In semiconductors, electrical conduction depends on the concentration of charge carriers, namely electrons and holes. When a semiconductor is in thermal equilibrium, the concentrations of electrons and holes are not independent of each other. Instead, they are related by a simple and very important law known as the Law of Mass Action.
This law plays a fundamental role in:
- Understanding intrinsic and extrinsic semiconductors
- Analysing carrier concentrations
- Explaining the effect of doping and temperature
- Studying p–n junctions and semiconductor devices
Statement of Law of Mass Action:
The law of mass action states that:
At a given temperature, the product of electron concentration and hole concentration in a semiconductor is constant and equal to the square of the intrinsic carrier concentration.
Mathematically,
$$np=n_i^2$$
where:
n = electron concentration in the conduction band
p = hole concentration in the valence band
ni = intrinsic carrier concentration
📌 Important:
This law is valid only under thermal equilibrium conditions.
Physical Meaning of the Law:
- If electron concentration increases, hole concentration must decrease
- If hole concentration increases, electron concentration must decrease
- Their product remains constant at a given temperature
This explains why, in doped semiconductors, increasing one type of carrier automatically reduces the other.
Derivation of Law of Mass Action:
From semiconductor statistics, the electron concentration is given by:
$$n= N_C exp\left[ -\left( \frac{E_C-E_F}{kT} \right) \right]$$ Where:
NC = effective density of states in the conduction band
EC = bottom of conduction band
EF = Fermi level
k = Boltzmann constant
T = absolute temperature
Similarly, hole concentration is given by:
$$p= N_V exp\left[ -\left( \frac{E_F-E_V}{kT} \right) \right]$$
Where:
NV = effective density of states in the valence band
EV= top of valence band
Multiplying the above two equations:
$$np= N_CN_V exp\left[ -\left( \frac{E_C-E_F+E_F-E_V}{kT} \right) \right]$$
Simplifying the exponent:
$$np= N_CN_V exp\left[ -\left( \frac{E_C-E_V}{kT} \right) \right]$$
But, $$E_C-E_V=E_g$$
Where Eg is the forbidden energy gap.
Thus, $$np= N_CN_V e^{-\left( \frac{E_g}{kT} \right)}———(1)$$
For an intrinsic semiconductor: $$n=p=n_i$$
So, $$n_i^2= N_CN_V e^{-\left( \frac{E_g}{kT} \right)}——–(2)$$
Comparing the two results (1) and (2): $$np=n_i^2$$
This is the law of mass action.
The expression does not impose any conditions that limit it to intrinsic semiconductors, as Eg remains unaffected by impurity concentration, while Nc and Nv are constants.
Hence, the above equation is equally valid for extrinsic semiconductors also. We can write it as for an extrinsic semiconductor…
For extrinsic semiconductors:-
If in n-type semiconductors, the electron concentration is denoted by nn and the hole concentration by pn, we can write it as
$$n_n=N_Ce^{-\left( E_C-E_F \right)/kT}$$ and
$$p_n=N_ve^{-\left( E_F-E_V \right)/kT}$$
Therefore, $$n_np_n=N_cN_ve^{-E_g/kT}$$
$$=> n_np_n=n_i^2$$
Similarly, if the electron and hole concentration in a p-type semiconductor is denoted by np and pp, then
$$p_p=N_ve^{-(E_F-E_V)/kT}$$ and $$n_p=N_ce^{-(E_C-E_F)/kT}$$
Therefore, $$p_pn_p=N_vN_ce^{-E_g/kT}$$
$$=> p_pn_p=n_i^2$$
As a result, the above two equations demonstrate that the product of majority and minority carrier concentrations in an extrinsic semiconductor at a given temperature is a constant equal to the square of the intrinsic carrier concentration.
Importance of Law of Mass Action:
The law of mass action:
- Explains the relationship between electrons and holes
- Helps to calculate minority carrier concentration
- It is essential for p–n junction analysis
- Forms the basis for semiconductor device equations
Important Examination Questions:
Short-Answer Questions:
- State the law of mass action.
- What is the significance of the law of mass action?
Long Answer / Derivation Questions:
- Derive the law of mass action in semiconductors.
- Explain the importance of law of mass action in intrinsic and extrinsic semiconductors.
⭐ Very frequently asked in university examinations
Conceptual Questions:
- Why does the product remain constant at a given temperature?
- How does temperature affect the law of mass action?
FAQs:
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1. What is the law of mass action in semiconductors?
The law of mass action states that, at thermal equilibrium, the product of the concentration of electrons and holes in a semiconductor remains constant.
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2. What is the mathematical expression of the law of mass action?
Here, n is the electron concentration, p is the hole concentration, and nᵢ is the intrinsic carrier concentration. -
3. Does the law of mass action apply to both intrinsic and extrinsic semiconductors?
Yes, it applies to both types, but only under thermal equilibrium conditions.
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4. How does temperature affect the law of mass action?
Temperature changes the value of nᵢ, but the relationship (n × p = nᵢ²) still holds at equilibrium.
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5. Why is the law of mass action important in semiconductor physics?
It helps in analyzing carrier concentrations and is essential for understanding the behavior of devices like diodes and transistors.
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6. Does the law of mass action hold under all conditions?
No, it is valid only under thermal equilibrium. It does not apply when external energy sources like electric fields or light disturb the system.