This document discusses the Drude model for explaining the optical and electric properties of metals using a free electron gas model. It describes how the Drude model relates the dielectric constant of metals to oscillations of free electrons in response to an applied electromagnetic field. It defines key terms like plasma frequency, damping frequency, and presents equations for the real and imaginary components of the dielectric function and how they describe polarization and energy dissipation in metals. Graphs are shown depicting how the real part of permittivity is negative at lower frequencies but becomes zero near the plasma frequency.

1. Drude Model- Dielectric constant of metals
Presented
by
R. Gandhimathi

2. Permittivity () and Permeability (µ)
➢ In an optical medium, how electromagnetic waves propagate is defined by the terms called permittivity and permeability
➢ I.e. Describe the interactions between the electromagnetic waves and materials
0
(1 )
r
r e
  
 
=
= +
0 -permittivity of free space-8.85418782 × 10-12 m-3 kg-1 s4 A2
r - relative permittivity
e -electric susceptibility, which is a measure of the extent to which an applied electric field to a dielectric material
causes polarization. 2 2
0 0 0
0 1 2
( )
( )
e eP E P E E
P E P P P
   = + + + − − − − − −
= + + + − − − −
where P(E) is the electric dipole moment per unit volume of the dielectric
P0-constant polarization
Permittivity () is related to electric field
➢ () -Measurement of resistance which is experienced whenever developing an electrical field inside a medium
➢ In other words,  is determined by how much a medium can polarize in response to an electric field
➢ Unit- Farads/meter

3. 1 0 eP E = Polarization is directly proportional to total electric flux density and direction
of E. Under the EM field material is polarized /magnetized.
Where
0
0 0
0
( )
(1 )
m
m
B H M
B H H
B H

  
 
= +
= +
= +
B
H
 =
➢ The term permeability is related to magnetic field
➢ It is defined as ratio of existing magnetizing field B within the material divided by the magnetic field strength H of the magnetizing field
With the applied EM field, Electric displacement can be written as
0
0
0
0
(1 )
e
e
D E P
D E E
D E

  
 
= +
= +
= +
0 rD E =
D-Number of flux lines crossing a surface normal to lines divided by the surface area
02
4
Q
D E
r


= =
Electric displacement (Without EM field)
Permeability ()
0 rB H
B H
 

=
=
m
M
H
 =Magnetic susceptibility
1r m = +
M-Magnetization of the material

4. For real materials  is function of frequency, Permittivity of real materials can be written in terms of complex numbers
( ) ( )     = +
( )  -Polarization
( )  -losses
The relative permittivity, permeability and refractive index of a material are defined by
0
1r e

 

= = +
0
1r m

 

= = +
n


=
Velocity of light
1
c

=

5. Drude model
➢ Drude model relates the optical and electric properties of metals with the behavior of their free electron gas density
➢ In metals, valence band is fully filled by electrons, nevertheless the conduction band is only partially filled
➢ According to this model, the electrons do not interact with each other and are scattered by the positive ions
(considers only collision time & mean free path)
➢ The linear response of these metals to electromagnetic field is determined by the dielectric function
Permittivity in the presence of an oscillating electric field (Without losses)
When free electrons travel in the field of an electromagnetic wave, electrons experience a force and the electron
motion in the field is given by Lorentz force. I.e. The force on a charge q moving with velocity v in the presence
of an electric and magnetic field E, B is called the Lorentz force and is given by
F qE qV B= + 
Equation of motion for an electron of the plasma sea subjected to an external electric field E
( ) ( )
2
2
, ,e
d r dr
m e E r t e B r t
dt dt
= − − 
em - Mass of the electron

6. 2
2
d r
dt
- acceleration
2
0 02
i t ikr i t ikr
e
d r dr
m e E e e B e
dt dt
 − + − +
= − −  
k-propagation constant
0 0[ ]i t ikr dr
e e E B
dt
− +
= − + 
2
02
i t
e
d r
m e E e
dt
−
= − 
dr
c
dt
since kr <<1,
The solution for the above equation is ( ) 2 0
i t
r t E
e
me
e 

−
=
I.e. electron oscillates in space with the frequency and phase of the external field
Total polarization is given by P=np
Dipole moment (p)

