Theoretical advances and future perspectives of all-inorganic germanium-based perovskites

Ziming Kuang, Baoyun Liang, Tengcheng Huang, Tingting Shi* and Weiguang Xie*
Siyuan Laboratory, Guangzhou Key Laboratory of Vacuum Coating Technologies and New Energy Materials, Department of Physics, College of Physics & Optoelectronic Engineering, Jinan University, Guangzhou 510632, China. E-mail: ttshi@jnu.edu.cn; wgxie@jnu.edu.cn

First published on 4th August 2025

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1. Introduction

Energy is essential for modern society, but the surging demand has caused scarcity and environmental harm, making sustainable alternatives a global priority. The shift to clean energy is now urgent for long-term development. Among various alternatives, solar energy has emerged as a front runner in renewable energy technologies due to its inherent advantages including environmental friendliness, abundance, and inexhaustibility. As the fundamental component of photoelectric conversion systems, solar cells operate based on the photovoltaic effect, directly transforming solar radiation into electrical energy. Recent projections indicate that solar photovoltaic generation could account for 25% of the global electricity demand by the mid-21^st century,¹ a transition that may fundamentally reshape the competitive landscape of the worldwide power market. Recognized as a highly promising energy solution, solar cell technology enables cost-effective and efficient clean energy production.² Therefore, intensifying research efforts in solar cell development and accelerating industrialization processes are not only crucial strategies for addressing current energy challenges and promoting sustainability, but also hold profound implications for long-term societal development. Perovskite materials are generally defined as compounds sharing the crystal structure of CaTiO₃, specifically ABX₃-type compounds. The structural versatility of perovskites enables diverse chemical substitutions, facilitating their applications in photovoltaics, photodetectors, and related optoelectronic devices. This structural adaptability has propelled perovskites as next-generation solar cell materials, sparking extensive research interest worldwide. The field achieved a milestone in 2009 when Tsutomu Miyasaka's team developed the first CH₃NH₃PbI₃ perovskite solar cell (PSC) with a power conversion efficiency (PCE) exceeding 3.8%.³ Remarkably, after just over a decade of intensive development, state-of-the-art PSCs have attained certified PCE values reaching 26.95%.⁴

However, perovskite solar cells (PSCs) still face multiple challenges in their path toward widespread commercialization, with film stability and toxicity issues posing significant barriers. Although organic–inorganic hybrid perovskites demonstrate impressive power conversion efficiencies (PCEs) in solar cells,^5–8 their organic cations exhibit proneness to decomposition under high-temperature and high-humidity conditions. This inherent instability severely compromises film integrity, thereby constraining the practical deployment of hybrid perovskites in photovoltaics.^9,10 Furthermore, the toxicity of lead—a critical component in high-performance perovskites—remains a major impediment to their commercial viability. To address this challenge, researchers have pursued extensive investigations focusing on replacing lead with non-toxic divalent cations that exhibit strong optical responses, thereby mitigating toxicity concerns. Promising candidate cations currently under investigation include Sn²⁺, Ge²⁺, Si²⁺, Sb²⁺, and Bi²⁺.^11–17 Tin has gained prominence as a lead substitute candidate due to its closely analogous electronic and structural characteristics to lead,^18,19 combined with the recent demonstration of tin-based perovskite solar cells achieving power conversion efficiencies above 13%.²⁰ However, tin-based perovskites face two critical challenges such as atmospheric oxidation of Sn²⁺ to Sn⁴⁺ that degrades stability,^21,22 alongside inherent high surface vacancy densities impairing luminescence properties in solar cell applications.²³

In the evolution of perovskite research, germanium-substituted all-inorganic perovskites have emerged as a scientific focal point, addressing the dual challenges of instability and lead toxicity inherent in conventional counterparts. Fig. 1 demonstrates remarkable breakthroughs in photoelectric conversion efficiency achieved by this material system over the past decade. Krishnamoorthy et al. computationally identified germanium as a lead alternative through density functional theory (DFT) screening. Their experimental work on CsGeI₃ crystals revealed a stable rhombohedral crystal structure maintaining phase integrity within operational temperature ranges. The derived solar cells exhibited photocurrent densities approximating 6 mA cm⁻²; yet, performance optimization was hindered by inadequate film-forming capability and atmospheric ambient instability.²⁴ This finding demonstrates the photovoltaic potential of germanium-based perovskites while revealing critical challenges requiring resolution. Montiel's team developed CsGeI₃ thin films via physical vapor deposition of CsI and GeI₂ precursors followed by annealing, achieving enhanced thermal stability albeit with a modest PCE of 0.58%.²⁵ While the efficiency remained suboptimal, this methodology pioneered new synthetic pathways for subsequent investigations. Chen et al. pioneered a simple solvothermal process for synthesizing lead-free cesium germanium halide perovskite quantum rods. The quantum rods (QRs), with larger particle volumes compared to quantum dots (QDs), provided enhanced per-particle absorbance cross sections, enabling CsGeI₃ QR-based devices to attain a PCE of 4.94%.²⁶ Liu et al. developed all-inorganic Ge-alloyed perovskite nanocrystals CsSn_0.6Ge_0.4I₃. The derived photovoltaic devices exhibited a PCE of 4.9%. These nanocrystals demonstrated significantly enhanced stability under environmental, thermal, and optical stresses compared to CsSnI₃ counterparts.²⁷ Yang et al. synthesized novel inorganic CsPb_1−xGe_xI₂Br perovskites under humid conditions. Planar solar cells based on the CsPb_0.8Ge_0.2I₂Br perovskite achieved a record PCE of 10.8%. Remarkably, the devices maintained nearly intact photovoltaic parameters after 7-hour exposure to 50–60% relative humidity.²⁸ This work represents dual advancements in both efficiency enhancement and stability improvement. Chen's team employed lead-free all-inorganic CsSn_0.5Ge_0.5I₃ solid-solution perovskites as light-harvesting layers in solar cells. The thin films were fabricated via one-step vapor-phase synthesis and integrated into planar heterojunction architectures. This configuration attained a PCE of 7.11%.²⁹ Persistent explorations in material systems, synthetic methodology innovation, and processing condition optimization have collectively driven progressive performance enhancements. These achievements systematically demonstrate incremental breakthroughs in photoconversion efficiency for germanium-based perovskites. They unequivocally confirm the exceptional research viability and substantial development potential of this material family.

However, it is crucial to note that, analogous to tin ions, germanium ions exist in a divalent state within germanium-based perovskites and exhibit a strong proneness to oxidation, as evidenced by multiple experimental studies.^30,31 To address this challenge, researchers have undertaken in-depth theoretical investigations into the stability of germanium-based perovskites. For instance, Liu et al. conducted formation energy calculations using the Vienna Ab Initio Simulation Package (VASP).³² Their computational results revealed that the formation energies of CsGeBr₃, CsGeI₃, and Cs₂GeCl₆ are all negative, indicating the thermodynamic stability of these materials. Furthermore, Sun et al. also performed formation energy calculations on MAMX₃ (M = Pb, Sn, and Ge and X = I, Br, and Cl).³³ They found that germanium-based perovskites are less susceptible to oxidation compared to their tin-based counterparts, and that MAPbI₃ and MAGeI₃ exhibit comparable stability. Moreover, Tian et al. employed DFT to design cerium (Ce) doping ratios.³⁴ Subsequent experimental synthesis demonstrated that the incorporation of Ce effectively mitigated moisture and air degradation of CsGeBr₃, thereby enhancing its relative stability.

