Parameter extraction using global particle swarm optimization approach and the influence of polymer processing temperature on the solar cell parameters

Over a past few years there has been a tremendous rise in interest of the scientific community in the field of organic/inorganic hybrid photovoltaic (PV) technology for its development as a low cost and scalable solar energy conversion platform.^1–3 In hybrid PV cells, the ease of solution processability and large absorption coefficient of the polymeric organic semiconductors in conjunction with the high carrier mobility and easy tunability of the band gap of the metal oxide offers them a unique advantage which makes them highly potential candidates for the energy conversion applications in third generation photovoltaic technology.^4–7 Intensive efforts are being put into developing this technology for practical applications and the performance parameters have been continuously improving.^8–10 An accurate estimation of the solar cell parameters is crucial not only to upgrade the quality of the PV cells by properly optimizing the fabrication steps but also for cell array and system simulation while using them in large PV modules in practical in-field applications. Moreover, the knowledge of the solar cell parameters is fundamental to gain an understanding of the physical processes occurring in the PV devices.^11–13 In the recent years, the development of evolutionary computational algorithms has led to a number of approaches towards accurate extraction of solar cell model parameters from the experimental I-V plots such as Genetic Algorithm(GA),^14,15 Simulated Annealing (SA),¹⁶ Harmony Search (HS)¹⁷ Pattern Search (PS),^18,19 Artificial Bee Swarm Optimization algorithm (ABSO),²⁰ Bird Mating Optimizer (BMO),²¹ and Particle Swarm Optimization (PSO).^22,23 Such metaheuristic algorithms possess the advantage of global search power and derivative-free approach in the algorithm. However, these approaches are sometimes limited by trapping at local points.²⁴ 

PSO method is based on the swarm intelligence optimization algorithm which mimics the social behaviour of a bird flock.¹³ A flock of bird while moving in search of food communicate information about their position, velocity, and fitness among them. In PSO algorithm, the words bird and particle are synonymous to each other and represents the probable solutions of the defined forward problem. Each individual particle possesses information about its position vector in the solution space and the velocity vector associated with it. All particles are set to change their position to new location in the solution space using the associated velocity vector at a particular time defined in the algorithm. The PSO algorithm stores the information about the best ever individual and global position attained by each particle during the optimization. Hence, with the knowledge of both these parameters, all particles collectively advance towards their global optimum position which represents the desired solution of the solar cell equation. PSO has shown promising results in the extraction of solar cell parameters with high accuracy as compared to GA, both for single and double diode models.²⁵ Hamid et al¹³ modified the traditional PSO technique and developed Nelder-Mead Particle Swarm Optimization (NM-PSO) technique to obtain unknown solar cell parameters from the experiment data. We would also like to highlight that the traditional PSO approach doesn’t take into account the values of solar cell parameters that were obtained in the previous iterations. Thus, in traditional PSO, chances are high that our solution may be trapped in the local minima, especially in solar cell parameter extraction problem due to high dimensionality of the problem.²⁶ 

In the present study, we have developed our own MATLAB based global PSO (GPSO) algorithm, which stores previous information at each iteration and increases the chances to obtain the global solution of solar cell parameters quicker than the traditional PSO. Further, we have utilized GPSO approach for accurately obtaining the PV parameters from the illuminated Current (I)-Voltage (V) data of hybrid solar cells (HSCs) with a wide search range used for each model parameter. Further, we utilize these parameters to study elaborately the role of polymer processing temperature on the performance of ZnO/PCDTBT co-polymer based hybrid PV system. The effect of polymer annealing on its charge transport properties, surface morphology, optical absorbance, and chemical compositional behaviour is investigated thoroughly to get an insight of the physical processes occurring in the PV system upon employing different processing conditions. The results are then used to explain the observed variation of the PV parameters of ZnO/PCDTBT solar cells obtained from the GPSO simulation of the experimental data.

A reduced and simplistic relation describing the I-V characteristics of a PV device under illumination was proposed by Zhang et al¹¹ using Lambert w function which can be explicitly written as,

where, η is the ideality factor, k_b is Boltzmann constant, T is device operating temperature, q is electronic charge, R_s is series resistance, R_sh is shunt resistance, lambertw is the numerically solvable Lambert W function, I_sc is short circuit current, and V_oc is open circuit voltage. The forward problem related to solar cell parameter estimation using Eq. 1 can generally be represented in a discrete form as,

where, F is a forward computation operator, S= [η, R_s, R_sh, V_oc, I_sc]^T is a solar cell parameter vector which needs to be estimated, and I^cal is a computed current vector. With the measured experimental data I^obs_, the general way of solving an ill-posed inverse problem for S in Eq. 2 is based on the minimization of data misfit function, which is typically written as,²⁴ 

where, N is the total number of measured current value in I^obs, and ϕ(S)^RMSE is the root-mean square error. In the present study, we have minimized ϕ(S)^RMSE using GPSO technique, subject to the certain conditions set for the solar cell parameters, i.e., η >0, R_s>0, R_sh>0, V_oc>0, and I_sc >0.

