First-Principles Investigation of the Structural Stability and Physical Properties of Lead-Free Ge-Based Halide Perovskites

Abstract

Inorganic lead halide perovskite semiconductor materials exhibit great potential in the optoelectronic field due to their excellent optical and electrical properties. However, lead toxicity and limited material stability hinder their commercial applications. Consequently, the pursuit of non-toxic, stable alternatives is imperative for the sustainable development of halide-perovskite semiconductors. Non-toxic germanium-based halide perovskites, as promising candidates, have attracted considerable attention. Here, we present a systematic first-principles investigation of the structural, electronic, elastic, and optical properties of cost-effective germanium-based halide perovskites NaGeX₃ (X = Cl, Br, I). Energy and phonon-spectrum calculations demonstrate that NaGeX₃ with the R3c space group exhibits the highest structural stability, rather than the commonly assumed cubic phase. Hybrid functional calculations reveal that the band gaps of R3c NaGeX₃ decrease monotonically with increasing halogen radius, that is, 4.75 eV (NaGeCl₃) → 3.76 eV (NaGeBr₃) → 2.69 eV (NaGeI₃), accompanied by a reduction in carrier effective masses. Additionally, mechanically stable R3c NaGeX₃ exhibits lower hardness and ductility than that of the cubic phase. Optical properties indicate that NaGeX₃ materials have strong absorption coefficients (>10⁶ cm⁻¹) and low loss in the photon energy range of 9–11 eV, suggesting that such cost-effective germanium-based halide perovskites can be used in various optoelectronic devices in the ultraviolet region.

1. Introduction

In recent years, lead halide perovskite semiconductor material APbX₃ (A is a univalent organic cation or alkali metal ion, X is a halogen ion such as Cl, Br, I) has attracted extensive attention due to its unique properties. APbX₃ presents excellent physical properties such as adjustable band gap [1], high defect tolerance [2], high absorption coefficient [3], carrier diffusion length [4] and long life [5], making it widely used in fields such as LEDs (LED) [6], photonic crystal lasers [7], photodetectors [8] and solar cell devices [9]. APbX₃ materials mainly consist of organic–inorganic and all-inorganic components according to the different A-site ions. For organic–inorganic mixtures such as MAPbX₃ (MA = CH₃NH₃⁺), although the combination of the different properties of organic and inorganic components makes it easier to achieve wideband gap adjustment [10], the poor thermal stability of the organic cation (which decomposes easily in air and ultraviolet light) limits its large-scale application [11]. As a result, researchers began replacing organic cations with the inorganic metal ion Cs⁺ to construct all-inorganic lead perovskite materials. It was found that all-inorganic perovskite materials exhibit higher stability than organic–inorganic hybrid lead perovskite materials [12]. As solar cell materials, their photoelectric conversion efficiency exceeds 22% [13]. Therefore, all inorganic materials have attracted widespread attention.

Although all-inorganic lead halide perovskites have made considerable progress over the past decade and exhibit a high conversion efficiency, they are harmful to the environment due to the toxic effects of Pb. Therefore, finding perovskite materials that can substitute for Pb while maintaining a comparable carrier mobility and light absorption coefficient is crucial for the application of halide perovskites. To address this issue, researchers have attempted to regulate the composition of APbX₃ with non-toxic or low-toxic metal elements (e.g, Ti, Sn, Sb, Ge, Bi) [14,15]. Among these elements, Ge (in the same group as Pb) is considered a promising candidate. On the one hand, Ge exhibits similar electronic properties to Pb, thus showing comparable optoelectronic characteristics. In contrast, Sn, another element from the same group, is harmful to the human body [16]. Moreover, ASnX₃ is prone to oxidation from Sn²⁺ to Sn⁴⁺ when exposed to air, thereby reducing the device’s photovoltaic efficiency [17]. It is noteworthy that the toxicity of Ge to the human body is lower than that of Sn and Pb. On the other hand, compared to Pb-based materials, germanium halide perovskite also has a better band gap value for solar cells [18,19,20,21], higher light absorption and higher light conductivity [22], showing greater potential in the photovoltaic field. For example, Chen [23] reported that CsGeX₃ with a uniform quantum rod (QR) structure achieves a power conversion efficiency of 4.92%, thereby enhancing its light absorption and charge transport capabilities.

In addition, CsGeX₃ can maintain the crystal structure and performance stability in high-temperature environments, is not prone to decomposition or phase transitions and exhibits good resistance to environmental factors such as oxygen and moisture in the air, reducing performance degradation caused by chemical reactions and extending the service life of the material [24]. Furthermore, the high light absorption coefficient and long carrier diffusion length of CsGeX₃ enable it to efficiently absorb photons and generate electron–hole pairs in optoelectronic devices. Meanwhile, these carriers can be efficiently transported within the material, thereby enhancing the photoelectric conversion efficiency of the devices [25]. Subsequently, numerous research groups have conducted modification research on CsGeX₃ materials. On the one hand, research teams have enhanced the crystalline quality, purity and electrical/optical properties of CsGeX₃ materials by optimizing the reaction system, precisely controlling synthesis parameters and introducing specific additives [26]; on the other hand, the band structure and carrier transport characteristics of the materials can be effectively adjusted by the proper doping of some elements in CsGeX₃. For example, doping the Ge element with equivalent or heterovalent ions introduces new energy levels, the electron distribution within the material is changed, and its optical and electrical properties are thereby affected [27].

