First-principles study of anomalous Nernst effect in half-metallic iron dichloride monolayer

Thermoelectric power generation based on the anomalous Nernst effect (ANE) is attracting attention owing to its flexible, simple nature and low generation cost.¹ The ANE allows for greater flexibility in device design since when the temperature gradient is applied, the material length along the temperature gradient is not required because the anomalous Nernst voltage enhances with the transverse length normal to both the magnetization and the temperature gradient. Thus, thermoelectric devices based on the ANE can be designed to be compatible with any heat source, and thin films and two-dimensional (2D) materials can be used as the base in these devices. In addition, in the case of the ANE, the electric current generates the Ettingshausen heat current from the low temperature side to the high temperature side, which can increase the efficiency conversion rate, because the directions of the Ettingshausen heat current and the electric current are perpendicular to each other.² In fact, this condition ensures that the efficiency of ANE-based thermoelectric devices is high.

The conversion efficiency of ANE thermoelectric materials is dependent on their anomalous Nernst coefficient (ANC). The primary challenge on the realization of thermoelectric devices is to ensure that the materials exhibit high ANC values. Generally, the magnitude of the ANC is proportional to the derivative of the anomalous Hall conductivity (AHC) as a function of energy.^3,4 The mechanism of the AHC itself is determined by the intrinsic and extrinsic contribution. The intrinsic contribution is given by the band structure where the interband coherence increases the electron anomalous velocity related to the Berry curvature,^5,6 while the extrinsic contribution is originated from skew scattering or side-jump due to the spin–orbit interactions.^5,7 Due to weak spin–orbit coupling in 3d transition metals, the intrinsic contribution might be dominant in 3d transition metal compounds within a moderately dirty regime.⁸ The weak spin–orbit interaction and/or nontrivial spin textures generate intrinsic contribution to the AHC.⁹ The 2D skyrmion material such as an EuO monolayer is predicted that it has shown a large ANC.^3,10 In 2D materials, the ANE is related to the quantum anomalous Hall effect; this effect, in turn, is determined by the quantized anomalous Hall conductivity (AHC), which can be expressed as $σxy=e2hC$⁠, where C is the Chern number. The quantized AHC was confirmed experimentally by Chang et al.¹¹ in a magnetically doped thin film of a topological insulator (Bi,Sb)₂Te₃, suggesting that it is possible to obtain a large ANC in 2D systems.

A high figure of merit can be achieved by 2D ANE thermoelectric materials based on the following several reasons. First, the discovery of ferromagnetism (FM) in 2D materials in 2017^12,13 has opened the way for the study of the ANE in 2D systems. Second, the use of a 2D material can also result in decreases in the thermal conductivity owing to surface phonon scattering,¹⁴ which is essential for increasing the figure of merit. In addition, the use of some 2D material such as graphene also ensures that the electrical conductivity is higher than that of the corresponding bulk structure while decreasing the material thickness.¹⁵ Besides that, the theoretical analysis also suggests that the thermoelectric coefficient for quantum-well structures increased linearly when the quantum-well thickness decreased.¹⁶ These advantages of 2D thermoelectric devices have been confirmed experimentally by Lee et al. using 2D SnS₂¹⁷ and Ohta et al. using 2D gas in SrTiO₃.¹⁸ 

In this study, by using density functional calculations, we explored the thermoelectric properties of a half-metallic 1T-FeCl₂ monolayer. In determining the magnitude of ANE, we focus in the intrinsic contribution of AHC based on the moderately dirty regime theory.⁸ We found that the 1T-FeCl₂ monolayer exhibits a high AHC, which, in turn, results in a large ANC. We show that the high AHC is attributable to the bands at the K-point near the Fermi level, where a large Berry curvature exists. In addition, we were able to tune the thermoelectric properties of the 1T-FeCl₂ monolayer through doping in order to ensure a large ANC at the Fermi level. Thus, based on the obtained results, it can be concluded that the 2D magnetic half-metallic materials can potentially be made to exhibit high ANC values.

