Transcript Lecture 19

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Electronic Materials and
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Demagnetization
• Say we wanted to demagnetize a sample
from point e by applying a reverse field
(i.e., -x direction)
• Then, magnetization would move along
from point e to point f
• Now, at point f, we suddenly turn off the
applied field
• We would find that B does not
remain zero, but recovers to along f
to point e’ and attains some value Br’
• The main reason is some that small
domain wall motions are reversible and
soon as the field is removed there is some
“bounce back” magnetization along f – e’
• However, we can anticipate this recovery
Fig 8.34
and remove the field intensity at some
point f’ so that magnetization is
Removal of the demagnetizing field at f does not
ultimately zero
necessarily result in zero magnetization as the
sample recovers along f-e'
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Demagnetization
• However, we don’t already know what f’
point we need to get to in order to
magnetization go to zero
• The simplest method to demagnetize the
sample is first to cycle H with ample
magnitude to achieve full saturation and
then continue to cycle H, but with
gradually decreasing magnitude
• This demagnetization process is
commonly known as deperming
Fig 8.35
A magnetized specimen can be demagnetized by
cycling the field intensity with a decreasing
magnitude, i.e. tracing out smaller and smaller B-H
loops until the origin is reached, H=0.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Soft and Hard Magnetic Materials
Fig 8.37
Soft and hard magnetic materials
• Engineered materials are typically classified into
soft or hard magnetic materials based on their BH behavior
• Soft magnetic materials are easy to magnetize
and demagnetize; therefore, they require
relatively low magnetic field intensities
• B-H loops are narrow, have a small area, so
hysteresis power loss per cycle is small
• Typically suited for applications where
repeated cycles are involved: electric
motors, transformers, and inductors
• Hard magnetic materials are difficult to
magnetize and demagnetize; therefore, they
require relatively large magnetic field intensities
• B-H loops are broad and almost rectangular
• They possess relatively large coercivities
which means they need large applied fields
to be demagnetized
• Their characteristics make them useful for
permanent magnets
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Types of Permeability
Fig 8.38
Definitions of (a) maximum permeability and (b) initial permeability.
• It is useful to characterize the magnetization of a material by a relative permeability, mr, since
is simplifies magnetic calculations
• The mr eqn above represents the slope of the straight line from the origin O to the point P
where this is a maximum when the line becomes a tangent to the B-H curve at P – any other
line is not a maximum
• The maximum relative permeability is denoted as mr,max
• Initial relative permeablity, mri, represents the initial slope of the initial B-H curve as the
materials is first magnetized from an unmagnetized state – this definition is useful for soft
magnets
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Superconductivity
Fig 8.44
• Certain materials have been found to
superconduct
• Some typical metals cannot
superconduct
• Superconductivity is when a material
is cooled below some critical
temperature, Tc, the material’s
resistivity totally vanishes, exhibiting
no resistance to current flow
• The resistivity of normal conductors
is limited by scattering from
impurities and crystal defects
ultimately resulting in a saturated
“residual” resistivity.
A superconductor such as lead evinces a transition to zero resistvity at a critical temperature Tc
(7.2 K for Pb) whereas a normal conductor such as silver does not, and exhibits residual
resistivity at the lowest temperatures.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Superconductivity and the Meissner Effect
Fig 8.45
The Meissner effect. A superconductor cooled
below its critical temperature expels all
magnetic field lines from the bulk by setting up
a surface current. A perfect conductor (σ = ∞)
shows no Meissner effect.
• Vanishing resistivity is not the only
characteristic of a superconductor
• A superconductor below its critical
temperature expels all the magnetic
field from the bulk of the sample as if it
were a perfectly diamagnetic substance
• This phenomenon is known as the
Meissner effect
• Above Tc, magnetic field lines will
penetrate the material
• However, when the superconductor is
cooled below Tc, it rejects all the
magnetic flux because it develops a
magnetization M by developing surface
currents such that M and the applied
field cancel everywhere inside the
sample
• In other words, moM is in the
opposite direction of applied field
and equal to its magnitude
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• Levitation of a magnet is the direct result of the Meissner effect: exclusion of the
