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(

*)2(*2

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