Introduction

In recent years, thermophotovoltaic (TPV) has been escalating as a promising technology for high power density generation. A TPV system converts thermal radiations from combustion of fuels, industrial waste heat, or nuclear energy into electricity. The advantages of noiseless, high reliability, mechanically stable without moving parts, and large power density, make TPV suitable for a vast range of real-world applications such as electrical generator1,2,3, aerospace applications1,4, submarine5, vehicle6, solar thermophotovoltaic (STPV) cell7,8,9, energy storage10,11, waste heat recovery system in metal-alloy industries2,12,13, power plant14,15 and fuel cell16. The TPV converters mainly utilize narrow bandgap (NB) semiconductor materials which allow them to harvest the maximum amount of infrared radiations (IRs). The advancement of nanotechnology and material science since 1990s have boosted the development of various NB TPV cells, such as germanium (Ge)17, indium arsenide (InAs)18, gallium antimonide (GaSb)19, indium gallium arsenide (InGaAs)20, indium gallium antimonide (InGaSb)21, indium gallium arsenide antimonide (InGaAsSb)22 and indium arsenide antimonide phosphate (InAsSbP)23. In the last 3 decades of research in TPV, most researchers focus on the utilization of GaSb cell due to its narrow bandgap of 0.72 eV. US company JX Crystals Inc has developed a high-performance GaSb TPV cell which is commercially available and is widely used in various TPV systems24. The GaSb TPV cell was reported to have an efficiency of 29% under radiation temperature of 1548 K13. On the other hand, InGaAs, which has similar bandgap energy and potential in achieving high TPV performance, was relatively more common for applications in telecommunication and sensing. In1-xGaxAs is a ternary semiconductor with bandgap energy (Eg) that can be engineered from 1.42 to 0.36 eV by varying the x composition of Ga atom, which corresponds to cutoff wavelengths (λc) from 0.87 to 3.34 µm25. At x = 0.47, In0.53Ga0.47As semiconductor material can be grown lattice-matched on an indium phosphide (InP) substrate, corresponds to Eg and λc of 0.74 eV and 1.68 µm, respectively. Moreover, In0.53Ga0.47As is a promising TPV cell due to its high crystal quality and the cost-effectiveness of InP substrate, making it suitable for large scale production as compared to other TPV materials.

It is worth mentioning that the existing epitaxy growth technology of metal–organic vapor-phase epitaxy (MOVPE) has the ability to produce In0.53Ga0.47As/InP heterojunction with high crystal quality and low defect density26,27. The main structure of In0.53Ga0.47As configuration includes the emitter, base, front surface field (FSF), back surface field (BSF), cap and buffer layers. In previous literature, the base thickness was reported between 1 and 5 µm, and emitter thickness was between 0.05 and 0.44 µm27,28,29. Several structures reported the use of highly doped In0.53Ga0.47As cap layer ~ 1 × 1019 cm−3 and highly doped InP BSF/buffer layer ≥ 1 × 1018 cm−3. Table 1 reviews the design structure and output performances of In0.53Ga0.47As photovoltaic (PV) cell under different testing conditions. Typical In0.53Ga0.47As cells have open-circuit voltage (Voc), short circuit current density (Jsc), fill factor (FF) and efficiency (η) ranging between 0.26 and 0.45 V, 18.8 and 64.5 mA/cm2, 59 and 74.2%, and 4.2 and 14.37%, respectively under air-mass 0 (AM0) and air-mass 1.5 (AM1.5) illuminations26,28,30,31. In 2019, Omair et al.32 reported a TPV cell efficiency of 29.1% under 1480 K radiation temperature, by recycling sub bandgap photons to the radiator. It is possible to achieve > 50% cell efficiency by improving the series resistance, material quality and reflectivity using chamber with high mirror reflectivity. Recently, Fan et al.33 presented an air-bridge structure that enables photons recycling, increasing the reflection of sub-bandgap photons to 99% and enhancing the efficiency up to 30%. As summarized in Table 1, the TPV testing conditions resulted in higher output performance as compared to solar spectrums. Moreover, commercially available TPV cell has the advantage of producing high output power density, ~ 26 times higher than the output power of solar PV cell34,35. All the reported work in the optimization of In0.53Ga0.47As cell are based on the alteration of single design variable36,37. Recently, a multi-variable optimization of solar cell was used to optimize the physical properties of electron transport materials, hole transport materials, and metal contact and layers thickness. It was accomplished using the MATLAB optimization toolboxes incorporated with one-dimensional (1D) Solar Cell Capacitance Simulator (SCAPS) for device simulation40. However, the study did not take into account the impact of doping concentration on the cell performance.

