Plasmons govern the high frequency optical properties of materials since they

determine resonances in the dielectric function ε(ω) and hence in the refraction

index

Often written as for non ferromagnetic materials.

Above the plasma frequency, electric fields penetrate into the matter which

becomes therefore transparent to electromagnetic radiation (ultraviolet

transparency of alkali metals).

)()()( ωµωεω =n

Plasmons, Surface Plasmons and Plasmonics

)()( ωεω =n

Surface plasmons determine the high frequency surface response function,

governing the screening of external fields and electron transmission at the

interface, determining e.g. photoemission intensities.

Plasmons exist also for nearly 2-dimensional systems, formed e.g. in charge

inversion layers and artificially layered materials [1]. Such 2D plasmons may

have low energy and govern many dynamical processes involving electrons and

phonons and mediate the formation of Cooper pairs in superconductors [2]

[1] M.H. March and M.P. Tosi, Adv. in Physics 44, 299 (1995)

[2] J. Ruvalds, Nature 328, 299 (1987)

Plasmons and Surface plasmons:

collective excited states of the 3D electron gas

Equation of motion for a free electron of mass m and charge e in presence of a

time varying electric field E=E(t) :

eEdt

xdm =

2

2ti

oeEE ω= ti

oexx ω=

2ωm

eEx o

o −= Em

neP

rr

2

2

ω−=and the polarization reads

ωm mω

EEm

neD

rrr)()41(

2

2

ωεω

π =−=

For ε=0 → E≠0 also when D=0 , i.e. when there are no external charges.

ε=0 determines therefore the condition for which self sustaining polarization waves

can exist at the frequency

m

nep

24πω =

2

2

1ω

ωε p−=For the free electron gas it follows:

while the electric displacement field is:

What determines the magnitude of plasmon frequency?

2

222

mead o

h==

oada 222 ==

6

63

333 8

2

8

224

h

em

adan

o

====

6

82

6

63222

2

2

8

244

hh

emem

m

e

m

nep

πππω ===

23

3632

23 emnk

ππ ==

22 )23(

3/1 mek π=

22

4 πω

hh

meE pp ==

distance between atoms in a solid (approx. twice the Bohr radius)

electron density

fcc lattice spacing containing 4 atoms per unit cell

28

33

6

23

h

emnkF

ππ ==

2

2

2)23(

3/1

h

mekF π=

2

4

3

2

2

4

42

3

2

22

8)23(

4)23(

2 hh

h meem

mEF ππ ==

in terms of the energy of the fundamental state of the H atom (Rydberg=0.5 Hartree)

2

4

2h

meEH = HHp EEE 98,2

2

4==

πH

HF E

EE 01,3

4)23( 3

2

2 == π

989.0423

1

2

43

2

2=⋅

⋅=

ππ

F

p

E

E

m

kE F

F2

22h

=

The ratio is slightly smaller than unity

because the plasmon is a collective

rather than a single particle excitation

Demonstration that nkF

23 3π=

Let’s start with the free electron gas model of the solid:m

k

m

pE

22

222h

==

For a linear chain of length L with N atoms separated by a lattice spacing a, L=Na we

have standing waves whenever the electron wavelength λ satisfies the requirement :

Ln

=2

λNa

n

L

nk

ππλπ

===2

π

i.e. n=1,2,3 ...N

The largest value of k is when n=N, i.e. , independent of N

(having more atoms implies a higher density of k points in the dispersion, not a

larger kmax since there are also more states)

ak

π=max

2

222

max

2

max22 mam

kE

πhh==The largest energy Emax is thus:

with κ continuous variable for very large n

2

2

22

2κ

πma

h=)(

2

2

3

2

2

2

12

22

nnnma

E ++=πh

For a particle in a square box quantization implies

Since n1, n2 and n3 are positive, the total number of

states N is then given by the volume of the octant of

a sphere

2

3

3

3

23

22

3 23

82

63

4

8

1)( Em

h

VmEVEN

ππ

ππκ =

=

=h

2

3

3

3

23

22

3 23

82

63

4

8

1)( Em

h

VmEVEN

ππ

ππκ =

=

=h

dEEgdEEmh

VEdN )(2

4)( 3

3==

π

Differentiating the above eq. we get the density of

states per energy interval

the number of electrons dn is twice as large, due to

the spin degenerationthe spin degeneration

and the total number of electrons is obtained by integrating dn from 0 up to Emax

∫∫ ===EE

Emh

VdEEm

h

VEdnn

0

233

3

3

302

3

162

8)(

ππ

2322

3

3 22

3

16

=

m

km

h

Vn Fhπ

nkF

23 3π=and hence

Volume plasmon dispersion depends on electronic polarizability α and is

quadratic in transferred momentum

Example Ag

At the surface, if σ is the charge density, the Maxwell equations read:

