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, increases and circuit becomes capable of handing high frequency – high speed.

So high speed/frequency low RC

Same idea in devices also 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, Rayliegh-Jeans, 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 , where = Wavelength (m) and ‘c' is the velocity of light in free space.

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 = and

Photon momentum .

Therefore an argument could be made that an electron with a mass and momentum also has a wavelength , and can then

be represented by wave, where for this particle momentum p can be associated with a

wave of wavelength , where particle momentum

Schrödinger and Heisenberg quantum mechanics to describe small particles like electrons, etc.

Postulates of Quantum Mechanics

•  There exist a wave function of a particle where x, y and z are the space variables and t is the time. From this one can find the dynamic behaviour of a particle in system, is complex.

•  The for a given system is determined by Schrödinger equation:

, where m is the mass of the particle and   is the potential energy operator for the system.

•  and 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    is given as

As examples, the space variables x, y, z are given as

The momentum are 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 mass m and a fixed total energy E.

Alone 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 , a one dimensional differential equation.Where

or or E has a parabolic dependence on k. General solution for the

simple equation can be written as where are unknown constants.

Therefore, total wave function (space and time dependence)

Compare this with a time-harmonic electromagnetic wave in free-space where

k = constant of propagation =

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, modes or eigen functions

for , 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,   and -x going wave Discrete points lie

on the continuous parabola of a free particle.

Fig.2.4

The integer n is called the quantum number

Value of from the equation that

Lecture4

Finite Potential well

Fig.2.5U(x) = 0 for 0 < x < a

U(x) = U 0 otherwise

We assume a particle with energy confined to the potential well

For 0 < x < a,

For x <0,

For x > 0,

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 , and therefore the coefficient of each power of  must vanish separately.

This gives in the recursion relation between    and a n ,

     (16)

It seems that knowing  and   can be calculated by using equation (16),

 

Thus we can write equation (12) as,

           (17)

If in the equation (16) should be zero for some value of the index n, then

. But since 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 . Usually

we take range .

•  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 , the determinant formed from coefficient should be equal to zero.

 

 

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. can thus take any value between +1 and -1. The

LHS with value between +1 and -1 gives allowed solutions for .

The plot of 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 modifies both and Q(x)

in such a way that the product of 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

One way of drawing is to k between in basically range. In the

eigenvalue equation you notice that increasing or decreasing by has

no effect on the allowed electron energy of E(k) is periodic with a period of .

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 constant E wave function grouped about a peak energy.

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 – classical idea.

wave packet also a mass – QM 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 work on the wave packet wave packet energy increases by

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

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, constant at the edge of the Brillouin zone/top or bottom of an energy band.

Therefore, constant near the edge of the Brillouin zone/top or bottom of an energy band.

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 2N electrons will fit the 2N states in the lowest 2 bands (valence bands).

Fig. 4.4

                                           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 But the band is

symmetric about . For every electron with a given direction, there will be

another electron with the same in –x direction.

Thus first band 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 varies from material to material.

2. Ge C Band minima at zone boundary (8 such min)

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 in Taylor series about a and keying only 1st significant term

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 = where q = wave no.

from equation (i).

 

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 Assume time harmonic solution.

Then

  

 

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 constants.

When 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 constant.

maximum pressure coming volume change . If compressibility

 

In a distance probability of scattering or reflection

= mean free path

= linear dimension of volume

If an electron is in state the probability that it will be in state after time t after a perturb potential is applied

Probability of transition from 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 probability of scattering is

Where = mean free path

In going a linear dimension the probability of reflection above

 

 

 

Assume is independent of velocity

mean free time between scattering/collisions.

From Boltzman's Theory of free particle in a gas.

 

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 change distribution gets distorted but the position of the atom remains the same

•  Rigid ion model ion if gets displaced the potential gets displaced.

•  Deformation potential model potential membrane gets distorted if the ion gets displaced.

Lecture15

Scattering in semiconductor

1. Phonon or lattice vibration scattering 2. Ionized (dopant) impurity scattering 3. Scattering by neutral impurity atoms and defects 4. Carrier-Carrier scattering 5. 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, = the effectives mass of the carrier,

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 and E are no longer directly proportional. Nonlinear behavior is seen.

Now geometry is often small (submicron) so even with 1 V across dimension,

 

So mobility concept may fail in such high field zones.

For very high E field saturates.

For Si at 300 K, for both electrons and holes.

Model,

 

 

For GaAs initially decreases with E after a critical field is 2 10 3 v/cm and then very

slowly increases, In a region of high which can be considered as saturated

velocity .

