2.Method of calculations

We have carried out first the principles calculations ^17,18 with both Full potential and linear augmented plane wave (FP-LAPW) methods ¹⁹ as implemented in the WIEN2k code ²⁰. The exchange-correlation effects were described with the parameterization of the generalized gradient approximation (GGA) ²¹. Well-converged basis sets are chosen with R_MT K_max = 8 while G_max =12Ry^1/2 for the Fourier charge density expansion of the potential in the interstitial region is used. A mesh of 104 special k-points was made in the irreducible wedge of the Brillouin zone. An energy threshold of - 6 Ry is used to separate the valance and core states for all calculations.

3.1.Structural properties

As is known, the so-called Full Heusler alloys crystallize in a highly ordered cubic structure, and their properties are related to the site preference of the atoms inside the structure. There are two types of highly ordered atomic arrangements ²²: AlCu₂Mn (space group 225 F -m3m) and CuHg₂Ti-type structures (space group 216 F-43m). For Ba₂AgZ (Z = As, Sb, Bi) alloys in the AlCu₂Mn-type structure, the atomic sites are described as follows in Wyckoff coordinates: Ba (0,0,0), Ba (0.5,0.5,0.5), Ag (0.25,0.25,0.25), and Z (0.75,0.75,0.75), and in the CuHg₂Ti-type structure, they are Ba (0,0,0), Ba (0.25,0.25,0.25), Ag (0.5,0.5,0.5), and Z (0.75,0.75,0.75). Structural optimizations on the Ba₂AgZ (Z = As, Sb, Bi) alloys were performed first to determine their equilibrium lattice constants. The total energy-lattice constant curves for these alloys are plotted in Fig.1. Clearly, the AlCu₂Mn-type structure state is energetically the most stable structure in comparison to the Hg₂CuTi-type at the equilibrium lattice constant for these three alloys. The values of the optimized parameters are given in Table I and are 8.06, 8.35, and 8.46 A for Ba₂AgAs, Ba₂AgSb and Ba₂AgBi, respectively. The equilibrium lattice constant of these alloys increases with the increasing covalent radii of the Z atoms (i.e., As→Sb→Bi). The formation energy E _f determines whether a compound can be experimentally synthesized or not ²³. E _f is the change in energy when a material is formed from its constituent elements in its bulk states and can be calculated for Ba₂AgZ (Z = As, Sb, Bi) compounds as Ef=(Etot-2EBa-EAg-EZ) with Z = Al, Si, Ge, As. Table I shows that the values of all compounds are negative, which confirms that these compounds are stable and can be synthesized experimentally. Among them, Ba₂AgAs is most easily synthesized because of its lowest formation energy. Thus, in order to consolidate our predictive results of our Ba₂AgZ (Z = As, Sb, Bi) Full-Heusler alloys, we have compared ours results in Table I with the other theoretically work for the same family of rattling Heusler Ba₂AuZ (Z = Sb, Bi) alloys (15). The obtained results stay in good agreement with this theoretical study. Based on this, all the further calculations on electronic, thermodynamic and thermoelectric properties of Ba₂AgZ (Z = As, Sb, Bi) were performed on in the type-Cu₂MnAl structure.

Figure 1 Total energy as a function of unit cell volume for the Ba₂AgZ (Z = As, Sb, Bi) compounds in both Cu₂MnAl and Hg2CuTi type structure. 

3.2.Electronic properties

It is well known that the band structure is vital for determining the thermoelectric properties because these properties depend greatly on the band structure. The calculated band structures of Ba₂AgZ (Z = As, Sb, Bi) R compounds have been illustrated in (2-4)a. As it can be seen, the general band structures of are similar for our R FH compounds. The valence band maximum (VBM) and conduction band minimum (CBM) are located at the low symmetry points L and Δ (between Γ and X) in the Brillouin zone indicate that Ba₂AgZ (Z = As, Sb, Bi) alloys are indirect bandgap semiconductors. The indirect energy gap E_g (L - Δ) of 0.566 eV for Ba₂AgAs, 0.548 eV Ba₂AgSb and 0.433 eV for Ba₂AgBi is obtained. Unfortunately, so far, no experimental measurements and theoretical data bandgap E_g for the investigated compounds are carried out to compare with. However, our results for bandgap E_g listed in Table I are in good agreement with the other theoretically work for the same family of rattling Full Heusler Ba₂AuZ (Z = Sb, Bi) ¹⁵.