magnet’s magnetic fields from the interior of the superconductor
Left: A magnet over a superconductor becomes levitated. The superconductor is a perfect
Diamagnet which means that there can be no magnetic field inside the superconductor.
Right: Photograph of a magnet levitating above a superconductor immersed in liquid nitrogen
(77 K). This is the Meissner effect. (SOURCE: Photo courtesy of Professor Paul C.W. Chu.)
Fig 8.46
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Type I Superconductor
Fig 8.47
The critical field vs. temperature in Type I
superconductors.
• Superconductivity below the
critical temp has been observed to
disappear in the presence of an
applied magnetic field that exceeds
a critical value, Bc
• This critical field depends on the
temperature and material
• Critical field is maximum when T
=0K
• As long as applied field is below
Bc at that temp, the material is
superconducting
• But, if field exceeds Bc, the
material reverts to the normal state
• The external field does actually penetrate the sample from the surface into the bulk; however,
the penetrating field decreases exponentially from the surface:
• Where Bo is the field at the surface, x is distance from the surface, and l is penetration depth
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Type I Superconductor
The critical field vs. temperature in three examples of Type I superconductors.
Fig 8.48
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 8.49
Characteristics of Type I and Type II superconductors. B = µoH
is the applied field and M is the overall magnetization of the
sample. Field inside the sample, Binside = µoH + µoM, which is
zero only for B < Bc (Type I) and B < Bc1 (Type II).
• Type I: abrupt transition
whereupon the
superconductivity
disappears
• Type II: transition does not
occur sharply:
• Bc1: lower critical field
• Bc2: upper critical field
• As magentic field
increases beyond Bc1,
magnetic flux lines are
no longer expelled, and
as field continues to
increase more flux lines
penetrate the sample
until Bc2 where all field
lines penetrate the
sample and
superconductivity
disappears
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Type II Superconductor
Fig 8.50
The mixed or vortex state in a
Type II superconductor.
• Transition does not occur sharply from Meissner state to
normal state
• Will go through an intermediate phase where the applied
field is able to pierce through certain local regions
• As magnetic field increases, initially the sample behaves as
a perfect diamagnet exhibiting Meissner and rejecting
magnetic flux
• Beyond Bc1, magnetic field flux lines are no longer totally
expelled
• Overall M in sample opposes field, but magnitude does not
cancel the field everywhere
• With increasing field, M gets smaller and more flux lines
pierce through the sample until Bc2 is reached where all
field lines penetrate the sample and superconductivity
disappears
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Type II Superconductor
Temperature dependence of Bc1 and Bc2.
Fig 8.51
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Critical Current Density
Fig 8.52
The critical surface for a niobium-tin alloy
which is a Type II superconductor.
• Any operating point inside this surface is in
the superconducting state
• When the current density through the
sample exceeds a critical value Jc, it is
found that superconductivity disappears
• The current through the superconductor
will itself generate a magnetic field and at
sufficiently high current densities, the
magnetic field at the surface of the sample
will exceed the critical field and extinguish
superconductivity.
• This plausible direct relation between Bc
and Jc is only valid for Type I
superconductors
• While for Type II, Jc depends in a
complicated way on the interaction
between the current and flux vortices
• Jc depends not only on temperature
and the applied magnetic field, but
also on the microstructure of the
superconducting material
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Superconductivity Origin
• The fundamental understanding of the
origin of superconductivity requires
A Copper Pair
QM and beyond the scope of the
course
• However, the primary idea is that at
sufficiently low temps, two oppositely
spinning and oppositely traveling
electrons can attract each other
indirectly through the deformation of
the crystal lattice of positive metal
ions
• Electron 1 distorts the lattice around it
and changes it vibrations as it passes
Fig 8.54
through the region. Low temps don’t
allow enough thermal randomization
A pictorial and intuitive view of an indirect attraction
between two oppositely traveling electrons via a lattice to randomize this induced lattice
distortion and vibration
distortion and vibration.
• The vibrations of this distorted region
now look differently to electron 2 that
is passing by
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Superconductivity Origin
