Tuning band gap and enhancing optical functions of AGeF₃ (A = K, Rb) under pressure for improved optoelectronic applications

Abstract

The current study diligently analyzes the physical characteristics of halide perovskites AGeF₃ (A = K, Rb) under hydrostatic pressure using density functional theory. The goal of this research is to reduce the electronic band gap of AGeF₃ (A = K, Rb) under pressure in order to improve the optical characteristics and assess the compounds’ suitability for optoelectronic applications. The structural parameters exhibit a high degree of precision, which correlates well with previously published work. In addition, the bond length and lattice parameters decrease significantly leading to a stronger interaction between atoms. The bonding between K(Rb)–F and Ge–F reveal ionic and covalent nature, respectively, and the bonds become stronger under pressure. The application of hydrostatic pressure demonstrates remarkable changes in the optical absorption and conductivity. The band gap becomes lower with the increment of pressure, resulting in better conductivity. The optical functions also predict that the studied materials might be used in a variety of optoelectronic devices operating in the visible and ultraviolet spectrum. Interestingly, the compounds become more suitable to be used in optoelectronic applications under pressure. Moreover, the external pressure has profound dominance on the mechanical behavior of the titled perovskites, which make them more ductile and anisotropic.

Introduction

Cubic perovskites have obtained substantial preference from researchers and scientists over the last few years. The versatile applicability of these perovskites in multiple sectors, e.g., semiconductors, sensors, superconductivity, photovoltaic cells, optoelectronic devices, and LEDs (light-emitting devices)^1–3 puts them in the center of attention. As a result, the researchers conducted both experimental and theoretical studies^4–7 on the physical behavior of perovskite materials to create new possibilities for their applications in various optoelectronic fields. Interestingly, the improvement of perovskite solar cells (PSCs) has been accelerated, resulting in power conversion efficiency (PCE) of 22.1%⁸. Till now, the best PCE of 25.2% is recorded for Pb-based perovskite solar cells⁹ but have less life span caused by humidity, moisture, temperature, and UV light¹⁰. On the other hand, the lead-free tin halide perovskite incorporated with ethylammonium iodide exhibited the PCE of ~ 13%¹¹. At initial stages, metallic Pb²⁺-based organic PSCs i.e., CH₃NH₃PbI₃ (or MAPbX₃) were developed^12–14. But the toxicity of these organic compounds created major disadvantages. Pb is a recognized toxin, creating a number of obstacles¹⁵. In addition, organic MA⁺ cations cause serious environmental drawbacks, making the manufacturing process extremely risky and harmful¹⁶. In order to nullify the toxicity, non-toxic cations, like Ge²⁺ and Sn²⁺ have replaced Pb²⁺ cation, and/or K⁺, Rb⁺, and Cs⁺ cations have used to replace the organic counterpart^15,17–19. Therefore, a new formation of ABX₃ has appeared in which A, B, and C denote the monovalent cation, divalent cation, and halogen anion, respectively. Inorganic perovskites based on Ge have emerged as a possible alternative of Pb, because they possess superior optical absorption and conductivity as compared to Pb-based perovskites²⁰. At ambient temperature, Ge-based perovskites do not exhibit any phase transformation^21,22. Besides, K and Rb have shown promising potential for photo-voltaic applications²³. Jain et al.²⁴ have utilized the first-principle calculations on RbSn(Cl,Br)₃ perovskites to evaluate prominent band gap suitable for photovoltaics. The monovalent cation K⁺-based perovskites have potential to be utilized in solar cells because of high absorption and configurable band gap^20,22,25. In recent studies, the inorganic halide perovskites have been recognized as a reliable material for solar cell applications^25,26. For the purpose of enhancing physical properties of halide perovskites, the application of hydrostatic pressure has demonstrated tremendous results^27–32. Usually, hydrostatic pressure modifies the lattice parameters³³, displacement of cation and anion^34,35, rotation of octahedral cages³⁶, phase transitions^37,38, etc. In the case of metal halides, structural properties, like lattice constants and unit cell volume decrease with increasing pressure^29,32. Identical behavior can be detected in halide perovskites as well. In recent works, inorganic halide perovskites, such as KCaCl₃³⁹, CsGeI₃³⁰, RbYbF₃⁴⁰, and CsGeI₃⁴¹ have shown reduction in band gap under hydrostatic pressure, resulting an improvement of conductivity. In addition, the application of pressure can remarkably develop the optical parameters of halide perovskites, enhancing the functionality in optoelectronic fields. Therefore, the motive of present work is to evaluate and examine the changes of various physical features of halide perovskites AGeF₃ (A = K, Rb) after applying hydrostatic pressure. More specifically, this study has analyzed the structural, electronic, optical, and mechanical properties of AGeF₃ (A = K, Rb) using first-principle calculations to observe whether the application of hydrostatic pressure has made them more appealing to optoelectronic fields or not.

