Theory and Application of Nanomaterials

Outline Optics with Metals Surface Plasmons Particle Plasmons

Theory and Application of NanomaterialsLecture 11: Plasmonics

S. Smith

SDSMT, Nano SE

FA17: 8/25-12/8/17

S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 1 / 24


Introduction

An excerpt from a recent paper in the journal Science shows a nano-metric ruler based on theextreme sensitivity of plasmon resonances to changes in the relative position of severalnano-sized noble metal “antennas”, which could be utilized in a nano-metric sensor (imagesfrom the journal Science1).

1Na Liu, et. al. Science 332 , 1407 (2011).S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 2 / 24


Outline

1 Optics with MetalsDielectric FunctionOptical Properties

2 Surface PlasmonsVolume PlasmonsSurface Plasmon Polariton

3 Particle PlasmonsMie Theory



Oscillator Model of Polarizable Media

We may assume the polarizability of materials originates from induced dipoles formed by theCoulomb force of an applied electric field on the bound charges resident in polarizable media.An imposed electric field distorts these charges as shown in figure 2:

Figure: (left): charge distortion δr in ponderable media upon application of an external field E.(center): Force vs radial distance of Coulomb force on charges in ponderable media. (right):Damped harmonic oscillator model of the electron motion in ponderable media, γ is defined asthe damping divided by m, and ω2

o = k/m.



Dielectric Function of Metals

For a small range of δr , the Coulomb force can be assumed to be linear and followHooke’s law: Fc = −keff x , thus we can write an equation of motion for the boundelectron:

mx + mγx + mω2ox = −eE

From which, if we assume time-harmonic dependence of the electric field, the resultingdisplacement δr ≡ x will also be time-harmonic:

x = − e

m

E

ω2o − ω2 − iγω

As the dipole moment is p = −ex, we need simply to include all N dipoles per unitvolume to get the polarization field:

D = εE = E + 4πP = E + 4πNe2

m

E


from which we can write:

ε(ω) = 1 +4πNe2

m

1


= 1 +ω2

p


−→ ω2p =

4πNe2

m

Where we have defined the plasma frequency, ωp, which is essentially the frequencyafter which the electrons can no longer completely screen the externally applied field.



Drude Form

In metals, the restoring force is extremely weak, and the term ω2o can be dropped at

frequencies of interest, equivalent to “cutting the spring” in the dampled oscillatormodel, giving:

ε(ω) = 1−ω2

p

ω2 + iγω= 1−

ω2p

ω(ω + iγ)≈ 1−

ω2p

ω2(small damping & high frequencies)

The above expressions do not take into account the possibility that the polarization maybe greater than unity at infinite frequency, which is usually accounted for by insertingε∞ into the above equations, and defining a new plasma frequency ωp = ωp/

√ε∞:

ε(ω) = ε∞ −ω2

p

ω2 + iγω−→ ε(ω)

ε∞= 1−

ω2p

ω(ω + iγ)≈ 1−

ω2p

ω2(small damping )



Optical Properties

The above expressions tell us that the dielectric constant becomes negative forfrequencies below ωp, which means the refractive index becomes imaginary:

n =√ε −→ n is a complex number for -ve ε(ω)

thus for plane waves of the form E(x , t) ∝ e i(kx−ωt), where k = n2π/λ, this will result inexponential decay:

E(x , t) ∝ e−|√ε(ω)| 2π

λxe−iωt

As conservation of energy requires that T + R + A = 1, where T is the transmission, Rthe reflection and A the absorption, and for weak damping A can be neglected, if thetransmission T is small for frequencies below ωp, the reflectivity must be high.



Reflectivity

Thus, as illustrated below, the metals described by the above analysis are highlyreflective for electromagnetic radiation with frequencies below the plasma frequency:

Figure: Reflectivity for a metal described by plasma frequency ωp, showing thatwaves below the plasma frequency cannot propagate in the metal and arereflected with near 100% efficiency.



Plasmons

A collective oscillation of electrons may be excited in a metal, which may be assumed to be aplasma, i.e. a “gas” of positive and negative charges, which is overall nuetral but the consituentcharges can still be influenced by applied fields. If an electric field is applied to a plasma,charges may move and an excitation called a plasmon can occur. As seen in the figure below,such an excitation is not an equilibrium state, and has some restoring force which drives theoscillation. These collective oscillations of the electron gas are termed plasmons.

Figure: (left): An electron gas surrounding the positive ions in a metal can be considered aplasma, a gas of charged particles which is overall electrically nuetral. (right): A collectiveoscillation in a plasma is termed a plasmon.



