Seebeck-induced anomalous Nernst effect in van der Waals MnBi2Te4 layers

Magnetic semiconductors with an anomalous Hall conductivity σ xy ≠ 0 near the Fermi energy are expected to have a large anomalous Nernst coefficient N owing to the Seebeck term, which is the product of the Hall angle ratio and Seebeck coefficient. In this study, we examined the typical cases of ∣N∣ ≥ 20 μV K–1 in the ferrimagnetic phase of semiconducting van der Waals layers MnBi2Te4 using first-principles calculations. A large enhancement in ∣N∣ was obtained by the Seebeck term for a wide range of carrier concentrations. The present results motivate further studies on the anomalous Nernst effect in intrinsically or doped magnetic semiconductors.

T hermoelectric conversion, which involves the conversion of waste heat into electric power, is an important technology from the perspective of energy conservation. The Seebeck effect (SE) produces an electric field E parallel to the temperature gradient ∇T, whereas the anomalous Nernst effect (ANE) produces E in the direction of the cross product of ∇T and magnetization M of magnetic materials. Several efforts have been made to apply SEbased thermoelectric modules because electric energy can be generated using thermal energy without any mediations. On the other hand, the ANE has attracted increasing attention in recent years because of the simpler and more flexible structure of ANE-based modules compared with that of SE-based modules and its ability to be applied to a larger area. 1,2) The anomalous Nernst coefficient N corresponding to the ANE is expressed using the electric conductivity tensorŝ and thermoelectric tensorâ from the linear response relation of charge currentˆˆ( the anomalous Hall conductivity σ xy produced by the intrinsic mechanism is expressed using the Berry curvature where e, ℏ, d, f, and ε are the elementary charge, Dirac constant, dimension of the system, Fermi-Dirac distribution function, and band energy, respectively, 4,5) and u nk is the periodic part of Bloch states | | y ñ = ñ ⋅ e u k k r k n i n with momentum space k and band index n. The thermoelectric conductivity α ij is expressed as an integral of σ ij : where μ and T are the chemical potential and temperature, respectively. For the anomalous Nernst coefficient N, the Berry curvature is important as both the pure ANE term N 0 (or α xy ) and the Seebeck term θ H S 0 (or σ xy S 0 ) include σ xy . Co 2 MnGa achieved a huge N ∼ 6 μV K -1 originating from a larger pure ANE term |α xy | than previously reported owing to its topological band structure. 6,7) Further studies to obtain a larger |N| are required because the minimum |N| for the application of ANE-based modules was estimated to be 20 μV K -1 . 1) In recent years, great progress has been made toward the application of the ANE with "Seebeck-effectinduced charge current". Zhou et al. achieved a large transverse thermoelectric effect N = 34.2 μV K -1 by fabricating a hybrid material FePt/Si, that combines transverse and longitudinal thermoelectric effects. 8) Even single materials can be designed by focusing on the Seebeck term θ H S 0 included in Eq. (1) to increase |N|. Based on this mechanism, a giant |N| was predicted in the monolayer model of a ferromagnetic semiconductor. 9) Recently, it has been predicted that van der Waals MnBi 2 Te 4 having an odd (⩾3) number of septuple layers (SLs) is a ferrimagnetic semiconductor with a large S 0 and nonzero σ xy , [10][11][12] and quantized σ xy has been experimentally observed at 5SLs. 13) Thus, a large |N| originating from |θ H S 0 | can be obtained. Furthermore, a low lattice thermal conductivity and a sizeable thermoelectric figure of merit ZT = N 2 σ yy T/κ, where κ is the thermal conductivity, are expected because Bi and Te are relatively heavy elements. However, there is no quantitative estimation of the anomalous Nernst coefficient in MnBi 2 Te 4 , although the ANE signals were experimentally observed. 14) In this study, we focused on MnBi 2 Te 4 with an odd number of SLs as a candidate material, with an anomalous Nernst coefficient N enhanced by the Seebeck term. Electronic structure calculations were performed for the most basic MnBi 2 Te 4 with 3SLs, which is a ferrimagnetic semiconductor, and N was calculated by combining the longitudinal coefficients σ xx and S 0 based on the semiclassical Boltzmann formalism with the transverse coefficients σ xy and α xy originating from the Berry curvature. We analyzed the |N| ⩾ 20 μV K -1 conditions near the Néel temperature of ∼25 K 15) in terms of the carrier concentration n. We observed that large |N| values were obtained robustly for a wide range of |n| = 1.8 × 10 10 − 4.7 × 10 12 cm −2 assuming the carrier doping by defect engineering and chemical doping, which were already confirmed in bulk MnBi 2 Te 4 , 16) or by electric-double-layer field-effect transistors. 17,18) In addition, we evaluated N for relaxation time τ = 10 fs, 100 fs, and 1 ps comparing theoretical and experimental electric conductivity. 19) The contribution of the Seebeck term |θ H S 0 | to |N| is always larger than that of the pure ANE term |N 0 |.