Results and discussion

Structural properties

The geometry optimization states that the selected compounds AGeF₃ (A = K, Rb) under study crystallized in cubic cell that have the space group Pm-3m (#221). In the unit cell, the A (= K, Rb), Ge, and F atoms are located at the corner, body center, and face center, respectively, with the Wyckoff positions 1a (0, 0, 0), 1b (0.5, 0.5, 0.5), and 3c (0, 0.5, 0.5), respectively. The optimized crystal structure of AGeF₃ (A = K, Rb) with crystallographic sites is illustrated in Fig. 1. The evaluated lattice constant of KGeF₃ at ambient pressure is 4.451 Å (Table 1), which is relatively closer to the reference study (4.46 Å)²⁰. The deviation value of 0.2% presents the high accuracy of this study. For RbGeF₃, the lattice constant is 4.490 Å (Table 1), showing no deviation from the previous work (4.49 Å)²⁰. The hydrostatic pressure ranging from 0 to 30 GPa is applied on both compounds to calculate the structural parameters as given in Table 1. The application of pressure demonstrates a significant effect on the structural parameters. The changes of relative lattice constants and unit cell volume with respect to the hydrostatic pressure are illustrated in Fig. S1a,b, respectively. The plotted graphs reveal the reduction of both lattice constant and unit cell volume under linear ascending of applied pressure. This decreasing tendency of lattice parameters under hydrostatic pressure indicates the reduction of bond length (Table 2) within the compounds. In order to justify the phase stability of AGeF₃ (A = K, Rb) under pressure, the formation energy (ΔE_f) is calculated using the following equation and recorded in Table 1.

$${\mathrm{\Delta }E}_{\mathrm{f}}\left({\mathrm{AGeF}}_3\right)=\frac{\left[E_{tot.},\left({\mathrm{AGeF}}_3\right),-,E_s,\left(\mathrm{A}\right),-,E_s,\left(\mathrm{Ge}\right),-,3,E_s,\left(\mathrm{F}\right)\right]}{N}.$$ 1

Here, E_s(A), E_s(Ge), and E_s(F) are the energy of A (= K, Rb), Ge, and F atoms, respectively, whereas, E_tot(AGeF₃) represents the unit cell total energy of AGeF₃, and N is the number of atoms in the unit cell. The negative values of ΔE_f at all applied pressures reveal the thermodynamic stability of titled halide systems⁴².

Figure 1.

The optimized crystal structure of halide perovskites KGeF₃ and RbGeF₃.

Table 1.

Calculated lattice constant (a), unit cell volume (V), and formation energy (ΔE_f) of AGeF₃ (A = K, Rb) at various applied pressures.

Compound	Calculated data	Pressure (GPa)
		0	10	20	30
KGeF3	a (Å)	4.451 [This work]	4.243	4.117	4.023
		4.4631
	V (Å3)	88.18	76.37	69.78	65.11
	ΔEf (eV/atom)	− 4.63	− 4.57	− 4.44	− 4.30
RbGeF3	a (Å)	4.490 [This work]	4.281	4.154	4.061
		4.4931
	V (Å3)	90.52	78.46	71.68	66.97
	ΔEf (eV/atom)	− 4.62	− 4.55	− 4.43	− 4.28

Table 2.

