<figure><img src="image_1.png" /></figure>

HAL
open science

# Hybridising photovoltaics and thermoelectrics: A detailed-balance analysis

Alexis Vossier, Etienne Blandre, Rodolphe Vaillon

► To cite this version:

Alexis Vossier, Etienne Blandre, Rodolphe Vaillon. Hybridising photovoltaics and thermoelectrics: A detailed-balance analysis. Solar Energy Materials and Solar Cells, 2025, 289, pp.113636. 10.1016/j.solmat.2025.113636. hal-05065902

HAL Id: hal-05065902

https://hal.science/hal-05065902v1

Submitted on 13 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

# Hybridising photovoltaics and thermoelectrics: A detailed-balance analysis

Alexis Vossier<sup>a ID,*</sup>, Etienne Blandre<sup>b ID</sup>, Rodolphe Vaillon<sup>c ID</sup>

## ARTICLE INFO

### Keywords:

Hybridisation

Photovoltaics

Thermoelectricity

Detailed-balance model

## ABSTRACT

The combination of photovoltaic and thermoelectric converters could potentially lead to an improvement in the efficiency with which solar energy is converted into electricity, thanks to the improved exploitation of the residual heat generated in the PV cells. Using the detailed balance formalism, we develop a simple model enabling to derive the ultimate efficiency limits of PV-TE systems on the basis of a restricted number of operating parameters, including the bandgap of the semiconductor materials used, the cell temperature and the concentration ratio of the solar flux to which the PV cells are subjected. It is shown that in the radiative limit, the added value of PV-TE systems remains relatively modest, with an improvement in the electrical power generated of the order of 5% relative to conventional PV systems, and up to 15% for CPV systems operating under concentrated solar flux. Secondly, the limited capacity of real photovoltaic cells to approach their own theoretical limits is taken into account. We demonstrate a substantial improvement in the performance of PV-TE systems incorporating realistic solar cells operating at only a fraction of their theoretical limit, compared with reference PV systems. The value of solar concentration, which provides an additional leverage in the quest for high temperatures, while mitigating the adverse effect of temperature on PV cell efficiency, is also highlighted. Finally, the paper discusses the practical limitations of the model and outlines the operating and material parameters that need to be taken into account in order to rigorously determine the added value of these systems compared with conventional PV modules.

## Introduction

Photovoltaic (PV) systems are currently the most economically attractive and technologically mature solar energy conversion technology. Nevertheless, most of the solar radiation absorbed by photovoltaic cells is converted into heat (typically 80% in commercial PV modules), due to the inability of the semiconductor materials used in PV cells to efficiently absorb and convert solar photons covering a wide range of wavelengths [1]. Exploiting the residual heat generated by PV cells in the hope of boosting the performance of solar energy conversion systems has raised many hopes, but also prompted a number of questions. Indeed, exploiting the heat generated by PV cells generally involves using them at temperatures significantly higher than ambient, logically resulting in a drop in performance [2] not necessarily compensated for by the integration of the residual heat conversion means. However, it has been shown on thermodynamics grounds that hybrid systems involving simultaneous photovoltaic and thermal conversion of solar energy are likely to lead to higher conversion efficiencies than conventional PV systems [3].

Among the solutions being considered to make better use of the thermal energy resulting from the partial conversion of solar energy by PV cells, the combination of thermoelectric (TE) converters and PV cells has instigated a broad range of research work in recent years [4]. Research efforts aimed at optimising the TE converter for this purpose include adapting its geometric properties [5], managing the temperature difference across the TE [6], improving heat transfer between the PV cell and the thermoelectric conversion element [7], and optimising operating conditions to ensure maximum conversion efficiency, both on numerical [8] and experimental grounds [9]. The interface between the photovoltaic (PV) and the thermoelectric (TE) converters has also motivated several optimisation studies, with particular emphasis placed on optimising the thermal management of the system through the integration of phase change materials [10] or by selecting materials with suitable thermal conductivities [11]. However, the maximum benefit one can expect from the association of PV and TE technologies remains unclear. In fact, most of the optimisation work conducted to date focus on the combination of one specific PV technology, such as Si or polymer cells [12], dye-sensitised solar cells [13], CIGS solar

<footnote><sup>a</sup> CNRS-PROMES, UPR 8521, 7 Rue Du Four Solaire, Odeillo, 66120, France</footnote>

<footnote><sup>b</sup> UMR 8520-IEMN, Université de Lille, CNRS, Centrale Lille, Université Polytechnique Hauts-de-France, Junia-ISEN, Lille, 59000, France</footnote>

<footnote><sup>c</sup> LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, 31000, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alexis.vossier@promes.cnrs.fr (A. Vossier).</footnote>
### Nomenclature

<table><tr><td><i>E</i></td><td>Photon energy, (eV)</td></tr><tr><td><i>f</i>(<i>λ</i>)</td><td>Spectral distribution, (W/m<sup>2</sup>/nm)</td></tr><tr><td><i>J</i></td><td>Current density, (A/m<sup>2</sup>)</td></tr><tr><td><i>P</i></td><td>Power, (W)</td></tr><tr><td><i>S</i></td><td>Seebeck coefficient, (V/K)</td></tr><tr><td><i>T</i></td><td>Temperature, (K (or °C))</td></tr><tr><td><i>V</i></td><td>Voltage, (V)</td></tr><tr><td><i>X</i></td><td>Concentration ratio of sunlight, (suns)</td></tr><tr><td><i>Z</i></td><td>Thermoelectric figure of merit, (K<sup>-1</sup>)</td></tr></table>

### Greek

<table><tr><td>η</td><td>Efficiency</td></tr><tr><td>κ</td><td>Thermal conductivity, (W/(mK))</td></tr><tr><td>λ</td><td>Wavelength, (nm)</td></tr><tr><td>σ</td><td>Electrical conductivity, (S/m)</td></tr><tr><td>ξ</td><td>Shockley-Queisser index</td></tr></table>

### Physical Constants

<table><tr><td><i>c</i></td><td>Speed of light in vacuum, 3 × 10<sup>8</sup> (m/s)</td></tr><tr><td><i>h</i></td><td>Planck constant, 6.62 × 10<sup>-34</sup> (J s)</td></tr><tr><td><i>k</i></td><td>Boltzmann constant, 1.38 × 10<sup>-23</sup> (J/K)</td></tr><tr><td><i>q</i></td><td>Elementary charge, 1.602 × 10<sup>-19</sup> (A s)</td></tr></table>

