Investigation of the half-metallicity signature and pressure-induced physical properties of cubic ACrO₃ (A = Si, Ge, Sn) multiferroic by DFT calculation

Abstract

We report a comprehensive first-principles investigation of the structural, electronic, elastic, magnetic, and optoelectronic properties and half-metallicity of ACrO₃ (A = Si, Ge, Sn) with space group Pm3m (no. 221), studied for the first time using density functional theory (DFT) in the pressure range of 0–30 GPa. The tolerance factor reveals that the compounds are cubic phase. SnCrO₃ is structurally stable. The optimized stable magnetic state of the compounds is ferromagnetic, and by applying Born stability criteria for elastic constants, it is found that ACrO₃ (A = Si, Ge, Sn) compounds are mechanically stable over the whole pressure range. The materials’ half-metallic nature is confirmed by band gaps and density of states (DOS) studies, which show that the majority spin-channel states are conducting, while the minority spin-down levels are semiconducting. Moreover, we calculated the bulk (B) and shear modulus (G), Poisson's ratio (v), Cauchy pressures, anisotropy (A), plasticity (B/C₄₄) and Young's modulus (E). The plasticity and bond distortion decrease as pressure increases. SnCrO₃ and SiCrO₃ are brittle at pressures from 0 to 10 GPa, and become ductile in the pressure range of 10 to 30 GPa. In contrast, GeCrO₃ is ductile over the whole pressure range. The spin-polarized band structure reveals that ACrO₃ (A = Si, Ge, Sn) maintains a direct band gap (R–R) over the whole pressure range. Several optical properties, including dielectric function, reflectivity, refractive index, conductivity, and absorption coefficient, are also calculated. It is revealed from the real part of the dielectric function that the compounds are optically metallic at lower energy and semiconducting at higher energy, and the plasma frequency rises with pressure. The optical properties and half-metallic response show that ACrO₃ (A = Si, Ge, Sn) has intriguing properties for spintronic and optoelectronic devices.

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1. Introduction

Modern electronic devices operate on the principle of electron control, which creates a binary system by considering whether or not electrons are flowing through a circuit. However, the computer industry believes that we have reached the computational limitations of such devices.^1,2 As a result, researchers around the world have been focusing on developing alternatives to conventional electronic devices, with spintronics emerging as a promising choice. Spintronic devices can improve device power by up to ten thousand times thanks to their octane coding based on the electron's spin.^3,4 Emerging technologies require the development of new materials that meet particular requirements. Specifically, magnetic materials with stable structures, good electrical properties, and the potential for thin-film application are needed for spintronic devices.^5,6 The vast majority of materials used in electronic devices are semiconductors. As magnetic behavior can be generated by doping with magnetic species, such materials can be applied in spintronics.^7–9 These materials combine high sensitivity with a remarkable ability to respond to environmental stimuli. Among these, multiferroic materials are an attractive option because they exhibit the coexistence of at least two of the three properties—ferroelectricity, ferromagnetism, and ferroelasticity—due to the interaction of spin, charge, lattice, and orbital degrees of freedom. The magnetoelectric effect, or coupling between electric and magnetic order parameters in multiferroic and magnetoelectric materials, offers an alternate method of electrically controlling the magnetic properties of dielectric materials. Numerous applications of the magnetoelectric effect have been proposed, such as modulators,¹⁰ field sensors, energy and frequency converters,¹¹ transformers and gyrators, transducers and memory devices,¹² and signal amplifiers.¹³ They have fascinating application possibilities in contemporary electronics and information technology, particularly in electromagnetic sensors and magnetic storage media. The development, performance enhancement, and use of multiferroic materials have emerged as critical areas of study for strongly correlated electronic systems and spintronics.¹⁴

A half-metallic compound exhibits the behaviors of both an insulator or a semiconductor in one spin channel and a metal in other spin channel.^15,16 de Groot et al.¹⁷ first introduced the idea of half-metallic compounds. This hybrid behavior culminates in complete (100%) spin polarization at the Fermi level. Subsequently, these substances generate spin-polarized currents, enhancing the efficiency of magnetoelectronic devices. Numerous materials have been proven to be half-metals, including transition metal pnictides and metal chalcogenides,¹⁸ oxides,^19,20 Heusler alloys,²¹ graphene nanoribbons,²² spinel ferrites,²³ and zinc blende magnetic semiconductors.²⁴ These magnetic materials have found significant applications in the field of spintronics.²⁵ However, more research is needed on half-metallicity signatures in multiferroic materials. ABO₃-type materials have been extensively studied for various technological applications—such as ferroelectric devices, gas sensors, and actuators—due to their ability to exhibit diverse physical and chemical properties, which depend on the specific metal atoms present at the A and B positions.²⁶ BiFeO₃, the most well-known multiferroic (MF) compound, has drawn much attention in recent decades due to its multiferroic characteristics; it exhibits a remarkable combination of electronic, optical, magnetic, and ferroelectric properties and can be readily synthesized via epitaxial growth on SrTiO₃ substrates. Other notable MF compounds include YMnO₃, BiMnO₃, PbNiO₃, FeTiO₃, PbVO₃, YMnO₃, TbMn₂O₅, Ca₃CoMnO₆, LuFe₂O₄, and BaNiF₄.^27–33 Jena et al.³⁴ investigated BiFeO₃ using DFT-based first-principles calculations using the pseudopotential method implemented in the Quantum Espresso (QE) package. They reported that the tetragonal phase goes through a half-metallic phase before evolving from a ferromagnetic semiconductor to a ferromagnetic metal as the c/a ratio drops from 1.264 to 1.016. The half-metallic phase in X = B/C/N/F-doped KNbO₃ was explored by Anar and Nazir.³⁵ They also looked into the band gap and estimated the Curie temperature in magnetically doped structures using the Heisenberg 2D Hamiltonian model, and they discovered that the temperature was high enough to be useful for practical purposes. Fang et al.^36,37 investigated the magnetic properties of BaMnO₃(Ba_1/3Sr_1/3Ca_1/3)TiO₃ and reported the existence of half-metallic properties using DFT. Using the LSDA+U scheme, Zhu et al.³⁸ investigated the half-metallic nature of the cubic BaCrO₃ structure and showed that the BaCrO₃ superlattice is a more stable interface than BaCrO₃. Ghosh et al.^39,40 conducted an experimental study on bismuth ferrite (BiFeO₃) and confirmed its ferromagnetic nature using X-ray photoelectron spectroscopy (XPS). Their findings were also validated by density functional theory (DFT+U), confirming that this material is feasible for spintronics applications. Recent studies on multiferroic (MF) materials have largely concentrated on modification strategies such as doping, co-doping, heterojunction engineering, the development of double-perovskite structures, and the tuning of intrinsic properties. In addition to these approaches, both experimental and theoretical efforts have been directed toward understanding key physical characteristics, including optical, vibrational, and elastic behavior. Among the theoretical methods, computational chemistry, particularly density functional theory (DFT), has proven to be a highly effective tool, offering a favorable balance between computational costs and predictive accuracy.^27,28 DFT, which is grounded in the Kohn–Sham formalism and incorporates the exchange–correlation functional, enables accurate modeling of the electronic structure and related properties. In this work, the B3LYP hybrid functional has been adopted owing to its demonstrated reliability and flexibility in the study of complex oxide systems.