Estimated bond lengths in AGeF₃ (A = K, Rb) at various applied pressures.

Pressure (GPa)	Bond length (Å)
	KGeF3		RbGeF3
	Ge–F	K–F	Ge–F	Rb–F
0	3.14714	2.22537	3.17491	2.24500
10	3.00043	2.12162	3.02725	2.14059
20	2.91083	2.05827	2.93745	2.07709
30	2.84527	2.01191	2.87188	2.03072

Electronic properties

The assessment a material’s electronic nature requires the understanding of its band structure and density of states. The band structures of KGeF₃ and RbGeF₃ at different applied pressures are shown in Figs. 2 and 3, respectively. The horizontal dotted line at 0 eV denotes the Fermi level (E_F). At 0 GPa, the valence band maximum (VBM) and conduction band minimum (CBM) indicated by green and red lines, respectively, of both compounds are noticed at R point of the Brillouin zone. Therefore, a direct band gap (E_g) of 1.98 eV is found for KGeF₃, while it is 2.012 eV for RbGeF₃. The E_g values found for AGeF₃ (A = K, Rb) are quite consistent with the theoretical values obtained using the GGA-PBE approximation¹⁹. With increasing pressure, the CBM of both compounds begin to move towards the E_F, resulting in a reduction of E_g. At 30 GPa, the E_g of KGeF₃ falls to 0.16 eV, while it is 0.26 eV for RbGeF₃. The reduction of E_g under pressure for both compounds is graphically presented in Fig. S2. There exists an inverse relationship between band gap and external pressure⁴³, which can increase the potential between electron and ion responsible for reducing lattice parameters (Table 1). The band gap at the Brillouin zone symmetry point shrinks when the lattice parameter is reduced. The reduction of E_g allows easy transport of electrons from valence band to conduction band. As a result, the optical absorption and conductivity may become higher beneficial for optoelectronic applications.

Figure 2.

Band structures of KGeF₃ under applied pressure.

Figure 3.

Band structures of RbGeF₃ under applied pressure.

Furthermore, the total density of states (TDOS) of AGeF₃ (A = K, Rb) are computed and illustrated in Fig. S3a,b to explicate the band structures. The vertical dashed line at 0 eV denotes the E_F. There observe no TDOS value at E_F for both compounds under all applied pressures, which also reflects the semiconducting nature of them. There is a significant pressure influence on TDOS in the conduction band, where all the sharp peaks gradually move towards the E_F as pressure increases. This peak shifting is responsible for the band gap shrinking under pressure, which is also appeared in the band structures at R point. However, the partial density of states (PDOS) is crucial to obtain the atomic contribution of a material for making its band structure. It is evident from Figs. 4 and 5 that the valence band of both compounds near the E_F mostly originate from Ge-4s and F-2p orbitals with small amount of Ge-4p orbital. On the other hand, the conduction band results from K-4s (Rb-5s), K-4p (Rb-4p), Ge-4s, Ge-4p, and F-2p states. It is apparent that the Ge-4p orbital is mainly responsible for narrowing the E_g in both compounds. The hybridization between Ge-4p and F-2p is promoted by raising external pressure, which raises the conduction bands towards the E_F and reduces the band gap. Furthermore, the shortening of Ge–F bond length in response to pressure (Table 2) could improve the hybridization between Ge-4p and F-2p orbitals in the conduction band, which lowers the CBM at R point of the Brillouin zone (Figs. 2, 3). Hence, the band gap of KGeF₃ (RbGeF₃) reduces from 1.98 (2.012 eV) to 0.16 eV (0.26 eV).

Figure 4.

Partial density of states of KGeF₃ under applied pressure.

Figure 5.

