Phys. 555/342: Ch 14A

Plasmons, Polaritons, Polarons, and Excitons

Chapter 14 in Kittell: Be careful he uses CGSTo map onto SI replace 4π with 1/ε0

We will discuss ε(ω,K)--energy and wavelength dependent

ε(ω,0)---describes collective excitation: Plasmons, etc.ε(ω,K)---describes electrostatic screening, e-e, e-P, e-

impurityD = E + 4πP = εED-displacementE-fieldP-polarization

∇ •D = ∇ •εE = 4πρext

∇ • E = 4πρ = 4π(ρext + ρind )

Phys. 555/342: Ch 14A

Dielectric Function for Electron GasPlasma Optics

ε(ω,k) → ε(ω,o) ≡ ε(ω)

λ → ∞

md2x

dt 2 = −eE

with

x ∝eiωt

E ∝eiωt

−ω 2mx = −eE x =eE

ω 2m

Dipole moment −ex = −e2E /ω 2m

P = −nex = −ne2

ω 2mE

Phys. 555/342: Ch 14A

Dielectric Function for Electron GasPlasma Optics

P = −nex = −ne2

ω 2mE ε(ω) =

D(ω)E(ω)

≡1+ 4πP(ω)E(ω)

ε(ω) =1+4πne2

mω 2

Plasma Frequency

ω p2 =

4πne2

m

ε(ω) =1+ω p

2

ω 2ω=ωp when ε=0

Phys. 555/342: Ch 14A

But the positive background also has a dielectric constantlabeled fi

Dielectric Function for Electron GasPlasma Optics

ε(∞)

ε(ω) = ε(∞) −4πne2

mω 2 = ε(∞) 1+˜ ω p

2

ω 2

⎡

⎣ ⎢

⎤

⎦ ⎥

˜ ω p = ω

ε = 0

Again---

˜ ω p2 =

4πne2

mε(∞)

Phys. 555/342: Ch 14A

Dielectric Function for Electron GasDispersion

∂ 2D∂t2 = c 2∇2EEM wave equation

But − −E ∝ei(K•r−ωt ) and − −D = ε(ω,K)E

ε(ω,K)ω 2 = c 2K 2

• ε real and >0: For ω real, K is real and a transverse EM wave propagates with phase velocity c/ε1/2

• ε real and <0: For ω real, K is imaginary and the wave is damped with a characteristic length 1/|K|

• ε is complex: For ω real, K is complex and the waves are damped in space.• ε is infinity: This means the system has a finite response in the absence of an

applied force; thus the poles of ε(ω,K) define the frequesncies of the free oscillations of the medium.

• ε=0: We shall see that longitudinally polarized waves are possible only at the zeros.

Phys. 555/342: Ch 14A

Transverse Optical Modes in a Plasmaε(ω,K)ω 2 = c 2K 2 ε(ω)ω 2 = ε(∞)(ω 2 − ˜ ω p

2 ) = c 2K 2

For ⋅ ⋅ω < ˜ ω p K2 <0 so K is imaginary E ∝e−|K |x

For ⋅ ⋅ω > ˜ ω p K2 >0 so K is real ω 2 = ˜ ω p2 + c 2K 2 /ε(∞)

This describes a transverse electromagnetic wave in a plasma.

Phys. 555/342: Ch 14A

Plasmons

Phys. 555/342: Ch 14A

Transparency of Alkalki Metals in the Ultraviolet

Metal should reflect below the plasma frequencyMetals should transmit above the plasma frequency

The Reflectance of Indium Antimonide with n=4x1018 cm-1

Phys. 555/342: Ch 14A

Longitudinal Plasma OscillationsZeros of dielectric function determine the frequency of the longitudinal modes of oscillation.

ε(ωL ) = 0

ε(ωL ) =1−ω p

2

ω 2 = 0

There is a free longitudinal oscillation mode (fig) of an electron gas at the plasma frequency. This is the low frequency cut off of the transfer mode

Phys. 555/342: Ch 14A

Longitudinal Plasma Oscillations

Electron charge

Positive background

The figure shows a longitudinal plasmon in a thin film with K=0.