Table 1 Structure design and the performance of reported heterojunction In0.53Ga0.47As cell.
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The optimization of TPV cell structure is critical in getting the highest achievable η. A slight increment in η will significantly increase the output power and total energy. The simplest method to optimize a structure is the single-layer/variable optimization, which optimizes only a single parameter at a time while other parameters are kept constant. Several attempts were made to optimize single variable, especially on the emitter and base layers of In0.53Ga0.47As cell36,37. However, device performance depends collectively on all the design variables45,46, and a more heuristic optimization that considers the effect of all important variables for the In0.53Ga0.47As TPV cell is necessary to achieve the optimum cell efficiency. Therefore, this study investigates the effect of each variable through single variable optimization and performs multi-dimensional (simultaneous multi-variables) optimization using real coded genetic algorithm (RCGA) to obtain the optimum configuration of In0.53Ga0.47As TPV cell.

Methods

In0.53Ga0.47As cell modelling and validation

The In0.53Ga0.47As cell was modeled using the computational numerical modeling TCAD Silvaco ATLAS software package. A 2-dimensional (2D) In0.53Ga0.47As model was constructed using DevEdit tool, while the computations on the electrical characteristics were mainly performed with ATLAS. The input–output transformation method is used to validate the In0.53Ga0.47As simulation model with similar experimental work reported by Sodabanlu et al.27. The input parameters are structure design and testing conditions. The structure design includes front and back gold contacts, thickness and doping concentration of the emitter, base, BSF, FSF, buffer and cap layers, as shown in Fig. 1a. No anti-reflective coating (ARC) was considered in the structure. The validation testing is performed under a standardized solar spectrum AM1.5 at room temperature (300 K). The output performance parameters that are of primary interest include JV-curve, Jsc, Voc, FF and η. The simulation took into consideration the Auger, radiative and Shockley–Read–Hall (SRH) recombination as well as the carrier’s lifetime and mobility concentration-dependent models. At a doping concentration of 1 × 1017 cm−3, lifetime of 16 ns for electrons and 40 ns for holes were used based on the model. While the In0.53Ga0.47As model and testing conditions remain constant over the validation process, the materials parameters of the In0.53Ga0.47As and InP were varied within the reported range in previous literature. Table 2 summarizes the electrical properties of In1-xGaxAs as a function of x composition and concentration-dependent models.

Figure 1
figure 1

Simulation to reported experimental of In0.53Ga0.47As (a) Baseline n-p structure27; the drawing was created using SketchUp58 (b) JV curves and performance parameters.

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Table 2 Material parameters of In1-xGaxAs and In0.53Ga0.47As at 300 K.
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Using the identified material parameters in Table 2, the current density–voltage (JV) characteristics of the In0.53Ga0.47As cell are obtained. The generated JV characteristics of the simulation model can be seen in Fig. 1b. It can be observed that a close agreement was obtained between the performance parameters of the simulation model and the reported experimental data. A percentage error of less than 1% was achieved for each parameter. For example, a percentage error of 0.61% between the experimental and simulation results was calculated for η. In addition, it was reported by other literature that the performance parameters of η, FF, Voc and Jsc, under similar testing condition (AM1.5), were in the range of 9.3–12.9%, 68–71%, 0.31–0.39 V and 21.5–42.8 mA/cm2, respectively30,41,42. The simulation results for the validation work achieved in this study are within the reported range; hence, validate the In0.53Ga0.47As cell model in this work.