• This 2nd electron feels a “net”
attractive force due the slight
A Copper Pair
displacements of the positive metal
ions from their equilibrium positions.
• This indirect interaction of the two
electrons at sufficiently low
temperatures is able to overcome the
mutual coulombic repulsion between
the electrons and hence bind to two
electrons together.
• The two electrons are called a Cooper
pair
• Opposite spins come from QM
Fig 8.54
• The net spin of a Cooper pair is zero
and their net linear momentum is also
A pictorial and intuitive view of an indirect attraction
between two oppositely traveling electrons via a lattice zero
distortion and vibration.
https://www.youtube.com/watch?v=fuloQcljFOs
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
These PowerPoint color
diagrams can only be used by
instructors if the 3rd Edition
has been adopted for his/her
course. Permission is given to
individuals who have
purchased a copy of the third
edition with CD-ROM
Electronic Materials and
Devices to use these slides in
seminar, symposium and
conference presentations
provided that the book title,
author and © McGraw-Hill are
displayed under each diagram.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Light is an electromagnetic wave
An electromagnetic wave is a traveling wave that has time-varying electric and magnetic
Fields that are perpendicular to each other and the direction of propagation z.
Fig 9.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Ex = Eo cos(tkz + )
Ex = electric field along x at position z at time t,
k = propagation constant, or wavenumber = 2/l
l = wavelength
 = angular frequency
Eo = amplitude of the wave
 is a phase constant which accounts for the fact that at t =
0 and z = 0 Ex may or may not necessarily be zero
depending on the choice of origin.
(tkz + ) =  = phase of the wave .
This equation describes a monochromatic plane wave of
infinite extent traveling in the positive z direction.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavefront
A surface over which the phase of a wave is constant is
referred to as a wavefront. A wavefront of a plane wave is a
plane perpendicular to the direction of propagation.
The interaction of a light wave with a non-conducting matter
(conductivity,  = 0) uses the electric field component Ex
rather than By.
The optical field refers to the electric field Ex.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
A plane EM wave traveling along z, has the same Ex (or By) at any point in a given xy plane
All electric field vectors in a given xy plane are therefore in phase. The xy planes are of
Infinite extent in the x and y directions.
Fig 9.2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Exponential Notation
Recall that
cos = Re[exp(j)]
where Re refers to the real part. We then need to take the real
part of any complex result at the end of calculations. Thus,
Ex(z,t) = Re[Eoexp(j)expj(tkz)]
or
Ex(z,t) = Re[Ecexpj(tkz)]
where Ec = Eoexp(jo) is a complex number that represents the
amplitude of the wave and includes the constant phase
information o.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavevector
We indicate the direction of propagation with a vector k, called
the wavevector, whose magnitude is the propagation constant,
k = 2/l. It is clear that k is perpendicular to constant phase
planes.
When the electromagnetic (EM) wave is propagating along
some arbitrary direction k, then the electric field E(r,t) at a
point r on a plane perpendicular to k is
E (r,t) = Eocos(tkr + )
If propagation is along z, kr becomes kz. In general, if k has
components kx, ky and kz along x, y and z, then from the
definition of the dot product, kr = kxx + kyy + kzz.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Wavevector
A traveling plane EM wave along a direction k.
Fig 9.3
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase Velocity
The time and space evolution of a given phase , for example
that corresponding to a maximum field is described by
 = tkz +  = constant
During a time interval t, this constant phase (and hence the
maximum field) moves a distance z. The phase velocity of
this wave is therefore z/t. The phase velocity v is
dz 
v
  l
dt k
where  is the frequency ( = 2π).
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase change over a distance z
The phase difference between two points separated
by z is simply kz since t is the same for each
point.
If this phase difference is 0 or multiples of 2 then
the two points are in phase. Thus, the phase
difference  can be expressed as kz or 2z/l.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index
When an EM wave is traveling in a dielectric
medium, the oscillating electric field polarizes the
molecules of the medium at the frequency of the
wave.
The stronger is the interaction between the field
and the dipoles, the slower is the propagation of
the wave.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Phase Velocity and r
The relative permittivity r measures the ease with which the