The particle swarm global optimization method is stochastic in nature; it utilizes the information of velocity vector alongside the location of best local and global qualities to update the current value of solar cell parameters of each particle in the swarm. In this way, it calculates the value of the optimization function for each swarm particle in their respective locations. Correspondingly, the associated velocity vector of each particle is updated based on the historical backdrop of each particle’s solar cell parameters and misfit-function value. This history stores the knowledge gained by each particle and also by the swarm as a whole, conceptually representing an autobiographical memory.²⁷ In this manner, the misfit-function estimation of every particle in the swarm is updated by utilizing the social behaviour of the swarm, which adjusts to its surroundings by coming back to previously found promising locales of the solution space via looking for optimal misfit function values overtime. Mathematically, the estimation of i^th solar cell parameter for j^th particle in the swarm at iteration $k+1$ is updated as,

where, $vi,jk+1$ is the corresponding updated velocity vector.²⁸ The velocity vector associated with each solar cell parameter is calculated as,

where,$vi,jk$ is the velocity vector; $Si,jk$ is the estimation of the i^th solar cell parameter for the j^th swarm particle at iteration k; $pi,bestk$ is estimation of the i^th solar cell parameter for the individual best misfit value recorded by the j^th swarm particle from initialization up to k^th iteration; $pi,gbestk$ is the i^th solar cell parameter for the global best misfit value recorded by the swarm up to k^th iteration from initialization; $r1,jk$ and $r1,jk$ represent a random numbers varying between [0, 1] at iteration k; w, c₁, and c₂ are the is the inertia weight,cognitive parameter, and social parameter respectively. Figure 1 schematically depicts the process of updating the estimated i^th solar cell parameter and the associated velocity vector for the j^th swarm particle as described in the Eq. 5. Here, it is important to highlight that the updated solar cell parameter is influenced not just by the local and global optimal solar cell parameters, it is additionally affected by the magnitude of the cognitive parameter c₁, social parameter c₂, and inertia weight w.

The implementation of GPSO in the present work has the following parameters: number of iterations, swarm size, velocity components, inertia weights, and acceleration coefficients. The number of iterations to get a decent result is always problem dependent. A small number of iterations may stop the search process prematurely, while too large value of number of iterations will add unnecessary computational complexity and computation time.²⁸ A large swarm size (i.e., number of particles in the swarm) may decrease the number of time steps required to get a decent optimization result, however it would increase the computational complexity of the problem.²⁴ The velocity components plays very important role while updating the particle’s velocity of each particle in the swarm. The right-hand side of Eq. 5 shows the three terms of velocity component which represent the inertia, cognitive, and social component, respectively. The inertial component will bias towards the current direction of the particle and at the same time it will also prevent the drastic change in the direction of the particles. Cognitive component measures the performance of each individual particle of the swarm relative to past performances whereas the social component measures the performance of the particular particle relative to its neighbour.²⁸ The random values r₁and r₂ along with the acceleration coefficients c₁ and c₂ will affect the cognitive and social component. The value of the constant c₁ and c₂ depend on the amount of confidence a particle has in itself and in its neighbours, respectively. The inertia weight w controls the effect of current velocity vector while updating velocity vector using in Eq. 5 (see Figure 1). The small inertia weight will reduce the effect of first term in Eq. 5 so that cognitive and social terms will influence more to updated velocity vector. As a result, it confines to the updated velocity vector in the nearby regions of the solar cell parameter space. The flowchart of the global particle swarm optimization is shown in Figure 2.

In contrary, large inertia weights amplify the updated velocity vector magnitude and reduce the influence of the cognitive and social velocity terms.²⁹ In the present study, we are linearly decreasing the value of inertial weight at each iteration as,²⁹ 

where, w_max and w_min are the maximum and minimum value of inertia weight respectively; k_max is the number of maximum iteration and k is the current iteration number.

