Temperature-Dependent Open-Circuit Voltage Measurements and
Light-Soaking in\
Hydrogenated Amorphous Silcon Solar Cells

Jianjun Liang,^1^ E. A. Schiff,^1^ S. Guha,^2^ B. Yan,^2^ and J. Yang^2^

^1^Department of Physics, Syracuse University, Syracuse, NY 13244-1130
USA

^2^United Solar Ovonic Corp., Troy, MI 48084 USA

# ABSTRACT

We present temperature-dependent measurements of the open-circuit
voltage *V~OC~*(*T*) in hydrogenated amorphous silicon *pin* solar cells
prepared at United Solar. At room-temperature and above, *V~OC~*
measured using near-solar illumination intensity differs by as much as
0.04 V for the as-deposited and light-soaked states; the values of
*V~OC~* for the two states converge below 250 K. Models for *V~OC~*
based entirely on recombination through deep levels (dangling bonds) do
not account for the convergence effect. The convergence is present in a
model that assumes the recombination traffic in the as-deposited state
involves only bandtails, but which splits the recombination traffic
fairly evenly between bandtails and defects for the light-soaked state
at room-temperature. Recombination mechanisms are important in
understanding light-soaking, and the present results are inconsistent
with models that assume an immediate connection between a recombination
process and defect generation.

# INTRODUCTION

The open-circuit voltage *V~OC~* is often the simplest solar cell
parameter to understand. Experimentally, *V~OC~* is relatively
independent of the thickness of a-Si:H *pin* solar cells. With ideal *p*
and *n* layers, *V~OC~* may be identified with bulk photocarrier
recombination processes in the intrinsic material.

A simple understanding of recombination processes in a-Si:H solar cells
would be valuable for two reasons. First, it would help establish which
materials parameters actually determine the efficiency of working cells.
Second, the metastable degradation of a-Si:H cells under illumination
(the Staebler-Wronski effect) is undoubtedly mediated by photocarrier
recombination -- so correctly identifying the recombination processes
occurring under solar illumination would be crucial to correctly
identifying the microscopic mechanism underlying metastability.

In the present work, we have studied the temperature-dependence of
*V~OC~* under strong illumination. The results exhibit an interesting
"convergence" effect: the differences in *V~OC~* for the as-deposited
and the light-soaked states at higher measurement temperatures
essentially disappear below 250 K. We show that a reasonably simple
"bandtail+defect" recombination model accounts for these *V~OC~*
measurements. In particular, the as-deposited state seems well-described
by valence bandtail recombination (and neglecting defect recombination).
The light-soaked state apparently involves a nearly equal combination of
bandtail and defect recombination.

We do not believe that this picture is a satisfactory model for
photocarrier recombination at very low excitation densities; more than
two decades of research at low excitation densities has revealed
daunting complexities that are certainly not accommodated by this
picture. Our view is that strong illumination strongly simplifies
recombination. Of course it is recombination under strong illumination
that is important in metastability.

We suspect that this near-equality of bandtail and defect recombination
traffic in light-soaked a‑Si:H is a consequence of "self-limitation" to
metastability: as recombination traffic switches to defects from
bandtails, the process which generates defects shuts down. This
viewpoint seems incompatible with a well-known model studied by
Stutzmann, *et al.* \[[^1]\] for the rate of growth of the defect
density *N~d~*(*t*) under illumination. A rough, widely-used empirical
guide is $N_{d}^{3}(t) = 3C_{\text{SW}}G^{2}t$, where *G* is the
photogeneration rate. The older model for the *t*^1/3^ behavior was
based on the assumption that defect recombination was completely
dominant. An alternative, "hydrogen-collision" perspective views the
*t*^1/3^ behavior as due to hydrogen kinetics \[[^2]\]. We propose
elsewhere that a modification of the hydrogen-collision model that
assumes that bandtail recombination is the event responsible for
hydrogen motion is consistent with the *N~d~*(*t*) experiments.

