Model Parameter Extraction for InGaN/GaN Multiple Quantum Well-Based Solar Cells Using Dynamic Programming

I.   Introduction

The bandgap of InGaN alloy, a third-generation semiconductor material, can be continuously adjusted from 0.7 eV to 3.4 eV [1], which nearly spans the entire solar spectrum. As a result, solar cells (SCs) based on InGaN alloys have the distinct advantage of high photoelectric conversion efficiency (η). In addition to the broad tunability of the bandgap, InGaN alloys also have high electron mobility [2], high absorption coefficient, high-temperature resistance [3], and strong radiation resistance [4]-[6], among other excellent properties. In order to improve the conversion efficiency of InGaN SCs, multi-quantum wells (MQWs) structures have been effectively utilized as active regions and have led to enhancement in the device performance [1]-[6]. In order to take full advantage of MQWs, it is necessary to adjust and optimize the photovoltaic circuit to improve the power conversion efficiency of InGaN/GaN MQWs SCs.

Compared to other conventional algorithms (the two-diode model, three-diode model, and multi-diode model), the single-diode model (SDM) is a relatively simple SCs model, which is widely used as it involves only a few unknown parameters, and can offer a reasonable accuracy [7]-[9]. The SDM is a transcendental equation and cannot express the I-V characteristics explicitly by elementary functions. Therefore, the extraction of photo-generated current, reverse saturation current of the diode, series resistance, parallel resistance, and ideal factor of the diode in the model is more complex and challenging.

Due to continuous improvement and innovation of parameter extraction algorithms for the diode model of SCs, several algorithms with higher accuracy have been proposed. For instance, Benayad et al. [10] used a hybrid genetic algorithm (TR-GA) and hybrid particle swarm optimization (TR-PSO) algorithm to extract SDM parameters of InGaN/GaN SCs by limiting the range of five parameters and their mean square errors (MSE) were 8.500E−9 and 5.980E−9, respectively. For commercial RTC France SCs, Liao et al. [11] proposed an improved algorithm that utilized the differential evolution of the difference vectors with an adaptive mutation strategy (DVADE) to extract the parameters of photovoltaic cells. The difference vector mechanism was repeatedly used to improve population diversity, and the optimal solution was found within the limited range of five parameters with a root mean square error (RMSE) of 9.860E−4. Diab et al. [12] proposed a tree growth algorithm (TGA), which defines four groups of different trees and replaces the optimal tree group according to the fitness function until it converges to the optimal solution within a limited range of five parameters. The RMSE of this method reached 9.750E−4. Although the aforementioned methods have demonstrated low RMSE, they have a high requirement for the ranges of the initial values of the five parameters. Large initial values can slow down the convergence in the extraction process, which increases the time complexity. To further improve the convergence speed and the calculation accuracy of this algorithm, Xu et al. [13] used a non-iterative parameter extraction method (NPEM) to extract the parameters of commercial photovoltaic cells. This method extracted five parameters from six independent transcendental equations by combining analysis of equivalent circuit and photovoltaic cell module data table. The NPEM method overcomes the limitations of low efficiency of the numerical iteration method, low accuracy of the semi-empirical method, and lack of physical significance of the optimization algorithm. As a result, the RMSE reached 6.710E−5. However, in their approach, Xu et al. simplify the equations by setting R_s = 0 and R_sh = ∞ (R_s is the series resistance, R_sh is the parallel resistance), and calculate the values of R_s and R_sh by using roughly calculated slope of the curve. The value calculated by the slope of any two points may cause R_s and R_sh to be too large or too small, which could introduce a certain deviation in the five parameters. Relatively, these slight deviations could have some impact on the parameter extraction of SCs. Hence, there is a requirement for a method for accurately extracting the parameters of InGaN SCs. So far, to the best of our knowledge, there is no accurate and fast dynamic algorithm for extracting relevant parameters in InGaN/GaN MQWs SCs using SDM.

This paper proposes the novel application of dynamic programming (DP) in SCs parameter extraction. Using the proposed algorithm, five parameters are extracted from the SDM of InGaN/GaN MQWs SCs with indium (In) compositions of 7% and 18%. Firstly, the SDM circuit model is combined with the semiconductor photovoltaic theory, and adaptive dynamic grouping is carried out based on the device series resistance and ideal factor range. Next, the optimal values of the five parameters are obtained using the RMSE of the curve and the η. The proposed algorithm has been verified to demonstrate better effectiveness and reliability when compared to the Fsolve function in MATLAB and other extraction methods.

