J Comput Electron (2014) 13:323–328DOI 10.1007/s10825-013-0533-0

Geometrical and physical optimization of a photovoltaic cellby means of a genetic algorithm

Giuseppe Alì · Francesco Butera · Nella Rotundo

Published online: 19 November 2013© Springer Science+Business Media New York 2013

Abstract A methodology for the geometrical and physicaloptimization of a photovoltaic cell is proposed, which makesuse of a detailed distributed model for the device simula-tion and a genetic algorithm. For the numerical simulationof the device, a TCAD simulator is used, appropriately in-terfaced with the genetic algorithm. Since the parametersto be optimized are geometrical, each simulation requiresa different mesh grid, which is automatically set within thegenetic algorithm optimization cycle. The evaluation of thefitness function requires the post-processing of the outputof the device simulation, which is performed by another ex-ternal software, also interfaced with the genetic algorithm.The feasibility of this methodology is assessed on a homo-geneous emitter solar cell, with some relevant free param-eters, related to the number of fingers in a cell and to thedoping profile of the emitter. The parameters which maxi-mize the efficiency of the cell are determined by using theproposed procedure.

Keywords Photovoltaics · Silicon · Solar cell · Geneticalgorithm · Optimization · Continuous and integer variables

G. Alì (B) · F. ButeraDepartment of Physics, University of Calabria, Arcavacatadi Rende, 87036 Cosenza, Italye-mail: giuseppe.ali@unical.it

G. AlìINFN, Gruppo Collegato di Cosenza, Arcavacata di Rende,87036 Cosenza, Italy

N. RotundoDepartment of Mathematics and Statistics, McGill University,Canada, Italy

1 Introduction

The last report on photovoltaic industry in Europe, publishedin October 2012, foresees that, within 2050, as much as the21 % of the power supply worldwide, that is, 6750 TWh,will be generated by photovoltaic cells [1]. Since the first so-lar cell was realized 55 years ago [2], the cost has decreasedof a factor 200, and this value is foreseen to decrease evenfurther [3]. These prevision refer to a well standardized tech-nology, with a well established structure, and any advantagecan be achieved only via a careful optimization of the exist-ing cells in order to maximize the efficiency.

The optimization of specific device parameters requiresthe use of a TCAD simulator and repeated simulations.Moreover, if the parameters have a geometrical nature, thatis, if the numerical grid used for each simulation is altered bychanging them, then each simulation requires a preliminarygrid generation. Now, performing this sequence of grid gen-erations and simulations can be computationally very chal-lenging. Even more so, if one wishes to automize the processand link the simulations to an optimization algorithm.

A possible alternative would be to reduce the complexityof the device model, by considering 1D versions of it [4],or by constructing some equivalent circuit formulations [5].We also mention two more recent papers [6, 7] which usethis approach to innovative PV cells. Then, a classical opti-mization algorithm can be successfully used, but eventuallyone has to check the validity of the results thus obtained,resorting to a detailed model and a TCAD simulator.

In this paper we present an optimization strategy, mak-ing use of a detailed distributed model for the PV cell, andbased on a genetic algorithm (GA) [8]. The idea of GA isinspired by population dynamics in biology. The parametersto be optimized are collected in an array, which is viewed asthe chromosome of an individual. Then a population of indi-viduals evolves, according to biology-inspired rules, so that

324 J Comput Electron (2014) 13:323–328

eventually the apter individuals survive, until the optimumis reached.

In the literature, the use of GA in PV cell optimization isgenerally performed with simplified models of the PV cell,and we have already mentioned some interesting examples.On the other hand, the use of GA in conjunction with de-tailed distributed models of the PV cell is not well devel-oped, although some results can be found in literature. Forinstance, this approach has been used for optimizing the fin-gers’ disposition on a cell [9]. Another interesting exampleconcerns the optimization of the output power in a multi-junction solar cell, which makes use of a TCAD device sim-ulator and parallel computing [10].

We adopt a modular approach, with different softwaretools coupled together, for a better reproducibility and ex-tension of the results. Thus, we interface the genetic algo-rithm with, on one side, the grid generation tool and theTCAD simulator, and on the other side the software used forthe post-processing of the results of each simulation, and forcomputing the fitness function. Here, the difficulty resides inthe interface between genetic algorithm, grid generation tooland TCAD simulator, and post-processing software, mainlydue to the use of proprietary software (Sentaurus [11], MAT-LAB [12]).

