Photovoltaic performance of lattice-matched gallium indium arsenide/germanium stannide dual-junction cell

Based on the photovoltaic properties and tandem solar cells theory, Gallium Indium Arsenide/Germanium Stannide (GaInAs/GeSn)-based double-junction (DJ) solar cells have been numerically simulated for the first time. In this study, we explore the band gap combination under lattice matching and obtain the content of In/Sn at optimal efficiency, which is expressed as Ga0.84In0.16As/Ge0.93Sn0.07 DJ solar cell (1.20/0.58 eV). Afterward, it is optimized in terms of variation in the doping contents and active layer thickness. To take full advantage of the electron mobility of the material, the optimal ‘inverted doping profile’ concentration N a(d) is 1.5(5)/5(20) × 1018 cm−3. In addition, the reasonable p(n) layer thickness could be comprised of 0.2–0.8(0.2–1)/0.5–3(1–4) μm of the DJ solar cells with less material consumption. When the p(n) layer thickness is 0.30(0.25)/0.9(1.35) μm, the tandem device can achieve an optimal efficiency of 31.00% with 28.98 mA cm−2 (J sc), 1.25 V (V oc) and 85% (FF). This study highlights that GeSn materials have the potential to combine with III–V materials to form low-cost and high-efficiency tandem devices.


Introduction
GaInP/GaInAs/Ge Multijunction (MJ) solar cells have attracted intensive attention due to their high conversion efficiency and high stability advantages [1]. At present, the highest conversion efficiency of GaInAs-based double-junction (DJ) solar cell is 32.8% [2], and it can reach 43.4% under concentrated light [3]. MJ solar cells in particular can experimentally enhance the maximum power conversion efficiency of solar cells from 31.8% to 47.6% [4]. However, it has not been popularized for civilian or commercial use because of the high material costs and low mechanical properties. Therefore, reducing the cost of MJ solar cells is a key problem to be resolved in the photovoltaic field. The GeSn material [5,6] has lower cost, longer carrier diffusion length and higher absorption coefficient compared to bulk Ge material in MJ solar cells, furthermore, the lattice constant and band gap of GeSn can be adjusted by controlling the Sn content.
Therefore, replacing the more expensive Ge material with GeSn material not only reduces the cost of materials but also broadens the absorption spectral range, which has very great prospects for applications in photovoltaics. Many research groups [7,8] have realized epitaxial growth of high-quality Ge x Sn 1−x semiconductor materials by using material growth technologies such as Molecular Beam Epitaxy (MBE), Metalorganic Chemical Vapor Deposition (MOCVD) and magnetron sputtering, and measured the related basic physical properties. In addition, GeSn devices such as detectors, tunneling field effect transistors, laser diodes and other optoelectronic devices are already fabricated and in production [9,10]. GeSn material has shown excellent photovoltaic properties, satisfying the requirements for potential application in thin film photovoltaic cells. In 2013, Benjamin R C et al [11] prepared the first GeSn layer with 0.7%-9% Sn concentration by RPCVD, and suggested the feasibility of GeSn in MJ solar cells. In the same year, Li Haofeng et al [12] realized a low lattice mismatch of GeSn used in In 0.65 Ga 0.35 P and In 0.17 Ga 0.83 As by adjusting the lattice constant of the materials in each layer of the solar cells. It was demonstrated that the GeSn is transformed into a direct band-gap material when the content of Sn is about 6.8% [13]. Kondratenko S V et al [14] studied that increasing the thickness of high Sn content in GeSn thin films can lead to strain relaxation and form a relaxed top layer, thereby reducing the band-gap, which is conducive to their application in GaInAs-based solar cells. At present, GeSn material is still in its nascent experimental stage in solar cells, and its theoretical research has made some progress. In 2013, Zhang et al [15] investigated the structural stability, elastic, lattice dynamic and thermodynamic properties of the ordered GeSn cubic alloy in zinc-blende (B3) structure and obtained the data of lattice dynamic dependent properties currently lacking for GeSn. In 2014, Jiang et al [16] developed the required accurate model for the direct gap transition and showed that it allows for the determination of the energy E 0 with meV precision. Following, Chang G E et al [17] developed theoretical models to calculate the composition-dependent band alignments, the band structures and the absorption coefficient for the proposed GeSn heterojunction phototransistors. In 2022, Ghosh et al [18] presented the theoretical models to calculate the carrier saturation velocities, the optical absorption coefficient of the GeSn vertical p-i-n homojunction waveguide photodetectors. Recently, to the best of our knowledge, simulation research on GeSn single-junction solar cell [19,20] is reported for the first time with the results showing prospects for GeSn application in solar cells. Nevertheless, research on GaInAs/GeSn-based DJ solar cells is seldom. Therefore, this paper studies its performance combined with the double-junction cell theory and experimental data, which is expected to be helpful for the MJ solar cells.
This study improves the theoretical model of solar cells by considering the effect of the internal parameters of the cell on the quantum efficiency. We have investigated the photovoltaic properties of lattice-matched GaInAs/ GeSn-based solar cells with the band-gap combination by tuning the content of In/Sn. The dependence of the photovoltaic properties of double junction solar cells on the light-doped layer doping concentration and the active layer thickness is evaluated. Therefore, the simulated results will be favourable for the experimental preparation of GaInAs/GeSn-based solar cells and the lower-cost application of the photovoltaic field.

