Unit 2
SEMICONDUCTOR PHYSICS
1 Question: Explain P-N Junction Diode and its I-V Characteristic?
Solution:
A P-N Junction Diode is formed by doping one side of a piece of silicon with a P-type dopant (Boron) and the other side with a N-type dopant (phosphorus). Ge can be used instead of Silicon. The P-N junction diode is a two-terminal device. This is the basic construction of the P-N junction diode. It is one of the simplest semiconductor devices as it allows current to flow in only one direction.
ZERO BIASED CONDITION
In this case, no external voltage is applied to the P-N junction diode; and therefore, the electrons diffuse to the P-side and simultaneously holes diffuse towards the N-side through the junction, and then combine with each other. Due to this an electric field is generated by these charge carriers. The electric field opposes further diffusion of charged carriers so that there is no movement in the middle region. This region is known as depletion width or space charge.
Figure 14: Unbiased or zero biased PN Junction Diode
FORWARD BIAS
In the forward bias condition, the positive terminal of the battery is connected to the P-Type material and the negative terminal of the battery is connected to the N-type material. This connection is also called as giving positive voltage.
Figure 15: Forward bias
Electrons from the N-region cross the junction and enters the P-region. Due to the attractive force that is generated in the P-region the electrons are attracted and move towards the positive terminal. Simultaneously the holes are attracted to the negative terminal of the battery. By the movement of electrons and holes current flows. In this condition, the width of the depletion region decreases due to the reduction in the number of positive and negative ions.
If this external voltage Vf becomes greater than the value of the potential barrier, approx. 0.7 volts for silicon and 0.3 volts for germanium, the potential barriers opposition will be overcome and current will start to flow.
This is because the negative voltage pushes or repels electrons towards the junction giving them the energy to cross over and combine with the holes being pushed in the opposite direction towards the junction by the positive voltage. This results in a characteristics curve of zero current flowing up to this voltage point, called the “knee” on the static curves and then a high current flow through the diode with little increase in the external voltage as shown in I-V characteristics.
REVERSE BIAS
In the reverse bias condition, the negative terminal of the battery is connected to the P-type material and the positive terminal of the battery is connected to the N-type material. This connection is also known as giving negative voltage.
Figure 16: Reverse bias
The positive voltage applied to the N-type material attracts electrons towards the positive electrode and away from the junction, while the holes in the P-type end are also attracted away from the junction towards the negative electrode.
The net result is that the depletion layer grows wider due to a lack of electrons and holes and presents a high impedance path, almost an insulator. The result is that a high potential barrier is created thus preventing current from flowing through the semiconductor material.
This condition represents a high resistance value to the PN junction and practically zero current flows through the junction diode with an increase in bias voltage. However, a very small leakage current does flow through the junction which can be measured in micro-amperes, ( μA ).
If the reverse bias voltage Vr applied to the diode is increased to a sufficiently high enough value, it will cause the diode’s PN junction to overheat and fail due to the avalanche effect around the junction. This may cause the diode to become shorted and will result in the flow of maximum circuit current, and this shown as a step downward slope in the reverse static characteristics curve in I-V characteristics.
I-V CHARACTERISTICS OF PN JUNCTION DIODE
The I-V Characteristic Curves, which is short for Current-Voltage Characteristic Curves or simply I-V curves of an electrical device
The application of a forward biasing voltage on the junction diode results in the depletion layer becoming very thin and narrow which represents a low impedance path through the junction thereby allowing high currents to flow. The point at which this sudden increase in current takes place is represented on the static I-V characteristics curve above as the “knee” point. The current starts increasing with increase in voltage. At knee voltage current shows a sharp increment in its magnitude. This behaviour is mentioned above. As large current flow in forward biasing so we measure this current in mA.
When a junction diode is Reverse Biased, the thickness of the depletion region increases and the diode acts like an open circuit blocking current flow. So only a very small leakage current will flow.
Figure 17: I-V characteristics
2 Question: Discuss conduction in intrinsic semiconductor?
Solution:
Intrinsic Semiconductor
- An intrinsic type of semiconductor material made to be very pure chemically. As a result it possesses a very low conductivity level having very few number of charge carriers, namely holes and electrons, which it possesses in equal quantities.
- The most commonly used semiconductor basics material by far is silicon. Silicon has four valence electrons in its outermost shell which it shares with its neighbouring silicon atoms to form full orbital’s of eight electrons. The structure of the bond between the two silicon atoms is such that each atom shares one electron with its neighbour making the bond very stable.
- As there are very few free electrons available to move around the silicon crystal, crystals of pure silicon (or germanium) are therefore good insulators. Silicon atoms are arranged in a definite symmetrical pattern making them a crystalline solid structure. A crystal of pure silica (silicon dioxide or glass) is generally said to be an intrinsic crystal (it has no impurities) and therefore has no free electrons.
- An extremely pure semiconductor is called as Intrinsic Semiconductor. On the basis of the energy band phenomenon, an intrinsic semiconductor at absolute zero temperature is shown below.
- Its valence band is completely filled and the conduction band is completely empty. When the temperature is raised and some heat energy is supplied to it, some of the valence electrons are lifted to conduction band leaving behind holes in the valence band as shown below.
- A hole is the absence of an electron in a particular place in an atom. Although it is not a physical particle in the same sense as an electron, a hole can be passed from atom to atom in a semiconductor material. It is considered to have positive charge. Holes are positive charge carriers.
- The electrons reaching at the conduction band move randomly. The holes created in the crystal also free to move anywhere.
- This behaviour of the semiconductor shows that they have a negative temperature coefficient of resistance. This means that with the increase in temperature, the resistivity of the material decreases and the conductivity increases.
- But simply connecting a silicon crystal to a battery supply is not enough to extract an electric current from it. To do that we need to create a “positive” and a “negative” pole within the silicon allowing electrons and therefore electric current to flow out of the silicon. These poles are created by doping the silicon with certain impurities.
3 Question: What is doping? Discuss its importance in semiconductor?
Solution:
Doping
The process by which an impurity is added to a semiconductor is known as Doping. The amount and type of impurity which is to be added to a material has to be closely controlled during the preparation of extrinsic semiconductor. Generally, one impurity atom is added to a 108 atoms of a semiconductor.
The purpose of adding impurity in the semiconductor crystal is to increase the number of free electrons or holes to make it conductive.
If a Pentavalent impurity, having five valence electrons is added to a pure semiconductor a large number of free electrons will exist. Which make a n-type extrinsic semiconductor.
If a trivalent impurity having three valence electrons is added, a large number of holes will exist in the semiconductor. Which make a p-type extrinsic semiconductor.
4 Question: What is P-type Extrinsic Semiconductor? Draw its energy level diagram and discuss the conduction mechanism in p type semiconductor?
Solution:
P-type Extrinsic Semiconductor
The extrinsic p-Type Semiconductor is formed when a trivalent impurity is added to a pure semiconductor in a small amount, and as a result, a large number of holes are created in it. A large number of holes are provided in the semiconductor material by the addition of trivalent impurities like Gallium and Indium. Such type of impurities which produces p-type semiconductor are known as an Acceptor Impurities because each atom of them create one hole which can accept one electron.
In a P-type semiconductor material there is a shortage of electrons, i.e. there are 'holes' in the crystal lattice. Electrons may move from one empty position to another and in this case it can be considered that the holes are moving. This can happen under the influence of a potential difference and the holes can be seen to flow in one direction resulting in an electric current flow. It is actually harder for holes to move than for free electrons to move and therefore the mobility of holes is less than that of free electrons. Holes are positively charged carriers.
A trivalent impurity like Aluminium, having three valence electrons is added to Silicon crystal in a small amount. Each atom of the impurity fits in the Silicon crystal in such a way that its three valence electrons form covalent bonds with the three surrounding Silicon atoms as shown in the figure below.
Energy Band Diagram of p-Type Semiconductor
The energy band diagram of a p-Type Semiconductor is shown below.
A large number of holes or vacant space in the covalent bond is created in the crystal with the addition of the trivalent impurity. A small or minute quantity of free electrons is also available in the conduction band.
They are produced when thermal energy at room temperature is imparted to the Silicon crystal forming electron-hole pairs. But the holes are more in number as compared to the electrons in the conduction band. It is because of the predominance of holes over electrons that the material is called as a p-type semiconductor. The word “p” stands for the positive material.
Conduction Through p Type Semiconductor
In p type semiconductor large number of holes are created by the trivalent impurity. When a potential difference is applied across this type of semiconductors.
The holes are available in the valence band are directed towards the negative terminal. As the current flow through the crystal is by holes, which are carrier of positive charge, therefore, this type of conductivity is known as positive or p type conductivity. In a p type conductivity the valence electrons move from one covalent to another.
The conductivity of n type semiconductor is nearly double to that of p type semiconductor. The electrons available in the conduction band of the n type semiconductor are much more movable than holes available in the valence band in a p type semiconductor. The mobility of holes is poor as they are more bound to the nucleus.
Even at the room temperature the electron hole pairs are formed. These free electrons which are available in minute quantity also carry a little amount of current in the p type semiconductors.
5 Question: For an intrinsic Semiconductor with a band gap of 0.7 eV, determine the position of EF at T = 300 K if m*h = 6m*e
Solution:
6 Question: What is N-type Extrinsic Semiconductor? Draw its energy level diagram and discuss the conduction mechanism in N type semiconductor?
Solution:
N-type Extrinsic Semiconductor
When a few Pentavalent impurities such as Phosphorus whose atomic number is 15, which is categorised as 2, 8 and 5. It has five valence electrons, which is added to silicon crystal. Each atom of the impurity fits in four silicon atoms as shown in the figure below.
Hence, each Arsenic atom provides one free electron in the Silicon crystal. Since an extremely small amount of Phosphorus, impurity has a large number of atoms; it provides millions of free electrons for conduction.
An N-type semiconductor material has an excess of electrons. In this way, free electrons are available within the lattices and their overall movement in one direction under the influence of a potential difference results in an electric current flow. This in an N-type semiconductor, the charge carriers are electrons.
Energy Diagram of n-Type Semiconductor
A large number of free electrons are available in the conduction band because of the addition of the Pentavalent impurity. These electrons are free electrons which did not fit in the covalent bonds of the crystal. However, a minute quantity of free electrons is available in the conduction band forming hole- electron pairs.
The Energy diagram of the n-type semiconductor is shown in the figure below.
