Carrier Lifetime And Photoconductivity: direct recombination
When excess electrons and holes are created in a semiconductor, there is a corresponding increase in the conductivity of the sample. If the excess carriers arise from optical luminescence, the resulting increase in conductivity is cabled photoconductivity.
Direct Recombination of Electrons and Holes:
- In direct recombination, an excess population of electrons and holes decays by electrons falling from the conduction band to empty states (holes) in the valence band.
- Energy lost by an electron in making the transition is given up as a photon.
- Direct recombination occurs spontaneously; that is, the probability that an electron and a hole will recombine is constant in time.
- As in the case of carrier scattering, this constant probability leads us to expect an exponential solution for the decay of the excess carriers.
- In this case the rate of decay of electrons at any time t is proportional to the number of electrons remaining at r and the number of holes, with some constant of proportionality for recombination, α_{r},.
- The net rate of change in the conduction band electron concentration is the thermal generation rate α_{r}n_{i}^{2}
equ (1)
- Let us assume the excess electron-hole population is created at t = 0, for example by a short flash of light, and the initial excess electron and hole concentrations Δn and Δp are equal.
- Then as the electrons and holes recombine in pairs, the instantaneous concentrations of excess carriers ∂n(t) and ∂p(t) are also equal.
- Thus we can write the total concentrations of the above equ in terms of the equilibrium values n_{0} and p_{0} and the excess carrier concentrations ∂n(t)=∂p(t).
we have
equ (2)
This nonlinear equation would be difficult to solve in its present form. Fortunately, it can be simplified for the case of low-level injection. If the excess carrier concentrations are small, we can neglect the ∂n^{2} term. Furthermore, if the material is extrinsic, we can usually neglect the term representing the equihbrium
minority carriers. For example, if the material is p-type (p_{0} > n_{0}), Eq. 2 becomes
The solution to this equation is an exponential decay from the original excess carrier concentration Δn:
Excess electrons in a p-type semiconductor recombine with a decay constant = (α_{n}p_{o})^{-1} called the recombination lifetime. Since the calculation is made in terms of the minority carriers, is often called the minority carrier lifetime.
The decay of excess holes in n-type material occurs with t_{p} = (α_{r}n_{0})^{-1}. In the case of direct recombination, the excess majority carriers decay at exactly the
same rate as the minority carriers. There is a large percentage change in the minority carrier electron concentration and a small percentage change in the majority hole concentration. Basically, the approximations of extrinsic material and low-level injection allow us to represent n(t) by the excess concentration
∂n(t) and ∂p(t) by the equilibrium value p_{0}. A more general expression for the carrier lifetime is
This expression is valid for n- or p-type material if the injection level is low.