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Modeling defect level occupation for recombination statistics

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

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Modeling defect level occupation for recombination statistics

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  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 Defect States Energy Valence Band

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

  4. Photoexcitation Photon

  5. Photoexcitation Photon

  6. Photoexcitation

  7. Photoexcitation

  8. Radiative Recombination.

  9. Photon Radiative Recombination.

  10. Photon Radiative Recombination.

  11. Electron Trapping.

  12. Electron Trapping.

  13. Defect Related Recombination.

  14. Heat Defect Related Recombination.

  15. Heat Defect Related Recombination.

  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-relateddensity of states.

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

  18. Band Density Of States Conduction Band Valence Band Energy 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 Ef Valence Band Defect States Conduction Band Energy

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

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

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

  28. Symmetric vs. Asymmetric defect distribution… • Symmetric Defect DOS: Ev Ec

  29. Symmetric Defect Fit…

  30. Asymmetric defect DOS… • Using 2 Gaussians…(fit for 2 Gaussians) Ev Ec

  31. 2-Gaussian Asymmetric Fit…

  32. 3-Gaussian Asymmetric Fit. Ev Ec

  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 <= Ev or 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,

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