Comprehensive Semiconductor Physics and the Fermi Level

Dynamics of Charge Carriers and Total Current

In the study of semiconductor physics, the movement of charge carriers is fundamental to understanding electrical conductivity. When an external electric field is applied to a semiconductor crystal, the movement of holes occurs in the same direction as the applied electric field. In contrast, free electrons move in the opposite direction of the influential electric field. This directional difference is a core characteristic of how charge behaves within the crystal lattice.

The total electric current flowing through a pure (intrinsic) semiconductor is defined as the sum of the electron current and the hole current combined. In this context, both electrons and holes are collectively referred to as charge carriers. This dual-carrier system distinguishes semiconductors from metallic conductors, where only electrons typically contribute to the current flow.

The Concept and Location of the Fermi Level

The occupation of energy levels by electrons is determined by a specific reference energy level known as the Fermi level. This level serves as the primary indicator for the probability of electron occupation relative to other available energy states. By definition, the Fermi level is the highest energy level that electrons can occupy at a temperature of absolute zero, denoted as $0\,K$. Typically, the energy levels that are actually occupied by electrons are situated below the Fermi level.

The specific location of the Fermi level varies depending on the type of material and its purity. In metallic conductors at a temperature of $0\,K$, the Fermi level is located at the top of the region filled with electrons; thus, the occupied energy levels reside directly beneath it. For pure (intrinsic) semiconductors, the Fermi level is positioned exactly in the middle of the forbidden energy gap, which is the space between the valence band and the conduction band.

Semiconductor Doping and Conductivity Control

When a semiconductor is doped with impurities—whether pentavalent (donor) or trivalent (acceptor)—it causes a displacement of the Fermi level. This level will shift either upward toward the conduction band or downward toward the valence band. The specific direction and magnitude of this shift are determined by the nature and concentration of the added impurity atoms.

While thermal excitation (increasing temperature) can increase electrical conductivity, it is often abandoned in favor of doping because thermal effects are difficult to control. Doping allows for the precise management of conductivity by introducing specific concentrations of impurities. A common industrial ratio for doping is the addition of one impurity atom for every $10^8$ Silicon atoms ($1:10^8$). This process is typically conducted at room temperature. Even at these low concentrations, doping significantly increases the number of charge carriers (electrons and holes), thereby vastly improving the material's ability to conduct electricity in a predictable and controllable manner.

Comparative Material Properties and Pair Production

It is observed that different semiconductor materials do not produce the same number of electron-hole pairs even when they are subjected to the same temperature. This variation occurs because the size of the forbidden energy gap is not uniform across different substances; each material has a unique energy gap that dictates how much energy is required to transition an electron from the valence band to the conduction band.

In terms of efficiency, the process of doping is considered superior to thermal excitation for increasing the electrical conductivity of semiconductors. The primary reason for this superiority is the ability to maintain precise control over the semiconductor's conductive properties. Doping provides a much larger increase in the concentration of charge carriers within the crystal compared to the relatively limited and uncontrollable effects generated by thermal energy alone.

Questions & Discussion

Question: Why is the movement of holes opposite to the movement of electrons in a semiconductor crystal? Response: This occurs because holes move in the direction of the applied electric field, while electrons move in the direction opposite to the influential field.

Question: What is meant by the total current flowing through a pure semiconductor? Response: It is the sum of the electron current and the hole current, where both electrons and holes are referred to as charge carriers.

Question: What determines the occupation of a specific energy level by electrons among the allowed energy levels? Response: The occupation of electrons in a specific level relative to a certain energy level is determined by the "Fermi level." The levels occupied by electrons are those below the Fermi level.

Question: Define the Fermi Level. Response: It is the highest energy level that electrons can occupy at a temperature of absolute zero ($0\,K$). (Note: This appeared in the 2024 Preliminary exams).

Question: Where is the Fermi level located in conductors at $0\,K$? Response: It is located at the region filled with electrons, and the energy levels occupied by these electrons are below the Fermi level.

Question: Where is the Fermi level located for pure semiconductors? Response: it is located in the middle of the forbidden energy gap between the valence band and the conduction band.

Question: What happens to the Fermi level when it is doped with pentavalent or trivalent impurities? Response: It shifts upward or downward, and that displacement is determined according to the type of impurity added.

Question: Why do we resort to doping semiconductors with pentavalent or trivalent impurities if thermal effect also increases electrical conductivity? Response: We do so because of the lack of control over conductivity via thermal means. We resort to adding trivalent or pentavalent impurities at appropriate concentrations at a ratio of one impurity atom for every $10^8$ Silicon atoms. This is done at room temperature and at low concentrations to increase the electrical conductivity by increasing the charge carriers (electrons and holes).

Question: Why is the same number of electron-hole pairs not formed at the same temperature for two different materials? Response: This is due to the difference in the forbidden energy gap for the two materials.

Question: Which is better for increasing the electrical conductivity of semiconductors: the doping process or the thermal effect? Explain. Response: The doping process is better. This is because it is possible to control the electrical conductivity of the semiconductor and increase it by a large percentage as a result of increasing the charge carriers in the crystal compared to what happens in the thermal effect.