Effect Of Temperature On Carriers, Fermi Level, Mobility

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

Mobility Vs 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

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

Fermi Level Explanation

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

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 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 position in intrinsic 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 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 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

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.

Our Globally Changing Climate

Climate is the usual weather of a place. Climate can be different for different seasons. A place might be mostly warm and dry in the summer. The same place may be cool and wet in the winter. Weather is the change we see and feel outside from day to day. Weather usually changes within hours. It might rain one day and might be sunny the next. Sometimes it is cold. Sometimes it is hot. Climate change is also referred to as global warming. Climate change is the change in the climate pattern due to increased atmospheric carbon dioxide (CO2); hence, increased atmospheric temperature. We breathe out (CO2) every moment, but this is not the reason for increased atmospheric temperature. The combustion and the burning of fossil fuels waste from industries, emit carbon dioxide and other greenhouse gases. Let's understand the phenomenon in simple words. A few decades back there were not many industries, cars. People used to live simple lives. With time men had developed comfortable ways of living. Men find a way of a comfortable life, but it will result in a polluted atmosphere. Now there are hundreds and thousands of industries, cars air crafts, and trains consuming natural energy resources and producing hazardous waste. During past times there were forests. As you know plants absorb carbon dioxide from the atmosphere. Deforestation also causes the proliferation of greenhouse gases, which contribute to global warming.

Greenhouse Effect:

The greenhouse effect is a natural phenomenon, warming the earth and the troposphere (the lowest layer of earth). The sun radiates the energy, and part of this energy reflects directly from the top of the earth. The remaining energy absorbs by the earth (land and oceans). The energy absorbed by the earth should be radiated to make the planet cooler. As the earth is much cooler than the sun. But this is not happening, the earth is not radiating the same amount of energy, it is trapped by the abundant gases (the greenhouse gases) in the atmosphere. This is called the greenhouse effect. Greenhouse gases are responsible for the hotter planet. These gases are

1.  Water vapors (H2O) 
2.  Carbon dioxide (CO2) 
3.  Methane (CH4) 
4.  Nitrous oxide (N2O) 
5.  CFC (chlorofluorocarbons)

 The IPCC (Intergovernmental Panel on Climate Change) forecasts the global temperature will continue to rise for decades and will rise to 2.5 to 10 degrees Fahrenheit. 

Impacts of Climate Change: 

Climate change is one of the biggest threats in the last few decades. It is the greatest threat our new generation will face. It is equally detrimental to all types of habitation. This article explains how much it is deadly in every aspect.

Public Health Problems:

SMOG: The word SMOG drives from SMoke and fOG. Fog is a natural phenomenon while smoke is produced from human activities. The photochemical reaction takes place when fog is mixed with smoke (gases present in the atmosphere like sulfur dioxide, and nitrous oxide) and other particles in the atmosphere and water vapors. This is visible air pollution yellowish, blackish. This usually occurs under high temperatures, sunshine, and calm winds. SMOG has serious effects on health. Like 

1. It harms your lungs. It contains ground-level Ozone which can harm your lungs even after symptoms disappear. It can irritate your respiratory system. Cough, throat, and chest irritation.
2. Dangerous for asthma patients. 
3. It can irritate your eyes. 
4. It is equally detrimental to animal species and green life.

Apart from SMOG, there are many other effects on the human environment which are given below.

1. Children, elder citizens, and pregnant women are at higher risk of climate-sensitive health issues. 
2. Waterborne diseases are directly proportional to warmer weather. 
3. Allergy seasons are getting longer, giving rise to infectious diseases and other mental and physical health problems. 
4. Crops yield becoming uneven and unpredictable. Which is responsible for problems like hunger, malnutrition, and undernutrition. 

Agricultural Sector: "There will be impacts on the quantity, quality, and location of the food we produce," said Dr. Sam Myers, a medical doctor and senior research scientist studying environmental health at the Harvard T.H. Chan School of Public Health. Ultimately, climate change will reduce the amount of food grown around the world, Myers told Live Science. Initially, some experts thought that rising carbon dioxide levels might act as a fertilizer and increase food yield, Myers said. In some regions, climate change may favorably increase crop yield. But this does not apply to all crops. Increased temperature harms crop production, weeds, insects, and pests benefit from this. Increased level of carbon dioxide in the atmosphere leads to a lower level of zinc iron and other nutrients in crops. Either flood or drought both extremes can destroy the crops. 

