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Appendix
Towards the maximum efficiency design of a perovskite solar cell by material properties tuning: A multidimensional approach
Manfred Georg Kratzenberg1,2*, Ricardo Rüther2 & Carlos Renato Rambo1
1 Electrical Materials Laboratory (LAMATE), Department of Electrical Engineering, Universidade Federal de Santa Catarina (UFSC), Trindade, PO Box 476, 88040-970 Florianópolis, Santa Catarina (SC), Brazil, 2 Fotovoltaica UFSC – Solar Energy Research Laboratory, Department of Civil Engineering, Universidade Federal de Santa Catarina (UFSC), Av. Luiz Boiteux Piaza, 1302, Florianópolis, SC, Brazil.
A. Further results and discussions A.1 Nonlinearities in two-dimensional optimizations A.2 Variable tuning in multidimensional optimizationsA.3 Combined variable contributions as a function of the variable improvement factorA.4 Ideal setups of the absorber layer thicknessA.5 Further working principles of the multidimensional optimizationA.6 Increase of the short circuit current density
B. Discussion about the adopted modeling
B.1 Applicability and accuracy of the present modeling B.1.1 Beer-Lambert lawB.1.2 Validation with the optical transfer matrix method B.1.3 Finite-difference time-domain modeling for light trapping B.1.4 Simulation of the short circuit current density for different absorber layer thicknessesB.1.5 Modeling with shape-optimized nanoparticlesB.2 Modeling of the drift-diffusion coefficients and the charge carrier mobilitiesB.3 Obtaining some of the material properties by regression
C. Flowchart of the optimization process
Supplementary References
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A. Further results and discussions
A.1 Nonlinearities in two-dimensional optimizations
If we optimized the cell’s PCE as a function of two model variables, we obtained the highest
efficiency value of 20 %, by the simultaneous reduction of the surface recombination velocities as
related to the back- and front charge conduction layers, sb, and sf, (Table A.1). Similar as in the
example of the multidimensional optimization, the two-dimensional optimization leads, with its PCE
growth of 20% - 15.7% = 4.3%, to a much higher efficiency increase as, in comparison, the summed
one-dimensional PCE increases of its model variables, which is 0.2% + 0.1% = 0.3% for sf and sb.
The substantial differences between the one- and the two-dimensional optimizations surprises and,
points to the strong nonlinearities as inherent to the drift-diffusion model, which is also visible, e.g.,
in Figs. 2c, 2e, 2f, and figure 4a in (Sun et al., 2015).
A.2 Variable tuning in multidimensional optimizations
In all of our optimizations, excluding Table A.4, we adopted in this section as initial conditions of
the nine-dimensional optimization scheme the material properties and absorber layer thickness as
obtained in (Sun et al., 2015), inclusively the short circuit current density. In the simplified fB-
constrained optimizations, we restricted the whole set of variables with a boundary-amplification
factors up to the value of fB = 160, inclusive t0 and ave, and we consider that advanced or combined
light trapping schemes (B.1.5) result to the increase of the short circuit current densities, even in cells
with thinly absorber layers. These simplified and overidealized constraints result in a higher
efficiency value of 29% (Table A.2). This table is also useful to understand better how the
optimization work and helps to identify dependencies in-between model variables.
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Table A.2
Optimization data and results related to the nine-dimensional optimization (boldface symbols) by use of the maximal boundary expansion of the model variables (fB = 160) for the whole set of nine optimization variables, considering a nearly ideal light-trapping scheme with shape-optimized nanoparticles; Jsc = 23 mA/cm2.
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Table A.1 Efficiency values and model variables, as obtained from the optimizations in the one- and two-dimensional spaces, considering different boundary - extensions factor fB, and using as setup configuration, the variable values obtained in (Sun et al., 2015), which present the initial efficiency of 15.7 %.