- The addition of pentavalent impurity results in a large number of free electrons.
- When thermal energy at room temperature is imparted to the semiconductor, a hole-electron pair is generated and as a result, a minute quantity of free electrons is available. These electrons leave behind holes in the valence band.
- Here n stands for negative material as the number of free electrons provided by the pentavalent impurity is greater than the number of holes.
Conduction Through n-Type Semiconductor
In the n-type semiconductor, a large number of free electrons are available in the conduction bands which are donated by the impurity atoms. The figure below shows the conduction process of an n-type semiconductor.
When a potential difference is applied across this type of semiconductor, the free electrons are directed towards the positive terminals. It carries an electric current. As the flow of current through the crystal is constituted by free electrons which are carriers of negative charge, therefore, this type of conductivity is known as negative or n-type conductivity.
The electron-hole pairs are formed at room temperature. These holes which are available in small quantity in valence band also consist of a small amount of current. For practical purposes, this current is neglected.
7 Question: Derive expression for carrier concentration for intrinsic semiconductors? Explain Fermi level for Intrinsic semiconductors and what is the effect of temperature on Fermi level?
Solution:
Intrinsic semiconductors—carrier concentration
Here we will calculate the number of electrons excited into conduction band at temperature T and also the hole concentration in the valence band. It is assumed that the electrons in the conduction band behave as if they are free particles with effective mass me* and the holes near the top of the valence band behave as if they are free particles with effective mass mh*.
Here we will calculate the electron concentration, hole concentration
Density of Electrons in Conduction Band
The number of free electrons per unit volume of semiconductor having energies in between E and E + dE is represented as N(E) dE
dE = width of Energy band
Therefore, we have:
N(E) dE = ge(E) dE fe(E) ……….(1)
ge(E) = The density of electron states per unit volume
fe(E) = Fermi-Dirac distribution function i.e. probability that an electron occupies an electron state
The number of electrons present in the conduction band per unit volume of material ‘n’ is obtained by integrating N(E) dE between the limits Ec and Ect
Where Ec = the bottom energy levels of conduction band
Ect = the bottom and top energy levels of conduction band
n = 
……….(2)
Can be written as
n = 
= 
n = 
- 
……….(3)
We know that above Ect, there is no electrons.
Hence, Equation (3) becomes
n = 
- 
n = 
……….(4)
The Fermi-Dirac distribution function fe(E) can be represented as:
(E) =
……….(5)
Compared to the exponential value, so the ‘1’ in the denominator can be neglected.
So 
>>1
Hence, 
(E) =
= 
……….(6)
The density of electron states ge(E) in the energy space from E = 0 to E can be written as:
(E) = 
[
]3/2 (E - 0) ½ ……….(7)
Where me* is the effective mass of an electron and h is Planck’s constant.
(E)dE = 
[
]3/2 (E - 0) ½dE ……….(8)
To evaluate n, the density of states is counted from Ec, since the minimum energy state in conduction band is Ec. So eq (8) can become
(E)dE = 
[
]3/2 (E - Ec) ½dE ……….(9)
Substituting Equations (6) and (9) in (4) gives
n = 
3/2
(E-
)1/2
dE
n =
[
]3/2 
1/2
dE………… (10)
The above equation can be simplified by the following substitution:
Put ɛ = E − Ec ………… (11)
So, dɛ = dE
In Equation (11), Ec is constant, as we change the variable E to ε in Equation (10), the integral limits also change.
In Equation (11), as E → Ec then ε → 0 and E → ∞, then ε also → ∞.
The exponential term in Equation (10) becomes:
= exp [
] = exp [
]
………… (12)
Substituting Equations (11) and (12) in (10), we get:
n =
[
]3/2 
dE
n =
[
]3/2
dE………… (13)
Above integral (I) can be simplified by substitution. Put ε = x2
So that dɛ = 2x dx
I =
2x dx
=
dx
=
=
T)3/2………… (14)
Substituting Equation (14) in (13) gives:
n =
[
]3/2
T)3/2
n =
n = 
n = 
………… (15)
The term 
is almost a constant compared with the exponential term as the temperature changes. So, it is a pseudo constant and is given by the symbol Nc. So, we have
n =Nc
………… (16)
Density of Holes in Valence Band
The number of holes per unit volume of semiconductor in the energy range E and E + dE in valence band is represented as P(E) dE. Proceeding same way( as in case of electrons) we have
Therefore, we have:
P(E) dE = gh(E) dE fh(E) ……….(17)
dE = width of Energy band
gh(E) = The density of holes states per unit volume
fh(E) = Fermi-Dirac distribution function i.e. probability that an hole occupies an electron state
The number of electrons present in the conduction band per unit volume of material ‘n’ is obtained by integrating P(E) dE between the limits Evb and EV
Where EV = the bottom energy levels of valence band
Evb = the bottom and top energy levels of valence band
The total number of holes present in the valence band per unit volume of material ‘p’ is obtained by integrating P(E) dE
……….(18)
Equation (18) can be represented as:
……….(19)
Now we know that below Evb no holes is present. Hence, Equation (19) becomes
……….(20)
We know hole can also be defined as absence of an electron.
Presence of a hole = the absence of an electron
Hence, the Fermi-Dirac function of holes fh(E) in the valence band is:
Compared to exponential, the ‘1’ in the denominator is negligible,
Hence,
……….(21)
The density of hole states between E and E + dE in valence band can be written similar to Equation (8.9) for electrons.
……….(22)
Where mh* is the effective mass of hole.
Substituting Equations (21) and (22) in (20),
……….(23)
The above equation can be simplified by the substitution:
Put ɛ = EV − E ............. (24)
So dɛ = − dE
In Equation (24), EV is constant, as we change the variable E to ε in Equation (23), the integral limits also change.
In Equation (24),
As E → EV then ε → 0
And E→ −∞, then ε → ∞
The exponential term in Equation (23) becomes:
............. (25)
Substituting Equations (24) and (25) in (23), we get:
………….(26)
From Equation (14), we know the integral value 
T)3/2 put here and we get
………….(27)
….The term 
is almost constant compared with the exponential term as the temperature changes. So, it is a pseudo constant and is given by the symbol NV. So, we have
…………… (28)
8 Question: Explain Fermi level for Intrinsic semiconductors and what is the effect of temperature on Fermi level?
Solution:
Fermi Level
We know that in intrinsic semiconductor
Electron concentration ‘n’ = Hole concentration ‘p’
Equating Equations (15) and (27), we get
Taking logarithms on both sides, we get
……………… (29)
Normally, mh* is greater than me*, since ln 
is very small so that EF is just lie in the middle of energy gap
Temperature effect on Fermi level
Fermi level slightly rises with increase of temperature.
But in case of pure intrinsic semiconductor like Si and Ge,
mh* ≈ me*
So in these cases Fermi level lies at the middle of energy gap.
9 Question: Explain Zener Diode and its I-V Characteristic?
Solution:
The satisfactory explanation of this breakdown of the junction was first given by the American scientist C. Zener. A zener diode is a special type of diode that is designed to operate in the reverse breakdown region.
An ordinary diode operated in this region will usually be destroyed due to excessive current. This is not the case for the zener diode.
It has already been discussed that when the reverse bias on a crystal diode is increased, a critical voltage, called breakdown voltage is reached where the reverse current increases sharply to a high value. The breakdown region is the knee of the reverse characteristic as shown in Figure 19.
The breakdown voltage is sometimes called zener voltage and the sudden increase in current is known as zener current.
The breakdown or zener voltage depends upon the amount of doping. If the diode is heavily doped, depletion layer will be thin and consequently the breakdown of the junction will occur at a lower reverse voltage.
On the other hand, a lightly doped diode has a higher breakdown voltage. When an ordinary crystal diode is properly doped so that it has a sharp breakdown voltage, it is called a zener diode. A properly doped crystal diode which has a sharp breakdown voltage is known as a zener diode.
Figure 18 shows the symbol of a zener diode. It may be seen that it is just like an ordinal, diode except that the bar is turned into z-shape.
Figure 18: Symbol of a zener diode
Figure 19: I-V characteristics of zener diode
As the curve reveals, two things happen when Vz is reached:
- The diode current increases rapidly.
- The reverse voltage Vz across the diode remains almost constant.
The following points may be noted
(i) A zener diode is like an ordinary diode except that it is properly doped so as to have a sharp breakdown voltage.
(ii) A zener diode is always reverse connected i.e. it is always reverse biased.
(iii) A zener diode has sharp breakdown voltage, called zener voltage Vz.
(iv) When forward biased, its characteristics are just those of ordinary PN junction diode.
(v) The zener diode is not immediately burnt just because it has entered the breakdown region. As long as the external circuit connected to the diode limits the diode current to less than burn out value, the diode will not burn out.
In other words, the zener diode operated in this region will have a relatively constant voltage across it, regardless of the value of current through the device. This permits the zener diode to be used as a voltage regulator or voltage stabiliser.
10 Question: Calculate the intrinsic concentration of charge carriers at 300 K given that m *e =0.12m o ,m *h =0.28mo and the value of brand gap = 0.67 eV.
Solution:
11 Question: Where Fermi level lie in case of p type and n type extrinsic semiconductor?
Or
Derive expression for Carrier Concentration in Extrinsic Semiconductors.
Solution:
Carrier Concentration in Extrinsic Semiconductors
The number of charge carriers present per unit volume of a semiconductor material is called carrier concentration.
Suppose donor and acceptor atoms are doped in a semiconductor.
At temperature T K,
n = number of conduction electrons
p = number of holes
N−A = number of acceptor ions
N+D = number of donor ions
We know that the below equation hold good in semiconductor. The total negative charge due to conduction electrons and acceptor ions is equal to holes and donor ions in unit volume of material.
So the material will be considered neutral if,
n + N−A = p + N+D ……….(34)
Equation (34) is called charge neutrality equation.
In the above equation
………..(35)
Concentration of acceptor ions N−A = acceptor concentration x probability of finding an electron in acceptor level
………..(36)
Similarly, the donor ions concentration is
………..(37)
In n-type material
Note
nn represents electrons in n-type material
pn represents holes concentration in n-type material.
There are no acceptor atoms so N−A = 0.
At 0 K, all the electron states in donor level are occupied by electrons.
As the temperature is increased from 0 K, some of the electrons jump from these donor states into the conduction band.