Shift with applied field

The electron gas is displaced by u.

Surface charge

There is an electric field E=4πneu: surface charge -neu

nmd2u

dt 2 = −eE = −4πn2e2u

d2u

dt 2 + ω p2u ω p

2 =4πne2

m

For small k we can use previous Eqn.

ω ≅ ω(1+3k 2vF

2

10ω p2 + ••)

Phys. 555/342: Ch 14A

Plasma Oscillations

A plasma oscillation is a collective longitudinal excitation of the conduction electrons. A plasmon is a quantum of plasma oscillations.

Phys. 555/342: Ch 14A

The Prediction of a Surface Plasmon

Rufus Ritchie

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Phys. 555/342: Ch 14A

History

• Ritchie and Birkhoff had submitted separate abstracts to a conference at the University of Maryland, but Rufus could not attend --.Birkhoff offered to present his paper, which met with some fierce criticism from Dr. Dennis Gabor (later a Nobel laureate), who said he had studies the problem and concluded that there was no such animal as the surface plasmon. Upon learning of this response, Rufus questioned what he should do. Colleague David Pines encouraged him to submit the paper anyway.

• It is now one of the most cited papers.

Phys. 555/342: Ch 14A

Ritchie’s 70th BirthdayThe impact of the concept of a surface plasmon: Nuc.

Inst. & Methods B96,448 (1995)

Paper referenced 435 times by 1985 in 1995 it had been referenced 620 times.

Phys. 555/342: Ch 14A

Plasmons

Phys. 555/342: Ch 14A

Surface Plasmons

Plasmons in the bulk: ωp determined by the free e- density and mass

Plasmons confined to surfaces: interact with light to form propagating surface plasmon polaritons (SPP)

Plasmons can be confined in nanoparticles: resonant SP modes that are localized

EM surface waves that exists at the interface of two media whose ε have opposite signs

Phys. 555/342: Ch 14A

EM surface waves that exists at the interface of two media whose ε have opposite signs

Fluctuations of the charge on the metal boundary give rise to the SP mode

Surface Plasmons

Phys. 555/342: Ch 14A

Surface Plasmons

Surface plasmons

dm

dm

|| )()(

ck

εωεεωεω

+=

2

22

⇒ Can exist only at εm < 00>||k

Dispersion relation and Drude

2

22 )(

ck ωωε=

2

2

1)(ωω

ωε p−=

&

Phys. 555/342: Ch 14A

When illuminated from within, the Lycurgus cup glows red due to embedded gold nanoparticles which have an absorption peak around 520 nm

Surface Plasmons- early uses

Colors in stained glass is due to metallic nanoparticles with different indices and aspect ratios

Phys. 555/342: Ch 14A

Phys. 555/342: Ch 14A

Phys. 555/342: Ch 14A

Phys. 555/342: Ch 14A

Phys. 555/342: Ch 14A

Photon STM Image of a Chain of Au nanoparticles [from Krenn et

al, PRL 82, 2590 (1999)]

Individual particles: 100x100x40 nm, separated by 100 nm and deposited on an ITO substrate. Sphere at end

of waveguide is excited using the tip of near-field scanning optical microscope (NSOM), and wave is

detected using fluorescent nanospheres.

Phys. 555/342: Ch 14A

Kittell Problem 10.1

Phys. 555/342: Ch 14A

Solution 10.1

1. kzx0E kA sin kx e

x∂ϕ

= − =∂

, and at the boundary this is equal to Exi. The normal

component of D at the boundary, but outside the medium, is ( )kA cos kx, where for aplasma ( ) = 1 Š p

2/ 2. The boundary condition is ŠkA cos kx = ( )kA cos kx, or ( ) = Š1, or p

2 = 2 2. This frequency p 2ω = ω is that of a surface plasmon.