Materials and methods
The device structure of the GaInAs/GeSn double-junction (DJ) solar cell is shown in figure 1. The bottom cell material is GeSn [20] and the top cell material is GaInAs [21,22]. To reduce optical losses and maintain higher conversion efficiency, a tunnel junction with low series resistance and a wider band-gap should be considered. In this paper, GaInAs/GeSn DJ solar cells are connected in series through the ideal tunnel junction p + /n + GaAs [23] to achieve optical transparency, high tunneling current and low voltage drop.
After Sunlight is incident on the surface of the solar cells, it decays along the thickness of the material. When series and parallel resistance losses and reflections are neglected, the short circuit current density J sci (i = 1, 2 represents the top and bottom cell parameters, respectively) can be described as equation (1) [24]: where q, l and f l ( ) denote elementary charge, wavelength and AM1.5 G Solar Spectra, respectively. Moreover, t is the top cell thickness, l ( ) IQE i is the internal quantum efficiency, and a l ( ) 1 is the top cell absorption coefficient.
The relationship between the optical absorption coefficient α and the photon energy of the semiconductor direct band-gap material obtained from the optical absorption theory [25] is:  Where n is the refractive index, e 0 is the vacuum dielectric constant, c is the speed of light, and h is the Planck constant. m 0 is the electron mass in free space, m r is the conversion mass which can be given by the electron effective mass * m e and hole effective mass * m h as Here v is the frequency of the photon and E g is the band gap. P cv is an optical transition matrix, and á ñ | | P cv 2 can be calculated as following [26]: where D is the spin-orbit splitting energy. Photon-generated carriers are generated in the electrically neutral p, n and depletion regions of the solar cells, and the generated photons recombine to form the photocurrents after reaching the space charge region. In the calculation, considering the influencing factors of the carrier generation, composition, diffusion and drift in each region, the internal quantum efficiency of the cells can be expressed as:  Where ( ) / L D e h is the minority electron (e)/hole (h) diffusion length (coefficient) in the p/n-type layer, / S F B is the surface recombination rate at the front /back surface, / W E B is the thickness of the quasi-neutral region of the p/n layer, and W d is the width of the space-charge region.
According to the Einstein relation, the diffusion coefficient can be expressed by the carrier mobility: Where k B and T denote the Boltzmann constant and temperature, respectively. And m / e h is the minority carrier mobility, which can be determined from the empirical Caughey-Thomas model [28] as:  (5), is represented as [30]: Where, e n and i s are the relative dielectric constant of the material and the intrinsic carrier concentration. The double-junction cells are a series structure, and the total current is limited by the minimum photocurrent, so the short-circuit current can be expressed as: The total current-voltage relationship [26] is expressed as follows: Where J i 0 indicates the reverse saturation current density, specifically expressed as follows [31]: The power of the double junction solar cell is expressed as:

d JV dJ
The FF of solar cell is the ratio of the maximum power output P m to the product of the V oc and J sc [32]: Photoelectric conversion efficiency of photovoltaic cells: According to experimental data, parameters of GaInAs and GeSn are shown in table 1.

Results and discussion
3.1. Band-gap combination during lattice matching Lattice constant matching [48] materials have low photon transport loss, which facilitates efficient light collection with minimal thermalization loss in a wide wavelength range. Lattice matching can also avoid strain and relaxation. The lattice matching across the interface demonstrates the high quality of the epitaxial growth. Therefore, this study explores the band-gap combination of GaInAs/GeSn solar cells in a lattice-matched system, which can provide a reference for the experimental preparation.
In the design and preparation of MJ solar cells, it is necessary to consider the band-gaps arrangement between the sub-cells. The broad and narrow band gap materials absorb higher and lower energy photons, Where a is the lattice constant, E g1 and E g2 are the band-gaps of -Ga In As The band-gap combination of Ga 1−x In x As/Ge 1−y Sn y DJ solar cells during lattice matching can be obtained from equations (15)- (18). As shown in figure 2 Based on the Ga 0.84 In 0. 16 As/Ge 0.93 Sn 0.07 DJ solar cell with lattice matching and band-gap matching, we continue to explore the impact of the material itself on the performance, focusing on the influence of the materials with the doping concentration and thickness on the device performance.

Doping concentration
Doping concentration is an important parameter that affects the performance of solar cells. To explore the influence of the top cell doping concentration on the efficiency of DJ solar cell at room temperature (300 K), we keep the doping concentration of Ge 0.93 Sn 0.07 [20] as N a2 is 10 19   above optimization result, the effect of the doping concentration of the Ge 0.93 Sn 0.07 bottom cell on the efficiency of the DJ solar cell is shown in figure 3(b). When the p(n) region doping concentration is 5 × 10 18 (2 × 10 19 ) cm −3 , the highest conversion efficiency reaches 29.28%. Thus, the optimum doping concentration in the diode front layer is lower than that for the rear layer, which is expressed as an 'inverted doping profile'. Physically, this contrary scenario can be attributed to the strong contrast in electron and hole minority carrier mobility. Specifically, the smaller hole mobility promises generally a shorter diffusion length in the n-type rear layer while the higher electron mobility leads to a much longer diffusion length in the p-type front layer. Which can more effectively collect the light carrier and improve the conversion efficiency of solar cells. Therefore, to fully utilize the advantage of electron mobility, an inverted doping profile is an alternative choice.
Since the cells' conversion efficiency can be affected by various photovoltaic parameters, it is worthwhile to evaluate the effect of doping on performance. And we have plotted in figure 3 the resulting J sc (c), V oc (d) and FF (e) as a function of doping concentration in a light-doped layer.
As shown in figure 3(c), there is nearly unchanged on J sc1,2 during increasing doping of N d1,2 , so we can flexibly select N d1,2 concentration in the experiment. However, it is easy to see that J sc1,2 reveals a reduction with increasing N a1,2 . Particularly, when N a1 is improved from 5 × 10 17 cm −3 to 5 × 10 19 cm −3 , J sc1,2 decreased from 34.55, 26.98 mA cm −2 to 9.41, 22.75 mA cm −2 , respectively. Physically, the upgrading doping promotes electron-hole pairs recombination and shortens the minority carrier lifetime (diffusion length), leading to the reduction of J sc . With the improving N a/d1,2 of each sub-cell, V oc1,2 enhance initially and then decline slowly (see figure 3(d)). To be specific, the maximal V oc1,2 of 1.109 V and 0.176 V can be composed of 5 × 10 18 cm −3 (N a/d1 ) and 10 19 cm −3 (N a/d2 ), respectively. V oc exhibits an enhancement at low doping, which is attributed to the improvement of carrier collection and the reduction of transport loss. At the same time, the recombination rate increases, leading to a lower voltage at elevated doping. Clearly, V oc1 is higher than V oc2 , because of the linear relationship between V oc with the band-gap of Ga 0.84 In 0.16 As (1.20 eV) wider than Ge 0.93 Sn 0.07 (0.58 eV).  shows that the trend of the FF 1,2 is sharply analogous to that of the V oc . When the doping N a(d) of Ga 0.84 In 0.16 As/Ge 0.93 Sn 0.07 is 1.5(5)/5(20) × 10 18 cm −3 , the FF 1 and FF 2 reach 90.9% and 73.4%, respectively. It is indicated that the 'inverted doping profile' of DJ solar cells is easy to maintain high V oc and FF.
In conclusion, when the doping N a(d) of Ga 0.84 In 0.16 As/Ge 0.93 Sn 0.07 is 1.5(5)/5(20) × 10 18 cm −3 , the maximal conversion efficiency of 29.28% is observed. Furthermore, we may adjust the doping concentration to improve the cells efficiency and reduce the production cost.
To maintain the optimal doping as described above, we investigate the influence of each sub-cells' active layer thickness on the performance before determining the optimal structure of the Ga 0.84 In 0.16 As/Ge 0.93 Sn 0.07 DJ solar cell, which may provide valuable guidance for the experimental preparation of solar cells.

