Ultimate Limit of Light Extinction by Nanophotonic Structures - Nano

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The ultimate limit of light extinction by nanophotonic structures Zhong-Jian Yang, Tomasz J. Antosiewicz, Ruggero Verre, F. Javier Garcia de Abajo, S. Peter Apell, and Mikael Käll Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b03512 • Publication Date (Web): 19 Oct 2015 Downloaded from http://pubs.acs.org on October 25, 2015

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The ultimate limit of light extinction by nanophotonic structures Zhong-Jian Yang,† Tomasz J. Antosiewicz,†,‡ Ruggero Verre,† F. Javier García de Abajo,§ S. Peter Apell,† and Mikael Käll*† †

Department of Applied Physics, Chalmers University of Technology, 41296 Göteborg, Sweden

‡

Center of New Technologies, University of Warsaw, Zwirki i Wigury 93, 02-089 Warsaw,

Poland §

ICFO, Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels

(Barcelona), Spain and ICREA, Institució Catalana de Reserca i Estudis Avançats, Barcelona, Spain

ABSTRACT: Nanophotonic structures make it possible to precisely engineer the optical response at deep sub-wavelength scales. However, a fundamental understanding of the general performance limits remains a challenge. Here we use extensive electrodynamics simulations to demonstrate that the so-called f-sum rule sets a strict upper bound to the light extinction by nanostructures regardless their internal interactions and retardation effects. In particular, we show that the f-sum rule applies to arbitrarily complex plasmonic metal structures that exhibit an extraordinary spectral sensitivity to size, shape, near-field coupling effects, and incident polarization. The results may be used for benchmarking light scattering and absorption

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efficiencies, thus imposing fundamental limits on solar light harvesting, biomedical photonics, and optical communications.

KEYWORDS: Nanophotonics, Plasmonics, Extinction, f-sum rule

The Thomas-Reiche-Kuhn sum rule, also known as f-sum rule1,2, states that the sum of all oscillator strengths associated with optically active transitions between initial and final electronic states i and j in a system is proportional to the total number of electrons Ne in that system, with the coefficient of proportionally simply expressed in terms of fundamental constants. The f-sum rule is well known to apply to sub-wavelength quantum systems, such as atoms and molecules, for which it has been extensively used to determine absolute x-ray scattering properties3,4. It is also an important ingredient in Fermi-liquid theory5. Perhaps a less known result is that it can be equally applied to classical systems, in this case expressed in terms of the frequency-dependent optical extinction cross-section σe(ω):

=

≈ 1.7 × 10 [! " # ].

(1)

Equation (1) implies that the number of electrons in the illuminated structure uniquely determines the light extinction, that is, the sum of absorption and scattering, integrated over all photon energies. The general applicability of Eq. (1) was, to the best of our knowledge, first proven by Sievers6, although his analysis explicitly refers to single ellipsoidal particles. We revisit the general proof of the f-sum rule in the Supporting Information (SI), including retardation, and show that it is valid for arbitrary systems composed of materials that can be described by a local, causal, and linear dielectric response function %&, , where a dependence


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on position and frequency is explicitly shown. We also note that additional sum rules can be obtained by weighting the extinction cross-section by different powers of the frequency , although these relations lack the simplicity of the f-sum rule, as they tend to be highly dependent on geometry. In particular, Purcell, in his analysis of scattering by non-absorbing interstellar grains7, derived a sum-rule that relates the extinction of a body to its static polarizability, as further discussed in the SI.

Figure 1. Numerical corroboration of the f-sum rule for free-electron systems. (A) Schematic of illumination and system configuration, which consists of Z cubes on a 1D lattice with 10 sites. (B) Four examples of extinction spectra for Z = 6 cubes illustrating the huge variation in plasmonic response. The different configurations are shown in the inset. The side-length of each cube is L = 20 nm. (C) Frequency integrated extinction, absorption and scattering for the 287 distinct configurations obtained for ( ∈ *1,10+ and L = 20 nm. (D) Log-log plot of integrated


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extinction, absorption and scattering. The plot includes data for L = 20, 30 and 50 nm (861 configurations in total). The integrated extinction scales linearly with the number of electrons (green line), while the average integrated scattering scales as (pink line) for the size range under consideration. All data are obtained from FDTD simulations based on a free electron dielectric response function with plasma frequency ћ- = 6.056 eV and scattering rate ћ. =0.132 eV.