Despite the numerous advantages of CsGeX₃ materials, the Cs element in CsGeX₃ is a rare metal priced at approximately 402.45 CNY/g (as shown in Figure 1), and the high cost will restrict its large-scale application in the semiconductor field. Compared with metallic Cs, MA (MA = CH₃NH₃⁺) is slightly lower in price, its price at approximately 0.1 CNY/g, offering greater advantages in terms of cost control. Most notably, Na, which belongs to the same main group as Cs, is one of the metal elements with abundant reserves on Earth. Its price is as low as 0.01 CNY/g (as shown in Figure 1), offering significant cost advantages and helping to reduce the production cost of Ge-based perovskite materials. Therefore, from the perspective of cost-effectiveness, especially in large-scale production and commercial applications, sodium has higher economic efficiency and feasibility. In addition, compared with CsGeX₃, NaGeX₃ has an impact on the lattice structure and ionic conduction due to the smaller ionic radius of Na than that of Cs, resulting in less hindrance during carrier transport in the NaGeX₃ system. This provides NaGeX₃ with higher ionic mobility and better performance in application scenarios requiring fast ionic conduction. However, research on NaGeX₃-type materials is scarce so far; in particular, there have been no experimental reports, and only a few studies have explored their geometric structures and physical properties [28,29]. The photoelectric properties of NaGeX₃ materials remain unclear, which significantly hinders the exploration of the diversity of NaGeX₃-based material systems and their application potential.

To expand the halide perovskite material system and predict the physical properties of novel halide perovskites, we conducted a systematic first-principles calculation based on density functional theory (DFT) to investigate the structural stability, electronic, elastic, and optical properties of NaGeX₃ (X = Cl, Br, I) perovskite materials. By calculating the decomposition energies of various space groups (Pm$\overline{3}$m, Pmna, R3m, R3c), we found that NaGeX₃ with the R3c space group exhibits the highest decomposition energy, thereby confirming its superior thermodynamic stability. Additionally, to unveil the physical characteristics of R3c NaGeX₃, we systematically computed their band structures, elastic modulus, dielectric constants, refractive indices, reflectance spectra, and absorption coefficients. The results suggest that NaGeX₃ materials hold significant potential for applications in ultraviolet sensors and photoelectric conversion devices.

2. Calculation Methods

All calculations in this work were performed using first-principles methods based on density functional theory (DFT), implemented with the VASP (Version 6.3.2) and DS-PAW (Version 2023A) software packages [30,31]. The interaction between ionic cores and valence electrons was described using the Projector-Augmented Wave (PAW) method [31], with a plane-wave cutoff energy of 500 eV. Electron–electron exchange and correlation effects were treated using the Perdew–Burke–Ernzerhof (PBE) [32] functional. Structural relaxation and static calculations for all systems were carried out using the DS-PAW software. Since the GGA method underestimates the band gap of materials, the more accurate hybrid density functional (HSE06) [33,34] was employed to calculate the density of states and band structures. k-point sampling of the first Brillouin zone was performed using the Monkhorst–Pack method [35] centered at the Γ point, and 7 × 7 × 7, 11 × 11 × 11, and 15 × 15 × 15 k-point grids were used for structural relaxation, electronic structure and optical properties, respectively. Phonon dispersion relations were computed using density functional theory (DFT) combined with the finite displacement method, utilizing a 2 × 2 × 2 supercell (containing 80 atoms). The valence electron configurations of Na, Ge, Cl, Br and I are 2p⁶3s¹, 4s²4p², 3s²3p⁵, 4s²4p⁵ and 5s²5p⁵, respectively. All atoms in the unit cell were fully relaxed, with a convergence precision of 10⁻⁵ eV for relaxation and a maximum interionic force of 0.01 eV/Å.