To the best of our knowledge, there have been no prior theoretical or experimental investigations of the physical properties of ACrO₃ (A = Si, Ge, Sn) multiferroic compounds. Understanding these materials is important, as multiferroics with tunable properties are of great interest for advanced electronic and spintronic applications. Our work aims to fill this gap by systematically investigating and comparing the structural, mechanical, electronic, and optical properties of ACrO₃ (A = Si, Ge, Sn) compounds under ambient and high-pressure conditions (up to 30 GPa). Such pressure-dependent studies can reveal important insights into the band gap behavior and Curie temperature evolution, which are crucial for designing materials that function reliably at higher temperatures. Therefore, this study provides both a novel theoretical foundation and potential practical implications for the design of future multifunctional devices, particularly in the fields of spintronics and high-pressure electronics.

2. Computational details

The DFT-based CASTEP computer software with the generalized gradient approximation (GGA) was used for first-principles calculations. The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional with projected augmented-wave (PAW) pseudopotentials was selected for the calculations,^41–43 and O-2s² 2p⁴, Cr-3s² 3p⁶ 3d⁵ 4s¹, Si-3s² 3p², Ge-3d¹⁰ 4s² 4p², and Sn-4d¹⁰ 5s² 5p² were used as valence electrons. For the lattice optimization, the convergence parameters were set as follows: (i) maximum ionic displacement of 0.002 Å, (ii) maximum stress component of 0.10 GPa, and (iii) maximum ionic force of 0.05 eV Å⁻¹. The electromagnetic wave function extension in a plane-wave basis set uses an energy cut-off of 400 eV, and the Brillouin zone is sampled using 8 × 8 × 8 k-points for all calculations. The Broyden method was applied to optimize the geometric structure using the Fletcher–Goldfarb–Shanno (BFGS) minimization technique.⁴⁴ The Vanderbilt-type ultrasoft pseudopotential (UPP) formalism was used to analyze the interactions between valence electrons and ions. UPPs achieve a substantially softer pseudo-wave function, requiring a significant reduction in plane waves for calculations with the same level of precision.⁴⁵ Additionally, spin polarization was only applied to the magnetic state in the lowest energy computation, and spin polarization was not applied to the non-magnetic state in all other computations. The elastic constants were determined using first-principles computations by carrying out a sequence of specified homogeneous distortions with finite parameters and calculating the resulting stress by optimizing the interior atomic freedom. With an upper limit of 0.5%, each strain element had three positive and three negative amplitudes. After that, the calculated stress as a function of strain was fitted linearly to obtain the elastic modulus. To facilitate the calculation of the optical properties for a dense mesh of evenly distributed k-points, BZ integration was performed using a 12 × 12 × 12 MP k-mesh.

3. Results and discussion

3.1. Ground state structural features

ACrO₃ (A = Si, Ge, Sn) adopts a cubic structure with the Pm3̄m space group, as shown in Fig. 1. These structures are optimized in ferromagnetic (FM) and non-magnetic (NM) states to verify the most stable state. All the investigated compounds are more stable in the ferromagnetic state, as shown in Table 1.

Fig. 1 The cubic crystallographic unit cell of ACrO₃ (A = Si, Ge, Sn).

Table 1 Calculated lattice constant a (A), equilibrium volume V₀, normalized lattice parameter, and volume cubic ACrO₃ (A = Si, Ge, Sn) multiferroic compounds at different pressure