Metal structures that support well-defined surface plasmons, that is, collective oscillations of conduction electrons, exhibit extraordinary but complicated light scattering, absorption and field-enhancement properties8-12. We therefore first put the validity of Eq. (1) to the test by a series of in silico experiments on structures characterized by a Lorentz-Drude dielectric function % = 1 + - ⁄ − − 1. , where - is the plasma frequency, . is the damping rate and is an intrinsic resonance frequency ( = 0 for the special case of a free-electron metal). Figure 1 shows results for a system comprising Z metal cubes, where Z runs from 1 to 10, placed along a one-dimensional (1D) lattice with 10 positions (Fig. 1a), yielding 287 distinct configurations (see Table S1 in the SI). Each cube has a side length L = 20 nm and nearestneighbor cubes are in metallic contact. We compute far-field spectra and optical cross sections using a commercial finite-difference time-domain (FDTD) software (FDTD Solutions; Lumerical Inc., Canada). Figure 1b shows four illustrative extinction spectra out of the 70 nondegenerate combinations available for Z = 6 (see Fig. S3 for more examples). The structures are illuminated by a plane wave polarized along the array direction. The spectra exhibit a huge variation in terms of the number of surface plasmon modes present, as well as in their corresponding resonance frequencies, line-widths, and intensities. However, when the extinction spectra are integrated over frequency, they all yield the same number, as predicted by Eq. (1) and as clearly seen in Fig. 1c, which presents results for all 287 configurations. We note the relation = - (34 !% /6 , which we use to produce the horizontal axis of Fig. 1d. We also calculate


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absorption spectra 7 and elastic scattering spectra 8 for the cube arrangements. However, the corresponding frequency-integrated quantities do not simply scale with , as clearly shown by the data presented in Figs. 1c and 1d (see also Fig. S4 in the SI). It is well known that absorption and scattering scale with volume as 7 ∝ : and 8 ∝ : , respectively, for particles in the Rayleigh regime ( : ≪ region through an analytical extrapolation (Fig. S9). This fully compensates for the deviation from the sum-rule expectation. The free-electron dielectric response function used for the simulations in Fig. 1 fulfills the causality property that is the basis for the application of the f-sum rule. To check the validity for real materials, we focus on aluminum and silicon, for which experimental dielectric data are available over a wide frequency range covering THz (meV) up to ~EHz (keV)3,4,17,18. We


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consider aggregates of spherical particles and use generalized Mie theory for the electrodynamics simulations because the FDTD method is impractical to simulate this very large frequency range. Mie theory is accurate but restricted to systems of spherical particles. Figures 3a and 3b show exemplary extinction spectra for Si and Al single spheres and dimers. The sharp peaks are dipolar surface plasmons with frequency close to - ⁄√3, the dipolar plasmon mode of a sphere, while the high-frequency spectral structures correspond to various core-electron excitations. However, the sum rule applies regardless the origin of the spectral features. This is clearly illustrated by Fig. 3c, where we compare the integrated extinction for a cutoff frequency of ≈ 10#A Hz (energy ~4 keV) with the prediction from the sum rule based on the total electron number density for the two materials. The results clearly confirm that the f-sum can be applied to real systems, at least as long as the dielectric data that are the basis for these calculations are trustworthy.

Figure 3. Application of the f-sum rule to aluminium and silicon nanostructures. (A) Extinction spectra of a single Al sphere (diameter D = 24 nm) and a heterodimer (D1 = 22 nm, D2 = 15 nm, gap distance d = 5 nm) for two orthogonal polarizations. (B) Extinction spectra for a Si sphere (D = 25 nm) and a homodimer (D = 20 nm, d = 5 nm). (C) Integrated extinction as a function of total volume for single spheres and dimers composed of Al and Si.


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The integration covers the 0.4-4000 eV region. The lines show the expected f-sum rule result (Eq. (1)) based on the total electron densities of Al and Si.

