Shockley-Read statistics

At steady state, the hole and electron fluxes into the trap state must balance and 

Consider the bulk region since we are dealing with an n-type semiconductor, n P nt, p, pt and 

We now apply the theory derived in the preceding paragraphs to the situation in which minority carriers are generated thermally. In the depletion region, the concentration of holes will be substantially reduced from the equilibrium value by the current flow in addition, n° is now substantially reduced from the bulk value. If we assume that nt is now dominant and that n x p n2, g(x) = Cn2jnt. Since this is independent of x, we may immediately integrate the flux equation to give 

The expression on the left-hand side of eqn. (212) is the hole flux. At the 

Equation (213) may be integrated further by the integrating factor exp — (W — x)2/2Lp = exp (e0 polkT) where / is the potential in the depletion layer. We find 

---

Fig. 60. Rate-limiting regime for Shockley Read statistics the Roman numerals denote the dominance of one particular term in eqn. (352). In terms of the denominator of eqn. (351), the regimes are defined by I, re > re, p, p, II, ret > re, p, pt II2> p, > re, p, re, and III, p > re, p, re,. The single arrows indicate the rate-limiting step in each case.

The competition between charge transfer and recombination via surface states can be treated exactly by Hall-Shockley-Read statistics, taking proper account of... 

Deep levels can be described by the Shockley-Read-Hall recombination statistics [5]. However, for a large number of deep states, the capture cross section for one type of carrier is many times larger than that for the other carrier. The state, therefore, interacts principally with only one of the band edges and can be characterised as either an electron or a hole trap. Capacitance techniques, such as DLTS (Deep Level Transient Spectroscopy), are particularly convenient for the determination of trap type and concentration. If additional experimental information is present to allow charge state determination, then the states can be characterised as deep acceptors or donors. 

The capability of drift-diffusion models can be increased by introducing localized states into the band gap. Recently, several studies have introduced single trap levels [83] as well as distributions of localized states in order to describe the results of transient and steady state experiments on polymer fullerene solar cells [54, 84-89]. Most of these models use a Shockley-Read-Hall type occupation statistics for the localized states, which we will discuss in more detail in Sect. 2.3 and the Appendix 2 before discussing some of the implications of this model in the case studies in Sect. 4.1. 

To find the occupation statistics for a trap - the Shockley-Read-Hall statistics [232,233] - we need to consider the four processes shown in Fig. 10. A single trap can capture and emit an electron and capture and emit a hole. If the same trap captures a hole and an electron, one recombination event happens. If a trap captures and emits an electron or a hole, the trap will have slowed down transport only. Table 3 summarizes the four rates that we need to consider. However, the four rates are not independent of each other in quasi-equilibrium. Because in equilibrium, detailed balance between inverse processes must be obeyed, the capture and emission processes must be connected. In addition, in thermal equilibrium the occupation function for all charge carriers (free or trapped, electrons or holes) must be the Fermi-Dirac function in thermal equilibrium, i.e. 

Fig. 10 Definition of the four rates of capture and emission of electrons and holes by a single trap level. These four rate equations are the basis of Shockley-Read-Hall statistics, which defines the occupation probability and the recombination rate via this trap...

The rates are defined in Fig. 10. Note that the occupation probability/is defined by Fermi-Dirac statistics in thermal equilibrium but by Shockley-Read-Hall statistics in non-equilibrium. The capture coefficients are denoted by p and the emission coefficients by e p... 

W. Shockley, W.T. Read, Statistics of the recombinations of holes and electrons. Phys. Rev. 

The influence of deep-level states or traps on the statistics of electron-hole recombination was first described by Shockley and Read and Hall. Deep-level states, as their name implies, lie close to the middle of the energy bandgap of the semiconductor. Due to the large energy separation from the valence-band and conduction-band edges, deep-level states are not fully ionized at room temperature. In contrast, shallow-level states are those considered to be fully iordzed at room temperature due to thermal excitation. 

---