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Supplemental Information for Experimental
Demonstration of Enhanced Photon Recycling in
Angle-Restricted GaAs Solar Cells
Emily D. Kosten, Brendan M. Kayes, and Harry A. Atwater
1 The Modified Detailed Balance Model
To model the cells in our experiments, we utilized a modified detailed balance model, which
included various forms of non-radiative recombination in addition to the radiative losses
from the cell. This allows our model to be much more realistic than a traditional idealized
detailed balance, where non-radiative losses are neglected. In addition, we simply input the
measured short-circuit current, to avoid issues with the variability of the solar simulator lamp
spectrum. Thus, the current at a given voltage,
J
(
V
) , in the modified model is expressed
as:
J
(
V
) =
J
sc
0
[
a
(
E
) +
n
2
a
(
E
)]
2
πq
h
3
c
2
E
2
e
(
E
qV
)
/kT
1
dE
qW
(
C
n
n
2
p
+
C
p
p
2
n
)
2
qSp
(1)
where
J
sc
is the measured short-circuit current, and the rest of the terms give the various
sources of loss from the cell. The first loss term includes radiative light emitted from the cell
or absorbed in the back reflector, where
a
(
E
) is the angle-averaged emissivity of the cell, and
a
(
E
) is the angle averaged absorption in the back reflector.
n
is the index of refraction in
GaAs, and is included because light only needs to be emitted into the cell, rather than air, to
1
Electronic
Supplementary
Material
(ESI)
for
Energy
&
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Science.
This
journal
is
©
The
Royal
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of
Chemistry
2014
be absorbed in the back reflector [1, 2].The next terms account for Auger recombination and
surface recombination where
C
n
and
C
p
are the Auger coefficients [3],
W
is the cell thickness,
and
S
is the surface recombination velocity, which we treat as an adjustable parameter.
n
and
p
, the electron and hole concentrations, are assumed to be constant across the cell and
are determined from the assumed base doping, the neutrality condition, the cell voltage, and
the law of mass action [4, 5].
We next develop an expression for
a
(
E
) and
a
(
E
). Because these cell are relatively thick,
we neglect modal structure within the cells, and utilize a multipass approach. For
a
(
E
) we
consider separately light within the fused silica escape cone, and light that lies outside this
escape cone. For light outside the escape cone, we extend Mart ́ı’s approach and imagine
light entering “through” the back reflector and then passing through the cell many times,
being absorbed in both the cell and back reflector [1, 2]. The fraction of light absorbed in
the back reflector is then expressed as:
a
(
E,θ
) =
(1
R
b
)(1
e
2
αW/
cos
θ
)
1
R
b
e
2
αW/
cos
θ
(2)
where
R
b
is the reflectivity of the back reflector,
α
is the absorption coefficient of GaAs,
and
θ
is the angle in GaAs. For light inside the escape cone, we use the same approach,
but consider the reflectivity,
R
c
, and transmissivity,
T
c
, of the cell surface, to find the back
reflector absorption:
a
(
E,θ
) =
(1
R
b
)(1
T
c
e
αW/
cos
θ
R
c
e
2
αW/
cos
θ
)
1
R
b
R
c
e
2
αW/
cos
θ
(3)
Finally, to calculate
a
(
E
) we evaluate
a
(
E,θ
) for all angles, and take an angle average
at each energy. We note that similar expressions have been derived by other authors for a
perfectly absorbing back reflector, and that these expressions are a straightforward extension
of the same approach. Furthermore, these results reduce to the previously derived results
[1, 2].
2
To calculate the emissivity of the cell,
a
(
E
), we use a multipass approach for light within
the fused silica control or substrate. First, we find the fraction of light returned to the cell
as a function of angle in the fused silica,
φ
, and the energy:
F
r
(
E,φ
) =
R
t
T
c
1
R
c
R
t
(4)
where
R
t
is the reflectivity at the top of the fused silica. For most angles,
R
t
is larger for the
angle restrictor than the control, so more light will be returned and less light will ultimately
escape the cell. Since light that is not recycled is ultimately emitted,
a
(
E,θ
) = (1
F
r
)
a
c
T
c
n
2
g
(5)
where we include the dual pass absorption of the cell,
a
c
, the transmissivity of the cell
surface, and the fact that emission occurs into fused silica, with refractive index,
n
g
, rather
than into air. Note that if the fused silica had an ideal AR coating,
F
r
would be zero and
the emissivity would simply be a function of the cell absorption and surface reflectivity as we
expect. Finally, we average the above expression over the angles in fused silica to find
a
(
E
).
(We could also do this calculation considering the angles in air rather than fused silica. While
the result is the same with appropriate accounting of total internal reflection, we present the
equations for fused silica as it is straightforward to generalize when accounting for light lost
from the sides, as discussed below.)
When considering the side loss as in figure 4 of the main manuscript, our simple multipass
expression for
F
r
is insufficient, as it neglects the cell edges. Therefore, we move to a ray-
tracing model, where we incorporate the cell edges, cell mount, measurement stage, and the
substrate geometry. In this ray tracing model, we place a source and receiver on the cell
area, and find the fraction of rays returned to the cell as a function of wavelength and angle
to determine
F
r
. We then proceed with the standard evaluation of
a
(
E
) as above.
Once
a
(
E
) is evaluated for each optical case, we use the measured Jsc and Voc values for
3
the control case to find a surface recombination velocity that describes the cell performance.
Then, we use the fitted value of
S
along with the measured Jsc value and
a
(
E
) for the
angle restrictor to predict the Voc of the cell under angle restriction, as in figure 3 of the
main manuscript. For figure 4 in the main manuscript, we simply include a separate set
of ray trace derived
a
(
E
) values for each optical setup. To determine the range for the
calculation, we use the uncertainties in the Jsc and Voc, as determined from the multiple
trials to determine a range for these values. We then use values for Jsc and Voc at the
edges of the range along such that the value for
S
is maximized or minimized. Finally, we
use these surface recombination values along with the observed temperature uncertainty and
uncertainty in the measured Jsc to determine a range of values for the predicted Voc under
angle restriction.
2 Rugate Angle Restrictor Design
As we noted in the main manuscript, one of the issues with the angle restrictor used in these
experiments is the reduction in current due to reflections near normal incidence of 3-5%.
Furthermore, a large second order reflecting band near 550nm would cause a very significant
current loss if our spectrum were not filtered to only include light with wavelengths longer
than 605nm. Thus, with the simplistic design used in the experiment, we would not expect
any efficiency increase under the full solar spectrum even with a nearly ideal planar cell, as
the current losses are too great. To achieve not only an increase in voltage, but also in overall
performance, these current losses must be addressed. Here, we present a rugate or graded
index design for angle restriction in GaAs, based on reference [6], which eliminates both the
second-order reflecting band and the smaller ripple-type reflections near normal incidence
observed in our experimental design [7, 8, 9]. While our experiment used an angle restrictor
deposited on fused silica and compared to a bare piece of fused silica, here we design an
angle restrictor to perform under glass, as in an installed solar array. Our concept is that
4