Modeling defect level occupation for recombination statistics

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

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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
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Photoexcitation
Photon
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Photoexcitation
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Photoexcitation
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Radiative Recombination.
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Radiative Recombination.
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Radiative Recombination.
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Electron Trapping.
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Electron Trapping.
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Defect Related Recombination.
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Defect Related Recombination.
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Defect Related Recombination.
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What do we measure?

• Recombination rate includes radiative and
  defect-related recombination.
• Measurements were taken of radiative efficiency
  vs. recombination rate.
• (radRate)/(radRatedefRate)
• vs. (radRate defRate)
• Objective Information about the defect-related
  density of states.

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

Conduction Band
Valence Band
Energy
Energy
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Looking at the Data
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Calculate x-Axis Use Rate value for y-Axis

• dP hole concentration in valence band
• dN electron concentration in conduction band

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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.

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Simple Theory Fit
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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

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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
Conduction Band
Defect States
Energy
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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
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Calculating band occupations

• dP and dN depend on QEFp and QEFn, respectively.

QEFn
QEFp
Conduction Band
Valence Band
dN
dP
Energy
Energy
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Calculating defect occupation

• dDp and dDn depend on Ef, and QEFs

Trapped hole occupation
Trapped electron occupation
QEFn
Ef
Ef
QEFp
Electron Traps
dDn
dDp
Hole Traps
Energy
Energy
Note graph represents an arbitrary midgap defect
distribution
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Symmetric vs. Asymmetric defect distribution

• Symmetric Defect DOS

Ev
Ec
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Symmetric Defect Fit
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Asymmetric defect DOS

• Using 2 Gaussians(fit for 2 Gaussians)

Ev
Ec
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2-Gaussian Asymmetric Fit
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3-Gaussian Asymmetric Fit.
Ev
Ec
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3-Gaussian Asymmetric Fit
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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

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In-depth look at the model

• Calculating DOS(e)
• DOS(e) ValenceBand(e) ConductionBand(e)
  defDos(e)
• ValenceBand(e) 0 if e gt Ev, if e gt Ev
• ConductionBand(e) 0 if e lt Ec, if e lt Ec
• defDos(e) is an arbitrary function denoting the
  defect density of states. defDos(e) 0 when e lt
  Ev or e gt Ec

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Fermi Function, and calculating Ef

• Fermi Function
• To calculate Ef, find Ef where

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Calculating QEFp/n

• QEFp denotes the point where
• QEFn denotes the point where

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Calculating Occupations

Note see slide 7 for rate value.
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Numerical Infinite Integrals

• Need a bijection
• And
• Then
• Using ArcTan,