Partial density of states of RbGeF₃ under applied pressure.

Charge density mapping helps to visualize the charge distribution around atoms and bonding nature of the compounds. Figures 6 and 7 illustrate the charge density mapping of AGeF₃ (A = K, Rb) along the crystallographic planes (100) and (200). The right side scales indicate the electron density, in which low and high intensity are imparted by blue and red colors, respectively. At ambient pressure, K(Rb) and F atoms exhibit spherical charge contours along the (100) plane, manifesting the existence of ionic bonding between them (Figs. 6a, 7a). In addition, a covalent bonding nature of Ge–F is predicted, as the elliptical shape of charge distribution is observed around Ge and F atoms along the (200) plane (Figs. 6b, 7b). Charge density is also estimated at an applied pressure of 30 GPa to understand the effect of pressure on charge distribution. There is no noticeable difference in the spherical charge contours around K(Rb) and F atoms along the (100) plane (Figs. 6c, 7c). However, the contours around Ge and F atoms become more elliptical along the (200) plane, intensifying the covalent bonding of Ge–F (Figs. 6d, 7d). The bond length of Ge–F is longer than that of K(Rb)–F (Table 1), indicating strong bonding between K(Rb) and F atoms than that of bonding between Ge and F atoms. Therefore, the weaker covalent bond of Ge–F and stronger ionic bond of K(Rb)–F are exhibited in the crystal structure of AGeF₃ (A = K, Rb), which validates the estimated results predicted by charge density maps. The bond length decreases monotonically as pressure increases (Fig. S4a,b) and hence, the ionic/covalent bonds become stronger.

Figure 6.

Charge density plots of KGeF₃ at 0 GPa and 30 GPa pressure.

Figure 7.

Charge density plots of RbGeF₃ at 0 GPa and 30 GPa pressure.

Optical properties

Metal halides without lead (non-toxic) have served as a source of interest due to their outstanding optical characteristics. They demonstrate much appreciated performance in optoelectronic devices and photovoltaic cells. This section deeply analyzes and discusses a few optical characteristics, namely absorption, conductivity, reflectivity, reflective index, and dielectric function. The findings are presented using an electronic polarization vector of [100] at 0 GPa and 30 GPa pressures.

It is necessary to determine dielectric function to obtain rest of the optical functions⁴⁴. It can be represented as ε(ω) = ε₁(ω) + iε₂(ω); where ε₁(ω) and ε₂(ω) denote the real and imaginary parts of dielectric function, respectively⁴⁵. According to Kramers–Kronig relation⁴⁶, ε₁(ω) is written as,

$${\epsilon }_1\left(\omega \right)=1+\frac{2}{\pi }P\underset{0}{\overset{\infty }{\int }}\frac{{\omega }^{{}^{\prime }{\epsilon }_2}\left({\omega }^{\prime }\right)}{{\omega }^{{}^{\prime }2}-{\omega }^2}d{\omega }^{\prime }.$$

On the other hand, ε₂(ω) may also calculate using the momentum tensors between the occupied and unoccupied wave functions^47,48.

$${\epsilon }_2\left(\omega \right)=\frac{{2e}^2\pi }{\mathrm{\Omega }{\epsilon }_0}\sum _{K,V,C}{\left|⟨,{\psi }_k^c,\left|\widehat{U}\right),·,\overrightarrow{r},\left|{\psi }_k^V\right),⟩\right|}^2\delta (E_K^C-E_K^V-E).$$

Here, ω signifies the light frequency. ${\psi }_k^c$ and ${\psi }_k^V$ denote the conduction and valance band wave function at k, respectively, e is the electronic charge, Ω represents the unit cell volume, and U indicates the unit vector along the polarization of the incident electric field. The delta function ensures energy and momentum conservation during a transition between occupied and unoccupied electronic states through the emission or absorption of photon energy, E. $E_K^C$ and $E_K^V$ denote the energy of electrons at a certain k-vector in the conduction and valence bands, respectively. The rest of the optical parameters are calculated using the expressions given elsewhere⁴⁹.

