f NP"eff = (g eff + 2) 4B r 23
(2)
where fNP is the filling fraction of the core-shell nanoparticles Ag@SiO2 or, in other words, the volume fraction of dipoles. By combining Equation 1 and 2 together, we get:
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COMMUNICATION Figure 3. Topological darkness measured in reflection from fabricated metamaterials. a) Ellipsometric reflection Ψ as a function of wavelength at the conditions of topological darkness observed near the wavelength of 400 nm. The inset shows the zoomed region near the plasmonic resonance with linear extrapolation lines. b) Ellipsometric phase Δ as a function of wavelength for four-, five-, and six-layer metamaterials in the conditions of topological darkness, same as (a). c) p-Polarized reflection Rp for four-layer metamaterial near conditions of darkness. d) s-Polarized reflection Rs for four-layer metamaterial at the same conditions as (c).
(g eff − 1) (g d − 1)(g m + 2g d ) + D (g m − g d ) (1 + 2g d ) = f NP (g eff + 2) (g d + 2)(g m + 2g d ) + D (2g d − 2)(g m − g d )
(3)
In fitting the experimental data, the diameter of the Ag core was chosen to be r1 = 25 nm for four and five layers of Ag@SiO2 particles and r1 = 24 nm for six layers. The coreshell nanoparticles were considered as spherical with a size of 82 nm taken from experimental measurements. The volume fraction fNP of the Ag@SiO2 particles was found from the best fit as fNP = 0.55. The electric permittivity of silver, εm, can be described by a sum of the intraband term (the Drude term for electronic response of the free electrons in the conduction band) and interband transitions (the Lorentz term for electronic transitions between the d valence band and the sp conduction band). Hence, we fit εm as:
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g m (T) = g 0 −
Tp2 T2
+ i T(
−
2
g j 2j
j =1
T2 − 2j + i T j
(4)
with ε0 = 1.45, បγ = 0.07 eV, បωp = 9 eV, Δεj = (1.0, 1.1) បΩj = (4.5, 5.7) eV, បΓj = (1.2, 2.7) eV, where j = 1,2, for details see work by Kravets et al.[43] In the fitting, we first used the Drude– Lorentz parameters extracted from the ellipsometric measurement for a thick (200-nm) Ag film on top of a glass substrate, and then adjusted the parameter ε0 which characterizes the contribution of interband transitions to the permittivity (this adjustment describes the changes in screening effects). The optical constant, εd, of a SiO2 film was parameterized by using a Cauchy function found from the ellipsometric measurements. The calculated effective complex refractive index of the core–shell metamaterial within the Maxwell–Garnett EMA approach agrees well with the refractive index extracted from
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Figure 4. Effective medium approximation. a) Comparison of dispersion curves for four-layer metamaterial calculated with Maxwell–Garnett theory (ntheor and ktheor) with the dispersion curves extracted from ellipsometric measurements (nexp and kexp). b) The calculated Ψ corresponding to experimental data of Figure 3a. c) The calculated p-polarized reflection Rp corresponding to Figure 3c. d) The calculated s-polarized reflection Rs which corresponds to Figure 3d. e) Ψ corresponding to conditions of Figure 3c and d. f) The calculated p-polarized zero-reflection curve and the dispersion curve for the four-layer core-shell metamaterial at angle of 48°. Notice that the intersection point at (n, k) = (1.2, 0.2) and wavelength of approxiamtely 400 nm which guarantees the absence of reflection (topological darkness). The inset shows the intersection of the zero-reflection curve and the dispersion curve of the four-layer metamaterial in 3D coordinates (n, k, λ).