Here, it is important to highlight that the efficacy of the GPSO is influenced by initial distribution of the swarm particle. The solution space which not initially covered by GPSO can only be found out if the momentum of swam particle (i.e. Eq. 5) takes away into such solution space domain and it is only possible when a particle finds a new local best position or if a new global best position is discovered by the swarm. In this study, to improve the probability of finding the global optimum misfit function value the particles are randomly distributed over the whole space of the misfit function. We have considered the population size in such a way that the entire misfit function domain is searched without leaving any unexplored solution space. This can be achieved by utilizing the following equations

where, $Si,j0$ and $vi,j0$ are the initial i^th solar cell parameter for the j^th swarm particle. S_min,j and S_max,j are the minimum and maximum search ranges of for the i^th solar cell parameter and r is the random number varying between [0, 1]. Perez and Behdinan in their work³⁰ illustrated that the particle swarm is stable and the system will converge only if the following conditions are satisfied,

However, it is still not guaranteed that the GPSO would provide global solution and thus its credibility as a global solution must be verified by misfit function values or the convergence criteria set in the algorithm.

We would like to highlight that there are several parameters which will affect the GPSO results. In the present work, we have performed several synthetic simulations with different values of population size, the number of generations, c₁, c₂ and w to obtain fast global optimum solution for solar cell parameters. The parameters with the following values provided the fast-global minima in our case: population size of 40; the number of generations kept as 200; and c₁ and c₂ were set to be 1.2 and 1.7, respectively. The maximum and minimum value of inertia weight ‘w’ was 0.9 and 0.4 respectively.

To test the accuracy of the present GPSO, we first calculated the synthetic I-V data using the single diode model with the representative solar cell parameters R_s = 0.0373 Ω, R_sh = 42 Ω, I_sc = 0.5737A, V_oc = 0.7604V, and η = 1.4561. We further added 2% and 5% Gaussian noise in the synthetic I-V data similar to work done by Zhang et al¹¹ to take into account the possibility of noisy experimental I-V data. Here, it is important to highlight that once sufficiently effective amount of noise is superimposed on synthetic I-V data; we can no longer estimate with accuracy the actual values of solar cell parameters. Therefore, initially the noise-free data is simulated to verify the efficiency of the GPSO in retrieving the actual solar cell parameters, following which the noisy data were simulated and the error in the parameter values was determined. Table I shows the values of the solar cell parameters used for calculating the synthetic data along with the simulated model parameters obtained using GPSO technique. The fitted synthetic and noisy I-V data is also shown in Fig. 2. A sufficiently small value of root mean square error (3.95 × 10^-5) was obtained to simulate the synthetic I-V data using a search range of over ±100% for each solar cell parameter. Conspicuously, the proposed GPSO method yields a very small rms error value when a sufficient number of iterations are performed. However, for noisy synthetic I-V data sets, the rms error value increases due to presence of noise, however it is still less than 0.63% of the 5% noise free data and is good enough to fit the measured noisy I-V data as shown in Fig. 3. With a population size (swarm size) of 40 used in the study for each model parameter, the histograms shown in Fig. 4 for synthetic and noisy data reveal a single peak for each solar cell parameter calculated indicating the particles converge uniquely towards their optimum value.

After the successful implementation of the GPSO approach to extract the model parameters from synthetic and noisy I-V data, we applied the algorithm to extract the solar cell parameters from the experimental I-V data obtained for the ZnO/PCDTBT based hybrid PV devices measured under AM 1.5 G solar irradiation. The details of the device fabrication scheme are provided in the section 1.1 of the supplementary material. To investigate the effect of the processing temperature on the PV parameters, a range of devices were fabricated and measured for different annealing temperatures employed for the PCDTBT co-polymer processing. The experimental and the simulated I-V plots for the set of devices fabricated with different processing temperature are shown in Fig. 5(a). The simulated curves show excellent agreement with experimental data.

The detailed calculated model parameters using the global PSO approach are shown in Table S1 in the supplementary material. Fig. 5 (b) graphically represents the variation of major solar cell parameters (J_SC, V_oc, FF, Efficiency, R_sh, and R_s) obtained for different annealing temperatures for processing the PCDTBT co-polymer films. Notably, the search range used for each model parameter in the GPSO approach has been kept sufficiently large to test the efficacy of the proposed algorithm which supersedes the other approaches in terms of ruling out the requirement of specific seed values to obtain the solution.¹¹ 

