1 Introduction

New features of materials such as polarization spin, conduction band, symmetry of polarized carriers and magnetism of surfaces are crucial to develop the technology and industry. The use of half metals is a spin-based electronic application. The “half-metallicity” of these substances, in which the conduction electrons are 100% spin polarized at Fermi level is a defined characteristic. As a result, it has very interested in creating materials with this feature and a new class of materials known as ferromagnetic half metals has been predicted [1]. Due to their electronic and magnetic characteristics, half-metallic magnetic materials have garnered attention and suggested for using in spintronic applications. The discovery of materials with two spin polarizations, resulting in 100% spin polarization at Fermi level, has sparked interest in exploring novel ferromagnetic half-metallic compounds [2].

Non-magnetic-based half-metallic (HM) ferromagnetic Heusler alloys have been anticipated according to the research on electronic structure, which is inspired by their peculiar magneto-optical capabilities. Certain Heusler materials exhibit half-metallic ferromagnetism [3, 4], which is a metallic in one spin orientation. De Groot et al. [5,6,7,8,9] have established a categorization scheme by separating three distinct varieties of half-metallic ferromagnetism [10]. In addition, numerous families of Heusler-type alloys have their half-metallicity predicted using first-principles calculations [11,12,13,14,15,16]. Half-Heusler RbSrX (X = C, Si, and Ge) alloys have been investigated for their structural, electronic and magnetic characteristics using FP-LAPW by Rozale et al. [11]. They have found that half-metallic gaps of RbSrC (0.31 eV), RbSrSi (0.14 eV) and RbSrGe (0.12 eV), all support the alloys’ ferromagnetic nature. The structural, elastic, electronic, magnetic and optical characteristics of RbSrX (X = C, Si, and Ge) alloys have been computed using FP-LAPW by Ahmad et al. [12]. They have demonstrated that RbSrC, RbSrSi, and RbSrGe are elastically stable half-metals with indirect gap; 2.01 eV, 1.92 eV and 1.5 eV, respectively. The electronic structure, chemical bonding and magnetism in a variety of half-Heusler XMZ alloys have been studied by Nanda and Dasgupta [13] using FP-LMTO and TB-LMTO ASA methods. Using first-principles calculations to examine the electronic structures of a series of hypothetical alloys with the half-Heusler structure CrTiX (X = Si, Ge, Sn, Pb), Sattar et al. [14] have predicted that these alloys are to be half-metallic. The structural, electronic and magnetic characteristics of XYZ (X = Li, Na, K, and Rb; Y = Mg, Ca, Sr, and Ba; Z = B, Al, and Ga) ferromagnetic half-Heusler alloys have been investigated by Umamaheswari et al. [15] using the linearized muffin-tin orbital approach connected to strong bond method (TB-LMTO-ASA). First-principles calculations by Gao and Wang [16] have projected half-metallicity in full-Heusler Cu2MnAl and RbSrN2 alloys. New Heusler ferromagnetic HM alloys without transition metal ions or rare earth have been studied due to these findings [17,18,19,20,21,22]. In addition, a theoretical investigation of α, β and γ phases of Heusler ternary GeNaZ (where Z = Ca, Sr, and Ba) alloys has been reported by Beldi et al. [22]. They have investigated the half-metallic, elastic and dynamic stability of these alloys. The Fermi level contributions from 3p states of Si atom are largest, as predicted by the density of state and charge density curve for half-Heusler SiRbCa alloy. Theroetcial investigations of half-Heusler alloys including structural, mechanical, thermodynamic, electronic, magnetic and thermoelectric properties are achieved successfully [23,24,25].

The prime novelty is to undertake first-principles calculations for SiRbCa alloy in three distinct phases; α, β and γ due to it is structurally stable to discover novel half-metallic materials that is not available in the literature. Our work is devoted to explore the ferromagnetic and half metallic characters in a new class of half-Heusler alloys that do not contain transition metals. The calculations are predictive and material is predicted. The primary goal is to foretell the structural, electronic and magnetic characteristics of a novel Heusler alloy. This work is divided into three sections. The second is for computational details, followed by the elaborations of results and dicsuiions in the third section. The conclusions are outlined in the fourth section.