MnBi 2 Te 4 with a space groupR m 3 is a layered compound stacking of Te-Bi-Te-Mn-Te-Bi-Te SLs in the [0001] direction by van der Waals forces. 15) Each Mn 2+ ion contributes mainly to a magnetic moment of ∼5 μ B . Furthermore, the intralayer exchange coupling in each SL has ferromagnetic order with an out-of-plane easy axis, whereas the interlayer exchange coupling between neighboring SLs has antiferromagnetic order, yielding an A-type antiferromagnetic ground state. 11,15,20) Using the reported lattice constants with a = b = 4.33370 Å and c = 40.91030 Å, we modeled 3SLs by doubling the c-axis and considering the vacuum region with more than 40 Å as shown in Figs. 1(a) and 1(b). Figure 1(c) shows the spin structure of Mn for each SL. The spin direction of Mn in the top and bottom SLs is opposite to that in the middle SL; therefore, the system has a finite net magnetic moment.
The anomalous Nernst coefficient was calculated in three steps: first, we performed first-principles calculations based on density functional theory (DFT) using OpenMX [21][22][23][24] to obtain the electronic structure of MnBi 2 Te 4 with 3SLs with spinorbit coupling. Second, the longitudinal quantities of σ xx and S 0 were obtained from BoltzTraP 25) based on the semiclassical Boltzmann transport formalism with the constant relaxation time approximation. Finally, the post-processing code 26) based on the Fukui-Hatsugai-Suzuki method 27) was performed to obtain the transverse quantities of σ xy and α xy .
In the DFT calculations, the Perdew-Burke-Ernzerhof generalized gradient approximation 28) was used for the exchange-correlation term. Considering the localization of 3d orbitals in Mn, the Hubbard U method was applied, and we used U = 4.0 eV based on a previous study. 12) Normconserving pseudopotentials and wave functions expanded by the linear combination of the pseudoatomic orbitals were used. Spin-orbit coupling was considered through the fully relativistic total-angular-momentum-dependent pseudopotentials. 29) The norm-conserving pseudopotentials included the 3s, 3p, 3d, and 4s electrons for Mn as valence electrons, while the 5d, 6s, and 6p for Bi; 4d, 5s, and 5p for Te were also considered. The pseudoatomic orbital basis sets were specified as Mn6.0-s3p3d3, Bi8.0-s3p3d2, and Te9.0-s3p3d1, where the number after each element represents the radial cutoff in units of Bohr, and the integers after s, p, and d indicate the numbers of s-, p-, and d-orbital sets, respectively. A cutoff energy of 300 Ry for the charge density and a 9 × 9 × 1 k-point grid were used.