Further, we critically examine the effect of the processing temperature of the PCDTBT co-polymer on the PV performance parameters obtained by the GPSO approach as depicted in Fig. 5 (b) in the previous section. The effect of annealing temperature on the performance of organic solar cells have been studied previously.^31–34 Evidently, increasing the annealing temperature of polymer results in enhancement of J_sc and FF which transcends into higher PCE in the devices. It is discernible from the figure, that the J_sc and the FF increases roughly following the trend of decreasing R_s value which indicates that the series resistances of the PV cells are primarily regulating the device efficiency. The R_s value decreases upon increasing the PCDTBT processing temperature and attains a minimum of 10 Ω/cm² at about 150 °C, preceding which incurs an increase in the R_s value resulting in a lowering of J_sc and FF. We believe an improvement of the polymer ordering and its interface with the ZnO upon increasing the annealing temperature is a major reason behind such enhancement of J_sc, R_s, and FF values of the PV devices. On contrary, the V_oc of the devices doesn’t record any significant variation which is apprehensible considering the fact that the HOMO energy level of the polymer films is not expected to change substantially upon increasing the annealing temperature.

Further, to extricate the reasons behind the improved J_sc, R_s and FF values of the PV devices upon processing the polymer at higher temperatures we studied the charge carrier transport phenomena inside the PCDTBT films as a function processing conditions. Field effect transistor (FET) structures using PCDTBT channel layer deposited under different annealing temperatures were fabricated following the standard device architecture as used by Cho et al.³⁵ Figure 6 shows the transfer characteristics of the OFET devices fabricated. The obtained curves show typical p-channel OFETs characteristics working under accumulation regime. The saturation field effect mobilities (μ_s) for holes were calculated using the sqrt I_d vs. V_g plots and are plotted against the annealing temperatures used for preparing the PCDTBT films as shown in Fig. 6(b). The mobility value increases steadily although moderately with the increase in annealing temperature. The maximum mobility value of 0.028 cm²V^-1s^-1 was obtained for the annealing temperature of 150 °C. Further increasing the annealing temperature marks a decrease of hole mobility to 0.006 cm²V^-1s^-1for 180 °C annealing. The exposure of a conjugated polymer to high temperature in air can typically trigger the oxidation of the polymer back bone, thereby breaking the electronic structure and semiconducting properties of the polymer as a result of which we can observe a reduced value of hole transport property above a characteristic annealing temperature.³⁵ Figure 3 shows transfer and output characteristics for the PCDTBT transistors with the optimized annealing temperature of 150 °C. Excellent linear and saturation region were exhibited within the applied voltage range in the output characteristics of the PCDTBT OFETs.

Also, an on/off ratio of ∼10³ was obtained exhibiting a decent transistor switching characteristics with the optimized annealing temperature of 150 °C. The rationale of the above can be understood by considering the fact that when PCDTBT co-polymer is allowed to dry at higher temperatures, we are deliberately making the polymer chains more mobile which manifest into an improved lamellar ordering and π-stacking in the polymer chains.^36,37 This increases the effective conjugation length in the polymer as a result of which the hole mobility value improves. Correspondingly, in the PV devices the increase in hole mobility with the annealing temperature results in the more efficient transport of the charges through the PCDTBT polymer towards the electrode subsequent to the photogeneration and exciton dissociation processes at the ZnO/PCDTBT interface and thus we register a lower value of R_s and a higher J_sc. At temperatures beyond 150 °C, the reduction in OFET mobility transcends into lower PCE of the PV devices by primarily affecting its series resistances. The degraded PV performance for the devices prepared under temperatures exceeding an optimum value may also arise from the change in surface morphology and deterioration at its intimate contact with the ZnO.

To corroborate our presumption, surface topography of the PCDTBT films were studied at representative processing temperatures between 60 °C and 180 °C. Figure 7 shows the AFM height scans of the polymer samples prepared under different annealing temperatures. The films show rather smooth surfaces with low rms roughness value of ∼0.6-1.0 nm at lower to moderate temperatures of annealing. However, the morphology showed some aggregations and rms roughness increased to 11.6 nm for higher temperatures of annealing (180 °C). Apparently, the higher surface aggregates can cause poor charge transport across the PV stack and abate its efficiency, thus requiring a proper optimization of the polymer processing to have a balanced effect of higher ordering of the polymer chains whilst maintaining high quality film morphology.^38,39

