PVRD-FASP: A Unified Solver for Modeling Carrier and ...hpcg.purdue.edu/bbenes/papers/Shaik19IJP.pdf · SHAIK et al.: PVRD-FASP: A UNIFIED SOLVER FOR MODELING CARRIER AND DEFECT TRANSPORT

1602 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 9, NO. 6, NOVEMBER 2019

PVRD-FASP: A Unified Solver for Modeling Carrierand Defect Transport in Photovoltaic Devices

Abdul R. Shaik , Member, IEEE, Daniel Brinkman, Igor Sankin, Christian Ringhofer, Dmitry Krasikov,Hao Kang , Member, IEEE, Bedrich Benes , Senior Member, IEEE, and Dragica Vasileska, Fellow, IEEE

Abstract—In this article, we present a simulator for modelingtransport of charge carriers and electrically active defect centersin solar cells by treating them on an equal footing, which allows usto address metastability and reliability issues. The exact nonlineardifferential equations set solved by our solver is presented. Theformulation of such differential equations, namely the continu-ity equations, drift-diffusion equation, and Poisson equation, forstudying charge and defect transport is explained. The parametersneeded for forming the differential equations are taken from firstprinciple calculations. The solver is verified with test cases builton PN heterojunctions, Cu diffusion in single crystal CdTe andcomparing Silvaco simulations with our numerical results.

Index Terms—Defect chemical reactions, drift-diffusion-reaction solver, implicit Euler with Newton iteration, PNheterojunction, transient solutions for continuity equation.

I. INTRODUCTION

THIN-FILM cadmium telluride (CdTe) photovoltaics arequickly maturing into a viable clean-energy solution

through demonstration of competitive costs and performancestability with existing energy sources. Over the last half decade,CdTe solar technology has achieved major gains in performance;however, there are still aspects that can be improved to progresstoward their theoretical maximum efficiency. Perhaps equallyvaluable as high photovoltaic efficiency and a low levelized costof energy is device reliability. Understanding the root causes forchanges in performance is essential for accomplishing long-termstability [1]. Understanding the underlying reasons of those

Manuscript received March 22, 2019; revised June 10, 2019 and August 5,2019; accepted August 6, 2019. Date of publication September 6, 2019; date ofcurrent version October 28, 2019. This work was supported by the U.S. Depart-ment of Energy’s Office of Energy Efficiency and Renewable Energy under SolarEnergy Technologies Office under Agreement DE-EE0007536. The work ofD. Vasileska was supported by the National Science Foundation under ContractECCS 1542160. (Corresponding author: Abdul R. Shaik.)

A. R. Shaik and D. Vasileska are with the School of Electrical, Computer andEnergy Engineering, Arizona State University, Tempe, AZ 85287-5706 USA(e-mail: [email protected]; [email protected]).

D. Brinkman is with the Department of Mathematics and Statistics, San JoseState University, San Jose, CA 95192 USA (e-mail: [email protected]).

I. Sankin and D. Krasikov are with the First Solar Inc., Perrysburg, OH 43551USA (e-mail: [email protected]; [email protected]).

C. Ringhofer is with the School of Mathematical and Statistical SciencesArizona State University, Tempe, AZ 85287 USA (e-mail: [email protected]).

H. Kang and B. Benes are with the Department of Computer GraphicsTechnology-Knoy Hall of Technology, Purdue University, West Lafayette, IN47907 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this article are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JPHOTOV.2019.2937238

instabilities is unclear due to the lack of the ability to charac-terize materials at atomistic levels and the lack of interpretationfrom the most fundamental material science point of view. The“migration of Cu” as a most common cause of metastabilityin CdTe solar cells has been investigated rigorously over thepast twenty years. Multiple hypotheses suggesting degradationof CdTe solar cell devices due to the interaction and evolutionof point defects and complexes were proposed, and none ofthem received strong theoretical and experimental confirmationuntil recently [1]. It should be emphasized that measured atomicimpurity profiles in CdTe provide very little intelligence onactive doping concentrations. The same elements could formdifferent energy states, which could be either donors or accep-tors, depending on their position in crystalline lattice. Defectsinteract with other extrinsic and intrinsic defects. For example,changing the state of an impurity from an interstitial donor toa substitutional acceptor often is accompanied by generationof a compensating intrinsic interstitial donor defect. Hence, asimulation tool is needed that addresses the above processesand treats the carriers and the defects on the same footing toaddress the problems of metastability and reliability. Also, thetool must use first principle parameters as most educated inputguess.

Several solvers [2]–[7] exist in the literature to study thecharge carrier transport but only a few solvers [6]–[8] addressthe problem of defect transport. The electrically active defectcenters obey the same fundamental physics equations as thecharge carriers, namely continuity, drift-diffusion, and the Pois-son equation. Thus, charge carriers and defects can be treatedon an equal footing. The generation–recombination term in thecontinuity equation for charge carriers includes radiative andnonradiative processes: Shockley–Read–Hall (SRH) process,Auger process, etc. The same term in the continuity equationfor arbitrary defects is not well known. Moreover, this termis strongly dependent on the interactions between both chargecarriers and other defect species. In this article, we modelthese interactions as defect chemical reactions and use reaction-kinetic theory to write the generation–recombination terms inthe continuity equation for both carriers and defects. The mainphysical parameters required for the continuity equation are thereaction rate constants, the diffusion constants, and the activationbarriers. These values are generally calculated using densityfunctional theory (DFT) and are taken from reference papersor from our own work [9]–[12].

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SHAIK et al.: PVRD-FASP : A UNIFIED SOLVER FOR MODELING CARRIER AND DEFECT TRANSPORT IN PHOTOVOLTAIC DEVICES 1603

Solving the semiconductor equations with these generationand recombination terms requires numerical integration of asystem of nonlinear stiff differential equations of order one intime and order two in space. In this article, we use numericalalgorithms for efficiently and accurately solving such a systemof differential equations.

