BREAKDOWN OF THE EFFICIENCY GAP TO 29% BASED ON
EXPERIMENTAL INPUT DATA AND MODELING
R. Brendel1,2, T. Dullweber1, R. Peibst1, C. Kranz1, A. Merkle1, D. Walter3
Institute for Solar Energy Research Hamelin (ISFH), Am Ohrberg 1, 31860 Emmerthal, Germany
2
Institute for Solid State Physics, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
3
Australian National University (ANU), Research School of Engineering, North Road, Acton, ACT 0200, Australia
1
ABSTRACT: We demonstrate a procedure for quantifying efficiency gains that treats resistive, recombinative, and optical losses on
an equal footing. For this we apply our Conductive Boundary (CoBo) model as implemented in the Quokka cell simulator. The
generation profile is calculated with a novel analytical light trapping model. This model parametrizes the measured reflection spectra
and is capable of turning the experimental case gradually into an ideal Lambertian scheme. Simulated and measured short circuit
current densities agree for our 21.2%-efficient screen-printed PERC cell and for our 23.4%-efficient ion-implanted laser-processed
IBC cell. For the loss analysis of these two cells we set all experimentally accessible control parameters (e.g. saturation current
densities, sheet resistances, and carrier lifetimes) one at a time to ideal values. The efficiency gap to the ultimate limit of 29% is
thereby fully explained in terms of individual improvements and in terms of their respective synergistic effects. This approach allows
comparing loss structures of different types of solar cells, e.g. PERC and IBC cells.
Keywords: Loss analysis, silicon solar cell; passivated emitter and rear cell; PERC; interdigitated back contact; IBC; Conductive
boundary model
1
INTRODUCTION
Having a quantitative breakdown of all losses greatly
helps optimizing solar cells [1]. Many useful approaches to
loss analysis are described in the literature. They range from
simply analyzing Voc, Jsc and FF losses with a two-diode
model to comprehensive predictive modeling [2] of specific
technologies.
The most basic approach to loss analysis
distinguishes “current losses”, “voltage losses” and “fill
factor losses”. They are, however, not independent. Any
current loss implied by recombination also impacts the open
circuit voltage and thus the fill factor. We introduced our
Free Energy Loss Analysis (FELA) to determine a set of
physically distinct recombination and transport losses [3].
Later FELA was extended to include optical losses [4].
However, this Free Energy Loss Analysis also has its
drawbacks. It does not tell us how far the device is from the
optimum since the free energy being generated in the devices
depends on the device parameters. In addition the free energy
dissipation by any larger loss (e.g. emitter recombination) is
not equal to the efficiency gain to be expected when avoiding
this loss. It is thus necessary to distinguish expected
efficiency gains and corresponding losses at the working
point. Summing all FELA losses does not fully explain the
gap to the ideal efficiency of 29%.
Our recently introduced Synergistic Efficiency Gain
Analysis (SEGA) [5,6] is capable of breaking down the full
efficiency gap to 29%. It idealizes the experimental device
stepwise rather than applying the FELA at the maximum
power point [3]. In our previous work [6] we used Sentaurus
transport simulations and ray tracing with SUNRAYS [7]. A
large number of input parameters were required: Doping
profiles for the transport simulation and thickness values as
well as the dielectric functions of various anti reflection
coatings for ray tracing. Not all of these data are readily
available for individual cells. Another disadvantage is that
this modeling is demanding in terms of computation time
(especially ray tracing) as well as hardware and software cost
(especially Sentaurus).
We therefore developed the conductive boundary (CoBo)
model [8,9] that facilitates less demanding transport
simulations. CoBo restricts the input parameters to
experimentally readily accessible data such as saturation
current densities, surface recombination velocities, sheet
resistances and carrier lifetime in the bulk. CoBo does not
spatially resolve the emitter and the back surface field (BSF)
regions. This enables fast numerical solutions of the coupled
carrier transport equations in the bulk of the device. This
speed comes at the price of not being able to model the
physics in the doped layers. We used Comsol to solve the
CoBo model [9]. Comsol is, unfortunately, becoming
increasingly expensive and every new device geometry
requires new Comsol programming.
The Quokka simulator by A. Fell [10] solves the cost
issue as it is freeware. It solves the equations of the CoBo
model in Matlab and makes the additional assumption of
neutrality in the bulk. The neutrality assumption comes at the
price of not being able to model the physics in the space
charge region of the bulk, which is, however, not necessary
for highly efficient cells. Quokka provides a script-based user
interface for modeling a wide range of cell geometries
without extra programming work.
Franklin et. al. already used Quokka for a loss analysis [1]..
Their approach combined our FELA [3] that is also
implemented in Quokka [10] with a step-wise idealization of
the optics in ray tracing simulations. With this approach the
authors were able to give a break-down of a major fraction,
but not the entirety of, the efficiency gap to 29%.