# Temperature-dependent *V~OC~* Measurements

For these experiments, six depositions of *nip* solar cells on stainless
steel substrates were done at United Solar Ovonic Corp.. The *n* and *p*
layers were the same in all depositions; the deposition time for the
intrinsic layer was chosen to give intrinsic layer thicknesses from 185
nm to 893 nm; the cells were not optimized for solar conversion
efficiency, but the individual layers are comparable to those used in
high-efficiency cells. Details of the deposition procedures have been
given elsewhere \[[^3]\]. As-deposited properties of the cells were
measured under a solar simulator. Further studies were done using a 30
mW, 685 nm wavelength near-infrared laser. We chose to use this laser
because its wavelength is absorbed fairly uniformly throughout the
intrinsic layer of the cells, which substantially simplifies modeling of
the measurements. We were able to achieve photocurrent densities in the
cells that were comparable to solar illumination.

The samples were mounted in a thermoelectric cryostat. For modeling of
the temperature-dependence measurements, it is essential to know the
temperature-dependent bandgap *dE~g~*/*dT* of the intrinsic layers. We
measured the temperature-dependent peak of the electroabsorption
spectrum in one of these cells \[[^4]\]. Equating *dE~g~*/*dT* with this
shift, we obtained *dE~g~*/*dT* = ‑4.7×10^‑4^ eV/K, which is comparable
to previous estimates based on interband optical absorption \[[^5]\].

The symbols in Fig. 1 represent the temperature-dependence measurements
of the open-circuit voltage under laser illumination for one cell
L14488-22. The several lines represent analytical models that will be
discussed subsequently along with computer modeling. The measurements
were done on a cell with 893 nm intrinsic layer thickness. In the
as-deposited state, the photocurrent density *J* measured at -2 V and
295 K was 4.7 mA/cm^2^; parameters measured under a solar illuminator
were: *V~OC~* = 0.982 V, *J~SC~* = 14.4 mA/cm^2^, and *P* = 7.9
mW/cm^2^, FF=0.66563, A=0.25 cm^2^. The light-soaked state corresponds
to 176 hours of illumination at open-circuit condition at the same
illumination intensity as the measurements. The sample's exposure to the
laser at each temperature was 9 seconds, so the total exposure time
during the temperature-dependence measurement was about 3 minutes.

Essentially the same results for *V~OC~* were obtained for several cells
spanning our thickness range; *V*~OC~'s very weak thickness dependence
is a well-known aspect of a-Si:H solar cells \[[^6]\]. Comparable
temperature-dependence measurements have been published previously for
the as-deposited state \[[^7],[^8]\]; the convergence of the *V~OC~*
measurements for the as-deposited and light-soaked states at lower
temperatures has not, to our knowledge, been noted before.

# A Shockley-Read Model for *V~OC~* in *pin* Solar Cells

One of the simplest models for the open-circuit voltage of a *pin* solar
cell has not, to the best of our knowledge, been published before. The
calculation serves a useful purpose in introducing a simple approach to
understanding open-circuit voltages based on the "quasi-Fermi levels" of
a homogeneous material, and it also appears to describe a-Si:H solar
cells in some regimes of temperature, light-soaking, and incident flux.

We first calculate the densities of electrons and holes based on the
assumption that electrons and holes recombine through a density *N~d~*
of donor-type deep levels (0/+); the defects are assumed to be neutral
in the dark. This is a version of "Shockley-Read" recombination
\[[^9]\]. Under illumination, with a bulk photogeneration rate *G*, we
write the rate equations governing the densities *n* and *p* of mobile
electrons and holes, respectively:\
$\frac{\text{dn}}{\text{dt}} = G - b_{\text{dn}}\text{nP}$ , (1)
$\frac{\text{dp}}{\text{dt}} = G - b_{\text{dp}}p\left( N_{d} - P \right)$,
(2)\
where *P* is the density of holes occupying the defects under
illumination, and *b~dn~* and *b~dp\ ~*describe the capture of electrons
and holes by positively charged and neutral defects, respectively. We
are neglecting the possibility of re-emission of carriers that have been
captured; the defect thus acts as a recombination center, and not as a
"trap." Charge neutrality requires $p + P = n$. If we assume
*p*\<\<*P\<\<N~d~* , we obtain for the steady-state solutions:\
$b_{\text{dp}}\text{pN}_{d} = b_{\text{dn}}\text{nP} = G$ ,\
$n = P = \sqrt{G/b_{\text{dn}}}$ , and
$p = G/\left( N_{d}b_{\text{dn}} \right)$ . (3)\
The approximation is valid for $G\text{<<}b_{\text{dn}}N_{d}^{2}$.

