Download Lecture1

Lecture1
Introduction
High Speed Semiconductor Devices
Where are they used?
At Microwave/Millimeter Wave/RF/Optical frequencies
For

COMMUNICATION
-Point to Point
-Broadcasting



SIGNAL PROCESSING
INTERCONNECTION
NETWORKING
Examples: RADAR, Satellite Communication, Mobile Communication, Metropolitan and
Wide Area Network
Using high speed devices



Oscillators
Amplifiers
Mixers/Modulations/Demodulation
For
Amplifiers/Oscillators/Filters/Mixers/Modulators

MESFETs, BJTs, HEMTs, Klystrons, Magnetrons, IMPATT Diodes, and Gunn
diodes.
Detector/Mixer

Schottky diodes, PIN diodes, Varactor diodes.
Now what makes a device high speed device?
Or
Why do we need special device considerations at high frequency?
Why cannot we use a BJT from a basic electronics lab, use it to make an
amplifier at 10 GHz?
Why a special BJT?
Well let us look at a simple circuit
Fig.1.1
Fig.1.2

no charge on C for t < 0

V in = unit step at t = 0



RC small
steady state is reached more quickly
We can go for frequency domain analysis to understand this in another way
Fig.1.3
Cut off frequency
As RC decreases,
high speed.
increases and circuit becomes capable of handing high frequency –
So high speed/frequency
Same idea in devices also
low RC
must reduce RC in devices.
There is a different delay mechanism know as transit time delay. We assume in normal
circuits that as soon as the voltage (Electric Field) is applied, current flows immediately
in the external circuit. Nature is, however not so kind. The carriers which are influened
by the electric field has to start from one terminal of the device and reach the other
terminal to show any observable current in the external circuit. Due to finite velocity of
the carriers, there is a delay, called the transit time delay as will be explained in the
following section.
Resistivity and cut off frequency
Think about a piece of semiconductor carrying current I .
Fig.1.4
A = area of cross-section
L = length
= resistivity
We know I = JA, J is the current density.
v = drift velocity, = mobility for (n) electrons and (p) holes, respectively and q the
electronic charge.
Hence,
Therefore, cut off frequency associated with this device is:
So to increase
we must increase
q and
are fundamental constants for Si.
If
So we must increase mobility and therefore drift velocity .

Increase doping. However, increase in doping reduces mobility & velocity.
Therefore optimization needed.
We need to study the basics of solid state physics to understand about
mobility, carrier concentration, etc.
Lecture2
To study this – we will review





Simple quantum mechanics
Energy band theory
Density of states for carriers
Recombination and generation of carriers
Carrier transport
Quantum Theory
Started in 1901 when Max Planck explained blackbody radiation.
Solid objects emit radiation when heated.
Black body radiation curves showing peak wavelengths at various temperatures:-
Fig.1.5
For more insight, play with Java Applets of Black body radiation from the following
links:
1. http://www.lon-capa.org/~mmp/applist/blackbody/ black.htm
2. http://webphsics.davidson.edu/alummi/MiLee/java/_mjl.htm
Many attempts were made to explain the Black Body radiation pattern i.e, RaylieghJeans, Wiens.
Fig.1.6
Finally Planck solved it.
Vibrating atoms in material could only radiate or absorb energy in discrete packets.
For a given atomic oscillator vibration at a frequency Hz. Planck postulated that the
energy of the oscillator was restricted to quantized values.
and
is the circular frequency
h = Planck's constant =
One quantum of energy
velocity of light in free space.
, where
= Wavelength (m) and ‘c' is the
Bohr's theory of atoms
To explain why atoms when energized (heat, electric discharge) emit discrete spectral
lines
, i = integer as shown in Fig.1.7
Fig.1.7
Bohr said that electrons in an atom move in well defined orbits, each orbit has a fixed
energy level and angular momentum.
Hydrogen atoms, atomic No. = 1, and has only one electron
Angular momentum of this atom
= electron rest mass,
= linear electron velocity
= radius of the orbit for a given n
Now at equilibrium, centrifugal force = electric force
for n th orbit
Total energy
Light energy emitted by a Hydrogen atom when heated or excited
discrete in nature and equal to
as shown in Fig .1.8
Fig.1.8
Quantum mechanics
To extend Bohr theory to He, Li, we need Quantum Mechanics which requires
Wave Particle Duality of De Broglie
Total energy
, where m is the mass of a photon.
Therefore, photon mass =
Photon momentum
and
.
Therefore an argument could be made that an electron with a mass
also has a wavelength , and can then
and momentum
be represented by wave, where for this particle momentum p can be associated with a
wave of wavelength
, where particle momentum
Schrödinger and Heisenberg
electrons, etc.
quantum mechanics to describe small particles like
Postulates of Quantum Mechanics
• There exist a wave function
variables and t is the time. From this
in system, is complex.
• The
of a particle where x, y and z are the space
one can find the dynamic behaviour of a particle
for a given system is determined by Schrödinger equation:
, where m is the mass of the particle and
the potential energy operator for the system.
•
and
is
must be finite, continuous and single-valued for all x, y, z and t.
•
probability that the particle is in a spatial volume element dV. So,
, integration over all
space.
• A mathematical operator
can be associated with each dynamic system variable i.e.
position, momentum, velocity, etc.
The expectation value, when this operator operates on
As examples, the space variables x, y, z are given as
The momentum
are given as
is given as
where
Similarly
Energy E is derived with an operator
Therefore the expected value of energy from postulate 5 is
If energy = E 0 (constant)
or
or
Schrödinger equation.
This is just like time harmonic case for EM field where the time dependence is
Therefore, now Schrödinger equation can be written as
or
where .
This is known as time independent Schrödinger equation and is much easier to
solve.
A few simple problems will be looked into the next lecture such as:
• Free Particle.
• Particle in a potential well.
• Infinite well.
• Finite well.
Lecture3
Solution of Schrödinger Equation
Free Particle
A particle (electron), alone in the universe.
Particle
Alone
mass m and a fixed total energy E.
no force on the particle.
constant potential energy everywhere
U(x, y, z) = constant, say zero
Let universe = one Dimensional only x variation
Time independent Schrödinger equation
which can be written as
or
, a one dimensional differential equation.Where
or E has a parabolic dependence on k. General solution for the
simple equation can be written as
unknown constants.
where
Therefore, total wave function (space and time dependence)
Compare this with a time-harmonic electromagnetic wave in free-space where
k = constant of propagation =
are
Hence wave function of free particle consists of a traveling wave. If particle moves in+x
then
and
Where
Normalizing,
= constant for all values of x.
Thus the probability of finding the particle in any dx is equal/same
If we take the universe to be infinite
Probability = 0
If we take the universe to be finite but large
no such difficulty
Now momentum
operator =
Therefore, expected value of momentum
Same as classical. Therefore,
DeBroglie relationship
Fig.2.1
Particle in an infinite potential well
Particle of mass m and fixed total energy E confined to a relatively small segment of one
dimensional space between x = 0 and x = a. In terms of the potential energy we can view
that the particle is trapped in an infinitely deep one dimensional potential well with
constant for 0 < x < a
Fig.2.2
Now time independent Schrödinger equation is
Boundary conditions
Solving
which can also be represented as
A + exp (+jkx) + A - exp (jkx)
Since
, as
vanishes for all x, so
discrete values.
Therefore,
for
modes or eigen functions
, Energy
So particle can have only a few discrete energy states/levels
Fig.2.3
Wave function
standing wave
particle is bouncing back and forth between the walls of the potential well.
average value of momentum at any x is 0. So to calculate momentum we have
to isolate forward wave going in +x or backward wave going in –x.
For +x wave,
on the continuous
and -x going wave
parabola of a free particle.
Discrete
points lie
Fig.2.4
The integer n is called the quantum number
Value of
from the equation that
Lecture4
Finite Potential well
Fig.2.5
U(x) = 0 for 0 < x <
a
U(x) = U 0 otherwise
We assume a particle with energy
For 0 < x < a,
For x <0,
For x > 0,
confined to the potential well
The three sets of solution are
.....(1)
Now we set up the boundary conditions.
Far from the potential well, wave functions must go to zero as the probability of finding
the particle away from potential region = 0
Therefore,
must be continuous at well boundary
must be continuous at well boundaries x = 0 & x = a
From boundary conditions
And we get four simultaneous equations
elimination
and
we obtain
if
identically.
Therefore,
or
This is a transcendental equation
We define a dimensionless quantity
is constant of the system.
and
from definition of k &
Eigen value equation becomes
.
Fig. 2.6
Let
then
if
intersect
at one points
intersect at 2 points
intersect
at 3 points
So for,
there are m points.
In the limit
(E finite).
.
Finite potential
In finite potential wells, we talked about Eigen value equation
(1)
for a given system constant
We have a discrete no. of solutions where LHS or RHS of (1) intersect.
Fig.2.7
Wave Function of Finite Potential Wells
For each intersection a value of
particular
and
outside the well of potential
One can clearly see from this that there is a finite probability of existence outside in the
classically forbidden region (classically a particle with an energy
cannot exist
outside the well). If the potential well is slightly modified as in the figure below.
Fig.2.8
The wave function will be nonzero in region C also.
Thus the particle will have a finite probability of existing or coming in region C past the
potential barrier B
In C the particle can appear as a free particle.
This quantum mechanical phenomenon of passing through a barrier is known as
tunneling. The wave function is different inside and outside.
