Module 5 part B3

Semiconductors with Graded Impurity Concentrations

Introduction to Doping

• Semiconductors can be modified through the process of doping, where impurities are introduced. This can significantly alter their electrical properties.

• Doping can introduce carriers (electrons or holes) into the semiconductor. The concentration of these carriers can vary across the material, leading to what is known as graded impurity concentrations.

Effects of Non-Uniform Doping

• When a semiconductor is doped non-uniformly, the distribution of dopants within it is not uniform. This non-uniformity leads to variations in the energy levels—specifically the Fermi level (EF)—with respect to the conduction band (EC) and valence band (EV).

• A higher concentration of carriers in one area leads to a corresponding change in the Fermi level in that region compared to areas with lower dopant concentrations. This can also induce an electric field within the semiconductor, resulting from the charge imbalance that develops.

Induced Electric Field

• The introduction of a graded impurity concentration creates an induced electric field within the semiconductor. This does not require an external applied electric field.

• The variation between the energy levels due to this doping causes a differential in charge distribution, leading to an induced electrical effect that can be quantitatively described with respect to the positions along the semiconductor.

Derivation of Carrier Concentration and Electric Field Relations

• From previous discussions, the concentration of electrons within the semiconductor can be linked to the energy difference between the Fermi levels (EF). This relationship can be expressed through derivative forms.

• Through calculations, we find that the derivative of charge concentration with respect to the position gives rise to the electric field present in the semiconductor. This relationship is derived from equations reflecting energy distributions and carrier mobilities.

Balancing Electric Field and Concentration Effects

• The balance between the mobility of carriers and their diffusion coefficients plays a substantial role. In conducting the calculations, if we denote the electron mobility as ( , \mu ) and the diffusion coefficient as ( \text{D} ), a specific relationship emerges:

  • The balance established between diffusion and mobility, such that ( \frac{D}{\mu} = \frac{kT}{e} ) provides insights into the dynamics of charge carriers within the semiconductor.

Application to P-Type Semiconductors

• The same principles applied to n-type semiconductors can be mirrored in p-type semiconductors, with the sign conventions adjusted appropriately.

• This leads us to the relationship ( \frac{dn}{dp} = \frac{D_p}{\mu_p} = kt ) where the diluting carriers create effective charge transport behaviors.

Conclusion

• Ultimately, the examination of semiconductor behavior under graded impurity concentrations provides crucial insights into their operational characteristics. The interactions of doping, induced electric fields, and carrier mobilities present rich opportunities for exploration.

• Additional examples and clarifications of these principles will be addressed in tutorial sessions to further solidify understanding.