? Why we need to study this concept?
Ans:- When we are finding electrical properties of semiconductors, we have to know that, the no. of charge carriers per cm³ are available.
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? To analyze the semiconductor devices, we need to know how many number of charge carriers are available.
To study this concept, we need to know about
(i) The Fermi - Dirac distribution function.
(ii) The Fermi Level
(iii) Electron and Hole Concentrations at equilibrium.
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Energy Band Diagram
To analyze the distribution of electrons over a range of allowed energy levels at thermal equilibrium, we can use Fermi Dirac Function f(E) and is given by
f(E) = 1 / (1 + e^((E-Ef)/kT))
Where Ef is called fermi level
FirstRanker.com k = Boltzmann constant.
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FirstRanker.com Distribution function also gives the probability of allowed energies occupied by the electrons. Hence, it is also called Probability Distribution Function.
Calculation of Electron Density :-
When the no. of electrons is very small compared to the available energy levels, the probability of an energy state being occupied by more than one electron is very small. Such a situation is valid when (E-Ef) >> 3kT.
Under this circumstance, the no. of available states in the C.B. is far larger than the no. of es in the band. Then, the F.D. Function can be approximated to Boltzmann function,
F(E) = e^(-(E-Ef)/kT)
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Let dn be the no. of es whose energy lies in the energy interval E and E+dE in the C.B. Then
dn = Z(E) F(E) dE
Where Z(E) dE is the density of states in the Energy Interval E and E+dE and F(E) is the probability that a state of energy is occupied by an electron.
FirstRanker.com Thus, the electron density in the C.B. can be found by integrating es between the limits Ec and 8. Ec is the energy corresponding to the bottom edge of the C.B. and 8, the energy corresponding to the top edge of the C.B.
n = ? Z(E) F(E) dE (limits Ec to 8)
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The density of states in the C.B. is given by
Z(E) dE = (4p / h³) * (2m?)^(3/2) * (E-Ec)^(1/2) dE
Here (E-Ec) be the kinetic energy of the conduction è at higher energy levels.
Thus, putting values from eq's in equation, we obtain
n = ? (4p / h³) * (2m?)^(3/2) * (E-Ec)^(1/2) * e^((-(E-Ef)/kT)) dE (limits Ec to 8)
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n = (4p / h³) * (2m?)^(3/2) * e^((-(Ec-Ef)/kT)) * ? (E-Ec)^(1/2) * e^((-(E-Ec)/kT)) dE (limits Ec to 8)
The integral in equation is of the standard form of type
? x^(1/2) * e^(-ax) dx = (v(p) / 2av(a)) Where a = 1/kT and x = (E-Ec)
FirstRanker.com n = (4p / h³) * (2m?)^(3/2) * e^((-(Ec-Ef)/kT)) * (v(p) / 2) * (kT)^(3/2)
Rearranging the terms, we get
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n = 2 * (2pm?kT / h²)^(3/2) * e^((-(Ec-Ef)/kT))
This is the expression for electron concentration in the C.B. of an Intrinsic Semiconductor.
n = N_c * e^((-(Ec-Ef)/kT)) Where N_c is a temperature dependent material constant known as the effective density of states in the C.B.
Equation shows the dependence of fermi-level on electron concentration.
FirstRanker.com Calculation of hole density :-
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In this case (Ev-E) will be the kinetic energy of the hole at lower energy levels. So, the F.D function can be approximated as
F(E) = e^((-(E-E))/kT)
and
P = ? Z(E) F(E) dE (limits -8 to Ev)
Here, Z(E) dE = (4p / h³) * (2m?)^(3/2) * (Ev-E)^(1/2) dE
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Put in equation
P = (4p / h³) * (2m?)^(3/2) * ? (Ev-E)^(1/2) * e^((-(E-E))/kT) dE (limits -8 to Ev)
P = (4p / h³) * (2m?)^(3/2) * e^((-(Ef-Ev))/kT) * ? (Ev-E)^(1/2) * e^((-(E-E))/kT) dE (limits -8 to Ev)
P = (4p / h³) * (2m?)^(3/2) * e^((-(Ef-Ev))/kT) * (v(p) / 2) * (kT)^(3/2)
Rearranging the terms, we get
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P = 2 * (2pm?kT / h²)^(3/2) * e^((-(Ef-Ev))/kT)
The above equation is the expression for the hole concentration in the V.B of an Intrinsic Semiconductor.
FirstRanker.com P = N_v * e^((-(Ef-Ev))/kT)
Where N_v is called the effective density of states in the V.B.
Equation shows the dependence of fermi-level on hole concentration.
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Intrinsic Density OR Intrinsic Concentration
A single event of bond breaking in a pure semiconductor leads to the generation of an electron-hole pair. At any temp. T, the no. of es generated will be equal to the no. of holes generated. As, the two charge carrier concentrations are equal, they are denoted by a common symbol n_i, which is called Intrinsic Density or Intrinsic Concentration. Thus,
n = p = n_i
n_i² = np = (N_c N_v) * e^((-(Ec-Ev))/kT)
n_i² = (N_c N_v) * e^((-Eg)/kT) [Ec-Ev = Eg]
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Putting the values of N_c and N_v from above
We obtain n_i = 2 * (2pkT / h²)^(3/2) * (m?m?)^(3/4) * e^((-Eg)/kT)
This is the expression for Intrinsic Carrier Concentration.
FirstRanker.com Variation of Intrinsic carrier Concentration with Temperature
The above equation can be written as
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n_i = 2 * (2pkT / h²)^(3/2) * (m?m?)^(3/4) * e^((-Eg)/2kT)
From equation, the following points can be analyzed:
(i) The intrinsic concentration is independent of Fermi Level.