Single layer/variable optimization

A single variable optimization gives an indication of the significance of a variable to the TPV cell performance, reveals the trend of performance variation for each variable and more importantly it identifies the range of the design parameters accurately, and that provides the RCGA with a much faster convergence speed as well as a higher solution accuracy. Based on the reported values for the thickness and doping concentration of each layer in Table 1, their maximum and minimum values were first estimated. However, since the majority of the In0.53Ga0.47As structures are used for PV application, several simulations were conducted to modify the upper and lower boundary conditions to suit TPV testing conditions. Table 3 shows the range of the design parameters for the In0.53Ga0.47As TPV cell. Individually, each variable was manipulated while the rest of the design parameters in Fig. 1a remained constant. In the BSF analysis, base thickness was fixed at 10 µm to reduce the absorption in BSF, thereby allowing the investigation of the layer functionality as field generator. The optimization process was conducted under radiation temperatures from 800 to 2000 K at 50% beam illumination intensity. The 50% beam intensity was preferred as selective radiator usually emitted ~ 50% less power than an ideal blackbody59. In addition, a low pass optical filter at 2 µm was employed in the simulation following the reported TPV system60,61,62. Fourspring et al.63 illustrated the use of the edge short-pass filter and emphasized that the optical interference material has high spectral efficiency which reduces the amount of energy that reaches the TPV cell for wavelengths higher than 2 µm. The cell temperature was maintained at 300 K, assuming that it is controlled using an effective thermal management system64. The consideration of heat transfer between the TPV cell and the environment can be a more crucial subject of investigation for TPV devices with nano-gap between the device and the heat emitter65.

Table 3 The range of thicknesses and doping concentrations for the single variable optimization In0.53Ga0.47As structure.
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Multi-dimensional optimization using real coded genetic algorithm method

The multi-dimensional optimization performs complete iterations for all possible combination of different variables to obtain the optimum values for all variables that achieve the highest efficiency. A flow chart of the multi-variable optimization of In0.53Ga0.47As structure under different radiation temperatures is presented in Fig. 2. The optimization process consists of both the device simulation module and the numerical optimization. Firstly, the device simulation module, where ATLAS was used to simulate the 2D In0.53Ga0.47As model. An initial population size of 50 input vector X was implemented in the final optimization version. It was demonstrated that a higher population size improves the accuracy and reduces the number of generation (Gen) required to allow the iterations to converge to the optimum In0.53Ga0.47As configuration. X is given in Eq. (1) as the six-layer/twelve design variables to be optimized.

$$X = X_{i} = [x_{i1} ,x_{i2} , \ldots ,x_{in} ] = [x_{i1} ,x_{i2} , \ldots ,x_{i12} ]$$
(1)

where i = 1–50 is the initial population size, n = 1–12 is the design variables based on some lower and upper physical constraints \({x}_{L}\le {x}_{in}\le {x}_{U}\) in Table 4. DBinternal tool is used to interfaced 50 sets of X to Deckbuild ATLAS, then solves the optically excited charge generation, 2D Poisson’s Equation, transport Equation, and continuity Equations and calculates the efficiency (η(X)) of the cell based on the inputs. Moreover, the numerical optimization of RCGA method was performed. Real coded is a direct representation of the variables, where no coding and encoding is required66. The objective function is to maximize the η of the In0.53Ga0.47As TPV cell, as presented in Eq. (2).

$$\eta_{\max } = \mathop {\max }\limits_{{x_{L} \le x_{in} \le x_{U} }} \eta (X)$$
(2)

where η is20,67:

$$\eta = \frac{{P_{out} }}{{Q_{emit} - Q_{filter} }}$$
(3)
Figure 2
figure 2

The flowchart that illustrates the hybridization of Silvaco TCAD with Real coded genetic algorithm.

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Table 4 The variables upper and lower boundary conditions for the multi-dimensional optimization In0.53Ga0.47As structure.
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Qemit is the thermal emission of the blackbody at a fixed temperature (TBB), and the Qfilter is the thermal reflected emission of the low pass optical filter. The photon flux of an emitting blackbody, Φ, as a function of the emitted λ and TBB, is calculated via Planck’s Law in Eq. (4).

$$\Phi (\lambda ,T_{BB} ) = \frac{{2\pi hc_{o}^{2} }}{{\lambda^{2} (\exp (\frac{{hc_{o} }}{{k_{B} \lambda T_{BB} }}) - 1)}}$$
(4)

where h is the Planck’s constant, kB is the Boltzmann’s constant and co is the speed of light.