experimental data using ellipsometry for N > 3, Figure 4a. We can see just one pronounced extinction peak near 400 nm in all keff curves. This peak can be associated with excitation of localized surface plasmons in the silver core. Outside the region of the localized plasmon resonance (350–450 nm) neither n nor k show significant changes. By using EMA dielectric constants and Fresnel coefficients, we calculated optical properties of the layers of core–shell particles with N = 4–6 and found good agreement between the measured and calculated values. For example, the calculated ellipsometric function Ψ in the 328
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condition of topological darkness, Figure 4b, shows very good agreement with the experimental data of Figure 3a. The same agreement can be seen for the calculated intensity reflection Rp and Rs for the four-layer metamaterial, Figure 4c and d, which correspond to experimental data shown in Figure 3c and d, respectively. The calculated data give a reasonable quantitative description for the optical properties of the sample except the region below λ = 300 nm where plasmonic resonances of higher orders should be taken into account. For completeness, we also provide the calculated value of Ψ for the four-layer
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metamaterial near the point of topological darkness, Figure 4e. It is worth stressing that, despite the fact that our metamaterials can be described within the Maxwell–Garnet EMA which is often used to describe random samples, the regular layered structure is very important for sensing applications as it reduces the number of unpredictable hot spots. The sensing properties of the fabricated metamaterials in a practical setup will depend on the setup and sensing protocol. The choice of the “operating point” is especially important for phase-sensitive detection. On the one hand, the closer this point to the point of total darkness the better the sensitivity of the method could be (the improvement of phase sensitivity over amplitude can be roughly approximated as 1/Ψmin, where Ψmin is the ellipsometry parameter in the minimum of the reflection curve expressed in radians[14]). This relation implies that the fabricated materials with topological darkness could in principle provide 100-fold increase of raw sensitivity as the measured minimum value of Ψ in our experiments was about 0.5°. On the other hand, the closer the “operating point” to the point of total darkness, the smaller the light intensity (which could affect the signal-to-noise ratio) and the smaller the dynamic range of measurements. For these reasons, in a practical setup, one normally detunes from the point of topological darkness by changing the angle of incidence.[14] At present, we are measuring the phase sensitivity of the fabricated materials by using graphene hydrogenation. The full description of these measurements lies beyond the scope of our manuscript and we are hoping to present them in some detail elsewhere. Finally, we show that the observed minima of reflection near 400 nm do indeed correspond to the topological darkness described in Figure 1. We consider the case of N = 4. Figure 4f shows a zero-reflection curve calculated with the help of Fresnel coefficients for a thin layer with a thickness of 304 nm placed on a silicon substrate at an incident angle of 48° (the color of the points on the curve corresponds to the wavelength at which zero reflection is achieved). This arrangement corresponds to the four-layer metamaterials.[27] When we plot the dispersion curve of our metamaterial (the curve neff(λ), keff(λ)) calculated using EMA, we see that this dispersion curve indeed intersects the zero-reflection curve in such a way that the wavelength of light at which the reflection vanishes coincides with the wavelength of the intersection point at the dispersion curve. This coincidence guarantees the topological darkness for our coreshell metamaterial. The intersection point is provided by the Jordan theorem, while the wavelength matching is achieved by changing the angle of incidence. This matching is simplified by the fact that all changes of the reflection along the dispersion curve happen within the range of the localized plasmonic resonance of the silver core (350–450 nm). This implies that the plasmonic effects are responsible for the observed darkness and the plasmonic enhancement of electromagnetic fields near 400 nm will contribute to enhanced phase sensitivity. The inset to Figure 4f illustrates the principle of topological darkness in 3D graph of (n, k, λ) where intersection of the zero-reflection curve and the dispersion curve of the fabricated metamaterial is guaranteed by a change of the angle of incidence. To conclude, we have experimentally demonstrated the total absence of reflection for self-assembled metamaterials under certain conditions following from topological arguments. The
observed topological darkness could be used to improve the phase-sensitive plasmonic sensing of the materials for chemical and biological applications[14,16,44,45] provided one could achieve sufficient stability of optical transduction signals.[17,19,46] The possibility to fabricate plasmonic nanosensors using selfassembly instead of slow and expensive methods based on nanolithography should dramatically enhance the area of applications of phase-sensitive plasmonic measurements.
