Supplementary Materials for

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1 advances.sciencemag.org/cgi/content/full/2/1/e151117/dc1 Supplementary Materials for Quantum Hall effect in a bulk antiferromagnet EuMni2 with magnetically confined two-dimensional Dirac fermions Hidetoshi Masuda, Hideaki Sakai, Masashi Tokunaga, Yuichi Yamasaki, Atsushi Miyake, Junichi Shiogai, Shintaro Nakamura, Satoshi Awaji, Atsushi Tsukazaki, Hironori Nakao, Youichi Murakami, Taka-hisa Arima, Yoshinori Tokura, Shintaro Ishiwata The PDF file includes: Published 29 January 216, Sci. Adv. 2, e (216) DOI: /sciadv Detailed analyses on resonant x-ray magnetic scattering for EuMni2 Detailed magnetic properties for EuMni2 Magnetic properties for EuZni2 Magnetization and transport features around 2 T for EuMni2 Hall angle and step size between the consecutive 1/ yx plateaus for EuMni2 Powder x-ray diffraction profile for each compound and typical geometry of the samples and electrodes Fig. S1. Resonant x-ray magnetic scattering for EuMni2 near the Eu L3 absorption edge. Fig. S2. Extinction rules and candidates of magnetic structure of Eu sublattice. Fig. S3. Determination of magnetic structure of Eu sublattice. Fig. S4. Detailed magnetic properties for EuMni2. Fig. S5. Magnetic properties for EuZni2. Fig. S6. Magnetization and transport features around 2 T. Fig. S7. Hall angle and step size between the consecutive 1/ yx plateaus. Fig. S8. Powder x-ray diffraction profile for each compound and geometry of the samples and electrodes. References (46, 47)

2 Detailed analyses on resonant x-ray magnetic scattering for EuMni 2 Figures S1A-C show resonant x-ray magnetic scattering profiles for EuMni 2 measured at 5 K at zero magnetic field. In order to study the origin of the ( 11) reflection shown in Fig. 1 in the main text, we performed energy scans near the Eu L 3 absorption edge (Fig. S1A). A single peak, suggesting resonant nature of the reflection, was found at kev, which arises from dipole resonant scattering involving transitions from the core 2p state to the empty 5d one (46). The 5d state should be polarized via the onsite 4f-5d mixing due to the locally broken inversion symmetry by crystal field at the Eu site. Additionally, the magnetic origin of the ( 11) reflection was evidenced by polarization analysis (Fig. S1). The magnetic form factor (f i mag ) of the ith Eu magnetic moment in the electric-dipole transition is described as f i mag (e e ) m i, (S1) where e and e denote unit vectors of the incident and scattered polarization, respectively, and m i unit vector of the ith Eu magnetic moment (47). For the measurement of the ( L) reflections with the σ-incident polarization (Inset to Fig. S1), magnetic scattering thus should appear only in the rotated σ-π channel, while charge scattering in the unrotated σ-σ channel. As shown in Fig. S1, the ( 11) reflection was observed in the σ-π configuration but not in the σ-σ configuration, which ensures the magnetic origin of this reflection. In order to determine the orientation of the Eu magnetic moments, we performed polarization analysis for the (4 1) magnetic reflection. Since the π-incident polarization is nearly parallel to the ab plane in measuring this reflection (Inset to Fig. S1C), the magnetic scattering by the c (ab) component of the magnetic moments should be observed in the π-π (π-σ ) channel. As shown in Fig. S1C, magnetic reflection was detected in the π-π ' channel but not in the π-σ channel, which suggests that the Eu magnetic moments are aligned along the c axis at 5 K at zero magnetic field. This result is consistent with the magnetization data for the AFM the phase (see Figs. 1A and 2A in the main text). Next, we discuss the magnetic structure of EuMni 2. From the observation of the ( 11) and (4 1) magnetic reflections, the magnetic order of Eu sublattice is characterized by a (1) reciprocal vector, which indicates that the Eu magnetic moments order ferromagnetically within the ab plane and antiferromagnetically along the c axis. However, there remain three types of stacking sequences of ferromagnetic layers along the c axis, as depicted in Fig. S2A. In the following, we show that type 2 in Fig. S2A is the most plausible, based on the comparison between the calculations and experimental data. The intensity of magnetic scattering I calc is expressed as

