2D-Berry-Curvature-Driven Large Anomalous Hall Effect in Layered Topological Nodal-Line MnAlGe

materials The nontrivial topological states arising from the crossing of valence and conduction the of exotic transport spectroscopic the advance-ment of topological practical a anomalous and thermal transport magnetic for quantized transport, high magnetic transition ferromagnetic (FM) materials, measurement of the Hall an additional component in Hall resistivity when the magnetic field is approxi-mately zero, which is known as the anomalous Hall effect. a topological the anomalous Hall effect be significantly to the of the Berry curvature of the topological band structure. observed Topological magnets comprising 2D magnetic layers with Curie temperatures ( T C ) exceeding room temperature for dissipationless quantum transport the identification a material with 2D ferromagnetic planes that exhibits an out-of-plane-magnetization remains a challenge. This study reports a ferromagnetic, topological, nodal-line, and semimetal MnAlGe composed of square-net Mn layers that are separated by nonmagnetic Al–Ge spacers. The 2D ferromagnetic Mn layers exhibit an out-of-plane magnetization below T C ≈ 503 K. Density functional calculations demonstrate that 2D arrays of Mn atoms control the electrical, magnetic, and therefore topological properties in MnAlGe. The unique 2D distribution of the Berry curvature resembles the 2D Fermi surface of the bands that form the topological nodal line near the Fermi energy. A large anomalous Hall conductivity of ≈ 700 S cm –1 is obtained at 2 K and related to this nodal-line-induced 2D Berry curvature distribution. The high transition temperature, large anisotropic out-of-plane magnetism, and natural heterostructure-type atomic arrangements consisting of magnetic Mn and nonmagnetic Al/Ge elements render nodal-line MnAlGe one of the few, unique, and layered topological ferromagnets that have ever been observed.

Topological magnets comprising 2D magnetic layers with Curie temperatures (T C ) exceeding room temperature are key for dissipationless quantum transport devices. However, the identification of a material with 2D ferromagnetic planes that exhibits an out-of-plane-magnetization remains a challenge. This study reports a ferromagnetic, topological, nodal-line, and semimetal MnAlGe composed of square-net Mn layers that are separated by nonmagnetic Al-Ge spacers. The 2D ferromagnetic Mn layers exhibit an out-of-plane magnetization below T C ≈ 503 K. Density functional calculations demonstrate that 2D arrays of Mn atoms control the electrical, magnetic, and therefore topological properties in MnAlGe. The unique 2D distribution of the Berry curvature resembles the 2D Fermi surface of the bands that form the topological nodal line near the Fermi energy. A large anomalous Hall conductivity of ≈700 S cm -1 is obtained at 2 K and related to this nodal-line-induced 2D Berry curvature distribution. The high transition temperature, large anisotropic outof-plane magnetism, and natural heterostructure-type atomic arrangements consisting of magnetic Mn and nonmagnetic Al/Ge elements render nodalline MnAlGe one of the few, unique, and layered topological ferromagnets that have ever been observed.
The discovery of materials with outstanding physical properties is important for both fundamental science and the development of next-generation technological devices. In this regard, topological materials have become the center of focus in solid-state The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.202006301.
in topological magnetic systems including Co 2 MnGa, Co 3 Sn 2 S 2 , Fe 3 Sn 2 , Fe 3 GeTe 2 , Co 2 MnAl, antiferromagnetic Mn 3 Sn, and Mn 3 Ge. [14,17,18,[26][27][28][29][30] Moreover, layered topological magnets that grow as stacks of 2D magnets, such as Fe 3 GeTe 2 , Co 3 Sn 2 S 2 , and Fe 3 Sn 2 , can be integrated into device applications because of their ease of processability and strong anisotropic magnetism. [31][32][33] Therefore, the utilization of 2D-type or quasi-2D-type magnetism in topological systems is interesting. In this study, we used symmetry and crystal-structure-based chemical design principles to choose a suitable layered magnet. Heusler compounds provided an ideal starting point because of the ease of tuning their structural and electronic features. Numerous Heusler compounds demonstrate interesting topological band structures containing Weyl crossings, topological insulating states, and other structures.
