Excellent thermoelectric performance of Fe₂NbAl alloy induced by strong crystal anharmonicity and high band degeneracy

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.

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 refrigeration^1,2. 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 (N_v)^6,7, resonance level^8,9,10 or incorporating magnetic nanoparticle^11,12 to decouple α and σ. The other is to minimize the ${\kappa }_{L}$ by forming a superlattices¹³, nanograins^14,15, alloying^16,17, point defects^18,19 or seeking compounds with intrinsically low ${\kappa }_{L}$ originating from liquid-like atomic behavior²⁰, or strong lattice anharmonicity^21,22.

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 performance^23,24,25, due to the relatively large α and high σ. Recently, Hinterleitner et al. confirmed that the PF of metastable Fe₂V_0.8W_0.2Al 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 materials^25,26,27. 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 substitution^30,31, 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 properties^33,34.

Fig. 1: Crystal structure.

Fe₂MAl (M = V, Nb, Ta) exhibits a regular cubic crystal structure, 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.

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 N_v 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% Fe_Al antisite defect, Fe_M antisite defect, M_Al 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

$$\Delta {E}={E}_{{({Fe}_{2}{MAl})}}-{2E}_{({Fe})-{E}_{(M)}}-{E}_{({Al})}$$

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 ${\kappa }_{L}$.

Fig. 2: Phonon properties.

Phonon band structures, phonon density of states and phonon dispersion curves along the Γ – X direction for (a) Fe₂VAl, (b) Fe₂NbAl, and (c) Fe₂TaAl. The colored lines represent contributions from individual atoms, and the width of the lines are proportional to their contributions.

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 anharmonicity^35,36,37. 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.

Fig. 3: Lattice thermal conductivity.

The calculated lattice thermal conductivity of Fe₂MAl (M = V, Nb, Ta) as a function of temperature.

Fig. 4: Group velocity, Grüneisen parameters and 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.

Table 1 The average transverse (${\rm{TA}}/{\rm{TA}}^{\prime}$), longitudinal (LA) and total Grüneisen parameters (${\bar{\gamma}}_{{TA}/{TA}^{\prime} /{LA}/{Total}}$) at 900 K

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 pairs^40,41. 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 materials^43,44. Meanwhile, avoided crossings between the acoustic and optical modes are also an indicator of rattling^45,46. 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.

Fig. 5: Accumulated lattice thermal conductivity.

The lattice thermal conductivities as a function of frequency for (a) Fe₂VAl, (b) Fe₂NbAl, and (c) Fe₂TaAl at 900 K. d The contribution of acoustic and optical branches to lattice thermal conductivity for Fe₂VAl, Fe₂NbAl, and Fe₂TaAl at 900 K, respectively.

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 materials^49,50. 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 Fe₂VAl^31,52,53. 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.

Table 2 Summary of Fe₂MAl band gap using different methods

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 atoms^56,57. 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 Fe₂NbAl^58,59. 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.

Fig. 6: Electronic band structures, carrier pocket shapes and projected density of states.

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. N_v, 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 N_v = 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 N_v = 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

$$\alpha =\frac{8{\pi }^{2}{k}_{B}^{2}}{3e{h}^{2}}{m}_{d}^{* }T{\left(\frac{\pi }{3n}\right)}^{\frac{2}{3}},$$

(1)

where ${m}_{d}^{* }$ is DOS effective mass and n is carrier concentration. σ is expressed as

$$\sigma ={ne}\mu ,$$

(2)

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 N_v 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 N_v, and the large N_v is favorable for high α and electrical transport performance^7,62. 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 N_v 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 N_v 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.

Fig. 7: Calculated electrical transport properties at 900 K.

Electrical conductivity σ of (a) p-type and (b) n-type, Seebeck coefficient α of (c) p-type and (d) n-type, and power factor PF of (e) p-type and (f) n-type. Different colored lines represent different compounds.

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 N_v.

Fig. 8: Thermoelectric performance.

a The calculated zT values of Fe₂NbAl as a function of carrier concentration at 300 and 900 K, respectively. b Temperature-dependent zT values for p-type and n-type Fe₂MAl.

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:

$${\kappa }_{L}\equiv {\kappa }_{\alpha \alpha }=\frac{1}{{NV}}\mathop{\sum}\nolimits_{\lambda }\frac{\partial f}{\partial {{{\rm T}}}}{{\hslash }}{\omega }_{\lambda }{\upsilon }_{\lambda }^{\alpha }{\upsilon }_{\lambda }^{\alpha }{\tau }_{\lambda },$$

(3)

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/τ^iso) and three-phonon anharmonic scattering (1/τ^anh). 1/τ^anh involves the sum over three-phonon transition probabilities ${\Gamma }_{\lambda {\lambda }^{{\prime} }{\lambda }^{{\prime} {\prime} }}^{\pm }$, which can be calculated as

$${\Gamma }_{\lambda {\lambda }^{{\prime} }{\lambda }^{{\prime} {\prime} }}^{\pm }=\frac{{{\hslash }}\pi }{8N}\left\{\frac{2({f}_{{\lambda }^{{\prime} }}-{f}_{{\lambda }^{{\prime} {\prime} }})}{{f}_{{\lambda }^{{\prime} }}+{f}_{{\lambda }^{{\prime} {\prime} }}+1}\right\}\frac{\delta \left({\omega }_{\lambda }\pm {\omega }_{{\lambda }^{{\prime} }}-{\omega }_{{\lambda }^{{\prime} {\prime} }}\right)}{{\omega }_{\lambda }{\omega }_{{\lambda }^{{\prime} }}{\omega }_{{\lambda }^{{\prime} {\prime} }}}{\left|{V}_{\lambda {\lambda }^{{\prime} }{\lambda }^{{\prime} {\prime} }}^{\pm }\right|}^{2},$$

(4)

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 package^69,70. 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 materials^71,72,73. The SPB model assumes that the τ is a function of energy as

$$\tau ={\tau }_{0}{E}^{r}.$$

(5)

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 by^74,75

$$\tau =\frac{{2}^{1/2}\pi {{{\hslash }}}^{4}\rho {\upsilon }_{l}^{2}}{3{E}_{d}^{2}{\left({m}^{* }{k}_{B}T\right)}^{3/2}}\frac{{F}_{0}(\eta )}{{F}_{1/2}(\eta )},$$

(6)

where ${\upsilon }_{l}$, E_d, 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 E_d is defined as ${E}_{d}=\Delta {\rm{E}}/(\Delta {\rm{V}}/{V}_{0})$. From the VASP code, the E_d 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

$${L}_{0}={\left(\frac{{k}_{B}}{e}\right)}^{2}\left\{\frac{\left(r+\frac{7}{2}\right){F}_{r+5/2}}{\left(r+\frac{3}{2}\right){F}_{r+1/2}}-{\left[\frac{\left(r+\frac{5}{2}\right){F}_{r+3/2}}{\left(r+\frac{3}{2}\right){F}_{r+1/2}}\right]}^{2}\right\}.$$

(7)