Table of Contents - 1 2 3 4 5 6 7 8 9 R S ¬ ®

In this Section:

- The full depletion approximation
- Calculation of the charge density
- Calculation of the electric field
- Calculation of the potential
- Calculation of the depletion layer width
- Calculation of the energy band diagram

The general analysis starts by setting up Poisson's equation:

- (pn5)

- (pn5a)

- (pn5b)

This second order non-linear differential equation can not be solved
analytically. Instead we will make the simplifying assumption that
the depletion region is fully depleted and that the neutral regions
contain no charge. This
*full depletion approximation* is the topic of the next section.

The full-depletion approximation assumes that the depletion region around
the
metallurgical junction has well-defined edges with an abrupt transition
between
the fully depleted region where no carriers are present and the
*quasi-neutral region*, a neutral region where
the carrier density is close to the doping density.

This approximation is justified by the fact that the carrier densities change exponentially with the position of the fermi energy relative to the band edges. For example as the distance between the fermi level and the conduction band edge is increased by 59 meV, the electron concentration at room temperature decreases to one tenth of its original value. The charge in the depletion layer is then quickly dominated by the remaining ionized impurities, yielding a constant charge density for uniformly doped regions.

We will therefore start our analysis using an abrupt charge density
profile, while
introducing two unknowns, namely the depletion layer width in the p-type
region,
*x _{p}*, and the depletion region width
in the n-type region,

- (pn17)

- (pn1)

pncharge.gif

The charge throughout the diode is given by the following equations:

- (pn1a)

- (pn1b)

- (pn1c)

- (pn1d)

The electric field is obtained from the charge density using
*Gauss's law*,
which states that the field gradient equals the charge density divided by
the dielectric constant or:

- (pn2)

An example which is calculated from the charge density shown in the figure above is provide below. Again the figure is calculated for a bias of -5 Volt (thick line) and 0 Volt (thin line).

pnfield.gif

- (pn6)

- (pn18)

- (pn17)

- (pn10)

- (pn11)

The potential in the semiconductor is obtained from the electric field using:

- (pn4)

pnpot.gif

- (pn7)

The depletion layer width is obtained by substituting the expressions
for *x _{p}* and

- (pn9)

- (pn12)

- (pn13)

pneb.gif

© Bart J. Van Zeghbroeck, 1996, 1997