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AFORS-HET 3.0: FIRST APPROACH TO A

TWO-DIMENSIONAL SIMULATION OF SOLAR CELLS

R. Stangl (1)(*), T. Geipel (1), M. Dubiel (1), M. Kriegel (2), Th. El-Shater (3), and K. Lips (1)

(1) Hahn-Meitner-Institute Berlin, Kekuléstr.5, D-12489 Berlin(2) Heriot Watt University, Edinburgh, United Kingdom

(3) Electronics Research Institute, Cairo, Egypt

(*) Tel: +49/30/8062-1312, Fax: +49/30/8062-1333, E-mail: [email protected]

ABSTRACT: We describe the new version 3.0 of AFORS-HET, a numerical computer simulation program for

modeling the electronic behavior of thin-film or wafer-based solar cells. This version is the first step from a one-

dimensional towards a two-dimensional (2D) solar cell simulation. Two approaches have been implemented: A rather

simple 2D network approach, describing majority carrier transport problems only, and the rigorous solution of the

semiconductor equations in two dimensions, describing minority and majority carrier transport adequately together

with the corresponding boundary conditions. Unfortunately, the rigorous approach still suffers from convergence

problems. The current capabilities of the two 2D calculation approaches are demonstrated. We present selected

results on the simulation of the newly developed Buried Grid RECASH (Rear Contact Crystalline/Amorphous Silicon

Heterojunction) Solar Cell [1].

Keywords: Simulation, Thin Film, c-Si, a-Si, Software, two dimensional (2D)

1 INTRODUCTION

There is an increasing necessity for two or even

three-dimensional (2D, 3D, respectively) solar cell

simulation. Most design features implemented in state-of-

the-art high efficiency Silicon wafer based solar cells can

only be described through 2D or even 3D approaches,

e.g. point or stripe contacts or the concept of placing both

contacts at the rear side of the solar cell (back contact

solar cells). In the field of thin-film solar cells, even the

standard series interconnection, which is widely used in

thin-film modules, is essentially a 2D problem.Furthermore, a variety of novel contact designs are

introduced, which have 2D or even 3D features, like for

example all kind if interdigitated contact grids, or the

CSG approach [2].

Of course, there is already a variety of 2D and 3D

semiconductor programs commercially available, like

DESSIS, COMSOL Multiphysics, or Crosslight APSYS.

However, most of these programs are complex, cost

intensive and have to be adapted to the needs of

photovoltaics. So far, none of the mentioned programs

allows the simulation of specific solar cell

characterization methods, which are widely used within

the solar cell community.

We want to bridge this gap by developing a simpleand easy to use 2D simulation program, which is

specifically adapted to the needs of photovoltaic.

AFORS-HET 3.0 is the first step towards this aim.

2 CAPABILITIES OF AFORS-HET

2.1 AFORS_HET 2.2 (old version)

AFORS-HET solves the one dimensional

semiconductor equations (Poisson´s equation and the

transport equation for electrons and holes) with the help

of finite differences under different conditions: (a)

equilibrium mode, (b) steady-state mode, (c) steady-state

mode with small additional sinusoidal perturbations, (d)

simple transient mode, that is switching external

quantities instantaneously on/off, (e) general transient

mode, that is allowing for an arbitrary change of external

quantities. The generation of electron-hole pairs can be

described by Lambert-Beer absorption or by taking

incoherent and coherent internal multiple reflections into

account. Radiative band-to-band recombination, Auger

recombination and Shockley-Read-Hall recombination

can be considered and super-bandgab and sub-bandgap

generation/recombination can be treated. Interface

currents are modelled to be driven either by drift-

diffusion or by thermionic emission. The metallic

contacts can be modelled as Schottky or Schottky-

Bardeen metal/semiconductor contacts or as

metal/insulator/semiconductor (MIS) contacts.

Thus, the internal cell characteristics, such as banddiagrams, local generation/recombination rates, carrier

densities, cell currents, and phase shifts can be

calculated.

Furthermore, a variety of solar cell characterisation

methods can be simulated: current-voltage characteristics

(I-V), quantum efficiency, transient or quasi-steady-state

photo conductance, transient or quasi steady-state surface

photovoltage, spectral resolved steady-state or transient

photo- and electroluminescence, impedance/admittance,

capacitance voltage, capacitance-temperature and

capacitance-frequency spectroscopy, and electrically

detected magnetic resonance.