### Subscripts

<table><tr><td><i>avg</i></td><td>Average</td></tr><tr><td><i>c</i></td><td>Cold</td></tr><tr><td><i>cell</i></td><td>Cell</td></tr><tr><td><i>exp</i></td><td>Experimental</td></tr><tr><td><i>g</i></td><td>Gap</td></tr><tr><td><i>h</i></td><td>Hot</td></tr><tr><td><i>hyb</i></td><td>Hybrid</td></tr><tr><td><i>in</i></td><td>Incident</td></tr><tr><td><i>max</i></td><td>Maximum</td></tr><tr><td><i>PV</i></td><td>Photovoltaic</td></tr><tr><td><i>rad</i></td><td>Radiative</td></tr><tr><td><i>sc</i></td><td>Short-circuit</td></tr><tr><td><i>SQ</i></td><td>Shockley-Queisser</td></tr><tr><td><i>TE</i></td><td>Thermoelectric</td></tr></table>

cells [14], or even GaAs [15], together with one TE technology, an approach that fails to provide a comprehensive understanding of the maximum performance that can be achieved by combining these two energy conversion systems.

The aim of this work is to shed new light on this issue by determining the maximum performance of combined PV-TE systems using the detailed balance formalism [16]. This approach enables the ultimate limits of photovoltaic cells to be derived on the basis of a very limited number of parameters. In the first section of this paper, the PV-TE system model developed in this study is explained in detail, as well as the main evaluation criterion considered here.

In the second section of this paper, the theoretical maximum limits of PV-TE hybrid systems incorporating PV cells operating at the radiative limit are derived. In particular, two distinct versions of PV-TE hybrid systems are considered, namely (1) PV-TE systems characterised by the use of conventional PV cells under direct illumination. The underlying motivation for the association with a thermoelectric converter is here to convert a fraction of the residual heat generated by the photovoltaic module (2) CPV-TE systems, which are based on the association of PV cells under concentrated solar flux and thermoelectric

converters. The motivation here is to take advantage of solar concentration, which offers an additional degree of freedom in obtaining high operating temperatures (> 100 °C) and, ultimately, better exploitation of the thermoelectric converter. In the third section of this paper, an attempt is made to evaluate the maximum performance of PV-TE systems incorporating realistic PV cells, operating at a fraction of their theoretical efficiency limit. On this basis, the performance of PV-TE systems integrating either state-of-the-art or commercially available TE and PV converters is then quantified. To conclude, a discussion is dedicated to the limitations of the model developed here, and to the scientific and technological challenges that will have to be addressed in order to develop PV-TE systems that significantly outperform conventional PV technologies.

## 1. Model

The ideal performance of hybrid systems based on the combination of PV cells and thermoelectric converters can simply be derived from the detailed balance formalism [16] that has been extensively developed over the past decades. In the case of a PV-TE hybrid system under solar concentration, the maximum electrical power that can be generated by the system is simply written as

$$P_{hyb} = P_{PV}(X, T_h, E_g) + P_{TE}(T_c, T_h, Z) \quad (1)$$

where $X, T_h, E_g$ refer respectively to the sunlight concentration ratio, the cell temperature (assumed to be equal to the thermoelectric converter's hot temperature), the bandgap of the semiconductor material used. Finally $Z$ and $T_c$ are the thermoelectric converter's figure of merit and the cold-side temperature of the thermoelectric converter.

In the radiative limit, the PV efficiency is simply calculated as:

$$\eta_{PV} = \frac{\max[V \times J(V)]}{P_{in}} \quad (2)$$

where $V$ is the voltage applied to the cell, $J$ is the current density and $P_{in}$ is the incident solar power density. The current-voltage curve of an ideal PV cell can simply be deduced from the equation:

$$J(V) = J_{sc} - J_{rad}(V) \quad (3)$$

where $J_{sc}$ is the short-circuit current density, while $J_{rad}$ is the diode current density, stemming from radiative recombination occurring in the cell. $J_{sc}$ can be written as [17]:

$$J_{sc} = q X \int_{0}^{\frac{hc}{E_g}} f(\lambda) \frac{\lambda}{hc} d\lambda \quad (4)$$

where $q$ is the elementary charge and $f(\lambda)$ refers to the spectral distribution of sunlight (taken equal to AM1.5D reference solar spectrum in this work).

The radiative recombination current density is calculated in the detailed balance limit as:

$$J_{rad}(V) = \frac{2\pi}{h^3 c^2} \int_{E_g}^{\infty} \frac{E^2 dE}{\exp\left(\frac{E-qV}{kT_{cell}}\right) - 1} \quad (5)$$

where $h$ and $k$ are respectively the Planck and Boltzmann constants, $c$ the speed of light and $E_g$ the bandgap of the PV cell. $E$ refers to the energy of photons, while $T_{cell}$ is the temperature of the PV cell, whose value is assumed to be equal to the hot side temperature of the thermo-element.

The thermoelectric efficiency is calculated as [18]:

$$\eta_{TE} = \frac{\Delta T}{T_h} \frac{\sqrt{1 + \bar{Z} \cdot T_{avg}} - 1}{\sqrt{1 + \bar{Z} \cdot T_{avg}} + \frac{T_c}{T_h}} \quad (6)$$

