IntRoduction to solid state physics

Week 1: Semiconductors – Intrinsic and Extrinsic Properties

Lecture Topics:

1. Introduction to Semiconductors

   • Definition of semiconductors:

     • Semiconductors are materials whose electrical conductivity lies between that of conductors (like metals) and insulators. Their conductivity can be modified by doping or temperature changes.

   • Energy band structure:

     • Semiconductors have a valence band (occupied by electrons) and a conduction band (where electrons move freely). The two bands are separated by a small energy gap, typically less than 3 eV.

     • Band gap: The energy difference between the top of the valence band and the bottom of the conduction band. Common semiconductors like silicon (Si) and gallium arsenide (GaAs) have band gaps of 1.12 eV and 1.42 eV, respectively.

2. Intrinsic Semiconductors

   • Intrinsic behavior:

     • In intrinsic semiconductors, there are no impurities, and electrical conductivity is purely dependent on the material’s properties.

     • At temperatures above absolute zero, some electrons gain enough thermal energy to jump from the valence band to the conduction band, creating electron-hole pairs.

   • Carrier concentration in intrinsic semiconductors:

     • The concentration of electrons in the conduction band is equal to the concentration of holes in the valence band.

• where ni is the intrinsic carrier concentration, NC​ and NV​ are the effective density of states in the conduction and valence bands, Eg​ is the energy band gap, and T is the temperature.

• Temperature dependence:

  1. As temperature increases, more electrons gain enough energy to move to the conduction band, increasing the carrier concentration exponentially.

Extrinsic Semiconductors

• Doping:

  • Doping involves adding impurities to a semiconductor to alter its electrical properties. Doped semiconductors are called extrinsic semiconductors.

• n-type semiconductors:

  • Formed by doping a semiconductor with an element that has more valence electrons than the host material (e.g., doping silicon with phosphorus).

  • The dopant introduces extra electrons, increasing the electron concentration in the conduction band.

  • Fermi level: The Fermi level moves closer to the conduction band in n-type semiconductors.

• p-type semiconductors:

  • Formed by doping with an element that has fewer valence electrons than the host material (e.g., doping silicon with boron).

  • The dopant creates holes in the valence band, increasing hole concentration.

  • The Fermi level moves closer to the valence band in p-type semiconductors.

Carrier Concentration in Extrinsic Semiconductors

• Electron and hole concentration:

  • In extrinsic semiconductors, the carrier concentration depends on the doping level and temperature.

• 1. For an n-type semiconductor, the electron concentration n is approximately equal to the dopant concentration, and the hole concentration p is reduced.

  2. For a p-type semiconductor, the hole concentration p is approximately equal to the dopant concentration, and the electron concentration n is reduced.

• Fermi level shifts:

  1. the Fermi level shifts depending on the doping type and concentration in doped semiconductors. In n-type semiconductors, the Fermi level shifts closer to the conduction band, and in p-type semiconductors, it shifts closer to the valence band.

Electrical Conductivity of Semiconductors

• Conductivity equation:

  • The electrical conductivity σ of a semiconductor is the sum of contributions from both electrons and holes:

where q is the charge of an electron, n is the electron concentration, p is the hole concentration, and μn\mu_nμn​ and μp​ are the mobilities of electrons and holes, respectively.

1. Doping effects:

   1. Doping increases the number of charge carriers, thereby increasing the conductivity of the semiconductor. The type of doping determines whether the semiconductor is primarily conducting through electrons (n-type) or holes (p-type).

Examples:

• Calculation of the intrinsic carrier concentration in a silicon sample at room temperature using the energy band gap and effective density of states.

• Determining the Fermi level shift for an n-type semiconductor doped with a known concentration of donors.

• Calculation of the electrical conductivity of a p-type semiconductor given the hole concentration, hole mobility, and electron mobility.

Homework/Exercises:

1. Calculate the intrinsic carrier concentration of a semiconductor with a band gap of 1.5 eV at room temperature.

2. For an n-type semiconductor doped with phosphorus at a concentration of 10^16/ cm^3, calculate the electron and hole concentrations.

3. Compare the Fermi level positions in intrinsic, n-type, and p-type semiconductors and explain how doping affects the position of the Fermi level.

4. Determine the electrical conductivity of a doped silicon sample given the carrier concentration and mobility data.

Suggested Reading:

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

• Research papers on semiconductor doping techniques and their impact on electronic device performance.

Key Takeaways:

• Semiconductors are materials with a small energy band gap, and their electrical properties can be tuned through temperature changes and doping.

• In intrinsic semiconductors, the carrier concentration is governed by temperature, while in extrinsic semiconductors, doping dominates the carrier concentration.

• The Fermi level in semiconductors shifts with doping, affecting the electrical conductivity and the behavior of devices like diodes and transistors.

• Understanding the behavior of semiconductors is essential for designing and optimizing electronic devices such as solar cells, LEDs, and transistors.

This week introduces fundamental semiconductor properties, focusing on intrinsic and extrinsic behavior, doping, and how these factors influence the electrical conductivity and performance of semiconductor devices.