Abstract
Full-Heusler alloys with earth-abundant elements exhibit high mechanical strength and favorable electrical transport behavior, but their high intrinsic lattice thermal conductivity limits potential thermoelectric application. Here, the thermoelectric transport properties of Fe-based Full-Heusler Fe2MAl (M = V, Nb, Ta) alloys are comprehensively investigated utilizing density functional theory. The results suggest that Fe2NbAl exhibits exceptionally low lattice thermal conductivity due to low phonon velocities and weakly bound Nb atoms. In Fe2NbAl, the underbonding of the Nb atoms leads large Grüneisen parameters and high anharmonic scattering rates of low-frequency acoustic phonon. Meanwhile, the high band degeneracy and large electrical conductivity lead to a maximum p-type power factor of 255.6 μW·K−2·cm−1 at 900 K. The combination of low lattice thermal conductivity and favorable electrical transport properties leads a maximum p-type dimensionless figure of merit of 1.7. Our work indicates Fe2NbAl, as a low-cost, environmentally friendly, is a potential high-performance p-type thermoelectric material.
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Introduction
Thermoelectric (TE) materials enable the direct conversion between heat and electricity without moving parts or emissions, have substantial application potential for waste-heat recovery, power generation, and electronic refrigeration1,2. The conversion efficiency of a TE material can be assessed by the dimensionless figure of merit zT = α2σT/κ, where α, σ, α2σ, κ, and T are the Seebeck coefficient, electrical conductivity, power factor (PF), thermal conductivity, and the absolute temperature, respectively3. The κ includes contributions from both electronic thermal conductivity (\({\kappa }_{e}\)) and lattice thermal conductivity (\({\kappa }_{L}\))4,5. It is extremely difficult to simultaneously increase α and σ while reducing κ because of the conflicting properties of TE materials. In the past few decades, strategies to optimize TE performance are mainly focused on two aspects. One is to enhance electrical transport performance by band degeneracy (Nv)6,7, resonance level8,9,10 or incorporating magnetic nanoparticle11,12 to decouple α and σ. The other is to minimize the \({\kappa }_{L}\) by forming a superlattices13, nanograins14,15, alloying16,17, point defects18,19 or seeking compounds with intrinsically low \({\kappa }_{L}\) originating from liquid-like atomic behavior20, or strong lattice anharmonicity21,22.
Full-Heusler (FH) alloys have general chemical formula X2YZ and crystallize in a cubic structure with four interpenetrating face-centered cubic (fcc) sublattices, as shown in Fig. 1. FH alloys are usually constructed from nontoxic, low-cost and earth-abundant elements and exhibit high mechanical strength, rendering them promising candidates for prospective applications in TE materials. Compared to traditional TE materials, many FH alloys exhibit outstanding electrical performance23,24,25, due to the relatively large α and high σ. Recently, Hinterleitner et al. confirmed that the PF of metastable Fe2V0.8W0.2Al thin film can reach a dramatically large value approximately 400 μW·K−2·cm−1, which is ten times larger than those obtained for Bi2Te3-based materials25,26,27. Garmroudi et al. reported that a maximum PF about 76 μW·K−2·cm−1 was achieved in Fe2VAl bulk alloy by driving the Anderson transition through the continuous disorder tuning28. Very recently, Gui et al. demonstrated the remarkable thermopower contribution from spin fluctuation and magnon drag in Co-based FH alloys Co2XAl (X = Ti, V, Nb), and the PF of Co2TiAl can reach 40 μW·K−2·cm−1, which is comparable to that of conventional semiconducting TE materials29. Nevertheless, the zT value of FH alloys is largely limited to about 0.1 because of the extremely high intrinsic \({\kappa }_{L}\). Although scientific researchers have made tremendous efforts to reduce \({\kappa }_{L}\), such as heavy element substitution30,31, nanostructuring32, the zT value is still less than 0.3. Thus, it is particularly important to theoretically search for a FH with low \({\kappa }_{L}\) and high electrical transport properties. Fortunately, the wide variety of FH compounds provides a large space for finding the desired properties33,34.