The Pout of the cell is defined as:

$$P_{out} = V_{m} J_{m}$$
(5)

where the photocurrent, Jm, as a function of voltage across the cell, Vm, is the difference between the short circuit current and recombination loss, given by:

$$J_{m} = J_{SC} - (R_{rad} + R_{SRH} + R_{Aug} )$$
(6)

where Rrad, RSRH and RAug, are the radiative, SRH and Auger recombination rates, respectively.

Based on the principle of detailed balance, the Jsc is calculated from the external quantum efficiency EQE(λ) and the Φ(λ):

$$J_{SC} = e\int\limits_{0}^{{\lambda_{{E_{g} }} }} {\Phi (\lambda )} EQE(\lambda )d$$
(7)

The fitness ratio is defined as the efficiency ratio (ηr(X)), and it is shown in Eq. (8). The chromosomes are arranged based on their fitness from higher to lower (i` = 1–50), and then some evolution mechanisms like survivor selection, crossover and mutation were used to build the next generation (Gen) using Excel and Visual basic. A 20% (10-best fitted chromosomes: Xi = X1 to X10) of the best chromosomes survived and directly passed to the next Gen as \({X}_{{i}^{^{\prime}}}\) while 80% (40-best fitted chromosomes) of the best-selected chromosomes go to the next step of crossover and mutation producing a new set of population68. This process is presented in Eq. (9).

$$\eta_{r} (X) = \left| {\frac{{\eta_{r} (X_{i} )}}{{\Sigma_{i = 1}^{50} \eta (X_{i} )}}} \right|$$
(8)
$$\left\{ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \right.\begin{array}{*{20}c} {X_{i} = X_{{i^{\prime}}} } \\ {p_{c} = 0.8,p_{m} = 0.01} \\ {X_{i} = 0} \\ \end{array} \begin{array}{*{20}c} {} & {} \\ \end{array} \begin{array}{*{20}c} {1 \le i^{\prime} \le 10} \\ {1 \le i^{\prime} \le 40} \\ {i^{\prime} > 40} \\ \end{array}$$
(9)

After some Gen’s simulations, child chromosomes were created from the best performing parent chromosomes producing the optimum In0.53Ga0.47As configuration. A stopping criterion was decided after observing no significant change in the efficiency and efficiency ratio within the Gen (ηr(X) = 2 ± 0.3). In this way, the optimization process was repeated for illumination source temperatures from 800 to 2000 K, with an interval of 200 K.

Results and discussion

Single layer/variable optimization

The effect of varying the thickness and doping concentration of cap, FSF, emitter, base, BSF and buffer layers on the performance parameters (Jsc, Voc, FF, η) are tabulated in Table 5. The variables are classified into three categories: insignificant where the variation in η is ≤ 0.4%, significant where the change in η is between 0.4 and 3% and highly significant where the change in η is ≥ 3%.

Table 5 The summary of the result for single variable optimization for blackbody temperature from 800 to 2000 K.
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As can be seen from Table 5, the Voc is not significantly affected by the doping concentration of the cap, FSF, BSF and buffer layers. Theoretically, Voc is influenced by dark current densities (J01 and J02), where J01 is contributed to the dark current due to surface and bulk recombination losses, and J02 is related to recombination due to traps in the space charge region (SCR)69. However, the implementation of InP in the front and rear side of the junction reduces the front surface recombination and back surface recombination, eliminating the dark current across the surface. The variation of cap, FSF, BSF and buffer doping concentration produces no effect on the Voc of the cell since no absorption and recombination have occurred. This is because the Type-II band alignment (staggered band) between InP/In0.53Ga0.47As and In0.53Ga0.47As /InP led to spatial separation of electrons and holes and passivated the surfaces70. However, the thickness increment of those layers will affect the Voc because of the losses due to the light aborption71.

The Jsc is mainly related to the absorption and diffusion length of the photo-generated carriers, as shown in Eq. (10).72.

$$I_{sc} = qG(L_{n} + L_{p} )$$
(10)

where G is the generation rate, q is the charge, and Ln and Lp are the electron and hole diffusion length, respectively.

