Metamodeling of numerical device simulations to

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ScienceDirect Energy Procedia 00 (2017) 000–000 www.elsevier.com/locate/procedia

7th International Conference on Silicon Photovoltaics, SiliconPV 2017

Metamodeling of numerical device simulations to rapidly create efficiency optimization roadmaps of monocrystalline silicon PERC cells Sven Wasmer*, Andreas A. Brand, Johannes M. Greulich Fraunhofer Institute for Solar Energy Systems ISE, Heidenhofstr. 2, 79110 Freiburg, Germany

Abstract In this contribution, we present an approach to simulate the energy conversion efficiencies of monocrystalline p-type silicon passivated emitter and rear cells (PERC) using a state-of-the-art design of experiment (DoE) and metamodeling approach. We preserve the accuracy of numerical device simulations whilst reducing the time for a simulation of cell efficiency from potentially hours to milliseconds. We show that only 1000 numerical simulations arranged in a space-filling DoE and metamodeling by Gaussian process regression are sufficient to cover a 13-dimensional input space, with a small mean absolute error of only 0.056%abs determined in a 10-fold cross validation. In the second part of this work, we apply the metamodel to iteratively scan this input space for the technological improvements that allow the largest efficiency gains. Simultaneously, we consider physical and technological constraints on the input parameters and optimize technologically freely changeable parameters after each step for a fair comparison. We present a roadmap that enables the production of more than 23% efficient monocrystalline p-type silicon PERC cells. Steps on this roadmap are the permanent regeneration of the boron-oxygen defect and the introduction of a selective emitter, followed by an improved front side metallization, an improved local back surface field and an improved rear passivation. © 2017 The Authors. Published by Elsevier Ltd. Peer review by the scientific conference committee of SiliconPV 2017 under responsibility of PSE AG. Keywords: Numerical Simulations; Metamodeling; PERC; Roadmap

* Corresponding author. Tel.: +49 (0)761 / 4588-5075; fax: +49 (0)761 / 4588-9250. E-mail address: [email protected] 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer review by the scientific conference committee of SiliconPV 2017 under responsibility of PSE AG.

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000

1. Introduction Numerical device simulations of crystalline silicon solar cells inherently provide the most accurate results, but are generally time-consuming and not easily accessible. Besides developing analytical models that may lack the accuracy and generality of numerical simulations, a solution can be to use a design of experiment (DoE) and metamodeling approach, as was done e.g. via polynomial regression in Ref. [1]. Such a metamodeling approach has the advantage of reducing the needed number of time-consuming numerical simulations, while preserving their accuracy to a sufficient extent, and using them more efficiently. In this work, we follow a similar approach, but use a space-filling DoE [2,3] covering wide input parameter ranges and apply Gaussian process regression, also known as Kriging in the field of geostatistics [4,5], specialised for computer experiments [6], capable of interpolating arbitrary smooth functions while needing very little assumptions on the interpolation model. Previous publications deal with the utilization of numerical device simulations to generate technological roadmaps on how to optimize cell efficiencies of monocrystalline p-type silicon passivated emitter and rear cells (PERC, [7]) most effectively [8,9]. Here, we make full use of the advantages of the metamodel concerning computation times and accessibility, as we consider analytically implemented physical and technological constraints and search for the optimal values of technological free parameters automatically by sweeping. Besides the application shown here, the advantages of the metamodel can also be used to analyse efficiency variations [10] and identify room for efficiency improvement and cost savings using current technologies [11]. 2. Approach 2.1. Numerical simulations We consider a two-dimensional symmetry element in the numerical device simulator Sentaurus TCAD [12], as depicted in Fig. 1a). The front side is modelled as in Ref. [13], thus using a transparent full-area front contact with homogeneous recombination and resistive properties. The series resistance contribution of the front side Rs,front as well as the dark saturation current density j02 of a global second diode are considered analytically. A summary of the electrical models and constants used is given in Refs. [14,15]. For the front side, we use a fixed phosphorus doping density profile and only vary the effective recombination velocity at the surface Seff,front in the numerical simulations. Concerning the effective dark saturation current density j0e,eff of the front side, we first calculate j0e,eff with j0e,eff = (1 – fmet,front)·j0e,pass + fmet,front·j0e,met

(1)

where j0e,pass and j0e,met are the dark saturation current densities of the passivated and metallized emitter,

Front passivation n+ emitter

BSF a)

Al

dBSF ry

p-type Si

Rear pass.