In systems with tens of electrically active defects, the amountof reactions and defect parameters is large and an ad hoc methodof incorporating the input parameters for defects is not suitable.To circumvent this problem, we developed a user-friendly soft-ware tool (called “PVRD-FASP” = FSLR, ASU, SJSU, andPurdue) that allows one to easily incorporate different defectspecies along with their specific properties.

In Section II of this article, we formulate and present thetheory we use to write the system of nonlinear differentialequations for modeling transport of charge carriers and defects.We also show that our generation–recombination formulation re-covers SRH processes in the corresponding limit. We also brieflycomment on the numerical algorithms employed in our solver.In Section III, we give four applications for solver validation.

1) Comparison of 0-dimensional (0-D) numerical simula-tions with analytical expressions for the generation–recombination process.

2) Calculation of copper point defect thermodynamic dopinglimit and comparison with known results.

3) Comparison of a light IV curve for a heterojunction CdTesolar cell with Silvaco.

4) Comparison of Cu diffusion in single crystal CdTe withexperiment using four different annealing recipes.

Finally, we present our conclusions, remarks, and futureworks.

II. THEORY

The partial differential equations for describing the transportof charge carriers and arbitrary point defects are mathematicallygiven as follows:

∂ui

∂t= −∇ · −→Ji +Gi −Ri (1)

−→Ji = −Di

kTui∇ (μi

ecp) (2)

−∇ · (ε∇φ) = q0

∑

i

ziui (3)

where i is the index of the species (either charge carrier or pointdefect), ui is the concentration,

−→Ji is the diffusion flux, μi

ecp isthe electrochemical potential, Gi −Ri is the net of generationand recombination rates,Di is the diffusivity, zi is the ionizationnumber of the charge state with sign of the ith species, k is theBoltzmann constant, T is the temperature, ε is the permittivity,φis the electrostatic potential, and q0 is the charge of an electron.The electrochemical potential μecp is defined as follows:

μecp = μcp + qφ = G0f + qφ+ kT ln

(u

Ns

)(4)

where G0f is the formation energy of the species, Ns is the

maximum volumetric number density of microstates the species

can occupy. The above definition for chemical potential is basedon the formation energies and configuration entropies in dilutelimit approximation [11], [13]. We can easily recover the stan-dard semiconductor equations for charge carriers from the abovedifferential equations (see Appendix A).

Next, we describe how to compute generation and recombi-nation terms for both charged carriers and defects using defectchemical reaction kinetic theory.

A. Generation and Recombination Term

As said earlier, the generation and recombination rate terms in(1) is well known for charge carriers but unknown for point de-fects. In this article, we represent the interactions between chargecarriers and defect species as a defect chemical reaction [14].We employ reaction kinetics to determine their time evolution.In this article, we consider only the defect chemical reactionsinvolving at most two reactants and at most two products (i.e.,bimolecular reactions). This approximation is always valid inthe dilute limit, and most of the higher order interactions can beapproximately modeled by chemical reactions with at most tworeactants or two products. With these assumptions we can writethe generation recombination rate term as follows:

Gi −Ri =dui

dt|reactions = UTQiU + P iU +Ki (5)

where U is the column vector with concentration of all species(point defects and charge carriers), Qi is the matrix correspond-ing to the reaction rate constants of second-order reactions, Pi

is the reaction rate constant row vector of first-order reactions,and Ki is rate constant of zero-order reactions involved by thespecies of index i.

With this form we can write the net rate as a reaction operatorR(U) as follows:

R(U) = UTQU + PU +K (6)

where Q is an M × M × M tensor, P is an M × M matrix, K is anM × 1 vector. One major advantage of writing R(U) in this formis that the Jacobian of the reaction operator can be explicitlycalculated in the form

JR(U) = (Q+QT )U + P. (7)

More details about such representation can be found inAppendix B.

Another advantage of using the point defect chemical reactionkinetics is that we can show generation and recombinationrate terms for charge carriers are easily recovered from theformulation using simple algebra and thermodynamic reactionequilibrium constant. For more details on this, see Appendix C.

B. Reaction Models

Although thermodynamics provides the ratio of forward andbackward rate constants (equilibrium constants), we still mustfind one of the two rate constants. For this we must use appropri-ate reaction models. For reactions with single reactant or productwe mostly use a barrier limited reaction. If the reactions involvetwo species, in either reactants or products, then the reaction


model will be based on the type of interacting forces occurringbetween the involved species. For a general reaction given as

AzA +BzBK−−−−→CzC + dDzD (8)

where d can be 0 (being single product) and 1 (being twoproducts), zX being the ionization of species X, the diffusion-controlled reaction rate constant [15] is given as follows:

K = 4πRcapt (DA +DB) exp

(−EA

kT

)(9)

whereRcapt is the capture radius,DX is the diffusivity of speciesX,EA is the activation energy of reaction (EA ≥ 0) representingthe probability of a collision resulting in the formation of prod-uct. The various types of reaction models, used in the solver, aredescribed below.

1) Diffusion Limited With Attraction Model: This model isused when the two reactants or products have opposite charges,i.e., zA × zB < 0. In such a case, there is Coulombic force ofattraction between the species and hence activation barrier iszero. Also, the capture radius is given by the Onsager radiuswhich is the radius at which the Coulombic energy is equalto the thermal energy. We consider an approximation of well-ordered lattice without significant perturbations of permittivityby high density of native defects or lattice disorder. Then, wecan calculate Onsager radius as follows:

ROnsager = q2 |zAzB |4πεkT

. (10)

Hence, the rate constant is given by

K = q2 |zAzB |εkT

(DA +DB) . (11)

2) Thermal Generation–Recombination Model: This modelis used when the reaction is given by

Null Kf−−−−−⇀↽−−−−−Kb

e−c + h+v . (12)

The backward rate is equated to the band-to-band recombi-nation rate coefficient B of the material. Currently, we assumethat coefficient B is independent of temperature, although fewpapers [16], [17] show it as being temperature dependent.

