SIMULATION OF PERIODIC NANOSTRUCTURES FOR DESIGN AND OPTIMIZATION OF PLASMONIC BIOSENSORS Niccolò Michieli CNISM and Department of Physics and Astronomy «G.Galilei» - University of Padua Nanostructures Group
LIGHT-MATTER INTERACTION BIOSENSORS Biosensor => transduce biological (and chemical) signal into • a set of useable informations (composition, concentration...) Light-matter interactions are the main physical • phenomena exploited to this aim. Particularly, Nanotechnology and Nanoscience • can yield important advantages: size of analytes: 0.1-1000 nm • energetic and roto-vibrational transitions => nanometric wavelenghts • • optical technology, sources and detectors (noble) metals nanostructures support coherent • free charges oscillations PLASMONICS
LIGHT-MATTER INTERACTION BIOSENSORS Biosensors Labeled Label-free Specific Refractive Surface SERS Fluorescence ... ... Absorption index modification Light Light Localized Surface Hot Spots Management Management Plasmons Plasmos Nanotriangles Nanoholes
PLASMONICS – SURFACE AND LOCALIZED PLASMONS Plasmons Surface Plasmons Localized Plasmons • Propagating modes Confined modes • Coupling by prism or grating • • Direct coupling • Poor Field Confinement Strong Field Confinement • Low absorption • High absorption • Extremely sensitive to • • Extremely sensitive to surrounding index changes (geometrically) small changes in surround
SURFACE PLASMON POLARITONS • Wave equation (Helmhotz’s Equation) at metal -dielectric interface: 𝜖 2 𝑭(𝑨) 2 𝑭 = 𝟏 2 𝜁 − 𝑙 𝑦 + 𝑙 0 𝜖𝑨 2 Putting the Boundary Conditions, the dispersion relation results: • 𝜁 1 𝜁 2 𝑙 𝑦 = 𝑙 0 𝜁 1 + 𝜁 2 Direct coupling forbidden!! • • Coupling methods:
LOCALIZED PLASMONS – QUASI-STATIC APPROXIMATION • Laplace Equation with Boundary Conditions: 𝛼 𝟑 𝝔 = 𝟏 Solutions given by a Spherical Harmonics Expansion: • ∞ 𝜚 𝑠, 𝜄 = 𝐵 𝑚 𝑠 𝑚 + 𝐶 𝑚 𝑠 − 𝑚+1 𝑄 𝑚 cos 𝜄 𝑚=0 Resulting Polarizability: • 𝛽 = 4𝜌𝑆 3 𝜁 − 𝜁 𝑛 𝜁 = −2𝜁 𝑛 Resonance!! 𝜁 + 2𝜁 𝑛 Resulting Electric Field in Quasi-Static (dipolar) Approximation: • 𝑙 2 𝒐 × 𝒒 × 𝒐 𝒇 𝒋𝒍𝒔 1 𝑠 3 − 𝑗𝑙 1 𝑠 2 𝒇 𝒋𝒍𝒔 𝑭 = 𝑭 𝟏 + + 3𝒐 𝒐 ∙ 𝒒 − 𝒒 4𝜌𝜁 0 𝜁 𝑛 𝒔 Far Field Near Field
BEYOND QUASI-STATIC APPROXIMATIONS Exact solutions of the electromagnetic problem only exists for a few particular cases: Spherical isolated particles: Mie theory • Multipole expansion (quadrupoles, ottupoles,...) Ellipsoidal particles: Gans theory • • Spherical interacting particles: Generalized Multiparticle Mie (GMM) For all other geometries: Discretization (DDA, FEM, FDFD, FDTD) Typical observed features: 14 0.6 12 0.5 Extinction Efficiency 10 0.4 Reflectance 8 0.3 6 0.2 4 0.1 2 0 0.0 500 600 700 800 900 1000 46 48 50 52 54 56 58 60 62 Wavelength (nm) Angle (°)
HOW TO EXPLOIT PLASMONICS FOR BIOSENSORS? Sensors Transmission/ Reflection SERS Extinction Resonances Propagating Hot Spots Modes Peaks Nano Triangles Nanohole Arrays
NANOSTRUCTURES SELF-ASSEMBLY - NSL Nano Triangles 2a. 1. SiO 2 NS (30-1000 nm)
NANOSTRUCTURES SELF-ASSEMBLY - NSL Nano Triangles 2a. 1. SiO 2 NS (30-1000 nm) Nano Hole Arrays 2b.
2D PERIODIC STRUCTURES: THE HONEYCOMB LATTICE All the structures taken in count are periodic, with a «honeycomb» lattice. The lattice can be decomposed in triangles or in exagons. • It can be seen as a sinlge plane of a HCP crystal. • The unit cell is a rhomb, which is one third of the unitary • exagon and is composed by two unit triangles.
MODELS SETUP: SOLVING MAXWELL’S EQUATIONS BY FINITE ELEMENTS • The geometry of the model contains: the nanostructure itself the substrate under the NS the medium over the NS PML domains to simulate infinite semi-spaces. • The domain is then divided in finite elements (meshing) Dielectric functions of glass and metals • from experimental data (J&C,Palik) • Helmholtz equation (FT of the wave equation) is solved point by point for each frequency: 𝛼 2 𝐹 + 𝑙 2 𝐹 = 0
NANOTRIANGLES • The unit cell contains 2 nanotriangles. These are placed in the centers of the two triangles forming the unit cell. • The tips of tips of the triangles are faced each-other. The exact shape of the monomer is determined by the technique • of deposition, the quantity of deposed metal, and by (optional) thermal annealing. Nanostructures have been modelized both as snipped prisms • and snipped thetrahedra.