Al

Si

rx b)

Fig. 1. a) Schematic cross section of a PERC solar cell with line-shaped rear contacts which is base for the 2D symmetry element used in the numerical simulations. The actual modelled rear contact is depicted with the dotted line, the front contact is modelled full-faced. In b), the implemented rear contact is shown together with the variable geometric parameters.

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000

respectively, and fmet,front is the metallization fraction. For each j0e,eff, the corresponding value for Seff,front is obtained via simulations of a refinement of the Kane-Swanson method [16] of symmetrical samples and used in the numerical device simulations of the solar cells. For the consideration of the local back surface field (BSF) and its dark saturation current density j0,BSF, we create an elliptical shaped local rear contact, as shown in Fig. 1b), accounting for the typical shape of a locally alloyed aluminium contact that forms on line-shaped ablated local contact openings of the rear side passivation. Thus, impacts on the generation rate and the current flow patterns are realistically considered. The contact is placed at the interface between the aluminium eutectics and the BSF. From the contact inwards to the silicon, we model the BSF with a linearly increasing concentration of aluminium dopants from 1.7∙1018 cm-3 to 5.4∙1018 cm-3 that ends almost abruptly after a variable thickness dBSF. We implement the abrupt doping profile (see e.g. Ref. [17]) using an analytical approximation of the Heaviside step function and allow an exponential decay rate of 22/dBSF, given in µm-1, at the interface between aluminium doping and silicon in order to smooth out meshing issues of the Al doped region. Further variable parameters of the local rear contacts are the semi-axis rx and ry of the elliptically shaped eutectics. For the sake of simplicity, we fix ry at 20 µm and only vary rx, later denoted as rBSF. Similar to the determination of the recombination parameters of the front side, we extract the dark saturation current density of the local BSF j0,BSF for varying dBSF via simulations of dark current-voltage (IV) curves. For each j0,BSF, the corresponding dBSF is then used in the device simulation of the solar cell. Concerning the optical simulations of the alkaline-textured cells featuring random pyramids at the front side, we use the ray-tracer of Sentaurus TCAD to model the depth-dependent generation profile for four different wafer thicknesses dSi, while keeping the thickness of the anti-reflection coating layer dSiNx fixed at 80 nm and the parameters of the Phong-model [18] for the reflection at the rear side at R0 = 0.935 and  = 2. In the electrical model using an effective front side, we scale the generation rate by (1-fmet,front) in order to account for the shading fraction due to the front side metallization fmet,front and by (1-Rfront) in order to have a free parameter Rfront if for example a dual-layer anti-reflection coating is considered. 2.2. Metamodeling Within our DoE and metamodeling approach based on Gaussian process regression, we arrange the input parameters in a so-called nearly-orthogonal latin hypercube (NOLH) design [2,3], with a design creator available online [19]. Compared to standard latin hypercube designs, these show low pair-wise correlation coefficients which Table 1. Input parameters and their design. Parameter

Unit

Explanation

Min.

Max.

Number of steps

Logarithmically distributed?

Start value

dSi wtot

µm µm

50 150

200 1500

4 1000

no no

150 553

rBSF

µm

5

60

1000

no

25

Ndop fmet,front Rfront

cm-3 -

4.7∙1015 0.005 -0.025

7.4∙1016 0.05 0.025

1000 1000 1000

yes no no

1.56∙1016 0.045 -0.015

Spass,rear

cm/s

1

100

1000

yes

40

bulk

µs

10

3000

1000

yes

58.6

Rs,front j02

Ωcm² nA/cm2

0 0

1 20

1000 1000

no no

0.40 0

j0,BSF

fA/cm²

104

3000

1000

yes

1500

j0e,pass j0e,met

fA/cm² fA/cm²

Wafer thickness (optical sim.) Half of the distance between two local rear contacts Radius of the elliptical local BSF Base doping concentration Shading ratio due to front met. Scaling of optical generation rate by (1-Rfront) Surface recombination at the passivated rear side Bulk minority charge carrier lifetime Series resistance front side Dark saturation current density of external diode Dark saturation current density of the local BSF Dark saturation current densities of the passivated (pass) and metallized (met) emitter

24 24

150 2500

1000 1000

yes yes

80 1500

process regression (%)

 predicted by Gaussian

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000

24

10-fold cross validation Mean absolute error: 0.056%abs.