3. Results and Discussion

3.1. Structural Stability

It is first necessary to evaluate the structural stability of the NaGeX₃ materials due to the absence of extant experimental data. Generally, the tolerance factor (τ) proposed by Goldschmidt [36] based on oxide perovskites, as shown in Equation (1), can be used to assess the stability of the perovskite structure in metal halide ABX₃ materials:

$$\tau ={\displaystyle \frac{R_{\mathrm{A}}+R_{\mathrm{X}}}{\sqrt{2}(R_{\mathrm{B}}+R_{\mathrm{X}})}}$$

(1)

where R_A, R_B, and R_X are the effective ionic radii of the A-site, B-site, and X-site ions, respectively. Generally, the lower the tolerance factor τ, the lower the structural symmetry index, and vice versa. A stable perovskite structure is formed when τ falls within the range of [0.75, 1]. In other words, if τ < 0.75 or τ > 1, it is difficult to form a stable perovskite structure due to the A-site cation being too small or too large, which will transform into ilmenite and calcite configurations, respectively. When 0.75 < τ < 1, a value of τ close to 1 results in an ideal cubic perovskite structure; the greater the deviation from 1, the more severe the distortion of the BX₆ octahedron and the lower the structural symmetry, which tends to form orthorhombic, rhombic, or trigonal crystal structures [37]. Typically, to ensure the stability and symmetry of the octahedral structure, the ionic radius of the B-site cation is smaller than that of the A-site cation. Six X-site anions surround the B-site cation to form a BX₆ octahedral crystal, and these crystals form a three-dimensional structure through apex-sharing. The calculated τ values of NaGeX₃ are listed in

Table 1. Tolerance factor (τ), lattice constants (LC), and band gap (E_g) of R3c NaGeX₃.

Halide	LC/Å					Eg/eV
	R3c	$\mathit{Pm}\overline{\mathbf{3}}$m	Pmna	R3m	Our Work GGA	Our Work HSE06	Other Work GGA [29]	τ
NaGeI3	a = 7.90, c = 22.48	a = 5.90	a = 4.41, b = 11.05, c = 16.32	a = 8.63, b = 8.56, c = 7.71	2.00	2.69	2.00	0.86
NaGeBr3	a = 7.50, c = 21.42	a = 5.49	a = 4.16, b = 11.00, c = 13.71	a = 8.41, b = 7.74, c = 9.23	2.83	3.76	2.79	0.88
NaGeCl3	a = 7.25, c = 20.44	a = 5.20	a = 3.98, b = 10.79, c = 12.65	a = 8.32, b = 7.58, c = 9.20	3.74	4.75	3.72	0.89

. All three compounds exhibit τ values within the range of [0.75, 1], indicating that they can all form stable perovskite structures. On the other hand, their τ values all deviate from 1, making it difficult to form a cubic structure. From this perspective, NaGeX₃ with the cubic phase reported in the literature [28,38] should, in principle, be unstable.

To determine the ground-state structure of the NaGeX₃, the stability of these phases was evaluated using decomposition energy ($E_{decomp}$) for four commonly reported space groups of halide perovskites, namely Pm$\overline{3}$m, Pmna, R3m and R3c [28,29,38,39]. $E_{decomp}$ can be calculated via Equation (2) as follows:

$$E_{\mathit{decomp}}=E_{eq}\left(C\right)-E\left(NaGe{X}_3\right)$$

(2)

where $E_{eq}\left(C\right)$ denotes the total energy of all decomposed components C at equilibrium and $E\left(NaGe{X}_3\right)$ represents the total energy of NaGeX₃ with different space groups. $E_{decomp}\ge 0\mathrm{e}\mathrm{V}$ indicates that the structure is stable at 0 K; otherwise, it tends to decompose into other component compounds, i.e., instability. However, generally, $E_{decomp}>-25$ meV/atom and the phase is considered a metastable phase with a high probability of experimental synthesis, considering the entropy contribution and the kinetic barrier to decomposition [40,41].

Based on the data from the Open Quantum Materials Database (OQMD) [42] and our DFT calculation results, the ternary phase diagrams of Na-Ge-X at 0 K were constructed, as shown in Figure 2a–c. Herein, only thermodynamically stable compounds are displayed. It can be seen from the Na-Ge-X ternary phase diagrams that the direct decomposition products of NaGeX₃ compounds are NaX and GeX₂. Therefore, for the calculation of the decomposition energy of NaGeX₃, we only consider the phase equilibrium reaction according to Equation (3):

$$NaGe{X}_3\to NaX+Ge{X}_2$$

(3)

To accurately evaluate the thermodynamic stability of NaGeX₃ across all phases, we first determined the ground-state structures and energies of the four phases. Using the DS-PAW software, we constructed models for all NaGeX₃ phases (Pm$\overline{3}$m, Pmna, R3m and R3c) and performed DFT relaxation for these structures. The relaxed lattice constants are listed in Table 1, and the atomic structures are shown in the inset of Figure 3a. As shown in Table 1, the lattice constants increase with the halide ionic radius due to the weakening of Ge-X covalent bonds. The calculated lattice constants for the Pm$\overline{3}$m and R3c phases using the GGA functional are consistent with literature values [28,29], validating the reliability of our computational models and parameters. Next, we calculated the energies of the corresponding decomposition products, NaX and GeX₂. For GeCl₂, which has not been experimentally synthesized, we used the C2/c space group structure from the OQMD database as the decomposition product for NaGeCl₃. For GeBr₂ and GeI₂, we obtained their structures from the Inorganic Crystal Structure Database (ICSD): P12₁/c1 phase for GeBr₂ and P$\overline{3}$m1 phase for GeI₂. The crystal structures of GeX₂ are illustrated in Figure 3a.