3) Capture Radius Limited Model: This model is used forreactions when zA × zB ≥ 0 and assumes the activation energyof the reaction to be zero. Thus, the rate constant is given asfollows:

K = 4πRcapt (DA +DB) . (13)

The user needs to specify the capture radius Rcapt.4) Capture Cross Section Limited Model: If one of the reac-

tant of (8) is charge carrier, then the sum of diffusivities can beapproximated as follows:

DA +DB ≈ De/h. (14)

Also, it is common to represent charge carrier capture reactionrate constant as product of capture cross section and thermalvelocities. Thus, the rate constant is given as follows:

K = σe/hvth,e/h. (15)

In case of diffusion limited with attraction model with chargecarrier the capture cross section of carrier is given as follows:

σe/h =4πROnsagerDe/h

vth,e/h. (16)

For capture radius limited model, with charge carrier as reac-tant, the user needs to input the capture cross section parameter.

5) Barrier Limited Model: This model is used for reactionswith single reactant and product. The rate constant is given asfollows:

K = ν exp

(−Eb

kT

)(17)

where ν is the attempted frequency of reaction (prefactor) andEb is the barrier height.

C. Other Models

Parameters, like diffusion coefficients, band gap, reactionrate constants, etc., involved in the differential equations aretemperature dependent. In the previous section, we showed thereaction rate models. Other models used in this article are givenbelow.

1) Varshini Model: This model is used to account the bandgap changes in semiconductor as a function of temperature. Themodel equation for the band gap as a function of temperature isas follows [18]:

Eg(T ) = E0 − αT 2

T + β(18)

where E0 is the 0K band gap, α and β are fitting parameters.2) Diffusion Barrier Model: The diffusion coefficient is as-

sumed to have Arrhenius relationship with temperature given asfollows:

D(T ) = D0 exp

(−ED

kT

)(19)

where ED is the diffusion barrier energy.This completes the description of physical models used in

solver.

D. Numerical Methods

Equations (1)–(3) are solved numerically by discretizing themin space and time. We use finite volume discretization for spaceand forward Euler method for time discretization. Furthermore,we employ time operator splitting for transport and reactionoperators. For transport operator, we apply Scharfetter–Gummeldiscretization and linearize the Poisson equation. The self-consistency is achieved through Gummel iteration. For reactionoperator, we use implicit time scheme with Newton iteration.

III. APPLICATION/VALIDATION OF SOLVER

In this section, we describe four types of simulations tovalidate the solver.


TABLE INUMERICAL VALUES USED IN THE SIMULATION

Fig. 1. Comparison of analytical versus numerical solution for time evolutionof electron and hole generated thermally.

A. 0-D Simulation of Thermally Generated Electronsand Holes

In this simulation, we take reaction (12) in CdTe material andrun a 0-D simulation with initial concentration of electrons andholes as 0, for a total time of 1e25 s. This simulation shows howfast thermal generation and recombination processes drive thesystem toward equilibrium. For this simulation, we can writethe exact analytical expression for the time evolution of theconcentrations (see Appendix G)

p(t) = n(t) = ni +2(

1ni

+ 2n(0)−ni

)exp(2Kbnit)− 1

ni

.

(20)

Here, n2i =

Kf

Kb, Kf , Kb are forward and backward rates of

(12), n(0) = p(0) is the initial electron and hole concentration,and t is the time. The numerical values are given in Table I.We can compare the numerical solution and analytical solutionand validate the numerical algorithm for the 0-D solver. Thecomparison results are shown in Fig. 1.

B. 0-D Simulation Study of Copper Defects in CdTe

In this article, we use the solver to study the thermodynamiclimits of copper defect concentrations in CdTe. The simulationfollows the work from reference [19] (APL). The authors, Yanget al., calculate the formation energies, defect transition levelsunder different conditions for Cui0, CuC0, and their chargestates, and by finding the solution of a constrained equilibriumproblem together with charge neutrality condition, they compute

the thermodynamic limits of doping concentration of copperdefects using analytical expressions. The equations applied in[19] are as follows:

[CuC0]=NCuC

× g0e−ΔHf (Cu0

C)/kBT

g0e−ΔHf (Cu0C)/kBT + g−1e−ΔHf (Cu−

C)/kBT

[CuC−]=NCuC

× g−1e−ΔHf (Cu−

C)/kBT

g0e−ΔHf (Cu0C)/kBT + g−1e−ΔHf (Cu−

C)/kBT

[Cui0] = NCui

× g0e−ΔHf (Cu0

i)/kBT

g0e−ΔHf (Cu0i)/kBT + g+1e−ΔHf (Cu+

i )/kBT

[Cui+] = NCui

× g+1e−ΔHf (Cu+

i )/kBT

g0e−ΔHf (Cu0i)/kBT + g+1e−ΔHf (Cu+

i )/kBT

[CuC0] + [CuC

−] + [Cui0] + [Cui

+] = NCuC+NCui

= NCu[CuC−] + [e−] = [Cui

+] + [h+] (21)

where NCu is the total copper concentration, NCuCis the total

copper at cation defect concentration,NCuiis the total interstitial

copper defect concentration, [X] and ΔHf (X) are the concen-tration and formation enthalpy of the defect species X and g’sare the degeneracy factors. With the calculated enthalpies fordefects and self consistently solving (21), the authors were ableto compute the thermodynamic doping limits for the Cu defectsin CdTe.