Since In0.53Ga0.47As material has a long electron and hole diffusion lengths, the increment of emitter, base and BSF thicknesses will significantly improve the absorption, carrier’s generation and Jsc. For instance, the generated electrons (holes) at absorber layers has a diffusion length of 18.89 (5.88) µm at 1 × 1017 cm−3 doping concentration. The increment of base layer thickness from 1 to 18.89 µm will increase the Jsc since majority of the generated minority carriers (electrons) are able to reach the SCR before they are recombined. The doping concentrations of the absorber (emitter, base and BSF layers) affect the probability of carrier recombination as the mobility, lifetime and diffusion length of carriers decrease with higher doping concentration. However, since the emitter layer and BSF layer were remained constant at 0.05 µm and 0.025 µm, the variation of doping concentration gives a minor effect on the cell Jsc.

The FF is defined as the ratio between the VocIsc to the actual operating condition of the cell Pmp = ImpVmp after considering the series resistance and shunt resistance in the structure73. The Pmp, Imp and Vmp are denoted as the cell maximum power, maximum current and maximum voltage, respectively. It can be seen that the FF and η performance parameters were affected by almost all of the design variables. This is due to the domination of resistance losses when manipulating the variable individually. Nevertheless, the manipulation of FSF and BSF doping concentration had no significant impact on cell performance since the thickness of those layers are kept at very thin (~ 0.02 µm), resulting in lower resistance losses and minor absorption.

Based on the single variable optimization, it was found that the manipulation of the thickness and doping concentration variables for the base layer significantly affects all performance parameters. In particular, the highest simulated η is achieved at an optimum base layer thicknesses of 5 and 16 µm with a blackbody temperature of 2000 K and 800 K, respectively. The variation of the optimal base thicknesses is related to the radiation spectrum and its peak wavelength (λp). Based on Wien’s displacement law, λp of the spectrum shifts toward shorter IRs for higher blackbody temperature hence thinner base layer is required. Low doping in the base layer led to a high minority carrier lifetime and long diffusion length, which could increase the probability of carriers reaching the contact before recombining. However, it could also reduce the conductivity of the layer, electric field, and the built-in potential at the junction that decreases the Voc performance, offsetting the improvement in Jsc. On the other hand, a thicker base layer will increase the absorption of IRs, which results in higher Jsc. However, it could also increase the recombination and shunt resistance74,75. Increasing the thickness is detrimental to cell performance since it causes higher recombination, especially if the structure has high SRH rate76. Moreover, the minority carriers generated in the lower region of the cell with diffusion length shorter than the thickness has a higher probability of recombining before reaching the SCR74. Therefore, it is worth exploring the tradeoff relationship of the base layer thicknesses and doping concentrations to acquire the optimum configuration of the base layer that produces the highest cell efficiency. The correlation of cell efficiency with the base layer thickness and base doping concentration is depicted in Fig. 3. The base thickness (doping concentration) was varied between 1 (6 × 1016) and 28 µm (1 × 1019 cm−3), while the rest of structure design variables were kept at their baseline values.

Figure 3
figure 3

The efficiency of In0.53Ga0.47As TPV cell vs thickness and doping concentration of the base layer under 1400 K blackbody temperature.

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Based on Fig. 3, it was found that an optimum η of 17.61% is obtained at an optimum base thickness (doping concentration) of 11 µm (1 × 1017 cm−3). The increment in base thickness improves the absorption of IR, especially the band-edge photons76. Based on the Einstein relationship, Caughey-Thomas model and SRH model, the diffusion length of electron and hole are calculated as a function of doping concentration. At base doping of 1 × 1017 cm−3, the diffusion length of electron (hole) is equal to 18.89 (5.88) µm. The electron diffusion length of 18.89 µm is longer than the 11 µm base thickness. Since the diffusion length is longer than the base thickness, the probability that photo-generated carriers can reach the SCR is quite high74. Furthermore, the high diffusion coefficient of In0.53Ga0.47As carriers permits the formation of the active junction at low doping concentration. In0.53Ga0.47As cell ability to produce active junction at low doping concentration allows it to obtained high cell efficiency with minimum recombination rate.

