Fermi Level of Intrinsic and Extrinsic Semiconductors

Definition

The Fermi level is the energy level at which the probability of finding an electron is 50% at thermal equilibrium. In a solid, it represents the highest occupied energy level at absolute zero temperature for electrons in a system. For semiconductors, the Fermi level is not a fixed physical boundary but an energy reference that indicates the tendency of electrons to occupy available states. Its position depends on temperature, impurity concentration, and the type of semiconductor. In intrinsic semiconductors, it lies near the middle of the forbidden energy gap, while in extrinsic semiconductors it shifts toward the conduction band or valence band depending on the doping type.

Main Content

1. Fermi Level in an Intrinsic Semiconductor

• In a pure semiconductor with no intentional impurities, the number of electrons in the conduction band equals the number of holes in the valence band, so the material is called intrinsic.
• In an intrinsic semiconductor, the Fermi level lies approximately at the middle of the band gap, though in real materials it may shift slightly depending on the effective masses of electrons and holes.

In an intrinsic semiconductor such as pure silicon or germanium, all atoms are identical and there are no donor or acceptor impurities added. At absolute zero, the valence band is completely filled, and the conduction band is empty. As temperature increases, some electrons gain enough energy to jump across the forbidden energy gap from the valence band to the conduction band, leaving behind holes in the valence band. Since each excited electron creates one hole, the concentration of electrons and holes remains equal.

Because of this symmetry, the Fermi level of an intrinsic semiconductor is usually located close to the center of the band gap. This position indicates that the probability of occupation is balanced between electrons and holes. For silicon, the intrinsic Fermi level lies very near the midpoint between the valence band maximum and the conduction band minimum. However, it is not always exactly at the center due to differences in effective mass of charge carriers.

The position of the intrinsic Fermi level can be written approximately as:

$$E_i \approx \frac{E_c + E_v}{2}$$

where $E_c$ is the conduction band edge and $E_v$ is the valence band edge. A more accurate expression includes temperature and effective density of states. This central position helps explain why intrinsic semiconductors have relatively low conductivity at room temperature compared with metals.

2. Fermi Level in an Extrinsic Semiconductor

• In an extrinsic semiconductor, impurities are intentionally added to increase conductivity, and the Fermi level shifts according to the type of dopant.
• In n-type semiconductors the Fermi level moves closer to the conduction band, while in p-type semiconductors it moves closer to the valence band.

Extrinsic semiconductors are formed by doping a pure semiconductor with a small amount of impurity atoms. If the added impurity has more valence electrons than the host atom, it donates extra electrons to the crystal and produces an n-type semiconductor. Common donor impurities in silicon include phosphorus, arsenic, and antimony. These donors create energy levels just below the conduction band. Because electrons can easily move from donor levels into the conduction band, the electron concentration becomes much larger than the hole concentration. As a result, the Fermi level rises and moves nearer to the conduction band.

In contrast, if the impurity has fewer valence electrons than the host atom, it creates holes and forms a p-type semiconductor. Common acceptor impurities include boron, aluminum, and gallium in silicon. These impurities introduce acceptor energy levels just above the valence band. Electrons from the valence band can move into these acceptor levels, leaving behind holes. Therefore, the hole concentration becomes greater than the electron concentration, and the Fermi level moves closer to the valence band.

This shift in the Fermi level is extremely important because it shows how doping changes the electrical properties of a semiconductor. The more heavily a semiconductor is doped, the closer the Fermi level moves toward the relevant band edge. In heavily doped semiconductors, the Fermi level can even enter the conduction band or valence band, producing degenerate semiconductor behavior.

3. Comparison and Significance of Fermi Level Position

• The location of the Fermi level determines the concentration of electrons and holes, which directly affects conductivity.
• The Fermi level is a key tool for predicting the behavior of semiconductor junctions, electronic devices, and charge flow in materials.

The Fermi level is not just a theoretical energy value; it is a practical indicator of how a semiconductor behaves. In an intrinsic semiconductor, the mid-gap Fermi level means electrons and holes are produced only by thermal excitation, so conductivity is limited. In extrinsic semiconductors, the shift in Fermi level indicates the presence of majority carriers. This makes conduction much easier because only a small amount of energy is needed to move carriers into available states.

The difference between intrinsic and extrinsic Fermi levels also explains the operation of p-n junctions. When p-type and n-type semiconductors are joined, electrons and holes diffuse across the junction until equilibrium is reached and the Fermi level becomes constant throughout the structure. This alignment is essential for the formation of the depletion region and built-in potential.

The concept is also used to determine carrier concentrations through statistical relations. For example, in non-degenerate semiconductors:

$$n = N_c e^{-(E_c - E_F)/kT}$$

$$p = N_v e^{-(E_F - E_v)/kT}$$

where $n$ is electron concentration, $p$ is hole concentration, $N_c$ and $N_v$ are effective density of states, $E_F$ is Fermi level, $k$ is Boltzmann constant, and $T$ is absolute temperature. These relations show how even a small shift in the Fermi level can cause a large change in carrier density.

Working / Process

1. In an intrinsic semiconductor, thermal energy excites some electrons from the valence band to the conduction band, creating equal numbers of electrons and holes, so the Fermi level stays near the middle of the band gap.
2. In an n-type semiconductor, donor impurities contribute extra electrons, increasing electron concentration and shifting the Fermi level upward toward the conduction band.
3. In a p-type semiconductor, acceptor impurities create holes by accepting electrons from the valence band, increasing hole concentration and shifting the Fermi level downward toward the valence band.

Advantages / Applications

• The Fermi level helps predict electrical conductivity and carrier concentration in semiconductors.
• It is essential in designing and analyzing p-n junctions, diodes, transistors, and integrated circuits.
• It is used in understanding doping effects, semiconductor equilibrium, and energy band diagrams in materials science and solid-state physics.

Summary

• The Fermi level indicates the energy distribution of electrons in a semiconductor.
• In intrinsic semiconductors, it lies near the middle of the band gap.
• In extrinsic semiconductors, it shifts toward the conduction band in n-type and toward the valence band in p-type materials.
• The concept is crucial for understanding conductivity, doping, and semiconductor device operation.