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Modeling defect level occupation for recombination statistics
Adam Topaz and Tim GfroererDavidson College
Mark WanlassNational Renewable Energy Lab
Supported by the American Chemical Society – Petroleum Research Fund
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A semiconductor:
Conduction Band
Valence Band
Defect States
Energ
y
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Electrons
Equilibrium Occupation in a Low Temperature Semiconductor.
Holes
Electron Trap
Hole Trap
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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.
Photon
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Radiative Recombination.
Photon
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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.
Heat
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Defect Related Recombination.
Heat
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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)/(radRate+defRate)
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
Valence Band
Defect States
0
0.2
0.4
0.6
0.8
1
1.2
-0.5 -0.3 -0.1 0.1 0.3 0.5
EnergyEv Ec
Energ
y
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Band Density Of States
energybandDOS
ConductionBand
ValenceBand
Energy Energy
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Looking at the Data…
Efficiency vs. Rate
0
0.2
0.4
0.6
0.8
1
1E+19 1E+20 1E+21 1E+22 1E+23Recombination rate (#/s/cm^3)
Eff
icie
ncy
77k
120k
165k
207k
250k
290k
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radB
rateefficiencydPdN
rate
dPdNradBefficiency
Rate vs. dPdN
1E+19
1E+20
1E+21
1E+22
1E+23
1E+25 1E+27 1E+29 1E+31 1E+33
dPdN ((#/cm^3)^2)
Rate
(#/s
/cm
^3)
77k
120k
165k
207k
250k
290kCalculate x-Axis
Use Rate value for y-Axis
•dP = hole concentration in valence band•dN = electron concentration in conduction band
Efficiency vs. Rate
0
0.2
0.4
0.6
0.8
1
1.2
1E+19 1E+20 1E+21 1E+22 1E+23
Recombination rate (#/s/cm^3)
Eff
icie
nc
y
77k
120k
165k
207k
250k
290k
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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.
2nradBndefArate
2|)log()log(| ltheoreticameasured raterateerror
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Simple Theory Fit…Rate vs. dPdN
1.00E+19
1.00E+20
1.00E+21
1.00E+22
1.00E+23
1.00E+25 1.00E+27 1.00E+29 1.00E+31 1.00E+33
dPdN ((#/cm^3)^2)
Rate
(#/s
/cm
^3)
77K
120K
165K
207K
250K
290K
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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
dPdNradBdDpdNdDndPdefArate )(
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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
Carrier states
Filled with Holes
Filled with electrons
ValenceBand
ConductionBand
DefectStates
Ef
Energy
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Non-Eq-filling
Already Filled states
Non-eqfilling
already filled states
Calculating QEFp and QEFn… Find QEFp and QEFn such that:
exN = increased occupation (red area)
Ef EfQEFp QEFn
exN exNFilledHole States
FilledElectronStates
Increased hole occupation Increased electron occupation
Energy Energy
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Calculating band occupations… dP and dN depend on QEFp and QEFn,
respectively.
Band States
dN
QEFn
dNBand States
dP
ConductionBandValence
Band
QEFp
dP
Energy Energy
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Calculating defect occupation… dDp and dDn depend on Ef, and QEF’s
defect states
non-eq-dDn
hole traps
Note: graph represents an arbitrary midgap defect distribution
defect states
non-eq-dDp
electron traps
QEFp QEFnEf Ef
ElectronTraps
dDp HoleTraps
dDn
Trapped hole occupation Trapped electron occupation
Energy Energy
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Symmetric vs. Asymmetric defect distribution…
Symmetric Defect DOS:
Defect DOS
0
2E+15
4E+15
6E+15
8E+15
1E+16
1.2E+16
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Energy (% of Eg -- 0 is midGap)
Nu
mb
er o
f S
tate
s
Ev Ec
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Symmetric Defect Fit…Rate vs. dPdN
1.00E+19
1.00E+20
1.00E+21
1.00E+22
1.00E+23
1.00E+25 1.00E+27 1.00E+29 1.00E+31 1.00E+33
dPdN ((#/cm^3)^2)
Rate
(#/s
/cm
^3)
77K
120K
165K
207K
250K
290K
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Asymmetric defect DOS… Using 2 Gaussians…(fit for 2 Gaussians)
Defect DOS
-1E+15
0
1E+15
2E+15
3E+15
4E+15
5E+15
6E+15
7E+15
8E+15
9E+15
1E+16
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Energy (% of Eg -- 0 is midGap)
Nu
mb
er o
f S
tate
s
Ev Ec
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2-Gaussian Asymmetric Fit…Rate vs. dPdN
1.00E+19
1.00E+20
1.00E+21
1.00E+22
1.00E+23
1.00E+25 1.00E+27 1.00E+29 1.00E+31 1.00E+33
dPdN ((#/cm^3)^2)
Rate
(#/s
/cm
^3)
77K
120K
165K
207K
250K
290K
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3-Gaussian Asymmetric Fit.
Defect DOS
0
2E+15
4E+15
6E+15
8E+15
1E+16
1.2E+16
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Energy (% of Eg -- 0 is midGap)
Nu
mb
er o
f S
tate
s
Ev Ec
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3-Gaussian Asymmetric Fit…Rate vs. dPdN
1.00E+19
1.00E+20
1.00E+21
1.00E+22
1.00E+23
1.00E+25 1.00E+27 1.00E+29 1.00E+31 1.00E+33
dPdN ((#/cm^3)^2)
Rate
(#/s
/cm
^3)
77K
120K
165K
207K
250K
290K
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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 > 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
62/319
32/3
10*)10*6.1(
*)2(*2
hMe
wcv
62/319
32/3
10*)10*6.1(
*)2(*2
hMh
wcc
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Fermi Function, and calculating Ef…
Fermi Function:
To calculate Ef, find Ef where:
)/)exp((1
1),(
kTfefeFF
Ef
Ef
dEEfEFFEDOSdEEfEFFEDOS *),(*)(*)),(1(*)(
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Calculating QEFp/n
QEFp denotes the point where:
QEFn denotes the point where:
dEQEFpEFFEfEFFEDOSexN *)),(),((*)(
dEEfEFFQEFnEFFEDOSexN *)),(),((*)(
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Calculating Occupations…
dEQEFpEFFEDOSdPEv
*),(1(*)(
Ec
dEQEFnEFFEDOSdN *),(*)(
Ec
Ev
dEQEFpEFFEfEFFEDOSdDp *)),(),((*)(
Ec
Ev
dEEfEFFQEFnEFFEDOSdDn *)),(),((*)(
Note: see slide 7 for rate value.
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Numerical Infinite Integrals…
Need: a bijection And
Then:
Using ArcTan,
),(: bag axg
x
)(lim bxgx
)(lim
b
a
dxxgg
xgfdxxf
))((
))(()(
1
1
2/
2/
))tan(1(*))(tan()(
dxxxfdxxf
2/
)arctan(
))tan(1(*))(tan()(
kk
dxxxfdxxf
)arctan(
2/
))tan(1(*))(tan()(kk
dxxxfdxxf