Digital Electronics Introduction

Introduction to Digital Electronics

Introduction To Digital Electronics:

Digital electronics is a sub-field of electronics engineering. Digital electronics circuits use digital signals instead of analog signals, and use binary numbers 1 and 0  for representing ON/OFF states respectively. Boolean Algebra is the basis of digital electronics. Today digital technology is not confined to computer systems. It is applied to a wide range of daily use appliances like such as TV, radar, medical instrumentation, military systems, and other consumer electronics.

      Key Questions:

     An overview of digital systems
     Define logic levels
     Define digital waveforms
     Logic families
     Building blocks of digital circuits

Digital Systems Overview:

The two binary digits are used to represent anything in digital systems like numbers, letters, symbols, instructions, logic, etc

Most systems use 1 as a high-level voltage and 0 as a low-level or ground voltage. This is called positive logic.
High=1, Low =0

Negative logic is less common in which systems use 1 as low-level or ground-level voltage whereas 0 represents high-level voltage.
High=0, Low=1

Logic Levels In Digital Electronics 

The binary 1 & 0 are specified by some voltage level. In a practical circuit, a HIGH (1) is voltage in between a specified minimum and specified maximum voltage range. There is a high voltage range for CMOS this range is between 3.5V-5V. Similarly, a LOW (0) is not the ground level or 0V. A LOW is a voltage in between a specified minimum voltage range and a specified maximum voltage range. For CMOS low voltage range is between 0V to 1.5 V.

Low and high logic levels in digital circuits
Logic levels in digital circuits


Digital Waveforms:

There are many kinds of waveforms, but if we are working with digital electronics we always deal with digital waveforms, that switch between only two logic levels 0 & 1.
It represents the two states of Boolean logic (high or low, true or false). At this point, we have to discuss some important parameters and concepts related to waveforms. I also tried to give pictorial representation as well where is possible.

Ideal waveforms: in an ideal waveform the transition from low to high or high to low is instantaneous.

Non-Ideal waveforms: in a nonideal waveform the transition from low to high or high to low takes some time due to stray capacitance and inductance.

Periodic waveforms: the waveform which repeats itself after a fixed interval (termed as Period T). And have an equal pulse width (tw)

Non-Periodic waveforms: the waveform that doesn't represent itself after fixed intervals and may have pulses with different pulse widths.
Periodic waveforms properties
Periodic and nonperiodic waveforms


Pulse train: digital signals are in form of series that are repeated and called pulse trains.

Rising edge: signal/waveform transition from low to high.

Falling edge: signal/waveform transition from high to low.

Leading edge: waveform/ signal edge that occurs first that is at t=0.

Trailing edge: waveform/signal edge that occurs at t=1.

Positive going pulse: the pulse that goes from LOW logic level (0) to HIGH logic level (1) or leading edge is a rising edge and the trailing edge is a falling edge.

Negative going pulse: the pulse that goes from HIGH logic level (1) to LOW logic level (0) or leading edge is the falling edge and the trailing edge is the rising edge.
Leading edge trailing edge falling edge rising edge
Positive going pulse and negative going pulse


Rise time (tr): the time required for a waveform to rise from 10% of its amplitude to 90% of its amplitude.

Fall time (tf): the time required for a waveform to fall from 90% of its amplitude to 10% of its amplitude.

Pulse width (tw): duration of a pulse, measured in between 50% of rise time to 50% fall time

Amplitude: the height of the waveform or the intensity of the waveform.

Period: time in which waveform repeats itself. T=1/f

Frequency: how many times a waveform repeats itself within one second f=1/T
Characteristics of non Ideal waveforms
Ideal and nonideal pulses


Duty cycle: ratio of pulse width (tw) to the period (T) and expressed in percentage
D.C = (tw/T) %

Logic Families:


Logic families include bipolar and metal oxide semiconductors.