Also the concentration of holes is extremely less compared with the concentration of conduction electrons [p << n]
From Equation (34) we have
n = p N+D
(Or) n ≈ N+D …………(38) [since p << N+D]
At temperature T K,
As the temperature increase Almost all the donor atoms donate electrons to conduction band.
So, in n-type material, the free electron concentration is almost equal to the donor atoms.
So we can rewrite the above equation as
nn ≈ ND …………….(39)
Where nn represents electrons in n-type material
Also the hole concentration in n-type material can be obtained by applying law of mass action nn pn =ni2
…………..(40)
Where pn represents holes concentration in n-type material.
In n-type material at 0 K, the Fermi energy level lies in the middle of Ec and ED
At temperature> 0K
……….(41)
With increase in temperature the Fermi level shifts upwards according to Equation (61) slightly due to ionization of donor atoms.
With further increase of temperature, electron-hole pairs are generated due to the breaking of covalent bonds, hence Fermi level shifts downwards.
In p-type semiconductor
Note
pp represents holes in p-type material
np represents electrons in p-type material
There are no donor atoms so means no ions present 
= 0.
At 0 K, all the acceptor levels are not occupied by electrons.
As the temperature is increased from 0 K, some electrons jump from top valence band energy levels to the acceptor states, leaving holes in the valence band and acceptor ions 
are formed.
At some room temperature T K, concentration of conduction electrons is extremely less compared with hole concentration.
∴ From Equation (34), we have
n + N−A = p ……………(42)
(or) N−A ≈ p ……………. (43) [since n << N−A]
At temperature T K, in p-type material,
The hole concentration is almost equal to the acceptor atoms in unit volume of the material.
So, Equation (43) can be written as
pp ≈ NA……………….(44)
Where pp represents holes in p-type material
The electron concentration in p-type material can be obtained by applying law of mass action as nppp = ni2
…………….(45)
Where np represent free electron concentration in p-type material.
In p-type material, the Fermi level lies in between EV and EA at 0 K
As the temperature is increased from 0 K, the Fermi level shifts downwards slightly as per Equation (46) due to ionization of acceptor atoms.
And with further increase of temperature, electron-hole pairs are generated due to the breaking of covalent bonds, so Fermi level shifts upwards.
12 Question: What is Recombination in semiconductor? Discuss any two different processes which lead to recombination?
Solution:
Recombination is the reverse process where electrons from the conduction and holes from valence band recombine and are annihilated.
These processes must conserve both quantized energy and momentum, and the vibrating lattice plays a large role in conserving momentum.
- Recombination and generation are always happening in semiconductors both optically and thermally.
- Their rates are in balance at equilibrium.
- If the product of the rate becomes greater than the recombination rate, again driving the system back towards equilibrium.
- As the electron and hole densities is a constant at equilibrium, maintained by recombination and generation occurring at equal rates.
- When there is a surplus of carriers the rate of recombination becomes greater than the rate of generation, driving the system back towards equilibrium.
In semiconductors several different processes exist which lead to recombination, the most important ones are:
1) Band-to-band recombination
2) Auger generation/recombination
3) Surface recombination
Note- You can discuss any two out of three.
1) Band-to-band recombination or Radiative recombination
Band-to-band recombination or Radiative recombination is the reverse process of photon absorption, where an electron drops back down to its equilibrium energy band and radiates a photon.
It is the process of electrons jumping down from the conduction band to the valence band in a radiative manner.
During band-to-band recombination the energy absorbed by a material is released in the form of photons. Generally these released photons contain the same or less energy than those initially absorbed.
The photon emitted may have the energy of the band gap difference or less, depending on how much energy is lost in the mechanism
Radiative recombination plays a more major role in direct band semiconductors.
The total radiative recombination rate, given below as R, is proportional to the product of the concentration of occupied states (electrons, n) in the conduction band and that of the unoccupied states in the valence band (holes, p)
R= Bnp
Band-to-band recombination depends on the density of available electrons and holes. Both carrier types need to be available in the recombination process. Therefore, the rate is expected to be proportional to the product of n and p. Also, in thermal equilibrium, the recombination rate must equal the generation rate since there is no net recombination or generation. As the product of n and p equals ni2 in thermal equilibrium, the net recombination rate can be expressed as:
U= B(np-ni2)
Where B is the bimolecular recombination constant for a given semiconductor. It can be calculated from the semiconductor’s absorption coefficient. ni2 is the familiar “intrinsic carrier concentration”
Auger recombination
Auger recombination involves three particles: an electron and a hole, which recombine in a band-to-band transition and give off the resulting energy to another electron or hole. The expression for the net recombination rate is therefore similar to that of band-to-band recombination but includes the density of the electrons or holes, which receive the released energy from the electron-hole annihilation.
In other words Auger recombination the energy is given to a third carrier which is excited to a higher energy level without moving to another energy band.
After the interaction, the third carrier normally loses its excess energy to thermal vibrations. Since this process is a three-particle interaction, it is normally only significant in non-equilibrium conditions when the carrier density is very high.
Auger recombination is most important at high carrier concentrations caused by heavy doping. In silicon-based solar cells, Auger recombination limits the lifetime and ultimate efficiency.
The more heavily doped the material is, the shorter the Auger recombination lifetime.
Surface recombination
Recombination at surfaces and interfaces can have a significant impact on the behavior of semiconductor devices.
Areas of defect, such as at the surface of solar cells where the lattice is disordered, recombination is very high.
This is because surfaces and interfaces typically contain a large number of recombination centers because of the abrupt termination of the semiconductor crystal, which leaves a large number of electrically active states.
In addition, the surfaces and interfaces are more likely to contain impurities since they are exposed during the device fabrication process.
The net recombination rate due to trap-assisted recombination and generation is given by:
The recombination is due to a two-dimensional density of traps, Nts, as the traps only exist at the surface or interface.
Understanding the impacts and the ways to limit surface recombination leads to better and more robust solar cell designs. Surface recombination is high in solar cells, but can be limited.
13 Question: The intrinsic carrier density is 1.5 × 1016 m–3. If the mobility of electron and hole are 0.13 and 0.05 m2 V–1 s–1, calculate the conductivity.
14 Question: What is Drift current and diffusion currents? Derive expression for total hole and total electron current?
Solution:
In case of semiconductors we observe two kinds of currents.
- Drift current
- Diffusion current
Drift current Definition:- The flow of electric current due to the motion of charge carriers under the influence of external electric field is called drift current.
When an electric field E is applied across a semiconductor material, the charge carriers attain a drift velocity vd
So drift velocity vd =μ .E
The relation between current density J and drift velocity vd is
J = Nqvd
Where N is the carrier concentration
q is the charge of electron or hole
From equations (1) and (2), we get
Jdrift = Nq μE
μ is the mobility of charge carrier.
The above equation shows the general expression for drift current density. Drift current density due to electrons is
Je(drift) = neμeE
Where n is the electrons carrier concentration
And μe is the mobility of electrons.
Drift current density due to holes is
Jh(drift) = peμhE
Where p is the carrier concentration of holes.
μh is the mobility of holes
So Total drift current density
Jdrift (total) = Je(drift) + Jh(drift) = neμeE + peμhE
= eE (nμe+pμh )
Diffusion current
Definition:-The flow of electric current due to the motion of charge carriers under concentration gradient is called diffusion current
Or
The motion of charge carriers from the region of higher concentration to lower concentration leads to a current called diffusion current.
Let ∆N be the excess electron concentration. Then according to Fick’s law, the rate of diffusion of charge carriers is proportional to concentration gradient
Rate of diffusion of charge ∝ -
= - D
Where D is the diffusion coefficient of charge carriers.
The negative sign indicates decrease of N with increase of x So,
The diffusion current density Jdiffu is
Jdiffu = - qD
Where q is the charge of the charge carrier
Diffusion current density due to holes is
Jdiffu (hole) = - eDh
Diffusion current density due to electrons is (as electron carry negative charge so we will get +sign here.
Jdiffu (electrons) = eDe
Jdiffu (total) = Jdiffu (hole) + Jdiffu (electrons)
Jdiffu (total) = - eDh
+ eDe
The expression for total current density due to holes is
Jh (total) = Jh(drift) + Jdiffu (hole) = peμhE - eDh
The expression for total current density due to electrons is
Je (total) = Je(drift) + Jdiffu (electrons) = neμeE + eDe
15 Question: The intrinsic carrier density is 1.5 × 1016 m–3. If the mobility of electron and hole are 0.13 and 0.05 m2 V–1 s–1, calculate the conductivity.
16 Question: Explain construction and principle of operation of bipolar junction transistor?
Solution:
Bipolar junction transistor or BJT is simply known as Transistor. A transistor consists of two pn junctions formed by sandwiching either p-type or n-type semiconductor between a pair of opposite types.
The prefix 'trans' means the signal transfer property of the device while 'istor ' classifies it as a solid element in the same general family with resistors.
Accordingly there are two types of transistors
- n-p-n transistor
- p-n-p transistor
An n-p-n transistor is composed of two n-type semiconductors separated by a thin section of p-type as shown in Figure 20 (i).
However, a p-n-p transistor is formed by two p-sections separated by a thin section of n-type as shown in Figure 20(ii).
Figure 20 (i) n-p-n transistor (ii) p-n-p transistor
In each type of transistor, the following points may be noted:
(i) Transistor may be regarded as a combination of two diodes connected back to back.
(ii) There are three terminals, one taken from each type of semiconductor.
(iii) The middle section is a very thin layer. The thin layer plays an important role in the function of a transistor.
A transistor has two pn junctions. One junction is forward biased and the other is reverse biased. The forward biased junction has a low resistance path whereas a reverse biased junction has a high resistance path. The weak signal is introduced in the low resistance circuit and output is taken from the high resistance circuit. Therefore, a transistor transfers a signal from a low resistance to high resistance.
A transistor (pnp or npn) has three sections of doped semiconductors. The section on one side is the emitter and the section on the opposite side is the collector. The middle section is called the base and forms two junctions between the emitter and collector.
Figure 21 (i) p-n-p transistor (ii) n-p-n transistor
Emitter: The section on one side that supplies charge carriers (electrons or holes) is called the emitter. The emitter is always forward biased w.r.t. Base so that it can supply a large number of majority carriers. In Figure 21 (i), the emitter (p-type) of pnp transistor is forward biased and supplies hole charges to its junction with the base. Similarly, in Figure 21 (ii), the emitter (n-type) of npn transistor has a forward bias and supplies free electrons to its junction with the base.
