1 Introduction

Intermetallic compounds have been studied extensively over the last few decades due to their applications in different areas such as in optoelectronics, magnetism, spintronics and thermoelectrics. Intermetallic compounds with AM2X2-type formula, where A is the rare earth element, M refers to a metal and X is the element of the main group III, IV or V of the periodic table, belonging to the Zintl phase, crystallize mainly into the ThCr2Si2-type (tetragonal) structure and sometimes into the CaAl2Si2-type (trigonal) structure. Some of the Zintl phase compounds are promising thermoelectric materials because of their excellent structural stability at high temperatures and low thermal conductivity [1,2,3,4,5,6]. AM2X2-type intermetallics also exhibit attractive magnetic properties, and therefore, considerable research has been done by many groups on the magnetic aspects [7,8,9,10]. Due to the strong interaction between magnetism and charge transport properties, a slight change in external parameters such as temperature, pressure and magnetic field may cause a remarkable modification of their physical properties [11].

Among the AM2X2 compounds, ThCr2Si2-type and CaCr2Si2-type structured materials have been investigated thoroughly in the field of high-Tc superconductivity [12,13,14,15,16,17], the most common being the ternary pnictides [18]. The most surprising behaviors of pnictides are that they tend to be antiferromagnetic [19], and superconductivity emerges with doping or under high pressures [18]. On the other hand, the Zintl phase with the CaAl2Si2-type structure has attracted a great deal of attention from the scientific community as thermoelectric materials. The transport properties of prototype CaAl2Si2 are controlled by both electrons and holes. Since their discovery, binary and ternary phases crystallizing in this structural pattern were challenging issues because of the stability limit [20]. Although it is very rare to develop CaAl2Si2-type structure, Pukas et al. [21] synthesized magnesium alumo-germanide (MgAl2Ge2) and confirmed its stable trigonal crystal structure by X-ray diffraction. The newly synthesized germanide is the third and last member in the Mg–Al–X (X = C, Si and Ge) system which belongs to the CaAl2Si2-type structure. In our previous work, we discussed the structural, electronic and elastic properties of this newly synthesized germanide in detail [22]. However, as far as we know, there is no report on the Fermi surface topography, charge density mapping, optical, thermodynamic and vibrational properties of MgAl2Ge2. Therefore, to bridge the existing research gap, we have undertaken the current project to investigate the above-mentioned unexplored properties in detail. Results obtained from the theoretical analysis reveal the suitability of MgAl2Ge2 as a potential material for future optoelectronic-, thermodynamic- and vibrational-based intermetallic research.

2 Computational methodology

The first-principles calculations are carried out via the plane-wave pseudopotential method based on density functional theory which is implanted in CASTEP (CAmbridge Serial Total Energy Package) code [23]. The generalized gradient approximation by Perdew–Burke–Ernzerhof (GGA-PBE) [24] has been employed for calculating the exchange–correlation contribution to the electron–electron interaction. Vanderbilt-type ultrasoft pseudopotential was used. The plane-wave cutoff energy was 750 eV. For sampling the Brillouin zone, a Monkhorst–Pack grid of 16 × 16 × 8 k-points was used. Geometry optimization was carried out using the convergence thresholds of 5 × 10−6 eV/atom for the total energy, 0.01 eVÅ−1 for the maximum force, 0.02 GPa for the maximum stress and 5 × 10−4 Å for the maximum atomic displacement. The vibrational properties were calculated using the DFPT linear response method [25]. The norm-conserving pseudo-approximation was used for obtaining temperature-dependent properties using DFPT. Different thermodynamic potential functions such as enthalpy, free energy and entropy of MgAl2Ge2 were calculated via quasi-harmonic Debye approximation.

The newly synthesized MgAl2Ge2 adopts a trigonal (space group \(P\overline{3}m1\)) crystal structure consisting of Mg atoms with a one-fold 1a Wyckoff position having fractional coordinates (0, 0, 0) which correspond to the Ca atoms in the structure of CaAl2Si2. On the other hand, Al and Ge atoms occupy two-fold sites in 2d Wyckoff position with fractional coordinates (1/3, 2/3, 0.6303) and (1/3, 2/3, 0.2404). The optimized crystal structure is shown in Fig. 1a. The lattice parameters of the optimized structure are a = 4.130 Å (\(a^{expt} =\) 4.117 Å) [21] and c = 6.839 Å (\(c^{expt} =\) 6.787 Å) [21]. Good correspondence between theoretical and experimental lattice parameters indicates the reliability of the present calculations.

