PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONS
Prog. Photovolt: Res. Appl. 2012; 20:51–61
Published online 23 March 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/pip.1109
RESEARCH ARTICLE
One-dimensional photogeneration profiles in silicon
solar cells with pyramidal texture
Simeon C. Baker-Finch* and Keith R. McIntosh
Centre for Sustainable Energy Systems, Australian National University, Canberra, ACT 0200, Australia
ABSTRACT
The key metric of surface texturing is the short-circuit current Jsc. It depends on front surface transmittance, light trapping
and the spatial profiles of photogeneration G and collection efficiency hc. To take advantage of a one-dimensional profile of
hc(z), where z is the shortest distance to the p–n junction, we determine G(z) via ray tracing. This permits rigorous optical
assessment of common pyramidal textures for various cell designs. When z is small, G(z) is largest beneath regular inverted
pyramids, upright pyramids (regular or random) and planar surfaces, respectively. This higher G(z) results in superior
collection of generated carriers in front-junction cells. In simulations of a conventional screen-print cell, 92.0% of
generated carriers are collected for inverted pyramids, compared to 91.4% for upright pyramids, and 90.0% for a planar
surface. Higher efficiency and rear junction devices are analysed in the paper. Despite differences in G(z) beneath textures,
inverted pyramids achieve the highest Jsc for all cell designs examined (marginally so for high-efficiency rear-contact cells)
due to superior front surface transmittance and light trapping. We assess a common one-dimensional model for
photogeneration beneath textured surfaces. This model underestimates G(z) when z is small, and overestimates G(z)
when z is large. As a result, the generation current determined is inaccurate for thin substrates. It can be computed to within
3% error for 250 mm thick substrates. However, errors in G(z) can lead to 7.5% inaccuracy in calculations of Jsc. Errors are
largest for lower efficiency designs, in which collection efficiency varies through the substrate. Copyright # 2011 John
Wiley & Sons, Ltd.
KEYWORDS
generation profile; optics; ray tracing; spatial collection efficiency; texture
*Correspondence
Simeon C. Baker-Finch, Centre for Sustainable Energy Systems, Australian National University, Canberra, ACT 0200, Australia.
E-mail: [email protected]
Received 2 November 2010; Revised 30 January 2011
1. INTRODUCTION
When the solar cell front surface is textured, rather than
planar, the light absorbing capability of the solar cell is
increased in two ways: (i) the front surface is less reflective
and (ii) the path length of light within the absorbing
substrate is increased. The first property (denoted a in
Figure 1) is achieved by designing the texture so as to
ensure that every incident ray of light meets the absorber
interface at least twice. The second property results
from the tendency of the texture to both lengthen the first
pass of the transmitted light through the substrate (see b
in Figure 1) and encourage multiple internal reflections
(denoted c and d in Figure 1).
Most commercial monocrystalline silicon solar cells
feature a front surface texture consisting of a random array
of pyramids of height between 1 and 10 mm. The highest
efficiency cells manufactured to date feature a regular
Copyright ß 2011 John Wiley & Sons, Ltd.
inverted pyramid texture [1,2]. In both cases, the texture is
achieved by the exposure of a planar {100} surface to an
anisotropic etch which reveals pyramidal structures having
{111} facets. Note that in this work, we restrict our analysis
to pyramidal features of sufficiently large size so as to
render diffractive effects negligible [3,4].
In a popular approach to the analysis of textured
surfaces, the value of the optical path length Z of light (or a
similar criterion such as the number of passes across
the substrate) is used as a figure of merit [5–8]. This
methodology is sufficiently general to be applied to a wide
range of structures, and it is particularly useful when
various texture morphologies are to be compared in terms
of their light trapping capabilities. When coupled with a
separate determination of the front surface transmittance, a
path length analysis can be used to calculate the sum total
photogenerated current density in the active region JG.
However, this approach offers limited insight into the
51
One-dimensional photogeneration profiles
S. C. Baker-Finch and K. R. McIntosh
Figure 1. Front surface texturing increases front surface
transmittance, and enhances the light trapping capabilities of
the solar cell.
spatial distribution of the photogeneration throughout
this active region. Hence, the impact of the textured surface
on Jsc is not easily determined.
Indeed, the critical performance indicator that is the
short circuit current density Jsc depends upon the spatial
profiles of the photogeneration G and the collection
efficiency of generated carriers hc according to
ZZZ
Gðx; y; zÞhc ðx; y; zÞdV
(1)
Jsc ¼ q
volume
where dV is a volume element and the integral is taken
over the 3-dimensional region occupied by the cell
substrate. (For brevity, we limit the definition of hc to
short-circuit conditions, although the collection of lightgenerated carriers can also depend on the cell voltage when
recombination in the device is injection-dependent.)
When modelling a solar cell in one dimension (which
is a very common and useful approximation), it is
necessary to convert Equation (1) into a one-dimensional
integral. An obvious choice of spatial variable is z, but in
this work, we instead choose z, where z denotes the shortest
distance between a point in 3-dimensional space and the
front surface. Figure 2 illustrates the difference between z
and z, in which the dashed blue lines represent z contours
and the solid grey lines represent z contours.
