Design and Simulation for Minimizing Non-Radiative Recombination Losses in CsGeI₂Br Perovskite Solar Cells

Abstract

CsGeI₂Br-based perovskites, with their favorable band gap and high absorption coefficient, are promising candidates for the development of efficient lead-free perovskite solar cells (PSCs). However, bulk and interfacial carrier non-radiative recombination losses hinder the further improvement of power conversion efficiency and stability in PSCs. To overcome this challenge, the photovoltaic potential of the device is unlocked by optimizing the optical and electronic parameters through rigorous numerical simulation, which include tuning perovskite thickness, bulk defect density, and series and shunt resistance. Additionally, to make the simulation data as realistic as possible, recombination processes, such as Auger recombination, must be considered. In this simulation, when the Auger capture coefficient is increased to 10⁻²⁹ cm⁶ s⁻¹, the efficiency drops from 31.62% (without taking Auger recombination into account) to 29.10%. Since Auger recombination is unavoidable in experiments, carrier losses due to Auger recombination should be included in the analysis of the efficiency limit to avoid significantly overestimating the simulated device performance. Therefore, this paper provides valuable insights for designing realistic and efficient lead-free PSCs.

1. Introduction

Hybrid organic–inorganic perovskite solar cells, as an efficient and low-cost photovoltaic technology for capturing solar energy, have emerged as one of the most active research fields in photovoltaic technology [1,2,3,4]. The power conversion efficiency (PCE) of single-junction PSCs has improved significantly over the past decade, increasing from the initial 3.8% to the latest certified value of 26.1% [5]. This advancement fully demonstrates the vast potential and rapid development of PSC technology. Currently, the highest PCE achieved by this technology has matched the efficiency of some commercial solar cells on the market, such as copper indium gallium selenide (CIGS) solar cells and crystalline silicon. In the four-terminal configuration of all-perovskite stacked PSCs, the PCE has reached an impressive level above 27% [6], and perovskite-silicon stacked solar cells have even surpassed the 34% efficiency threshold [7]. The notable advancements are attributed to the superior characteristics of metal halide perovskite, as well as the rapid assimilation of lessons from other photovoltaic technologies, such as dye-sensitized solar cells, organic photovoltaics, and silicon solar cells. However, despite the remarkable breakthrough in performance, the stability issue of PSC technology and lead toxicity still limits its commercialization. To overcome this challenge, researchers have begun to explore new material systems, among which lead-free cesium compounds have attracted much attention for their excellent photovoltaic performance and tunable energy-band structure. Cesium lead-free halide PSCs not only have higher photovoltaic conversion efficiency but also have significant advantages in terms of cost and material safety, which are expected to promote the sustainable development of the photovoltaic industry [8,9].

CsGeI₂Br has attracted considerable interest within the perovskite industry due to its non-toxic composition, together with its 1.579 eV band gap, high absorption coefficient (over 10⁵ cm⁻¹) in the visible range, and conductivity [10,11]. Recent studies have shown that compared to organic–inorganic hybrid perovskites, cesium lead-free halide perovskites exhibit higher thermal stability. Among all lead-free inorganic PSCs, CsSnI₃ achieves the highest PCE, reaching 11.21% [12]. Without considering non-radiative recombination, the simulation by Joy Sarkar and others using the solar cell capacitance simulator-one dimension (SCAPS-1D) version 3.3.10 employed Spiro-OMeTAD as the hole transport layer (HTL), TiO₂ as the electron transport layer (ETL), and CsGeI₂Br as the perovskite layer, achieving a device efficiency of 25.15%, open-circuit voltage (V_oc) of 1.23 V, short-circuit current density (J_sc) of 26.42 mA/cm², and fill factor (FF) of 77.74% [11]. Nonetheless, it is inescapable that non-radiative recombination losses are present across all categories of solar cells, mainly due to phenomena such as tail recombination, electron–phonon coupling, Auger recombination, and recombination facilitated by defects [13]. The performance characteristics of the record device, including the V_oc, J_sc, FF, and PCE, are caused to deviate from the radiation limit by these losses [14,15,16,17]. Additionally, non-radiative recombination centers usually accelerate the degradation of perovskite absorbers, which hinders their development [18,19]. In order to further improve the performance of the device within the Shockley–Queisser (SQ) radiation limit and to improve the long-term stability of solar cells, it is extremely important to reduce all types of non-radiative recombination losses.

