Thermodynamic modeling of a spectrum split perovskite/silicon solar cell hybridized with thermoelectric devices

To facilitate the attainment of higher performance in the tandem perovskite/silicon solar cell, this work seeks to conduct the thermodynamic modeling and analysis of a spectrum splitting tandem perovskite/silicon solar hybridized with thermoelectric (TE) devices. A dichroic beam splitter is employed as the solar concentrator for the utilization of the entire solar spectrum and the effects of temperature dependency and leg geometry alteration are considered in the TE device pins. Additionally, a TE cooler (TEC) is lapped behind the backplate of the tandem solar cell to reduce overheating and allow for higher power generation from the hybrid system. Finally, novel equations are developed to study the effects of varying the halide composition of the perovskite on the overall system performance. Among several fascinating results obtained, it was disclosed that increasing the bromine composition only improved the system's performance when a halide composition of 0.2 was used for hybrid organic‐inorganic perovskite thickness ranging from 300 to 400 nm. It was also forecasted that as the solar cells are exposed to higher concentrated solar irradiances, the effects of increasing the halide composition on the system's performance will become more notable. Additionally, utilizing a TEC to maintain the tandem solar cell backplate temperature at 293K is not beneficial to the system's performance when a beam splitter is used. Finally, it was found that the highest system efficiency of 42% was obtained at a split wavelength of 800 nm which is considerably higher than the 23.6% reported for standalone perovskite/Si. These results are sufficient to provide useful insights regarding the operation of spectrum splitting tandem perovskite/silicon solar cells with TE devices.


| General overview on photovoltaics
The use of photovoltaic (PV) modules is on the increase due to the clean energy they provide relative to greenhouse gas emissions from fossil fuel sources like coalfired power plants. 1 Additionally, the fear that fossil fuel sources will soon be exhausted, due to the exponentially increasing human population, has accelerated the search for efficient alternate energy sources. 2 However, the sensitivity of solar cells to light and temperature causes a severe reduction in the PV module efficiency. 3 This is why previous works sought to mitigate the efficiency reduction of PVs by combining them with different solidstate energy devices which convert the waste heat to useful electricity. 4 Due to the adverse effects of high temperature on the PV module, spectrum splitters were utilized to split the concentrated beam into individual energy components. 5 The major components are short wavelengths for the PV module (such as ultraviolet and visible light) and long wavelengths (near-infrared and infrared) for the solid-state energy converters. 6

| Benefits of tandem perovskite/ silicon solar cells
Research has shown that higher efficiencies are obtainable by increasing the number of stalked PV cells. 7 The major objective in the design of PV solar cells is to maximize the efficiency-to-cost ratio. Perovskite solar cells (PSCs) have reasonably met that requirement. 8 Efficiencies of lead halide-based PSCs has rapidly been improved, reaching 20% on small-area devices making this material a competitive thin-film PV technology. 9 Furthermore, recent research has shown that introducing a mixture of bromide and iodide in the halide composition of lead methylammonium perovskites allows for continuous fine-tuning of the band gap which is a fine property for multi-junction solar cells. 10 The major variations of tandem device configuration for solar cells are the perovskite-perovskite and perovskite-silicon tandem solar cells. 11 The perovskite-perovskite configuration has a lower greenhouse gas emission factor and energy payback time compared to the silicon benchmark. Also, the thermodynamic efficiencies of the perovskite-perovskite and perovskite-silicon tandem solar cells are very high (exceeding 30%), thus, paving the way for upgrading the silicon cell performance at little extra cost in short-term and long-term basis, respectively. 12 Perovskite/Si combination as tandem solar cells is of great interest due to the possibility of boosting efficiencies above 30% while reducing the cost per kilowatt hour (kWh). 13 Futscher and Ehrler 14 indicated that perovskite/Si at different locations may require different tandem configuration and/or perovskite band gaps to minimize the cost per kWh. They also showed that by using a perovskite top cell with the ideal band gap for the respective tandem configuration, perovskite/Si with power conversion efficiency limits above 41% is possible for all three tandem configurations even at non-ideal climate conditions. Al-Alshouri et al. 15 reported a tandem monolithic perovskite/Si solar cell with a record-breaking efficiency of 29.15%. The absorber of the perovskite cell was made to remain stable when exposed to illumination via a series of very fast extraction of holes and minimization of nonradiative recombination at selected hole interfaces. Kim et al. 16 designed and developed a bifacial 4-terminal tandem perovskite/Si heterojunction solar cell that generated 30% efficiency. The efficiency recorded by the hybrid bifacial solar cell was greater than the 29.43% Schockley-Quiesser limit for crystalline-based Silicon solar cells. He et al. 10 17 noted that the high conversion efficiencies obtained from tandem perovskite/ Si solar cells were due to the advantages of perovskite materials such as their low cost, tunable bandgaps, and easy fabrication.