In Figure 1a, we display the evolution of a layered crystal structure of a 3D Heusler compound (Cu 2 MnAl-type structure, Fm m 3 ). [34] Tetragonal PbFCl, a ternary ordered variant of the Cu 2 Sb-type structure (space group P4/nmm), was obtained by removing every second layer of atoms in the tetrahedral sites of the Cu 2 MnAl-type structure. Many ternary Mn-based compounds such as MgMnGe, CaMnSn, SrMnSn, SrMnGe, CaMnGe, and MnAlGe adopt this structure type. [35] Except MnAlGe, all of the other mentioned compounds display antiferromagnetic ordering and are semiconductors. MnAlGe is a special example in this class of compounds because it contains an extra valence electron that imparts metallic character and exhibits ferromagnetism at a very high temperature (T C ≈ 503 K). In the structure, the Mn atoms (denoted by blue spheres; Figure 1a) formed in 4 4 square-net layers that were separated by nonmagnetic Al-Ge spacers. A previous study discovered a topological electronic structure in PbFCl-type, nonmagnetic ZrSiS and HfSiS with Si-square nets. [36][37][38] The intensively studied superconductor LiFeAs with Fe-square nets is also an isostructural compound. [39][40][41] Herein, we discuss the topological properties of MnAlGe based on first-principle calculations as well as angle-resolved photoemission spectroscopy (ARPES) measurements, and their relevance to the electrical and Nernst thermoelectric transport properties. We found that MnAlGe is a topological, nodal-line, ferromagnet with multiple nodal lines in its electronic structure. The application of spin-orbit www.advmat.de www.advancedsciencenews.com coupling resulted in a strong Berry curvature near the Fermi level, leading to a large experimental anomalous Hall conductivity (AHC) of ≈700 S cm -1 at T = 2 K. Interestingly, we were able to show that the AHC of the individual Mn layers is close to the quantum conductance for a single electron channel of e h 2 (3.874 × 10 −5 S), where e is the electronic charge and h is the Planck's constant. We ascribe this result to the out-of-plane FM order of the layered magnetic structure and the inherent topological nature of the band structure.
The layered, intermetallic MnAlGe was crystallized in a ternary variant of the Cu 2 Sb-type structure (i.e., an anti PbFCl structure) (Figure 1b 2 z′) and ( 1 2 0 z′), respectively. [42][43][44][45] The structure could be viewed as a natural heterostructure consisting of magnetic and nonmagnetic layers stacked along the c-axis (Figure 1c). Specifically, the magnetic Mn atoms aligned within a 2D plane (001) and were separated by two nonmagnetic atomic layers of Al and Ge. The Mn-Mn and Mn-Al/Ge distances in the compound were ≈2.765 and ≈2.778 Å, respectively. The distance between the neighboring Mn planes was equal to the c-parameter, i.e., 5.933 Å.
MnAlGe exhibited a metallic state with both spin-up and spin-down states contributing near the Fermi level (Figure 2a). The bands near the Fermi level were dominated by the Mn-d orbitals for both the spin-up and spin-down channels ( Figure S1, Supporting Information). Hence, the electrical and magnetic properties of MnAlGe were governed by the squarenet sheets of Mn atoms, while the other two elements (Al and Ge) stabilized the crystal structure. Due to the mirror planes of {M x |(0.5,0,0)} and {M y |(0,0.5,0)}, the spin-down channel formed two nodal lines at band n and band n+1 (where n is the total electron number in the primitive cell) in the k x = 0 and k y = 0 planes, respectively, as highlighted in the band structure plot shown in Figure 2b,c. The intrinsic AHC was determined by the integral of the Berry curvature across the entire Brillouin zone (BZ). Therefore, the best way to understand the intrinsic AHE was to consider the Berry curvature distribution in k-space. Considering the magnetic moment along the c-axis and spin orbit coupling (SOC), the double degeneracy of the nodal line lifted and a bandgap opened ( Figure S2c, Supporting Information). Meanwhile, a large, 2D-type cylindrical Berry curvature was generated near this bandgap due to