RECOMBINATION OF CARRIERS
In the previous section we examined the effects of the application of an external electric field to a metal or a semiconductor. This is the most common way to deviate from thermodynamic equilibrium. But there are other ways, notably the application of an electromagnetic field (light) and the injection of carriers from one side of the sample to the other one.
FIGURE 3.4 Composite relaxation time and relaxation times for impurities Tjmp and lattice vibrations tph as a function of temperature T.
The former creates electron-hole pairs, whereas the latter simply adds more electrons or holes into the system, thereby destroying the relation
that we have seen in section 2.7. Key to understanding both processes is the concept of recombination of electrons with holes in semiconductors. The theory we are going to give is adequate, provided that the condition of “low injection” holds, which means that the majority carriers remain almost unaltered (percentagewise), whereas it is the minority carriers that change significantly.
Let us take an arithmetic example to see how this is possible. Consider a sample of Si doped with No = 1015/cm3 donors. Since the intrinsic concentration of Si и, «10lo/cm3 at room temperature we deduce that
Now imagine an injection of Ю10/cm3 of holes. This will change the number of holes tremendously (by 5 orders of magnitude), but even if we assume that all these holes recombine with electrons, the precentage change in the number of electrons is insignificant. Therefore in what follows we will assume that the number of majority carriers remains essentially unaltered and the process of injection or the incidence of light do not change the character of the semiconductor, i.e. change it from N to P and vice versa.
FIGURE 3.5 (a) Arrows G8 and Req portray the thermal generation and recombination of carriers
respectively between the valence and conduction bands of a semiconductor at equilibrium, (b) a non-equilibrium case with Gф indicating the excitation of electrons from the valence band to the conduction band due to an electromagnetic field of the appropriate frequency.
Of course the idea that we have just expressed that all injected or photogenerated minority carriers will recombine is not representative of reality and was only used to show that the majority carrier concentration remains essentially unaltered. Both the photogeneration of carriers and the injection of minority carriers create a new state of things that is not given by equation 2.61. This new state forms the basis for the understanding of the PN junction, which is the basic building block of both the bipolar junction transistor and the metal-oxide-semiconductor field effect transistor.
Let us begin with an N-type semiconductor in thermodynamic equilibrium. Under this condition, equation 2.61 holds true, but it must be interpreted as a case of dynamic equilibrium in which electrons constantly recombine with holes and new electrons in equal amounts are created thermally. These two processes are shown schematically in figure 3.5a by the corresponding arrows connecting the band edges Ec, Ev. Since we have an equilibrium, i.e. a time-independent set of variables we must have (the subscript о denoting the equilibrium)
where G0 is the rate of thermally activated electrons (producing holes) and Req is the rate of electron recombination in equilibrium. Therefore, as expected,
Ge is determined solely by temperature, whereas the recombination rate R is proportional to both n and p. Hence we can write quite generally (not only for thermodynamic equilibrium)
where C is a constant. For the case exclusively of thermodynamic equilibrium we get
Now consider as an example of a deviation from thermodynamic equilibrium, an electromagnetic field incident on the N-type semiconductor. This field will excite electrons from the valence band to the conduction band creating electron-hole pairs at a constant rate G,,, as shown in figure 3.5b. We no longer have a time-independent situation as light keeps piling up electrons in the conduction band creating holes in the valence band. How will a new steady state (not a thermodynamic equilibrium) be reached if electron-hole pairs keep being created? Answer: recombination will increase.
The net thermal generation minus recombination rate at non-equilibrium is
where we have used the fact that the number of majority carriers is not appreciably changed. But given equation 3.17, for the net rate of whole recombination away from equilibrium we have
The quantity xp is called the recombination time and although the symbol is the same, it is not the same as the relaxation time for collisions which usually bears the same symbol.
Then to get the net rate of creation of electron-hole pairs, we must subtract from the
above net rate of recombination
where in equation 3.20 we have included a subscript n to indicate that the equation refers to holes in an N-type semiconductor. This will constitute a standard notation from this point onwards. From the above equation we can see that the incident light (of hoi ^ Eg) will create a new steady state after a time much longer than xp, where the number of holes would have increased by
In fact, the complete solution of 3.20 (a 1st order differential equation) under the initial condition that Ap„ (0) = 0 is
FIGURE 3.6 Time evolution of the excess holes Ap(t) after an electromagnetic field has been switched on.
The graph of this function is shown in figure 3.6 for two values of xp. For a P-type semiconductor, by exactly the same kind of arguments we get
and for its solution
We have so far considered generation and recombination processes which consist of only transitions from the top of the valence band Ev to the bottom of the conduction band Ec and vice versa. However, there are other indirect and more intricate generation and recombination processes that occur through what we call “states in the gap” of the semiconductor. These are states in the gap which derive from unwanted impurity atoms in the semiconductor just like the As and Ga atoms which are intentionally implanted in Si to create donors and acceptors respectively. These impurities are usually metal atoms that are left over after the purification process of the semiconductor.
Let us call Et the energy of one such state. Let us also distinguish whether the energy state in the gap is occupied or empty by the symbols £,(occ) and £,(emp) respectively. Then the following 4 electron processes may occur
At steady state the capture (cap) and emission (em) processes must balance out for each type of carrier separately Then in a notation with capital R standing for rate and subscripts n, p standing for electrons and holes respectively we must have
This problem was tackled by Shockley, Read, and Hall and hence bears their names. Under certain simplifying assumptions the recombination rate R, of electrons and holes through a state E, whose energy is equal to the mid-gap energy is
where xn and xp are constants with the dimension of time.