The reason for choosing z as the spatial variable is that
hc depends most strongly on the distance to the p–n
junction (which tends to conform to the surface geometry),
rather than z. Hence, in most cases, G(z)hc(z) is a more
accurate representation of a textured solar cell in onedimension than G(z)hc(z).
In this work, we determine G(z) within silicon waferbased solar cells with four distinct surface textures:
regular upright pyramids, regular inverted pyramids,
random upright pyramids and grooves. The function
G(z) is coupled with an appropriate one-dimensional
device simulation (namely PC1D [9], which incorporates a
model for hc(z)) to calculate Jsc for each texture and a range
of cell designs. This approach is the most suitable means
to determine the precise impact of front surface texture
upon device performance.
Both surface texture and internal reflections tend
to complicate the trajectory of light rays through the
substrate, and hence render the determination of G(z)
52
Figure 2. Two-dimensional representation of contours of constant values of the variables z (solid grey lines) and z (dashed blue
lines) beneath (a) upright pyramid texture and (b) inverted pyramid texture. Also shown are typical generation profiles plotted
against z (top) and z (bottom). As discussed in the text, G(z) is
characterized by a kink resulting from the slow increase in cell
volume at low values of z.
difficult. In this work, we employ Monte Carlo ray tracing
(a technique that is widely used in the photovoltaics
research community [10–12]) to aid in the computation of
G(z). Using a previous application [10], it is possible to
export a relationship G(z) (where constant contours of the
spatial variable z are shown as solid lines in Figure 2).
However, since the cumulative volume of the simulated
region depends strongly upon z for z h (where h is the
pyramid height), the available generation profile displays a
characteristic kink at z h (see Figure 2). Furthermore, as
mentioned earlier, z, rather than z, is the most relevant
parameter for one-dimensional analyses. In a second
application, it is possible to determine G(z) for a periodic
groove texture [12]. Zechner et al. [13] employ a combined
ray tracing and path classification technique to determine
accurate 2- and 3-dimensional generation profiles throughout V-grooved solar cells with one or both sides textured.
Despite the significant role played by ray tracing research
in the improvement of silicon solar cell optical design, G(z)
beneath a range of pyramidally textured surfaces has, to
our knowledge, never been presented.
A second approach to deal with the dual complications
of texture and internal reflections is to approximate G(z)
with an analytical function. A simple model for G(z)was
proposed by Basore [14,15] and extended by Brendel et al.
[16]. This model and variations are applied in a range of
IQE and hc analyses (see for example References [17–21]).
However, as suggested by both Basore [15] and
Brendel et al. [16], the model provides a relatively poor
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
S. C. Baker-Finch and K. R. McIntosh
One-dimensional photogeneration profiles
approximation to the generation profile within the pyramid
region. Via ray tracing simulation, we quantify this error,
and assess its impact on the accuracy of Jsc determined by
one-dimensional device simulation.
2. RAY TRACING TO DETERMINE
G(z)
We employ a 3-dimensional polarization ray tracing
technique [22,23] coupled with a Monte Carlo approach
to optical interactions in order to determine the generation
profiles beneath a range of surface textures that are
common on monocrystalline silicon solar cells. In all
modelling, we assume that the bulk of the cell consists of
intrinsic silicon (having the refractive index and extinction
coefficient given by Green and Keevers [24]), and that no
free-carrier absorption takes place.
2.1. Impact of texture morphology
In Figure 3, the generation profile G(z) is plotted for a
range of pyramidal textures under the AM1-5g spectrum.
In each case, the modelled structure consists of a silicon
substrate of 250 mm thickness, coated with an optimally
thick layer of amorphous SiNx in air (refractive index and
thickness as per Reference [23]). The feature height of
each regular texture is 10 mm, and there is no flat area
present between features. Each of the randomly distributed
pyramids has a height of between 5 and 10 mm, where
the pyramid heights vary randomly, following a
uniform distribution within this range. The model used
to approximate the random upright pyramid morphology
is based on that discussed by Rodriguez et al. [25], and
is described in detail (and referred to as the morphology
in which pyramids are allowed to ‘overlap’) in Reference
[23].
The profiles shown in Figure 3 take into account
internal reflections and successive traversals of the
substrate by each light ray. We assume unity, specular,
internal reflectance at the rear surface. These assumptions
are reasonably consistent with the optical behaviour of
the oxidized, aluminium-coated rear surface of a high
efficiency silicon solar cell: for the predominant angles
of incidence, at least 95% of rays are reflected when
the oxide thickness is more than 100 nm [26]. This is
consistent with Kray et al. [27], who determined a value of
96% for the internal reflectance of such a surface when
examining a 180 mm thick substrate. We add that high
efficiency cell designs to date feature a planar rear surface
[1,2,28]. Moreover, any variation in rear surface properties
is relatively unimportant in this work, in which the critical
outcomes arise from a comparison between textures, rather
than from absolute values. In any case, for a cell thickness
of 250 mm, the maximum overestimate in photogenerated
current incurred by assuming unity rear reflectance (rather
than 96%) is 0.2 mA cm2.
