Magnetic exchange induced Weyl state in a semimetal EuCd₂Sb₂

A recent research frontier in condensed matter physics is the realization of various fermionic particles and related phenomena in solids, naturally bridging with those predicted in high-energy physics.¹ Three varieties of fermions chronologically realized in topological semimetals (TSMs) and superconductors, i.e., Dirac,^2–4 Weyl,^5–7 and Majorana,^8–16 have captured immense interest because of the associated exotic quantum features.^17–23 The Dirac and Weyl fermions found in three-dimensional (3D) TSMs are usually characterized by a nontrivial bulk band crossing at discrete nodal points protected by a certain symmetry against gap formation. The low-energy excitation around the nodal points, behaving as an electronic band linear dispersion that can be satisfyingly described by the massless relativistic Dirac/Weyl equation along all three momentum directions, can be viewed as quasiparticle or emergent relativistic fermion.^2–7,24,25 The nodal point in 3D Dirac semimetal (DSM), i.e., the Dirac point (DP), is fourfold degenerate formed by the crossing of two doubly degenerate bands.^2–4,24,25 Once the spin degeneracy is lifted by either breaking the time-reversal symmetry (T) or the spatial inversion symmetry (P), the DP characterized by the chiral symmetry will split into a pair of Weyl points (WPs) behaving as the monopoles of the Berry curvature.^5–7 The WPs emerged in pairs are well separated in momentum space under the conservation of the opposite chirality of the Berry curvature, which are consequently topologically stable and will not be annihilated unless they could be moved to the same k-point in the Brillouin zone (BZ). The existence of WPs can produce extraordinary intriguing properties, such as the chiral anomaly effect, gravitational effect, strong intrinsic anomalous and spin Hall effects, and large magnetoresistance (MR).^19–25 

The P-breaking Weyl fermions, which are realized entirely by the crystal structure, have been established in a number of Weyl semimetals (WSMs) such as the TaAs family,^6,7,26 (W/Mo)Te₂,^27,28 and some photonic crystals.²⁹ As a contrast, T-breaking Weyl fermions are still very sparse despite the numerous theoretically predicted candidates.^30–33 The experimental detection of the WPs in these materials has encountered difficulties mainly in which the WPs are usually far away from the Fermi level (Ef) or charge carrier density of trivial Fermi surfaces (FSs) is very large. More ideal WSMs are therefore strongly desired for the Weyl physics. To find a T-breaking WSM, one can either apply a finite external magnetic field (B) to break T, for example, in GdPtBi,³⁴ or use the spontaneous magnetism to break T, for example, in YbMnBi₂.³² However, these proposed candidates are still waiting for identification. We herein report a very different way to create T-breaking Weyl fermions, that is, in EuCd₂Sb₂ with the type IV magnetic space group, via switching the antiferromagnetic (AFM) structure into a spin polarized one by applying external B.

The EuCd₂Sb₂ single crystals were grown by using tin as the flux.³⁵ Compositions of the crystals were examined by using energy-dispersive X-ray (EDX) spectroscopy, confirming that the compositions are actually rather close to the nominal values. The phase and quality examinations of EuCd₂Sb₂ were performed on a single-crystal x-ray diffractometer equipped with a Mo Kα radioactive source (λ = 0.710 73 Å). Magnetic properties of EuCd₂Sb₂ were characterized on a commercial magnetic property measurement system (MPMS-3). The temperature dependent magnetization [M(T)] along the out-of-plane (B∥c) direction was measured in the zero-field cooling (ZFC) and field-cooling (FC) mode in the temperature range of 1.8–300 K. The T dependent longitudinal resistivity ρ_xx(T) was measured by a four-probe configuration at various B up to 9 T on a commercial physical property measurement system (PPMS) apparatus. High magnetic field magnetotransport measurements up to B ∼ 55–60 T were carried out on a pulsed magnet installed at the Wuhan National High Magnetic Field Center (WHMFC-Wuhan). High-resolution angle-resolved photoemission spectroscopy (ARPES) was used to directly measure the band structure of the paramagnetic EuCd₂Sb₂. The data were measured at 20 K at BL4 and BL10 of ALS, USA. Fresh EuCd₂Sb₂ surfaces for measurements were obtained by in situ cleaving of the crystals at 20 K. The overall energy and angle resolutions were 15 meV and 0.2°, respectively.