To unlock the full potential of CsGeI₂Br-based PSCs, we have investigated their working mechanisms and optimization outcomes utilizing numerical simulations. The performance of PSCs is comprehensively assessed through a detailed examination of the optical/electrical parameters, which include perovskite thickness, bulk defect density, series and shunt resistance, and Auger recombination. The simulation results suggest that (1) bulk defects are identified as important factors limiting the device performance and (2) high shunt resistance, low series resistance, and appropriate Auger recombination coefficients reduce non-radiation recombination losses and improve a device’s V_oc and FF, making it approach the theoretical calculated SQ limit efficiency. Therefore, this study provides guidance for researchers to optimize PSCs and ultimately approach the photovoltaic impact of the SQ radiation limit.

2. Theoretical Methods

SCAPS-1D is a widely used simulation software designed specifically for studying the electrical and optical properties of thin-film solar cells [20]. Detailed analyses of critical solar cell processes, such as carrier transport and recombination, are supported in one-dimensional structures. In this study, SCAPS-1D simulation software (Version 3.3.10) was employed. Given the increasing importance of numerical simulation for understanding, designing, and optimizing high-efficiency solar cells, the software was utilized to intuitively analyze cell parameters, enabling rapid and accurate predictions of cell performance. This, in turn, guided experiments and optimized the research and development process [21]. SCAPS-1D simulations are primarily based on Poisson’s equation, the continuity equations for electrons and holes (Equations (1)–(3)), constitutive equations (Equations (10) and (11)), and boundary conditions for semiconductor devices [22].

$${\displaystyle \frac{\partial }{\partial x}}\left({\epsilon }_0\epsilon {\displaystyle \frac{\partial \psi (x)}{\partial x}}\right)=-q\left(p-n+N_D^{+}-N_A^{-}+p_t-n_t\right)$$

(1)

$$-{\displaystyle \frac{\partial J_n}{\partial x}}-U_n+G={\displaystyle \frac{\partial n}{\partial t}}$$

(2)

$$-{\displaystyle \frac{\partial J_p}{\partial x}}-U_p+G={\displaystyle \frac{\partial p}{\partial t}}$$

(3)

where q is the amount of element charge; ε and ε₀ stand for relative permittivity and free space permittivity, respectively; n and p are the electron and hole concentrations; Ψ is the potential; p_t and n_t are the hole and electron density distributions; N_D⁺ and N_A are the donor and acceptor concentrations; J_n and J_p are the electron and hole current density; G is the carrier generation rate; and U_n and U_p are the electron and hole recombination rate.

In general, the boundary conditions of hole and electron flow density and device potential are determined, the concentration of electron and hole and electric field can be solved, and other relevant working parameters of cell devices can be further obtained. Set the potential at the back contact to 0, then the boundary conditions at the back contact are as follows:

$$\phi (L)=0$$

(4)

$$J_n(L)=-q{S}_{n^{\prime }}^b[n^{\prime }(L)-n_{eq}(L)]$$

(5)

$$J_p(L)=q{S}_{p^{\prime }}^b[p^{\prime }(L)-p_{eq}(L)]$$

(6)

The boundary conditions at the front contact are as follows:

$$\phi (0)={\phi }_f-{\phi }_b+V_{app}$$

(7)

$$J_n(0)=q{S}_{n^{\prime }}^f[n^{\prime }(0)-n_{eq}(0)]$$

(8)

$$J_p(0)=-q{S}_{p^{\prime }}^f[p^{\prime }(0)-p_{eq}(0)]$$

(9)

The superscripts f and b denote the front and back electrodes, respectively, while V_app represents the applied bias across the device. ψ_f and ψ_b indicate the work functions of the front and back contacts. n_eq(L), p_eq(L), n_eq(0), and p_eq(0) refer to the equilibrium concentrations of free electrons and holes at x = L and x = 0, respectively. p’ and n’ are the carrier concentrations at the contacts, and the surface recombination rates of holes and electrons at the front and back electrodes are denoted as S_p′^f, S_n′^f, S_p′^b, and S_n′^b, respectively.

The constitutive equations are as follows:

$$J_n=-{\displaystyle \frac{{\mu }_{n}n}{q}}{\displaystyle \frac{\partial E_{Fn}}{\partial x}}$$

(10)

$$J_p=+{\displaystyle \frac{{\mu }_{p}p}{q}}{\displaystyle \frac{\partial E_{Fp}}{\partial x}}$$

(11)

The electron and hole quasi-Fermi levels are represented by E_Fn and E_Fp; μ_n and μ_p are electron and hole mobility. Together with appropriate boundary conditions at the interfaces and contacts, this results in a system of coupled differential equations in (Ψ, n, p) (Equations (1)–(3)) or (Ψ, E_Fn, E_Fp) (Equations (10) and (11)).