As can be seen from Figure 3b, among the four structures of NaGeX₃, the type of halogen has almost no effect on the stability trend of the materials. Specifically, for NaGeCl₃, NaGeBr₃ and NaGeI₃, the R3c phase exhibits the largest of the decomposition energies, which are 0.09 meV/atom, 0 meV/atom and −0.13 meV/atom, respectively. The corresponding decomposition energies of the other three phases are all smaller than that of the R3c phase, with the Pm$\overline{3}$m phase showing the lowest value. This indicates that the R3c phase is the most stable, especially for NaGeCl₃ and NaGeBr₃, which is consistent with the calculation results of the tolerance factor, i.e., it is difficult to form a cubic structure. For the R3c phase of NaGeI₃, although its decomposition energy is negative, the value of −0.13 meV/atom indicates that it is metastable. Considering the contribution of entropy, it will become stable at finite temperatures. To further confirm the stability of the R3c phase, we calculated the phonon spectra of the R3c phase, as shown in Figure 3c–e. Generally speaking, the imaginary frequencies in the phonon spectrum are specifically manifested in that the frequency values of some phonon modes appear as negative numbers. It can be seen that there are no imaginary frequencies in the entire Brillouin zone, indicating that the R3c phase of NaGeX₃ is dynamically stable. Therefore, in subsequent studies, we only consider NaGeX₃ with the R3c phase.

To clearly reveal the geometric structure of NaGeX₃ with the R3c phase, we present its crystal structure and Ge-X(X = Cl, Br, I) bond lengths in Figure 4a. The R3c-phase NaGeX₃ belongs to the trigonal crystal, where X atoms and Ge atoms form an octahedron [GeX₆] with the Ge atom at the center. The Ge-X and Na-X bonds exhibit covalent and ionic characteristics, respectively. In this structure, adjacent [GeX₆] octahedra share a halogen X atom and extend continuously in three directions through corner-sharing, forming a three-dimensional perovskite framework. Na⁺ cations are interspersed in the interstitials of the octahedral framework [GeX₆]. The radius of Na⁺ cations (1.02Å) is less than 2.6Å, satisfying Goldschmidt’s empirical rule, thus enabling NaGeX₃ to maintain the three-dimensional perovskite structure [43]. As illustrated in Figure 4a, the [GeX₆] octahedra hosts in the trigonal phase undergo significant lattice distortion, which consists of three longer Ge–X bonds and three shorter Ge–X bonds. For comparison, the geometry of the cubic phase is shown in Figure 4b, in which the six Ge–X (X = Cl, Br, I) bond lengths of the [GeX₆] octahedron are equivalent, namely, 2.59 Å for Ge-Cl, 2.74 Å for Ge-Br and 2.95 Å for Ge-I. It can be speculated that this distortion of the [GeX₆] octahedra hosts should induce a transition from the cubic phase to a more stable trigonal phase. It can be qualitatively explained from the perspective of the Na-X bonds. Taking NaGeI₃ as an example, in the cubic phase, Na and I form a dodecahedron with a Na-I bond length of 4.17 Å, while forming more stable [NaX₆] octahedra with a Na-I bond length of 3.2–3.3 Å in the trigonal phase, indicating that the Na-I bonds in the trigonal phase are stronger, leading to a more stable structure. Consequently, compared with the highly symmetric cubic phase, the trigonal phase system exhibits a lower total enthalpy, ultimately demonstrating higher structural stability. Given that three-dimensional perovskites exhibit strong and broad optical absorption, generate free carriers upon photoexcitation and possess high carrier mobility and long carrier diffusion lengths, these properties make NaGeX₃ structurally suitable for optoelectronic devices, particularly photovoltaic devices.

3.2. Electronic Structure

For semiconductor materials, electronic structure is a key factor determining their properties, and it is also crucial for understanding the nature of interatomic bonding, electrical conductivity and optoelectronic properties. As is well-known, the GGA method underestimates the band gap of materials. To accurately evaluate the band gap of NaGeX₃ (X = Cl, Br, I), we calculated the band structures using the hybrid density functional (HSE06). The calculated band structures and density of states (DOS) are shown in Figure 5. The red horizontal dashed line at 0 eV represents the Fermi level (E_f). In Figure 5, the energy range displayed is −4 to 8 eV, which includes the valence band maximum (VBM) and conduction band minimum (CBM) regions. For all NaGeX₃ structures, the VBM is located at the Γ point. The difference is that the CBM of NaGeBr₃ and NaGeCl₃ is located at the F point, as shown in Figure 5a,b, while the CBM of NaGeI₃ is also located at the Γ point. This indicates that the former two have indirect band gaps, whereas NaGeI₃ has a direct band gap. Therefore, the luminous efficiency of NaGeI₃ should be higher than that of the other two materials.

