How Far Does the Fermi Level Move When You Dope Silicon?
Only logarithmically: with donors at concentration ND, the Fermi level rises above the intrinsic level by EF − Ei = kT ln(ND/ni), which at 300 K is 59.5 meV for every factor of 10 in doping. Silicon with 10¹⁶ donors per cm³ has its Fermi level 0.347 eV above Ei and 0.205 eV below the conduction band edge, and even a hundred times more doping moves it only another 0.119 eV.
What you'll learn
- Why the Fermi level moves with the logarithm of the doping, not the doping itself
- The two forms of the formula, measured from Ei and from EC, and which constants each needs
- A worked example for 10¹⁶ donors per cm³, with the holes checked two ways
- How compensation and very heavy doping change the answer
- The mistakes that cost marks: wrong signs, wrong reference level and leaving the formula's range
Conventions and assumptions
Silicon at T = 300 K, so kT = 25.85 meV. Values used: ni = 1.5 × 10¹⁰ cm⁻³ (a common textbook value; some books use 1.0 × 10¹⁰, which shifts every answer here by 0.010 eV), effective density of states NC = 2.8 × 10¹⁹ cm⁻³ and band gap 1.12 eV. Every dopant is ionised, and the semiconductor is non-degenerate: EF stays at least about 3kT away from both band edges. Energies are in eV, and Ei is the intrinsic level, close to the middle of the gap.
Why does the Fermi level move logarithmically?
In equilibrium the electron concentration depends exponentially on where the Fermi level sits: n0 = ni e^((EF − Ei)/kT). Turn that round and the position of EF is a logarithm of the concentration: EF − Ei = kT ln(n0/ni). Adding donors raises n0 to about ND, so the Fermi level climbs towards the conduction band, but a factor of 10 in ND adds only kT ln 10 = 59.5 meV. The same relation measured from the band edge is n0 = NC e^(−(EC − EF)/kT), which gives EC − EF = kT ln(NC/ND). Both forms describe one level; they differ only in the reference, so pick the one whose constants the question gives you. For p-type silicon everything mirrors: EF sits kT ln(NA/ni) below Ei.
Worked example: 10¹⁶ donors per cm³
- Electrons: every donor is ionised and ND ≫ ni, so n0 = 10¹⁶ cm⁻³.
- From Ei: EF − Ei = 0.02585 × ln(10¹⁶ / 1.5 × 10¹⁰) = 0.02585 × 13.41 = 0.347 eV.
- From EC: EC − EF = 0.02585 × ln(2.8 × 10¹⁹ / 10¹⁶) = 0.02585 × 7.94 = 0.205 eV, so EF is well inside the gap and the non-degenerate formulas hold.
- Holes by mass action: p0 = ni²/n0 = (1.5 × 10¹⁰)² / 10¹⁶ = 2.25 × 10⁴ cm⁻³.
- Holes from the Fermi level: p0 = ni e^(−0.347/0.02585) = 2.25 × 10⁴ cm⁻³ as well, which confirms the position of EF.
| ND (cm⁻³) | EF − Ei (eV) | EC − EF (eV) |
|---|---|---|
| 10¹⁴ | 0.228 | 0.324 |
| 10¹⁵ | 0.287 | 0.265 |
| 10¹⁶ | 0.347 | 0.205 |
| 10¹⁷ | 0.406 | 0.146 |
| 10¹⁸ | 0.466 | 0.086 |
What changes with both kinds of dopant?
When a sample has donors and acceptors, the acceptors take electrons from an equal number of donors, so only the difference counts. For ND = 10¹⁶ and NA = 4 × 10¹⁵ cm⁻³, n0 = (ND − NA)/2 + √(((ND − NA)/2)² + ni²) = 6.00 × 10¹⁵ cm⁻³, and EF − Ei = 0.02585 × ln(6 × 10¹⁵ / 1.5 × 10¹⁰) = 0.333 eV. Removing 40% of the net donors moved the Fermi level by only 13 meV, which is the logarithm again. The square root matters only when ND − NA is close to ni, for instance in nearly compensated or very lightly doped material.
Where does the formula stop working?
Two limits. At the heavy end, the derivation uses Boltzmann statistics, which need EF at least about 3kT below EC. That distance falls to 3kT when ND = NC e^(−3) = 1.39 × 10¹⁸ cm⁻³. Beyond that the semiconductor heads towards degeneracy, where EF can enter the conduction band, and the Fermi-Dirac integral is needed. At the light end, when ND approaches ni, n0 is no longer equal to ND and the intrinsic carriers count. Temperature moves both limits, because ni grows quickly with T: at high temperature ni can exceed ND, and the Fermi level falls back towards Ei even though the doping hasn't changed.
How big is a 0.1 eV move?
Because the relation is exponential, a modest shift in energy is a large change in concentration: moving EF by 0.1 eV at 300 K multiplies n0 by e^(0.1/0.02585) = 47.9. That is why band diagrams drawn to scale show the Fermi level moving only slightly even between lightly and heavily doped regions.
Where this sits in the GATE syllabus
The GATE 2027 EC syllabus lists formation of energy bands, energy bands in intrinsic and extrinsic semiconductors and equilibrium carrier concentration first in Section 3, Electronic Devices. Questions ask for the Fermi level position from the doping, the carrier concentrations from the Fermi level, or a concentration at another temperature, and each uses the formulas here.
Common mistakes
- Measuring from the wrong level. kT ln(ND/ni) is the distance from Ei; kT ln(NC/ND) is the distance from EC. Mixing them gives a number that fits neither.
- Getting the sign wrong for p-type. Acceptors move EF below Ei, by kT ln(NA/ni).
- Using ND when acceptors are present. The net doping, ND − NA, sets the electrons.
- Changing temperature but not kT, ni or NC. All three change, and ni changes the most.
- Using the non-degenerate formula above about 10¹⁸ cm⁻³ in silicon, where EF is within 3kT of the band edge.
Check your understanding
- Q1. Silicon at 300 K has 10¹⁷ donors per cm³. How far is EF above Ei? Answer: 0.02585 × ln(10¹⁷ / 1.5 × 10¹⁰) = 0.406 eV, which is 59.5 meV more than at 10¹⁶.
- Q2. A p-type sample has NA = 10¹⁵ cm⁻³. Where is EF? Answer: 0.287 eV below Ei, the mirror image of n-type with the same doping.
- Q3. By what factor must ND rise to move EF up by 0.119 eV at 300 K? Answer: e^(0.119/0.02585), about 100, since each 59.5 meV is one factor of 10.
Related
Official source
Last verified
The GATE 2027 EC syllabus lists energy bands in intrinsic and extrinsic semiconductors and equilibrium carrier concentration in Section 3, Electronic Devices (checked against the official PDF from IIT Madras on 29 October 2026). Every Fermi level, concentration and limit in this post was computed in docs/verification/ece/fermi_doping.py.