1 Introduction

New semiconductors with characteristics that might possibly enhance performance and energy efficiency in electronic devices are the subject of continuing research and development. In recent years, there has been a growing interest among researchers to investigate the potential of Heusler alloys as semiconductors for use in electronic and optoelectronic devices. Heusler compounds, named after Fritz Heusler [1], are a family of materials that have gained significant attention due to their diverse properties [2, 3] and potential technological applications [4,5,6,7,8,9,10,11,12,13,14,15,16]. They are classified into two categories [3]: half-Heusler (HH) with a general chemical formula of XYZ and full-Heusler (FH) represented by X2YZ, with quaternary Heusler (QH) with XX’YZ composition closely associated with the full-Heusler alloys. X, X’ and Y can be a transition, alkali and alkaline earth metals and Z is a main group element. Most HH compounds with 18 valence electrons per formula unit have a semiconducting behavior [17, 18] and have received great attention as promising semiconductors for various applications such as thermoelectrics, transparent conductors, topological insulators, and buffer layers of solar cells. On the other hand, this behavior is rare in quaternary and full-Heusler alloys. However, a semiconductor character is observed for the SH and QH compounds, which have 24 valence electrons per formula unit [19,20,21,22,23,24,25,26]. Recent studies [27,28,29,30,31,32] have focused on exploring the structural, electronic, vibrational, and thermoelectrical properties of various quaternary Heusler compounds containing 18 valence electrons per formula unit using first-principles calculations. These studies have shown the investigated materials are mechanically and dynamically stable semiconductors with band gaps in the range 0.2 to 2.5 eV. In their study, Jiangang He et al. [33] predict new stable full-Heusler compounds with 10 valence electrons showing semiconducting behavior with excellent thermoelectric properties. A new family of full-Heusler compounds without transition metals has appeared recently [2], such as Li2CaC, Li2SrC [34] and Li2MgSi [35], which have been shown to have semiconducting properties with band gaps of 1.127 eV, 0.979 eV and 0.2 eV, respectively.

In the context of the search for new semiconductor materials, investigations related to the structural, electronic, elastic and optical properties of the full-Heusler alloys Na2SrSi and Na2SrGe were carried out. As far as we know, there are no experimental or theoretical studies of the compounds Na2SrSi and Na2SrGe available in the literature. Therefore, our findings may be useful as a prediction for future research, especially for experimentalists.

2 Crystal structure and computational details

The conventional full-Heusler alloys X2YZ crystallize in the cubic structure in two prototype structures: Cu2MnAl and Hg2CuTi prototypes, as shown in Fig. 1. In the Cu2MnAl-type (Fm3m, space group no. 225), the X, Y and Z atoms are located at A (0, 0, 0) and B (0.5, 0.5, 0.5) sites, C (0.25, 0.25, 0.25) sites, and D (0.75, 0.75, 0.75) sites, respectively. The Hg2CuTi-type (216, F43m) is obtained by placing X atoms at A (0, 0, 0) and C (0.25, 0.25, 0.25) sites, and Y atoms at B (0.5, 0.5, 0.5). The calculations of the structural, elastic, electronic, magnetic, and optical properties were performed using DFT [36, 37] based on the FP-LAPW method [38], as implemented in WIEN2K package [39]. The electronic exchange–correlation potential is described with the generalized gradient approximation (GGA) [40]. To obtain better and more accurate electronic and optical properties, we have used the modified Becke-Johnson (mBJ) scheme [41]. The convergence of the basis set was controlled by a cutoff parameter RMTKmax = 9 where RMT is the smallest of the muffin-tin sphere radii and Kmax is the largest reciprocal lattice vector. The RMT spheres of 2.00 bohr for Na and Si, and 2.20 bohr for Sr and Ge, were adopted. For the integrations over the Brillouin zone, 2000 k-points with a k-mesh of 12 × 12 × 12 are introduced. The cut-off energy, which defines the separation of valence and core states, was chosen as -6 Ry. The energy and charge convergence values were set to 10–4 Ry and 0.001 e, respectively. The optical properties and the total density of states (DOS) were performed using a dense Monkhorst–Pack grid of 27 × 27 × 27 k-points [42].

