1Romme 2012

Plasmonics

fundamentals and applications in biosensing

Andreas B. DahlinDiv. of BionanophotonicsDept. of Applied Physics

Chalmers University of Technology

March 26, 2012

March 26, 2012 Romme 2012 2

Outline

++++

----

----

++++

metal

e- cloud

dielectric

metal

dielectric

e- cloud

• Some basic optics.

• Nanoparticle plasmons.

• Surface plasmons.

• Applications.

+ - + - + - + -- + - + - + - +

• No math derivations.

• Basic physical assumptions.

• Limits of analytical theory.

http://en.wikipedia.org/wiki/File:James_Clerk_Maxwell.png

March 26, 2012 Romme 2012 3

Light as a Plane Wave

( ) ( ) ( )xk-xktEtxE ImiReiω0 eee, −=

• Amplitude (E0)

• Frequency (ω)

• Propagation (real part of wacevector k)

• Decay (imaginary part of the wavevector k)

We assume linear polarization.

The intensity (power, energy flux) is:

n refractive index of the mediumε0 permittivity of free spacec speed of light in vacuum

20

0

2EcnI ε=

March 26, 2012 Romme 2012 4

Optical Properties of Materials

Materials are described by their dielectric function ε (the relative permittivity), withfrequency dependence and energy absorbtion: ε(ω) = Re(ε(ω)) + iIm(ε(ω))

Simplifications:

• Permeability: We assume no magnetic activity (μ = 1).

• Homogeneity: We assume homogenous materials (no spatial dependence of ε).

• Anisotropy: We assume no directional dependence for ε (birefrigrence).

• Linearity: ε does not vary with incident field strength.

These assumptions are not always reasonable, but work well for the materials and frequency regions one usually deals with in plasmonics.

Dielectrics can often be well described as non-absorbing(Im(ε) = 0) and non-dispersive (no dependence on ω).

This gives a single refractive index value n = ε1/2, e.g. water normally has n = 1.333 and ε = 1.777.

http://upload.wikimedia.org/wikipedia/commons/0/0b/Dispersive_Prism_Illustration_by_Spigget.jpg

silver gold copper

March 26, 2012 Romme 2012 5

Metals

Metals have ε with strong ω dependence. The free electrons can be described by a classical Drude model:

• ε∞ is the contribution from other factors than free electrons.

• τ is the lifetime of the electron movement.

• ωp is the plasma frequency.

• Ne is the free electron density.

• meff is the effective electron mass.

( )⎥⎦⎤

⎢⎣⎡ +

−= ∞

τωω

ωωε

i

2pε

0eff

2e

p εmeN

ω =

Interband transitions can also be included. They give colors to some metals!

Wikipedia: Drude model

http://upload.wikimedia.org/wikipedia/commons/1/16/Electrona_in_crystallo_fluentia.png

March 26, 2012 Romme 2012 6

Experimental Data

Metals normally reflect light, but all metals become dielectrics when ω > ωp, which gives Re(ε) > 0 in Drude model.

Example: Au has one conduction e- per atom, density 19.3g/cm3 and atomic weight 197u, so Ne = 5.9×1028m-3 and for meff = m0 we get ωp = 1.38×1016Hzcorresponding to λ0 =137nm, i.e. UV light.

There are transparent conductors such as indium tin oxide, which has a low charge carrier density so that ω > ωp for visible light (but not IR).

ITO glass

March 26, 2012 Romme 2012 7

Nanoparticle Plasmons

++++

----

----

++++

metal

e- cloud

dielectric

March 26, 2012 Romme 2012 8

Far Field Properties

collimated light

goblin

The extinction cross section σ is the effective area of the shadow of the particle.

For common objects σ is equal to the geometrical cross section area under collimated illumination.

• Normal Goblin (>>λ) – shadow has same size.

• Nanogoblin (

March 26, 2012 Romme 2012 9

Scattering and Absorption

The extinction cross section is acquired from the imaginary part of the polarizability (according to the optical theorem):

The extinction cross section has two contributions, scattering and absorption. The Rayleigh (elastic) scattering cross section of an oscillating dipole is:

The absorption (loss of light) is by energy conservation:

We need expressions for α to model the far field optical properties!

24

sca π6αkσ =

scaextabs σσσ −=

( )αkσ Imext =

Ancient Roman nanotechnology!Blue light is absorbed.Green light is scattered.Red light is transmitted.

Ag, Au, Cu (60%, 30%, 10%) in glass.

I. Freestone et al. Gold Bulletin 2007.

March 26, 2012 Romme 2012 10

The Electrostatic Approximation

A plane wave encounters a nanoparticle.