  • Bipolar ICs include

    • DTL (Diode Transistor Logic)

    • TTL (Transistor Transistor Logic)

    • ECL (Emitter Coupled Logic)

    • IIL (Integrated Injection Logic)

  • Metal Oxide semiconductors (MOS) ICs include

    • CMOS (Complementary MOS)

    • NMOS (N-channel MOS)

    • PMOS (P-channel MOS)

    • QMOS (Quick MOS)


Building Blocks of Digital Electronics:


Logic gates:

Logic gates perform basic logic operations on one or more binary inputs and produce a single binary output. They implement Boolean logic. Logic gates are fundamental building blocks of digital Integrated circuits. There are three types of logic gates: AND, OR, NOT. The other logic gates are derived from primary gates. Like NOR, NAND, XOR, and XNOR.

Latches and Flip-flops:

A latch is a simple memory element that can hold a bit as long as power is supplied. It has two stable states (as in a bistable multivibrator) that can use to store information. They are asynchronous and work on clock levels.

Latches and flip-flops are basic building blocks of sequential circuits.
Flip-flops are similar to latches. But they are synchronous and work on clock edges.
The output of both latches and flip-flops depends not only on current input but also on previous inputs and outputs.

Timers:
Timers are used in industries or used in other appliances to control process, and automatically starts/stops appliances after a predetermined interval of time. Quartz timers are more accurate than others.

Counters:

As the name implies counter is for counting electronic events, such as pulses. It usually consists of several flip-flops. There are synchronous and asynchronous counters.

Encoders:
An encoder is a device that converts information from one format to another. In digital electronics, it is a combinational circuit that converts information into a coded form, such as binary or BCD. It has 2n input lines and n-bit output lines.

Decoders:
It has the opposite functionality as the encoder. It converts information from n-bit coded input lines and produces 2n unique outputs.

Multiplexers (Mux):

It is also a combinational circuit and has many input lines and a single output. It is used to select one input from many and feed it to the output. It has 2n  input lines and n select lines. And only one output line.

Demultiplexer (Demux):
It is the exact opposite of multiplexers. It is also called a data distributor. It has a single input. 2n output and n select lines. Select lines decide which output line will be available for incoming input. Input is routed to the selected output lines.

Adders:
A digital circuit that performs addition. There are two types of adder circuits. Half adder and full adder.

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Effect Of Temperature On Carriers, Fermi Level, Mobility

Effect of Temperature on Semiconductor Properties
In this post impact of temperature on various properties of semiconductors used for electronic devices is examined.  The semiconductors are highly influenced by temperatures. In this post temperature dependence of carrier concentration is explained. the variations in Fermi level, and how the mobility of carriers is affected by temperature and doping.

Key Questions:

What is the effect of temperature on carriers' concentration?
What is the effect of temperature on the Fermi level?
What is the effect of temperature and doping on mobility?


What is the effect of temperature on carriers' concentration?
An increase in the temperature of a semiconductor can result in a substantial increase in the number of free electrons, as a result, the number of holes. Electrons and holes are created in pairs called EHP (electron-hole pair). At higher temperatures inter-atomic bonds become weak. So the thermal energy is used to break the bonds.  As a result electron and hole pairs are created. As electrons gain more energy make their way to the conduction band and leave a hole in the valence band. The electrons in the conduction band quickly lose energy fall back to the valence band and recombine with a hole. The merging of free electrons and holes is called Recombination.

The temperature dependence of electron concentration in a doped semiconductor can be visualised in figure 1. At low temperature (1/T is large) or only a few EHP exists. As the temperature increases, ionization occurs, and all the donor atoms donated electrons to the conduction band. After ionization, the conduction band concentration becomes no=Nd. Finally, at higher temperatures thermally generated EHP or intrinsic carriers are much greater than Nd.
Carrier concentration varies with temperature


What is the effect of temperature on the Fermi level?
The temperature and Fermi level have no direct relation or dependence. Fermi level is a theory that helps to understand semiconductor physics concepts. Fermi level falls indefinitely as the temperature increases. 
As we know the Fermi level changes as the doping concentration changes. For n-type semiconductors, it lies just below the conduction band. For p-type semiconductors, it lies just above the valence band.
So, as the temperature increases, EHP is created. As the temperature is increased at a level where thermally generated EHP are much greater than doping concentration. The temperature dependence of the Fermi level can be seen from the following equation:
no=Nce-(Ec-Ef)/kT
Solving this equation
Ef=Ei+kT ln(no/ni)