Acknowledgements The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007– 2013) under grant agreement No 228762 and was carried out in the frame of the METACHEM project (2009–2013). Received: July 23, 2013 Revised: August 20, 2013 Published online: October 18, 2013
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Topological Darkness in Self-Assembled Plasmonic Metamaterials Ludivine Malassis, Pascal Massé, Mona Tréguer-Delapierre, Stephane Mornet, Patrick Weisbecker, Philippe Barois, Constantin R. Simovski, Vasyl G. Kravets, and Alexander N. Grigorenko* Artificial nanostructured metamaterials can demonstrate new exciting properties not easily achieved in natural materials. With the help of nanostructuring, one can realize surface plasmon subwavelength optics,[1] efficient plasmonic nanotrapping,[2] negative index of refraction,[3] visible light magnetic permeability,[4,5] negative optical phase,[5,6] a plasmonic blackbody based on properties of Fresnel coefficients[7] and connected to coherent absorbers,[8] plasmonic waveguides,[9] quantum plasmon tunneling,[10] quantized plasmonic absorption,[11] optical cloaking,[12,13] and many other unusual phenomena important for applications. Recently, we have shown that nanostructured plasmonic materials can demonstrate socalled topological darkness[14] which could be important for phase-sensitive plasmonic detection[15,16] based on localized surface plasmons,[17–19] which have attracted a lot of attention recently.[20] Topological darkness was achieved with the help of collective plasmon resonances[21–26] exploiting light diffraction in nanoparticle arrays to attain the desired optical properties. Here we show that topological darkness can be realized in bulk plasmonic metamaterials produced by inexpensive self-assembly methods[27] which could make its application to phase-sensitive detection[14] practical. Figure 1 briefly illustrates the concept of topological darkness. Consider light reflection from a thin layer characterized by a refractive index nˆ = n + ik placed on a dielectric substrate. For a given angle of incidence and given film thickness there is a set of values (n, k) which guarantees the absence of Dr. L. Malassis, Prof. P. Barois CNRS, CRPP, UPR8641, F-33600, Pessac, France Dr. L. Malassis, Prof. P. Barois Univ. Bordeaux, CRPP, UPR8641, F-33600, Pessac, France Dr. P. Massé, Dr. M. Tréguer-Delapierre, Dr. S. Mornet CNRS, ICMCB, UPR9048, F-33600, Pessac, France Dr. P. Massé, Dr. M. Tréguer-Delapierre, Dr. S. Mornet Univ. Bordeaux, ICMCB, UPR9048, F-33600, Pessac, France Dr. P. Weisbecker CNRS, LCTS, UMR 5801, F -33600, Pessac, France Prof. C. R. Simovski Dept. Radio Sci. Eng., School ELEC Aalto University FI 00076, Aalto, Finland Dr. V. G. Kravets, Prof. A. N. Grigorenko School of Physics and Astronomy University of Manchester Manchester, M13 9PL, UK E-mail: [email protected]
DOI: 10.1002/adma.201303426
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reflection from the structure (obtained by tuning the light wavelength). This set can be found by analyzing the Fresnel coefficients and is usually represented by zero-reflection curves (the thicker the film, the more curves could provide zero reflection for a given spectral range). Figure 1 shows the p-polarized zeroreflection curve (i.e., for the light polarized parallel to the plane of incidence) computed for a 200 nm film on a glass substrate at an angle of incidence 63°. Normally, continuous metal films cannot provide the optical constants necessary for achieving zero reflection. For example, the curve A–B of Figure 1 shows the dispersion curve (n(λ), k(λ)) for silver and the top inset shows the measured reflection from a 200 nm silver film at an incident angle of 63° which is quite large and never reaches zero. The situation changes dramatically for nanostructured films. In this case, the effective dispersion curve (neff(λ), keff(λ)) shown in Figure 1 as the curve P–Q can have endpoints in two different areas separated by the zero-reflection curve. Hence the effective dispersion curve has to intersect the zero-reflection curve at some point due to Jordan curve theorem.[28] (The Jordan curve theorem states that the line connecting points inside and outside of an enclosed area intersects the boundary of this area at some point. For 3D, the Jordan curve theorem suggests that when two interlocking loops are deformed to remove their interlock they will intersect each other at some stage.) By changing the angle of incidence, one can adjust the intersection point in such a way that it will happen at the same wavelength for both curves (the zero-reflection curve and the curve P–Q), which results in zero reflection for a nanostructured material, see the bottom inset of Figure 1. We refer to this absence of reflection as topological darkness because the effect survives the imperfections of the internal nanostructure geometry due to the curve topology.[14] We have also experimentally showed that it can drastically improve the phase sensitivity of localized plasmon resonances.[14] For this reasons, we believe that materials which demonstrate topological darkness could find a wide scope of sensing applications. Before describing the main result of our work, it is worth putting the concept of topological darkness into a historical context. The absence of reflection was previously observed in various optical systems. For example, light reflection from a single interface at the Brewster angle should in theory provide zero reflection due to the nature of a dipole radiation pattern[29] (the absence of radiation in the direction of an oscillating dipole). Also, light reflection in the Kretschmann configuration (for a thin metal film deposited on a dielectric substrate) should in theory provide zero reflection.[30] The closest analogue of topological darkness can be found in the plasmonic