The optical absorption coefficient (α) denotes the amount of energy absorbed by a substance per unit length. The efficiency of a material's optimal solar energy conversion can simply be described by it. From Fig. 8a, the absorption does not begin at 0 eV, since both KGeF₃ and RbGeF₃ contain a band gap at ambient pressure. In the ultraviolet region, KGeF₃ and RbGeF₃ show three sharp peaks in the range of ~ 8–22 eV and ~ 8–18 eV, respectively. So, both compounds work as good absorbers in the ultraviolet region at 0 GPa. When 30 GPa pressure is applied both compounds also show peaks within the ultraviolet region similar that exhibit at 0 GPa. In this case, both compounds show an additional peak at ~ 15 eV. Interestingly, the absorption spectra almost start from 0 eV (but not 0 eV due to having very small band gap) at 30 GPa (inset of Fig. 8a). The absorption in the visible light region is explicitly shown in Fig. 8b. The absorption of both compounds in the visible region is explicitly higher at 30 GPa than that observed at 0 GPa. As a result, the studied perovskites are expected to use visible light energy for photovoltaic conversion at a pressure of 30 GPa, potentially increasing the efficiency of solar cells.

Figure 8.

Calculated pressure induced optical (a) absorption vs. energy, (b) absorption vs. wavelength, (c) conductivity vs. energy, and (d) conductivity vs. wavelength of AGeF₃ (A = K, Rb).

Photoconductivity is another term of optical conductivity (σ). It refers to the conductivity of photons in a substance⁵⁰. Figure 8c illustrates the σ of AGeF₃ (A = K, Rb) at 0 GPa and 30 GPa pressures. At 0 GPa, the perovskites exhibit σ in the visible area. The σ achieves its maximum value at negative ε₁(ω) (Fig. 9c). Therefore, KGeF₃ and RbGeF₃ show maximum σ in the energy region ~ 22–24 eV and ~ 18–22 eV, respectively. At 30 GPa, both compounds show increased σ in the visible region (Fig. 8d). Just like at 0 GPa, the highest σ peak of KGeF₃ is in the energy region ~ 22–24 eV. But the highest σ peak slightly shifts to the energy range ~ 19–21 eV for RbGeF₃. However, the σ of both compounds significantly increases in the visible region under pressure as can be seen in Fig. 8d because of increased absorption.

Figure 9.

Calculated pressure induced spectra of optical (a) reflectivity, (b) refractive index, (c) real part of dielectric function, and (d) imaginary part of dielectric function of AGeF₃ (A = K, Rb).

The reflectivity (R) is a critical optical feature for material’s photovoltaic applications. When exposed to photon with very low energy, KGeF₃ and RbGeF₃ reflect around 10% of the incident light (Fig. 9a). It rises for both substances when they transit from the infrared to the visible range. After transiting to the ultraviolet region, KGeF₃ and RbGeF₃ exhibit their highest R peak at ~ 23 eV and ~ 20 eV, where the dielectric function’s real part is negative. Under 30 GPa pressure, the R of both compounds spikes to 20% at zero energy. The application of pressure increases the R of both materials throughout almost whole energy regions. However, the relatively lower R (with or without the application of pressure) at low energy region indicates the potentiality of both compounds in solar cell applications. Additionally, both compounds should be applied as coating material to minimize solar heating because of their higher R at high energy region⁵¹.

The refractive index (n) is used to determine the amount of light bent or refracted as it enters into a substance. Furthermore, the phase velocity of an electromagnetic wave in a medium can also be calculated by n. According to Fig. 9b, both compound’s n is prominent at low energy and showing a fluctuating nature in the high energy region. When 30 GPa pressure is applied, the n of both compounds significantly enhances at 0 eV. This implies that AGeF₃ (A = K, Rb) should be preferable for optical devices, like photonic crystals and waveguides⁵².