where $T_h$ and $T_c$ respectively refer to the hot-side and cold-side temperatures of the thermo-element, $\Delta T$ is the temperature difference between
<table><caption>**Table 1**
Typical efficiencies of TE technologies.</caption><thead><tr><th></th><th>Symbol</th><th>Z<sub>max</sub> (K<sup>-1</sup>)</th><th>T@Z<sub>max</sub> (K)</th><th>ZT@Z<sub>max</sub></th></tr></thead><tbody><tr><td colspan="5">Low-temperature TE (T≤ 100 °C)</td></tr><tr><td>Bi<sub>0.5</sub>Sb<sub>1.5</sub>Te<sub>3</sub> [19]</td><td>○</td><td>5.8×10<sup>-3</sup></td><td>320</td><td>1.86</td></tr><tr><td>Bi<sub>0.48</sub>Sb<sub>1.52</sub>Te<sub>3</sub> + 0.05 wt% PbTe [20]</td><td>×</td><td>3×10<sup>-3</sup></td><td>300</td><td>0.9</td></tr><tr><td>Ge doped Bi<sub>0.4</sub>Sb<sub>1.6</sub>Te<sub>3</sub> [21]</td><td>*</td><td>4.6×10<sup>-3</sup></td><td>300</td><td>1.36</td></tr><tr><td>Bi<sub>2</sub>Te<sub>2.7</sub>Se<sub>0.3</sub> [22]</td><td>□</td><td>2.9×10<sup>-3</sup></td><td>330</td><td>0.96</td></tr><tr><td colspan="5">High-temperature TE (T≥ 100 °C)</td></tr><tr><td>(Ge<sub>0.87</sub>Pb<sub>0.13</sub>Te)<sub>0.95</sub>(Bi<sub>2</sub>Te<sub>3</sub>)<sub>0.05</sub> [23]</td><td>◇</td><td>3.1×10<sup>-3</sup></td><td>570</td><td>1.77</td></tr><tr><td>0.1Nb/Mg<sub>3</sub>Sb<sub>1.5</sub>Bi<sub>0.49</sub>Te<sub>0.01</sub> [24]</td><td>△</td><td>3.1×10<sup>-3</sup></td><td>430</td><td>1.33</td></tr><tr><td>Ge<sub>0.61</sub>Ag<sub>0.11</sub>Sb<sub>0.13</sub>Pb<sub>0.1</sub>Bi<sub>0.01</sub>Te [25]</td><td>☆</td><td>3.8×10<sup>-3</sup></td><td>680</td><td>2.58</td></tr><tr><td>Br doped SnSe single crystal [26]</td><td>*</td><td>3.6×10<sup>-3</sup></td><td>770</td><td>2.77</td></tr></tbody></table>

the hot side and the cold side, $\bar{Z}$ is the thermoelectric figure of merit averaged over the temperature range of interest, and $T_{avg}$ the average temperature, taken here as $(T_c + T_h)/2$. The thermoelectric figure of merit Z can be written:

$$Z = \frac{S^2 \sigma}{\kappa} \qquad (7)$$

where S is the Seebeck coefficient, σ is the electrical conductivity and κ is the thermal conductivity (all these physical quantities being temperature-dependent). In this work, Z is considered as an input variable, whose value dictates the combined PV-TE efficiency.

Table 1 summarises the maximum values of Z associated with different thermoelectric materials reported in recent literature, together with the associated temperature (the corresponding value of ZT is also given for information). Thermoelectric materials with Z values greater than $3×10^{-3}$ K$^{-1}$ have been developed at the laboratory scale in recent years. Z values in excess of $5×10^{-3}$ K$^{-1}$ remain rare although a record value of $5.8×10^{-3}$ K$^{-1}$ was reported for Bi$_{0.5}$Sb$_{1.5}$Te$_{3}$ at a temperature of 320 K [19].

The thermoelectric performance associated with a particular material is estimated here on the basis of the average value of Z integrated over the temperature range to which the thermoelectric converter is exposed. This quantity, denoted $\bar{Z}$, can be calculated as [27]:

$$\bar{Z} = \frac{1}{T_h - T_c} \int_{T_c}^{T_h} Z(T) dT \qquad (8)$$

The temperature dependence of the parameters Z and $\bar{Z}$ for the various thermoelectric materials considered here is shown in Figure S1 (Supplementary Material).

Finally, assuming that thermal losses are neglected (i.e. the solar power not converted into electricity is fully usable as thermal power by the thermoelectric converter), the maximum electrical power that can be extracted from the hybrid system is derived from:

$$P_{TE} = \eta_{TE} \times (P_{in} - P_{PV}) \qquad (9)$$

## 1.1. Evaluation method

The deployment of PV-TE hybrid systems is justified from a technical point of view if the conversion efficiency of the system exceeds that of a PV system optimised to demonstrate maximum performance under comparable operating conditions. For given conditions of solar concentration, the optimisation of the two families of systems consists of determining the bandgap $E_g$ and the cell temperature $T_c$ (assumed here to be equal to the hot temperature of the thermoelectric converter) that will maximise the power generated. The PV power delivered by the hybrid system is initially evaluated as a function of the bandgap of the cell used and its temperature, as outlined in Eqs. (2) to (5). The available thermal power that can be harnessed by the thermoelectric converter is simply expressed as the difference between the incident solar power and the PV power (Eq. (9)). The electrical power generated by the thermoelectric converter is then calculated as the product of the latter and the thermoelectric conversion efficiency, described by

Eqs. (6) to (8). A schematic representation of the logical structure of the model developed in the context of this work is shown in Fig. 1 (this model was developed using the MATLAB software).

A PV-TE hybrid system will be considered relevant if the Power Improvement Factor (denoted PIF here) satisfies the following condition:

$$PIF = \frac{P_{hyb}(X, E_{g,PV-TE}, T_{PV-TE}, \bar{Z}) - P_{PV}(X, E_{g,PV}, T_{PV})}{P_{PV}(X, E_{g,PV}, T_{PV})} > 0 \qquad (10)$$

where X is the typical concentration ratio to which the PV and the PV-TE systems are exposed, $E_{g,PV-TE}$ and $E_{g,PV}$ are the optimal bandgap leading to the highest power output of the PV and PV-TE systems, while $T_{PV-TE}$ and $T_{PV}$ refer to the optimal temperature at which each respective systems delivers the highest power output. In this work, it is assumed that the hot and cold temperatures of the thermoelectric converter and cell are fixed, in the spirit of a detailed balance analysis. The limitations of this assumption will be discussed in the final section of this work.

# 2. Results

## 2.1. Ideal PV cells

### 2.1.1. Conventional PV modules

First, we quantify how integrating a thermoelectric converter into an ideal PV module operating at the radiative limit increases the electrical power generated by the system. We first consider the case of a conventional PV system, operating at an illumination of 1000 W/m$^2$, and characterised by an operating temperature comprised between 25 and 75 °C (a sketch of which is represented in Fig. 2(a)). This temperature level is typically characteristic of the temperatures actually reached by PV modules in real operating conditions [28]. The integration of a thermoelectric converter is relevant on a technical level if the extra power provided by the thermo-element allows counterbalancing the reduction in photovoltaic power associated with the high temperature of the PV module in real conditions, compared with the reference temperature of 25 °C at which PV modules are typically designed. Fig. 3 illustrates the increase in power of a PV-TE hybrid module relative to the electrical power generated by a PV module operating over the temperature range between 25 and 75 °C. The cold source is taken equal to the ambient temperature, assumed equal to 25 °C.