While organic–inorganic hybrid germanium-based perovskites have been extensively studied and demonstrate excellent potential for photovoltaic applications,^35,36 their stability issues, particularly the inherent instability of organic–inorganic hybrid systems, remain a critical bottleneck hindering the development of perovskite materials. Regarding toxicity concerns, the detrimental effects of lead have been well-documented. Upon degradation and dissolution in water, lead-based perovskites release highly toxic Pb²⁺ ions that pose significant bioaccumulation risks.³⁷ The toxicity profile of tin-based perovskites remains contentious: while Conings et al. initially suggested that tin halide perovskites might exhibit greater toxicity than lead-containing analogues,³⁸ subsequent experimental validation by Xiao et al. confirmed their safety under practical operating conditions.³⁹ In contrast, germanium's biocompatible nature renders Ge-based perovskites the most environmentally benign alternative in this materials landscape. In this context, germanium-based all-inorganic perovskites demonstrate promising potential in addressing stability and environmental concerns of conventional perovskites, making their systematic investigation crucial for advancing perovskite technologies. Theoretical approaches have emerged as pivotal tools in this domain due to their rapid implementation and methodological accessibility. Against this backdrop, this review concentrates on theoretical advancements in germanium-based all-inorganic perovskites. Structural analyses encompass a systematic examination of crystal structure stability, atomic configuration patterns, and lattice parameters, elucidating their deterministic roles in macroscopic material behavior. Optoelectronic investigations prioritize band structure engineering and gap modulation mechanisms. The scope further extends to multifunctional performance metrics including mechanical robustness, thermoelectric response, and ferroelectric functionality. Concluding with a comprehensive and in-depth analysis, we present forward-looking perspectives on the future development of germanium-based all-inorganic perovskites.

2. Structural characteristics of all-inorganic germanium-based perovskites

In perovskite crystals, the A-site cation occupies the center of cubic unit cells, surrounded by twelve X anions forming a cub-octahedral coordination polyhedron with coordination number 12. The B-site cation is located at cube vertices enveloped by six X anions in octahedral coordination (coordination number 6). These materials predominantly crystallize in the cubic system, where the comparable ionic radii of A-site cations and X anions facilitate cubic close packing. The ideal perovskite structure adopts cubic symmetry with equivalent lattice parameters (a = b = c) and orthogonal bond angles (α = β = γ = 90°), belonging to the Pmm space group.⁴⁰ This section concentrates on theoretical investigations elucidating the structural uniqueness of all-inorganic germanium-based perovskites and their intrinsic structure–stability relationships. The Goldschmidt tolerance factor (t_G), a pivotal indicator for predicting structural stability in halide perovskites, plays a decisive role in analyzing these germanium-based systems. Building upon this descriptor, we systematically categorize and summarize the structural characteristics of all-inorganic germanium-based perovskites.

2.1 Structural characteristics of all-inorganic perovskites with germanium ions

In the research domain of germanium-based perovskite structures, numerous scientists have conducted systematic experimental investigations. In 1965, Christensen et al. first synthesized CsGeCl₃ compounds.⁴¹ Later in 1982, building upon this foundation, Guen et al. synthesized CsGeI₃ crystals and comprehensively investigated their properties, revealing a phase transition to the R3m symmetry at 290 °C.⁴² Subsequently, Thiele et al. successfully prepared trigonal R3m-phase CsGeI₃ crystals under ambient conditions.⁴³ Despite these advancements, the most stable configuration of the CsGeI₃ system remained undetermined. In 2021, Luo et al. provided novel insights through first-principles calculations. As shown in Fig. 2(a) and (b), contrasting with the orthorhombic P4/mbm space group configuration of CsSnI₃, R3m-phase CsGeI₃ exhibits a lower total energy of −14.18 eV per formula unit compared to its Pmmm-phase counterpart (−14.08 eV per formula unit). This demonstrates that the most stable configuration of CsGeI₃ adopts the non-centrosymmetric R3m phase, where Ge²⁺ deviates from octahedral centers, while the Pmmm phase exists as a metastable structure.⁴⁴ Subsequently, Huang et al. conducted further investigations into the non-centrosymmetric stability of germanium-based perovskites. As illustrated in Fig. 2(c) and (d), using CsGeCl₃ as a prototype, they performed first-principles calculations through VASP, enabling meticulous comparative analysis of lattice parameters and Bravais angles between R3m and Pmm configurations in CsGeX₃ systems. This study revealed that, in R3m-phase CsGeX₃, Ge ions undergo displacement along the body diagonal direction. This displacement induces octahedral distortion in GeX₆ polyhedra, manifesting as compressive strain in three Ge–X bonds and tensile strain in the remaining three. In striking contrast, Pmm-phase CsGeX₃ manifests identical lattice parameters (a = b = c) and orthogonal Bravais angles (90°), conforming to a prototypical cubic configuration.⁴⁵ Synthesis of these findings demonstrates that the B-site cation off-centering in germanium-based perovskites exhibits marked divergence from conventional perovskites, conferring unique stability advantages.

In 2019, Chen et al. reported a Ge–Sn hybrid perovskite solar cell achieving 7.11% PCE with exceptional operational stability.²⁹ This breakthrough demonstrated the viability of germanium substitution for partial tin replacement in enhancing perovskite optoelectronic performance. Luo et al. focused on the energy differences between centrosymmetric and non-centrosymmetric states in CsSnI₃ and CsGeI₃ systems. Through first-principles calculations, they systematically compared the structural energies of CsSn_xGe_1−xI₃ in R3m and P4/mbm phases across varying doping ratios (x). Their results revealed near-degenerate energy states between the two space groups at x = 0.5, leading to a structural phase transition. Furthermore, they proposed a stabilized distorted structural model for CsSn_0.5Ge_0.5I₃ based on computational evidence. In this configuration, both Sn²⁺ and Ge²⁺ deviate from octahedral centers, forming distorted octahedral units. Researchers concluded that octahedral distortion combined with non-centrosymmetric displacement plays a critical role in stabilizing the low-energy state of the structure.⁴⁴ In a corroborative study, Wu et al. calculated the structural energies of CsSn_0.5Ge_0.5I₃ across multiple space groups. As shown in Fig. 2(e), they discovered a novel P1-phase Sn–Ge distorted perovskite structure exhibiting superior stability compared to other space groups. The atomic arrangement of this P1-phase CsSn_0.5Ge_0.5I₃ is detailed in Fig. 2(f).⁴⁶

2.2 Tolerance factor and structural stability

The Goldschmidt tolerance factor (t_G) serves as a pivotal empirical indicator for assessing the structural stability of perovskite compounds. For perovskites with the general formula ABX₃, the t_G is defined by the following relationship:

	(1)

where r_A, r_B, and r_x denote the Shannon ionic radii⁴⁷ of the A-site cation, B-site cation, and anion X, respectively.⁴⁸ This parameter empirically quantifies the geometric compatibility between the A-site cation and the cavity formed by the BX₆ octahedral framework. A t_G value within 0.9 ≤ t_G ≤ 1 suggests optimal structural compatibility, favoring the formation of cubic perovskite phases. For 0.71 ≤ t_G ≤ 0.9, cooperative tilting of BX₆ octahedra induces structural distortions, leading to orthorhombic or rhombohedral structures. Values outside this range (t_G ≤ 0.71 or t_G ≥ 1) typically result in non-perovskite architectures.⁴⁹