Collector: The section on the other side that collects the charges is called the collector. The collector is always reverse biased. Its function is to remove charges from its junction with the base. In Figure 21 (i), the collector (p-type) of pnp transistor has a reverse bias and receives hole charges that flow in the output circuit. Similarly, In Figure 21 (ii), the collector (n-type) of npn transistor has reverse bias and receives electrons.
Base: The middle section which forms two pn junctions between the emitter and collector is called the base. The base emitter junction is forward biased allowing low resistance for the emitter circuit. The base-collector junction is reverse biased and provides high resistance in the collector circuit.
Before studying action of transistor, some important facts about transistor should be keep in mind.
- The transistor has two pn junctions i.e. it is like two diodes. The junction between emitter and base may be called emitter-base diode or simply the emitter diode. The junction between the base and collector may be called collector-base diode or simply collector diode.
- The emitter diode is always forward biased whereas collector diode is always reverse biased.
- Three regions of transistor are emitter, base and collector. The base is much thin as compared to emitter while "collector is wider than both. To keep it simple it is assumed emitter and collector is considered to be of same size.
- The emitter is heavily doped so that it can inject a large number of charge carriers (electrons or holes) into the base. The base is lightly doped and very thin it passes most of the emitter injected charge carriers to the collector. The collector is moderately doped.
- The resistance of emitter diode (forward biased) is very small as compared to collector diode (reverse biased). Therefore, forward bias applied to the emitter diode is generally very small whereas reverse bias on the collector diode is much higher.
Working of Transistor
(i) Working of npn transistor: As discussed in the npn transistor emitter-base junction is forward biased to emitter-base junction and collector-base junction is reverse biased. The forward bias causes the electrons in the n-type emitter to flow towards the base. This constitutes the emitter current Ie.
Ie = Ib + Ic
Figure 22: n-p-n transistor
As these electrons flow through the p-type base, they tend to combine with holes. As the base is lightly doped and very thin, therefore, only a few electrons (less than 5%) combine with holes to constitute base current Ib. The remainder i.e. more than 95% cross over into the collector region to constitute collector current Ic In this way, almost the entire emitter current flows in the collector circuit. It is clear that emitter current is the sum of collector and base currents i.e.
(ii) Working of pnp transistor: In the pnp transistor emitter-base junction is forward biased to emitter-base junction and collector-base junction is reverse biased. The forward bias causes the holes in the p-type emitter to flow towards the base. This constitutes the emitter current Ie. As these holes cross into n-type base, they tend to combine with the electrons. As the base is lightly doped and very thin, therefore, only a few holes (less than 5%) combine with the electrons.
Figure 23: p-n-p transistor
The remainder (more than 95%) cross into the collector region to constitute collector current Ic. In this way, almost the entire emitter current flows in the collector circuit. It may be noted that current conduction within pnp transistor is by holes. However, in the external connecting wires the current is still by electrons.
Thus the input circuit (i.e. emitter-base junction) has low resistance because of forward bias whereas output circuit (i.e. collector base junction) has high resistance due to reverse bias. As we have seen, the input emitter current almost entirely flows in the collector circuit. Therefore a transistor transfers the input signal current from a low resistance circuit to a high-resistance circuit.
17 Question: Find the resistivity of an intrinsic semiconductor with intrinsic concentration of 2.5 × 1019 per m3. The mobilities of electrons and holes are 0.40 m2/ V-s and 0.20 m2/ V-s.
Solution:
Given data are:
Intrinsic concentration (ni) = 2.5 × 1019/m3
Mobility of electrons (μn) = 0.40 m2/V-s
The mobility of holes (μp) = 0.20 m2/V-s
The conductivity of an intrinsic semiconductor (σi) = nie[μn + μp]
18 Question: Derive hall coefficient? Show how hall coefficient and mobility are related?
Solution:
When the magnetic field is applied perpendicular to a current-carrying conductor, then a voltage is developed in the material perpendicular to both the magnetic field and current in the conductor. This effect is known as Hall Effect and the voltage developed is known as Hall voltage (VH).
Hall Effect is useful to identify the nature of charge carriers in material and hence to decide whether the material is an n-type semiconductor or p-type semiconductor, also to calculate carrier concentration and mobility of carriers.
Hall Effect can be explained by considering a rectangular block of an extrinsic semiconductor in which current is flowing along the positive X-direction and magnetic field B is applied along Z-direction as shown in Figure.
11 Figure: Hall Effect
Suppose if the semiconductor is n-type, then mostly the carriers are electrons and the electric current is due to the drifting of electrons along negative X-direction or if the semiconductor is p-type, then mostly the carriers are holes and the electric current is due to drifting of the holes along positive X-direction.
As these carriers are moving in the magnetic field in the semiconductor that means they experience Lorentz force represented by FL
FL = Bevd
Where vd is the drift velocity of the carriers. (already explained in the previous section).
We can obtain the direction of this force by applying Fleming’s left-hand rule in electromagnetism.
Fleming’s left-hand rule can be explained as If we stretch the thumb, forefinger, and middle finger in three perpendicular directions so that the forefinger is parallel to the magnetic field and the middle finger is parallel to the current direction, then the thumb represents the direction of the force on the current-carrying carriers.
So the Lorentz force is exerted on the carriers in the negative Y-direction. Due to Lorentz force, more and more carriers will be deposited at the bottom face (represented by face 1in the figure) of the conductor.
The deposition of carriers at the bottom face is continued till the repulsive force due to accumulated charge balances the Lorentz force.
After some time of the applied voltage, both the forces become equal in magnitude and act in opposite direction, then the potential difference between the top and bottom faces is equal to Hall voltage and that can be measured.
At equilibrium, the Lorentz force on a carrier
FL = Bevd……………..(1)
And the Hall force
FH = eEH ……………..(2)
Where EH is the Hall electric field due to accumulated charge.
At equilibrium, FH = FL
EEH = Bevd
∴ EH = Bvd ……………..(3)
If ‘d’ is the distance between the upper and lower surfaces of the slab, then the Hall field
EH = 
……………..(4)
In n-type material, Jx = –nevd
vd = - 
……………..(5)
Where n is free electron concentration, substituting (5) in (3), we have
∴ EH = -B 
……………..(6)
For a given semiconductor, the Hall field EH is proportional to the current density Jx and the intensity of magnetic field ‘B’ in the material.
i.e. EH ∝ JxB
(or) EH = RHJxB ……………..(7)
Where RH = Hall coefficient
Equations (6) and (7) are the same so, we have
RHJxB =-B 
RH = - 
= - 
……………..(8)
Where ρ is the charge density
Similarly for p-type material
RH = 
=
……………..(9)
Using Equations (8) and (9), carrier concentration can be determined.
Thus, the Hall coefficient is negative for n-type material. In n-type material, as the more negative charge is deposited at the bottom surface, so the top face acquires positive polarity and the Hall field is along negative Y-direction. The polarity at the top and bottom faces can be measured by applying probes.
Similarly, in the case of p-type material, a more positive charge is deposited at the bottom surface. So, the top face acquires negative polarity and the Hall field is along positive Y-direction. Thus, the sign of the Hall coefficient decides the nature of (n-type or p-type) material.
The Hall coefficient can be determined experimentally in the following way:
Multiplying Equation (7) with ‘d’, we have
EHd = VH = RHJxBd ……………..(10)
From (Figure 15) we know the current density Jx
Jx = 
Where W is the width of the box. Then, Equation (10) becomes
VH = RH
Bd = RH
RH = 
……………..(11)
Substituting the measured values of VH, Ix, B, and W in Equation(11), RH is obtained. The polarity of VH will be the opposite for n- and p-type semiconductors.
The mobility of charge carriers can be found by using the Hall effect, for example, the conductivity of electrons is
n = neμn
Or we can rewrite it as
μn = 
=n RH……………..(12)
By using equation (11)
μn = n
……………..(13)
19 Question: Discuss the application of the Hall Effect?
Solution:
• Using magnetic flux leakage – To properly inspect items such as pipes or tubes, Hall Effect probes work with something called magnetic flux leakage. This is a way of testing such items, and being able to spot potential corrosion, erosion, or pitting. This is specifically used in steel items and can give important information about lifespan or safety.
• Sensors to detect rotation speed – A Hall Effect probe can be used in bicycle wheels, speedometers in the automotive world, electronic types of ignition systems, and gear teeth.
• Used to detected movement – You will often find a Hall Effect probe used in such items as Go-Kart controls, smartphones, paintball guns, or airsoft guns, as well as some GPS systems.
• Ferrite Toroid Hall Effect current transducers – This is mainly used in electronic compasses, making use of the magnetic field to show direction.
• Split-ring clamp-on sensors – These types of Hall Effect probes are used to test equipment without having to take the whole circuit board apart, e.g. Complex items.
• Analog multiplication – Anything which needs a power measurement, e.g. Sensing, and is also used in small computers.
• General power measurement – Any device which needs to be tested for its power input can be done by a Hall Effect probe.
• Position and motion sensors – This is mainly used in a DC motor, often the brushless type.
• The automotive world – Hall Effect probes are used widely in the automotive world, especially in fuel injection and ignition. Wheel rotation sensors also use Hall Effect probes, e.g. For anti-lock braking.
- For determination of the type of given semiconductor.
- For N-type, Hall coefficient RH= negative
- For P-type, Hall coefficient RH= Positive
- To determine carrier concentration n and p; that is n=p=1/e𝑅𝐻
- Determination of mobility of charge carriers μn =
=n RH. Where 𝜎= electrical conductivity - To determine the sign of charge carriers whether the conductivity is due to electrons or holes.
Main Advantages of Using Hall Sensors
Why is a Hall Effect probe advantageous in all of these instances? Because the probes are not affected by outside influences, e.g. Water or dirt. They can also easily sense the measure of the output they need when they are placed in the right position. On top of this, Hall Effect probes are safer, because the voltage never actually makes it directly to the sensor/probe. This makes this type of measurement overall so much safer than other methods.
