Ray optical light trapping in silicon
microwires: exceeding the 2
n
2
intensity
limit
Emily D. Kosten,
1
Emily L. Warren,
2
and Harry A. Atwater
1
,
3
,
∗
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology,
Pasadena, California 91125, USA
2
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
3
Kavli Nanoscience Institute, California Institute of Technology,
Pasadena, California 91125, USA
*haa@caltech.edu
Abstract:
We develop a ray optics model of a silicon wire array geometry
in an attempt to understand the very strong absorption previously observed
experimentally in these arrays. Our model successfully reproduces the
n
2
ergodic limit for wire arrays in free space. Applying this model to a wire
array on a Lambertian back reflector, we find an asymptotic increase in
light trapping for low filling fractions. In this case, the Lambertian back
reflector is acting as a wide acceptance angle concentrator, allowing the
array to exceed the ergodic limit in the ray optics regime. While this leads
to increased power per volume of silicon, it gives reduced power per unit
area of wire array, owing to reduced silicon volume at low filling fractions.
Upon comparison with silicon microwire experimental data, our ray optics
model gives reasonable agreement with large wire arrays (4
μ
m radius), but
poor agreement with small wire arrays (1
μ
m radius). This suggests that the
very strong absorption observed in small wire arrays, which is not observed
in large wire arrays, may be significantly due to wave optical effects.
© 2011 Optical Society of America
OCIS codes:
(080.0080) Geometric optics; (350.6050) Solar energy; (000.6590) Statistical
mechanics; (030.5630) Radiometry; (260.6970) Total internal reflection.
References and links
1. M. Kelzenberg, S. Boettcher, J. Petykiewicz, D. Turner-Evans, M. Putnam, E. Warren, J. Spurgeon, R. Briggs,
N. Lewis, and H. Atwater, “Enhanced absorption and carrier collection in si wire arrays for photovoltaic appli-
cations,” Nat. Mater.
9
, 239–244 (2010).
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, 1082–1087 (2010).
3. L. Tsakalakos, J. Balch, J. Fronheiser, M. Shih, S. LaBoeuf, M. Pietrzykowski, P. Codella, B. Korevaar,
O. Sulima, J. Rand, A. Davuluru, and U. Ropol, “Strong broadband absorption in silicon nanowire arrays with a
large lattice constant for photovoltaic applications,” J. Nanophoton.
1
, 013552 (2007).
4. B. Tian, X. Zheng, T. Kempa, Y. Fang, J. Huang, and C. Lieber, “Coaxial silicon nanowires as solar cells and
nanoelectronic power sources,” Nature
449
, 885–889 (2007).
5. E. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells,” J. Am. Chem. Soc.
130
, 9224–9225
(2008).
6. B. Kayes, H. Atwater, and N. Lewis, “Comparison of the device physics principles of planar and radial p-n
junction nanorod solar cells,” J. Appl. Phys.
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, 114302–114311 (2005).
7. M. Putnam, S. Boettcher, M. Kelzenberg, D. Turner-Evans, J. Spurgeon, E. Warren, R. Briggs, N. Lewis, and
H. Atwater, “Si microwire-array solar cells,” Energy Environ. Sci.
3
, 1037–1041 (2010).
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EXPRESS
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8. L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,”
Nano Lett.
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, 3249–3252 (2007).
9. C. Kenrick, H. Yoon, Y. Yuwen, G. Barber, H. Shen, T. Mallouk, E. Dickey, T. Mayer, and J. Redwing, “Radial
junction silicon wire array solar cells fabricated by gold-catalyzed vapor-liquid-solid growth,” Appl. Phys. Lett.
97
, 143108 (2010).
10. K. Peng and S. Lee, “Silicon nanowires for photovoltaic solar energy conversion,” Adv. Mater.
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, 1–18 (2010).
11. O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, “High performance wire-array silicon
solar cells,” Prog. Photovolt. Res. Appl.
