semiconductor physics and Devices

Week 2: Extrinsic Semiconductors - Doping and Carrier Dynamics

Lecture Topics:

1. Introduction to Extrinsic Semiconductors

   • Definition of extrinsic semiconductors: Semiconductors whose electrical properties have been modified by introducing impurities (dopants).

   • Purpose of doping: To increase the number of charge carriers (electrons or holes) and enhance electrical conductivity.

   • Types of doping:

     • n-type: Introduces donor impurities that provide extra electrons.

     • p-type: Introduces acceptor impurities that create holes in the valence band.

2. N-type semiconductors

   • Donor impurities (e.g., phosphorus in silicon) donate electrons to the conduction band.

   • The Fermi level in an n-type semiconductor moves closer to the conduction band.

   • Electron concentration in n-type semiconductors: 

n≈ND (at temperatures well below the ionization energy of the donor), 

where ND​ is the concentration of donor atoms.

2. • Temperature dependence: At low temperatures, donor electrons are bound to the impurity atoms, while at higher temperatures, they are excited into the conduction band.

3. P-type semiconductors

   • Acceptor impurities (e.g., boron in silicon) create holes in the valence band.

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

   • Hole concentration in p-type semiconductors: 

p≈NA (at temperatures well below the ionization energy of the acceptor), 

, where NA is the concentration of acceptor atoms.

3. • Temperature dependence: At low temperatures, holes remain bound to acceptor atoms, and at higher temperatures, they are free to move in the valence band.

4. Fermi Level in Extrinsic Semiconductors

   • The position of the Fermi level depends on the type and concentration of dopants.

   • In n-type semiconductors, the Fermi level is closer to the conduction band.

   • In p-type semiconductors, the Fermi level is closer to the valence band.

   • As temperature increases, the Fermi level shifts slightly but remains near the dopant energy level for extrinsic semiconductors.

5. Carrier Concentration in Extrinsic Semiconductors

   • Charge neutrality condition: The total number of positive and negative charges must be equal in a semiconductor.

   • For n-type semiconductors: n≈ND​ and 

   • for p-type semiconductors: p≈NA

   • Minority carriers: Even in doped semiconductors, there are still a small number of opposite charge carriers (e.g., holes in n-type and electrons in p-type), but their concentration is much lower than the majority carriers.

   • Mass action law: The product of the electron and hole concentrations is constant at a given temperature:

where Nc and Nv are the effective density of states in the conduction and valence bands, Eg​ is the energy band gap, kB​ is Boltzmann’s constant, and T is the temperature.

1. Temperature dependence of carrier concentration and the importance of band gap size in determining the material’s behavior at different temperatures.

Fermi Level in Intrinsic Semiconductors

• Introduction to the Fermi level: The energy level at which the probability of an electron occupying a state is 50%.

• For intrinsic semiconductors, the Fermi level lies near the middle of the band gap.

• Temperature dependence of the Fermi level position in intrinsic semiconductors.

Electrical Conductivity of Intrinsic Semiconductors

• Calculation of the electrical conductivity σ\sigmaσ for an intrinsic semiconductor

where ni is the intrinsic carrier concentration.

Electrical Conductivity of Extrinsic Semiconductors

• Electrical conductivity σ\sigmaσ of extrinsic semiconductors is dominated by the majority carriers:

• where n and p are the electron and hole concentrations and μe​ and μh​ are the mobilities of electrons and holes, respectively.

• The conductivity increases significantly with doping because the number of charge carriers increases.

Examples:

• Calculate the Fermi level shift in an n-type semiconductor with a given donor concentration.

• Estimating electron and hole concentrations in an extrinsic semiconductor at a given temperature.

• Calculate the electrical conductivity for both n-type and p-type semiconductors, given the carrier concentration and mobility.

Homework/Exercises:

1. Explain how the Fermi level shifts in an n-type and p-type semiconductor when doped with donor and acceptor atoms.

2. Calculate the electron concentration in an n-type semiconductor with a donor concentration of 10^17 /cm^3 at room temperature.

3. Derive the expression for electrical conductivity in a p-type semiconductor given the hole concentration and mobility.

Suggested Reading:

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

Key Takeaways:

• Doping a semiconductor introduces donor or acceptor atoms, which modify the carrier concentration and shift the Fermi level.

• n-type semiconductors have an excess of electrons, while p-type semiconductors have an excess of holes, and these majority carriers dominate the electrical conductivity.

• The mass action law governs the relationship between electron and hole concentrations, ensuring charge neutrality in the material.

This week introduces extrinsic semiconductors, focusing on the effects of doping on carrier concentration, Fermi level positioning, and electrical conductivity. In future lectures, understanding these concepts will be crucial for analyzing semiconductor devices such as diodes and transistors.