We apply this procedure to the optimization of some geo-metrical parameters of a solar cell, in order to maximize theefficiency of the cell. In particular, we wish to optimize thedoping profile of the emitter, modeled by a double Gaus-sian characterized by two parameters, and the number offingers on the solar cell. It is easy to understand that if thenumber of fingers increases then the portion of solar cell ex-posed to sunlight decreases, which leads to the phenomenonof “shading”, thus reducing the efficiency of the cell [13].On the contrary, by decreasing the number of fingers of thesolar cell, the surface exposed to the sunlight increases, butthe series resistance will increases and the photo-generatedcharges will recombine with a lower probability. Thus, thenumber of fingers is a discrete parameter which needs to beoptimized in order to maximize the efficiency.

We wish to stress that the resulting optimization probleminvolves both continuous and integer variables. The resultsshow that the approach by genetic algorithm is feasible andleads to an optimal efficiency, in line with similar resultsobtained by non-authomatic methods.

The paper is organized as follows. In Sect. 2 we describethe solar cell structure and the geometrical parameters to beoptimized. The PV cell simulations for each optimizationcycle are performed by using a TCAD simulator, namelySentaurus, interfaced with external software. In Sect. 3 wediscuss the interface with the external software used forcomputing the emitter doping profile, when the mesh is cre-ated, and for computing the efficiency of the cell from the

Fig. 1 Cross section solar cell structure

results of each device simulation. Section 4 details the opti-mization procedure. Finally, in Sect. 5 we present the resultsof our optimization process.

2 Description of the PV cell

We consider a homogeneous emitter (HE) single printing so-lar cell [14], in which the emitter is doped through a uniquediffusion process. The structure of the PV cell is shownin Fig. 1. The 180 µm p-type silicon substrate (Dsub) isdoped with boron acceptor atoms with a concentration of1×1016 cm−3, leading to a substrate resistivity of 1.5 � cm.The back surface field (BSF) region is modeled by a Gaus-sian function with a doping peak of 6 × 1019 cm−3 and ajunction depth of 1 µm.

The depth of the n-doped emitter region is denoted byDem, and its kink-and-tail profile is modeled by a doubleGaussian profile, which takes into account the diffusion pro-cess by which the emitter is realized, and the ionization phe-nomenon [15, 16]. The double Gaussian doping profile ofthe emitter is one of the geometrical characteristics of thePV cell which we wish to optimize. More precisely, the dou-ble Gaussian is characterized by two concentration peaks,Cpeak1 = 5 × 1020 cm−3 and Cpeak2 = 3 × 1019 cm−3, lo-cated at depth Dpeak1 = 0 and Dpeak2, respectively. The val-ues of the depth of the second peak, Dpeak2, and of the thick-ness of the emitter, Dem, are not specified, but they need tobe optimized within some known expected ranges.

The width of the solar cell, Wsub (see Fig. 1), that is, thedistance between two fingers, depends on the number of fin-gers N (see Fig. 2), which is an integer parameter to be op-timized.

For symmetry reasons, it is sufficient to perform a de-tailed simulation of a half cell only, as shown in Fig. 2. Thisis essentially a 2D structure, and the 3D behavior can bereconstructed by post-processing the results of a 2D simula-tion, as shown in the next section.

2D numerical simulations are performed by using Sen-taurus, a state-of-the-art TCAD device simulator, in whichphysical models are calibrated in order to account for typi-cal c-Si PV technologies [17, 18]. The solar cell is modelled

J Comput Electron (2014) 13:323–328 325

Fig. 2 The simulation domain

by the standard drift-diffusion equations, as implemented inSentaurus, accounting for a constant lattice temperature of298 ◦K. We do not consider temperature variations in thissimulation.

The following refinement models have been adopted: theband-gap narrowing (BGN) model by Schenk to account forthe effective intrinsic carrier density [19], the Auger recom-bination model with the parameterization adopted by Al-termatt in [20], the mobility model proposed by Klaassen[21, 22], and the bulk Shockley-Read-Hall (SRH) lifetimemodel in boron-doped Cz-Si according to Glunz’s param-eterization [23]. We have also considered the Fermi-Diracstatistics in order to model the highly-doped regions. Weadopted the parameterization proposed by Kimmerle et al.in [24]. The surface recombination velocity (SRV) at SiNx-passivated front surface is calculated by considering thechemical phosphorus surface concentration. By consideringthe standard AM1.5G spectrum and by accounting for ran-dom pyramids textured SiNx front surfaces, 1D optical gen-eration rate profiles are calculated [18] and embedded inthe TCAD simulation flow by means of an external code.We have considered a solar radiation at ground level of1000 W/m2.