The simulation uses an AM1.5G light source of 1000 W/cm² for constant lighting, and the temperature is set to 300 K. The device structure type used in the simulation is n-i-p, typically consisting of an FTO substrate, an ETL, a perovskite layer, an HTL, as well as a back-contact electrode system, and it is illustrated in Figure 1a, where Cu₂O is chosen as the HTL and TiO₂ as the ETL. Figure 1b shows the energy-level arrangement of the designed CsGeI₂Br with the charge-transport layer. While series resistance (R_s) is 5 Ω cm², shunt resistance (R_sh) is 5000 Ω cm². The parameters for CsGeI₂Br, Cu₂O, and TiO₂ are displayed in

Table 1. Material parameters of HTL, ETL, and perovskite required in the device modeling.

Material Parameters	FTO [23]	TiO2 [24]	Cu2O [25,26]	CsGeI2Br [11]
Thickness (nm)	500	30	350	800
Band gap, Eg (eV)	3.5	3.2	2.17	1.579
Electron affinity (eV)	4.0	3.9	3.2	3.76
Dielectric permittivity (relative)	9.0	9.0	7.1	18.0
CB effective density of states (1/cm3)	2.2 × 1018	2 × 1018	2.02 × 1017	9.65 × 1017
VB effective density of states (1/cm3)	1.8 × 1019	1.8 × 1019	1.1 × 1019	1.04 × 1018
Electron thermal velocity (cm/s)	1 × 107	1 × 107	1 × 107	1 × 107
Hole thermal velocity (cm/s)	1 × 107	1 × 107	1 × 107	1 × 107
Electron mobility (cm3/Vs))	20	20	200	20
Hole mobility (cm3/Vs)	10	10	80	20
Shallow uniform donor density ND (1/cm3)	1.8 × 1019	9 × 1016	0	0
Shallow uniform acceptor density NA (1/cm3) Total density (1/cm3)	0	0	1 × 1018	2 × 1016
	1 × 1015	1 × 1015	1 × 1015	1 × 1015

. Additionally,

Table 2. Material parameters of interface defect layers.

Interface	Defect Type	Capture Cross Section: Electrons/Holes (cm2)	Energetic Distribution	Reference for Defect Energy Level	Total Density (cm−3)
ETL/CsGeI2Br	neutral	1 × 10−19/1 × 10−19	single	above the VB maximum	1 × 1010
CsGeI2Br/HTL	neutral	1 × 10−19/1 × 10−19	single	above the VB maximum	1 × 1010

presents the parameters for the interface defects between the CsGeI₂Br layer and the transport layers, and these parameters are based on previous experiments and the published literature.

3. Results and Discussion

3.1. Dependence of Perovskite Thickness

The adjustment of perovskite material thickness significantly impacts its electrical, optical, and stability properties. Being too thin or too thick can reduce carrier transport efficiency, cause energy-level mismatches, degrade interface charge-transfer efficiency, and compromise stability. Precise thickness control is crucial for optimizing the performance of PSCs, aiming for the optimal PCE and stability [27,28]. This study examined thickness variations between 400 and 1200 nm, as depicted in Figure 2, which suggest the following: (i) Figure 2a reveals that as thickness increases, the V_oc decreases from 1.37 V to 1.40 V due to an enhancement in the spectral response in the long-wavelength region [29]. (ii) With increasing thickness, both the J_sc and PCE show an upward trend, and the PCE increases from 25.09% to 28.25%. (iii) Conversely, the FF decreases as the thickness increases.

In detail, this is an important indicator of the conversion efficiency of a solar cell or electronic device. There is a specific mathematical relationship between the fill factor and parameters such as the V_oc, J_sc, V_mmp, and J_mmp, and usually the FF is calculated as follows:

$$FF={\displaystyle \frac{V_{mpp}\times J_{mpp}}{V_{oc}\times J_{sc}}}$$

(12)

where V_mpp is maximum power voltage and J_mpp is maximum power current. Specifically, as the perovskite thickness increases, there is an increase in light scattering and absorption within the device, leading to a decrease in the V_oc and J_sc. Additionally, with the increased thickness, the carrier transport distance within the device also increases, resulting in a reduction in V_mmp and an increase in J_mmp. These changes affect the calculated FF; thus, the overall trend is a decrease in the FF with increasing thickness.