We applied our solver to do the same computation but withthe reaction formalism. The reactions employed to reproducethe results are as follows:

Cui0 −−−−−⇀↽−−−−−CuC

0

CuC0 −−−−−⇀↽−−−−−CuC

− + h+

CuC− −−−−−⇀↽−−−−−CuC

0 + e−

Cui0 −−−−−⇀↽−−−−−Cui

+ + e−

Cui+−−−−−⇀↽−−−−−Cui

0 + h+. (22)

The formation energies, degeneracies, capture rates are allthe same as that of [19] for the Cd-poor case and we rana simulation varying total copper concentration from 1e15 to1e19. We got the comparison results as shown in Figs. 2 and 3.The slight differences can be accounted for the finite precisionarithmetic errors as evident from Fig. 4. Thus, this comparisonresults show that the reaction formalism assumed in our solveris valid and encompasses relevant defect chemistry results fromthe literature.

C. 1-D Simulation of Heterojunction CdTe Device

In this article, we use the solver to study a CdTe-based solarcell and compare the results with Silvaco simulations. The devicestructure is shown in Fig. 5. The structure consists of fourlayers: 0.2 μm of zinc telluride (ZnTe), 1 μm of CdTe, 0.4 μmof cadmium sulfide (CdS), and finally 0.4 μm of transparent


Fig. 2. Comparison of hole concentration versus total Cu concentration.

Fig. 3. Comparison of Cu defect concentrations versus total Cu concentration.

Fig. 4. Comparison of total mass and charge errors versus total Cuconcentration.

conducting oxide (TCO). The doping of each layer is shownin Fig. 5. The ZnTe layer has p-type doping of 1.7e18 cm−3,CdTe has p-type doping of 1e14 cm−3, CdS has n-type dopingof 1e16 cm−3 and TCO has doping of 1.2e17 cm−3. The biasapplied is with respect to TCO at ZnTe. The light is AM1.5Gand is illuminated from TCO side.

We built the same structure in Silvaco and with our solverusing the same material parameters. In our solver we specify the

SRH processes and radiative processes through the reactionsgiven as follows:

Null−−−−−⇀↽−−−−− ec− + h+

v

N 0D−−−−−⇀↽−−−−−N+

D + e−cN+

D−−−−−⇀↽−−−−−N 0

D + h+v

N 0A−−−−−⇀↽−−−−−N−

A + h+v

N−A−−−−−⇀↽−−−−−N 0

A + e−c .

(23)

The donor dopant (ND) are in CdS and TCO and acceptordopant (NA) are present in ZnTe and CdTe layers, respectively,with the concentration described above.

The capture rates and all material parameters are kept samefor both our simulation and Silvaco simulation. We computedthe IV curves and compared them, as shown in Fig. 6, withSilvaco simulations of an identical structure. We see that wehave achieved very close match in short-circuit current betweenour simulation (27.26 mA/cm2) and that of Silvaco simulation(27.01 mA/cm2).

The open-circuit voltage in our simulation is very close to0.86 V, whereas that of Silvaco simulation is around 0.86 V. Thiscomparison further validates our formalism and methodology tostudy point defects and charge carriers on equal footing.

D. 1-D Simulation of Cu Diffusion in Single Crystal CdTe

In this article, we perform copper diffusion in single crystalCdTe and compare it with the experimentally measured Cu SIMSdata as given in [20]. We perform the numerical simulation withfour different annealing recipes namely 3 min anneal at 350 °C,6 min anneal at 350 °C, 10 min anneal at 300 °C, and 20 minanneal at 300 °C. The simulation structure consists of a 0.5 μmZnTe layer doped with Cu and an acceptor concentration of1e19 cm−3 and 19.5 μm sx-CdTe layer. The Cu species presentin ZnTe are assumed to form a neutral complex [CuC–Cui]0 withconcentration 2.5e20 cm−3. Previous simulations [20] assumedan effective model in the ZnTe layer that maintains an acceptordoping of 1e19 cm−3. The current solver does not need theeffective model and the Cu complex is closer to the real structureof the Cu:ZnTe layer. The reactions considered for this study aregiven in Table II.

The reactions in ZnTe are complex dissociation, radiativegeneration and recombination, and SRH reactions for acceptors.Reaction 3 in CdTe is the Cd knock off reaction which is themain reaction involved in the Cu acceptor formation in CdTeconsidered in this article. The other reactions are the carriercapture reactions involved with defects that resulted in theknock-off reaction and dissociation reaction. The first principlenumbers are taken from [10] and [21]. Since first principleparameters are not available for the point defects considered inZnTe, we assumed similar numbers as that of CdTe. We variedthe formation energies in ZnTe and CdTe for Cdi defect and[CuiCuC]° complex formation energy in ZnTe to match theexperimental secondary ion mass spectrometry (SIMS) data.


Fig. 5. Schematics of heterojunction CdTe device simulation.

TABLE IIREACTIONS AND THEIR MODELS CONSIDERED IN THE Cu DIFFUSION IN SINGLE CRYSTAL CdTe SIMULATION

DLWA: diffusion limited with attraction, TGR: thermal generation recombination, CXL: capture cross section limited, CRL: capture radius limited.

Fig. 6. Comparison of IV curves for CdTe solar cell.

Our simulation shows that we need not vary the diffusion barriersas was done in [20].

Comparisons of measured copper profiles and numericalsimulations for samples prepared by 350 °C 3 min anneal recipe,350 °C 6 min anneal recipe, 300 °C 10 min anneal recipe,and 300 °C 20 min anneal recipe are shown in Fig. 7, Fig. 8,Fig. 9, and Fig. 10, respectively. We see that our simulationresult closely matches the experimentally measured profiles in

Fig. 7. Copper profile comparison with 3 min 350 °C anneal recipe.

the CdTe layer. The measurements near the start of the ZnTe layershow a decrease in Cu concentration which can be attributed asan artifact of SIMS measurement.

The numerical simulation results are not taken after the cooldown to room temperature although the measurements are takenat room temperature. Simulations of cooldown process is still inprogress, however we do not expect the resulting Cu profiles tochange significantly from those presented in Figs. 7–10.