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blackbody[7] (which realizes maximal light absorption from a thin layer of an optical material placed on a dielectric substrate) also known as a coherent absorber.[8] However, all these examples of systems with zero reflection can be easily affected by sample disorder, inhomogeneity, or irregularities. As a result, light reflected under Brewster conditions is not exactly zero and is elliptically polarized due to a transitional layer;[31] light reflection at the level of 1–5% is still detected in the Kretschmann configuration for surface plasmon resonance in gold films due to surface corrugation[16]; nonzero reflection can be observed for a plasmonic blackbody due to, e.g., sample disorder. Topological darkness suggested by Kravets et al.[14] develops the concept of zero reflection further and allows one to elevate problems of sample disorder, inhomogeneity, or irregularities. This concept suggests that a wide class of optical systems with a mild restriction on the effective dispersion curve (neff(λ), keff(λ)) demonstrates exactly zero reflection at certain angles of incidence and light wavelength. This mild restriction on the effective dispersion curve of a structure only implies that the dispersion curve should start and end in two different areas separated by a zero-reflection curve. Then, exactly zero reflection can be achieved due to the Jordan curve theorem. More importantly, this zero reflection is topologically stable and cannot disappear in the presence of disorder. (A simple analogy of such topological stability can be found in the spectrum of massless Dirac fermions in graphene.[32] This spectrum is topologically stable and cannot be affected by the presence of disorder as an
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Figure 1. Topological darkness in plasmonic metamaterials. The graph plots: p-polarized zero-reflection curve for a thin layer (thickness 200 nm) placed on a glass substrate (for an angle of incidence 63°), the dispersion of the bulk silver (the curve A–B), the dispersion curve of a hypothetic metamaterial (the curve P–Q). The color of the points corresponds to the light wavelength. The top inset shows the p-polarized reflection for a silver 200-nm film (angle of incidence 63°). The bottom inset shows the reflection for a metamaterial with the dispersion curve P–Q. Zeroreflection point near 420 nm corresponds to the topological darkness.
intersection point between two curves cannot disappear after small deformations of the curves.) It is worth stressing that topological darkness provides “inevitable” exactly zero reflection for a large class of optical systems. The important application for “dark” metamaterials is phase-sensitive optical transduction. There are other techniques which enjoys the benefits of enhanced phase sensitivity: interferometry,[33] whisperinggallery modes,[34] polarimetry[35] (e.g., based on Pancharatnam phase[36]), etc. However, these methods are either not compatible with high-throughput analysis, or do not enjoy plasmonic enhancement of the sensitivity, or are much more complicated. In the work by Kravets et al.,[14] the topological darkness was observed in a variety of regular plasmonic nanoarrays made by electron-beam lithography. Here, we report the topological darkness in bulk plasmonic metamaterials made by self-assembly which are more attractive for applications due to relatively low fabrication costs. Three-dimensional samples made of N layers of core-shell Ag@SiO2 nanoparticles were prepared by repeating N times a single-layer procedure based on the inverse Langmuir–Shaefer method developed by Lee et al.[37] This method enabled the fabrication of dense assemblies of nanoparticles with a uniform coverage of the whole substrate.[27] Figure 2a and b show a monolayer and a three-layer structure of fabricated metamaterials, respectively, while Figure 2c shows schematics of the studied samples. Core–shell nanoparticles Ag@SiO2 were prepared in two steps. First, silver nanoparticles were made and centrifuged to achieve a monodisperse solution. Then, monodisperse silver particles have been covered with a silica layer using the well-established Stöber method.[38] The size distribution of the nanoparticles was characterized by using a Hitachi H600 transmission electron microscope. The diameter of the Ag core was 27 ± 2 nm. The nanoparticles have a pseudospherical shape with an average size of 82 ± 3 nm. Figure 2d shows the TEM image of the Ag@SiO2 nanostructures with N = 6 layers. The absorption spectrum of the colloidal suspension in ethylene glycol exhibits a sharp band at 425 nm that is characteristic of the dipole resonance of the silica-coated Ag nanospheres. The details of the sample fabrication and characterization can be found in our previous work.