Fig. 1
figure 1

a Crystal structure and b electronic band structure of MgAl2Ge2

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3 Results and discussion

3.1 Energy band structure and Fermi surface

Fermi surface is a key characteristic to understand the charge transport properties of a compound. The Fermi surface topography can be derived from the electronic band structure. The electronic energy band structure of MgAl2Ge2 is presented in Fig. 1b. The energy band structure of MgAl2Ge2 has been displayed in the first Brillouin zone along high symmetry directions. From Fig. 1b, it is seen that four valence bands (bands 9, 10, 11 and 12) cross the Fermi level (marked by the dark pink dashed line). Therefore, MgAl2Ge2 would display metallic behavior. Fermi surface of MgAl2Ge2 is shown in Fig. 2a–e. Both electron- and hole-like sheets are seen in these figures. Four Fermi sheets are produced for the four bands crossing the Fermi level. Band 9 produces an electron-like cylindrical Fermi sheet around the G point only. Band 10 produces another electron-like distorted spherical sheet along the G-A line only. Band 11 also produces a distorted spherical electron-like sheet along the G-M, G-K and G-A lines, and band 12 produces a hole-like sheet around the M-point and the K-point in the Brillouin zone. The deviation of Fermi sheets from free-electron sphere may be as a result of the effective Coulomb potential (attractive and repulsion) and non-isotropic crystalline state of MgAl2Ge2 [26]. However, the presence of both electron- and hole-like Fermi surfaces indicates the manifold band nature of intermetallic MgAl2Ge2. These multi-sheets are favorable for the enhancement of conductivity of MgAl2Ge2. The overall topography of the Fermi surface implies that charge conduction in the c-direction has electronic nature, while the conduction in the ab-plane is expected to show hole-like character. In addition, the pressure-dependent Fermi surface topological transition can cause the volume-dependent anomalies of phonon spectrum which may affect the thermodynamic properties of MgAl2Ge2.

Fig. 2
figure 2

a–e Fermi surface topology and f charge density distribution map in the (001) plane of MgAl2Ge2

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3.2 Charge density distribution

The electron densities or accumulation of charges in different crystallographic planes are closely related to the nature of the atomic bonding in a compound. Therefore, charge density mapping can yield valuable information regarding the bonding character of MgAl2Ge2. The charge density mapping of MgAl2Ge2 based on charge density difference at different atomic sites is shown in Fig. 2f in the (001) crystallographic planes. The red and blue colors of the adjacent scale bar indicate the high and low density of electronic charge, respectively. The charge density map reveals the presence of Al–Ge, Mg–Ge and Al–Al covalent bonds in MgAl2Ge2. Among these covalent bonds, Al–Ge and Al–Al bonding seem to be stronger than the Mg–Ge bond. The lower covalent radii of Al and Ge may accumulate more electronic charges between themselves and generate stronger covalent bonds than that between Mg and Ge atoms.

3.3 Optical properties

The study of the optical properties of a material is crucial to explore its potential for optoelectronic device applications. The optical response of a material to incident electromagnetic radiation is entirely determined by means of the various energy-dependent optical parameters such as absorption coefficient, reflectivity, refractive index, optical conductivity, loss function and dielectric constants. Figure 3a–f shows the calculated optical parameters of MgAl2Ge2 for photon energies up to 25 eV with electric field polarization vectors along [100] and [001] directions. We readily see from Fig. 3 that the optical parameters under study are fairly isotropic in nature; there are only slight differences in the positions and heights of the peaks with respect to the state of polarization of the incident electric field. A Gaussian smearing of 0.5 eV was used for all the calculations which integrates the k-points on the Fermi surface effectively. Since MgAl2Ge2 is metallic, both interband and intraband electronic transitions contribute to dielectric functions. Therefore, the Drude term with an unscreened plasma frequency 3 eV and damping 0.05 eV has been used for the low-energy part of the spectrum.