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
Figure 3. G(z) beneath various surface textures determined by
ray tracing.
When compared to an equivalent structure with a planar
front surface, all textures increase the density of carrier
generation in the several microns of the substrate closest
to the front surface. This effect, illustrated in Figure 3, is
attributed to the reduction in front surface reflectance
due to the multiple meetings of the ray with the absorber
(see a in Figure 1), as well as the tendency of the texture to
refract rays away from the local surface normal (see b in
Figure 1). A regular array of inverted pyramids causes
the largest increase in G within the near-surface region.
For z < 3 mm, the generation rate beneath this texture is
higher than the rate beneath other morphologies. This is
attributable to the superior transmittance of the inverted
pyramids as well as to the refraction of rays into oblique
angles of traversal. At z ¼ 35 mm, G beneath this
morphology decreases steeply because, on average, for
a given interval of distance along the ray trajectory, z
increases by a larger amount when that interval is in the
‘bulk region’ of the cell than in the region of the pyramid.
The location of this effect is predicted by feature size, and
is discussed below. An equivalently rapid change in G is
not seen for any other front surface morphology.
53
One-dimensional photogeneration profiles
S. C. Baker-Finch and K. R. McIntosh
The generation profile is perhaps more easily assessed
when represented as the cumulative generation current
density
JG;cum ¼ q
Z
z
Gðz0 Þdz0
(2)
0
In Figure 4, we plot JG,cum for each texture morphology.
For most z, the regular upright pyramids, random upright
pyramids and grooves exhibit similar JG,cum. For all
textured surfaces, at least 31% of the total generation
current results from generation within the first micron
below the front surface (a typical emitter depth). The
proportion of current generated there is highest (37%)
beneath a regular array of inverted pyramids. Importantly,
the considerable generation within this region of high
recombination activity highlights the need for optimization
of front side diffusions and passivation, and indicates
the likely magnitude of advantages to be attained by
selective emitter designs.
The value of JG,cum at z ¼ 5 mm provides the clearest
elucidation of the impact of the various surface textures
upon the generation profiles; whilst 75% of current generation occurs within the first 5 mm of substrate below a
regular array of inverted pyramids, the proportion is only
64–65% for the other textures, and just 56% when the front
surface is planar. For a front-contact cell, this near surface
region is one of particular importance because it contains
the collecting p–n junction and because the surface is
heavily doped and is always a source of significant
recombination. Note in Figure 4 that with an inverted
pyramid texture, increases in substrate thickness offer
diminishing returns in regions beyond the characteristic
kink at z ¼ 5 mm. This particular observation indicates that
the inverted pyramid morphology would be particularly
preferable for very thin front-contact silicon solar cells.
Figure 4. JG,cum (z) beneath various surface textures determined by ray tracing.
54
2.2. Impact of feature size
The feature size of a regular surface texture is
readily controlled during manufacture (see, for example,
Reference [29]). Although variations in light trapping
capability resulting from favourable ratios of feature size
and substrate thickness (as described in Reference [6])
cause only negligible variation in the generation profiles
for the 250 mm thick substrates observed in this work,
the variation of texture feature size h does cause minor
perturbations in G(z) and JG,cum (z), particularly for
inverted pyramids.
Figure 5 plots JG,cum (z) for (a) upright and (b) inverted
regular pyramids. A slight dependence on feature size is
noticeable for regular upright pyramids, and similar results
were observed for a random array of upright pyramids and
for a groove texture. A more significant dependence on
feature size is observed for the inverted pyramids. This
Figure 5. JG,cum (z) beneath (a) regular upright pyramid texture
and (b) regular inverted pyramid texture. Derived from G(z) for
each case determined by ray tracing. Feature heights of 5, 10 and
20 mm are plotted.
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
S. C. Baker-Finch and K. R. McIntosh
One-dimensional photogeneration profiles
effect is explained by the relationship between z and the
distance travelled by an average ray in the substrate
beneath a front surface textured with inverted pyramids: a
kink in the profile of JG,cum at z h=2 is caused by the
tendency of the ray to travel for a distance within close
proximity (z < h=2) of a front surface facet, before
continuing along a trajectory which takes it quickly away
from the front surface.
texture that achieves the highest Jsc. The random array of
upright pyramids outperforms the regular array of upright
pyramids, with groove-textured cells achieving slightly
lower values of Jsc. The reasons for these differences are
now discussed with the aid of Table III, which assesses the
textures in terms of three losses: (i) front surface
transmission, (ii) light-trapping (iii) and carrier collection
in the short-circuit condition. They are based on the
incident photon current Ji between 300 and 1200 nm in the
AM1-5g spectrum.