fB = 1.1 fB = 1.5 sf sb n p Vbi Dn Dp t0 ave sf sb n p Vbi Dn Dp t0 ave
sf 15.8 15.9 15.8 15.8 16.8 15.9 15.9 15.9 15.9 sf 15.9 16.0 15.9 15.9 17.6 16.5 15.9 16.2 16.2sb 15.8 15.8 15.8 16.7 15.8 15.8 15.8 15.8 sb 15.8 15.8 15.8 17.2 16.1 15.8 15.8 15.8
n 15.7 15.7 16.7 15.8 15.8 15.8 15.8 n 15.7 15.7 17.2 16.1 15.8 15.8 15.8p 15.7 16.7 15.8 15.8 15.8 15.8 p 15.7 17.2 16.1 15.8 15.8 15.8Vbi 16.7 16.8 16.7 16.7 16.8 Vbi 17.2 17.4 17.2 17.2 17.3Dn 15.8 15.8 15.9 15.9 Dn 16.1 16.1 16.1 16.2Dp 15.8 15.8 15.8 Dp 15.8 15.8 15.8t0 15.8 15.8 t0 15.8 16.1
ave 15.8 ave 15.8
fB = 5 fB = 20
sf sb n p Vbi Dn Dp t0 ave sf sb n p Vbi Dn Dp t0 ave
sf 15.9 17.3 15.9 15.9 18.3 17.8 15.9 17.1 17.3 sf 15.9 18.5 15.9 15.9 18.4 18.1 15.9 17.8 18.1sb 15.8 15.8 15.8 17.3 17.2 15.8 15.8 15.8 sb 15.8 15.8 15.8 17.3 18.3 15.8 15.8 15.8
n 15.7 15.7 17.2 17.0 15.8 15.8 15.8 n 15.7 15.7 17.2 17.7 15.8 15.8 15.8p 15.7 17.2 17.0 15.8 15.8 15.8 p 15.7 17.2 17.7 15.8 15.8 15.8Vbi 17.2 17.9 17.3 17.2 17.4 Vbi 17.2 18.2 17.3 17.2 17.4Dn 17.0 17.2 17.0 17.2 Dn 17.7 18.2 17.7 18.0Dp 15.8 15.8 15.8 Dp 15.8 15.8 15.8t0 15.8 17.2 t0 15.8 18.2
ave 15.8 ave 15.8 fB = 40 fB = 60 sf sb n p Vbi Dn Dp t0 ave sf sb n p Vbi Dn Dp t0 ave
sf 15.9 19.0 15.9 15.9 18.4 18.1 15.9 18.0 17.0 sf 15.9 19.3 15.9 15.9 18.4 18.0 16.0 18.0 17.0sb 15.8 15.8 15.8 17.3 18.7 15.8 15.8 15.8 sb 15.8 15.8 15.8 17.3 19.0 15.8 15.8 15.8
n 15.7 15.7 17.2 17.9 15.8 15.8 15.8 n 15.7 15.7 17.2 18.0 15.8 15.8 15.8p 15.7 17.2 17.9 15.8 15.8 15.8 p 15.7 17.2 18.0 15.8 15.8 15.8Vbi 17.2 18.3 17.3 17.2 17.4 Vbi 17.2 18.3 17.3 17.2 17.4Dn 17.9 18.7 17.9 18.3 Dn 18.0 19.0 18.0 18.3Dp 15.8 15.8 15.8 Dp 15.8 15.8 15.8t0 15.8 18.7 t0 15.8 19.0
ave 15.8 ave 15.8 fB= 80 fB = 100 sf sb n p Vbi Dn Dp t0 ave sf sb n p Vbi Dn Dp t0 ave
sf 15.9 19.5 15.9 15.9 18.4 18.0 19.5 18.1 17.0 sf 15.9 19.7 15.9 15.9 18.4 18.0 16.0 18.1 17.0sb 15.8 15.8 15.8 17.3 19.2 15.8 15.8 15.8 sb 15.8 15.8 15.8 17.3 19.4 15.8 15.8 15.8
n 15.7 15.7 17.2 18.0 15.8 15.8 15.8 n 15.7 15.7 17.2 18.0 15.8 15.8 15.8p 15.7 17.2 18.0 15.8 15.8 15.8 p 15.7 17.2 18.0 15.8 15.8 15.8Vbi 17.2 18.4 17.3 17.2 17.4 Vbi 17.2 18.4 17.3 17.2 17.4Dn 18.0 19.2 18.0 18.4 Dn 18.0 19.3 18.1 18.4Dp 15.8 15.8 15.8 Dp 15.8 15.8 15.8t0 15.8 19.2 t0 15.8 19.3
ave 15.8 ave 15.8 fB = 120 fB = 160 sf sb n p Vbi Dn Dp t0 ave sf sb n p Vbi Dn Dp t0 ave
sf 15.9 19.8 15.9 15.9 18.4 18.0 16.0 18.1 17.0 sf 15.9 20.0 15.9 15.9 18.4 18.1 19.9 18.1 17.0sb 15.8 15.8 15.8 17.3 19.5 15.8 15.8 15.8 sb 15.8 15.8 15.8 17.3 19.7 15.8 15.8 15.8
n 15.7 15.7 17.2 18.0 15.8 15.8 15.8 n 15.7 15.7 17.2 18.1 15.8 15.8 15.8p 15.7 17.2 18.0 15.8 15.8 15.8 p 15.7 17.2 18.1 15.8 15.8 15.8Vbi 17.2 18.4 17.3 17.2 17.4 Vbi 17.2 18.4 17.3 17.2 17.4Dn 18.0 19.4 18.1 18.5 Dn 18.1 19.6 18.1 18.5Dp 15.8 15.8 15.8 Dp 15.8 15.8 15.8t0 15.8 19.5 t0 15.8 19.6
ave 15.8 ave 15.8
The model variables are: sf, surface recombination velocity related to the front charge conduction layer; sb, surface recombination velocity as related to the back charge conduction layer; n, number of excess electrons per unit volume that are available for the recombination process within the p-type layer; p, number of excess holes per unit volume that are available for the recombination process within the n-type layer; Vbi, built-in voltage; Dn, diffusion coefficients of electrons; Dp, diffusion coefficient of holes; t0, absorber layer thickness; ave, average optical decay length.