Dispersion Relation of the Plasmon

A close look at the figure shows that the plasmon is describable by a plane wave, with anassociated wavelength λp and wavenumber kp . Thus we may solve a wave-equation for thepropagation of the plasmon, assuming a lossless dielectric constant as derived above:

∂2D

∂t2= c2∇2E −→ ε(ω)ω2E = c2k2E −→ ε∞(1−

ω2p

ω2)ω2 = c2k2 −→ ω2 = ω2

p +k2c2

ε∞

The last form is the dispersion relation for the plasmon in bulk metal, that is the volumeplasmon, plotted below:

Figure: The dispersion relation for the volume plasmon, compared to the dispersion relation forlight.



Surface Plasmon Polariton

At surfaces, the momentum mis-match can be fulfilled by imparting higher spatialfrequencies on the light field, as is done in the prism below (so-called Kretschmanngeometry)

Figure: Excitation of the surface plasmon in a thin-metal film deposited on thesurface of a prism.



Electromagnetic Theory

We can assume we have a plane wave incident on a hemispherical prism, as shown in the figure.The fields have the form: Ei = Eoi e

i(kyi y±kzi z−ωt), and Hi = Hoi ei(kyi y±kzi z−ωt), with

Hoi = Hxi x and Eoi = Eyi y + Ezi z, i = 1, 2, corresponding to ε1 and ε2. Putting these intoMaxwell’s vector equations:

∇× E = −1

c

∂B

∂tand ∇×H =

1

c

∂D

∂t

and assuming time-harmonic fields yields:

∇× E =

∣∣∣∣∣∣i j k∂x ∂y ∂z

0 Eyi Ezi

∣∣∣∣∣∣ = (ikyi Ezi ∓ ikzi Eyi )i = −1

c

∂B

∂t= i

ω

cHxi i (1)

∇×H =

∣∣∣∣∣∣i j k∂x ∂y ∂z

Hxi 0 0

∣∣∣∣∣∣ = (±ikzi Hxi )j− (ikyi Hxi )k =1

c

∂D

∂t= −iεi

ω

c(Eyi j + Ezi k) (2)



Dispersion Relation for the SPP

Equating the vector components yields the system of equations:

iω

cHxi = ikyi Ezi ∓ ikzi Eyi ∓ for i = 1, 2 (3)

±ikzi Hxi = −iεiω

cEyi ± for i = 1, 2 (4)

−ikyi Hxi = −iεiω

cEzi (5)

From the above equations and the boundary conditions E1t = E2t , D1n = D2n and H1t = H2t ,we can show:

kz2

ε2+

kz1

ε1= 0 and εi

ω2

c2= k2

yi + k2zi (i = 1, 2) giving: ky = ksp =

ω

c

√ε1ε2

ε1 + ε2



Frequency Dependent Dispersion for the SPP

Inserting the above form(s) for ε(ω) into the expression for ksp yields:

ksp =ω

c

√√√√√√√ ε1(ε2b −ω2

p

ω2)

ε1 + ε2b −ω2

p

ω2

=ω

c

√ε1(ε2bω2 − ω2

p)

(ε1 + ε2b)ω2 − ω2p

−→ ksp =ω

c

√ω2 − ω2

p

2ω2 − ω2p



Particle Plasmons and Mie Theory

The classic treatment of the modes excited by light at the surface of a spherical particledue to Mie2 which begins with the scattering geometry shown below:

2Principles of Optics, 7th edition, Max Born and Emil Wolf (1999).S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 15 / 24


Mie Analysis

The analysis is sketched as follows: first, incident field is taken to be Ei = e ikIz x and

Hi = ikI

kI2e ikIz y and considered to be time-harmonic, thus the vector Maxwell’s equations

can be written as:

∇×H = −k1E where k1 =iω

c

(ε+ i

4πσ

ω

)and ∇× E = k2H where k2 = i

ω

c

The boundary conditions are: HIt = HII

t , and E It = E II

t at r = a, the radius of thesphere. Writing out the first set of Maxwell’s equations for the curl of H givesexpressions for the vector components of E:

−k1Er =1

r 2 sin θ

[∂(rHφ sin θ)

∂θ− (rHθ)

∂φ

](6)

−k1Eθ =1

r sin θ

[∂Hr

∂φ− ∂(rHφ sin θ)

∂r

](7)

−k1Eφ =1

r

[∂(rHθ)

∂r− ∂Hr

∂θ

](8)

and similar equations for H.



Mie Analysis continued ...