At the time being, approximately 600 AFORS-HET

downloads have been recorded, coming mainly fromuniversities and research institutions worldwide. The

current download rate (August 2007) is four downloads

per week. So far, six versions have been published at

major solar energy conferences [3-8] with the version 1.2

being awarded at the 31st IEEE conference in Lake

Buena Vista, USA, January 2005 [5]. AFORS-HET has

intensively been used by various groups within the

amorphous/crystalline Silicon (a-Si:H/c-Si) solar cell

community, mainly in order (i) to evaluate maximum

obtainable efficiencies for a-Si:H/c-Si solar cells, (ii) to

derive design criteria for those solar cells, and (iii) to

develop and quantify characterization methods for

monitoring the a-Si:H/c-Si interface recombination. For

more information about AFORS-HET and also aboutsimulation studies performed with AFORS-HET visit our

website:

www.hmi.de/bereiche/SE/SE1/projects/aSicSi/AFORS-HET.

22nd European Photovoltaic Solar Energy Conference, 3-7 September 2007, Milan, Italy

82



2.2 AFORS_HET 3.0 (new version)

The 1D program AFORS-HET 2.2 has been extended

towards a new version, AFORS-HET 3.0, which is

additionally able to describe solar cells in 2D. Two

approaches in order to perform 2D calculations have

been implemented: A rather simple 2D networkapproach, describing majority carrier transport problems

only, and the rigorous solution of the semiconductor

equations in two dimensions, describing minority and

majority carrier transport adequately together with the

corresponding boundary conditions.

However, at the time being, there are still some

discretisation-dependent convergence problems if

pursuing the second approach. AFORS-HET will be

further developed. If the convergence problems within

the second 2D approach (2D solution of the

semiconductor equations) are solved, a complete 2D

version of AFORS-HET 3.0 shall be launched.

3 2D SIMULATION WITHIN AFORS-HET

The current capabilities of the two new 2D

calculation modes of AFORS-HET 3.0 are demonstrated,

showing selected results on the simulation of a new

Silicon wafer based high efficiency solar cell concept,

the Buried Grid RECASH (Rear Contact

Crystalline/Amorphous Silicon Heterojunction) Solar

Cell [1].

Figure 1: Sketch of the cross section of a Buried GridRECASH solar cell [1] and a corresponding symmetry

element.

This back contact solar cell essentially consists of an

electrically insulated Al grid, contacting the p-doped

crystalline c-Si absorber (Silicon wafer) and buried

within a rear side deposited, n-doped emitter of

amorphous Silicon, a-Si:H(n) [1].

In order to perform a two dimensional solar cell

simulation, it is helpful to divide the solar cell into

symmetry elements and solve the problems under

investigation only within this symmetry element using

appropriate boundary conditions. A possible symmetry

element of this solar cell is sketched in Fig. 1. Due tosymmetry reasons, there are no internal cell currents

across the symmetry lines (symmetric boundary).

Specifying a fixed 2D array of grid points of arbitrary

spacing (discretisation) within this symmetry element,

the internal cell parameters are calculated at the discrete

positions given by the grid points.

3.1 2D network simulation

The 2D network simulation of AFORS-HET uses anequivalent circuit model to describe the majority carrier

transport within the symmetry element (see Fig. 2). The

grid points of the symmetry element are interconnected

with transport resistances, R T, which are calculated from

the semiconductor properties, when the two neighboring

grid points under consideration are placed within the

same material (Fig. 2). If two neighboring grid points are

placed within different materials, a local p/n junction is

assumed when the two semiconductor materials are of

opposite doping type, and a contact resistance, R c, can be

specified when both semiconductor materials are of the

same doping type. The local p/n junctions within the

symmetry element are described by a simple one-diode

model (reverse saturation current I0, diode ideality factorn1, photocurrent I ph, shunt resistance R sh are interface

input parameters) (see Fig. 3. Furthermore, a contact

resistance to the metal contacts can be specified.

Figure 2: Corresponding RECASH symmetry element

using the 2D network approach of AFORS-HET.

Figure 3: Network elements used within the 2D network

approach of AFORS-HET: (left) contact of two

semiconductor materials of opposite doping type, (right)

contact of two semiconductor materials of the samedoping type and contact to a metal boundary.

Applying a voltage between the specified metallic

contacts, the program solves for the internal potentials at

the grid points. Thus the internal cell currents and the total

currents into the metallic boundaries and, therefore, also

complete I-V characteristics of the solar cell can be

calculated.

The intension of this rather simple approach is to get a

rough approximation on 2D effects using a limited amount

of input parameters and to study the influence of additional

series resistances due to 2D contacting schemes.

