# 4.4 The electrostatic analysis of a p-n diode

Table of Contents - 1 2 3 4 5 6 7 8 9 R S ¬ ­ ®
In this Section:
1. The full depletion approximation
2. Calculation of the charge density
3. Calculation of the electric field
4. Calculation of the potential
5. Calculation of the depletion layer width
6. Calculation of the energy band diagram

## 4.4 The electrostatic analysis of a p-n diode

The electrostatic analysis of a p-n diode is of interest since it provides knowledge about the charge and field in the depletion region. It is also required to obtain the capacitance-voltage characteristics of the diode.

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

(pn5)
where the charge density is written as a function of the electron density, the hole density and the donor and acceptor densities. The electron and hole densities can then be expressed as a function of the potential yielding:
(pn5a)
with
(pn5b)
where the potential is taken to be zero in the n-type region.

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.

## 4.4.1 The full-depletion approximation

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, xp, and the depletion region width in the n-type region, xn. The sum of the two depletion layer widths in each region is the total depletion layer width w, or:

(pn17)
From the charge density profile we then calculate the electric field and the potential across the depletion region. A first relationship between the two unknowns is obtained by setting the positive charge in the depletion layer equal to the negative charge. This is required since the electric field in the quasi-neutral regions must be zero. A second relationship between the two unknowns is obtained by relating the potential across the depletion layer width to the applied voltage. The combination of both relations yields a solution for xp and xn, from which all other parameters can be obtained.

## 4.4.2 Calculation of the charge density

Once the full-depletion approximation is made it is easy to find the charge density profile: It equals the sum of the charges due to the holes, electrons, ionized acceptors and ionized holes as given by:
(pn1)
where it is assumed that within the depletion region no free carriers (i.e. electrons or holes) are present. An example is shown in the figure below. The figure shows the charge density at -5 Volt bias (thick line) and at 0 Volt bias (thin line).

pncharge.gif

Fig.4.4.1 Charge density in a p-n junction at -5 Volt bias (thick line) and at 0 Volt bias (thin line). The p-type region is on the left with Na = 1016 cm-3 and the n-type region is on the right with Nd = 2 x 1016 cm-3
As can be seen from the figure, the charge density is constant in each region, as dictated by the full-depletion approximation. The depletion region width also increases as a more negative bias is applied.

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

(pn1a)
(pn1b)
(pn1c)
(pn1d)

## 4.4.3 Calculation of the electric field

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)
As a result one finds that the electric field changes linearly with position. The integration constant is obtained by assuming that the electric field is zero at both ends of the depleton region, or at -xp and xn. The electric field has to be zero outside the depletion region since any field would cause the free carriers to move so that the associated charge eliminates that field. The fact that the electric field is zero on both ends of the depletion region also implies that the total positive charge per unit area in the depletion layer equals the total negative charge.

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

Fig.4.4.2 Electric field in a p-n junction at -5 Volt bias (thick line) and at 0 Volt bias (thin line) (Na = 1016 cm-3 and Nd = 2 x 1016 cm-3)
The maximum electric field occurs at x = 0 and can be obtained by integrating from -xp to 0 or by integrating from xn to 0 yielding:
(pn6)
Both calculations must yield the same result since the dielectric constant is the same in both regions. This provides the first relationship between the two unknowns, xp and xn, namely:
(pn18)
This equation expresses the fact that the total positive charge in the n-type depletion region exactly balances the total negative charge in the p-type depletion region. We can then combine this equation with the expression for the total depletion layer width, w,
(pn17)
and rewrite the depletion layer width in each region as a function of the total depletion layer width, yielding:
(pn10)
and
(pn11)

## 4.4.4 Calculation of the potential

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

(pn4)
We therefore integrate the electric field yielding a piece-wise parabolic potential versus position. An example is shown in the figure below, again for an applied bias of -5 Volt (thick line) and 0 Volt (thin line).

pnpot.gif

Fig.4.4.3 Potential in a p-n junction at -5 Volt bias (thick line) and at 0 Volt bias (thin line) (Na = 1016 cm-3 and Nd = 2 x 1016 cm-3)
The total potential across the semiconductor must equal the difference between the built-in potential and the applied voltage, which provides a second relation between xp and xn, namely:
(pn7)

## 4.4.5 Calculation of the depletion layer width

The depletion layer width is obtained by substituting the expressions for xp and xn into the expression for the potential across the depletion region, yielding:

(pn9)
from which the solutions for the individual depletion layer widths, xp and xn are obtained:
(pn12)
(pn13)

## 4.4.6 Calculation of the energy band diagram

pneb.gif

Fig.4.4.4 Energy band diagram of a p-n junction at -5 Volt bias (upper set of curves) and at 0 Volt bias (lower set of curves)

4.3 ¬ ­ ® 4.5

© Bart J. Van Zeghbroeck, 1996, 1997