22 20 18 16 14 14

16

18

20

22

24

 by numerical simulations (%) Fig. 2. Prediction accuracy of the metamodel for the cell efficiency  via Gaussian process regression determined in a 10-fold cross validation. A very low mean error of 0.056%abs is achieved, confirming the validity of the metamodel for the whole input space.

guarantees that no main, quadratic, and two-way interaction effects of the regression model are confounded with each other. The core property of latin hypercube designs is that one-dimensional projections are perfectly uniform, with the values of each input dimension only occurring once and being distributed equidistantly. The input parameters together with their ranges are given in Table 1 and are evenly distributed in the DoE on linear or logarithmical scale (last column), the latter one proving to be able to smoothen steep gradients as occurring for example for bulk lifetimes smaller than 100 µs. We carry out altogether 1000 numerical device simulations of PERC solar cells in the NOLH design and interpolate the results for the solar cell conversion efficiency  with Gaussian process regression. For that purpose, we utilize the user-friendly implementation of Gaussian processes in the Python package “scikit-learn” [20] which we later use in our scripts for the creation of the roadmap. Concerning the parameters of the Gaussian process, we optimize the hyperparameters isotropically, that is allowing different variances for each dimension of the 13-dimensional Gaussian kernel functions. A constant underlying regression model is assumed such that the interpolation is solely done by the addition of the 1000 kernel functions. A small uncertainty of 0.3%rel is allowed for these supporting points and the metamodel is tested in a 10-fold cross validation, see Fig. 2. A very low mean absolute error of 0.056% abs for  is achieved, confirming the high accuracy of the metamodel for the whole input space. 3. Roadmap In order to standardize the creation of roadmaps, we distinguish between input parameters that take a fixed value, free parameters (e.g. pitch of the local contacts on the rear side) and constrained input parameters (e.g. bulk doping concentration and bulk lifetime). The free and constrained parameters are being optimized for highest  after each step on the roadmap using the advantages of the metamodel, while the fixed parameters remain constant. This prevents parameters with monotonic impact on  such as bulk or Rs,front from being pushed to the extreme values. We carry out the global optimization by iteratively conducting one-dimensional optimizations until convergence is achieved. 3.1. Constraints on input parameters As we simulate within our DoE every combination of input parameter values, certain constraints on these have to be implemented before the creation of the roadmap, as input parameters can be correlated. These correlations can be challenging in terms of optimization problems, but are fully taken care of in the presented approach. In our case here, on the one hand, we consider the dependency of the bulk lifetime on the bulk doping concentration [21,22] as shown in Fig. 3a), illustrating maximum achievable efficiencies of 20.6% in the degraded ([Oi] = 6∙1017 cm-3, f = 2) and 21.5% in the regenerated state of the boron-oxygen defect at the starting point of our analysis. On the other hand, the parameters of the assumed full-area front contact are not changeable independently, namely the series

+ 0.96%abs.

100

10 5 6 7 8 910

a)

20

21.8 21.5 21.2 20.9 20.6 20.3 20.0 19.7 19.4 19.1 18.8

30 40 50 6070

Base doping concentration Ndop (1015 cm-3)

b)

1.0 0.8

Finger width wfinger / resistance Rfinger

23.5 23.3 23.1 22.9 22.7 22.5 22.3 22.1 21.9 21.7 21.5

Standard (50 µm / 46 /m) Improved (20 µm / 143 /m)

Efficiency  (%)

BO deactivated (Walter 2016)

Front series resistance Rs,front (cm2)

1000

BO activated (Bothe 2005) [Oi] = 61017 cm-3, f = 2

Efficiency  (%)

Bulk lifetime bulk (µs)

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000

0.6 0.4

+ 0.45%abs.

0.2 Selective emitter 0.0 2.5

3.0 3.5 4.0 4.5 Shading ratio fmet,front (%)

5.0

Fig. 3. a) Simulated cell efficiencies at the starting point dependent on the bulk lifetime bulk and the base doping concentration on a grid of 50x50 equidistant points. For every point, the front metallization ratio and the pitch of the local contacts on the rear side are optimized. The constraints given by the degraded (solid) and regenerated parametrization (dashed) of the boron-oxygen defect are depicted as lines. b) Similar plot after the implementation of the selective emitter for the constraint between the series resistance contribution of the front side Rs,front and the shading ratio due to the front metallization fmet,front for standard and improved finger widths.