>

−<=

πσπσ

2:0

2:0

z

zEz

>

−<=

πεσπσ

2:0

2:0

z

zDz

Surface plasmons

Surface plasmons are the normal modes of charge fluctuation at a metallic surface

and govern the long range interaction between the metal and the rest of the world.

vacuum ε=1 metal ε= -1

πσ2−=zE

πσ2=zE+

+

+

>=

πεσ2:0zDz

Dz is continuous at the interface so that -2πσ = 2πεσ → ε(ω)= -1

nearly free electrons

2

p

sp

ωω =

The frequency of a proper surface plasmon is thus determined by

its volume dielectric function

z

+

Photoemission spectroscopy

Plasmon and surface plasmon are observed

in photoemission spectra. Their relative

intensity depends on the kinetic energy of the

electrons. The bulk plasmon can be excited

while the electron is inside the solid, the

surface plasmon when the electron leaves

the surface on the trajectory to the analyser

The photoemission probability may be strongly affected by the surface response

function as shown for the case of the surface Shockley state of Al and Be.

No photoemission is observed at photon energies coinciding with the plasma

resonance since then the crystal becomes transparent to the electric fields

Right: In phase (solid) and out of phase

(dashed) contributions to the normal

component of the electric field near a jellium

surface

1

Contrary to its frequency, the dispersion of the

surface plasmon is determined by surface

properties and in particular by the position of

the centroid of the screening charge with

respect to the geometric surface, defined by

the d parameters which correspond to the

centroid of the screening charge for electric

fields vertical and parallel to the surface.

Surface plasmon dispersion

))())((2

11)(0()( 2

|||||| qoqddq spsp +−−= ⊥ ωωω

The field associated to the surface plasmon oscillates

along the surface and decreases exponentially

towards the bulk.

The charge density felt by the surface plasmon

depends thus on q||

zqrqi

o eer ||||)(−⋅Φ=Φ

rr

The position of the centroid of induced charge is located outside of the surface

in the low density electron spill out region since there the electron gas is more dilute

and thus more compressible

d(ωsp)>0

→ dispersion slope is

negative for free electron

metals

The position of the centroid of the

screening charge vs frequency

diverges towards the interior of the

metal at ωp . At the surface palsmon

frequency it is still positive.

Surface plasmon dispersion for free electron metals

The dispersion is initially linear,

the quadratic terms dominates at

large q||

Measurement of surface plasmon dispersion by HREELS (high resolution

electron energy loss spectroscopy) and ELS-LEED

Energy conservation:

Momentum conservation:

Dipolar energy loss cross section:

HREELS High Resolution Electron Energy Loss Spectroscopy

measurement of surface plasmon dispersion

Energy conservation:

Momentum conservation:

M..Rocca, Surface Science Reports 22,1 (1995)

Multipole plasmon mode+++ - - - +++ - - -

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

only the

multipole

mode is

observed in

photoyield photoyield

experiments

In presence of d-electrons, the plasmon frequency is displaced to lower

frequencies due to the contribution of interband transitions to the dielectric

function )()()( 21 ωεωεωε i+=

For Ag, plasmon damping is small at ωp

but ε=0 is shifted from 9 eV to 3.8 eV

Surface plasmon dispersion in presence of d-electrons: Ag and Au

H. Raether, Springer tracts in Mod. Phys. Vol 88, 1980 J. Daniels, Z.Phys. 203, 235 (1967)

Surface plasmon dispersion in presence of d-electrons:

Ag and Au

Liebsch model: the d-electrons are schematized by a dielectric medium which

extends up to a distance zd from the geometric surface

The interaction of the electric potential, associated with the plasmon, with the

dielectric medium causes the shift of the plasmon energy

ωs 6.5 eV → 3.7 eV

at large q|| the induced fields penetrate less and the shift is smaller and hence ωs

higher if this effect overcompensates the negative slope due to the position of the

centroid of induced charge centroid of induced charge

→ POSITIVE DISPERSION

M. Rocca, Low energy EELS investigation of

electronic excitations at metal surfaces, Surf. Sci.

Rep. 22, 1 (1995)

A. Liebsch, Electronic excitations at metal surfaces,

Plenum Press (1997)

Surface plasmon dispersion for Ag

M. Rocca et al. PRL 64, 2398 (1990)

Multipole Surface plasmon at Ag surfaces

F. Moresco et al. PRB 54, 14333 (1996)

confirmed by photoyield measurements in 2001

Mie resonance

t

Plasmons in thin films

of thickness t

R

d

m

d

))(Re(

2

3 0

0

ω

εεω

ω ⊥

+=

∆h

h

ω= ω(q||) on surfaces

ω= ω(1/t) on thin films

ω= ω(1/r) on clusters

t

thin films of thickness t

The same holds true in presence of d-electrons

Mie Resonance

t εdεd

The influence of the polarizable medium scales with

the surface to volume ratio!!