The band gap values of all R3c phase NaGeX₃ calculated using HSE06 range from 2.69 to 4.75 eV, as listed in Table 1. For comparison with the literature, we also calculated the band gaps of all NaGeX₃ using the GGA method (Table 1). The GGA results are consistent with those reported in the literature [29], confirming the reliability of our calculations. Additionally, the GGA band gap values of all NaGeX₃ are approximately 1 eV smaller than those from HSE06, further verifying the underestimation of band gaps by GGA. Additionally, compared to the cubic phase with a band gap less than 1 eV for NaGeX₃ [29,39], the band gaps of all trigonal NaGeX₃ are much larger. This can be explained structurally, as the longer Ge-X bonds in the trigonal phase tend to weaken the orbital overlap between Ge p orbitals and X p orbitals. Since the VBM is mainly contributed by X p orbitals, the weakening of this overlap results in a significant decrease in the VBM energy. On the other hand, shorter Ge-X bonds can enhance the local orbital hybridization between Ge p orbitals and X p orbitals. Given that the CBM is mainly dominated by Ge p orbitals, the strengthening of hybridization leads to an increase in the CBM energy of the trigonal phase. The aforementioned synergistic effect arises from the decrease in VBM energy and the increase in CBM energy, resulting in a significant widening of the bandgap in the trigonal phase compared to the cubic phase.

For different halogens, it is found that the band gaps increase as the halogen atomic radius decreases, i.e., from 2.69 eV for NaGeI₃ to 3.76 eV for NaGeBr₃ and 4.75 eV for NaGeCl₃. This trend is mainly determined by the outermost p-orbital energy levels and electronegativity of the halogens. In perovskite material systems, an ideal band gap usually needs to balance efficient absorption of sunlight with effective generation and transport of carriers. Compared with some traditional perovskite materials, NaGeI₃, with a direct band gap of 2.69 eV, has a maximum achievable power conversion efficiency (PCE) of approximately 20% based on the Shockley–Queisser (SQ) limit, which is lower than the most widely used hybrid FAPbI₃ perovskite with about 33% PCE [10,13]. However, the slightly wider band gap avoids the rapid non-radiative recombination that reduces photoelectric conversion efficiency in narrow-bandgap materials, while still maintaining sufficient sunlight absorption. These characteristics lay a solid theoretical foundation for the application of NaGeI₃ in optoelectronic devices such as perovskite solar cells and photodetectors, making it a promising candidate for next-generation high-performance perovskite materials. To gain a better understanding of the electronic properties of NaGeX₃, we calculated their total density of states (TDOS) and partial density of states (PDOS). As shown in Figure 5, all NaGeX₃ exhibit a similar electronic distribution. The valence bands are primarily contributed by Ge (4p) atoms and halogen X atoms (Cl 3p, Br 4p, I 5p), with the contribution of X to the VBM increasing as the halogen ionic radius increases. The CBMs are also mainly contributed by Ge and halogen X atoms, with Ge being the dominant contributor, while the 3s and 3p states of Na make little contribution. Notably, significant hybridization is observed between Ge and X in all halides, confirming the covalent nature of their chemical bonds. Additionally, based on the DOS of Na, it is expected that ionic bonds will form between Na and halogen X, which helps maintain the stability of the perovskite structure rather than participating in electronic transition processes. Thus, it can be concluded that the electronic transitions during photoexcitation primarily occur between the p orbitals of halogens and Ge atoms.

3.3. Effective Mass

Carrier effective mass ($m^{*}$) is another key parameter describing the electronic properties of semiconductor materials, and it is of great significance for understanding the electrical conductivity, optoelectronic properties and thermoelectric properties of semiconductor materials. According to Equation (4) [44], $m^{*}$ is inversely proportional to the curvature of the material’s energy bands. Therefore, based on the parabolic approximation, we calculated the effective masses of electrons (m_e) and holes (m_h) near the bottom of the CBM and the top of the VBM for all NaGeX₃ by fitting the dispersion relations as follows:

$$m^{*}={\hslash }^2{\left({\displaystyle \frac{{\partial }^{2}E\left(k\right)}{\partial k^2}}\right)}^{-1}$$

(4)

where ℏ, k and E(k) are the reduced planck constant, wave vector and energy eigenvalue at the band edge, respectively. For the calculation of m_e and m_h, we performed a fine fitting on the band structures of the CBM and VBM around the Γ point, as these band structures exhibit obvious parabolic characteristics, as shown in Figure 5. The calculated average effective masses of carriers (with m₀ being the electron mass) for all NaGeX₃ along the Z → Γ and Γ → X directions are listed in

Table 2. The average carrier effective masses of NaGeX₃ along the Z → Γ and Γ → X directions.

