Generation-Recombination of Carrier

In a semiconductor under thermal equilibrium, carriers possess an average thermal energy corresponding to the ambient temperature. This thermal energy excites some VB electrons to reach the CB. This upward transition of an electron from the VB to CB leaves behind a hole in the VB and an electron-hole pair is created. This process is called the carrier generation (G). On the other hand, when an electron makes a transition from the CB to VB, an electron-hole pair is annihilated. This reverse process is called the carrier recombination (R). Under thermal equilibrium, G = R so that the carrier concentration remains the same and the condition pn = nj is maintained. The thermal G-R process is shown in Figure 2.10.

The equilibrium condition of a semiconductor is disturbed by optically or electrically introducing free carriers exceeding their thermal equilibrium value resulting in pn > nj or electrically removing carriers resulting in pn < n~. The process of introducing carriers in excess of thermal equilibrium values is called the carrier injection and the additional carriers are called the excess carriers. In order to inject excess carriers optically, we shine light with energy E = hv > E_g on an intrinsic semiconductor so that the valence electrons can be excited into the CB by the excess energy ДE = (hv - where h and v are Planck’s constant and frequency of light, respectively. In this band-to-band tunneling process, we get optically generated excess electrons (nj and holes (p,) in the semiconductor as shown in Figure 2.10. Therefore, the total non-equilibrium values of carrier concentration are given by

FIGURE 2.10 Band-to-band generation of electron-hole pairs under optical illumination with photon energy hv, where h and v are Planck’s constant and the frequency of incident light, respectively; the symbol represents electrons and “O” represents holes.

Injection Level

In the case of carrier injection into the semiconductor, we observe from Equation 2.62 that both n and p are greater than the intrinsic carrier concentration of the semiconductor and therefore, pn > nf. If the injected carrier density is lower than the majority carrier density at equilibrium so that the latter remains essentially unchanged while the minority carrier density is equal to the excess carrier density, then the process is called low level injection. If the injected carrier density is comparable to or exceeds the equilibrium value of the majority carrier density, then it is called high level injection.

To illustrate the injection levels, we consider an n-type extrinsic semiconductor with N_d= lx 10¹⁵ cm"³. Then from Section 2.2.4.1, the equilibrium majority carrier electron concentration is given by n„₀ = 1 x 10¹⁵ cm"³, whereas from Equation 2.21, the minority carrier hole concentration is given by p_n0 = 1 x 10⁵ cm"³. Here, n„₀ and p„₀ define the equilibrium concentrations of electrons and holes, respectively, in an n-type material. Now, we shine light on the sample so that 1 x 10¹³ cm"³ electron- hole pairs are generated in the material. Then using Equation 2.62, the total number of electrons n„ s 1 x 10¹⁵ cm'³ = n„₀ and p„= lx 10¹³ cm"³. Thus, the majority carrier concentration n„ remains unchanged, whereas the minority carrier concentration p„ is increased significantly. This is an example of low level injection. On the other hand, if 1 x 10¹⁷ cm"³ electron-hole pairs are generated by incident light, then from Equation 2.62, we get n„ s 1 x 10¹⁷ cm"³ and p„ = 1 x 10¹⁷ cm'³ changing both the electron and hole concentrations in the semiconductor resulting in a high level injection. The mathematics for high level injection is complex and therefore, we will consider only low level injection.

Recombination Processes

The semiconductor material returns to equilibrium through recombination of injected minority carriers with the majority carriers in the case of carrier injection, or through generation of electron-hole pairs in the case of extraction of carriers.

The electron-hole recombination process occurs by transition of electrons from the CB to the VB. In a direct bandgap semiconductor like GaAs where the minimum of the CB aligns with the maximum of the VB, an electron in the CB can give up its energy to move down to occupy the empty state (hole) in the VB without a change in momentum as shown in Figure 2.11(a). Since the momentum (к) must be conserved in any energy level transition, an electron in GaAs can easily make direct transition from E_c to £j, across E. This is called the direct or band-to-band recombination. When the direct recombination happens, the energy given up by electron will be emitted as a photon which makes it useful for light-emitting diodes [Figure 2.11(b)].

