In one line
As temperature rises, the Fermi level of both n-type and p-type semiconductors moves towards the middle of the gap (the material becomes intrinsic-like), and as doping increases it moves towards the conduction band (n-type) or valence band (p-type).
Explained simply
Doping gives the material a "personality" (n or p). Heating gradually washes that personality out: so many electron-hole pairs are created across the gap that they swamp the dopant carriers, and the material starts behaving like pure silicon. Adding more dopant strengthens the personality, pushing the Fermi level closer to its band.
How it works
- n-type, temperature: at 0 K, \(E_F = (E_c + E_d)/2\). As \(T\) rises, donors ionise and the term \(kT\ln(N_c/N_D)\) grows (both \(kT\) and \(N_c \propto T^{3/2}\) increase), so \(E_F\) moves down. At high temperature intrinsic carriers dominate (\(n_i \gg N_D\)) and \(E_F\) approaches the intrinsic level \(E_i\) near mid-gap.
- p-type, temperature: at 0 K, \(E_F = (E_v + E_a)/2\). With rising \(T\), \(E_F\) moves up towards mid-gap and approaches \(E_i\) at high temperature.
- n-type, doping: \(E_F = E_c - kT\ln(N_c/N_D)\). Larger \(N_D\) makes the log smaller, so \(E_F\) moves up towards \(E_c\). At very heavy doping it can reach or enter the conduction band (degenerate semiconductor).
- p-type, doping: \(E_F = E_v + kT\ln(N_v/N_A)\). Larger \(N_A\) moves \(E_F\) down towards \(E_v\).
- A graph of \(E_F\) versus \(T\) shows the n-type curve falling and the p-type curve rising, both converging on \(E_i\).
Key formula
\(E_c - E_F = kT\ln\dfrac{N_c}{N_D}\) (n-type), \(E_F - E_v = kT\ln\dfrac{N_v}{N_A}\) (p-type). Symbols as in the previous topic; \(T\) in K.
Quick example
For n-type Si at 300 K with \(N_c \approx 2.8\times10^{25}\) m−3: if \(N_D = 10^{22}\), \(E_c - E_F \approx 0.21\) eV; if \(N_D = 10^{24}\), \(E_c - E_F = 0.0259\times\ln28 \approx 0.086\) eV. A hundredfold increase in doping moves \(E_F\) about 0.12 eV closer to the conduction band.
Common mistakes
- Saying the Fermi level moves towards the band edge with increasing temperature; it moves towards mid-gap.
- Forgetting that very high temperature makes any doped semiconductor behave intrinsically.
- Drawing n-type and p-type curves moving the same way with doping.
How MU asks it
Asked as "Explain the variation of Fermi level with temperature and impurity concentration in extrinsic semiconductors" (5 marks), which matches CO6 of the syllabus. Draw \(E_F\) versus \(T\) for both types, show band diagrams for low and high doping, and give the reason in one line for each trend.