The dielectric function characterizes the interaction of a material with incoming electromagnetic radiation. As a result, it is critical to have knowledge about dielectric function for optoelectronic device applications. The real (ε₁) and imaginary (ε₂) parts of dielectric function of AGeF₃ (A = K, Rb) are shown in Fig. 9c,d, respectively. The static dielectric function, ε₁ (0) is an important quantity, which measures the efficiency of an optoelectronic device¹⁵. A material with a greater ε₁ (0) has a lower rate of charge recombination, which results in improved performance of optoelectronic devices⁵². At ambient pressure, both compounds show identical ε₁ low energy, which enhances in the infrared–visible region and declines upon entering into the ultraviolet region. In addition, both KGeF₃ and RbGeF₃ show negative ε₁ at energy ranging from ~ 22 to 24 eV to ~ 18 to 22 eV, respectively. This implies that the compounds show high reflectivity at that energy region, which is evident in Fig. 9a. When 30 GPa pressure is applied, the ε₁ (0) is remarkably increased owing to the lower recombination of charges, which makes the compounds even more suitable for optoelectronic devices applications. At 0 GPa, the ε₂ is higher in the visible and early ultraviolet region, conveying high absorption at that regions⁵³. But the spectrum of the ε₂ shifts to the low energy region at high pressure. Specifically, the larger ε₁ and ε₂ at low energy together with smaller ε₁ and ε₂ at high energy areas evident the feasibility of both compounds in microelectronics and integrated circuits³⁹, and the superiority is greatly enhanced under pressure.

Mechanical properties

The elastic constants (C_ij) are usually used to determine the structural stability and mechanical characteristics of a material. The elastic nature describes how a material deforms under strain before recovering and returning to its original shape once the load is removed. It is important to reveal information about the binding properties between adjoining atomic planes, the anisotropic nature, and structural stability⁵⁴. A cubic compound has three independent elastic constants: C₁₁, C₁₂, and C₄₄. Table 3 lists the computed C₁₁, C₁₂, C₄₄, and Cauchy pressure (C₁₂ − C₄₄) for these two compounds under pressure. The elastic constants at ambient pressure are comparable with reported study²⁰ but rise linearly as pressure increases (Fig. S5a). Since the well-known Born stability requirements (C₄₄ > 0, C₁₁ − C₁₂ > 0, and C₁₁ + 2C₁₂ > 0)⁵⁵ are nicely matched by all the calculated elastic constants, both the studied compounds are mechanically stable even under applied pressure. In addition, C₁₂ − C₄₄ can identify the brittleness and ductility of materials. If C₁₂ − C₄₄ possesses a positive (negative) value, the material should be ductile (brittle)⁵⁶. Therefore, the titled compounds are expected to be ductile because of having positive values of C₁₂ − C₄₄ (Table 3). However, KGeF₃ is slightly more ductility than that of RbGeF₃ (Table 3).

Table 3.

Calculated elastic constants C_ij (GPa) and Cauchy pressure C₁₂ − C₄₄ (GPa) of AGeF₃ (A = K, Rb) at various applied pressures.

Pressure (GPa)	Compound	C11	C12	C44	C12 − C44
031	KGeF3	80.67	35.57	5.96	29.61
	RbGeF3	82.61	34.28	12.53	21.75
0	KGeF3	93.66	30.17	13.07	17.10
	RbGeF3	91.35	33.25	16.53	16.72
10	KGeF3	165.95	52.47	10.22	42.25
	RbGeF3	163.40	57.57	16.45	41.12
20	KGeF3	233.93	77.25	6.66	70.59
	RbGeF3	227.88	81.06	15.51	65.55
30	KGeF3	293.88	98.42	2.51	95.91
	RbGeF3	292.15	107.79	13.99	93.80