NANOTRIANGLES – RESONANCE TAYLORING The resonance position (and shape) is strongly dependent on the geometry of the triangles, interactions and surrounding media characteristics. To taylor the resonance, the following parameters are considered: The geometric properties of the structures used for • parametrization: Lattice constant (dependent on PS nanospheres) Triangles side and height (dependent on deposition parameters) Snipping of the tips (dependent on deposition and annealing) Dielectric properties of environment have been considered • Presence of interfaces (substrates) •
NANOPRISMS - GEOMETRY Aspect Ratio Closer Base 16 L=200 nm to 1 => Blueshift L=300 nm 14 h=30 nm Larger => Redshift h=45 nm Extinction Efficiency 12 10 8 6 4 2 0 400 500 600 700 800 900 1000 1100 1200 Wavelenght (nm) Base Prism: L=100 nm; h=10 nm; snip=10 nm, silver
NANOPRISMS – GEOMETRY AND INTERACTION Single Snipping Interactions d=120nm c-c 14 d=240nm c-c 12 d=360nm c-c Extinction Efficiency exagon l=120nm 10 snip=20nm snip=30nm 8 6 4 2 0 400 500 600 700 800 900 Wavelength (nm) Base Prism: L=100 nm; h=10 nm; snip=10 nm, silver
NANOPRISMS – SURROUNDING MEDIA/INTERFACE Air 2 ) -14 m 8 Silica Glass Quartz Glass 7 Extinction Cross Section (10 Silica Glass Interface 6 5 n=2.05 4 n=1.45 3 2 1 0 600 800 1000 1200 Wavelength (nm)
NANOTRIANGLES – HOT SPOTS • Surface Enhanced Raman Spettroscopy (SERS) efficiency is strogly dependent on local field: = 𝐹 2 𝜕 𝑗𝑜 ∙ 𝐹 2 𝜕 𝑝𝑣𝑢 ≈ 𝐹 4 𝑇𝐹𝑆𝑇 = 𝐽 𝑇𝐹𝑆𝑇 𝐺 4 4 𝐽 0 𝐹 0 𝐹 0 • The presence of plasmonic hot spots can enhance Raman signal up to 14 orders of magnitude. • Given the strong dependance, almost all the signal comes from the narrow hot spot regions. To predict the experimental Enhancement Factor, an average over the whole surface is • needed. If the analytes are coupled to group that bound preferentially to metals (Sulphur groups), • the average can neglect the (low enhancement) glass surface.
NANOTRIANGLES – HOT SPOTS Max(E)/E 0 1.0 Ave(E)/E 0 4 )/E 4 Ave(E 0 0.8 Normalized Values 0.6 0.4 0.2 0.0 400 600 800 1000 1200 Freespace incident Wavelenght (nm) 1E7 SERS Enhancement Factor 𝐹𝐺 = 1.5 ∙ 10 5 100000 1000 10 400 500 600 700 800 900 1000 1100 1200 Freespace incident Wavelenght (nm)
NANOPRISMS - SENSITIVITY • Localized Plasmons => strong Near Field, decay ≈ 10 − 20 𝑜𝑛 Sensitivity close to the surface => also a monolayer shifts the resonance • • Simulation of the resonance for different thicknesses and geometries of functionalization Functionalization Functionalization Prism Prism T=5, Absorption 2 ) t=0nm T=10, Absorption -14 m t=1nm T=15, Absorption 1.0 4 t=2nm Extinction Cross Section (10 T=20, Absorption Absorption (Arb. Units) t=5nm T=30, Absorption 0.8 t=10nm 3 0.6 2 0.4 1 0.2 0 0.0 600 650 700 750 800 850 600 700 800 900 1000 1100 Wavelenght (nm) Freespace incident Wavelenght (nm) 𝜇 𝑆 = 𝑏 − 𝑐 ∙ 𝑑 𝑈 𝜖𝜇 𝜖𝜇 Optimal Coupling to Plasmons => 𝜖𝑈 = 3.0 𝜖𝑈 = 5.8 𝜖𝜇 = −𝑐 ∙ ln (𝑑) 𝜖𝑈 𝑈=0 𝑈=0 𝑈=0
NANOTRIANGLES - SUMMARY • The effects of the geometric properties of the nanostructure have been investigated. The resonance can be taylored by tuning 4 parameters: • Parameters Lattice Constant Side Lenght Height Snip/Shape Controlled by NS diameter Deposition setup Annealing Effect of Increase Redshift Redshift Blueshift Blueshift Less curvature and interaction Reason Larger Structures AR closer to 1 Optimization tips for the two applications: • Application SERS Refractive Index SENSING Lattice Constant Dependent on Raman Spectrum of Analytes Dependent on the desired position of resonance, once fixed other parameters Side Lenght Longest possible => better t-t coupling Height Low structures => higher avg field High structures => better coupling to func. Snip/Shape No snip, sharp edges => Hot Spots Moderate snip => more homogeneity Tips Collimation => longer sides, sharper Higher Aspect Ratio, Annilation to lower edges. Evaporation => homogeneity defects and strongest t-t coupling (no planets)
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