What is the effect of temperature and doping on mobility?
Either increase in temperature or doping level tends to reduce the carrier mobility.
At higher temperatures, a carrier moving through the lattice experiences vibration of the lattice. These vibrations decrease the mobility of carriers. Or you can say carriers are scattered by lattice vibrations and hence mobility decreases. Collective vibrations of atoms in the crystal are called lattice vibrations or phonons (phonons are considered particles). The frequency of such scattering increases as temperature increases and hence mobility decreases.

On the other hand, at a lower temperature, a phenomenon called Impurity scattering is dominant. The thermal motion of the carrier is also slow at low temperatures. Now, what are ionized impurities? When a semiconductor is doped either with p-type or n-type, it will leave a hole or donate an electron and leaves behind ionized charged impurity. Slow-moving carriers are more likely to be scattered by these ionized impurities and hence decrease mobility. The phenomenon is dominant at low temperatures because at higher temperatures carriers are moving with greater momentum and are less likely to be scattered by ionized impurity.
Impurity scattering and lattice scattering


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Mobility Vs Conductivity of Semiconductors

Mobility and Conductivity of Semiconductors
Carriers' mobility and conductivity are important parameters for the operation of electronic devices. Let's have a look at these properties.

Mobility:
As the name implies it describes a movement or mobile property. How quickly an electron or hole moves through a metal or semiconductor in the presence of an electric field.

The capability of movement is connected with a parameter called drift velocity. When an external electric field is applied across a piece of semiconductor, free electrons and holes are accelerated by the electric field and acquire a velocity component (superimposed on their thermal motion) called drift velocity. Drift velocity (V) is directly proportional to the electric field (E).
V=uE
where u = quantitative parameter called mobility and units are cm2/(V.s)

  • Mobility depends upon temperature, electric field (E), impurity concentration, defect concentration, electron and hole concentration
  • Electrons are faster particles than holes for Si un=1500 up=475
  • At higher temperatures, mobility decreases because collisions are inelastic, due to this average energy decreases, speed decreases and the number of carriers increases due to ionization
  • Mobility increases as electric field intensity decrease u=V/E
  • Higher mobility leads to better performance in electronic devices

Conductivity:
A property of charge carriers describing its capacity of conduction.
Doping or impurity concentration enhances the number of charge carriers and hence, the electrical conductivity of semiconductors.
It is similar to the conductivity of metals. But conductivity for semiconductors depends upon entirely different parameters. The conductivity of semiconductors is directly proportional to the following factors
  • Charge carriers (q)
  • The concentration of carriers (n)
  • Mobility of carriers (u)

Conductivity σ = qnu

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Carriers Concentration in Semiconductors Using Fermi Dirac's Distribution Function

Carriers Concentration in Semiconductors Using Fermi Dirac's Distribution Function

How to calculate carriers concentration in a semiconductor using Fermi Dirac's distribution function:


We have calculated the carrier concentration using the mass action law (Electrons and Holes Concentration In Semiconductors). 
Before reading it is better to read about the Fermi Level concept,  which is here. 
Explanation of Fermi Level In Semiconductors

 Key Questions

  • Evaluate equation for carrier concentration in semiconductors using Fermi Dirac's function

Now there is another way of calculating carrier concentration by using Fermi Dirac's function and density of states. 


Where N(E)dE is the density of states in the energy range dE. And can be calculated by quantum mechanics and the Pauli exclusion rule.

A closer examination of Fermi Dirac's function f(E) shows that Fermi function becomes extremely small for larger energies E. That is energy bands that are present far above the conduction band. and hence the product of (E)N(E) decreases for energy levels above Ec. So mostly electrons occupy energy states at the edge of the conduction band at equilibrium. And only a few electrons occupy energy states far above the conduction band. Or the probability of occupation of energy states far above the conduction band is very low.