We use standard conditions [11] for the Ohmic contactsat the terminals of the simulation domain, and homogeneousNeumann conditions, that is, insulating conditions in theother parts of the simulation domain.

Figure 3 shows the mesh used in Sentaurus, with about26.000 grid points. The mesh is more dense (approximately2.5 times denser) in the regions where the doping and op-tical generation rate gradients are higher and close to thediscontinuities of the structure.

3 Doping profile and computation of the efficiency

The device simulator is interfaced with an external software,for simple data processing. The double Gaussian dopingprofile of the emitter is obtained using a MATLAB routine.Figure 4 shows a typical diffusion profile obtained by theroutine as function of the diffusion depth. The figure alsoshows the two depth parameters Dpeak2 and Dem.

MATLAB routines are also used for the computation ofthe efficiency of the solar cell, which is the fitness functionof the optimization process.

Fig. 3 Solar cell mesh realized using Sentaurus

Fig. 4 The double Gaussian emitter doping profile, for Dpeak2 =0.055, Dem = 0.2

Fig. 5 Simplified equivalent circuit model for solar cell

After each simulation step, a post-processing of the out-put data is performed. Specifically, a first MATLAB routineextracts all the data required for the computation of the effi-ciency, from the output files produced by the device simula-tor.

Next, in order to recover the behavior of the original3D structure, a second MATLAB routine corrects the volt-age and current values by taking into account some resis-tive losses of the cell, as reported in [17]. Figure 5 showsa simple circuit model of the solar cell [25], upon whichthe above post-processing is based. The series resistanceRS accounts for resistive losses due to fingers, bus-bar,and metal/semiconductor contact, while the Shunt resistance

326 J Comput Electron (2014) 13:323–328

RSH accounts for the current losses in the junction, due toleakage.

Once the corrected values of voltage and current are ob-tained, the efficiency η can be computed by the well-knownformula [25]

η = VOCISCFF

Pin,

where VOC is the open-circuit voltage, ISC the short-circuitcurrent, FF the fill factor, and Pin the input power providedto the cell.

4 Optimization procedure

Our goal is maximizing the efficiency of the silicon solarcells. To do this we used an optimization strategy based ona genetic algorithm (GA) applied to determined geometricparameters of the solar cell. We briefly describe the basicGA [8] used in our optimization process, summarizing themain steps.

4.1 Initialization of GA population

The initial step of a GA is building the initial populationof individuals, determined by their “chromosomes”, that is,by randomly generated arrays. We consider an initial pop-ulation of 100 individuals. In our case every chromosomecontains a set of three parameters that are the number of fin-gers, N , and two emitter doping depths, Dpeak2, Dem. Theseparameters are chosen in the following ranges, provided byARCES and based on previous practical experience:

N : from 65 to 165,

Dpeak2: from 0.05 µm to 0.15 µm,

Dem: from 0.2 µm to 1.0 µm.

These ranges ensure that an optimal value exist, which hasbeen confirmed by our algorithm.

4.2 Evaluation of the GA population

The next step of the algorithm is the evaluation of a fitnessfunction for each individual, to determine the individualsmore suitable for survival and evolution. In our case, the fit-ness function is the efficiency of the cell. This require a setof device simulations for each individual, in order to deter-mine the I–V characteristic of the corresponding cell.

Since the chromosome of an individual encodes geom-etry-related information, the physical structure of the solarcell to be simulated changes at variance with each individ-ual. Thus each simulation must be preceded by a mesh gen-eration. Figure 6 shows an example of geometry variation of

Fig. 6 Variation of W as function of fingers during the optimizationprocess. The first structure refers to 70 fingers and the second to 82fingers

the simulated cell due to a change of the number of fingers,which is related to the length of the computational domain.

The mesh generator of the TCAD simulator, and the sim-ulator itself were interfaced with the GA written in C, via awrapper capable of restarting them with geometrical param-eters provided by the GA. Also the post-processing MAT-LAB software was interfaced with the GA, computing theefficiency for each simulated solar cell.

At the end of this step, each individual of the populationis associated with a fitness value, that is, the efficiency of thecorresponding solar cell.

4.3 Evolution of the GA population

These fitness values are used for ranking the population, sothat only the most fit chromosomes may evolve and be prop-agated in future generations. This leads to the subsequentstep, that is, the evolution of the GA population.