Furthermore, multiple bands have energies close to the VBM and CBM and lead to a sharp increase in the density of states around the Fermi level, which is favorable for large Seebeck coefficients. Bilc et al. ¹³ recently showed that bands that are flat along one direction and highly dispersive along others lead to a high Power factor (PF) in FH materials. In fact, such bands are present in ours new R compounds Ba₂AgZ (Z = As, Sb, Bi) as well, where the valence band along L-Γ is flat, whereas it is highly dispersive along with all other directions, as depicted in the red circle of band structures. The total (TDOS) and partial (PDOS) densities of states are plotted in Fig. (2 -4)b for these three compounds.

Figure 2 Band structure and total and partial density of states for the Ba₂AgAs compound. 

Figure 3 Band structure and total and partial density of states for the Ba₂AgSb compound. 

Figure 4 Band structure and total and partial density of states for the Ba₂AgBi compound. 

We observe that above the Fermi level, the conduction band is heavily dominated by Ba-4d and (As-4p*, Sb-5p*, Bi-6p*) states for all R Heusler compounds, while below the Fermi level, in the top of the valence band represented by the three bands between 0 and -2 eV, TDOS is predominant by (As-4p, Sb-5p, Bi-6p) orbitals with a minor contribution from Ag-5𝑠 represented by the single valence band around -2.5 eV for Ba₂Ag(As, Sb, Bi) compounds, respectively. The two most electropositive atoms Ba donate their four 5s electrons to the electronegative Ag and (As, Sb, Bi) atoms. Thus, the Fully occupied 4d (represented by the flat valence bands around -4.5 eV) and 5s states of the Ag atom are extremely localized and far below the Fermi level, such that Ag only weakly interacts with its neighboring atoms. Also, the As-4s, Sb-5s, and Bi-6𝑠 electron lone pairs are deep below the Fermi level and are chemically inactive, represented by the single valence band around -9 eV. Therefore, as it is presented in Figs. (2-4)c, although the R compounds have in total ten valence electrons per f.u., these latter inactive (As-4s, Sb-5s, and Bi-6s) lone pairs lead effectively to an eight-electron system, resulting in filled bonding and empty antibonding states following the electron counting rule to obtain a semiconductor.

3.3.Thermodynamic properties

Thermodynamic properties of materials are closely related to their volume; therefore, examining the volume change under different working conditions of temperature and pressure is very important to design materials. Thermodynamic properties of the Ba₂AgZ (Z = As, Sb, Bi) R compounds have been determined using the GIBBS2 code, which is an implementation of the quasi-harmonic Debye model ²⁴, in which the non-equilibrium Gibbs function G*(V;P,T) can be written in the next form:

where E(v) is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, θD is the Debye temperature, and Avib is the vibrational term, which can be written using the Debye model of the phonon density of states as ²⁵:

where n is the number of atoms per formula unit, D(θ/T) represents the Debye integral, and for anisotropic solid, θ is expressed as (25):

where M is the molecular mass per unit cell and B_s the adiabatic bulk modulus, approximated by the static compressibility ²⁶:

where f(σ) is given by Ref. ²⁶; σ is the Poisson ratio. Therefore, the non-equilibrium Gibbs function G * (V;P,T) as the function of (V; P,T) can be minimized with respect to volume V,

By solving Eq. (18), one can obtain the thermal equation of state (EOS) V(P,T). The heat capacity C_V and the thermal expansion coefficient α are given by ²⁷,

where γ is the Grüneisen parameter.

In this work, properties are calculated in a wide range of temperature and pressure, from 0 to 1000 K and from 0 to 20 GPa, respectively. Heat capacity C_V is related to the lattice vibrations and internal energy, which are responsible for many physical properties of materials. The variation of Ba₂AgZ (Z = As, Sb, Bi) heat capacity C_V as a function of temperature at fixed pressures is displayed in Fig. 5a). It is important to mention that at temperature 0 K, the heat capacity always takes zero value, implying the absence of lattice vibrations in the material.

Figure 5 a) Heat capacity C _V , b) thermal expansion coefficient α, c) Debye temperature θ _D , and d) entropy S vs temperature and pressure for Ba₂AgZ (Z = As, Sb, Bi) alloys calculated by quasiharmonic Debye approximation. 