| Thermoelectric devices applied in tandem PVs
Thermoelectric (TE) modules are solid-state energy conversion devices that directly convert thermal energy to electricity based on TE effects. 18 These devices offer several desirable perks such as solid-state and noiseless operation, environmental friendliness, and zero maintenance costs. 19 Due to their harnessing of the TE effects to convert thermal energy, they have found various applications in converting waste heat in several power systems to electricity. 20 More specifically, they have proven to be good companions with PV systems in hybrid designs such as the PV-thermoelectric generators (TEGs) 21 and PV-TE coolers (TECs). 22 However, PV-TEC sandwiched module has shown improved the efficiency and the overall power output of the hybrid system due to the Peltier cooling process. 23 In addition, PV-TEG hybrid system with a beam splitter demonstrated higher efficiency in comparison to the sandwiched system 24 due to the conversion of waste heat from the back plate to electric output. In addition, some authors 25 hybridized the PV-TEG system with single junction PV and beam splitters to enable spectrum management and reduce the thermal losses encountered in the direct coupling/sandwiching method.
The emerging high-efficiency PSC has also been combined with TEGs to form hybrid generating systems capable of utilizing the broad solar spectrum. 26 Zhang et al. 27 estimated the features and feasibility of directly coupling a PSC with TE modules and reported that the efficiencies obtained from the hybrid and standalone systems were 18.6% and 17.8%, respectively, for a temperature coefficient that was lower than 2%. It was further shown that the PSC is a very suitable option for the PV-TEG system. Xu et al. 28 optimized the performance of a splitting PSC integrated with a TEG and recorded an efficiency of 20.3% after optimizing the performance of the device. The hybrid device was operated under AM 1.5G conditions, an open-circuit voltage of 1.29 V, and an irradiance of 100 mW cm À2 . Liao et al. 29 recorded efficiencies of 20.4%, 5.16%, and 20.8% for a standalone PSC, TEG, and hybrid PSC-TEG system, respectively. After optimizing the operating parameters of the PSC-TEG system, the hybrid device was able to realize a maximum efficiency of 22.9% for an optimum layer thickness of 449.7 nm. It was finally stated that incorporating the TEG in the PSC facilitated the reduction of waste heat emission and improvement of the power and conversion efficiency of the PSC. Zhou et al. 7 utilized a TEG in reducing the heat generation process in a concentrated PV system made of metal-halide PSC. At a concentrated solar irradiance of 3 suns, the hybrid system was able to achieve a maximum efficiency of 35%, which was 4.7% higher than that of the standalone PSC. Finally, Lorenzi et al. 30 showed that relative to the amorphous Silicon and Gallium Indium Phosphide cells, the PSC showed a maximum efficiency enhancement of 3.1% when a bismuth-telluride based TEG was operated in tandem. Based on their results, they further concluded that the PSC has a greater potential of harnessing waste heat via TEGs compared to the popular Si solar cells. However, no efforts have been made to combine the hybrid PSC/Si system with TE devices so as to improve the system efficiency by reducing the overheating of the tandem cells.

