The Fermi Level in Doped Semiconductors

Introduction

A band diagram has more than a valence edge and a conduction edge. It also needs a measure of how likely the available states are to contain electrons. That measure is the Fermi level E F, the electron chemical potential at equilibrium. Doping changes E F even if the host's basic band gap is little changed. Reading its position helps predict whether electrons or holes dominate, but treating it as a literal occupied energy state inside the gap creates confusion.

Core explanation

The Fermi–Dirac occupation probability is f(E) = 1/[1 + exp((E − E F)/(k BT))]. At E = E F, f = 1/2. A semiconductor can have E F inside a gap with no allowed bulk states at that exact energy; the level is a chemical-potential reference, not necessarily an occupied electron orbital. For a nondegenerate semiconductor, n = N c exp[−(E c − E F)/(k BT)] and p = N v exp[−(E F − E v)/(k BT)] . Raising E F toward E c raises n and lowers p; lowering it toward E v has the opposite effect. MIT's semiconductor fundamentals lecture gives these band-edge formulas.

At a given temperature, shallow donor doping tends to move E F upward toward the conduction band, while acceptor doping moves it downward toward the valence band. The actual position comes from charge neutrality and dopant ionisation, not from drawing it arbitrarily. If donors and acceptors compensate, the shift is smaller than total donor count suggests. Under the nondegenerate approximation, E c − E F = k BT ln(N c/n) . A tenfold change in n changes E F by k BT ln 10, provided N c and T are fixed and the approximation remains valid. This logarithmic relation explains why carrier concentrations can vary by huge factors across a relatively modest portion of the gap.

Temperature complicates a simple “Fermi level always moves up when heated” rule. Heating changes effective state densities, intrinsic pair generation and dopant ionisation. In an n-type sample, E F may approach the intrinsic level at high temperature as pairs dominate; at low temperature donor freeze-out changes its position in another way. The intrinsic level need not be exactly at midgap when conduction and valence densities of states differ. Sketches should label the temperature regime rather than imply a universal path.

For a material in true thermal equilibrium, there is one Fermi level throughout a contacted system, including across a p–n junction, after charge redistributes. Band edges can bend spatially while E F remains flat. Under illumination or bias, the electron and hole populations can be described by separate quasi-Fermi levels; calling either one a single equilibrium Fermi level would be misleading. These distinctions become central when analyzing junctions and solar cells. MIT's microelectronic-device notes treat equilibrium statistics and junction electrostatics as connected topics.

Step-by-step reasoning

1. Draw E v and E c, with E g between them. 2. Determine whether dopants make electrons or holes the majority carriers. 3. At a specified temperature, solve carrier and ionised-dopant neutrality. 4. Use the appropriate carrier-statistics relation to place E F. 5. Check nondegeneracy: a Fermi level inside a band requires more complete statistics.

Visual explanation

Make three identical band-gap columns labelled p-type, intrinsic and n-type. Place E F toward E v, near the intrinsic energy and toward E c respectively. On a separate junction diagram, draw E c and E v bending across position but one horizontal E F at equilibrium. This prevents confusing a spatially varying band edge with a varying equilibrium chemical potential.

Real-world analogy

Think of E F as a setting on a statistical occupancy dial: turning it upward makes high-energy electronic seats more likely to be occupied. The dial does not require a seat exactly at its indicated level. The analogy works only for equilibrium probabilities; a driven sample may need separate settings for electrons and holes.

Real-world example

Fabricators measure carrier densities to infer whether a doped semiconductor is suitable for a contact or a junction. A donor-doped region may have an E F closer to E c than a neighbouring acceptor-doped region before contact. When they touch and equilibrate, charge moves until a common E F is established, creating band bending and a depletion region. The process does not demand that the chemical dopants physically cross the junction.

Why?

Why does raising E F increase n? In the occupation function, moving E F nearer E c makes conduction-band states less energetically costly relative to the electron chemical potential. More of those allowed states are occupied. The same shift makes empty valence-band states less common, so p falls. Their equilibrium product remains tied to E g and temperature in the simple nondegenerate model.

Common misconception

“E F in the gap means a real electron occupies the middle of the gap” is false for an ideal gap without states. “A flat E F means flat bands” is also false: electrostatic potential can bend the band edges while the equilibrium electrochemical potential remains constant. Heavy doping can move E F into a band, where the elementary Boltzmann expression should be replaced by Fermi–Dirac integrals.

Worked example

At one temperature take k BT = 0.0259 eV and N c = 2.0 × 10¹⁹ cm⁻³. For nondegenerate electron concentration n = 2.0 × 10¹⁶ cm⁻³, E c − E F = k BT ln(N c/n) = 0.0259 ln(1000) = 0.179 eV . If n rises by a factor of ten at the same temperature with N c unchanged, the distance shrinks by k BT ln 10 ≈ 0.0596 eV . The method assumes the band structure and nondegenerate regime are still appropriate.

Quick check

1. In a nondegenerate semiconductor, does donor doping generally move the equilibrium Fermi level toward E c or E v? Answer: Toward the conduction-band edge E c, because it increases the equilibrium electron population.

Exam focus

Use the word “equilibrium” when asserting a single Fermi level. Locate E F relative to band edges and state the carrier type implied. Do not mistake Fermi level for a real impurity level. If computing from an exponential formula, use consistent energy units for k BT and the band energies.

Advanced insight

The derivative of carrier concentration with respect to E F is related to electronic compressibility and screening. This is why heavily doped regions can screen electric fields over short distances. In a device under bias, separate quasi-Fermi levels quantify nonequilibrium electron and hole populations; their separation can be interpreted as a local electrochemical driving force for recombination and light emission.

Summary

The equilibrium Fermi level is an electron chemical potential governing occupancy. Donors tend to move it toward E c and acceptors toward E v, while temperature and compensation modify the precise position. One equilibrium Fermi level can coexist with spatial band bending; driven systems require additional concepts.

Practice questions

1. What is f(E) when E = E F at finite temperature? Answer: One-half, by the Fermi–Dirac occupation formula. 2. Can E F lie in a gap containing no allowed states? Answer: Yes. It is a chemical-potential parameter, not necessarily an occupied state. 3. If n increases tenfold at fixed T in the nondegenerate model, by how much does E F move toward E c? Answer: By k BT ln 10, assuming N c and the band edges stay fixed. 4. At equilibrium across a p–n junction, is E F flat or must each side keep its original separate level? Answer: E F is flat after equilibration; band edges bend as charge redistributes.