1. Modeling defect level occupation for recombination statistics Adam Topaz and Tim Gfroerer Davidson College Mark Wanlass National Renewable Energy Lab Supported by the American Chemical Society – Petroleum Research Fund

2. A semiconductor: Conduction Band Valence Band Defect States Energy

3. Electrons Equilibrium Occupation in a Low Temperature Semiconductor. Holes Electron Trap Hole Trap

4. Photoexcitation Photon

5. Photoexcitation Photon

6. Photoexcitation

8. Radiative Recombination.

9. Photon

10. Radiative Recombination. Photon

11. Electron Trapping.

13. Defect Related Recombination.

14. Heat

15. Defect Related Recombination. Heat

16. What do we measure?  Recombination rate includes radiative and defect-related recombination.  Measurements were taken of radiative efficiency vs. recombination rate. (radRate)/(radRate+defRate) vs. (radRate + defRate)  Objective: Information about the defect-related density of states.

17. The Defect-Related Density of States (DOS) Function Conduction Band Valence Band Defect States Energy Ev Ec Energy

18. Band Density Of States  Conduction Band Valence Band Energy

19. Looking at the Data…

20. Calculate x-Axis Use Rate value for y-Axis dP = hole concentration in valence band dN = electron concentration in conduction band

21. The simple theory…  Assumptions: dP = dN = n Defect states located near the middle of the gap  No thermal excitation into bands.  Fitting the simple theory: radB is given. Find defA to minimize logarithmic error   defA is the defect related recombination constant  radB is the radiative recombination constant.

22. Simple Theory Fit…

23. A Better Model…  Assumptions: defA independent of temperature (and is related to the carrier lifetime)  Calculations: Calculate Ef for a given temperature, bandgap and defect distribution Calculate QEfp / QEfn for a given exN (the value of exN is chosen to match experimental dPdN) Calculate occupations (dP, dN, dDp, and dDn)  dDp = trapped hole concentration  dDn = trapped electron concentration  Ef is the Fermi energy  QEFp/n is the quasi-Fermi energy for holes and electrons respectively  exN is the number of excited carriers

24. Calculating Ef…  The Fermi energy Ef is the energy where: (# empty states below Ef) = (# filled states above Ef) Red area = Blue area Valence Band Conduction Band Defect States Ef Energy

25. Calculating QEFp and QEFn…  Find QEFp and QEFn such that: exN = increased occupation (red area) Ef QEFp QEFn exN Filled Hole States Filled Electron States Increased hole occupationIncreased electron occupation Energy

26. Calculating band occupations…  dP and dN depend on QEFp and QEFn, respectively. QEFn dN Conduction Band Valence Band QEFp dP Energy

27. Calculating defect occupation…  dDp and dDn depend on Ef, and QEF’s Note: graph represents an arbitrary midgap defect distribution QEFp QEFnEf Electron Traps dDpHole Traps dDn Trapped hole occupationTrapped electron occupation Energy

28. Symmetric vs. Asymmetric defect distribution…  Symmetric Defect DOS: EvEc

29. Symmetric Defect Fit…

30. Asymmetric defect DOS…  Using 2 Gaussians…(fit for 2 Gaussians) EvEc

31. 2-Gaussian Asymmetric Fit…

32. 3-Gaussian Asymmetric Fit. EvEc

33. 3-Gaussian Asymmetric Fit…

34. Conclusion…  Simple Theory  Defect slope is too steep and theory does not allow for temperature dependence!  Temperature dependence and shallow defect slope can be modeled using: An occupation model that allows for thermal defect-to-band excitation. An asymmetric defect level distribution

35. In-depth look at the model…  Calculating DOS(e) DOS(e) = ValenceBand(e) + ConductionBand(e) + defDos(e)  ValenceBand(e) = 0 if e > Ev, if e >= Ev  ConductionBand(e) = 0 if e < Ec, if e <= Ec  defDos(e) is an arbitrary function denoting the defect density of states. defDos(e) = 0 when e = Ec

36. Fermi Function, and calculating Ef…  Fermi Function:  To calculate Ef, find Ef where:

37. Calculating QEFp/n  QEFp denotes the point where:  QEFn denotes the point where:

38. Calculating Occupations…         Note: see slide 7 for rate value.

39. Numerical Infinite Integrals…  Need: a bijection  And  Then:  Using ArcTan,