3. TEXTURE CHOICE FOR CELL
DESIGN
3.1. Front surface transmission loss
As stated in Equation 1, the cell short circuit current Jsc
depends on both generation profile and spatial collection
efficiency. In the following, we couple the function G(z)
with a function hc(z) that has been calculated within
PC1D [9] in order to model the impact of texture
morphology upon Jsc attained by each of three typical
solar cell structures. The parameters used to define a high
efficiency front junction cell (HE FC), a high efficiency
rear junction cell (HE RC) and an industry-standard
screen-printed cell (SP FC) are given in Table I.
We use the simulations to elucidate compatibilities
between cell design and the front surface morphology.
In particular, if a relatively large proportion of the total
photogenerated current JG is retained in Jsc, we conclude
that the spatial profile of the generation is well matched to
the spatial collection profile hc.
For each cell design and front surface texture, the
modelled Jsc is given in Table II. Regardless of cell type,
a regular array of inverted pyramids is the front surface
The front surface transmittance TFS ðlÞ was determined for
each morphology and antireflection coating in accordance
with the method outlined in Reference [23].
In Table III, the current lost due to front surface reflection
and antireflection coating absorption is given by Ji–JT.
This loss is, as expected, lower for textured surfaces.
The spectrum-weighted transmittance of the front surface
increases from around 89% for planar front surfaces to
between 96.7 and 97.1% for textured equivalents. Lowest
transmission loss is achieved by a regular array of inverted
pyramids. More details can be found in Reference [23].
3.2. Light trapping loss
Losses due to imperfect light trapping properties can be
evaluated by the difference between current transmitted
and generated JT–JG. When assessed in this way, the light
Table I. PC1D parameters defining the three cell designs studied in this work. Each cell has an area of 1 cm2. All other parameters
(carrier mobilities, Auger parameters, etc.) are set to PC1D defaults for crystalline silicon.
Device
Cell design
HE FC
HE RC
SP FC
Bulk
t (mm)
Emitter contact (V)
r (V cm)
tn ¼ tp (ms)
250
250
250
0.1
0.1
1
1 ( p-type)
5 (n-type)
1 ( p-type)
5000
5000
50
Front
Cell design
HE FC
HE RC
SP FC
rsheet (V/sq)
Depth (mm)
Profile
Sn ¼ Sp (cm s1)
150
150
50
0.7
0.7
0.8
erfc
erfc
erfc
500
5000
50000
Rear
Cell design
HE FC
HE RC
SP FC
rsheet (V/sq)
Depth (nm)
Profile
Sn ¼ Sp (cm s1)
—
150
—
—
0.7
—
—
erfc
—
50
50
50000
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
55
One-dimensional photogeneration profiles
S. C. Baker-Finch and K. R. McIntosh
Table II. Results of one-dimensional cell simulations with ray traced generation profiles as inputs. Error is determined by performing
simulations with generation profiles above and below the mean profile by the 95% confidence interval of that profile.
Short circuit current Jsc (mA cm2)
Texture morphology
Planar
Regular upright
Regular inverted
Random upright
Groove
HE FC
HE RC
SP FC
37.45
41.35 0.16
41.84 0.16
41.67 0.16
41.07 0.15
37.03
40.83 0.16
41.28 0.16
41.15 0.16
40.51 0.14
33.84
37.98 0.14
38.63 0.10
38.19 0.18
37.97 0.14
trapping capacity of the various pyramidal textures under
observation is very similar. Very small advantages appear
to be enjoyed when the front surface features a regular
array of inverted pyramids, which traps 92.9% of
transmitted light. The advantages of this formation over
a regular array of upright pyramids (92.1% trapping) stem
from the tendency of the inverted pyramids to cause light to
cross the substrate at a more oblique angle, rather than
improved internal reflectance. The random array of upright
pyramids (92.8%) displays similar light trapping capacity
to the inverted pyramid morphology. The poor performance of the groove texture (91.5%) derives from its
tendency to scatter light into only two dimensions [8];
recall that the rear surface is planar.
The poor light trapping performance of planar structures
in general is screened by the large substrate thickness used
for these simulations. As well, the poor transmittance of
planar surfaces in the long-wavelength region limits the
potential for a high JG and simultaneously increases the light
trapping capacity of the morphology as quantified by JT JG.
In any case, planar surfaces (91.2% trapping) offer slightly
decreased performance compared with textured surfaces.
We stress that in this work, the light trapping capacity is
defined as equivalent for all cell designs—even for the
simulated ‘industry standard’ cell, rear internal reflectance
is assumed to be unity. Thus, the magnitude of JG may
be overestimated by up to 2.5 mA cm2 for this cell
type.
3.3. Collection loss
We classify the photogenerated current that is not collected
at the p–n junction as a ‘collection loss’. The magnitude of
this loss (JGJsc) indicates the capacity of a given cell
design to convert a particular spatial distribution of
photogenerated carriers into external current. Thus, a low
collection loss implies that the surface texture drives
generation in favourable regions of the cell. This loss is
summarized for the various front surface textures and cell
designs in Table III. In Table IV, its value is presented as a
percentage of JG.
For high efficiency cell designs, very little of the
generated current is lost to recombination. The poorer bulk
quality and a lack of surface passivation results in larger
collection loss for the SP FC cell for all textures.