We simplify the incident field:

External field is static and infinite – use electrostatic theory!

Even if the field oscillates, the theory holds as long as there are no dynamics in the polarization process, i.e. if the free electrons react without delay.

This is actually reasonable if the particle is small (tens of nm) compared to the wavelength of light!

( ) ( ) ( ) 0ImiReiω0 eee, EEtxE xk-xkt == −

++++

----

++

--

E0induced dipole

× × ×R

March 26, 2012 Romme 2012 11

Electrostatic Sphere Theory

For electrostatic case, Maxwell’s equations are reduced to LaPlace’s equation and the field inside the sphere is:

The polarization density (P, generally a vector field) is acquired from the applied field and the depolarization field:

The induced dipole moment (p), which defines the polarizability, is the volume integral of P inside the sphere:

We can then get α (unit is volume):

0m

in 23 Eεε

E+

=

[ ] inm EεεP −=

[ ] [ ] 0m

m

3

inm

33

0 23

3π4

3π4

3π4 E

εεεεREεεRPRαEp

+−=−===

m

m0 2

3εεεεVα

+−

=

ε(ω)

εm(ω)

J.D. Jackson, Classical Electrodynamics (Wiley 1999)

March 26, 2012 Romme 2012 12

First Spectral Model

Gold nanospheres in water exhibit a plasmon resonance around λ = 520nm.

Resonance approximately when Re(ε(ω)) = -2εm if Im(ε(ω))

March 26, 2012 Romme 2012 13

Other Shapes

We can also model ellipsoids using electrostatic theory.

We introduce shape factors L1, L2 and L3:

L1+L2+L3 = 1 and the polarizability (for each L) is now:

12

3

2

2

2

1

=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡Rz

Ry

Rx

[ ] [ ][ ][ ]∫∞

++++=

023

22

21

2

321 d12

sRsRsRsRs

RRRLj

j

[ ]mmm

0 εεLεεεVα−+

−=

A. Dmitriev et al. Nano Letters 2008.M.A. El-Sayed Accounts of

Chemical Research 2001.

March 26, 2012 Romme 2012 14

Ellipsoid Resonances

We now get one resonance for each shape factor (ellipsoid axis)!

Resonance shifts to longer wavelengths the more the shape deviates from a sphere.

One resonance will be twofold degenerate for spheroids.

Excitation requires a polarization component along the resonance axis.

one resonance

two resonances

Drastic color changes, shape matters more than size!

March 26, 2012 Romme 2012 15

Shell Coatings

We can also introduce a shell of a third material on the spheres:

Also possible for ellipsoids:

[ ][ ] [ ][ ][ ][ ] [ ][ ]mssmss

mssmssout0 222

223εεεεεεεεεεεεεεεεVα−−−+++−−−+

=ζ

ζ

[ ] [ ][ ][ ] [ ][ ][ ][ ] [ ][ ] [ ]ssoutmsoutmoutinss

ssoutinssmsout0 εεεLεεLεLLεεε

εεεLLεεεεεVα−+−+−−+

−+−−+−=

ζζζζ

We now need both inner and outer radius which gives the filling fraction: ζ = Rin/Rout

This is very useful for modelling how the spectrum changes when molecules bind to the nanoparticle!

March 26, 2012 Romme 2012 16

Metallic Shells

The same equations also hold for a metallic shell on a dielectric core.

Resonance shifts to longer wavelength for higher filling fractions (less metal).

Mode splitting into two completely new resonances with different charge distributions!

H. Wang et al. Accounts of Chemical Research 2007.

H. Wang et al. Nano Letters 2006.

March 26, 2012 Romme 2012 17

Smaller Particles

When particles are smaller than ~10 nm, surface scattering of electrons comes into play.

Resonance damping by modification of Drude model with Fermi velocity vF:

( )⎥⎦⎤

⎢⎣⎡ ++

−= ∞

Rv

τωω

ωωε

F

2p

iiε

The electrons encounter the surface before the field alternates (within 1 fs). The electron mean free path (Drude) is not causing this effect (it is accounted for in τ).

Even smaller particles around ~1 nm become quantum dots instead.

March 26, 2012 Romme 2012 18

Larger Particles

For particles larger than ~50nm, the quasistatic approximation does not work andcharge dynamics come into play.

Exact solutions (Mie and Gans theories) possible for spheroids, although complicated.

