Drude Model-Dielectric constant of metals

Frequency dependent permittivity of metals

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

2. Permittivity ()and Permeability(µ)
➢Inanopticalmedium,howelectromagneticwavespropagateisdefinedbythetermscalledpermittivityandpermeability
➢I.e.Describetheinteractionsbetweentheelectromagneticwavesandmaterials0
(1)
r
re


=
=+

0
-permittivity of free space-8.85418782 ×10
-12
m
-3
kg
-1
s
4
A
2r

-relativepermittivity

e
-electricsusceptibility,whichisameasureoftheextenttowhichanappliedelectricfieldtoadielectricmaterial
causespolarization.22
0 0 0
0 1 2
()
()
ee
PEPEE
PEPPP
=+++−−−−−−
=+++−−−−
whereP(E)istheelectricdipolemomentperunitvolumeofthedielectric
P
0-constantpolarization
Permittivity()isrelatedtoelectricfield
➢()-Measurementofresistancewhichisexperiencedwheneverdevelopinganelectricalfieldinsideamedium
➢Inotherwords,isdeterminedbyhowmuchamediumcanpolarizeinresponsetoanelectricfield
➢Unit-Farads/meter

3. 10 e
PE= Polarization is directly proportional to total electric flux density and direction
of E. Under the EM field material is polarized /magnetized.
Where 0
00
0
()
(1)
m
m
BHM
BHH
BH



=+
=+
=+ 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 themagnetic field strengthHof the magnetizing field
WiththeappliedEMfield,Electricdisplacementcanbewrittenas0
0
0
0
(1 )
e
e
D E P
D E E
DE

  

=+
=+
=+ 0r
DE=
D-Number of flux lines crossing a surface normal to lines divided by the surface area02
4
Q
DE
r


==
Electric displacement (Without EM field)
Permeability ()0r
BH
BH


=
= m
M
H
=
Magnetic susceptibility1
rm
=+
M-Magnetization of the material

4. Forrealmaterialsisfunctionoffrequency,Permittivityofrealmaterialscanbewrittenintermsofcomplexnumbers( ) ( )     =+ ()
-Polarization()
-losses
Therelativepermittivity,permeabilityandrefractiveindexofamaterialaredefinedby0
1
re



= = + 0
1
rm



= = + n


=
Velocity of light 1
c

=

5. Drude model
➢Drudemodelrelatestheopticalandelectricpropertiesofmetalswiththebehavioroftheirfreeelectrongasdensity
➢Inmetals,valencebandisfullyfilledbyelectrons,neverthelesstheconductionbandisonlypartiallyfilled
➢Accordingtothismodel,theelectronsdonotinteractwitheachotherandarescatteredbythepositiveions
(considersonlycollisiontime&meanfreepath)
➢Thelinearresponseofthesemetalstoelectromagneticfieldisdeterminedbythedielectricfunction
Permittivity in the presence of an oscillating electric field (Without losses)
Whenfreeelectronstravelinthefieldofanelectromagneticwave,electronsexperienceaforceandtheelectron
motioninthefieldisgivenbyLorentzforce.I.e.Theforceonachargeqmovingwithvelocityvinthepresence
ofanelectricandmagneticfieldE,BiscalledtheLorentzforceandisgivenbyF 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
= − −  e
m
-Mass of the electron

6. 2
2
dr
dt -acceleration 2

002

i t ikr i t ikr
e
d r dr
m e E e e B e
dt dt
− + − +
= − −  
k-propagation constant00
[ ]
i t ikr dr
e e E B
dt
−+
= − +  2
02
it
e
dr
m e E e
dt
−
= −  dr
c
dt

since kr<<1,
The solution for the above equation is()
20
it
r t E
e
me
e


−
=
I.e.electronoscillatesinspacewiththefrequencyandphaseoftheexternalfield
Totalpolarizationisgivenby P=np
Dipole moment (p)

7. p-dipole moment of single electron = r(t).e ()
2
02
e
it
P n e r t
Ee
ne
P
m


−
=
=−
−
n-free electron gas or concentration of electron gas, 0
it
EeE
−
= 2
2
e
neP
Em
−= 0
,
e
PE=
Weknowthat0e
P
E
= 0
2
2
e
e
ne
m
=− 2
2
2
0
2
e
p
e
e
ne
m




=−
−= 2
0
p
e
ne
m


−=
Plasma frequency

p
-
Dependsmass,concentrationandchargeofcarriersandcorrespondstointernalelectrostaticoscillationsofplasma
Above
ptherealpartofthedielectricfunctionbecomespositiveandthemetalstartstobehave
likeanon-absorbingdielectricmedium
Plasmafrequencyisconsideredasmaximumfrequencyofplasmaresponse

8. ( ) 1
e
  =+ e
 isviewedslightlydifferentinaconductivematerial,whichgivesthemodifieddielectricfunction
inmetalsas
Theelectricsusceptibility2
2
1()
p



−=
-dielectricfunctionoftheundampedfreeelectronplasma
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 m
eand charge e to an external
electric field can be described as:2
02e
it
e
d r dr
m e E e
dt dt
m


−
= − − 
-Mean Free path time1


= Damping frequency
The damping frequency plays an important role, governing the magnitude of the resonance

9. ()
0
()
it
e
e
mi
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
it
r t E
e
m
e


−
= 2
0
2
0
2
e
e
e
e
ep
ne
m
ne
m





  
=−
=−
=− 2
2
2
0
it
e
e
e
P np
ne
P
m
ne
P
Ee
E
E
m
neP
m




−
=
=−
=−
=− 2
2
1
pp
i
 



−+=
Total polarization
The solution for equation of motion of electrons

10. Re()andIm()componentsofcomplexdielectricfunctionε(ω)=ε
1(ω)+iε
2(ω)aregivenby22
2 2 2 2
1
)(
pp
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

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Matter 29 (2017) 203002
References

12. Thank you