Chapter 2: Semiconductor Fundamentals |
Recombination of electrons and holes is a process by which both carriers annihilate each other: electrons occupy - through one or multiple steps - the empty state associated with a hole. Both carriers eventually disappear in the process. The energy difference between the initial and final state of the electron is released in the process. This leads to one possible classification of the recombination processes. In the case of radiative recombination, this energy is emitted in the form of a photon. In the case of non-radiative recombination, it is passed on to one or more phonons and in the case of Auger recombination it is given off in the form of kinetic energy to another electron. Another classification scheme considers the individual energy levels and particles involved. These different processes are further illustrated with Figure 2.8.1. |
Figure 2.8.1 : | Carrier recombination mechanisms in semiconductors |
Band-to-band recombination occurs when an electron moves from its conduction band state into the empty valence band state associated with the hole. This band-to-band transition is typically also a radiative transition in direct bandgap semiconductors. |
Trap-assisted recombination occurs when an electron falls into a "trap", an energy level within the bandgap caused by the presence of a foreign atom or a structural defect. Once the trap is filled it cannot accept another electron. The electron occupying the trap, in a second step, moves into an empty valence band state, thereby completing the recombination process. One can envision this process as a two-step transition of an electron from the conduction band to the valence band or as the annihilation of the electron and hole, which meet each other in the trap. We will refer to this process as Shockley-Read-Hall (SRH) recombination. |
Auger recombination is a process in which an electron and a hole recombine in a band-to-band transition, but now the resulting energy is given off to another electron or hole. The involvement of a third particle affects the recombination rate so that we need to treat Auger recombination differently from band-to-band recombination. |
Each of these recombination mechanisms can be reversed leading to carrier generation rather than recombination. A single expression will be used to describe recombination as well as generation for each of the above mechanisms. |
In addition, there are generation mechanisms, which do not have an associated recombination mechanism, such as generation of carriers by light absorption or by a high-energy electron/particle beam. These processes are referred to as ionization processes. Impact ionization, which is the generation mechanism associated with Auger recombination, also belongs to this category. The generation mechanisms are illustrated with Figure 2.8.2. |
Figure 2.8.2 : | Carrier generation due to light absorption and ionization due to high-energy particle beams |
Carrier generation due to light absorption occurs if the photon energy is large enough to raise an electron from the valence band into an empty conduction band state, thereby generating one electron-hole pair. The photon energy needs to be larger than the bandgap energy to satisfy this condition. The photon is absorbed in this process and the excess energy, E_{ph} - E_{g}, is added to the electron and the hole in the form of kinetic energy. |
Carrier generation or ionization due to a high-energy beam consisting of charged particles is similar except that the available energy can be much larger than the bandgap energy so that multiple electron-hole pairs can be formed. The high-energy particle gradually loses its energy and eventually stops. This generation mechanism is used in semiconductor-based nuclear particle counters. As the number of ionized electron-hole pairs varies with the energy of the particle, one can also use such detector to measure the particle energy. |
Finally, there is a generation process called impact ionization, the generation mechanism that is the counterpart of Auger recombination. Impact ionization is caused by an electron/hole with an energy, which is much larger/smaller than the conduction/valence band edge. The detailed mechanism is illustrated with Figure 2.8.3. |
Figure 2.8.3: | Impact ionization and avalanche multiplication of electrons and holes in the presence of a large electric field. |
The excess energy is given off to generate an electron-hole pair through a band-to-band transition. This generation process causes avalanche multiplication in semiconductor diodes under high reverse bias: As one carrier accelerates in the electric field it gains energy. The kinetic energy is given off to an electron in the valence band, thereby creating an electron-hole pair. The resulting two electrons can create two more electrons which generate four more causing an avalanche multiplication effect. Electrons as well as holes contribute to avalanche multiplication. |
2.8.1. Simple recombination-generation model |
A simple model for the recombination-generation mechanisms states that the recombination-generation rate is proportional to the excess carrier density. It acknowledges the fact that no net recombination takes place if the carrier density equals the thermal equilibrium value. The resulting expression for the recombination of electrons in a p-type semiconductor is given by: |