resistance contribution of the front side Rs,front and the shading ratio due to the front metallization fmet,front. The contributions to Rs,front include the finger resistance Rfinger, specific contact resistance c and the sheet resistance of the emitter Rsh, while fmet,front is given by the number #finger and widths wfinger of the fingers and a constant part by the busbars. Given these parameters, an analytical relation Rs,front(fmet,front) can be deduced [23]. We allow the optimization along that curve because the number of fingers is technologically freely changeable. Two exemplary constraints are given in Fig. 3b) after the implementation of the selective emitter for standard screen printed metallization (wfinger = 50 µm, Rfinger = 46 /m) and for an assumed improvement by thinner fingers (wfinger = 20 µm, Rfinger = 143 /m). An efficiency increase of +0.45%abs can be observed, while the optimal numbers of fingers are 103 and 156 for the standard and improved metallization, respectively. In case of the homogeneous emitter, the efficiency increase turned out to be +0.3%abs, highlighting an increased benefit of thinner fingers with the introduction of a selective emitter. Detailed input parameter values will be given in the next chapter. 3.2. Results and discussion We start with a typical parameter set for a monocrystalline silicon PERC cell with degraded p-type base material, screen-printed and fired contacts on a homogeneous emitter and orient ourselves by Ref. [24]. The values of the chosen single parameters are given in the last column of Table 1. Concerning the optical and resistive properties of the front side at the starting point, we choose experimentally determined values of wfinger = 50 µm, Rfinger = 46 /m, c = 4 mcm² and Rsh = 80 /sq. The tested scenarios and references that were used as guidance are listed in Table 2. The efficiency improvement of each scenario is evaluated in every step of the roadmap. The scenario with the largest improvement is then chosen and set as new starting point for the next iteration of the evaluation of the remaining scenarios. The suggested improvements ranked according to their efficiency gain are (1) regeneration of the base material, (2) selective emitter, (3) improved front side metallization, (4) improved local BSF and (5) improved rear passivation as shown in Fig. 4a), resulting in a total  increase of 2.8%abs, pointing out that efficiencies well above 23% are possible with the PERC concept on monocrystalline p-type silicon using industrial-feasible technological improvements. Another advantage of our approach is that the optimal base resistivity after each step can be determined conveniently and we find it to change significantly in the roadmap, directly related to the bulk lifetime and dependent on the overall level of recombination in the cell.

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000 Table 2. Tested scenarios. Number

Name

New parameters

1 2a)

Regeneration [22] Selective emitter [25]

2b)

Homogen. Emitter [26]

3

Front metallization [27]

4 5

Local BSF [28] Rear side passivation [29]

Parameterization bulk(Ndop) j0e,pass = 25 fA/cm² j0e,met = 180 fA/cm² Rsh = 135 /sq c = 0.5 mcm² j0e,pass = 38 fA/cm² j0e,met = 1500 fA/cm² Rsh = 94 /sq c = 4 mcm² wfinger = 20 µm Rfinger = 143 /m j0,BSF = 300 fA/cm² Spass = 10 cm/s

1.0

22.5 0.8

22.0 21.5

0.6

21.0 20.5

int

r on S F at i on ion t te ati lB mi rat e e l liz ca assiv n n i e a o t e t v r L i g e t p m Sta Re elec ar nt S Re Fro a) o gp

0.4

23.5 23.0

With improved homogeneous emitter Cell efficiency Optimal base resistivity

1.2 1.0

22.5 22.0 21.5

0.8 0.6

21.0 20.5

r int on on SF ation t te po erati lizati lB mi g e l ca assiv n . n i a o t n e t r L e g e p g Sta Re nt m ar mo Re Ho Fro b)

0.4

Optimal base resistivity b (cm)

Cell efficiency  (%)

23.0

1.2

Cell efficiency  (%)

With selective emitter Cell efficiency Optimal base resistivity

23.5

Optimal base resistivity b (cm)

The results underline the findings of another work [24], where the front side of current PERC cells was determined to be responsible for the major losses after a permanent deactivation of the boron-oxygen defect. Another key result is the identification of the need for an improved front side metallization, i.e. thinner fingers, after the implementation of the selective emitter, accounting for the increased sheet resistance in the homogeneous area of these emitters. As pointed out in the end of the last chapter, the improved front side metallization would not be as effective with a homogeneous emitter. However, the implementation of a selective emitter is not straightforward concerning the alignment process of the selective region and the front side metallization as well as the additional process steps needed for the formation of the lowly and of the highly doped regions. We therefore also give in Fig. 4b) the roadmap in case the step of the introduction of the selective emitter is omitted and only the scenario of an improved homogeneous emitter is tested (see Table 2, scenario 2b). Here, we find the order of the steps to be very similar to the case with the selective emitter and identify a maximum possible efficiency of 22.6%. Note that for that purpose we assume the front contact properties of the saturation current density beneath the contacts j0e,met and the contact resistivity c of the improved emitter to remain comparable to the ones of the more highly doped emitter at the starting point. Especially j0e,met poses a challenge to future silver pastes development, while c seems to be able to be kept low [26]. For an assumed higher j0e,met = 3000 fA/cm², we find the efficiency increase due to the improved homogeneous emitter at each step of the roadmap to be below 0.1%abs and the maximum efficiency to saturate at 22.3% (not shown here).