Fig. 1
figure 1

The crystal structure of the full-Heusler X2YZ (a) Hg2CuTi-type structure and (b) Cu2MnAl-type structure (blue-Na, Red – Sr, and green-Si/Ge)

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The phonon spectra are calculated by using density functional perturbation theory (DFPT) [43,44,45,46,47] as implemented in the Quantum ESPRESSO code [48, 49]. The Perdew-Zunger (LDA) exchange–correlation pseudopotentials available in the GBRV (Garrity-Bennett-Rabe-Vanderbilt) database [50] are exploited.

3 Results and Discussion

3.1 Structural properties and phase stability

The first very important step is to obtain information about the structural parameters of Na2SrX (Si and Ge), particularly the lattice constant (a0), the compressibility modulus B0 and its derivative B'. This first procedure predicts the most stable phase of the material. Figure 2 represents the energy-volume change curves for Na2SrSi and Na2SrGe in the magnetic (FM) and non-magnetic (NM) states for the two atomic arrangements (Cu2MnAl and Hg2CuTi). As a result, the more stable structure of the two compounds under study corresponds well to the Hg2CuTi-type structure with a nonmagnetic character. The structural parameters are deduced by fitting the curve of variation of total energy as a function of volume by the Murnaghan equation of state [51]. The results obtained are given in Table 1.

Fig. 2
figure 2

Total energy as a function of volume of Na2SrSi and Na2SrGe compounds

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Table 1 Optimized lattice parameters of Na2SrX (Si and Ge) alloys in the nonmagnetic (NM) state of the Hg2CuTi structure
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In order to confirm the structural stability and the possibility of synthesizing the Na2SrX (Si and Ge) alloys experimentally, we calculated the formation (Ef) and cohesive (Ec) energies according to the following formulas [52, 53]:

$${E}_{f}={E}_{Total}^{{Na}_{2}SrX}-\left({2E}_{Na}^{bulk}+{E}_{Sr}^{bulk}+{E}_{X}^{bulk}\right)$$
(1)
$${E}_{c}=\left(2{E}_{Na}^{iso}+{E}_{Sr}^{iso}+{E}_{X}^{iso}\right)-{E}_{Total}^{{Na}_{2}SrX}$$
(2)

where \({E}_{Total}^{{Na}_{2}SrX}\) is the equilibrium total energy of the studied compounds under their equilibrium lattice constant, \({E}_{Na}^{bulk}, {E}_{Sr}^{bulk}\) and \({E}_{X}^{bulk}\) correspond to the total energies per atom of each element in the bulk, \({E}_{Na}^{iso}, {E}_{Sr}^{iso}and{E}_{X}^{iso}\) are the total energies of the atomic components in the isolated state. The results obtained in Table 1 confirm the thermodynamic stability of these compounds as well as the possibility of their experimental synthesis.

3.2 Mechanical and dynamical stabilities

The elastic properties are deduced from the elastic constants Cij [54]. These coefficients are coupled to the stability and stiffness of the compounds. As both Na2SrX (Si and Ge) compounds have cubic symmetry, they are characterized by three independent elastic constants, namely: C11, C12 and C44, which must meet Born Huang's stability criteria [55] if these alloys are mechanically stable:

$$\left\{\begin{array}{ll}{\mathrm{C}}_{11}-{\mathrm{C}}_{12} > 0, {\mathrm{C}}_{11} > 0, {\mathrm{C}}_{44} > 0 &and \\ {\mathrm{C}}_{11}+2{\mathrm{C}}_{12} > 0 &and \\ {\mathrm{C}}_{12} < B < {\mathrm{C}}_{11} & \end{array}\right.$$
(3)