(2.8.1) |
and similarly for holes in an n-type semiconductor: |
(2.8.2) |
where the parameter t can be interpreted as the average time after which an excess minority carrier recombines. |
We will show for each of the different recombination mechanisms that the recombination rate can be simplified to this form when applied to minority carriers in a "quasi-neutral" semiconductor. The above expressions are therefore only valid under these conditions. The recombination rates of the majority carriers equals that of the minority carriers since in steady state recombination involves an equal number of holes and electrons. Therefore, the recombination rate of the majority carriers depends on the excess-minority-carrier-density as the minority carriers limit the recombination rate. |
Recombination in a depletion region and in situations where the hole and electron density are close to each other cannot be described with the simple model and the more elaborate expressions for the individual recombination mechanisms must be used. |
2.8.2. Band-to-band recombination |
Band-to-band recombination depends on the density of available electrons and holes. Both carrier types need to be available in the recombination process. Therefore, the rate is expected to be proportional to the product of n and p. Also, in thermal equilibrium, the recombination rate must equal the generation rate since there is no net recombination or generation. As the product of n and p equals n_{i}^{2} in thermal equilibrium, the net recombination rate can be expressed as: |
(2.8.3) |
where b is the bimolecular recombination constant. |
2.8.3. Trap assisted recombination |
The net recombination rate for trap-assisted recombination is given by: |
(2.8.4) |
The derivation of this equation is beyond the scope of this text. |
This expression can be further simplified for p >> n to: |
(2.8.5) |
and for n >> p to: |
(2.8.6) |
were |
(2.8.7) |
2.8.4. Surface recombination |
Recombination at surfaces and interfaces can have a significant impact on the behavior of semiconductor devices. This is because surfaces and interfaces typically contain a large number of recombination centers because of the abrupt termination of the semiconductor crystal, which leaves a large number of electrically active states. In addition, the surfaces and interfaces are more likely to contain impurities since they are exposed during the device fabrication process. The net recombination rate due to trap-assisted recombination and generation is given by: |
(2.8.8) |
This expression is almost identical to that of Shockley-Hall-Read recombination. The only difference is that the recombination is due to a two-dimensional density of traps, N_{ts}, as the traps only exist at the surface or interface. |
This equation can be further simplified for minority carriers in a quasi-neutral region. For instance for electrons in a quasi-neutral p-type region p >> n and p >> n_{i} so that for E_{i} = E_{st}, it can be simplified to: |
(2.8.9) |
where the recombination velocity, v_{s}, is given by: |
(2.8.10) |
2.8.5. Auger recombination |
Auger recombination involves three particles: an electron and a hole, which recombine in a band-to-band transition and give off the resulting energy to another electron or hole. The expression for the net recombination rate is therefore similar to that of band-to-band recombination but includes the density of the electrons or holes, which receive the released energy from the electron-hole annihilation: |
(2.8.11) |
The two terms correspond to the two possible mechanisms. |
2.8.6. Generation due to light |
Carriers can be generated in semiconductors by illuminating the semiconductor with light. The energy of the incoming photons is used to bring an electron from a lower energy level to a higher energy level. In the case where an electron is removed from the valence band and added to the conduction band, an electron-hole pair is generated. A necessary condition is that the energy of the photon, E_{ph},is larger than the bandgap energy, E_{g}. As the energy of the photon is given off to the electron, the photon no longer exists. |
If each absorbed photon creates one electron-hole pair, the electron and hole generation rates are given by: |
(2.8.12) |
where a is the absorption coefficient of the material at the energy of the incoming photon. The absorption of light in a semiconductor causes the optical power to decrease with distance. This effect is described mathematically by: |
(2.8.13) |
The calculation of the generation rate of carriers therefore requires first a calculation of the optical power within the structure from which the generation rate can then be obtained using (2.8.12). |
Example 2.11 | Calculate the electron and hole densities in an n-type silicon wafer (N_{d} = 10^{17} cm^{-3}) illuminated uniformly with 10 mW/cm^{2} of red light (E_{ph} = 1.8 eV). The absorption coefficient of red light in silicon is 10^{-3} cm^{-1}. The minority carrier lifetime is 10 ms. |
Solution | The generation rate of electrons and holes equals:
where the photon energy was converted into Joules. The excess carrier densities are then obtained from: The excess carrier densities are then obtained from: So that the electron and hole densities equal: |
ece.colorado.edu/~bart/book | Boulder, August 2007 |