Fig. 3
figure 3

a Absorption coefficient, b reflectivity, c optical conductivity, d refractive index, e dielectric constant and f loss function of MgAl2Ge2 electric field polarizations along [100] and [001] directions

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The energy-dependent absorption spectrum of MgAl2Ge2 is shown in Fig. 3a. The figure shows that α(ω) increases sharply in the visible spectral range. The major peak in this spectrum, due to interband transition, is observed at around 4.0 eV i.e., in the near-ultraviolet (UV) region. The absorption of electromagnetic radiation begins from 0 eV reflecting the metallic character of MgAl2Ge2. It is interesting to note that, although the Ni-based germanide [27] absorbs the UV radiation quite effectively, the MgAl2Ge2 absorbs both the visible and UV radiation. Therefore, this material can be a promising absorber of electromagnetic radiation in both visible and UV regions.

The reflectivity, R(\(\omega\)), profile of MgAl2Ge2 is shown in Fig. 3b. It is observed that R(\(\omega\)) is above 90% and 44% in the infrared region and visible region, respectively. Li et al. [28] concluded that materials having the reflectivity ~ 44% or above in the visible spectral range are capable of minimizing solar heating. In the UV part of the electromagnetic spectrum, the reflectivity is non-selective over an extended energy range (5 eV–14 eV) and sustains a high value around 80%. A further increase in photon energy results in a sharp decrease in R(\(\omega\)), and it becomes zero for energies greater than 15 eV. Therefore, MgAl2Ge2 can be used as a coating material for avoiding solar heat.

Figure 3c shows the photoconductivity σ(ω) spectrum of MgAl2Ge2. The photoconductivity is seen to start at zero photon energy, indicating again that MgAl2Ge2 does not have a band gap in the electronic band structure. It is also observed that σ(ω) decreases at photon energy < 0.5 eV, grows up after and shows a sharp peak at ~ 2.50 eV. The decrease in σ for < 0.5 eV, its subsequent increase at higher energies and finally peaking at ~ 2.50 eV roughly follow the absorption coefficient spectrum within the same range of energy. This is an effect arising from the matrix element of photon-induced transitions of electrons between different energy states, controlled mainly by the product of electronic density of states at the energy levels involved.

The study of the refractive index is very important for materials due to its wide functional roles in the operations of different optical tools such as photonic crystals and waveguides [29]. The real, n(ω), and imaginary, k(ω), parts of the refractive index are displayed in Fig. 3d. The static value of the real part n(0) is found to be 6.0. This particular parameter measures the phase velocity of light inside the compound. On the other hand, the imaginary part k(ω) measures the attenuation of light as it moves through the material. Metallic systems are characterized by high values of k(ω) for frequencies below the plasma edge.

Figure 3e illustrates the real, ε1(ω), and imaginary parts, ε2(ω), of the dielectric constant of MgAl2Ge2. As seen, both dielectric constants vanish at ~ 15 eV that corresponds to the photon energy at which the reflectivity (Fig. 3b) demonstrates a quick drop and energy loss function, L(ω) (describing the energy loss of a fast electron traversing the material), exhibits a very sharp peak as exposed in Fig. 3f. Thus, the above facts fulfill the condition for plasma resonance at 15 eV. The negative value of the dielectric constant reflects the Drude-like behavior that is very common in metallic systems.

3.4 Thermodynamic properties

The correlations of thermodynamic parameters such as Debye temperature and specific heat with the elastic constants of materials allow us a profound understanding of the relationships between the mechanical and phonon structures. We have investigated the thermodynamic properties of MgAl2Ge2 by using the quasi-harmonic Debye model. The details regarding this model can be found elsewhere [30, 31]. In this section, we have calculated the Debye temperature, specific heat and volume thermal expansion coefficient at different temperatures (T = 0−1000 K) and pressures (P = 0−50 GPa). To do so, the Birch–Murnaghan equation of state [32] at zero temperature and pressure was used to fit the E–V data. In this work, the structure of MgAl2Ge2 has been optimized by varying the cutoff and k-points. At first, we have noted the energy and volume by taking a k-point mesh at different cutoff energy. After plotting, E–V data we have fixed the cutoff energy and noted the E and V for different k-point mesh. From these two plots, we have set the cutoff energy and k-point grid at minimum energy of the structure. The lattice of this particular structure is used for further calculations. The thermodynamic properties of a material can be obtained from the non-equilibrium Gibbs function \(\it {\text{G}}^{*} {{(V; P, T)}}\) given by [30]

