**NUMERICAL MODELING OF CIGS AND CdTe SOLAR CELLS:**

**SETTING THE BASELINE**

M. Gloeckler, A.L. Fahrenbruch, and J.R. Sites

Physics Department, Colorado State University, Ft. Collins, CO 80523 USA

# ABSTRACT

Numerical modeling of polycrystalline thin-film solar cells is an
important strategy to test the viability of proposed physical
explanations and to predict the effect of physical changes on cell
performance. In general, this must be done with only partial knowledge
of input parameters. Nevertheless, for consistent comparisons between
laboratories, it is extremely useful to have a common starting point, or
baseline. We will discuss guidelines that should be considered assigning
input parameters for numerical modeling. Consequently specific baseline
parameters for CIGS and CdTe are proposed. The modeling results for
these baseline cases are presented and it is discussed how the baseline
cases serve to describe some of the most important complications that
are often found in experimental CIGS and CdTe solar cells.

**1. INTRODUCTION**

The major applications for modeling in solar cell research are: testing
the viability of proposed physical explanations, predicting the effect
of changes in material properties and geometry on cell performance, and
fitting of modeling output to experimental results. The input parameter
sets used to fit only experimental J-V data may not be unique.
Therefore, fitting of experimental data is only conclusive if a wide set
of data, i.e. J-V at different temperatures, and QE at different biases,
is used. Input parameters that are well known should not be changed at
any time, whereas parameters that have only marginal effect on the
output can be tested and then not changed. These parameters excluded,
the remaining parameters are available for fitting purposes.

**2. SELECTION OF INPUT PARAMETERS**

**2.1 Front and Back Contacts and Surfaces**

In general, contacts can be assumed ohmic or, depending on the focus of
the modeling, assigned a Schottky barrier height consistent with
experimental observations. The reflection at the back surface has only
minor influence on the achievable short-circuit current density (J~sc~),
and these influences only become noticeable if the absorber is chosen to
be fairly thin. Many modeling tools support a constant multiplicative
reflection factor for the front surface (i.e. R­~F~ = 0.1, 10%
reflection). Quantum efficiency (QE) is then reduced by this fraction
and, if interference effects are neglected, QE will show a fairly flat
response at intermediate wavelengths of \~1-R~F~.

**2.2 Material Parameters**

Carrier mobilities for polycrystalline material should be chosen lower
than the values reported for crystalline material. Effective masses of
m~e~^\*^ = 0.2m~0~ for electrons and m~h~^\*^ = 0.8m~0~ for holes, which
are numbers typical for direct band gap material, are recommended unless
more specific data is available. The ratio of the carrier mobilities
(~e~/~h~) should be approximately inversely proportional to the ratio of
the effective masses (m~h~^\*^/m~e~^\*^).

The effective density of states, N~C~, can be calculated using eqn. (1)
\[1\] and similarly for N~V~. Note the direct temperature dependence in
N~C~ and N~V~, which should be taken into account for temperature
dependent modeling.

$N_{C} = 2\left( \frac{2{\pi m}_{e}^{}\text{kT}}{h^{2}} \right)^{\frac{3}{2}}$
(1)

Carrier concentrations can be determined from capacitance-voltage
analysis. Typical numbers are in the authors' experience the order of
10^16^ cm^-3^ for CIGS and 10^14^ cm^-3^ for CdTe. The band gaps of the
semiconductors are known (see Tables I & II), and for the
Cu(In~1-x~Ga~x~)Se~2~ alloys an approximate expression can be used (2)
\[2\]. Band offsets at the interfaces will be discussed within the
specific material sections below.

E(x) = 1.02 + 0.67x + 0.11x(x-1) (2)

**2.3 Defect States**

Most numerical simulators use the Shockley-Read-Hall (SRH) model \[3\]
to describe carrier recombination currents. There are two approaches to
do this: either the model assumes a constant minority carrier lifetime
or the input parameters are capture cross sections, ~e~ and ~p,~ and the
defect distributions N~def~(E). An estimate of the lifetime (LT) can be
calculated from the defect density (DD) parameters by

$\tau \simeq \left( \sigma\ v_{\text{th}}N_{\text{def}} \right)^{- 1}$
(3)

v~th~ 10^7^ cm/s is the thermal velocity of the electrons. However, in
general the lifetime depends on the cross sections for both carriers and
the Fermi level.