The physical effects that need to be accounted for are:

• higher order modes

• dynamic depolarization

• radiative damping

Example of analytic approximation:

0

2

0

30

4π6πi1 α

Rkαk

αα

j

−−=

dynamicsradiation

March 26, 2012 Romme 2012 19

The Near Field

E0

(y, n

m)

z = 0λ0 = 526 nm

(x, nm)

field enhancement (|E|/E0)

Au

H2O

( ) ( ) [ ] ⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ ++−−= zyx3xRexy,, 5300

rrrr

rr zyxrx

rEzxE α

Important in many applications!

If we know the extension of the field we can estimate when coupling effects come into play.

Usually requires numerical methods, but there are some analytic models, e.g. for a static dipole.

March 26, 2012 Romme 2012 20

Interaction Effects

So far we have looked at single particle properties. If particles get close to each other (near field overlap), we have short-range coupling effects.

We can model the orientation averaged sphere pair polarizability:

For separation distance dsep → ∞ we get αpair as the sum of the individual polarizabilities.

If many particles are ordered in arrays, we have long range diffractive coupling effects in scattering.

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

+⎥⎦

⎤⎢⎣

⎡+

+−

+⎥⎦

⎤⎢⎣

⎡+

=

6sep

22part1part

3sep

2part3

1

2

6sep

22part1part

3sep

2part3

1

2

1partpair

16π1

π22

4π1

π1

3d

dRR

d

dRR

ααα

α

αα

α

α

March 26, 2012 Romme 2012 21

Surface Plasmons

metal

dielectric

e- cloud+ - + - + - + -- + - + - + - +

March 26, 2012 Romme 2012 22

The Single Interface Surface Plasmon

Can a wave at the interface of two materials satisfy Maxwell’s equations?

Continuity of fields gives boundary condition:

The wavevectors are related through:

One can derive the dispersion relation between frequency and wavevector:

( ) ( )( ) 0

00 zm

z =<

+>

ωεεzkzk

( )02z2x20m >+= zkkkε( ) ( )02z2x20

March 26, 2012 Romme 2012 23

Equations for the Wavevectors

The wave must be transverse magnetic to satisfy Maxwell’s equations:

What are the wavevectors? If we assume nice material properties:

• Metal is ”quite metallic”: -Re(ε(ω)) > εm

• Metal is ”not very lossy”: -Re(ε(ω)) >> Im(ε(ω))

• Loss in dielectric is ”insignificant”: Im(εm) ≈ 0

We get simplified expressions for:

• Propagation: Re(kx)

• Dissipation of the surface plasmon: Im(kx)

• Decay of the evanescent field: Im(kz)

( ) ( )( ) mm

0x ReReRe

εεεε+

= kk

( ) ( )( )( )( )[ ]2

3

m

m0x Re2

ImReReIm

εε

εεεε

⎥⎦

⎤⎢⎣

⎡

+= kk

( ) m

2m

0z Re εεε+

= kk

( ) [ ]ye,, ωi0 zxrr tzkxkHtzxH −±=

March 26, 2012 Romme 2012 24

Surface Plasmon Near Field

The wavevectors give us the near field distribution of the electric field vector components Ex(x,z) and Ez(x,z):

Time dependence omitted and a ”snapshot” of the wave is modelled for λ0 = 633nm.

( ) [ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛∝> + zkxk

kzxE zxim

zx eRe0, ωε

( ) [ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛∝> + zkxk

kzxE zxim

xz eRe0, ωε

March 26, 2012 Romme 2012 25

Excitation of Surface Plasmons

We now have a continuum of frequencies! Not like nanoparticles which have one or a few resonance frequencies.

Photon dispersion: k = εm1/2ω/c

The dispersion relation shows that light cannot couple so easily to surface plasmons because frequency and wavevector cannot both be matched.

We need to get some additional photon momentum: Δkx

March 26, 2012 Romme 2012 26

Grating Excitation

Momentum added by diffraction from a grating.

Excitation is possible by simply illuminating the metal surface if it has a periodic structure.

But the grating itself influences the dispersion relation!

( )( ) ( ) Λ

jn 0mm0

m0 sinRe λθελεελε

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

+

approximation

resonance condition

( ) ( )Λ

jkk π2sinRe x += θ

March 26, 2012 Romme 2012 27

Total Internal Reflection Excitation

Momentum added by illuminating through a high refractive index material prism.

Excitation possible in reflection configuration.

But the fact that the film is so thin influences the dispersion relation!

approximation

( )( ) ( )θεωε

εωε sinRe pm

m n=⎟⎟⎠

⎞⎜⎜⎝

⎛

+

( ) ( )θsinRe p0x nkk =

resonance condition

March 26, 2012 Romme 2012 28

Mode Splitting in Thin Films

The single interface surface plasmon splits into a bondingand an antibonding mode with different field distributions.