10
, 1002 (2010).
12. J. Zhu, Z. Yu, G. Burkhard, C. Hsu, S. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical
absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett.
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, 279–282 (2009).
13. C. Lin and M. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant
for photovoltaic applications,” Nano Lett.
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, 3249–3252 (2007).
14. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am.
72
(7), 899–907 (1982).
15. M. Putnam, D. Turner-Evans, M. Kelzenberg, S. Boettcher, N. Lewis, and H. Atwater, “10
μ
m minority-carrier
diffusion lengths in si wire synthesized by cu-catalyzed vapor-liquid-solid growth,” Appl. Phys. Lett.
95
, 163116
(2009).
16. M. Born and E. Wolf,
Principles of Optics, 7th Ed.
(Cambridge University Press, 1999).
17. We find our model very slightly exceeds the ergodic limit across all aspect ratios for the smallest filling fraction.
This is observed across aspect ratios, with no trend with increasing aspect ratios. The maximum amount by which
the ergodic limit is exceeded is approximately 1% and is likely due to small inaccuracies in the model.
18. This should not be confused with the areal filling fraction of the wire array. In solar cells, the power can be
calculated by multiplying the short circuit current, the open circuit voltage, and the fill factor, where the fill
factor accounts for the fact that the current-voltage curve is not square in the power-producing region.
19. K. Plass, M. Filler, J. Spurgeon, B. Kayes, S. Maldonado, B. Brunschwig, H. Atwater, and N. Lewis, “Flexible
polymer-embedded si wire arrays,” Adv. Mater.
21
, 325–328 (2009).
20. C. Bohren and D. Huffman,
Absorption and Scattering of Light by Small Particles
(Wiley-VCH, 2004).
1. Introduction
Silicon nanowire and microwire arrays have attracted significant interest as an alternative to
traditional wafer-based technologies for solar cell applications [1–12] . Originally, this interest
stemmed from the device physics advantages of a radial junction, which allows for the decou-
pling of the absorption length from the carrier collection length. In a planar cell, both of these
lengths correspond to the thickness of the cell, and high quality material is necessary so that the
cell can absorb most of the light while successfully collecting the carriers. In contrast, a radial
junction offers the possibility of using lower quality, lower cost materials without sacrificing
performance [5,6]. More recently, such arrays have been found to exhibit significant light trap-
ping and absorption properties [1–3], and this absorption has been modeled using the transfer
matrix formalism in the nanowire regime [8,13].
Enhancing the light trapping and absorption within a solar cell leads to increased produc-
tion of electron hole pairs, and a corresponding increase in short circuit current. Light trapping
schemes, such as texturing, are particularly important in the case of silicon and other low ab-
sorbing materials. For textured planar substrates in the ray optics regime the light trapping limit
under isotropic illumination will be referred to here as the ergodic limit. In this limit, the in-
tensity of light inside the substrate is
n
2
times the intensity of light incident upon the substrate,
or 2
n
2
for the case of a back-reflector, where
n
is the index of refraction for the substrate. This
effect is due to the randomizing effect of the texturing and total internal reflection [14]. Some
very recent experimental results have suggested that nano and microwire arrays can exceed the
ergodic limit [1, 2]. To explore this further, we follow the approach used to derive the ergodic
limit in the planar case to find the expected light trapping and absorption for wires in the ray
optics limit. This allows us to compare to the ergodic limit and consider wires of a different
scale than those considered in previously.
While much of the previous work has been done with nanowires in the subwavelength
regime, far below the ray optics limit, large diameter microwires can be grown by vapor-liquid-
solid (VLS) techniques [1]. Previous device physics modeling suggests that for efficient car-
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rier collection wires should have diameters similar to the minority carrier diffusion length, [6]
and experimental measurements show diffusion lengths for VLS grown microwires of 10 mi-
crons [15]. Because wires with such diameters could approach the ray optics limit for solar
wavelengths, it seems sensible to model these structures in the ray optics regime. In addition,
comparison of the ray optics model with experimental data provides insight into the relative
importance of wave optics effects for wires of various diameters.