Halide	me (m0)			mh (m0)			$\mathbf{D} = \mathit{m}\mathit{h}/{\mathit{m}}_{\mathit{e}}$
	Z-Γ	Γ-X	AVG	Z-Γ	Γ-X	AVG	Z-Γ	Γ-X	AVG
NaGeI3	0.39	0.45	0.42	0.65	0.50	0.58	1.67	1.11	1.39
NaGeBr3	0.76	0.82	0.79	0.64	1.90	1.27	0.84	2.32	1.58
NaGeCl3	1.51	1.46	1.49	0.86	3.56	2.21	0.57	2.44	1.50

. The results show that the average effective electron mass m_e of all NaGeX₃ materials is smaller than the effective hole mass m_h, indicating that the electron mobility is significantly higher than that of holes. Moreover, as the halogen ionic radius increases, the $m^{*}$ of NaGeX₃ decreases. NaGeI₃ exhibits the smallest effective electron and hole masses, with average values of 0.42m₀ and 0.58m₀, respectively. This significant mass difference indicates that electrons are more mobile than holes, contributing to their high carrier mobility. In addition, D = m_h/m_e is often used to describe the degree of carrier recombination; a higher D value indicates that electron–hole pairs are more likely to separate [45], thereby reducing the carrier recombination rate. According to the data in Table 2, NaGeI₃ has the smallest average D value (1.39). This result indicates that, compared with the other two materials, the electron–hole pairs in NaGeI₃ are more difficult to effectively separate, that is, it has the highest carrier recombination rate. This phenomenon is associated with the fact that NaGeI₃ has the smallest carrier effective mass.

3.4. Mechanical Properties

The mechanical properties of solids not only help explain the response mechanism of interatomic potentials to external forces, but also play a crucial role in various fundamental phenomena of solid-state materials (such as stability, strength, ductility, brittleness and other characteristics). These properties include parameters such as Young’s modulus (Y), bulk modulus (B), shear modulus (G), anisotropy factor and other elastic parameters, providing an important basis for exploring the behavioral characteristics of materials under different conditions. The elastic constants and modulus of all NaGeX₃ were calculated and listed in

Table 3. Calculated elastic properties of R3c-phase NaGeX₃(X = Cl, Br, and I).

Elastic Parameters	This Work for R3c			$\mathbf{Other}\mathbf{Work}\mathbf{for}\mathit{Pm}\overline{\mathbf{3}}$m [28]
	NaGeCl3	NaGeBr3	NaGeI3	NaGeCl3	NaGeBr3	NaGeI3
C11 (GPa)	10.52	8.91	13.65	57.89	50.78	41.11
C12 (GPa)	3.80	2.55	4.05	10.96	8.64	6.32
C13 (GPa)	2.32	1.70	2.94	/	/	/
C14 (Gpa)	−0.90	−0.50	−1.16	/	/	/
C33 (GPa)	6.72	5.33	9.70	/	/	/
C44 (GPa)	4.03	3.21	4.68	6.11	5.79	5.57
B (GPa)	4.79	3.75	6.21	20.26	22.69	17.92
E (GPa)	8.33	7.10	10.88	28.57	22.11	23.00
G (GPa)	3.44	3.00	4.50	10.86	10.03	8.98
B/G	1.39	1.25	1.38	2.45	2.26	2.00
$\vartheta$	0.21	0.18	0.21	0.32	0.31	0.27

based on the relationship between the two [46]. For comparison, the counterparts of cubic NaGeX₃ calculated by Pingak et al. [28] were also listed in Table 3.

Based on the mechanical stability criteria proposed by Gao et al. [47], all trigonal NaGeX₃ materials are mechanically stable because their C_ij values satisfy the following criteria: C₄₄ > 0, C₁₁ − |C₁₂| > 0, C₃₃(C₁₁ + C₁₂) − 2C₁₃² > 0 and C₄₄(C₁₁ − C₁₂) − 2C₁₄² > 0. Additionally, for all trigonal NaGeX₃, NaGeI₃ possesses the highest hardness and stiffness, owing to the maximum Young’s modulus (E = 10.88 GPa), Bulk Modulus (B = 6.21 GPa) and shear modulus (G = 4.5 GPa), indicating that NaGeI₃ has the strongest resistance to mechanical deformation among the three materials, while NaGeBr₃ is the most sensitive to external stress due to the lowest elastic modulus (E = 7.1, B = 3.75 and G = 3). However, compared with the corresponding cubic structures, the elastic moduli of all trigonal NaGeX₃ are smaller, as listed in Table 3, indicating that the trigonal cases are soft materials with lower hardness and stiffness, more suitable for scenes requiring slight mechanical stability (e.g., thin-film devices).