In the radiative limit, the improvement in electrical power delivered by a PV-TE system relative to that delivered by a PV module operating at above ambient temperature is modest (the improvement in electrical power delivered is no more than 5% for module temperatures exceeding 70 °C and for Z values close to $6×10^{-3}$ K$^{-1}$). The efficiency improvement associated with the selected state-of-the-art thermoelectric materials is modest and lower than 5%. In real operating conditions, PV-TE hybrid modules will necessarily be exposed to significant variations in temperature, which cannot be captured by the evaluations provided here. A rigorous estimate of the electrical power supplied under real conditions by a PV-TE module should thus necessarily take into account the characteristic temperature variations over a given period.
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic representation of the PV-TE hybrid system model: a hybrid PV-TE system outperforms its PV counterpart if the sum of the PV and TE power optimised with respect to the cell temperature and its bandgap, exceeds the electrical power generated by a conventional PV cell (optimised with respect to $E_g$, and considering a fixed cell temperature).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Schematic representation of the two categories of PV-TE hybrid systems studied: (a) PV-TE hybrid system based on the integration of a conventional PV module. The thermo-element, identical in size to the PV module, is optimised to exploit the waste heat generated by the PV module. (b) CPV-TE hybrid system, based on the combination of CPV cells/modules and a thermo-element, which are not necessarily of equal size. Here, solar concentration offers an additional degree of freedom in the quest of higher hot temperature and, consequently, enhanced thermoelectric performance.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Improvement in the power output of a hybrid PV-TE system relative to the power output of a conventional PV module, both operating at the radiative limit. The improvement provided by the integration of a thermo-element is expressed as a percentage of the electrical power delivered by a conventional PV module operating at a characteristic temperature of between 25 and 75 °C. $\bar{Z}$ values associated with state-of-the-art thermoelectric materials are also represented as symbols.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. Improvement in the power output of a hybrid PV-TE system relative to the power output of a conventional CPV module operating at (a) 10 suns (b) 100 suns and (c) 1000 suns. The improvement provided by the integration of a thermoelectric module is expressed as a percentage of the electrical power delivered by a conventional CPV module operating at a characteristic temperature of 70 °C. The Z̄ values associated with different state-of-the-art thermoelectric materials are reported as colour symbols.</figcaption></figure>

### 2.1.2. Concentrator PV modules

Concentrating photovoltaic modules could possibly benefit from the integration of thermoelectric modules with the aim of converting all or part of the residual heat dissipated by the latter. The operating conditions characterising PV systems under concentrated solar flux differ significantly from the operating conditions of conventional PV modules and could further justify the use of PV-TE hybrid modules: (1) the hot temperature is likely to be significantly higher, given the increased solar power density absorbed by the cell; (2) the ability of photovoltaic cells to withstand high temperatures improves significantly under concentrated solar flux [29]. A major difference in concentrated PV systems is that the characteristic temperature of the PV cells, which is the heat source of the thermoelectric module, can be adjusted over a wide temperature range. Characterisation of III-V cells has been carried out up to temperatures of 400 °C [30] and 600 °C [31], demonstrating promising performance.

The added value of such systems should be judged on their ability to generate an electrical power exceeding the typical power delivered by a conventional CPV module, whose characteristic temperature is kept as low as possible (operating temperatures of CPV modules typically comprised between 40 and 70 °C have been demonstrated in real-life conditions [32]). Fig. 4 shows the improvement in electrical power of a CPV-TE hybrid system, expressed as a percentage increase in electrical power compared with a conventional CPV module whose temperature has been set at 70 °C. Three concentration factors characteristic of the main families of optical systems for concentrating solar flux are considered, namely (1) low solar concentration (X=10); (2) medium solar concentration (X=100); (3) high solar concentration (X=1000).

Several conclusions can be drawn from the analysis of Fig. 4: we observe a large region of T-Z combinations for which the performance of PV-TE systems is lower than that of the reference CPV system. The solar concentration is used to generate a temperature increase that can be partially exploited by the thermoelectric converter. If the performance of the TE converter (characterised by the value of Z̄) is too low to counterbalance the negative effect of temperature on the PV cells, then the overall performance of the system decreases, resulting in PV-TE electrical power generation lower than that of the reference CPV system. There is a characteristic value of Z̄ for which the performance of the hybrid system is independent of the temperature of the PV cell. This value, which depends on the solar concentration factor to which the hybrid system is exposed, is of the order of ~4 × 10⁻³ K⁻¹ for a system characterised by a concentration factor of 10 suns, and drops to ~2.5×10⁻³ K⁻¹ for a system operating at 1000 suns. The concentration dependence of this limiting value of Z̄ can be explained by the reduced

temperature sensitivity of PV cells exposed to increasing solar irradiances. Below this threshold Z̄ value, an increase in temperature leads to a decrease in the power generated by the hybrid system, compared with the reference power generated by the CPV system. On the other hand, hybrid systems incorporating thermoelements characterised by Z̄ values above the limit value show an increase in the power generated by PV-TE hybrid systems for increasing temperatures.

Overall, the power improvement associated with PV-TE hybridisation is modest (it barely exceeds 10% over the range of parameters studied) and essentially corresponds to a combination of high Z̄ values and relatively high cell temperatures. The analysis of the performance of hybrid systems associated with the different thermoelectric materials selected, illustrated by coloured symbols, is also instructive. Improvements in the performance of PV-TE hybrid systems incorporating state-of-the-art thermoelectric materials remain limited, with power gains compared with conventional CPV systems of no more than 5%. The most promising thermoelectric materials are not necessarily those showing the highest $Z_{max}$ or $ZT_{max}$ values, but rather those with the smallest possible variations in the average Z̄ value over the temperature range likely to be exploited by a given PV-TE converter. In this respect, the example of Br doped SnSe single crystal is particularly interesting since this material, despite showing very high $Z_{max}$ and $ZT_{max}$ values, does not lead to any improvement in the electrical power generated, compared with a conventional CPV system. On the contrary, materials with maximum values of $Z_{max}$ and $ZT_{max}$ that are significantly lower than the latter, such as $(Ge_{0.87}Pb_{0.13}Te)_0.95(Bi_2Te_3)_0.05$ or $0.1Nb/Mg_3Sb_{1.5}Bi_{0.49}Te_{0.01}$, demonstrate significantly higher performance, thanks to the smaller variations in Z̄ throughout the temperature range of interest that characterises these materials.