**Defect Density (DD) Model.** Since variations of the energetic defect
distribution show only negligible effects on the output, it is
recommended to position recombinative defect states in a narrow
distribution close to the middle of the band gap (generic "mid-gap"
states).

In the SRH formalism \[3\], a defect state can change its charge state
only by one elementary charge; therefore, one can always make the
following distinctions: A donor-like (acceptor-like) defect state is
likely to give away (accept) an additional electron. The two possible
charge states for donors (acceptors) are positive and neutral (negative
and neutral). It follows that free electrons (holes) will be coulomb
attracted to the ionized donor-like (acceptor-like) defect state,
whereas holes (electrons) will

have no strong interaction with the donor-like (acceptor-like) defects,
giving very small hole (electron) cross-sections. The transition of a
hole into the ionized donor-like defect, or the transition of an
electron into a neutral donor-like defect is rare. One problem with the
SRH picture is that physical impurities often generate several possible
defect levels that have different charge states; however, a
cross-linking between two or more energy levels is not contained in the
SRH formalism. We assign the attractive cross-section corresponding to a
radius at which the coulomb potential energy of the carrier in the field
of the charged trap equals kT (4) \[4\] (k Boltzman constant, q
elementary charge).

$\sigma_{\text{att}} = \frac{q^{4}}{\text{16}\text{πε}_{s}^{2}k^{2}T^{2}}\  \approx \ \text{10}^{- \text{13}} - \ \text{10}^{- \text{12}}\ \text{cm}^{2}$
(4)

Neutral cross-sections can range between 10^-18^ and 10^-15^ cm^2^; the
latter number corresponds to the physical size of an atom and is often
referred to as "geometric" cross-section. Smaller values account for
quantum mechanical transition probabilities or compensate for strong
field effects in the region of interest.

In the DD model, free carriers can be trapped in defect states, which
alters the space charge distribution and leads to a deformation of the
band structure. Illumination and/or voltage bias can alter the trapped
carrier density, giving rise to transient effects. Mid-gap acceptor-like
defect states in n-type material or mid-gap donor-like defect states in
p-type material will completely ionize in thermal equilibrium unless the
layer is strongly depleted or inverted. Often it is useful to compensate
for the changes in free carriers and space charge by adjusting shallow
donor (acceptor) densities.

**The Lifetime (LT) model** is in general easier to use since it
requires only one input parameter. Lifetimes calculated by eqn. (3) do
not always correspond to DD model simulations due to the approximation
that went into the derivation of eqn. (3); in general lifetimes have to
be chosen somewhat higher. Also, DD models usually calculate lifetimes
that vary over several orders of magnitude within the same material,
which violates the basic assumption of uniform lifetimes.

**Surface recombination** velocity at the front and back contact is
chosen as v~th~, which effectively recombines every minority carrier
that reaches the contacts.

**Tail states** are not incorporated in the baseline cases. They were
found to be negligible contributors to the recombination current unless
their density is several orders of magnitude larger than that of the
mid-gap defect states.

**3. CIGS AND CdTE BASELINES**

The CIGS and CdTe baseline cases are intended to serve as a starting
point for more specific and more complete simulations. They have only
the minimum number of layers necessary to model a typical device and are
free of secondary features not common for these cells. Parameters are
chosen to reflect state-of-the-art experimental devices.

**3.1 CIGS Baseline**

Three layers: n-ZnO transparent contact, n-CdS window, and p-CIGS
absorber. Baseline case parameters are shown in Table I.

The band alignment is chosen to be Type I for the CdS-CIGS interface and
Type II for the ZnO-CdS interface. The offset at the CdS-CIGS interface,
E~C~ = 0.3 eV, is based on theoretical \[5\] and experimental \[6\]
results. Besides the band alignment, device performance is essentially
independent of ZnO parameters.

