 Extended Drude model (Lorentz-Drude model)

Optical properties of materials

Insulating media (dielectric) : Lorentz model

Conducting media (Metal, in free-electron region) : Drude model

Conducting media (Metal, in bound-electron region) : Drude-Sommerfeld model

 Extended Drude model (Lorentz-Drude model)

( ) 0 ( ) ( ) ( )

r

r

D     E 

 

 (spatially local response of media)

: relative dielectric constant

= relative permittivity

= dielectric function

Let’s start from Maxwell equations

Flux densities in a medium

Total electric flux density = Flux from external field + flux due to material polarization

in most of optical materials.



= permittivity of material

(external) (external)

Ampere-Maxwell’s Law

Stokes theorem (very general)

Faraday’s Law (Faraday 1775-1836)

EM wave Equation in bulk media ()

: EM wave equation in bulk media without net charge ( _ and =0)

EM waves in dispersive media : )

Relation between E and P (J) is dynamic:

In ideal case, assuming an instantaneous response,

But, in real life,

P(r,t) results from response to E over some characteristic time τ:

Function x(t) is a scalar function lasting a characteristic time τ

Convolution theorem

Response to E is DISPERSIVE

 Therefore, the temporally dispersive response of a medium induces a frequency dependence in permittivity.

A. Propagation in insulating media (dielectric : P, J = 0)

Consider a linear, homogeneous, isotropic medium .

: all the materials properties resulting from P : EM wave equation in insulating media (dielectric)

( linear, homogeneous, isotropic, J=0, _ , and =0)

Note 3 : In inhomogeneous media:

2 2

2

( dispersion rel ation ), where 1 ( )

r r

k c

    

  

2 2

2

0 0 2 2 2

1

r r

E E

E    ^ t  c ^ t

  

 

 



The wave equation is satisfied by an electric field of the form of:

 

0

exp

E  E ^  i k r    t ^ 



  

B. Propagation in conducting media (free-electron region: J, P = 0)

Consider an ideally free electron (not bounded to a particular nucleus)  Drude model

J

linearly proportional to

E

:

J =  E



is the conductivity

The wave equation is satisfied by an electric field of the form of:

 

0

exp

E  E ^  i k r    t ^ 



  

₂ ^{2 2}

2 2

0

0

dispersion r

1 ( ) ( )

( ) ( )

elati 1

r

on

r

k i

c c

i

    

 

   

 

 

    

 

 

2 2

2 2 2

0

1 E E

E c t c t





 

  

 

 



: EM wave equation in conducting media (free-electron region) ( linear, homogeneous, isotropic, P=0, _ , and =0)

 

_r

  ¹   ( ) for dielectric 

Consider a bounded electron (not ideal free-electron due to interband transition) Modified Drude model

P

b induces internal current density ^:

The wave equation is satisfied by an electric field of the form of:

 

0

exp

E  E ^  i k r    t ^ 

   

 

2 2

2

2 2

0

0

1 ( )

( ) ( ) 1 ( )

b r

r b

k i

c c

i

  

  

 

       



 



 

     

 

   

 

2

2 2

2

0 0 0

2 2 2 2 2

1

_b

1

b

P

E J E

E J J

c ^ t  ^ t  ^ t c ^ t  ^ t

      

    



  

  

: EM wave equation in conducting media (bound-electron region) ( linear, homogeneous, isotropic, _ , and =0)

C. Propagation in conducting media (bound-electron region: J, P = 0)

For the noble metals* (e.g. Au, Ag, Cu), the filled d-band close to Fermi surface causes a residual polarization, P_b, due to the positive back ground of the ion cores.

Ag⁴⁷: 1s²… 4d¹⁰ 5s¹ Au⁷⁹: 1s²… 5d¹⁰ 6s¹

Cu²⁹: 1s²… 3d¹⁰ 4s¹ Al¹³: 1s²… 3s² 3p¹(different)

*In physics the definition of a noble metal is strict. It is required that the d-bands of the electronic structure are filled.

Taking this into account, only copper, silver and gold are noble metals, as all d-like band are filled and don't cross the Fermi level.