Various essential mechanical characteristics, such as bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (v), Pugh’s ratio (B/G), and Zener anisotropy index (A) of AGeF₃ (A = K, Rb) are determined using the estimated C_ij and presented in Table 4 with available reported data²⁰. The B and G are determined using the Voigt–Reuss scheme. The Voigt and Reuss coefficients describe the upper and lower bounds of the effective modulus, respectively. For cubic lattices, the Voigt bulk modulus (B_V) and Voigt shear modulus (G_V) as well as the Reuss bulk modulus (B_R) and Reuss shear modulus (G_R) are described by the well-known expressions^57,58. According to Hill’s theory⁵⁹, the B and G are the arithmetic mean of Voigt and Reuss expressions. Furthermore, the E and v are provided by the equations reported elsewhere⁵⁹. The B and G stand for fracture resistant and plastic deformation, respectively. Because of having greater B and G, RbGeF₃ is more fracture and plastic deformation resistant than KGeF₃ (Fig. S5b). E is a measure of material’s stiffness and has proportional relationship. As a result, RbGeF₃ will be stiffer than KGeF₃. However, the application of pressure induces more resistance to fracture and plastic deformation as well as makes them stiffer than that exhibited by the compounds without pressure. The variation of elastic moduli under applied pressure is graphically represented in Fig. S5b.

Table 4.

The calculated bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio v, Pugh’s ratio B/G, and Zener anisotropy index A of AGeF₃ (A = K, Rb) at various applied pressures.

Pressure (GPa)	Compound	B	G	E	B/G	v	A
031	KGeF3	50.61	10.52	29.53	4.81	0.40	0.26
	RbGeF3	50.39	16.35	44.27	3.08	0.35	0.52
0	KGeF3	51.34	18.81	50.30	2.73	0.366	0.412
	RbGeF3	52.62	20.76	55.03	2.49	0.326	0.569
10	KGeF3	90.30	22.01	61.08	4.10	0.387	0.180
	RbGeF3	92.97	26.85	73.47	3.46	0.368	0.311
20	KGeF3	129.4	22.92	64.93	5.65	0.416	0.085
	RbGeF3	130.00	30.67	85.29	4.24	0.391	0.211
30	KGeF3	163.57	22.35	64.14	7.32	0.434	0.026
	RbGeF3	169.24	33.22	93.55	5.09	0.407	0.152

The critical value of v to distinguish a materials’ ductile or brittle nature is 0.26⁶⁰. A material is said to be ductile if v is larger than 0.26. Thus, both KGeF₃ and RbGeF₃ are concluded as ductile materials (Table 4). Another essential feature is B/G, which has a crucial value of 1.75 to divide solid materials into ductile or brittle⁶¹. The calculated values of B/G also reveal the ductile behavior of both compounds (Table 4). However, the ductility of KGeF₃ is slightly higher than that of RbGeF₃. The ductility of the studied compounds at 0 GPa has previously been predicted²⁰, which is consistent with this study. It seen from Figs. S6a,b that both v and B/G, respectively, are increased with increasing pressure, which exhibit more ductile nature of the studied compounds under pressure. Interestingly, the outcome of v and B/G completely resemblances the data of C₁₂ − C₄₄ (Tables 3, 4).

In applied engineering, the ability to observe the influence of elastic anisotropy on these features is critical⁶². The properties of a system may be directionally dependent and anisotropic index is used to calculate it. For example, the shear anisotropic factor is utilized to determine the degree of anisotropy in the bonding strength of atoms along different crystallographic planes. Three shear anisotropic factors A₁, A₂, and A₃ have been found along the (100), (010), and (001) planes, respectively⁶³. For cubic systems, these are similar to the Zener anisotropy factor (A) and can be determined by the empirical formula⁶⁴. An isotropic material must have the unit value of A and the departure of unity denotes anisotropy⁶⁵. Both the compounds exhibit anisotropic nature, which are enhanced under pressure (Table 4). However, KGeF₃ show more anisotropy as compared to KGeF₃. Figure 10a–c show the direction dependence of E, G, and v, respectively, at 0 and 30 GPa pressure to highlight the anisotropic character of KGeF₃ and RbGeF₃. The isotropy of is represented by the spherical 3D plots, whereas anisotropy is revealed by non-spherical plots⁷. The elastic anisotropy of studied perovskites is appeared in all directions, as indicated by the non-spherical 3D contour plots. The deviation of spherical plots is more extreme at 30 GPa pressure than that exhibited at 0 GPa pressure, manifesting that the applied pressure may promote the anisotropy of AGeF₃ (A = K, Rb).