A single deep acceptor (donor) trap is used for the CdS (CIGS) layer.
The high number of defects in the CdS layer is necessary to generate the
typical superposition failure in CIGS solar cells which is addressed
below. The model shows only small effects of other CdS parameters. Most
CIGS parameters have significant effects on the results and agree on a
qualitative level with the basic diode model. The defect density is
chosen low compared to most experimental devices; typical densities in
the authors' experience are the order of 10^14^ -- 10^15^ cm^-3^. In the
CdS (CIGS) the deep defects are compensated through increased shallow
donor (acceptor) density.

The CIGS mobilities are assigned a factor of three below crystalline
material values. Default values of 0.2m~0~/0.8m~0~ were used for the
effective masses in all layers. Absorption is defined based on
measurements on polycrystalline CdS \[7\] and CIGS \[8\].

Table I: CIGS baseline case. e/h refers to electron/hole properties. ~b~
barrier height (~bn~ = E~C~ -- E~F,~ ~bp~ = E~F~ -- E~V)~, S surface
recombination velocity, W layer width, dielectric constant, mobility,
n/p electron/hole density, E~g~ band gap energy, N~C~ and N~V~ effective
density of states, E~C~ conduction band offset, N~DG(AG)~ acceptor-like
(donor-like) defect density, E~A(D)~ peak energy in, W~G~ distribution
width, capture cross section.

+----------------------+--------+-----+-------+-----+---+---------------+
| ### Gener            |        |     |       |     |   |               |
| al Device Properties |        |     |       |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
|                      |        | **F |       |     | * |               |
|                      |        | ron |       |     | * |               |
|                      |        | t** |       |     | B |               |
|                      |        |     |       |     | a |               |
|                      |        |     |       |     | c |               |
|                      |        |     |       |     | k |               |
|                      |        |     |       |     | * |               |
|                      |        |     |       |     | * |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~b~ \[eV\]           |        | ~   |       |     | ~ |               |
|                      |        | bn~ |       |     | b |               |
|                      |        | =   |       |     | p |               |
|                      |        | 0.0 |       |     | ~ |               |
|                      |        |     |       |     | = |               |
|                      |        |     |       |     | 0 |               |
|                      |        |     |       |     | . |               |
|                      |        |     |       |     | 2 |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| S~e~ \[cm/s\]        |        | 10  |       |     | 1 |               |
|                      |        | ^7^ |       |     | 0 |               |
|                      |        |     |       |     | ^ |               |
|                      |        |     |       |     | 7 |               |
|                      |        |     |       |     | ^ |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| S~h~ \[cm/s\]        |        | 10  |       |     | 1 |               |
|                      |        | ^7^ |       |     | 0 |               |
|                      |        |     |       |     | ^ |               |
|                      |        |     |       |     | 7 |               |
|                      |        |     |       |     | ^ |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| Reflectivity R~f~    |        | 0   |       |     | 0 |               |
| \[1\]                |        | .05 |       |     | . |               |
|                      |        |     |       |     | 8 |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| Layer Properties     |        |     |       |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
|                      | *      |     | **    |     |   | **CIGS**      |
|                      | *ZnO** |     | CdS** |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| W \[nm\]             | 200    |     | 50    |     |   | 3000          |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~0~ \[1\]            | 9      |     | 10    |     |   | 13.6          |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~e~ \[cm^2^/Vs\]     | 100    |     | 100   |     |   | 100           |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~h~ \[cm^2^/Vs\]     | 25     |     | 25    |     |   | 25            |
+----------------------+--------+-----+-------+-----+---+---------------+
| n, p \[cm^-3^\]      | n:     |     | n:    |     |   | p: 2x10^16^   |
|                      | 10^18^ |     | 1     |     |   |               |
|                      |        |     | 0^17^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| E~g~ \[eV\]          | 3.3    |     | 2.4   |     |   | 1.15          |
+----------------------+--------+-----+-------+-----+---+---------------+
| N~C~ \[cm^-3^\]      | 2.2x   |     | 2.2x1 |     |   | 2.2x10^18^    |
|                      | 10^18^ |     | 0^18^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| N~V~ \[cm^-3^\]      | 1.8x   |     | 1.8x1 |     |   | 1.8x10^19^    |
|                      | 10^19^ |     | 0^19^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| E~C~\[eV\]           | \- 0.2 |     |       | 0.3 |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| **Gaussian (midgap)  |        |     |       |     |   |               |
| Defect States**      |        |     |       |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
|                      | *      |     | **    |     |   | **CIGS**      |
|                      | *ZnO** |     | CdS** |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| N~DG~, N~AG~         | D:     |     | A:    |     |   | D: 10^14^     |
| \[cm^-3^\]           | 10^17^ |     | 1     |     |   |               |
|                      |        |     | 0^18^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| E~A~, E~D~ \[eV\]    | midgap |     | m     |     |   | midgap        |
|                      |        |     | idgap |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| W~G~ \[eV\]          | 0.1    |     | 0.1   |     |   | 0.1           |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~e~ \[cm^2^\]        | 1      |     | 10    |     |   | 5x10^-13^     |
|                      | 0^-12^ |     | ^-17^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+
| ~h~ \[cm^2^\]        | 1      |     | 10    |     |   | 10^-15^       |
|                      | 0^-15^ |     | ^-12^ |     |   |               |
+----------------------+--------+-----+-------+-----+---+---------------+