P

_b also linearly proportional to E:

P

_b

  

_{0 b}

E

0 b

b b

P E

J  ^ t    ^ t

 

2 2

2

2 2 2 2 2

0

1 E

_b

E E

E c t c t c t

 



  

   

  

  



*Interband transition: excitation of electrons from deeper bands into the conduction band

Quasi-free-electron model

Drude-Sommerfeld model

Now, how to determine  ^and  _b () ^?

Microscopic origin of -response of matter

Professor Vladimir M. Shalaev, Univ of Purdue

The equation of motion of the oscillating electron,

: of each electron

= : dampling constant (or, collision freq 1 ( )

effect ( )

uency ive ma

) ( is kno

( ) ss

2 2

 

 

 

 

  

E

r

F v

x

d r d r

m C r m e E

dt dt m

F r

_

F E 

 



      

14

wn as the relaxation time of the free electron gas) ( is typically 100 THz = 10 Hz) 

+

No external field

Applied external electric field

+ - _r(t)

E

_x

Electron cloud

- F

_r

r(t)

E

_x

F

_E

F

_

Classical Electron Oscillator (CEO) Model = Lorentz model

A.  of insulating media (dielectric : P, J = 0)

A. Insulator

A. Insulator

A. Insulator

Assume _j ‘s are the same for all atoms

( ) N

_j

  

 

A. Insulator (single atomic gas)

A. Insulator (single atomic gas)

1

A. Insulator (single atomic gas)

A. Insulator (realistic gas with different atoms)

A. Insulator (solids)

Local field at an atom

A. Insulator (solids)

유도해보시오

!







 

i m

e

j o

j

  



 



 

² ₂

1

₂

A. Insulator (solids)

 

0

3

3 3

2 2

4 .

2

embed embed

r r

solid j j embed embed

j r r

embed embed

V N V V

a if the medium is a sphere with radius a

   

  

   

  

 

   

       

                        

  

     



 The Polarizability of a solid with volume V given by the Clausius-Mossotti relation is



 





 

m i Ne m

m C Ne N

N

o

   

 



 

 

 



 

 

2 0

2 2

31

3 1 1

For a simple case when C

_j

= C for all atoms in solid,







 

i m

e

j o

j

   

 



 

² ₂

1

₂





 

j

j j j

j j

N N







13

1

^{where and 2}

j j

C C

m m

  

2

2 2

0

( )

^p

i

  

  

   ^{, where}

^o2

2

3

_o

C Ne m m

 ^{ } 

A. Insulator (solids)

This is the same form as the single atomic gasses, except the different definition of ₀.

In general when C

_j

are not identical,

 

( )

2

2 2

p j

j j j

f i

  

  

   

f

_j is the oscillation strength (the fraction of dipoles having



_j^.

B. Drude model

B.  in free-electron region: Drude model = Free-electron model

 Drude model : Lorenz model (Harmonic oscillator model) without restoration force (that is, free electrons which are not bound to a particular nucleus)

2 2

: v A C

The current density is defined J N e

m s m

 

      

 



The equation of motion of a free electron (not bound to a particular nucleus; ),

====> ( : relaxation time )

2 14

2

0

1 10

  

  

 

C

d r m d r d v

m e E m m v e E s

dt dt

d C r

t  

 

^



        

Lorentz model

(Harmonic oscillator model)

Drude model (free-electron model) If

C = 0

  

m d v m v e E

dt     ^dJ _J ^{N e}

²

_E

dt     ^ m ^ 

 

  

   

 

 

0 0

0

0 0

:

( ) ( ) exp ( ) ( ) exp

:

exp exp exp

Assume that the applied electric field and the conduction current density are given by

E r E r i t J r J r i t

Substituting into the equation of motion we obtain

d J i t

J i t i J

dt

 

   

   

  

     

       

  

   

 

 

 

0

2 0

2

0 0

exp exp

exp :

,

e

e

i t J i t

N e E i t

m

Multiplying through by i t

i J N e E or equivalently

m

  





 

  

 

   

 



 

    

 





 

Local approximation to the current-field relation

2

e

dJ N e

J E

dt     ^ m ^ 

 

  

 

²

e

i J N e E

  ^ m ^

    

 

 

B. Drude model

 

 

 

