Magnetic generation and switching of topological quantum phases in a trivial semimetal $\alpha{\mathrm{-EuP}}_3$

Topological materials have drawn increasing attention owing to their rich quantum properties, as highlighted by a large intrinsic anomalous Hall effect (AHE) in Weyl and nodal-line semimetals. However, the practical applications for topological electronics have been hampered by the difficulty in the external control of the band topology. Here we demonstrate a magnetic-field-induced switching of band topology in $\alpha{\mathrm{-EuP}}_3$, a magnetic semimetal with a layered crystal structure derived from black phosphorus. When the magnetic field is applied perpendicular to the single mirror plane of the monoclinic structure, a giant AHE signal abruptly emerges at a certain threshold magnetization value, giving rise to a prominently large anomalous Hall angle of $\left|\Theta_{\mathrm{AHE}}\right| \sim 20^{\circ}$. When the magnetic field is applied along the inter-layer direction, which breaks the mirror symmetry, the system shows a pronounced negative longitudinal magnetoresistance. On the basis of electronic structure calculations and symmetry considerations, these anomalous magneto-transport properties can be considered as manifestations of two distinct topological phases: topological nodal-line and Weyl semimetals, respectively. Notably, the nodal-line structure is composed of bands with the same spin character and spans a wide energy range around the Fermi level. These topological phases are stabilized via the exchange coupling between localized Eu-4$f$ moments and mobile carriers conducting through the phosphorus layers. Our findings provide a realistic solution for external manipulation of band topology, enriching the functional aspects of topological materials.


I. INTRODUCTION
Topological semimetals (TSs) are a new class of quantum materials hosting non-trivial massless fermionic states, which are derived from linearly dispersing bands crossing at the Fermi level [1][2][3][4][5][6][7][8]. Such band crossings occur in the bulk and are formed at either isolated or continuous points in momentum space, the latter being commonly known as topological nodal-line semimetals. TSs are further categorized into two groups-Weyl and Dirac typesdepending on whether they lack or hold spin-degeneracy throughout their Brillouin zone, respectively. Owing to their relativistic nature, such band crossings generate a local but large internal magnetic flux, known as the Berry curvature, that can give rise to intriguing transport phenomena such as the intrinsic anomalous Hall effect (AHE) [9][10][11][12]. By controlling the time and space inversion symmetries, one such topological phase can in principle be turned into another. Externally this can be achieved by applying a magnetic field or via mechanical distortion, enabling practical manners in which to exploit the band topology and embed it into devices with novel spintronic functionalities of relevance for future energy and information technologies.
Thus far, various TSs containing 3d and 4f-elements have been discovered in nature or grown under controlled conditions [4][5][6][7][8][13][14][15][16][17][18][19][20], incorporating magnetism as a possible controlling parameter. However, the expectations for the magnetic tunability of their electronic states and functionalities have not been met, owing to multiple factors. For example, the magnetic states in 3d-TSs are generally irresponsive to the magnetic field due to the strong chemical hybridization of 3d orbitals with the rest of the electronic structure. More critically, in most of these systems, the relevant topological nodes are distant from the Fermi level, EF. As such, they are untraceable in transport measurements unless extensive band tuning is performed, which is not always possible, or at least leads to collateral damages that can affect the desired properties of the system. Therefore, significant effort is being dedicated towards the discovery of TSs with tunable topological nodes in immediate proximity to their Fermi level. In this paper, we introduce a magnetic semimetal, α-EuP3, with a simple, yet profoundly tunable electronic structure that ideally meets these criteria.

II. RESULTS
α-EuP3 is a layered magnetic semimetal with a monoclinic crystal structure (space-group type C2/m) whose b-vector serves as a two-fold rotational axis normal to a mirror plane [21][22][23] (Figs. 1(a) and (b)). The bulk of α-EuP3 is comprised of two-dimensional (2D) puckered polyanionic layers of phosphorus that stack along the c-axis, with Eu 2+ cations in between. The structure of the P-layers is closely related to that of orthorhombic black phosphorus [24][25][26] and can be derived by removing 1/4 of its P-centers. The remaining P-layers form a closed shell by receiving valence electrons from the 6s-orbitals of Eu cations, to the extent that the whole system gains a semi-metallic or semiconducting character with half-filled Eu-4f shells. This black-phosphorus-derived quasi-2D nature coordinated with localized f-moments is the key aspect of α-EuP3 which enables large spin-polarization in the intrinsically non-magnetic P-layers via energetical proximity of their carriers with the immobile Eu-4f moments. The locality of the Eu-4f moments and their close proximity to EF further allows a directional control of the resulting spin-charge coupling that can be externally enforced by a magnetic field. Such highly field-controllable magnetic layers enable us to increasingly induce a large internal field by aligning the local moments, which, as will be shown shortly, yields a topologically non-trivial electronic structure that can similarly be controlled using a magnetic field.