Optimization of the active layer thickness
3.3.1. Ga 0.84 In 0.16 As Top cell thickness To reveal the active layer-controlled electricity generation, taking the W p2/n2 as 0.1/3 μm for Ge 0.93 Sn 0.07 thickness [20], we have systematically simulated the J sc , V oc , FF and η versus the thickness of the active layer for the Ga 0.84 In 0.16 As top cells, as shown in figures 4(a)-(d).
In figure 4(a), with the increasing p-GaInAs thickness, J sc rises rapidly and then decreases gradually, while the n-GaInAs thickness changes, J sc remains stable virtually. It may be affected by the recombination of the minority carriers. Theoretically, the photon-generated carriers are produced in the p, n, and depletion (electro neutrally) regions of the solar cells. A photocurrent can only be initiated when the carriers recombine before entering the space charge region. As the thickness increases, the photons can be absorbed to produce electron-hole pairs, but they are too distant from the space charge region to recombine before diffusing to the depletion region, which is unable to form an effective current. Figure 4(b) demonstrates that V oc enhances during the p-GaInAs increasing thickness from 1.244 V to the maximum 1.260 V at 0.5 μm. Subsequently, the V oc is reduced to 1.200 V. However, as the thickness of the n-GaInAs changes, it remains constant practically. This may be due to the increasing thickness result in the enhancement of the concentration of photo-generated charge carriers, which raises the internal electric field. Unfortunately, a relatively thick absorbent layer may minish the electric-field strength because of the collected electron-hole pairs recombination on the surface. In figure 4(c), FF changes between 82.9% and 89.6% with the growing thickness of the GaInAs top cell. It may be influenced by the current of the series cells. FF is enlarged when the J sc of the top and bottom cells are in similar ranges and is slightly lower otherwise.
The trend of conversion efficiency (figure 4(d)) affected by thickness is consistent with J sc , which indicates that J sc is a determinant of series device performance. The thickening GaInAs active layer can increase effectively the absorption of photon carriers, leading to the rise of J sc and η. When the Ga 0.84 In 0.16 As thickness W p/n is optimized, the maximum η of Ga 0.84 In 0. 16  The results are similar to the effect of the top thickness, J sc is enhanced from 28.50 mA cm −2 to 29.00 mA cm −2 with the p-GeSn layer thickness at 0.5-3 μm and the n-GeSn layer thickness at 1-4 μm in figure 5(a). The reason is that the optical absorption rate rises as the active layer's thickness increases because of the improvement of photon absorption at longer wavelengths. Figure 5(b) indicates that the V oc is enlarged from 1.200 V to the maximum 1.257 V with 0.9 μm p-GeSn layer thickness. However, the V oc is approximately constant when varying the thickness of the GeSn layer. The improvement of V oc may be attributed to the increase of the photo-generated carriers' concentration. For the same reason there is an enhancement of FF before 0.5 μm ( figure 5(c)). On the contrary, the V oc and FF begin to decline when the increasing thickness of the GeSn cell approaches 1 μm. It may be due to the increasing thickness results in the enhancement of the recombination rate, which would decrease the internal electric field. Considering the influence of the three parameters, η of the device presents evident enhancement first and then gradual decline, which is relatively high with 30.00% ∼ 31.00% of the p/n-GeSn layer thickness at 0.5-3/1-4 μm in figure 5(d). The results show that the optimized W p/n of Ge 0.93 Sn 0.07 cell is 0.9/1.35 μm, the η of Ga 0.84 In 0. 16 As/Ge 0.93 Sn 0.07 DJ solar cells reaches 31.00%.
To achieve a high conversion efficiency (30.00% ∼ 31.00%) and improve the film quality, the active layer should be selected between 0.5-3 μm of the p-GeSn and 1-4 μm of the n-GeSn.
In summary, the structure and performance parameters of the Ga 0.84 In 0. 16 (2)) and the internal quantum efficiency model (equation (5)), the FF of the top cell and the bottom cell reaches 90% and 71%, respectively. It is amazing that the system's FF approaches 85%. Based on the properties of the current density-voltaic model, the η of Ga 0.84 In 0.16 and Ge 0.93 Sn 0.07 can achieve 29.16% and 3.32%, respectively, improving the η of Ga 0.84 In 0. 16 As/Ge 0.93 Sn 0.07 DJ solar cell to 31.00%. This is an exciting result that will provide guidance for the preparation of MJ solar cells.

