Part 2 : Nanostuctures Module 4 : Transport
Lecture 3 : Indirect Recombination
 
Under a low injection condition for $ n$-type semiconductor (for which $ n>>p$), one may write using $ p=p_o+\Delta_p$, etc.,, that pn − ni2 ≈ p0n0 − ni2 + n0Δp + p0Δn = (n0 + p0)Δp, since n0 p0 = ni2 and Δn = Δp because of overall charge neutrality. This means that pn − ni2 ≈ (n0 + p0)Δp ≈ n0 Δp. Using this, U can be rewritten as,
$\displaystyle U=v_{th}\sigma_o N_t \, \frac{(p-p_o)}{1+2(n_i/n_o) \cosh {\beta(E_t-E_{Fi})}}=\frac{\Delta_p}{\tau_p},$
 (32)
defining $ \tau_p=[1+2(n_i/n_o) \cosh {\beta(E_t-E_{Fi})}]/[v_{th}\sigma_o N_t]$. Thus the recombination rate of trap assisted indirect recombination is given by the same expression as that for direct recombination, except that $ \tau_p$, the life time of the excess minority carrier now depends on the location of the trap level relative to the intrinsic Fermi-energy.
Surface Recombination

On the surface, because of the abrupt discontinuity of the lattice structure, a large number of localized energy states or generation-recombination centers appear (in the form of dangling bonds) in the surface region, where the recombination rates can be enhanced by these so called surface states. The kinetics of recombination at the surface is analogous to the bulk case, with $ N_t$ replaced by $ N_{ts}$ the number of dangling bonds per unit area at the surface, $ E_t$ replaced by $ E_{ts}$, the energy level of the trap at the surface ($ ts$), i.e, the surface recombination center, and $ p$ by $ p_s$, the hole concentration at the surface. For the limiting case, where the electron concentration at the surface is essentially same as that in the bulk, the total number of carriers recombining at the surface per unit area per unit time is

$\displaystyle U_s \approx v_{th} \sigma_p N_{ts}(p_s-p_o) = (p_s-p_o)S_{lr}\,,$
 (33)
where $ S_{lr}= v_{th} \sigma_p N_{ts}$, has the dimension of velocity and is called the low ($ l$) injection surface recombination ($ r$) speed ($ S$).
Auger Recombination
Here the energy and momentum generated by the recombination of an electron-hole pair get transferred to a third particle (either so called Auger electron or Auger hole). The Auger recombination rate is $ R_{Aug}$ which is given by $ B n^2 p$ for Auger electron, and $ B n p^2$ for Auger hole, where $ B$ is a constant having strong temperature dependence. The Auger particle eventually loses its energy to the lattice by scattering processes. Usually this recombination process is important, when carrier concentration is high due to either high doping or high injection level. It is easy to show that its contribution to $ U$ is again of the form $ U =\Delta_p /\tau_p$ as before, where $ \tau_p$ is now determined by the Auger processes.