The obtained elastic constants Cij for Na2SrX (Si and Ge) alloys are presented in Table 2. From the results obtained, we can draw the following conclusions for each of the compounds examined: (i) The mechanical stability in the Hg2CuTi-type structure with a nonmagnetic (NM) ground state of both Na2SrSi and Na2SrGe is confirmed from the positive calculated values of elastic constants (C11, C12, C44) and the required condition for stability. (ii) The value of the bulk modulus derived from the elastic constant is very close to the value computed from the third-order Birch-Murnaghan state equation, indicating the reliability of the theoretical method. (iii) The elastic constants Cij of Na2SrSi are comparable to those of Na2SrGe, which implies that these compounds have the same resistance to external stress. (iv) Since C11 > C44, these compounds display higher resistance to unidirectional deformation compared with shear deformation.

Table 2 The calculated elastic properties of Na2SrX (Si and Ge) compounds
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Bulk modulus (B0), Shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν), and elastic anisotropy factor (A) are calculated from the calculated Cij values using the Voigt-Reuss-Hill approximation [56,57,58]:

$${B}_{0}=\frac{1}{3}\left({C}_{11}+2{C}_{12}\right)$$
(4)
$$G=\frac{1}{2}\left({G}_{V}+{G}_{R}\right)$$
(5)
$${G}_{V}=\frac{1}{5}\left({C}_{11}-{C}_{12}+3{C}_{44}\right)$$
(6)
$${G}_{R}=\frac{5{C}_{44}\left({C}_{11}-{C}_{12}\right)}{{4C}_{44}+3\left({C}_{11}-{C}_{12}\right)}$$
(7)
$$E=\frac{9BG}{3B+G}$$
(8)
$$\upsilon =\frac{3B-2G}{2\left(3B+G\right)}$$
(9)
$$A=\frac{{2C}_{44}}{{C}_{11}-{C}_{12}}$$
(10)

From the calculated values of the above parameters, we can conclude the following: The Pugh ratio B/G used to predict the brittle (B/G < 1.75) or ductile (B/G > 1.75) character of materials is about 1.29 for Na2SrSi and 0.96 for Na2SrGe, so Na2SrX (Si and Ge) alloys reveal their brittle nature. This behavior is confirmed by the Cauchy pressure (CP = C12- C44) because the brittle character occurs when CP < 0, clearly this is the case in our situation. The Poisson ratio ν is a useful parameter for characterizing the mechanical behavior of materials. In the case of brittle materials Poisson ratio is less than 0.26. The calculated values of ν for Na2SrSi and Na2SrGe, which are 0.19, further support their brittle nature as determined by Pugh's ratio and Cauchy's pressure. Moreover, the Poisson's ratio also provides insights into the bonding nature of crystalline materials. Typically, materials with ionic bonds have a Poisson ratio around 0.25, while those with covalent bonds have a value around 0.1. Na2SrX (Si and Ge) alloys appear to have covalent bonding properties based on the predicted value of ν. Young’s modulus E measures the stiffness of a compound. We found that Young’s modulus values for Na2SrSi and Na2SrGe are similar. The anisotropy factor A has an important role in engineering science. If A is equal to 1.0, the crystal is elastically isotropic, while any value other than one indicates anisotropy. The calculated value of A for the Na2SrX (Si and Ge) alloys suggests that these compounds have a profound anisotropy.

The phonon dispersion curves of Na2SrSi and Na2SrGe compounds have been investigated and plotted in Fig. 3. Since these compounds have four atoms in their primitive cell, there are twelve phonon modes for each alloy. Three of them with lower frequency values are the acoustic modes and the rest are the optical modes. At their equilibrium structures, Na2SrSi and Na2SrGe crystals show no negative frequency. Thus, these compounds are dynamically stable.