We systematically reviewed computational studies on t_G values for all-inorganic germanium-based perovskites, providing insights into their structural stability. It should be emphatically noted that for germanium-based perovskites exhibiting significant octahedral distortions (e.g., asymmetric Ge–X bond stretching), the tolerance factor (t_G) remains a preliminary screening indicator for structural stability; yet, it must not be considered an exclusive criterion. Comprehensive stability assessment necessitates multidimensional evaluation incorporating formation energy, cohesive energy, and lattice dynamics parameters. As summarized in Fig. 3, 11 out of 15 germanium-based perovskites exhibit t_G values within the ideal cubic phase prediction range (0.9–1, shaded in pale yellow), highlighting their structural compatibility. Jong et al. conducted comprehensive theoretical investigations on AGeX₃ (A = Cs and Rb and X = I, Br, and Cl) systems using VASP. Their computational results revealed that all six compounds exhibit t_G within the range of 0.90 < t_G < 0.97, suggesting their propensity to crystallize into stable perovskite phases. Furthermore, employing the PBEsol functional, they systematically investigated the lattice constants and bond angles of CsGeX₃ (X = I, Br, and Cl). The calculations demonstrated a progressive reduction in lattice constants and a convergence of bond angles toward 90° as the ionic radius of the X-site halide anions decreases (from I to Cl). These observations indicate lattice contraction accompanied by a rhombohedral-to-cubic phase transition. This structural evolution aligns with the observed monotonic increase in t_G values from X = I to X = Cl, where t_G = 1 corresponds to an ideal cubic perovskite configuration. Researchers attributed this trend to the synergistic effects of the off-centering displacement of Ge atoms and the strengthening of Ge–X chemical bonds.⁵⁰ These findings corroborate the structural characteristics of germanium-based perovskites discussed in preceding sections. Similarly, Pak et al. performed DFT calculations on ABF₃ systems (A = Na and K and B = Si and Ge). Their results indicated that all examined structures except KSiF₃ possess tolerance factors consistent with stable BF₆ octahedra.⁵¹ Hasan et al.'s theoretical investigation on indium-based halide germanium perovskites revealed tolerance factors of 0.79 and 0.78 for InGeCl₃ and InGeBr₃, respectively. While these values fall outside the ideal tolerance factor range (0.9–1.0), they remain in close proximity to this stability window. The authors conclude that such near-optimal values suggest that these compounds retain some propensity for perovskite formation.⁵²

Beyond the conventional Goldschmidt tolerance factor, researchers have pursued alternative approaches to assess structural stability in all-inorganic germanium-based perovskites. Ruddlesden–Popper halide perovskites (RP-HPs), characterized by distinctive two-dimensional layered architectures, have garnered significant scientific interest. Pioneering work by Li et al. demonstrated that all-inorganic two dimensional Ruddlesden–Popper halide perovskites (2DRP-HPs) exhibit reduced binding energies, enhanced optical responses, and superior stability compared to hybrid counterparts.⁵³ To address lead toxicity in 2DRP-HPs, Haq et al. employed the full-potential linearized augmented plane wave (FPLAPW) method within the Wien2K code to investigate Cs₂BX₂Y₂ systems (B = Ge and Si and X/Y = Cl/Cl, Br/Cl, I/Cl, Br/Br, I/Br, and I/I), systematically analyzing their structural, electronic, and optical properties. These systems crystallize in tetragonal symmetry with the space group I4/mmm (No. 139). For enhanced stability prediction, the authors introduced a modified tolerance factor τ defined as follows:⁵⁴

	(2)

where r′ represents the averaged Shannon ionic radii of X and Y anions.⁵⁵ Detailed computations revealed τ > 1 for all investigated Cs₂BX₂Y₂ configurations. According to established stability criteria, these τ values confirm the structural integrity of these compounds.⁵⁶ Liu et al. calculated the formation energies (ΔH) of CsGeX₃ and Cs₂GeX₆ systems. In materials chemistry, ΔH serves as a critical stability indicator—negative values signify thermodynamic stability under ambient conditions. Their computations revealed negative ΔH values for CsGeBr₃ (−0.32 eV f.u.⁻¹), CsGeI₃ (−0.08 eV f.u.⁻¹), and Cs₂GeCl₆ (−0.18 eV f.u.⁻¹), confirming their ambient stability. In contrast, positive ΔH values for Cs₂GeBr₆ (+0.15 eV f.u.⁻¹) and Cs₂GeI₆ (+0.54 eV f.u.⁻¹) indicate thermodynamic instability.³² Bouhmaidi's team investigated CsGeX₃ stability through cohesive energy analysis. Cohesive energy quantifies the interatomic binding strength—higher values correlate with enhanced bond energies and structural robustness. Their calculations yielded cohesive energies of 2.96 eV per atom (CsGeBr₃), 2.68 eV per atom (CsGeCl₃), and 3.81 eV per atom (CsGeF₃). These results demonstrate strong intramolecular bonding in CsGeX₃ systems, with CsGeF₃ exhibiting superior structural stability.⁵⁷ Huang et al. elucidated the relationship between phonon characteristics and structural stability in CsGeX₃ (X = Cl, Br, and I) through first-principles calculations. This study innovatively proposed the “imaginary frequency coefficient” metric. Phonon spectrum analysis revealed that the R3m phase exhibits superior dynamic stability compared to the Pmm phase—its imaginary frequencies are localized near specific high-symmetry points in the acoustic branches, whereas those in the Pm3m phase are widely distributed across optical branches. Synergistic analysis with the tolerance factor (t_G) demonstrated that as halide ions transition from I to Cl, the t_G increases while the imaginary frequency coefficient decreases, attributable to strengthened Ge–X bonding driven by enhanced halogen electronegativity. Thermodynamic validation further showed that R3m-CsGeCl₃ possesses the widest chemical potential stability window. The negative formation energy and high cohesive energy establish a multidimensional verification framework, collectively confirming that the R3m phase—induced by Ge off-centering—represents a dual thermodynamic/kinetic stable configuration.⁵⁸

Moreover, in experimental research, the stability of materials is closely linked to the synthesis method employed. Chen et al. first synthesized CsGeX₃ QRs via a facile solvothermal method. Transmission electron microscopy (TEM) images and energy-dispersive X-ray spectroscopy (EDS) analysis confirmed the uniform morphology and good crystallinity of these QRs. They further compared the stability of CsGeI₃ and CH₃NH₃PbI₃ perovskite thin films; testing revealed that the CsGeI₃ film decomposed only about 45% after 48 hours in air, significantly superior to the CH₃NH₃PbI₃ film, which exhibited a decomposition rate as high as 98% within 24 hours.⁵⁹ Montiel et al. fabricated CsGeI₃ thin films using physical vapor deposition (PVD) technology, demonstrating enhanced thermal stability compared to MAPbI₃.²⁵ However, research efforts focused on the synthesis of all-inorganic germanium-based perovskites remain relatively scarce.