2 2

0 0

0 0

2

, , :

:

( ) ,

1 /

( ) /

1

:

0

/

N e N e

J E E where static conductivity

m m

of an oscillating applied field J

For static fields For t

E E

i

dynamic

N e m

i i

he general cas

conduc

e

 

 

 

  

   

 





 

    

 

 

   

 



 







  

  

tivity

 

²

e

i J N e E

  ^ m ^

    

 

 

0

( ) 1 ( )

r

i  

  ^{ }  

   

2 2 2

0 0 0 0 0 0

2

2 2

2 2 2

0 0 0

0

2 2

2 2

0

( ) 1 1 1

1 / 1 /

,

( ) 1 ,

r

p

p

r p

c i c c

i i

i i

i i

N e N e

The plasma frequency is defined c c

m m

N e

i m

       

           

     

 

   

   

   

   

                            

 

    

 

  



B. Drude model

 

 

0

0

( ) 1 /

, ( ) ~ :

, 1

,

. i

For very low frequencies

As the frequency of the applied field increase

and the electrons follow the electric field

the inertia of electrons introduces a phase l purely r

ag in the electron response t eal

s

  

 

  

 

 



 

   

²

^{ }

0 0 0

, .

, , ( )

( ) ~ / /

90 / :

.

, 1

i

is complex

i purely imag

o the field and

J i E e E

and the electron oscillations are out of phase with For very hi

the appli gh freq

ed field uenc s

inary ie





 

         

  

 



  

Note : Dynamic conductivity 

0

( ) 1 ( )

r

i  

  ^{ }  

B. Drude model

2 2 2

2 2 2 3 2

( ) 1

^p

1

^{p p}

r

i

i

   

       

   

                   

2 2 2 2

2 3 2 3

( ) 1 1

/

p p p p

r

 i   i 

       

       

                              (1) For an optical frequency, 

_visible



Dielectric constant of free-electron plasma (Drude model)

(2) Ideal case : metal as an undamped free-electron gas

• no decay (infinite relaxation time)

• no interband transitions

( )

0 r



  

^_



_r

( ) 1 

²₂^p

  ^{ } 

B. Drude model

2

0

N e :

static conductivity

 m

 

2 2 2

0 0

0 p

c N e

    m

   

If the electrons in a plasma are displaced from a uniform background of ions,

electric fields will be built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their original positions.

Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions with a characteristic frequency known as the plasma frequency.

/ ( ) / : electrostatic field by small charge separation exp( ) : small-amplitude oscillation

( )

( )

2 2 2

2 2

2

s o o o

o p

s p p

o o

E Ne x x

x x i t

d x Ne Ne

m e E m

dt m

    

  

  

 

 

 

       

Note : What is the actual meaning of  _p

Note : plasma wavelength  _p

p

²

p

 c

 ^ 

B. Drude model

Note : What happens in  when  _p ?

B. Drude model

2 2

0 2

2

2 ( )

2 0

( ) ( 0)

( ) ( , ) :

^{j k r t}

E E E D assume J

t

k k E k E k E E E e

c





  

^{ }

         



    

(1) For transverse waves, k E   0 k

²

( , ) k

₂²

c

  



(2) For longitudinal waves, k k E (  )  k E

²

 0  ( , ) 0 k  

0

0

D    E  P

Therefore, at the plasma frequency 

_p

,

 E is a pure depolarization field (No transverse field strength in media).

( , k

_p

) 0

    

The quanta of these longitudinal charge oscillations are called plasmons (or, volume plasmons)

Volume plasmons do not couple to transverse EM waves (can be only excited by particle impact)

 means that at the plasma frequency only a longitudinal wave can propagate through.

B. Drude model

( ) 0,

_inc

If    K  V

Note : What happens in  when  _p ?

Optical potential

Incident energy

K

_inc

 k

₀²

, the incident wave is completely reflected.