We begin by assuming there is no absorption in the wires and examine the case for isotropic
illumination so that we can compare to the ergodic light trapping limit for textured, weakly
absorbing, planar dielectric substrates. To make this comparison, it is also necessary to postu-
late textured surfaces for the wires. We then examine the case of wires on a Lambertian back
reflector, which are illuminated isotropically over the upper half sphere. Finally, we add a weak
absorption term and find the absorption as a function of wavelength and angle of incidence,
allowing us to compare with experimental data.
2. Modeling wire array intensity enhancement under isotropic illumination
We base our model on the principle of detailed balance, as was done to derive the ergodic limit
for textured planar sheets [14]. In detailed balance, the light escaping from the wires is set
equal to the light entering the wires. To illustrate our approach and show proof of concept for
the model, we first imagine a hexagonal array of wires suspended in free space and isotropically
illuminated. Furthermore, we assume that the wire surfaces are roughened such that they act as
Lambertian scatterers. In other words, the brightness of the wire surfaces will be equal regard-
less of the angle of observation [16]. This fully randomizes the light inside the wires in the limit
of low absorption, just as the roughened surfaces of planar solar cells do. The randomization of
light within the wires serves to trap the light inside by total internal reflection.
With these assumptions in mind, we find the governing equation by simply balancing the
inflows and outflows of light within a single wire.
I
inc
2
A
end
̄
T
end
+
I
inc
A
sides
̄
F
=
I
int
2
A
end
̄
T
end
n
2
+
I
int
A
sides
̄
L
n
2
(1)
Above,
I
inc
is the intensity of the incident radiation,
I
int
is the the intensity of light within the
wires,
A
sides
is the area of the wires sides,
A
end
is the area of one wire end, and
n
is the index
of refraction of the wire. In addition,
̄
T
end
is the average transmission factor through the end,
̄
L
is light from the sides which escapes the array, and
̄
F
is the incident light which enters through
the sides.
The terms on the left hand side represent the energy entering the wire array, with the two
terms representing the incident light which enters through the side and tops of the wire respec-
tively. For the top of the wire, the calculation is quite simple because there is no shadowing
or multiple scattering, assuming that the wires are all the same height. Thus, we only need to
average the transmission into the top over the incident angles to find
̄
T
end
. For light entering
through the sides, we take into account transmission into the wire in addition to shadowing and
multiple scattering. Thus, for a given incident angle, we determine
̄
F
, which gives the fraction
of light transmitted through the sides, averaged over the angles of the incident radiation.
On the right hand side, we have the energy outflows. Once again, the outflows from the top
are quite simple, as all light that leaves the top is lost to the array. The factor of 1
/
n
2
,isdue
to total internal reflection of the randomized light inside the wire, as Yablonovitch previously
demonstrated for ergodic structures [14]. Due to the isotropic incident radiation, the averaged
transmission fractor
T
end
is the same for incident and escaping light. For losses through the
sides, much of the emitted light will be transmitted into other wires, and not lost from the array.
Thus, an average loss factor,
̄
L
, is found, which gives the side losses which are not transmitted
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into other wires.
We rearrange the above equation to find the degree of light-trapping, or
I
int
/
I
inc
.
I
int
I
inc
=
n
2
(
2
A
end
̄
T
end
+
A
sides
̄
F
)
2
A
end
̄
T
end
+
A
sides
̄
L
(2)
Note, that in the limit where the area of the sides goes to zero, the light trapping factor is
n
2
,
which reproduces the ergodic limit for a planar textured sheet, isotropically illuminated, as we
expect. If
̄
F
is larger than
̄
L
, the light trapping in this structure could exceed the ergodic limit.
