Crystal structure, stability, and transport properties of Li2BeAl and Li2BeGa Heusler alloys: a DFT study

In this study, the structural, elastic, electronic, and thermoelectric properties of full Li2BeAl and Li2BeGa Heusler alloys were explored using density functional and the Boltzmann transport theories. The GGA and HSE approximations have been used for the exchange–correlation potential. Results indicated that these two compounds are more energetically stable in the inverse Heusler structure. Additionally, both Li2BeAl and Li2BeGa Heusler alloys were found to be mechanically stable due to the positive values of the elastic constants. Also, the high values of the Young's modulus indicate that these compounds are stiff and exhibit a semi-metallic nature. The band gaps were determined to be 0.13 eV and − 0.22 eV for Li2BeAl and Li2BeGa alloys, respectively, using the GGA approximation. By employing the HSE hybrid functional, however, the band gap for Li2BeAl increased to 0.26 eV, and for Li2BeGa, it decreased to − 0.16 eV. Regarding thermoelectric properties, Seebeck coefficient, electrical conductivity, electronic and lattice thermal conductivities, power factor, and the figure of merit have been calculated for both Li2BeAl and Li2BeGa Heusler alloys at different temperatures. Seebeck coefficient in both alloys decreases with increasing the temperature and has the highest value at 300 K. Thermal conductivity and electrical conductivity increase with increasing the temperature, which confirms the intermetallic behavior of the Heusler alloys. The results obtained for both alloys show that n-type doping has better thermoelectric properties than p-type doping. The maximum value of the figure of merit (ZT) was obtained for n-type doping, which was 1.43 at 660 K for Li2BeAl and 0.39 at 1000 K for Li2BeGa alloy. The high values of ZT especially for electron-dopped Li2BeAl suggest the great potential of this material for use in thermoelectric devices. This study suggests that the proposed materials have potential applications in spintronic devices and thermoelectric materials due to their intermetallic character and effective thermoelectric coefficients.


Methodology
In this study, the electronic and thermoelectric properties of Li 2 BeAl and Li 2 BeGa full Heusler alloys were calculated.The Quantum Espresso (QE) package 30 was implemented in all electronic calculations by implementing the generalized gradient approximation (GGA) based on the correlation exchange function of Perdew, Burke, and Ernzerhof (PBE) 31 and the hybrid functional of Heyd-Scuseria-Ernzerh (HSE) 32 .Also, Thermo-pw 33 package was used to calculate network dynamics and to calculate the phonon scattering curves and phonon density of states.In addition, The semi-classical Boltzmann transport equations and generating maximally-localized Wannier functions (MLWF), 34 as implemented in Boltztrap2 35 and Wannier90 36 computational packages, were www.nature.com/scientificreports/used to investigate thermoelectric properties such as the Seebeck coefficient, thermal conductivity, and electrical conductivity.Xcrysden 37 software was also used to visualize crystal structures.
The kinetic energy cut-off of the wave function was set to be 80 Ry and a 350 Ry cutoff energy was used for the kinetic energy.Also, the lattice structures were optimized with the energy uncertainty of 1 × 10 -10 Ry.The first Brillouin zone was modeled using a 24 × 24 × 24 k-point mesh.For calculation of the phonon dispersion curves and the density of phonon states, we employed a q-point mesh of 16 × 16 × 16 using Thermo-pw code.Additionally, we used an 8 × 8 × 8 K-point mesh in the Wannier 90 code to calculate the thermoelectric properties.The Boltzmann density of states were obtained with the k-mesh of 300, and the energy steps of 1 × 10 -4 was used.A dense k-mesh of 24 × 24 × 24 has also been used to calculate the thermoelectric properties using Boltztrap2 code.
Dynamical stability of the studied alloys was studied using Car-Parrinello 38 ab-initio molecular dynamics (CPMD) simulation in canonical ensemble (NVT) formalism, as implemented in QE.Calculations are performed on a 2 × 2 × 2 supercell containing 32 atoms at temperatures of 500 and 1000 K. Nose-Hoover thermostat was used to control the temperature, and the time-step of 4 a.u. with a total run of 55,000 steps was done.
The lattice thermal conductivity of the crystals was also calculated using the finite displacement method in a relaxation time approximation solution of the linearized Boltzmann transport equation as implemented in Phono3py 39 code in the DFT framework on 2 × 2 × 2 supercells.Total of 452 supercells were produced by Pho-no3py without any cut-off, and the second and third order interatomic force constants are calculated using QE with 5 × 5 × 5 k-point mesh.A 50 × 50 × 50 q-mesh then was used to calculate the lattice thermal conductivity using Phono3py code.