In this work, the TE transport properties of Fe-based Fe2MAl FH alloys are comprehensively investigated based on first-principles density functional calculations. The results reveal that p-type Fe2NbAl alloy has an exceptionally low \({\kappa }_{L}\) (4.85 W·m−1·K−1 at 900 K) and high PF (255.6 μW·K−2·cm−1). We discover that the low \({\kappa }_{L}\) is due to the high anharmonic scattering rates of Fe2NbAl induced by the weakly bound Nb atoms. Meanwhile, due to the influence of Nb atoms, the top of the valence band of Fe2NbAl changes from Γ point to X point, achieving multiple valence Nv and a decoupling of α and σ. Those lead to high zT values that reach 0.66 at 300 K and 1.7 at 900 K for p-type Fe2NbAl, and the corresponding optimum carrier concentrations are 9.17 × 1018 cm−3 and 1.13 × 1020 cm−3, respectively.
Results and discussion
Crystal structure and phonon properties
According to the relative positions of the X and Y atoms, ternary intermetallic FH compounds X2YZ can be divided into two types of structures: regular cubic and inverse cubic. Figure 1 exhibits a regular cubic crystal structure of Fe2MAl with space group \({Fm}\bar{3}m\), where the Fe, M, and Al atoms are located in the Wyckoff positions of 8c (1/4, 1/4, 1/4), 4b (1/2, 1/2, 1/2) and 4a (0, 0, 0), respectively. The inverse cubic structure can be obtained by exchanging the positions between four Fe atoms and four M atoms or four Al atoms. In addition, to simulate the disordered state, the energy values of 12.5% FeAl antisite defect, FeM antisite defect, MAl antisite defect and pure phase of Fe2MAl are calculated, as listed in Supplementary Table 1. The results show that the fully ordered L21 structure of Fe2VAl, Fe2NbAl, and Fe2TaAl have the lowest energy value of −30.87 eV, −32.04 eV and −33.94 eV at 0 K, respectively. Garmroudi et al. also suggest ternary Fe2VAl is more likely to form a fully ordered L21 structure below 1350 K28. The regular cubic crystal structures of Fe2VAl, Fe2NbAl, and Fe2TaAl exhibit an optimized lattice constant of 5.6916 Å, 5.909 Å, and 5.897 Å, respectively.
To investigate the three compounds dynamical stability, ab initio molecular dynamics (AIMD) simulations have been performed at 900 K, as shown in Supplementary Fig. 1. Clearly, the total energy and temperature have only small fluctuations during 10 ps AIMD simulations at 900 K, and there is no evidence of broken bond or structural change during the AIMD simulation, indicating that the regular cubic crystal structure Fe2MAl is thermal stability, and can exist in principle at 900 K. In addition, the formation energy is calculated by
where \({E}_{({{Fe}}_{2}{MAl})}\) is the total energy of Fe2MAl. \({E}_{(M)}\), \({E}_{(V)}\), \({E}_{({Nb})}\), \({E}_{({Ta})}\) and \({E}_{(M)}\) are the energy of formation per atom. The basic crystal parameters and lowest energy of Fe, V, Nb, Ta and Al elements are listed in Supplementary Table 2. For Fe2VAl, Fe2NbAl and Fe2TaAl, the calculated formation energies are −2.03, −1.96 and −2.19 eV, respectively, which indicates they are all energetically stable.
The phonon dispersions and the projected phonon density of states (PHDOS) of Fe2MAl are displayed in Fig. 2. The real frequencies of all modes further confirm that the three crystals are dynamically stable. Furthermore, the phonon frequency of Fe2MAl gradually decreases as the M changes from light V to heavy Ta. This is also observed from the PHDOS of Fe2MAl. In Fe2TaAl, the states of heavy Ta are located in the lower part of the spectrum. However, in Fe2VAl, the states of V are strongly mixed with the states of Fe in lower and middle part of the spectrum due to their small mass difference. This suggests that the heavy atoms will lead to low phonon frequencies and phonon velocities. We know that the low phonon velocities favor low lattice thermal conductivities, thus, it may seem reasonable to expect that the Fe2TaAl alloy possesses the lowest \({\kappa }_{L}\).