Compound	Parameter	0 GPa	10 GPa	20 GPa	30 GPa
SiCrO3	Energy EFM (eV)	−3885.06	−3884.90	−3884.77	−3884.68
	Energy ENM (eV)	−3884.74	−3884.74	−3884.64	−3884.47
	Spin-polarized lattice constant a (A)	3.79	3.70	3.65	3.60
	Non-spin-polarized lattice constant a (A)	3.75	3.69	3.65	3.61
	Volume (FM) V0	54.57	50.52	48.79	46.93
	Volume (NM) V0	53.07	50.49	48.73	47.11
	Normalized lattice parameter a/a0	1	0.97	0.96	0.94
	Normalized volume V/V0	1	0.92	0.89	0.85
GeCrO3	Energy EFM (eV)	−6253.95	−6253.88	−6253.76	−6253.70
	Energy ENM (eV)	−6253.75	−6253.71	−6253.62	−6253.57
	Spin-polarized lattice constant a (A)	3.81	3.74	3.69	3.66
	Non-spin-polarized lattice constant a (A)	3.78	3.72	3.67	3.63
	Volume (FM) V0	55.63	52.72	50.58	49.09
	Volume (NM) V0	54.40	51.58	49.61	47.94
	Normalized lattice parameter a/a0	1	0.98	0.968	0.960
	Normalized volume V/V0	1	0.94	0.90	0.88
SnCrO3	Energy EFM (eV)	−5765.26	−5765.08	−5764.89	−5764.56
	Energy ENM (eV)	−5764.98	−5764.89	−5764.74	−6764.44
	Spin-polarized lattice constant a (A)	3.87	3.81	3.72	3.68
	Non-spin-polarized lattice constant a (A)	3.84	3.77	3.72	3.68
	Volume (FM) V0	58.29	55.60	51.78	49.85
	Volume (NM) V0	56.91	53.95	51.77	49.85
	Normalized lattice parameter a/a0	1	0.98	0.96	0.950
	Normalized volume V/V0	1	0.953	0.88	0.855

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To investigate how pressure affects the crystal structure of ACrO₃ (A = Si, Ge, Sn), we examined the variation in unit cell volume and lattice constant under pressures ranging from 0 to 30 GPa. The results in Fig. 3(a) and (b) indicate that the lattice constant and volume decrease as pressure increases, although they do not decrease at the same rate. As pressure increases, the ratios of a/a₀ and the normalized volume V/V₀, where V₀ and a₀ stand for the volume at zero pressure, and the equilibrium lattice parameter decrease, as shown in Fig. 3(c). This normalization allows for a more precise comparison of how much the lattice parameter and volume change with pressure. Fig. 3 provides a comprehensive view of how ACrO₃ (A = Si, Ge, Sn) responds to applied pressure, confirming that the material retains its cubic phase while exhibiting significant changes in its structural parameters.

Fig. 2 Cohesive energy of cubic ACrO₃ (A = Si, Ge, Sn) compounds at different pressure.

Fig. 3 Variation of (a) lattice parameter, (b) cell volume and (c) normalized lattice parameter and volume of ACrO₃ (A = Si, Ge, Sn) as a function of pressure.

We also calculated the cohesive energy (E_coh) of the ABO₃ compounds to explore more about their structure stability using the equation:⁴⁶

where is the total equilibrium energy of ABO₃ and E^iso_A, E^iso_B(Cr) and E^iso_O are the energies of the isolated atoms, respectively.

A material with high cohesive energy is more stable due to stronger bonds (e.g., ionic, covalent) that require more energy to break. High cohesive energy correlates with greater hardness, strength, and resistance to deformation, as well as tightly bound electrons that influence conductivity and electronic properties. These results suggest that pressure has a non-linear effect on the cohesive energy of ACrO₃ (A = Si, Ge, Sn) compounds as shown in Fig. 2. The initial reduction in cohesive energy under pressure could be due to increased repulsion or distortion in the lattice. The slight recovery at higher pressures may indicate a more compact or stabilized structure. All of our compounds possess high cohesive energy, so we can infer that they are more resistant to mechanical stress and chemical corrosion, making them suitable for harsh environments (Table 2).

Table 2 The cohesive energy of cubic ACrO₃ (A = Si, Ge, Sn) multiferroic compounds at different pressure

	Structure	0 GPa	10 GPa	20 GPa	30 GPa
Cohesive energy	SiCrO3	24.98	19.97	20.07	20.33
	GeCrO3	23.34	20.94	20.87	21.01
	SnCrO3	25.37	21.79	21.81	21.91

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3.2. Bonding analysis

The atomic charges, chemical bond lengths, and bond populations determined using the Mulliken technique are displayed in Table 3. Regarding the O atoms, the charge transfer from the A (= Si, Ge, Sn) and Cr atoms is around 0.90, 1.22, 1.06 and 0.94 electrons, respectively. The minimum and maximum values for the bond populations suggest that the chemical bond has strong iconicity and covalency, respectively.^47,48 Consequently, we conclude that ACrO₃ (A = Si, Ge, Sn) exhibits covalent and ionic bonding behavior. Table 3 displays the predicted O–A (A = Si, Ge, Sn) and Cr–O bond lengths based on Mulliken population analysis. A multiferroic compound tolerance factor can be determined by measuring the bond lengths between the constituent atoms, which allows for the identification of the crystal symmetry given by ref. 49:where (A–O) and (Cr–O) are the bond lengths. The computed tolerance factors are 0.97, 1.009, and 0.999 for SiCrO₃, GeCrO₃, and SnCrO₃, respectively. The compounds all fall within the range of 0.93–1.02,⁵⁰ which validates that they are cubic.

Table 3 Mulliken charges, bond lengths, and bond populations analysis of ACrO₃ (A = Si, Ge, Sn) obtained using the GGA method

	Species	s	P	d	Total	Charge (electron)	Bond	Population	Bond length (Å)	Tolerance factor (t)
SiCrO3	Si	2.04	1.06	0	3.10	0.90	O–Si	0.03	2.60	0.97
	Cr	2.28	6.39	4.44	13.10	0.89	O–Cr	0.85	1.88
	O	1.85	4.75	0	6.60	−0.60
GeCrO3	Ge	1.71	1.07	10	12.78	1.22	O–Ge	−0.16	2.69	1.009
	Cr	2.44	6.58	4.31	13.33	0.67	O–Cr	1.05	1.90
	O	1.85	4.78	0	6.63	−0.63
SnCrO3	Sn	1.98	0.95	10.0	12.94	1.06	O–Sn	0.06	2.74	0.999
	Cr	2.31	6.45	4.30	13.06	0.94	O–Cr	0.82	1.93
	O	1.85	4.82	0	6.67	−0.67