Structural properties
Full Heusler alloys, which have the X 2 YZ chemical formula, exhibit an L2 1 crystal structure with a space group of Fm3m (225).The X, Y, and Z elements occupy the Wyckoff positions of (0, 0, 0) and ( 1 www.nature.com/scientificreports/Heusler structures and inverse Heusler structures.Then, the total energy (E) as a function of the primitive unit cell volume (V) was fitted with the third-order Birch-Murnaghan equation of state (EOS) 40 : to estimate the total energy (E 0 ) and the volume of the equilibrium primitive cell (V 0 ), as well as the bulk modulus (B) and its derivative (B′).The fitted and DFT results were shown in Fig. 2, and the calculated parameters are reported in Table 2. From Fig. 2, it is evident that the Birch-Murnaghan EOS was well fitted to the DFT data, and therefore, the obtained parameters are consistence.Furthermore, it is seen that the inverse Heusler structure exhibits a lower total energy, which indicates that it is a more stable and favorable structure for both Li 2 BeAl and Li 2 BeGa Heusler alloys.Cohesive energy 41 (E coh ) is the total energy of a single atoms arranged in a solid state.It enables the analysis of the structural stability of a compound following experimental synthesis.The following equation is used to calculate the cohesive energy, for a compound with the X 2 YZ chemical formula: where, E total is the total energy of the structure, and E X atom , E Y atom , and E Z atom are the total energy per atom for X (Li), Y (Be), and Z (Al or Ga), respectively.Table 1 represents the total energy values and energy per atom for Li 2 BeAl and Li 2 BeGa Heusler alloys.The negative cohesive energy values confirm that these phases are thermodynamically stable from an energetic standpoint.Also, Li 2 BeAl is slightly more stable than the Li 2 BeGa due to the more negative E coh value. (1)

Dynamical stability
To investigate the dynamical stability of Li 2 BeAl and Li 2 BeGa Heusler alloys, we also calculated the phonon dispersion curves and phonon density of states.Since Heusler alloys have four atoms per unit cell, each wave vector has twelve phonon states, including three phonon states, three acoustic states, and nine optical states.Figure 3 displays the phonon scattering curves, and Fig. 4 shows the phonon density of states of Li 2 BeAl and Li 2 BeGa Heusler alloys.Acoustic modes with an energy value of 0 cm -1 are considered as phonon modes.Two modes with a lower energy value are transverse acoustic (TA) modes, and one mode with a higher energy value is longitudinal acoustic (LA) 42 .Moreover, optical modes with lower energy values in each branch are transverse optical (TO) modes, while modes with higher energy values in each branch are longitudinal optical (LO) modes 1 .It can be seen from Figs. 3 and 4 that, in both Heusler alloys there are two distinct band regions, and the acoustic bands were completely separated from the optical bands.In other words, the phonon-photon scattering is not permitted in both studied Heusler compounds.
Phonons, especially the TA modes, can be utilized to analyze a compound's thermal conduction properties.If the lowest frequency TO mode has a lower frequency than the TA mode, the thermal conductivity falls.Since low-frequency TO modes scatter significant amounts of TA modes, they reduce the material's ability to conduct heat 43 .In Fig. 3, we observed that there is no interaction between the LA and TO modes; hence, they do not affect the thermal conductivity of Li 2 BeAl and Li 2 BeGa Heusler compounds.Furthermore, the phonon spectra obtained for the Li 2 BeAl and Li 2 BeGa Heusler alloys in all directions exhibit a positive phonon frequency in the Brillouin zone, indicating that these compounds are dynamically stable in the inverse Heusler structure.

Thermal stability
In previous sections, thermodynamic stability and dynamic stability were proved for Li2BeAl and Li2BeGa Heusler alloys.In this section, the thermal stability of the 2 × 2 × 2 supercells of Li2BeAl and Li2BeGa Heusler alloys at the temperatures of 500 and 1000 K was performed using CPMD calculations in canonical ensemble (NVT) formalism.It can be seen from Figs. 5 and 6 that, the although the final geometrical structure of the alloys after the simulation at different temperatures shows small distortion compared to the initial structure, the cell structure was remained almost unchanged.Also, in Figs. 5 and 6, it can be seen that the total energy fluctuations have a

