Paolo Biagioni, January 10th, 2019

Optical properties of metal

NFO-9, Lausanne – September 10-15, 2006 2
Optical response in terms of the
complex dielectric constant

 ( )  1 ( )  i 2 ( )
Electric field E
Polarization P (electric dipole moment per unit
volume inside the material)
P   0 E
Dielectric displacement
D   0E  P   0 (1   )E   0E
P ( ) D ( )
 ( )  1  
 0 E ( )  0 E ( )
Paolo Biagioni
 Relation between  and n
NFO-9, Lausanne – September 10-15, 2006 3

Complex refractive index n( )  n1 ( )  in2 ( )

 ()  n () 2

 2 1 1
n1   1   2
2 2

 2 2
 
n2  2

 2n1

Paolo Biagioni
Influence of  () on wave propagation
NFO-9, Lausanne – September 10-15, 2006 4

For a time-harmonic wave the group velocity is given by

c
f  v 
n

  2f

 2  ()  c k
2 2 2

 k
 
(dispersion relation)

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 5

How do we describe the optical response of a metal?

1) Free electron gas

2) Drude-Sommerfeld model

include damping due to carrier scattering

3) Drude-Lorentz model

include damping due to interband transitions

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 6
Free electron Fermi gas (1)

• A gas of free electrons moving against a fixed

background of positive ion cores

• Works well for alkali metals up to the near UV and

for many transition metals up to the visible-near IR.

• Why can the electrons be considered “free”?

1) Matter waves propagate freely in a periodic structure (Bloch’s theorem)
2) Electron-electron scattering is quite inefficient (Pauli principle)

• Works well for few-nm nanostructures as well, since

the spacing between energy levels is <<kBT
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 7
Free electron Fermi gas (2)

How do free electrons respond to a time-harmonic field?

 it
Look for solutions
E (t )  E0e
x(t )  x0eit
d 2x
me 2  qe E (t )
dt qe
x(t )  E (t )
me 2

Displaced electrons generate a macroscopic polarization

nqe2
P(t )  nqe x(t )   E (t )
me 2

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 8
Free electron Fermi gas (3)
free
So what about the dielectric constant?
p 

P( ) nq 2
 2

 free ( )  1   1  1 2
e p
nqe2
 0 E ( )  0 me 2
 with  
2

 0 me
p

(plasma frequency)

Li Na K Rb Au Ag Cu Al
p(nm) 155 210 315 340 136 138 134 80

At optical frequencies, <p

Electrons oscillate out of phase with the optical field

Large reflectivity of metals at optical frequencies

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 9
Free electron Fermi gas (4)

And the dispersion relation?

 free ( )  c k      c k
2 2 2 2 2
p
2 2

(forbidden frequency gap)

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 10
Bulk plasma oscillations (plasmons)

• The plasma frequency is related to one of the most

fundamental oscillations of a free electron gas moving in
the background of positive ions:

d 2x
2
  px  0
2

dt
(longitudinal mode
at    p )

s = -nqex

E=s/0
s = nqex
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 11
Drude-Sommerfeld model (1)

• Include damping effects due to electron-electron

and electron-phonon scattering:
d 2x dx
me 2  qe E (t )  me 
dt dt

 p2  p2  p2
 Drude ( )  1   1 i
  i
2
 2 2
( 2   2 )

Significant imaginary part, related to Ohmic losses and

therefore to a certain penetration depth (skin depth) of
optical fields inside the material. Typical penetration depths
for metals of interest in the visible are of a few 10s nm
(e.g. 31 nm for Au or13 nm for Al at 620 nm wavelength)
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 12
Drude-Sommerfeld model (2)

Au: fit of experimental data with Drude-Sommerfeld model:

Missing interband transitions

Missing offset

Integrated
Au
contribution from core
electrons

Drude-Lorentz
model
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 13
Drude-Lorentz model (1)
• Describe interband transitions as oscillations
of bound electrons:
Effective mass of bound electrons
Spring constant
d 2x dx
m 2  qe E (t )  m  x
dt dt
Damping of bound electrons

~p2 First interband abs. in Au

 Lorentz ( )  1  2
(0   2 )  i
Lorentz absorption
resonance

Re()
~ n~qe2
 p  0
 m 0
with 
 
0  m

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 14
Drude-Lorentz model (2)

 ()      Drude ()   Lorentz ()

Interband peak Nearly-free

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 15
Drude-Lorentz model (3)

Au: extended fit of experimental data

 ()      Drude ()   Lorentz ()

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 16
Optical properties of gold

1

2

Nearly-free Drude
Onset of interband
behavior above 600 nm
transitions around
500-550 nm
Good for red and near-IR photons
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 17
Optical properties of silver

1

2

Nearly-free Drude
Onset of interband behavior above 450 nm
transitions around
400 nm Better than Au for green spectral region,
but degradation in air
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 18
Optical properties of copper

1

2

Very similar to Au,

but degradation in air
Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 19
Optical properties of aluminum

1

2

Interband transition
Larger (negative) around 800 nm
real part

Good metal in the 400-600 nm region,

cannot be used in near IR
Paolo Biagioni
 20
…and what about nanostructures (5-100 nm)?
NFO-9, Lausanne – September 10-15, 2006

• As already discussed in the beginning, the concept of a

bulk dielectric constant does not break up
However

• The role of surface scattering is not negligible anymore

(large surface/volume ratio):

additional damping term (significant for size<10 nm)

𝑣𝐹
Γ 𝑟 = Γ0 + 𝐴
𝑟
(r radius of the particle, vF Fermi velocity)

Paolo Biagioni
NFO-9, Lausanne – September 10-15, 2006 21
References

• L. Novotny and B. Hecht,

Principles of Nano-Optics (Cambridge University Press),
Ch. 12 “Surface plasmons”

• S. A. Maier
Plasmonics – Fundamentals and Applications (Springer),
Ch. 1 “Electromagnetics of metals”

• C. Kittel
Introduction to Solid State Physics – 7th ed. (John Wiley & Sons),
Ch. 6 “Free electron Fermi gas”, Ch. 10 “Plasmons, polaritons, and
polarons”

Paolo Biagioni