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3.3. Electronics properties

A thorough understanding of a material's electronic structure is essential for determining its potential applications across various fields of scientific research. The energy band gap, in particular, plays a key role in defining the suitability of a material for use in smart devices, spintronics, and memory technologies. Furthermore, theoretical quantum-level analyses provide detailed insights into the electronic nature of materials. Based on their electronic properties, materials can be broadly classified as metals, insulators, semiconductors, spin-gapless semiconductors, half-metals, and more.⁵¹ The electronic properties of a material are determined by the contributions of various electronic states in the valence and conduction bands. Understanding the electrical structure is crucial for providing helpful information regarding their experimental modeling and potential use in fabricating numerous energy devices. It has been reported that GGA typically underestimates the electronic band gap, especially in the case of semiconductors and insulators. This discrepancy is primarily explained by the GGA functional's simplicity and the lack of the necessary flexibility to precisely capture exchange–correlation energy, which results in a comparatively lower electronic band gap energy value. Therefore, we studied the band structure, DOS and PDOS using the GGA+U functional.⁵² The electrical band structures of ACrO₃ (A = Si, Ge, Sn) at varying pressures are shown in Fig. 4. Fig. 4 indicates that the top of the valence band (VB) and the bottom of the conduction band (CB) do not overlap; both occur at the R symmetry point. This suggests that ACrO₃ (A = Si, Ge, Sn) exhibits half-metallic behavior.

Fig. 4 Predicted spin-polarized band structures of ACrO₃ (A = Si, Ge, Sn) under hydrostatic pressure of 0–30 GPa using the GGA+U functional.

Fig. 5 Density of states (DOS) for GGA in the cubic structure of ACrO₃ (A = Si, Ge, Sn).

It has been discovered that under ambient conditions, the top of the VB and the bottom of the CB are located at the R symmetry point. Remarkably, it was found that for SnCrO₃, elevating the pressure caused the top of the VB at the R and Γ symmetry locations to move closer to the Fermi level and at 20 and 30 GPa, the top of the valence band at the R and Γ symmetry points crosses the Fermi level. Conversely, no relative shift was seen when looking at the band structures of SiCrO₃ and GeCrO₃, suggesting that the electrical characteristics of these materials are stable under external pressure. Due to this band structure stability, SiCrO₃ and GeCrO₃ may retain consistent electrical conductivity and other electronic properties. Closely examining Fig. 4 reveals that the energy band edges at the R and Γ points have flat bands, while the remaining points exhibit slightly bouncy curvature. This indicates that ACrO₃ (A = Si, Ge, Sn) has higher effective masses at the L and Γ points than at other locations.⁵³

The spin-polarized total and partial DOS, displayed in Fig. 5 and 6, explain contributions of various electronic states to the band structure. When considering the spin-up channel, the ACrO₃ (A = Si, Ge, Sn) compounds possess occupied states at the Fermi level and show metallic characteristics. Conversely, ACrO₃ (A = Si, Ge, Sn) exhibits semiconducting behavior for the spin-down channel and unoccupied states at the Fermi level. Fig. 5 shows that as the pressure is increased to 30 GPa, the conduction bands move downward, the valence bands shift upward, and the band gap eventually contracts for SiCrO₃ and SnCrO₃. Fig. 5 indicates that at 30 GPa, there is no band gap at the Fermi level, and the material exhibits metallic behavior. Thus, SiCrO₃ and SnCrO₃ undergo a phase shift at 30 GPa. The semiconducting gap in GeCrO₃ is almost constant throughout the pressure range, suggesting that the electronic bandgap is highly stable and is not affected by variations in pressure.

Fig. 6 Spin-polarized partial density of states (PDOS) of cubic ACrO₃ (A = Si, Ge, Sn) at various pressures.

Fig. 7 The relation between Curie temperature and pressure.

Fig. 6 clearly shows that in the spin-up channel, the O-2p state contributes to the bottom of the VB at −2.4 eV, while the Cr-3d state at the Fermi level contributes significantly. The plot indicates that the hybridization of O-2p and Cr-3d states is responsible for the top of the VB and that this hybridization contributes more to the spin-up channel than the spin-down channel, making the ACrO₃ (A = Si, Ge, Sn) compounds metallic in the spin-up channel. Interestingly, the O-2p state at the bottom of the valence band drifts leftwards with increasing pressure. In the CB, the highest contribution comes from the Sn-s state at 2.2 eV and the O-2p state at 4.5 eV. In the case of a down channel, polarization occurs at the Fermi level, and the 3d state of the Cr atom jumps into the CB and is localized at 2.2 eV.

3.4. Spin magnetic moment and spin polarization

The temperature at which spontaneous magnetization in magnetic material ceases is called the Curie temperature, sometimes referred to as the Curie point or the magnetic transition point. The Curie temperature is calculated using the Heisenberg mean field approximation:⁵⁴where ΔE is the energy difference between non-magnetic E_NM and ferromagnetic state E_FM and represents the energy barrier that must be overcome to transition from the non-magnetic to the ferromagnetic state. Fig. 7 illustrates that the Curie temperature did not fall linearly with pressure but rapidly decreased, especially for SiCrO₃, which fell faster than the other investigated compounds. Leger et al.⁵⁵ investigated the effect of pressure on the T_C of alloys with transition atoms and found similar trends, so our results are consistent with the literature. We found that the Curie temperatures at normal pressure are 1236.71 K, 772.94 K, and 1082.12 K for ACrO₃ (A = Si, Ge, Sn) compounds, respectively, which are relatively higher than room temperature and only seen in full-Heusler alloys.⁵⁶ ACrO₃ (A = Si, Ge, Sn) has a high Curie temperature, making it a promising material for spintronics applications⁵⁷ (Fig. 8).

Fig. 8 Effect of pressure on the magnetic moments at O₃, Cr, Si, Ge, and Sn atomic sites for ACrO₃ (A = Si, Ge, Sn) multiferroic materials in GGA calculations.