Figure 10.

Anisotropic 3D representation of (a) Young’s modulus, (b) shear modulus, and (c) Poisson’s ratio of AGeF₃ (A = K, Rb) at 0 and 30 GPa pressure.

Conclusions

The physical characteristics of lead-free halide perovskites AGeF₃ (A = K, Rb) under hydrostatic pressure are investigated using DFT. The lattice constant and cell volume reveal similarities with available studies, but decrease with the application of pressure. With increased pressure, the band gap narrows considerably, resulting in improving optical functions and make the compounds suitable for solar cell applications. The ionic/covalent bonds in the compounds also become stronger under pressure. Both compounds exhibit ductile nature at ambient pressure, as determined by their Cauchy pressure, Poisson’s ratio, and Pugh’s ratio. The compounds become more ductile because of pressure effect. The anisotropic nature of both compounds demonstrates similar nature as ductility. At last, it can be expected that this literature will shed fresh light on the improvement of perovskite solar cells and their prospective applications.

Computational method

The present computations are done by Cambridge Serial Total Energy Package (CASTEP) grounded on density functional theory (DFT)⁶⁶. The orbital shape approximations are not taken into account in the CASTEP code⁶⁷. Though the compounds KGeF₃ and RbGeF₃ are yet to synthesize their crystal structure is constructed by taking the reported crystallographic data determined by the theoretical investigation²⁰. Houari et al.²⁰ predicted that both compounds may have cubic perovskite-type structure with the space group Pm-3m (#221) as well as the lattice constant is 4.46 Å and 4.49 Å for KGeF₃ and RbGeF₃, respectively. The generalized gradient approximation (GGA) combined with Perdew–Berke–Emzerhof (PBE) functional is chosen to perform the exchange–correlation effect⁶⁸. To evaluate the electron–ion interaction, the Vanderbilt-type ultrasoft pseudopotential is selected⁶⁹. The cut off energy is 900 eV following a k-point grid of 12 × 12 × 12. To sample the Brillouin zone, the Monkhorst–Pack scheme⁷⁰ is considered. The convergence tolerance factors are set as 5 × 10^–6 eV/atom for total energy, 5 × 10^–4 Å for maximum displacement, 0.01 eV/Å for maximum force, and 0.02 GPa for maximum stress. The Broyden–Fletcher–Goldferb–Shanno (BFGS) algorithm⁷¹ is employed to optimize the crystal structure. The relatively similar approximations were also employed to optimize the crystal structure of experimentally synthesized Pb-based and Pb-free cubic halide perovskites^39,72–74. In this study, the hydrostatic pressure up to 30 GPa with an interval of 10 GPa is applied during structural optimization. The optimized crystal structure is constructed by VESTA software⁷⁵. The electronic and optical properties are calculated by using the same parameters that are utilized in structural optimization. The elastic constants and elastic moduli are determined by “stress–strain” method⁷⁶ embodied in the CASTEP code. The ELATE program⁷⁷ is used to create the three-dimensional (3D) anisotropic contour plots of Young’s modulus, shear modulus, and Poisson’s ratio.

Author contributions

M.S.A.: Investigation, methodology, data curation, writing the original draft. M.S.: Formal analysis, conceptualization, writing the original draft, supervision, reviewing, and editing. A.B., T.A., A.S.: Formal analysis, data curation. K.M.H.: Formal analysis, validation, writing the original draft, supervision, reviewing, and editing.

Data availability

The datasets generated and/or analyzed in this study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-022-12713-4.