medium becomes polarized and hence it indicates the extent
of interaction between the field and the induced dipoles.
For an EM wave traveling in a nonmagnetic dielectric
medium of relative permittivity r, the phase velocity v is
given by
v
1
 r o m o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index
If the frequency  is in the optical frequency range then r
will be due to electronic polarization as ionic polarization will
be too sluggish to respond to the field. However, at the
infrared frequencies or below, the relative permittivity also
includes a significant contribution from ionic polarization and
the phase velocity is slower. For an EM wave traveling in free
space, r = 1 and vvacuum = 1/[omo] = c = 3108 m s–1, the
velocity of light in vacuum. The ratio of the speed of light in
free space to its speed in a medium is called the refractive
index n of the medium,
c
n   r
v
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index and Wavevector
Suppose that in free space ko is the wavevector (ko = 2π/l)
and lo is the wavelength, then the wavevector k in the
medium will be nko and the wavelength l will be lo/n.
Indeed, we can also define the refractive index in terms of the
wavevector k in the medium with respect to that in vacuum
ko ,
k
n
ko
In non-crystalline materials such as glasses and liquids,
the material structure is the same in all directions and n
does not depend on the direction. The refractive index is
then isotropic.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index and Isotropy
Crystals, in general, have nonisotropic, or anisotropic,
properties.
Typically noncrystalline solids such as glasses and liquids,
and cubic crystals are optically isotropic; they possess only
one refractive index for all directions.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Example: Relative permittivity and refractive
index
Relative permittivity r or the dielectric constant of
materials is frequency dependent and further it depends
on crystallographic direction since it is easier to polarize
the medium along certain directions in the crystal. Glass
has no crystal structure, it is amorphous. The relative
permittivity is therefore isotropic but nonetheless
frequency dependent.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The relationship n = (r) between the refractive index n
and r must be applied at the same frequency for both n
and r.
At low frequencies all polarization mechanisms present
can contribute to r whereas at optical frequencies only
the electronic polarization can respond to the oscillating
field.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
RECALL: Relative Permittivity and Polarizability
Factors that affect n – consider expression for relative permittivity
 r  1
N e
o
r = relative permittivity
N = number of molecules per unit volume
e = electronic polarizability
o = permittivity of free space
Assumption: Only electronic polarization is present
• Therefore, both atomic concentration (or density) and polarizability increase n
• Ex: Glasses of a given type with a greater density usually have higher n
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive Index Comparison
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Electronic Polarization, r and n
Fig 9.4
Electronic polarization of an atom. In the presence of a field in the +x direction, the
electrons are displaced in the –x direction (from O), and the restoring force is in the +x
direction.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Electronic Polarization, r and n
Fig 9.4
• Refractive index of
materials in general depends
on the frequency, or
wavelength, l
• This l dependence follows
directly from the frequency
dependence of the relative
permittivity, r.
• Above can also represent what happens to an atom in the presence of an oscillating
E-field caused by a light wave passing through
• Recall: In the presence of a field in the +x direction, the electrons are displaced in
the –x direction (from O), and the restoring force is in the +x direction
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Induced Polarization
First, consider applying a dc field. The restoring force Fr
acting on the electron shell is x. In equilibrium, the net
force on the negative charge shell is zero or ZeE + (x) = 0
from which x is known. Therefore the magnitude of the
induced electronic dipole moment is
pinduced = (Ze)x = (Z2e2/)E
Suppose we suddenly remove the applied electric field
polarizing the atom. The equation of motion of the negative
charge center is then (force = mass  acceleration)
Recall: the restoring force always acts to pull the electrons toward
the nucleus O.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Resonant or Natural Frequency
Always acts to pull the electrons toward the nucleus O
Restoring force = –x = Zmed2x/dt2
Solving this Diff. Eqn., we can show that:
The displacement at any time is a simple harmonic motion
x(t) = xocos(ot)
where the angular frequency of oscillation o is
  