II.   Algorithm Model

1   The idea of dynamic programming algorithms

The basic algorithm of DP [14] treats the problem to be solved as a set of interrelated subproblems. The optimal value of the subproblem is solved by the bottom-up method, and each subproblem in the set is solved only once. The optimal solution of each subproblem and its objective function value are stored. The optimal objective function is selected from the set of stored objective functions. Finally, the solution corresponding to this objective function is chosen as the overall solution. The algorithm increases the space complexity and decreases the time complexity.

2   Single diode model of SCs

SDM is adopted as the model for solving the parameters for the InGaN/GaN MQWs SCs [15]. The single-diode equivalent circuit model of SCs is composed of a constant current source, a single diode, a parallel resistance, and a series resistance, as shown in Figure 1.

Figure  1.  Single diode equivalent circuit model of the SCs.

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The current-voltage (I-V) characteristic in the SDM of the SCs can be expressed by

where i is the load current, i_ph is the photo-generated current, i₀ is the reverse saturation current of the diode, T is the operating temperature of the battery, n is the ideal factor of the diode, and q, which is the charge of the electron, is equal to 1.60218E−19 C. The Boltzmann constant k is 1.380649E−23 J/K. V is the voltage across the load.

Since the relation between I and V is in the form of an implicit transcendental equation containing five parameters, namely, I_ph, I₀, R_s, R_sh, and n, three basic expressions can be obtained according to (1).

When V = 0 V, Equation (2) is obtained as follows:

When I = 0 A, equation (3) is obtained as follows:

When I = I_m, V = V_m, (I_m, V_m) is the maximum power point, equation (4) is given by

If the slope at the maximum power point in the P-V curve is taken as 0, equation (5) can be obtained.

The derivative of V is obtained from (5) as follows:

Simplifying (6) gives

Finally, by derivation of (7), equation (8) can be obtained.

Equation (9) can also be calculated from (6).

Generally, the five parameters of the SCs model can be solved by combining (2), (3), (4), (8) and (9). In the study of the SDM of SCs, the magnitude of R_sh is typically in the order of kΩ, whereas R_s is generally in the order of mΩ. Hence, R_s/R_sh is nearly 0, and I_ph is approximately equal to I_sc [16]–[19]. Since $I_0 \ll I_{sc}$, the influence of $I_0$ on the equivalent circuit can be ignored. Moreover, R_s can make the I-V characteristic curve shift to the left, where the shift is directly proportional to the value of R_s. In contrast, R_sh has the property of making the I-V characteristic curve shift downward. The smaller the R_sh, the more downward the curve shifts [20]. The shifting curves for SCs are shown in Figure 2.

Figure  2.  The shifting curves for SCs.

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The left and right deviation of the curve will cause the decrease of I_sc, I_m, V_oc, and V_m. R_s and R_sh significantly affect the fill factor (FF) and the η of the SCs.

FF is the ratio of the maximum output power P_max of the SCs to I_sc and V_oc, which can be expressed as

η is the ratio of P_max of the SCs to the total power of incident light irradiated on the surface of the SCs, which can be expressed as

3   The objective functions set

The accuracy of the extracted parameters was demonstrated by calculating the RMSE between the initial (experimental) I-V data and the I-V data calculated by extracting the parameters, which is the conventional method used in most literature [7], [11]-[13].

Herein, the accuracy as determined by RMSE and the error in η are adopted, and the corresponding solution vector satisfying the minimum value of RMSE and ∆η is taken as the optimal solution of the parameters to be solved. The two objective functions are as follows。

The RMSE is defined as

where N is the number of sample points measured by SCs I-V data, I_i indicates the measured current value, and ${\widehat{I}}_{i}$ indicates the current value calculated.

The ∆η is defined as

where, η_meas is the measured η, and $\widehat{\eta }$ is the calculated η.

III.   The Structure of SCs

1   The structure of InGaN/GaN MQWs SCs

In this work, the InGaN/GaN MQWs SCs materials were prepared by MOCVD (metal-organic chemical vapor deposition). The structure of the InGaN/GaN MQWs SCs (1 mm × 1 mm) is shown in Figure 3.

Figure  3.  The structure of InGaN/GaN MQWs SCs (1 mm × 1 mm).