The Pugh’s ratio (B/G) and Poisson’s ratio ($\vartheta$) are related to the ductility and brittleness of materials. A material is ductile if the Pugh’s ratio is greater than 1.75; otherwise, it is brittle [48]. Similarly, a higher Poisson’s ratio (>0.26) indicates better ductility [46]. As shown in Table 3, all three trigonal materials exhibit brittleness due to their B/G and $\vartheta$ being less than 1.75 and 0.26, respectively, with NaGeBr₃ having the highest brittleness. In contrast, all cubic cases possess ductility, with NaGeCl₃ having the highest ductility owing to the greatest B/G (2.45) and $\vartheta$ (0.32). This result indicates that, compared to cubic structures, trigonal NaGeX₃ materials are not suitable as flexible devices.

3.5. Optical Properties

The dielectric function ($\epsilon$) is a crucial parameter for the optical properties of materials, as it contains information about the material’s response to incident photons. It serves as a core bridge connecting the microscopic electronic responses of materials to macroscopic optical properties such as absorption (α), reflection (R) and refraction (n). The complex form of the dielectric function [49] can be expressed as follows:

$$\left(\omega \right)={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right)$$

(5)

ε₁(ω) and ε₂(ω) represent the real and imaginary parts of the dielectric function, respectively. Other optical properties such as absorption (α), reflection (R), refraction (n) and the extinction coefficient (k) can be calculated from the obtained ε₁ and ε₂. For detailed calculation formulas, please refer to the literature [50].

Figure 6 presents the calculated dielectric constants, absorption coefficients, refractive indices/extinction coefficients and reflectivity/loss of the three NaGeX₃ materials as functions of photon energy within the energy range of 0–20 eV. Figure 6a,b show the variation curves of the real part ε₁(ω) and imaginary part ε₂(ω) of the dielectric function with incident photon energy. It can be seen from the figures that the static dielectric constant $\epsilon$₁(0) increases with increasing halogen atomic radius. The $\epsilon$₁(0) values of NaGeCl₃, NaGeBr₃ and NaGeI₃ are 3.34, 4.01 and 5.68, respectively, as listed in

Table 4. Calculated frequency independent dielectric constant ε₁(0), refractive index n(0), and reflectivity R(0) of NaGeX₃ (X = Cl, Br, I).

Compounds	ε1(0)	n(0)	R(0)
NaGeI3	5.68	2.38	0.17
NaGeBr3	4.01	2.03	0.12
NaGeCl3	3.34	1.83	0.09

. This result is reasonable because according to the Penn model [51], ${\epsilon }_1\left(0\right)=1+({{h\omega }_p/E_g)}^2$, where a smaller band gap corresponds to a larger $\epsilon$₁(0). Moreover, starting from the zero-frequency limit, the $\epsilon$₁ of all NaGeX₃ begins to increase, reaches a maximum within the energy range of [2, 5] eV, then starts to decrease, drops to a negative value around 10 eV, and finally fluctuates slowly with higher photon energies. Overall, in the energy range of 0–10 eV, the $\epsilon$₁ of NaGeX₃ materials changes rapidly as the incident light energy gradually increases; when the incident light energy exceeds 10 eV, the change rate of the real part of the dielectric function slows down significantly and gradually becomes stable.

The value of the imaginary part of the dielectric function, $\epsilon$₂, arises from electron transitions from the top of the valence band to the bottom of the conduction band, determining the linear response of the system to light at small wave vectors. Its peak value can reflect the intensity of electron excitation transitions in the material; a larger peak indicates a stronger electron absorption capacity. As shown in Figure 6b, the absorption edge of $\epsilon$₂ for all compounds is located at the band gap position of their respective materials, and the absorption edge increases with the increase in halogen radius, indicating that NaGeI₃ has the strongest absorption. As the incident light energy increases, the number of states where electrons can undergo transitions increases. Within the energy range of 0–13 eV, the $\epsilon$₂ of NaGeX₃ changes rapidly, showing first and second peaks, and the subsequent changes are relatively slow. Combined with the analysis of electronic structure, it is known that the first peak originates from electron transitions between the p orbitals of halogen X and the 4p orbitals of Ge.