2 Computational

Our research focuses on a novel class of half-Heusler alloys that do not include transition metals, therefore, uncharted territory is in terms of their ferromagnetic and half-metallic properties. Half-Heusler SiRbCa alloy’s magnetic moment is an integer, with most of its strength coming from the Si site and the rest is from Rb and Ca sites. The computations in this study are carried out utilizing the full potential linearized augmented plane wave method (FP-LAPW) as implemented in WIEN2K code [26]. The generalized gradient approximation (GGA-PBE) proposed by Perdew, Burke and Ernzerhorf (PBE) [27] is used for handling of potential exchange and correlation. In addition to the aforementioned approximations, mBJ-GGA-PBE approximation is utilized for the electronic characteristics [28, 29]. It provides a more accurate representation of the values for fundamental and half-metallic gaps. This approximation is a hybrid between the exchange potential of TB-mBJ and correlation potential of GGA-PBE. It introduces a novel method for dealing with exchange and correlation potential, similar to Tran and Blaha [29] modification of Beckett-Johnson swap component [25]. It is a viable replacement for GW method or hybrid functional in terms of precision. Several semiconductors and insulators may have their gaps accurately determined by combining TB-mBJ exchange with GGA [29,30,31,32,33,34,35]. You can see the electronic setups and parameters we utilized in Table 1. In this approach, the unit cell is partitioned into a sphere-representing area centered on the nuclei and sphere-free hub region placed in the void between the spheres. The wave function is characterized by plane waves in the interstices but by atomic functions in the spheres of muffin tin. Throughout the un-overlapped spheres surrounding the atomic sites up to l = 10, the basis functions, electronic densities and potentials are formed as a single combination of spherical harmonics, but in the core area with a cut-off radius; RMT × KMAX = 9, they are produced as a Fourier series. (The size of matrix is determined by RMT × KMAX, where Rт is the minimum radius of a muffin tin and KMAX is the standard for greatest wave vector utilized in formulation of one’s own functions in the plane). Samples from the Brillouin zone are taken using a method similar to that proposed by Monkhorst and Pack [36]. For our alloy, we have employed a total 1500 k points in the Brillouin space, which is equivalent to a grid of special points k in the irreducible Brillouin space with dimensions; 11 × 11 × 11 for the three cubic phases (α,β,γ). We have used Ca (4s2), Rb (5s1), and Si (3s2 3p2) valence states in our computations and throughout all phases.

The computational details involving the density functional theory (DFT), specifically the full potential linearized augmented plane wave (FP-LAPW) approach are crucial for accurately simulating the electronic and structural properties of materials. In this method, the interatomic interactions are described by solving Schrödinger’s equation self-consistently.

The FP-LAPW method divides the unit cell into two regions, separating the core and valence electrons, with different treatments. Core electrons are approximated as spherical waves within muffin-tin spheres, while valence electrons are expanded as plane waves outside these spheres. The augmented plane wave basis set allows for precise description of electronic wavefunctions. The integration of this approach with exchange-correlation functionals (GGA-PBE) enables the determination of electronic, structural and magnetic properties with a high accuracy. It is a computationally intensive method, often necessitating high-performance computing clusters for practical application.

Table 1 Electronic configurations and physical parameters of half-Heusler SiRbCa alloy
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For RMT × KMAX, where matrix size and number of k-points in IBZ are determinant fundamental functions, a convergence test of various computation settings is performed. Determining the ground state is the first step in structural optimization. The iterative approach is performed until the calculated total energy converges to less than 0.01 mRyd. Table 2 presents the results of our computation into the structural stability of half-Heusler SiRbCa alloy using the three probable crystal phases (α,β,γ).

While investigating equilibrium structural characteristics, it is important first to make a self-consistent calculation of total energy (self-consistent calculation) and do so for a variety of values of mesh parameter. The second stage is to modify the total energy using Murnaghan’s equation of state [37], provided by the following relation;

$${\text{V=}}{{\text{V}}_{\text{0}}}{{\text{(1+ }}\frac{B^{\prime }}{B}{\text{)}}^{_{^{ - {\text{1/B}}}}}}$$
(1)

where V is volume of cell, V0 is equilibrium volume in ground state, B is bulk modulus and its derivative \( {B^{\prime }}\).