In the BoltzTraP calculation, a Fourier series interpolation was conducted to obtain the energy eigenvalues at any k point. We checked the convergence of the k-point dependence of the band interpolation and subdivided the k-point mesh into 45 × 45 × 1 for longitudinal properties (see supplementary materials). To calculate the transverse properties, a 500 × 500 × 1 k-point grid was employed as a result of the convergence. Figure 2(a) shows the electronic structure near the Fermi energy. The band gap was opened by magnetic ordering, and the magnitude of the band gap was 57 meV. The theoretical and experimental band gaps reported in previous studies were 66 meV 10) and 71 ± 15 meV, 30) respectively. Therefore, our results are consistent with those of previous studies, and we confirmed that U = 4.0 eV was reasonable. The total density of states (DOS) and projected DOS (PDOS) of the p-orbitals of Bi and Te are shown in Fig. 2(b). The p-orbital contribution of Te was dominant in the valence band near the Fermi energy, whereas the p-orbitals of Bi and Te were dominant in the conduction band.
The calculated chemical-potential dependence of the transport properties is shown in Fig. 3. Figure 3(a) shows the anomalous Hall conductivity σ xy at T = 0 K. The plateau area arises near the Fermi energy and indicates a quantum anomalous Hall effect. This result is consistent with that of a previous study. 11) The longitudinal electric conductivity σ xx with a constant relaxation time τ is shown in Fig. 3(b). As we will describe later, several cases of τ were considered: 10 fs, 100 fs, and 1 ps. According to the above equation, σ xx depends on the relaxation time τ. α xy and −σ xy S 0 at T = 15 K are shown in Fig. 3(c). These two values are independent of the τ. At sufficiently low temperatures, α xy is approximated by the Mott relation where k B is the Boltzmann constant. 31,32) From this approximation, a large α xy is obtained when the energy derivative of σ xy (0, ε) is large. σ xy in Fig. 3(a) had a peak at approximately −0.07 eV, which originated from the Berry curvature of the two lower bands from the highest occupied band (see supplementary materials). As the differential coefficient of σ xy was large at this peak, the α xy was also large at approximately μ = -0.07 eV from the Mott relation, as shown in Fig. 3(c). Focusing on −σ xy S 0 , it can be understood that this value almost increased N near the band edge because S 0 diverged as shown in Fig. 3(d), and α xy and −σ xy S 0 had the same sign. Away from the band edge, in regions where more carriers were doped, −σ xy S 0 had the same order of magnitude as α xy . Thus, the Seebeck term −σ xy S 0 was dominant over N in the low-carrier limit near the band edge and worked to enhance N in cooperation with the pure ANE term α xy in the region where more carriers were doped. At μ = ± 0.2 eV, −σ xy S 0 was small because S 0 was close to 0, but N was obtained from α xy . These results indicate how the Seebeck term −σ xy S 0 and the pure ANE term α xy contributed to changes in N with the amount of carrier doping. The relaxation time τ must be determined for the anomalous Nernst coefficient N because σ xx and θ H in Eq. (1) include τ. We estimated τ by comparing the experimental resistivity ρ xx with our calculations. It was estimated to be τ ∼ 10 fs for the carrier concentration |n| ∼ 10 20 cm −3 , 19) which corresponds to ∼10 13 -10 14 cm −2 in two dimensions in our system. In addition, it was estimated to be τ ∼ 100 fs for |n| ∼ 10 12 cm −2 and τ ∼ 1 ps for |n| ∼ 10 11 cm −2 by linear extrapolation. As τ is expected to vary with |n|, τ was chosen such that τ was realistic for |n|: three kinds of τ of 10 fs, 100 fs, and 1 ps were considered.  Table I classifies the typical cases with |N| ⩾ 20 μV K -1 in terms of the carrier concentration n and relaxation time τ at T = 20 K, which is close to the Néel temperature of ∼25 K. 15) This clarifies the discussion of Fig. 3(c). The anomalous Hall conductivity σ xy was considered in the intrinsic mechanism and had a value of e 2 /h near the Fermi energy. This value corresponds to ∼1.0 × 10 2 S cm -1 in our system. Onoda et al. reported that σ xx > 10 3 S cm -1 when σ xy originating from the intrinsic mechanism was ∼10 2 S cm -1 . 33) Therefore, we consider the case of θ H < 1. The Seebeck term −θ H S 0 was always larger than the pure ANE term N 0 and had the same sign for all cases where |N| exceeded 20 μV K -1 , which is referred to as the Seebeckinduced anomalous Nernst effect in this study. For the hole carrier concentration of n h = 4.7 × 10 12 cm −2 and τ = 10 fs, |θ H S 0 | and |N 0 | had the same order of magnitude and contributed to |N| = 24.0 μV K -1 . At τ = 100 fs, |θ H S 0 | was one order of magnitude larger than |N 0 |. When |θ H | greatly exceeded 1, the |N| was reduced by q + 1 H 2 as expressed in Eq. (1). For τ = 1 ps, a large |N| dominated by |θ H S 0 | with a large |S 0 | ∼ 10 2 μV K -1 was realized for the low-carrier region corresponding to the vicinity of the band edge. Therefore, it is expected that |N| is enlarged by the Seebeck term in magnetic semiconductors with |θ H | > 0.1 and |S 0 | ∼ 10 1 -10 2 μV K -1 .