Therefore there exists a finite probability of reflection at the well walls, called quantum
mechanical reflection at the well walls. In analogy to optics it may be looked at as two
partially reflecting mirrors, where an infinite potential well could be visualized as two
100% reflecting mirrors.
It is used in Tunnel diodes and operation of many other solid-state devices. One of the
usage of this phenomenon for high frequency decive is called Resonant Tunnelling Diode
(RTD), which would be used as an oscillator and even as an amplifier.
Lecture5
LINEAR HARMONIC OSCILLATOR
A harmonic oscillator is a particle which is bound to an equilibrium position by a force
which is proportional to the displacement from that position.
Thus we have,
(1)
where
is the spring constant.
The potential is expressed as,
(2)
The linear harmonic oscillator can then be visualized on a mass connected to a spring of
spring constant on shown in Fig. 2.8
Fig.2.8
The time independent Schrödinger equation is given by,
or,
To solve equation (3), we consider a dimension less quantity,
(4)
and
(5)
using (5) and (4)
(6)
For large values of y we can neglect
we get equation (6) on,
(7)
Equation (7) is satisfied approximately by the solution,
(8)
Substituting equation (8) in (7) we get,
(9)
This indicates that equation (8) satisfied equation (7) approximately and hence we
consider the exact solution as,
(10)
Putting the value of
from (10) in (6)
(11)
The trick next, is to linearize the above equation.
Equation (11) can be solved by using the power series method.
Let the trail solution be,
(12)
(13)
(14)
(15)
Putting equation (13), (14) and (15) in (11),
This equation must hold for all values of
of
, and therefore the coefficient of each power
must vanish separately.
This gives in the recursion relation between
and a n ,
(16)
It seems that knowing
(16),
and
can be calculated by using equation
Thus we can write equation (12) as,
(17)
If in the equation (16)
. But since
should be zero for some value of the index n, then
is a multiple of
so on, all the succeeding coefficients
which are related to
by the recursion relation (16) would vanish, and one or the other
bracketed series in equation (17) would terminate to become a polynomial of degree n.
This occurs, when,
or,
(n = 0, 1, 2 ..) (18)
Energy Quantization :
We have obtained the condition when the wave function is acceptable as
(n = 0, 1, 2 . .
.)
again
was
(19)
The variation of the energy levels is shown in the Fig.2.10
Fig.2.10
Reference:
1. R.F. Pierret, Advanced Semiconductor Fundamentals, ( Volume VI in Modular Series
on Solid devices),
Addison Wesley Publishing House Reading , MA , 1989.
2. Solid State and Semiconductor Physics, 2 nd Edn. J.P.Mckelvey, 1966, Harper and
Row, New Y
3. Introduction to Solid State Physics, 7 th Edition by C. Kittel (John Wiley & Sons,
1996).
Lecture6
Energy Band Theory
Semiconductors Si, Ge, GaAs, InP, CdTe…. - all are crystalline solids.
This is a 3D regular arrangement of atoms called the lattice. Some elemental
semiconductors such as Si & Ge have only one kind of material at the lattice sites, where
as GaAs, InP,…are compound semiconductors and are composed of two interpenetrating
3D lattices.


phycomp.technion.ac.il/~sshaharr/nonbravias.html
www.chem.lsu.edu/.../MERLO/flattice/20zns.html
There may be defects in the lattice and each atom in a lattice vibrates.

http://simple-semiconductors.com/
Defects and vibrations are second order phenomenon which would be discussed
separately.
If we have a charged ion at x = 0 and an electron outside the attractive force between the
ion and the electron is
The PE of the
and PE =0 at
Fig.
3.1
If we consider two ions separated by ‘a', the potential profile in on shown in Fig.3.2
Fig.3.2
To get a feel for the electron energies in a crystal semiconductor let us consider an
electron in a 1D lattice of atoms. The simplest lattice structure is a 1 D lattice.
A regular array of atoms placed periodically.
Fig.3.3
At each lattice site (x = 0, a, 2a, ….) there exists an ion with some net charge
Bloch Theorem
Relates value of wave function within any unit cell of a periodic potential to an
equivalent point in any other unit cell.
Concentrate on the behavior of any unit cell in the whole array. For 1D system the Bloch
theorem says that if
(1) if U(x) is the periodic such that U(x + a) = U(x)
then
(2) Equivalently,
where
= wave function for an unit cell. and
Boundary conditions are imposed at end points of the periodic potential.
Now, the wave number k in periodic potential set has several properties as:
• It can be shown that and only two distinct values of k exist for each and every allowed
values of E i.e.
.
• For a given E, values of k differing by a multiple of
give rise to one and same
wave function solution. As k is periodic or multiple-valued with a range
we take
range
. Usually
.
• If the periodic potential is assumed to be in extent, running from
to
then there are restrictions on k. k can take a continumm of values. k must be real
otherwise exp(jkx) or thus
will blow up at either
or
.
• In dealing with crystals of finite extent, information about the boundary conditions on
crystal surface may be lacking. To avoid this we assume a periodic boundary condition
assuming the lattice is a ring of N atoms.
Fig. 3.4
Therefore,
or
So for the 1D lattice of ions the periodic set of finite potential wells, for large array x may
be assumed to go from
.
Thus for a finite crystal k can only assume a set of discrete values, but as N is large
therefore one has closely spaced discrete values.
Lecture7
A simplified picture of the periodic potential is given in Fig. 3.5
Fig. 3.5
It is quite similar to the finite potential analysis
periodicity = a + b
, where k is continuous
We will first consider the case: 0 < E < U 0
Inside well wave function =
Outside well wave function =
Schrödinger equation for 0 < x < a:
Schrödinger equation for -b < x < 0:
The general solutions to equations (1) and (2) are:
Now applying the continuity condition on wave function and its derivative at x = 0
Fig. 3.6
Fig. 3.6
continuity requirement
Note that x = -b is the same boundary as that of
x = a.
Now wave function and its derivative must observe Bloch Theorem
With
periodicity requirements.
Similarly,
Applying Boundary and Periodicity conditions
Eliminating
and
by using first two equations in last two equations
2 eqns in 2 unknown constants. For non trivial values of
formed from coefficient should be equal to zero.
, the determinant
Lecture8
Eigen value equation
We get the following eigen value equation by simplifying the determinant:
Now, say:
(system constant)
(normalized energy)
For
Putting these values in our eigen value equation:
Now consider the 1D crystal lattice.
Fig 3.3
does not necessarily mean that the electron is outside the crystal lattice. Therefore,
we can also consider the case of
here.
For
Putting these values in the eigen value equation:
Now we assumed infinitely long lattice in K-P model. Therefore, k can take any value.
RHS of eigen value eqn.
LHS
The plot of
with value
can thus take any value between +1 and -1. The
between +1 and -1 gives allowed solutions for
as a function of
.
is shown in Fig 3.7
Fig
3.7
Note that the allowed values of
are shown by the shaded regions where
Therefore there are allowed bands of
i,e. allowed bands of E(energy bands).
.
Lecture9
Energy in Brilloun Zone representation.
We learned about energy bands
E or
values for which
has a
solution and between these bands where it is not possible to find a value of E or are
called forbidden gaps. Discrete number of such bands are separated by band gaps.
Now for
If we plot the allowed values of energy as a function of k, we obtain the E-k diagram for
the one dimensional lattice.
Brillouin zone
When we plot the expanded E-k diagram of Periodic potential perturbation we notice the
dissimilarity with the free space E-K diagram (given by the dotted line),
Free Particle Solution
How in particular can the periodic potential solution with an adjustable k approach the
free particle solution with a fixed k in the limit where E > >U 0 ?
In this regard it must be remembered that the wave function for an electron in a crystal is
the product of two
and Q(x) where Q(x) is the wave function in the unit cell. Q(x) is
also a function of k. Increasing or decreasing k by
in such a way that the product of
modifies both
approach the free particle.
The k-value associated with given energy band is called a Brillouin Zone.
1 st Brillouin Zone
2 nd Brillouin Zone
and Q(x)
One way of drawing is to k between
in basically
eigenvalue equation you notice that increasing or decreasing
range. In the
by
no effect on the allowed electron energy of E(k) is periodic with a period of
has
.
Fig. 3.8
Fig.
3.8
Therefore all the electron energies can be represented within
, by
changing the k values by
, where n is an integer. This representation of the electron
is called the reduced Brillouin zone representation as shown in Fig. 3.9 where the bands
of energies are identified. As the number of electrons in the system increases the bands
starts to be filled up from the lowest available energies. Normally most of the bands are
completely full of electrons as allowed by Pauli's Exclusion Principle. At low
temperature, there could be a band completely empty. The one below it is usually
completely full, called the valence band. None of these electrons can now conduct
electricity. If now a condition arises that some of the electrons from the completely filled
band can be excited into the completely empty band, then current can be conducted by
the electrons in the empty band called the conduction band.
Also according to the Bloch theorem there are two and only two k values associated with
each allowed energy, one for the electron moving in the +ve direction and the other for
the electron moving in the –ve direction.
Also note that
at
& k = 0. This is a property of all E-k plots.
Fig.
3.9
Lecture10
Bloch parameter k:-
For free particles k = wave number =
expected value of momentum
For a particle bound to a periodic potential or crystal ,
= crystal momentum.
is not the actual momentum but the momentum related to the constant of motion
which incorporates the crystal interaction.