Variable specification sf sb n p Vbi Dn Dp t0 ave
Units [cm / s] [cm / s] [1 / cm 3] [1 / cm 3] [V] [cm ² / s] [cm ² / s] [nm] [nm] [%]
1 - Values obtained in (Sun et al. 2015) 2.00E+02 1.92E+01 8.43E+06 1.30E+08 0.78 5.00E-02 5.00E-02 450.0 100.0 15.7
2 - Lower boundary values 1.25E+00 1.20E-01 5.27E+04 8.13E+05 0.00 3.13E-04 3.13E-04 2.81 0.63 -
3 - Lower constraint factor (eq. 5) 1 / 160 1 / 160 1 / 160 1 / 160 1 / 160 1 / 160 1 / 160 1 / 160 -
4 - Upper boundary values 3.20E+04 3.07E+03 1.35E+09 2.08E+10 1.40 8.00 8.00 1000 1.60E+04 -
5 - Upper constraint factor (eq.5) 160 160 160 160 1,8 160 160 2.22 160 -
6 - Optimized values 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 3.54 0.63 28. 98
7 – Actual modification factors 1 / 160 1 / 160 n. m. n.m. 1/8 160,0 160 1 / 127 1 / 160 -
Row 1, Configuration as obtained from the one-dimensional thickness optimization in (Sun et al. 2015), which represent the initial conditions;Row 2, The constraining lower boundary limits specified for the optimization, as related to the constraint multiplication factors in Line 3; Rowe 4, The constraining upper boundary limits specified for the optimization, as associated with the constraint multiplication factor in Line 5; Rowe 6, The ideal model variable values, as obtained from the multidimensional optimization process; Rowe 7, The actual modification factors, as calculated with the optimized values from Line 6 (n.m. means not modified).
The setup variables in Table A.2 do appear in row 1, and for a maximal boundary-expansion factor
of fB = 160, the optimization algorithm searches for the ideal properties and the ideal absorber layer
thickness, which leads to the highest simulated efficiency, as presented in row 6. The actual
modification factor shows to what extent the optimization algorithm modifies a variable for an
optimized configuration. The optimization algorithm improved most of the variables as much as
possible, adjusting its lower and upper constraints to the limits of the boundaries as set up by the
constraint factors 1/160 or 160 (Table A.2 - rows 3 and 5). Solely the variable t0 is modified
different, with the ideal value of 3.54 nm; instead, its lower constraint limit of t0 = 2.81 nm, as
defined by the fB = 1/160, which results, therefore, in the actual modification factor of fB = 1/127.
Stepwise efficiency growths, up to the maximum value of 29 % in Table A.2, are presented in Tables
A.3.1 and A.3.2. In the first table, we specify the constraints of all variables with the fB factor. In the
second table, we accomplish this variable tuning only for t0 and ave, while the remaining variables
are set up to its assumed ideal values with fB = 160, similar as in Fig. 2f. While these tables consider
a fB-constrained specification of the lower limit of ave, the used constraints in (Figs. 3 and 4 and
Tables 1 to 3), do consider, in a more conservative simulation, a higher value of ave, as a constrained
by Eqn. (9), leading therefore to a somewhat lower efficiencies.
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Table A.3.1Optimized model variables (boldface variable symbols), as obtained from eleven optimizations in a nine-dimensional function space considering a different fB factor in each of the optimizations. In the best optimization, we achieved an ideal PCE of 29% in setting up nearly perfect light trapping with shape-optimized nanoparticles in an extra thin absorber layer of t0 =3.5 nm; qGmax = 23 mA/cm2.