This seems unlikely, however, as time-reversal invariance would suggest that
̄
L
=
̄
F
because
each path into the array much also be an equally efficient path out of the array. Furthermore,
from a thermodynamics perspective, we expect that the light trapping in this structure should be
exactly
n
2
. This is because the equipartition theorem states that all the states or modes should
be equally occupied in thermodynamic equilibrium, and the density of states is
n
3
the of states
in free space. (When calculating the intensity, it is necessary to multiply by the group velocity
which goes as 1
/
n
, such that the intensity is increased by
n
2
) [14]. Thus, this case will allow us
to assess the accuracy of the model and the assumptions necessary to simplify the calculation.
Fig. 1. (a) Schematic of the wire array for isotropic illumination. The blue wires illustrate
how light escaping from the side of a wire impinges on a neighboring wire a given distance
away. The orange wires illustrate how the sides of the wires are shadowed by neighboring
wires for a given distance and angle of incidence. (b) A top-down view of the wire array
illustrates the radial escape approximation. The arrows show the directions of light escape
being considered, and the yellow areas give the in-plane angle subtended by the neighbor-
ing wires, with the distinct shades indicating neighboring wires at two distinct distances.
The wires farther away will have greater loss associated than the closer wires.
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Averaging over all solid angles, with an appropriate intensity weighting gives,
̄
T
end
:
̄
T
end
=
∫
2
π
0
∫
π
/
2
0
T
(
φ
)
cos
(
φ
)
sin
(
φ
)
d
φ
d
θ
∫
2
π
0
∫
π
/
2
0
cos
(
φ
)
sin
(
φ
)
d
φ
d
θ
=
∫
2
π
0
∫
π
/
2
0
T
n
cos
2
(
φ
)
sin
(
φ
)
d
φ
d
θ
∫
2
π
0
∫
π
/
2
0
cos
(
φ
)
sin
(
φ
)
d
φ
d
θ
=
2
3
T
n
(3)
where
φ
is the angle of incidence and
T
n
is the transmission factor at normal incidence, and
where we have used the transmission factor associated with a Lambertian surface (
T
n
cos
(
φ
)
)
[16].
To calculate
̄
L
, we determine the fraction of light,
g
, escaping from the sides of a given wire
that impinges on neighboring wires. Then, we determine the transmission into those neighbor-
ing wires and the effect of multiple scattering from neighboring wires. To find
g
,weinvoke
a radial escape approximation where we treat each wire as if it were a line extending upward
from the plane of the array. This approximation will be more accurate for low filling fraction
arrays, because with greater distance between the wires means that neighboring wires more
closely approximate line sources. The radial escape approximation serves to significantly sim-
plify the treatment of the in-plane shadowing. With this assumption, we only need to calculate
the portion of the in-plane angle that is subtended by wires at a given distance, and the losses
associated with each distance in order to find
g
. As Fig. 1(b) illustrates, the in-plane angle
subtended by neighboring wires at a given distance is calculated geometrically.
The fraction of light that impinges on a wire a given distance away,
f
(
h
)
is easily calculated
from geometrical arguments and the properties of Lambertian surfaces, as Fig. 1(a) illustrates.
To simplify the calculation we ignore the increase in wire to wire distance as the wires curve
away from each other. As before, this approximation will be more accurate for lower filling
fractions, where the wires are farther apart and this effect will be smaller.
f
(
h
)=
∫
θ
T
−
θ
B
cos
(
θ
)
d
θ
∫
π
/
2
−
π
/
2
cos
(
θ
)
d
θ
=
sin
(
θ
T
)+
sin
(
θ
B
)
2
(4)
To find
g
(
d
)
, we integrate
f
(
h
)
over the height of the wire and normalize.
g
(
d
)=
∫
l
0
sin
(
θ
T
)+
sin
(
θ
B
)
dh
2
l
=
√
l
2
+
d
2
−
d
l
(5)
Then,
g
is an average of
g
(
d
)
weighted by the angles subtended at each distance.