This crystal momentum parameter k is also periodic with a period of
. The E-K
diagram is therefore the Energy versus crystal momentum characteristics of an electron in
the crystal.
Energy Band Solution
Energy band solution indicates only the allowed energy and momentum states but not
about time evolution of electron's position etc.given E, k gives possible values of position
of finding the electron with a certain probability i.e, position is uncertain.
Heisenberg Uncertainty Principle.
If E is known exactly, uncertainty in t is infinity, we can not find anything about the
electron's position.
Therefore for a particle motion we need wave packets
grouped about a peak energy.
constant E wave function
Probability of finding the represented particle in a given region of space = 1 for some
specified time.
Center of mass of a particle moving with a velocity
wave packet also a mass – QM idea
– classical idea.
Packet of traveling wave with center frequency and center wave number k then
describes the particle motion represented by this group.
group velocity,
E, k gives the center values of energy and crystal momentum.
Lecture11
Effect of External Force
External force F acting on the wave packet would be any force other than the crystalline
force associated with periodic potential.
External force may arise from dopants or external electric field.
Force F acting over a short distance dx
energy increases by
work on the wave packet
we know Rate of change of momentum =
= Mass x Acceleration
=
therefore, Effective mass
At k = 0, curvature of (b) band is more than the curvature of (a), i.e.
Fig.4.1
wave packet
Fig.4.1
Heavy mass
slower movement, larger transit time.
Mobility of a carrier
curvature of
seen in Fig.4.2
Fig.4.2
Now consider the band segment of the Kronig-Penny model as shown in Fig. 4.3
Fig. 4.3
near the bottom or minimum of bands.
near the top part of each band.
For
In response to applied force the particle/electron will accelerate in a direction opposite to
than expected from purely classical calculation.
In most cases we do not know the exact equation of the E-k diagram (difficult even for
the Kronig-Penny model) but it been found that top or bottom of the band edge/Brillouin
zone edge of the E-k relation is approximately parabolic as shown in Fig. 4.3
where A is constant.
Therefore,
band.
constant at the edge of the Brillouin zone/top or bottom of an energy
Therefore,
band.
constant near the edge of the Brillouin zone/top or bottom of an energy
Current Flow
If N atoms are there in 1 dimensional crystal then the distinct k values in each band = N
spaced apart by
For the sake of discussion we assume each atom gives 2 electrons to the crystal
a total
of 2N free electron in the crystal. 2N electrons will be distributed among the available
energy states.
At temperature
bands).
Fig. 4.4
2N electrons will fit the 2N states in the lowest 2 bands (valence
Fig. 4.4
At room temperature sufficient thermal energy is there and a few electrons from the top
of the 2 nd band will move up to the bottom of the 3 rd band as shown in Fig. 4.5.
Fig. 4.5.
If a voltage applied to the crystal a current will flow through the crystal.
4 th band – no electron at room temperature – no current as totally empty bands do not
contribute to the charge transport process.
Concept of Holes
Now consider the 1st band where N states are filled by N electrons at room temperature.
With external voltage, electrons will move with velocity
symmetric about
. For every electron with a given
another electron with the same
Thus first band
But the band is
direction, there will be
in –x direction.
no current,
So totally filled energy band do not contribute to the charge transport process.
Thus first band
on current,
So totally filled energy band do not contribute to the charge transport process.
Asymmetry
applied electric field.
For nearly empty third band current
where L
length of 1D crystal and
For the nearly filled second band current
This summation is more difficult to evaluate as we have very large number of states
filled.
Now if the band was completely filled,
Therefore, we can write
This current is the same as if a positively charge particle is placed on the empty states and
the remaining states are unoccupied.
Overall motion of electrons in nearly filled energy band can be described by the empty
electronic states provided that the effective mass of the empty states is taken to be
negative of
, i.e.
Now in this example empty energy states are at the top of the bands
effective mass
there. Therefore, Empty energy states at the top of a band
+ve charge with +ve
Normally the electron or holes stay mostly at the edge of a band (holes in top of Valence
band and electrons in the bottom of conduction band) where parabolic band
approximation can be applied. So in semiconductors electrons and holes generally have
constant effective mass and the classical treatment/behaviour is a good approximation.
Valence Band
1. Maxima occurs at zone center at k = 0
2. Valence band =
three sub bands
Two bands are degenerate (have same allowed energy) at k = 0. Third band max
at reduced energy k =0.
1. At k = 0, the shape and curvature are orientation independent.
Conduction Band
1. Somewhat similar, but minimum where electrons will gather
material to material.
2. Ge C Band minima at zone boundary (8 such min)
varies from
3. Si C Band minima at
from the zone center.
4. GaAs C Band min at k = 0. Direct bandgap good for light emission
conservation of momentum transition from V Band to C Band.
5. 3D structure
is tensor.
Band gap Energy
It is the range from Valence Band maximum to Conduction Band minimum.
At
Si
GaAs
A decrease in T results in a contraction of crystal lattice
stronger atomic bonds
increase in Band gap energy
is a good model
E G 0)
Ge
0.7437 eV
235
Si
1.170 eV
636
GaAs
1.519 eV
204
Lecture 12
Mobility
Mobility is the measure of ease of carrier motion within a semiconductor crystal. Motion
is impeded by collision which results in decreased mobility.
Current Density
holes
electrons
where E is the applied electric field and
is the mobility. q the
electronic charge and n & p are the respective carrier concentrations.
For Si with
at
k,
and
Where GaAs with
at
k
and
Lattice Vibrations
Consider the ID lattice Fig.5.1
Fig.5.1
Hooke's law is obeyed exactly
If one atom is displaced by x from equilibrium position
Potential energy of the atom is a function of distance from neighboring atoms Displaced
position potential energy is changed by
(1)
potential energy when distance from neighbour =x
Expanding
term
in Taylor series about a and keying only 1st significant
potential function of a harmonic oscillator
But this
assumption that only one atom oscillates since atoms are bond to each other we expect
that vibration of one atom in lattice will set other atom to vibration
let
displacement of
and
atom from equilibrium
Fig.5.2
Fig.5.2
l = integer
force acting on l th atom, a distance from adjacent atoms and for two adjustment atoms
putting it to themselves.
Force F on l th atom
2 nd law to
atom
…............(i)
Then (i) become
wave equation (ii)
Complex oscillation possible
let regular frequency =
from equation (i).
where q = wave no.
proper Sign should be chosen to make
Fig.5.3
Fig.5.3
Classical single harmonic oscillator there is a single frequency
Lattice vibration are characterized by a continum of frequencies with a limiting
maximum value
But repetition is there for
or
So Brillouin zone concept is coming out.
Now We consider a Crystal of finite member of atoms N
Boundary condition for the displacement of lattice atom at the end of crystal may be
taken periodic boundary condition to be no displacement.
Periodic boundary condition
The lattice vibration we talked about applicable to material like
in which all atoms
identical and one atom/unit cell. Material with 2 atoms/unit cell or 2 types of atoms/unit
cell=Semiconductors.
Double periodicity
Fig.5.4
Fig.5.4
Two equations describes the vibrations (2 atom 2 period case)
Right hand side is wavelike
Oscillation Frequency
Frequency =
and wave vector
Then
Assume time harmonic solution.
A&B constants
Substituting
2 equations 2 unknowns,
or
For nontrivial A & B
Dispersion relation between
Solving for
where,
Two values of
for any wave number q.
Fig.5.5
Solving for
When
constants.
The ratio of amplitude
for lower branch.
Vibrations of neighboring atoms are in phase Similar to those when an acoustic wave
propagation in the crystal, optical mode lower frequency.
3-D crystal: 2 distinct modes of transverse vibration also along with optical or acoustic
longitudinal mode
Actual vibration
If we associate a vibration mode of wave number q
quantum mechanical wave function
Schrödinger equ. For a harmonic oscillator each mode of vibration different energy state
with energy give by
particles Picture of lattice vibrations
Each mode of vibration
number of particles each with energy
State of vibration changes by creation or annihilation of such particles
q = Pseudo momentum
because of periodicity
Thus the vibration of the lattice with a wave vector
collection of particles called
phonons which associated by the energy
and pseudo momentum
. At a particular
temp. Lattice atoms execute random vibrations and phonons of various energies &
characteristics would be present.
Hamiltonian of harmonic oscillator
Hamiltonian of lattice system is
and normalized
is
Kronig penney model with stationary lattice atoms is not entirely accurate as lattice
vibration Hamiltonian of vibrating lattice + electrons
individual electrons and lattice atoms
The eigen value equation is
eigenvalues are modified energy instant due to changing position of lattice atoms
is the periodic component of Block function out of perturbed lattice.
Lecture 14
Lattice Scattering for Mobility
The position of the conduction band and valance bond extrema in a semiconductor
depends on the Lattice spacing. These spacing. changes during vibration and there is
perturbation in the
and
.
Fig.5.6
Fig.5.7
Effect of lattice vibration is small for bound electrons with lower energy. But the effect is
more on conduction electrons which are loosely bound to atom/nucleus.
Acoustic Vibration Compression & expansion in lattice crystal. In compressed position
the Energy band is and altered, So that forbidden gap is increased. If expanded forbidden
band gap width decreases.
Potential Energy out of electron-electrons interaction is neglected as it is small under
adiabatic condition.
The incident wave
Electron may be reflected at step
or it may be transmitted.
