Why does the Fermi level lie at the middle of the band gap in an intrinsic semiconductor?

In one line

In an intrinsic semiconductor the number of electrons in the conduction band equals the number of holes in the valence band, and this balance places the Fermi level at (nearly) the centre of the band gap.

Explained simply

The Fermi level is the "50% occupancy" energy. In a pure semiconductor, every electron that jumps to the conduction band leaves exactly one hole behind. Since electrons and holes are equal in number, the 50% point must sit symmetrically between the two bands, like the pivot of a balanced see-saw.

How it works

1. Electron concentration in the conduction band: \(n = N_c\,e^{-(E_c - E_F)/kT}\).
2. Hole concentration in the valence band: \(p = N_v\,e^{-(E_F - E_v)/kT}\).
3. Intrinsic condition: \(n = p\), so \(N_c\,e^{-(E_c - E_F)/kT} = N_v\,e^{-(E_F - E_v)/kT}\).
4. Take natural logs: \(-(E_c - E_F) = -(E_F - E_v) + kT\ln(N_v/N_c)\).
5. Rearranging: \(E_F = \dfrac{E_c + E_v}{2} + \dfrac{kT}{2}\ln\dfrac{N_v}{N_c}\). Since \(N_v/N_c = (m_h^*/m_e^*)^{3/2}\), this equals \(\dfrac{E_c + E_v}{2} + \dfrac{3kT}{4}\ln\dfrac{m_h^*}{m_e^*}\).
6. If \(m_h^* = m_e^*\) (or at 0 K), \(E_F = \dfrac{E_c + E_v}{2}\): exactly mid-gap.
7. Since \(kT\) is small and the log term is small, \(E_F\) stays very close to mid-gap at ordinary temperatures (shifting slightly towards the band with the lighter effective mass).

Key formula

\(E_F = \dfrac{E_c + E_v}{2} + \dfrac{3kT}{4}\ln\dfrac{m_h^*}{m_e^*}\); for equal effective masses, \(E_F = \dfrac{E_c + E_v}{2}\). \(E_c\): bottom of conduction band, \(E_v\): top of valence band, \(m_e^*, m_h^*\): effective masses, \(N_c, N_v\): effective densities of states (m⁻³).

Quick example

For silicon, \(E_g = 1.12\) eV. Taking \(E_v = 0\), \(E_c = 1.12\) eV, the intrinsic Fermi level is at about \(0.56\) eV, in the middle of the gap. At this energy no electron state exists (it lies in the forbidden gap); the Fermi level is a reference energy, not an occupied level.

Common mistakes

• Saying electrons occupy the Fermi level in a semiconductor; it lies in the forbidden gap.
• Forgetting to state the equal effective mass assumption for the exact mid-gap result.
• Dropping the factor of 2 when solving for \(E_F\).

How MU asks it

Asked as "Show that the Fermi level lies at the centre of the forbidden gap in an intrinsic semiconductor" (5 marks). Write \(n\) and \(p\) expressions, equate them, solve for \(E_F\), state the assumption, and draw the band diagram with \(E_F\) at mid-gap.