As assessed by the collection loss metric, the regular
inverted pyramid and groove morphologies are best suited
to application in an SP FC cell. For this design, it is critical
that a large proportion of carriers are generated near the
junction. For both front contact designs (HE FC and SP
FC), the groove texture drives generation to occur at the
most favourable locations, as attested by the low collection
losses. On the other hand, examining the simulation results
for the HE RC cell, we find that the planar and regular
upright pyramid morphologies are most suitable.
The definition of the collection loss facilitates an
interesting direct comparison between a regular array of
Table III. Analysis of current loss mechanisms in various simulated cell structures with various front surface textures. Error in
collection loss is determined from simulations of generation profiles offset from the mean profiles by the 95% confidence interval of
each profile. The incident photon current represents the AM1.5-g spectrum limited to the range 300–1200 nm.
Texture morphology
Planar
Regular upright
Regular inverted
Random upright
Groove
56
Incident
current Ji
(mA cm2)
46.27
Front surface
transmission loss
Ji–JT (mA cm2)
5.11
1.33
1.11
1.26
1.33
Light trapping
loss JT–JG
(mA cm2)
3.61
3.54 0.15
3.19 0.07
3.23 0.20
3.80 0.15
Collection loss
JG–Jsc (mA cm2)
HE FC
HE RC
SP FC
0.10
0.05 0.01
0.13 0.09
0.11 0.04
0.07 0.02
0.52
0.57 0.01
0.69 0.09
0.63 0.05
0.63 0.01
3.71
3.42 0.01
3.34 0.03
3.59 0.04
3.17 0.02
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
S. C. Baker-Finch and K. R. McIntosh
One-dimensional photogeneration profiles
Table IV. Collection losses listed as a percentage of the total
current generated throughout the cell substrate.
Texture morphology
Planar
Regular upright
Regular inverted
Random upright
Groove
Collection loss (JG–Jsc)/JG (%)
HE FC
HE RC
SP FC
0.27
0.12
0.30
0.27
0.16
1.38
1.38
1.64
1.52
1.52
9.88
8.26
7.95
8.60
7.70
inverted pyramids and a random array of upright pyramids.
It is perhaps counterintuitive that the inverted pyramid
morphology is not inherently compatible with high
efficiency designs. In the case of a high efficiency front
or rear contact cell, some 0.3 or 1.64% of generated carriers
are lost to recombination. Compare this with the loss of
0.27 or 1.52% of carriers in a similar cell with a surface
texture consisting of a random array of upright pyramids.
These differences in collection loss are small, and can be
accounted for by the simulation uncertainties estimated.
Hence, we suspect that the extra complexity and cost
involved with the formation of inverted pyramids is not
justified by the preferential distribution of generation into
regions of high spatial collection efficiency. Instead, the
advantage of inverted pyramids is superior front surface
transmittance.
3.4. Summary
Overwhelmingly, a relatively high Jsc is attributable to the
capacity of a texture to improve front surface transmission.
Further small advantages are offered by light trapping.
Thirdly, for certain cell designs, losses due to recombination (namely ‘collection losses’) depend on the front
surface morphology. For example, there is negligible
improvement attained by a front surface of inverted
pyramids rather than random upright pyramids in rear
junction cells, in which case the generation of carriers near
the front surface tends not to be preferable.
4. ASSESSMENT OF THE BASORE
MODEL FOR G(z)
Although ray tracing offers an accurate means to determine
the profile of photogeneration beneath textured surfaces, it
is computationally intensive and time consuming. As an
alternative, Basore presented an analytical method that
approximates G(z) [14,15]. In this section, we assess the
model by comparing it to the results of ray tracing.
In Basore’s model, the first pass of a ray through a
textured cell is described by a piecewise continuous
function having two parts. As depicted in Figure 6, the ray
passes through a near-surface region of thickness we at an
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
Figure 6. Schematic description of the Basore model for light
travelling through substrates with textured surfaces. Illustration
at top (as in Reference [14]) defines the various angles of ray
traversal after the ray enters via the textured front surface. At
below, a two-dimensional representation of a textured surface is
shown as a guide to the determination of the angles w and #1 .
Note that only rays entering at the first optical interaction (where
ui ¼ 54:7 ) form the basis of the model.
angle w with respect to the surface normal, and continues
through the substrate at an angle W1. Note that this model
assumes that all light is coupled into the substrate at the
first interaction (when the angle of incidence is equal to
54.78, the characteristic angle of the texture). Subsequent
passes of light (at angles W2 and Wn) occur for light reflected
at the back and front internal surfaces (with reflectance Rb1,
Rf1, Rbn and Rfn). The profile function is given as Equation
10 in Reference [16]. Note that in this work we omit the
extension of Brendel et al. [16], that is, we assume that the
angle #n ¼ 60 displays no wavelength-dependence. This
omission has minimal impact upon simulations of thick
cells.
