1. Chemistry 140 a Lecture 11 Surface, Bulk, and Depletion Region Recombination

2. Quasi Fermi Levels For calculations, it would be convenient to assume flat QFLs within a certain region, Δx, under study. Then, the driving force for recombination would be equal everywhere. When are QFLs flat? When Δn or Δp is constant within Δx.

3. Flat QFLs QFLs are flat in the bulk when: Recombination at any position x is slow compared to thermal diffusion E F,n E F,0 E F,p

4. Flat QFLs QFLs are flat in the bulk when: Light excitation is uniform or light excitation is not uniform but diffusion of carriers flattens QFLs e- e- e- e - h+ h+ h+ h + hνhν e- e- e- e - h+ h+ h+ h + t = 0 non-uniform QFLs t = t 1

5. Flat QFLs e- e- e- e - h+ h+ h+ h + hνhν e- e- e- e - h+ h+ h+ h + t = 0t = t 1 If t 1, the time it takes for a uniform distribution of carriers to occur via diffusion, is less than τ br, τ sr, or τ dr, then the QFLs are flat.

6. Diffusion Time in Si e- e- e- e - h+ h+ h+ h + d = 300 μm For recombination greater than about 25 μs, we can assume flat QFLs.

7. Depletion Region Recombination E F,0 E F,n E F,p E F,0 WWW’ V app Let’s examine the depletion region after applying a bias V app : V app The new effective depletion region is W’ < x < 0. W to W’ is a quasi-neutral region, and there is no field there. It is like the bulk, except E F,p changes with x. Quasi Fermi levels are flat within this new depletion region.

8. QFLs Not Flat For flat QFLs in the depletion region, recombination at the surface must be slow relative to diffusion of carriers. If surface recombination is fast relative to diffusion of carriers, QFLs will not be flat: E F,n E F,p

9. n(x) and p(x) Vary with x E F,n E F,p W ETET The recombination rate in the depletion region is not like in the bulk or on the surface. We now need to plug in n(x) and p(x) and integrate over 0 to W.

10. Depletion Region Recombination Most general form: Before, we just plugged in n b = n(x) or n s = n(x) and p b = p(x) or p s = p(x). Now n(x) and p(x) change from 0 to W because of band bending. We have to integrate over all n(x) in 0 < x < W:

11. Depletion Region Recombination Assumptions: N T (x) = N T n 1 and p 1 are constants with respect to x and do not change with V bi for a given trap: n 1 = N C exp(-(E C -E T )/kT) p 1 = N V exp(-(E T -E V )/kT) k n and k p are constant (this can fail since σ(E T ) may vary due to ionized or unionized trap states, e.g. Zn 2+  Zn + ): n(x) = N C exp[-(E C (x)-E F,n )/kT] p(x) = N V exp[-(E F,p -E V (x))/kT]

12. n(x)p(x) Not Dependent on x If E F,n and E F,p are constant with x, then n(x)p(x) is a constant with x. n(x)p(x) = N C N V exp[-(E C (x)-E F,n +E F,p -E V (x))/kT] = N C N V exp[-(E C (x)-E V (x))/kT] exp[-(E F,p -E F,n )/kT] E C (x)-E V (x) = E g (x) = E g everywhere n(x)p(x) = N C N V exp[-E g /kT] exp[-(E F,p -E F,n )/kT] n(x)p(x) = n i 2 exp[-(E F,p -E F,n )/kT] -(E F,p -E F,n ) = qV app n(x)p(x) = n i 2 exp[-qV app /kT] No dependence on x. Increases in –V app result in n(x)p(x) > n i 2.

13. Depletion Region Recombination Returning to U(x)… We may ignore the “1” when V app > 0.75 V (V app > 3kT/q). A harder assumption is to ignore n 1 in the denominator for significant band bending. We must have either high-level injection or large V app.

14. Determination of U total Our assumptions: p 1 and n 1 are negligible k n = k p = σν

15. Determination of U max There will be some U max in this region (the depletion region) where n(x) = p(x). This is where the denominator is a minimum.

16. Determination of U max

17. Analysis for x x max E F,n E F,p W’xmxm 1 2 0 1 2 x < x m exp(…) is negative n(x) < n(x m ) extra band bending x > x m exp(…) is positive n(x) > n(x m ) V = ε m x = (V/cm)*cm

18. U(x) in Terms of U max

19. U total in Terms of U max For normally doped semiconductors, the maximum is strongly peaked away from W, so extend the integral to .

20. U total Replace exp(…) with sinh(…) if you want to include the –n i 2 term we neglected. Important term. Not like thermionic emission, where it went like exp(-qV app /kT). The e - and h + recombination is as if we lost half of the voltage to the other carrier.

21. U vs. x U x …a dimensionless quantity that is the ratio of the thermal to applied voltage.

22. End Surface, Bulk, and Depletion Region Recombination