As you can see, understanding how something is put into practice in the real world helps you to understand it in real terms. The Hall Effect is certainly very commonplace these days, in much more methods and applications than we realize. While certainly very useful in the automation world, even basic items such as a compass make large use of this scientific approach.
s
Unit 2
SEMICONDUCTOR PHYSICS
1 Question: Explain P-N Junction Diode and its I-V Characteristic?
Solution:
A P-N Junction Diode is formed by doping one side of a piece of silicon with a P-type dopant (Boron) and the other side with a N-type dopant (phosphorus). Ge can be used instead of Silicon. The P-N junction diode is a two-terminal device. This is the basic construction of the P-N junction diode. It is one of the simplest semiconductor devices as it allows current to flow in only one direction.
ZERO BIASED CONDITION
In this case, no external voltage is applied to the P-N junction diode; and therefore, the electrons diffuse to the P-side and simultaneously holes diffuse towards the N-side through the junction, and then combine with each other. Due to this an electric field is generated by these charge carriers. The electric field opposes further diffusion of charged carriers so that there is no movement in the middle region. This region is known as depletion width or space charge.
Figure 14: Unbiased or zero biased PN Junction Diode
FORWARD BIAS
In the forward bias condition, the positive terminal of the battery is connected to the P-Type material and the negative terminal of the battery is connected to the N-type material. This connection is also called as giving positive voltage.
Figure 15: Forward bias
Electrons from the N-region cross the junction and enters the P-region. Due to the attractive force that is generated in the P-region the electrons are attracted and move towards the positive terminal. Simultaneously the holes are attracted to the negative terminal of the battery. By the movement of electrons and holes current flows. In this condition, the width of the depletion region decreases due to the reduction in the number of positive and negative ions.
If this external voltage Vf becomes greater than the value of the potential barrier, approx. 0.7 volts for silicon and 0.3 volts for germanium, the potential barriers opposition will be overcome and current will start to flow.
This is because the negative voltage pushes or repels electrons towards the junction giving them the energy to cross over and combine with the holes being pushed in the opposite direction towards the junction by the positive voltage. This results in a characteristics curve of zero current flowing up to this voltage point, called the “knee” on the static curves and then a high current flow through the diode with little increase in the external voltage as shown in I-V characteristics.
REVERSE BIAS
In the reverse bias condition, the negative terminal of the battery is connected to the P-type material and the positive terminal of the battery is connected to the N-type material. This connection is also known as giving negative voltage.
Figure 16: Reverse bias
The positive voltage applied to the N-type material attracts electrons towards the positive electrode and away from the junction, while the holes in the P-type end are also attracted away from the junction towards the negative electrode.
The net result is that the depletion layer grows wider due to a lack of electrons and holes and presents a high impedance path, almost an insulator. The result is that a high potential barrier is created thus preventing current from flowing through the semiconductor material.
This condition represents a high resistance value to the PN junction and practically zero current flows through the junction diode with an increase in bias voltage. However, a very small leakage current does flow through the junction which can be measured in micro-amperes, ( μA ).
If the reverse bias voltage Vr applied to the diode is increased to a sufficiently high enough value, it will cause the diode’s PN junction to overheat and fail due to the avalanche effect around the junction. This may cause the diode to become shorted and will result in the flow of maximum circuit current, and this shown as a step downward slope in the reverse static characteristics curve in I-V characteristics.
I-V CHARACTERISTICS OF PN JUNCTION DIODE
The I-V Characteristic Curves, which is short for Current-Voltage Characteristic Curves or simply I-V curves of an electrical device
The application of a forward biasing voltage on the junction diode results in the depletion layer becoming very thin and narrow which represents a low impedance path through the junction thereby allowing high currents to flow. The point at which this sudden increase in current takes place is represented on the static I-V characteristics curve above as the “knee” point. The current starts increasing with increase in voltage. At knee voltage current shows a sharp increment in its magnitude. This behaviour is mentioned above. As large current flow in forward biasing so we measure this current in mA.
When a junction diode is Reverse Biased, the thickness of the depletion region increases and the diode acts like an open circuit blocking current flow. So only a very small leakage current will flow.
Figure 17: I-V characteristics
2 Question: Discuss conduction in intrinsic semiconductor?
Solution:
Intrinsic Semiconductor
- An intrinsic type of semiconductor material made to be very pure chemically. As a result it possesses a very low conductivity level having very few number of charge carriers, namely holes and electrons, which it possesses in equal quantities.
- The most commonly used semiconductor basics material by far is silicon. Silicon has four valence electrons in its outermost shell which it shares with its neighbouring silicon atoms to form full orbital’s of eight electrons. The structure of the bond between the two silicon atoms is such that each atom shares one electron with its neighbour making the bond very stable.
- As there are very few free electrons available to move around the silicon crystal, crystals of pure silicon (or germanium) are therefore good insulators. Silicon atoms are arranged in a definite symmetrical pattern making them a crystalline solid structure. A crystal of pure silica (silicon dioxide or glass) is generally said to be an intrinsic crystal (it has no impurities) and therefore has no free electrons.
- An extremely pure semiconductor is called as Intrinsic Semiconductor. On the basis of the energy band phenomenon, an intrinsic semiconductor at absolute zero temperature is shown below.
- Its valence band is completely filled and the conduction band is completely empty. When the temperature is raised and some heat energy is supplied to it, some of the valence electrons are lifted to conduction band leaving behind holes in the valence band as shown below.
- A hole is the absence of an electron in a particular place in an atom. Although it is not a physical particle in the same sense as an electron, a hole can be passed from atom to atom in a semiconductor material. It is considered to have positive charge. Holes are positive charge carriers.
- The electrons reaching at the conduction band move randomly. The holes created in the crystal also free to move anywhere.
- This behaviour of the semiconductor shows that they have a negative temperature coefficient of resistance. This means that with the increase in temperature, the resistivity of the material decreases and the conductivity increases.
- But simply connecting a silicon crystal to a battery supply is not enough to extract an electric current from it. To do that we need to create a “positive” and a “negative” pole within the silicon allowing electrons and therefore electric current to flow out of the silicon. These poles are created by doping the silicon with certain impurities.
3 Question: What is doping? Discuss its importance in semiconductor?
Solution:
Doping
The process by which an impurity is added to a semiconductor is known as Doping. The amount and type of impurity which is to be added to a material has to be closely controlled during the preparation of extrinsic semiconductor. Generally, one impurity atom is added to a 108 atoms of a semiconductor.
The purpose of adding impurity in the semiconductor crystal is to increase the number of free electrons or holes to make it conductive.
If a Pentavalent impurity, having five valence electrons is added to a pure semiconductor a large number of free electrons will exist. Which make a n-type extrinsic semiconductor.
If a trivalent impurity having three valence electrons is added, a large number of holes will exist in the semiconductor. Which make a p-type extrinsic semiconductor.
4 Question: What is P-type Extrinsic Semiconductor? Draw its energy level diagram and discuss the conduction mechanism in p type semiconductor?
Solution:
P-type Extrinsic Semiconductor
The extrinsic p-Type Semiconductor is formed when a trivalent impurity is added to a pure semiconductor in a small amount, and as a result, a large number of holes are created in it. A large number of holes are provided in the semiconductor material by the addition of trivalent impurities like Gallium and Indium. Such type of impurities which produces p-type semiconductor are known as an Acceptor Impurities because each atom of them create one hole which can accept one electron.
In a P-type semiconductor material there is a shortage of electrons, i.e. there are 'holes' in the crystal lattice. Electrons may move from one empty position to another and in this case it can be considered that the holes are moving. This can happen under the influence of a potential difference and the holes can be seen to flow in one direction resulting in an electric current flow. It is actually harder for holes to move than for free electrons to move and therefore the mobility of holes is less than that of free electrons. Holes are positively charged carriers.
A trivalent impurity like Aluminium, having three valence electrons is added to Silicon crystal in a small amount. Each atom of the impurity fits in the Silicon crystal in such a way that its three valence electrons form covalent bonds with the three surrounding Silicon atoms as shown in the figure below.
Energy Band Diagram of p-Type Semiconductor
The energy band diagram of a p-Type Semiconductor is shown below.
A large number of holes or vacant space in the covalent bond is created in the crystal with the addition of the trivalent impurity. A small or minute quantity of free electrons is also available in the conduction band.
They are produced when thermal energy at room temperature is imparted to the Silicon crystal forming electron-hole pairs. But the holes are more in number as compared to the electrons in the conduction band. It is because of the predominance of holes over electrons that the material is called as a p-type semiconductor. The word “p” stands for the positive material.
Conduction Through p Type Semiconductor
In p type semiconductor large number of holes are created by the trivalent impurity. When a potential difference is applied across this type of semiconductors.
The holes are available in the valence band are directed towards the negative terminal. As the current flow through the crystal is by holes, which are carrier of positive charge, therefore, this type of conductivity is known as positive or p type conductivity. In a p type conductivity the valence electrons move from one covalent to another.
The conductivity of n type semiconductor is nearly double to that of p type semiconductor. The electrons available in the conduction band of the n type semiconductor are much more movable than holes available in the valence band in a p type semiconductor. The mobility of holes is poor as they are more bound to the nucleus.
Even at the room temperature the electron hole pairs are formed. These free electrons which are available in minute quantity also carry a little amount of current in the p type semiconductors.
5 Question: For an intrinsic Semiconductor with a band gap of 0.7 eV, determine the position of EF at T = 300 K if m*h = 6m*e
Solution:
6 Question: What is N-type Extrinsic Semiconductor? Draw its energy level diagram and discuss the conduction mechanism in N type semiconductor?
Solution:
N-type Extrinsic Semiconductor
When a few Pentavalent impurities such as Phosphorus whose atomic number is 15, which is categorised as 2, 8 and 5. It has five valence electrons, which is added to silicon crystal. Each atom of the impurity fits in four silicon atoms as shown in the figure below.
Hence, each Arsenic atom provides one free electron in the Silicon crystal. Since an extremely small amount of Phosphorus, impurity has a large number of atoms; it provides millions of free electrons for conduction.
An N-type semiconductor material has an excess of electrons. In this way, free electrons are available within the lattices and their overall movement in one direction under the influence of a potential difference results in an electric current flow. This in an N-type semiconductor, the charge carriers are electrons.
Energy Diagram of n-Type Semiconductor
A large number of free electrons are available in the conduction band because of the addition of the Pentavalent impurity. These electrons are free electrons which did not fit in the covalent bonds of the crystal. However, a minute quantity of free electrons is available in the conduction band forming hole- electron pairs.
The Energy diagram of the n-type semiconductor is shown in the figure below.