Multi-dimensional optimization using real coded genetic algorithm method

The results of multi-dimensional optimization on In0.53Ga0.47As TPV cell model under 800–2000 K blackbody temperatures are summarized in Table 6. It was found that the optimized cell presented in this work has thicker base layer as compared to those TPV cells reported in Table 1. At 1800 K blackbody temperature, the optimized cell has an efficiency of 20.48%, compared to 15% reported for TPV cell tested under similar testing conditions40. Compared to the reported In0.53Ga0.47As TPV cell efficiencies which can be as high as 30%32,33,77, the optimized structure in this paper does not consider photons recycling and ARC. Instead of utilizing photon recycling, this work focuses on the TPV cell design structure. A thicker base layer between 16 and 18 µm is preferred to enhance the current density as it increases the absorption of near band-edge photons and free-carrier absorption (FCA), which highly impact TPV cell efficiency78,79. At base layer thickness of 16–18 µm, the maximum cell efficiency of 23.18% was reported at 1400 K radiation temperature, while the minimum efficiency of 18.41% was achieved at 800 K. The electrical losses due to the high photogenerated carriers increased at radiation temperatures > 1400 K, which reduced the η. Although maximum efficiencies were achieved with base layer thickness of 16–18 µm, considering the technical difficulties and cost in growing high-quality In0.53Ga0.47As layer at > 10 µm, the base layer thickness was reduced to 8 µm. With this reduction in base layer thickness, the efficiency of the TPV cell is reduced only by an average of 1% (Maximum reduction of 1.9% at 1000 K and minimum reduction of 0.1% at 2000 K). RCGApractical represents the RCGA results after taking into account the practicality in the growth of semiconductor wafer by reducing the base layer thickness to 8 µm. A thinner cap and buffer layers are designed with a higher doping concentration to form a better front and rear ohmic contacts, resulting in higher FF and lower series resistance. Besides, a thin FSF and BSF layers are designed with a higher doping concentration, resulting in a better front and rear junction passivation with minimum optical absorption.

Table 6 The optimum In0.53Ga0.47As TPV configuration from real coded genetic algorithm method (RCGApractical).
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A proper cell design has to consider both optical and electronic losses. The optical losses were the dominant factor in determining the change in efficiencies of the non-optimized cell. Optimized cell efficiency, on the other hand, were dominated by the electrical losses, especially at high radiation temperature with high energy spectrum density. The In0.53Ga0.47As TPV cell has to be optically thick (i.e. to absorb all or most of the incident illumination) and electronically thin (i.e. to collect the photoexcited electron–hole pairs with little or no losses). These two requirements lead to an optimal configuration that maximizes efficiency. At 1400 K blackbody temperature, Jsc and FF are increased respectively from 755.01 to 1719.35 mA/cm2 and 62.65 to 68.63% after optimizing the entire cell configuration. Most of the optimization works focused on optimizing the electrical losses of cell to improve the Voc performance by reducing the thickness of the absorber layer27,80. However, the absorption of near band-edge photons is neglected. Alharbi et al.81 highlighted that the leading cause of the reduced efficiency of a solar cell below the theoretical limit is the drop in the estimated Voc while usually, the obtained Jsc is around the theoretically maximum values. It is important to mention that the solar spectrum is mainly concentrated around the visible region, and these photons do not require thicker base absorber. On the other hand, TPV illumination flux is usually concentrated at infrared wavelengths, and thicker absorber is needed to improve the absorption of IRs and significantly increase Jsc. This statement was supported by the optimization study reported by Baudrit and Algora46, where increasing the absorber thickness of the bottom cell in GaInP/GaAs dual-junction increases the value of photocurrent density. Furthermore, the Jsc and η were increased respectively from 13.85 to 15.62 A/cm2 and 32.6 to 36.4% under 1000 suns concentration.

For the validation of optimization, the In0.53Ga0.47As TPV structure reported from the RCGA optimization method was validated by two validation methods (VM), which are single-variable method (VM 1) and iteration method (VM 2). VM 1 utilized the results from the single-variable optimization where the optimum values of the 12 design parameters obtained through single-variable optimization were used to determine the efficiency of the TPV cell. VM 2 utilized the iteration method to simulate all possibilities for the 12 design parameters. However, it was estimated that the total number of iterations are 1210 simulation runs. To reduce the computation time to a manageable level, we used a bigger step and only varied the important design variables while the insignificant variables were maintained at their baseline values. Due to these reasons, it is expected that the cell efficiencies obtained through VM 1 and VM 2 are slightly lower than those obtained through RCGA method.