For example, in another system, a theoretical model of EuO, which is a ferromagnetic semiconductor, was reported to obtain |N| ∼ 20 μV K -1 for τ = 100 fs. 9) In this case, θ H S 0 was dominant to N, close to the τ = 1 ps cases in our system. In a ferromagnetic semiconductor, Ga 1−x Mn x As, N ∼ 10 μV K -1 was observed. 34) In this system, the giant Hall angle ratio of θ H ∼ 10 can be implied from the longitudinal and transverse resistivity. On the other hand, the anomalous Nernst coefficient of |N| ⩾ 20 μV K -1 with the Hall angle ratio of θ H ∼ 0.1 -1 was predicted in our system. Finally, we estimated the power factor (PF) and the figure of merit ZT. Hereafter, the PFs in SE and ANE are denoted as PF SE and PF ANE , respectively. The maximum values at T = 20 K and τ = 1 ps were PF SE = 13.4 μW cm −1 K −2 and PF ANE = 1.05 μW cm −1 K −2 for n h = 1.5 × 10 11 cm −2 . The PF SE was reported to be approximately 10 1 μW cm −1 K −2 in typical thermoelectric materials. [35][36][37][38][39] In addition, the PF ANE of typical magnetic materials, such as Ni, Fe, Co 2 MnGa, and FePt, was estimated at ∼10 −3 -10 −1 μW cm −1 K −2 . 6,8,40) Therefore, the PF SE of our system had the same order of magnitude as that of typical thermoelectric materials, [35][36][37][38][39] and the PF ANE was large compared with that of other magnetic materials. 6,8,40) It was reported that the thermal conductivity κ of a thin film is lower than that of the bulk in Bi 2 Te 3 . 41) Thus, we make an assumption of κ = 1.4 W m −1 K −1 in our system, although κ ; 2.80 W m −1 K −1 42) was reported for bulk MnBi 2 Te 4 . Under this assumption, the ZTs of SE and ANE (denoted as ZT SE and ZT ANE ) for T = 20 K were ZT SE = 0.0192 and ZT ANE = 0.00150 for n h = 1.5 × 10 11 cm −2 , respectively. Compared with our previous study on CoMnSb, which is a ferromagnetic half-Heusler alloy with ZT ANE = 6 × 10 −5 at T = 490 K, 43) MnBi 2 Te 4 with 3SLs was considered to have a large ZT ANE regardless of the low temperature.
We investigated the ANE of the ferrimagnetic semiconductor MnBi 2 Te 4 with 3SLs expected to yield a large anomalous Nernst coefficient N owing to the Seebeckinduced ANE. 9) N was obtained by first-principles calculation combined with longitudinal quantities from the Boltzmann transport equation and transverse quantities from the Berry curvature. The Seebeck term |θ H S 0 | was always larger than the pure ANE term |N 0 | and enhanced |N| with various carrier concentrations. The results suggest that to achieve a large ZT, the design of thermoelectric devices may be required in terms of not only a large anomalous Nernst coefficient but also high electric conductivity and low thermal conductivity, by adjusting the carrier concentration. Such minute tuning can be realized by the elaborate fabrication of two-dimensional materials, and van der Waals materials may be candidate materials because of their flexible lamination.