Fig. 9 Spin moment versus lattice constants for ACrO₃ (A = Si, Ge, Sn) compounds

At T = 0 K, the spin-polarized computation directly obtains the magnetic moment. Utilizing the actual magnetic moments per formula unit, the magnetic moment is calculated as follows:

μ = 3O_oxygen + Cr + A(Sn, Ge, Si) (4)

The calculated magnetic moment is listed in Table 4, and it is observed that at pressure up to 10 GPa, the total magnetic moment is almost the same for all the compounds. However, above 10 GPa, the total magnetic moment changed for SnCrO₃. Additionally, we looked into how pressure affects the magnetic moment per atom of each of the three compounds between 0 and 30 GPa. Moulay et al.'s⁵⁸ investigation of the pressure dependence of magnetic moments for multiferroic compounds revealed that the individual elements’ magnetic moments do not significantly change in response to pressure.

The constant compression of its lattice directly impacts the physical characteristics of a material. These characteristics influence the compound's magnetic, structural, electrical, and optical qualities. The robust shift lattice parameter (RTLC) is the critical lattice parameter at which the total magnetic moment of a half-metal substance abruptly shifts to a non-integer value, and the electronic nature of the compound changes from half-metal to metal at that lattice parameter, making it a fundamental lattice constant to study.⁵⁹Fig. 9 shows that the magnetic moment for SnCrO₃ changed from an integer (≅1.99μ_B) to a non-integer value as the lattice constant decreased with an increase in external pressure. At 3.78 Å, there is a sudden transition, and the compound loses its integer magnetic moment (2μ_B to 0.50μ_B). This sudden shift verifies that the robust lattice constant for SnCrO₃ is 3.78 Å.

Fig. 10 Elastic constant (left) and bulk modulus (right) of ACrO₃ (A = Si, Ge, Sn) at different hydrostatic pressures.

The spin polarization of ACrO₃ (A = Si, Ge, Sn) is reflected by the bandgap present at the Fermi level, which is essentially the difference in spin density between the spin-up and -down channels near the Fermi level (E_F). Spin polarization can be obtained from the following relation:

where N↑(E_F) and N↓(E_F) are the DOS for the spin-up and spin-down channels near the Fermi level, respectively. For these two channels, when any of the states of the Fermi level is zero, 100% spin polarization will occur. From the band diagram, it is observed that the DOS along the down channel near the Fermi level is zero, which makes ACrO₃ (A = Si, Ge, Sn) a half-metallic material (Table 4).

Table 4 Spin magnetic moment and Curie temperature of ACrO₃ (A = Si, Ge, Sn) at various pressures

	Magnetic moment	0 GPa	10 GPa	20 GPa	30 GPa
SiCrO3	Moment μO	0.03	0.09	0.12	0.12
	Moment μSi	0.06	0.07	0.07	0.07
	Moment μCr	1.90	1.85	1.83	1.81
	Total μ	1.99	2.01	2.02	2.01
	Curie temperature TC (K)	1236.71	616.35	502.41	386.47
GeCrO3	Moment-μO	0.03	0.09	0.12	0.12
	Moment-μGe	0.06	0.07	0.07	0.07
	Moment-μCr	1.90	1.85	1.81	1.81
	Total μ	1.99	2.01	2.0	2.0
	Curie temperature TC (K)	772.94	657	541.06	502.41
SnCrO3	Moment μO	−0.03	0.03	0.03	0.03
	Moment μSn	0	0	0	0
	Moment μCr	2.02	1.98	0.47	0.47
	Total μ	1.99	2.01	0.50	0.50
	Curie temperature TC (K)	1082.12	734.29	579.70	463.76

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3.5. Mechanical properties

Elastic constants are necessary to obtain a more thorough theoretical comprehension of the characteristics of materials that are determined by the electron–phonon interaction processes and the DOS of phonons. As the stress–strain curve in many minerals exhibits low nonlinearity, practical applications can assume a linear relationship between stress and strain.⁶⁰ Elastic constants can be used to calculate essential physical parameters like melting point, hardness, sound velocities, Debye temperature, and shear modulus. The elastic constants are the total energy second derivative for different lattice deformations. More thorough computations of the total energy are necessary⁶¹ for the elastic constants. The elastic constants can be calculated from the formula:where E is the internal energy, V₀ indicates the equilibrium volume, and e_i and e_j are the stress and strain components.

As ACrO₃ (A = Si, Ge, Sn) is a cubic structure with a high degree of symmetry, the elastic constant reduces to three independent elastic constants: C₁₁, C₁₂, and C₄₄.⁶² The following is an expression of the conventional mechanical stability conditions in cubic crystals at equilibrium in terms of elastic constants:^63,64

C₄₄ > 0, C₁₁ > |C₁₂|, C₁₁ + 2C₁₂ > 0 (7)