Here we introduce an analytical optical model for extracting
an approximate photogeneration profile from an experimental
reflectance spectrum. Our optical model treats front and rear
surface as being partially light scattering. Fitting
experimental reflectance data with this model is much faster
than fitting by ray tracing. Following Franklin [1] we also
apply a stepwise idealization of the optics but do the same
also for the transport.
We perform a Synergetic Efficiency Gain Analysis
(SEGA) of a Passivated Emitter and Rear Cell (PERC) [11]
and an Interdigitated Back Contacted (IBC) cell [12] using
the CoBo implementation in Quokka and the new optical
model presented here. We find that for the PERC cell fiftyeight percent of the efficiency gap to 29% is due to
recombination losses and forty-six percent of this gap is due
to synergistic effects of the various individual improvements.
CELLS INVESTIGATED
Details on our industrial type PERC cell were
reported previously [13]. It is 6” pseudo square and screenprinted on both sides. The cell has five busbars and no
soldering pads on the rear side. It was manufactured at the
ISFH research line SolarTeC. The independently confirmed
efficiency is 21.2 %. Saturation currents, carrier lifetime,
sheet resistances, surface recombination velocities and other
parameters were all measured on co-processed or similarly
processed test structures. The emitter has a sheet resistance of
75 Ω and a saturation current density of 100 fA/cm2 in the
non-metalized region. These and more data were reported in
[6].
The IBC cell that we reported in reference [14] has an area of
4 cm2. All structures were defined by laser processing. It was
processed without photolithography. Ion implantation was
used to form the emitter and the back surface field (BSF)
layer. The independently confirmed efficiency is 23.4%.
Saturation currents, carrier lifetime, sheet resistances, surface
recombination velocity and most other parameters were
measured on co-processed or similarly processed test
structures. It is only the saturation current at the metal contact
to the P-doped layer that we estimated using experimental
results from other processes.
The experimental parameters of the one-sun illuminated
current voltage curves of both cells are listed in Table 1.
Table 1: Comparison of experimental and simulated
parameters of the current-voltage curve under one-sun
illumination with an AM1.5G spectrum at 0.1 W/cm2 and
25°C.
Voc
Jsc
FF
η
[mV] [mA/cm^2] [%]
[%]
PERC Experiment 662
39.8
80.6
21.2*
PERC CoBo
668
39.8
81.0
21.5
IBC
Experiment 693
41.3
81.9
23.4*
IBC
CoBo
696
41.2
82.8
23.7
* Independently confirmed by Fh-ISE
3
OPTICS
3.1
Reflectance measurements
The symbols in Figure 1 show the experimental
hemispherical reflection of the PERC cell (circles) and the
IBC cell (diamonds). The reflection is measured at room
temperature under 8° of incidence with an integrating sphere
that is 15 cm in diameter, using a spot size of 2 cm x 2 cm.
For the PERC cell, the measured data are corrected for the
reflectance of the metal fingers by forcing the total
reflectance to zero at its minimum. By this approach, the
reflectance of the inter-finger regions is estimated.
3.2
Optical model
Figure 2 sketches the optical model that we develop to
parametrize the measured reflectance and the corresponding
depth-dependent carrier generation profile. Specular light of
unity intensity impinges on the wafer surface. We distinguish
specular (black) and diffusive (red) light intensities Ij,k,l
(index j =s or d) at the top and bottom of the wafer (k = t or
b), and we distinguish light propagating downwards and
upwards (l = d or u). The model has thus ten variables
(circles in Figure 2): Four light intensities Ij,t,l at the top of the
wafer, four intensities Ij,b,l at the bottom of the wafer, plus the
diffusive reflectance Rd and the specular reflectance Rs. The
front surface of the wafer is characterized by its Lambertian
character Λf and the front reflectance Rf =1-Tf, with Tf
denoting the transmittance of the front. We thus assume that
dielectric coatings on the front do not absorb light. Similarly
the rear side is characterized by Λb and Rb.
0.6
Measurement
Extrapolated Rf
0.5
IBC
Model
0.4
Reflection R
2
PERC
0.3
0.2
0.1
0
-0.1
200
400
600
800
1000
Wavelength λ [nm]
1200
1400
Figure 1: Measured (symbols) and modelled (lines)
hemispherical reflectance spectra of the PERC (diamonds)
and the IBC (circles) cell.
Rs
Rd
Λf
1− Λ f 1− Λ f
Rf
Tf
1− Λ f
Tf
1 − Tf
Tf
Rf
1− Λ f
0
1 

1 − T f 2 
n 

I d ,t ,d
I d ,t ,u
Ts
Td
Td
I s ,b,u
I d ,b,d
(1 − Λ b )
Rb
Λf
1
n2
I s ,t ,u
Ts = e −αW
I s ,b,d
Tf
Rf
Tf
Λf
I s ,t ,d
Λf
W
z
I d ,b,u
Λb
Rb
Rb
Figure 2: Power flows of the light trapping model. Black
lines denote specular light and red lines denote light
propagating in random direction. We consider power flux
upwards and downwards. The blue lines denote power
transfer from the specular channel to the diffusive channel
according to the Lambertian characters of the front and the
rear surface. Collect the factors along the lines to derive the
system of equations of this optical model. Specular light of
unity intensity impinges on the wafer surface. The model has
10 variables Ij,k,l. We distinguish specular (black) and
diffusive (red) light intensities (index j =s or d) at the top and
bottom of the wafer (k = t or b), and we distinguish light
propagating downwards and upwards (l = d or u).
