IntRoduction to solid state physics

Week 2: Carrier Concentration, Fermi Levels, and Electrical Properties in Semiconductors

Lecture Topics:

1. Carrier Concentration in Semiconductors

   • Intrinsic carrier concentration (Revisited):

     • The carrier concentration in an intrinsic semiconductor is equal for electrons and holes, and it depends on the temperature and the band gap of the material.

     • Formula for intrinsic carrier concentration:

where NC​ and NV​ are the effective density of states in the conduction and valence bands, Eg is the energy band gap, T is the temperature, and kB​ is the Boltzmann constant.

• Extrinsic carrier concentration:

  1. In doped semiconductors, the carrier concentration is influenced by the concentration of dopants:

     1. n-type semiconductors: Electron concentration (n) increases due to donor atoms.

     2. p-type semiconductors: Hole concentration (p) increases due to acceptor atoms.

  2. Ionization of dopants:

     1. Donors and acceptors are typically fully ionized at room temperature, meaning almost all dopant atoms contribute to the carrier concentration.

Fermi Level in Intrinsic and Extrinsic Semiconductors

• Fermi level in intrinsic semiconductors:

  • In an intrinsic semiconductor, the Fermi level (EF​) is located near the middle of the energy band gap.

  • Formula for the intrinsic Fermi level:

• • Fermi level in extrinsic semiconductors:

    1. In n-type semiconductors, the Fermi level shifts closer to the conduction band as more donor atoms are introduced.

    2. In p-type semiconductors, the Fermi level shifts closer to the valence band as more acceptor atoms are introduced.

    3. The position of the Fermi level can be determined by the doping concentration and temperature.

Effect of Temperature on Carrier Concentration

• Low-temperature behavior:

  • At low temperatures, the carrier concentration in doped semiconductors decreases as thermal energy is not sufficient to ionize the dopant atoms.

• High-temperature behavior:

  • At higher temperatures, intrinsic carrier generation dominates, and the semiconductor behaves similarly to an intrinsic material, with both electron and hole concentrations increasing.

• Freeze-out, extrinsic, and intrinsic regions:

  • Freeze-out region: At very low temperatures, dopants are not ionized, and carrier concentration is low.

  • Extrinsic region: At moderate temperatures, dopants are ionized, and the carrier concentration is determined by the doping level.

  • Intrinsic region: At high temperatures, intrinsic carriers dominate over the doped carriers.

Electrical Conductivity and Mobility in Semiconductors

• Conductivity formula:

  • The electrical conductivity (σ\sigmaσ) of a semiconductor depends on both electron and hole concentrations:

• where q is the electron charge, n and p are the electron and hole concentrations, and μn​ and μp​ are the mobilities of electrons and holes, respectively.

• Carrier mobility:

  1. Electron mobility (μn​) and hole mobility (μp​) represent how easily carriers move through the semiconductor when an electric field is applied.

  2. Factors affecting mobility:

     1. Lattice scattering: Carriers are scattered by the vibrating lattice atoms, reducing mobility, especially at high temperatures.

     2. Impurity scattering: Scattering due to ionized impurities (dopants), which dominates at lower temperatures in heavily doped semiconductors.

Drift and Diffusion Currents

• Drift current:

  • The motion of charge carriers (electrons and holes) under the influence of an applied electric field.

  • Drift velocity (vd): The average velocity of charge carriers due to an electric field, given by:

where μ is the carrier mobility and E is the electric field.

Diffusion current:

• Carriers move from regions of high concentration to regions of low concentration, creating a diffusion current.

• Diffusion coefficient (D) and mobility are related by the Einstein relation:

Mass Action Law

• Mass action law:

  • The product of electron concentration (n) and hole concentration (p) in a semiconductor is constant at a given temperature:

where ni is the intrinsic carrier concentration. This relationship holds in both intrinsic and extrinsic semiconductors.

1. Implication of doping:

   1. In n-type semiconductors, n≫p, while in p-type semiconductors, p≫n. The minority carrier concentration can still be calculated using the mass action law.

Examples:

• Calculation of the Fermi level in an intrinsic semiconductor at room temperature.

• Calculation of the electron and hole concentrations in an n-type semiconductor doped with donor atoms.

• Calculation of the conductivity of a p-type semiconductor given the doping concentration and mobility values.

• Application of the Einstein relation to find the diffusion coefficient for electrons in a semiconductor.

Homework/Exercises:

1. Calculate the intrinsic carrier concentration of a semiconductor with a band gap of 1.1 eV at a temperature of 300 K.

2. For a silicon semiconductor doped with 10^16 /cm^3 donor atoms, calculate the electron and hole concentrations at room temperature.

3. Determine the drift velocity of electrons in an n-type semiconductor when an electric field of 100 V/m is applied, given the electron mobility.

4. Explain the relationship between electron and hole concentrations in extrinsic semiconductors and how temperature affects this relationship.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 8: Semiconductors (continued).

• Research papers on the effects of doping and temperature on carrier mobility and conductivity in advanced semiconductor materials.

Key Takeaways:

• The Fermi level in semiconductors plays a crucial role in determining carrier concentration and electrical properties.

• Doping shifts the Fermi level, making semiconductors either n-type or p-type, with a corresponding increase in electron or hole concentration.

• The electrical conductivity of semiconductors depends on carrier concentration, mobility, and the type of doping.

• Temperature has a significant effect on carrier concentration and mobility, influencing the performance of semiconductor devices.

This week focuses on deepening the understanding of carrier concentration, Fermi levels, and electrical properties in semiconductors, with practical applications in designing and analyzing semiconductor devices.