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

PLASMONICS

• 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

LIGHT-MATTER INTERACTION BIOSENSORS

Biosensors

Labeled

SERS

Hot Spots

Fluorescence

Light Management

Specific Absorption

Light Management

...

Label-free

Refractive index

Localized Plasmons

Surface Plasmos

Surface modification

...

Nanotriangles Nanoholes

PLASMONICS – SURFACE AND LOCALIZED

PLASMONS

Plasmons

Surface Plasmons Localized Plasmons

• Propagating modes

• Coupling by prism or grating

• Poor Field Confinement

• Low absorption

• Extremely sensitive to

surrounding index changes

• Confined modes

• Direct coupling

• Strong Field Confinement

• High absorption

• Extremely sensitive to

(geometrically) small changes

in surround

SURFACE PLASMON POLARITONS

• Wave equation (Helmhotz’s Equation) at metal-dielectric interface:

𝜕2𝑬(𝑧)

𝜕𝑧2 + 𝑘02𝜀 − 𝑘𝑥

2 𝑬 = 𝟎

• Putting the Boundary Conditions, the dispersion relation results:

𝑘𝑥 = 𝑘0

𝜀1𝜀2

𝜀1 + 𝜀2

• Direct coupling forbidden!!

• Coupling methods:

LOCALIZED PLASMONS –

QUASI-STATIC APPROXIMATION

• Laplace Equation with Boundary Conditions:

𝛻𝟐𝝓 = 𝟎

• Solutions given by a Spherical Harmonics Expansion:

𝜙 𝑟, 𝜃 = 𝐴𝑙𝑟𝑙 + 𝐵𝑙𝑟

− 𝑙+1 𝑃𝑙 cos 𝜃

∞

𝑙=0

• Resulting Polarizability:

𝛼 = 4𝜋𝑅3𝜀 − 𝜀𝑚

𝜀 + 2𝜀𝑚

• Resulting Electric Field in Quasi-Static (dipolar) Approximation:

𝑬 = 𝑬𝟎 +1

4𝜋𝜀0𝜀𝑚𝑘2 𝒏 × 𝒑 × 𝒏

𝒆𝒊𝒌𝒓

𝒓+ 3𝒏 𝒏 ∙ 𝒑 − 𝒑

1

𝑟3 −𝑖𝑘

𝑟2 𝒆𝒊𝒌𝒓

𝜀 = −2𝜀𝑚 Resonance!!

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:

500 600 700 800 900 10000

2

4

6

8

10

12

14

Extin

ctio

n E

ffic

ien

cy

Wavelength (nm)

46 48 50 52 54 56 58 60 620.0

0.1

0.2

0.3

0.4

0.5

0.6

Re

fle

cta

nce

Angle (°)

HOW TO EXPLOIT PLASMONICS FOR

BIOSENSORS?

Sensors

SERS

Hot Spots

Transmission/

Extinction

Resonances

Peaks

Reflection

Propagating Modes

Nanohole Arrays Nano Triangles

NANOSTRUCTURES SELF-ASSEMBLY - NSL 2a.

1. SiO2 NS (30-1000 nm)

Nano Triangles

NANOSTRUCTURES SELF-ASSEMBLY - NSL 2a.

1. SiO2 NS (30-1000 nm)

2b.

Nano Triangles

Nano Hole Arrays

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:

𝛻2𝐸 + 𝑘2𝐸 = 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

400 500 600 700 800 900 1000 1100 12000

2

4

6

8

10

12

14

16E

xtin

ction

Eff

icie

ncy

Wavelenght (nm)

Base

L=200 nm

L=300 nm

h=30 nm

h=45 nmLarger => Redshift

Aspect Ratio Closer

to 1 => Blueshift

Base Prism: L=100 nm; h=10 nm; snip=10 nm, silver

400 500 600 700 800 9000

2

4

6

8

10

12

14

Single

d=120nm c-c

d=240nm c-c

d=360nm c-c

exagon l=120nm

snip=20nm

snip=30nm

Ext

inct

ion

Effi

cien

cy

Wavelength (nm)

NANOPRISMS – GEOMETRY AND INTERACTION

Interactions Snipping

Base Prism: L=100 nm; h=10 nm; snip=10 nm, silver

NANOPRISMS – SURROUNDING

MEDIA/INTERFACE

600 800 1000 12000

1

2

3

4

5

6

7

8E

xtin

ction

Cro

ss S

ection

(10

-14 m

2)

Wavelength (nm)

Air

Silica Glass

Quartz Glass

Silica Glass Interface

n=1.45

n=2.05

NANOTRIANGLES – HOT SPOTS

• Surface Enhanced Raman Spettroscopy (SERS) efficiency is strogly dependent on local

field:

𝐹𝑆𝐸𝑅𝑆 =𝐼𝑆𝐸𝑅𝑆

𝐼0=

𝐸2 𝜔𝑖𝑛 ∙ 𝐸2 𝜔𝑜𝑢𝑡

𝐸04 ≈

𝐸4

𝐸04

• 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

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

Norm

aliz

ed V

alu

es

Freespace incident Wavelenght (nm)