3. Optoelectronic properties of all-inorganic germanium-based perovskites

3.1 Shockley–Queisser limits and theoretical bandgap values

According to the Shockley–Queisser limit theory, the ideal bandgap value for single-junction solar cells is approximately 1.3–1.4 eV.⁶² Following this criterion, we compiled computational bandgap data for all-inorganic germanium-based perovskite analogs from multiple studies, as summarized in Fig. 4, with explicitly detailed numerical values and their primary sources are provided in Table S1. Considering the well-documented phenomenon that the inherent self-interaction error (SIE) in semi-local exchange–correlation functionals (e.g., LDA/GGA) systematically underestimates bandgap values,⁶³ we annotated the specific functional types used in each study during statistical analysis. The results reveal that, among the 15 surveyed ABX₃-type all-inorganic germanium-based perovskites, only approximately 6 materials exhibit bandgap values within the optimal range. In Fig. 4, the dark gray area (1.3–1.4 eV) represents the ideal bandgap range, while the light gray area (0.9–1.6 eV) corresponds to materials theoretically capable of achieving about 25% Shockley–Queisser efficiency.⁴⁹ Jong et al. investigated the electronic structures of AGeX₃ (A = Cs and Rb and X = I, Br, and Cl) using HSE06 hybrid functional-based DFT calculations. Their results demonstrated that CsGeI₃, with a bandgap of 1.64 eV, is more suitable as a light absorber, while RbGeI₃ (bandgap: 1.78 eV) is applicable as a top-cell material for tandem solar cells. Further studies revealed that substituting I atoms with Br or Cl significantly increases the bandgap to over 2 eV. Beyond bandgap engineering, carrier mobility and exciton behavior are equally critical for photovoltaic performance. Through band structure analysis, Jong et al. revealed low effective masses in CsGeI₃ (electron: 0.15m_e, hole: 0.16m_e), indicating superior charge transport capability. More notably, calculations based on the Wannier–Mott model yielded an exciton binding energy of only 19.24 meV, significantly lower than that of MAPbI₃ (about 50 meV), which is attributed to enhanced screening from GeX₆ octahedral distortion, thereby promoting room-temperature exciton dissociation.⁵⁰ In 2022, Hasan et al. calculated the bandgaps of FrBX₃ compounds using the PBE functional under the generalized gradient approximation (GGA). Their research focused on bandgap variations induced by substitutions at the B-site (cation) and X-site (anion). The results indicated that replacing Ge with Sn at the B-site reduces the bandgap, which they attributed to the increased lattice constants of FrSnX₃ compared to FrGeX₃. When modifying the halogen atoms at the X-site, researchers observed that for both FrGeX₃ and FrSnX₃, the material's bandgap decreases as the halogen atomic size increases.⁶⁴ These findings highlight the tunable bandgap of halogenated germanium-based perovskites, demonstrating their promising potential for photovoltaic applications. For two-dimensional Ruddlesden–Popper germanium-based perovskites, Haq et al. revealed exceptional carrier transport properties through first-principles calculations: Cs₂GeI₂Cl₂ exhibits remarkably low effective masses of 0.185m_e for electrons and 0.236m_e for holes, significantly lower than conventional silicon. More critically, the exciton binding energy calculated via the Wannier–Mott model (113.44 meV) is substantially reduced compared to its three-dimensional counterparts. This reduction is attributed to enhanced quantum confinement effects and dielectric screening in the 2D structure, thereby facilitating more efficient room-temperature exciton dissociation and charge extraction.⁵⁶

In theoretical investigations of material bandgaps, the substantial computational discrepancies arising from functional selection merit particular attention. The systematic bandgap underestimation inherent to standard DFT methodologies necessitates the careful selection of computational approaches. Jong et al. employed four computational schemes combining PBEsol/HSE06 functionals with/without spin–orbit coupling (SOC) effects to evaluate CsGeX₃ (X = I, Br, and Cl) bandgaps. Comparative analysis with experimental data revealed stark contrasts: while PBE/PBEsol accurately reproduce bandgaps in organic systems like MAPbI₃,^65,66 these functionals systematically underestimate values in all-inorganic germanium-based perovskites. SOC inclusion exacerbates this underestimation, whereas HSE06 hybrid functionals achieve remarkable agreement with experimental measurements. Band structure analysis attributes this accuracy to the HSE06's proper treatment of conduction band uplifting and valence band depression.⁵⁰ Liu et al.'s computational results revealed pronounced bandgap underestimation in CsGeX₃ systems using the PBE functional, particularly exacerbated in CsGeCl₃ and CsGeBr₃ configurations. HSE06 functional calculations yielded bandgap values closely aligned with experimental reports for CsGeBr₃ and CsGeI₃,⁴⁰ while persistent underestimation remained in CsGeCl₃.³² In 2023, given the systematic PBE-based bandgap underestimation in all-inorganic halide perovskites, Pak's team adopted the HSE06 hybrid functional (without SOC) for ABF₃ (A = Na and K; B = Si and Ge) systems to achieve enhanced computational accuracy. Parallel PBE functional calculations were systematically conducted for comparative methodology evaluation. Their results demonstrated systematically larger bandgap values obtained from HSE06 compared to PBE in fluoride perovskite systems. This discrepancy originates from HSE06-induced conduction band uplifting and valence band depression, collectively enlarging the bandgap.⁵¹ Haq et al. further demonstrated the superior accuracy of GGA-mBJ potential over conventional GGA in bandgap calculations for all-inorganic 2DRP-HPs. Standard GGA approaches exhibited substantial bandgap underestimation. Remarkably, GGA-mBJ-derived bandgaps showed consistency with HSE hybrid functional results regardless of SOC inclusion.⁵⁶ Beyond DFT calculations, the GW approximation has been demonstrated to accurately describe the electronic structures of materials. It provides high-precision predictions for semiconductor bandgaps and other electronic properties. Numerous research groups have applied this method to predict the electronic structures of perovskites.^37,38,67 However, its computational cost remains prohibitively high.³⁹ Consequently, it is primarily recommended for theoretical investigations where precision is paramount.

3.2 The optoelectronic properties of all-inorganic germanium-based perovskites modulated by Ge incorporation

The optical and electronic characteristics of materials encode critical information about light–matter interactions, serving as pivotal metrics for evaluating photovoltaic applicability and elucidating their energy conversion potential. As evidenced by preceding structural analyses, germanium-based perovskites exhibit distinct atomic configurations. This section therefore concentrates on theoretical insights into germanium's unique optoelectronic modulation mechanisms.