Elastic properties
Elastic properties of the studied compounds were also calculated to check the mechanical stability, ductility, and hardness of Li 2 BeAl and Li 2 BeGa Heusler alloys.There are three independent elastic constants C 11 , C 12 , and C 44 for a cubic crystal which are calculated using the thermos-pw package 33 , and are listed in Table 2 at T = 0 K and  All calculated values for these constants are positive for both compounds and are consistent with Born-Huang stability conditions.Therefore, Li 2 BeAl and Li 2 BeGa Heusler alloys are mechanically stable and resistant to deformation and hardness.Also, Young's modulus (E), bulk modulus (B), shear modulus (G), and Poisson's ratio (ν) were obtained using the Voigt-Reuss-Hill average 46,47 , and were listed in Table 2.The bulk and shear modulus were measured to check the resistance of a crystalline structure to the volume stresses and plastic deformation, respectively.The bulk modulus values were 51.7 and 53.14 GPa for Li 2 BeAl and Li 2 BeGa compounds, respectively.Young's modulus is one of the main elastic constants that measure the stiffness of materials, which were obtained to be 108.77and 112.41 GPa for Li 2 BeAl and Li 2 BeGa Heusler alloys, respectively.The high values of the Young's modulus indicate the existence of a strong covalent bond as well as the stiffness of the compounds.The Poisson's ratio (ν) is a dimensionless material property that predicts how much a material will laterally contract when experiencing longitudinal strain.The Poisson's ratio has a value between 0 and 0.5.If ν of a compound is close to zero, the composition does not deform elastically, and if it is close to 0.5, the composition has a large elastic deformation.The value of Poisson's ratio for Li 2 BeAl alloy is about 0.15 and for Li 2 BeGa alloy is 0.146, which shows that both compounds may not deform elastically and have brittle behavior.

Electronic properties
This section focuses on the investigation of the electronic properties of Li 2 BeAl and Li 2 BeGa Heusler alloys through the calculation of the electronic band structure and density of states (DOS).Figures 7 and 8 show  the band structure and DOSs for both alloys.For the band structure calculation, high-symmetry points were considered while the Fermi energy level was set to be 0.0 eV.As can be seen from these figures, the minimum value of the band gap occurs at X and Γ symmetry points.The values obtained for the indirect band gap using PBE and HSE approximations are 0.13 eV and 0.26 eV, respectively, for Li 2 BeAl composition.The indirect band gap values obtained for the Li 2 BeGa compound were found to be − 0.22 eV and − 0.16 eV from the PBE and HSE approximations, respectively.These results indicate that both alloys have intermetallic properties.
To better understand the electronic band structure, the total and projected density of states (PDOS) for both alloys were also plotted.To visualize the contribution of electron orbitals, a Gaussian broadening function of FWHM = 0.05 eV has been applied to broaden the DOS lines.Figure 9a shows that the Li, Be, and Al atoms participate in both valence and conduction bands.Moreover, Fig. 9b shows s-orbital electrons contribute significantly to the conduction band, and p-orbitals have involvement in both conduction and valence bands.Similarly, Fig. 10a shows that the Li, Be, and Ga atoms participate in both valence and conduction bands.Additionally, Fig. 10b shows the s and p-orbitals contribute to the conduction band, while the p-orbital contributes to both valence and conduction bands.These results complement the band structure calculations and show that d-orbital electrons have less contribution to the conduction band.
Maximally-Localized Wannier Functions (MLWF) using the Wannier90 package and the semi-classical Boltzmann transport theory as implemented in Boltztrap2 package have been used to interpolate the band structure and calculate the thermoelectric properties.The interpolated band structure of Wannier90 code with the band structure obtained from the PBE approximation as well as the interpolated band structure of Boltztrap2 with the HSE functional are shown in Figures S.1 and S.2 of the supplementary information.The band structure obtained  The contributions of s, p, and d orbitals of each band have been calculated through the Wannier90 code for Li 2 BeAl and Li 2 BeGa Heusler alloys.Figure 11 shows that s and p-orbital electrons have significant contributions to the conduction band and p-orbital have greater involvement in both bands.The results from the Wannier functions complement the DOS and band structure calculations.