IV. CONCLUSION

Photovoltaic solar cells based on CdTe represent the largestsegment of commercial thin-film module production worldwide.Recent improvements have matched the efficiency of multicrys-talline silicon while maintaining cost leadership. Further inves-tigations on the role of defects (point-like or grain boundaries)can shed light on the chemical and physical processes occurringwithin the cell, thus enabling one to improve efficiency, mini-mize metastabilities, and help slow down aging of the cells andmodules.

For this purpose, a drift-diffusion-reaction simulator (PVRD-FASP) was developed within our team. The simulator allowsmodeling of the charged carriers and active defects on equalfooting. The theoretical model used in this simulator is presentedhere as well as key numerical details. Our new reaction simulatorwas validated through several simulation examples. The 0-Dsolver was verified via comparison with available published dataon thermodynamic calculations of copper defects in CdTe. Thesolver was verified on the simulation of CdTe/CdS heterojunc-tion solar cell made. The dark and light IV characteristics ofthe cell calculated with our solver were compared with Silvacosimulation results. Perfect matching between the two simulationexperiments was obtained. Finally, the drift-diffusion-reactionsolver was tested on real experimental data of Cu diffusion intosingle-crystal CdTe. Again, very good agreement with SIMSprofiles with different annealing recipes was obtained. We con-clude that the PVRD-FASP solver can robustly perform relevantand diverse simulations to study diffusion/activation of dopants,device performance, and other physically relevant processes.

APPENDIX ASTANDARD SEMICONDUCTOR EQUATIONS FOR CHARGE

CARRIERS FROM ELECTROCHEMICAL FORMULATION

For charge carriers, we can simplify the differential equationsgiven in (1)–(3) as follows:

∂n

∂t= −∇ · −→Jn +Gn −Rn

−→Jn = −Dn

kTn∇

(G0

f,n − q0φ+ kT ln

(n

NC

))

∂p

∂t= −∇ · −→Jp +Gp −Rp

−→Jp = −Dp

kTp∇(G0

f,p + q0φ+ kT ln

(p

NV

))

∇ · (ε∇φ) = −p+ n−ND +NA. (24)

Using the formation energies of the free carriers [see (62)],we can simplify the flux equations as follows:

−→Jn = −Dn

kTn∇ (−χ− kT ln Nc)− μnn E −Dn∇n

−→Jp = −Dp

kTp∇ (χ+ Eg − kT ln NV ) + μpp E −Dp∇p

(25)

where we used the definition of electric field and Einstein rela-tionship between diffusivity and mobilities. The above equationsare the standard semiconductor equation for heterostructures(see [22]) and in case of homogeneous structures the first gradi-ent term is zero. Thus, the equation reduces to

−→Jn = −μnn E −Dn∇n

−→Jp = μpp E −Dp∇p (26)

which are the standard equations for charge carriers insemiconductors.


APPENDIX BDETAILS ON GENERATION RECOMBINATION RATES FROM

DEFECT CHEMICAL REACTIONS

The rate equation for a general defect chemical reaction (27)with at most two reactants or products

aA+ bB Kf−−−−−⇀↽−−−−−Kb

cC + dD (27)

is given as

−1a

d[A]

dt= −1

b

d[B]

dt=

1c

d[C]

dt=

1d

d[D]

dt

= Kf [A]a[B]b −Kb[C]c[D]d. (28)

The above rate equation is still complex, and we furtherrestrict ourselves to stoichiometric coefficients a, b, c, and dof either d = 1 or d = 0. Although this looks like a strongsimplification to the domain of defect chemical reactions, itis still enough to cover the defect chemical interactions ofinterest.

If any of the species is involved in more than one reaction, thenthe rate equation for that species will have contributions fromreaction rates of both the first reaction and the other reaction.For example, if the reaction system is given as follows:

A+B Kf1−−−−−⇀↽−−−−−Kb1

C +D (29)

A Kf2−−−−−⇀↽−−−−−Kb2

E + F (30)

Null Kf3−−−−−⇀↽−−−−−Kb3

A+G (31)

then the rate equation for the time evolution of species A is givenas follows:

d[A]

dt= −Kf1[A][B] +Kb1[C][D]−Kf2[A]

+ Kb2[E][F ] +Kf3 −Kb3[A][G]. (32)

The above rate equation can be elegantly represented asfollows:

d[A]

dt= UTQAU + PAU +KA (33)

where U is a concentration (column) vector of length 7 given asfollows:

U = ([A] , [B] , [C] , [D] , [E] , [F ] , [G])T . (34)

QA is a 7 × 7 sparse matrix with nonzero elements

qA12 = qA21 = −Kf1

2

qA34 = qA43 =Kb1

2

qA56 = qA65 =Kb2

2

qA17 = qA71 = −Kb3

2. (35)

PA is a row vector of length 7 with nonzero elements

pA1 = −Kf2 (36)

and KA is the constant term given as Kf3.The representation given by (33) can be extended to any

species and we can write the corresponding Q, P, K variables asfunctions of reaction rate constants. Consider a general reactionrate equation

dui

dt=

M∑

k=1

M∑

j=1

aijkujuk +

M∑

j=1

bijuj + ci (37)

for i = 1, 2, …, M, where M is the total number of species anda, b, c are the reaction rate constants in which the species i isinvolved. Then, the parameters Qi, P i, Ki can be calculated asfollows:

qijk = qikj =

⎧⎪⎨

⎪⎩

12a

ijk, if j �= k

aijk, if j = k

0, otherwise

(38)

pij = bij , K = ci. (39)

Thus, the generation–recombination term can be expressed asfollows:

Gi −Ri =dui

dt|reactions = UTQiU + P iU +Ki. (40)

APPENDIX CSYSTEMATIC CORRESPONDENCE WITH RADIATIVE AND SRH

GENERATION RECOMBINATION RATES

For a radiative process, the generation and recombination termin the continuity equation of charge carriers can be expressed asfollows:

G−R = b(n2i − np). (41)

Considering the radiative process interactions, the equivalentreaction is as follows:

Null Kf−−−−−⇀↽−−−−−Kb

e−c + h+v (42)

where e−c and h+v denote a free electron in conduction band and

a free hole valence band of the material. Hence, the rate equationis given as follows:

d[e−c ]dt

=d[h+

v ]

dt= Kf −Kb[e

−c ][h

+v ]

= Kb

(Kf

Kb− [e−c ][h

+v ]

). (43)

If the ratio of forward rate to backward rate is n2i , we can easily

see that (41) and (43) are equivalent. The thermodynamics ofreactants and products results in exactly this condition in theratio. See (63).