Lattice thermal conductivity
The \({\kappa }_{L}\) of Fe2MAl was calculated by the ShengBTE code, as shown in Fig. 3. The \({\kappa }_{L}\) of the three alloys decreases correspondingly with increasing temperature from 300 to 900 K. The \({\kappa }_{L}\) values of Fe2VAl, Fe2NbAl and Fe2TaAl are 13.8, 4.85 and 8.29 W·m−1·K−1 at 900 K, respectively. Interestingly, Fe2NbAl exhibits the lowest \({\kappa }_{L}\), which is inconsistent with the expected results mentioned above. Figure 4a–f show the group velocity and Grüneisen parameters of the Fe2MAl. Obviously, the group velocity of the three alloys decreases correspondingly with increasing M atomic masses from V to Ta. However, the Grüneisen parameters for Fe2NbAl are obviously higher than those of Fe2VAl and Fe2TaAl, which means that Fe2NbAl has a stronger crystal anharmonicity. Usually, a large Grüneisen parameter reflects a strong crystal anharmonicity35,36,37. To quantitatively evaluate the anharmonicity, the average transverse (TA and TA’, for the lowest and second lowest transverse), longitudinal (LA) and total (Total) Grüneisen parameters (\({\bar{\gamma }}_{{TA}/{TA}{\prime} /{LA}/{Total}}\)) were calculated by \({\bar{\gamma}}=\sqrt{\left\langle {\gamma }_{{\mathcal{i}}}^{2}\right\rangle}\), as listed in Table 1. The average acoustic Grüneisen parameters of Fe2NbAl are \({\bar{\gamma }}_{{TA}}\) = 2.75, \({\bar{\gamma }}_{{TA}{\prime} }\) = 3.68, \({\bar{\gamma }}_{{LA}}\) = 2.94, and \({\bar{\gamma }}_{{Total}}\) = 3.15, respectively. Although the Grüneisen parameters of Fe2NbAl are lower than that of SnSe with ultralow \({\kappa }_{L}\) (\({\bar{\gamma }}_{{TA}}\) = 5.1, \({\bar{\gamma }}_{{TA}{\prime} }\) = 1.1, \({\bar{\gamma }}_{{LA}}\) = 5.9, and \({\bar{\gamma }}_{{Total}}\) = 4.1)38,39, it is significantly higher than that of Fe2VAl (\({\bar{\gamma }}_{{TA}}\) = 2.45, \({\bar{\gamma }}_{{TA}{\prime} }\) = 2.38, \({\bar{\gamma }}_{{LA}}\) = 2.45, and \({\bar{\gamma }}_{{Total}}\) = 2.43) and Fe2TaAl (\({\bar{\gamma }}_{{TA}}\) = 1.26, \({\bar{\gamma }}_{{TA}{\prime} }\) = 1.34, \({\bar{\gamma }}_{{LA}}\) = 1.39, and \({\bar{\gamma }}_{{Total}}\) = 1.33). This suggests that Fe2NbAl will have a higher three-phonon anharmonic scattering rates.
The group velocity of different phonon modes as a function of frequency for (a) Fe2VAl, (b) Fe2NbAl and (c) Fe2TaAl. The Grüneisen parameter of different phonon modes as a function of frequency for (d) Fe2VAl, (e) Fe2NbAl and (f) Fe2TaAl. The TA TA’, and LA modes are shown as green, red, and blue points, respectively. The anharmonic scattering rates of (g) Fe2VAl, (h) Fe2NbAl and (i) Fe2TaAl.