Fig.5.7
Fig.5.7
Reflected
perfectly elastic collision Transmitted over the barrier
But lattice wave causing
moving with acoustic velocity Doppler shift which make
reflected momentum different from incident momentum – can be neglected assuming
electron moves with high velocity.
Transmitted looses energy.
is continuous at
We also have
is continuous
or
is continuous
b or
Deformation Potential and Mean force path
with step height
to be small
is related to strain as
= deformation potential constant
= shift of conduction per unit dilational strain.
Since thermal energy is the major come of vibration
maximum pressure coming volume change
constant.
. If compressibility
In a distance
probability of scattering or reflection
= mean free path
= linear dimension of volume
If an electron is in state
after a perturb potential
Probability of transition from
the probability that it will be in state
after time t
is applied
per unit time is Transition prob. is max if
. In collision of energy is conserved. A real change of electron state from
to
may
occur if energy associated with lattice vibration may change by creation or absorption of
a phonon.
The jump –
1st approximation
strain is volume
Shift of conduction band edge per unit strain
Acoustic phonon scattering
Strain is produced only be longitudinal vibration shear associated with transverse
vibrations does not affect energy eigenvalues.
Passage of longitudinal classic wave/phonon
= coherent regions of compression/extension which are of the order of
in linear extent
l = length of disturbance.
Volume
of this linear extent is subject to a dilatorily stress producing a max pressure
and volume change
stored Strain energy =
Source of strain energy is thermal then stored strain energy
Or
Compressibility
Reflection coefficient
In a distance
Where
probability of scattering is
= mean free path
In going a linear dimension
the probability of reflection
Assume
is independent of velocity
mean free time between scattering/collisions.
From Boltzman's Theory of free particle in a gas.
above
Shockley has shown from a full Quantum Mechanical treatment that
where C 11 = clastic constant or young's modulus for longitudinal extension <110>
The mobility
in
Due to lattice Scattering
Valence band deformation potential constant.
Conduction band deformation potential constant
Much more rigorous analysis is possible with perturbation theory and 3 possible models
• Deformable lattice ion model
the atom remains the same
• Rigid ion model
change distribution gets distorted but the position of
ion if gets displaced the potential gets displaced.
• Deformation potential model
displaced.
potential membrane gets distorted if the ion gets
Lecture15
Scattering in semiconductor
1.
2.
3.
4.
5.
Phonon or lattice vibration scattering
Ionized (dopant) impurity scattering
Scattering by neutral impurity atoms and defects
Carrier-Carrier scattering
Piezo electric scattering
Of these, (1) and (2) dominates. For most high speed devices where large carrier
concentrations area not involved.
Where the mobility due to each scattering mechanism
written separately, the overall
mobility is
For Phonon or lattice vibration scattering,
C 11 = average longitudinal elastic constant of semiconductor,
of the carrier,
= the effectives mass
E ds = displacement of edge of the band per unit dilation of lattice, and T = the absolute
temperature.
For lon scattering
N 1 = ionized dopant or impurity density
= permittivity
Therefore overall mobility is
only for GaAs
Mobility parameters
Specifically for Si,
N = dopant concentration i.e
at room temperature
are determined experimentally
Parameters for Si:
1358
461
1352
459
1345
458
1298
448
1248
437
986
378
801
331
Parameters for GaAs:
0.4
1100
7100
0.542
0.8
200
8000
0.551
0.9
100
8100
0.594
Temperature Dependence
Again from experiment we find
all have a temperature dependence of
the form
For Si we have:
Electrons
Holes
N ref (cm -3 )
2.4
92
54.3
-0.57
1268
406.9
-2.33(electrons),-2.23(holes)
0.91
0.88
0.146
GaAs behavior slightly different
High field effect
We know that drift velocity
But linear proportionality is valid only at low temperatures.
For E field intensity
Nonlinear behavior is seen.
and E are no longer directly proportional.
Now geometry is often small (submicron) so even with 1 V across
So mobility concept may fail in such high field zones.
For very high E field
saturates.
dimension,
For Si at 300 K,
for both electrons and holes.
Model,
For GaAs
initially decreases with E after a critical field is 2
slowly increases, In a region of high
velocity
10 3 v/cm and then very
which can be considered as saturated
.
However, in GaAs for holes the velocity saturates and
Independent
of E field. The variation of the drift velocity with electric field for holes and electrons are
shown in Fig 5.8 and 5.9
Fig
5.8
Fig
5.9
Intervalley Electron Scattering
GaAs conduction band minimum is at
Secondary minimum of conduction band is at L (<111>)
L valley is sparsely populated at room temperature.
With E field, valley electrons gain energy between scattering events. If valley electron
gain energy
intervalley transfer becomes possible and the population at L
valley becomes enhanced at the expense of valley.
At center of the
valley
. At L valley
. Effective mass
increases in L valley mobility and Drift velocity decreases in L valley as
Lower valley mobility
population
; upper valley mobility
total electron
.
Arrange
is strong function of E.
is strong function of E.
for
Lecture16
Carrier Density
Density of states
Between the energies E 1 and E 2
no. of allowed states available to electron/holes in
the cited energy range per unit volume of the crystal.
From Band theory
difficult
A good approximation can be made from band edges, the regions of bands normally
populated by carriers.
Fig.6.1
For electrons near the bottom of C Band the band forms a pseudo-potential well. The
bottom lies at E c and termination of the band at the crystal surface forms the walls of the
well. The energy of electrons is relatively small compared with surface barriers. One can
think being in a 3 Dimensional box. The density of states at the band edges density of
states available to a particle of mass m* in a box with dimensions of the potential box
Schrödinger equation
Consider a particle of mass a m and total energy E. Size of box
U(x, y, z) = Constant everywhere = 0
Time independent Schrödinger equation,
(1)
(2)
Solve using separation of variables
Put in equation 1.
(3)
Since k is a constant
such that
Thus
for x = 0, x =
a
Thus
Similarly
(no. of modes in a
wave resonator)
Thus
Such that
where
area integers.
Thus a few discrete energy values are allowed inside.
Allowed solutions/energy/levels
If abc is large, small increments in to
large no. of states allowed
to
.
k space and draw.
Fig.6.2
Fig.6.2
k space vector
end points of all such vectors be dots.
k space unit cell of volume =
Thus
contains one allowed solution.
This is not complete.
Now for
= (-1, 1, 1). . . . . six other possible states for which E is same.
If one counts all points on the k space it is thus necessary to divide by eight to obtain the
number of independent solutions.
Thus
But electrons have 2 spin states
Thus
Now number of states with k value between arbitrary k and
k+dk
=
Now
Going from k to E space
No. of electron energy states with energy between E and
E+dE =
Density of energy states with energy between E and E+dE
Now for actual CB or VB densities of state near band edges m
If E c = min CB energy and
= average effective mass (
or Ge m is complicated.
= max VB energy
effective mass
Bias
for GaAs or compound semiconductors) for Si
Fermi Dirac statistics
- Assumption not all allowed states are filled
- Electrons are indistinguishable.
- Each state one electron (Pauli exclusion principle)
- Total no. of electrons = fixed
= constant, total energy
Electrons are viewed as indistinguishable “balls” which are placed in allowed state
“boxes”.
Each box one single ball.
Total energy of system is fixed.
Balls are grouped in rows
energy level
no. of boxes in each energy level
energy. Fermi function
E F = Fermi energy
k = Boltzman's constant =
For closely spaced levels
Continuous variable E
Fig 6.3
no. of states of allowed electronic states at a given
Fig
6.3
f(E)
occupancy factor for electrons at energy E.
g(E)
density of states at energy E
1-f(E)
occupancy factor for holes at energy E
Distribution of Electron
The distribution of electrons in conduction band
No. of electrons is CB with energy between E and + dE (E > E C )
The distribution of holes in the valence band =
The total carrier concentration in a band: conduction,
The total no. of holes in valence band,
If
for CB
and
for VB
[(Fermi Dirac integral of order 1/2)]
Where, N c = effective density of conduction band states, N v = effective density of
valence states.
At room temperature (300k) we have:
Semiconductor
Ge
Si
GaAs
Properties of
function
gamma function.
here
with a max error of
Thus
for
From Fig 6 (d)
is closely approximated by exp
Thus
Fermi level lies the band gap more than 3kT from either band edge
said to be nondegenerate.
Fig.6.5
Semiconductor is
Lecture17
Maxwell-Boltzman approximation
Simplified form
Maxwell-Boltzman distribution (energy distribution at high temp for
molecules in a low density gas.)
Intrinsic semiconductor
Thus we can write
Also we know
Where E G = Bandgap energy.
Fig 6.6 shows
vs T.
Carrier Concentration variation with Energy
It is the plot of allowed electron energy states as a function of position along a direction
Fig. 7.1
Carrier distribution vs. E for electrons can be graphed by multiplying g(E) and f(E).
Carrier distribution vs holes can be graphed by multiplying
This is shown in Fig. 7.2
and (1-f(E)).
Fig. 7.2
When we have an electric field energy band diagram, bands bend with x.
potential energy
Therefore,
Kinetic energy is (E- Ec) in conduction band and
Change Neutrality Equation
Maxwell's equation gives
semiconductor dielectric constant.
in valence band
Lecture18
Extrinsic Doping :
Imperities could be incorporated into the crystal which could either contribute extre
electrons or could accept electrons.In the former,afer the contribution of the electron/s the
impurity itself becomes positively charged(as it was neutral to start with)and into
concentration is called ND+.In the latter case the impurity after accepting electron/s
becomes negatively charged and its concentration is called NA -
Assuming uniform doping under equilibrium
, Donors produce +ve ions, Acceptors produce –ve ions.
if
, this is the charge neutrality relation.
