Where does the Fermi level lie in n-type and p-type extrinsic semiconductors, and why?

In one line

In an n-type semiconductor the Fermi level lies close to the conduction band (between the donor level and \(E_c\) at 0 K), and in a p-type semiconductor it lies close to the valence band (between \(E_v\) and the acceptor level at 0 K).

Explained simply

The Fermi level tracks where the electrons "are". Adding donors (like phosphorus in silicon) puts many extra electrons near the conduction band, so the 50% occupancy point rises towards it. Adding acceptors (like boron) creates many holes near the valence band, so the 50% point drops towards it.

How it works

1. n-type: pentavalent donors create a donor level \(E_d\) just below \(E_c\) (about 0.01 to 0.05 eV below in Si and Ge).
2. At 0 K, electrons sit in the donor level and the conduction band is empty. The Fermi level is midway between them: \(E_F = \dfrac{E_c + E_d}{2}\).
3. At room temperature, nearly all donors are ionised, so \(n \approx N_D\). From \(n = N_ce^{-(E_c - E_F)/kT}\): \(E_F = E_c - kT\ln\dfrac{N_c}{N_D}\).
4. p-type: trivalent acceptors create an acceptor level \(E_a\) just above \(E_v\).
5. At 0 K: \(E_F = \dfrac{E_v + E_a}{2}\).
6. At room temperature, \(p \approx N_A\), so \(E_F = E_v + kT\ln\dfrac{N_v}{N_A}\).

Key formula

n-type: \(E_F = E_c - kT\ln\dfrac{N_c}{N_D}\); p-type: \(E_F = E_v + kT\ln\dfrac{N_v}{N_A}\). \(N_D, N_A\): donor and acceptor concentrations (m⁻³), \(N_c, N_v\): effective densities of states in the conduction and valence bands (m⁻³), \(kT\) in eV.

Quick example

Silicon with \(N_D = 10^{22}\) m⁻³, \(N_c \approx 2.8\times10^{25}\) m⁻³, \(kT = 0.0259\) eV: \(E_c - E_F = 0.0259\times\ln(2800) \approx 0.0259\times7.94 \approx 0.21\) eV. So \(E_F\) is 0.21 eV below the conduction band, well above mid-gap (0.56 eV below \(E_c\)), as expected for n-type.

Common mistakes

• Placing the n-type Fermi level near the valence band (swapping n and p).
• Writing \(E_F = (E_c + E_d)/2\) as valid at all temperatures; it holds at 0 K.
• Forgetting that the donor level lies inside the band gap, just below \(E_c\).

How MU asks it

Asked as "Explain the position of Fermi level in n-type and p-type semiconductors with band diagrams" (5 marks). Draw both band diagrams with \(E_c\), \(E_v\), \(E_d\) or \(E_a\) and \(E_F\) marked, write the 0 K positions, and give the \(kT\ln\) expressions with a sentence on why \(E_F\) moves.