Resulting J-V and QE output (T = 300 K) using the Table I parameters are
shown in Fig. 1 and 2. J~sc~ = 34.6 mA/cm^2^, V~oc~ = 0.64 V, ff =
79.5%, = 17.7%, and diode quality factor A 1.4. Light and dark J-V
curves show a reasonable superposition. The efficiency is close to the
record efficiency cell \[9\], = 18.8%. The cell in reference \[9\] has a
somewhat higher V~oc~ due to band gap grading in the space charge
region. Quantum efficiency (Fig. 2) shows a peak response of (1 -- R~f~)
95% and falls off in the range below 520 nm due to absorption and
recombination in the CdS layer.

![](media/image1.wmf)

> Fig. 1 CIGS baseline: Light and dark J-V

![](media/image2.wmf)

Fig. 2: Quantum efficiency - CIGS and CdTe baseline

**3.2 CIGS - Discussion**

Several issues are frequently addressed with modeling tools. Most
prominent are the explanation of superposition failure \[10-12\] and
absorber grading \[13-15\].

**Superposition.** Fig. 3 shows the conduction band diagram for the
baseline in the dark and under illumination at 0V bias. The choice of
electron affinities generates a barrier at the CdS-CIGS interface; under
illumination this barrier is considerably reduced due to trapping of
holes into the deep acceptor states. The barrier height that is chosen
for the baseline case is not sufficiently high to influence J-V results
at room temperature. However, it has been shown elsewhere \[12\] that an
increased offset, 0.4 eV instead of 0.3 eV, generates the superposition
failure often observed with CIGS cells. This type of non-superposition
can only be described with DD models since it requires charge trapping
in defect states. To a certain extent, manually adjusted free carrier
concentrations, between light and dark simulation, can model trapping
effects in LT models.

![](media/image3.wmf)

Fig. 3: CIGS baseline: Conduction band diagram - 0V Bias

**Absorber Grading.** Preferred software tools are those that allow a
continuous variation of material parameters within a layer. A stepwise
grading has to be done with care since it generates unrealistic
discontinuities in the band structure. Often only the band gap is
adjusted and changes in other material parameters are neglected due to
insufficient experimental data. Numerical simulations have shown that
grading can increase the calculated efficiency by several percent
\[13,14\].

**3.3 CdTe Baseline**

Three layers: n-SnO~2~ transparent conducting oxide (TCO), n-CdS window,
and p-CdTe absorber. Baseline case parameters are listed in Table II.

The SnO~2~ layer can be omitted in some cases, since device performance
is almost independent of all its material parameters. The conduction
bands at the SnO~2~-CdS interface were aligned since there appears to be
no experimental evidence of a detrimental band offset there. For other
TCOs, such as indium tin oxide, ∆E~C~ may have to be considered. For
SnO~2~ and CdS default values of effective masses m^\*^~e~ (m^\*^~h~) =
0.2 (0.8)m~0~ are used.