Fig.6.6
Relationship for
and
Now let N D = no. of donor atoms
and
= no. of donor atoms ionized
is the ratio of no. of empty states to total no. of states in donor energy
g D = 2(standard value)
Similarly,
g A = 2(standard value)
In the above expressions, g D and g A are the degeneracy factors
From charge neutrality condition.
We have
solving this equation given N V , N c , N A , E c , E v , T, E D , E A
F etc
Free-out/Extrinsic T
Suppose N D > > N A and N D > > n:
electron concentration > > hole concentration. Also
.
we can find E
where
(a computable constant at a given T)
Therefore,
Solving this quadratic equation,
, (+ve root chosen as
)
or,
Similar result can be obtained for acceptor doped material.
is typically much grater than N D
, p doped Si, T = 300 k
n = 0.9996
, So 99.96% of P atoms are ionized at room temperature
At T = 77 K i.e Liquid N 2 temperature,
Special case of High Temperature
When a semiconductor is kept at a high temperature, most dopants area ionized
But we know that
Therefore,
and
donor doped
acceptor doped
Position of Ei
1) Exact position of E i
Intrinsic semiconductor
. In this case, n = p, and E F = E i
E i in Si is 0.0073 eV below midgap.
GaAs is 0.0403 eV above midgap, at 300 k.
2) Freeze out/Extrinsic T.
this is useful for low temperature
calculations.
3) Extrinsic
Therfore if the doping is a function of position, the bands will bend as shown in Fig. 7.2
Lecture19
Generation Recombination process in semiconductors
When a semiconductor is perturbed from equilibrium state – carrier numbers are
modified. Recombination generator process – order restoring mechanism – carrier excess
or deficit inside the semiconductor stabilized or removed.
Perturbation
optical excitation, electron bombardment
current injection from a contact
Drift and diffusion currents are also there along with process of recombination
Band to Band Recombination
Direct Thermal recombination
Fig. 8.1
Fig. 8.1
It is Direct annihilation of CB electron and a VB hole.
An electron falls from an allowed CB state to hole in VB.
The process of recombination is usually radiative with the production of a photon.
Band to Band Generation
Thermal or photon energy is absorbed
Fig. 8.2
Fig. 8.2
Band to Band Recombination
Photons are almost mass less
very small momentum
Therefore, photon assisted transition
vertical on E-k plot
In GaAs direct bandgap semiconductor, there is little change in momentum is needed for
recombination process to proceed. Conservation of energy and momentum is simply met
by release of photon.
Fig. 8.3
Fig. 8.3
In an indirect bandgap semiconductor there is a change of crystal momentum associated
with a recombination process. The emission of a photon will conserve energy but not
momentum. For Band to Band recombination in an indirect bandgap semiconductor to
proceed a photon must be emitted or absorbed - lattice vibration.
Fig. 8.4
Fig. 8.4
Band to Band process in indirect semiconductors is complicated. This means that a
diminished rate band to –band recombination in indirect semiconductors. In indirect
semiconductors R-G, center recombination dominates.
All info on band to band radiative recombination optical absorption coefficient
(Direct BG)
Fig. 8.5
.
Fig. 8.5
Photon
electron hole.
generation rate
Planck's radiantion law
Radiative recombination rate
Thermal equilibrium
Therefore,
The excess carrier Radiative recombination rate
Let
and
and
rate at which electrons and holes disappear.
has units of sec-1 and so we can define
as carriers lifetime (minority carrier)
Radiative recombination lifetime
n type semiconductor Donor
.
At high level injection
.
Ge
Si
GaAs
Gap
at 300K.
R-G Centers/Traps
Deep Energy level
Impurity Atoms
Crystal defects
allowed energy levels in the midgap region.
deep level states
from CB and hole from VB comes to the RG center and gets annihilated. It can also be
the considered on capture of having one electron from CB to deep state and then the same
electron can jump to VB canceling a
RG recombination
non radioactive
Heat/lattice vibration produced
Fig. 8.6
Fig.
8.6
Fig. 8.7
Fig. 8.7
Dominant mechanism in G- R in semiconductors.
rate of change of electron concentration due to RG recombination + generation
rate of change of holes
No of R-G traps/cm 3 filled with
No of R-G traps/cm 3 empty.
Total no of traps or RG centers per cm 3 .
A mechanism in trap recombination or generation.
Electron capture
no. of electrons to be captured
empty trap states.
more captured n
Constant.
The probability that the recombination-generation center is occupied by an electron
-ve sign to indicate that electron capture acts to reduce the no. of electrons in CB
units of cm 3 /sec.
= thermal velocity capture cross section.
Lecture20
Diffusion and Continuity Equations
Diffusion is a process, in which a particle tend to spread out a redistribute as a result of
random thermal motion migrating from regions of high particle concentration to low
particle concentration
to produce uniform distribution.
Electrons and holes are charged so when they diffuse
diffusion current.
Derivation of Diffusion current
Assumptions
• One dimensional only
• All carriers move with the same velocity
(in practice a distribution of velocity)
• The distance moved by carriers between collisions is a fixed length L. (L is actually
mean distance moved by carrier between collisions).
Randomness of thermal motion
equal no. of particles moving in +x and –x
Fig.9.1
Fig.9.1
Derivation
Within and
section
equal outflow of particles per second from any interior section
to neighbouring sections on the right and left. But because of concentration gradient the
no. of particles moving from right to left second is greater than no. of particles moving
from left to right.
Fig.9.2
Fig.9.2
Of the holes in a volume LA on either side of x = 0
Will move in proper direction so as to cross x = 0 plane
= holes moving in +x which cross x=0 plane in time
= holes moving in +x which cross x = 0 plane in time
But
= net no of +x directed holes with cross x = 0 plane in time
Net current cross x=0 plane.
We define
Generalizing
are constants in cm 2 /sec
Einstein relation
Fermi level inside a material is not dependent on position.
Under equilibrium, if there is no current.
non zero electric field is established inside a nonuniformly doped semiconductor
under equilibrium.
Under equilibrium drift and diffusion currents balance.
or
general form of Einstein relationship.
When
Then
Similarly
Continuity Equations
Current in semiconductor
Under ac and transit condition we need to add displacement current also.
Now in general we need to include carrier generation and recombination also.
There will be a change in carrier concentration within a given small region if
there is an imbalance between total carrier currents in and out of the region.
Recombination
Generation
For holes
Continuity equations.
Lecture21
Diodes
We use varactor diodes, GUNN, Tunnel, IMPATT, Schottky (metal+semiconductor) and
PIN diodes in Microwave Apllications such as
mixers (heterodyning) (RF-IF conversion) as in Fig 10.1
Fig
10.1
Detectors as in Fig 10.2
Fig
10.2
Switches
Lecture22
P-N junction Diodes
Abrupt p-n junction
Fig.10.3
Fig.10.3
Built-in potential=
At equilibrium
Thus
Thus
minority carrier in n side
minority carrier in p side
Thermal equilibrium E field in neutral regions=0
Thus Total-ve charge in P side-total + ve charge in n side
Fig.11.4
Fig.11.4
Poissons Equation
In n side
.
Integrating
for
for
Maxwell's field at x=0(junction)
Also we obtain
W=Total depletion region width
Eliminating
from previous 3 equations.
More accurate expression for depletion region width
If one side is heavily
doped the depletion region is in weakly doped place) Depletion –layer capacitance per
unit area
,
Incremental increase in charge per unit area for a voltage increase of
Inside depletion layer assuming the following
• Boltzman relation is an approximation
• Abrupt Depletion Layer
• Low injection (injected minority carrier < majority carrier)
• No generation current inside depletion layer
where
and
When voltage is applied the minority carrier densities on both sides are changed and
are imrefs or quasi Fermi level for electrons and holes (E Fn and E Fp ) under
nonequilibrium
condition as applied voltage/bias or optical field as shown in Fig. 10.4
……………..
Forward bias
Reversed bias
Now
=
Similarly
Electron and hole current densities is proportional to gradient of quasi Fermi level.
Now
applied voltage across the junction. Now at the boundary of
depletion layer at p side i.e. at
We have from (A) & (B)
Where
is the equilibrium electron density on the p side.
Similarly
Where
is the equilibrium hole density on the p side.
n side Continuity equation in steady state
Let
recombination rate
Then in 1-D we obtain
(steady
state)
for n side
Charge neutrality holds approximately
ultiplying first by
we get
and second equation by
and taking Einstein relation as
=lifetime
as
type
in n-
Low injection assumption
Neutral region
Now Boundary condition
(injection)
assuming large structure
exponential decay of electron and
hole current components in the Depletion region as shown in Fig. 10.5
Fig. 10.5
Diode Currents
Where
At
Similarly we obtain at the p side
…..Total current
…..Total current
as shown in Fig 10.6
Fig 10.6
Abrupt junction with P + doping voltage drop in p + is neglected
V = Constant for one sided abrupt junctions.
For Ge ideal equation is valid.
For GaAs and Si only qualitative agreement as shown in Fig 10.7
Fig
10.7
This variation is due to:• surface effects.
• generation and recombination.
• tunneling of carriers between states in the bandgap.
• High injection conditions.
• Series resistance effect.
