Module 2:PN Junction (III)Reverse Characteristics

The equation for current flowing through the diode, derived earlier, is given by the expression:

This expression was derived under the assumptions:

(i) Low Level Injection

(ii) Negligible recombination within the SCR

Although the equation was derived in the context of forward bias, much of the derivation remains valid in reverse bias also
In reverse bias, instead of injection of minority carriers in P and N-regions, there is extraction of minority carriers from them.

Holes now flow from and electrons from .
As a result, N-region gets depleted of holes and P-region gets depleted of electrons

Since can have a maximum value of and respectively,

These are the conditions for low level injection if "injection" is interpreted as having a negative value in this case

Similarly, for the N-side

The spatial variation of electron and hole density is shown below:
The electrons diffuse from the bulk of the P-region to the edge of the depletion region after which they are swept away by the junction field. Similarly, the holes diffuse from the bulk of the N-region to the depletion region edge after which they are swept by the electric field to the P-region.
The source of electrons in P-region and holes in N-region is thermal generation of carriers.

It was shown earlier, in the context of forward bias, that :
This expression is equally valid in reverse bias also, with the difference that the last term now represents generation of carriers within the space charge region, instead of recombination.
In Forward bias, we had neglected this term but as we shall see, this term is the dominant term under reverse bias for Silicon PN junction diodes.

Since within the space charge region:

The negative sign indicates generation !
Over a large fraction of the depletion width, the electron and hole densities are much smaller than the intrinsic carrier density so that

Approximation:
This allows the generation current to be written as :

The net reverse bias current can be written as:

The first tem represents the current due to minority carrier diffusion and the second due to generation within the space charge region.

Example 3.1 Calculate the reverse leakage current for a Silicon PN Junction with

Solution :

For the reverse bias of 1 Volts, depletion width W=0.26µm
The magnitude of the diffusion current is
The magnitude of the generation current is
The generation current is several orders of magnitude larger than the diffusion current !

The above example shows that for silicon PN junction diodes,

Because the depletion width varies as , the reverse bias current would increase slowly with increase in the reverse bias.

Example 3.2 In example 3.1 suppose a similar PN junction is made but on a semiconductor with a bandgap of 0.7 eV. Other things remaining the same, will it still be true that the reverse leakage current is dominated by generation current within the depletion region?

Solution : The generation current would increase by a factor , while the ideal diode saturation current would increase by a factor . The two currents are now comparable. For even smaller bandgaps, the reverse leakage current will be determined entirely by the ideal diode saturation current.

Breakdown:
The reverse current increases slowly with increase in reverse bias till impact ionization induced breakdown begins to occur within the space charge region.

Impact Ionization: An electron or a hole travelling through a region of high electric field can acquire enough energy to create another electron-hole pair.

Impact ionization is characterized by a parameter called ionization coefficient:

= probability that an electron causes an impact ionization within dx
= probability that a hole causes an impact ionization within dx
It is natural to expect that the ionization coefficients would a function of carrier energy and therefore the electric field.

There are a variety of models for impact ionization coefficient, simplest of which is :

for Silicon (22)
As the reverse bias increases, the electric field within the junction also increases thereby increasing the probability of impact ionization.
An electron or hole generated due to impact ionization within the depletion region can acquire enough energy again to cause another impact ionization. The new electron-hole pairs generated can in turn generate further electron-hole pairs.
As a result of this process, a single carrier entering the depletion region can get multiplied many times over. This process of multiplication is known as Avalanche Multiplication.
The normal reverse current gets multiplied by the avalanche multiplication process. When avalanche multiplication becomes large, very large reverse current begins to flow and breakdown is said to occur.
To obtain an expression for breakdown voltage, it has to be precisely defined. This is explained using the Figure below:

Suppose a single electron enters the depletion region at . Due to avalanche multiplication, , number of electrons will come out at the end .

Breakdown :
The number of electrons generated within will come from impact ionization caused by the electrons and holes in this region so that

where n(x) is the number of electrons at x travelling right to the N-region and p(x) is the number of holes travelling left towards the P-region.
Since no holes are assumed to enter the depletion region, p(x) must be due to impact ionization in the region .

An equal number of electrons also must have been generated also so that, the number of electrons that would come out of the depletion region must be:

This allows Eq. (24) to be re-written as:

On Integrating across the depletion region:

The breakdown condition for can be now written as:

where

The computation of breakdown voltage is simpler if we take a one sided junction such as a P+N junction.