To further illustrate the practical relevance of the f-sum rule, we measured and simulated extinction and absorption spectra for disk-shaped gold nanoparticles (Fig. 4). The samples were produced on cover glass slides by Hole-mask Colloidal Lithography (HCL) according to the procedures described in Ref.19. The particles have different diameters D but the same height h, which means that their in-plane dipolar plasmon resonance shifts with the aspect ratio D/h in a similar way as in oblate spheroidal metal nanoparticles. Varying D from ~54 nm to ~216 nm results in a rapid plasmon redshift and a concomitant increase in extinction, primarily due to increasing scattering losses (Fig. 4a). Both of these trends are reproduced by FDTD simulations (Fig. 4c). We integrated the extinction spectra over the investigated wavelength range and found that the data points fall on a straight line through the origin when plotted against particle volume, except for a small deviation for the smallest particle sizes (Fig. 4d). The integrated absorption, in contrast, falls off rapidly with increasing disk diameter similar to the data in Fig. 1. The f-sum rule expectation that the integrated extinction should scale linearly with volume is thus essentially fulfilled even though a very limited spectral region actually contributes to the integrand in this case. In fact, the integrated extinction obtained from Fig. 3d only accounts for ~0.5% of the total electron density of gold. To understand this, we recall that both free and bound electrons contribute to the gold dielectric response within the considered photon energy range. The former contribution is manifested through plasmon resonances, strongly dependent on particle morphology, whereas the latter is determined through the accessible interband transitions and therefore essentially independent of particle size and shape. The bound electrons will thus contribute to the integrated extinction in proportion to volume and spectral range as long as the


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incident excitation field is able to probe the complete structure. This is the case for the samples in Fig. 4 because the nanodisks height h = 20 nm is smaller than the optical penetration depth (d > ~30 nm) for energies in the interval between the “interband threshold” at ~2.34 eV (~530 nm) and the spectral cutoff at ~3.54 eV (350 nm). Additionally, the dipolar plasmon contribution is almost completely covered by the integration range, except for the smallest nanodisk, for which the plasmon excitations overlap the interband excitations and therefore broadens to the extent that part of the response is pushed beyond the high frequency cutoff20 (Fig. S10).

Figure 4. Experimental and simulated extinction and absorption of gold nanodisks with varying diameters (D = 54-216 nm) but the same height (h = 20 nm). (A) Experimentally measured extinction spectra of h = 20 nm high gold nanodisks fabricated by colloidal lithography with various disk diameters D. (B) Scanning electron micrographs of the nanodisk samples. (C) Simulated extinction spectra based on the same structure parameters as in the experiments and tabulated data for the Au dielectric function17. (D) Integrated extinction and absorption over the


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spectral range shown in A and B (350 − 950 nm wavelength) as a function of nanodisk volume. The line is a linear regression through the origin.

Plasmonic structures are well known for their use as optical antennas in applications such as molecular surface-enhanced Raman scattering and fluorescence spectroscopy21, as well as light harvesting based on hot-electron generation22, or heat generation23. We investigated the antenna effect in the context of the f-sum rule for a prototypical dimer antenna system, which we simulated as two Drude metal cuboids separated by a gap in which strong field-enhancement is produced (Fig. 5a). A dielectric cuboid, acting as a receiver and characterized by a Lorentz dielectric function with resonance frequency matched to the antenna resonance, is gradually inserted into the gap while the integrated extinction and absorption contributions are monitored through FDTD calculations. As expected from the f-sum rule and the above results, the total integrated extinction is found to be independent of the specific location of the dielectric and antenna components (Fig. 5d). However, the corresponding absorption contributions show drastic changes (Fig. 5b), that is, the absorption in the receiver increases by a factor ~6 as it is moved from a position outside the range of the antenna near-fields into the “hot” gap region, while the absorption in the metal exhibits a concomitant reduction (Fig. 5c). The antenna effect thus effectively redistributes absorbed photon energy between the two materials, but the net light extinction ability of the composite system remains the same.


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Figure 5. The f-sum rule applied to a plasmonic antenna-reciever system. (A) Schematic of the system, consisting of a dielectric “receiver” (red) and a gap antenna formed by two free-electron cuboids. (B) Integrated absorption cross-section for the receiver as a function of distance from the antenna gap. (C) Integrated absorption in the antenna as a function of distance. (D) Area plot of the integrated absorption in the receiver (red) and antenna (grey) together with the integrated scattering cross-section of the composite system (green) as a function of distance (see A). The sum of the absorption and scattering contributions is independent of separation. The receiver has a volume of 10×12×12 nm3, plasma frequency ћ- =3.87 eV, resonance frequency ћ = 2.764 eV and damping rate ћ. = 0.033 eV, while each antenna element has a volume of 22×12×12 nm3, plasma frequency ћ- = 9.87 eV and damping rate ћ. = 0.92 eV. The gap separation is 16 nm. Note that the absorption contributions can be calculated by enclosing each individual element in a power monitor box surounding the respective element, while the scattering contribution is a property of the coupled system and cannot therefore be separated into antenna and receiver contributions.