Under Reverse Bias the generation current, for a generation rate of G per unit volume is
Therefore, one sided
diode (N A >>N D ) the reverse bias current
J R is dominated by the diffusion and Shockley equation is followed.
For Si and GaAs,
dominant.
may be small and the generation term may be comparable or
Under Forward bias the major R-G process in the depletion region is a capture process.
The Recombination current density is given by
where
is the effective recombination lifetime.
for
and
Lecture23
Diffusion Capacitance
Deflection layer capacitance-reverse junction when forward biased another contribution
from rearrangement of minority carrier density-Diffusion capacitance Applied voltage
Current
Forward Bias
Small signal amplitude
Small signal as component of hole density in n
For
.
Similar expression for electron density in p side.
Now for holes n in n side we know that continuity equation is S
Thus
Or
Non Time varying case
So here
effect of frequency dependence
Small signal current
admittance due to diffusion only
low frequency
low frequency
and
are function of bias frequency.
will dominate at high frequencies.
so deflection layer
capacitance
Transient response of Diode
The diode will not respond to the reverse voltage until excess minority carriers in neutral
n and p regions have been withdrawn.
Model of diode
If we apply reverse bias suddenly then the diode passes a reverse current higher than
reverse saturation current
for some t. The current then falls as the stored minority
carriers are withdrawn, eventually reaching
.
One can use charge storage model to understand this transient behaviour.
Consider a
is p only.
junction-depletion region on the n side then the essential minority carrier
Fig 10.7 depicts the hole distribution in neutral n-region.
Fig.10.7
Multiplying continuity equation by qAdx and integrating it over the entire neutral n
region from x n to w 1 , we get
which can be written as
where
is the excess minority carrier stored in a n region at any
time.
is hole current at x = w 1 and is related to the transit time and neutral n
region of width w 1
A time constant
is defined by the relation,
is related to the passage time through the neutral n-region of width W n , so we can
write
where
the switching trajectory is shown in Fig. 10.12
Fig.10.12
Now in storage phase
almost constant so the equation
has the solution
where
at t = 0 ,
hence
is a constant and is determined by the condition that
.
Now we assume a triangular hole distribution where the charge remains constant in the nregion at
, as depicted in Fig. 10.13
Fig.10.13
and
where
is the time till the diode remains forward biased
Thus
The stored charge is equal to the area of the triangle with base
, so
writing
and solving for
we get
Solving
More accurate derivation of involves solution of time-dependent continuity equation
with appropriate boundary conditions and is given as
Lecture 24
Varactor Diode Structure
The word varactor comes from variable reactor, it is a device whose reactance can be
varied in a controlled manner with a bias voltage. The symbol for this diode is
It is used widely in amplifier and harmonic generator, few examples are
-mixing
-detection
- variable voltage tuning
Let the doping distribution be
The abrupt doing profile is achieved by epitaxy or ion-implantation, where m=0
In the varactor diode one side is heavily doped and on the other side the doping and the
impurity concentration decreases with distance as shown in Fig
11.1[http://www.vias.org/feee/varactor_diodes.html]
Fig 11.1
[http://www.vias.org/feee/varactor_diodes.html]
where m is a number < 0 like -5/3, -3/2,-1 etc
Poission's equation in the n-side is given by
Applying the boundary conditions
and
(applied voltage)
the depletion width, we get the capacitance as
Where A is the area of junction and
Therefore the slope of this variation is given by
The variation of C vs. V is shown in Fig 11.2
(built-in voltage), where W is
Fig
11.2
If
then
(Hyperabrupt junction), when
then
(abrupt
junction)and when m=0 then S=1/2(graded junction). This is depicted in Fig 11.2
The larger the value of S, larger is the variation of
with biasing.
The equivalent circuit of varactor diode is shown in Fig 11.3
Fig
11.3
Where,
is the junction capacitance,
is the series resistance,
is the parallel
equivalent resistance of the generation-recombination current, diffusion current, and the
surface leakage current.
Both
voltage.
decreases with reverse bias voltage and
increases with reverse bias
The Quality factor Q of the varactor
The maximum Quality Factor therefore is
The variation of Q with frequency is shown in Fig 11.4
Fig
11.4
Therefore major considerations for a varactor diodes are (i) Capacitance, (ii) Voltage, (iii)
Variation of capacitance with voltage, (iv) Maximum working voltage, (v) Leakage
current and at high frequencies for a good varactor diode the equivalent circuit can be
represented with lumped elements as
Varactor application
Varactor voltage
, where i(t) is the current flowing, where,
This is difficult to solve and therefore the analysis is done in the frequency domain where
a set of coupled non-linear algebraic equation are solved. In this case
and
where m and n are harmonic numbers and P m,n is the average power flowing into nonlinear harmonics n
An important application of pumped varactors is the parametric amplifier. When the
varactor is pumped at a frequency F p and a signal is introduced as F c then at F s the
varactor behaves as an impedance with a negative real parts. The negative part can be
used for amplification. The series resistance limits the frequencies F p and F s and
introduces noise.
For losses reactances the power is
Therefore the Manley-Rowe frequency-power formula for the case of lossless reactance
are
The output voltage is
If the parametric amplifier is designed such that only power can flow at input frequency
and output frequency is available at nF p
, frequency dividers, rational fraction generators. Hence
, P 1 corresponds to power at f p .
Parametric small signal amplifiers and frequency converters
If the RF signal at frequency
is small compared to pump signal at frequency
power exchanged at the side band frequencies
and
for
, the
is negligible.
The corresponding Manley-Rowe equation is
Under this condition the optimum gain is
where,
is the modulation ratio, f c is the dynamic cut-off frequency given by
, R s being the series resistance and S max -S min is the elastance swing
Lecture 25
PIN Diode
In PIN diode the i region is sandwiched between the p and n region as shown in Fig 11.5
Fig
11.5
The i-region is either a high resistivity
layer.
PIN diodes are fabricated by
-epitaxial process
-Diffusion of p and n in high resistability substrate
-ion drift method
The concentration, charge density and electric field profiles are shown in Fig 11.6
Fig 11.6
PIN diodes are used widely in microwave wave circuits such as microwave switch with
constant depletion layer and high power.
The switching speed
Where W is the total depletion region width and
region.
is the saturation velocity across i
In addition the PIN diode can be used as
-variable attenuator by varying device resistance that change approximately likely with
forward current
-Modulate signals up to GHz range.
-Photo detection of internal modulated light in reverse bias.
Under Riverse Bias the junction capacitance is
and the series resistance is
Where
is the i-region resistance and
is the contact resistance.
The reverse bias current is
Where
is the ambipolar life time. The I-V characteristics is shown in Fig 11.7
Fig 11.7
PIN application
PIN diodes finds application in switching circuits.
The diode admittance(Y r ) in the reverse bias state and impedance (Z f ) in the forward
bias can be expressed as
and
where,
is the diode cut-off frequency and
is the reverse-bias series resonance frequency
Beam-lead PIN diodes are usually used in such circuits
Lecture26
When a metal-semiconductor junction is formed such that the carries see a barrier to flow
from one terminal to the other,it as called a schottky barries,as shown in 12.6.
When the Schottky diode is forward biased (negative with respect to metal) by a voltage
the barrier for electrons in Semiconductor decreases from
electrons flows from Semiconductor to metal
to
. More
increases greatly as shown in Fig 12.6
Fig
12.6
But
remains unchanged because no voltage drop across metal and
unchanged. Reverse bias drops across semiconductor increasing the barrier
remains
to
where V R is negative now,
decreases more but
remains almost
unchanged. Small reverse current flows from Semiconductor to metal as shown in Fig
12.7
Fig
12.7
Now suppose we have a semiconductor. with
as shown in Fig 12.4
Fig
12.4
No depletion layer is formed in Semiconductor, no barrier exists in semiconductor or in
metal.
Metal +n-type is rectifying for
Opposite is true for p-type
Fig12.5
and non rectifying
rectifying contact otherwise ohmic as shown in
Fig12.5
For rectifying contact
Maximum field
occurs at
The space charge per unit area
Or
throughout
Fig10.5 ps
. Thus if
is constant
Fig10.5
Schottky effect lowering of the barrier
When an electron is at a distance of x from metal surface +ve charge induced on metal
surface. The force of attraction between electron and induced +ve charge.=force that will
exist between electron and (positive charge at –x) image charge. Attractive force (image
force)
transfer from
. The work done on electron in the course of its
to x
Potential energy of an electron at a distance x from the
metal surface. This energy must be added to barrier energy
potential energy of electron
This is shown in Fig 12.8
.
to obtain total
Fig
12.8
Putting
So in effect the barrier height varies with
field
Current Transport in Metal-semiconductor Diodes
The Metal-Semiconductor diodes is a majority current device. The mechanisms of current
transport in M-S diodes are
• Transport of electrons from semiconductor over the potential barrier into the metal
(dominating factor for moderately doped semiconductor with
at room temperature).
operated
• Quantum mechanical tunnelling of electrons through the barrier (important for heavily
semiconductor and responsible for most of ohmic contacts.
• Electron hole recombination in their depletion region.
• Electron hole recombination in the neutral semiconductor region.
Transport over potential barrier
For high mobility materials
thermionic emission theory
For low mobility materials
diffusion theory
processes. thermionic emission theory
Actually a combination of two
• Barrier ht.
• Thermal equilibrium
• Net current flow does not effect this equilibrium
2 currents flux one from Metal to
Semiconductor, other from semiconductor to metal, shape of the barrier profile is not
important current flow depends only on
.
current density
concentration of electrons with energies
min energy needed to thermionically emit into metal.