The C_V, along with the temperature of all compounds, increases rapidly up to 200 K. Beyond this temperature, the rising rate decreases considerably, and the heat capacity tends to the saturation value, corresponding to the Dulong-Petit limit, indicating the thermal excitation of all phonon modes in all ours R compound. Results suggest that the lattice vibrations will become stronger by increasing temperature, and a contrary behavior can be obtained when raising pressure. A stronger dependence on temperature of the heat capacity is obtained than on pressure. The predicted values of specific heat of Ba₂AgAs, Ba₂AgSb, and Ba₂AgBi alloys at 300 K and 0 GPa are 98.08, 98.35, and 98.56 J ⋅ mol^-1⋅ K^-1, respectively. Another important thermodynamic parameter for both theoretical study and practical design of materials is the thermal expansion coefficient α. In Fig. 5b), the Ba₂AgZ (Z = As, Sb, Bi) thermal expansion coefficient is depicted as a function of temperature at various pressures. We see that α increases as T³ at low temperature, and above 200 K, an exponential increase is seen. Also, α decreases as the pressure increases where variation is more significant in the case of Ba₂AgAs as compared with Ba₂AgSb and Ba2AgBi.

Debye temperature is another thermodynamic parameter, which can be used to characterize the strength of covalent bonds. High values of Debye temperature also suggests the good hardness of the material, while low values are frequently found in soft materials. Figure 5c) shows the change of the Debye temperature under the effect of temperature (at several constant pressures). From these plots, it is found that the θD decreases with temperature in all ours compounds at different pressures. The variation suggests the effect of the anharmonic behavior of the solids. The calculated value of the Debye temperature θD at zero pressure and 300 K is found to be equal to 175.55, 161.26, and 148.61 K for Ba₂AgAs, Ba₂AgSb, and Ba₂AgBi, respectively, suggesting that Ba₂AgAs is harder than Ba₂AgSb and Ba₂AgBi. We have used the variation of entropy (S) to infer the nature of order and disorder, regarded as the main thermodynamic properties of a crystal. It is noticed from 5d) that at fixed pressures P, entropy S increases monotonously with the temperature T. The current investigations demonstrated that the lattice entropy behaves with strong pressure and temperature dependence, at 0 GPa and 300 K, S= 185.35, 195.68 and 203.73 J⋅mol^-1⋅ K^-1, respectively.

3.4.Thermoelectric properties

In order to estimate the thermoelectric performance of ours compounds, we have computed the transport properties such as Seebeck coefficient S, thermopower factor PF, electrical conductivity σ, electronic thermal conductivity κe, and figure of merit ZT using the semiclassical Boltzmann theory and rigid band approach with accurate Brillouin zone (BZ) sampling with a dense grid of 100000 k-points to obtain convergence ²⁸.

The transport coefficients σ , κ, S, ZT and PF as a function of temperature (T) and chemical potential (μ) are obtained from the following equations:

where Ω is the volume of the unit cell and f₀ is the Fermi-Dirac distribution function, and Ξi,k is the transport distribution kernel defined elsewhere ²⁹.

The results are plotted in Fig 6(a-f). An efficient thermoelectric material is required to have high electrical conductivity, low thermal conductivity, and a large Seebeck coefficient. The variation of the electrical conductivity σ/τ as a function of the temperature is plotted in Fig. 6a). The electrical conductivity (σ/τ) increases slowly up to 500 K and then after it increases sharply and at 1000 K attains a value of 2.33, 2, and 1.95 (10 ⋅ 19 Ω-1⋅ m^-1⋅s^-1) for Ba₂AgAs, Ba₂AgSb, and Ba₂AgBi, respectively. This behavior indicates the semiconducting nature of ours compound and thus supports the band structure.

Figure 6 The variation of electrical conductivity σ / τ a), electronic thermal conductivity ·e, lattice thermal conductivity · κ _L , total thermal conductivity · κ _tot b)-d), Seebeck coefficient S e) and figure of merit ZT f) as a function of temperature for Ba₂AgZ (Z = As, Sb, Bi) alloys. 

Figures 6(b-d) displays the temperature dependences of total thermal conductivity (κtot=κe+κL), the lattice thermal conductivity κ_L, and the electronic thermal conductivity κ_e contributions. The lattice thermal conductivity was determined using Slack’s equation ³⁰:

where θD is the Debye temperature, γ is the Grüneisen parameter, V is the volume per atom, n is the number of atoms in the primitive unit cell, M is the average mass of the atoms in the crystal, and T is an absolute temperature. The constant A is defined as:

It is clear that the temperature dependence of the electronic thermal conductivity κ_e is similar to the behavior of electrical conductivity. The lattice thermal conductivity decreased as the temperature increased in all materials because, at high temperatures, the phonon scattering is dominated by Umklapp processes and may stand it as a potential candidate for thermoelectrics. All ours R Heusler compound exhibit extremely low κ_L of 0.275, 0.323 and 0.384 W.m^-1⋅ K^-1 at 300 K for Ba₂AgAs, Ba₂AgSb, and Ba₂AgBi, respectively. The discovery of such low values of κ_L is all the more exciting since only complex crystal structures with large unit cells, e.g., skutterudites and clathrates, which are well known for their low thermal conductivities where guest atoms rattle in the host cage structures and scatter the heat-carrying phonons (31),(32). Analogously to clathrates, the Ag atoms rattling are in a pseudocage structure composed of the alkali and main group atoms (Z). The strongly localized Ba-6𝑠 state leads to weak interactions with the 6p states of the Z-site atoms, such that Ag behaves like the inert alkaline (earth) metals in clathrate cages. So, in strong contrast to the common perception that only complex or host-guest structures with large unit cells can lead to such low thermal conductivity, R. Heusler compounds demonstrate that ultralow thermal conductivity can be realized in very simple crystal structures.

