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 Valence Band Defect States Energy

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

Photoexcitation Photon

Photoexcitation Photon

Photoexcitation

Radiative Recombination.

Photon

Radiative Recombination. Photon

Electron Trapping.

Defect Related Recombination.

Heat

Defect Related Recombination. Heat

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.

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

Band Density Of States  Conduction Band Valence Band Energy

Looking at the Data…

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

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.

Simple Theory Fit…

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

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

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

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

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

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

Symmetric Defect Fit…

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

2-Gaussian Asymmetric Fit…

3-Gaussian Asymmetric Fit. EvEc

3-Gaussian Asymmetric Fit…

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

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

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

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

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

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