The intensity of specular reflectance from the
incoming light is Rf (1-Λf). The corresponding diffusive
reflectance Rf Λf contributes to the total diffuse reflectance
Rd. Intensities transferred from the specular (black) channels
to the diffusive (red) channels are denoted as blue dotted
lines in Figure 2. The Intensity of the first specular
Rs = (1 − Λ f )(I s ,t ,u T f + R f ) .
Solving (1) through (10) for the ten intensities we find the
total reflectance R
= Rs + Rd to be
RbT f2 ( − (Λ b − 1)(Λ f − 1)Ts2 (RbTd2 (n 2 − T f ) − n 2 ) − Λ bTd (Λ f − 1)Ts + Td2 Λ f )
RbTd2 (T f − n 2 ) + n 2
R=
+ R f ((Λ b − 1)Rb (Λ f − 1)(T f − 1)Ts2 + 1)
. (11)
(Λ b − 1)Rb (Λ f − 1)(T f − 1)Ts2 + 1
transmission through the front surface is Tf (1−Λf) while the
transmitted diffusive intensity is Tf Λf.
The symbol Ts = e−αW in Figure 2 denotes the normal
transmittance of specular light through the wafer of thickness
W. For every wavelength the wafer has the absorption
coefficient α and refractive index n. Similarly Td denotes the
transmission factor of fully randomized light. See [15] for an
analytic expression for Td.
The upward and downward light flows shown in the
Figure 2 define ten linear equations. We explain only one of
them. The diffusive intensity (upper left red circle)
(
(10)
)
=
I d ,t ,d I d ,t ,u 1 − T f / n 2 + Λ f (I s ,t ,u (1 − T f ) + T f )
The carrier generation rate
z
1−
ln(Td )
g (z ) =
(I d ,b,u Td W + I d ,t ,d Td z /W )
−
W
z
1−
ln(Ts )
(I s ,b,u Ts W + I s ,t ,d Ts z /W )
−
W
depends in the four intensity variables Ij,k,l. For understanding
(12) please note that the absorption coefficient for specular
light is −ln(Ts ) / W while diffusive light has an apparently
larger absorption coefficient −ln(Td ) / W due to oblique
traversal. Inserting the solutions for these four variables
yields the easy to program fully analytic depth-dependent
generation profile
(1)
at the top of the wafer propagating downwards has three
contributions: (i) The diffusive part Λ f T f of the transmission
−
T f ( − ln [Ts ]Ts
z
W
−
2z
(TsW − RbTs2 ( − 1 + Λ b ))( − 1 + Λ f ) −
g( z ) = −
(12)
z
2z
2z
ln [Td ]Td W (n 2TdW Λ f + RbTd ( − (n 2 + TdW (n 2 − T f ))Ts Λ b ( − 1 + Λ f ) + n 2Td Λ f ))
−n 2 + RbTd2 (n 2 − T f )
W (1 + Rb ( − 1 + T f )Ts2 ( − 1 + Λ b )( − 1 + Λ f ))
)
.
(13)
into the wafer (blue), (ii) the diffusive part
I s ,t ,u R f Λ f = I s ,t ,u (1 − T f )Λ f of specular light that is reflected
A Matlab code for equations (11) and (13) will be made
available [16].
internally (blue), (iii) the internally reflected diffusive
intensity I d ,t ,u (1 − T f / n 2 ) (red). In (iii) the divisor n 2
3.3
describes the restriction of the internal transmission for
diffuse light by total internal reflection. Collecting the factors
along the black, red, and blue lines in Figure 2 generates the
other nine equations
I d ,b,d = I d ,t ,d Td ,
I d ,b,u= I d ,b,d Rb + I s ,b,d Λ b Rb ,
(2)
(3)
I d ,t ,u = I d ,b,u Td ,
=
Rd
and
I d ,t ,u T f
n2
+ Λ f (I s ,t ,u T f + R f ) ,
(4)
(5)
I s ,t ,d = (1 − Λ f )(I s ,t ,u (1 − T f ) + T f ) ,
(6)
I s ,b,d = I s ,t ,d Ts ,
(7)
=
I s ,b,u I s ,b,d (1 − Λ b ) Rb ,
(8)
I s ,t ,u = I s ,b,u Ts ,
(9)
Application to the cells investigated
At every wavelength we use (11) for modeling the
measured reflectance. This requires knowledge of the front
reflectance Rf. We take Rf to be equal to the measured
reflectance R for short wavelengths and extrapolate the
measurement with a second order polynomial (dashed blue
line in Figure 1) into the near infrared spectral range. The
three coefficients of the polynomial are chosen to reproduce
the experimental reflectance R at 800, 850 and 900 nm. More
elaborate extrapolations schemes that account for the Fresnel
oscillations yield the same results here.