3.2 Solving the 2D semiconductor equationsEfforts have been undertaken in order to solve the

semiconductor equations (Poisson´s equation and the

transport equation of electrons and holes) in two

dimensions under equilibrium and steady-state

RTA

RTB

n1 I0

Iph RSh

material A

material B

Vext

RC RTAy

RTAx

RTBx

RC

material B

metalcontact

material A

ligh

Al2O3 isolation

a-Si:H(n) emitter

c-Si(p) absorber

SiNx

Al absorber contact

Al emitter contact

symmetricboundary

symmetricboundary

I=0 I=0

c-Si(p) absorber

a-Si(n):H emitter

metal contact

transportresistance

contact resistance

one-diode-model

metal contact


83



conditions. Within an arbitrarily rectangular shaped

space region, consisting of different semiconductors (for

example, the symmetry element of Fig. 4), these

equations are solved at the grid points in order to yield

the local potentials and the local electron and hole

concentrations within the symmetry element. Radiative band-to-band recombination, Auger recombination and

Shockley-Read-Hall recombination is considered, super-

bandgap and sub-bandgap generation / recombination can

be treated. Interface currents are modeled to be driven

either by drift diffusion or by thermionic emission. The

boundaries of the symmetry element can be modeled as

metallic Schottky or Schottky-Bardeen boundaries, as

insulator boundaries, as metal/insulator boundaries or as

symmetric boundaries.

Figure 4: Corresponding RECASH symmetry element

using the AFORS-HET approach to solve the 2D

semiconductor equations on a rectangular grid.

Fig. 4 shows a typical discretisation scheme for the

symmetry element of the RECASH solar cell, together

with the appropriate boundary conditions: In order todescribe the electrically insulated absorber contact (plus

pole), which is buried in the a-Si:H(n) emitter, a metal

boundary towards the c-Si(p) absorber and a

metal/insulator boundary towards the a-Si:H(n) emitter

was used. The grid points are located at the intersections

of the grid.

4 CURRENT 2D PERFORMANCE OF AFORS-HET

In order to demonstrate the current state-of-the-art of

both implemented 2D modes within AFORS-HET 3.0,

some screenshots describing a typical 2D input / output

of the program are shown.

4.1 Input of a 2D structure

Fig. 5 shows a general 2D input window of

AFORS-HET. In order to specify the rectangular shaped

space region (the symmetry element) in which the cell

parameters under investigation are to be solved, an

arbitrary number columns and rows have to be defined,

which are needed for the construction of the symmetry

element. The properties of the resulting blocks, internal

interfaces, and boundaries can then be specified (see

Fig. 5). A different semiconductor can be appointed at

each block, using the same semiconductor material input

parameter window as in the 1D versions of AFORS-

HET. The same situation also applies for the internalinterfaces and for the boundaries. At the time being, the

incident light is always assumed to be perpendicular to

the specified structure as shown in Fig. 5, i.e. only

normal incidence of the incoming light can be

considered.

Fig. 6 shows the corresponding input specifications

of the RECASH symmetry element, in case of using the

AFORS-HET approach to solve the 2D semiconductor

equations.

Figure 5: Screenshot of the general 2D input within

AFORS-HET

Figure 6: 2D input specifications of the RECASH

symmetry element, in case of using the AFORS-HET

approach to solve the 2D semiconductor equations.

4.2 Output of 2D cell parameters

Figure 7: Screenshot of the 2D output of the internal cell parameters under equilibrium conditions, modeling the

RECASH symmetry element. Errors in the solution are

indicated by arrows.

metal/insulator

boundary(positive pole)

metal boundary

(positive pole)

insulator boundary

metal boundary

(negative pole)

symmetric

boundary

a-Si:H(n)

c-Si(p)

column

block

row

light

boundary

interface

c-Si(p)

light

symmetric

boundary

no interface

c-Si(p)

a-Si:H(n)

insulator

boundary

SiN

Schottky

flatband boundary

Al/c-Si

plus pole

symmetric

boundary

MIS

boundary

Al/Ox/a-Si

lus ole

Schottky

flatband

boundary

Al/a-Si

minus pole400 µm 15 µm

TE interface

a-Si/c-Si

µm275

µm0.03


84



A typical 2D AFORS-HET output window is

demonstrated in Fig. 7, showing the two dimensional

internal cell parameters (internal band diagrams, internal

cell currents, internal carrier concentrations and internal

generation / recombination rates) under equilibrium

conditions (no illumination, no external voltage) for theRECASH symmetry element in case of using the


equations.

As can be seen from Fig. 7, there is still an error in

the discretisation of the boundary conditions: The

displayed solution of the internal cell parameters under

equilibrium conditions is ill behaved, as the there is an

artificially high recombination rate at the corner grid

point located at the end of the a-Si:H(n) emitter. This

leads to internal cell currents unequal to zero. This error

is dependent on the chosen discretisation of the

symmetry element and has to be fixed before the


equations is fully functional.