The equations for mechanical stability under pressure are given by ref. 65:From the data reported in Table 5, the dynamic stability of the compounds ACrO₃ (A = Si, Ge, Sn) was assessed using pressure-modified elastic criteria. SiCrO₃ and SnCrO₃ are dynamically stable at 0, 10, and 30 GPa, but both exhibit instability at 20 GPa, primarily due to a negative C₄₄ − P, indicating shear softening under pressure. In contrast, GeCrO₃ is unstable at ambient pressure but becomes dynamically stable at pressures above 10 GPa, suggesting pressure-induced stabilization. The bulk modulus (B), shearing modulus (G), Young's modulus (E), anisotropy (A), Poisson's ratio (v), Pugh's ratio (B/G), plasticity measurement (B/C₄₄), and Kleinman parameter (ς) were calculated using the following expressions, respectively:^66–70As the shear modulus indicates the rigidity of a compound, the mechanical failure in ACrO₃ (A = Si, Ge, Sn) should be revealed by the shear component under the criteria G < B in this case. The bulk modulus of a material determines the counteraction against the volume change due to applied pressure, compared to the shear modulus, which defines the counteraction against plastic distortion.⁷¹ The bulk modulus values of SiCrO₃, GeCrO₃ and SnCrO₃ exhibit a clear increasing trend with applied pressure, indicating enhanced resistance to volume compression. For SiCrO₃, the bulk modulus increases from 674.13 GPa at 0 GPa to 840.77 GPa at 30 GPa, suggesting improved mechanical rigidity under compression. GeCrO₃ shows a substantial increase from 185.12 GPa to 852.15 GPa over the same pressure range, which may indicate a pressure-induced structural stabilization. In the case of SnCrO₃, the bulk modulus rises steadily from 616.97 GPa to 840.77 GPa, reflecting consistent stiffening behavior under pressure. The stiffness of a solid can be determined by calculating its Young's modulus (E), which is the ratio of tensile stress to tensile strain, and a low E value indicates greater plasticity. In the current investigation, the Young's modulus (E) for GrCrO₃ and SnCrO₃ is relatively low at 0 and 10 GPa, and the compounds significantly lose flexibility at higher pressures. Conversely, SiCrOSiCrO₃ has a maximum value at pressure of 10 GPa.

Table 5 Bulk modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), hardness (GPa), plasticity measurement (B/C₄₄), Pugh's ratio (B/G), (C₁₁–C₄₄) Cauchy pressure (GPa), Poisson's ratio (v), Kleinman parameter (ζ) and Zener anisotropy factor (A) of ACrO₃ (A = Si, Ge, Sn)

Compound	Elastic parameters	0 Pa	10 Gpa	20 Gpa	30 Gpa
SiCrO3	C 11	651.77	744.04	711.45	782.30
	C 12	348.24	393.47	369.52	449.62
	C 44	60.88	68.21	2.47	56.31
	Bulk modulus (B)	674.12	765.49	725.24	840.77
	Shear modulus (Voigt) GV	72.88	83.75	68.88	77.79
	Shear modulus (Reuss) GR	80.05	90.26	4.07	76.56
	Shear modulus (Hill) G	76.46	87.01	36.47	77.18
	Young's modulus (Hill) E	221.05	251.50	107.63	224.67
	Cauchy pressure (C12–C44)	287.36	325.26	367.05	393.31
	Kleinman parameter (ζ)	0.65	0.64	0.64	0.68
	Pugh's ratio (B/G)	8.81	8.79	19.88	10.89
	Poisson ratio (v)	0.44	0.44	0.47	0.45
	Plasticity measurement (B/C44)	11.07	11.22	293.62	14.93
	Anisotropy factor (A)	0.40	0.38	0.01	0.33
	Dynamic stability	Stable	Stable	Unstable	Stable
GeCrO3	C 11	102.49	680.65	806.98	830.71
	C 12	133.87	382.18	396.60	436.79
	C 44	75.06	89.95	85.52	86.39
	Bulk modulus (B)	185.11	722.5	800.09	852.14
	Shear modulus (Voigt) GV	8.73	77.68	99.18	96.06
	Shear modulus (Reuss) GR	−57.18	106.94	111.54	111.14
	Shear modulus (Hill) G	−24.20	92.31	105.36	103.73
	Young's modulus (Hill) E	−75.91	265.62	302.79	299.06
	Cauchy pressure (C12–C44)	58.81	292.23	311.08	350.4
	Kleinman parameter (ζ)	1.19	0.67	0.61	0.64
	Pugh's ratio (B/G)	−7.64	7.82	7.59	8.21
	Poisson ratio (v)	0.56	0.43	0.43	0.44
	Plasticity measurement (B/C44)	2.46	8.03	9.35	9.86
	Anisotropy factor (A)	−4.78	0.6	0.41	0.43
	Dynamic stability	Stable	Stable	Stable	Stable
SnCrO3	C 11	677.05	707.66	711.45	782.30
	C 12	278.44	282.71	369.52	449.62
	C 44	126.53	92.78	2.47	56.31
	Bulk modulus (B)	616.96	636.54	725.24	840.77
	Shear modulus (Voigt) GV	105.02	103.54	68.88	77.79
	Shear modulus (Reuss) GR	148.17	119.76	4.07	76.56
	Shear modulus (Hill) G	126.59	111.65	36.47	77.18
	Young's modulus (Hill) E	355.48	316.46	107.63	224.67
	Cauchy pressure (C12–C44)	151.91	189.93	367.05	393.31
	Kleinman parameter (ζ)	0.54	0.53	0.64	0.68
	Pugh's ratio (B/G)	4.87	5.70	19.88	10.89
	Poisson ratio (v)	0.40	0.41	0.475	0.455
	Plasticity measurement (B/C44)	4.87	6.86	293.62	14.93
	Anisotropy factor (A)	0.63	0.43	0.01	0.33
	Dynamic stability	Stable	Stable	Unstable	Stable

---

Poisson's ratio, also known as the transverse deformation coefficient, is an elastic constant that describes how a material deforms sideways in response to longitudinal force. Poisson's ratio is a characteristic with a minimum value of approximately 0.25⁷² in most metals. We can infer the ductile or brittle behaviors of GeCrO, SiCrO, and SnCrO under various pressure situations from Fig. 11. The graph illustrates how each compound responds differently to pressure variations. All of the compounds exhibit ductility across the whole pressure range. The decreasing trend of Poisson's ratio in GeCrO₃ under pressure may suggest a transition toward a more brittle nature, which could be due to pressure-induced changes in bonding or structure.