For SRH processes, the generation and recombination on asingle defect in steady-state is given as in [23]

G−R =(n1p1 − np)n+n1Cp

+ p+p1Cn

. (44)


The equivalent reaction representing the SRH process in caseof acceptor type species can be written as follows:

A0 Kf1−−−−−⇀↽−−−−−Kb1

A− + h+v

A− Kf2−−−−−⇀↽−−−−−Kb2

A0 + e−c . (45)

The rate equations can be written as follows:

d[A0]

dt= − d[A−]

dt= −Kf1[A

0] +Kb1[A−][h+

v ]

+Kf2[A−]−Kb2[A

0][e−c ] (46)

d[e−c ]dt

= Kf2[A−]−Kb2[A

0][e−c ] (47)

d[h+v ]

dt= Kf1[A

0]−Kb1[A−][h+

v ]. (48)

From (46) we note that [A0] + [A−] is constant in time(say [A]tot). Hence, in steady-state, we have

[A0] =Kf2 +Kb1p

Kf1 +Kf2 +Kb1p+Kb2n[A]tot

[A−] =Kf1 +Kb2n

Kf1 +Kf2 +Kb1p+Kb2n[A]tot (49)

where [h+v ] = p and [e−c ] = n is used for notational simplicity.

Using (49) in (47) and (48) with few algebraic simplificationswe can write

dn

dt=

dp

dt=

Kf1Kf2 −Kb1Kb2np

Kf1 +Kf2 +Kb1p+Kb2n. (50)

Now if the ratio of the reaction rates is restricted asKf1

Kb1= p1 = Nve

EV −ETkT ,

Kf2

Kb2= n1 = Nce

−EC−ETkT (51)

then (50) can be simplified further as follows:

G−R =(n1p1 − np)n+n1Kb1

+ p+p1Kb2

[A]tot (52)

which is equivalent to (44). Equation (51) will hold based onthe thermodynamics of reactants and products as shown in thefollowing section. A similar equivalence can be shown for donortype centers as well.

A. Thermodynamic Relationship Between Forward andBackward Rate Constants

Consider a general reaction

N∑

i=1

xiXiKf−−−−−⇀↽−−−−−Kb

M∑

j=1

yjYj (53)

with N reactants and M products. At thermodynamic equilib-rium the difference in the chemical potentials of reactants andproducts should be zero. Thus, we have

ΔrG =

N∑

i=1

xiμXi−

M∑

j=1

yjμYj(54)

where ΔrG is the Gibb’s free energy change of reaction, μ isthe chemical potential of the species which can written from (4)neglecting the electrostatic potential as follows:

μXi= G0

f,Xi+ kT ln

([Xi]

Ns,Xi

). (55)

Note that the chemical potential here is defined as the sum ofspecies formation energy and its configurational entropy in thedilute limit as expressed in [12] and [13]. Thus, at equilibrium,we have

N∑

i=1

(xiG

0f,Xi

+ kT ln

([Xi]

xieq

Nxi

s,Xi

))

=N∑

i=1

(yjG

0f,Yj

+ kT ln

([Yj ]

yjeq

Nyj

s,Yj

)). (56)

Upon simplification, we obtain

M∏j=1

[Yj ]yjeq

N∏i=1

[Xi]xieq

=

M∏j=1

Nyj

s,Yj

N∏i=1

Nxi

s,Xi

exp

(−ΔrG

0

kT

)(57)

where ΔrG0 is the standard enthalpy change of reaction or

standard reaction Gibb’s energy defined as follows:

ΔrG0 =

N∑

j=1

yjG0f,Yj

−M∑

i=1

xiG0f,Xi

. (58)

We also know that at equilibrium, we have

Kf

N∏

i=1

[Xi]xieq −Kb

M∏

j=1

[Yj ]yjeq = 0. (59)

Hence, using (57) and (59), we can write the equilibriumconstant as follows:

Keq =Kf

Kb=

M∏j=1

Nyj

s,Yj

N∏i=1

Nxi

s,Xi

exp

(−ΔrG

0

kT

). (60)

Therefore, the reaction rate constants follow a fixed ratio.Applying this for the recombination reaction (12), we have

Kf

Kb= Ns,e−cNs,h+

vexp

(−G0

f,e−c+G0

f,h+v

kT

). (61)

The number of microstates for electrons in the conductionband and holes in the valence band areNC andNV , respectively.The formation energies of electrons in the conduction band andholes in the valence band (referenced from vacuum level) is

G0f,e−c = −χ, G0

f,h+v= χ+ Eg (62)

whereχ is the electron affinity of the material andEg is the bandgap of the material. Thus, we have the ratio

Kf

Kb= NcNv exp

(−Eg

kT

). (63)


For the reactions in (45), we have

Kf1

Kb1=

Ns,A−Ns,h+v

Ns,A0exp

(−G0

f,A− +G0f,h+

v−G0

f,A0

kT

)

(64)

Kf2

Kb2=

Ns,A0Ns,e−c

Ns,A−exp

(−G0

f,A0 +G0f,e−c

−G0f,A−

kT

)(65)

where the formation energies are given as follows:

G0f,A0 = Ef,A0 − χ− Eg

G0f,A− = Ef,A0 − sign(A−)ET − χ− Eg

G0f,e−c = −χ, G0

f,h+v= χ+ Eg. (66)

Here sign(A−) is the sign of the charge on the species, ET isthe trap level (transition level from 0/-; see [10]) with respect tovacuum and Ef,A0 is defect formation energy of neutral speciescalculated through DFT (see Appendix D for details on howtransition levels of species are related to the formation energiesof its corresponding neutral species). Assuming the microstatesfor neutral and charged species are nearly same, we arrive at

Kf1

Kb1= Nv exp

(−χ+ Eg + ET

kT

)= Nv exp

(Ev − ET

kT

)

Kf2

Kb2= Nc exp

(−−χ− ET

kT

)= Nc exp

(−Ec − ET

kT

)

(67)

where we have used the relations

Ec = −χ, Ev = −χ− Eg. (68)

Hence, thermodynamics guarantees that the SRH process isrecovered from our reaction formulation, i.e., the thermodynam-ics guarantees the equivalence as a chemical defect reaction.