Naturally, not all of the light which strikes a neighboring wire will be transmitted into the
wire. As before, we calculate a transmission factor as a function of distance,
T
int
(
d
)
and take a
weighted average to find the overall internal transmission factor,
T
int
. Here, however, we must
account for the curvature of the wire because this significantly affects the angle the transmitted
light makes with the wire surface. Assuming equal brightness for the allowed in plane and out
of plane angles, the expression for
T
int
(
d
)
is:
T
int
(
d
)=
∫
l
0
∫
θ
T
−
θ
B
∫
α
2
−
α
1
T
n
cos
2
(
φ
)
d
α
d
θ
dl
∫
l
0
∫
θ
T
−
θ
B
∫
α
2
−
α
1
cos
(
φ
)
d
α
d
θ
dl
(6)
where the
θ
’s give the bounds of the out of plane angles, the
α
’s the bounds of the in-plane
angles, and
φ
is the overall angle made with the wire.
To find
̄
L
we sum the losses in each pass through the wire array. For the first pass through the
wire array, 1
−
g
of light which left the wire side is lost, because it does not impinge on any of
the other wires, and escapes. This is multiplied by
̄
T
end
because the light must leave the side of
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the wire before it can escape the array. On the second pass, the losses,
L
2
, are as follows:
L
2
=
̄
T
end
g
(
1
−
T
int
)(
1
−
g
)
(7)
This assumes that the reflected light has a uniform height distribution. In reality, more of the
light emitted from the sides of the wires will impinge on the middle of the neighboring wire
than either end, owing to the Lambertian distribution of light from the emitting wire. Thus,
this assumption will overestimate the losses on succeeding passes through the array, but greatly
reduces the computational intensity of the calculation by allowing for a generalization of the
losses on the
i
th pass through the array as:
L
i
=
̄
T
end
(
g
(
1
−
T
int
))
i
−
1
(
1
−
g
)
(8)
This can easily be summed to give
̄
L
.
̄
L
=
̄
T
end
(
1
−
g
)
∞
∑
n
=
0
(
g
(
1
−
T
int
))
n
=
1
−
g
1
−
g
(
1
−
T
int
)
(9)
In calculating
̄
F
the main additional phenomenon we must address is shadowing. As Fig. 1(a)
illustrates, the shadowing fraction,
u
, as a function of wire to wire distance and angle of inci-
dence is:
u
(
d
,
β
)=
l
−
s
l
=
d
cot
(
β
)
l
(10)
We then take a weighted average over the angle subtended at each distance to find
u
(
β
)
, and
also find the transmission factor for the incoming light as a function of
β
by averaging over all
in-plane angles
α
.
T
0
(
β
)=
∫
π
/
2
−
π
/
2
T
n
cos
2
(
φ
)
d
α
∫
π
/
2
−
π
/
2
cos
(
φ
)
d
α
(11)
As before
φ
is the overall angle the incoming ray makes with the wire, which will depend
on both
α
and
β
. Finally, we modify the multiple scattering model because light will only be
reflected off the unshadowed portion of the wire, which will vary as a function of
β
. For the
losses on the first pass through the array:
L
1
(
β
)=
u
(
β
)(
1
−
T
0
(
β
))(
1
−
g
1
(
β
))
(12)
For
i
>
1,
L
i
(
β
)=(
1
−
g
)
u
(
β
)(
1
−
T
0
(
β
))
g
1
(
β
)(
1
−
T
1
(
β
))[
g
(
1
−
T
int
)]
i
−
2
(13)
where
L
i
gives the losses on the
i
th bounce, as before, and
T
1
and
g
1
give the transmission and
impingement factors associated with the light reflected from the unshadowed portion of the
wires. Summing to find the total losses:
L
t
(
β
)=
u
(
β
)(
1
−
T
0
(
β
))
(
1
−
g
1
(
β
)+
(
1
−
g
)
g
1
(
β
)(
1
−
T
1
(
β
))
1
−
g
(
1
−
T
int
)
)
(14)
Thus, for a given angle,
β
, the amount of light which is transmitted into the wires,
F
(
β
)
,
accounting for multiple scattering and shadowing is:
F
(
β
)=
u
(
β
)
−
L
t
(
β
)
(15)
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Averaging over all the angles of incidence gives
̄
F
.