The dispersion relations should now satisfy:

For the bonding mode as d becomes smaller:

• Higher Im(kz) (quick decay).

• Higher Im(kx) (high dissipation).

• Higher Re(kx) (shorter wavelength).

( ) ( )

( ) ( ) dkzzzz

zzzz

z

kε

kε

kε

kε

kε

kε

kε

kε

2,i2

3,

3

2,1,

1

2,

3,

3

2,1,

1

2,

eiiii

iiii

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

=⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

ωω

ωω ( )[ ]2x2

1z,1 ωωε kc

k −⎥⎦⎤

⎢⎣⎡−=

( ) ( )[ ]2x2

z,2 ωωωε kc

k −⎥⎦⎤

⎢⎣⎡±=

( )[ ]2x2

3z,3 ωωε kc

k −⎥⎦⎤

⎢⎣⎡−=

March 26, 2012 Romme 2012 29

Long Range Surface Plasmons

The antobonding mode is exactly the opposite to the bonding mode in all respects. It gives long rangesurface plasmons.

The antibonding mode is usually leaky into the higher index material, but becomes bound under certain conditions.

Excitation always possible when dielectrics are index matched, e.g. water together with magnesium fluoride or teflon (n ≈ 1.3).

March 26, 2012 Romme 2012 30

Fresnel Calculations

Problem: Even if we know the wavevectors and the near field of the surface plasmons, we cannot model the full spectrum, e.g. from spectroscopy in reflection mode.

The full spectrum of an arbitrary multilayer system can be calculated by the transfer matrix method.

Inputs:

• Angle of incidence.

• Wavelength of incident light.

• Thicknesses of the layers.

• Refractive indices of the layers.

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Applications

March 26, 2012 Romme 2012 32

Overview of Applications

• Optical circuits.

• Enhanced solar cells.

• Superlenses.

• Nanoscale lasers.

• Catalysis.

• Bioapplications: • Biology tools: • Sensors for investigating biomolecule function.

• Biocompatible labels.

• Medical: • Disease diagnostics sensors.

• Photothermal therapy.

• Sensors for drug discovery.

gold and silver colloidsWikipedia: Argyria

March 26, 2012 Romme 2012 33

Refractometric Detection

When the refractive index of the surrounding environment changes, the resonance condition for plasmon excitation changes too.

Any molecule that binds to the surface can be detected, but there are extreme demands on the chemical functionalization of the surface.

Surface Plasmon Reonance (SPR) for biomolecular interaction analysis is now an established technology.

March 26, 2012 Romme 2012 34

Nanoparticle Coupling Assays

Based on plasmonic coupling, different from refractometric detection.

Large spectral changes gives high sensitivity!

”Cleavage assays” limited to special analytes such as restriction enzymes or proteases.

”Aggregation assays” require mobile particles and the possibility to form sandwich complexes with two different binding events.

March 26, 2012 Romme 2012 35

Other Spectroscopy Techniques

Spectroscopy techniques for biological applications:

• Plasmonic enhancement of fluorescence.

• Plasmon resonance energy transfer.

• Surface enhanced Raman scattering.

Often inelastic: The emitted light does not have the same wavelength as the incident light!

Spectroscopic fingerprints for identifying molecules.

A. Kinkhabwala et al. Nature Photonics 2009.

G.L. Liu et al. Nature Methods 2007.

G.R. Alvarez-Puebla et al. Small 2010.

March 26, 2012 Romme 2012 36

Nanoparticles as Labels

Nanoparticles can be used as labels for enhancement of signal in biosensing applications or just for visualization.

Only succesful commercial nanoplasmonic biosensor (not SPR) so far is the lateral flow device for detection of babies.

March 26, 2012 Romme 2012 37

Summary

• Optical properties of materials. • Typical assumptions in plasmonics.• Drude free electron model for metals.

• Difference between far field and near field optical properties.

• Nanoparticle plasmons. • Electrostatic approximation.• Ellipsoidal particles.• Shell particles.• Very small or slightly larger particles.• Basics of interaction effects.

• Surface plasmons. • Field distribution for single interface.• The dispersion relation.• Excitation with gratings or total internal reflection.• Mode splitting in thin films.• Spectral modelling by Fresnel equations.

• Principles of plasmonic biosensors, especially those operating by refractometric detection.

March 26, 2012 Romme 2012 38

The End

My research webpage: http://www.adahlin.com/research/

Questions are welcome now or later: adahlin@chalmers.se

Anyone interested in a Masters project or Ph.D. Position???

A.B. Dahlin, Plasmonic Biosensors(IOS Press 2012)

Several figures reproduced with permission from my book.

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