Now, if we generate excess carriers (An, Ap) at a rate G_L due to the incident light, then for low level injection, we get Ap = An = Ux = G,x; where U is the net

FIGURE 2.11 Bandgap in semiconductors: (a) direct bandgap; (b) band-to-band recombination in a direct bandgap semiconductor; and (c) indirect bandgap.

recombination rate and т is the excess carrier lifetime. If p₍₎ and n₀ are the equilibrium concentrations of electrons and holes, respectively, and p and n are the respective total concentrations due to generation, then Дp = p - p₀ and An = n- n_a and the net recombination rate due to direct recombination is given by

where:

x„and t_;> are the excess carrier electron and hole lifetime, respectively

It is to be noted that for band-to-band recombination, the excess carrier lifetime for an electron is equal to that of a hole since the single phenomenon annihilates an electron and a hole simultaneously.

For indirect bandgap semiconductors such as silicon and germanium (Figure 2.11(c)), the probability of direct recombination is very low. Physically, this means that the minimum energy gap between E_c and E_v does not occur at the same point in the momentum space as shown in Figure 2.11(c). In this case, for an electron to reach the VB, it must experience a change of momentum as well as energy to satisfy the conservation principle. This can be achieved by recombination processes through intermediate trapping levels, called the indirect recombination as shown in Figure 2.12.

In indirect bandgap semiconductors, the impurities that form electronic states deep in the energy gap assist recombination of electrons and holes. Here, the word deep indicates that the states are far away from the band-edges and near the center of the energy gap. These deep states are commonly referred to as the recombination centers or traps. Such recombination centers are usually unintentional impurities, which are not necessarily ionized at room temperature. These deep level impurities have concentrations far below the concentration of donor or acceptor impurities which have shallow energy levels. As an example, gold (Au) is a deep level impurity intentionally used in silicon to increase the recombination rate. This recombination via deep level impurities or traps is often referred to as the indirect recombination as shown in Figure 2.12. The G-R processes shown in Figure 2.12 consist of “1” electron capture by an empty center, “2” electron emission from an occupied center, “3” hole capture by an occupied center, and “4” hole emission by an empty center.

FIGURE 2.12 Generation and recombination in an indirect bandgap semiconductor; E, is the trap level deep into the bandgap; 1, 2,3, and 4 represent the generation and recombination

processes; here, represents electrons and “O” represents holes.

Now, let us consider the following example where an impurity like Au is introduced to provide a trapping level or a set of allowed states at energy £,. The trap level E, is assumed to act like an acceptor (it can also be neutral or negatively charged). Recombination is accomplished by trapping an electron and a hole. (The analysis can be easily extended to the case where the trap acts like a donor, i.e., positively charged or neutral charge states). The indirect recombination process was originally proposed by Shockley and Read [27] and independently suggested by Hall [28] and is therefore, often referred to as the Shockley-Read-Hall (SRH) recombination. By considering the transition processes shown in Figure 2.12, Shockley, Read, and Hall showed that for low level injection, the net recombination rate is given by

where:

v_lh = carrier thermal velocity (w 1 x 10⁷ cm sec^-1) о = carrier capture cross-section (« 10^-15 cm²)

N, is the density of trap centers

v„,CiV, is the capture probability or capture cross-section From Equation 2.64, we observe the following:

• 1. The “driving force” or the rate of recombination is proportional to (pn - nf j, that is, the deviation from the equilibrium condition
• 2. U = 0 when [np = nf j, that is, under the equilibrium condition
• 3. U is maximum when E, = £,, that is, trap levels near the midgap are the most efficient recombination centers

Now, for simplicity of understanding, let us consider the case when £, = £,. Then, from Equation 2.64, the net recombination rate is given by

Then for an n-type semiconductor with low level injection, n » p + 2n_r then denoting p = p_n as the total excess minority carrier concentration and (p_n0 = n~ / n J as the equilibrium minority carrier concentration, we get after simplification of Equation 2.65

where:

т_р is the minority carrier hole lifetime in an /г-type semiconductor and is given by

In an n-type material, lots of electrons are available for capture. Therefore, Equation 2.66 shows that the minority carrier hole lifetime x_p is the limiting factor in the recombination process in an n-type material.

Similarly, for a p-type semiconductor, we can show from Equation 2.65 that the net recombination rate for electrons is given by

where:

т_я is the minority carrier electron lifetime in a p-type semiconductor and is given by

Thus, for a p-type semiconductor, the minority carrier electron lifetime is the limiting factor in the recombination process.

The other recombination process in silicon that does not depend on deep level impurities and which sets an upper limit on lifetime is Auger recombination. In this process, the electrons and holes recombine without trap levels and the released energy (of the order of energy gap) is transferred to another majority carrier (a hole in a p-type and electron in an n-type silicon). Usually, Auger recombination is important when the carrier concentration is very high (> 5 x 10¹⁸ cm^-3) as a result of high doping concentration or high level injection.