Mie resonance for clusters

K

• experiment

∗ □ theory

Ag

Similar result for thin Ag films Y. Borensztein Eur. Phys. Lett.31, 34 (1995)

Tiggesbaeumker et al. 1992

The bright colors of stained

glasses of gothic cathedrals

were obtained by nanosized

gold particles which resonate at

the Mie resonance.

This phenomenon corresponds

to light confinement

Notre Dame Paris

Plasmon confinement in nanostructured Ag films

t2r

Deposition at 300 K yields isolated flat clusters

at 90 K a percolated layer

Plasmons are however localized in both cases as

demonstrated by the absence of dispersion at small q||

STM image

F. Moresco et al. Phys. Rev. Lett. 83, 2238 (1999)

Plasmons in nanostructured ultrathin Ag/Si(111)7x7

→ slope depends on

surface to volume ratio

t2r

F. Moresco et al. Phys. Rev. Lett. 83, 2238 (1999)

Confinement in percolating layer

due to frequency mismatch at the

touching points of the clusters

The interaction with light: surface plasmon polaritons

Unlike the case of volume plasmons light can interact with surface plasmons due

to the lower symmetry giving rize to an avoided crossing of the dispersion curve.

The mixed mode is called surface plasmon polariton.

The light cone does not cross the surface plasmon dispersion curve unless it has

an imaginary q//z (evanescent waves) or if the missing momentum is provided by

surface roughness or by a nanometric superlattice.

Attenuated total reflection

Measurement of Surface Plasmon Polaritons:

2D surface plasmons: Ag monolayer on Si √3x √ 3 structure

N2D areal density of electrons

ω ∝√q||

ε(Si) dielectric constant of Si

= 10.5 - 11.5

m* effective electron mass

in the film

Similar plasmons observed in

charge inversion layers at

semiconductor surfaces

March and Tosi Adv. Phys. (1995)

2D surface plasmons: Ag monolayer on Si √3x √ 3 structure

Two dimensional electron gases exist an all metal surfaces

supporting Shockley surface states in band gaps.

2D surface plasmons at bare metal surfaces

Such states may generate plasmons with acoustic linear dispersion

The linear rather than squareroot dispersion

implies possible applications to devices since

no distortion occurs when converting light into

the plasmon and backwards (nanooptical

devices and metamaterials with adjustable

dielectric properties)

Plasmons govern the high frequency optical properties of materials since they

determine resonances in the dielectric function ε(ω) and hence in the refraction

index

)()()( ωµωεω =n

Plasmons, Surface Plasmons and Plasmonics

ε1

1The plasmon resonances

causes a strong variation of the

ε2

causes a strong variation of the

dielectric function which may

become very small or even

negative.

When both ε(ω) and µ(ω)

become negative then also

n(ω) may be negative since the

dephasing introduced both by

ε(ω) and µ(ω) corresponds to

positive angles (delays)

More recent concepts: Plasmonics and plasmonic materials

Plasmon resonances are used to build metamaterials, i.e. artificial materials with

very small (ENZ - Epsilon Near Zero) or even negative index of refraction (NIR).

ENZ were proposed for the construction of invisibility cloaks by which

even massive objects could be made perfectly transparent to light.

R. A. Shelby et al., Science 292, 77 -79 (2001)

Metamaterials with negative refraction (NIR)

NIR act as a sort of optical antimatter and were recently used for the construction

of perfect superlenses with no limit to the resolving power.

To keep in mind

Surface plasmons control the high frequency optical response of materials.

For usual 3D surface plasmons, the frequency is a bulk property, but the

dispersion is determined by the surface electronic structure. The dispersion is

then negative for simple metals and positive for d-metals. At very small

wavevectors the dispersion cuts the light cone giving rise to surface plasmon

polaritons.

For clusters surface plasmons correspond to the Mie resonance showing the

same properties vs inverse cluster size. For thin film the surface plasmon

frequency scales with inverse film thickness. frequency scales with inverse film thickness.

Surface plasmons exist also for 2D electron gases and are then acoustic with

squareroot like dispersion. For intrinsic two dimensional surface states the

interaction with the underlying electron gas implies a surface plasmon with

acoustic linear dispersion which can couple with light via surface roughness or

surface nanostructures and looks promising for applications

Further reading:

• M. Rocca, “Low energy EELS investigation of surface electronic

excitations”, Surf. Sci. Rep. 22, 1 (1995)

• A. Liebsch, “Excitations at metal surfaces” (Plenum Press London (1977))

• J. Pendry, “Playing tricks with light”, Science 285, 1687 (1999)

• Subsequent Science and Nature articles 1999-2007