fB sf sb n p Vbi Dn Dp n p t0 ave [-] [cm/s] [cm/s] [1/cm3] [1/cm 3] [V] [cm²/s] [cm²/s] [cm²/Vs] [cm²/Vs] [nm] [nm] [%]
5.00 40.00 3.84 8.43E+06 1.30E+08 0.80 0.25 0.25 9.65 9.65 99.63 20.00 20.09
10.00 20.00 1.92 8.43E+06 1.30E+08 0.82 0.50 0.50 19.30 19.30 334.98 10.00 20.77
20.00 10.00 0.96 8.43E+06 1.30E+08 0.86 1.00 1.00 38.61 38.61 351.59 5.00 22.33
30.00 6.67 0.64 8.43E+06 1.30E+08 0.90 1.50 1.50 57.91 57.91 18.22 3.33 24.96
40.00 5.00 0.48 8.43E+06 1.30E+08 0.94 2.00 2.00 77.22 77.22 13.79 2.50 25.71
60.00 3.33 0.32 8.43E+06 1.30E+08 1.01 3.00 3.00 115.82 115.82 9.28 1.67 26.69
80.00 2.50 0.24 8.43E+06 1.30E+08 1.09 4.00 4.00 154.43 154.43 7.00 1.25 27.37
100.00 2.00 0.19 8.43E+06 1.30E+08 1.17 5.00 5.00 193.04 193.04 5.62 1.00 27.89
120.00 1.67 0.16 8.43E+06 1.30E+08 1.25 6.00 6.00 231.65 231.65 4.70 0.83 28.31
140.00 1.43 0.14 8.43E+06 1.30E+08 1.32 7.00 7.00 270.26 270.26 4.04 0.71 28.67
160.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 3.54 0.63 28.98
Observation: The values in italic formatted numbers correspond to the optimized configuration in Table A.2, line 6.
Table A.3.2Optimized model variables (boldface variable symbols) as obtained from eleven multidimensional optimizations, which consider the boundary adjustments in a two-dimensional in function space, as controlled by specific boundary-extension factors fB, and nearly ideal light trapping based on shape-optimized nanoparticles; qGmax = 23 mA/cm2.
fB sf sb n p Vbi Dn Dp n p t0 ave [-] [cm/s] [cm/s] [1/cm3] [1/cm 3] [V] [cm²/s] [cm²/s] [cm²/Vs] [cm²/Vs] [nm] [nm] [%]
1.25 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 443.89 80.00 26.16
1.50 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 370.22 66.67 26.26
2.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 277.96 50.00 26.43
5.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 111.67 20.00 26.96
9.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 62.18 11.11 27.31
20.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 28.07 5.00 27.77
30.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 18.75 3.33 28.01
40.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 14.08 2.50 28.17
60.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 9.41 1.67 28.41
80.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 7.06 1.25 28.58
160.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 3.54 0.63 28.98
Observation: (i) The values in italic formatted numbers correspond to the vertex line in Fig. 2f; (ii) The values formatted in boldface correspond to the optimized configuration in Table A.2, line 6.
A.3 Combined variable contributions as a function of the variable improvement factor
The advantage of multidimensional optimizations we also elucidate by the consecutive inclusion of
optimization variables in the optimization process, as presented in Fig. A.1. Similar to the two-
dimensional optimization, we obtained the highest efficiency increase by the inclusion of the two
surface recombination velocities. Since the average optical decay length must stand in closed
relationship to the absorber layer thickness, the addition of ave lead to the feeblest efficiency
increase, which is, however, raised after the inclusion of the optimized absorber layer thicknesses.
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A.4 Ideal setups of the absorber layer thickness
Perovskite solar cells stand out by the fact that they can be mass-produced, by several different low-
cost manufacturing techniques (Brittman et al., 2015; Burschka et al., 2013; Rong et al., 2017). Some
of these methods, e.g., the solvent-solvent extraction method, allow perovskite coatings with
thickness tunable coating thicknesses (Zhou et al., 2015). The minimum absorber layer thickness
with this technique is 20 nm, presents a very homogeneous surface, and does not present visible
pinholes. However, the thinnest possible layer coating of other methods, especially printing
techniques (Rong et al., 2017), is much higher.
Fig. A.1. Optimized efficiency from 160 optimizations as a function of an variable boundary-expansion factor (fB = 1…160, ℕ) which include the following sets of optimization variables: (i) only the diffusion coefficient of electrons (marked with Dn); (ii) the combination of Dn and Dp (shown as +Dp); (iii) the variables Dn , Dp and Vbi (+Vbi); (iv) the variables Dn , Dp , Vbi and sf (+sf); (v) the variables Dn , Dp , Vbi , sf and sb (+sb); (vi) the variables Dn , Dp ,Vbi , sf , sb and ave (+ave); and finally (vii) the variables Dn , Dp ,Vbi , sf , sb , ave and t0 (+t0), presenting the highest efficiency values.