| Scientific novelty
The literature review shows the potential of the PSC in utilizing the broad solar spectrum by incorporating TE devices. Furthermore, the emerging tandem PSC/Si system is showing very promising power densities and efficiencies due to the deposition of perovskite in the Si layers. However, it has been found that the efficiency of the tandem PSC/Si system is strongly affected by spectral and temperature changes as well as the composition of the halide. Up to date, no effort has been made to mitigate the effects of spectral, temperature, and halide composition on the efficiencies of tandem PSC/Si systems.
This work is novel in many ways. First, a thermodynamic modeling of a concentrated tandem PSC/Si system combined with a spectrum splitting TEG having trapezoidal leg geometry is presented. The spectrum splitting system used is a dichroic beam splitter which allows the TEG to utilize the infrared region which is detrimental to the efficiency of the hybrid PSC/Si system. Second, in a bid to modify and increase the utilizable spectrum, the hybrid PSC/Si system is coupled to a TEC to reduce the thermal heating of the system for improved power generation and demonstration of the halide proportion. Finally, the effects of varying the halogen composition are studied so as to better understand the effect of its tunable band gap energy.

| METHODOLOGY
In this section, the detailed descriptions of the hybrid PSC/Si tandem system operated with TE devices are presented in Section 2.1. Then, Section 2.2 shows the model equations used to describe the performance of the various components making up the spectral splitting hybrid device.

| Description of the conceptualized system
The proposed hybrid system as shown in Figure 1 consists a dichroic spectrum splitter, concentrator, perovskite/Si tandem solar cells, trapezoidal TEG, and TEC. These components are added to improve the performance of the overall system. TEG can directly convert thermal energy into electric power. Radiation energy near the bandgap is directly converted to electricity by PV panel and simultaneously, infrared energy is utilized by the TEG to convert heat to electricity. Consequently, more electricity can be produced by the hybrid system than the electricity produced by a single PV or TE system.
The tandem PV device is sandwiched with the thermoelectric cooler and a co-receiver of the concentrated split light beam with the trapezoidal leg generator. As certain fraction of the incoming solar irradiance, incidents on the surface of both solid-state converters, heat is being lost to the atmosphere through convection and radiation from both top and back surfaces.

| Modelling equation formulation
For a wavelength range of 280 4000 nm and the cutoff or split wavelength λ s , the total concentrated irradiance into the entire system is given 31 as where all terms in Equation (1) are defined in the nomenclature.

| Analysis of the trapezoidal TEG
The footprints of the TEG are known to be of great importance in the modeling and performance of a TEG. Geometric parameters of the TEG (eg, thermoelement leg length and cross-sectional area) can influence the performance of a hybrid system. For the trapezoidal geometry under study, the length which is dependent the crosssectional area of the TEG is shown in Figure 2 is expressed by 20 as in Equation (2).
The one-dimension energy balance of the trapezoidal leg for an infinitesimal element, yields for the p and ntype leg, respectively, as given by Equation (3). 32,33 where the parameters in all equations are defined in the nomenclature. The thermal conductivity can be replaced with a mean value within the junction temperature range and treating it as a constant while differentiating and rearranging Equations (2)- (4) gives The differential Equations (5) and (6) can be solved by applying the principle of superposition using the Euler-Cauchy and Wronskian approach for the complementary and particular solutions. These two equations result to The constants c 1 , c 2 , c 3 , and c 4 in Equations (7) and (8) can be evaluated using the following boundary conditions: For the trapezoidal leg, considering that the geometrical parameters of both n and p-type legs are same, the internal resistance for a uni-couple or a pair is derived and expressed as Since the process occurs at steady state and isolates a TEG leg, the conductive heat transfer rate in x-direction of Figure 2 is written as The negative sign indicates that the heat transfer direction is in opposite direction to the x-axis. Rearranging Equation (11) becomes Substituting, integrating, and incorporating the mean thermal conductivities of the pair, Equation (12) yield From Equation (13), the thermal conductance for the trapezoidal leg is given as The overall energy balance for the entire solar TEG in Figure 1 gives, For the selective absorber, energy balance is also given as The heat absorbed at the TEG hot junction is given by equation (18) The radiation and convective heat terms can be replaced with an equivalent heat quantity. Substituting Equation (18) into Equation (17) gives the expression in Equation (19) α TG The electrical current of the TEG with Thomson influence considered is given as Substituting Equation (20) into Equation (19) and simplifying gives F I G U R E 2 Cross section of the trapezoidal TEG leg. TEG, thermoelectric generator where ζ 000 , ζ 00 , and ζ 0 are respectively expressed as The component efficiency of the trapezoidal TEG is given by Equation (25) The thermal conductivity and electrical conductivity of the mid/high-temperature semiconductor (Skutterudite) used to model the TE legs are provided in Table 1, while Figure 3 shows the variation of the Seebeck conductivity with operating temperature.