Most of the limited experimental and theoretical results for ∆E~C~ at
the CdS-CdTe interface lie between zero and -0.3 eV, so it is assumed to
be Type II, ∆E~C~ = -0.1 eV, based on theoretical results \[16\].
Although this particular case is not sensitive to this range of ∆E~C~,
it is probably best to treat ∆E~C~ as a variable.

The model shows only weak dependences on CdS parameters, except for
layer thickness. The absorption curves used were measured on
polycrystalline CdS and CdTe material \[7\].

The CdTe mobilities chosen are approximately half of the crystalline
values and m^\*^~e~ (m^\*^~h~) = 0.1 (0.8)m~0~, determined as an average
from several published sources. A single deep acceptor (donor) trap is
used for the CdS (CdTe) layer.

Table II: CdTe baseline case. Description of the symbols is given in the
caption to Table I.

+----------------------+--------+-----+-------+----+---+---------------+
| ### Gener            |        |     |       |    |   |               |
| al Device Properties |        |     |       |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
|                      |        | **F |       |    | * |               |
|                      |        | ron |       |    | * |               |
|                      |        | t** |       |    | B |               |
|                      |        |     |       |    | a |               |
|                      |        |     |       |    | c |               |
|                      |        |     |       |    | k |               |
|                      |        |     |       |    | * |               |
|                      |        |     |       |    | * |               |
+----------------------+--------+-----+-------+----+---+---------------+
| ~b~ \[eV\]           |        | ~   |       |    | ~ |               |
|                      |        | bn~ |       |    | b |               |
|                      |        | =   |       |    | p |               |
|                      |        | 0.1 |       |    | ~ |               |
|                      |        |     |       |    | = |               |
|                      |        |     |       |    | 0 |               |
|                      |        |     |       |    | . |               |
|                      |        |     |       |    | 4 |               |
+----------------------+--------+-----+-------+----+---+---------------+
| S~e~ \[cm/s\]        |        | 10  |       |    | 1 |               |
|                      |        | ^7^ |       |    | 0 |               |
|                      |        |     |       |    | ^ |               |
|                      |        |     |       |    | 7 |               |
|                      |        |     |       |    | ^ |               |
+----------------------+--------+-----+-------+----+---+---------------+
| S~h~ \[cm/s\]        |        | 10  |       |    | 1 |               |
|                      |        | ^7^ |       |    | 0 |               |
|                      |        |     |       |    | ^ |               |
|                      |        |     |       |    | 7 |               |
|                      |        |     |       |    | ^ |               |
+----------------------+--------+-----+-------+----+---+---------------+
| Reflectivity R~f~    |        | 0.1 |       |    | 0 |               |
| \[1\]                |        |     |       |    | . |               |
|                      |        |     |       |    | 8 |               |
+----------------------+--------+-----+-------+----+---+---------------+
| Layer Properties     |        |     |       |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
|                      | **Sn   |     | **    |    |   | **CdTe**      |
|                      | O~2~** |     | CdS** |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| W \[nm\]             | 500    |     | 25    |    |   | 4000          |
+----------------------+--------+-----+-------+----+---+---------------+
| ~0~ \[1\]            | 9      |     | 10    |    |   | 9.4           |
+----------------------+--------+-----+-------+----+---+---------------+
| ~e~ \[cm^2^/Vs\]     | 100    |     | 100   |    |   | 320           |
+----------------------+--------+-----+-------+----+---+---------------+
| ~h~ \[cm^2^/Vs\]     | 25     |     | 25    |    |   | 40            |
+----------------------+--------+-----+-------+----+---+---------------+
| n, p \[cm^-3^\]      | n:     |     | n:    |    |   | p: 2x10^14^   |
|                      | 10^17^ |     | 1     |    |   |               |
|                      |        |     | 0^17^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| E~g~ \[eV\]          | 3.6    |     | 2.4   |    |   | 1.5           |
+----------------------+--------+-----+-------+----+---+---------------+
| N~C~ \[cm^-3^\]      | 2.2x   |     | 2.2x1 |    |   | 8x10^17^      |
|                      | 10^18^ |     | 0^18^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| N~V~ \[cm^-3^\]      | 1.8x   |     | 1.8x1 |    |   | 1.8x10^19^    |
|                      | 10^19^ |     | 0^19^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| E~C~\[eV\]           | 0      |     |       | -0 |   |               |
|                      |        |     |       | .1 |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| **Gaussian (midgap)  |        |     |       |    |   |               |
| Defect States**      |        |     |       |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
|                      | **Sn   |     | **    |    |   | **CdTe**      |
|                      | O~2~** |     | CdS** |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| N~DG~, N~AG~         | D:     |     | A:    |    |   | D: 2x10^14^   |
| \[cm^-3^\]           | 10^15^ |     | 1     |    |   |               |
|                      |        |     | 0^18^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| E~A~, E~D~ \[eV\]    | midgap |     | m     |    |   | midgap        |
|                      |        |     | idgap |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| W~G~ \[eV\]          | 0.1    |     | 0.1   |    |   | 0.1           |
+----------------------+--------+-----+-------+----+---+---------------+
| ~e~ \[cm^2^\]        | 1      |     | 10    |    |   | 10^-12^       |
|                      | 0^-12^ |     | ^-17^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+
| ~h~ \[cm^2^\]        | 1      |     | 10    |    |   | 10^-15^       |
|                      | 0^-15^ |     | ^-12^ |    |   |               |
+----------------------+--------+-----+-------+----+---+---------------+