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The 80 nm thick GaN nucleation layer (NL-GaN) was grown on a (0001)-oriented patterned sapphire substrate (PSS) at 650 ℃, and an unintentionally doped GaN buffer (U-GaN) with a thickness of 2.5 μm was subsequently grown at 980 ℃. An N-GaN layer with a thickness of 2 μm was deposited on the U-GaN at 1070 ℃ with a Si doping concentration of 1E+19 cm⁻³ to form an N-type region. Subsequently, two types of InGaN/GaN (3/9 nm) MQW with In compositions of 7% and 18% along with 10 QW periods were deposited on U-GaN as the active layer. The temperatures of the quantum well and the barrier were 750 ℃ and 800 ℃, respectively. Finally, 40 nm P⁺GaN and 40 nm P⁺⁺GaN with Mg doping concentrations of 1E+19 cm⁻³ and 1E+20 cm⁻³ were grown at 920 ℃ and 900 ℃, respectively. The device was prepared by evaporation of In tin oxide (ITO) and metal electrode (Cr/Ni/Au) on P⁺⁺GaN. The P⁺⁺GaN layer in the material was used for doping to form the P-type region, and the built-in electric field was established with the N-GaN layer to facilitate the transmission of light-generated carriers.

2   System architecture

The overall structure of the I-V data acquisition system and parameter extraction module is shown in Figure 4. The data acquisition system includes xenon lamp, InGaN/GaN MQWs SCs, and semiconductor characterization equipment (Keithley Model 4200-SCS). The xenon lamp simulates an AM1.5 solar light source that illuminates the InGaN/GaN SCs. The Keithley Model 4200-SCS is used to collect I-V data of SCs. The data is processed by a parameter extraction module, which is based on the MATLAB development environment design parameter extraction algorithm for parameter extraction and objective function.

Figure  4.  Overall test process of InGaN/GaN MQWs SCs.

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3   Parameter extraction algorithm based on DP

The parameter extraction algorithm is developed based on the concept of DP. Here, converged the range of R_s, and n, combined with (2), (3), and (4) are utilized to solve the three parameters of I_ph, I_0, and R_sh, as shown in Figure 5.

Figure  5.  The main idea of the parameter extraction algorithm.

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It is challenging to extract the coefficients of R_s and n when they are the arguments of an exponential function (e^x). In this algorithm, I_ph, I₀, and R_sh are taken as unknowns. Combining (2), (3), and (4) establishes the (14).

Its matrix form is expressed as:

Here, the rank of the coefficient matrix is equal to the rank of the augmented matrix, indicating that the equation set has a unique solution. Therefore, the values of I_ph, I₀, and R_sh can be determined by R_s and n.

Since the range of R_s values is quite large, it is necessary to narrow and determine the range of R_s to reduce the time complexity. To achieve this, R_s is divided into three intervals, namely, [0.001,0.01], [0.01,0.1], and [0.1,1], and take the lowest order of magnitude of each interval. In the ideal case of n = 1, the ergodic solution is carried out for the three intervals and the average value of RMSE under each interval is calculated. The interval with the smallest RMSE is taken as the interval of R_s. Because the R_s interval is discretized and sampled, there may be errors in the results of parameter extraction. The smaller the sampling interval, the longer the calculation time, but the RMSE tends to be consistent through testing, and the error of parameter extraction results could be small and negligible.

The limitation of the method proposed in this paper is that the range of n and R_s of the device still needs to be determined before parameter extraction, and then the five parameters can be extracted according to the range of the two parameters.

Since there is a positive correlation between the number of defects and n [4], the range of n is set to twice that of silicon cells, which is [1,4]. The remaining three parameters are accurately solved by combining the range of R_s and n. In the solution algorithm, 121 fixed values are collected in the range of n with 0.025 as the sampling interval. The value of 1 to 11 times the minimum order of magnitude of the R_s interval is taken at a sampling interval of 0.5. As shown in Figure 6, FW represents three orders of magnitude of R_s. After determining the order of magnitude of R_s, the values of 1 time, 1.5 times, 2 times, 2.5 times until 11 times of the order of magnitude of R_s are obtained, and 20 intervals are assigned, such as [1,1.5], [1.5,2] until [10.5,11]. A random R_s real value of 20 is obtained by randomly taking a value from each interval. The range of n is divided into 121 large groups. Each of the large groups is further divided into 20 groups. Each group is assigned a non-repeating R_s value and calculated 2420 times. In each large group, the values of I_ph, I₀, R_sh, RMSE, and ∆η are calculated according to the values of n and R_s. The minimum RMSE and ∆η values form the criteria for screening the optimal solution for each large group. Then, the optimal solution in 121 groups is screened by the same criteria as the final solution for the values of the five parameters. The flowchart of the algorithm is shown in Figure 6.