APPENDIX DRELATIONSHIP BETWEEN TRANSITION LEVEL AND FORMATION

ENTHALPY FOR DEFECTS

Suppose the formation enthalpies and transition levels of adefect are as shown in Fig. 11. Then, it is clear that

ΔH+f = ΔH0

f − (+1)(E+/0 − 0

)

ΔH ′f = ΔH0

f − (+1)(E+/0 − E2+/+

)

ΔH2+f = ΔH ′

f − (+2)(E2+/+ − 0

)

ΔH−f = ΔH0

f − (−1)(E0/− − 0

)(69)

by simple rules of coordinate geometry for straight lines withslopes. For further details please see [10]

APPENDIX ETIME EVOLUTION OF THERMALLY GENERATED CARRIERS

The reaction representing thermally generated carriers isgiven in (12). The differential equation for this system is (43).

Fig. 11. Defect formation energy w.r.t valence band maxima (VBM).

Choosing charge neutrality for the initial condition we have

n(0) = p(0), n(t) = p(t) (70)

where n(t) and p(t) represent the concentration of electrons andhole at time t. Thus, the differential equation to be solved isgiven as follows:

dp

dt=

dn

dt= Kb

(n2i − n2

)(71)

which is a Ricatti equation with the constant value n(t) = ni asone particular solution. Obviously, this solution cannot be thephysically correct solution to our system. The other solution canbe found by writing the solution as follows:

n(t) = ni +1z. (72)

Substituting this into (71) and simplifying we obtain

− 1z2

dz

dt= Kb

(n2i −

(ni +

1z

)2)

⇒ 1z2

dz

dt= Kb

(2ni

z+

1z2

)

⇒ dz

dt− 2Kbniz −Kb = 0

⇒ d

dt

(ze−2Kbnit

)= Kbe

−2Kbnit

⇒ z(t)e−2Kbnit − z(0)e0 = Kbe0 − e−2Kbnit

2Kbni

⇒ z(t) = z(0)e2Kbnit +e2Kbnit − 1

2ni. (73)

The value of z(0) can be found as follows:

n(0) = ni +1

z(0)

⇒ z(0) =1

n(0)− ni. (74)


Hence, the final solution can be given as follows:

n(t) = ni +1

z(t)

= ni +1(

1n(0)−ni

+ 12ni

)e2Kbnit − 1

2ni

= ni +2(

2n(0)−ni

+ 1ni

)e2Kbnit − 1

ni

. (75)

ACKNOWLEDGMENT

The authors would like to thank Dr. Yang for providing theirsimulation data on Cu defects in CdTe to us for comparison.

REFERENCES

[1] D. Guo, D. Brinkman, A. R. Shaik, C. Ringhofer, and D. Vasileska,“Metastability and reliability of CdTe solar cells,” J. Phys. D. Appl. Phys.,vol. 51, no. 15, Apr. 2018, Art. no. 153002.

[2] R. Varache et al., “Investigation of selective junctions using a newlydeveloped tunnel current model for solar cell applications,” Sol. EnergyMater. Sol. Cells, vol. 141, pp. 14–23, Oct. 2015.

[3] M. Burgelman, P. Nollet, and S. Degrave, “Modelling polycrystallinesemiconductor solar cells,” Thin Solid Films, vol. 361, pp. 527–532,Feb. 2000.

[4] T. T. et al., “Analysis of microeletronic and photonic structures (AMPS1D).” [Online]. Available: http://www.ampsmodeling.org/. Accessed on:Apr. 13, 2019.

[5] D. A. Clugston and P. A. Basore, “PC1D version 5: 32-bit solar cell mod-eling on personal computers,” in Proc. Conf. Rec. 26th IEEE Photovolt.Spec. Conf., 1997, pp. 207–210.

[6] “Silvaco TCAD,” [Online]. Available: https://www.silvaco.com/products/tcad.html. Accessed on: Apr. 13, 2019.

[7] “Synopsys TCAD.” [Online]. Available: https://www.synopsys.com/silicon/tcad.html. Accessed on: Apr. 13, 2019.

[8] M. Burgelman, P. Nollet, and S. Degrave, “Modelling polycrystallinesemiconductor solar cells,” Thin Solid Films, vol. 361, pp. 527–532, 2000.

[9] D. Krasikov, A. Knizhnik, B. Potapkin, S. Selezneva, and T. Sommerer,“First-principles-based analysis of the influence of Cu on CdTe electronicproperties,” Thin Solid Films, vol. 535, no. 1, pp. 322–325, 2013.

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Abdul R. Shaik (M’18) received the B.Tech. degreein electrical engineering from the Indian Institute ofTechnology Patna, Patna, India, in 2012, and the M.S.degree in electrical engineering from Arizona StateUniversity, Tempe, AZ, USA, in 2016. He is currentlyworking toward the Ph.D. degree in electrical engi-neering from Arizona State University.