Fig. 4g–i present the calculated anharmonic scattering rates of Fe2MAl at 900 K. The anharmonic scattering rates of the low-frequency acoustic branch of Fe2NbAl is close to twice those of Fe2VAl and Fe2TaAl, indicating the strong crystal anharmonicity of Fe2NbAl. To uncover the origin of the strong crystal anharmonicity of Fe2NbAl, we calculated the electron localization function (ELF) for the three compounds, as shown in Supplementary Fig. 2a–c. The ELF can be expressed as ELF(r) = \((1+{\left\{K(r)/{K}_{h}[\rho (r)]\right\}}^{2})\)−1, where K is the curvature of the electron pair density for electrons of identical spin, \(\rho (r)\) the density at r, and \({K}_{h}[\rho (r)]\) the value of K in a homogeneous electron gas with density ρ. An ELF of 0 corresponds to no localization as in regions with no electrons. A value of 0.5 reflects the behavior of a homogeneous electron gas, with values near 0.5 being characteristic of metallic bonding. An ELF value of 1 indicates full localization, with high values characterizing covalent bonds, core shells, and lone pairs40,41. The ELF values can be used to identify the differences of chemical bonding between compounds41.
In the three compounds Fe2VAl, Fe2NbAl and Fe2TaAl, the ELF values between M and Fe are 0.76, 0.49 and 0.77, respectively. The ELF values between M and Al are 0.88, 0.71 and 0.89, respectively. The results suggest Fe2NbAl has a relatively soft lattice that favors low sound velocity, strong anharmonicity and high phonon-phonon scattering42. However, the sound speed is greatly affected by the atomic mass in Fe2MAl, resulting in Fe2TaAl having the lowest sound speed. In Fe2NbAl, the Nb atoms have weak bonding with other atoms. This is reminiscent of guest atoms “rattling” in the host cages. As shown in Supplementary Fig. 3, around each Nb atom, there is a hexadecahedron network consisting of 8 Fe atoms and 6 Al atoms in the crystal structure of Fe2NbAl. From Fig. 2, the Nb modes in Fe2NbAl exhibit relatively dispersionless behavior around 4.5 to 7.5 THz, which is one of the characteristics associated with rattling in thermoelectric materials43,44. Meanwhile, avoided crossings between the acoustic and optical modes are also an indicator of rattling45,46. There are avoided crossings between the acoustic and optical modes along Γ – X in Fe2NbAl43, as shown in Fig. 2b. Then, the localized ratting-like42 motion of Nb atoms lead to large anharmonic scattering rates and Grüneisen parameters.
Higher anharmonic scattering rates are associated with shorter phonon lifetimes, resulting in reduced contributions of these phonons to \({\kappa }_{L}\)46. Figure 5a–c show the \({\kappa }_{L}\) values of Fe2MAl as a function of frequency were calculated at 900 K. It can be clearly seen that the \({\kappa }_{L}\) increases rapidly for the low frequency acoustic modes in Fe2VAl (1–8.3 THz) and Fe2TaAl (1–4.9 THz). The phenomenon is more obvious in Fe2TaAl, where acoustic phonons contribute 83% of \({\kappa }_{L}\). However, the \({\kappa }_{L}\) of Fe2NbAl increases slowly for the low frequency acoustic modes (1–6.9 THz). This part of the phonons only contributes 63% of \({\kappa }_{L}\), which is the result of the ratting-like of Nb atoms. Anharmonic vibration of Nb atoms can effectively scatter low frequency phonons and results in a lower \({\kappa }_{L}\) for Fe2NbAl. Indeed, from the calculated \({\kappa }_{L}\) cumulative values of acoustic and optical branches, as shown in Fig. 5d, it is clearly seen that the contribution of \({\kappa }_{L}\) from low-frequency acoustic phonons is significantly reduced in Fe2NbAl. This can also be seen from the calculated phonon density of states and anharmonic scattering rates. The low-frequency acoustic branch of Fe2NbAl is mainly contributed by Nb atoms. Notably, the anharmonic scattering rates of the low-frequency acoustic branch of Fe2NbAl are significantly higher than those of Fe2VAl and Fe2TaAl. Hence, it is reasonable to conclude that the bound Nb atoms inside the cage induces strong anharmonic scattering and consequently reduces the \({\kappa }_{L}\) of Fe2NbAl.