As a result, we consider that each coating technique requires an inherent minimum thickness ( t0-min)
for the deposition of the pinhole-free absorber layers. Absorber layer thicknesses below t0-min would
increase the shunt resistance because of the appearing pinholes (Qiu et al., 2016) and, consequently,
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+ t0
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decrease the PSC’s efficiency. Furthermore, residual conduction effects, as related to nanoparticles in
thin absorber layers, as discussed in (Atwater and Polman, 2010), can also reduce the cell's PCE.
Therefore, in manufactured PSCs, the t0-min thickness limit should be specified to values, which
knowingly avoid or suppress undesired pinhole- and conduction-effects, leading thus to non-
dominant losses related to these effects.
A.5 Further working principles of the multidimensional optimization
Beyond the avoidance of recombination sites by passivation, which reduce the surface recombination
velocities, and the reduction of ave, the PSC’s efficiency can also be increased by (i) higher built-in
voltages, which increase the built-in electric field, and therefore, output voltage (Green, 2009); (ii)
by higher diffusion coefficients, which are related to longer diffusion lengths or shorter lifetimes of
electrons and holes (Eqns. A.7 and A.8); and (iii) an adequate relationship between ave and t0 (Fig.
2f, Tables A.2, A3.1, and A.3.2).
A diffusion length is the average diffusion distance between a charge carrier’s generation and its
recombination. Higher diffusion coefficients are also related to higher charge carrier mobilities
(Eqns. A.5 and A6), which lead to the effect that a higher number of charge carriers can be converted
into current generating electrons and holes, as a smaller number of charge carriers recombine in the
absorber layer before reaching the charge conduction layers. A thinner absorber layer is also of
advantage in this concern, as a lower diffusion length of the charge carriers is necessary to reach the
charge conduction layers.
A.6 Increase of the short circuit current density
The highest optimized efficiency of 27.6% in Table 3, is obtained because of the cell’s superior: (i)
open-circuit voltage, (ii) fill factor, and (iii) short circuit current density. However, some authors
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present cells with an higher Jsc, of e.g. 24.88 mA/cm2 (Jung et al., 2019), even without light trapping,
which is higher than the highest value Jsc of 24.5 mA/cm2, as presented in (Cai et al., 2015) for a
400 thick absorber layer and light trapping. Using this higher Jsc as fixed setup configuration in our
optimizations, we obtained without light trapping (ave = 100 nm) and a t0 = 450 nm a maximum
efficiency of 28.13% (Table A.4), because of the V0C and FF increase. Light trapping should increase
this simulated efficiency significantly, but presently, there are no FDTD simulations available, which
show how the Jsc increases further as a function of light trapping in these cells. Furthermore, (Jung et
al., 2019) use a unique concept with two absorber layers, combining a wide-bandgap absorber layer
with a narrow-bandgap absorber layer. Therefore, we cannot accomplish further optimizations
considering light trapping for this cell.
Table A.4Optimized model variables (boldface symbols) as obtained from eleven optimizations in the nine-dimensional function space considering a specific boundary-extension factor ( fB) in each optimization. Here we set up an increased short circuit current density of Jsc = 24.88 mA/cm2, which leads to an optimized PSC of 28.13% without light trapping.
fB sf sb n p Vbi Dn Dp n p t0 ave [-] [cm/s] [cm/s] [1/cm3] [1/cm 3] [V] [cm²/s] [cm²/s] [cm²/Vs] [cm²/Vs] [nm] [nm] [%]
5.00 40.00 3.84 8.43E+06 1.30E+08 1.40 0.25 0.25 9.65 9.65 450.00 100.00 21.66
10.00 20.00 1.92 8.43E+06 1.30E+08 1.40 0.50 0.50 19.30 19.30 450.00 100.00 22.95
20.00 10.00 0.96 8.43E+06 1.30E+08 1.40 1.00 1.00 38.61 38.61 450.00 100.00 24.25
30.00 6.67 0.64 8.43E+06 1.30E+08 1.40 1.50 1.50 57.91 57.91 450.00 100.00 25.00
40.00 5.00 0.48 8.43E+06 1.30E+08 1.40 2.00 2.00 77.22 77.22 450.00 100.00 25.54
60.00 3.33 0.32 8.43E+06 1.30E+08 1.40 3.00 3.00 115.82 115.82 450.00 100.00 26.30
80.00 2.50 0.24 8.43E+06 1.30E+08 1.40 4.00 4.00 154.43 154.43 450.00 100.00 26.83
100.00 2.00 0.19 8.43E+06 1.30E+08 1.40 5.00 5.00 193.04 193.04 450.00 100.00 27.25
120.00 1.67 0.16 8.43E+06 1.30E+08 1.40 6.00 6.00 231.65 231.65 450.00 100.00 27.59
140.00 1.43 0.14 6.02E+04 1.30E+08 1.40 7.00 7.00 270.26 270.26 450.00 100.00 29.15
160.00 1.25 0.12 8.43E+06 1.30E+08 1.40 8.00 8.00 308.87 308.87 450.00 100.00 28.13