̄
F
=
∫
2
π
0
∫
π
/
2
0
F
(
β
)
sin
2
(
β
)
d
β
d
η
∫
2
π
0
∫
π
/
2
0
sin
2
(
β
)
d
β
d
η
(16)
Above,
η
is the polar angle, sin
(
β
)
d
β
d
η
is the differential solid angle, and the additional factor
of sine gives the change in intensity with angle of incidence.
3. Results for wire array intensity enhancement under isotropic illumination
Inserting the expressions found above into Eq. (2), we calculate the light trapping factor across
a range of areal filling fractions, the fraction of the array covered by wires, for various wire
aspect ratios. The results are given in Fig. 2 and are indicated by the curves labeled “no back
reflector”. For very large filling fractions we approach the ergodic limit, because the terms
involving the wire sides become very small. We also reproduce the ergodic for very low filling
fractions, where the radial escape approximation will be most accurate [17]. In between the
results fall below the ergodic limit, likely because the side loss factor,
̄
L
, is overestimated in
the radial escape approximation. Because we expect thermodynamically that the result should
be
n
2
, this suggests that our approximations are reasonable, especially for low filling fractions,
which are more likely to be of experimental interest. We also note that our results are closer
to the ergodic limit for smaller aspect ratios. This is likely because the terms involving the
wire sides are relatively smaller, and thus inaccuracies in those terms, such as overestimating
̄
L
,
will have less impact. Thus, our approach reasonably approximates the result we expect from
thermodynamics, and the inaccuracies introduced by the radial escape approximation are well
understood.
4. Modeling wire array intensity enhancement with a Lambertian back reflector
We now investigate the effect of having a Lambertian back reflector with isotropic illumination
in the upper half-sphere. In this case, no light will enter or escape through the bottom ends
of the wires, which are covered by the back-reflector, and light that strikes the reflector will
be scattered. In the planar case, the ergodic light trapping limit for such a geometry is 2
n
2
,
owing to the back reflector. Additionally, it seems that this geometry would give optimal scatte-
ring, as can be understood by basic physical arguments. Experimentally, it has been found that
placing scatterers within the wire array can, in combination with a back-reflector, improve the
performance of the array [1, 7]. This is because scatterers prevent light which is at normal or
nearly normal incidence from going between the wires and bouncing off a planar back-reflector
and out of the array. Imagine that we could place scatterers at any height level within the wire
array. The light that scatters upward from the scatterers near the bottom of the array will be
more likely to impinge on a wire, as Fig. 3 shows. For optimal scattering, then, the scatterers
should be placed at the bottom of the array. Since a Lambertian back reflector is similar to
placing scatterers on a planar back reflector, this geometry allows us to investigate an optimal
scattering regime as well as providing an interesting comparison to the planar case.
The governing equation for this case once again relies on detailed balance, as shown below.
I
inc
A
end
̄
T
end
+
I
inc
A
sides
̄
F
+
I
inc
A
re f l
̄
R
=
I
int
A
end
̄
T
end
2
n
2
+
I
int
A
sides
̄
L
2
n
2
(17)
The terms on the left give the light entering a wire, and the terms on the right give the amount
of light escaping. Note that a factor of 1
/
2
n
2
replaces the 1
/
n
2
factor because the back reflector
doubles the intensity of the light within the wires [14]. In addition,
̄
L
and
̄
F
are replaced with
̄
L
#138520
- $15.00
USD
Received
22
Nov
2010;
accepted
25
Jan
2011;
published
4 Feb
2011
(C)
2011
OSA
14
February
2011
/ Vol.
19,
No.
4 / OPTICS
EXPRESS
3322