$$\it G^{*} \left( {V;P,T} \right) = E\left( V \right) + {{PV}} + A_{{{{vib}}}} \left[ {\varTheta \left( V \right);T} \right]$$
(1)

where \(\it {\text{E}}\left( {\text{V}} \right)\) is the energy per unit cell, PV represents the pressure–volume product, \({\theta }\left( {\text{V}} \right)\) is the Debye temperature and the vibrational term Avib can be expressed as

$$\it A_{{{\text{vib}}}} \left( {\theta ,T} \right) = nkT\left[ {\frac{{9\varTheta }}{8T} + 3\ln \left( {1 - e^{{ - \frac{{\varTheta _{D} }}{T}}} } \right) - D\left( {\frac{{\varTheta _{D} }}{T}} \right)} \right]$$
(2)

where n is the number of atoms per formula unit, k is the Boltzmann constant and \( {{D}}\left( {\frac{{{\varTheta }_{{{D}}} }}{{{T}}}} \right)\) is the standard Debye integral.

The Debye temperature (\({\varTheta }_{{\text{D}}}\)) is an important lattice dynamical parameter. From the magnitude of \({\varTheta }_{{\text{D}}}\), one can infer to a number of physical parameters, namely specific heat, electrical and thermal conductivities, thermal expansion coefficient, etc. It is also related to the bonding strength among the atoms within the solids. The temperature- and pressure-dependent graphical presentations of \( {\varTheta }_{{\text{D}}}\) at different pressure and temperature are shown in Fig. 4a and b, respectively. It is seen from these figures that the \( {\varTheta}_{{\text{D}}}\) decreases with increasing temperature, whereas \( {\varTheta}_{{\text{D}}}\) increases with increasing pressure which reflects that the thermal vibrational frequency of the atoms in this compound changes with temperature and pressure. One point is remarkable that at very low temperatures, \( {\varTheta}_{{\text{D}}}\) exhibits an overall saturation to a plateau. The existence of such a plateau is usually thought to be associated with zero-point vibration [33, 34]. We have calculated the Debye temperature using the Anderson method based on elastic constant in our previous study and found a value of 347 K [22]. This value is comparable with the value obtained from the quasi-harmonic Debye model.

Fig. 4
figure 4

Variation of Debye temperature of MgAl2Ge2 a with temperature at different pressures and b with pressure at different temperatures

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The isothermal bulk modulus is a thermodynamic quantity which specifies the resistance to compression. The temperature dependence and pressure dependence of bulk modulus (B) of MgAl2Ge2 at different pressures and temperatures are shown in Fig. 5a and b, respectively. From Fig. 5a, it is observed that at zero pressure the bulk modulus of MgAl2Ge2 is found to decrease below 600 K; after that, it increases linearly. The increase in bulk modulus after 600 K is found to be altered in the presence of pressure. Moreover, bulk modulus diminishes slowly with temperature due to high pressure. The bulk modulus, in contrast, is found to increase significantly with increasing pressure at different fixed temperatures which is shown in Fig. 5b. The decrease in inter-atomic distance due to pressure increases in bulk modulus of MgAl2Ge2. This indicates that a decrease in pressure or an increase in temperature reduces the hardness of MgAl2Ge2.

Fig. 5
figure 5

Variation of bulk modulus of MgAl2Ge2 a with temperature at different pressures and b with pressure at different temperatures

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The specific heat is an intensive property of material strongly correlated with temperature and pressure. The dependence of specific heat at constant volume (CV) at different pressures is shown in Fig. 6a. It is observed that CV increases as a function of temperature caused by phonon thermal softening. It is seen that pressure has a small influence on heat capacity. At sufficiently low temperatures, CV follows the Debye-T3 power law. However, at high temperature (T > 500 K), CV approaches the classical Dulong–Petit limit (125 Jmole−1 K−1) which is common to all solids at high temperature. The same trend is observed for specific heat at constant pressure (CP) and is shown in Fig. 6b.

Fig. 6
figure 6

Temperature dependence of a specific heat CV and b specific heat CP of MgAl2Ge2 at different pressures

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Figure 7a and b shows the variation of volume thermal expansion coefficient (αV) with temperature and pressure, respectively. It is seen from Fig. 7a that αV increases rapidly with temperature up to 250 K but above 250 K the increment is gradual, suggesting that αV is more sensitive to temperature at low temperature than at high temperature. However, the volume thermal expansion coefficient decreases strongly with pressure at a constant temperature as shown in Fig. 7b.