Carrier velocity in x direction
electron density in energy range E and E+dE
density of states occupancy function
We assume that all energy in conduction band=Kinetic energy
Then
is the number of electrons /unit volume that have speed between v and v +
dv over all direction as shown in Fig 12.9
Fig
12.9
Rectangular coordinates from radial v
is the minimum velocity required in x direction to go over the potential barrier
Again,
Effective Richardson Const.
Free electrons Richardson const.
For GaAs,
isotropic in the lowest minimum of conductive band.
rest mass.
Multi vally semiconductors
are direction cosines of the normal to the emitting plane relative to the
principal axes ellipsoid
For
are the components of the effective mass tensor.
transverse mass
longitudinal mass
Barrier height for electrons moving from metal to semiconductor remains the same,
unaffected by a voltage. It must be therefore equal and opposite to the current
V=0 at room temperature
Thus
Thus Total current density for a = majority carriers.
Diffusion theory (Schottky) for current
It is for the current from semiconductor to the metal again
• Barrier
• Effect of electron collision within depletion region is included
• Carrier conductance at x=0 and x=w are unaffected by current flow
• Impurity concentration of semiconductor is non nondegenerate
Current in depletion region depends on local field and concentration gradient
with
Steady state, current density is independent of x
Thus Integrating both sides with an integrating factor
Boundary condition are
Putting these Boundary conditions
Similar to thermoinic expression- drift depends on Temperature T
Diffusion explanation is dominant if n semiconductor is heavily doped.
Tunneling current
When semiconductor is highly doped then depletion region is very narrow and electrons
can tunnel through it.
Lecture27
MESFET (Metal Semiconductor Field Effect Transistor)
MESFETs are Mostly made with GaAs
Analog and digital applications.
contenders for GaAs IC 10 5 transistor mark
-Communication technology satellite and fibre optic
-Cell phone
- Oscillator and 168 GHz for amplifier
New materials for MESFET
SiC and GaN wide band gap semiconductor
-higher breakdown voltage 100kV
-higher thermal conductivity
-GaN has higher electron velocity than GaAs
Also SiGe are used
MESFETs are suitable for compound semiconductor where oxide making is difficult.
Silicon dioxide material are easy to obtain in Si, and therefore mostly MOSFETs are used
in Si/SiGe.n-channel is always better as faster electron transport than holes in most
materials.
Gate channel Metal-Semiconductor/ Schottky diode
Normally ON MESFETs (Depletion Mode) with zero gate (–ve threshold) voltageis used.
Normally OFF MESFETs (Enhancement Mode) have positive threshold voltage on the
gate.
Ion implantation and electrode metallization are needed for the fabrication of MESFETs
on a Semi-insulating GaAs substrate. Simpler technology than MOSFETs or HEMTs.
Basic FET operation
Normally on type
From the simplified diagram shown in Fig 13.1 we have
Fig
13.1
Resistance of channel
L=length
Z=width
and
Channel acts like a constant resistor
Now if
increases-voltage distribution over channel changes as V(x) and the with of
the depletion region becomes
Therefore,
when source is grounded
Along the length of the channel the Resistance changes and the overall resistance of the
channel now i
However as V D increases, after a certain voltage, I D saturates at
Where V P is called the pinch-off voltage as shown in Fig 13.2
Fig
13.2
When
the additional voltage appears across the depletion region, so the
depletion region widens, this is shown in Fig 13.3
.
Fig
13.3
For W=a,
,
Lecture28
Drain Current
In deriving the current components the following assumptions are made
1. Constant mobility assumed
2. Uniformly doped channel
Considering Fig 13.4
Fig
13.4
we get
Electron drift velocity
Put
and W(x) expression together and integrate from source
to drain
for
For small drain voltage
Expand in a taylor series and retian 1 st two terms
This represents the 1-V characteristics of linear region given
more negative i.e,
Finally
as
is made more and
decreases.
, transistor to turn off.
Turn off voltage or threshold voltage.
Transconductance for small
in linear region.
in saturation region.
Drain current in saturation region.
Independent of
Short channel MESFETs
depends on E.
In the above equation the value of V G should be placed with sign.
Field Dependant Mobility.
V s = electron saturation velocity
Integrating from source to drain
The I-V characteristics are shown in Fig 13.6 Fig. 2
Fig
13.6
Reduction in
due to field dependent mobility.
is same as before.
1. Drain saturation voltage
2.
is reduced,
Saturated velocity model
The variation of velocity with electric field are shown in Fig 13.7 and 13.8
Fig 13.7
Fig.13.8
i ndependent of
In practice a combination of
and
Better explaination of experimental results of
High Frequency Performance:
Smalll signal input current
and output current
is used.
characterstics.
So cut-off frequency
In the saturated velocity model as shown in Fig 13.8
Fig
13.8
Transconductance is
And cut-off frequency is
Lecture 29
Semiconductor Heterojunction .
For faster transit time, RC should be decreased. This is used in making HBTs/HEMTs.
Two semiconductors with two different bandgaps can be grown one on top of the other or
a material can be grown with variable band as shown in Fig 14.1 For a constant bandgap
semiconductor electrons and holes move in opposite direction with the application of the
electric field. In a heterostructure as shown both can move same direction !!
Fig
14.1
Material that has a higher Band Gap is denoted by
and that having lower
Band Gap is denoted by n or p as usual. So we can have pN, nP or p+ N heterojunction.
Few applications of heterostructures are HBT, diode laser, LED, Photodetector, Quantum
well devices, solar cells.
The materials that are grown together must be latticed matched. Few lattice matched
compound semiconductors are
and
and
and
on
GaAs.
is basically solid solution (alloy) of Al, As and GaAs with no interface traps
We use molecular beam Epitaxy (MBE) or metal organic chemical vapour deposition
(MOCVD) to growth these structures. Liquid phase epitaxy is also used.
Calibration is done with Reflection high energy electron diffraction (RHEED)
For
Bandgap minimum remains at
keeping the alloy as direct bandgap and
for
Bandgap minimum occurs at X making the material indirect bandgap as
shown in Fig 14.2
Fig
14.2
The energy diagram of the direct and indirect bandgap materials are shown in Fig 14.3
Fig 14.3
Now consider an abrupt p-n junction p type GaAs and N type AlGaAs
First we draw their energy band diagram side by side as shown in Fig 14.4 then the two
energy band diagrams are brought in contact, keeping the E F same on both sides as
shown in Fig 14.5 Electron flows from N side to
and a depletion region is formed.
Fig 14.4
side and holes from p side to N side
Fig
14.4
Fig.14.5
Fig.14.5
Hence at thermal equilibrium Fermi level lines up on both sides.
Hence we get
We note that the two sides has two different permittivity.
The Poisson Equation are
Now theE field is zero at
and
Thus
and
The potential profile is obtained by integrating
potential is given by
The built-in potential on the p-side is
and assuming
, the
And the built-in potential on the N-side is
The electric flux density is continuous across the jiunction, hence
Also
, (the charge conservation relationship)
Considering all these we can show that for applied
Also
So,
Current transport
Junction diode and M-S diode too more complex. minority + majority current flow also.
Now think of an
junction
or
Now think of an
junction Fig. 14.6
Fig. 14.6
Effective electron mass of
Where
Effective hole mass is
and
Where,
and
The Bandgap is given by
for
and
and
The dielectric constant is
The electron affinity is
for
and
The conduction band density of states is related to
Where
and
The valence band density of states is
where
for
by
Lecture 30
is the forward transit time which is equal to the average time an electron spends in the
Base and is
Where
is the physical base width,
diffusion constant.
is the Fudge factor
and
is the electron
In practice we can make
Ist order model of BJT (CE-forward active)
Now we can define a frequency
short circuit current gain is
Where
is finite base current
And emitted base current
So we have the collector current as
being the low frequency current gain
The first order model is given in Fig 15.4
at which
Fig
15.4
Current gain is modeled as
In case of electrons diffusing across base,
The transit time is given by
And the cut-off frequency is given by
where is the time required to change the
base- junction resistance.
Space charge transit time
junction.
by charging up the capacitances through
is time required to drift through depletion region in BC
being the Base-Collector depletion region width.
Collector charging time
Where
and
are emitter and collector resistances.
A more detailed BJT model is the Gummel-Poon model, shown in Fig 15.5, in this model
we have
The Early effect is depicted in Fig 15.5
Fig 15.5
Lecture 31
BJT: Current Model
Now we go back to the current model of BJT where we have stated that electrons injected
from the emitter will diffuse across the Base. One aim in making BJT is always to reduce
the base width to reduce this unwanted base current. Good designs tries to make electron
transfer from emitter to collector to be maximum and I b (due to the hole back injection)
to be minimum.
Electron-Hole diffusion
Electron diffusing across the base and collected in the collector gives rise to the collector
current I c given by
where
is the emitter area,
thickness,
is the minority diffusion coefficient,
is the intrinsic concentration in base and N base is the base doping level.
The base current due to hole back injection is
now the ratio of desired to undesired current component is
The ratio
is the Base
is roughly of the order of 1 and
can be controlled by doping.
To make
, the base doping should be lower than the emitter doping.
This works for most transistors, however for high speed operation the base resistance R B
and the junction capacitance C BE are required to be low. R B decreases with increase in
N base and C BE decreases with decreasing N emit . Therefore
is difficult to
achieve. So high speed(f T ) and high (
ratio) cannot be both achieved
simultaneously and an optimization is needed. A recourse to this is to keep N emit low
and N base moderate, but increase the electron injection efficiency by using a hot
electron injection and field assisted base transport as done in HBT.
