The generation and recombination of electrons and holes in a semiconductor play an important role in their electrical and optical behavior. These processes are defined as:
Recombination: A process whereby electrons and holes are annihilated or destroyed.
Generation: A process whereby electrons and holes are created.
These process are illustrated in the figure below (Pierret Fig. 5.1)
The recombination processes can be reversed resulting in generation processes. Band-to-Band generation occurs when an electron from the valence band is excited by light or heat to the conduction band. Generation of an electron and hole by an R-G center intermediary can be done in a couple of ways including and electron from the valence band being excited to the trap state and then to the conduction band creating a hole in the valence band and electron in the conduction band. Also, an electron in the trap state can be excited to the conduction band while the hole is excited to the valence band. Another common carrier generation is carrier generation via impact ionization where a high energy electron losses energy to produce an electron-hole pair. These processes are illustrated in the following figure (Pierret Fig. 5.2).
Following Pierret, the following definitions will be used:
The time rate of change in the electron concentration due to both R-G center recombination and R-G center generation
The time rate of change in the hole concentration due to both R-G center recombination and R-G center generation
Number of R-G centers per cm3 that are filled with electrons
Number of empty R-G centers per cm3
Total number of R-G centers per cm3, NT=nT+pT
The term will be positive if the electron concentration in the conduction band is increasing and negative if it is decreasing. Likewise for .
The possible electronic transitions between a single-level R-G center and the energy bands are illustrated in Fig. 5.5 of Pierret:
In the figure, transitions (a) and (b) effect conduction band electron concentration and transitions (c) and (d) effect valence band hole concentrations. Hence, these concentrations can be written as:
It is easily seen that transition (a) is directly proportional to the conduction band electron concentration n and the concentration of empty R-G centers given by pT. This can be written as:
where cn is the constant of proportionality and is called the electron capture coefficient. It is also easily seen that transition (b) is directly proportional to the concentration of filled R-G centers given by nT and the concentration of empty conduction band states. However, because the vast majority of conduction band states are empty, the dependency of transition (b) on the concentration of empty conduction band states drops out resulting in the following equation:
where en is the constant of proportionality and is referred to as the electron emission coefficient.
The transitions (c) and (d) effect the valence band hole concentration. Transition (c) will be proportional to the valence band hole concentration and the concentration of filled R-G centers:
where cp is the hole capture coefficient. Transition (d) is proportional to the electron concentration in the valence band (this dependency drops out because there are virtually unlimited electrons in the valence band) and the concentration of empty R-G centers:
We can then write down equations for the conduction band electron recombination rate (rN) and valence band hole recombination rate (rP):
Under equilibrium conditions, the Principle of Detailed Balance can be invoked. The Principle of Detailed Balance states that under equilibrium conditions, each fundamental process and its inverse must self-balance independent of any other process that may be occurring inside the material. Hence, it is seen that rN=0 and rP=0 and we can eliminate one of the constant of proportionalities in each equation. Introducing the subscript “0” to refer to equilibrium, the electron and hole emission coefficients are easily solved for:
where n1 and p1 are computable constants given by:
To a good approximation, the emission and capture coefficients under nonequilibrium conditions equal their values under equilibrium conditions, hence:
Hence, the recombination rates under nonequilibrium conditions can be written as:
The steady-state condition is much different than the equilibrium condition. In equilibrium, there is the principle of detailed balance where every carrier recombination process is in equilibrium with its inverse process. In the steady state situation, there is not a detailed balance but an overall tradeoff between the various processes that produced time independent observables, such as time independent conduction band electrons, valence band holes, filled and empty trap states… Pierret uses the fact that the concentration of filled trap states is independent of time. Thus we can write:
Hence, under steady-state conditions rN = rP and we can solve for nT:
Substituting this result into the equations for rn and rp, we have:
where and are defined as the minority electron lifetime and the minority hole lifetime respectively.
Low Level Injection
Low level injection means that the perturbation in the conduction band electron concentration (Dn) and valence band hole concentration (Dp) are small with respect to the equilibrium majority carrier concentration, either no for n-type material or po for p-type material. Consider n-type material for the calculation below. Substituting n = no +Dn and p = po +Dp in the recombination equation, we have:
Multiplying everything out and using the facts that , , we get:
For p-type material and low level injection, the recombination rate becomes:
For high level injection where , the recombination rate becomes:
For an R-G depletion region where and we obtain the recombination rate:
This result is a negative quantity indicating that generation of electron-hole pairs occur to reestablish equilibrium. This can be expressed as:
where is the generation lifetime.