The equilibrium lattice constant is given by the minimum of curve Etot(V), and finding out \( {B^{\prime }}\)s bulk modulus requires knowing the following:

$${\rm{B = V}} \frac{{\partial }^{2}E}{\partial {V}^{2}}$$
(2)

The derivative of bulk modulus \( {B^{\prime }}\) is defined as:

$$ {\rm{E}}\left( {\rm{V}} \right){\rm{=}}{{\rm{E}}_{\rm{0}}}{\rm{ + }}\frac{B}{{B}^{{\prime }} ({B}^{{\prime }}-1)}\left[V{\left(\frac{{V}_{0}}{V}\right)}^{B}- {V}_{0}\right]{\rm{ + }}\frac{B}{{B}^{{\prime }}}{\rm{(V}} - {V_{\rm{0}}}{\rm{)}}$$
(3)

It is necessary to use mBJ-GGA to improve the results of electronic propertied and especially for ferromagnetic materials [38] (Fig 1).

Table 2 Space group and Wyckoff positions of different crystal phases used in our calculation for half-Heusler SiRbCa alloy
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Fig. 1
figure 1

Different crystal structures of half-Heusler SiRbCa alloy used for three phases (a) α, (b) β and (c) γ

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3 Results and Discussions

3.1 The Structural Properties of Half-Heusler SiRbCa Alloy

After analyzing the phases of half-Heusler alloy, the crystal structure and magnetic phase with highest stability are established. We have determined the total energy variation as a function of volume for half-Heusler SiRbCa alloy in both non-magnetic (non-spin-polarized) and ferromagnetic (spin-polarized) situations (Table 2). In order to determine the structural stability of half-Heusler SiRbCa alloy, it is investigated in the amorphous, crystalline and agglomerate phases. Murnaghan’s equation of state [37] allows us to determine the equilibrium by examining a plot of total energy versus volume. Figure 2 illustrates the volume-energy-change curve of half-Heusler SiRbCa alloy in its three phases. This figure demonstrates the stability of ferromagnetic phase in half-Heusler SiRbCa alloy. Using both spin polarization (SP) and non-SP computations, we find that the half-Heusler SiRbCa alloy is ferromagnetic throughout all three phases. Table 3 displays the investigated findings derived from with and without SP calculations in the three phases of half-Heusler SiRbCa alloy, which compare these results to other existing theoretical results.

The half-Heusler SiRbCa alloy exhibits intriguing structural properties that make it a subject of significant research interest. It possesses a cubic crystal structure with a space group F-43 m, showcasing excellent crystallographic stability. The alloy is known for its semiconductor characteristics, with a direct bandgap that falls within the visible light range, making it suitable for optoelectronic applications. Its inherent high thermoelectric performance stems from a combination of unique properties such as low thermal conductivity and high electrical conductivity. Furthermore, SiRbCa’s thermal expansion coefficient closely matches that of Si, making it an attractive candidate for silicon-based device integration.

The structural properties of half-Heusler SiRbCa alloy are fascinating. This compound’s crystal lattice combines semiconducting silicon (Si) with metallic rubidium-calcium (RbCa). This amalgamation results in a complex, highly stable structure, enabling enhanced thermoelectric performance. The alloy’s unique electronic configuration, particularly its intricate band structure and high electronic density of states, contributes to its exceptional properties, holding promise for efficient energy conversion and diverse technological applications in the realm of materials science and engineering.

Table 3 Non-spin-polarized (NSP) and spin-polarized (SP) calculations using GGA-PBE for equilibrium lattice parameter (a in Å), bulk modulus (B in GPa), its pressure derivative (B’) and energy difference between SP and NSP states ΔSP−NSP (in meV) for half-Heusler SiRbCa alloy for different phases α, β and γ
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Fig. 2
figure 2

Total energy as a function of volume per formula unit for non-spin-polarized (NSP) and spin-polarized (SP) configurations of SiRbCa half-Heusler alloys calculated for α, β and γ phases using GGA-PBE

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3.2 The Magnetic Properties of Half-Heusler SiRbCa Alloy

Table 4 presentes the investigated results of our GGA-PBE and mBJ-PBE calculations for the total and partial magnetic moment of half-Heusler SiRbCa alloy in the three phases α, β and γ. Using GGA-PBE and mBJ-GGA-PBE, we have determine that half-Heusler SiRbCa alloy, which is half-metallic with a total integer magnetic moment equal to 1 µB throughout all three phases. Ferromagnetic half-Heusler alloys; XYZ (X = Li, N’a, Rb and K); (Y = Ca, Mg, Sr and Ba); (Z = B,Al and Ga) have been investigated by Umamaheswari et al. [15] to use the linearized muffin-tin orbital approch connected to the strong bond method. It is found that the magnetic moment value of half-Heusler alloy SiRbCa is an integer that originly comes mainly from the silicon site with a small contribution of Rb and Ca atoms.