To demonstrate the effect of Eu-4f moments on the electronic structure, Figure 1 equal to zero, meaning that the system is topologically trivial under zero magnetic field in the PM phase ( Fig. 1(g)). The colored solid lines are the corresponding energy bands when the Eu 4f-moments are fully aligned along the c*-axis, thus, represent the field-induced FM phase. This makes the 4f orbitals of divalent Eu ions half-filled 4f 7 in the highest possible magnetic state S = 7/2 (i.e., 7 μB), due to the implications of the Hund's rule. Our calculations reveal that these filled f-bands lie ~ 1.5 eV below EF. Such proximity to EF allows the localized f-moments to be coupled to each other through a carrier-mediated exchange interaction mechanism [27][28][29]. As a result of such interaction with itinerant carriers, the valence and conduction bands near EF undergo opposite spin splitting depending on how the carriers couple to the localized f-moments [30][31][32]. The valence bands, consisting of occupied P-p orbitals, show a Kondolike interaction with an antiparallel coupling to the Eu-f moments, whereas the conduction bands, consisting of unoccupied Eu-s orbitals, undergo a Hund-like parallel coupling. Consequently, spinup valence bands are forced to stay above their spin-down counterparts, whereas conduction bands follow an opposite spin splitting. Another important consequence of such spin splitting is the immunity of particular types of crossings between the conduction and valence bands, that could lead to the formation of a nodal line or Weyl nodes depending on whether the magnetic field breaks the mirror plane or respects it. These scenarios are schematically illustrated in Fig. 1(g) and will be discussed in more detail in the context of magneto-transport, after a description of the experimental data.  configurations. Here, it is noteworthy that the negative MR is pronounced for the configuration with I || B || c, whereas it is nearly identical for the others as shown in Fig. 2(h). This characteristic will be further discussed in section III.
The Hall resistivity in magnetic systems is described by the empirical formula ρH = ρH N + ρH A + ρH T . The first term is the normal Hall contribution ρH N = R0B due to the Lorentz force, the second is the AHE, and the third is the topological Hall effect derived from spin textures associated with a topological number, such as skyrmions [35,36]. The topological Hall effect is unlikely to be the case for α-EuP3 because the deviation from the normal Hall effect is persistent at T > TN, where the system has no long-range spin ordering. This leads us to describe the Hall effect in α-EuP3 as ρH = ρH N + ρH A . Further on, we will mainly focus our analysis on the T > TN region. In order to extract ρH A , ρH N must be determined from experimental data, which is not necessarily definitive and thus requires a certain degree of ad hoc assumptions. Several approaches for extracting ρH A are presented in the Supplemental Material [37].
Since each approach gives reasonably similar results in the qualitative level, here we estimate ρH N by using ρH T = 150 K , which is far above TN and approximately reflects the normal contribution to the Hall effect.
Since magnetization M characterizes the proximityinduced exchange splitting, it is meaningful to investigate the relation between M and the AHE. Figure 3(a) shows ρc*a A = ρc*aρc*a T = 150 K in the space of M and T, showing a T-independent relationship between M and ρc*a A such that the anomaly clearly emerges at a certain threshold of M ~ 1.8 µB/Eu (= MC). In other words, the trigger of the anomaly is the magnitude of M rather than a specific spin texture, which strongly suggests that the band splitting plays a critical role in the emergence of the AHE. Figure   3 Although the σc*a A -M curve shows a variation depending on how we estimate ρH N , as mentioned above, the overall behavior reasonably follows the colored guide-to-eyes shown in Fig. 3(b) (see Fig. S9 [37]). We can also quantify the AHE using the anomalous Hall angle ΘAHE = tan -1 (σc*a A /σaa). As presented in Table 1 and in Fig. S11 [37], the value of |ΘAHE| in this work reaches 14˚ ~ 28˚ at low temperatures, which is notably large even when compared with the maximum values observed in previously reported materials [4,6,7,[38][39][40][41][42][43] including topological candidates. These experimental results suggest an unconventional origin of the anomaly.
