Energy band structure
The minority carrier “lifetime” (τ) measures how long a carrier is likely to stay around for before recombining and is one of the most important parameters for the characterization of semiconductor wafers used in the preparation of power electronic devices and photovoltaic solar cells. Stating that "a silicon wafer has a long lifetime" usually means minority carriers generated in the bulk will persist for a long lifetime before recombining.
Figure: Lifetime map of a Silicon wafer and photoconductivity map of a
Figure: Lifetime map of a Silicon wafer and photoconductivity map of a silicon wafer
The lifetime is related to the recombination rate by:
where τ is the minority carrier lifetime, Δn is the excess minority carriers concentration and R is the corresponding recombination rate.
In order to understand practical measurements of the lifetime it is, first of all, essential to realize that usually we measure relatively thin slices of material and, therefore, the recombination process don’t occur only in the bulk but we should expect that the surfaces of the wafer may play an important role in the global recombination processes that we are measuring. Electrons and holes can recombine at the surfaces of a silicon wafer, and the speed at which they do is characterized traditionally by a parameter called the surface recombination velocity, S.
Therefore, recombinations occur in the bulk as well as on both surfaces of the samples and the measured lifetime is in fact an effective lifetime (τeff) which depends on the bulk lifetime (τb) and on the recombination velocity S of the surfaces (τs). The effective lifetime value is given by:
The lifetime is quite unpredictable and difficult to control. It can vary by several orders of magnitude, from approximately 1 ns to 1 ms in common silicon solar cell materials. The highest value ever measured is 32ms, for undoped silicon, and the lowest 10-9 s, for heavily doped silicon. In the same way that the life expectancy is an indicator of the quality of life in a country, the lifetime says the quality of the silicon material. This quality depends, primarily, on the methods used to purify and grow crystalline silicon.
In the bulk of the material the carriers recombine by either radiative (also known as band-to-band) recombination, Auger recombination or Shockley-Read-Hall (SRH) via traps within the energy gap. The lifetime of carriers in the material bulk τb is composed of radiative lifetime τrad, Auger lifetime τA and a SRH lifetime τSRH with the relation:
Radiative (Band-to-Band) recombination is the recombination mechanism that dominates in direct bandgap semiconductors. The light produced from a light emitting diode (LED) is the most obvious example of radiative recombination in a semiconductor device. The key characteristics of radiative recombination are:
An electron directly combines with a hole in the conduction band and releases a photon;
The emitted photon has an energy similar to the band gap and is therefore only weakly absorbed such that it can exit the piece of semiconductor.
For an indirect bandgap semiconductor such as silicon τrad is very large and usually neglected.Recombination through defects, also called Shockley-Read-Hall or SRH recombination, does not occur in perfectly pure and undefected material. The two steps involved in SRH recombination are:
An electron (or hole) is trapped by an energy state in the forbidden region which is introduced through defects in the crystal lattice. These defects can either be unintentionally introduced or can have been deliberately added to the material, for example in doping the material;
If a hole (or an electron) moves up to the same energy state before the electron is thermally re-emitted into the conduction band, then it recombines.
The rate at which a carrier moves into the energy level in the forbidden gap depends on the distance of the introduced energy level from either of the band edges. Therefore, if an energy is introduced close to either band edge, recombination is less likely as the electron is likely to be re‑emitted to the conduction band edge rather than recombine with a hole which moves into the same energy state from the valence band. For this reason, energy levels near mid-gap are very effective for recombination. The effect of such recombination centres can be adequately described with the Shockley-Read-Hall model (Hall, 1952; Shockley and Read, 1952). For the common case that the recombination centre is located near the middle of the energy gap, it is possible to simplify the SRH expression for the minority carrier lifetime to show more clearly its dependence on carrier injection level:
An Auger Recombination involves three carriers. An electron and a hole recombine, but rather than emitting the energy as heat or as a photon, the energy is given to a third carrier, an electron in the conduction band. This electron then thermalizes back down to the conduction band edge. Auger recombination is most important in heavily doped or heavily excited material.
The bulk lifetime for extrinsic silicon can be determined using semi-empirical models based on lifetime measurements of float-zone silicon with very low defect levels. The lifetime is dependent on the excess carriers and doped atoms concentrations. Most silicon wafers have higher levels of contaminants and so lower lifetimes than calculated here.
Figure 1. Hole and Electron Bulk Lifetime for different Silicon Doping [3-4]
However, recombinations occur in the bulk as well as on both surfaces. The defects at a semiconductor surface are caused by the interruption to the periodicity of the crystal lattice, which causes dangling bonds at the semiconductor surface. The reduction of the number of dangling bonds, and hence the recombination, is achieved by growing a silicon nitride layer on top of the semiconductor surface which ties up some of these dangling bonds. This reduction of dangling bonds in known as surface passivation.
Let us consider a common case where the two surfaces of the wafer are identical and characterized by a certain surface recombination velocity S. The effective lifetime can be written with the following expression:
where W is the thickness of the sample.
According to the last expression, the effective lifetime would be zero when the surface recombination velocity is very high. In reality there is a limit on how low the effective lifetime can be because electrons and holes have to travel towards the surfaces by the relatively slow diffusion mechanism in order to recombine. It can be calculated that for uniform photogeneration the minimum effective lifetime is:
For a typical p-type 0.03 cm thick wafer, and a diffusion coefficient for electrons Dn = 27 cm2 s-1, the lifetime that can be expected for a non-passivated surface is τeff=2.8 ns. Nevertheless, the surface-limited lifetime can be even lower if the source of light is predominantly of short wavelength, because electrons that are photogenerated by visible or blue light (such as a white flash lamp) are very close to the surface and can diffuse to it almost instantaneously.
O. Palais “High resolution lifetime scan maps of silicon wafers”, Materials Science and Engineering B71 (2000) 47–50
Andres Cuevas, Daniel Macdonald, "Measuring and interpreting the lifetime of silicon wafers, Solar Energy", Volume 76, Issues 1-3 (2003) 255-262
Alamo, J. A., and R. M. Swanson, "Modeling of Minority Carrier Transport in Heavily Doped Silicon Emitters", Solid-State Electronics 30, 1127, November 1987
M. S. Tyagi and R. Van Overstraeten, Minority carrier recombination in heavily-doped silicon, Solid-State Electron.,vol. 26, pp. 577 1983.