The first-principles calculations on the electronic structure of EuCd₂Sb₂ were carried out by using the projector augmented wave (PAW) method as implemented in the VASP package.^36,37 The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof was employed for the exchange-correlation functional.³⁸ The kinetic energy cutoff of the plane wave basis was set to be 340 eV. The experimental lattice constants and atomic positions of EuCd₂Sb₂ were adopted.³⁵ A 16 × 16 × 10 k-point mesh for the BZ sampling of the primitive cell and the Gaussian smearing technique with a width of 0.05 eV for the FS broadening were utilized. The spin-orbit coupling (SOC) effect was taken into consideration. The correlation effect among Eu 4f electrons was incorporated by using the GGA+U formalism of Dudarev et al.⁴⁰ with an effective Hubbard interaction U. According to a previous study,³⁵ the energies of Eu 4f bands relative to the Ef are sensitive to U, which was chosen as 4.5 eV herein, as determined from the ARPES measurements presented later.

The clean crystals with the typical size of 1.2 × 1 × 0.2 mm³ are shown by the picture in Fig. 1(a). The perfect reciprocal space lattice without any other miscellaneous points in the single crystal XRD, as shown in Figs. 1(e)–1(g), indicates pure phase and high quality crystals used in this study. The diffraction pattern could be satisfyingly indexed on the basis of a trigonal structure with the lattice parameters a = 4.6926(6) Å and c = 7.7072(8) Å in the space group P-3m1 (No. 164), consistent with those values reported previously.^35,39 Based on a careful refinement, the crystal structure was precisely solved, as shown in Fig. 1(b), where the arrows denote the spin directions. Two Sb-Cd zigzag chains are interlaced and superimposed along the c-axis between two Eu atom layers. The structure views from other different directions are also illustrated in Figs. 1(c) and 1(d).

The M(T) along the out-of-plane direction (B∥c) is shown in Fig. 2(a). The Curie-Weiss plot to the paramagnetic region above 50 K gives the Weiss temperature of θ_p = 4.6(5) K and an effective magnetic moment of about 7.94 μ_B, which is consistent with the expectation from Eu²⁺ (4f⁷, S = 7/2, L = 0). An anomaly is observed at T_N ∼ 7.5 K, signifying the AFM ordering at this temperature. By increasing B, the AFM order is apparently suppressed, consistent with the fact that the spins of Eu²⁺ are fully polarized along the c-axis and the system eventually enters into the spin polarized state (SPS) when B > 3.2 T. The inset in Fig. 2(a) shows that the saturation moment at 2 K is close to 7 μ_B, suggesting that spins of the localized Eu 4f electrons are actually saturated. Unlike the two-dimensional (2D) DSM EuMnBi₂,⁴¹ we did not observe any spin-flop transitions in EuCd₂Sb₂.