In Figures 7(a) and (b), we compare G(z) determined by
ray tracing and the Basore model. Inputs to the model
are listed in Table V. Light trapping parameters were
identical for the two textures tested, namely, regular
upright pyramids and regular inverted pyramids. The
morphologies only differ in terms of TFS ðlÞ (note that the
antireflection coatings are as above). Since G(z) is similar
for regular upright pyramids, random upright pyramids and
57
One-dimensional photogeneration profiles
Figure 7. Comparison of generation profiles determined by ray
tracing (RT) and the Basore model (top), and quantification of the
model error as a percentage of G(z) determined by ray tracing.
grooves, the results of the following analysis can be
extended to all of the textures investigated in this work.
It is clear in Figure 7(a) that the Basore model provides a
reasonable assessment of G(z) beneath upright pyramids.
The difference between it and ray tracing can be readily
assessed with Figure 7 (b), which plots ðGmodel GRT Þ=GRT
as a function of z. It shows that at worst, the model
underestimates the simulated G(z) by 20% and overestimates it by 13%. The Basore model is a poorer model of
G(z) beneath inverted pyramids, particularly when
3 < z < 50 mm. The simple approximation to the ray
trajectory results in a relatively poor approximation to the
generation in this region, overestimating G(z) as determined by ray tracing by up to 135%.
The cumulative generation current functions resulting
from the profiles determined by ray tracing and the Basore
model are plotted in Figure 8(a). Figure 8(b) illustrates the
relative difference in the modelled curves for JG,cum. When
the front surface is textured with a regular array of upright
58
S. C. Baker-Finch and K. R. McIntosh
Figure 8. Comparison of cumulative current density profiles
determined by ray tracing and the Basore model (top), and
quantification of the model error as a percentage of JG,cum (z)
determined by ray tracing.
pyramids, the modelled JG,cum is within 10% of the raytraced function across the range of z. Larger discrepancies
between modelled and ray-traced profiles in the case of the
inverted pyramid texture are manifested in the form of a
relative difference in JG,cum larger than 25% at z 3 mm.
Fortunately, for a wafered solar cell, the combination of
under- and over-estimates of G(z) results in a reasonable
model of JG (i.e., the error in JG,cum at around 100 mm is
less than 10%).
We note that the accuracy of the Basore model depends
on the accuracy of several input parameters that are
difficult to ascertain in an experimental sample. Consequently, it is common to adjust parameters such as Rb1,
Rb2 and Rf1 to attain a good match between the model and
experiment. Indeed, errors in JG as well as local errors in
G(z) could be reduced by adjustment of these parameters
but the disadvantage of this approach is that it decreases the
physical significance of those parameters.
To assess the Basore model, we compare with the more
accurate 3D ray tracing approach, which accounts for the
complex geometry in full. Where possible, we choose the
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
S. C. Baker-Finch and K. R. McIntosh
One-dimensional photogeneration profiles
Table V. List of model parameters used to approximate the spatial generation profile beneath textured surfaces.
Model parameter
Value
Justification
w
#1
#2
#n
Rb1
Rbn
Rf1
l-dependent
54:7 ’
#1
608
1
1
0.65 0.05
Rfn
0.93 0.05
Calculated from Snel’s law, assuming an incident angle of 54.78
See Figure 6
Specular rear internal reflectance (to match ray tracing setting)
Effective angle of ‘randomized’ light
Unity rear internal reflectance
As above
Approximated by simulation of rays of long wavelength (l ¼ 1000 nm)
arriving at internal surface texture with #2 ¼ 41:5 to macroscopic
surface normal (extension to non-normal incidence of a previous work [23])
As above, with isotropic incidence. Note that this parameter ranges
between approximately 0.921 and 0.938 depending on ARC [16]
The model described by Baker-Finch and McIntosh [30] was used.
Transmittance varies depending on surface texture morphology
TFS ðlÞ
Various
derives from the inaccuracy in JG, rather than the local
errors in G(z) that are observed above.
On the other hand for the SP FC cell, the divergence
between Jsc derived from the two techniques is greater.
With upright (regular or random) pyramid texture, an
error of 2–3% in modelled JG becomes an error of 6.5%
in Jsc. Similarly, when the front surface features a regular
array of inverted pyramids, the local errors in G(z)
observed above appear to drive the 3% error in JG to
become a 7.5% error in Jsc. It is demonstrated here that
when the spatial collection efficiency varies throughout the
bulk of the device, the accuracy of the generation profile
becomes critical.
Overall, we conclude that the Basore model provides
a useful approximation to the generation profile beneath
textured surfaces for limited applications. In particular,
it can be used to estimate the sum total photogenerated
current within wafer-based, thick solar cells. In a
rudimentary test of its application to thinner devices,
we find that the Basore model underestimates Jsc by up
to 15% when the component of G(z) occurring within the
first 30 mm is applied in PC1D simulation. We add that due
to local errors in G(z) as determined by the Basore model,
parameters of the model to be physically accurate and to
match the inputs to the three-dimensional ray tracing
simulation. Values chosen for each parameter, with
justifications, are given in Table V.