Max(E)/E0

Ave(E)/E0

Ave(E4)/E

4

0

400 500 600 700 800 900 1000 1100 1200

10

1000

100000

1E7

SE

RS

Enhan

cem

ent

Facto

r

Freespace incident Wavelenght (nm)

𝐸𝐹 = 1.5 ∙ 105

• Localized Plasmons => strong Near Field, decay ≈ 10 − 20 𝑛𝑚

• Sensitivity close to the surface => also a monolayer shifts the resonance

• Simulation of the resonance for different thicknesses and geometries of functionalization

Functionalization

NANOPRISMS - SENSITIVITY

600 650 700 750 800 8500

1

2

3

4

Extinctio

n C

ross S

ectio

n (

10

-14 m

2)

Freespace incident Wavelenght (nm)

t=0nm

t=1nm

t=2nm

t=5nm

t=10nm

600 700 800 900 1000 11000.0

0.2

0.4

0.6

0.8

1.0

Absorp

tion (

Arb

. U

nits)

Wavelenght (nm)

T=5, Absorption

T=10, Absorption

T=15, Absorption

T=20, Absorption

T=30, Absorption

Prism

Functionalization

Prism

𝜕𝜆

𝜕𝑇 𝑇=0

= 3.0 𝜕𝜆

𝜕𝑇 𝑇=0

= 5.8 Optimal Coupling to Plasmons => 𝜆𝑅 = 𝑎 − 𝑏 ∙ 𝑐𝑇 𝜕𝜆

𝜕𝑇 𝑇=0

= −𝑏 ∙ ln (𝑐)

NANOTRIANGLES - SUMMARY

• The effects of the geometric properties of the nanostructure have been investigated.

• The resonance can be taylored by tuning 4 parameters:

• Optimization tips for the two applications:

Parameters Lattice Constant Side Lenght Height Snip/Shape

Controlled by NS diameter Deposition setup Annealing

Effect of Increase Redshift Redshift Blueshift Blueshift

Reason Larger Structures AR closer to 1 Less curvature and interaction

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

edges. Evaporation => homogeneity

(no planets)

Higher Aspect Ratio, Annilation to lower

defects and strongest t-t coupling

NANOHOLE ARRAYS • Nanohole Array Parameters:

• Substrate: silica glass (n=1.445)

Computed quantities:

• Transmittance (normal, various

functionalizations)

• Reflectivity (wavelenght and angle

sweeps)

• Local fields

Parameters: Value (nm):

Lattice Constant (𝒂𝟎) (HCP) 535

Holes Radius (𝑅) 205

Film height (ℎ) 46

87

𝑎0

ℎ 2𝑅

Substrate

Functionalization Film

Holes

NANOHOLE ARRAY

REFLECTANCE – SENSITIVITY MAPS

Angle (deg)

Wav

elen

ght (

nm)

Ref

lect

ance

ℎ = 46 𝑛𝑚

𝜆 = 1140

𝜃

NANOHOLE ARRAY

REFLECTANCE – SENSITIVITY MAPS

Angle (deg)

Wav

elen

ght (

nm)

Ref

lect

ance

ℎ = 87 𝑛𝑚

𝜆 = 1120

𝜃

NANOHOLE ARRAY - SENSITIVITY MAPS

Reflectance – h = 46 nm

Surface: Nanohole Array on Silica (As)

Point Plot: Nanohole with Functionalization

REFLECTANCE - ℎ = 46 𝑛𝑚

45 50 55 60 65 700.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

Re

flecta

nce

Angle (deg)

T=0 nm

T=3 nm

T=4 nm

T=5 nm

T=7 nm

T=10 nm

T=15 nm

Functional-

ization T (nm)

Dip Angle

(deg)

0 57.0

3 55.7

4 55.0

5 54.3

7 53.6

10 52.3

15 50.4

𝑆𝑙𝑜𝑝𝑒 = −0.44 𝑑𝑒𝑔/𝑛𝑚

REFLECTANCE - ℎ = 87 𝑛𝑚

Functional-

ization T (nm)

Dip Angle

(deg)

0 59.1

3 58.2

4 58.0

5 57.7

7 57.0

15 54.9

45 50 55 60 65 700.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Re

flecta

nce

Angle (deg)

T=0 nm

T=3 nm

T=4 nm

T=5 nm

T=7 nm

T=15 nm

𝑆𝑙𝑜𝑝𝑒 = −0.29 𝑑𝑒𝑔/𝑛𝑚

Experimental h = 100nm: 𝑆𝑙𝑜𝑝𝑒 = −0.18 𝑑𝑒𝑔/𝑛𝑚

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

Rhole

=200nmRhole

=150nm

Wavelenght

Transmittance

Reflectance

Absorbance

Rhole

=100nm

Nanohole Array. Lattice constant= 535 nm

Wavelenght Wavelenght

Wavelenght Wavelenght Wavelenght

Wavelenght Wavelenght Wavelenght

NANOHOLE ARRAY -

FAR FIELD AND LOCAL FIELD MAPS

Holes

T=

50

nm

T=

100

nm

T=

150

nm

650 700 750 800 850 900 950 1000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Tra

nsm

itta

nce

Wavelenght (nm)