The result of integration is obvious. Neglecting the energy states far above conduction band (because the probability of finding electrons above Ec is very low, as discussed above). For calculation of electrons concentration at equilibrium we only consider energy states at the bottom of the conduction band where the probability of finding an electron is maximum. Thus the electrons concentration at equilibrium is simply
no=Ncf(Ec)
f(E)=1/(1+e(E-Ef)/kT)
f(E)=e-(Ec-Ef)/kT… By approximation
no=Nce-(Ec-Ef)/kT

Similarly, most holes occupy energy states near the top of the valence band. Or the probability of finding an empty state below the valence band decreases. Or (1-f(E)) decreases rapidly below Ev. The product (1-f(E))NE decreases for energy levels below Ev. We discussed the symmetry property in between f(E) and 1-f(E), and Ev and Ec. So the holes concentration in the valence band at equilibrium is
po=Nv(1-f(E))
1-f(Ev)=1- 1/[1+e(Ev-Ef)/kT]
1-f(Ev)=e-(Ef-Ev)/kT… By approximation
po=Nve-(Ef-Ev)/kT
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Fermi Level Explanation

Fermi level and doping effect
Fermi Level in Semiconductors:
The concept of Fermi level is of cardinal importance in semiconductor physics.
The topic is not so easy to understand and explain. After reading again and again from different books and other resources I was able to understand and write… Now I aim to share my knowledge. I tried to make it simple. The concepts of Fermi Dirac's Function will be useful in deriving the equation for carrier concentration. Which is explained later.  First I will explain all the terms used in this context.

Forbidden gap/Forbidden energy band: In insulators and semiconductors, the energy bands within which no electrons may not be located. In metals, there are no such energy bands.

Valence band: It is simply the outermost orbit of an atom of any element.

Conduction band: The band of orbitals that is higher in energy.

Fermi level: It is the imaginary energy level that lies at the top of the available electrons energy levels at absolute zero. The probability of finding an occupied space at the fermi level is ½.

Fermi energy: The energy at the position of the Fermi level.

Bandgap: The energy difference between the top of the valence band and the bottom of the conduction band.

Fermi Dirac statistics: Electrons in solids follows Fermi Dirac Distribution function f(E). It gives the probability that an energy state with energy E will be occupied by the electron.  
f(E)=1/(1+e(E-Ef)/kT)

The density of states N(E): The number of electrons state per unit energy per unit volume.  
Simplified energy band diagram
Simplified energy band diagram

A Closer examination of Fermi Distribution Function:
The Fermi distribution function can be used to calculate the concentration of electrons and holes in a semiconductor if the density of states in the valence and conduction band is known.

f(E)=1/(1+e(E-Ef)/kT)
Where,
Ef is fermi energy or Fermi level where the probability of occupancy of an electron is 50%
T is the temperature in Kelvin
K is a Boltzmann constant 8.62*10-5 eV/K

At T=0  … f(E)=1
At E<Ef … exponent becomes negative f(E)=1
At E>Ef … exponent becomes positive f(E)=0

It explains that T=0 all the electrons are below Fermi level as the temperature increases the probability of finding an electron at a higher energy level also increases.
At higher temperatures, the probability of occupation of energy states near the conduction band is higher and the energy states near the valence band are most likely to be emptied.

Graphical representation of Fermi Dirac's function for various temperatures
Graphical representation of Fermi Derac's Function

The symmetry of f(E) around Ef
f(E)=Probability of filled states above Ef
1-f(E)=Probability of filled states below Ef

f(Ef+∆E)=1-f(Ef-∆E)
∆E any state above or below Fermi level has the same probability to be filled.


Where N(E)dE is the density of states
According to the above equation, total electron concentration is the integral over the entire conduction band. The probability values (Ev and Ec) for intrinsic material are quite small. In Si at 300K n=p=1010cm-3. The density of available states is much higher at this temperature like 1019. The probability of occupancy f(E) for an individual state in the conduction band and the hole probability (1-f(E)) for a state in the valence band is quite small. Because of the large density of states in each band, small changes in f(E) can result in significant changes in carrier concentration.