4.1. Application of the Basore model
generation profiles to one-dimensional
cell simulation
We input modelled functions for G(z) for various surface
textures into the PC1D software in order to determine the
extent to which the local errors in modelled G(z) impact
upon Jsc. The results summarized in Table VI indicate that
the divergence in Jsc from results based on ray traced
functions for G(z) derive chiefly from the difference in JG
identified above.
For the high efficiency cell structures, throughout which
the spatial collection efficiency is very high, Jsc values
derived from modelled generation profiles agree with those
derived from ray traced profiles to within 3%. This error
in Jsc derived from the application of the Basore model
Table VI. A comparison of ray traced (RT) and Basore model (model) results. Generation current and short circuit current are given for a
range of cell structures and front surface textures. In the bottom half of the table, the values in parentheses are the ratio of modelled and
ray traced values for the variable.
Morphology and technique
Regular upright (RT)
Regular inverted (RT)
Random upright (RT)
Regular upright (model)
Regular inverted (model)
Random upright (model)
JG (mA cm2)
41.40 0.15
41.97 0.07
41.78 0.20
40.57 (98.0%)
40.76 (97.1%)
40.61 (97.2%)
Jsc (mA cm2)
HE FC
HE RC
SP FC
41.35 0.16
41.84 0.16
41.67 0.16
40.41 (97.7%)
40.59 (97.0%)
40.45 (97.1%)
40.83 0.16
41.28 0.16
41.15 0.16
40.03 (98.0%)
40.21 (97.4%)
40.07 (97.4%)
37.98 0.14
38.63 0.10
38.19 0.18
35.62 (93.7%)
35.76 (92.6%)
35.64 (93.3%)
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
59
One-dimensional photogeneration profiles
reasonable results for cell short circuit current can
be garnered only when hc is weakly dependent on z.
5. CONCLUSIONS
The shortest distance to the front surface z is a logical
choice of parameter for one-dimensional solar cell
simulations because, typically, the p–n junction conforms
to the front surface. Calculation of carrier generation as
a function of z (rather than z) is therefore a critical
precursor to accurate analyses. Such a calculation,
although trivial for planar surfaces, is complicated by
surface texturing since light can enter at a range of surface
facets, and be refracted to a range of modes as it travels
through the absorber. Despite the prevalence of ray tracing
tools for solar cell optical analyses, this work was the first
to present generation profiles as a function of the distance
to the nearest facet of a pyramid textured front surface.
We found that a front surface texture consisting of
a regular array of inverted pyramids exhibits different
behaviour than those texture morphologies consisting of
upright pyramids. In particular, it enhances photogeneration in the near surface (near junction) region. Simulations
indicated that this behaviour is not inherently critical
in high efficiency cell designs. Instead, larger gains are
attainable when instilling a front surface texture of inverted
pyramids on an industry standard screen-printed cell,
in which it is important that carriers are generated near
the p–n junction. Replacing a random array of upright
pyramids with inverted pyramids increases the short
circuit current of such a cell by between 0.2 and
0.5 mA cm2. For rear junction designs, there is minimal
optical benefit attained by a front surface of inverted
pyramids rather than random upright pyramids. In general,
we found that the crucial impact of surface texture is to
enhance front surface transmittance, rather than to drive
generation into favourable regions of collection efficiency.
Finally, we assessed the accuracy of a popular
approximation to the generation profile beneath textured
surfaces. We concluded that, despite inaccuracies of up to
135% in the first few microns of substrate below the textured
surface (i.e. within the pyramids themselves), this model
provides a useful means of determining generation current
achieved within textured-wafer silicon solar cells. Indeed,
sum total generation current can be approximated with less
than 5% error provided that the cell substrate is thicker than
150 mm. However, local errors in the generation profile
limit the application of the model in device modelling to
situations in which the spatial collection efficiency varies
only weakly throughout the device.
REFERENCES
1. Zhao J, Wang A, Altermatt P, Green MA. Twenty-four
percent efficient silicon solar cells with double layer
antireflection coatings and reduced resistance loss.
Applied Physics Letters 1995; 66(26): 3636–3638.
60
S. C. Baker-Finch and K. R. McIntosh
2. Zhao J, Wang A, Green MA, Ferrazza F. 19.8%
efficient ‘honeycomb’ textured multicrystalline and
24.4% monocrystalline solar cells. Applied Physics
Letters 1998; 73(14): 1991–1993.
3. Plá JC, Durán JC, Skigin DC, Depine RA. Ray tracing
vs. electromagnetic methods in the analysis of antireflective textured surfaces. Proceedings of 26th IEEE
Photovoltaic Specialist Conference, Anaheim, 1997;
187–190.
4. Llopis F, Tobı́as I. Influence of texture feature size on
the optical performance of silicon solar cells. Progress
in Photovoltaics: Research and Applications 2005; 13:
27–36.
5. Yablonovitch E. Statistical ray optics. Journal of
the Optical Society of America 1982; 72(7): 899–907.
6. Campbell P, Green MA. Light trapping properties of
pyramidally textured surfaces. Journal of Applied
Physics 1987; 62(1): 243–249.