The results obtained through RCGA, VM 1 and VM 2 are tabulated in Table 7. For different radiation temperatures, the average percentage differences between RCGA and VM 1, and between RCGA and VM 2, were 5.17% and 6.95%, respectively. On the other hand, RCGA method is able to converge to the optimum structure in a faster manner as compared to the long simulation time needed by the iteration method.

Table 7 The summary of the results obtained from RCGA, VM 1 and VM 2 with their percentage differences.
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Figure 4 illustrates the EQE of non-optimized and RCGApractical-optimized In0.53Ga0.47As TPV cells as a function photon λ. It can be seen from Fig. 4 that the EQE(λ) of the multi-variable optimized cell is higher than that of the non-optimized TPV cell. This is due to the enhancement of photocurrent generation at a wavelength higher than 0.85 µm. A possible explanation for this is that the optimized In0.53Ga0.47As TPV configuration tends to have higher absorption and collection of IRs photo-generated carriers. Despite the reduction in Voc by 6.12% after the optimization process, Jsc is significantly increased by 127.73%, resulted in the improvement in the TPV cell efficiency by 138.23%.

Figure 4
figure 4

The external quantum efficiency vs wavelength for non-optimized and RCGApractical optimized In0.53Ga0.47As TPV cell under 1400 K blackbody temperature.

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Based on the optimized results, a significant increase in η is attained when the base layer increased from 1 to 8 µm. This finding can be supported by an analysis of the absorption coefficient and the absorption length of the In0.53Ga0.47As cell. The absorption coefficient describes the light penetration in a semiconductor before being absorbed and can be obtained using Kramers–Kronig Dispersion relation as follows:

$$\alpha (\lambda ) = \frac{4\pi k(\lambda )}{\lambda }(\mu {\text{m}}^{ - 1} )$$
(11)

where k(λ) is the extinction coefficient of In0.53Ga0.47As82,83. On the other hand, the absorption length (α−1) is given as the inverse of the α and describes the penetration length of majority photons in semiconductor before being absorbed. Due to absorption in the material, the illumination intensity weakened with increasing penetration length and can be described by means of a decaying exponential function, as shown in Eq. (12).74.

$$\alpha_{Abs} = (1 - R).(1 - e^{ - \alpha d} )$$
(12)

where ηAbs is absorption efficiency, the effective cell thickness (d) can be determined from Eq. (13) by maximizing the absorption in the cell ηAbs =  ~ 99% and R = 074.

$$0.99 = (1 - e^{ - \alpha d} ) = 1 - 0.99 \to d = \frac{\ln (1 - 0.99)}{{ - \alpha }}\mu {\text{m}}$$
(13)

Figure 5 shows the result of the α, α−1and d of In0.53Ga0.47As as a function of λ. It can be seen that thicker absorber layer is needed to absorb IRs. For instance, photons with λ of 1.65 μm have an absorption length of 4.25 μm. To effectively absorb ~ 99% of the photons, the cell thickness will therefore be approximately 19.59 μm.

Figure 5
figure 5

The absorption coefficient, absorption length and effective thickness at different wavelengths.

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The In0.53Ga0.47As has a spectral response with wavelength at around 1.75 μm. It can be seen that infrared light requires a thick absorber, so that majority of the photons are absorbed. The illumination intensity of the blackbody is mainly concentrated at λ > 1.0 μm. For instance, 1400 K blackbody temperature has 69.25% of power density for wavelengths between 1 and 1.8 μm as compared to 3.43% of power density for wavelengths between 0.2 and 1 µm. An optical BSR is used to improve the light absorption, as the optical path distance can be doubled due to the back reflection20. Therefore, although the effective thickness to absorb photons up to 1.75 μm is approximately 32–36 μm (based on Fig. 5), the effective device thickness can be reduced to approximately 16–18 μm. However, based on additional simulation work upon completing the multi-dimensional optimization, an effective thickness higher than 8 μm (up to 18 μm) will only increase the efficiency by an average of about 1% for various blackbody temperatures due to the sharp decline of the absorption coefficient for λ ≥ 1.63 µm. Other than that, the light trapping method such as Lambertian rear reflector and textured surface can be used to improve the light absorption in the cell74,84.