To further comprehend the reason for the variation in the V_oc, quasi-Fermi level splitting (QFLS) is a key factor impacting the V_oc [30], with Equation (9) illustrating its relationship with the Voc. QFLS is defined as the difference between the electron quasi-Fermi level (E_Fe) and the hole quasi-Fermi level (E_Fh) (Figure 2e) [30].

$$V_{oc}={\displaystyle \frac{QFLS}{q}}={\displaystyle \frac{1}{q}}(E_{Fe}-E_{Fh})$$

(13)

Figure 2f shows that as thickness increases, E_Fn remains constant while E_Fp decreases, leading to reduced QFLS and V_oc. Thus, perovskite layer thickness significantly affects both QFLS and V_oc. While greater thickness improves light absorption, it also increases non-radiative recombination and series resistance, further reducing QFLS and V_oc. Studies suggest the optimal thickness is below 1000 nm. Based on experimental validation, 900 nm offers an ideal balance, enhancing absorption efficiency while minimizing recombination losses, ensuring both efficiency and stability [31,32].

3.2. Dependence of Perovskite Bulk Defect Density

One of the primary impediments to the enhancement of PSC performance is the non-radiative recombination loss of bulk carriers. To enhance the device’s performance and stability, it is crucial to efficiently minimize the surface-related non-radiative recombination losses [33,34]. The bulk defect density (N_t) in CsGeI₂Br-based PSCs is adjusted from 10¹² cm⁻³ to 10¹⁷ cm⁻³ (Figure 3). As shown in Figure 3a, with the increase in N_t, both the V_oc and J_sc decrease; however, the reduction in the J_sc is relatively small. This effect is due to the increase in the concentration of defects, which accelerates the recombination process of charge carriers inside the device, consequently decreasing the count of functional carriers and, consequently, a decrease in both the V_oc and J_sc.

To further verify that lower N_t reduces interfacial charge recombination, electrochemical impedance spectroscopy (EIS) is used for measurements [35,36,37]. There are two processes that can contribute to EIS.

$$Z=Z^{\prime }-j{Z}^{″}$$

(14)

The impedance is a complex number Z that can be expressed as two parts of the real part Z′ and the imaginary part Z″.

$$Z^{\prime }=R_S+{\displaystyle \frac{R_{CT}}{1+{\omega }^2{C}^2{R}_{CT}^2}}$$

(15)

Here, R_s is the series resistance; R_CT is the charge-transfer resistance, corresponding to the diameter of the semicircle in the Nyquist plot; and C is the capacitance. ω is the frequency.

Figure 3c shows the Nyquist curve for the CsGeI₂Br device. In the context of EIS, key parameters such as the R_s, the R_CT, and the recombination resistance (R_rec) can be derived from the measurement data. R_s is typically calculated from the initial intercept on the Z′ axis of the high-frequency arc in a Nyquist plot. At high frequencies, R_tr and the low-frequency R_rec correspond to the charge transfer at the HTL/perovskite interface and the recombination at the perovskite/ETL interface, respectively. Under light and high photoconductivity conditions, the R_rec predominantly contributes, given the negligible contribution of the R_CT to the impedance response in high-performance PSCs [10,38]. The dominant semicircle in the Nyquist plot shown in Figure 3c is associated with the R_rec. As the N_t increases, the R_rec values corresponding to CsGeI₂Br devices gradually decrease. The smaller the R_rec, the easier it is for carrier recombination within the perovskite, resulting in fewer electrons and holes being collected by the carrier layer, thereby reducing device performance [39]. Considering low doping is challenging, we chose N_t = 10¹⁴ cm⁻³ as the optimal value for enhanced device performance.

3.3. Dependence of Perovskite Series and Shunt Resistance

In PSCs, R_s and R_sh lead to energy loss, thereby diminishing overall efficiency. This encompasses the resistances contributed by the back contacts (Au), HTL, perovskite layer, ETL, and front contacts (FTO). These resistance effects are challenging to eliminate completely within solar cells. Therefore, it is important to conduct a thorough examination of these factors in order to enhance the efficiency of PSCs [40,41].