Figure  6.  The algorithm flow of this work.

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4   Parameter extraction method of Fsolve function

Since the equation of SDM is a transcendental equation, the general method is to iterate the transcendental equation [7], [11], [12] by setting the initial values of the five parameters. The Fsolve function of MATLAB can be used as the comparison method for parameter extraction, which uses the least square method to solve nonlinear equations. The initial values of the parameters need to be set in the Fsolve method. Although the Fsolve method has high accuracy, it has stringent requirements on the initial value of parameters. Therefore, it often causes non-convergence of solutions as a result of the initial value problem. The Fsolve function needs to use the initial values of five parameters and also requires five nonlinear equations. Equations (2), (3), (4) and (9) can be connected to establish the equations to meet the solving conditions.

The initial value of the five parameters is debugged through the traversal method under different functional tolerances in advance, and have made efforts to tune them. The Function tolerance was found to have almost no effect on the five-parameter extraction results and RMSE. Finally, the optimized Fsolve function will be compared with the algorithm in this paper. The flowchart of parameter extraction using the Fsolve function is shown in Figure 7.

Figure  7.  The algorithm flow of the Fsolve function.

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IV.   Results and Discussion

The Keithley Model 4200-SCS equipment under AM1.5 standard sunlight conditions was used to measure the I-V characteristic curves of InGaN/GaN MQWs SCs with In compositions of 7% and 18% at 298 K. The test environment is shown in Figure 8(a). The measurement results of the I-V characteristic curves of InGaN/GaN MQWs SCs with In compositions of 7% and 18% are shown in Figure 8(b).

Figure  8.  (a) Test environment for InGaN/GaN MQWs SCs; (b) I-V characteristic curves of InGaN/GaN MQWs SCs.

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The Five parameters are extracted from the SDM of two In components SCs by using the proposed algorithm and the Fsolve function in MATLAB. The extraction results as listed in Table 1. The I-V characteristic curves for InGaN/GaN MQWs SCs with In compositions of 7% and 18% obtained using five parameters are shown in Figure 9(a) and (b), respectively, where a good agreement can be observed between the calculated and measured curves. The curves calculated by the Fsolve method begin to deviate from the measured curves at the corner after the maximum power point, and the calculated V_oc decreases. The I_sc, V_oc, and RMSE data results under the two conditions are listed in Table 2 and Table 3. Combined with the data in Table 1, it can be observed that the increase of the In concentration causes an increase in the values of I_ph, R_s, and n, while I₀ and R_sh show a decline. This is attributed to the increase in the light absorption coefficient of InGaN due to increased In content, which leads to an increase in the number of photocarriers in the material. This in turn affects the parameters such as I_ph, I_sc, R_sh, and R_s. Since R_sh is proportional to V_oc of the device, V_oc also decreases with the increase of In content [21], [22]. The increase of In content leads to the increase of lattice mismatch between InGaN and GaN and the increase of internal defects, which leads to the deterioration of material quality and affects n of the diode. In order to further verify the validity of parameter extraction, errors of I_sc, V_oc, FF, and goodness of fit (R²) are used to ascertain the accuracy of the results. R² can be used to verify the similarity between the calculated curve and the actual curve. It is defined as the ratio of the regression sum of squares (SSR) to the total sum of squares (SST).

Table  1.  Five parameters of InGaN/GaN MQWs SCs.

Method	Iph (A/cm2)	I0 (A/cm2)	Rs (kΩ)	Rsh (kΩ)	n
This work (7% In)	1.635E−4	7.400E−29	0.449	75.618	1.600
Fsolve (7% In)	1.626E−4	1.310E−35	0.813	70.791	1.296
This work (18% In)	5.883E−4	1.670E−15	0.624	8.975	3.225
Fsolve (18% In)	5.915E−4	1.080E−25	0.928	10.183	1.676

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Figure  9.  Comparison of I-V curves between the two methods under (a) 7% In and (b) 18% In.