Thermoelectric properties
Currently, the world is facing an energy crisis, and there is a need for suitable and efficient alternatives of fossil fuels that are compatible with the environment.Suitable thermoelectric materials can be an adequate substitute for fossil fuels by converting waste energy into electricity.The efficiency of converting thermal energy into electricity in thermoelectric materials depends on the transport coefficients, namely Seebeck coefficient(S), electrical conductivity (σ), thermal conductivity (κ e + κ l ), and figure of merit (ZT).
To calculate the thermoelectric properties of Li 2 BeAl and Li 2 BeGa Heusler alloys, the semi-classical Boltzmann transport equation with constant relaxation time and rigid approximation, as implemented in Boltztrap2 code, was used.To test the reliability of the obtained results, the Seebeck coefficient has also been calculated with Maximally-Localized Wannier Functions using the Wannier90 code, and compared with the results of the  Furthermore, Boltztrap2 package was implemented to calculate thermoelectric properties such as Seebeck coefficient(S), electronic thermal conductivity (κ e ), electrical conductivity (σ), and the power factor (PF).These properties were calculated as functions of chemical potential, temperature, and carrier concentration using the HSE functional.The results are given in 12 eV can be seen, and the S quickly tends to be zero when moving away from this range.One of the peaks was related to the p-type (hole with µ < 0) and the other is related to the n-type (electrons with µ > 0) transport conditions.As can be seen, these peaks are very close to the maximum of the valence band and the minimum of the conduction band, respectively.Similar results for Li 2 BeGa alloy can also be seen in Fig. S.5a.In general, with an increase in the temperature the S decreases for both alloys, due to an increase in the thermal energy (i.e., an increase in the electron/hole carrier concentration).The S values of Li 2 BeGa alloy is higher than Li 2 BeAl, so that, the highest value of the S is 929 µV/K at T = 300 K and decreases to 341 µV/K at T = 900 K for Li 2 BeAl.Corresponding values for Li 2 BeGa are 1580 and 859 µV/K, respectively.These values confirms that both compounds and especially Li 2 BeGa alloy have good thermoelectric performance.It is also noticeable mentioning that, the semiconductor nature of the Li 2 BeGa alloy makes the S to be zero for certain values of the chemical potential at low temperatures, which depends on the electrical band gap, and was observable at 300 K.
The Seebeck coefficient (S) in Boltzmann transport calculations is independent of the relaxation time, τ, but the electrical conductivity (σ) is linearly dependent on the τ, under the relaxation-time approximation.Additionally, the Wiedemann-Franz equation, κ e = σLT (L: Lorentz factor, 2.45 × 10 -8 WΩ/Κ 2 ), yields the electrical thermal conductivity (κ e ); hence κ e and τ are related.Because of the intricate dispersion mechanism, the relaxation times in bulk materials are typically difficult to ascertain.Yabuuchi et al. 48, however, used a fixed value of τ = 1 × 10 -14 s after comparing the computed results with the experimental values, which is also adopted in this work.In Figs.S.4b and S.5b, unlike S, the σ exhibit an almost similar behavior at different temperatures.Also, the σ was zero in the range of µ = − 1.7 to − 0.8 eV at all temperatures for Li 2 BeAl alloy, whereas, for Li 2 BeGa compound the zero σ values are observed in the µ = − 0.93 to 2.15 eV range.Also, the behavior of σ against µ was completely opposite to S, so that, where S is maximum σ has the lowest value, and vice versa.Also, by moving away from the Fermi level, the σ increases, because, the σ is directly proportional to the charge carrier density.Also, in both alloys, σ for n-type doping interval (positive μ values) is higher than the p-type doping (negative μ values).
Figures S.4c and S.5c show κ e as a function of µ at constant temperatures of 300, 500, 700, and 900 K. κ e specifies the ability to transfer heat by electrons and phonons.The electronic thermal conductivity depends on the electrical conductivity based on the Wiedemann-Franz Law.As can be seen in these figures, unlike σ which behaved similarly at all temperatures, the electronic thermal conductivity increases significantly with increasing the temperature from 300 to 900 K.The lowest value of κ e for Li 2 BeAl alloy is in the range of µ = − 1.7 to − 0.8 eV for all temperatures, whereas, for Li 2 BeGa alloy lowest value of κ e are in the range of µ = − 0.95 to 2.2 eV.In both alloys the κ e of positive µ is higher than negative µs, which indicate that n-type doping results in a higher κ e than p-type doping.Because, the better the thermoelectric performance of the material was obtained at lower κ e , p-doping is more favorable in both alloys.
Figures S.4d and S.5d shows the power factor (PF) as a function of chemical potential.PF is one of the important quantities to check the efficiency of a thermoelectric material, and can be obtained from the calculated values of S and σ (PF = S 2 σ).As can be seen, PF increases with increasing the temperature, and the lowest value of PF was observed at T = 300 K.For Li 2 BeAl alloy, two peaks of 0.015 and 0.2 W/mk 2 at T = 900 K can be seen in the p-type doping region, namely, µ = − 2.05 and − 0.22 eV.However, for Li 2 BeGa alloy, the maximum values of PF at T = 900 K are 0.016 and 0.013 W/mK 2 which were observed at µ = 2.81 and -1.07 eV, respectively.Also, in contrast to the Al containing alloy, Li 2 BeGa alloy shows higher PF values for n-type doping, so, to have a better thermoelectric performance, electron doping will be better than hole doping in this alloy.