The resulting J-V and QE output (T = 300 K) is shown in Figs. 4 and 2.
J~sc~ = 24.6 mA/cm^2^, V~oc~ = 0.87 V, ff = 76%, = 16.4%, and the diode
quality factor A 1.8. The efficiency is equal to record of 16.4% \[17\].
The quantum efficiency shows a peak response of (1 -- R~f~) 90%. The CdS
layer thickness determines the loss below 520 nm; increasing CdS
thickness, will lead to larger QE losses and a reduction of J~sc~ by up
to a few mA/cm^2^.

![](media/image4.wmf)

> Fig. 4: CdTe baseline: Light and dark J-V

**3.4 CdTe Discussion**

**Fig. 5** shows the conduction band at 0 V and 0.8 V forward bias. Due
to the small carrier concentration, the device is heavily depleted at
zero bias and there are strong band structure changes caused by
photogenerated carriers. This type of band deformation can lead to
non-superposition of light and dark J-V curves. Another contribution to
non-superposition is the field aided collection of photo-generated
carriers. Hence in CdTe, non-superposition is caused within the absorber
layer, whereas in CIGS devices it is created in the CdS layer.

![](media/image5.wmf){width="2.734027777777778in" height="1.7625in"}

Fig. 5: CdTe baseline: Conduction band diagram

Two significant CdS/CdTe modeling results are the effects of the back
contact barrier height ~b~ on carrier transport and the transition in
cell behavior from fully to partially depleted CdTe \[18\]. Fig. 6 shows
calculated J-V data for ~b~ = 0.3 and 0.5 eV and two acceptor densities,
exhibiting cross-over and current-limiting (roll-over for V \> V~oc~)
anomalies. Experimental J-V curves often show these anomalies,
especially after elevated-temperature stress. The variation of ~b~ can
explain both cross-over and roll-over; the contact diode has a reversed
polarity from the main diode and limits the hole current flow. Assuming
sufficiently high bulk recombination, the net forward current also
becomes limited above V~oc~. When the CdTe is totally depleted,
roll-over vanishes and V~oc~ is shifted to smaller values with an
increase in cross-over.

A final comment: the CdTe layer is highly compensated and the balance
between large donor and acceptor densities is very critical, while
electronic properties for the defects are not well known, so modeling
must be compared with a wide range of experimental data.

![](media/image6.wmf){width="2.8243055555555556in"
height="1.9861111111111112in"}

Fig. 6: J-V characteristics at 300 K (light: open symbols, dark: filled
symbols, W~CdTe~ = 4 m)

**4. ACKNOWLEDGEMENTS**

The modeling calculation for this work used the AMPS software developed
at the Pennsylvania State University by S. Fonash et al. supported by
the Electric Power Research Institute. This research was supported by
the U.S. National Renewable Energy Laboratory.

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