Hasan et al. computationally compared the wavelength-dependent absorption spectra of FrBX₃ (B = Ge and Sn and X = Cl, Br, and I) series, with key findings visualized in Fig. 5(a) and (b). Within the solar spectrum (380–750 nm wavelength range, accounting for 43% of total energy), comparative analysis of absorption spectra revealed that substituting germanium with tin at the B-site induces significant attenuation of absorption peaks. This conclusively demonstrates the superior visible-light energy conversion capability of germanium-based FrGeX₃ perovskites over their tin-based FrSnX₃ counterparts.⁶⁴ Wu et al. computationally mapped the bandgap evolution of CsGe_1−xSn_xI₃ across varying doping ratios (x) under different space groups, as shown in Fig. 5(c). Remarkably, the R3m phase exhibits the widest bandgap tunability (1.02–0.49 eV). Density of states (DOS) analysis reveals that the higher energy level of Sn-5s orbitals compared to that of Ge-4s orbitals drives progressive bandgap reduction with increasing x (i.e., decreasing Ge content).⁴⁶ Employing PBE functional calculations, Pak et al. evaluated carrier effective masses in ABF₃ (A = Na and K and B = Si and Ge). Notably, germanium-based AGeF₃ compounds demonstrate higher electron and hole effective masses than silicon-based ASiF₃ analogues. While this suggests superior light absorption in silicon-based systems, the sufficiently low effective masses in AGeF₃ remain viable for charge transport applications. Calculations of the optical absorption coefficients further indicate that silicon-based ASiF₃ (A = Na and K) perovskites have slightly lower photon energy absorption onsets than germanium-based counterparts. However, both materials exhibit peak absorption coefficients of 11–18 × 10⁴ cm⁻¹ (Fig. 5(d)), demonstrating adequate light-harvesting capabilities. These characteristics are advantageous for improving solar cell efficiency.⁵¹ Experimentally, Yang et al. developed Ge–Pb perovskite luminescence films by partially substituting lead (Pb) with germanium (Ge). Their research revealed that at Ge molar fractions of 10–20 mol%, the photoluminescence quantum efficiency (PLQE) of the films significantly increased from 53% for Ge-free reference samples to approximately 71%. Further transient photoluminescence (PL) decay kinetics analysis indicated a distinct concentration threshold effect associated with Ge incorporation: within the optimal Ge content range (10–20 mol%), radiative recombination predominates, effectively suppressing non-radiative losses and thereby achieving high PLQE; when the Ge molar fraction exceeds 30 mol%, trap-assisted recombination becomes dominant, resulting in a sharp decline of PLQE below 40%.⁷⁰

3.3 Exploration of the optoelectronic properties of all-inorganic germanium-based perovskites

All-inorganic germanium-based perovskites demonstrate remarkable structural stability, providing a solid foundation for their applications in materials science. However, significant room for improvement remains in their optoelectronic performance. For instance, in the ABF₃ system (A = Na and K and B = Si and Ge), germanium-based fluorinated perovskites exhibit suitable optoelectronic properties; yet, their performance still lags behind silicon-based counterparts within the same system.⁵¹ In the AGeX₃ system (A = Cs and Rb and X = I, Br, and Cl), the theoretical bandgap values significantly deviate from the ideal range.⁵⁰ Given these challenges, researchers worldwide have conducted extensive studies to improve their optoelectronic properties.