7. p-dipole moment of single electron = r(t). e
( )
2
02
e
i t
P n e r t
E e
n e
P
m


−
=
= −
−
n-free electron gas or concentration of electron gas,
0
i t
E eE −
=
2
2
e
n eP
E m 
−=
0 ,eP E =We know that 0 e
P
E
 =
0
2
2
e
e
n e
m 
 =−
2
2
2
0
2
e
p
e
e
n e
m 




 = −
−=
2
0
p
e
n e
m


−=
Plasma frequency
p
-Depends mass, concentration and charge of carriers and corresponds to internal electrostatic oscillations of plasma
Above p the real part of the dielectric function becomes positive and the metal starts to behave
like a non-absorbing dielectric medium
Plasma frequency is considered as maximum frequency of plasma response

8. ( ) 1 e  = +
e is viewed slightly different in a conductive material, which gives the modified dielectric function
in metals as
The electric susceptibility
2
2
1( ) p
 


−= -dielectric function of the undamped free electron plasma
Permittivity of electron gas is determined by p.
Primitivity of metals (accounting collisions of electron)
Accounting collisions of electrons with lattice, The response of a free electron of mass me and charge e to an external
electric field can be described as:
2
02e
i t
e
d r dr
m e E e
dt dt
m 
 −
= − −
- Mean Free path time
1


=Damping frequency
The damping frequency  plays an important role, governing the magnitude of the resonance

9. ( ) 0
( )
i t
e
e
m i
r t E e 
  
−
=
+
2
1 p
permitivity
  = −
1
( )i

  
=
+
the dielectric function of the free electron gas:
2
( )
1 p
i



 +
−=
( )i −Multiply and divide with
( ) 0
e
i t
r t E
e
m
e  −
=
2
0
2
0
2
e
e
e
e
e p
n e
m
n e
m

 



  
= −
= −
= −
2
2
2
0
i t
e
e
e
P np
n e
P
m
n e
P
E e
E
E
m
n eP
m



−
=
= −
= −
= −
2
2
1 p p
i
 



− +=
Total polarization
The solution for equation of motion of electrons

10. Re() and Im() components of complex dielectric function ε(ω) = ε1(ω) + iε2(ω) are given by
2 2
2 2 2 2
1
)(
p p
i
  
 

+
+ +
−=
➢ Re() –describes the polarization and the negative dielectric constant leads to a strong
imaginary part of the refractive index n = √ ε. Thus light can penetrate a metal only to a
very small extent
➢ Im() describes the dissipation of energy associated with the motion of electrons in the
metals (radiative damping, electron gas confinement, structural imperfections, and metal
heating losses)
Model graph
➢ At lower frequencies the permittivity of metals is negative. i.e. frequencies lower
than plasma frequency(p)
➢ Negative permittivity is characteristics of metals it results in higher reflectivity of
metals
➢ In the vicinity of plasma frequency real part of permittivity becomes zero
➢ Above plasma frequency there is no difference between metals and dielectrics

11. 1. M. Born and E. Wolf. Principles of Optics. Cambridge University Press, Cambridge, sixth edition, 1980.
2. C.F. Bohren and D.R. Huffman.Absorption and scattering of light by small particles.Wiley science paperback series.
John Wiley & Sons, Inc., New York, 1983.
3. R.E. Hummel.Optische Eigenschaften von Metallen und Legierungen.Number 22 in Reine und angewandte
Metallkunde in Einzeldarstellungen. Springer Verlag, Berlin, Heidelberg, New York, 1971.
4. J. Aukong, Electromagnetic Wave Theory, John Wiley & Sons, New York (1986).
5. P. Šolín, Partial Differential Equations and the Finite Element Method, John Wiley & Sons, New York (2006).
6. Theocaris P.S., Gdoutos E.E. (1979) Electromagnetic Theory of Light. In: Matrix Theory of Photoelasticity. Springer
Series in Optical Sciences, vol 11. Springer, Berlin, Heidelberg
7. Nannapaneni Narayana Rao, Fundamentals of Electromagnetics for Engineering, Pearson Education (2008)
8. Vincenzo Amendola, Roberto Pilot, Marco Frasconi, Onofrio M Maragò, Maria Antonia Iatì, J. Phys.: Condens.
Matter 29 (2017) 203002
References

12. Thank you