PN Junction Equations

Applying the Basic Equations to a PN Junction

The basic equations are:

Poisson's Equation which describes the elementary relationship between the charge and strength of the electric field strength. Transport Equation which describes the movement of carriers or the flow current. Continuity Equation which tracks all the carriers in relation to generation, movement and recombination. Those equations could be labelled “book keeping” equations because they account for every carrier.

Requirements for applying Basic Equations

The Basic equations for electric field, transport and continuity of carriers are not easy to solve in closed form except by making a number of assumptions. Other than the assumption of a device that is one dimensional, the depletion approximation is the utmost vital simplifying assumption in determination of a closed form solution to the basic equations. The depletion approximation makes the assumption that the electric field in a device is being confined to a certain region of this device. As per this assumption, the device can be accordingly divided into regions which have an electric field and other regions that do not. Where regions which do not have an electric field  are named QNR (quasi neutral regions) and other regions that has an electric field are named depletion region  or space-charge.So approximations are used that assumes that the electric field is only constrained to certain regions of the device. A device which is one-dimensional device will be divided into three different regions with an electric field or without an electric field. The two regions which have no electric field in a PN junction are on the outer ends of the device and are commonly called quasi neutral regions. The region that has an electric field is the depletion region and this region is positioned where the junction is located.

General Procedure using the depletion approximation:

The device is divided into two regions, ones which have an electric field and others that don’t have an electric field.

1. Start solving for the depletion region’s electrostatic characteristics. Such a solution will depend on the assumed doping profile. Hereby the calculations will be restricted an abrupt junction and to constant doping.
2. After that solve for the carrier current and concentration in the quasi-neutral regions under conditions of steady state. The followed steps for this are detailed below:
   1. Determining the general solution for the specific device. The general solution will only depend on the device’s generation and types of recombination.
   2. Find the specific solution that relies on the depletion region’s surfaces and the conditions at the edges.
3. Determine the relationship that links the currents on one side of the depletion region and the currents on the other side of the region. This would depend on the mechanisms of recombination and generation in the depletion region.