A semiconductor with a band gap of 0.8 eV and an intrinsic carrier concentration of 10^12 cm^-3

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A semiconductor with a band gap of 0.8 eV and an intrinsic...

A semiconductor with a band gap of $0.8 \mathrm{eV}$ and an intrinsic carrier concentration of $10^{12} \mathrm{~cm}^{-3}$ is doped with $10^{18} \mathrm{~cm}^{-3}$ donors on the left half and $10^{17} \mathrm{~cm}^{-3}$ acceptors on the right half. Draw the equilibrium band diagram. Calculate the junction potential and the position of the Fermi level, and indicate them on the band diagram. Suppose an electron at the conduction band edge on the $\mathrm{p}$ side goes over the $\mathrm{n}$ side without scattering. Assuming parabolic band structure, calculate its wavevector there. The effective mass of the carriers is $0.2 m_{0}$.

Key Concepts

Wavevector Calculation

The wavevector (k) of a carrier in a semiconductor can be calculated using the parabolic band relation, which connects the kinetic energy of the carrier to k via the effective mass. Specifically, the relation E = (?²k²)/(2m*), where ? is the reduced Planck constant, allows for the extraction of k when the energy difference from the band edge is known. This calculation is crucial in understanding the momentum of electrons that traverse the junction without scattering.

Parabolic Band Structure

A parabolic band structure is an approximation where the energy of carriers near the band edges varies quadratically with the wavevector. This model simplifies the analysis of carrier dynamics by allowing the use of effective mass in calculations and is particularly valid for electrons at the bottom of the conduction band or holes at the top of the valence band.

Fermi Level Alignment

At equilibrium, the Fermi level, which represents the chemical potential for electrons, remains constant throughout the semiconductor structure. The alignment of the Fermi level across regions with different doping levels drives the bending of bands, thereby modulating the barrier height for carrier transport across the junction.

Effective Mass

The effective mass is a parameter that accounts for the interaction between charge carriers and the periodic crystal lattice. It modifies the inertia of the carriers, influencing their acceleration under an electric field and their quantum mechanical behavior. The effective mass is a key factor when using the parabolic band approximation to compute quantities like the carrier wavevector.

Doping in Semiconductors

Doping introduces impurities into an intrinsic semiconductor to create n-type (donors) or p-type (acceptors) materials, thereby shifting the position of the Fermi level and altering the carrier concentrations. Heavily doped regions lead to a significant band bending near the junction, which gives rise to a built-in electric field and the formation of depletion regions.

Energy Band Diagram

An energy band diagram represents the distribution of energy levels in a semiconductor, showing the conduction band, valence band, and the Fermi level. In equilibrium, the Fermi level is constant across the device, even though the bands bend due to variations in doping levels. This diagram is essential for visualizing how carriers behave in different regions of the semiconductor, especially at junctions where different doping types meet.

Junction (Built-In) Potential

The junction potential, or built-in potential, is the voltage difference that develops across a p-n junction due to the diffusion of carriers and the resulting charge imbalance. It plays a crucial role in establishing equilibrium by aligning the Fermi levels across the junction and determines the extent of the band bending observed in the energy band diagram.

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