He worked as a Software Intern in LG soft India,Bangalore, India. From 2012 to 2014, he workedas an Analog Design Engineer with Silabtech Pvt.Ltd. (later acquired by Synopsys Inc.), Bangalore. He

worked as a Mixed Signal and Analog IP Design Intern in Invecas Inc., San Jose,CA, USA. He is currently working as a Research Assistant with ECEE, ASU,Tempe, AZ, USA.

Mr. Shaik has won the Best Student Oral Presentation at the Spring MRSMeeting (2019).

Daniel Brinkman received the B.S. degree in mathe-matics and physics from the University of Minnesota,Twin Cities, Minneapolis, MN, USA, in 2009, andthe Ph.D. degree in applied mathematics and the-oretical physics from the University of Cambridge,Cambridge, U.K., in 2013.

From 2013 to 2016, he was a Visiting AssistantProfessor with the School of Mathematical and Sta-tistical Sciences, Arizona State University, Tempe,AZ, USA. Since 2016, he has been an AssistantProfessor with the Department of Mathematics and

Statistics, San Jose State University, San Jose, CA, USA. His main researchinterests include modeling and numerical simulation with partial differentialequations, novel electronic structures such as graphene, organic semiconductors,and thin-film solar cells.

Igor Sankin received the M.S. degree from KievPolytechnic Institute, Kiev, Ukraine, in 1993, and thePh.D. degree from Mississippi State University, Mis-sissippi State, MS, USA, in 2006, both in electricalengineering.

From 2001 to June 2008, he worked withSemiSouth Laboratories, Inc., Starkville, MS, devel-oping power SiC switches and rectifiers. Currently,he is a Fellow with First Solar, Inc., Perrysburg, OH,USA, working with CdTe-based thin-film PV.

http://www.ampsmodeling.org/

https://www.silvaco.com/products/tcad.html

https://www.synopsys.com/silicon/tcad.html


Christian Ringhofer received the Ph.D. and Ha-bilitation degrees in applied mathematics from theTechnical University of Vienna, Vienna, Austria, in1981 and 1986, respectively.

He has been a Professor with Arizona State Uni-versity, Tempe, AZ, USA, since 1984. He has heldvisiting positions with the University of Vienna (Aus-tria), the Technical University of Hamburg (FRG),the Universidad de Buenos Aires (Argentina) andJiadong University, Shanghai, China. He has writtenand coedited six books and more than 140 refereed

journal articles.Dr. Ringhofer is a member with the Society for Industrial and Applied

Mathematics (SIAM) and an Associate Editor for the SIAM Journal of AppliedMathematics.

Dmitry Krasikov received the Ph.D. degree inphysics from Lomonosov Moscow State University,Moscow, Russia, in 2008.

He was a Researcher with an R&D Company,Kintech Lab, Moscow, where he worked on theo-retical analysis of materials and devices. He is cur-rently a Development Engineer with the First Solar,Inc., Perrysburg, OH, USA, working on the revealingphysical mechanisms behind the solar cell perfor-mance limitations and performance instabilities usingadvanced multiscale models of solar cell fabrication

and operation under different stress conditions.

Hao Kang (M’18) received the B.E. degree in soft-ware engineering from Beijing Jiaotong University,Beijing, China, in 2000, and the Engineer’s degreein computer science from the Stevens Institute ofTechnology, Hoboken, NJ, USA, in 2015. He is cur-rently working toward the Ph.D. degree in computergraphics technology with Purdue University, WestLafayette, IN, USA, in 2019.

Since 2015, he is a Research Assistant with PurdueHigh Performance Computer Graphics Laboratory.He worked as a Research Intern with Siemens Health-

ineers in 2017 summer and Adobe Research in 2018 spring. His research interestsinclude software engineering, human–computer interaction, three-dimensionalcomputer graphics and vision, and machine learning and robotics.

Bedrich Benes (SM’17) received the M.S. and Ph.D.degrees in computer science from Czech TechnicalUniversity in Prague, Prague, Czechia, in 1991 and1998, respectively.

He is a Professor of Computer Graphics Tech-nology with the Purdue University and ComputerScience (by courtesy), West Lafayette, IN, USA. Heis the Director of the High Performance ComputerGraphics Laboratory, Purdue University. He was amember of numerous program committees of vari-ous conferences including ACM Siggraph and Euro-

graphics and he was a Paper Chair of Eurographics 2017. He is the Editor ofComputer Graphics Forum and an Associate Editor of Computers and Graphics.He is known for his work in geometric modeling, procedural modeling, scientificvisualization, and software optimizations in digital manufacturing.

Dragica Vasileska (F’19) received the B.S.E.E. andM.S.E.E. degrees in electrical engineering and infor-mation technologies from the University Sts. Cyriland Methodius Skopje, Republic of North Macedo-nia, in 1985 and 1992, respectively, and the Ph.D.degree from Arizona State University, Tempe, AZ,USA, in 1995.

From 1995 to 1997, she held a Faculty ResearchAssociate position within the Center of Solid-StateElectronics Research, Arizona State University. Inthe fall of 1997, she joined the Faculty of Electri-

cal Engineering, Arizona State University. In 2002, she was promoted as anAssociate Professor, and in 2007 as a Full Professor. Her research interestsinclude semiconductor device physics and semiconductor device modeling, withstrong emphasis on quantum transport and Monte Carlo particle-based devicesimulations. Recently, her focus includes modeling metastability and reliabilityof solar cells. She has published more than 180 publications in prestigiousscientific journals, more than 200 conference proceedings refereed papers, 25book chapters, which has given numerous invited talks and authored five books.

Dr. Vasileska was the recipient of the 1998 NSF CAREER Award. Herstudents have won the Best Paper and Best Poster Award at LDSD confer-ence in 2004, Best Oral Presentation Award at ISDRS Conference in 2011,two best poster awards at 42nd PVSC Conference in 2015, third Best PosterPlace at NREL/SNL/BNL PV Reliability Workshop in 2017, Best Student OralPresentation at Spring MRS Meeting in 2019, etc.

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