We performed first-principles electronic structure simulations based on the collinear and noncollinear density functional theory (DFT)¹⁹ using the OpenMX code.²⁰ We used the generalized gradient approximation with the Perdew–Burke–Ernzerhof functional to treat the exchange–correlation potential.²¹ For this, we used the norm-conserving pseudopotential method.²² Moreover, we extended the wave functions using a linear combination of multiple pseudoatomic orbitals²³ with specification Fe6.0 S-s2p3d3f1 and Cl7.0-s3p3d2 where 6.0 and 7.0 is the cutoff radius (in bohrs) of Fe and Cl atoms, respectively, S means soft pseudopotential, and the number after s, p, d, and f is the radial function multiplicity of each angular momentum component. The charge density cutoff energy was set at 500.0 Ry. A (20, 20, 1) k-point mesh was used for the self-consistent field (SCF) calculations. For the noncollinear density functional calculations, the spin–orbit interaction (SOI) was included.²⁴ We checked for convergence during k-point sampling as well as for the cutoff energies. All the atomic positions and lattice parameters of the 1T-FeCl₂ monolayer were optimized using the eigenvector following quasi-Newton algorithm until all the forces were smaller than 10⁻⁶ hartrees/bohr.²⁵ The lattice constant was determined based on the total energy minimum.

The structure of the 1T-FeCl₂ monolayer is shown in Fig. 1. The side view of the 1T-FeCl₂ monolayer structure is shown in Fig. 1(a), while the top view is shown in Fig. 1(b). These figures show that each Fe atom is surrounded by six Cl atoms. It also shows the primitive unit cell of the hexagonal lattice, wherein the lattice constant, the magnitude of $a⃗$ is equal to that of $b⃗$ with the vacuum region, c = 17.26 Å. After structural relaxation, the lattice constant of the 1T-FeCl₂ monolayer was 3.48 Å. The distance between the Fe and Cl atoms (d_Fe–Cl) is 2.44 Å, while the vertical distance between the Cl atoms (d_Cl–Cl) is 2.78 Å. Furthermore, the angle of Fe–Cl–Fe (θ) is 89.90°. These calculated structural parameter values of the monolayer were in good agreement with those reported in previous theoretical studies on FeCl₂.^26–29 The calculated lattice constants for monolayer FeCl₂ were similar to those of bulk FeCl₂, a = 3.6 Å.^30,31

The thermoelectric coefficient can be obtained based on the linear response of the charge current, which is given as j = σ_ijE + α_ij(−∇T), where σ_ij and α_ij are the conductivity tensors and thermoelectric conductivity tensors, respectively, E is the electric field, and ∇T is the temperature gradient. The Nernst coefficients are related to the conductivity tensor, and these relationships can be represented as follows:

where S₀ = α_xx/σ_xx, N₀ = α_xy/σ_xx, and θ_H = σ_xy/σ_xx are the pure Seebeck coefficient, pure ANC, and Hall angle, respectively. Furthermore, the electrical conductivity can be calculated from $σxx=e2τ∑n∫vxn(k)2(−∂f∂εnk)dk$⁠, while the AHC can be computed from $σxy=−e2ℏ∑n∫Ωzn(k)f(εnk)dk$⁠. The thermoelectric conductivity tensors can be represented as $αij=kBe∫σij(ε)∣T=0ε−μT(−∂f∂ε)dε$⁠, where e, τ, $vxn$⁠, f, $εnk$⁠, ℏ, $Ωzn(k)$⁠, T, k_B, and μ are the elementary charge, relaxation time, group velocity of electrons, Fermi–Dirac distribution function, eigenenergy, reduced Planck constant, Berry curvature, temperature, Boltzmann’s constant, and chemical potential, respectively. The n is the band index, and k is the wave vector. The band decomposition of the Berry curvature can be estimated from $Ωn(k)≡i∇kunk∣×∣∇kunk$⁠. The Berry curvature and σ_xy for each band were calculated by using the Berry connection defined on a discretized Brillouin zone.³² 

To obtain the thermoelectric properties, the Wannier90 code was used to construct maximally localized Wannier functions (MLWFs) based on the results of the DFT calculations.³³ We construct 22 Wannier bands within the range of −15 eV to 15 eV for the outer window energy and −4 eV to 4 eV for the inner window energy. We also simulated the transport properties based on the MLWFs using the semiclassical Boltzmann transport theory³⁴ within constant relaxation time approximation, τ = 10 fs, with k-point mesh (300 300, 1). The relaxation time originated in electron–phonon interaction can be evaluated from ab initio theory by computing the imaginary part of electron–phonon self-energy;³⁵ however, in magnetic materials, the relaxation time depends on electron–magnon interaction and electron–electron interaction including spin fluctuation. Here, we neglected these effects and assumed the constant relaxation time based on our previous study on a half-metallic material.³⁶ We have previously used this method successfully to study the thermoelectric properties of half-Heusler compounds and skyrmion crystals.^3,10,36