Please note that we include free carrier absorption
[17] by majority carriers in the bulk and in the doped surface
layers. For this we increase the absorption coefficient α
above its band to band value. For the doped layers this
enhancement is calculated from the sheet resistances of the
emitter and the BSF. We assume a rectangular doping profile,
apply an estimated layer thickness and choose the doping
concentration to reproduce the measured sheet resistance
when applying Klaassen’s unified mobility model [18]. The
areal fraction and the respective layer thickness values are
accounted for by appropriately “diluting” the corresponding
free carrier absorption into the bulk of the Si wafer.
The optical model has three free parameters Λf, Λb,
and Rb. We find that different pairs of Λf, Λb can give very
similar results. Fixing Λf to 0.335 makes the fitting more
stable. The value Λf = 0.335 makes the average path length of
the first traversal equal to the path length of the spectral
transmission through a {111} pyramidal surface texture.
The solid lines in Figure 1 show the fit (lines) of
equation (11) to the measurements (symbols). We find a
minimum deviation of model and experiment with (Rb =
0.897, Λb= 0.907) for the PERC cell and with (Rb = 0.944,
Λb=0.865) for the IBC cell. The back reflectance of the IBC
cell is thus considerably higher than that of the PERC cell.
We speculate that this is mainly due to a higher reflectance of
evaporated aluminum when compared to screen-printed
aluminum. In addition the area of direct metal to silicon
contact is less than half for the IBC cell when compared to
the PERC cell.
The value of this parametrization of the reflectance
and the carrier generation rate for the SEGA is that on the
one hand it describes the reflectance spectra well and on the
other hand it can also model ideal Lambertian light trapping
when taking Rf = 0, Rb = 1 and Λf = 1. There is no specular
light in the cell for unity Lambertian character Λf at the front
surface. In this case the Lambertian character Λb of the rear
side does not influence the simulation result.
4
TRANSPORT MODEL
We use the software program Quokka [10] version
2.2.2 for the transport modeling of the cells. A lumped series
resistance accounts for ohmic losses. The value of this
resistance is chosen to have a power dissipation that equals
the sum of the power dissipation in the fingers and in the
busbars. We calculate this value analytically by assuming that
the series resistance is sufficiently small to allow for a
laterally constant current density in the cell. This is a valid
assumption for efficient cells.
Figure 3 illustrates half of the two unit cells with ntype layers in red, p-type layers in green, dielectric surfaces
in blue and metal contacts in grey. The CoBo model requires
saturation current densities Jox and sheet resistances Rx as
input parameters for all doped surface layers [9]. Quokka
further requires the contact resistances ρx as input parameters
for the contacted boundaries [10]. Here x = e, ec, BSF, BSFc
stands for the emitter, the emitter contact, the BSF layer, and
the BSF contact layer, respectively. Dielectric coated surfaces
are described by a surface recombination velocity S.
The unit cell of the PERC cell is two dimensional.
We select the parameters of the Shockley-Read-Hall
recombination in the bulk to describe the experimental
lifetime data of a corresponding test structure. See reference
[5] for this procedure. The shading of the busbars is
considered by enhancing the shading of the fingers. We
model the dielectric rear surface of the PERC cell as a
conductive boundary layer, however with a high sheet
resistance of 1 MΩ. The reasons for this will become
apparent later.
The unit cell of the IBC cell is three dimensional,
thus Figure 3 only shows a cross section through the contacts.
Here we assume an injection independent carrier lifetime of 5
ms in the bulk. Please note that we assume unity collection
efficiency in all doped surface layers. This introduces an
error of the short circuit current density of typically less than
0.2 mA/cm2 for efficient screen-printed cells. This
assumption could be relaxed in a more detailed analysis that
also evaluates experimental quantum efficiencies.
PERC
Emitter metal
Emitter cont.: Joec Rec r ec
Emitter: Joe Re
Bulk: r b, t b
BSF cont.: JoBSFc RBSFc r BSFc
Dielectric surface: S
BSF metal
IBC
Dielectric surface: S
Bulk: r b, t b
Emitter cont.:
Joec Rec r ec
Emitter metal
Emitter: Joe Re
BSF: JoBSF RBSF
BSF cont.:
JoBSFc RBSFc r BSFc
BSF metal
Figure 3: Two-dimensional schematic illustration of half of
the unit cell of the PERC and the IBC cell, respectively. Red
color indicates p-type doping, green color is n-type doping,
blue is dielectric surface passivation, and gray is the
contacting region of the metallization. The PERC structure is
modelled in 2D while the IBC structure is modelled in 3D.
Thus the figure only shows a cross section through the 3D
unit cell of the IBC cell.
