our wafer will be doped with phosphorus to a conc. at which the material exhibits extrinsic behaviour.
To determine the minimum concentration of phosphorus required for the wafer to exhibit extrinsic behavior, you would need to calculate the critical doping concentration or the concentration at which the material transitions from being intrinsic (pure) to extrinsic (doped). This concentration is known as the intrinsic carrier concentration, denoted as ni.
To calculate the intrinsic carrier concentration, you can use the following equation:
ni = √(Nv * Nc * exp(-Eg / (2 * k * T)))
where:
- Nv is the effective density of states in the valence band
- Nc is the effective density of states in the conduction band
- Eg is the energy gap between the conduction and valence bands
- k is the Boltzmann constant (8.617333262145 x 10^-5 eV/K)
- T is the temperature in Kelvin
The intrinsic carrier concentration provides a reference point for determining the minimum concentration of phosphorus needed for extrinsic behavior. Once you have the intrinsic carrier concentration, you can then choose a phosphorus doping concentration higher than this value to ensure extrinsic behavior in the material. The specific concentration of phosphorus will depend on the material properties and desired electrical characteristics.
It's important to note that the actual doping concentration for extrinsic behavior may vary depending on the desired performance criteria, the wafer fabrication process, and other factors. Additional characterization and experimentation might be necessary to fine-tune the doping concentration for specific applications.