The AHE has been well studied in FM metals [9,44,45] and can be classified into the following three mechanisms. The first mechanism is known as the intrinsic contribution, related to r (real) space and k (momentum) space Berry curvature b(R) (R = r, k). The former arises in non-coplanar spin textures with finite scalar spin chirality χijk = Si • (Sj × Sk), where Sn are spins, which is again unlikely to be the case here. In contrast, the latter originates from the multi-bands in k-space and is described as where φ is the electronic wave function [11]. The second and third mechanisms can be described by the skew-scattering and side-jump contributions, which are both related to spin-orbit coupled impurity scattering. While the skew-scattering contribution can be ruled out because it only arises in highly conductive metals, the side-jump contribution can be present. However, since the atomic spin-orbit coupling term is expected to be generally small in the relevant bands [46], the side-jump contribution is also unlikely to dominate the large AHE. Moreover, these contributions are expected to scale with  [2,47,48] (Fig. 4(c)). Interestingly, this nodal-ring around the Γ-point can span a large energy window, encompassing EF (Fig. 4(d)), and hence contribute directly to the anomalous Hall part of the transport. This energy spanning is due to the layered nature of α-EuP3, which causes anisotropic electronic dispersions, ranging from steep ones for any k-vector in the intralayer direction, to nearly flat dispersions for the k-vectors in the inter-layer direction. As shown in Fig. 4(b), and also in the schematic panel of Fig. 3(a) share the same mirror eigenvalues [2]. This is the only way to avoid hybridization while respecting all existing symmetries.

III. DISCUSSION
Let us discuss the origin of the anomalous transport based on the calculated band characteristics. The contribution of the Berry curvature on the Hall resistivity depends on the measurement configuration, as seen in the following relation [1] Fig. S15 [37]). Note that the positive MR forming a kink at low magnetic fields is found in a wide temperature range as shown in Fig. 2(g). Although this kink can be a characteristic feature of the magnetic generation of the Weyl states, further studies are necessary to reveal the origin of this behavior.
When B || b (FM-b), which is the case for Figs. 2(a)-(c), the f-moments are aligned normal to the mirror plane. In this measurement configuration, the mirror symmetry is preserved and nodal-rings emerge within the mirror plane. In the low M region, the topological band crossings are wholly above EF; thus, the k-space  Fig. 2(c). As for the prominently large |ΘAHE|, not only the large numerator σH A but also the small denominator σaa is presumed to be an essential factor, which can be understood as a result of few trivial bands composing the Fermi surface rather than the dirtiness of the material (see Fig. S11 [37]).
In other words, the combination of the topological nodes and the simple band manifold at EF enables the coexistence of a large σH A and small σaa in α-EuP3. Combining all the experimental and theoretical evidence, it is reasonable to presume that the topological nodes are generated under a sufficient net magnetic moment, strong enough to break or respect the mirror symmetry, and then can be tuned towards the Fermi level by increasing the degree of spin polarization of the localized f-moments.
Our experimental and theoretical work on α-EuP3 presents a case for magnetic-field-triggered generation and switching of topological states, as illustrated in Fig. 1(g). The gigantic AHE in the nodal-line phase persists up to high magnetic fields, which is consistent with our theoretical expectation of the existence of tilted nodal-rings spanning a wide energy range around EF. Crucially, these findings lead to a more general concept: that the combination of two critical components-a highly anisotropic crystal structure with a relevant mirror plane and proximity-induced f-p couplingis a key guiding principle for designing novel magnetic TSs suitable for external control of topological phases and transport. This  a, b, c, where we define a*, b*, c* as a* || b × c, b* || c × a (|| b), and c* || a × b, respectively. (c) The electronic structure of α-EuP3 in the paramagnetic (B = 0, dashed lines) and field-induced ferromagnetic (colored lines) phases. The localized 4f-bands lie approximately 1.5 eV below the Fermi level. In the FM phase calculation, Eu 2+ f-moments are fully aligned along the c*-axis, corresponding to M = 7 µB/Eu, and the color of the bands represents the spin polarization.     Fe film 0.63 300 [43]