Table 1 shows that a fair agreement of the measured
one-sun current-voltage (IV)-curve parameters with the
simulated ones. Please note that we did not do any fitting
with the transport model. The differences between
experiment and modeling are sufficiently small to determine
the relative size of the various loss channels. The modelled
short circuit current densities agree with the experimental
values to within 0.1 mA/cm2. This agreement justifies the use
of the new optical model in our SEGA. The voltages are
slightly overestimated which could be due to neglecting
perimeter losses at the edge of the cells.
4.1 SEGA of the 21.2% PERC cell
The synergistic efficiency gain analysis (SEGA)
analysis of the PERC cell is shown in Figure 4. This SEGA
plot is generated automatically by our Matlab code that
modifies the script containing the cell definition, runs
Quokka, and evaluates the results. Fitting of the reflectance
spectra, all 24 device simulations and generating the SEGA
plot takes about 4 min on a laptop (Lenovo X201). The plot is
generated as follows.
We first simulate the reference case using the
experimental input parameters. The simulated reference cell
has an efficiency of 21.5%. Starting from the reference case
we set the diode saturation current of the emitter and the
emitter contact to zero while leaving all other parameters as
measured. This gives a simulated efficiency improvement of
0.698% (first red bar, marked “Emitter recombination”).
Then we set the saturation currents back to their experimental
values and reduce the sheet resistance of the emitter to zero.
This adds 0.298% (first blue bar marked “Emitter lateral
resistance”). Zero emitter sheet resistance in the PERC cell
does approximately exclude lateral electron transport losses
in the cell. For excluding the lateral transport losses of holes
we set the sheet resistance of the BSF and the sheet resistance
of the dielectric surface to zero. This adds 0.118% of
efficiency (blue bar marked “BSF lateral resistance”). Lateral
transport loss by electrons is thus more important in our
PERC cell than lateral transport loss by holes. Next we reset
the sheet resistance to the experimental value and proceed by
assuming that the cell had optically transparent fingers on the
front. This adds 0.454% to the efficiency (first green bar,
marked “Grid shading”) of the reference cell. Then we reset
the grid shading to the experimental value. In this fashion we
vary every model parameter to their ideal values and then set
them back. Figure 4 sorts 18 individual efficiency
enhancements by size and cumulates them in a Pareto-type of
plot. There is negligible loss due to the lateral resistance by
the metal on the openings of the BSF since the PERC cell is
fully metallized on the rear side (zero bar marked “BSF metal
resistance”). We also neglect the small area of the BSF of the
PERC cell that is not metallized (zero bar marked “BSF
recomb.”).
Figure 4: SEGA analysis of a PERC cell. A simulation that
uses experimentally determined input parameters serves as
the reference. Ohmic gains are marked as blue, optical gains
as green, and gains by less recombination as red bars.
Efficiency changes by closing individual loss channels are
labeled upwards. Synergistic changes are labeled
downwards.
Resetting the parameters to their experimental
reference values makes the various efficiency changes
directly comparable and indicates the maximum gain that the
experimentalist can expect for his device from improving
only this parameter. We sort all efficiency improvements by
size to produce a Pareto-type plot shown in Figure 4. The
largest individual loss channel in our PERC cell is thus
emitter recombination. With a hypothetical ideal emitter (Joe
= J0ec = 0) we can expect a gain in efficiency of 0.7%. The
four next largest losses are all of optical nature due to grid
shading (0.45%), front reflection (0.44%), back reflection
(0.43) and busbar shading (0.37).
All 18 individual improvements (labeled upwards)
sum up to an efficiency of 25.5% rather than to the limiting
efficiency of 29% [19]. Thus, we perform four more
simulations (labeled downwards in Figure 4) to account for
the rest of the efficiency gap to 29%.
(i) We first model the reference case with all six
resistive losses simultaneously switched off. This gives
-0.006% extra efficiency (blue bar marked downwards as
“Synergy resistance”) when compared the sum of
improvements of all individual resistive losses (blue bars
labeled upwards). A linear behavior is expected here since
resistances of various origins basically add up to the total
resistance. We speculate that the small synergistic effect is
negative since holes reach the recombinative rear contacts
more easily in a device with zero lateral BSF resistance.
(ii) The synergistic efficiency enhancement over the
sum of all optical improvements (green bars labeled upwards)
is also small (“Synergy optics” is 0.078%). Here we do
expect a positive effect since improving the optics not only
improves the short circuit current but also the open circuit
voltage causing a non-linearity.
(iii) A reference cell with all recombination channels
turned off has a simulated efficiency of 25.85%. This value is
2.98% larger than the efficiency of the reference cell plus the
sum of all individual recombinative improvements (red bar
marked “Synergy recombination“).
(iv) In a final simulation we model the reference cell
with all parameters set to ideal values. The result is 29% and
exceeds the sum of improvements by the three groups
(resistance, optics, and recombination) by 0.432% (black bar
marked “Synergy of groups”).