### 2.2. Realistic PV cells

The estimates developed in the previous section are based on the strong assumption that the PV cells used operate at the radiative limit, neglecting the existence of various limiting mechanisms that affect their performance in reality. These limiting mechanisms lead to a degradation in the electrical performance of these cells and, ultimately, to the generation of additional thermal losses. In such circumstances, the integration of a thermoelectric converter could enable this additional heat input to be converted. The limiting mechanisms likely to affect the performance of PV cells are numerous, their nature and intensity varying widely with the type of PV cell used (direct or indirect gap semi-conductor material, organic or inorganic solar cell, etc.) and with
<table><caption>**Table 2**
Typical efficiencies of PV technologies.</caption><thead><tr><th></th><th>η<sub>exp</sub> (%)</th><th>η<sub>SQ</sub> (%)</th><th>ξ</th><th>E<sub>g</sub> (eV)</th><th>T (K)</th><th>X (suns)</th></tr></thead><tbody><tr><td colspan="7">Conventional PV technology</td></tr><tr><td>Si (record) [34]</td><td>26.8</td><td>33.6</td><td>0.8</td><td>E<sub>g</sub> = 1.12 eV</td><td>T = 298 K</td><td>1</td></tr><tr><td>GaAs (record) [34]</td><td>29.1</td><td>32.8</td><td>0.89</td><td>E<sub>g</sub> = 1.42 eV</td><td>T = 298 K</td><td>1</td></tr><tr><td>Si (commercial)</td><td>21.5</td><td>33.6</td><td>0.64</td><td>E<sub>g</sub> = 1.12 eV</td><td>T = 298 K</td><td>1</td></tr><tr><td>CdTe (commercial)</td><td>18.6</td><td>31.6</td><td>0.59</td><td>E<sub>g</sub> = 1.12 eV</td><td>T = 298 K</td><td>1</td></tr><tr><td>CIGS (commercial)</td><td>15.1</td><td>33.5</td><td>0.45</td><td>E<sub>g</sub> = 1.12 eV [35]</td><td>T = 298 K</td><td>1</td></tr><tr><td colspan="7">Concentrator Photovoltaics</td></tr><tr><td>Si [36]</td><td>27.6</td><td>38.6</td><td>0.716</td><td>E<sub>g</sub> = 1.12 eV</td><td>T = 298 K</td><td>92</td></tr><tr><td>GaAs [34]</td><td>30.8</td><td>36</td><td>0.855</td><td>E<sub>g</sub> = 1.42 eV</td><td>T = 298 K</td><td>61</td></tr><tr><td>CIGS [37]</td><td>23.3</td><td>36.4</td><td>0.64</td><td>E<sub>g</sub> = 1.15 eV</td><td>T = 298 K</td><td>14</td></tr><tr><td colspan="7">High-temperature PV cells</td></tr><tr><td>GaAs [31]</td><td>7.7</td><td>36</td><td>0.31</td><td>E<sub>g</sub> = 1.34 eV</td><td>@T = 497 K</td><td>1</td></tr><tr><td>GalnP/GaAs [38]</td><td>16.3</td><td>34</td><td>0.48</td><td>E<sub>g</sub> = 1.83/1.25 eV</td><td>@T = 673 K</td><td>350</td></tr><tr><td>InGap/GaAs/InGaAs [39]</td><td>33.7</td><td>54</td><td>0.62</td><td>E<sub>g</sub> = 1.87/1.39/0.97 eV</td><td>@T = 399 K</td><td>460</td></tr></tbody></table>

the type of PV system considered (conventional photovoltaic panel or CPV module). As a first approximation, it is possible to take into account the capacity of “real” PV cell technologies to approach their own theoretical limits to a greater or lesser extent through the Shockley-Queisser index, proposed in [33], which globally translates the capacity of a given photovoltaic technology to approach its own theoretical limit. The SQ index is simply written as:

$$ \xi_{SQ} = \frac{\eta_{exp}}{\eta_{SQ}} \qquad (11) $$

where $\eta_{exp}$ refers to the experimental efficiency of PV cells and $\eta_{SQ}$ is the upper theoretical efficiency limit derived using the Shockley-Queisser formalism. $\xi_{SQ}$ values approaching 1 indicate cell technologies performing close to their theoretical limit, while low $\xi_{SQ}$ are typically representative of PV cells strongly penalized by one or several limiting mechanism. Table 2 summarises the experimental efficiencies and $\xi_{PV}$ values associated with several PV cell technologies at the state-of-the-art or commercial deployment stage. The table also includes characteristic values of these parameters for concentrating PV cells, but also for “high-temperature” PV cells, developed with the aim of converting solar radiation at high temperatures as efficiently as possible.

### 2.2.1. Conventional PV modules

Fig. 5 shows the maximum conversion efficiency (left) and the power improvement (right) of a PV-TE hybrid system based on conventional PV technology (i.e. illumination equivalent to 1 sun) and considering PV modules operating at a fraction of their radiative limit, and for thermoelectric modules characterised by $\bar{Z}$ values between 0.3 and $5 \times 10^{-3}$ K$^{-1}$. The temperature of the PV module is set equal to 70 °C (the curves corresponding to a module temperature of 45 °C are shown in figure S2 in the supplementary material section).

Taking into account the limited capacity of real PV cells to approach their theoretical limits further motivates the development of hybrid systems. The power gain associated with thermoelectric conversion typically represents 5 to 8% for state-of-the-art cells demonstrating a very good ability to approach their theoretical limit, and for values of $\bar{Z}$ greater than $2.5 \times 10^{-3}$ K$^{-1}$. This power gain is likely to be as high as 10%–20% for commercial Si or CIGS modules, and could be even higher for low-cost PV modules with low conversion efficiencies.

### 2.2.2. Concentrator PV modules

Fig. 6 shows the improvement in electrical power delivered by a PV-TE hybrid system compared to a CPV system characterised by a concentration factor of 1000 suns, for values of $\xi_{SQ}$ between 0.1 and 1 and for values of $Z$ between 0.3 and $5 \times 10^{-3}$ K$^{-1}$ (the curves corresponding to illuminations of 10 and 100 suns are respectively reported in figures S3 and S4 in the supplementary material section).

For each calculation, the optimum bandgap and cell temperature to maximise the conversion efficiency of the hybrid system are identified. Several lessons can be drawn from the analysis of this figure:

* While the PV-TE combination offers an increase in the electrical power generated of around 15% in the case of PV cells operating at the radiative limit, the increase in electrical power of hybrid systems incorporating PV cells operating at a fraction of their radiative limit is considerably greater: it can reach 30 to 50% for state-of-the-art HT solar cells operating at 50 or 60% of their radiative limit, if combined with sufficiently high $\bar{Z}$ thermoelectric materials.