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Identifying ideal structure parameters for specific functionalities, such as light absorption, scattering, field-enhancement, or antenna performance24,25 is not straightforward because solving Maxwell’s equations for arbitrary nanostructures is in general so computationally demanding that optimization algorithms fail to converge within reasonable time frames. However, recent results based on evolutionary schemes26,27 and convex optimization approaches28 are highly promising and the f-sum rule can provide useful upper boundaries for such approaches. Consider, for example, the antenna enhancement of the integrated absorption in the receiver of Fig. 5 and let O and P be the total number of electrons in the uncoupled antenna and receiver, respectively. The maximum absorption in the receiver in isolation is dictated by its maximum integrated extinction and therefore proportional to P . Likewise, the absorption of the receiver in the coupled system can never be higher then the total integrated extinction of that system, which is proportional to O + P . An upper bound for the antenna enhancement is therefore given by the f-sum rule as Q7R = O + P ⁄P ≈ O ⁄P , where the approximation holds when O ⁄P ≫ 1 . For the particular parameters chosen, we have O ⁄P ≈ 36 , indicating that considerably higher enhancement factors than the factor of six obtained here could be reached through further optimization of the antenna geometry. The f-sum rule also has other appealing applications. For example, it directly provides a lower limit to the amount of material needed to absorb a given fraction of a broadband light source. We first note that Eq. (1) can be rewritten to yield the total integrated extinction induced by a single electron in terms of only natural constants. Multiplying by ħ to convert frequency into energy, Eq. (1) yields a value of ~1.12 eVÅ2 per electron, that is, the average extinction cross-section in an energy interval ∆E will be of the order 1 eVÅ2/∆E per electron if the total extinction provided by the sum rule is exhausted within this energy interval. Reasoning in this way, we can estimate the minimum number of


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electrons per unit area needed to absorb a fraction of the solar spectrum, as illustrated in Fig. 6. As an application of this result, we note that the wavelength range covered by the plasmonic region of the gold particle spectra in Fig. 3 (~500-1000 nm, ∆E ≈ 1.5 eV) essentially matches the range specified by the red line in the inset of Fig. 6, which in turn corresponds to ~40% of the total solar irradiance. From the main graph in Fig. 6, we see that one would need an electron density of at least ~7.5×1015 cm-2 to completely absorb this fraction. Since the plasmonic region in Fig. 4 exhaust ~20% of the total valence electron density of gold (U VW ≈ 5.9×1022 cm-3), a nanostructured gold surface designed for “perfect absorption” within the specified spectral region has to have an average thickness of at least ~1.27 nm. This type of analysis can of course be easily extended to other materials and light sources, as well as to applications that involve both scattering and absorption.

Figure 6. Minimum number of electrons per unit area needed to absorb a given fraction of the solar spectrum. The graph is constructed as follows: we first draw a horizontal line crossing the solar spectrum for normal incidence29, as shown in the inset. The spectral region corresponding to solar irradiance higher than the line defines a certain fraction of the total solar intensity incident on a surface. The f-sum rule dictates the minimum number of electrons per unit area needed to absorb this fraction. The main curve is obtained by scanning the line


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from top to bottom in the solar irradiance spectrum and calculating the corresponding electron density needed to exhaust the sum rule.

In summary, we have presented a comprehensive analysis and numerical test of the f-sum rule and shown that it applies to a variety of complex system configurations and material parameters. There seems to be a widespread belief that the f-sum rule only applies to isolated atomic-scale objects and this might be the main reason why it has been largely overlooked by the nanophotonics community. We hope that the present results demonstrate its usefulness in providing strict limits to optical performance.