Rising Sea Levels:

The sea level is rising at an accelerating rate. Core samples, tide gauge readings, and most recently, satellite measurements tell us that over the past century, the Global Mean Sea Level (GMSL) has risen by 4 to 8 inches (10 to 20 centimeters). However, the annual rate of rising over the past 20 years has been 0.13 inches (3.2 millimeters) a year, roughly twice the average speed of the preceding 80 years. The two major causes of the rise in sea level are shrinking land ice and the thermal expansion of seawater. Shrinking land ice, i.e. mountain glaciers, and polar ice sheets are releasing water into the sea. As the atmospheric temperature rises the sea water absorbs the heat and water molecules expand due to thermal expansion, which results in a rise in sea level. The oceans are acting as the planet's heat sink. Most of the heat and atmospheric carbon dioxide are absorbed by the ocean. The oceans are getting acidified as they absorb carbon dioxide. As the oceans get warmer and acidified will lead to species migration. Marine mammals such as polar bears are especially endangered because of melting glaciers. They want ice to survive. The aftermath of the rising sea level are: threats to coastal areas Saltwater intrusion 

Livestock On the same risk as well:

Climate change is also a major threat to the sustainability of livestock. Climate factors like increased temperature, humidity, and wind speed affect the well-being of livestock and will lead to a decline in their growth, reproduction, and milk production efficiency. Too many rainfalls cause the disease breakout and increase the risk of animal diseases. Weather extremes like floods and drought may lead to severe loss.

Conclusion:

This insidious creep has ruined our planet. The Concentration of greenhouse gases continuously increases. Global temperature will continue to rise due to human activities such as fossil fuel combustion, irrigated agriculture, oil extraction, and deforestation. Heat-trapping gas disturbs mankind's animals and is responsible for more intense and stronger hurricanes, more drought and heat waves, blizzards, tornadoes, rising sea levels, and changed precipitation patterns.

Elementary Electronics Concepts

I compiled some elementary electronics concepts. I hope it will helpful for those students who are newbie in this field. Please send your feedback and suggestions if you find this post helpful.

What is insulator?

An object that doesn't allow heat,light sound and electricity to pass through it.

An insulator doesn't have free electrons to move freely or we can say the highest energy band is filled with electrons. Example wood,rubber glass ceramic etc.

What is conductor?

An object that allows the flow of current because it has free electrons present in its valence band. Different materials have different level of of conductivity.

Example all metals are good conductor.

What is semiconductor?

An object whose electrical properties and conductivity lies in between conductors and insulators. They are neither good conductor nor good insulator. There is a threshold voltage at which semiconductor start to conduct.

Example group IV elements in the periodic table like silicon,germanium etc

What is resistor?

It is a two terminal electrical component and passive in nature. As the name implies it creates resistance to regulate the flow of electric current in electronic circuits. The amount of current flows through the circuit is inversely proportional to the resistance. The lower the resistance the higher will be the current flows through the circuit.

It can be used to provide specific voltage to an active device like transistor. This little device keeps your precious component from damage.

What is capacitor?

It is a two terminal electrical component and passive in nature. It consists of metal plates which store electric charges and have potential difference between these plates. The two plates are separated by a dielectric material. One plate accumulate positive charge and the second plate accumulate negative charge.

It is widely used in electronic circuits for example coupling capacitor, in smoothing circuit, in timing circuit, in tuning circuits etc.

What is inductor?

It is a two terminal electrical component and passive in

 nature. It is also called choke or coil. It consists of a conductor in the form of loop or coil. When AC current is passed through it, magnetic field is produced. It stores energy in the form of magnetic field.

It is widely used in surge protectors, oscillators, low pass filters, tuning circuits etc.

What is a diode?

A diode is an electronic component made up of semiconductor material. It is a two terminal pn- junction device. The fundamental property of a diode is to allow current in one direction only (forward bias) while blocking in reverse direction (reverse bias). It has non linear current voltage characteristics. There is a threshold voltage or cut-in voltage to drive the diode in forward bias. It is used in rectification circuits because it allows the current to flow in one direction only. 

What is a transistor?

A transistor is an electronic device made up of semiconductor material. It is a three terminal device. N-type is sandwiched between two p-type materials (PNP configuration) or p-type material is sandwiched between two N-type materials. One terminal acts as an input terminal, one acts as an output terminal the middle one is used for controlling. The three terminals are named as emitter collector and base.

It has three modes of operation cut-off region active region and saturation region. It is used in switching application (when drive in cut-off and saturation region) and audio application as an amplifier (active region)

What is an amplifier?
An amplifier is the basic building block of any electronic device or system. As the name implies, amplifier is used for amplification purpose. When an input is applied to an amplifier, it enhances the magnitude of the applied signal. 

Output = gain X input