The right part of Figure 4 summarizes the breakdown
of the efficiency gap by summing all individual resistive,
recombinative and optical contributions, respectively (blue,
green, and red bar marked “Sum …”). Recombination
accounts for 4.33% (= 1.35% sum of individual
recombination channels +2.98% synergy). This is fifty-eight
percent of the total efficiency gap. Thus recombination
clearly dominates over optical gains of 1.95%
(=1.877%+0.078%) and resistive gains of 0.75%
(=0.752%−0.006%). It is also interesting to note that the
synergy contributions (−0.006% for resistances, 0.078% for
optical effects, 2.977% for recombination processes, and
0.432% for the synergy of the three groups) add up to 3.48%.
Synergy effects thus account for forty-six percent of the
efficiency gap to 29%.
Origin of the synergy of recombination
When switching off one recombination mechanism,
the carrier concentration is increased. Therefore, the
recombination currents for the other recombination
mechanisms (product of the carrier concentration and the
recombination parameter) increase. Switching off the second
recombination mechanism generates a greater efficiency
improvement than if it were switched off the first time. The
importance of efficiency improvement by switching off any
recombination mechanism increases as the carrier
concentration and thus the cell voltage increases.
Figure 5: Simulated open circuit voltage Voc as a function of
the reduction factor x of all recombination parameters
The circles in Figure 5 are simulated open circuit
voltages of the PERC reference cell with all recombination
rates reduced by factor x. The solid line is a fit with the
equation Voc =Vth ln[40 mA/cm2/(Ji +Jo)] where Vth is the
thermal voltage. We find Ji = 8.3 fA/cm2 for the intrinsic
saturation current and Jo = 213 fA/cm2 for the extrinsic
recombination current. We could thus also understand the xaxis as showing the saturation current Jo = x 213 fA/cm2. The
increase in voltage is non-linear with decreasing
recombination and so is the efficiency of the device.
Figure 6 displays the simulated efficiencies as a
function of the saturation current density Jo = x 213 fA/cm2.
The reference cell (x = 1) has an efficiency of 21.5% (lower
solid line). Reducing recombination at a smaller Jo (higher
voltage and higher carrier concentration) gives a larger
impact on the efficiency. Simply summing the improvements
by switching off all extrinsic recombination individually
leads to an expected efficiency of 22.9% (middle solid line
line). However, switching off all channels together reduces
the extrinsic saturation current to zero and enhances the
efficiency to 25.8% (upper solid line). The smaller the
saturation current already is, the faster the efficiency rises
when further decreasing the saturation current. This causes
what we term the synergistic effect of recombination here.
Figure 6: Simulated efficiency as a function of the saturation
current density Jo = x 213 fA/cm2. The black horizontal lines
indicate the efficiency of the reference cell, the latter
enhanced by the sum of improvements due to avoiding all
individual recombination losses, and the latter enhanced by
the synergy of recombination losses.
The significance of the order of switching off loss
channels becomes even more apparent if we now consider the
ideal cell as the reference and switch on every loss channel
individually. This yields Figure 7. The synergistic
contributions are determined just as explained above. We find
that the sum of all efficiency losses would reduce the
efficiency to 13%, a value that is much smaller than the
21.5% of the simulated reference cell. This is because we
introduce every loss at maximum voltage where the
efficiency is much more sensitive to all the individual losses
than it is in the experimental device.
It is interesting to note that all resistive losses in our
PERC cell would still support a hypothetical 28.2%-efficient
cell in which there is no extrinsic recombination and that is
optically ideal. Optically ideal means no reflection losses and
Lambertian light trapping. Resistive losses are thus a
relatively minor target for enhancing the efficiency of our
PERC cells.
Figure 7: SEGA-type of plot of the PERC cell but now with
the ideal cell as the reference case. We calculate the
efficiency losses when switching on individual loss channels.
Ohmic gains are marked as blue, optical gains as green, and
gains on less recombination as red bars. Efficiency changes
by closing individual loss channels are labeled upwards.
Synergistic changes are labeled downwards.
Difference of two SEGA plots
We also calculate the SEGA plot (not shown) of our
reference PERC cell but now with an improved emitter. For
this we assume that we were able to reduce the saturation
current density of the emitter region from 100 to 35 fA/cm2
and that the emitter has a sheet resistance that increased from
75 to 95 Ω. Figure 8 shows the difference between the SEGA
plot of the reference PERC cell minus the SEGA plot of the
same PERC cell but exhibiting this improved emitter. The
improved emitter enhances the efficiency of our PERC cell
from 21.5% to 21.8%. The difference SEGA plot explains
how this improvement is distributed over various effects. The
largest positive effect (+0.389%) is the reduction of emitter
recombination (first red bar marked “Emitter recomb.”). This
enhances the voltage and thus increases the recombination in
all other channels (2nd to 4th red bar labeled upwards). The
enhanced sheet resistance (blue bar marked “Emitter lateral
resist.”) also has a negative effect (-0.066%). Thus individual
positive and individual negative effects almost level out. It
finally is the enhanced voltage or the synergy of
recombination (red bar marked “Syn. recomb.”) that makes
the overall effect positive.