Electronic structure and thermoelectric performance
The electronic band structures of Fe2MAl are calculated by Perdew-Burke-Ernzerhof (PBE)47 potential without Spin-orbit coupling (SOC) and with SOC, Tran-Blaha modified Becke-Johnson (TB-mBJ)48 potential without SOC and with SOC, respectively, as listed in Table 2. We note that spin-orbit coupling leads to a decrease in the band gap, regardless of the PBE potential or the TB-mBJ potential. Thus, considering the effect of SOC is necessary to accurately predict the relevant properties of new materials49,50. Indeed, Okamura et al. suggest Fe2VAl with an energy gap of 0.1–0.2 eV by using photoconductivity measurements51. The temperature-dependent electrical resistivity ρ(T) obtained on various Fe2VAl samples also indicates the non-metallic behavior of Fe2VAl31,52,53. The band gap of Fe2VAl (0.21 eV) calculated using the TB-mBJ potential with SOC is closer to the experimental value. The band gap (~ 0.3 eV) of Fe2NbAl is calculated by Hamad et al54. using PBE functional. The band gap (0.8 eV) of Fe2TaAl is calculated by Khandy et al. 55 using mBJ potential. These are consistent with our calculation results. However, there are no experimental data for Fe2NbAl and Fe2TaAl. Thus, the calculated electronic structures by TB-mBJ functional and SOC are used to obtain electrical transport properties.
From Fig. 6a–c, the band gap of Fe2VAl and Fe2NbAl is significantly smaller than that of Fe2TaAl due to the orbital hybridization. From the atom-projected density of states (DOS), as shown in Fig. 6j–l, the Fe-V DOS curve shows an obvious overlap between the Fe 3d and V 3d states near the Fermi level. The relevant hybridization peaks of Fe 3d and V 3d appear at −2.1, −1.7, 1.0, and 1.4 eV, indicating a strong interaction between Fe and V atoms56,57. The DOSs of Fe-Nb are similar to that of Fe-V, but the overlapped DOSs of their Fe 3d and Nb 4d orbitals are smaller than that of Fe-V. However, there is no obvious overlap between Fe 3d and Ta 5d states in Fe2TaAl. These results suggest the Fe-3d and V-3d of Fe2VAl have strong orbital hybridization. Fe-3d and Nb-4d in Fe2NbAl have relatively weak orbital hybridization, and no evident orbital hybridization phenomenon in Fe2TaAl. Based on the hybridizations between atomic orbitals (Fig. 6j, k), we can roughly distinguish the bonding and antibonding regions of two orbitals: the bonding (−2.5 to −1.7 eV) and antibonding [from conduction band minimum (CBM) to 1.7 eV] regions of Fe-d and V-d for Fe2VAl, and the bonding (−2.8 to −2.0 eV) and antibonding (from CBM to 1.3 eV) regions of Fe-d and Nb-d for Fe2NbAl58,59. The antibonding states of Fe2NbAl and Fe2VAl are located at the bottom of the conduction band, which can depress the energy value of the bottom of the conduction band and form a narrow bandgap semiconductor60. Indeed, both Fe2VAl and Fe2NbAl are narrow bandgap semiconductors, and the bandgap of Fe2NbAl is only 0.18 eV, which also means that Fe2NbAl is easy to obtain the optimal zT value by doping.
Band structures of (a) Fe2VAl, (b) Fe2NbAl and (c) Fe2TaAl as determined using the modified Becke-Johnson potential with spin-orbit coupling. Carrier pocket shapes as given by isosurfaces 0.12 eV above the conduction band for electrons of (d) Fe2VAl, (f) Fe2NbAl and (h) Fe2TaAl and 0.12 eV below the valence band maximum for holes of (e) Fe2VAl, (g) Fe2NbAl and (i) Fe2TaAl, projected density of states of (j) Fe2VAl, (k) Fe2NbAl and (l) Fe2TaAl.