The ρ_xx(T) at various B presented in Fig. 2(b) shows several prominent features: above 40 K, ρ_xx(T) displays an almost linear behavior with the residual resistivity ratio ρ_xx(300 K)/ρ_xx(40 K) being approximately 1.5, indicative of an essential semimetallic nature. Below 40 K, ρ_xx starts to gradually increase upon cooling until it reaches a peak at T_N and subsequently exhibits a sudden drop with further decrease in the temperature to 2 K. The low temperature behavior of ρ_xx, totally resembling that of the sister compound EuCd₂As₂, could be reasonably understood in connection with the scattering of conduction electrons by the localized Eu²⁺, which is coupling with the Cd and As orbitals and the gradual spin alignment in external B. The disappearance of such a peak at B > 3.2 T, the critical magnetic field (B_c) at which the spins are completely aligned along the c-axis, also supports this argument. Since the easy alignment of Eu spins along the c-axis by B, it naturally reminds us of the magnetotransport measurements on EuCd₂Sb₂ with B∥c to experimentally map out the FS in the spin polarized structure. The B dependence of ρ_xx in the temperature range of 2–30 K measured up to 55 T with the current (I) in the ab-plane (B∥c⊥I) is depicted in Fig. 3(a). The Shubnikov-de Hass (SdH) oscillations are clearly visible at large B after subtracting a cubic polynomial background from the experimental data. The extracted relative amplitudes of the SdH oscillations plotted against the reciprocal magnetic field 1/B are presented in Fig. 3(b). It is clear that the SdH oscillations, which are striking at B > 14 T, remain discernible at temperatures up to at least 30 K, but their amplitudes systematically decrease with the increase in the temperature. The fast Fourier transform (FFT) of the SdH oscillations, as shown in Fig. 3(c), discloses two features at the two fundamental frequencies, 30 T and 117 T. This is consistent with the result from the ab initio calculations, which suggest that two hole bands around the Γ point cross the Ef, as shown in Fig. 7. However, the FFT amplitude of 117 T is apparently much larger than that of 30 T and the transverse Hall resistivity [the inset in Fig. 8(b)] shows near linear behavior with a positive slope, implying that the hole band of 117 T plays a dominant role. Assuming circular FS cross sections of the hole bands within the ab-plane, the external cross-sectional area A_F of the FS could be evaluated by using the Onsager relation F = (ℏ/2πe)A, which gives A_F1 = 1.11(4) nm⁻² and A_F2 = 0.28(5) nm⁻², corresponding to the frequencies of 117 T and 30 T, respectively. The effective mass m^* of quasiparticles on the Fermi pocket could be obtained by fitting the temperature dependence of the FFT amplitude by the temperature damping factor of the Lifshitz-Kosevich (L-K) equation expressed as $Δρxx∝X/ sinh(X)cos2πFB−γ+δ$⁠,^42,43 where X = 14.69m^*T/B′ and B′ is the average magnetic field. The fitted results shown by the main panel of Fig. 3(d) give the cyclotron mass m^* of 0.237m₀ for the 117 T hole pocket and 0.355m₀ for the 30 T hole pocket, where m₀ denotes the mass of a free electron. The analysis of ρ_zz in Fig. 4 also gives the harmonic values for the two hole-pockets, suggesting the reliability of the analysis. It is worth noting that the two values are larger than that of the gapless DSM Cd₃As₂ but are comparable with that of SrMnBi₂.^44,45 The analysis also yields the Fermi vectors k_F of 0.596 nm⁻¹ and 0.302 nm⁻¹ for the 117 T and 30 T hole pockets, respectively, associated with very large fermion velocities v_F = 2.9 × 10⁵ m s⁻¹ and 9.8 × 10⁴ m s⁻¹, respectively, estimated through v_F = $ℏ$k_F/m^* in the case of an ideal linear dispersion.

Generally speaking, the pseudospin rotation under a magnetic field in a Dirac/Weyl system will produce a nontrivial φ_B, which could be accessed from the Landau level (LL) index fan diagram or a direct fit to the SdH oscillations by using the L-K formula.^42,43,46 The phase shift is generally a sum as −γ + δ, where γ is the phase factor expressed as 1/2—φ_B/2π and δ represents the dimension-dependent correction to the phase shift. In a 2D case, δ amounts to zero, while in a 3D case, δ is ±1/8 where the sign depends on the type of charge carriers and the kind of cross section extremum.^47,48 To provide more evidence for the nontrivial topological state in EuCd₂Sb₂, the Berry phase φ_B was examined. The inset in Fig. 3(d) shows the plot of LL index N as a function of 1/B for the 117 T hole pocket. Here, the Δρ_xx valley positions (closed circles) in 1/B were assigned to be integer indices, and the Δρ_xx peak positions (open circles) were assigned to be half-integer indices. All the points almost fall on a straight line. According to the L-K formula $Δρxx∝ cos2πFB−γ+δ$⁠, we can obtain $FB=γ−δ+0.5+N$⁠, thus allowing a linear fitting that gives an intercept 0.41(1), which demonstrates a π Berry phase. Importantly, since the observed lowest LL index is 3 at 45 T, the uncertainty associated with the fitted intercept is very small, suggesting that the determination of φ_B is reliable. The B dependence of ρ_zz up to 60 T measured with the B∥c∥I in the temperature range of 2–30 K is presented in Fig. 4 along with the analysis. The results uncovering the nontrivial Berry phase for both frequencies at 30 T and 117 T are similar to those given above, further guaranteeing the reliability of our analysis.