T=0 nm

T=3 nm

T=4 nm

T=5 nm

T=7 nm

T=10 nm

T=15 nm

NANOHOLE ARRAY

TRANSMITTANCE – ℎ = 46 𝑛𝑚

640 645 650 655 6600.40

0.42

0.44

0.46

0.48

0.50

Tra

nsm

itta

nce

Wavelenght (nm)

665 670 675 680 6850.00

0.01

0.02

0.03

0.04

0.05

Tra

nsm

itta

nce

Wavelenght (nm)

860 880 900 920

0.43

0.44

0.45

0.46

0.47

Tra

nsm

itta

nce

Wavelenght (nm)

640 645 650 655 6600.40

0.42

0.44

0.46

0.48

0.50

Tra

nsm

itta

nce

Wavelenght (nm)

665 670 675 680 6850.00

0.01

0.02

0.03

0.04

0.05

Tra

nsm

itta

nce

Wavelenght (nm)

860 880 900 920

0.43

0.44

0.45

0.46

0.47

Tra

nsm

itta

nce

Wavelenght (nm)

NANOHOLE ARRAY

TRANSMITTANCE – ℎ = 46 𝑛𝑚

Thickness

(nm)

Peak 1 (nm) Dip (nm) Peak 2 (nm)

0 647.1 670.1 872.5

3 648.0 672.3 877.0

4 648.2 672.9 879.0

5 648.4 673.5 880.5

7 648.8 674.5 884.0

10 649.5 675.8 888.0

15 650.5 677.7 897.0

Feature 𝝀𝟎 (nm) Sensitivity Feature width (nm)

648 0.21 ~40

672 0.44 ~30

877 1.63 ~100

NANOHOLE ARRAY – WHICH IS THE BEST?

0 5 10 15

50

52

54

56

58

60

Slope = -0.44 deg/nm

Dip

Angle

(de

g)

Functionalization Thickness (nm)

h=46 nm

h=87 nm

Slope = -0.29 deg/nm

Transmittance, ℎ = 46 𝑛𝑚 Transmittance, ℎ = 87 𝑛𝑚

Reflectance Method Peak/dip Sensitivity (/nm)

T, ℎ = 46 𝑛𝑚 P1, 𝝀𝟎 = 647 nm 0.21 nm

T, ℎ = 46 𝑛𝑚 D1, 𝝀𝟎 = 670 nm 0.19 nm

T, ℎ = 46 𝑛𝑚 P2, 𝝀𝟎 = 872 nm 2.09 nm

T, ℎ = 87 𝑛𝑚 P1, 𝝀𝟎 = 646 nm 0.21 nm

T, ℎ = 87 𝑛𝑚 D1, 𝝀𝟎 = 666 nm 0.44 nm

T, ℎ = 87 𝑛𝑚 P2, 𝝀𝟎 = 823 nm 1.63 nm

R, ℎ = 46 𝑛𝑚 𝝀𝟎 = 1140 nm -0.44 deg

R, ℎ = 87 𝑛𝑚 𝝀𝟎 = 1120 nm -0.29 deg

0 2 4 6 8 10 12 14 16

650

660

670

870

880

890

900 Peak 1

Dip

Peak 2

Wavele

nght (n

m)

Thickness (nm)

0 2 4 6 8 10 12 14 16640

650

660

670

820

830

840

850

860

Peak 1

Dip

Peak 2

Wavele

nght (n

m)

Thickness (nm)

NANOHOLE ARRAYS - SUMMARY

• NHA can be used both in transmittance and in reflectance to build biosensors.

• SERS cannot use NHA due to low enhancement factor (~ 4-5)

• The response can be taylored by tuning 3 parameters:

Parameters Lattice Constant Hole Radius Height

Controlled by NS diameter RIE Deposition setup

Mode Transmittance Reflectance

Hole Radius Larger holes => lower absorption Smaller holes => lower dissipation

Height Thin film => higher transmittance and

better sensitivity

Thick film => lower dissipation

Thin film => better sensitivity

Tips Thin film and large holes Small holes => better conduction =>

narrower dip

Thin film => better sensitivity

• Optimization tips for the two operative modes:

CONCLUSIONS • Plasmonic Bio-Sensors design can take great advantage from nanostructures obtained by

NSL

• The design of the nanostructures can be controlled by tuning a small set of parameters, and

simulations give hints on how to obtain best performaces.

• Nanotriangles can be used both for Refractive Index-sensitive and SERS substrates. In

both cases, the agreement with experiment is pretty good, keeping in count the defects on

the real samples

• Nanohole arrays can be used both in transmittance and in reflectance

• Reflectance give the best (and most stable against defects) results

• Transmittance is far easier to measure, and it’s usable too

• Simulations show how to tune parameters to get the desired result

Thank You for your Attention!

Thanks to people, present and past members, of my group:

Boris, Carlo, Giovanni P, Giovanni P, Valentina B, Valentina M, Valentina R,

Marco, Martina

And

Tiziana and Giovanni M