Liu et al. employed the HSE06 hybrid functional to calculate the bandgap of CsGeI₃ under varying strains. The results revealed that the bandgap could be tuned from 0.73 eV to 2.30 eV within a strain range of −4% to 4%.⁶⁰ This aligns with the bandgap variation trend observed by Xu et al. under −2% to 2% strain. Notably, their comparative analysis revealed that CsPbI₃ exhibits bandgap widening (from 1.53 eV to 1.64 eV) under compressive strain (−2% to 0%), whereas it shows bandgap narrowing (from 1.64 eV to 1.61 eV) under tensile strain (0% to 2%). In striking contrast, the CsGeI₃ system demonstrates the monotonic bandgap increase across the full −4% to 4% strain range, signifying superior bandgap tunability.⁷¹ Furthermore, Liu et al. discovered that, under 0.52 GPa pressure, CsGeI₃ exhibits a bandgap of 1.36 eV, falling within the 1.3–1.4 eV range. This meets the Shockley–Queisser Limit criteria, indicating optimal photovoltaic performance. Using the PBE functional with a scissor operator, they also calculated the optical absorption spectra of CsGeI₃ under various strains, as shown in Fig. 6(a). The figure clearly shows that compressive strain induces a significant red shift of the absorption edge, while tensile strain leads to a blue shift. Moreover, strained structures exhibit enhanced absorption intensity in the visible light region.⁶⁰ Furthermore, strain engineering has also yielded significant successes in the experimental modulation of germanium-based perovskites. Yang et al. developed low-toxicity perovskite light-emitting diodes (PeLEDs) by partially substituting Pb with Ge. However, they observed that at Ge contents exceeding 30 mol%, intense spontaneous strain (particularly shear strain) exacerbated non-radiative recombination, leading to a drastic deterioration in device performance.⁷⁰ To overcome this limitation, Zhou et al. proposed an A-site strain engineering strategy, which involved substituting small Cs⁺ ions with larger FA⁺ (formamidinium) and MA⁺ (methylammonium) ions. This approach effectively suppressed octahedral tilting and reduced lattice distortion. Experimental results confirmed that the triple-cation structure [FA_0.25(Cs_0.95MA_0.05)_0.75]Ge_xPb_1−xBr₃, with 30 mol% Ge content, achieved a PLQE of 48% and a maximum external quantum efficiency (EQE) of 8.5%. The underlying mechanism was attributed to the reduction of shear strain components e₄ to <1% and e_tx to −0.1%.⁷² These findings demonstrate that strain engineering offers a viable strategy for optimizing the optoelectronic properties of all-inorganic germanium-based perovskites. Previous studies have demonstrated that SnI, as the primary defect in CsSnI₃, generates a deep in-gap level that severely limits the PCE.⁷³ Zhou et al. computationally revealed that Ge_I defects in CsGe_1−xSn_xI₃ alloys exhibit comparable defect formation energies to Sn_I. Their investigation of both defects in CsGe_1−xSn_xI₃ showed that neither generates deep levels at x = 0.5 (CsGe_0.5Sn_0.5I₃), whereas at x = 0.25 or 0.75, at least one defect introduces the deep level. As shown in the defect energy level diagram, Ge_I and Sn_I induce two distinct defect states: one singly occupied (occ1) near the CBM, and another doubly occupied (occ2) adjacent to the VBM. Band structure analysis of CsGe_xSn_1−xI₃ with varying x (Fig. 6(b)) reveals that Sn alloying passivates the occ2 defect state by elevating the VBM, thereby pushing it out of the bandgap. Additionally, researchers specifically investigated the evolution of occ1 with x values. As shown in Fig. 6(e), the occ1 defect state induced by Ge_I in CsGe_0.5Sn_0.5I₃ exhibits delocalized and shallow characteristics. Orbital hybridization analysis (Fig. 6(d)) reveals that the average bond length between Ge_D and neighboring Ge_NN/Sn_NN atoms increases at x = 0.5 compared to x = 0.25/0.75, indicating weakened orbital hybridization. Consequently, the occ1 defect state induced by Ge_I becomes energetically elevated and shallow in CsGe_0.5Sn_0.5I₃. These findings collectively suggest that CsGe_0.5Sn_0.5I₃ with benign defects (Sn_I and Ge_I) represents a promising candidate for high-performance photovoltaic applications.⁷⁴ Wu et al. systematically investigated defect characteristics in the CsSn_0.5Ge_0.5I₃ alloy system. Their calculations on all intrinsic defects in P1-phase CsSn_0.5Ge_0.5I₃ revealed that only eight point defects (I_Cs, I_Sn, I_Ge, Cs_I, Sn_I, Ge_I, Sn_i, and Ge_i) induce deep transition levels, while others exhibit shallow-level behavior. Using the chemical potential diagrams in Fig. 6(c) and (f), they selected two representative growth conditions: point A (I-poor/Cs-rich) and point G (I-rich/Cs-poor), calculating formation energies for all point defects under these conditions. The results demonstrate that, under I-poor conditions, iodine vacancies (V_I) exhibit the lowest formation energy across the Fermi level range, acting as dominant defects. The Fermi level becomes pinned by V_I and V_Ge, enabling tunable conductivity from p-type to n-type. Under I-rich conditions, the formation energies of the dominant acceptor V_Cs and donor V_I intersect at p-type Fermi levels. Compensation effects between V²⁺_Cs and V⁻_I charge defects pin the Fermi level, resulting in compensated p-type conductivity. Based on these computational and thermodynamic analyses, the research team proposed that CsSn_0.5Ge_0.5I₃-P1 thin films should be synthesized under iodine-rich conditions to achieve optimal p-type conductivity and maximize photovoltaic performance, providing critical guidance for subsequent experimental fabrication.⁴⁶ Tian et al. theoretically determined that Ce and Eu doping can substitute Ge to form stable CsGe_1−xLn_xBr₃ structures using the plane-wave ultrasoft pseudopotential (PWUS) method with GGA and PW91 exchange–correlation functional. Furthermore, the bond lengths of Ge/Ln–Br and Cs–Br exhibit variations depending on the type and concentration of Ln, providing theoretical guidance for subsequent experimental synthesis of CsGeBr₃:Ln³⁺ with enhanced luminescence properties. In addition, the research team conducted high throughput computing on the material systems. As revealed in Fig. 6(g)–(j), with increasing Eu³⁺ concentration, the luminous efficacy of radiation (LER) decreases, whereas both SLER and correlated color temperature (CCT) exhibit significant enhancement. Most importantly, through the combination of CsGeBr₃:Ce and Eu³⁺, it is possible to obtain high-quality white light with a color rendering index (R_a) as high as 94 and a color temperature of 5608 K, and this result clarifies the performance targets for experimental construction of white light-emitting diode (WLED) devices.³⁴ In current photovoltaics, a significant gap remains between the practical device performance of perovskite solar cells and their theoretical limits. Numerical simulation software serves as a powerful theoretical tool for efficient material screening, device structure optimization, and performance limit prediction, providing critical guidance for experimental research and development. Saikia et al. simulated CsGeI₃ solar cells with a conventional n–i–p structure (FTO/TiO₂/CsGeI₃/CuI/Ag) using SCAPS-1D software. Through meticulous optimization of charge transport layer materials and thicknesses (TiO₂: 10 nm and CuI: 30 nm), perovskite layer thickness (800 nm), doping concentrations (CsGeI₃ acceptor doping: 10¹⁶ cm⁻³; ETL/HTL donor doping: 10¹⁹ cm⁻³), and defect densities (bulk/interface defects: 10¹⁵ cm⁻³), they achieved a simulated efficiency of 10.8%. This validates the feasibility of CsGeI₃ as a light-absorbing layer and defines critical parameter ranges.⁷⁵ Subsequently, to further break through efficiency bottlenecks and reduce costs, Zhang et al. innovatively proposed and simulated an HTL-free all-inorganic CsGeI₃ cell structure (FTO/ETL/CsGeI₃/metal back electrode). They discovered that when ZnOS is employed as the ETL, its unique conduction band depression structure effectively enhances electron extraction and suppresses interface recombination; simultaneously, selecting high-work-function tungsten (W) as the back electrode creates a back-contact barrier of about 0.3 eV at the CsGeI₃/W interface, which effectively blocks electron backflow and promotes hole collection. This approach eliminates the need for expensive and unstable organic HTLs, establishing a new paradigm for developing low-cost, high-efficiency lead-free germanium-based perovskite solar cells.⁷⁶ This demonstrates that, guided by software simulations, advanced device structure design and precise interface/bulk-phase modulation offer substantial room for enhancing the performance of all-inorganic germanium-based perovskite solar cells. Most recently, Das et al. employed SCAPS-1D simulations with CsGeI₃ as the absorber layer, predicting a promising PCE of 21.6%.⁷⁷ In a parallel advancement, Dash's team innovatively utilized RbGeI₃ as the light-harvesting material in all-inorganic perovskite solar cells, achieving an optimized PCE of 25.76% with a fill factor (FF) of 79.81% through systematic device parameter refinement.⁷⁸ Besides DFT calculations and software simulations, machine learning (ML), as a highly efficient data mining tool, has demonstrated considerable potential in the field of materials science in recent years. The traditional “trial-and-error” approach is not only highly resource-intensive but also struggles to extract key structure–property relationships from complex high-dimensional parameter spaces. In contrast, ML, by analyzing historical experimental data to build predictive models, can transcend the limitations of empirical knowledge and significantly accelerate the processes of novel material discovery and device optimization.⁷⁹ Currently, ML is gaining widespread application in perovskite research, utilizing real experimental data to train ML models to guide the design of perovskite materials and the development of high-performance solar cells.^80–88 In 2024, Gao et al. pioneered the construction of a machine learning model based on an artificial neural network (ANN) algorithm, predicting a wide bandgap (∼1.6 eV) tin-based perovskite material; combined with the optimized interfacial energy level alignment (near-zero energy level mismatch), this could enable a PCE exceeding 20%.⁸⁹ Similarly, Kim et al. proposed a computational framework integrating DFT with ML; they employed a crystal graph convolutional neural network (CGCNN) to conduct high-throughput screening and computational analysis of 41,400 candidate compounds, identifying CsGe_0.3125Sn_0.6875I₃ as the optimal absorber layer material for single-junction solar cells.⁹⁰ Although ML research specifically targeting all-inorganic germanium-based perovskites remains in its infancy, it has already exhibited substantial application potential in predicting the properties and designing perovskite materials.

4. Additional characteristics

In the exploration of all-inorganic germanium-based perovskites, we have thoroughly elucidated their structural and optoelectronic characteristics, which establish a seminal foundation for understanding their application potential in optoelectronic devices. However, given the exceptional semiconductor properties of perovskite materials, the investigation of their multifaceted material properties is crucial for pushing the boundaries of practical applications. Specifically, for all-inorganic germanium-based perovskites, research focus has extended to their distinctive properties in mechanical, ferroelectric, and thermoelectric domains.

