Fermi level

Fermi level

1. What Fermi Level Is

Fermi level $E_F$ = chemical potential for electrons — energy where Fermi-Dirac occupation probability

equals 1/2. It is the energy of the highest filled state at $T=0$ for metals, and the energy where electron and hole probabilities equal for semiconductors.

At $T=0$: $f=1$ for $E<E_F$, $0$ for $E>E_F$ (sharp step). At $T>0$, step smears over few $kT$.

2. Position in Bands (what exam expects to draw)

Material	$E_F$ Location
Intrinsic semiconductor	mid-gap $(E_c+E_v)/2$ plus small $kT ln(m_h/m_e)$ correction
N-type	near $E_c$:

(donor pushes up) |
| P-type | near $E_v$:

|
| Metal | inside conduction band (conduction electrons fill to $E_F$ ~ few eV) |

So doping moves $E_F$ toward the majority band.

3. Meaning for Devices

• Equilibrium: $E_F$ is flat (constant) across a system in thermal equilibrium (diffusion vs drift balance in p-n junction results in built-in potential $V_{bi}=(E_{Fc}-E_{Fv})/e$).
• Non-equilibrium (forward bias, illumination): quasi-Fermi levels $E_{Fn}, E_{Fp}$ split.

4. Worked Formula

Si: Nc≈2.8e25 m⁻³, N_D=1e22 ⇒ E_c - E_F =kT ln(Nc/ND)=0.026·ln(2800)≈0.21 eV ⇒ E_F 0.21 below Ec, 0.91 above Ev (since Eg=1.12) ✓

Intrinsic: Ef≈0.56 eV above Ev (mid) ✓

5. Exam Points

• Define $E_F$ via $f(E_F)=1/2$ and diagram — 5 marks, very frequent.
• Show positions: intrinsic mid-gap, n-type near $E_c$, p-type near $E_v$ — 7 marks, almost certain (the classic band diagram with $E_F$).
• Formula $E_F=E_c -kT ln(Nc/ND)$ for n-type — 5 marks, frequent.
• Mention flat $E_F$ in equilibrium, split quasi-Fermi under bias — 3 marks.