Fig. 7
figure 7

Variation of volume thermal expansion coefficient of MgAl2Ge2 a with temperature at different pressures and b with pressure at different temperatures

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3.5 Vibrational properties

The phonon dispersion curves have been calculated along the direction of G-A-H-K-G-M-L-H in the first Brillouin zone, as shown in Fig. 8a. The phonon dispersion consists of three acoustic and twelve optical branches. Since the mass of Ge atom is much higher than Mg and Al, it can be said that Ge dominates the low-frequency phonon modes, while the contributions of Mg and Al are found at high frequencies. The most striking feature of this spectrum concerns the lowest phonon branches which reach negative phonon frequencies at several points of the Brillouin zone. The negative values of the phonon frequencies correspond to dynamical failure of the crystal due to time-dependent perturbation. It should be noted that the computed phonon dispersion spectrum corresponds to absolute zero temperature. The situation at higher temperature can be different. Therefore, we predict that MgAl2Ge2 becomes dynamically unstable at low temperature, but it becomes stable at higher temperature. The imaginary modes of frequencies do not appear at high temperature since the compound has been already synthesized and no indication of lattice dynamical instability has been found [35]. The imaginary phonon modes at low-T indicate that MgAl2Ge2 might be susceptible to structural phase transition at low temperature. As far as we know, there is no available experimental or theoretical data on temperature-dependent structural parameters of MgAl2Ge2; therefore, conclusive remarks cannot be made. In the future, the study of possible temperature or pressure dependence of structural phase transition could be an interesting topic.

Fig. 8
figure 8

a Phonon dispersion curves and b temperature-dependent enthalpy, free energy and entropy of MgAl2Ge2

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Furthermore, thermodynamic behavior of MgAl2Ge2 can be described by some temperature-dependent potential functions such as enthalpy H(T), Helmholtz free energy F(T) and entropy S(T). These thermodynamic parameters have been calculated at zero pressure using the quasi-harmonic approximation, where the Helmholtz free energy can be defined as F(T) = H(T)−TS(T). The variation of enthalpy, free energy and entropy with temperature is shown in Fig. 8b. All these potential functions show almost zero values below 50 K. Above this temperature, both enthalpy and entropy increase significantly with increasing temperature, but Helmholtz free energy of MgAl2Ge2 varies in a reverse way.

4 Conclusions

MgAl2Ge2 is a new representative of CaAl2Si2-type structure among a variety of alumo-germanides. Previously, we disclosed the structural, mechanical and electronic properties of MgAl2Ge2. In this paper, we have studied Fermi surface topography, charge density distribution, optical, thermodynamic and lattice dynamical properties of intermetallic MgAl2Ge2 using the first-principles method based on density functional calculations. Electronic energy band structure confirms that MgAl2Ge2 is metallic. The Fermi surface shows both electron- and hole-like sheets indicating manifold band nature of MgAl2Ge2. Charge density mapping of MgAl2Ge2 indicates the presence of covalent bonds. Optical parameters do not exhibit significant optical anisotropy. The optical absorption and photoconductivity spectra started from zero photon energy as well as the large negative value of the dielectric constant indicates the metallic nature of MgAl2Ge2 which is also verified by energy band structure. The reflectivity spectra exhibit a sharp drop, and the energy loss function demonstrates a very sharp peak at 15 eV which is known as the plasma edge. From thermodynamic study, it is found that Debye temperature exhibits an overall saturation to a plateau at very low temperature which is usually thought to be associated with zero-point vibration. The hardness of MgAl2Ge2 is seen to vary both with temperature and with pressure. The specific heat at constant volume follows the Debye-T3 power law at temperature below 500 K, after which approaches the classical Dulong–Petit limit of 125 Jmole−1 K−1. The specific heat at constant pressure also follows the same trend. According to the phonon dispersion curves, indications regarding lattice instability were found around the Brillouin zone boundary at low temperature.

Since there are no experimental data on the studied properties of MgAl2Ge2 in this work, we anticipate that the obtained optoelectronic, thermal and vibrational properties of MgAl2Ge2 would serve as useful reference and stimulate future experimental and theoretical work to validate our results.