* The incorporation of PV cells capable of approaching their theoretical limit leads to a lower optimum hot source temperature over a wider range of $\bar{Z}$ values. Although the power gain associated with PV-TE hybridisation is greater for PV cells with medium or low $\xi_{PV}$ values, the optimum hot source temperature reaches 400 °C over a wider range of $\bar{Z}$ values. Therefore, the use of medium or low efficiency PV cells implies a preference for high PV cell operating temperatures regardless of the $\bar{Z}$ of the thermoelectric material, whereas the use of high efficiency PV cells will offer more flexibility in terms of operating temperature, particularly in the case of thermoelectric materials offering modest $\bar{Z}$ values. These trends can be explained by the combination of the negative effect of temperature on PV cells (resulting in a significant reduction in the PV power generated) and the increase in thermoelectric power associated with a greater temperature difference between the hot and cold sources. The optimum temperature logically results from the greater or lesser ability of the two converters to exploit the available solar or thermal resource, and also appears to vary with the typical illumination level to which the PV cell is submitted (as can be seen if Fig S3 and S4, demonstrating a wider range of low optimum temperatures for concentration ratios of 10 and 100 suns).

## 2.3. Discussion

The evaluation method proposed here enables the contribution of PV-TE hybridisation to be quantified in the case of systems integrating PV cells operating at a fraction of their radiative limit and thermoelectric converters with $\bar{Z}$ values representative of state-of-the-art thermoelectric converters. Nevertheless, this approach suffers from several limitations that must be mentioned.

First, the evaluation method discussed here is based on the comparison of two PV and PV-TE systems operating under comparable conditions and characterised by identical capacities to approach their own theoretical limits. In reality, optimal operation of a PV-TE hybrid system may in some cases expose the system to significantly
<figure><img src="image_8.png" /><figcaption>Fig. 5. Conversion efficiency (left), and power improvement (right) of a hybrid PV-TE system operating at 1 sun, encompassing PV cells operating at a fraction $\xi_{PV}$ of their theoretical limit. The improvement provided by the integration of a thermoelectric module is expressed as a percentage of the electrical power delivered by a conventional PV module operating at a characteristic temperature of 70 °C, and with the same ability to approach its own theoretical limit.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Conversion efficiency (left), Power improvement (centre) and optimal PV cell temperature (right) of a hybrid PV-TE system operating at 1000 suns, encompassing PV cells operating at a fraction $\xi_{PV}$ of their theoretical limit. The improvement provided by the integration of a thermoelectric module is expressed as a percentage of the electrical power delivered by a conventional CPV module operating at a characteristic temperature of 70 °C, and with the same ability to approach its own theoretical limit. The PV cell temperature explored covers the 70–400 °C range.</figcaption></figure>

higher temperatures than those resulting in best performance of the standalone PV system. Under such conditions, the assumption that the two systems are comparably close to their own theoretical limits is questionable. Figure S5 shows the temperature-dependence of the $\xi_{PV}$ factor for 3 distinct III-V solar cell technologies. Whatever the technology considered, a regular decrease in the $\xi_{PV}$ factor is observed for increasing temperatures, with coefficients of variation of the $\xi_{PV}$ factor as a function of temperature ($\frac{d\xi_{PV}}{dT}$) typically varying between ~$-6 \times 10^{-4}$/K (tandem and triple junction cells) and ~$-14 \times 10^{-4}$/K (single junction GaAs cells). Two hypotheses can explain these trends: firstly, the cell architecture developed for high-temperature conversion of solar energy must necessarily be adapted to take into account the temperature dependence of important material parameters likely to affect the operation of solar cells (electrical resistance, carrier diffusion coefficient, lifetimes, etc.). The use of conventional cell architectures optimised for ambient temperature operation will inevitably lead to

sub-optimal conversion of solar energy at high temperatures. The efficient conversion of solar energy under such extreme conditions is likely to require the adaptation of the geometry, thickness and doping of the different material layers and the contact grid, as well as the preference of alternative materials with high thermal stability at the targeted temperature levels [40]. Nevertheless, it is conceivable that even a cell structure adapted for optimal conversion at high temperature will still be limited in its ability to approach its own theoretical limits, given the deleterious effect of temperature on certain important limiting mechanisms such as carrier recombination [41] or resistive losses [42], that naturally affect its efficiency.

A second point of concern is related to the estimation of the electrical power delivered by a thermoelectric converter subjected to a given temperature difference. Considering the integrated mean value $\bar{Z}$ does not enable the temperature dependence of the material parameters to be rigorously taken into account [27]. A more rigorous formulation that better describes the temperature dependence of the S, $\sigma$ and $\kappa$
parameters has been suggested by the same authors, and a precise knowledge of the latter is therefore a necessary prerequisite for a more accurate assessment of the added-value of PV-TE hybridisation.

Third, the evaluation method proposed here is based on the assessment of the electrical power gain associated with PV-TE hybridisation for a given solar power density and a given hot source temperature. In real operating conditions, these two parameters are likely to vary significantly depending on the season, the location of the system and the weather conditions. Such an assessment therefore provides only a partial view of the contribution of hybridisation, which needs to be studied into more detail, taking into account the temporal variability of the available solar resource and the characteristic temperature of the system.

Finally, the assessment neglects the heat losses of the system, assuming that all the solar energy not converted by the PV cell is available in the form of heat and can be used by the thermoelectric converter. In practice, it is necessary to take into account the thermal balance of the system in order to predict the temperature of the various sub-systems more accurately [43], and to optimise the system accordingly [44]. The addition of a thermal balance to evaluate the PV operating temperature requires the introduction of several additional parameters, making the evaluation much more complex and moving it further away from the principle of searching for optimum conditions by means of a “simple” detailed balance approach.