 o  

Zme 

1 /2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
x(t) = xocos(ot)
  1 / 2

 o  

Zme 

This is the oscillation frequency of the center of mass of the electron
cloud about the nucleus and xo is the displacement before the removal of
the field. After the removal of the field, the electronic charge cloud
executes simple harmonic motion about the nucleus with a natural
frequency o; o is also called the resonance frequency.
The oscillations of course die out with time because there is an
inevitable loss of energy from an oscillating charge cloud. An oscillating
electron is like an oscillating current and loses energy by radiating
electromagnetic waves; all accelerating charges emit radiation.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Forced Oscillations by the Optical Field
Consider now the presence of an oscillating electric field due an
electromagnetic wave passing through the location of this atom.
The applied field oscillates harmonically in the +x and x directions,
that is E = Eoexp(jt).
There is again, a restoring force acting on the displaced electrons to
bring back e- to equilibrium placement around the neucleus
Newton’s second law for Ze electrons with mass Zme driven by E is
given by,
2
d x
Zme 2   ZeEo exp( jt )  x
dt
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Displacement of Electrons by the Oscillating Optical Field
d 2x
Zme 2   ZeEo exp( jt )  x
dt
The solution of this equation gives the instantaneous
displacement x(t) of the center of mass of electrons from the
nucleus (C from O),
eEo exp( jt )
x  x(t )  
2
2
me (o   )
The induced electronic dipole moment is then simply
given by pinduced = (Ze)x.
The electronic polarizability e is the induced dipole
moment per unit electric field,
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Induced Polarization and Electronic Polarizability
2
pinduced
Ze
e 

E
me (o2   2 )
The displacement x and hence electronic polarizability e increase as
 increases. Both become very large when  approaches the natural
frequency o. In practice, this is not the case because the two factors
impose a limit – large x is no longer linear; there is always energy loss
The simplest (and a very “rough”) relationship between the relative
permittivity r and polarizability e is
 r  1
N
o
e
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Dispersion Relation
where N is the number of atoms per unit volume. Given that the
refractive index n is related to r by n2 = r, it is clear that n must
be frequency dependent, i.e.
2 

NZe
1
2

n  1  

 2
2

m



 o e  o
We can also express this in terms of the wavelengthl. If lo =
2c/o is the resonance wavelength, then
2
2 
2


NZe
l
l
2
o




n  1  

 2

2

m

2

c

l

l
 o e 
o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Dispersion Relation
The relationship between n and the frequency , or wavelength
l, is called the dispersion relation.
n will always be wavelength dependent and will exhibit a
substantial increase as the frequency increases towards a natural
frequency of the polarization mechanism.
We considered the electronic polarization of an isolated atom
with a well-defined natural frequency o. In the crystal,
however, the atoms interact and further we also have to
consider the valence electrons in the bonds. The overall result is
that n is a complicated function of the frequency or the
wavelength.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
There are no perfect monochromatic waves in practice.
We have to consider the way in which a group of waves
differing slightly in wavelength will travel along the zdirection.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
When two perfectly harmonic waves of frequencies 
and  +  and wavevectors kk and k + k interfere, they
generate a wave packet which contains an oscillating field at
the mean frequency  that is amplitude modulated by a
slowly varying field of frequency . The maximum
amplitude moves with a wavevector k and thus with a group
velocity that is given by
d
vg 
dk
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Two slightly different wavelength waves traveling in the same direction result in a
wave packet that has an amplitude variation that travels at the group velocity.
Fig 9.5
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The group velocity therefore defines the speed with which
energy or information is propagated.
Since  = vk and the phase velocity v = c/n, the group
velocity in a medium can be readily evaluated. Suppose
that v depends on the wavelength or k by virtue of n being
a function of the wavelength as in the case for glasses.
Then,
 c 2 
 