Fig. 3
figure 3

Calculated phonon dispersion curves for Na2SrSi and Na2SrGe alloys

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3.3 Electronic properties

Figure 4 shows the band structure of Na2SrX (Si and Ge) alloys calculated within GGA-mBJ approaches. As can be seen, the calculated electronic band structure with the optimized equilibrium lattice constant of the studied compounds shows similarities and exhibits a semiconductor behavior with an indirect band gap (L—X) of 0.652 eV and 0.629 eV for Na2SrSi and Na2SrGe, respectively. The band structure can be better understood using total (TDOS) and partial (PDOS) densities of states. From Fig. 5, it is evident that the profiles of TDOS and PDOS of Na2SrSi and Na2SrGe compounds are similar. The Si-s band (Ge–s band) of Na2SrSi (Na2SrGe) is positioned at -5.5 eV (-6.36 eV). The region just below the Fermi energy level is related to Sr-d (d-t2g) states and X-p states, but the major impact comes from Si-p (Ge-p). On the contrary, in the region above the Fermi energy level, the main contribution is dominated by Sr-d (d-t2g) states, and the contributions of the X-p states are very low. The contribution of Na in the two regions is very small compared to Sr and Si (Ge) atoms.

Fig. 4
figure 4

Band structures and total densities of states (TDOS) of Na2SrX (Si and Ge) alloys calculated within GGA-mBJ approaches

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Fig. 5
figure 5

Total (TDOS) and partial densities of states of states (PDOS) of Na2SrX (Si and Ge) alloys calculated within GGA-mBJ approaches

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Another significant quantity that affects the optoelectronic characteristics of materials and devices is the exciton binding energy (Eb) [59]. We computed the exciton binding energies of Na2SrSi and Na2SrGe using the Mott-Wannier model (WM) [60], which is applicable for inorganic semiconductors with a narrow band gap and a high dielectric constant. Eb is obtained from:

$${E}_{b}=\frac{13.6 {\mu }_{r}^{*}}{{m}_{0}{\varepsilon }_{0}^{2}}$$
(11)

where \({m}_{0}\) is the free electron mass, \({\varepsilon }_{0}\) is the static dielectric constant and \({\mu }_{r}^{*}\) is the reduced effective mass:

$$\frac{1}{{\mu }_{r}^{*}}=\frac{1}{{m}_{e}^{*}}+\frac{1}{{m}_{h}^{*}}$$
(12)

\({m}_{e/h}^{*}\) are the effective masses of the hole and electron. Thus, the effective mass can be computed using the relation:

$$\frac{1}{{m}^{*}}=\frac{1}{{\hslash }^{2}}\frac{{\partial }^{2}E(k)}{\partial {k}^{2}}$$
(13)

The exciton binding energies are about 8.58 meV for Na2SrSi and 6.91 meV for Na2SrGe. The lower exciton binding energies obtained are comparable to those determined experimentally for GaAs (4.0 meV) [61] and GaSb (2.1 meV) [62], suggesting that both compounds are suitable for photovoltaic applications.

3.4 Optical properties

The interaction of light with a solid is described by its macroscopic optical properties. The optical properties (refractive index, reflectivity, energy loss spectra, optical conductivity, and optical absorption) are determined by the complex dielectric function \(\varepsilon \left(\omega \right)\). Where, \(\varepsilon \left(\omega \right)\) represents the linear response of matter to incident electromagnetic radiation of various frequencies. This function is given by:

$$\varepsilon \left(\omega \right)={\varepsilon }_{1}\left(\omega \right)+i{\varepsilon }_{2}\left(\omega \right)$$
(14)