4.1 Mechanical properties

Mechanical stability serves as a critical metric for evaluating mechanical properties. Enhanced mechanical stability not only ensures structural integrity under stress but also correlates with material reliability and operational lifespan and the fabrication of homogeneous thin films. Elastic constants (C_ij) provide quantitative descriptors for mechanical stability in crystalline materials. These tensor components govern the crystal's response to external stress fields, with their precise measurement delivering crucial insights into material stiffness and structural stability.⁶¹ For cubic crystals, three independent elastic constants (C₁₁, C₁₂, and C₄₄) are typically employed to comprehensively characterize their stiffness properties. Herein, C₁₁ quantifies the material's stiffness under deformation along the crystallographic plane normal direction. C₁₂ represents the shear deformation resistance perpendicular to the principal crystallographic planes. And, C₄₄ denotes the shear rigidity parallel to the primary crystallographic planes.⁵² The Born stability criteria,⁹¹ a fundamental principle for evaluating mechanical stability, are defined as follows:

C11 > 0, C44 > 0, C11 − C12 > 0, C11 + 2C12 > 0	(3)

A material is deemed mechanically stable when its elastic constants satisfy these criteria. From these three fundamental elastic constants, critical mechanical parameters can be derived, including the bulk modulus (B), shear modulus (G), and Young's modulus (E), all measured in GPa. The mathematical formulations are expressed as follows:

	(4)

The scientific community has established robust theoretical frameworks for distinguishing between brittle and ductile material behaviors. In 1954, Pugh proposed the ratio of the bulk modulus to the shear modulus (B/G) as a critical indicator for materials characterization. Materials demonstrating a Pugh ratio (B/G) exceeding 1.75 exhibit enhanced ductility, whereas ratios below this threshold signify brittle characteristics.⁹² Moreover, Frantsevich established Poisson's ratio (v) as an additional discriminative parameter. His research identified 0.26 as the critical Poisson's ratio: materials with v < 0.26 manifest brittleness, while those with v > 0.26 display ductile behavior.⁹³ v is mathematically defined as follows:

	(5)

Furthermore, the Cauchy pressure (C₁₂ − C₄₄) has emerged as a third critical parameter in this analytical framework. Empirical evidence confirms that positive Cauchy pressure values correspond to ductile behavior, whereas negative values reliably predict brittle characteristics.⁶⁴

Based on the aforementioned evaluation criteria, we extensively collected and systematically analyzed numerous academic papers on the elastic constant calculations of all-inorganic germanium-based perovskites, with the statistical results summarized in Table S2. Research by Hasan et al. demonstrates that FrGeI₃ exhibits brittle characteristics, whereas FrGeBr₃ resides in the critical state between brittleness and ductility.⁶⁴ Among other investigated all-inorganic germanium-based perovskite materials, InGeCl₃, InGeBr₃,⁵² GaGeF₃, InGeF₃,⁶¹ KGeF₃, NaGeF₃,⁵¹ and Li₂GeZ₆ (Z = Cl, Br, and I)⁹⁴ all demonstrate both mechanical stability and ductility. Evidently, the majority of studied all-inorganic germanium-based perovskites exhibit outstanding mechanical properties.

4.2 Pyroelectric performance

The pyroelectric phenomenon refers to the process where a change in external temperature induces alterations in the internal polarization state of a crystal, thereby generating macroscopic bound charges on the crystal surface to achieve thermal-to-electrical energy conversion. Pyroelectric materials play a pivotal role in promoting green energy development, consequently receiving extensive and in-depth research over the past decades, with current widespread application across multiple domains.⁹⁵ The pyroelectric performance is quantified by the figure of merit ZT:

	(6)

This figure of merit reflects the material's electronic transport characteristics. In practical applications, achieving high-efficiency thermoelectric conversion necessitates materials with high Seebeck coefficient (S), high electrical conductivity (σ), and low thermal conductivity (K_e). Recent studies have demonstrated exceptionally high ZT values in selenides (e.g., SnSe, PbSe, and CuSe₂)^96–102 and tellurides (e.g., GeTe, SnTe, and PbTe).^103–107 However, the scarcity of Se and Te elements renders these materials cost-prohibitive, thereby constraining their large-scale applications to some extent. Emerging research indicates that halide perovskites exhibit ultralow thermal conductivity^108–110 and high charge carrier mobility.^111,112 Concurrently, the relatively low cost of halide perovskite materials endows them with significant potential for thermoelectric conversion applications. Regarding all-inorganic germanium-based perovskites, they have already demonstrated promising prospects in photovoltaic applications. Building upon this foundation, numerous researchers have redirected their focus to investigating the thermoelectric properties of these materials.

Using the single-mode relaxation time approximation, Jong et al. solved the Boltzmann transport equation to calculate the constant-volume heat capacity, relaxation time, and group velocity, subsequently determining the temperature-dependent lattice thermal conductivity of CsGeX₃ (X = I, Br, and Cl) through relevant theoretical formulations. As shown in Fig. 7(a), the thermal conductivity exhibits a gradual decrease with increasing temperature, while at fixed temperatures, it sequentially increases from X = I to Br to Cl. At 300 K, the thermal conductivity values of CsGeI₃, CsGeBr₃, and CsGeCl₃ are 0.10, 0.16, and 0.18 W m⁻¹ K⁻¹, respectively, which are significantly lower than those of conventional thermoelectric materials such as GeTe¹⁰³ and PbTe.¹¹³ Simultaneously, employing deformation potential theory (DPT), Jong et al. estimated charge carrier mobilities of 1,677, 1,401, and 863 cm² V⁻¹ s⁻¹ for CsGeI₃, CsGeBr₃, and CsGeCl₃, respectively, comparable to state-of-the-art thermoelectric materials. Remarkably, these values significantly surpass those of conventional perovskite counterparts, including FASnI₃ (1.6 cm² V⁻¹ s⁻¹),¹¹⁴ MASnI₃ (1.6 cm² V⁻¹ s⁻¹),¹¹⁵ MAPbBr₃ (24 cm² V⁻¹ s⁻¹), and MAPbI₃ (67.2 cm² V⁻¹ s⁻¹).¹¹⁶ The combined characteristics of ultralow thermal conductivity and high carrier mobility suggest promising potential for CsGeX₃ in high-performance pyroelectric applications.¹¹⁷ Bouhmaidi et al. conducted an in-depth investigation on the thermoelectric performance of the CsGeX₃ system by calculating the figure of merit ZT using the Quantum Espresso software package based on DFT. As shown in Fig. 7(b), the ZT value of CsGeCl₃ remains nearly zero at low temperatures, gradually increases with increasing temperature, and stabilizes at 0.64 above 800 K. In contrast, the ZT value of CsGeBr₃ starts at 0.99 under low-temperature conditions (T = 180 K), progressively decreases with increasing temperature, and drops to 0.1 at high temperatures (>900 K), indicating that its superior thermoelectric performance is limited to the low-temperature regime. Notably, the ZT value of CsGeF₃ remains consistently above 0.9 throughout the investigated temperature range, demonstrating minimal temperature dependence. Collectively, these three materials exhibit significant potential for pyroelectric applications.⁵⁷ Alburaih et al. investigated the thermoelectric properties of the double perovskite Li₂GeZ₆ (Z = Cl, Br, and I) through first-principles calculations based on DFT, combined with the semi-classical Boltzmann equation. Taking Li₂GeCl₆ as an example, the results shown in Fig. 7(c) and (d) reveal that its power factor (S²σ) increases with temperature, reaching a ZT value of 0.74 at 300 K, with an overall minimal variation in ZT across the temperature range. The same trend is observed for Li₂GeBr₆ and Li₂GeI₆, with ZT values approaching unity, suggesting the promising potential of this material series for pyroelectric device applications.⁹⁴

4.3 Ferroelectric properties

Ferroelectrics exhibit spontaneous polarization within their structure, and the direction of this polarization can be reversed under an applied electric field. At the microscopic level, this property originates from the unique ionic arrangement in ferroelectric crystals. In the absence of an external field, relative displacements of ions in the crystal lattice generate intrinsic electric dipole moments, thereby forming spontaneous polarization. The switchable polarization orientation enables the widespread application of ferroelectric materials in switching devices, sensors, and related fields.^118,119 However, conventional ferroelectrics typically exhibit wide band gaps. Consequently, exploring ferroelectric semiconductors with suitable band gaps offers a novel approach for designing optoelectronic switches, ferroelectric photovoltaic devices, and other applications beyond traditional ferroelectric material boundaries. The key requirements for realizing such applications lie in their pronounced polarization capability and efficient light absorption. Previous studies have reported ferroelectric responses in certain organic–inorganic halide perovskites.^120,121 Nevertheless, given the inherent instability and volatility of organic components, researchers have shifted the focus toward investigating the ferroelectricity of all-inorganic perovskites to develop more stable materials.