You may assume the solutions come in two forms (akin to TE and TM):

( eE, eH) and ( mE, mH)

transversality of the solutions requires that eEr = Er , eHr = 0 and mEr = 0, mHr = Hr .Thus, we can simplify the above expressions to come up with the following set ofequations:

−k1eEr =

1

r 2 sin θ

[∂(r eHφ sin θ)

∂θ− ∂(r eHθ)

∂φ

](9)(

∂2

∂r 2+ k2

)(r eHθ) = − k1

sin θ

∂ eEr

∂φ(10)(

∂2

∂r 2+ k2

)(r eHφ) = k1

∂ eEr

∂θ(11)

1

k21 r

2 sin θ

∂

∂r

[∂(r sin θ eHθ)

∂θ+∂(r eHφ)

∂φ

]= 0 (12)

where the last equation comes from ∇ · eH = 0.



Mie Analysis continued ... continued ...

To solve these equations, Mie used potential theory, and introduced the Debyepotentials eπ and mπ which can be found by the method of separation of variables:

m(e)π = R(r)Θ(θ)Φ(φ)

Yielding Bessel’s equation, Legendre’s equation and the harmonic equation:

d2(rR(r))

dr 2+(k2 − α

r 2

)rR(r) = 0 (13)

1

sin θ

d

dθ

(sin θ

dΘ(θ)

dθ

)+

(α− β

sin2 θ

)Θ(θ) = 0 (14)

d2Φ(φ)

dφ2+ βΦ(φ) = 0 (15)

with well known solutions:

Φ(φ) = am cos(mφ) + bm sin(mφ) (16)

Θ(θ) = P(m)l (ξ) = P

(m)l (cos θ) m = −l ,−l + 1, ..., l − 1, l (17)

R(r) =1

rζ

(1)l (kr) where ζ

(1)l (ρ) =

√πρ

2H

(1)

l+ 12

(ρ) (18)

where H(1)

l+ 12

(ρ) is the Hankel function of the first kind.



Mie Analysis continued ... continued ... continued ...

The potentials can thus be assembled and the field obtained according to the generatingequations:

Er = eEr + mEθ =∂2(r eπ)

∂r 2+ k2r eπ (19)

eEθ + mEθ =1

r

∂2(r eπ)

∂r∂θ+ k2

1

r sin θ

∂(r mπ)

∂φ(20)

Eφ = eEφ + mEφ =1

r sin θ

∂2(r eπ)

∂r∂φ− k2

r

∂(r mπ)

∂θ(21)



Mie Analysis (continued ... )4

With the solutions (16)-(18), we can construct eπ and mπ and with the assistance of(19)-(21) calculate the fields (shown here are the electric fields only):

E (s)r =

1

(k I)2

cosφ

r 2

∞∑l=1

l(l + 1) eBlζ(1)l (k Ir)P

(1)l (cos θ) (22)

E(s)θ = − 1

k I

cosφ

r

∞∑l=1

[eBl ζ

(1)′

l (k Ir)P(1)′

l (cos θ) sin θ − i mBl ζ(1)l (k Ir)P

(1)′

l (cos θ)]

(23)

E(s)φ = − 1

k I

sinφ

r

∞∑l=1

[eBl ζ

(1)′

l (k Ir)P(1)′

l (cos θ) sin θ]

(24)




Thus we see the various combinations of special functions found in (22)-(24) areweighted by the coefficients eBl and mBl , similar to a multi-pole expansion. The e(m)Bl

are termed the partial-wave amplitudes for the l th partial wave. In many circumstances,one or a few terms in this expansion can reproduce all the observed properties. Eachterm has unique symmetry, and it is often useful to examine these waves.

Figure: First two partial waves and the lines of electric and magnetic force.Nodes are represented by “o”.




If we introduce the scaling parameter q = 2π aλ

, and if q << 1, we may assume only thefirst partial wave is non-zero:

eBl=1 = iq3 n2 − 1

n2 + 2= i

(2πa

λ

)3n2 − 1

n2 + 2, where n2 =

(k II

k I

)2

=εII

εI+ i

4πσ

ωεI

thus for high ω or low σ, eBl=1 = i

(2πa

λ

)3εII − εI

εII + 2εI−→ resonance @ |εII+2εI| = 0

We see that the amplitude of the first partial wave can become very large for a specificfrequency which makes εII(ω) = −2εI, this is the plasmon frequency.




As seen in the figure below, this freqency depends on the size of the particle due to theso-called Fermi-damping, which modifies the intrinsic material damping:

ε(ω) = 1−ω2

p

ω2 + iγωwhere γ = γbulk +

vF

a

Figure: Energy-losses due to electrons with finite mean-free path λmfp and velocities vF .



Calculate Mie Resonances

Figure: Extinction cross section for Au nano particles as a function of λ .

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