- The addition of pentavalent impurity results in a large number of free electrons.
- When thermal energy at room temperature is imparted to the semiconductor, a hole-electron pair is generated and as a result, a minute quantity of free electrons is available. These electrons leave behind holes in the valence band.
- Here n stands for negative material as the number of free electrons provided by the pentavalent impurity is greater than the number of holes.
Conduction Through n-Type Semiconductor
In the n-type semiconductor, a large number of free electrons are available in the conduction bands which are donated by the impurity atoms. The figure below shows the conduction process of an n-type semiconductor.
When a potential difference is applied across this type of semiconductor, the free electrons are directed towards the positive terminals. It carries an electric current. As the flow of current through the crystal is constituted by free electrons which are carriers of negative charge, therefore, this type of conductivity is known as negative or n-type conductivity.
The electron-hole pairs are formed at room temperature. These holes which are available in small quantity in valence band also consist of a small amount of current. For practical purposes, this current is neglected.
7 Question: Derive expression for carrier concentration for intrinsic semiconductors? Explain Fermi level for Intrinsic semiconductors and what is the effect of temperature on Fermi level?
Solution:
Intrinsic semiconductors—carrier concentration
Here we will calculate the number of electrons excited into conduction band at temperature T and also the hole concentration in the valence band. It is assumed that the electrons in the conduction band behave as if they are free particles with effective mass me* and the holes near the top of the valence band behave as if they are free particles with effective mass mh*.
Here we will calculate the electron concentration, hole concentration
Density of Electrons in Conduction Band
The number of free electrons per unit volume of semiconductor having energies in between E and E + dE is represented as N(E) dE
dE = width of Energy band
Therefore, we have:
N(E) dE = ge(E) dE fe(E) ……….(1)
ge(E) = The density of electron states per unit volume
fe(E) = Fermi-Dirac distribution function i.e. probability that an electron occupies an electron state
The number of electrons present in the conduction band per unit volume of material ‘n’ is obtained by integrating N(E) dE between the limits Ec and Ect
Where Ec = the bottom energy levels of conduction band
Ect = the bottom and top energy levels of conduction band
n = 
……….(2)
Can be written as
n = 
= 
n = 
- 
……….(3)
We know that above Ect, there is no electrons.
Hence, Equation (3) becomes
n = 
- 
n = 
……….(4)
The Fermi-Dirac distribution function fe(E) can be represented as:
(E) =
……….(5)
Compared to the exponential value, so the ‘1’ in the denominator can be neglected.
So 
>>1
Hence, 
(E) =
= 
……….(6)
The density of electron states ge(E) in the energy space from E = 0 to E can be written as:
(E) = 
[
]3/2 (E - 0) ½ ……….(7)
Where me* is the effective mass of an electron and h is Planck’s constant.
(E)dE = 
[
]3/2 (E - 0) ½dE ……….(8)
To evaluate n, the density of states is counted from Ec, since the minimum energy state in conduction band is Ec. So eq (8) can become
(E)dE = 
[
]3/2 (E - Ec) ½dE ……….(9)
Substituting Equations (6) and (9) in (4) gives
n = 
3/2
(E-
)1/2
dE
n =
[
]3/2 
1/2
dE………… (10)
The above equation can be simplified by the following substitution:
Put ɛ = E − Ec ………… (11)
So, dɛ = dE
In Equation (11), Ec is constant, as we change the variable E to ε in Equation (10), the integral limits also change.
In Equation (11), as E → Ec then ε → 0 and E → ∞, then ε also → ∞.
The exponential term in Equation (10) becomes:
= exp [
] = exp [
]
………… (12)
Substituting Equations (11) and (12) in (10), we get:
n =
[
]3/2 
dE
n =
[
]3/2
dE………… (13)
Above integral (I) can be simplified by substitution. Put ε = x2
So that dɛ = 2x dx
I =
2x dx
=
dx
=
=
T)3/2………… (14)
Substituting Equation (14) in (13) gives:
n =
[
]3/2
T)3/2
n =
n = 
n = 
………… (15)
The term 
is almost a constant compared with the exponential term as the temperature changes. So, it is a pseudo constant and is given by the symbol Nc. So, we have
n =Nc
………… (16)
Density of Holes in Valence Band
The number of holes per unit volume of semiconductor in the energy range E and E + dE in valence band is represented as P(E) dE. Proceeding same way( as in case of electrons) we have
Therefore, we have:
P(E) dE = gh(E) dE fh(E) ……….(17)
dE = width of Energy band
gh(E) = The density of holes states per unit volume
fh(E) = Fermi-Dirac distribution function i.e. probability that an hole occupies an electron state
The number of electrons present in the conduction band per unit volume of material ‘n’ is obtained by integrating P(E) dE between the limits Evb and EV
Where EV = the bottom energy levels of valence band
Evb = the bottom and top energy levels of valence band
The total number of holes present in the valence band per unit volume of material ‘p’ is obtained by integrating P(E) dE
……….(18)
Equation (18) can be represented as:
……….(19)
Now we know that below Evb no holes is present. Hence, Equation (19) becomes
……….(20)
We know hole can also be defined as absence of an electron.
Presence of a hole = the absence of an electron
Hence, the Fermi-Dirac function of holes fh(E) in the valence band is:
Compared to exponential, the ‘1’ in the denominator is negligible,
Hence,
……….(21)
The density of hole states between E and E + dE in valence band can be written similar to Equation (8.9) for electrons.
……….(22)
Where mh* is the effective mass of hole.
Substituting Equations (21) and (22) in (20),
……….(23)
The above equation can be simplified by the substitution:
Put ɛ = EV − E ............. (24)
So dɛ = − dE
In Equation (24), EV is constant, as we change the variable E to ε in Equation (23), the integral limits also change.
In Equation (24),
As E → EV then ε → 0
And E→ −∞, then ε → ∞
The exponential term in Equation (23) becomes:
............. (25)
Substituting Equations (24) and (25) in (23), we get:
………….(26)
From Equation (14), we know the integral value 
T)3/2 put here and we get
………….(27)
….The term 
is almost constant compared with the exponential term as the temperature changes. So, it is a pseudo constant and is given by the symbol NV. So, we have
…………… (28)
8 Question: Explain Fermi level for Intrinsic semiconductors and what is the effect of temperature on Fermi level?
Solution:
Fermi Level
We know that in intrinsic semiconductor
Electron concentration ‘n’ = Hole concentration ‘p’
Equating Equations (15) and (27), we get
Taking logarithms on both sides, we get
……………… (29)
Normally, mh* is greater than me*, since ln 
is very small so that EF is just lie in the middle of energy gap
Temperature effect on Fermi level
Fermi level slightly rises with increase of temperature.
But in case of pure intrinsic semiconductor like Si and Ge,
mh* ≈ me*
So in these cases Fermi level lies at the middle of energy gap.
9 Question: Explain Zener Diode and its I-V Characteristic?
Solution:
The satisfactory explanation of this breakdown of the junction was first given by the American scientist C. Zener. A zener diode is a special type of diode that is designed to operate in the reverse breakdown region.
An ordinary diode operated in this region will usually be destroyed due to excessive current. This is not the case for the zener diode.
It has already been discussed that when the reverse bias on a crystal diode is increased, a critical voltage, called breakdown voltage is reached where the reverse current increases sharply to a high value. The breakdown region is the knee of the reverse characteristic as shown in Figure 19.
The breakdown voltage is sometimes called zener voltage and the sudden increase in current is known as zener current.
The breakdown or zener voltage depends upon the amount of doping. If the diode is heavily doped, depletion layer will be thin and consequently the breakdown of the junction will occur at a lower reverse voltage.
On the other hand, a lightly doped diode has a higher breakdown voltage. When an ordinary crystal diode is properly doped so that it has a sharp breakdown voltage, it is called a zener diode. A properly doped crystal diode which has a sharp breakdown voltage is known as a zener diode.
Figure 18 shows the symbol of a zener diode. It may be seen that it is just like an ordinal, diode except that the bar is turned into z-shape.
Figure 18: Symbol of a zener diode
Figure 19: I-V characteristics of zener diode
As the curve reveals, two things happen when Vz is reached:
- The diode current increases rapidly.
- The reverse voltage Vz across the diode remains almost constant.
The following points may be noted
(i) A zener diode is like an ordinary diode except that it is properly doped so as to have a sharp breakdown voltage.
(ii) A zener diode is always reverse connected i.e. it is always reverse biased.
(iii) A zener diode has sharp breakdown voltage, called zener voltage Vz.
(iv) When forward biased, its characteristics are just those of ordinary PN junction diode.
(v) The zener diode is not immediately burnt just because it has entered the breakdown region. As long as the external circuit connected to the diode limits the diode current to less than burn out value, the diode will not burn out.
In other words, the zener diode operated in this region will have a relatively constant voltage across it, regardless of the value of current through the device. This permits the zener diode to be used as a voltage regulator or voltage stabiliser.
10 Question: Calculate the intrinsic concentration of charge carriers at 300 K given that m *e =0.12m o ,m *h =0.28mo and the value of brand gap = 0.67 eV.
Solution:
11 Question: Where Fermi level lie in case of p type and n type extrinsic semiconductor?
Or
Derive expression for Carrier Concentration in Extrinsic Semiconductors.
Solution:
Carrier Concentration in Extrinsic Semiconductors
The number of charge carriers present per unit volume of a semiconductor material is called carrier concentration.
Suppose donor and acceptor atoms are doped in a semiconductor.
At temperature T K,
n = number of conduction electrons
p = number of holes
N−A = number of acceptor ions
N+D = number of donor ions
We know that the below equation hold good in semiconductor. The total negative charge due to conduction electrons and acceptor ions is equal to holes and donor ions in unit volume of material.
So the material will be considered neutral if,
n + N−A = p + N+D ……….(34)
Equation (34) is called charge neutrality equation.
In the above equation
………..(35)
Concentration of acceptor ions N−A = acceptor concentration x probability of finding an electron in acceptor level
………..(36)
Similarly, the donor ions concentration is
………..(37)
In n-type material
Note
nn represents electrons in n-type material
pn represents holes concentration in n-type material.
There are no acceptor atoms so N−A = 0.
At 0 K, all the electron states in donor level are occupied by electrons.
As the temperature is increased from 0 K, some of the electrons jump from these donor states into the conduction band.