Here, \({\varepsilon }_{1}\left(\omega \right)\) and \({\varepsilon }_{2}\left(\omega \right)\) represent the real and imaginary parts of the dielectric function, respectively, and they are related by the Kramers–Kronig relations [63]. \({\varepsilon }_{1}\left(\omega \right)\) and \({\varepsilon }_{2}\left(\omega \right)\) of Na2SrSi and Na2SrGe under photon energies are shown in Fig. 6. We note that the spectra of the real and imaginary parts of the dielectric function of the two materials under study are almost identical and have the same peaks, with a slight difference in intensity. From the imaginary component, the threshold energy for optical transition between the top of the valence band and the bottom of the conduction band, which represent the fundamental absorption edge, is found to be consistent with our calculated energy gap for these compounds. The electronic transitions from the p states of Si/Ge atoms to the unoccupied d states of the Sr atom are manifested as several peaks in the curve of \({\varepsilon }_{2}\left(\omega \right)\). Two main peaks of the \({\varepsilon }_{2}\left(\omega \right)\) spectrum of Na2SrSi and Na2SrGe occur approximately at the same position in the visible region, at 1.7 eV and 2.90 eV, which can be attributed to the same transitions. At zero frequency limits, we define the static dielectric constant \({\varepsilon }_{1}\left(0\right)\). The values of \({\varepsilon }_{1}\left(0\right)\) listed in Table 1 are inversely proportional to the band gap, which is consistent with the Penn model [64]. At low energies, where \({\upvarepsilon }_{1}\left(\upomega \right)\) is positive, the electromagnetic wave propagates through the materials. The region where \({\upvarepsilon }_{1}\left(\upomega \right)<0\), is characterized by strong reflectivity and absorption. Figure 7 shows the refractive index dependence on energy of the incident radiation of Na2SrX (Si and Ge) alloys. The variation of \(n\left(\omega \right)\) with energy exhibits a similar trend to \({\varepsilon }_{1}\left(\omega \right)\), as both parameters are related to the amount of light that is absorbed by a material. For Na2SrSi and Na2SrGe, the maximum value of n(ω) is observed in the infrared domain of light spectrum at 0.8 eV (4.07) and 0.82 eV (4.26), respectively. The static values of refractive index \(n\left(0\right)\) are about 3.62 for Na2SrSi and 3.74 for Na2SrGe. Our calculated static values of refractive index \(n\left(0\right)\) and dielectric constant \({\varepsilon }_{1}\left(0\right)\) are consistent with those of some binary semiconductors such as GaAs and GaSb [61, 65, 66]. Hence, Na2SrSi and Na2SrGe have the potential for photovoltaic applications. From the spectrum of the energy dependence of the absorption (Fig. 4) for the two compounds, we notice that the absorption edges correspond very well to the band gaps. Na2SrSi and Na2SrGe are good absorbers in the ultraviolet region; there is no absorption peak in the IR and visible regions. The reflectivity follows a similar behavior to absorption; \(R\left(\upomega \right)\) shows also two bands; the first band is up to 13 eV and the second lies in the energy range (17.75–31 eV). Therefore, in the energy range (13–17.75 eV), the absorption is very low and the reflectivity is negligible, then these full Heulser compounds exhibit transparent properties. The obtained results suggest that these materials are beneficial for several applications, like optoelectronics.

Fig. 6
figure 6

Calculated real (\({\varepsilon }_{1}\left(\omega \right)\)) and imaginary (\({\varepsilon }_{2}\left(\omega \right)\)) parts of the dielectric function, reflectivity and absorption for the Na2SrX (Si and Ge) materials

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Fig. 7
figure 7

Calculated refractive index for the Na2SrX (Si and Ge) Heusler alloys

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4 Conclusion

In this paper, the FP-LAPW method based on density functional theory (DFT) was used to study the properties of Na2SrX (X = Si and Ge) full-Heusler alloys. Structural, electronic, and optical properties at optimized equilibrium lattice constants are investigated. Calculations show that the Hg2CuTi-type structure is energetically the most favorable for the two compounds. The electronic results showed that these compounds are semiconductors with an indirect band gap (L—X). The exciton binding energy of Na2SrSi and Na2SrGe alloys was found to be relatively small. The phonon dispersion calculations proved that in the ground state, Na2SrGe and Na2SrSi compounds are dynamically stable. Optical properties showed that Na2SrSi and Na2SrGe are promising materials for photovoltaic applications. The studied compounds are good absorbers in the ultraviolet region and exhibit a maximum reflectivity of about 60%. Therefore, we expect that the presented results can provide reliable data for further work and that the studied materials can be used in optoelectronic devices in the ultraviolet range.