A further explanation for the high EQE is due to the effective separation and collection of generated carriers. In0.53Ga0.47As structure is frequently constructed with FSF and BSF layers to reduce the surface recombination and enhance the Voc41,85. High to low doping concentration between the FSF or BSF and the active junction is vital to generate a SCR similar to the SCR between n-p junction. For instance, holes diffused out of this highly doped BSF layer into the lower-doped base layer, leaving site-fixed negatively charged acceptor atoms behind74. The generated electrical field that acts like an electric mirror that returns the electrons generated through absorption in the direction of the SCR74. The probability of undesired recombination at the rear of the cell is thus significantly reduced. Belghachi et al.86 described a mathematical model on the importance of high-low junction in the front and rear sides of GaAs cell, which play a crucial role in enhancing the light-generated free carriers’ collection. The thickness of FSF and BSF layers should be as thin as possible while the doping concentration should be as high as 1 × 1019 cm−386. The band diagram is an alternative way to view the effect of FSF and BSF layers. The band diagram of the In0.53Ga0.47As/InP cell was extracted from TCAD model, with the FSF and BSF layers.

Figure 6 presents the band diagram of In0.53Ga0.47As heterojunction. The band offset between In0.53Ga0.47As and InP forms type–II band (staggered) leads to spatial separation of electrons and holes. A discontinuity in conduction for base/(BSF/buffer) interface prevents the electrons from moving further up to the back contact while allowing the flow of holes. The discontinuity in valance for the FSF/emitter interface prevents the holes from moving further up to the front contact while allowing the flow of electrons74. The band alignment of In0.53Ga0.47As structure was investigated in several studies87,88. The n-InP will form a barrier to holes, but a sink for electrons at the FSF and visa verse at the BSF/buffer. Besides that, cell total voltage is now divided into the potential level at n-p junction and additional level at n+n and pp+ interfaces74. Hence, FSF layer improves the EQE of shorter wavelengths photo-generated carriers, and the BSF/buffer layer increases the EQE of longer wavelengths photo-generated carriers86. It should be stressed that BSF/buffer layer is very significant to enhance the Jsc and Voc of the TPV cell since the long-wavelength photons tend to absorb at the deeper region of the structure. The combination of both thicker absorber layer and BSF/buffer significantly improves the collection of photocurrent collection of λ near to the band-edge of In0.53Ga0.47As TPV cell.

Figure 6
figure 6

The band diagram of In0.53Ga0.47As cell heterojunction.

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Conclusion

In summary, the simulation model is validated with reported experimental data, generating a low η percentage error of 0.61% between experimental and simulation. Research gap was identified based on comprehensive comparison of previous structure designs and performance of In0.53Ga0.47As cell. Single variable and multi-dimensional optimization (RCGA) of heterojunction In0.53Ga0.47As cell are developed and applied at 800–2000 K TPV radiation temperatures. The single variable optimization is used to investigate the effect of thickness and doping concentration of layers, which demonstrated the significant impact of the base layer to achieve high performance In0.53Ga0.47As TPV cell. Meanwhile, under radiation temperatures ranging from 800 to 2000 K, optimized In0.53Ga0.47As heterojunction TPV cell using RCGA increases the η by an average of 11.86% as compared to the reference structure. It was found that the incorporation of a thicker absorber with effective barrier layer BSF/Buffer layer improves the absorption and collection of photo-generated carriers near the band-edge, which produced higher output performance. The increment of the In0.53Ga0.47As cell efficiency leads to a significant increase in the generated output power, demonstrating great potential of TPV for industrial waste heat harvesting. Finally, the method of hybridizing the Silvaco TCAD software with RCGA for multi-dimensional optimization can be readily adopted in optimization work for semiconductor devices, such as solar cell, TPV cell and photodetectors.