  vk  

n( l) 
l 
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Velocity and Group Index
where n = n(l) is a function of the wavelength. The group
velocity vg in a medium is approximately given by,
d
c
v g (medium) 

dk n  l dn
dl
This can be written as
c
v g (medium) 
Ng
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Index
dn
Ng  n  l
dl
is defined as the group index of the medium.
In general, for many materials the refractive index n and
hence the group index Ng depend on the wavelength of
light. Such materials are called dispersive.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Group Index
Refractive index n and the group index Ng of pure SiO2
(silica) glass are important parameters in optical fiber design
in optical communications. Both of these parameters depend
on the wavelength of light. Around 1300 nm, Ng is minimum
which means that for wavelengths close to 1300 nm, Ng is
wavelength independent. Thus, light waves with wavelengths
around 1300 nm travel with the same group velocity and do
not experience dispersion. This phenomenon is significant in
the propagation of light in glass fibers used in optical
communications.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Refractive index n and the group index Ng of pure SiO2 (silica) glass as a
function of
wavelength.
Fig 9.6
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Magnetic Field, Irradiance and Poynting Vector
The magnetic field (magnetic induction) component By
always accompanies Ex in an EM wave propagation.
If v is the phase velocity of an EM wave in an isotropic
dielectric medium and n is the refractive index, then
c
Ex  vBy  By
n
where v = (ormo)1/2 and n = r
• Thus, the two fields are simply and intimately related for am EM
wave propogating in an isotropic medium
• Any process that alters Ex also intimately changes By in
accordance with the equation above
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 9.7
A plane EM wave traveling along k crosses area A at right angles to the direction of
propagation. In time t, the energy in the cylindrical volume At (shown dashed) flows
through A.
• As the EM wave moves in the direction of the wavevector, k,
there is an energy flow in this direction.
• The wave brings electromagnetic energy.
• A small region of space where the electric field is Ex, has an
energy density (energy/volume)
• The same goes for the magnetic field By
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Energy Density in an EM Wave
As the EM wave propagates in the direction of the
wavevector k, there is an energy flow in this direction. The
wave brings with it electromagnetic energy.
The energy densities in the Ex and By fields are the same,
1
1 2
2
 o  r Ex 
By
2
2 mo
The total energy density in the wave is therefore orEx2.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 9.7
• Suppose an ideal “energy meter” is placed in the path of the EM
wave so that the receiving area A of this meter is perpendicular
to the direction of propogation
• In a time interval t, a portion of the wave of spatial length v t
crosses A
• Thus, a volume of A v t of the EM wave crosses A in time t
• The energy in this volume consequently becomes received
From Principles of Electronic Materials and Devices, Third Edition , S.O. Kasap (© McGraw-Hill, 2005)
Poynting Vector and EM Power Flow
If S is the EM power flow per unit area,
S = Energy flow per unit time per unit area
( Avt )( o r E x2 )
S
 v o r E x2  v 2 o r E x B y
At
In an isotropic medium, the energy flow is in the direction
of wave propagation. If we use the vectors E and B to
represent the electric and magnetic fields in the EM wave,
then the EM power flow per unit area can be written as,
S = v2orEB
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Poynting Vector and Intensity
where S, called the Poynting vector, represents the energy flow per
unit time per unit area in a direction determined by EB (direction of
propagation). Its magnitude, power flow per unit area, is called the
irradiance (instantaneous irradiance, or intensity).
• The field Ex at the receiver location varies sinusoidally which
means energy flow is also sinusoidal
• If we write Ex = Eo sin(t) and then calculate the average
irradiance by averaging S over one period, we would obtain
average irradiance:
I  Saverage  v o r E
1
2
2
o
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Average Intensity
Since v = c/n and r = n2 we can write
I  Saverage  c o nE  (1.3310 )nE
1
2
2
o
3
2
o
The instantaneous irradiance can only be measured if the
power meter can respond more quickly than the
oscillations of the electric field. Since this is in the optical
frequencies range, all practical measurements yield the
average irradiance because all detectors have a response
rate much slower than the frequency of the wave.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)