In semiconductors, electrical conduction depends on the number of charge carriers available for motion. These carriers are:
- Electrons in the conduction band
- Holes in the valence band
To quantitatively understand semiconductor behavior, we must derive expressions for:
- Electron concentration in the conduction band
- Hole concentration in the valence band
- Intrinsic carrier concentration
These derivations are based on:
- Energy band theory
- Density of states
- Fermi–Dirac statistics
Electron Concentration in the Conduction Band
Physical Meaning:
The electron concentration (n) is defined as:
The number of electrons present in the conduction band per unit volume.
Electrons contribute to conduction only when they are in the conduction band.
Basic Idea of Derivation:
To find electron concentration:
- We count how many allowed energy states exist in the conduction band
- We calculate the probability that an electron occupies each state
Mathematically:
Electron concentration = ∫ (Density of states) × (Probability of occupation)
Let dn be the number of electrons whose energy lies in the energy interval E and E + dE in the conduction band.
Then, dn = D(E) f(E) dE
where:
D(E) dE = Density of states in the energy interval E and E + dE
F(E) = Fermi Function (probability of occupation of energy level E)
Density of States in the Conduction Band
The density of states in the conduction band is given by:
$$D(E) = \frac{4\pi(2m_e^{*})^{3/2}}{h^{3}}\sqrt{E – E_c}$$
where:
- me*= effective mass of electron
- Ec = conduction band minimum energy
- h = Planck’s constant
This tells us how many energy states are available at energy.
Probability of Occupation (Fermi–Dirac Function)
The probability that an electron occupies an energy level is:
$$f(E)=\frac{1}{1 + e^{\frac{(E – E_F)}{kT}}}$$
where:
- f(E) = Fermi Function (probability of occupation)
- EF = Fermi energy
- k = Boltzmann’s constant
- T = Temperature
For non-degenerate semiconductors (normal engineering physics assumption):
$$f(E) \approx e^{-\frac{E – E_F}{kT}}$$
Expression for Electron Concentration
Electron concentration is:
$$n = \int_{E_c}^{\infty}D(E)f(E)dE$$
Substituting value of D(E) and f(E) in above expressions, we get:
$$n = \int_{E_c}^{\infty}\frac{4\pi(2m_e^{*})^{3/2}}{h^{3}}\sqrt{E – E_c} e^{-(E – E_F)/kT}dE$$
$$n = \int_{E_c}^{\infty}\frac{4\pi(2m_e^{*})^{3/2}}{h^{3}}\sqrt{E – E_c}e^{-(E – E_F + E_c – E_c)/kT}dE$$
$$n = \frac{4\pi(2m_e^{*})^{3/2}}{h^{3}}e^{-(E_c – E_F)/kT}\int_{E_C}^{\infty}{(E – E_C)}^{1/2}e^{-(E – E_C)/kT}dE$$
Let $$\frac{E-E_c}{kT}= x$$
=> dE = kT dx
Hence,
$$n = \frac{4\pi(2m_e^{*})^{3/2}}{h^{3}}e^{-(E_c – E_F)/kT}\int_{0}^{\infty}{(kTx)}^{1/2}e^{-x}kTdx$$
$$n = \frac{4\pi(2m_e^{*}kT)^{3/2}}{h^{3}}e^{-(E_c – E_F)/kT}\int_{0}^{\infty}{(x)}^{1/2}e^{-x}dx$$
Since
$$
\int_{0}^{\infty}{x}^{1/2}e^{-x}dx = \left(\frac{\pi}{4}\right)^{1/2}$$
Therefore,
$$n = 2\left(\frac{2\pi m_e^{*}kT}{h^{2}} \right)^{3/2}e^{-(E_c – E_F)/kT}$$
$$n = N_c e^{-(E_c – E_F)/kT}$$
Where
$$N_c = 2\left( \frac{2\pi m_e^*kT}{h^2} \right)^{3/2}$$
is called the effective density of states in the conduction band. It is temperature dependent constant.
In silicon at 300K,
$$N_c = 2.8 \times 10^{25}/m^3$$
Physical Interpretation
- Electron concentration increases when:
- Fermi level moves closer to conduction band
- Temperature increases
- This explains why conductivity increases with temperature
Hole Concentration in the Valence Band
Physical Meaning:
The hole concentration (p) is defined as:
The number of holes present in the valence band per unit volume.
A hole represents the absence of an electron.