These solutions can be rewritten in terms of the electron and hole
quasi-Fermi levels, which are defined
$E_{\text{Fn}} \equiv E_{c} + k_{B}T\text{ln}\left( n/N_{c} \right)$ and
$E_{\text{Fp}} \equiv E_{v} - k_{B}T\text{ln}\left( p/N_{v} \right)$ ,
where *N~c~* and *N~v~* are the effective densities of states at the
conduction bandedge *E~c~* and valence bandedge *E~v~* , respectively.
We obtain:\
$E_{\text{Fn}} = E_{c} - \left( k_{B}T/2 \right)\text{ln}\left( b_{\text{dn}}N_{c}^{2}/G \right)$ ,
(4)\
$E_{\text{Fp}} = E_{v} + \left( k_{B}T \right)\text{ln}\left( b_{\text{dp}}N_{v}N_{d}/G \right)$
(5)

For *pin* solar cells with uniform photogeneration and "ideal" *p* and
*n* layers, the open-circuit voltage may be approximated by
$\text{eV}_{\text{oc}} = E_{\text{Fn}} - E_{\text{Fp}}$. We know of no
proofs of the validity of this relationship, which would require a
quantitative definition of "ideal" *p* and *n* layers; the relationship
has been validated by numerical studies in several cases \[[^10]\]. We
obtain:\
$\text{eV}_{\text{oc}} = E_{g} - \left( k_{B}T/2 \right)\text{ln}\left( b_{\text{dn}}N_{c}^{2}/G \right) + k_{B}T\text{ln}\left( b_{\text{dp}}N_{v}N_{d}/G \right)$
(6)\
It is worth noting that the position of the (0/+) level of the defect in
the gap does not appear in these expressions, which simply reflects the
neglect of emission processes.

# Defect Parameters

Table I presents the defect parameters used in the solar cell modeling
in this paper. For the most part these have been taken from Street's
early work \[[^11]\]. Street measured deep-trapping mobility-lifetime
products *μτ~t~* using photocarrier time-of-flight, and deep-level
densities *N~d~* using infrared absorption and electron spin-resonance.
To the best of our knowledge, later measurements (including those by
some of us) do not represent a significant improvement on his estimates
of the product (*μτ~t~*)*N~d\ ~*for these photocarrier capture
processes.

We have used only the parameters for the (0/+) level of the defect, and
we have neglected the *D*^-^ state (and the (0/-) level) of this defect.
This neglect may seem surprising, since it has been established from
transient photocurrent measurements \[Error: Reference source not
found,[^12]\] that electrons are rapidly trapped onto neutral states
*D*^0^ to form negatively charged *D*^-^ states. Many experiments have
explored the density-of-states associated with this (0/-) level.
Essentially all of these measurements have been done at fairly low
photoexcitation levels; among other reasons, experimenters have been
interested in reducing light-soaking during measurements. For
measurements done under strong (nominally solar) illumination, electron
deep-trapping becomes undetectable in a-Si:H \[[^13],[^14]\]. There is
no well-accepted explanation for this quenching of the deep-trapping
process by illumination, although one of us (EAS) favors the "defect
relaxation" concept.