[27] The optical measurements were performed in the 240– 1000 nm wavelength range with a variable-angle spectroscopic Woollam ellipsometer M2000F. The spot size was down to 30 μm which was small enough to ensure sensing applications and large enough to have small angular variation of the angle of incidence. In addition to the intensity reflection coefficients Rp and Rs, we also measured the ellipsometric parameters Ψ and Δ defined as rp/rs = tan(Ψ)exp(iΔ) where rp = Ep/Ei and rs = Es/Ei are the amplitude reflection coefficients for light polarization parallel (p), and perpendicular (s) to the plane of incidence, and Ei denotes the electrical field of the incident light. From the definition it follows that Ψ goes to zero whenever the p-polarized reflection goes to zero and Ψ is equal to 90° whenever the s-polarized reflection is zero. Figure 3 shows the main result of our work – the topological darkness observed in self-assembled core–shell plasmonic metamaterials. For the self-assembled metamaterial with N > 3 we found zero reflection for p-polarized light. Figure 3a shows the ellipsometric reflection Ψ for four, five, and six-layer metamaterials in the conditions of topological darkness observed
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Figure 2. Self-assembled core-shell metamaterials. a) Schematics of one layer. b) Three-layer core–shell metamaterial. c) Cross-section of the structure. d) TEM image of six-layer metamaterial shown at two magnifications ×21333 and ×42667 in inset (TEM operated at 10 kV).
near the wavelength of 400 nm at angles of incidence of 48°, 54°, and 63°, respectively. The ellipsometric reflection is just a convenient way to represent the p-reflection coefficient when the reflection becomes very small: Ψ = 0 if and only if Rp = 0. In addition, the ellipsometric data are measured with a better accuracy than normal reflection as they are stable under intensity variation of the light source and perturbations in ambient media. From Figure 3a–d, we see that minimal measured ellipsometric reflection was at the level Ψmin = 0.5° which gives the minimum light intensity reflected from our sample for p-polarization as low as Rpmin = tan2(Ψmin) Rsmin = 5 × 10−5% (where we used Rsmin = 0.7% from Figure 3d). In our installation, the spectral detection was performed by using a photodiode-array spectrophotometer and hence we measured digitized spectra with the wavelength step of about 1 nm. This procedure means that the zero-reflection point could be missed during digitization of the spectrum. The inset to Figure 3a displays the zoomed region of the spectra near 400 nm which shows that data points (corresponding to measured spectral data) can miss the zero-reflection point. In this case, the linear extrapolation (see the straight lines of the inset to Figure 3a) can be used to estimate the position of the theoretically expected topological darkness. As one would expect,[14,16,30,39] the phase Δ shows fast changes in the conditions close to topological darkness which can be used for plasmonic biosensing, see Figure 3b. For completeness, we provide the spectral dependence of intensity reflection Rp and Rs for the four-layer metamaterial under the conditions close to the topological darkness at angles of incidence of 45°, 48°, and 50°. It is worth noting that despite
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the fact that experimental structures (Figure 2d) are not as regular as ideal 3D structures shown in Figure 2a–c, the zero reflection for our “imperfect” samples survives due to topological nature of the suggested phenomenon, see the discussion above. We found that the optical properties of the self-assembled metamaterials with N > 3 are described well within the Maxwell–Garnett effective medium approximation (EMA).[40,41] When the incident light wavelength is much larger than the size of the particle, the core–shell sphere can be approximated by a dipole with polarizability as shown in Equation 1:[42]
"eff = 4B r 23
(g d − 1)(g m + 2g d ) + D (g m − g d )(1 + 2g d ) (g d + 2)(g m + 2g d ) + D (2g d − 2)(g m − g d )
(1)
where r2 is the outer radius of core-shell particle, ρ = (r1/r2)3 is the fraction of the total particle volume occupied by the metalcore, and εm and εd are the permittivity of the metal core and the dielectric shell, respectively. Consequently, a core–shell metamaterial can be described by an effective dielectric constant given by the Clausius–Mossotti formula:[40] (g eff − 1) f NP"eff = (g eff + 2) 4B r 23
(2)
where fNP is the filling fraction of the core-shell nanoparticles Ag@SiO2 or, in other words, the volume fraction of dipoles. By combining Equation 1 and 2 together, we get:
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COMMUNICATION Figure 3. Topological darkness measured in reflection from fabricated metamaterials. a) Ellipsometric reflection Ψ as a function of wavelength at the conditions of topological darkness observed near the wavelength of 400 nm. The inset shows the zoomed region near the plasmonic resonance with linear extrapolation lines. b) Ellipsometric phase Δ as a function of wavelength for four-, five-, and six-layer metamaterials in the conditions of topological darkness, same as (a). c) p-Polarized reflection Rp for four-layer metamaterial near conditions of darkness. d) s-Polarized reflection Rs for four-layer metamaterial at the same conditions as (c).