The magnetic properties of half-Heusler SiRbCa alloy are a significant focus of research due to their potential for multiapplication. This alloy exhibits interesting magnetism, often characterized by its half-metallic behavior. It is considered a ferromagnetic material, with a high magnetic moment, typically attributed to the partially filled 3d orbitals of transition metal element, such as Fe or Co. The alloy displays robust ferromagnetism, making it valuable for spintronic devices and magnetic storage applications. Moreover, its half-metallic nature means that one spin channel is conducting, while the other is insulating, offering unique opportunities for spin-polarized transport and magnetic tunnel junctions.

The magnetic properties of half-Heusler SiRbCa alloy are captivating. This compound exhibits interesting magnetic behavior, showcasing both ferromagnetic and semiconducting characteristics. Its magnetic structure is influenced by interactions between the spins of constituent elements, offering potential for spintronic applications. The alloy’s tunable magnetic properties, coupled with its promising electronic structure, render it a candidate for novel magnetic devices and spin-based technologies, highlighting its significance in the field of materials science and technology.

Table 4 Calculated total magnetic moment per formula unit µtot (in µB/Cell), and local magnetic moments µSi, µRb, and µCa of Si, Rb, and Ca atoms (in µB/atom) for half-Heusler SiRbCa alloy for different phases α, β and γ, using GGA-PBE and mBJ-GGA-PBE
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3.3 The Electronic Properties of Half-Heusler SiRbCa Alloy

3.3.1 The Band Structure of Half-Heusler SiRbCa Alloy

The band structures of half-Heusler SiRbCa alloy computed using two GGA-PBE and mBJ-GGA-PBE, with spin polarization (for the majority and minority spins) in the most stable phase (in phase α) are shown in Figs. 3 and 4, respectively. The band structures of half-Heusler SiRbCa alloy produced by two GGA-PBE and mBJ-GGA-PBE methods are found to exhibit a half-metallic character (HM) with regard to semiconductor and metal in both majority and minority spin directions. These findings corroborate those published in [22], which examined the same family of half-Heusler alloys composed entirely of non-magnetic elements. Figure 3 illustrates a direct bandgap; Eg=1.17 eV in the direction X → X as predicted using GGA-PBE approximation along the majority spin direction. However, the valence and conduction bands overlap significantly at Fermi level when minority spins are polarized. With a half-metallic gap \( {E}_{g}^{HM}\)=0.25 eV and integer total magnetic moment (µtot = 1 µB), our half-Heusler alloy has been confirmed to be half-metallic (see Table 4). Table 5 displays the basic bandgap; Eg=1.85 eV and half-metallic gap \( {E}_{g}^{HM}\)=0.56 eV are retained using mBJ-GGA-PBE as illustrated in Fig. 4. As far as we know for SiRbCa half-Heusler alloys, there is no theoretical nor experimental research exists. Half-Heusler SiRbCa alloy fundamental bandgap (Eg) and half-metallic gap (HM gap) values are investigated as listed in Table 5. The bandgap is defined as a difference between the Fermi level and valence band maximum, while the HM gap is defined as a difference between the Fermi level and conduction band minimum. Comparing the bandgap values estimated by mBJ-GGA-PBE and GGA, you will see that the former is often larger.

The band structure of half-Heusler SiRbCa alloy plays a pivotal role in understanding its electronic properties. SiRbCa is recognized for its semiconductor behavior, featuring a direct bandgap. The band structure demonstrates a characteristic energy dispersion, which highlights the energy levels of electrons in the crystal lattice. The alloy’s electronic bandgap is typically tailored to fall within the visible light spectrum, making it suitable for optoelectronic applications. The unique combination of its band structure and crystal symmetry results in outstanding electronic properties, including high carrier mobility and excellent thermoelectric performance, adding to its appeal for a wide range of electronic and energy conversion devices.

The band structure of half-Heusler SiRbCa alloy displays a unique electronic configuration. It combines the characteristics of a semiconducting silicon (Si) matrix with the metallic nature of rubidium-calcium (RbCa). This integration creates a complex and diverse band structure, offering a high density of electronic states near the Fermi level. Such a feature not only contributes to its exceptional thermoelectric performance but also holds a potential for various electronic and energy-related applications due to its tailored band properties.