The ground states of the 1T-FeCl₂ monolayer are half-metallic, as can be seen from the band structure and density of states. Figure 2 shows the electronic band structure and projected density of states (PDOS) of the 1T-FeCl₂ monolayer in the FM configuration; it can be seen that the minority states cross the Fermi level, while the majority states have a large gap. These results are in fairly good agreement with those of earlier studies.^26,37

Most of the states near the Fermi level of the 1T-FeCl₂ monolayer were formed by the 3d orbital of the Fe atom and the 2p orbitals of the Cl atoms, as shown in the density of states of two atoms in Fig. 2(b). These states are anti-bonding states of Fe 3d and Cl 2p. According to the ligand field theory,³⁸ the octahedral geometry of Cl atoms around a Fe atom leads to the splitting of the energy between the d orbitals. This is ascribable to the electron–electron repulsion between the Fe and Cl orbitals. The higher energy orbital, e_g (⁠$dx2−y2,d3z2−r2$⁠), is directly connected to six Cl atoms and has more Coulombic interaction energy, while the lower energy orbital, t_2g (d_xy, d_xz, d_yz), is relatively stable as it lies between the Cl atoms. The d orbital states, in particular, t_2g, determine the magnitudes of the thermoelectric coefficients of the 1T-FeCl₂ monolayer.

Table I shows the thermoelectric properties of the 1T-FeCl₂ monolayer without carrier doping. As stated in Eq. (1), the Seebeck and Nernst effects cannot be separated. The main component of N consists of the pure ANC (N₀), the Hall angle (θ_H), and the pure Seebeck coefficient (S₀). The contributions of S₀ to N are around 0.2% each. However, N₀ and θ_HS₀ weaken each other since the sign of N₀ and θ_HS₀ are the same at both 50 K and 100 K. N can be increased if we tune the chemical potential μ by carrier doping, as can be seen from the rigid band approximation (RBA) in Fig. 3.

The chemical potential dependences of the Nernst, N, coefficients of the 1T-FeCl₂ monolayer at 50 K and 100 K are shown in Fig. 3(a). As shown in Fig. 3(a), the value of N is small at μ = 0 at both 50 K and 100 K. On the other hand, a large value of N appears at approximately μ = 0.16 eV, μ = 0.31 eV, and μ = 0.35 eV, which is described by peaks 1, 2, and 3, respectively. By using self-consistent field carrier doping methods, we confirmed that the RBA is quite well for 1T-FeCl₂.

The origin of peaks 1, 2, and 3 is contributed by N₀ and θ_HS₀ in Figs. 3(b) and 3(c), respectively. N₀ has a positive magnitude and θ_H S₀ has a negative magnitude in peak 1. According to Eq. (1), N₀ and θ_H S₀ strengthen each other. In Table II, it is shown that S₀ contributes 2% for N, which generates N magnitude more than 1 μV/K. The different case occurred in peak 2 and peak 3. According to Figs. 3(b) and 3(c), (n)₀ and θ_HS₀ have the same sign, which indicate that those variables weaken each other. However, the N is still large, as shown in Table II.

To elucidate the properties of coefficients N, the chemical potential dependence of the AHC (σ_xy) and the longitudinal electrical conductivity (σ_xx) with τ = 10 fs is shown in Fig. 3(d). The sign and magnitude of S₀ depend on σ_xx as a function of chemical potential. As shown in Fig. 3(d), σ_xx exhibits a positive slope at energy values lower than $ε$ = −0.2 eV, which results in S₀ exhibiting a negative value. This is also the case for N₀. The sign and magnitude of N₀ depend on σ_xy as a function of energy. When there is a slope at an energy in σ_xy(⁠$ε$⁠), the N₀ is large around the energy. It is well known that the N₀ is determined by the slope of σ_xy as stated from Mott’s formula, which is $αij=−(πkB)23e∂σij(ε)∂εT∣ε=μ$⁠. The slope of σ_xy on μ = −0.3 eV to μ = −0.1 eV did not contribute to N₀ due to large σ_xx. However, the slope of σ_xy around μ = −0.1 eV to μ = −0.18 eV contributes to large N₀. In addition, the large value of N around μ = 0.3 eV and μ = 0.35 eV, which is described by peak 2 and 3 as shown in Fig. 3(a), is not mainly from σ_xy. It is from the σ_xx that has magnitude approaching to zero.