In the Fe2MAl compound, as the M atom changes from V to Ta, the valence band top of the compound also changes from Γ to X point, leading to a higher symmetric degeneracy of Fe2NbAl and Fe2TaAl. Nv, including both symmetry and orbital degeneracies, is determined by the effective total number of independent carrier pockets or valleys in the Brillouin zone (BZ)7. We plot the carrier pockets of the three compounds as given by isoenergy surfaces 0.12 eV above CBM for electrons and 0.12 eV below the valence band maximum (VBM) for holes, as displayed in Fig. 6d–i, respectively. Clearly, for the valence band of Fe2VAl, the isoenergy surface has only one pocket at the Γ point. While, there is an effective degeneracy of Nv = 3 for the valence band of Fe2VAl because the maximum points of the three energy bands are located at the same point Γ. For Fe2TaAl, the maximum point of valence band is located at X point, and there is only one energy band. However, as the top of the valence band changes from Γ point to X point, the isoenergy surfaces changes from one pocket at Г point to 6 half-pockets at X point. Thus, the full number of the degeneracy for Fe2TaAl is also 3. There is an effective degeneracy of Nv = 6 for valence band of Fe2NbAl. This arises from a light hole band maximum at the X point, and a heavy hole band separated by a very small spin-orbit gap of 0.05 eV.
The α for the acoustic phonon scattering (metals or degenerate semiconductors) is given by refs. 3,25,59
where \({m}_{d}^{* }\) is DOS effective mass and n is carrier concentration. σ is expressed as
where e is electron charge and μ is mobility. The μ can be given as \(\mu \propto {({m}^{* })}^{-1}{({m}_{b}^{* })}^{-3/2}\) for the charge carriers predominantly scattered by acoustic phonons61. Here, \({m}^{* }\) is transport effective mass, and \({m}_{b}^{* }\) is single-band effective mass (the relationship between \({m}^{* }\) and \({m}_{b}^{* }\) is shown in Supplementary Note 1). From Eq. (1), a large \({m}_{d}^{* }\) is favorable for a large α. large α can be obtained by either a local, flat single-band effective mass (\({m}_{b}^{* }\)) or a large Nv since \({m}_{d}^{* }\) = \({N}_{v}^{2/3}{m}_{b}^{* }\). Increasing the \({m}_{b}^{* }\) of the energy band will enhance the α, but also leads to the deterioration of the σ. When the intervalley scattering is negligible, μ is not a function of Nv, and the large Nv is favorable for high α and electrical transport performance7,62. Meanwhile, we note that the energy band near the VBM form a steep peak at X point for Fe2NbAl while that around VBM at Γ point for Fe2VAl is flat, which means Fe2NbAl will have small hole effective mass and large σ. Compared to Fe2NbAl, the energy band near the VBM of Fe2TaAl form a relatively gentle peak, indicating Fe2TaAl has a moderate hole effective mass and electrical conductivity.
Figure 7 shows the calculated σ, α, and PF of the three compounds at 900 K using the BoltzTraP code. The σ of p-type Fe2NbAl is significantly higher than that of p-type Fe2VAl, and p-type Fe2TaAl have a moderate σ. Due to the small σ, the p-type Fe2VAl has the highest α. Note that, although the σ of p-type Fe2NbAl is much higher than that of p-type Fe2TaAl, the α of the two alloys are similar or identical. This is because the high Nv suppresses the deterioration of the α in Fe2NbAl. For n-type Fe2VAl, Fe2NbAl and Fe2TaAl, it shows a general trend that the σ gradually decreases and the α gradually increases. Due to the high Nv of n-type Fe2TaAl, the α of n-type Fe2TaAl is superior to that of n-type Fe2VAl and Fe2NbAl. From the calculated σ and α, the PFs of the three compounds at 900 K are obtained. For p-type, Fe2NbAl has the highest PF (255.6 μW·K−2·cm−1), while for n-type, Fe2VAl has the highest PF (128.4 μW·K−2·cm−1), and the electrical transport properties of p-type alloys are better than those of n-type alloys.
Combining the high electrical transport and low κ, we obtained zT values of Fe2NbAl as a function of carrier concentrations at 300 and 900 K, as shown in Fig. 8a. The highest zT values of p-type and n-type Fe2NbAl at room temperature are about 0.66 and 0.71, and the corresponding optimum carrier concentrations are 9.17 × 1018 cm−3 and 8.30 × 1018 cm−3, respectively. At 900 K, the maximum zT of p-type and n-type Fe2NbAl are approximately 1.7 and 1.15, and the corresponding carrier concentrations are 1.13 × 1020 cm−3 and 3.78 × 1020 cm−3, respectively. To compare the thermoelectric properties of the three compounds more clearly, the maximum ZT value of Fe2MAl as a function of temperature from 300 to 900 K are plotted in Fig. 8b. Whether it is p-type or n-type materials, Fe2NbAl has the highest zT value. The favorable TE properties of p-type Fe2NbAl are the consequences of the strong anharmonic vibration of Nb atoms and large Nv.
