Applying the Basic Equations to a PN Junction

Overview

1. Approximation that assumes that the electric field is confined to some region of the device.
2. The one-dimensional device is divided into three regions with or without an electric field.
3. In a P-N junction, 2 regions having no electric field are on the ends of the device and are termed quasi-neutral regions. The depletion region has an electric field and is located where the junction is.

Depletion approximation

The equations on the previous page are difficult to solve in closed form unless several simplifying assumptions are made. In addition to the assumption of a one-dimensional device, the most central simplifying assumption in determining a closed form solution to the above equations is the depletion approximation. The depletion approximation assumes that the electric field in the device is confined to some region of the device. According to this assumption, the device can then be broken up into regions that have an electric field and those that do not. This is shown below for a pn junction, where Regions I and III do not have an electric field (called quasi-neutral regions or QNR) and Region II has an electric field (which is called space-charge or depletion region).

Schematic showing the regions with and without the electric field according to the depletion approximation. Later in the derivation a change in coordinate system is used where the origin is at the edge of the depletion region and leads into the quasi-neutral regions.

General Procedure using the depletion approximation:

Divide the device into regions with an electric field and without an electric field.

1. Solve for electrostatic properties in the depletion region (Region II on the diagram). This solution depends on the doping profile assumed. Here we will restrict the calculations to constant doping and an abrupt junction.
2. Solve for the carrier concentration and current in the quasi-neutral regions (Regions I and III on the diagram) under steady-state conditions. The steps in this are:
1. Determine the general solution for the particular device. The general solution will depend only on the types of recombination and generation in the device.
2. Find the particular solution, which depends on the surfaces and the conditions at the edges of the depletion region.
3. Find the relationship between the currents on one side of the depletion region and the currents on the other side. This depends on the recombination/generation mechanisms in the depletion region.