breaking of the band degeneracy by SOC with opening band, anti-crossing bandgaps (see Figure 2d). Interestingly, the band that constituted the nodal line formed a set of narrow, open, cylindrical Fermi surfaces (FS-1) (see Figure 2e) extending along the c-axis that exactly matched the Berry curvature distribution. In other words, these 2D Fermi surfaces almost entirely contributed to the AHE with large anisotropy. We theoretically estimated the energy-dependent AHC of MnAlGe, considering the contribution from the Berry curvature (Figure 2f). A large AHC was observed in the system near the Fermi level. In a MnAlGe with perfect stoichiometry, our ab initio calculation estimates an AHC value of ≈300 S cm -1 . We further found that electron doping in the system had a large effect on the AHC value and led to a peak value of ≈1000 S cm -1 , assuming a 60 meV shift of chemical potential. Therefore, the theoretical results indicated that a large AHC can occur in charge neutral or electron-doped MnAlGe. In addition to the abovementioned, narrow, 2D, Fermi surface responsible for the AHE, two additional sets of Fermi surfaces were observed (FS-2 and FS-3) with larger volumes ( Figure S3, Supporting Information). Open FS-3, in particular, covered a significant volume of the BZ and dispersed in all three directions. Due to its large volume, this Fermi surface is expected to dominate the normal electrical transport without significant anisotropy, which will be discussed later here. Since SOC induced a bandgap along the nodal line inside the BZ and a Weyl point was not observed in the system, the band structure in MnAlGe can be viewed as geometrically equivalent to a 3D AHE insulator and we can define an effective, quantized AHC in this bandgap.
We have carried out ARPES measurements to investigate the electronic structure of MnAlGe. Figure 2g (also see Figure S5, Supporting Information) represents the band dispersion along the Γ-X direction measured by ARPES for surface and projected bulk bands from the [001] surface termination. The ARPES data were taken using 60 eV of circularly polarized light at 14 K. The measured band structure agrees well with the first-principle calculations with the exception of a slight downshift of the nodalline energy. Within the energy resolution of the experiment, we determined the band crossing point to be ≈46 ± 18 meV below the E F . The intense surface state (SS) that extended to the nodal point (NP) further confirmed the topological origin of the band crossing. The slight discrepancy between the measured and calculated SS dispersion was attributed to the details of surface termination.
MnAlGe is a ferromagnet with a high Curie temperature (T C ≈ 503 K) and its ordered structure exhibits strongly anisotropic magnetism along the c-axis (i.e., the easy axis). The observed, ordered magnetic moment is 1.81 μ B per Mn atom. [44] The low magnetic moment of the Mn atoms is due to covalent bonding between the Mn and Al/Ge atoms, which facilitates the super-exchange and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions of Mn atoms with the Al/Ge layer. The observation also indicates a strong itinerant nature of d electrons in this ferromagnet. [42,43] The temperature-dependent magnetization measurement with an applied magnetic field of 0.1 T (inset of Figure 3a) demonstrated a paramagnetic to FM transition near 500 K, consistent with previous reports. [42][43][44][45] In the magnetic-fielddependent magnetization measurement at 2 K, the magnetization saturated rapidly at a value of 2.03 μ B per formula unit at an applied magnetic field of 0.3 T along the tetragonal c-axis, as shown in Figure 3a. When the magnetic field was applied in the ab-plane, saturation of the magnetization required a large magnetic field of ≈ 6 T. Hence, MnAlGe is a highly anisotropic ferromagnet, where the c-axis is the easy axis and the ab-plane is the hard plane for magnetization. Similar behavior was observed at 300 K with a slightly higher saturation magnetization that could be related to the quality of the synthesized crystal.