The thermoelectric properties of Li 2 BeAl and Li 2 BeGa alloys were shown as a function of carrier concentration in Figs.S.6 and S.7, respectively.The positive values of the carrier concentration correspond to the hole doping (p-type) and the negative values correspond to the electron doping (n-type) in these figures.Figures S.6a and S.7a show S as a function of carrier concentration.S has a strong dependence on the carrier concentration of the charge.As can be seen, two peaks were obtained for S, one peak has a positive value and the other one has a negative value.Both peaks for both alloys are observed in electron doping region (i.e., carrier concentration of ~ 2 × 10 22 cm -3 ), however, the magnitude of S for Li 2 BeAl alloy is greater than that of the Li 2 BeGa.
Figures S.6b and S.7b show the electrical conductivity as a function of carrier concentration.It is seen that, in both Heusler alloys n-type doping has more σ than p-type doping.The carrier concentration dependence of the electrical thermal conductivity was shown in Figures S.6c and S.7c.As can be seen, the obtained values of κ e in n-type doping is more than p-type doping for both alloys.Furthermore, both σ and κ e values in the n-type doping region are approximately zero at the carrier concentrations of − 1.92 × 10 22 and − 2.0 × 10 22 cm -3 for Li 2 BeAl and Li 2 BeGa alloys, respectively.PF was also plotted versus carrier concentration in Figures S.6d and S.7d.The maximum values of PF were obtained in the n-type doping region and T = 900 K for both alloys.
The temperature dependence of the thermoelectric properties for Li 2 BeAl and Li 2 BeGa compounds were shown in Figures S.8 and S.9, respectively, at fixed carrier concentrations for hole and electron doping.Positive and negative S values indicate p-type and n-type doping, respectively.It is observed that S increases rapidly with increasing the temperature.The maximum value of |S| was obtained at the temperatures of about 850 and 1000 K for Li 2 BeAl and Li 2 BeGa compounds, respectively.The obtained results are in good agreement with the reported results for LiScGe composition 49 .κ e and σ also increased with increasing temperature in both alloys, because electrons receive more energy and their flux increases with the temperature raise.Also, the non-linear behavior of σ with temperature indicates that both alloys are semi-metallic.Power factor (PF) results can also be seen in Figs.S.8d and S.9d.It has a behavior similar to that of the Seebeck coefficient and increases with increasing the temperature.For electron doping, it reaches a maximum value of 0.030 and 0.014 mW/mK 2 at 1000 K for Li 2 BeAl and Li 2 BeGa alloys.For hole doping, PF increases with increasing the temperature to reach maximum values of 0.004 and 0.003 mW/mK 2 for Li 2 BeAl and Li 2 BeGa alloys at 1000 K, indicating that for both Li 2 BeAl and Li 2 BeGa alloys, electron doping performance was more profound than hole doping.Furthermore, PF shows the performance of a thermoelectric material and the higher PF values obtained for Li 2 BeAl alloy show the better performance of this compound as a thermoelectric material.
Lattice thermal conductivity (κ l ) was also calculated as a function of temperature for Li 2 BeAl and Li 2 BeGa alloys.The results shown in Fig. 12 exhibit that κ l decreases with increasing the temperature due to the phenomenon of phonon scattering and reaches its lowest value at temperatures above 800 K, which is in good agreement with the results obtained for Li 2 BeSi, Li 2 BeGe, Li 2 BeSn alloys 50 .Also, Li 2 BeGa alloy has higher thermal conductivity than Li 2 BeAl.
The performance of a thermoelectric material and its capacity to effectively produce electrical energy from a thermal source are evaluated by the figure of merit (ZT).For appropriate thermoelectric applications, materials with a ZT close to or greater than unity are sufficient and viable options.ZT can be calculated using, Figure 13 represent the calculated ZT for Li 2 BeAl and Li 2 BeGa Heusler alloys as a function of temperature at constant carrier concentrations.The maximum value of ZT for n-type doping is 1.43 at 660 K for Li 2 BeAl alloy, and is 0.39 at 1000 K for Li 2 BeGa alloy.However, for p-type doping the maximum ZT values of 0.33 for Li 2 BeAl alloy at 710 K and 0.10 for Li 2 BeGa alloy at 950 K were obtained.The obtained results show that electron doping (3) ZT = S 2 Tσ (κ e + κ l ) in both alloys have better efficiency than hole doping and Li 2 BeAl alloy is better than Li 2 BeGa alloy for use in thermoelectric devices.
Figure 13b and d also show ZT as a function of carrier concentration for Al and Ga containing alloys at different temperatures.Kindly check and confirm inserted volume number is correct for the reference 2 .In this figure, it is seen that the ZT obtained for electron doping has higher values than hole doping in both alloys.