Furthermore, we performed UV-Vis absorption and X-ray photoelectron spectroscopic (XPS) studies to investigate any changes in the optical absorption and chemical bonding in the PCDTBT thin films for different processing temperatures employed. The absorption spectra of the PCDTBT films annealed at different temperatures in air ambient are shown in Fig. 8 We observe two distinct absorption bands at 395 nm corresponding to the π-π^* transition and at 570 nm attributable to charge transfer between donor and acceptor units of the polymer chain.^40,41 We do not observe any substantial change in the absorption bands for annealing temperatures up to 150 °C apart from a slight red shift of the spectrum shown in the inset. The red shift of the absorption peak has been previously reported,^42,43 and suggests an improvement in long range ordering and increment in effective conjugation length in the semiconducting polymer which is consistent with the results obtained from OFET measurements. Further exposure to high temperatures resulted in a reduced absorption of the higher wavelength band indicating a possible degradation of the electronic band structure of the PCDTBT polymer. The XPS spectra of the polymer thin films Fig. 9, obtained under different annealing temperatures shows the typical evolution of the C 1s, N 1s, and S 2p core level. The C 1s core level spectra of the PCDTBT films exhibits an asymmetric broadening towards higher B.E and can be resolved into three different components. The peak at ∼284.5 eV originates from the C-C bonding in the polymer chain, while the higher B.E peak at 285.4 eV can be assigned to the C-H bonds occurring in groups like -CH₂, aromatic carbon etc. The signal with peak at ∼287.6 eV may result from the bonds between C-N and C-S groups.³⁵ The annealing of the films at different temperature doesn’t seemingly alter the spectral shape and position of the C 1s main peak; however, a slight broadening is observed when the samples annealed at higher temperature of 180 °C. N 1s XPS spectra of PCDTBT films can be deconvoluted into two components; one centering at lower binding energy ∼399.3 eV and the other high energy peak at ∼400.0 eV. The lower B.E peak originates from the N core levels in the electron DTBT accepting unit and the higher B.E peak is due to the N in the electron donating carbazole unit.^35,44 We observe no significant shifting of the peak position of N 1s spectra in the samples annealed up to 150 °C, however, a ∼0.13 eV shift toward lower B.E was observed for the annealing temperature of 180 °C without any difference in the spectral shape. Fig. 9(c) shows the corresponding S 2p spectra of PCDTBT polymer films which consist of 2p_1/2 and 2p_3/2 peaks which arise due to spin-orbit splitting. We resolved the S 2p main peaks into four components which can be attributed to arise from two S containing species, i.e. thiophene and benzothiadiazole. The first doublet at ∼163.8 eV and ∼164.6 eV originates from sulphur in thiophene group whereas the second doublet in the spectra at ∼165.3 eV and ∼166.6 eV results due to the benzothiadiazole group.^35,44 Similar shifting of the main S 2p peak towards lower B.E observed for the sample annealed at high temperature of 180 °C indicates that the PCDTBT co-polymer is stable under annealing in air up to a temperature of ∼150 °C. Further increasing the annealing temperature results in the modification of the electronic structure of the co-polymer as indicated from obtained peak shifts in the XPS spectra. Thus, the reduction in the absorption spectra for samples annealed at 180 °C in air is consistent with the XPS study and marks the degradation of electronic structure of the polymer.

Hence, it is apparent from the results that the processing conditions during active polymer deposition play a very crucial role in obtaining better PCEs in PV devices and hence needs to be critically optimized. The annealing of the PCDTBT polymer at higher temperatures enhances its charge transport property due to enhanced ordering in the system and results in superior device performances. Although beyond a critical temperature the polymer electronic structure starts deteriorating along with its surface morphology, hence a balance between the two competing processes needs to be reached.

In conclusion, we have proposed a GPSO approach for accurately identifying the solar cell parameters and presented its viability in extracting the various model parameters (J_sc, V_oc, η, R_s, R_sh) from the experimental I-V curves of practical ZnO/PCDTBT HSCs overwide search range values. Further, we examined the variation of the extracted parameters as a function of different processing temperatures used for the HSCs and unveiled the physical processes affecting the device performance parameters. We observe that annealing the polymer at higher temperature causes deliberate motion of the polymer chains resulting in enhanced ordering in the system. As a result, the polymer charge transport property increases as exhibited from the OFET measurements which leads to lower series resistances and higher FF values in the PV devices. However, after an optimum temperature the polymer’s electronic structure deteriorates as confirmed from XPS, UV-Vis absorption spectra and surface aggregations in AFM scans. Thus, our study reveals the feasibility and global search power of GPSO approach in studying the performance parameters of ZnO/PCDTBT HSCs and also captures the need of using proper processing conditions to enhance the charge transport PCE of the PV devices.

See supplementary material for the details of the device fabrication scheme are provided in the section 1.1 of the supplementary material for the manuscript. The detailed calculated model parameters using the global PSO approach are shown in Table S1 in the supplementary material.