To understand the effect of the topological states on the physical properties of the sample, we performed electronic transport measurements on the as-grown MnAlGe crystal. In Figure 3b, we show the zero-field resistivity (ρ xx ) of the MnAlGe crystal. For both the I || ab-plane and I || c-axis (inset of Figure 3b) configurations, ρ xx remained relatively constant. ρ xx of the sample decreased when the temperature decreased, indicating the metallic nature of the compound. For the I || ab-plane at 300 K, a ρ xx of ≈148 µΩ cm was measured, and reached ≈31 µΩ cm at 2 K, resulting in a residual resistivity ratio (RRR = ρ 300K /ρ 2K ) of ≈5. The nearly isotropic behavior of the resistivity was due to the dominating influence of the open FS-3 level that provided the majority of the charge carriers for conduction. Now, we focus on the Hall effect of MnAlGe. We measured the Hall resistivity in two configurations: 1) magnetic field B || c-axis and electrical current (I) in the ab-plane; and 2) B || abplane and I in the c axis ( Figure S10, Supporting Information). The field dependence of the Hall resistivity (ρ yx ) for both of the configurations at 2 K is shown in Figure 3c. The Hall resistivity, when B was parallel to the ab-plane, demonstrated linear behavior without any saturation. When B was applied along the c-axis, the Hall resistivity increased rapidly in the low magnetic field regime followed by saturation upon further increase of the magnetic field. This result indicated a strong, anisotropic anomalous Hall response in MnAlGe. In the following, we will focus on the transport measurements with B || c-axis and I in the ab-plane where we observe strong anomalous Hall effect. Figure 3d represents ρ yx as a function of B at various temperatures. The total Hall conductivity (σ xy ) was estimated from the measured Hall resistivity (ρ yx ) and the longitudinal resistivity (ρ xx ) as [25] www.advmat.de www.advancedsciencenews.com (1) The estimated Hall conductivity is presented in Figure 4a (for additional data, see Figures S12 and S13, Supporting Information). In order to estimate the AHC, the high-field, linear Hall effect was projected towards zero magnetic field. The linear intercept in the Hall conductivity axis is the AHC ( Figure S15, Supporting Information). Following a similar process, we estimated the AHC from the σ xy (B) data at various temperatures. A maximum σ H A value of ≈700 S cm -1 with an anomalous Hall angle (AHA = σ σ xx H A ) of ≈2.18% was measured at 2 K. Furthermore, to understand the large anisotropic outof-plane magnetization and 2D nature of Berry curvature, we have carried out angular-dependent measurements of the AHE in various current configurations at 2 K (see discussion in Supporting Information and Figure S16, Supporting Information). The observation indicates a nearly constant AHE (ρ σ yx xy , A A ) at θ ≠ 90° configuration and supports the large anisotropic out-ofplane magnetization in MnAlGe.
When a temperature gradient was applied along the sample length, the transverse voltage in the presence of a magnetic field measured the Nernst thermoelectric response. Unlike the Hall conductivity, which accounted for all of the occupied electronic states, Nernst thermopower (S xy ) is sensitive to the electronic states near the Fermi energy. Therefore, it is an important tool for studying topological materials wherein the topological features exist near the Fermi energy, like in the present system of MnAlGe. Figure 4b displays the field-dependent S xy of MnAlGe at different temperatures, confirming its anomalous behavior. The temperature gradient was created in the ab-plane and the magnetic field was applied along the c-axis. The shape of the S xy versus B data matches that of the Hall resistivity. We obtained a large, anomalous Nernst thermopower value of S xy ≈ 1.3 µV K −1 at 52 K. The measured Nernst signal was much larger than predicted by the magnetization value (see Supporting Information). This result associates MnAlGe with a number of nontrivial materials that exhibit large anomalous Nernst effects.
We have established that, although MnAlGe is not a van der Waals compound, due to the large separation between the magnetic square net layers of Mn atoms, the AHE is dominated by the 2D electronic features in the band structure. There are three important points we would like to emphasize. First, the electronic states near the Fermi energy were dominated by Mn-d orbitals, while the s and p orbitals of the nonmagnetic Al-Ge subunit only contributed far from the Fermi energy. The fat bands observed in Figure S2, Supporting Information, clearly show the relative contribution of the orbitals in the band structure. Hence, the transport properties of MnAlGe were entirely determined by the Mn-square net layers. Second, the unique 2D distribution of the Berry curvature in MnAlGe that arose from the band