The distortion of a material (solid) that experiences irreversible shape changes in response to an applied force is called plasticity in material science.⁷³ The link between applied pressure and the plasticity of SiCrO, GeCrO, and SnCrO is depicted in Fig. 10 (right). The plasticity of SiCrO₃, GeCrO₃, and SnCrO₃ exhibits a clear dependence on applied pressure, reflecting changes in their mechanical response under stress. For SiCrO₃, plasticity increases from 11.07 at 0 GPa to 14.93 at 30 GPa, indicating improved capacity for permanent deformation. GeCrO₃ initially shows low plasticity (2.47 at 0 GPa), which rises steadily to 9.86 at 30 GPa, suggesting pressure-induced enhancement in ductility, likely due to changes in bonding or structural arrangement. SnCrO₃ follows a similar trend, with plasticity increasing from 4.87 to 14.93 over the same pressure range, pointing to significant improvement in deformability under compression. These trends highlight the potential of these materials for high-pressure applications where mechanical adaptability is critical.⁷⁴

Fig. 11 Ductile-brittle behavior of ACrO₃ (A = Si, Ge, Sn) at different pressures.

Anisotropy is a crucial parameter for compounds used in engineering applications. A material with an anisotropy value equal to one is considered isotropic, while any other deviation from unity indicates the degree of elastic anisotropy present in the crystal. The value of parameter A for the compounds ACrO₃ (A = Si, Ge, Sn) is either less than or higher than unity, making the crystal anisotropic.

3.6. Optical properties

Knowing a material's optical characteristics is essential for figuring out its optoelectronic nature. Furthermore, examining optical properties is crucial because semiconductors frequently display metal-like activity at extremely low frequencies and insulating behavior at very high frequencies.⁷⁵Fig. 12(a) and (b) display the computed ε₁(ω) and ε₂(ω) components of the dielectric function in the pressure range of 0–30 GPa in the 0–30 eV range for ACrO₃ (A = Si, Ge, Sn) with electric field polarization vectors along the [100] direction. As the resulting photon energy increased in the low-energy zone, the dielectric functions (both the real part ε₁ and the imaginary part ε₂) were significantly lowered, as shown in Fig. 12(a). The observed dielectric constant trend suggests that the compounds under investigation are appropriate for use in microelectronic applications and integrated circuits.⁷⁶ The dielectric function is a complex function that is given by ref. 77:

ε(ω) = ε₁(ω) + iε₂(ω) (17)

One of the key properties of half-metallic materials is the plasma frequency, ω_p. It has a real component of the dielectric function with zero value.⁷⁷ Based on Fig. 12(a), we can see that SiCrO₃ and GeCrO₃ have zero values of ε₁ over the entire pressure range in the ultraviolet region at 5–10 eV. In contrast, SnCrO₃ also has zero values of ε₁ in the visible energy range for 10–30 GPa and at pressures of 0 to 20 GPa in the ultraviolet region, where transitions occur from metallic to dielectric. In the IR spectrum region, the real component of the dielectric function shows a distinctive peak. The intra-band Drude-like characteristic of ACrO₃ (A = Si, Ge, Sn) is the negative frequency of the real part of the dielectric function. The imaginary part of the dielectric function has a value that is very close to zero, and the real part tends to be united in the high-energy zone. It verifies that the material is optically isotropic by presenting it as a virtually transparent substance with minimal absorption in the infrared spectrum. The static dielectric constant ε₁(0) for SnCrO₃ε₁(0) at pressures 0, 10, 20, and 30 GPa are 0.5, 5, 5.3 and 5.5 eV, respectively SnCrO₃. It is evident from Fig. 12(a) and (b) that the real and imaginary components of the dielectric functions both have relatively greater values at low energies and begin to decline as energy increases.

Fig. 12 Energy-dependent optical parameters of ACrO₃ (A = Si, Ge, Sn) at various pressures. (a) Real, (b) imaginary part of dielectric function, (c) refractive index, (d) conductivity, (e) absorption co-efficient, (f) reflectivity and (g) loss function.

Refractive index n(ω) is a crucial optical characteristic that may be found in terms of the complex dielectric function as shown in ref. 78:

The spectrum of refractive index with incident energy is displayed in Fig. 12(c). The presence of free electrons in metallic compounds is responsible for the maximum value of the ACrO₃ (A = Si, Ge, Sn) refractive index in the low-energy regions. For SiCrO₃ and GeCrO₃, the refractive index shows significant peaks in the visible energy range, remains approximately constant across the complete pressure range, and gradually drops in the ultraviolet energy region. SnCrO₃ has refractive index peaks in the infrared region and progressively drops in the visible and ultraviolet wavelength ranges.

The optical conductivity σ(ω) of ACrO₃ (A = Si, Ge, Sn) is shown in Fig. 12(d). It is an important optical parameter that may be expressed as follows:

where W_CV is the transition probability per unit of time.⁷⁹Fig. 12(d) shows that for SiCrO₃ and GeCrO₃, conductivity increases quickly in the visible zone and reaches its maximum value in the ultraviolet range as incident energy increases. SiCrO₃ and GeCrO₃ have the highest conductivity among these compounds at 20 GPa, with no peaks seen in the visible or infrared spectrum. The spectrum for SnCrO₃ differs significantly from those of the other compounds, with two minor peaks occurring in the energy ranges of 4.8–7.5 eV and 1–2.5 eV. For the second significant peak, the conductivity in the ultraviolet energy region increases as pressure rises. Because of the hybridization of the valence and conduction bands, conductivity begins at zero photon energy, indicating that there is no band gap at the Fermi level for the up channel, which was confirmed during the investigation of the DOS.