ASSOCIATED CONTENT Supporting Information. Proof of generalized f-sum rule including retardation. Discussion about Purcell’s static sum rule. The f-sum rule for a number of other compositional and geometrical parameters. Effect of interband excitations on the integrated extinction for Au disks. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Author Contributions MK initiated and coordinated the project. ZJY and JGA performed numerical simulations. JGA and PA developed analytical theory. RV made samples and optical measurements. All authors contributed with analysis, discussions and in writing the manuscript. Notes


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The authors declare no competing financial interests. ACKNOWLEDGMENT This work was supported by the Knut and Alice Wallenberg Foundation and the Swedish Foundation for Strategic Research. We thank Philippe Tassin and Peter Johansson for stimulating discussions. TOC:

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Figure 1. Numerical corroboration of the f-sum rule for free-electron systems. (A) Schematic of illumination and system configuration, which consists of Z cubes on a 1D lattice with 10 sites. (B) Four examples of extinction spectra for Z = 6 cubes illustrating the huge variation in plasmonic response. The different configurations are shown in the inset. The side-length of each cube is L = 20 nm. (C) Frequency integrated extinction, absorption and scattering for the 287 distinct configurations obtained for Z∈⌈1,10⌉ and L = 20 nm. (D) Log-log plot of integrated extinction, absorption and scattering. The plot includes data for L = 20, 30 and 50 nm (861 configurations in total). The integrated extinction scales linearly with the number of electrons Ne (green line), while the average integrated scattering scales as Ne2 (pink line) for the size range under consideration. All data are obtained from FDTD simulations based on a free electron dielectric response function with plasma frequency ωp= 9.2×1015 s-1 and scattering rate γ= 2×1014 s-1. 112x97mm (300 x 300 DPI)


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Figure 2. Extinction by rectangular free-electron structures with constant volume (2x105 nm3) but varying height. The parameters describing the dielectric function are the same as in Fig. 1. (A) Extinction spectra for structures of square section and height 20, 50 and 80 nm. The inset shows a log-log plot of the high-energy region. (B) Frequency-integrated extinction versus height (red dots). The small deviation from the f-sum rule (solid line) is caused by a finite cutoff in the integration at ωc=15ωp, which can be compensated by including the high-frequency extinction tail through an analytical extrapolation (blue dots). 139x256mm (300 x 300 DPI)


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Figure 3. Application of the f-sum rule to aluminium and silicon nanostructures. (A) Extinction spectra of a single Al sphere (diameter D = 24 nm) and a heterodimer (D1 = 22 nm, D2 = 15 nm, gap distance d = 5 nm) for two orthogonal polarizations. (B) Extinction spectra for a Si sphere (D = 25 nm) and a homodimer (D = 20 nm, d = 5 nm). (C) Integrated extinction as a function of total volume for single spheres and dimers composed of Al and Si. The integration covers the 0.4-4000 eV region. The lines show the expected f-sum rule result (Eq. (1)) based on the total electron densities of Al and Si. 64x29mm (300 x 300 DPI)


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Figure 4. Experimental and simulated extinction and absorption of gold nanodisks with varying diameters (D = 54-216 nm) but the same height (h = 20 nm). (A) Experimentally measured extinction spectra of h = 20 nm high gold nanodisks fabricated by colloidal lithography with various disk diameters D. (B) Scanning electron micrographs of the nanodisk samples. (C) Simulated extinction spectra based on the same structure parameters as in the experiments and tabulated data for the Au dielectric function16. (D) Integrated extinction and absorption over the spectral range shown in A and B (350-950 nm wavelength) as a function of nanodisk volume. The line is a linear regression through the origin. 109x91mm (300 x 300 DPI)


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Nano Letters

Figure 5. The f-sum rule applied to a plasmonic antenna-reciever system. (A) Schematic of the system, consisting of a dielectric “receiver” (red) and a gap antenna formed by two free-electron cuboids. (B) Integrated absorption cross-section for the receiver as a function of distance from the antenna gap. (C) Integrated absorption in the antenna as a function of distance. (D) Area plot of the integrated absorption in the receiver (red) and antenna (grey) together with the integrated scattering cross-section of the composite system (green) as a function of distance (see A). The sum of the absorption and scattering contributions is independent of separation. The receiver has a volume of 10×12×12 nm3, plasma frequency ωp= 5.88×1015 s-1, resonance frequency ω0= 4.2×1015 s-1 and damping rate γ= 5×1013 s-1, while each antenna element has a volume of 22×12×12 nm3, plasma frequency ωp= 1.5×1016 s-1 and damping rate γ= 1.4×1015 s-1. The gap separation is 16 nm. Note that the absorption contributions can be calculated by enclosing each individual element in a power monitor box surounding the respective element, while the scattering contribution is a property of the coupled system and cannot therefore be separated into antenna and receiver contributions. 111x103mm (300 x 300 DPI)


Nano Letters

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Nano Letters

TOC figure 34x14mm (300 x 300 DPI)


Ultimate Limit of Light Extinction by Nanophotonic Structures - Nano

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