Figure 5 plots the measured band dispersion along the high symmetry $K¯$-$Γ¯$-$K¯$ direction, which contains the dispersive bands near the Ef and flat Eu 4f bands sitting at 1.7 eV below the Ef. The FS at the BZ center is dominated by hole-type carriers. By comparing the ab initio calculations with different Hubbard interactions U in the AFM and ferromagnetic (FM) phase, we found the best agreement at U = 4.5 eV. To further validate the choice of U, we studied the magnetic properties of EuCd₂Sb₂ in several typical magnetic configurations of Eu spins, as shown in Fig. 6. Depending on U, three different types of collinear magnetic states can be stabilized. The A-type AFM configuration was found to be the most energetic favorable state for U between 3 and 8 eV, which validates the choice of the effective U value (4.5 eV) from the comparison with ARPES measurement. The magnetic ground state of EuCd₂Sb₂ in the A-AFM configuration with weak coupling of interlayer Eu spins also agrees with the experimentally measured Néel temperature of 7.5 K and is similar to the case of EuCd₂As₂.³⁵ Further calculations with the different easy axis of magnetization show that Eu spins prefer to lie in the ab-plane. Nevertheless, the magnetic anisotropy energy (MAE), i.e., the energy difference (ΔΕ) between the in-plane and out-of-plane spin directions, is modest (ΔΕ = 0.2 meV/Eu). The weak interlayer interaction and the small MAE indicate that EuCd₂Sb₂ is prone to the external magnetic field, which is in good accordance with the measured low saturation field in the magnetization measurements.

The calculated electronic structures of EuCd₂Sb₂ at AFM and SPS are shown in Fig. 7. Below T_N ∼ 7.5 K, EuCd₂Sb₂ orders antiferromagnetically with the Eu spins lying in the ab plane.³⁵ Due to the broken C₃ symmetry along the c-axis, the DP near the Ef along the Γ-Z path of BZ shown in Fig. 7(a) is no longer protected by the combined inversion (P) and nonsymmorphic time-reversal (T′) symmetries,^35,49,50 but rather opens tiny band gaps, as is seen from Figs. 7(b) and 7(c). Due to the band inversion around G point, EuCd₂Sb₂ may be a nontrivial topological insulator with a momentum-dependent chemical potential. The topological property of EuCd₂Sb₂, which has nonsymmorphic time-reversal symmetry, is further confirmed from the evaluation of the Z₂ invariant. Since EuCd₂Sb₂ also owns inversion symmetry, the topological invariant Z₂=1 can be easily calculated by the parity product of all occupied states at the time-reversal invariant momenta (Table I). Once B > B_c, all spins of Eu ions become polarized along the c-axis and consequently form a spin polarized long-range order, which breaks the PT′ symmetry of the AFM phase and results in a completely different topology of EuCd₂Sb₂. The calculated electronic structure at the SPS is shown in Fig. 7(d) in which WPs near the Ef can be clearly observed. We note that all the WPs in this system are along the Γ-Z path. Focusing on an energy window of −0.1 to 0.1 eV around the Ef, as shown in Fig. 7(e), we found five pairs of WPs with their specific positions and energies in the BZ being summarized in Table II.

The magnetotransport properties of EuCd₂Sb₂ are intimately related to the nontrivial topological band structure and magnetism. Since both the gapped DP in the AFM state and the WPs in the FM state locate along the Γ-Z path of BZ, as seen in Fig. 7, we therefore concentrate on the magnetoresistance (MR) measured with B∥c∥I. At relatively low magnetic field below B_c, the nontrivial band structure near the Ef could produce the initial negative MR (n-MR), as shown in Figs. 4(a) and 8(a). When Eu spins are all polarized along the c-axis when B > B_c, the band structure will reconstruct with more bands crossing the Ef. The reduced scattering by parallel spins along the c-axis would further diminish the resistivity. It is worth noting that the critical points at which the MR measured with B∥I and B⊥ I start to rise are different, those are, 6.7 T and 4.5 T, respectively. The difference might be caused by the n-MR induced by the chiral anomaly, which is usually generated in the Dirac/Weyl semimetal when B∥I due to a population imbalance between Weyl fermions with opposite chiralities.⁵¹ The combination of chiral anomaly and band structure reconstruction in EuCd₂Sb₂ can thus cause more prominent n-MR than that in nonmagnetic WSM TaAs.¹² However, due to the magnetic nature of EuCd₂Sb₂, the origin of the n-MR in EuCd₂Sb₂ needs a further examination.⁴⁸ Alternatively, the large anomalous Hall effect (AHE) caused by the Berry curvature is generally conceived to be a fingerprint of the WPs around the Fermi level in WSMs. We therefore tried the isolation of the anomalous component of the Hall signal at the SPS by subtracting the linear positive background [the red line shown by the inset of Fig. 8(b)], and the result is shown in Fig. 8(b). The large value of approximately 0.1 mΩ mm corresponding to the Hall platform is comparable to those of Co₂MnGa and Co₃Sn₂S₂, which have been widely recognized as magnetic WSMs candidates.^52–54 