4.2 SEGA of the 23.4% IBC cell
The SEGA analysis of the IBC cell is shown in
Figure 9. In this cell front reflection (first green bar) is the
largest individual effect that costs 0.571% of efficiency.
Almost as large is the emitter recombination (first red bar).
As for the PERC cell it is recombination (see red bars in the
summary on the right) that accounts for 1.425% (“Sum
Recombination”) + 2.261% (“Synergy of recombination”) =
3.686%.
Figure 8: Difference of the SEGA of our PERC cell and the
SEGA of the same cell with an improved emitter. Ohmic
gains are marked as blue, optical gains as green, and gains
by less recombination as red bars. Efficiency changes by
closing individual loss channels are labeled upwards.
Synergistic changes are labeled downwards.
This is seventy-one percent of the efficiency gap to 29%.
Again, recombination losses are dominating the gap. A plot
similar to Figure 6 for the PERC cell can be calculated for the
IBC cell and confirms that practical limits of reducing Jo [20]
will finally limit the achievable efficiency potential.
Figure 9: SEGA plot of the IBC cell. Ohmic gains are marked
as blue, optical gains as green, and gains by less
recombination as red bars. Efficiency changes by closing
individual loss channels are labeled upwards. Synergistic
changes are labeled downwards.
4.3 Comparing losses of different cell types
Please note, that we use corresponding loss categories
for the PERC and the IBC cell as illustrated in Figure 3. It is
therefore possible to calculate the difference between the
SEGA plot of the PERC and the IBC cell. Figure 10 shows
this difference. The IBC cell largely gains over the PERC cell
by avoided grid and busbar shading (first two green bars).
The optical advantage is 0.785% (“Sum optics”) + 0.035%
(“Synergy optics”) = 0.82% as summarized on the right hand
side of Figure 10. The smaller pitch gives the IBC cell an
advantage in the lateral transport in the emitter (first blue bar
marked “Emitter lateral resist.”). The overall saturation
current density of the IBC cell is smaller than for the PERC
cell and this gives an efficiency advantage of
0.639%= -0.077% (“Sum recombination”) + 0.716%
(“Synergy recombination”) by avoided recombination.
•
•
•
•
it gives a breakdown of the full efficiency gap up to
the ultimate efficiency limit of 29% by including
synergy effects between the various groups of
losses,
it does not anticipate a specific technological
roadmap for efficiency improvements since it sets
the control parameters stepwise to ideal values,
it can be done within minutes on a lap top due to
CoBo and avoiding ray tracing,
and it does not require expensive hard- or software
due to using Quokka.
The authors very much hope that many readers will find the
Synergistic Efficiency Gain Analysis presented here helpful
in bringing their cells a little bit closer to the physical limit.
6
ACKNOWLEDGEMENTS
This work was supported by the State of Lower Saxony, the
project HighPERC, and the project Hercules. We
acknowledge the funding of these projects by the German
Federal Ministry for Economic Affairs and Energy and by the
European Union, respectively. We thank A. Fell (ANU) for
writing such a nice piece of software that can make traveling
with a laptop a delight. We also thank K. Bothe and P.
Altermatt for helpful discussions on the loss analysis of solar
cells.
REFERENCES
Figure 10: Difference of SEGA plots of PERC and IBC.
Ohmic gains are marked as blue, optical gains as green, and
gains by less recombination as red bars. Efficiency changes
by closing individual loss channels are labeled upwards.
Synergistic changes are labeled downwards.
5
[1]
E. Franklin, K. Fong, K. McIntosh, A. Fell, A. Blakers,
T. Kho, D. Walter, D. Wang, N. Zin, M. Stocks, E.-C.
Wang, N. Grant, Y. Wan, Y. Yang, X. Zhang, Z. Feng,
P. Verlinden, 29th EUPVCEC 2014, and submitted to
Progress in Photovoltaics, (2014). DOI:
10.1002/pip.2556.
[2]
Wagner H, Dastgheib-Shirazi A, Chen R, Dunham ST,
Kessler M, Altermatt PP. Improving the predictive
power of modeling the emitter diffusion by fully
including the phosphsilicate glass (PSG) layer. In 37th
IEEE Photovoltaic Specialists Conference (PVSC),
2011, pp. 002957-002962. DOI
10.1109/PVSC.2011.6186566.
[3]
Brendel R, Dreissigacker S, Harder NP, Altermatt PP.
Theory of analyzing free energy losses in solar cells.
Applied Physics Letters 2008; 93: 173503. DOI
10.1063/1.3006053.
[4]
Greulich J, Hoffler H, Wuerfel U, Rein S. Numerical
power balance and free energy loss analysis for solar
cells including optical, thermodynamic, and electrical
aspects. Journal of Applied Physics 2013; 114: 204504.
DOI 10.1063/1.4832777.