Lecture 32
HBT : Heterojunction Biplar Transister
In normal homo junction BJT we have
In HBT the emitter is of wider band gap as shown in Fig 16.1
Fig.16.1
The electron can be easily injected from the emitter to the base but holes from p to Nemitter see a much larger energy barrier
and hence the current I b due to hole back
injection is reduced. Thus the base region can be made highly doped which reduces the
base resistance R B . This is one of the advantage of HBT over BJT.
In abrurpt HBT, we have
Instead of using abrupt HBT we can use graded HBT made of the materials
and
Here the aluminium concentration is graded and hence called Graded Band Gap
heterostructure.
In a Graded Band Gap Heterojunction the barrier for the hole can be made larger then
electron barrier by
and we have
Thus
can be made even large than abrupt Heterojunction.
Here we consider an example
For
As/GaAs graded hetero junction
With no Heterojunction
And with abrupt Heterojunction
(useless device)
is a good material for Heterojunction
Other good Hetero junction are
The HBT structure is shown in Fig 16.2
Fig.16.2
Components of the Base currents
The various components of the Base currents are shown in Fig 16.3, they are
Fig
16.3
• Back injection of holes
• Extrinsic base surface recombination current.
• Base contact surface recombination current
• Bulk recombination current in Base layer
• Depletion region recombination current in B-E depletion region.
Bulk recombination current in the base region (I Bbulk ) is the dominant base current and
the current gain is given by
Where, is the minority electron recombination lifetime in Base and
carrier transit time in Base.
Now
is the minority
when the Base is too thin.
can be decreased by an E field in base arising due to the linearly grading the bandgap
in the Base region.
A linearly graded Si/SiGe HBT is shown in Fig 16.4
Fig
16.4
The Poisson Equation is given by
Base-Collector junction where the Base doping is greater than the Collector Doping the
depletion region is mostly inside the Collector, where the Electric field is high and
electrons travel mostly with saturation velocity.
The electron carrier concentration
The electron carrier concentration inside the collector is given by
is the Collector current density.
Thus the Electric field in Base-Collector depletion region is given by
.
When
is small
,
where N C is the Collector doping concentration
As
increases
slope becomes more negative as shown in Fig 16.5
Fig
16.5
While the current density increases and the area under the field profile should remain
constant the depletion region thickness would continue to increase until it reaches x= X c
as shown in Fig 16.6
Fig
16.6
As the current density increases to a level such that x=X c , the net charge inside the
junction becomes zero and the field profile is constant as shown in Fig 16.7
Fig
16.7
When J c increases further such that x>X c the net charge inside the junction becomes
negative, the electric field takes negative values as shown in Fig 16.8
Fig
16.8
When there is no more field to prevent holes from spilling, base pushout or Kirk effect
occurs as shown in Fig 16.9 The current gain decreases as the transit time associated with
the thickened base layer increases
Fig
16.9
Emitter-Collector transit time
Emitter-Collector transit time is
Where
is the time required to change the base potential by charging the
capacitances
(B-E junction capacitance) and
the differential Base-Emitter junction resistance.
(B-C junction capacitance) through
is the base transit time.
is the transit time through B-C depletion region and
is the Collector charging time
Hence the cut-off frequency is
Lecture33
HFET (hetrojunction FET) OR HEMT –high electron mobility transistor or
(two dimensional electron gas(2DEG))or MODFET
A potential is formed at junction of two dissimilar semiconductors(AlGaAs/GaAs), wher
is higher than the occupation levels of the electrons in the conduction band.
The electrons accumulated in this potential well and form a sheet of electron similar to
the inversion layer in a MOS structure. The thickness of this sheet is 10nm, smaller
than the De-broglie wavelength of the electron in that material. This sheet of free
electrons behaves like free atoms in a gas and hence it is called electron gas..
In this structure
has been observed.
This structure is similar to FET with 2DEG(Two Dimentional Electron Gas )as channel.
Application of a bias voltage to gate modulates charges in the 2DEG and thus channel
conducts current, this is similar to FET which is faster than MESFET operation. The
formation of 2DEG is shown in Fig 17.1
Fig.17.1
QUANTUM WELL : PICTURE
It can be seen that if the temperature is high(seldom encountered) or the applied field(V
DS ) is high, the electrons are excited to high energies and may escape from the
triangular well. It may also be scattered into conduction band of the barrier(spatial
transfer) and the carriers would be lost from the channel. To avoid this a quantum well
may be introduced at the interface.
Actually in this case we work at several possibilities of hetrostructures.
Charge transfer occurs leading to a conducting channel within single quantum well or
multiple quantum well. The energies of the electrons are quantized in the step like density
of states. Therefore many carriers can be put in the channel with a narrower dispersion of
energies.
Electron wave inside well is
This is Kane model.
Where x is the growth direction and k is the transverse electron wave vector.
block wave form and
is the envelope wave function which is the solution of
Where
is the effective mass,
energy of carriers.
is the potential and
is the confinement
Boundary conditions are
should be continuous at the interfaces.
The triangular potential well is shown in Fig 17.1
The triangular quantum well Potential
is linear for
and
at x = 0
The Schrödinger equation is
The boundary condition is
The above equation has two independent solutions.
One solution that is nonsingular at
function is
is AIRY function (Ai), so the resulting wave
as shown in Fig 17.2
Fig.17.2
The quantized energy levels are
Where,
is the nth zero of Ai(x), hence
So
The basic idea of HFET or HEMT also known as MODFET- Modulation Doped Field
Effect Transistor is that at equilibrium charge transfer occurs at the heterojunction to
equalize the Fermi level on both side. Doping the N side gives wide base. Electrons are
transferred to the GaAs side until an equilibrium is reached, this occurs because electron
transfer raises the Fermi level on the GaAs side due to filling of the conduction band by
electrons and also raises the electrostatic potential of the interface region because of the
more numerous ionizer donors in the AlGaAs side. This charge transfer effect makes
possible an old dream of semiconductor technologist, ie getting conducting electrons in a
high mobility, High purity semiconductor without having to introduce mobility limiting
donor impurities.
The various charge transfer mechanisms in heterojunctions are
1. Electric charges and field near the interface determine the energy band bendings
in the barrier and in conducting channel.
2. The quantum calculation of the electron energy levels in the channel determines
the confined conduction band levels
3. The thermodynamic equilibrium conduction determine the density of transferred
electrons.
Assuming that before the charge transfer the potential is flat band. After charge transfer
of
electrons the electric field in the potential well created can be taken as constant to
first order and is given by
Gauss' Electrostatic potential is given by
Hence the Schrödinger equation for the electron envelope wave function is
The energy level in infinite triangular potential well for the ground state
is
Where
and is usually determined experimentally.
As charge transfer increases, potential created by transferred electrons also increases,
leading to the lowering of the bottom of the Conduction . A 2DEG is formed when the
Conduction band goes below the Fermi level, hence we get
and the energy in channel is given by
In AlGaAs Fermi energy level is pushed downwards by electrostatic potential
up at the interface, where
Where W is the width of depletion region(
Where
) in AlGaAs
is the donor on AlGaAs
Calculating energies from the bothom of Conduction Band, we get
and
built
Where
is the donor binding energy in AlGaAs.
The donor concentration
is equal to
i.e, the number of electron transferred.
Hence we have
From the above equation N S can be calculated if the other parameters are known. The
Fermi level is determined empirically by the model given by
In practice undoped AlGaAs spacer layer of thickness
is used to separate Donor
atom from channel electrons (2DEG) to prevent coloumb interactions resulting in an
increased mobility
is decreased as shown in Fig 17.2
The charge in the conduction band is
Where
We have calculated the 2DEG charge density N S and it can be related to the gate voltage
by
with
Where ,
Where
is the 2DEG capacitance per unit area as given by
and is usually determined experimentally.
As charge transfer increases, potential created by transferred electrons also increases,
leading to the lowering of the bottom of the Conduction . A 2DEG is formed when the
Conduction band goes below the Fermi level, hence we get
and the energy in channel is given by
In AlGaAs Fermi energy level is pushed downwards by electrostatic potential
up at the interface, where
Where W is the width of depletion region (
) in AlGaAs
built
Where
is the donor on AlGaAs
Calculating energies from the bothom of Conduction Band, we get
Where
usually
is the distance of the centroid of 2DEG from x = 0 as shown in Fig 17.3 and is
Fig 17.3
Again the threshold voltage or pinch off voltage is given by
Where
is the Schottky barrier height on the donor layer as shown in Fig 17.4
Fig 17.4
At room temp
also modulates the bound carrier density in Donor layer and free
electron in Donor layer.
For a simplest model
independent of
is so large that all electrons in 2DEG channel move with
as shown in Fig 17.5 and
Fig 17.5
Where
is the electron density per surface area and is given by
width and
is the gate length.
The transconductance is
So we have
, z is the gate
Simplest model.
With
we have
and with
The voltage in the channel
we have
and the current is given by
Hence current is given by
Where
is the position of entrance to channel on the source side.
The saturation current is the current for which field on the drain side at
reaches
Where,
just
is
is the source resistance
For
we get
a linear behavior for a short highly conductive channel. The I-V characteristic is shown in
Fig 17.6
Fig
17.6
for
we get