Thickness dependence of the anomalous Nernst effect and the Mott relation of Weyl-semimetal Co2MnGa thin films

We report a robust anomalous Nernst effect in Co2MnGa thin films in the thickness regime between 20 and 50 nm. The anomalous Nernst coefficient varied in the range of -2.0 to -3.0 uV/K at 300 K. We demonstrate that the anomalous Hall and Nernst coefficients exhibit similar behavior and fulfill the Mott relation. We simultaneously measure all four transport coefficients of the longitudinal resistivity, transversal resistivity, Seebeck coefficient, and anomalous Nernst coefficient. We connect the values of the measured and calculated Nernst conductivity by using the remaining three magneto-thermal transport coefficients, where the Mott relation is still valid. The intrinsic Berry curvature dominates the transport due to the relation between the longitudinal and transversal transport. Therefore, we conclude that the Mott relationship is applicable to describe the magneto-thermoelectric transport in Weyl semimetal Co2MnGa as a function of film thickness.


INTRODUCTION
The anomalous Nernst effect (ANE) experimentally manifests as an anomalous transversal voltage, that is perpendicular to both the heat current and magnetization. It represents the thermoelectric counterpart of the anomalous Hall effect (AHE). The origin of both the AHE and ANE have contributions from the extrinsic and intrinsic (i.e. Berry phase) mechanisms [1][2][3]. When the AHE and the ANE effects are dominated by the momentum space Berry curvature of the electronic structure, it allows for an advantageous prediction of compounds with large electrical and thermal effects based solely on the topology of the electronic structure [4][5][6].
The AHE and the ANE are connected via the Mott relation [2,3]. In general, the Seebeck coefficient can be expressed via the Mott relation: = ( ) , where is the Boltzmann constant, e is the elementary charge, and the energy (E) derivative of the electrical conductivity ( ) at the Fermi level ( ) [6,7]. Generally, the Mott relation can be applied in materials where each charge carrier acts independently [8,9]. Additionally, the Mott relation holds to explain the dominant intrinsic character in the anomalous transport in ferromagnetic materials such as the spinel feature of CuCr 2 Se 4-x Br x [10], diluted magnetic semiconductors (DMS) [11], and with Berry phase or curvature [6,12,13]. However, recent theoretical work [12] questions the validity of the Mott relation in materials with a nontrivial topology of the electronic bands and suggests that the ANE could be sensitive to electronic states invisible to the AHE. The relationship between the ANE and the AHE was systematically studied and confirmed by Pu et al. [11] in the ferromagnetic semiconductor GaMnAs, in which the AHE arise from the intrinsic spin-orbit coupling. A systematic study of the Mott relation in thin films with a nontrivial topology of the electronic states is missing, mostly because, typically the ANE coefficient is small and not easy to measure and quantify. In addition, a series of samples is required since a simple temperature dependent measurement of the nontrivial AHE and ANE in one sample is not conclusive.
Co 2 MnGa, which is a member of the Co 2 YZ-based full Heusler family, has very interesting properties, such as a high Curie temperature of 700 K and a high spin polarization [14]. Furthermore, Co 2 MnGa has been regarded as a promising material, since it is a magnetic Weyl semimetal and has an unconventional topological surface state [15][16][17]. Additionally, recent work has identified the Berry curvature as the origin of the large AHE and suggested that the topology can be tuned by selecting the magnetic space group [1]. More recently, Belopolski et al. systematically measured the anomalous Hall response by considering the Berry curvature field and linked it with the evaluated topological Weyl fermion lines [18].
Experimentally, the magneto-thermoelectric properties of bulk Co 2 MnGa have demonstrated a record value for the anomalous Nernst coefficient (ANC or ) of -6 µV/K [6].
Additionally, we reported an of approximately -2 µV/K at 300 K [19] in a 50 nm thin film of Co 2 MnGa.
Here, we report a systematic study of the ANE in Co 2 MnGa thin films that exhibit large values. We confirm the robustness of the reported value above -2 µV/K in the thickness range of 20 to 50 nm. We further employ the thickness series to systematically study the relationship between the ANE and AHE. We reveal the validity of the Mott relation in this particular material at 300 K by comparing the measured values of the anomalous Nernst conductivity with the calculated values. investigation conducted by X-ray techniques revealed Co 2 MnGa Bragg peaks [20] revealing high degree of atomic order, consistent with previous work [19].  (Th1 and Th2). An external magnetic field was applied perpendicular to the sample plane. An Oxford Instruments cryostat with two thermometers is used to monitor the sample's base temperature. We define the thermal gradient as ∇T=ΔT/L, where ΔT is the temperature difference and L is the distance between the on chip thermometers Th1 and Th2 (1.3 mm). The measured voltage at each thermometer as a function of the base temperature is shown in Fig. 2(b). This serves as a calibration curve for the evaluation of the thermal gradient. Fig. 2  To quantify , we swept the external magnetic field from -5 to 5 Tesla and measured the at the transverse contacts #1, Please note that the difference of the measured value between contact #1 and #2 is negligible (~0.13 µV/K). The ANC of all thicknesses of the Co 2 MnGa thin films at various temperatures are presented in Fig. 3(a). The ordinary Nernst effect is very small in Co 2 MnGa thin films. Consequently, , at 300 K shows only a small change after saturation ( Fig. 3(b)). Therefore, the coefficient was evaluated in the following way: = − × ∇#, where and m are the electric field induced by the ANE and the magnetization vector, respectively. Before applying ∇T, the base temperature of the setup was obtained at a specific temperature. Then, a current was applied to a heater to generate a temperature gradient. Two thermometers were employed to quantify ∇T, afterwards, the values are evaluated. In the measured temperature range from 10 K to 300 K, gradually increases with increasing temperatures, as expected for a magneto-thermal effect far below the Curie temperature [22]. All samples in the thickness range between 20 to 50 nm exhibit large above -2 µV/K. Interestingly, the 40 nm films exhibit even higher values of -3 µV/K. Similarly, gradually decreased with increasing temperature from 10 K to 300 K, as shown in Fig. 4(a). Fig. 4(b) shows as a function of the applied external field µ 0 H. The saturation field at 300 K is 1.4 T. The ANE and AHC are both weakly reduced in the 50 nm In this study, the ANE of Ni thin films increased to 25 nm and decreased at higher thicknesses, revealing that the enhancement of the ANE in thin films is mostly dominated by the intrinsic and side-jump mechanisms [25]. Notably, thickness variations of the ANE cannot be explained by the variation in the magnetization, as shown in Fig. 1(a). As already presented, the saturation magnetization of all studied samples is very similar. The longitudinal conductivity is shown in Fig.4 (c). In Fig. 4(d), we quantified the Seebeck coefficient is the Seebeck voltage at 300 K and measured the longitudinal resistivity $ . The variation of and between various samples can be caused by a weak stoichiometry variation, as shown for example by Sato et al. [26]. In Table I, we summarized the | |, | |, , |$ |, and |$ | with thickness from 20 to 50 nm. In the following we discuss the character of the AHE. Different contributions to the AHE can be studied by estimating the dependency between the longitudinal and transversal conductivity [23]. This approach was employed to study many materials in order to distinguish intrinsic and extrinsic contributions to the AHE [7]. In the simplest scenario, three regions can be identified as illustrated in Fig. 5. First, in the poorly conducting regime ( ,~3 × 10 Ω 12 3 12 ), the dependence of on the residual resistivity is well described by 4 ∝ 2.7 (experimentally) [7,30]. Secondly, in the intermediate region ( ~3 × 10 − 5 × 10 9 Ω 12 3 12 ), the behavior can be explained by the intrinsic Berry-phase contribution [31]. For Co 2 MnGa thin films, the combination of and is located in this region. Accordingly, the AHE in Co 2 MnGa thin films may include a Berry curvature contribution. Thirdly, in the extremely conducting case ( : 5 × 10 7 Ω 12 3 12 ), depends on the constituents of the compounds and on Landau-level formation at low magnetic field due to the high mobility of the charge carriers [31].  Table II. We employed an approach described by Guin and Pu et al. [6,11]; because the is related to other transport coefficients, we can determine the transverse thermoelectric conductivity as ? (= −? ) by  Table III. To date, many studies have been conducted to search for the origin of the AHE and the ANE [1,5]. High values were achieved by tuning the chemical composition [35,36] and band structure [6] and predicted in simulations [37]. These studies were not only motivated by the understanding of the nature of the unconventional topological states and their consequences for magneto-thermal transport but were also motivated by a search for a path towards spin-caloritronic devices [32,38]. Recently, the origin of this anomaly was explained by topological Weyl fermion lines in the Berry curvature in bulk systems [18].
However, extrinsic scattering effects are expected to be small in the thin film regime, because of the contributions from the surface states. Therefore, we believe it is justified to take the intrinsic contributions into account. Notably, the present study offers a promising thin film material with a large room temperature , that is robust over a thickness range of 20 to 50 nm. We present various values for ferromagnetic thin films in Fig. 6(d). Therefore, it can be seen that our Co 2 MnGa thin films have outstanding values and represent a record value within the ferromagnetic thin film experiments.

III. SUMMARY
Nernst coefficients, we employed this system to study the Mott relation in thin films with a nontrivial topology inherent to band structure. We observed that the ANE is largest -3 µV/K for the 40 nm thin film. An analog trend is observed when studying the AHE. By comparing various thicknesses with magnetometry measurements, we observe that this trend is independent in the variation of magnetization. In agreement with recent reports, we believe that both the ANE and the AHC have contributions arising from not only finite Berry phase curvature. Furthermore in this thinner regime, intrinsic behavior plays a dominant role. We show that the Mott relation is valid in this material with a nontrivial topology of the band structure.