Also the concentration of holes is extremely less compared with the concentration of conduction electrons [p << n]
From Equation (34) we have
n = p N+D
(Or) n ≈ N+D …………(38) [since p << N+D]
At temperature T K,
As the temperature increase Almost all the donor atoms donate electrons to conduction band.
So, in n-type material, the free electron concentration is almost equal to the donor atoms.
So we can rewrite the above equation as
nn ≈ ND …………….(39)
Where nn represents electrons in n-type material
Also the hole concentration in n-type material can be obtained by applying law of mass action nn pn =ni2
…………..(40)
Where pn represents holes concentration in n-type material.
In n-type material at 0 K, the Fermi energy level lies in the middle of Ec and ED
At temperature> 0K
……….(41)
With increase in temperature the Fermi level shifts upwards according to Equation (61) slightly due to ionization of donor atoms.
With further increase of temperature, electron-hole pairs are generated due to the breaking of covalent bonds, hence Fermi level shifts downwards.
In p-type semiconductor
Note
pp represents holes in p-type material
np represents electrons in p-type material
There are no donor atoms so means no ions present 
= 0.
At 0 K, all the acceptor levels are not occupied by electrons.
As the temperature is increased from 0 K, some electrons jump from top valence band energy levels to the acceptor states, leaving holes in the valence band and acceptor ions 
are formed.
At some room temperature T K, concentration of conduction electrons is extremely less compared with hole concentration.
∴ From Equation (34), we have
n + N−A = p ……………(42)
(or) N−A ≈ p ……………. (43) [since n << N−A]
At temperature T K, in p-type material,
The hole concentration is almost equal to the acceptor atoms in unit volume of the material.
So, Equation (43) can be written as
pp ≈ NA……………….(44)
Where pp represents holes in p-type material
The electron concentration in p-type material can be obtained by applying law of mass action as nppp = ni2
…………….(45)
Where np represent free electron concentration in p-type material.
In p-type material, the Fermi level lies in between EV and EA at 0 K
As the temperature is increased from 0 K, the Fermi level shifts downwards slightly as per Equation (46) due to ionization of acceptor atoms.
And with further increase of temperature, electron-hole pairs are generated due to the breaking of covalent bonds, so Fermi level shifts upwards.
12 Question: What is Recombination in semiconductor? Discuss any two different processes which lead to recombination?
Solution:
Recombination is the reverse process where electrons from the conduction and holes from valence band recombine and are annihilated.
These processes must conserve both quantized energy and momentum, and the vibrating lattice plays a large role in conserving momentum.
- Recombination and generation are always happening in semiconductors both optically and thermally.
- Their rates are in balance at equilibrium.
- If the product of the rate becomes greater than the recombination rate, again driving the system back towards equilibrium.
- As the electron and hole densities is a constant at equilibrium, maintained by recombination and generation occurring at equal rates.
- When there is a surplus of carriers the rate of recombination becomes greater than the rate of generation, driving the system back towards equilibrium.
In semiconductors several different processes exist which lead to recombination, the most important ones are:
1) Band-to-band recombination
2) Auger generation/recombination
3) Surface recombination
Note- You can discuss any two out of three.
1) Band-to-band recombination or Radiative recombination
Band-to-band recombination or Radiative recombination is the reverse process of photon absorption, where an electron drops back down to its equilibrium energy band and radiates a photon.
It is the process of electrons jumping down from the conduction band to the valence band in a radiative manner.
During band-to-band recombination the energy absorbed by a material is released in the form of photons. Generally these released photons contain the same or less energy than those initially absorbed.
The photon emitted may have the energy of the band gap difference or less, depending on how much energy is lost in the mechanism
Radiative recombination plays a more major role in direct band semiconductors.
The total radiative recombination rate, given below as R, is proportional to the product of the concentration of occupied states (electrons, n) in the conduction band and that of the unoccupied states in the valence band (holes, p)
R= Bnp
Band-to-band recombination depends on the density of available electrons and holes. Both carrier types need to be available in the recombination process. Therefore, the rate is expected to be proportional to the product of n and p. Also, in thermal equilibrium, the recombination rate must equal the generation rate since there is no net recombination or generation. As the product of n and p equals ni2 in thermal equilibrium, the net recombination rate can be expressed as:
U= B(np-ni2)
Where B is the bimolecular recombination constant for a given semiconductor. It can be calculated from the semiconductor’s absorption coefficient. ni2 is the familiar “intrinsic carrier concentration”
Auger recombination
Auger recombination involves three particles: an electron and a hole, which recombine in a band-to-band transition and give off the resulting energy to another electron or hole. The expression for the net recombination rate is therefore similar to that of band-to-band recombination but includes the density of the electrons or holes, which receive the released energy from the electron-hole annihilation.
In other words Auger recombination the energy is given to a third carrier which is excited to a higher energy level without moving to another energy band.
After the interaction, the third carrier normally loses its excess energy to thermal vibrations. Since this process is a three-particle interaction, it is normally only significant in non-equilibrium conditions when the carrier density is very high.
Auger recombination is most important at high carrier concentrations caused by heavy doping. In silicon-based solar cells, Auger recombination limits the lifetime and ultimate efficiency.
The more heavily doped the material is, the shorter the Auger recombination lifetime.
Surface recombination
Recombination at surfaces and interfaces can have a significant impact on the behavior of semiconductor devices.
Areas of defect, such as at the surface of solar cells where the lattice is disordered, recombination is very high.
This is because surfaces and interfaces typically contain a large number of recombination centers because of the abrupt termination of the semiconductor crystal, which leaves a large number of electrically active states.
In addition, the surfaces and interfaces are more likely to contain impurities since they are exposed during the device fabrication process.
The net recombination rate due to trap-assisted recombination and generation is given by:
The recombination is due to a two-dimensional density of traps, Nts, as the traps only exist at the surface or interface.
Understanding the impacts and the ways to limit surface recombination leads to better and more robust solar cell designs. Surface recombination is high in solar cells, but can be limited.
13 Question: The intrinsic carrier density is 1.5 × 1016 m–3. If the mobility of electron and hole are 0.13 and 0.05 m2 V–1 s–1, calculate the conductivity.
14 Question: What is Drift current and diffusion currents? Derive expression for total hole and total electron current?
Solution:
In case of semiconductors we observe two kinds of currents.
- Drift current
- Diffusion current
Drift current Definition:- The flow of electric current due to the motion of charge carriers under the influence of external electric field is called drift current.
When an electric field E is applied across a semiconductor material, the charge carriers attain a drift velocity vd
So drift velocity vd =μ .E
The relation between current density J and drift velocity vd is
J = Nqvd
Where N is the carrier concentration
q is the charge of electron or hole
From equations (1) and (2), we get
Jdrift = Nq μE
μ is the mobility of charge carrier.
The above equation shows the general expression for drift current density. Drift current density due to electrons is
Je(drift) = neμeE
Where n is the electrons carrier concentration
And μe is the mobility of electrons.
Drift current density due to holes is
Jh(drift) = peμhE
Where p is the carrier concentration of holes.
μh is the mobility of holes
So Total drift current density
Jdrift (total) = Je(drift) + Jh(drift) = neμeE + peμhE
= eE (nμe+pμh )
Diffusion current
Definition:-The flow of electric current due to the motion of charge carriers under concentration gradient is called diffusion current
Or
The motion of charge carriers from the region of higher concentration to lower concentration leads to a current called diffusion current.
Let ∆N be the excess electron concentration. Then according to Fick’s law, the rate of diffusion of charge carriers is proportional to concentration gradient
Rate of diffusion of charge ∝ -
= - D
Where D is the diffusion coefficient of charge carriers.
The negative sign indicates decrease of N with increase of x So,
The diffusion current density Jdiffu is
Jdiffu = - qD
Where q is the charge of the charge carrier
Diffusion current density due to holes is
Jdiffu (hole) = - eDh
Diffusion current density due to electrons is (as electron carry negative charge so we will get +sign here.
Jdiffu (electrons) = eDe
Jdiffu (total) = Jdiffu (hole) + Jdiffu (electrons)
Jdiffu (total) = - eDh
+ eDe
The expression for total current density due to holes is
Jh (total) = Jh(drift) + Jdiffu (hole) = peμhE - eDh
The expression for total current density due to electrons is
Je (total) = Je(drift) + Jdiffu (electrons) = neμeE + eDe
15 Question: The intrinsic carrier density is 1.5 × 1016 m–3. If the mobility of electron and hole are 0.13 and 0.05 m2 V–1 s–1, calculate the conductivity.
16 Question: Explain construction and principle of operation of bipolar junction transistor?
Solution:
Bipolar junction transistor or BJT is simply known as Transistor. A transistor consists of two pn junctions formed by sandwiching either p-type or n-type semiconductor between a pair of opposite types.
The prefix 'trans' means the signal transfer property of the device while 'istor ' classifies it as a solid element in the same general family with resistors.
Accordingly there are two types of transistors
- n-p-n transistor
- p-n-p transistor
An n-p-n transistor is composed of two n-type semiconductors separated by a thin section of p-type as shown in Figure 20 (i).
However, a p-n-p transistor is formed by two p-sections separated by a thin section of n-type as shown in Figure 20(ii).
Figure 20 (i) n-p-n transistor (ii) p-n-p transistor
In each type of transistor, the following points may be noted:
(i) Transistor may be regarded as a combination of two diodes connected back to back.
(ii) There are three terminals, one taken from each type of semiconductor.
(iii) The middle section is a very thin layer. The thin layer plays an important role in the function of a transistor.
A transistor has two pn junctions. One junction is forward biased and the other is reverse biased. The forward biased junction has a low resistance path whereas a reverse biased junction has a high resistance path. The weak signal is introduced in the low resistance circuit and output is taken from the high resistance circuit. Therefore, a transistor transfers a signal from a low resistance to high resistance.
A transistor (pnp or npn) has three sections of doped semiconductors. The section on one side is the emitter and the section on the opposite side is the collector. The middle section is called the base and forms two junctions between the emitter and collector.
Figure 21 (i) p-n-p transistor (ii) n-p-n transistor
Emitter: The section on one side that supplies charge carriers (electrons or holes) is called the emitter. The emitter is always forward biased w.r.t. Base so that it can supply a large number of majority carriers. In Figure 21 (i), the emitter (p-type) of pnp transistor is forward biased and supplies hole charges to its junction with the base. Similarly, in Figure 21 (ii), the emitter (n-type) of npn transistor has a forward bias and supplies free electrons to its junction with the base.