| Analysis of the Hybrid Tandem PV-TEC
The analysis of the Hybrid Tandem PV-TEC with an irradiance within 280 nm to the cutoff wavelength incident on the perovskite/Si cells, powering a TEC and expected subjectsthe multi-junction cells to reasonable operating temperature. There would be thermal losses through radiation by reflection and convection to the atmosphere, while giving a net power output (after accounting for the necessary power to drive the TEC). Isolating the hybrid system and taking an energy balance of the system, the energy balance is given as The radiation and convection heat losses can be expressed as 36 Empirically, the combined transfer coefficient is represented by h 0 ¼ 5:7 þ 3:8v. The quantity of heat from the TEC hot junction is given as In this study, TECs are basically defined with specified characteristics such as I max , V max , Q max , and DT max . Imax is the input current that can produce the maximum temperature change, DT max across the TEC module. Vmax is the DC voltage at the maximum temperature difference of DT max and Q max is the maximum amount of heat absorbed at the TEC cold side at I = I max and DT max = 0. Utilizing these parameters, the module properties can be expressed as follows: From Equation (28), the hot junction temperature is given by The electrical characteristics and geometry dimensions of the TEC used in this work are provided in Table 2.
The net power output of the PV-TEC is given by Equation (33) with the TEC powered by tandem module. 38 To improve the conversion efficiency and enable the sub cells respond to a range of the spectrum band, multijunction cells are interconnected in series as shown in Figure 4. As a result, the open circuit voltage and short circuit current density for multi-junction devices are given by Equations (34) and (35). 39 where the spectral response is given by Equation (36) Substituting Equations (27), (32), (34), (35), and the last term of (28), the expression gives Equation (37)The fill factor is expressed as where β ¼ qV oc nk B T pv . From series expansion, natural logarithmic series of β þ 1 is given as The combinations of Equations (38) and (39) can be simplified by applying numerical and analytical approaches. The fill factor can be fitted to a polynomial function using the expression where α, γ, and ℘ are constants. Substituting the expression in Equation (40) and β ¼ qV oc nk B T pv into Equation (37) and assuming the cold junction temperature of the TEC cooler to equal the tandem module temperature because the module is very thin in thickness in nanometric unit. The expression can be rearranged and simplified to give where ϑ 000 , ϑ 00 , and ϑ 0 are respectively expressed as: The overall efficiency of the entire system comprising the multi-junction perovskite/Si, TEG, and coolers is given as

| Analysis of the TE fin
For evaluating the TE fin thermal parameter, the fin thermal resistance is given as: where h cv and A eff are respectively defined as the convective heat transfer coefficient and the effective heat transfer area of the fin. h cv can be estimated using:  where P rf and R e are the Prandtl number of the fluid and the Reynolds number of the flow, respectively. They can be evaluated by: where μ f , c pf , k f , v ef , and ν f are the kinematic viscosity, constant pressure heat capacity ratio, thermal conductivity, kinetic velocity, and the dynamic viscosity of the fluid, respectively. The effective heat transfer area is given by: where the fin efficiency η f is given by The fin geometrical properties are depicted in Figure 5 while the geometrical and thermal parameters used to define the fin in the numerical modeling equations are provided in Table 3.

| Modeling the halide composition variation
However, it is essential to note that the maximum wavelength, λ max of the solar spectrum which can be used for the photoelectric effect differs with the computation expression as the split wavelength, λ s varies 34 : For the perovskite (CH 3 NH 3 Pb 3(1Àx) Br 3x , for 0 ≤ x ≤ 1), the bandgap energy is dependent on the halide composition, x, and expressed as E g ¼ 1:55 þ 0:75x; in this case, the halide is Bromine.