  0

(N. Garcia, et. Al, “Zero permittivity materials”, APL, 80, 1120 (2002))

Wave propagation in this material can happen only with phase velocity being infinitely large satisfying the “static-like” equation

(A. Mario, et. Al, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern”, PR B, 75, 155410, 2007)

C.  _b  in bound-electron region : Drude-Sommerfeld model

= Quasi-free-electron model

 

0

1 1

( ) ( ) 1 ( ), usually 0

r

i  

b

      

 

^

 

     

2 2 2

2 2 2 3 2

( )

^{p p p}

r

i

i

   

   

     

 

   

                   

C. Drude-Sommerfeld model

Plasmons in metal nanostructures, Dissertation, University of Munich by Carsten Sonnichsen, 2001 silver

gold

Measured data and modified Drude model for Ag:

 

 



 '  1  

^p₂²

, " 

₃^2p

 

 



 



₃

2 2

2

"

,

' 

_



^p



^p

Drude model:

Modified Drude model:

Contribution of bound electrons

3.7

p

9.1 eV











200 400 600 800 1000 1200 1400 1600 1800

-150 -100 -50 0 50

Measured data:

 '

 "

Drude model:

 '

 "

Modified Drude model:

 '

 "



Wavelength (nm)

 '

-

_d

Bound SP mode : ’

_m

< -

_d



Wavelength (nm)

2.48 eV

500 nm

1.24 eV

1000 nm

0.62 eV

1500 nm

 For Ag, good matched bellow 2.5 eV

For Ag,

C. Drude-Sommerfeld model

Let’s compare the modified Drude model with experimental measurement

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals”, Phys. Rev. B, 6, pp. 4370-4379 (1972).

3 2

2 2 2

( )

2 ^p

r

p

i

 

 

   



 

   

               



Regions of interband transitions

 Not good matched in interband regions!  Need something more.

C. Drude-Sommerfeld model

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals”, Phys. Rev. B, 6, pp. 4370-4379 (1972).

C. Drude-Sommerfeld model

D. bound-electron region : Extended Drude (Drude-Lorentz) model



²



2

2 2 2

( )

^p

L r

L

i

i L

   

 



  



   

 

     

 

        



( )

2

2 2

j p

j j j

f i

  

  





 

Since the Lorentz terms of insulators have a general form of

The Drude-Lorentz model consists in addition of one Lorentz term to the modified Drude model.

For gold (Au),

Alexandre Vial, et.al, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method”, Phys. Rev. B, 71, 085416 (2005).

Excellent agreement within 500 nm ~ 1 m for gold (Au)

D. Extended Drude model

1.4 1.6 1.8 2.0 2.2 2.4 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0

1000900 800 700 600 500

Im Permittivity

E [eV]

J & C Exp Drude DrudeLorentz

[nm]

1.4 1.6 1.8 2.0 2.2 2.4 -80-70

-60-50 -40-30 -20-10100

Relative error in %

E [eV]

Drude DrudeLorentz

1.4 1.6 1.8 2.0 2.2 2.4 -40

-35 -30 -25 -20 -15 -10 -5

1000900 800 700 600 500

Re Permittivity

E [eV]

J & C Exp Drude DrudeLorentz

 [nm]

1.4 1.6 1.8 2.0 2.2 2.4 -2

0 2 4 6 8 10

Relative error in %

E [eV]

Drude DrudeLorentz

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -40

-35 -30 -25 -20 -15 -10 -5 0

1800515001200900 600 300

Re Permittivity

E [eV]

J & C Exp Drude DrudeLorentz

 [nm]

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0

1 2 3 4 5 6 7

8180015001200900 600 300

Im Permittivity

E [eV]

J & C Exp Drude DrudeLorentz

 [nm]

But, not correct at higher energy

Extended Drude (Drude-Lorentz) model

 

2 2

2 2 2

( )

^{p L}

r

L L

i i

 

  

   



 

     

                  Modified Drude model for metal in bound-electron region

2 2 2

2 2 2 3 2

( )

^{p p p}

r

i

i

   

   

     

 

   

                    Drude model for metal in free-electron region

2 2 2

2 2 2 3 2

( ) 1

^p

1

^{p p}

r

i

i

   

       

   

                   

2

2 2

/ ,

p j

j j

N i

 

  

  

13

( ) 1 ,

1

j j j

r

j j j

N N



  ^{ }  



 ^

^2j

^{ C m}

^j

Lorentz model for dielectric (insulator)

In summary :

Also, keep in mind…..