In summary, we have theoretical studied the \({\kappa }_{L}\), electronic structures, and electrical transport properties of the isostructural natural compounds, Fe2VAl, Fe2NbAl and Fe2TaAl. The results show that Nb atoms serve as generalized rattlers in the cages formed by Fe and Al atoms. The anharmonic vibration of weakly bound Nb atoms result in a high anharmonic scattering rates and low \({\kappa }_{L}\) (4.85 W·m−1·K−1 at 900 K) of Fe2NbAl. Meanwhile, the top of the valence band of Fe2NbAl changes from Γ point to X point due to the effect of Nb atoms, leading to a high electrical conductivity and a high PF (255.6 μW·K−2·cm−1). The highest zT value of p-type Fe2NbAl can reach 1.7 at 900 K due to the low \({\kappa }_{L}\) and the high PF. Our work suggests that Fe2NbAl, as a low-cost, environmentally friendly FH alloy, is expected to become a high-performance high-temperature TE material.
Computational methods
Density functional theory calculations
The crystal structures optimization of Fe2MAl are implemented in the Vienna ab initio simulation package (VASP)63 based on density functional theory. We use the exchange-correlation potential of generalized gradient approximation (GGA) within the PBE for the optimization. For electron-ion interactions, we use the projector augmented wave (PAW) method with a plane wave kinetic energy cut-off of 500 eV64, and a regular 15 × 15 × 15 k-point mesh is used to sample the Brillouin zone (BZ). The electronic self-consistent energy convergence criterion is 10−8 eV. In addition, a 3 × 3 × 3 supercell (with 108 atoms in total) is employed for the ab initio molecular dynamic (AIMD) calculations, and the temperature is kept at 900 K. Following the structure optimization, we calculated electronic structure and transport using the full-potential linearized augmented plane wave (LAPW) method as implemented in the WIEN2k code65. Here, we use a unit cell of four atoms for performance calculations. Meanwhile, we use the TB-mBJ potential to avoid the classical underestimate of the band gap from the GGA functionals, which generally gives band gaps for semiconductors in agreement with experiment. The SOC is included in these calculations. The computed electronic structures are used to obtain electrical transport properties. The transport properties are carried out in the framework of semiclassical Boltzmann theory as implemented in the BoltzTraP code66 based on first principles eigenvalues at approximately 50000 k-points.
Lattice thermal conductivity
Based on the first principles, \({\kappa }_{L}\) along the α axis can be calculated via linearized Boltzmann equation67 with harmonic and anharmonic interatomic force constants as the sum of contributions over all the phonon modes \(\lambda\) with branch p and wave vector q:
where N, V, and f are the number of uniformly spaced q points in the BZ, the volume of the unit cell, and the Bose-Einstein distribution function depending on the phonon frequency \({\omega }_{\lambda }\), and \({\upsilon }_{\lambda }^{\alpha }\) is the phonon velocity, respectively. The phonon lifetime \({\tau }_{\lambda }\), an inverse of the total scattering rate, is limited by two processes: two-phonon scattering from isotopic disorder (1/τiso) and three-phonon anharmonic scattering (1/τanh). 1/τanh involves the sum over three-phonon transition probabilities \({\Gamma }_{\lambda {\lambda }^{{\prime} }{\lambda }^{{\prime} {\prime} }}^{\pm }\), which can be calculated as
where the upper row of curly brackets strokes and + sign represent the absorption process, and the lower row of curly brackets strokes and - sign represent the emission process. The scattering matrix elements \({V}_{\lambda {\lambda }^{{\prime} }{\lambda }^{{\prime} {\prime} }}\) are limited by the third-order interatomic force constants (IFCs).