The number of holes per unit volume in the energy range E and E + dE can be written as
dp = D(E) [1 – f(E)] dE
Probability of Hole Formation
The probability that a valence band state is empty (i.e., contains a hole) is:
1 – f(E)
For non-degenerate semiconductors:
$$1-f(E)≈ e^{-(E_F-E)/kT}$$
Density of States in the Valence Band
The density of states in the valence band is given by:
$$D(E) = \frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}\sqrt{E_v – E}$$
where:
- mh*= effective mass of hole
- Ev = valence band maximum energy
Expression for Hole Concentration
Hence, the number of holes in the energy interval E and E + dE of the valence band is
$$dp =\frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}\sqrt{E_v – E}\times e^{-(E_F – E)/kT}$$
The hole concentration in the valence band is obtained by integrating the above equation with a suitable limit. Hence, the hole concentration is given by
$$p = \frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}\int_{-\infty }^{E_v}\sqrt{E_v – E}\times e^{-(E_F – E)/kT}dE$$
$$p = \frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}\int_{-\infty }^{E_v}\sqrt{E_v – E}\times e^{\frac{-(E_F – E + E_v – E_v)}{kT}}dE$$
$$p = \frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}e^{\frac{-(E_F-E_v)}{kT}}\int_{-\infty }^{E_v}\sqrt{E_v – E}\times e^{\frac{(E – E_v)}{kT}}dE$$
Let $$\frac{E_v-E}{kT} = x$$
$$=>dE = -kTdx$$
Substituting values:
$$p = \frac{4\pi(2m_h^{*})^{3/2}}{h^{3}}e^{\frac{-(E_F-E_v)}{kT}}\int_{\infty }^{0}{(kTx)^{1/2} e^{-x}}{(-kTdx)}$$
$$p = \frac{4\pi(2m_h^{*}kT)^{3/2}}{h^{3}}e^{\frac{-(E_F-E_v)}{kT}}\int_{0}^{\infty }{x^{1/2} e^{-x}}{dx}$$
Using the value of integration, we get:
$$p = \frac{4\pi(2m_h^{*}kT)^{3/2}}{h^{3}}e^{\frac{-(E_F-E_v)}{kT}}\left(\frac{\pi}{4} \right)^{1/2}$$
$$p = 2\left( \frac{2\pi m_h^{*}kT}{h^{2}}\right)^{3/2}e^{\frac{-(E_F-E_v)}{kT}}$$
$$p = N_v e^{-(E_F – E_v)/kT}$$
Where,
$$N_v = 2\left( \frac{2\pi m_h^{*}kT}{h^{2}}\right)^{3/2}$$
is the effective density of states in the valence band. For silicon at 300K,
$$N_v=10^{25}/m^{3}$$
Physical Interpretation
Hole concentration increases when:
- Fermi level moves closer to the valence band
- Temperature increases
Intrinsic Carrier Concentration
Physical Significance
In an intrinsic semiconductor:
$$n-p=n_i$$
Multiplying electron and hole concentrations:
$$np=N_cN_v\times e^{-\left( \frac{E_c-E_v}{kT} \right)}$$
But,
$$E_c – E_v=E_g$$
Hence:
$$np=N_cN_v\times e^{-E_g/kT}$$
$$=>n_i^{2}=N_cN_v\times e^{-E_g/kT}$$
$$=>n_i=(N_cN_v)^{1/2}e^{-E_g/2kT}$$
Since,
$$N_c\propto T^{3/2}$$ and
$$N_v\propto T^{3/2}$$
Therefore,
$$n_i\propto T^{3/2}e^{-E_g/2kT}$$
- Smaller band gap → larger intrinsic carrier concentration
- Higher temperature → rapid increase in
- Germanium has higher than silicon due to smaller band gap
Important examination questions
Basic Concept Questions
What is intrinsic carrier concentration?
Why are electrons and holes equal in an intrinsic semiconductor?
What is the role of the Fermi level in carrier concentration?
Understanding-Based Questions
Why does electron concentration increase with temperature?
Why is the intrinsic carrier concentration strongly temperature dependent?
What happens to carrier concentration if the band gap increases?
Derivation-Based Questions (Very Important for Exams)
- Derive the expression for electron concentration in the conduction band:
$$n=N_ce^{−(E_c−E_F)/kT}$$
- Derive the expression for hole concentration in the valence band:
$$p=N_ve^{−(E_F−E_v)/kT}$$
- Derive the expression for intrinsic carrier concentration:
$$n_i=(N_cN_v)^{1/2}e^{-E_g/2kT}$$
- Show that:
$$np=n_i^2$$
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