anti-crossing of the nodal line perfectly matched the 2D FS obtained from the band constituting the nodal line. Therefore, the AHE observed in this compound originated from the 2D feature of the bands that contained nodal lines. Moreover, the scaling of ρ yx A versus xx ρ 2 indicates www.advmat.de www.advancedsciencenews.com the intrinsic contribution to AHE. It should be noted that the relation of ρ yx A versus xx ρ 2 is not strictly linear in the temperature dependent AHE data. Therefore, a temperature range 2-100 K has been used for linear fitting to obtain the intercept ≈560 S cm -1 as the intrinsic contribution (see discussion in Supporting Information for the departure from linearity, Figure S17, Supporting Information). Based on the ARPES study, the nodal-line anticrossings was situated just below the Fermi energy that enhanced the AHC. This can be understood from Figure 2f, which shows that shifting the nodal line from above to below the E F moves AHC from the base of a peak in the σ xy A versus E data to almost the peak. This explains the observation of a large AHC in experiments compared to the value predicted by DFT, when the Fermi energy is placed below the nodal line. Finally, the part of the FS that contributed the majority of the charge carriers for conduction exhibited dispersion in all three dimensions. Thus, the electrical resistivity and the ordinary Hall effect will demonstrate a small anisotropy, which is consistent with experimental observations. Next, we discuss how MnAlGe compares with other well-known 2D, FM, anomalous Hall systems. Quasi-2D kagome lattice compounds Co 3 Sn 2 S 2 and Fe 3 Sn 2 have recently been established as topological Weyl and massive Dirac semimetals. Interestingly, the separation between the Mn layers in MnAlGe (5.93 Å) was larger than the separation between the two Co-kagome layers in Co 3 Sn 2 S 2 (4.39 Å) and comparable to that of the Fe-kagome layers in Fe 3 Sn 2 (6.60 Å). Notably, while the FS in Fe 3 Sn 2 is open and quasi-2D, the FSs in Co 3 Sn 2 S 2 are 3D. The value of the AHC in MnAlGe was higher than in Fe 3 Sn 2 (170 S cm −1 ) and comparable to that of Co 3 Sn 2 S 2 (505-1130 S cm −1 ). In a true, van der Waals, topological nodalline, compound Fe 3 GeTe 2 , the separation between the two magnetic layers was significantly larger (8.16 Å), while in Fe 1/4 TaS 2 , which forms by the Fe intercalation of the van der Waals gap of TaS 2 , the separation between the Fe layers (6.07 Å) was approximately equal to MnAlGe. Importantly, the AHC and FM transition temperature of MnAlGe were among the highest reported, as shown in Table S2, Supporting Information.
As the AHE in MnAlGe resulted from the 2D band, we estimated the anomalous Hall conductance per FM Mn layer ( G H A ) by multiplying the bulk AHC by the separation between the two Mn layers. The value of G H A at 2 K was 4.153 × 10 −5 S, which was close to the quantum conductance for a single electron channel of e h 2 (3.874 × 10 −5 S), as shown in Figure 4c. For comparison, we also estimated this value for the layered compounds Co 3 Sn 2 S 2 and Fe 3 GeTe 2 . The calculated charge carrier density per sheet of magnetic layer in MnAlGe was ≈ 2.54 × 10 14 cm −2 at 2 K (see Supporting Information) that was one order higher than graphene, which is known to show quantum effects up to room temperature. [3] The zero-field electrical conductivity of MnAlGe in ≈10 4 S cm -1 , further specifying that the anomalous Hall transport in this compound, is in the intrinsic regime. The σ H A value decreased with increased temperature and reaches ≈190 S cm -1 at 300 K ( Figure S12b, Supporting Information).
We have demonstrated that FM MnAlGe that is composed of Mn-square net layers is a topological nodal-line metal. The electrical transport properties and magnetism in this compound are solely controlled by the Mn layers while the nonmagnetic Al-Ge spacers provide the structural stability. Nodal-line band anticrossings near the Fermi energy were responsible for a large Berry curvature that was reflected by one of the largest reported AHCs. Importantly, the Berry curvature distribution was uniquely 2D, matching the 2D Fermi surface obtained from the bands constituting nodal lines near the Fermi energy. Hence, the anomalous Hall effect in this compound was mostly 2D. This brings MnAlGe into a class of recently discovered, layered ferromagnets that exhibit large anomalous Hall effects with an additional advantage of a very high FM transition temperature. Our work should motivate the search of new layered topological magnets for anomalous transport properties. b) Magnetic field dependence of the Nernst thermopower (S xy ) for MnAlGe at selected temperatures. The magnetic field is applied along the c-axis and the temperature gradient along the ab-plane. c) Temperature dependence of AHC per magnetic layer for MnAlGe, Co 3 Sn 2 S 2 , and Fe 3 GeTe 2 .