## Conclusion

The combination of a thermoelectric converter and a PV system is a potentially promising way of making better use of the waste heat generated by PV cells and increasing the efficiency of solar-to-electricity conversion. The detailed balance model is a valuable tool for determining the optimum parameters leading to maximum conversion efficiencies, based on a small number of input parameters. In this work, a simple PV-TE hybrid converter model based on the detailed balance formalism has been developed, taking into account the ability of PV cells to approach their theoretical limit and the quality factor $\bar{Z}$ of the thermoelectric converter. It has been shown that the contribution of PV-TE hybridisation remains modest for PV cells operating at the radiative limit, but increases significantly for realistic PV cells operating at only a fraction of their own theoretical limit. Solar concentration offers an additional degree of freedom in the quest for high efficiency, by giving access to wider temperature ranges, potentially favourable for thermoelectric conversion, and by limiting the negative effect of temperature on PV cell efficiency. It is demonstrated that such an approach could lead to an increase in the power generated by the system of the order of 15% for cells operating at the radiative limit, and significantly more for realistic cells at today's state of the art. Despite its simplicity, the approach proposed here nevertheless has a number of limitations that should be addressed in future work: the performance metric used, which is based on the increase in electrical power generated using an optimised PV system as a reference, does not take into account the greater or lesser ability of PV cells to approach their own theoretical limits as a function of temperature. This potentially introduces a bias in favour of systems working at high temperatures. The failure to take into account the temperature dependence of thermoelectric materials, as well as the fact that thermal losses are neglected, is likely to lead to an overestimation of the performance of hybrid systems: a rigorous estimate will therefore have to take these two aspects into account. It should be underlined, however, that the approach developed here, by neglecting the material parameters, offers a global view of the parameter space within which the performance of PV-TE systems will eventually be able to outperform that of conventional PV systems. Finally, an inherent limitation of this generic type of model is the difficulty of quantifying uncertainties and providing error bars. These would require the consideration of material or geometric parameters, which the model deliberately omit (precisely in order to preserve the

general character, independent of the choice of materials or configuration). Consequently, a more precise evaluation of the performance of a given configuration and choice of materials, which is outside the scope of this work, will require the model to be developed by integrating an increasing number of material and geometric parameters that are specific to a given PV-TE system.

## CRediT authorship contribution statement

**Alexis Vossier:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Etienne Blandre:** Validation. **Rodolphe Vaillon:** Writing – review & editing, Validation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

We would like to thank the members of the PV-TE group, and in particular Evelyne Martin and Inès Revol, for the constructive discussions on these issues.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.solmat.2025.113636.

## Data availability

Data will be made available on request.

## References

[1] L.C. Hirst, N.J. Ekins-Daukes, Fundamental losses in solar cells, Prog. Photovolt., Res. Appl. 19 (3) (2011) 286–293.

[2] O. Dupré, R. Vaillon, M.A. Green, Physics of the temperature coefficients of solar cells, Sol. Energy Mater. Sol. Cells 140 (2015) 92–100.

[3] A. Luque, A. Marti, Limiting efficiency of coupled thermal and photovoltaic converters, Sol. Energy Mater. Sol. Cells 58 (2) (1999) 147–165.

[4] J. Tang, H. Ni, R.-L. Peng, N. Wang, L. Zuo, A review on energy conversion using hybrid photovoltaic and thermoelectric systems, J. Power Sources 562 (2023) 232785.

[5] H. Hashim, J.J. Bomphey, G. Min, Model for geometry optimisation of thermoelectric devices in a hybrid PV/TE system, Renew. Energy 87 (2016) 458–463.

[6] S. Shittu, G. Li, X. Zhao, X. Ma, Series of detail comparison and optimization of thermoelectric element geometry considering the PV effect, Renew. Energy 130 (2019) 930–942.

[7] E. Yin, Q. Li, Y. Xuan, Thermal resistance analysis and optimization of photovoltaic-thermoelectric hybrid system, Energy Convers. Manage. 143 (2017) 188–202.

[8] O. Rejeb, S. Shittu, C. Ghenai, G. Li, X. Zhao, M. Bettayeb, Optimization and performance analysis of a solar concentrated photovoltaic-thermoelectric (CPV-TE) hybrid system, Renew. Energy 152 (2020) 1342–1353.

[9] S. Mahmoudinezhad, A. Rezania, D.T. Cotfas, P.A. Cotfas, L.A. Rosendahl, Experimental and numerical investigation of hybrid concentrated photovoltaic-thermoelectric module under low solar concentration, Energy 159 (2018) 1123–1131.

[10] Y. He, Y. Tao, C. Zhao, X. Yu, Structure parameter analysis and optimization of photovoltaic-phase change material-thermoelectric coupling system under space conditions, Renew. Energy 200 (2022) 320–333.

[11] J. Zhang, H. Zhai, Z. Wu, Y. Wang, H. Xie, M. Zhang, Enhanced performance of photovoltaic-thermoelectric coupling devices with thermal interface materials, Energy Rep. 6 (2020) 116–122.

[12] D. Kraemer, L. Hu, A. Muto, X. Chen, G. Chen, M. Chiesa, Photovoltaic-thermoelectric hybrid systems: A general optimization methodology, Appl. Phys. Lett. 92 (24) (2008).
[13] N. Wang, L. Han, H. He, N.-H. Park, K. Koumoto, A novel high-performance photovoltaic-thermoelectric hybrid device, Energy & Environ. Sci. 4 (9) (2011) 3676–3679.

[14] R. Bjørk, K.K. Nielsen, The performance of a combined solar photovoltaic (PV) and thermoelectric generator (TEG) system, Sol. Energy 120 (2015) 187–194.

[15] X. Ju, Z. Wang, G. Flamant, P. Li, W. Zhao, Numerical analysis and optimization of a spectrum splitting concentration photovoltaic-thermoelectric hybrid system, Sol. Energy 86 (6) (2012) 1941–1954.

[16] W. Shockley, H.J. Queisser, Detailed balance limit of efficiency of pn junction solar cells, J. Appl. Phys. 32 (3) (1961) 510–519.

[17] T. Tiedje, E. Yablonovitch, G.D. Cody, B.G. Brooks, Limiting efficiency of silicon solar cells, IEEE Trans. Electron Devices 31 (5) (1984) 711–716.

[18] S. Angrist, Direct energy conversion, Allyn and Bacon Chemistry Series, Allyn and Bacon, 1965.

[19] S.I. Kim, K.H. Lee, H.A. Mun, H.S. Kim, S.W. Hwang, J.W. Roh, D.J. Yang, W.H. Shin, X.S. Li, Y.H. Lee, et al., Dense dislocation arrays embedded in grain boundaries for high-performance bulk thermoelectrics, Science 348 (6230) (2015) 109–114.

[20] S. Liang, J. Xu, H. Wang, X. Tan, G.-Q. Liu, H. Shao, B. Yu, S. Yue, J. Jiang, Investigation on structure and thermoelectric properties in p-type bi 0.48 sb 1.52 te 3 via PbTe incorporating, J. Mater. Sci., Mater. Electron. 29 (2018) 7701–7706.

[21] Y. Wang, L. Huang, C. Zhu, J. Zhang, D. Li, H. Xin, M. Danish, X. Qin, Simultaneously enhanced power factor and phonon scattering in Bi0.4Sb1.6Te3 alloy doped with germanium, Scr. Mater. 154 (2018) 118–122.