B. Discussions about the adopted modeling
B.1 Applicability and accuracy of the present modeling
B.1.1 Beer-Lambert law
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The Beer-Lambert modeling calculates the optical absorption of solar irradiance, and its conversion
in free charge carriers, as a function of the absorber layer thickness, and as a function of its related
optical decay length, which is, by its own a function of the irradiance wavelength. For PSC cells, the
Beer-Lambert model was already used in (Sun et al., 2015; Adinolfi et al., 2017; Ball et al., 2015 and
Ren et al., 2017), in-between other references. For the model as proposed in (Sun et al., 2015), the
elaborated simplification of the average optical decay length (ave) accounts for the whole set of
optical decay lengths as appear for the solar irradiance spectrum. In this formulation, the cell’s short
circuit current density, of qGmax = 23 mA/cm2 (Sun et al., 2015), is proportional to the total or
maximum generation of charge carriers per cell area Gmax [m-2s-1], which we calculate as follows
. (A.1)
In Eqn. (A.1), Geff [m-3s-1] is the effective generation of charge carriers per volume unit of the
absorber layer, and ave [m] the average optical decay length. The authors obtained a simplified
solution of Eqn. (A.1) by the integration of this Beer-Lambert expression, which results in the
following equation
Gmax = Geff ave . (A.2)
For the case without light trapping, the material-specific properties, defined as Geff = 1.4356 x 1013
cm-3s-1 and ave = 100 nm, were validated with the OTM modeling method and measurements of Jsc, in
(Sun et al., 2015). The Beer-Lambert model considers that approximately 99% of the light is
absorbed in the forward direction, while 1% reflects on the back reflector or anode (Fig. 1b). For
example, when using ave = 100 nm and t0 = 450 nm, this 1% corresponds to the part of the charge
carrier generation curve, which appears for penetration depths beyond t0 (x > 450 nm) in Fig. 1b.
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/
0max )(t
x
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B.1.2 Validation with the optical transfer matrix method
A simulation method that presents a higher resolution than the Beer-Lambert model in the modeling
of the short circuit current density is the cavity-modeling based on the OTM method. This
formulation calculates the photocurrent density as a function of the (i) extinction coefficient and (ii)
refractive index. Both of these indexes are a function of the solar irradiance’s wavelength, and in its
conjunction, they form the complex refractive index. The modeling of the short circuit current
density without light trapping, as used in our model, was validated in (Sun et al., 2015) by (i) the
OTM method and (ii) the measurements with a manufactured solar cell, which result in low
deviations of only 0.1%. Exact values of the extinction coefficient and the complex refractive index
for the PSC, as used in (Sun et al., 2015), were also validated in (Jiang et al., 2015).
B.1.3 Finite-difference time-domain modeling for light-trapping
In comparison to the OTM model, an even higher-resolution simulation model of Jsc is defined by the
finite-difference time-domain (FDTD) modeling (Cai et al., 2015). Therefore, this model presents the
highest accuracy, in comparison to the Beer-Lambert, and OTM modeling, in the modeling of the
short circuit current density. The FDTD is a numerical method of computational electrodynamics,
which uses time-dependent partial Maxwell differential equations in the discretization of a tiny but
finite-volume section of the absorber layer. This method pertains to the group of finite-difference
modeling, a simulation procedure, which is most known from the finite-element method (FME). The
objective of the FDTD is to model the cell’s absorption, as accomplished over a broad frequency
spectrum of the electromagnetic solar irradiance. In the case of light trapping with plasmonic
nanoparticles, it simulates the omnidirectional scattering of incoming light, an effect inherent to
plasmonic nanoparticles, when inserted into the absorber layers of a PSC. The authors (Cai et al.,
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2015) accomplished a FDTD simulation for a PSC, which presents the same absorber material
(CH3NH3PBI3) as used in (Sun et al., 2015), and therefore, its simulation lead in a very similar short
circuit current density, considering a cell with t0 = 400 nm without light trapping. As a result, we
assume a high accuracy for our optimizations, using the short circuit current densities as simulated in
(Cai et al., 2015) for the cases with light trapping.
