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
A semiconductor:
Conduction Band
Valence Band
Defect States
Energ
y
Electrons
Equilibrium Occupation in a Low Temperature Semiconductor.
Holes
Electron Trap
Hole Trap
Photoexcitation
Photon
Photoexcitation
Photon
Photoexcitation
Photoexcitation
Radiative Recombination.
Radiative Recombination.
Photon
Radiative Recombination.
Photon
Electron Trapping.
Electron Trapping.
Defect Related Recombination.
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
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
Band Density Of States
energybandDOS
ConductionBand
ValenceBand
Energy Energy
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
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
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
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
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 )(
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
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
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
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
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
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
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
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
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
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
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 <= Ev or e >= Ec
62/319
32/3
10*)10*6.1(
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hMe
wcv
62/319
32/3
10*)10*6.1(
*)2(*2
hMh
wcc
Fermi Function, and calculating Ef…
Fermi Function:
To calculate Ef, find Ef where:
)/)exp((1
1),(
kTfefeFF
Ef
Ef
dEEfEFFEDOSdEEfEFFEDOS *),(*)(*)),(1(*)(
Calculating QEFp/n
QEFp denotes the point where:
QEFn denotes the point where:
dEQEFpEFFEfEFFEDOSexN *)),(),((*)(
dEEfEFFQEFnEFFEDOSexN *)),(),((*)(
Calculating Occupations…
dEQEFpEFFEDOSdPEv
*),(1(*)(
Ec
dEQEFnEFFEDOSdN *),(*)(
Ec
Ev
dEQEFpEFFEfEFFEDOSdDp *)),(),((*)(
Ec
Ev
dEEfEFFQEFnEFFEDOSdDn *)),(),((*)(
Note: see slide 7 for rate value.
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
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