| RESULTS AND DISCUSSION
The thermodynamic modeling of a tandem perovskite/Si-TEC hybridized with a TEG module using dichroic beam  as a spectrum splitter was carried out. The study used a blend of analytical and numerical approaches utilizing the combination of MATLAB, python, and Excel spreadsheet environment. The computation was performed under a solar spectral irradiance of 1.5 air mass AM, solar concentrations ranging from 1 to 3, and fractions of the TEC maximum current as input such as 0:2I max , 0:4I max , 0:6I max , and 0:8I max . However, since perovskite cells are still undergoing material modification and this work emphasizes on what its performance would be if applied to complex system. Through numerical computations and fittings, the fill factor constants are α, γ, and ℘ are À0.002, 0.0706, and 0.2051. To utilize the wavelength influence on the spectral irradiance for the mathematical computation, Simpson's 3/8 rule was used to perform numerical integration on the 1.5AM spectral irradiance and the result is presented in Figure 6. However, for this hybrid system thermodynamic numerical analysis, the bandgap energy of the perovskite (CH 3 NH 3 Pb 3(1Àx) Br 3x (for 0 ≤ x ≤ 1)) is dependent on the halide composition, x, which in this case is Bromine. The result of influence of the halide composition (Bromine) to the output current density of the perovskite/Si-TEC at different cell temperatures of 30 C, 35 C, 40 C, 45 C, and 50 C, concentration ratio, C = 3 and 0.2 of the TEC maximum current (I c ¼ 0:2I max ) is shown in Table 4.
From Table 4, it was observed that the Bromine composition affects the output current density of the system. For all temperatures used (303K, 308K, 313K, 318K, and 323K), current density, J sc increases as the temperature increases and reduces as the Bromine composition, x decreases. For 0.5 ≤ x ≤ 1.0 (1.93 ≤ Eg ≤ 2.3), the value of the tandem module current density seems to be constant except for 303K, 318K, and 323K which showed equal absolute variation of 0.001.
The system performance for current density variation with solar irradiance and concentration factor for different TEC input current (a) I c ¼ 0:2I max ; (b) I c ¼ 0:4I max ; (c) I c ¼ 0:6I max ; and (d) For all values of G and I c (0:2I max , 0:4I max , 0:6I max , and 0:8I max ) with the tandem module held at a temperature of 293K with a split wavelength of 800 nm Figure 7ad, the current density J sc , progressively decreases as the concentration ratio increases through a nonlinear polynomial path.
The plot of current density variation with solar irradiance within a short temperature interval for different TEC input current and concentration factor I c ¼ 0:2I max , I c ¼ 0:4I max , I c ¼ 0:6I max , and I c ¼ 0:8I max is shown in Figure 8a-d. Unlike the plots in Figure 7a-d, the group of plots shows minute variation in the temperature of the cell. Regardless of the variation in temperature, the unit value concentration still remains in the plot with the highest perovskites/Si current density values for all values of IC utilized in the computation (ie, 0.2Imax, 0.4Imax, 0.6Imax, and 0.8Imax).
The plot however is quadratic and the difference in values for the different concentration plot is small. For clarity of this statement, a plot with wider temperature range is needed as shown in Figure 9, for 0:8I max .
The combined system efficiency dependence on the perovskite/Si-TEC current density and voltage is shown in Figure 10. From the 3D graphical plot, the plot with concentration ratio C = 1 has the highest efficiency amidst the three-concentration ratio (C = 1-3). From conventional PV module design, it is at least expected that the module efficiency increases after certain concentration ratio values before it begins to drop. But in this case, the graphical nature is contrary to the conventional and the reason is that beyond C = 1 much electrical power is required by the TEC to bring the perovskite/Si to a temperature of 293K. Carefully observing the plot, it is also seen that the planes of J sc and V oc have an edge trajectory that is akin to an exponential function. Viewing the plot through the efficiency and J sc plane, the graph is seen to be twisted F I G U R E 6 1.5AM integrated spectrum T A B L E 3 Geometrical and thermal properties of the fin 40 faintly and the efficiency drops as the current density increases across 6 V.  perovskite/Si module to run the TEC to sustain the PV cell temperature at 293K. Viewing the graphical plot from the η sys À G plane, it is seen that the efficiency drops asymptotically without converging to zero as the irradiance increases. The reason for this is that at high irradiance much heat needs to be removed from the PV module to bring its temperature to 293K. For this reason, much power must be extracted and utilized by the TEC. The V oc also increased progressively with a maximum combined system efficiency of about 42%. However, beyond 400 W m À2 , the plots were all asymptotic for all V oc values. This is because the beam was split into UV and IR, and the range under consideration on the perovskite/Si, is the UV. The UV not having thermal effect but light will have temperature at the module surface within very close and low range. Therefore, F I G U R E 8 (A) Current density variation with solar irradiance within a short temperature interval for different TEC input current and concentration factor I c ¼ 0:2I max . (B) Current density variation with solar irradiance within a short temperature interval for different TEC input current and concentration factor I c ¼ 0:4I max . (C) Current density variation with solar irradiance within a short temperature interval for different TEC input current and concentration factor I c ¼ 0:6I max . (D) Current density variation with solar irradiance within a short temperature interval for different TEC input current and concentration factor I c ¼ 0:8I max . TEC, thermoelectric cooler F I G U R E 9 Wider temperature plot for clarity of curve nature of Figure 8 small amount of electrical power would be responsible for bringing the cells under the required operating condition. Figure 12 presents a 2D graphical plot of the perovskite/Si-TEC current density against the split/cutoff wavelength of the dichroic spectrum splitter. The plot analysis was evaluated with the perovskite/Si kept at room temperature and the TEC operating with 20% less of its maximum design current. The plot demonstrated that the perovskite/Si-TEC current density for concentration factor of 1 is higher than that of 2 and 3 with maximum values of 5.1545, 4.9458, and 4.8122 A/cm 2 , respectively. Uniquely, all plots show a sharp downward slope from values range of 400 nm to 526 nm of the split wavelength. This slope emanated from the sharp increase in the solar spectrum within wavelength range of 400-526 nm (see Figure 6) and manifested downwardly because of the electric power expended by the TEC to maintain the cell at operating temperature.
In comparison with previous studies that focused on monolithic standalone perovskite/silicon solar cells that recorded maximum efficiency of 23.6%, 35 it is noticed that the proposed tandem system hybridized with TE device had a maximum efficiency of 42%. This signified that the maximum efficiency of the proposed system was 1.81 times higher than that of the previous standalone system due to the inclusion of TE devices in the hybrid system design.