[22] B. Zhu, Z.-Y. Huang, X.-Y. Wang, Y. Yu, N. Gao, F.-Q. Zu, Enhanced thermoelectric properties of n-type direction solidified Bi2Te2. 7Se0.3 alloys by manipulating its liquid state, Scr. Mater. 146 (2018) 192–195.

[23] L. Wang, J. Li, C. Zhang, T. Ding, Y. Xie, Y. Li, F. Liu, W. Ao, C. Zhang, Discovery of low-temperature GeTe-based thermoelectric alloys with high performance competing with Bi 2 Te 3, J. Mater. Chem. A 8 (4) (2020) 1660–1667.

[24] J.-W. Li, Z. Han, J. Yu, H.-L. Zhuang, H. Hu, B. Su, H. Li, Y. Jiang, L. Chen, W. Liu, et al., Wide-temperature-range thermoelectric n-type Mg3 (Sb, Bi) 2 with high average and peak zT values, Nat. Commun. 14 (1) (2023) 7428.

[25] B. Jiang, W. Wang, S. Liu, Y. Wang, C. Wang, Y. Chen, L. Xie, M. Huang, J. He, High figure-of-merit and power generation in high-entropy GeTe-based thermoelectrics, Science 377 (6602) (2022) 208–213.

[26] C. Chang, M. Wu, D. He, Y. Pei, C.-F. Wu, X. Wu, H. Yu, F. Zhu, K. Wang, Y. Chen, et al., 3D charge and 2D phonon transports leading to high out-of-plane zT in n-type SnSe crystals, Science 360 (6390) (2018) 778–783.

[27] H.S. Kim, W. Liu, G. Chen, C.-W. Chu, Z. Ren, Relationship between thermoelectric figure of merit and energy conversion efficiency, Proc. Natl. Acad. Sci. 112 (27) (2015) 8205–8210.

[28] O. Dupré, R. Vaillon, M.A. Green, Thermal behavior of photovoltaic devices, Phys. Eng. 10 (2017) 3–978.

[29] A. Braun, B. Hirsch, A. Vossier, E.A. Katz, J.M. Gordon, Temperature dynamics of multijunction concentrator solar cells up to ultra-high irradiance, Prog. Photovolt., Res. Appl. 21 (2) (2013) 202–208.

[30] E.E. Perl, J. Simon, J.F. Geisz, M.L. Lee, D.J. Friedman, M.A. Steiner, Measurements and modeling of III-V solar cells at high temperatures up to 400°C, IEEE J. Photovoltaics 6 (5) (2016) 1345–1352.

[31] A. Maros, S. Gangam, Y. Fang, J. Smith, D. Vasileska, S. Goodnick, M.I. Bertoni, C.B. Honsberg, High temperature characterization of GaAs single junction solar cells, in: 2015 IEEE 42nd Photovoltaic Specialist Conference, PVSC, IEEE, 2015, pp. 1–5.

[32] Y. Ota, H. Nagai, K. Araki, K. Nishioka, Temperature distribution in 820X CPV module during outdoor operation, in: AIP Conference Proceedings, 1477, (1) American Institute of Physics, 2012, pp. 364–367.

[33] A. Vossier, F. Gualdi, A. Dollet, R. Ares, V. Aimez, Approaching the Shockley-Queisser limit: General assessment of the main limiting mechanisms in photovoltaic cells, J. Appl. Phys. 117 (1) (2015).

[34] M.A. Green, E.D. Dunlop, M. Yoshita, N. Kopidakis, K. Bothe, G. Siefer, X. Hao, Solar cell efficiency tables (version 63), Prog. Photovolt., Res. Appl. 32 (1) (2024) 3–13.

[35] J. Keller, K. Kiselman, O. Donzel-Gargand, N.M. Martin, M. Babucci, O. Lundberg, E. Wallin, L. Stolt, M. Edoff, High-concentration silver alloying and steep back-contact gallium grading enabling copper indium gallium selenide solar cell with 23.6% efficiency, Nat. Energy (2024) 1–12.

[36] A. Slade, V. Garboushian, 27.6% efficient silicon concentrator solar cells for mass production, in: Technical Digest, 15th International Photovoltaic Science and Engineering Conference, Beijing, 2005.

[37] J. Ward, K. Ramanathan, F. Hasoon, T. Coutts, J. Keane, M. Contreras, T. Moriarty, R. Noufi, A 21.5% efficient Cu (In, Ga) Se2 thin-film concentrator solar cell, Prog. Photovolt., Res. Appl. 10 (1) (2002) 41–46.

[38] E.E. Perl, J. Simon, D.J. Friedman, N. Jain, P. Sharps, C. McPheeters, Y. Sun, M.L. Lee, M.A. Steiner, (Al) GaInP/GaAs tandem solar cells for power conversion at elevated temperature and high concentration, IEEE J. Photovoltaics 8 (2) (2018) 640–645.

[39] M.A. Steiner, J.F. Geisz, D.J. Friedman, W.J. Olavarria, A. Duda, T.E. Moriarty, Temperature-dependent measurements of an inverted metamorphic multijunction (IMM) solar cell, in: 2011 37th IEEE Photovoltaic Specialists Conference, IEEE, 2011, pp. 002527–002532.

[40] R. Vaillon, S. Parola, C. Lamnatou, D. Chemisana, Solar cells operating under thermal stress, Cell Rep. Phys. Sci. 1 (12) (2020).

[41] E.E. Perl, D. Kuciauskas, J. Simon, D.J. Friedman, M.A. Steiner, Identification of the limiting factors for high-temperature GaAs, GaInP, and AlGaInP solar cells from device and carrier lifetime analysis, J. Appl. Phys. 122 (23) (2017).

[42] J.R. Wilcox, J.L. Gray, A distributed emitter model for solar cells: Extracting a temperature dependent lumped series resistance, in: 2012 38th IEEE Photovoltaic Specialists Conference, IEEE, 2012, pp. 002122–002127.

[43] S. Hanauer, L. Boudier, A. Mlayah, I. Revol, Multiphysics modeling tool for photovoltaic-thermoelectric hybrid devices integrating a photothermal interface, Energy Convers. Manag.: X 23 (2024) 100665.

[44] B. Lorenzi, G. Chen, Theoretical efficiency of hybrid solar thermoelectric-photovoltaic generators, J. Appl. Phys. 124 (2) (2018).