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 to .

large no. of states allowed 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 = contains one allowed solution.

Thus 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 effective mass

If E c = min CB energy and = max VB energy Bias

 

= average effective mass ( for GaAs or compound semiconductors) for Si or Ge m is complicated.

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 no. of states of allowed electronic states at a given energy. Fermi function

 

E F = Fermi energy

k = Boltzman's constant =

For closely spaced levels

Continuous variable E

Fig 6.3

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 Semiconductor is said to be nondegenerate.

 

Fig.6.5

 

 

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 and (1-f(E)).

This is shown in Fig. 7.2

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 in valence band

Change Neutrality Equation

Maxwell's equation gives

semiconductor dielectric constant.

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 we can find E F etc

Free-out/Extrinsic T

Suppose N D > > N A and N D > > n:

electron concentration > > hole concentration. Also .

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 allowed energy levels in the midgap region.

Crystal defects 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 and second equation by and taking Einstein relation as

we get

 

 

=lifetime

 

as in n-type

 

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, may be small and the generation term may be comparable or dominant.

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. so deflection layer capacitance will dominate at high frequencies.

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 junction-depletion region on the n side then the essential minority carrier is p only.

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 is a constant and is determined by the condition that

at t = 0 ,

hence

.

Now we assume a triangular hole distribution where the charge remains constant in the n-

region 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) (built-in voltage), where W is 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

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 decreases with reverse bias voltage and increases with reverse bias voltage.

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 non-

linear 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 , the

power exchanged at the side band frequencies and for 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 is the saturation velocity across i region.

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 to . More

electrons flows from Semiconductor to metal increases greatly as shown in Fig 12.6

Fig 12.6

But remains unchanged because no voltage drop across metal and remains

unchanged. Reverse bias drops across semiconductor increasing the barrier 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 and non rectifying

Opposite is true for p-type rectifying contact otherwise ohmic as shown in Fig12.5

 

Fig12.5

For rectifying contact

Maximum field occurs at

 

The space charge per unit area

 

 

Or . Thus if is constant throughout     

Fig10.5 ps

 

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) . The work done on electron in the course of its transfer from to x

Potential energy of an electron at a distance x from the

metal surface. This energy must be added to barrier energy to obtain total

potential energy of electron .

This is shown in Fig 12.8

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 operated at room temperature).

•  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 Actually a combination of two processes. thermionic emission theory

•  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 are the components of the effective mass tensor.

For

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 with 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

 

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 contenders for GaAs IC 10 5 transistor mark Analog and digital applications.

-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 as is made more and

more negative i.e, decreases.

Finally , 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 is used.

Better explaination of experimental results of characterstics.

High Frequency Performance:

Smalll signal input current

and output current

 

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 side and holes from p side to N side and a depletion region is formed.

Fig 14.4

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  and assuming , the potential is given by

The built-in potential on the p-side is

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 and

Effective hole mass is

 

 

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 by

 

Where for

and

The valence band density of states is

where

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, is the Fudge factor and is the electron diffusion constant.

In practice we can make

Ist order model of BJT (CE-forward active)

Now we can define a frequency at which 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

 

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 by charging up the capacitances through base- junction resistance.

Space charge transit time is time required to drift through depletion region in BC junction.

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, is the minority diffusion coefficient, is the Base

thickness, 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 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 N-

emitter 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 (useless device)

And with abrupt Heterojunction

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 is the minority carrier transit time in Base.

Now 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 (B-C junction capacitance) through the differential Base-Emitter junction resistance.

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, is the potential and is the confinement energy of carriers.

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 is AIRY function (Ai), so the resulting wave function is

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 built up at the interface, where

Where W is the width of depletion region( ) in AlGaAs

Where is the donor on AlGaAs

Calculating energies from the bothom of Conduction Band, we get

and

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 , is the 2DEG capacitance per unit area as given by

 

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   built up at the interface, where

Where W is the width of depletion region ( ) in AlGaAs

Where is the donor on AlGaAs

Calculating energies from the bothom of Conduction Band, we get

Where is the distance of the centroid of 2DEG from x = 0 as shown in Fig 17.3 and is usually

 

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 is so large that all electrons in 2DEG channel move with

independent of as shown in Fig 17.5 and

Fig 17.5

 

Where is the electron density per surface area and is given by , z is the gate

width and is the gate length.

The transconductance is

So we have

Simplest model.

With we have and with we have

The voltage in the channel 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 just

reaches is

Where, 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