For this case:

At breakdown:

The Maximum electric field at the junction when breakdown occurs can be expressed as:

The max. electric field at breakdown is a weak function of doping:
It can therefore be said that whenever the maximum electric field at the junction acquires a critical value of , breakdown would occur.

Taking at breakdown allows an estimate of the breakdown voltage to be determined rapidly for any PN junction diode.

Example 2.3 Determine breakdown voltage for a PN junction shown below Assume that = 0.9 Volts

Solution : We first perform a check whether at breakdown, the depletion width still lies in the lightly doped region or not. If it does then, This shows that depletion width will extend into the higher doped N-region as well resulting in the following diagram.

The electric field at is obtained using the expression .
Using the Poisson's equation: .
The area under the electric field curve will be equal to + BV so that BV = 40.7 Volts .

Example 2.4 Keeping in mind that electron ionization coefficient is larger than hole ionization coefficient , which diode or is likely to have a higher breakdown voltage with identical doping values.

Solution : The question can be answered by examining the electron and hole density profiles within the depletion region generated due to impact ionization. These are shown below:

In the junction, the electron density is maximum near the high field region at the junction and hole density is minimum. As a result most of impact ionization is done by electrons, while the reverse holds true for junction. Therefore junction will have lower breakdown voltage.

Example 2.5 Obtain an expression for the breakdown voltage of a cylindrical PN junction. This is of interest because junctions have a curvature near the periphery which can be considered as cylindrical.

Solution : The Figure below shows the junction.

The Poisson equation in cylindrical coordinates can be written as

Integration of Poisson's equation with the boundary condition that electric field at the depletion edge is zero we obtain

Further integration gives

The expressions above can be used to find the breakdown voltage by using the fact that at breakdown, the electric field is equal to the critical field. The table below shows the breakdown voltages computed for a doping of and different radii of curvature.

As a comparison, the breakdown voltage for a planar junction turns out to be 31 Volts.

The expression for multiplication factor derived earlier suggests that multiplication can be empirically modeled as

The parameter n varies with the structure of the PN junction, with

n=6 for diode

n=4 for diode
The avalanche breakdown is the most common mechanism of breakdown in PN junction diodes.
There is another mechanism called Zener breakdown that comes into play in diodes with heavily doped P and N regions.
As noted earlier, in reverse bias, the holes are required to flow from the P-side to the N-side and electrons from P-side to the N-side. The reverse current is normally small because there are so few holes in N-region and electrons in P-region.
However, there are plenty of electrons in valence band of P-side and plenty of empty states in the conduction band of N-side. Except via tunneling, the electrons from the valence band of P-region cannot flow to empty states in the conduction band of N-side due to presence of a potential barrier
When the probability of tunneling becomes significant, large reverse current begins to flow and Zener breakdown is said to occur.

The Figure below depicts the tunneling process:

The barrier that the electron sees while tunneling, can be approximated as a triangular barrier as shown below:

The tunneling property can be written as

Use of the triangular barrier approximation gives :

As expected, the transmission probability increases exponentially with the thickness of the barrier which can be expressed as

where is the electric field within the junction.
As doping increases, the electric field increases causing barrier to become narrower and tunneling probability to increase.
To achieve significant tunneling, the barrier width should be only a few tens of Angstroms.
The field calculated for Avalanche breakdown was , which is lower than that required for Zener breakdown. It appears, therefore, that avalanche breakdown would always precede Zener breakdown !
However, it is not the electric field but the carrier energy that is really important for impact ionization. A very high electric field in a very narrow region may not allow a carrier to gain enough energy so that impact ionization becomes significant.
As a result, Zener breakdown occurs in very heavily doped junctions only with small depletion widths. Because of the small depletion widths, the breakdown voltage, despite the high electric field, is often Volts.
Diodes which have breakdown voltages larger than 7-8 Volts break down due to Avalanche multiplication process. In the intermediate range both the processes may be active.
It is possible to determine the breakdown mechanism by measuring the temperature sensitivity of the breakdown voltage. Diodes which break down via avalanche multiplication have a positive temperature coefficient, while those that breakdown via tunneling have a negative temperature coefficient.
The increase in avalanche breakdown voltage with temperature occurs due to increased scattering which makes it more difficult for carriers to acquire energy from the electric field.
The decrease of Zener breakdown voltage with increase in temperature occurs because of increased carrier velocity which increases the flux of carriers attempting to cross the barrier. Since transmission probability remains unchanged, the tunneling current increases with temperature.
Most of the diodes that go under the name Zener diodes have a breakdown via avalanche multiplication rather than tunneling.