4.3 Typical 2D calculation time

At the moment, the program is not optimized for

minimum calculation time. Assuming a typical

discretisation with 50x50 grid points, a solution will take

approximately 2 seconds for the 2D network approach

and 2 minutes in case of solving the 2D semiconductor

equations. If a reduced grid size of 10x30 grid points is

used, also the second approach is solved within some

seconds.

5 SIMULATION OF RECASH SOLAR CELL

5.1 RECASH efficiency potential

In order to estimate the efficiency potential of our

novel Silicon wafer based high efficiency concept, the

Buried Grid RECASH Solar Cell, a one dimensional

approximation had to be done, as the approach to solve

the 2D semiconductor equations was not always

convergent: The absorber contact (Al-grid) was not

modeled explicitly, assuming a fictive front contact with

surface recombination velocity of 30 cm/s (due to the

SiN passivation). The corresponding AFORS-HET 2.2

simulations demonstrated a RECASH efficiency

potential of >24%, see [1].

5.2 Influence of the Al-grid finger distanceIn order to study the influence of the Al-grid finger

distance on the solar cell performance, the simple 2D

network approach can be used. The Al-grid finger

distance was varied, while keeping the fraction of

absorber metallization constant (6 %). The one-diode

input parameters of the 2D network program can be

obtained from a 1D reference solar cell. In this specific

case, a fit of the one-diode model to the published I-V

characteristic of a Sanyo HIT solar cell [9] was

performed, in order to extract the corresponding input

parameters. The resulting simulated I-V characteristics of

the AFORS-HET 2D network program “measurement

I-V_2D_network” are shown in Fig. 8. Obviously, an

Al-grid finger distance larger than 400 µm is alreadycausing a significant loss in the filling factor, assuming a

270 µm thick c-Si(p) wafer with a resistivity of 1 Ω cm

and a 30 nm thick a-Si:H(n) emitter with a resistivity of

0,125 Ω cm.

0 100 200 300 400 500 600 7000

10

20

30

40

1D reference

50µm

100µm

250µm

400µm 500µm

750µm

1mm

2mm

3mm

4mm

5mm

j [ m A / c m

2 ]

V [mV]

decreasing

Al-grid finger distance

Figure 8: simulated RECASH I-V characteristics,

according to the 2D network program of AFORS-HET,

as a function of the Al-grid finger distance

In order to extract the resulting series resistance of

the RECASH solar cell structure as a function of theAl-grid finger distance, the simulated I-V characteristics

of Fig. 8 were once more fitted to a one-diode model, as

is shown in Fig. 9.

1 2 3 40,00

0,25

0,50

0,75

1,00

1,25

1,50

s e r i e s r e s i s t a n c e R

S

[ Ω ]

gridfinger distance d [mm]

Figure 9: Series resistance of the I-V characteristics of

Fig. 8 due to the 2D majority carrier transport as a

function of the Al-grid finger distance.

6 SUMMARY

A new version AFORS-HET is presented, offering a

simple 2D network approach in order to model the 2D

behavior of solar cell structures. The approach to solve

the 2D semiconductor equations within AFORS-HET isnot fully functional at present.

7 REFERENCES

[1] R.Stangl, M.Bivour, E.Conrad, I.Didschuns, L.Korte,

K.Lips, and M.Schmidt , this conference, 1AO.6.4

[2] P.A.Basore, Proc. IEEE-29, New Orleans, USA,

2002, 49-52

[3] A.Froitzheim, R.Stangl, M.Kriegel, L.Elstner,

W.Fuhs, Proc. WCPEC-3, Osaka, Japan, May 2003

[4] R.Stangl, A.Froitzheim, M.Kriegel, T.Brammer,

S.Kirste, L.Elstner, H.Stiebig, M.Schmidt, W.Fuhs,

Proc. PVSEC-19, Paris, France, June 2004, 1497[5] R.Stangl, M.Kriegel, S.Kirste, M.Schmidt, W.Fuhs,

Proc. IEEE-31, Lake Buena Vista, USA, January

2005


85



[6] R.Stangl, M.Kriegel, M.Schmidt, Proc. PVSEC-20,

Barcelona, Spain, June 2005

[7] R.Stangl, M.Kriegel, D.Schaffarzik, M.Schmidt,

Proc. PVSEC-15, Shanghai, China, October 2005

[8] R.Stangl, M.Kriegel, M.Schmidt, Proc. WCPEC-4,

Hawaii, USA, May 2006[9] E.Maruyama, A.Terakawa, M.Taguchi, Y.Yoshimine,

D.Ide, T.Baba, M.Shima, H.Sakata, M.Tanaka,

Proceedings WCPEC-4, (2006), Hawaii, USA, 1455


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