Phys. 555/342: Ch 14A

Kittell: Problem 10,2

Phys. 555/342: Ch 14A

Kittell, solution 10.2

2. A solution below the interface is of the form kz( ) A cos kx eϕ − = , and above the interface kz( ) A cos kx e−ϕ + = , just as for Prob. (1). The condition that the normal component of D be continuous across the interface reduces to 1( ) = Š 2( ), or

2 2p1 p2 2 2 2

p1 p22 2

11 1 , so that ( ) .2

ω ω− = − + ω = ω + ω

ω ω

Phys. 555/342: Ch 14A

Kittell, Problem 10.5

Phys. 555/342: Ch 14A

Kittell, solution 10.5

5. 2 2 2 2md /dt m e 4 e /3 4 ne /3= − ω = − = π = − πr r E P r . Thus 24 ne 3m.2

οω = π

Phys. 555/342: Ch 14A

Quantum treatment of Surface Plasmons

Quantum Calculation of the induced charge near a surface as a result of applying a constant electric field.

4π3

rsa0[ ]3 =1/n

a0 is Bohr Radii

Analog of“classical Surface Charge”

Friedel Oscillations: 2kF

Phys. 555/342: Ch 14A

A view Equations

ωP =4πne2

m

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

δn(z,w) → induced − ch arge σ(ω) → induced − Total − ch arge

σ(ω) = dzδn(z,ω) =ε(ω) −1

ε(ω) +1

⎡

⎣ ⎢ ⎤

⎦ ⎥ ∫ E(ω)

2π

ε(ω) =1−ωP

2

ω 2In “Jellium Model”

Phys. 555/342: Ch 14A

Physical properties of the simple metals and of Ag calculated from the free electron model: ωSP is surface plasmon energy

When there is a difference between theory and Exp. It means the band structure is important.

Phys. 555/342: Ch 14A

A calculation for finite ω• Fundimentally different

from the static case. The magnitude of oscillations much larger and the wavelength is much longer. The “surface charge peak” is not well defined.

• The system is not responding adiabatically to the applied field at these frequencies.

• These long wavelength oscillations have been seen experimentally

Phys. 555/342: Ch 14A

A comparison

Phys. 555/342: Ch 14A

Work by Feibelman in the 1970’s showed that the response of the surface could be characterized by the d(ω) function.

d(ω) =dz • z •δn(z,ω)

σ (ω)∫

First moment of induced charge.

• Red(ω) develops pole at ~0.8ωP

• No features at ωSP• Imd(ω) gives photo

absortion.

Phys. 555/342: Ch 14A

Surface Photoelectric Effect

Phys. 555/342: Ch 14A

New term in excitation matrix dσdΩ

∝|< Ψf | 2

A •

P − i ∇ •

A | Ψi >|2 {δ(E f − Ei − ω)}

Spatial variation in A

Phys. 555/342: Ch 14A

Variation in A

Phys. 555/342: Ch 14A

A few more Equations from a paper by Persson and Zaremba

g(q,ω) = dz •eqz •δn(z,ω)∫The surface response function g(q,ω)

g(q,ω) determins the amplitude of the reflected component of the electrostatic potential interacting with the surface. Q is the absolute value of the parallel componendt of the momentum q||. In the small q limit, we have.

g(q,ω) =ε(ω) −1

ε(ω) +11+

2ε(ω)

ε(ω) +1wd(ω)

⎛

⎝ ⎜

⎞

⎠ ⎟

Substitute in Drude dielectric ω(q) = ωSP 1− d(ωSP )q /2( )

Phys. 555/342: Ch 14A

How do you measure dispersion

Phys. 555/342: Ch 14A

Instrument

Phys. 555/342: Ch 14A

Data Surface plasmon

Phys. 555/342: Ch 14A

Phys. 555/342: Ch 14A

New mode: Multiple Plasmon

Phys. 555/342: Ch 14A

Comparison Exp. Vs. Theory

Phys. 555/342: Ch 14A

MM Energy vs. electron density