(g eff − 1) (g d − 1)(g m + 2g d ) + D (g m − g d ) (1 + 2g d ) = f NP (g eff + 2) (g d + 2)(g m + 2g d ) + D (2g d − 2)(g m − g d )
(3)
In fitting the experimental data, the diameter of the Ag core was chosen to be r1 = 25 nm for four and five layers of Ag@SiO2 particles and r1 = 24 nm for six layers. The coreshell nanoparticles were considered as spherical with a size of 82 nm taken from experimental measurements. The volume fraction fNP of the Ag@SiO2 particles was found from the best fit as fNP = 0.55. The electric permittivity of silver, εm, can be described by a sum of the intraband term (the Drude term for electronic response of the free electrons in the conduction band) and interband transitions (the Lorentz term for electronic transitions between the d valence band and the sp conduction band). Hence, we fit εm as:
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g m (T) = g 0 −
Tp2 T2
+ i T(
−
2
g j 2j
j =1
T2 − 2j + i T j
(4)
with ε0 = 1.45, បγ = 0.07 eV, បωp = 9 eV, Δεj = (1.0, 1.1) បΩj = (4.5, 5.7) eV, បΓj = (1.2, 2.7) eV, where j = 1,2, for details see work by Kravets et al.[43] In the fitting, we first used the Drude– Lorentz parameters extracted from the ellipsometric measurement for a thick (200-nm) Ag film on top of a glass substrate, and then adjusted the parameter ε0 which characterizes the contribution of interband transitions to the permittivity (this adjustment describes the changes in screening effects). The optical constant, εd, of a SiO2 film was parameterized by using a Cauchy function found from the ellipsometric measurements. The calculated effective complex refractive index of the core–shell metamaterial within the Maxwell–Garnett EMA approach agrees well with the refractive index extracted from
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Figure 4. Effective medium approximation. a) Comparison of dispersion curves for four-layer metamaterial calculated with Maxwell–Garnett theory (ntheor and ktheor) with the dispersion curves extracted from ellipsometric measurements (nexp and kexp). b) The calculated Ψ corresponding to experimental data of Figure 3a. c) The calculated p-polarized reflection Rp corresponding to Figure 3c. d) The calculated s-polarized reflection Rs which corresponds to Figure 3d. e) Ψ corresponding to conditions of Figure 3c and d. f) The calculated p-polarized zero-reflection curve and the dispersion curve for the four-layer core-shell metamaterial at angle of 48°. Notice that the intersection point at (n, k) = (1.2, 0.2) and wavelength of approxiamtely 400 nm which guarantees the absence of reflection (topological darkness). The inset shows the intersection of the zero-reflection curve and the dispersion curve of the four-layer metamaterial in 3D coordinates (n, k, λ).