(ii) The Intrinsic concentration has an exponential dependence on the Band Gap Value Eg.
(iii) It strongly depends on temperature.
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(iv) The factor 2 in the exponent indicates that two charge carriers are produced for one Covalent Bond broken.
For Numericals: The Intrinsic Charge Carrier Concentration may be approximated to
n_i = 10^(21.7) * T^(3/2) * 10^(-2500Eg/T)
Note :- n.p = n_i² is also called as Mass-Action Law.
FirstRanker.com The probability that an electron being thermally promoted to the C.B. is given by
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F(E) = 1 / (1 + e^((E-Ef)/kT)) = 1 / (1 + e^(Eg/2kT))
[Ec-Ef = Eg/2]
Fraction of es in the C.B:
n / N = 1 / (exp(Eg/2kT)) = e^(-Eg/2kT)
n ? no. of es excited to C.B. Levels.
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N ? total no. of es available in the V.B. initially.
Note :- These formulas are Important for Numerical Problems.
FirstRanker.com Variation of Fermi Level with Temperature in an Intrinsic Semiconductor
With an increase in temperature, the fermi level gets displaced upward to the bottom edge of the C.B. if m? > m? or downward to the top edge of the V. B. if m? < m? (as shown in fig.)
In most of the materials, the shift of Fermi Level on account of m? ? m? is insignificant.
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The Fermi Level in an Intrinsic Semiconductor may be considered as staying in the middle of the Band gap.
FirstRanker.com We know that the thermal equilibrium concentration of electrons and holes are:
n = N_c * e^((-(Ec-Ef)/kT)) and
p = N_v * e^((-(Ef-Ev)/kT))
Where Ef - Intrinsic fermi Energy
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But we know that at thermal equilibrium for intrinsic Semiconductor, we have
n = p
N_c * e^((-(Ec-Ef)/kT)) = N_v * e^((-(Ef-Ev)/kT))
Taking Natural Logarithm on both sides, we get
Ln N_c + (-(Ec-Ef)/kT) = Ln N_v + (-(Ef-Ev)/kT)
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kT Ln N_c - Ec + Ef = kT Ln N_v - Ef + Ev
2Ef = (Ec + Ev) + kT Ln (N_v / N_c)
Ef = (Ec + Ev) / 2 + (kT / 2) Ln (N_v / N_c)
Ef = EmidBand Energy + (kT / 2) Ln ((m? / m?)^(3/2))
Ef = EmidBand Energy + (3 / 4) kT Ln (m? / m?) (By Putting N_c Values)
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FirstRanker.com From equation following things are obtained:
(1) if m? = m? then Ef = Emidgap. Thus, the fermi level is exactly at the center of the band gap.
(2) if m? > m? then Feemi Energy Level Ef is slightly above the center of the band gap.
(3) if m? < m? then Ef is slightly below the Center of the band gap.
Extrinsic Semiconductor
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n-type Semiconductor
1. Intrinsic + Pentavalent = n-type
2. Electrons are majority charge Carrier (Holes are in Minority)
3. n > p
4. n > n_i and p < n_i
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p-type semiconductor
Intrinsic + trivalent = P-type
Electrons are Minority Charge Carrier (Holes are in majority)
p > n
p > n_i and n < n_i
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FirstRanker.com We know that n = N_c * e^((-(Ec-Ef)/kT))
Taking Logarithm on Both the sides, we get
Ln n = Ln N_c + (-(Ec-Ef)/kT)
Ec-Ef = Ln N_c - Ln(n)
Ec-Ef = kT Ln (N_c / n)
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In n-type semiconductor, we have N_d > n_i and n = N_d
Ec-Ef = kT Ln (N_c / N_d) [N_d - Effective density of states of Donor coumponents]
Equation represents the information regarding Ef for n-type Semiconductor w.r.t. Ec. It is clear that Ef lies below Ec.
Similarly, in p-type Semiconductor
Ev - Ef = kT Ln (N_v / N_a) [N_a - for Acceptor components]
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Equation represents the information reparing Ef for P-type Semiconductor w.r.t Ev. It is clear that Ef Lies above Ev.
FirstRanker.com Variation of Ef with Doping Concentration
We know that Ec-Ef = kT Ln (N_c / N_d)
As N_d increases, (Ec-Ef) decreases. Thus, Ef Shifts towards the Conduction Band for n-type Semiconductor.
Similarly, Ef shifts towards the Valence Band for p-type semiconductor as acceptor impurity Concentration increases.
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In general as Doping level increases the fermi level Shift's towards the band.
FirstRanker.com We know that, for n-type semiconductor the Position of the Farmi Level is given by
Ec-Ef = kT Ln (N_c / N_d) Thus,
As the temperature increases, N_c increases and hence (Ec-Ef) is also increases. Thus, Ef moves away from the Conduction Band.
Similarly for p-type semiconductor, we have
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Ev-Ef = kT Ln (N_v / N_a) Thus, as the temperature increases, N_v increases and hence (Ef-Ev) also increases. Thus, Ef moves away from the Valence Band.
Note : At higher temperature, the semiconductor material Coses its extrinsic characteristics and begins to behave more Like an Intrinsic Semiconductor.
Because on increasing the temperature thermal generation will occur that leads to the same Concentrations of both the carriers ice, the material becomes Intrinsic again. Thus, for a Particular temperature value (Chitical temp. Tc), the Semiconductor completely behaves as an intrinsic one and its conductivity increases with temperative.
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