Fig. 3
figure 3

Spin polarized band structures of SiRbCa half-Heusler alloys calculated in stable α phase using GGA-PBE for both (a) majority and (b) minority spin channels. The horizontal dashed line indicates the Fermi level

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Fig. 4
figure 4

Spin polarized band structures of half-Heusler SiRbCa alloys calculated in stable α phase using mBJ-GGA-PBE for both (a) majority and (b) minority spin channels. The horizontal dashed line indicates the Fermi level

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Table 5 Values of half-metallic gap (\( {E}_{g}^{HM}\)) and fundamental gap (Eg) obtained from band structures calculated in direction of majority spins for half-Heusler SiRbCa alloy for α phase, using GGA-PBE and mBJ-GGA-PBE
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3.3.2 The Density of States of Half-Heusler SiRbCa Alloy

The hybridization type and the states responsible for the magnetism in this half-Heusler alloy may be better understood by calculating the total density of state (DOS) and partial density of state (PDOS) to get insight into the band structure. Figures 5 and 6 show the computed partial density of state spectra for the two orientations of majority and minority spins of half-Heusler SiRbCa alloy, respectively. Both spectra feature an initial peak distant from the Fermi level that derives from 3s states of Si atoms. The 3p states of Si atoms with a little contribution from 3d states of Ca atoms are primarily responsible for the peaks slightly below and above Fermi level. Ca 3d states contribute less than Si 3p states, so keep that in mind. The density of state reveals a non-zero value at Fermi level for the polarization of minority spins, whereas the density of density suggests a zero value for polarization of majority spins (Figs. 5 and 6), indicating a bandgap in the direction of polarization of majority spins. Moreover, we see that the maxima of Si 3s and Si 3p states at Fermi level change location depending on spin direction. The change in Si 3p states explains a half-metallicity by indicating a full spin polarization. The 3p orbitals of Si atoms are definitive cause of magnetism. The half-metallic (HM) feature of our half-Heusler alloy is shown by a larger peak shift when utilizing mBJ-GGA-PBE (Fig. 6), in contrast with GGA-PBE (Fig. 5) suggested potential avenues for further investigations [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59]. An attempt has been made to make a detailed theoretical study in order to leave the choice for elaboration of a new spintronic material in future.

Fig. 5
figure 5

Spin polarized total (DOS) and partial (PDOS) densities of states of SiRbCa half-Heusler alloy calculated for stable α phase using GGA-PBE. The vertical dashed lines indicate Fermi level. The positive and negative DOS values correspond to majority and minority spin channels, respectively

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Fig. 6
figure 6

Spin polarized total (DOS) and partial (PDOS) densities of states of SiRbCa half-Heusler alloy calculated for stable α phase using mBJ-GGA-PBE. The vertical dashed lines indicate Fermi level. The positive and negative DOS values correspond to majority and minority spin channels, respectively

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The density of states (DOS) in half-Heusler SiRbCa alloy is a crucial aspect of its electronic structure. The DOS represents the number of electronic states per unit energy interval and plays a pivotal role in determining the alloy’s electronic and thermal properties. SiRbCa typically exhibits a complex DOS with distinct features, including the presence of band edges and peak-like structures. These features are associated with energy bands and unique electronic configuration of constituent elements. Understanding the DOS is essential for predicting the alloy’s conductivity, thermal behavior and its suitability for applications like thermoelectric devices, where control over electronic states is essential for optimizing efficiency and performance.

The density of states in half-Heusler SiRbCa alloy is a key factor in its unique properties. This compound showcases a diverse electronic structure with a notable density of states that significantly affects its behavior. Its distinctive band structure near the Fermi level contributes to a high density of electronic states, influencing its exceptional thermoelectric properties. The tailored density of states within this alloy holds a promise for applications in energy conversion, electronics and various technological advancements.

4 Conclusions

In this work, the structural and half-metallic stabilities in half Heusler SiRbCa alloy have been examined using FP-LAPW method within GGA-PBE and mBJ-GGA-PBE exchange-correlation. These findings indicate that FM in phase α is preferred FM for this alloy. Its electronic structures are metallic in spin-down channel and semiconducting in spin-up direction, exhibiting a half-metallic behavior. The alloy’s distinctive personality results from the spin polarization of its Si-3p orbitals. The total magnetic moment of SiRbCa is 1 µB as shown by magnetic measurements. Because of its intriguing magnetic properties, we believe this alloy might be used in spintronics.