Doping: Discussed Fermi distribution and density of states.
Numericals: Solved simple problems regarding P and N doping (majority and minority carriers).
Key Terminology:
Majority carriers: Type of carrier predominant in a doped semiconductor.
Minority carriers: Lesser type of carrier in the doped semiconductor.
Current Lecture Overview
Focus on:
Temperature dependence of carrier concentration.
High doping effects.
Scattering and mobility (to be discussed in future lectures).
Doping and Charge Neutrality
Doping Types:
N-type: Doped with phosphorous or arsenic (donor atoms).
P-type: Doped with boron (acceptor atoms).
Equations of Carrier Concentrations:
Majority carrier concentration for N-type doping (e.g., n-type silicon with doping density of 10^17 cm^-3):
[ n = 10^{17} \text{ cm}^{-3} ]
[ p = \frac{n_i^2}{n} \approx 10^{3} \text{ cm}^{-3} ]
Charge Neutrality Condition:
Total positive charge = Total negative charge
Equation:
[ p + N_D^+ = n + N_A^- ]
Where N_D and N_A represent ionized donor and acceptor concentrations respectively.
Donor and Acceptor Dynamics
N-type Semiconductor:
Donors provide electrons, leaving positively charged ionized atoms.
Total number of free electrons and positively charged ions must balance.
P-type Semiconductor:
Acceptors create holes, leading to negatively charged ionized cores.
Equilibrium Condition:
The number of free carriers is determined by doping types and their concentrations.
Temperature Dependence of Carrier Concentration
As temperature decreases, the carrier concentration also decreases.
Carrier Freeze-Out:
Occurs at low temperatures when thermal energy is insufficient to ionize donor/acceptor atoms.
In this state, fewer electrons are available for conduction.
Plotting Carrier Concentration:
A log plot of carrier concentration versus inverse temperature shows:
At low temperatures, the carrier concentration decreases exponentially.
At moderate temperatures, it stabilizes (extrinsic region) before becoming intrinsic at high temperatures, where intrinsic carrier concentration surpasses dopant levels.
Donor Ionization Energy
Donor Ionization Energy (E_D):
Close to conduction band, typically much smaller than thermal energy (KT) at room temperature (e.g., 10 meV vs. 26 meV).
With decreasing temperature, the fraction of ionized donors decreases, resulting in reduced free electron concentration.
High-Temperature Electronics
Wide Band Gap Materials:
Suitable for high-temperature applications due to lower intrinsic carrier concentrations (e.g., Gallium Nitride and Silicon Carbide).
High temperatures can lead to significant increases in intrinsic carrier concentration in narrower band gap materials like silicon.
Conclusion and Future Topics
Recap of key points: charge neutrality, temperature dependence of carrier concentration, and implications of doping.
Next lecture topics to include:
Incomplete Ionization: Not all dopants become ionized even at room temperature.
Drift and Conductivity Concepts: Mobility, diffusion, and their relations to semiconductor behavior.