We have estimated the electrical conductivity as shown in Figs. 6a). As the temperature increases, an exponentially increasing trend can be seen, which may trigger its thermoelectric efficiency. This increasing conductivity is attributed to the semiconducting behavior of the alloys.

The variation of total thermal conductivity κ is shown in Figs. 6b)-d), where the electronic thermal conductivity κ_e increases with temperature, while the lattice part κ_L decreases with an increase in temperature. In the case of electronic thermal conductivity κ_e, as the temperature increases from 300 to 1000 K, the exponential increase occurs from 0.041, 0.0382, and 0.0373 W/mK to 0.825, 0.71, and 0.72 W/mK for Ba₂AgAs, Ba₂AgSb and Ba₂AgBi, respectively. In the case of lattice thermal conductivity κ_L, there is a decrease from 0.83, 0.97 and 1.15 W/mK at 100 K to 0.0825, 0.097, and 0.16 W/m Kat 1000 K for Ba₂AgAs, Ba₂AgSb, and Ba₂AgBi, respectively. These values are very small as compared to traditional Full Heusler alloys ^6,10,13. The capability of showing lower lattice thermal conductivity may stand it as a potential candidate for thermoelectrics.

The total thermal conductivity κ_tot shows an exponential decaying trend in all compounds up to 400 K due to the abrupt increase of large phonon vibrations before increasing slightly with temperature. We also computed the total Seebeck coefficient S variation calculated to designate its nature as shown in 6e). As one can see, the Seebeck coefficient of our R. Heusler compound is positive. The positive sign of S explains that the holes are dominant charge carriers. Therefore, our R. Heusler compounds are p-type materials. The calculated transport coefficients are now used to estimate the thermoelectric efficiency through the figure of merit ZT measurement. The materials are considered as good elements for thermoelectric devices if their ZT is about or greater than unity ³³. The variation of ZT is shown in 6f), which shows linearly that ZT increase with increasing temperature. At room temperature, these values are respectively 0.175, 0.143 and 0.121 for Ba₂AgAs, Ba₂AgSb and Ba₂AgBi. At 1000 K, its respective maximum values are 0.76, 0.73 and 0.52 respectively. It is clear that a higher temperature results in a higher figure of merit for the compound Ba₂AgAs with respect to Ba₂AgSb and Ba₂AgBi but Ba₂AgAs is regarded as a promising thermoelectric material. The value of electrical conductivity, thermal conductivity, Seebeck coefficient, and figure of merit ZT at room temperature are summarized in and compared with other R. Heusler ^14-16 and Full Heusler compounds ^6,7,9.

Finally, the applicability of material for thermoelectric technology is reflected in terms of power factor PF. The high power factor of material suggests high efficiency. The variation of power factor PF at 300 K as a function of chemical potentials (μ) are plotted in Fig. 7. The compound is considered pure at the zero chemical potential (μ = 0 eV), which is located at the middle of the bandgap. The positive value of μ represents the n-type doping, while the negative μ describes the p-type doping. The optimal doping level at which the power factor reaches the highest value is one of the main aspects in the domain of thermoelectric compounds. From 7, it is clear that the PF of hole-doping (p-doping) is larger than that of electron-doping (n-doping); this signifies that p-type is better than n-type doping. Following the work of Bilc et al., ¹³ and He et al. ¹⁴, a constant relaxation time of τ=3.4×10-14 s was used. For p-type doping, an even larger PF of 15 m⋅W⋅ m^-1⋅K^-2 in Ba₂AgAs can be observed due to the “flat-and-dispersive” band at the L-point compared to both other R Heusler Ba₂AgSb and Ba₂AgBi. The maximum value of the power factor and the corresponding optimal doping levels for our alloys are presented in and compared with other R. Full Heusler ¹⁴ and F. Heusler ¹³ compounds.

Figure 7 Power Factor (PF) of the Ba₂AgZ (Z = As, Sb, Bi) alloys at 300 K as a function of the chemical potential μ.