We performed first-principles simulations to elucidate the transverse thermoelectric effect (anomalous Nernst effect) of the half-metallic FeCl_{2} monolayer. We analyzed its thermoelectricity based on the semiclassical transport theory including the effect of Berry curvature and found that carrier-doping induced a large anomalous Nernst effect that was ∼6.65 *μ*V/K at 100 K if we assumed 10 fs for the relaxation time. This magnitude originates in a large Berry curvature at the K-point of a hexagonal Brillouin zone. These results suggest that two-dimensional ferromagnetic half-metallic materials can potentially be used in thermoelectric devices.

## I. INTRODUCTION

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 $\sigma xy=e2hC$, 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)_{2}Te_{3}, 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 SnS_{2}^{17} and Ohta *et al.* using 2D gas in SrTiO_{3}.^{18}

In this study, by using density functional calculations, we explored the thermoelectric properties of a half-metallic 1T-FeCl_{2} 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-FeCl_{2} 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-FeCl_{2} 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.

## II. METHODS

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 orbitals^{23} 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-FeCl_{2} 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-FeCl_{2} monolayer is shown in Fig. 1. The side view of the 1T-FeCl_{2} 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 $a\u20d7$ is equal to that of $b\u20d7$ with the vacuum region, *c* = 17.26 Å. After structural relaxation, the lattice constant of the 1T-FeCl_{2} 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 FeCl_{2}.^{26–29} The calculated lattice constants for monolayer FeCl_{2} were similar to those of bulk FeCl_{2}, *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*_{ij}*E* + *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 *S*_{0} = *α*_{xx}/*σ*_{xx}, *N*_{0} = *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 $\sigma xx=e2\tau \u2211n\u222bvxn(k)2(\u2212\u2202f\u2202\epsilon nk)dk$, while the AHC can be computed from $\sigma xy=\u2212e2\u210f\u2211n\u222b\Omega zn(k)f(\epsilon nk)dk$. The thermoelectric conductivity tensors can be represented as $\alpha ij=kBe\u222b\sigma ij(\epsilon )\u2223T=0\epsilon \u2212\mu T(\u2212\u2202f\u2202\epsilon )d\epsilon $, where *e*, *τ*, $vxn$, *f*, $\epsilon nk$, *ℏ*, $\Omega zn(k)$, *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 $\Omega n(k)\u2261i\u2207kunk\u2223\xd7\u2223\u2207kunk$. 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 theory^{34} 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}

## III. RESULTS AND DISCUSSION

The ground states of the 1T-FeCl_{2} 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-FeCl_{2} 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-FeCl_{2} monolayer were formed by the 3*d* orbital of the Fe atom and the 2*p* 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 3*d* and Cl 2*p*. 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} ($dx2\u2212y2,d3z2\u2212r2$), 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-FeCl_{2} monolayer.

Table I shows the thermoelectric properties of the 1T-FeCl_{2} 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 (*N*_{0}), the Hall angle (*θ*_{H}), and the pure Seebeck coefficient (*S*_{0}). The contributions of *S*_{0} to *N* are around 0.2% each. However, *N*_{0} and *θ*_{H}*S*_{0} weaken each other since the sign of *N*_{0} and *θ*_{H}*S*_{0} 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.

T (K) . | S_{0} (μV/K)
. | N_{0} (μV/K)
. | θ_{H} (×10^{−2})
. | N (μV/K)
. |
---|---|---|---|---|

50 | 0.46 | 0.24 | 0.19 | 0.24 |

100 | 5.51 | 0.49 | 0.21 | 0.48 |

T (K) . | S_{0} (μV/K)
. | N_{0} (μV/K)
. | θ_{H} (×10^{−2})
. | N (μV/K)
. |
---|---|---|---|---|

50 | 0.46 | 0.24 | 0.19 | 0.24 |

100 | 5.51 | 0.49 | 0.21 | 0.48 |

The chemical potential dependences of the Nernst, *N*, coefficients of the 1T-FeCl_{2} 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-FeCl_{2}.

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

Peak . | μ
. | N_{0} (μV/K)
. | θ_{H}
. | S_{0} (μV/K)
. | N (μV/K)
. |
---|---|---|---|---|---|

1 | 0.16 | 0.62 | −0.02 | 14.2 | 1.35 |

2 | 0.31 | −3.96 | −0.02 | 77.2 | −2.32 |

3 | 0.35 | 8.13 | −0.03 | −44.6 | 6.65 |

Peak . | μ
. | N_{0} (μV/K)
. | θ_{H}
. | S_{0} (μV/K)
. | N (μV/K)
. |
---|---|---|---|---|---|

1 | 0.16 | 0.62 | −0.02 | 14.2 | 1.35 |

2 | 0.31 | −3.96 | −0.02 | 77.2 | −2.32 |

3 | 0.35 | 8.13 | −0.03 | −44.6 | 6.65 |

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 *S*_{0} 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 $\epsilon $ = −0.2 eV, which results in *S*_{0} exhibiting a negative value. This is also the case for *N*_{0}. The sign and magnitude of *N*_{0} depend on *σ*_{xy} as a function of energy. When there is a slope at an energy in *σ*_{xy}($\epsilon $), the *N*_{0} is large around the energy. It is well known that the *N*_{0} is determined by the slope of *σ*_{xy} as stated from Mott’s formula, which is $\alpha ij=\u2212(\pi kB)23e\u2202\sigma ij(\epsilon )\u2202\epsilon T\u2223\epsilon =\mu $. The slope of *σ*_{xy} on *μ* = −0.3 eV to *μ* = −0.1 eV did not contribute to *N*_{0} due to large *σ*_{xx}. However, the slope of *σ*_{xy} around *μ* = −0.1 eV to *μ* = −0.18 eV contributes to large *N*_{0}. 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*_{xy}*S*_{0} 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=\alpha xy\u2212\sigma xyS0\gamma $, where $\gamma =(1+\theta H2)\sigma xx$, from Eq. (1). According to Fig. 4, peak 1 in Fig. 3(a) is contributed by the different sign of *α*_{xy} and *σ*_{xy}*S*_{0}. Peak 2 and peak 3 have the same sign and almost the same magnitude of *α*_{xy} and *σ*_{xy} *S*_{0} at the chemical potential. In this condition, *α*_{xy} and *σ*_{xy}*S*_{0} 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-FeCl_{2} monolayer is comparable to the result of Fe_{3}GeTe_{2} (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 *μ*V/K at 100 K, while our calculated values for those of the 1T-FeCl_{2} monolayer are *θ*_{H} = 0.072, *α*_{xy} = 0.35 A/mK, and N = 6.65 *μ*V/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.

## IV. CONCLUSIONS

The density functional calculations were performed for a half-metallic 1T-FeCl_{2} monolayer. The anomalous Nernst coefficient has a value as high as 6.65 *μ*V/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-FeCl_{2} 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-FeCl_{2} monolayer possess high ANC magnitudes, which can potentially be applied in thermoelectric devices.

## ACKNOWLEDGMENTS

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.

## REFERENCES

_{3}

_{2}(X = Cl, Br, I): Stability and tunable magnetism from first principles and Monte Carlo simulations

_{2}

_{2}at high pressures

_{2}, CoBr

_{2}, FeCl

_{2}, and CoCl

_{2}

_{2}monolayer by applying strain: First-principles study

_{3}GeTe

_{2}

_{2}