# Discussion: Analytical models and *V~OC~*(*T*) Measurements

Fig. 1 shows calculations for three analytical models along with the
measurements. The "no trap" model assumes bimolecular, interband
recombination \[Error: Reference source not found\]. The Shockley-Read
model uses *N~d~* = 10^16^ cm^-3^ and parameters from Tables I & II. The
valence bandtail model assumes that holes are trapped into an
exponential bandtail, and that electrons recombine with these bandtail
trapped holes \[Error: Reference source not found\]. The modeling
parameters that do not involve defects are summarized in Table II. They
are unchanged from previous work \[[^15]\], which did rely on
temperature-dependent *V~OC~* measurements on similar samples to obtain
*E~g~* and *N~v~*. Quite different parameters have been used by other
modelers \[[^16],[^17]\]; we cannot give a full discussion of our
choices here. In preparing Fig. 1, we assumed that the linear change of
bandgap with temperature persist to 0 K. In reality there is little
temperature-dependence below 150 K.

We have neglected the temperature-dependence of all of these parameters
excepting the bandgap. Because *V~OC~* depends logarithmically on most
of these parameters, this neglect isn't likely to lead to significant
error for the present work. Temperature-dependent parameters may be
important for studies of the power.

While the analytical modeling has a number of deficiencies compared to
full computer calculations that we shall present shortly, it does give
useful insights. In particular, we believe that they largely exclude one
viewpoint on the origins of the light-soaking effect on *V~OC~*. In this
viewpoint, *V~OC~* is dominated by defect recombination in both the
as-deposited and light-soaked states. In Fig. 2, we have regraphed the
measurements of Fig. 1, and shown two calculations of *V~OC~*(*T*) based
on the Shockley-Read model with defect densities chosen to offer
reasonable agreement with the measurements for higher *T*. The
Shockley-Read calculations are essentially parallel on these scales, and
converge at *T* = 0 K.

# Computer Modeling of *V~OC~*(*T*)

Fig. 3 presents a full computer calculation of *V~OC~*(*T*) using the
AMPS-1D computer program \[[^18]\] and the parameters given in Table I
and II; the *p* and *n* layers for the computer calculations were
"ideal," and varying them substantially had no noticeable effect on
*V~OC~*. For these calculations we included the temperature-dependent
absorption coefficient of the samples as well as the
temperature-dependent bandgap; we inferred the absorption coefficient at
685 nm from photocurrent density measurements on samples with five
different thicknesses at ‑2V bias. We estimate that the photogeneration
rate across the intrinsic layer varied about 30% for the 893 nm sample.
The solid line in the figure represents the model incorporating the
valence bandtail, but neglecting defects altogether. The dashed line
represents the model incorporating *N~d~* = 5×10^15^ cm^-3^ defects as
well as the bandtail. For completeness, we note that we placed this
level 1.1 eV below the conduction bandedge, but (as expected) the exact
level position had little effect on the calculation. A similar
"bandtail+defect" computer model was studied previously by some of us to
account for the experimental correlation of the fill-factor for cells
and *V~OC~* as light-soaking proceeds \[[^19]\]. It is worth noting
that, for the bandtail+defect model, the hole quasi-Fermi level varied
about 0.01 eV across the *i*-layer. This result implies that an
analytical version of this model assuming a constant quasi-Fermi level
would have been noticeably different. On the other hand, computer
calculations of the bandtail model and of the Shockley-Read were
consistent with the analytical results.

The experimental results are somewhat lower than the model predictions
at lower temperatures. A very clear saturation of *V~OC~* for
temperatures below 200 K was reported previously, and attributed to the
onset of *V~OC~*-limitation by the *p*-layer \[Error: Reference source
not found\]; we did not attempt to fit our data by modifying the
*p*-layer parameters.

The authors thank Steven Hegedus for discussions and access to
unpublished measurements. This work has been supported through the Thin
Film Photovoltaics Partnership of the National Renewable Energy
Laboratory (NDJ-2-30630-24 at Syracuse and ZDJ-2-30630-19 at United
Solar).

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