Since the peaks of N are not only from σ_xy, the chemical potential dependence of α_xy and σ_xyS₀ at 100 K in Fig. 4 is shown for better understanding the origin of large N in Fig. 3(a). Figure 4 also explains the intrinsic thermoelectric properties that are independent from relaxation time τ, and it is also accessible in the experiment. Without assuming relaxation time τ, it is possible to compare experimental results with theoretical results for the α_xy.³⁹ The ANC can be written as $N=αxy−σxyS0γ$⁠, where $γ=(1+θH2)σxx$⁠, from Eq. (1). According to Fig. 4, peak 1 in Fig. 3(a) is contributed by the different sign of α_xy and σ_xyS₀. Peak 2 and peak 3 have the same sign and almost the same magnitude of α_xy and σ_xy S₀ at the chemical potential. In this condition, α_xy and σ_xyS₀ weaken each other. N should be approached zero at these chemical potentials. However, because around the chemical potential in those peaks has a bandgap, it resulted in small σ_xx, which generates peak 2 and peak 3 in Fig. 3(a). Furthermore, α_xy tends to be stable between μ = 0.04 eV and μ = 0.16 eV. It comes from a large slope of σ_xy in Fig. 3(c). This result fits with Mott’s relation. It also indicates that the large N can be reached without the change in α_xy. Besides that, α_xy of the 1T-FeCl₂ monolayer is comparable to the result of Fe₃GeTe₂ (FGT)⁴⁰ although their electronic structures are different. FGT has large Hall angle θ_H = 0.07, thermoelectric conductivity α_xy = 0.3 A/mK, and N = 0.3 μV/K at 100 K, while our calculated values for those of the 1T-FeCl₂ monolayer are θ_H = 0.072, α_xy = 0.35 A/mK, and N = 6.65 μV/K [see Figs. 3(c) and 4].

Next, we will discuss the origin of the peaks of σ_xy, which contributes to large N. Figure 5(a) shows the band structure with SOI. The sign change of σ_xy can be related with band filling near the Fermi level. The blue band in Fig. 5(a) is about 3/4 filling, while the red band in Fig. 5(a) is about 1/4 filling. If charge doping is introduced, the Fermi level will be shifted, and it will change the charge filling near the Fermi level between the red and blue band. The effect of this band filling on σ_xy is shown in Fig. 5(b). The blue band has a negative sign of σ_xy, and the red band has a positive sign of σ_xy. The total σ_xy, which is marked by purple lines, is calculated by the sum of red and blue band contributions. The α_xy is shown in Fig. 5(c). The almost constant α_xy, which is described by the purple line in Fig. 3(d) near the Fermi level, is generated by the slope of the red and blue band at chemical potential.

The origin of the red and blue band contribution can also be explained in the Berry curvature, as shown in Fig. 5(d). The large Berry curvature is revealed around the K-point in the Brillouin zone at μ ∼ 0.2 eV. The detailed explanation of the Berry curvature, which is related with the red and blue band, is shown in Fig. 5(e). Berry curvatures in the Γ–K line have a magnitude, which are contributed by the red and blue band. The magnitudes appear because of broken mirror symmetry in the Γ–K line, resulting in unusual SOI, which is called Ising SOI.^41–43 Ising SOI originated from the d-orbitals of the transition metal, giving rise to a large Berry curvature in the Γ–K line. However, Berry curvatures of the red and blue bands on the Γ–K line around the Γ-point cancel each other, while it reinforced around the K-point. The difference in the signs of the Berry curvature also affects σ_xy. The red band contributes to the positive peak of σ_xy, while the blue band leads to σ_xy exhibiting a negative peak. These facts suggest that the origin of σ_xy can be ascribed to the Berry curvature around the K-point mentioned above.

The density functional calculations were performed for a half-metallic 1T-FeCl₂ monolayer. The anomalous Nernst coefficient has a value as high as 6.65 μV/K for chemical potentials of 0.33–0.36 eV based on the rigid band approximation. The high value of the ANC can be attributed to the large Berry curvature, which is induced by the bands around the K-point of the Brillouin zone of the 1T-FeCl₂ monolayer. The ANC can be increased by electron doping, but it is decreased by hole doping based on self-consistent field calculation. These results suggest that the ferromagnetic half-metallic 1T-FeCl₂ monolayer possess high ANC magnitudes, which can potentially be applied in thermoelectric devices.

This work was supported by a Grant-in-Aid for Scientific Research on Innovative Area “Nano Spin Conversion Science” (Grant No. 17H05180). This work was also supported by a JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric Analysis for Materials Design” (Grant No. 18H04481). This work was partially supported by Grants-in-Aid on Scientific Research under Grant No. 16K04875 from the Japan Society for the Promotion of Science. The computations in this research were performed using the ISSP, University of Tokyo.