Collector: The section on the other side that collects the charges is called the collector. The collector is always reverse biased. Its function is to remove charges from its junction with the base. In Figure 21 (i), the collector (p-type) of pnp transistor has a reverse bias and receives hole charges that flow in the output circuit. Similarly, In Figure 21 (ii), the collector (n-type) of npn transistor has reverse bias and receives electrons.
Base: The middle section which forms two pn junctions between the emitter and collector is called the base. The base emitter junction is forward biased allowing low resistance for the emitter circuit. The base-collector junction is reverse biased and provides high resistance in the collector circuit.
Before studying action of transistor, some important facts about transistor should be keep in mind.
- The transistor has two pn junctions i.e. it is like two diodes. The junction between emitter and base may be called emitter-base diode or simply the emitter diode. The junction between the base and collector may be called collector-base diode or simply collector diode.
- The emitter diode is always forward biased whereas collector diode is always reverse biased.
- Three regions of transistor are emitter, base and collector. The base is much thin as compared to emitter while "collector is wider than both. To keep it simple it is assumed emitter and collector is considered to be of same size.
- The emitter is heavily doped so that it can inject a large number of charge carriers (electrons or holes) into the base. The base is lightly doped and very thin it passes most of the emitter injected charge carriers to the collector. The collector is moderately doped.
- The resistance of emitter diode (forward biased) is very small as compared to collector diode (reverse biased). Therefore, forward bias applied to the emitter diode is generally very small whereas reverse bias on the collector diode is much higher.
Working of Transistor
(i) Working of npn transistor: As discussed in the npn transistor emitter-base junction is forward biased to emitter-base junction and collector-base junction is reverse biased. The forward bias causes the electrons in the n-type emitter to flow towards the base. This constitutes the emitter current Ie.
Ie = Ib + Ic
Figure 22: n-p-n transistor
As these electrons flow through the p-type base, they tend to combine with holes. As the base is lightly doped and very thin, therefore, only a few electrons (less than 5%) combine with holes to constitute base current Ib. The remainder i.e. more than 95% cross over into the collector region to constitute collector current Ic In this way, almost the entire emitter current flows in the collector circuit. It is clear that emitter current is the sum of collector and base currents i.e.
(ii) Working of pnp transistor: In the pnp transistor emitter-base junction is forward biased to emitter-base junction and collector-base junction is reverse biased. The forward bias causes the holes in the p-type emitter to flow towards the base. This constitutes the emitter current Ie. As these holes cross into n-type base, they tend to combine with the electrons. As the base is lightly doped and very thin, therefore, only a few holes (less than 5%) combine with the electrons.
Figure 23: p-n-p transistor
The remainder (more than 95%) cross into the collector region to constitute collector current Ic. In this way, almost the entire emitter current flows in the collector circuit. It may be noted that current conduction within pnp transistor is by holes. However, in the external connecting wires the current is still by electrons.
Thus the input circuit (i.e. emitter-base junction) has low resistance because of forward bias whereas output circuit (i.e. collector base junction) has high resistance due to reverse bias. As we have seen, the input emitter current almost entirely flows in the collector circuit. Therefore a transistor transfers the input signal current from a low resistance circuit to a high-resistance circuit.
17 Question: Find the resistivity of an intrinsic semiconductor with intrinsic concentration of 2.5 × 1019 per m3. The mobilities of electrons and holes are 0.40 m2/ V-s and 0.20 m2/ V-s.
Solution:
Given data are:
Intrinsic concentration (ni) = 2.5 × 1019/m3
Mobility of electrons (μn) = 0.40 m2/V-s
The mobility of holes (μp) = 0.20 m2/V-s
The conductivity of an intrinsic semiconductor (σi) = nie[μn + μp]
18 Question: Derive hall coefficient? Show how hall coefficient and mobility are related?
Solution:
When the magnetic field is applied perpendicular to a current-carrying conductor, then a voltage is developed in the material perpendicular to both the magnetic field and current in the conductor. This effect is known as Hall Effect and the voltage developed is known as Hall voltage (VH).
Hall Effect is useful to identify the nature of charge carriers in material and hence to decide whether the material is an n-type semiconductor or p-type semiconductor, also to calculate carrier concentration and mobility of carriers.
Hall Effect can be explained by considering a rectangular block of an extrinsic semiconductor in which current is flowing along the positive X-direction and magnetic field B is applied along Z-direction as shown in Figure.
11 Figure: Hall Effect
Suppose if the semiconductor is n-type, then mostly the carriers are electrons and the electric current is due to the drifting of electrons along negative X-direction or if the semiconductor is p-type, then mostly the carriers are holes and the electric current is due to drifting of the holes along positive X-direction.
As these carriers are moving in the magnetic field in the semiconductor that means they experience Lorentz force represented by FL
FL = Bevd
Where vd is the drift velocity of the carriers. (already explained in the previous section).
We can obtain the direction of this force by applying Fleming’s left-hand rule in electromagnetism.
Fleming’s left-hand rule can be explained as If we stretch the thumb, forefinger, and middle finger in three perpendicular directions so that the forefinger is parallel to the magnetic field and the middle finger is parallel to the current direction, then the thumb represents the direction of the force on the current-carrying carriers.
So the Lorentz force is exerted on the carriers in the negative Y-direction. Due to Lorentz force, more and more carriers will be deposited at the bottom face (represented by face 1in the figure) of the conductor.
The deposition of carriers at the bottom face is continued till the repulsive force due to accumulated charge balances the Lorentz force.
After some time of the applied voltage, both the forces become equal in magnitude and act in opposite direction, then the potential difference between the top and bottom faces is equal to Hall voltage and that can be measured.
At equilibrium, the Lorentz force on a carrier
FL = Bevd……………..(1)
And the Hall force
FH = eEH ……………..(2)
Where EH is the Hall electric field due to accumulated charge.
At equilibrium, FH = FL
EEH = Bevd
∴ EH = Bvd ……………..(3)
If ‘d’ is the distance between the upper and lower surfaces of the slab, then the Hall field
EH = 
……………..(4)
In n-type material, Jx = –nevd
vd = - 
……………..(5)
Where n is free electron concentration, substituting (5) in (3), we have
∴ EH = -B 
……………..(6)
For a given semiconductor, the Hall field EH is proportional to the current density Jx and the intensity of magnetic field ‘B’ in the material.
i.e. EH ∝ JxB
(or) EH = RHJxB ……………..(7)
Where RH = Hall coefficient
Equations (6) and (7) are the same so, we have
RHJxB =-B 
RH = - 
= - 
……………..(8)
Where ρ is the charge density
Similarly for p-type material
RH = 
=
……………..(9)
Using Equations (8) and (9), carrier concentration can be determined.
Thus, the Hall coefficient is negative for n-type material. In n-type material, as the more negative charge is deposited at the bottom surface, so the top face acquires positive polarity and the Hall field is along negative Y-direction. The polarity at the top and bottom faces can be measured by applying probes.
Similarly, in the case of p-type material, a more positive charge is deposited at the bottom surface. So, the top face acquires negative polarity and the Hall field is along positive Y-direction. Thus, the sign of the Hall coefficient decides the nature of (n-type or p-type) material.
The Hall coefficient can be determined experimentally in the following way:
Multiplying Equation (7) with ‘d’, we have
EHd = VH = RHJxBd ……………..(10)
From (Figure 15) we know the current density Jx
Jx = 
Where W is the width of the box. Then, Equation (10) becomes
VH = RH
Bd = RH
RH = 
……………..(11)
Substituting the measured values of VH, Ix, B, and W in Equation(11), RH is obtained. The polarity of VH will be the opposite for n- and p-type semiconductors.
The mobility of charge carriers can be found by using the Hall effect, for example, the conductivity of electrons is
n = neμn
Or we can rewrite it as
μn = 
=n RH……………..(12)
By using equation (11)
μn = n
……………..(13)
19 Question: Discuss the application of the Hall Effect?
Solution:
• Using magnetic flux leakage – To properly inspect items such as pipes or tubes, Hall Effect probes work with something called magnetic flux leakage. This is a way of testing such items, and being able to spot potential corrosion, erosion, or pitting. This is specifically used in steel items and can give important information about lifespan or safety.
• Sensors to detect rotation speed – A Hall Effect probe can be used in bicycle wheels, speedometers in the automotive world, electronic types of ignition systems, and gear teeth.
• Used to detected movement – You will often find a Hall Effect probe used in such items as Go-Kart controls, smartphones, paintball guns, or airsoft guns, as well as some GPS systems.
• Ferrite Toroid Hall Effect current transducers – This is mainly used in electronic compasses, making use of the magnetic field to show direction.
• Split-ring clamp-on sensors – These types of Hall Effect probes are used to test equipment without having to take the whole circuit board apart, e.g. Complex items.
• Analog multiplication – Anything which needs a power measurement, e.g. Sensing, and is also used in small computers.
• General power measurement – Any device which needs to be tested for its power input can be done by a Hall Effect probe.
• Position and motion sensors – This is mainly used in a DC motor, often the brushless type.
• The automotive world – Hall Effect probes are used widely in the automotive world, especially in fuel injection and ignition. Wheel rotation sensors also use Hall Effect probes, e.g. For anti-lock braking.
- For determination of the type of given semiconductor.
- For N-type, Hall coefficient RH= negative
- For P-type, Hall coefficient RH= Positive
- To determine carrier concentration n and p; that is n=p=1/e𝑅𝐻
- Determination of mobility of charge carriers μn =
=n RH. Where 𝜎= electrical conductivity - To determine the sign of charge carriers whether the conductivity is due to electrons or holes.
Main Advantages of Using Hall Sensors
Why is a Hall Effect probe advantageous in all of these instances? Because the probes are not affected by outside influences, e.g. Water or dirt. They can also easily sense the measure of the output they need when they are placed in the right position. On top of this, Hall Effect probes are safer, because the voltage never actually makes it directly to the sensor/probe. This makes this type of measurement overall so much safer than other methods.
As you can see, understanding how something is put into practice in the real world helps you to understand it in real terms. The Hall Effect is certainly very commonplace these days, in much more methods and applications than we realize. While certainly very useful in the automation world, even basic items such as a compass make large use of this scientific approach.
s