Fermi Level In Semiconductors:

Intrinsic Semiconductors: As we know that the concentration of holes in the valence band (p) is equal to the concentration of electrons in the conduction band (n).  n=p it implies that the chances of finding an electron near the conduction band edge are equal to the chances of finding a hole near the valence band edge. (i.e. symmetry property which is explained above). f(Ec)=1-f(Ev)
Thus Fermi level lies at the middle of the bandgap. Intrinsic Fermi level With is shown by:
Ei=Ec-Eg/2=Ev+Eg/2
Fermi level at the middle of conduction band and valence band
Fermi level position in intrinsic semiconductors
Carrier concentration in n-type semiconductors

Fermi Level In Extrinsic/ Doped Semiconductors:


 How does the Fermi Level shift as doping concentration increases?

n-type Semiconductors: There are more electrons in the conduction band than there are holes in the valence band. Also, the probability of finding an electron near the conduction band edge is larger than the probability of finding a hole near the valence band edge. Thus Fermi level for n-type Semiconductors lies near the conduction band.
Fermi energy level is located just below conduction band in n-type semiconductors
Fermi level position in n-type semiconductors

p-type semiconductors: 
There are more holes in the valence band than there are electrons in the conduction band. Also, the probability of finding a hole near the valence band edge is larger than the probability of finding an electron near the conduction band edge.

Fermi level is located just above valence band in p-type semiconductors
Fermi level position in p-type material



Fermi Dirac's function Application:
I hope you find this post helpful. The Fermi Dirac's function and Fermi level concept will helpful for the evaluation of the equation for carriers concentration in semiconductors. The link to the post is given below.

 to evaluate equation for carriers concentration using Fermi Dirac's function

Reference: Solid State Electronic Devices 6th edition  Ben G Streetman and Sanjay Kumar Banerjee



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The Mass Action Law

The Mass Action Law
When we are working with doped semiconductors we have to know about the concentration of minority carriers. Since the concentration of majority carriers is known by the amount of doped impurity. The mass action law provides an easy way to derive equation for minority carriers concentration in semiconductors.

Key Questions:

  • What is the mass action law?
  • How it is helpful in determining the equation for carrier concentration in semiconductors

The Mass Action Law:
This law is used to derive an important relationship between concentration of minority carriers and majority carriers at a constant temperature.

The addition of n-type impurities decreases the number of holes. Similarly, Doping with p-type impurities decreases the concentration of free electrons.
A theoretical analysis lead to the result that under thermal equilibrium, the product of positive and negative charge carriers is a constant, irrespective of the amount of donor and acceptor impurities. This relationship is called the mass action law.

ni2=np ….. (equation 1)

Where ni the intrinsic Carrier concentration is a function of temperature.


Electron and Hole Concentration:
The  mass action law (equation 1) is used to derive the carrier concentration in n-type or p-type material. The amount of majority carriers is approximately equal to the amount of impurity doping is added. The mass action law is used to find out the minority carrier concentration.

According to the principle of electrical neutrality, overall charge on any material should be equal.
So ND+p=NA+n….(equation 2)

Where ND= donor concentration
             NA=Acceptor concentration
             p=hole concentration
             n=electron concentration

Intrinsic Semiconductors

For intrinsic semiconductors ND=NA=0

From equation 2
p=n
By mass action law
np=ni2
p2=n2=ni2
p=n=ni

n-type Semiconductors
For n-type Semiconductors NA=0 and ND=n
From the mass action law
pn=ni2
p=ni2/n
                       p=ni2/ND     equation 3

The above equation is used to calculate the minority carriers concentration (in this case holes) in an n-type semiconductor.

p-type Semiconductors:
For p-type semiconductors ND==0 and NA=p
From the mass action law
np=ni2
n=ni2/p
                         n=ni2/NA       equation 4

The above equation is used to calculate the minority carriers concentration (in this case electrons) in a p-type semiconductor.

According to the law if we increase doping level in the semiconductor material, the concentration of minority carriers would decrease.


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