In order to set the next generation of individuals, we pro-ceed as follows. First we select the “elite” of the population,that is, the 10 % of individuals with the best fitting values.The “elite” is kept also in the new population.

Another 85 % of individuals are generated by reproduc-tion from couples of individuals of the elite. The main repro-ductive mechanism is the crossover, which means the gen-eration of individuals by a recombination of genes of theparental couple, allowing for a mutation rate of 3 %.

Finally the last 5 % or the new population is made of new,randomly generated individuals.

4.4 Stop criterion

Once a new population is created, a new evaluation of thenew individuals is required, with the exception of the elite,of which we already know the efficiency. In particular, thebest individual, with the highest fitness value, is known.

J Comput Electron (2014) 13:323–328 327

Fig. 7 Flow chart of the optimization process by means of a geneticalgorithm

We use a simple stop criterion: the GA cycle ends whenthe difference of the efficiency of the new and the old bestindividuals is less than a given threshold, which we choose0.5×10−6. When this criterion is satisfied, a last check pop-ulation is created and evaluated, and the GA cycle terminatesif the stop criterion is confirmed.

Figure 7 shows the flow chart of the above-described op-timization algorithm. A wrapper written in C supervises thecommunication between Sentaurus, MATLAB and the ge-netic algorithm. The entire process was run under UNIX,which provides an ideal platform for these kinds of algo-rithms.

5 Results

We have implemented the optimization algorithm describedin the previous section, obtaining promising results. The al-gorithm converges after 4 steps. The evaluation of the fit-ness values of a single population, of 100 individuals, re-quires approximately 10 hours, with a processor AMD Phe-nom Quad-Core Processor (1.8 GHz) with 8 GB RAM. Thecomputational cost is mainly due to the mesh generation andthe TCAD simulation, which require about 9 hours per pop-ulation. The full optimization cycle was completed in ap-proximately 55 hours.

Table 1 Optimal values givenby the genetic algorithm Variable Value

Dpeak2 (µm) 0.055696

Dem (µm) 0.20000

N 82

Efficiency 18.2999

JSC (A/cm2) 0.0362

VOC (V) 0.6193

FF 81.5707

We have found an optimal efficiency η = 18.2999, whichis in line with similar results obtained for Si-based solarcell. In the considered range of the parameters, the effi-ciency varies approximately from 15 to the found optimalvalue. The geometric values Dpeak2, Dem, N correspondingto the optimal efficiency are shown in Table 1. The table alsoshows the short-circuit current JSC, the open-circuit voltageVOC and the fill factor FF.

We notice that the depth of the emitter coincides with thelower bound of the range given for this parameter. Also thelocation of the second peak of the double Gaussian dopingprofile of the emitter is close to the lower bound of the range,but it is slightly shifted above it. The above comments sug-gest that an additional sensitivity analysis on these two pa-rameters should be performed, extending the range of theparameters. This is beyond the scope of the present paper,which focuses on assessing the feasibility of the proposedmethodology.

The present work shows that the use of GA algorithmscan be successfully applied to the optimization of geomet-rical parameters of solar cell, combined with an appropri-ately interfaced TCAD device simulator. In order to obtainmore significant result, a more accurate model structure ofthe solar cell is required, and possibly a simulation mod-els which takes into account variations of temperature. Westress that this approach can also be applied to a full 3Dstructure, which would allow for a better study of the effectof the bus-bar on the efficiency of the cell. Finally, we ob-serve that our method can be easily parallelized, since theevaluation of each individual in a population is independentfrom the others, thus reducing the computational time.

Acknowledgements The authors wish to thank R. De Rose andP. Magnone of ARCES (University of Bologna) for their support andhelp. The second author acknowledges financial support from MaTe-RiA PON a3_00370.

References

1. Kerr, M.J., Campbell, P., Cuevas, A.: Lifetime and efficiency lim-its of crystalline silicon solar cells. In: Conference Record ofthe Twenty-Ninth IEEE Photovoltaic Specialists Conference 2002,pp. 438–441 (2002)

328 J Comput Electron (2014) 13:323–328

2. Swanson, R.M.: Approaching the 29 % limit efficiency of sili-con solar cells. In: Proc. 31st IEEE Photovolt. Spec. Conf., LakeBuena Vista, USA, pp. 889–894 (2005)

3. del Canizo, C., del Coso, G., Sinke, W.C.: Crystalline silicon solarmodule technology: towards the 1 euro per watt-peak goal. prog.photovolt. Res. Appl. 17, 199–209 (2009)