In all-inorganic systems, germanium-based perovskites have emerged as a key focus for ferroelectricity research due to the off-centering displacement of B-site ions. Using first-principles calculations with the Berry phase approach, Zhang et al. performed interpolation between the undistorted cubic structure and the polar rhombohedral structure to determine the spontaneous polarization (P_s) of CsGeBr₃. The computational results are presented in Fig. 8(a). The results reveal a spontaneous polarization value of 19.7 μC cm⁻² along the 〈111〉 direction in the distorted phase at room temperature. This value exceeds the estimated P_s (12.2 μC cm⁻²) derived from multiplying the Born effective charges (BECs) of the displaced ions by their displacements in the polar structure, indicating pronounced charge transfer between cations and anions – a hallmark of ferroelectric polarization. To elucidate the origin of ferroelectricity, the real-space charge density distribution in the polar R3m structure was calculated, revealing non-centrosymmetric charge density along the 〈111〉 direction that correlates with Ge-ion off-centering.¹²² This theoretical finding provides critical insights into the microscopic mechanism underlying ferroelectric behavior in CsGeBr₃. Through DFT calculations, Chen et al. also theoretically investigated the ferroelectricity of CsGeI₃. The calculations revealed a substantial ferroelectric polarization of about 15 μC cm⁻² and an energy barrier of 0.067 eV (Fig. 8(b)), which are consistent with their experimental measurements. Regarding the piezoelectric response, the calculated piezoelectric modulus d₃₃ of CsGeI₃ reached 34.95 pC N⁻¹ (Fig. 8(c)), comparable to conventional ferroelectrics. Notably, under 5% tensile strain, the d₃₃ value was significantly enhanced to 190 pC N⁻¹. Furthermore, as shown in Fig. 8(d), the applied strain notably increased both the polarization magnitude and the switching energy barrier, indicating effective stabilization of ferroelectricity.¹²³ This work demonstrates that strain engineering serves as a promising strategy for tailoring the ferroelectric properties of all-inorganic germanium-based perovskites. Recently, Kashikar et al. developed an effective Hamiltonian parameterization strategy based on hybrid exchange–correlation functionals. This successfully addresses the underestimation of Curie temperature (T_C) in halide perovskite ferroelectrics CsGeX₃ (X = Cl, Br, and I) by conventional functionals. Theoretically, the experimental T_C values of these materials are reproduced with high accuracy. Through molecular dynamics simulations, the finite-temperature properties of these materials are first predicted. These properties are confirmed to rival those of oxide ferroelectrics.¹²⁴

5. Conclusions and perspectives

All-inorganic germanium-based perovskites have emerged as a prominent research focus in addressing the inherent instability of organic perovskites and toxicity concerns of lead-based counterparts. Theoretical computation has played a pivotal role in material exploration, enabling the effective prediction of material properties through advanced computational methodologies. This review provides an in-depth and comprehensive analysis of theoretical investigations encompassing the structural characteristics, optoelectronic properties, mechanical stability, thermoelectric performance, and ferroelectric behavior of all-inorganic germanium-based perovskites. Current computational achievements demonstrate remarkable progress across multiple dimensions of germanium-based perovskite research. For instance, the central displacement of germanium ions exhibits a strong correlation with the exceptional structural stability of these materials. Many all-inorganic germanium-based perovskite materials exhibit excellent mechanical stability and favorable ductility; their thermoelectric and ferroelectric performances are comparable to those of conventional thermoelectric and ferroelectric materials, indicating potential application prospects. However, there remains a noticeable gap in photoelectric conversion efficiency compared to mainstream perovskite materials or other solar cell materials, suggesting substantial room for improvement. Furthermore, the selection of computational methods is of critical importance. Particularly, for calculating electronic characteristics such as bandgaps, it is essential to carefully choose computational methods that are better suited for the studied system and cross-validate with experimental results to obtain more accurate theoretical data. To promote further development of all-inorganic germanium-based perovskite materials, the following research directions warrant in-depth exploration:

(1) Exploration of novel compounds. The AGeX₃ perovskite structure exhibits high flexibility in elemental substitution, where A-site elements can be replaced with In, K, Na, Ga, etc., and X-site elements can be varied among different halogens. Furthermore, investigating special-structured perovskites such as 2D perovskites and double perovskites may lead to the discovery of all-inorganic germanium-based perovskites with superior performance.

(2) Composition engineering: Through theoretical studies on perovskite doping, exploring the variation patterns of material properties under different doping ratios will facilitate the screening of high-performance all-inorganic germanium-based perovskites, thereby achieving precise performance modulation.

(3) Defect investigation: Systematic studies on the defect characteristics of all-inorganic germanium-based perovskites should be conducted to identify materials with benign defects. Concurrent exploration of optimal environmental conditions for material synthesis will provide theoretical guidance for fabrication processes.

(4) Expansion of application fields: Continuous exploration of the latent properties of all-inorganic germanium-based perovskites should be pursued, with active investigation into their potential applications in ferroelectrics, thermoelectrics, and other emerging domains. This will broaden application boundaries and enhance the comprehensive value of these materials.

(5) Synergistic integration of multiple theoretical methodologies with experimental investigations. Beyond DFT calculations, advanced theoretical approaches such as software simulations and machine learning should be integrated to enable multifaceted screening and optimization of materials and devices, while GW approximation should be employed for the high-precision analysis of electronic properties. Concurrently, bridging theoretical predictions with experimental validation, particularly in domains like doping and strain engineering where experiments should be guided by theoretical frameworks will significantly enhance research accuracy and efficiency.

Conflicts of interest

There are no conflicts to declare.

Data availability

No primary research results, software or code have been included, and no new data were generated or analyzed as part of this review.

Supplementary information available: Table S1 lists bandgap values of all-inorganic Ge-based perovskites in Fig. 4 across computational methods. Table S2 provides elastic constants, critical mechanical parameters, Pugh ratio and Poisson's ratio for these compositions. See DOI: https://doi.org/10.1039/d5tc01917f

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant no. 61674070, 62174072 and 21973034), the Guangdong Basic and Applied Basic Research Foundation (grant no. 2019B151502049), and The Uniqueness and Innovation Projects for the Universities in Guangdong Province (grant no. 2022KTSCX010). Computer time at the National Supercomputer Center in Guangzhou (NSCCGZ) and the High-Performance Computing Platform of Jinan University are gratefully acknowledged. T.S. also acknowledges the Guangzhou Key Laboratory of Vacuum Coating Technologies and New Energy Materials (no. 201605030008) and the National Innovation and Entrepreneurship Training Program for Undergraduate (no. 202410559008).

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