In this simulation study, R_s was varied from 1 to 6 Ω cm² and R_sh from 1000 to 7000 Ω cm², with the changes in photovoltaic parameters being examined, as depicted in Figure 4a,b from which we can conclude the following: (i) When the series resistance R_s increases, the V_oc of the device only experiences a slight change, going from 1.4345 V to 1.4355 V, while there is a slight increase noted in the J_sc, moving from 24.95 mA/cm² to 24.97 mA/cm². Conversely, the reductions in the FF and PCE are notably more pronounced, decreasing from 80.00% to 88.01% and from 28.64% to 31.53%, respectively. As illustrated in Figure 4c, the increase in R_s results in a reduction in the semicircle radius in the Nyquist plot, indicating a weakening of the electric field strength. This leads to an increased likelihood of electron and hole recombination before reaching the electrodes due to interactions with defects, impurities, or interfaces within the device, thereby diminishing the overall efficiency of the PSC, so we chose R_s = 1 Ω cm² as the optimal value for enhanced device performance. (ii) With the increase in R_sh (R_s = 1 Ω cm²), the V_oc and J_sc are slightly affected, with the V_oc dropping from 1.433V to 1.435V and the J_sc rising from 24.95 mA/cm² to 24.97 mA/cm². In contrast, R_sh increases cause more significant changes in the FF and PCE, with the FF decreasing from 84.37% to 88.28% and the PCE from 30.17% to 31.62%. Figure 4f illustrates that the Nyquist plot’s semicircle expands with rising R_sh, reflecting an increasing R_rec for the CsGeI₂Br device. A higher R_rec reduces recombination in the perovskite, leading to more collected electrons and holes in the charge-transport layer, improving device performance, so the optimal research parameter determined in this simulation is R_sh = 7000 Ω cm². Therefore, under conditions of high R_sh and low R_s, the device performance of the CsGeI₂Br-based PSCs is better. Under this condition, the V_oc = 1.44V, J_sc = 24.97 mA/cm², FF = 88.28%, and PCE = 31.62%.

3.4. Dependence of Perovskite Auger Recombination

Auger recombination represents a crucial limiting factor in the efficiency of PSCs, precipitating a range of detrimental effects, including carrier recombination losses, interface energy-level mismatches, the augmentation of defect states, and limitations on charge transport. The final section will provide a comprehensive investigation of the impact of Auger recombination on device performance [42,43]. For Auger recombination process, U_Auger is expressed as follows:

$$U_{Auger}=(c_n^{A}n+c_p^{A}p)(np-n_i^2)$$

(16)

The parameters C_n^A and C_p^A represent the Auger recombination coefficients for electrons and holes, respectively. Here, n and p denote the densities of electrons and holes, while n_i is the intrinsic carrier density.

When analyzing the device characteristics using the SCAPS-1D program, the radiative recombination coefficient (3 × 10⁻¹¹ cm³ s⁻¹) and Auger electron/hole capture coefficient are taken into account; we specialize in the effect of Auger recombination on the performance of PSCs. In our research, the radiative recombination coefficient is found to be less than 10⁻²⁰ cm⁶ s⁻¹. The variation in PSC photovoltaic parameters with the changes in Auger electron/hole capture coefficients (C_n/C_p) is depicted in Figure 5. This study reveals that the photovoltaic performance parameters remained essentially unchanged within the range of 10⁻³¹ to 10⁻²⁹ cm⁶ s⁻¹, with the PCE dropping from 29.11% to 29.10% However, as the Auger recombination coefficient exceeded 10⁻²⁷ cm⁶ s⁻¹, these parameters began to decline, with the PCE dropping from 28.78% to 11.20%. This decline is attributed to the increased Auger recombination and carrier recombination, which in turn limited the carrier transport properties. The reduction in carrier transport properties can lead to an increase in the device’s R_s, consequently diminishing cell efficiency. Hence, it is advised to maintain the C_p of PSCs at a level less than 10⁻²⁹ cm⁶ s⁻¹ to reduce the effect of Auger recombination on photovoltaic efficiency. This value aligns with the findings of Ali K. Al-Mouso et al., indicating its feasibility [43].

4. Conclusions

In this study, we elucidated the working mechanisms and optimization outcomes of CsGeI₂Br-based PSCs through rigorous numerical simulation. A detailed explanation was given on how parameters such as perovskite thickness, bulk defect density, series and shunt resistance, and Auger recombination affect device performance by using a TiO₂/CsGeI₂Br/Cu₂O/Au device structure and a series of studies. Based on the simulation results, a low bulk defect density, high R_sh, low R_s, and appropriate Auger recombination coefficient are crucial for the high efficiency of CsGeI₂Br-based PSCs. Additionally, when the Auger capture coefficient is increased to 10⁻²⁹ cm⁶ s⁻¹, the efficiency drops sharply from 31.62% (without considering Auger recombination) to 29.10%. The results suggest that the carrier lifetime in the PSCs should be reduced to below 10⁻²⁹ cm⁶ s⁻¹ to minimize the Auger recombination. Only by suppressing interfacial defect recombination can the full potential of the perovskite absorber be achieved. The ultimate goal is to push this technology toward its radiative limit by suppressing defects in the perovskite bulk or at grain boundaries.