The characteristic feature of the WSMs lies in their existence of Fermi arc connecting a pair of nodes with opposite chirality. The WPs of EuCd₂Sb₂ around the Ef embed in the other bulk bands generally hinder the observation of their arc as they would ultimately hybridize with the bulk bands. Fortunately, some pairs of the WPs in EuCd₂Sb₂ clearly demonstrate visible arcs spreading over a large territory of the surface BZ, which was not often seen in other WSMs. The calculated states on the (100) and (010) surfaces are illustrated in Fig. 9. These two surfaces are different, as EuCd₂Sb₂ is only 3-fold rotationally invariant about the c-axis. As clearly seen from the surface states comparison shown in Figs. 9(a)–9(d), although the bulk states (red shaded area) demonstrate strong similarity as the states originating from the surfaces (red lines), they behave clearly different. Besides some clear trivial surface bands forming closed circles around the vertical BZ boundary, other surface states stably reside on both surface BZs despite the different surface potentials, indicating their topological stability. Although they hybridize with the bulk bands around Γ making the identification of clear Fermi arc difficult, one can still attribute them to different pairs of WPs. The observation of Fermi arcs further evidences the WPs appeared in the electronic structure of the SPS. Generally, one pair of DPs will split into two pairs of WPs when T is broken, and the emergence of the WPs in EuCd₂Sb₂ could be interpreted by the fact that in the band structure of the SPS reconstructs with emerging topology [Fig. 7(d)], which is not just due to the lift of band degeneracy. The surface states of EuCd₂Sb₂ are presented at constant energy values of [Figs. 9(a) and 9(b)] E = Ef and [Figs. 9(c) and 9(d)] E = Ef − 0.06 eV. The two columns, i.e., (a), (c) and (b), (d), correspond to the states at (100) and (010) surfaces with the surface BZ indicated by light-red and light-blue surfaces in the middle subset. Inside each plot, the locations of the projected bulk WPs are marked with the same color code as used in Fig. 7(e) to represent their chirality. Mirrored by the horizontal axis, these WPs form pairs with possible Fermi arc connected in between.

In summary, by performing high magnetic field magnetotransport measurements and ab initio calculations, we have demonstrated a rare example showing AFM topological insulator to SPS WSM transition in EuCd₂Sb₂, which belongs to the type IV magnetic space group. The appearance of T-breaking WPs induced by such a magnetic exchange manipulation has never been observed in other materials. It provides an ideal material not only for investigating the relation between magnetic order and band topology but also for realizing different topological states in a single material via manipulating the magnetic exchange. Furthermore, the SPS WSM provides the opportunity to study other exotic physics, such as the anomalous Nernst and thermal Hall effects.^{52–57,61,62} Further investigations on these types of materials will definitely open new horizons for the band topology theory and applications of TSMs in devices.

Note added in proof. During the preparation of this paper, we became aware that several related works appeared. One is the theoretical prediction of a single pair of WPs in EuCd₂As₂ but lacks experiments.⁵⁸ The other one reported the theoretical and experimental studies on our EuCd₂As₂ crystals, which claimed the discovery of an ideal Weyl state induced by magnetic exchange.⁵⁹ A very recent study by using ARPES on EuCd₂As₂ claimed dynamic WSMs induced by intrinsic FM spin fluctuations.⁶⁰ The information about the Fermi arcs that are crucial for a WSM was absent in both work.

The authors acknowledge the support by the Natural Science Foundation of Shanghai (Grant No. 17ZR1443300), the Shanghai Pujiang Program (Grant No. 17PJ1406200), the National Key R&D Program of China (Grant Nos. 2017YFA0302903 and 2017YFA0305400), and the National Natural Science Foundation of China (Grant Nos. 11774424, 11674229, 11874263, and 11874264). Gong and Liu wish to thank Zhong-Yi Lu and Jian-Feng Zhang for helpful discussions. Computational resources were supported by the Physical Laboratory of High Performance Computing at Renmin University of China, the HPC Platform of ShanghaiTech University Library and Information Services, and the School of Physical Science and Technology.