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I was told that the mass action law $$np = n_i^2$$ only applies to compensated semiconductors and intrinsic semiconductors, and it does not apply to n-type or p-type semiconductors. Is this true? I also heard the argument that the hole concentration decreases as the electron concentration increases in case of a n-type doping, therefore the mass action law is still applicable. I am not sure which is right, but I am sure I am missing something here.

When solving problems such as the one below

The electron concentration in silicon at $T=300\text{ K}$ is $n_0 = 2\times 10^5 \text{ cm} ^{-3}$.

(a) Determine the position of Fermi level with respect to the valence band energy level.

(b) Determine $p_0$.

(c) Is this n- or p-type material?

Source: Semiconductor physics and devices, Donald A. Neamen

I am not sure whether I can apply the mass action law or not. For example, could part (b) solved using the mass action law since the electron concentration is given? $$p_o = n_i^2/n_o$$

Please help me understand when I can or cannot apply the mass action law.

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Law of mass action is true for both, intrinsic and extrinsic semiconductors. Due to doping, one of the carrier concentrations decreases while the other increases such that they obey the mass action law. Mass action is the only alternative to solving the kind of problems that you state. You should also see this Wikipedia article for more on the mass action law.

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The only type of semiconductors in which law of mass action doesn't apply is for degenerate semiconductors.

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