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Table  2.  Comparison of I_sc, V_oc of the two methods for 7% In

	Isc(A/cm2)	Voc(V)	RMSE
Test data	1.640E−4	2.390	–
Fsolve	1.626E−4	2.375	1.401E−5
This work	1.635E−4	2.390	9.880E−6

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Table  3.  Comparison of I_sc, V_oc of the two methods for 18% In

	Isc(A/cm2)	Voc(V)	RMSE
Test data	5.500E−4	2.170	–
Fsolve	5.915E−4	2.136	2.168E−5
This work	5.883E−4	2.170	1.162E−5

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where, $\overline{I}$ is the average value of measured current I. An R² that is closer to 1 indicates that the calculated curve is nearly identical to the actual curve, thereby proving the validity of the results. Figures 10(a) and (b) show the ∆η, ∆FF, and R² between the measured curve and calculated curve determined by the proposed method and Fsolve method after extracting parameters of SCs with In compositions of 7% and 18%. The ∆FF is defined as

Figure  10.  Comparison of data errors between the two methods under (a) 7% In and (b) 18% In.

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where FF is the measured fill factor, and $\widehat{FF}$ is the calculated fill factor.

Using the proposed method, the ∆η, ∆FF, and R² are found to be 3.633E−4, 3.633E−4, and 0.926 for the SC with a 7% In component, respectively. Whereas the ∆η is 2.679E−4, ∆FF is 2.679E−4 and R² is 0.909 for the SC with an 18% In component. The Fsolve method calculates the ∆η, ∆FF, and R² to be 1.042E−2, 1.520E−2, and 0.794, respectively, for the SC with an In component of 7%. Whereas the ∆η is 1.499E−2, ∆FF is 2.985E−2 and R² is 0.847 for the SC with an In component of 18%. From Figures 10(a) and (b) it can be seen that the proposed method is superior to the Fsolve method in ∆η, ∆FF, and R².

The curves of the number of calculations with RMSE in the parameter extraction process of the Fsolve method and the proposed method are shown in Figure 11. The Fsolve method has been calculated more than 7000 times to achieve the above error results, and the calculation time is 3.922 h and 4.944 h, respectively. The method in this paper has only been calculated 2420 times to obtain better results than the Fsolve method, and the calculation time is 1.535 h and 1.119 h, respectively. The result shows that the method proposed in this paper is simpler and faster than Fsolve in computational complexity.

Figure  11.  (a) The Counts-RMSE curves of the Fsolve; (b) The Counts-RMSE curves of this work.

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To further verify the accuracy of the proposed method, the RMSE calculated by the proposed method is compared with the current mainstream GA, PSO, and other SCs extraction methods and the results are listed in Table 4. Compared with other parameter extraction algorithms [10]-[13], [23], [24], the proposed algorithm has a smaller MSE and RMSE, which indicates a higher accuracy for SCs parameter extraction.

Table  4.  Comparison of accuracy between this work and past work

Algorithm	SCs	MSE	RMSE
This work (7%In)	InGaN/GaN MQW SC	9.761E−12	9.880E−6
This work (18%In)	InGaN/GaN MQW SC	1.350E−10	1.162E−5
2019 TR-GA [10]	InGaN/GaN SC	8.500E−9	/
2019 TR-PSO [10]	InGaN/GaN SC	5.980E−9	/
2020 DVADE [11]	Commercial RTC France SC (Si-based)	/	9.860E−4
2020 TGA [12]	Commercial RTC France SC (Si-based)	/	9.750E−4
2021 NPEM [13]	Simulated Data (Si-based and other commercial SCs)	/	6.710E−5
2019 NTPB [23]	Commercial RTC France SC (Si-based)	/	8.129E−4
2018 MLBSA [24]	Commercial RTC France SC (Si-based)	/	9.860E−4

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V.   Conclusion

In this paper, a parameter extraction method for an SDM of InGaN/GaN MQWs SCs based on dynamic programming is proposed. The parameters of InGaN/GaN MQWs SCs with an in composition of 7% and 18% are extracted. The calculated values of I_sc and V_oc obtained by this method are consistent with measured data. ∆η and ∆FF are superior to the Fsolve method, and R² reached 0.926 and 0.909.

The validity and reliability of the proposed algorithm are verified by comparing the Fsolve function of MATLAB and other extraction methods. Compared with earlier reports, this method has a smaller error value of about 9.880E−6 and 1.162E−5. The proposed extraction algorithm can provide a reference for power optimization of semiconductor photovoltaic cell systems in the future.