4.14 Sspd_chapter_2.2.9-2.2.10-2.2.11:diffusion,life-time and  (Page 2/3)

Under very high injection condition, when minority carriers concentration is comparable to majority carriers concentration then Direct Recombination takes place called Auger Recombination as shown in Figure 2.2.42. This situation is encountered in Silicon Control Rectifier when it fires and goes in Saturation mode. It also occurs under large forward bias of PN Junction diode.

2.2.10.2. Decay of Perturbations in equilibrium values of electron and holes. Under normal conditions, conducting electrons and holes are in equilibrium conditions. There are two mechanisms which are perpetually at work within a semi-conductor:

Generation of EHP by thermal process or optical process and represented by ‘G’.

Recombination of EHP by indiect recombination process and represented by ‘R’.

Under equilibrium condition: G = R and n and p achieve an equilibrium value which is time –invariant.

There can always be external perturbing factors.

If external perturbing factors persist, n and p will acquire a new equilibrium value.

If external perturbing factor is transitory then perturbation in n and p will be a decaying type transient.

This decaying type transient is a solution of the following differential equation which beautifully embodies the Generation and Recombination process:

Excess carrier decay time constant is always decided by the minority carrier life-time hence in the above Equation both decay rates are controlled by τ _n because in this case electrons are the minority carriers.

In the above equation, LHS gives the rate of change of the carriers and RHS gives the excess carrier divided by carrier Life-Time. If excess carrier is positive, rate of change is negative and if excess carrier is negative then rate of change is positive.

Phyical implication is the following:

1. if the actual carrier is greater than the thermal equilibrium value then the excess carrier decays to equilibrium value by recombination.
2. If the actual carrier is less than the thermal equilibrium value then the deficit carrier builds up to equilibrium value by generation.
3. In both cases the perturbed values exponentially decay and actual values settle to equilibrium values.

Equation 2.2.10.2.1. is a first order ordinary linear differential equation. Its variable is the perturbed values:

Rewriting Equation 2.2.10.2.2. in terms of perturbed values:

Or

Therefore the solution of the first order ordinary linear differential equation is:

A is an arbitrary constant determined by one boundary condition:

At time t=0,

Substituting this boundary condition we get: A = n ^{^} ₀ ;

Theefore the solution is rewritten as:

If n>n ^- then excess carrier recombines exponentially with time constant τ _n ;

If n<n ^- then deficit in concentration is made up by net generation and carrier concentration exponentially increases with time constant τ _n ; until the thermal equilibrium value is reached.

Exponential decay of perturbation in thermal equilibrium value of carrier concentration is shown in Figure 2.2.43. for different values of Life-Time.