Based on the dielectric function, the absorption coefficient α(ω) of NaGeX₃ was further calculated, which can directly quantify the material’s ability to absorb light. The variation in α(ω) with photon energy is illustrated in Figure 6c. As shown in the figure, within the energy range of 0–20 eV, the absorption coefficient of NaGeI₃ reaches a maximum value of 1.34 × 10⁶ cm⁻¹ at ℏω = 9.77 eV, while NaGeBr₃ and NaGeCl₃ reach the same maximum value of 1.38 × 10⁶ cm⁻¹ at ℏω = 10.27 eV and 11.37 eV, respectively. The three materials exhibit weak absorption in the visible light range (ℏω = 1.65–3.26 eV). When the photon energy exceeds that of visible light, the absorption coefficient reaches a large order of magnitude (10⁶ cm⁻¹) over a wide range of photon energies, mainly showing ultraviolet absorption. Figure 6d presents the relationship between the absorption spectrum and wavelength. When Cl is replaced by I, the absorption spectrum undergoes a red shift, and NaGeI₃ exhibits a higher absorption coefficient, which exceeds 7 × 10⁵ cm⁻¹ near a wavelength of 280 nm. In the figure, the region with a wavelength less than 380 nm is the ultraviolet region, and that greater than 380 nm is the visible light region. It can be inferred that these three materials may be ideal for manufacturing ultraviolet sensors, and NaGeBr₃ and NaGeI₃ may be applicable to the fabrication of blue-light lasers. Compared with the other two, NaGeI₃ has superior light absorption performance in the visible range. In addition, the refractive index n(ω) and extinction coefficient k(ω) are important optical parameters describing the interaction between light and matter. The refractive index n determines the path and direction of light propagation in different media, while the extinction coefficient k represents the energy attenuation of light propagating in the material. Figure 6e shows the refractive index n(ω) and extinction coefficient k(ω) of NaGeX₃. As can be seen from Figure 6e and Table 4, the static refractive indices n(0) of NaGeCl₃, NaGeBr₃ and NaGeI₃ are 1.83, 2.03 and 2.38, respectively. The photon energies corresponding to the maximum peaks of the refractive index n are 4.2 eV, 3.4 eV and 2.74 eV, respectively, and the refractive index gradually decreases with further increase in photon energy. A larger static refractive index n(0) essentially reflects a stronger polarization ability of the material under static conditions, which is specifically manifested as more significant electron cloud distortion, easier ion displacement or higher free carrier concentration. Thus, NaGeI₃ exhibits more excellent properties. Furthermore, the extinction coefficient k of NaGeI₃, NaGeBr₃ and NaGeCl₃ is greater than the refractive index n in the energy ranges of 7.79–14.75 eV, 10.12–13.87 eV and 11.26–13.35 eV, respectively. This is because, within these energy ranges, the real part of the dielectric function ε₁ < 0, meaning that the wave vector K is imaginary, i.e., light in these ranges cannot propagate in NaGeX₃. Corresponding to the absorption coefficient, the extinction coefficients of NaGeCl₃, NaGeBr₃ and NaGeI₃ all gradually decrease when the energy exceeds around 10 eV, and eventually approach zero near 20 eV. At this point, the incident light frequency is greater than or equal to the natural oscillation frequency, and the optical quantities characterizing solid absorption all approach zero. The refractive index varies with frequency as normal dispersion (the refractive index decreases with increasing frequency), and the extinction coefficient exhibits strong absorption characteristics at the band edge.

To further investigate the energy loss during electron excitation under photon interaction, we calculated the relationships of the loss function and reflectivity of NaGeX₃ with photon energy, as shown in Figure 6f. The results indicate that although the loss functions of the three materials contain two loss peaks within the range of material light absorption (9–11 eV), their values are relatively low. Moreover, the maximum reflectivity of NaGeX₃ in the energy range of 9–11 eV, i.e., the ultraviolet region, does not exceed 36%. This suggests that NaGeX₃ materials have little impact on the absorption of photons; in other words, ultraviolet light can more easily enter such germanium-based halide perovskite materials.

In summary, the low-cost Ge-based halide perovskite material NaGeX₃ exhibits strong absorption and low loss in the ultraviolet region, showing potential for application in ultraviolet sensing and optoelectronic devices. Compared with the other two materials, NaGeI₃ requires lower photon energy to reach the maximum light absorption intensity, with its absorption peak redshifted and the wavelength of absorbed light lengthened. This indicates that NaGeI₃ can more efficiently capture photons in the long-wavelength or visible light regions, providing sufficient carriers for optoelectronic devices and thus holding certain potential as a perovskite solar cell material. However, the widespread application of NaGeX₃ with the R3c space group is limited by the large band gap. Future studies can further regulate the electronic structure and optical properties of NaGeX₃, especially NaGeI₃, by adjusting the band gap, such as by performing partial substitution or doping modification on A-site cations (e.g., Cs⁺, K⁺) and B-site cations (e.g., Ca²⁺, Mg²⁺). Regulating the cationic radii can alter the tolerance factor (τ) of the system, thereby regulating the structural phase of the material.

4. Conclusions

In this paper, first-principles calculations based on density functional theory were employed to systematically investigate the geometric structures, electronic, elastic and optoelectronic properties of NaGeX₃ (X = Cl, Br, I). Among the four space groups, the R3c phase NaGeX₃ structure exhibits the highest thermodynamic and dynamic stability. Studies on the electronic and optical properties of R3c phase NaGeX₃ reveal that the valence band maximum and conduction band minimum of NaGeX₃ (X = Cl, Br, I) are dominated by the p-orbitals of halogens and Ge, respectively. Additionally, the band gap of the material decreases with increasing halogen atomic radius. Furthermore, the absorption coefficient reaches a maximum of 1.38 × 10⁶ cm⁻¹ within the photon energy range of 9–11 eV, and the absorption spectrum redshifts as the halogen radius increases. This study indicates that NaGeX₃ holds promising application prospects in fields such as ultraviolet sensing and photoelectric conversion devices.