7. Campbell P. Light trapping in textured solar cells.
Solar Energy Materials 1990; 21: 165–172.
8. Campbell P. Enhancement of light absorption from
randomizing and geometric textures. Journal of the
Optical Society of America 1993; 10(12): 2410–
2415.
9. Clugston DA, Basore PA. PC1D version 5: 32-bit solar
cell modelling on personal computers. Proceedings of
the 26th IEEE Photovoltaics Specialists Conference,
1997; 207–210.
10. Brendel R. Sunrays: a versatile ray tracing program
for the photovoltaic community. Proceedings of the
12th European Photovoltaic Solar Energy Conference, 1994; 1339–1342.
11. Cotter JE. RaySim 6.0: a free geometrical ray tracing
program for silicon solar cells. Proceedings of the 31st
IEEE Photovoltaics Specialists Conference, 2005;
1165–1168.
12. Smith AW, Rohatgi A, Neel SC. Texture: a ray tracing
program for the photovoltaic community. Proceedings
of the 21st IEEE Photovoltaics Specialists Conference,
1990; 426–431.
13. Zechner C, Fath P, Willeke G, Bucher E. Two- and
three-dimensional optical carrier generation determination in crystalline silicon solar cells. Solar Energy
Materials and Solar Cells 1998; 51: 255–267.
14. Basore PA. Extended spectral analysis of internal
quantum efficiency. Proceedings of the 23rd IEEE
Photovoltaics Specialists Conference, 1993; 147–152.
15. Basore PA. Numerical modelling of textured silicon
solar cells using PC-1D. IEEE Transactions on
Electron Devices 1990; 37(2): 337–343.
16. Brendel R, Hirsch M, Plieninger R, Werner JH.
Quantum efficiency analysis of thin-layer silicon solar
cells with back surface fields and optical confinement.
IEEE Transactions on Electron Devices 1996; 43(7):
1104–1113.
17. Abenante L, Izzi M. An accurate analytical approach
including internal reflectance for calculating quantum
efficiency of planar solar cells. Proceedings of the
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
S. C. Baker-Finch and K. R. McIntosh
18.
19.
20.
21.
22.
23.
23rd European Photovoltaic Solar Energy Conference, 2008; 302–304.
Abenante L, Izzi M. A new carrier-generation profile
function for planar p–n junction solar cells. Proceedings of the 23rd European Photovoltaic Solar Energy
Conference, 2008; 305–307.
Kittidachachan P, Markvart T, Bagnall DM, Greef R,
Ensell GJ. A detailed study of p–n junction solar cells
by means of collection efficiency. Solar Energy
Materials and Solar Cells 2007; 91: 160–166.
Yang WJ, Ma ZQ, Tang X, Feng CB, Zhang WG, Shi
PP. Internal quantum efficiency for solar cells Solar
Energy 2008; 82(2): 106–110. doi 10.1016
Al-Omar AS, Ghannam MY. Optimum two-dimensional short circuit collection efficiency in thin multicrystalline silicon solar cells with optical confinement.
Solar Energy Materials and Solar Cells 1998; 52(1–
2): 107–124.
Yun G, Crabtree K, Chipman R. Properties of the
polarisation ray tracing matrix. Proceedings of SPIE
Vol. 6682 (Polarisation Science and Remote Sensing
III), 2007; 66820Z.
Baker-Finch SC, McIntosh KR. Reflection of normally
incident light from silicon solar cells with pyramidal
texture. Progress in Photovoltaics, 2010. DOI:
10.1002/pip.1050
Prog. Photovolt: Res. Appl. 2012; 20:51–61 ß 2011 John Wiley & Sons, Ltd.
DOI: 10.1002/pip
One-dimensional photogeneration profiles
24. Green MA, Keevers MJ. Optical properties of intrinsic
silicon at 300 K. Progress in Photovoltaics 2007; 3(3):
189–1192.
25. Rodriguez JM, Tobias I, Luque A. Random pyramid
modelling. Solar Energy Materials and Solar Cells
1997; 45: 241–253.
26. Green MA. Silicon Solar Cells: Advanced Principles
and Practice. Centre for Photovoltaic Devices and
Systems, University of New South Wales: Sydney,
1995; 96.
27. Kray D, Hermle M, Glunz SW. Theory and experiments on the back side reflectance of silicon wafer
solar cells. Progress in Photovoltaics 2008; 16: 1–16.
28. Mulligan WP, Rose DH, Cudzinovic MJ, De Cuester
DM, McIntosh KR, Smith DD, Swanson RM. Manufacture of solar cells with 21% efficiency. Proceedings
of the 19th European Photovoltaic Solar Energy Conference, 1994; 387–390.
29. King DL, Buck ME. Experimental optimization of an
anisotropic etching process for random texturization
of silicon solar cells. Proceedings of the 22nd IEEE
Photovoltaics Specialists Conference, 1991; 303.
30. Baker-Finch SC, McIntosh KR. A freeware program
for precise optical analysis of the front surface of a
solar cell. Proceedings of the 35th IEEE Photovoltaics
Specialists Conference, 2010.
61