experimental data using ellipsometry for N > 3, Figure 4a. We can see just one pronounced extinction peak near 400 nm in all keff curves. This peak can be associated with excitation of localized surface plasmons in the silver core. Outside the region of the localized plasmon resonance (350–450 nm) neither n nor k show significant changes. By using EMA dielectric constants and Fresnel coefficients, we calculated optical properties of the layers of core–shell particles with N = 4–6 and found good agreement between the measured and calculated values. For example, the calculated ellipsometric function Ψ in the 328
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condition of topological darkness, Figure 4b, shows very good agreement with the experimental data of Figure 3a. The same agreement can be seen for the calculated intensity reflection Rp and Rs for the four-layer metamaterial, Figure 4c and d, which correspond to experimental data shown in Figure 3c and d, respectively. The calculated data give a reasonable quantitative description for the optical properties of the sample except the region below λ = 300 nm where plasmonic resonances of higher orders should be taken into account. For completeness, we also provide the calculated value of Ψ for the four-layer
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metamaterial near the point of topological darkness, Figure 4e. It is worth stressing that, despite the fact that our metamaterials can be described within the Maxwell–Garnet EMA which is often used to describe random samples, the regular layered structure is very important for sensing applications as it reduces the number of unpredictable hot spots. The sensing properties of the fabricated metamaterials in a practical setup will depend on the setup and sensing protocol. The choice of the “operating point” is especially important for phase-sensitive detection. On the one hand, the closer this point to the point of total darkness the better the sensitivity of the method could be (the improvement of phase sensitivity over amplitude can be roughly approximated as 1/Ψmin, where Ψmin is the ellipsometry parameter in the minimum of the reflection curve expressed in radians[14]). This relation implies that the fabricated materials with topological darkness could in principle provide 100-fold increase of raw sensitivity as the measured minimum value of Ψ in our experiments was about 0.5°. On the other hand, the closer the “operating point” to the point of total darkness, the smaller the light intensity (which could affect the signal-to-noise ratio) and the smaller the dynamic range of measurements. For these reasons, in a practical setup, one normally detunes from the point of topological darkness by changing the angle of incidence.[14] At present, we are measuring the phase sensitivity of the fabricated materials by using graphene hydrogenation. The full description of these measurements lies beyond the scope of our manuscript and we are hoping to present them in some detail elsewhere. Finally, we show that the observed minima of reflection near 400 nm do indeed correspond to the topological darkness described in Figure 1. We consider the case of N = 4. Figure 4f shows a zero-reflection curve calculated with the help of Fresnel coefficients for a thin layer with a thickness of 304 nm placed on a silicon substrate at an incident angle of 48° (the color of the points on the curve corresponds to the wavelength at which zero reflection is achieved). This arrangement corresponds to the four-layer metamaterials.[27] When we plot the dispersion curve of our metamaterial (the curve neff(λ), keff(λ)) calculated using EMA, we see that this dispersion curve indeed intersects the zero-reflection curve in such a way that the wavelength of light at which the reflection vanishes coincides with the wavelength of the intersection point at the dispersion curve. This coincidence guarantees the topological darkness for our coreshell metamaterial. The intersection point is provided by the Jordan theorem, while the wavelength matching is achieved by changing the angle of incidence. This matching is simplified by the fact that all changes of the reflection along the dispersion curve happen within the range of the localized plasmonic resonance of the silver core (350–450 nm). This implies that the plasmonic effects are responsible for the observed darkness and the plasmonic enhancement of electromagnetic fields near 400 nm will contribute to enhanced phase sensitivity. The inset to Figure 4f illustrates the principle of topological darkness in 3D graph of (n, k, λ) where intersection of the zero-reflection curve and the dispersion curve of the fabricated metamaterial is guaranteed by a change of the angle of incidence. To conclude, we have experimentally demonstrated the total absence of reflection for self-assembled metamaterials under certain conditions following from topological arguments. The
observed topological darkness could be used to improve the phase-sensitive plasmonic sensing of the materials for chemical and biological applications[14,16,44,45] provided one could achieve sufficient stability of optical transduction signals.[17,19,46] The possibility to fabricate plasmonic nanosensors using selfassembly instead of slow and expensive methods based on nanolithography should dramatically enhance the area of applications of phase-sensitive plasmonic measurements.
Acknowledgements The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007– 2013) under grant agreement No 228762 and was carried out in the frame of the METACHEM project (2009–2013). Received: July 23, 2013 Revised: August 20, 2013 Published online: October 18, 2013
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