B.1.4 Simulation of the short circuit current density for different absorber layer thicknesses
In this paper, we set up Jsc(t0) as a function of different absorber layer thickness, using the values as
obtained in (Cai et al., 2015) for light trapping with spherical nanoparticles, as obtained by FDTD
simulations. The authors show that efficient light trapping, by the use of spherical nanoparticles,
increases the cell’s Jsc for a large range of absorber layer thicknesses form t0 = 10…400 nm. For t0 =
50 nm, an almost complete restoration of the cell’s short circuit current the authors obtained, if
compared to the case without light trapping. Already for t0 > 50 nm, light trapping increases the Jsc of
a cell without light trapping with t0 = 400 nm. Therefore, we adopted t0 = 50…400 nm in our
optimizations that results in Figs. 3 and 4.
Considering that an improved light trapping, using shape optimized nanoparticles (Appendix B.1.5),
results in a full restoration of Jsc even for t0 < 50 nm, the higher efficiency values as presented in
(Tables A.2, A.3.1, and A.3.2) can be obtained. E.g., an efficiency of 28.4% and 29% can be
achieved for cells with an absorber layer thickness of 9.4 nm and 3.54 nm, considering fB = 60 and
160 in Table A.3.2. These results we based on a fB-constrained absorber layer thickness and a qGmax
of 23 mA.
In a more conservative setup of our optimizations, as presented in Tables 1 to 3 and Figs. 3 and 4, we
did not use such thin absorber layers and used the higher minimum values of ave, as defined in the
equality constraint (Eqn. 9), which lead to lower efficiencies. As a result, we adopted in our
modeling the same relationship in-between ave and t0 as presented in (Sun et al., 2015), while the
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short circuit current density (Jsc = q Gmax) is the same as obtained by FDTD simulations (Cai et al.,
2015). Consequently, we obtain carefully selected setup conditions, which result in the maximal PCE
values of 27.8% and 28.1% in m-constrained and fB-constrained optimizations (Table 2).
B.1.5 Modeling with shape-optimized nanoparticles
Shape-optimized nanoparticles and further related up-conversion effects of these nanoparticles lead
to an improved light trapping effect in comparison to spherical nanoparticles*. Since it is presently
not known how the short circuit current density behaves with these nanoparticles, we set up in an
optimistic simplification a constant short circuit current density of qGmax = 23 mA/cm2, even for extra
thin absorber layers in Tables A.2, A.3.1, and A.3.2. This superior property is based on the
circumstance that shape-optimized nanoparticles are more effective than spherical nanoparticles
(Kakavelakis et al., 2017), and also considers, eventually, a combination of these nanoparticles with
further light trapping concepts. However, the effectiveness of shape-optimized nanoparticles in tiny
absorber layers should be qualified by a correct quantification of the attainable Jsc, e.g., by the use of
an FDTD simulation, as already presented for spherical nanoparticles in (Cai et al., 2015).
B.2 Modeling of the drift coefficients and the charge carrier mobilities
The mobilities of electrons and holes n and p, also named as drift coefficients are also improved in
our multidimensional optimizations as to see from Tables 1, 2, 3, A.2 to A5. However, we do not
specify these coefficients as optimization variables in our modeling, because these drift coefficients
* In light-trapping structures, nanoparticles are utilized to confine resonant photons by an induced coherent oscillation of electrons in an atom’s conduction band. This process is the so-called localized surface plasmon resonance (LSPR). Furthermore, specific nanoparticles can even produce an up-conversion effect of photons, transforming a part of the lower energy photons to photons with higher inherent energy, matching, therefore, the solar cell’s bandgap, and leading, thus, to the generation of free charge carriers (Kakavelakis et al., 2017). While the plasmonic effect leads to the increased absorption of the available photons and its conversion to free charge carriers, the up-conversion effect ensues, additionally, a larger generation of free charges. As an outcome of these improvements, nanoparticles reduce the solar cell’s internal series resistance; and increase its internal shunt or parallel resistance, also known as the recombination resistance. In such a purpose, the effectiveness of the plasmonic nanoparticles can be tuned by the nanoparticles constitution, size, shape, and its surrounding environment in which the nanoparticles are inserted (Kakavelakis et al., 2017). For these reasons, we consider that the results as obtained in Table 2, can be improved further, which, therefore, leads to the model applicability for absorber layers with thicknesses of t0 < 50 nm.
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are a function of the diffusion coefficients, as shown by Eqns. (A.3) to (A.5). We used these Einstein
expressions (Eqns. A.3 and A.4), to reduce the total number of model variables in the
multidimensional optimization space. In Eqns. (A.3) and (A.4), kB is the Boltzmann constant with the
value of kB = 1.38064852 × 10-23 J/K, and q is the electric charge of an electron or a hole.