The absorption coefficient I(ω) spectra for ACrO₃ (A = Si, Ge, Sn) are shown in Fig. 12(e). This spectrum gives information on light absorption and the final disappearance of an incident electromagnetic beam that passes through an absorbing material. In the infrared region, at 0 and 30 GPa, compounds ACrO₃ (A = Si, Ge, Sn) absorb incident light and show semiconducting characteristics. When the pressure increases, the first noticeable peaks for SiCrO₃ and GeCrO₃ are seen in the 6–9 eV energy range. These peaks move to the right with increasing pressure. SnCrO₃, on the other hand, has a modest peak in the infrared region and reaches its most significant value in the ultraviolet. ACrO₃ (A = Si, Ge, Sn) show recurrent peaks in the 10–20 eV energy region. The absorption edge was altered with pressure because of the decrease in the band gap. Our analysis shows that SnCrO₃ has an excellent absorption coefficient in the high-energy range, with the most prominent peaks occurring at 27.5 eV.

Reflectivity is the ratio of incident power to reflected power. It describes a material surface characteristic and is calculated using the following equation⁸⁰,

Fig. 12(f) shows the reflectivity as a function of incident energy. SiCrO₃ and GeCrO₃ have reflectivity of between 40 and 45% at zero-incident energy, and their spectra are nearly identical. The reflectivity decreases in the visible energy range before increasing to a high of roughly 48% at 0 GPa and a roughly constant value of 50% between 0 and 30 GPa. For SnCrO₃, reflectivity starts from 0.51, 0.78, and 0.85 eV at 0, 10, 20 and 30 GPa, respectively, for zero-incident energy. Reflectivity falls significantly with increasing incident energy at 10 and 30 GPa, dropping by about 55%, and at 0 and 10 GPa. A reduction of about 30% within the visible energy range, prior to reaching 1.5 eV, with some minor fluctuations in the 4–7 eV range. In the ultraviolet radiation range, the maximum reflectance is roughly 52% for 0 and 10 GPa and 44% and 46% for 20 and 30 GPa, respectively. It is seen that the reflectivity peaks for all the compounds shift to the right, undergoing a significant form change as pressure increases from 0 to 30 GPa. Fig. 12(f) reveals that ACrO₃ (A = Si, Ge, Sn) is not a good reflector in the visible region, and the R(0) value is relatively high in the IR region compared to other areas. Thus, ACrO₃ (A = Si, Ge, Sn) can be used in IR mirrors as a coating for optical instruments.⁸¹ The loss per unit area resulting from heating, plasma resonance, and scattering is measured by the optical energy loss factor L(ω). The electron energy loss function L(ω) can be obtained from the real and imaginary components of the complex dielectric function:Fig. 12(g) illustrates the electron energy loss spectrum (EELS). We observed negligible energy loss of electrons for ACrO₃ (A = Si, Ge, Sn) compounds in the IR and visible energy ranges. The significant losses occur in the ultraviolet energy region (25–30 eV). Therefore, from the energy loss spectra, the significant peaks of plasmon energy (a collective oscillation of valence electron energy) for SnCrO₃ are situated at 27, 27.5, 28 and 29 eV, respectively, at 0, 10, 20, and 30 GPaSnCrO₃, which indicates that the plasmon energy spectrum moves towards the higher energy region as pressure increases. Similar features were observed for SiCrO₃ and GeCrO₃, where major peaks move toward the right with increasing pressure. The highest peak in the energy loss spectrum occurs because of bulk plasmonic excitation. This happens at a specific incident photon energy and its corresponding frequency, known as the bulk plasma frequency.⁸² Rather than absorbing high-energy photons, we detected some modest peaks in the energy range of 15–20 eV, which suggest surface plasmonic excitations at the metal–dielectric interface due to optical transition. This loss spectroscopy indicated that the substances can be used for non-linear optics, biosensors, near-field optics (light guiding below the diffraction limit), and other applications.⁸³

4. Conclusion

In this study, we used the DFT framework with a GGA to investigate ACrO₃ (A = Si, Ge, Sn) compounds. Our results were consistent with those reported in the literature. We found that the lattice constant and bond length decrease with increasing hydrostatic pressure. The investigated substances retained their cubic phases over a range of pressures. Determining the tolerance factor confirmed the stability of the cubic phase under different pressure regimes. ACrO₃ (A = Si, Ge, Sn) was found to be a half-metal based on spin-polarized electronic band structures, the DOS, and the integer value of the magnetic moment (μ_B) per unit cell. We observed that the structure is more stable in the ferromagnetic state than in the non-magnetic state. An analysis of Mulliken charges showed that ACrO₃ (A = Si, Ge, Sn) exhibits covalent and ionic bonding behavior. SnCrO₃ exhibited metallic behavior at the critical lattice constant after losing its integer magnetic moment, while the other compounds maintained constant magnetic moments. We also examined the ductile and brittle behavior and bonding characteristics of the compounds using Poisson's ratios. Our analysis indicated that SiCrO₃ and SnCrO₃ are brittle at 0 and 10 GPa and ductile in other pressure regions, while GeCrO₃ exhibited ductile behavior at all pressures. The Kleinman parameter decreased linearly, suggesting a decrease in plasticity with pressure. Additionally, we investigated the optical behavior of ACrO₃ (A = Si, Ge, Sn) by calculating the dielectric function, refractive index, optical conductivity, absorption, reflectivity, and energy loss function. We observed maximum absorption and minimum energy loss in the ultraviolet region. The real and imaginary parts of the dielectric function showed that the compound is metallic in the infrared and visible range, while exhibiting semiconducting behavior in the ultraviolet energy region. As there is no literature data available for these ACrO₃ (A = Si, Ge, Sn) materials, the data reported in this study can serve as a reference for future research and will be valuable for understanding and developing effective optoelectronic and spintronics devices.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Data availability

The data supporting this article have been included as part of the ESI.†

Acknowledgements

We thankfully acknowledge the Ministry of Science and Technology, Bangladesh for their financial support in this project conducted by the members of the Condensed Matter Physics lab at Jahangirnagar University through their R&D fund allocation program. We are also thankful to the Jahangirnagar University-funded Project for the financial assistance to conduct the research work in the Computational Physics (CP) Research Lab in the Department of Physics at Jahangirnagar University.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ma00169b

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