Fig. 4. Evolution of  and optimal base resistivity by iteratively using the improvement that leads to the largest increase of , each for the roadmap a) with selective and b) with improved homogeneous emitter.

Sven Wasmer et al. / Energy Procedia 00 (2017) 000–000

4. Summary and conclusion We presented a metamodeling approach to efficiently create optimization roadmaps of monocrystalline p-type silicon passivated emitter and rear solar cells (PERC). Base for our metamodel were numerical device simulations, where we introduced the implementation of the elliptical shape of the cross-section of the local line contacts of the rear side of standard PERC cells. We showed the ability of the chosen metamodeling via Gaussian process regression to reliably interpolate 1000 numerical simulations conducted in a space-filling design of experiment of 13 typical solar cell modeling parameters, each varied on a broad range of technologically realistic values. For the creation of the roadmap, we chose a scenario-based approach, hence comparing each possible improvement by its actual efficiency increase and iteratively choosing the improvement leading to the highest one. In the process, we considered physical and technological constraints on the input parameters and optimized freely variable parameters for a fair comparison. Steps on the roadmap presented here are the permanent regeneration of the boron-oxygen defect and the introduction of a selective emitter, followed by an improved front metallization, an improved local back surface field and an improved rear passivation, increasing the efficiency from 20.6% at the starting point to 23.4% altogether. If the challenging step of introducing a selective emitter is omitted, we still found an efficiency of 22.6% possible in the case of an improved homogeneous emitter. Using the advantages of the metamodel concerning applicability and speed, changes to input parameter values and scenarios can be conducted conveniently and hence a quick transfer to different PERC production lines is possible. Furthermore, the approach can basically be transferred to other solar cell concepts. This work, together with cost assessments of the different tested scenarios, lays ground for a realistic evaluation of incremental improvements of the PERC technology. Another interesting application, extending the scope of this work, would be to couple the metamodeling approach with process simulations to optimize e.g. the emitter diffusion profile and emitter diffusion process parameters. Acknowledgements This work was conducted within the project “CUT-B” (0325910A), supported by the German Ministry for Economic Affairs and Energy. The authors would like to thank E. Lohmüller, P. Saint-Cast and S. Werner of “Project Management PERC” for fruitful discussions and experimental input. Sven Wasmer gratefully acknowledges the support by scholarship funds from the State Graduate Funding Program of Baden-Württemberg. References [1] Müller M, Altermatt PP, Wagner H, Fischer G. Sensitivity analysis of industrial multicrystalline PERC silicon solar cells by means of 3-D device simulation and metamodeling. IEEE Journal of Photovoltaics 2014;4:107–13, doi:10.1109/jphotov.2013.2287753. [2] MacCalman AD, Vieira H, Lucas T. Second-order nearly orthogonal Latin hypercubes for exploring stochastic simulations. Journal of Simulation 2016;58:371, doi:10.1057/jos.2016.8. [3] MacCalman AD. Flexible space-filling designs for complex system simulations. Dissertation, Naval Postgraduate School. Monterey, California; 2013. [4] Krige DG. A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Southern African Institute of Mining and Metallurgy 1951;52:119–39. [5] Matheron G. The intrinsic random functions and their applications. Adv. Appl. Probab. 1973;5:439–68, doi:10.1017/S0001867800039379. [6] Sacks J, Welch WJ, Mitchell TJ, Wynn HP. Design and analysis of computer experiments. Statist. Sci. 1989:409–23, doi:10.1214/ss/1177012413. [7] Blakers AW, Wang A, Milne AM, Zhao J, Green MA. 22.8% efficient silicon solar cell. Applied Physics Letters 1989;55:1363–5. [8] Min B, Wagner H, Müller M, Neuhaus H, Brendel R, Altermatt PP. Incremental Efficiency Improvements of Mass-Produced PERC Cells Up to 24%, Predicted Solely with Continuous Development of Existing Technologies and Wafer Materials. In: 31st European Photovoltaic Solar Energy Conference and Exhibition; 2015, p. 473–6. [9] Altermatt PP, McIntosh KR. A roadmap for PERC cell efficiency towards 22%, focused on technology-related constraints. Energy Procedia 2014;55:17–21, doi:10.1016/j.egypro.2014.08.004.

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