[5]
Petermann JH. Prozessentwicklung und
Verlustanalysen für dünne monokristalline
Siliziumsolarzellen und deren Prozessierung auf
SUMMARY
This paper presented a new optical model for
parameterizing the experimental hemispherical reflectance
and the depth-dependent photogeneration. We combine this
analytical model with numeric transport simulations using the
CoBo model as implemented in Quokka. With these
simulations tools we conduct a Synergistic Efficiency Gain
Analysis (SEGA) of a PERC and an IBC cell. We consider
SEGA to be advantageous for experimentalists seeking to
improve their cells because
•
•
•
•
it solely relies on experimentally easily accessible
input parameters (e.g. sheet resistances) due to the
CoBo model,
it directly quantifies the impact of these measurable
parameters on the cell efficiency,
it treats resistive, optical and recombinative losses
on equal footing,
it does not blur the relative sizes of losses by
prejudicing a specific sequence of loss elimination,
Modullevel. Ph.D. Thesis, Fakultät für Mathematik und
Physik, Leibniz University Hannover, 2014.
[6]
[7]
[8]
[9]
C. Kranz, T. Dullweber, and R. Brendel, Simulationbased efficiency gain analysis of 21.2%-efficient
screen-printed PERC solar cells. Presented at
SiliconPV 2015, in Konstanz, March 23-25, 2015.
Brendel R. Sunrays: A versatile ray tracing program for
the photovoltaic community. in 12th European
Photovoltaic Solar Energy Conference. WIP:
Amsterdam, 1994, pp. 1339 -1442.
R. Brendel, S. Hermann, S. Eidelloth, P. Altermatt, T.
Neubert, A. Merkle, T. Brendemühl. Laser processing
and loss analysis of emitter-wrap-through solar cells.
19th Workshop on Crystalline Silicon Solar Cells &
Modules: Materials and Processes, August 9 - 12, 2009;
Vail Cascade Resort: Vail, Colorado.
Brendel R. Modeling solar cells with the dopantdiffused layers treated as conductive boundaries.
Progress in Photovoltaics 2012; 20: 31-43. DOI
10.1002/pip.954.
[10] Fell A. A free and fast three-dimensional/twodimensional solar cell simulator featuring conductive
boundary and quasi-neutrality approximations. IEEE
Transactions on Electron Devices 2013; 60: 733-738.
DOI 10.1109/TED.2012.2231415.
[11] Blakers AW, Wang A, Milne AM, Zhao JH, Green
MA. 22.8-PERCENT EFFICIENT SILICON SOLARCELL. APPLIED PHYSICS Letters 1989; 55: 13631365. DOI Doi 10.1063/1.101596.
[12] Lammert MD, Schwartz RJ. The interdigitated back
contact solar cell: A silicon solar cell for use in
concentrated sunlight. IEEE Transactions on Electron
Devices 1977; 24: 337-342. DOI 10.1109/TED.1977.18738.
[13] Hannebauer H, Dullweber T, Baumann U, Falcon T,
Brendel R. 21.2%-efficient fineline-printed PERC solar
cell with 5 busbar front grid. physica status solidi
(RRL) – Rapid Research Letters 2014; 8: 675–679. DOI
10.1002/pssr.201409190.
[14] Merkle A, Peibst R, Brendel R. High efficient fully ionimplanted, co-annealed and laser-structured back
junction back contacted solar cells. in Proc. 29th
European Photovoltaic Solar Energy Conference. WIPRenewable Energies: Munich, 2014, pp. 954 - 958.
DOI 10.4229/EUPVSEC20142014-2AV.2.61.
[15] Brendel R, Scholten D. Modeling light trapping and
electronic transport of waffle-shaped crystalline thin-
film Si solar cells. Applied Physics A-Materials Science
& Processing 1999; 69: 201-213. DOI DOI
10.1007/s003390050991.
[16] This paper was selected and accepted for publication in
the peer reviewed EU PVSEC 2015-Special Issue of
Progress in Photovoltaics. The complementary material
of this publication contains the Matlab code for Eq.
(11) and (13). Please cite the special issue rather than
this conference contribution.
[17] Rudiger M, Greulich J, Richter A and Hermle M.
Parameterization of free carrier absorption in highly
doped silicon for solar cells. IEEE Transactions on
Electron Devices 2013; 60: 2156-2163. DOI
10.1109/TED.2013.2262526.
[18] Klaassen DBM. A unified mobility model for device
simulation - I. Model equations and concentration
dependence. Solid-State Electronics 1992; 35: 953-959.
DOI http://dx.doi.org/10.1016/0038-1101(92)90325-7.
[19] Richter A, Hermle M, Glunz SW. Reassessment of the
limiting efficiency for crystalline silicon solar cells.
IEEE Journal of Photovoltaics 2013; 3: 1184-1191.
DOI 10.1109/jphotov.2013.2270351.
[20] Smith DD, Cousins P, Westerberg S, De JesusTabajonda R, Aniero G, Yu-Chen S. Toward the
practical limits of silicon solar cells. IEEE Journal of
Photovoltaics 2014; 4: 1465-1469. DOI
10.1109/JPHOTOV.2014.2350695.