(A.3)
(A.4)
From (A.3) and (A.4) follows that a constant semiconductor temperature T [K] results in a fixed
thermal voltage Vt [V]. As a result, Eqns. (A.3) and (A.4) can be converted and used to calculate the
mobilities µn and µp by Eqns. (A.5) and (A.6)
µn = Dn / Vt (A.5)
µp = Dp / Vt (A.6)
It follows that the mobilities are a direct function of the diffusion coefficients considering fixed
values of q, kB, and T. Therefore, n and n have not to be used as optimization variables in our
multidimensional optimization approach.
In the formulation of the drift-diffusion model, the calculation of the diffusion coefficients of
electrons and holes (Eqns. A.3 to A.6) is specified in (Park et al., 2012) by the following two
equations.
Dn = Ln /n ½ (A.7)
Dp = Lp /p ½ (A.8)
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Where Ln and Lp are the diffusion lengths and n and p are the lifetimes of electrons and holes. The
Ln and Lp define the pathway lengths the charge carriers diffuse, until being subjected to its
recombination at the end of this pathway. The lifetimes, n and p, represent the time interval in which
the generated charge carrier moves in the absorber layer by the diffusion process until it recombines
at the end of this pathlength.
B.3 Obtaining some of the material properties by regression
We observed that the improvement of different material properties in Sun’s model (Sun et al., 2015)
result in redundant shape improvements of the modeled J-V curve. While ave and qGmax modify the
directly the short circuit current density, improvements in the (i) diffusion coefficients and (ii) the
surface recombination velocities lead to the cell’s voltage increase, presenting similar shape
improvements. Furthermore, the built-in voltage increase results also in a voltage increase, showing,
however, a slightly different shape. Because of these redundancies, we conclude that with a
regression, it is only possible to extract some of the model’s material properties, while measurements
must determine the other material properties.
The authors (Sun et al., 2015) used fixed values for (i) the short circuit current density qGmax, and (ii)
the average optical decay length ave, which the authors both obtained from the OTM modeling. As
diffusion coefficients, the authors adopted the values as measured in (Stranks et al., 2013). Using a
regression of the J-V curve (Sun et al., 2015) estimates the remaining material properties, as based on
the J-V curves in the dark and under exposition to light.
As a result of this redundancy, we recommend that most of the material properties should be
obtained by measurements, e.g., by use of the capacitance-voltage characteristics for the
determination of Vbi (Sun et al., 2015), while the variables which are difficult to measure, or, which
present high measurement uncertainties, should be estimated by regressions.
C. Flowchart of the optimization process
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Fig. A.2 Flowchart of the NP optimization process, with (i) configuration of the considered boundary-expansion factor; (ii) inner loop, for the validation if the optimization process has the variable value increased, or decreased, beyond its upper or above its lower limits as defined by the boundary-expansion factor fB; (iii) outer loop, for the calculation of the J-V curve, the MPP-power; and validation of the convergence criterion, which specifies the minimal necessary efficiency gain as being 0.001%.
Supplementary References
Adinolfi V., Peng W., Walters G., Bakr O.M., Sargent E.H., The electrical and optical properties of organometal halide perovskites relevant to optoelectronic performance Adv. Mater. 2017.
Ball J.M., Stranks S.D., Hörantner M.T., Hüttner S., Zhang W., Crossland E.J., Ramirez I., Riede M., Johnston M.B., Friend R.H., Optical properties and limiting photocurrent of thin-film perovskite solar cells Energy Environ. Sci. 2015 602-609.
Brittman S., Adhyaksa G.W.P., Garnett E.C., The expanding world of hybrid perovskites: Materials properties and emerging applications MRS Commun. 2015 7-26.
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Burschka J., Pellet N., Moon S.-J., Humphry-Baker R., Gao P., Nazeeruddin M.K., Grätzel M., Sequential deposition as a route to high-performance perovskite-sensitized solar cells Nature 2013 316-319.
Jiang Y., Green M.A., Sheng R., Ho-Baillie A., Room temperature optical properties of organic–inorganic lead halide perovskites Sol. Energy Mater. Sol. Cells 2015 253-257.
Jung E.H., Jeon N.J., Park E.Y., Moon C.S., Shin T.J., Yang T.-Y., Noh J.H., Seo J., Efficient, stable and scalable perovskite solar cells using poly (3-hexylthiophene) Nature 2019 511.
Park J.-K., Kang J.-C., Kim S.Y., Son B.H., Park J.-Y., Lee S., Ahn Y.H., Diffusion length in nanoporous photoelectrodes of dye-sensitized solar cells under operating conditions measured by photocurrent microscopy J. Phys. Chem. Lett. 2012 3632-3638.
Rong Y., Hou X., Hu Y., Mei A., Liu L., Wang P., Han H., Synergy of ammonium chloride and moisture on perovskite crystallization for efficient printable mesoscopic solar cells Nat. Commun. 2017 1-8.
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