The harmonic second-order IFCs and the phonon dispersions are obtained by the PHONOPY package68. We use 3 × 3 × 3 supercells (with 108 atoms in total) and 2 × 2 × 2 k-point meshes to obtain the dynamic matrix, and the same supercell and k-meshes are used to calculate the anharmonic third-order IFCs, as implemented in the thirdorder.py code61. Then, combining second-order IFCs and third-order IFCs at 0 K, the \({\kappa }_{L}\) is calculated by iteratively solving the linearized Boltzmann-Peierls transport equation via the ShengBTE package69,70. In the calculation, a phonon momentum q mesh of 25 × 25 × 25 is used, and a Gaussian smearing (scalebroad parameter) of 0.05 is chosen to simulate the phonon wave vectors.
Electrical transport properties
The electrical transport parameters (α and σ/τ) of Fe2MAl can be calculated by the BoltzTraP code. To obtain σ, it is imperative to figure out the carrier relaxation time τ. The τ can obtain by the single parabolic band (SPB) model, which is successfully applied to many materials71,72,73. The SPB model assumes that the τ is a function of energy as
Here, r is called the scattering parameter. In this paper, the acoustic-phonon scattering for electrons is assumed, \(r=-1/2\) is thus used for subsequent calculations. The τ for the acoustic-phonon scattering is calculated by74,75
where \({\upsilon }_{l}\), Ed, and ρ are the longitudinal sound velocity, the deformation potential constant, and the mass density, respectively. The \({\upsilon }_{l}\) can be obtained from the bulk modulus (B) and shear modulus (G) by \({v}_{l}=\sqrt{(B+4/3G)/\rho }\)76, and the \({F}_{x}\left(\eta \right)\) can be calculated by, \({F}_{x}\left(\eta \right)={\int }_{0}^{\infty }{E}^{x}/[1+\exp (E-\eta )]{dE}\), where η (=\({E}_{F}/{k}_{B}T\)) is the reduced chemical potential. The deformation potential Ed is defined as \({E}_{d}=\Delta {\rm{E}}/(\Delta {\rm{V}}/{V}_{0})\). From the VASP code, the Ed for holes and electrons can be obtained based on the energy changes of VBM and CBM with volume change ΔV/V0. These detailed parameter values are listed in Supplementary Table 3. The transport effective mass (\({m}^{* }\)) is calculated by \({\left({m}^{* }\right)}^{-1}=\sigma /{e}^{2}\tau n\), as shown in Supplementary Fig. 4a, b, where n is the carrier density77, and can be obtained directly using BoltzTraP code. Then, we can obtain the τ values of p-type and n-type for Fe2MAl, as shown in Supplementary Fig. 4c, d. Finally, the \({\kappa }_{e}\) is calculated by \({\kappa }_{e}={L}_{0}\sigma T\), where L0 is the Lorenz number, as shown in Supplementary Fig. 5. It is defined as
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The codes used during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by National Key Research and Development Program of China (Nos. 2019YFA0704900, 2023YFB3809400), National Natural Science Foundation of China (Nos. 52130203, 52102301, 52102298, 52172232, 92163122, 52202034, 52201256), Natural Science Foundation of Hubei Province (No. 20231j0209), and Basic and Applied Basic Research Foundation of Guangdong Province (No. 2022B1515120005).
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W.Z. and J.Y. conceived and designed the study. X.Y. and S.K. performed theoretical calculations and physical analysis. D.L. and T.C. drew the crystal structure diagram and analyzed the thermal stability. C.L., W.X., and L.L. analyzed the electrical and thermal transport properties. Wanting Zhu, X.N., P.W., and Q.Z. corrected the grammar mistakes. All authors read the paper and commented on the text.
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Ye, X., Yu, J., Ke, S. et al. Excellent thermoelectric performance of Fe2NbAl alloy induced by strong crystal anharmonicity and high band degeneracy. npj Quantum Mater. 9, 60 (2024). https://doi.org/10.1038/s41535-024-00671-1
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DOI: https://doi.org/10.1038/s41535-024-00671-1
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