4. Girardini, K., Jacobsen, S.E.: Optimization and numerical modelsof silicon solar cells. Solid-State Electron. 34, 69–77 (1991)

5. Chen, M.-J., Wu, C.-Y.: A new method for computer-aided opti-mization of solar cell structures. Solid-State Electron. 28, 751–761(1985)

6. Chen, Y.-M., Lee, C.-H., Wu, H.-C.: Calculation of the optimuminstallation angle for fixed solar-cell panels based on the geneticalgorithm and the simulated-annealing method. IEEE Trans. En-ergy Convers. 20, 467–473 (2005)

7. Jervase, J.A., Bourdoucen, H., Al-Lawati, A.: Solar cell parameterextraction using genetic algorithms. Meas. Sci. Technol. 12, 1922–1925 (2001)

8. Affenzeller, M., Winkler, S., Wagner, S., Beham, A.: Genetic Al-gorithms and Genetic Programming: Modern Concepts and Prac-tical Applications, 1nd edn. Chapman & Hall/CRC Press, Lon-don/Boca Raton (2009)

9. Faizabadi, E., Khamechi, M.A., Toosi, K.N.: Optimization of grid-lines and fingers of solar cells by a new numerical method. In:Joint 32nd International Conference on Infrared and MillimeterWaves, 2007 and the 2007 15th International Conference on Tera-hertz Electronics. IRMMW-THz. IEEE Press, New York (2007)

10. Utsler, J.: Genetic algorithm based optimization of advanced so-lar cell designs modeled in Silvaco ATLASTM. Master’s thesis(2006)

11. Sentaurus Device User Guide, Version C-2009.06 SYNOPSYS(2009)

12. MATLAB, Version 8.0.0.783 (R2012b), The MathWorks Inc.,Natick, Massachusetts (2012)

13. Quaschning, V.: Influence of shading on electrical parameters ofsolar cells. In: Conference Record of the Twenty Fifth IEEE,Photovoltaic Specialists Conference, 13–17 May, pp. 1287–1290(1996)

14. Stem, N., Cid, M.: Physical limitations for homogeneous andhighly doped n-type emitter monocrystalline silicon solar cells.Solid-State Electron. 48, 197–205 (2004)

15. Fenning, D., Bertoni, M., Buonassini, T.: Predictive modeling ofthe optimal phosphorus diffusion profile in silicon solar cell. In:24th European Photovoltaic Solar Energy Conference, Hamburg,Germany, 21–25 september (2009)

16. Simon, M.S.: Semiconductor Devices: Physics and Technology,pp. 470–475. Wiley, New York (1985)

17. De Rose, R., Zanuccoli, M., Magnone, P., Frei, M., Sangiorgi, E.,Fiegna, C.: Understanding the impact of the doping profiles onselective emitter solar cell by two-dimensional numerical simula-tion. IEEE J. Photovolt. 3(1), 159–167 (2013)

18. Zanuccoli, M., De Rose, R., Magnone, P., Sangiorgi, E., Fiegna,C.: Performance analysis of rear point contact solar cells by three-dimensional numerical simulation. IEEE Trans. Electron Devices59(5), 1311–1319 (2012)

19. Schenk, A.: Finite-temperature full random-phase approxima-tion model of band gap narrowing for silicon device simulation.J. Appl. Phys. 84(7), 3684–3695 (1998)

20. Altermatt, P.: Models for numerical device simulations of crys-talline silicon solar cells a review. J. Comput. Electron. 10(3),314–330 (2011)

21. Klaassen, D.: A unified mobility model for device simulation:I. Model equations and concentration dependence. Solid-StateElectron. 35(7), 953–959 (1992)

22. Klaassen, D.: A unified mobility model for device simulation:II. Temperature dependence of carrier mobility and lifetime. Solid-State Electron. 35(7), 961–967 (1992)

23. Glunz, S., Rein, S., Lee, J., Warta, W.: Minority carrier lifetimedegradation in boron-doped Czochralski silicon. J. Appl. Phys.90(5), 2397–2404 (2001)

24. Kimmerle, A., Wolf, A., Belledin, U., Biro, D.: Modelling carrierrecombination in highly phosphorus-doped industrial emitters. In:Proc. of SiliconPV 2011 Conf., Energy Procedia, vol. 8, pp. 275–281 (2011)

25. Green, M.A.: Solar Cells: Operating Principles, Technology andSystem Applications, 2nd edn. Prentice Hall, New York (1998)