| CONCLUSION
The thermodynamic modeling and analysis of a spectrum splitting concentrated perovskite/silicon tandem solar cell integrated with TE devices was carried out in this work to further improve the performance of the standalone perovskite/silicon solar cell. A dichroic beam splitter was used as the solar concentrator to enable the broad utilization of the entire solar spectrum. The light component from the Sun was utilized by the tandem perovskite/silicon solar cell for energy conversion while the heat component was channeled to a TEG for power generation. A TEC was directly lapped to the backplate of the tandem solar cell for temperature regulation; and consequent efficiency enhancement. The accuracy of the numerical model was ensured by accounting for temperature dependency in the variable cross-sectional area TE legs and all convective and radiative losses were fully accounted for in the developed equations. The analysis was made comprehensive enough to show the effects of F I G U R E 1 0 The system efficiency variation of the current density and voltage for concentration factor of C, 1-3 F I G U R E 1 1 The system efficiency variation of the PV voltage for concentration factor of 1-3 F I G U R E 1 2 The variation of current density with cutoff wavelength varying the halide composition on the combined system's efficiency. Based on the results obtained, the following conclusions can be made: • A combined system efficiency of 42% was obtained from the proposed tandem perovskite/silicon solar cell integrated with TE devices when a unity solar concentration ratio was used at a split wavelength of 800 nm. This was 1.8 times higher than the efficiencies reported for standalone tandem perovskite/ silicon solar cells. • For the temperature range considered, the Bromine composition exerted a significant effect on the hybrid system's performance only for Bromine compositions varying from 0.2 to 0.5. • For a Bromine composition of 0.2, it was observed that varying the halide composition only yielded a higher system efficiency for a wavelength range of 300-400 nm. It was also forecasted that when the hybrid system is exposed to higher concentrated solar irradiances, the significance of varying the Bromine composition becomes very notable. • Maintaining the system's backplate temperature at 293K using a TEC was disadvantageous to the system's efficiency when a dichroic beam splitting concentrator was used. Furthermore, the power extracted from the solar cell to power the TEC was disadvantageous to the system's performance. Hence, it becomes unnecessary to use a TEC if the solar cell is operating within a functional working temperature range.