Conclusion
The crystal structure, electronic, elastic, and thermoelectric properties of Li 2 BeAl and Li 2 BeGa Heusler alloys were investigated using DFT and Boltzmann transition theory methods.Results show that compounds in the inverse Heusler structure are more energetically stable than their full Heusler structure counterparts, and these compounds are also mechanically and chemically stable.The electronic properties of both alloys reveal an intermetallic behavior.Car-Parrinello ab-initio molecular dynamics simulations confirm the thermal stability of both crystals at T = 1000 K, and therefore, the thermoelectric properties were calculated for both alloys up to

Figure 2 .
Figure 2. Energy-Volume optimization of the lattice parameters of (a) Li 2 BeAl and (b) Li 2 BeGa alloys.

Figure 5 .
Figure 5. Molecular dynamics (MD) simulation of Li 2 BeAl Heusler alloys at temperatures of 500 K and 1000 K, i and f show the initial and final geometric structure of Li 2 BeAl Heusler alloys.

Figure 6 .
Figure 6.Molecular dynamics (MD) simulation of Li 2 BeGa Heusler alloys at temperatures of 500 K and 1000 K, i and f show the initial and final geometric structure of Li 2 BeAl Heusler alloys.

Figure 7 .
Figure 7.The PBE (blue) and HSE (red) band structure (a), and total density of states (TDOS) (b) of Li 2 BeAl Heusler alloy.The Fermi level is set to be zero.

Figure 8 .
Figure 8.The PBE (blue) and HSE (red) band structure (a), and total density of states (TDOS) (b) of Li 2 BeGa Heusler alloy.The Fermi level is set to be zero.

Figure 9 .
Figure 9. (a) Total density of states (TDOS) and projected density of states (PDOS) of atoms, and (b) PDOS of orbitals for Li 2 BeAl Heusler alloy.The Fermi level is set to be zero.

Figure 10 .
Figure 10.(a) Total density of states (TDOS) and projected density of states (PDOS) of atoms, and (b) PDOS of orbitals for Li 2 BeGa Heusler alloy.The Fermi level is set to be zero.
Figs. S.4 to S.9 of the supplementary information.S, σ, κ e , and PF for Li 2 BeAl alloy are shown in Fig. S.4 as a function of chemical potential (µ) at constant temperatures of 300, 500, 700, and 900 K.The thermoelectric properties of Li 2 BeGa alloy are also shown in Fig. S.5.The chemical potential (µ) is a crucial parameter affects the transports features of a material.Indeed, the electrons in the valence or conduction band that participate in the electronic transport are determined by the position of µ in the band structure, which affects both the conductivity and the Seebeck coefficient.In Fig. S.4a, two peaks at µ = − 1.32 eV and − 1.

Figure 11 .
Figure 11.Projected band structure for (a) s-orbitals, and (b) p-orbitals of Li 2 BeAl Heusler compound, as well as (c) s-orbitals and (d) p-orbitals, and (e) d-orbital for the Li 2 BeGa Full Heusler alloys.

Figure 12 .
Figure 12.Lattice thermal conductivity as a function of temperature for Li 2 BeAl (blue) and Li 2 BeGa (red) Full Heusler alloys.

Figure 13 .
Figure 13.The figure of merit as a function of temperature (left), and carrier concentration (right) for Li 2 BeAl (a and b) Li 2 BeGa (c and d) Full Heusler alloys.

Table 1 .
The total energy (E total ), cohesive energy (E coh ) and the energies of each individual atom.All energies are in eV.