Conclusions
Based on the principle of meticulous balance and combining the available material parameters, this paper calculates that the band-gap combination of the GaInAs/GeSn-based DJ solar cells are 1.1-1.23 eV/ 0.45-0.63 eV. We explore the band-gap combination (1.20/0.58 eV) under lattice matching (0.572 nm) and obtain the In/Sn content at the optimal efficiency, which is expressed as Ga 0.84 In 0. 16 As/Ge 0.93 Sn 0.07 DJ solar cell. The effect of doping layer concentration and activated layer thickness on the performance of the Ga 0.84 In 0.16 As/Ge 0.93 Sn 0.07 DJ solar cell has been numerically simulated for the first time. To take full advantage of the electron mobility of the material, the optimal 'inverted doping profile' concentration N a(d) is 1.5(5)/5 (20) × 10 18 cm −3 . Moreover, when the p(n) region thickness is 0.30(0.25)/0.9(1.35) μm, the optimal efficiency reaches 31.00%. At this point, J sc , V oc and FF are 28.98 mA cm −2 , 1.25 V and 85%, respectively. To reduce material costs, the active layer thickness could be selected between 0.2-0.8(0.2-1)/0.5-3(1-4) μm of Ga 0.84 In 0.16 As/Ge 0.93 Sn 0.07 DJ solar cell. To explore the lower-cost solar cells, the simulations of combining GeSn with III-V solar cells may provide valuable guidance for the experimental preparation of the cells.

Data availability statement
No new data were created or analysed in this study.

Conflicts of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.

Funding statement
This work was supported by National Natural Science Foundation of China Youth Found (12204026) and Beijing Science and Technology New Star Program (Z211100002121079).