3 I calc e iq r mag i if i 2 = (e e ) e iq r i im i 2 (S2) where r i is the position of the ith Eu site taken from Ref. (15) and f i mag is the magnetic form factor of the ith Eu moment given in Eq. (S1). Considering the extinction rules obtained by calculating I calc for the three types of magnetic order (Fig. S2), the observation of ( 11) magnetic reflection suggests that the structures type 2 and type 3 remain as possible candidates. On the other hand, no signal was observed at around (3 ) (Fig. S3), which suggests that only the structure type 2 satisfies the extinction rules. In addition, we have measured the intensities (I obs) of several ( L) (L=odd) and (1 L) (L=even) magnetic reflections and compared them with the calculated intensities I calc for type 2 and type 3 (Fig. S3C). Measured intensities are in excellent agreement with the calculation results for type 2, which again supports type 2 as the most probable Eu moment arrangement. A C kev E=6.975 kev s c s p E=6.975 kev c s p p Fig. S1: Resonant x-ray magnetic scattering for EuMni 2 near the Eu L 3 absorption edge. (A) Energy scans at the ( 1) ragg reflection and ( 11) magnetic reflection with σ-incident polarization. (), (C) Polarization analysis of the ( 11) and (4 1) magnetic reflections at 5 K at kev. The broad peak denoted by arises from an unknown powder line. Experimental setup is shown schematically in each inset.

4 A type 1 type 2 type 3 Eu i Mn i c a ( 11) (4 1) (3 ) type 1 type 2 type 3 Fig. S2: (A) Three types of Eu moment arrangement at 5 K. () Extinction rule obtained by calculating I calc for each magnetic structure. Allowed and forbidden reflections are denoted by o and, respectively.

5 (4 1) 5 K A (3 ) 5 K C Fig. S3: Determination of magnetic structure of Eu sublattice for EuMni 2. (A, ) Scans around the (4 1) and (3 ) reflections at 5 K at kev with π-incident polarization. (C) Comparison between the observed (I obs) and calculated (I calc) intensities of the ( L=odd) and (1 L=even) magnetic reflections at 5 K.

6 Detailed magnetic properties for EuMni 2 The magnetic properties around the antiferromagnetic transition temperature of Eu sublattice (T N 22 K) are summarized in Figs. S4. We show in Fig. S4A the Curie-Weiss plot, the fitted results of which are consistent with those reported previously (15). ased on both temperature (Fig. S4) and field (Fig. S4D) scans, we have determined the magnetic phase diagram for the Eu sublattice for the field parallel to the c axis, as shown in Fig. 1F. In Fig. S4C and S4E, we also plot the temperature and field profiles of derivative of magnetization (dm/dt and dm/dh) at fixed fields and temperatures, respectively. In Fig. S4C, the transition temperatures to the AFM and spin-flop AFM are clearly discerned as peak or cusp structures in the dm/dt profiles. The transition fields, H f and H c, can also be unambiguously determined as clear peaks in the dm/dh profiles shown in Fig. S4E.

7 dm/dt (a.u.) dm/dh (a.u.) M ( /f.u.) M ( /f.u.) A C T ( 5) 1 T 2 T 3 T 4 T 4.5 T 5.3 T 5.5 T 5.7 T 6 T 7 T D E H f H f H c 25 K 2 K 17 K 13 K 1 K 7 K 4.2 K 1.4 K H c T (K) H (T) 3 Fig. S4: (A) Inverse magnetic susceptibility versus temperature. The temperature profile of M/H above T N is well fitted by conventional Curie-Weiss law with minimal anisotropy, the results of which are shown in the plot. The contribution from the Mn sublattice that antiferromagnetically orders at 315 K is included as a constant susceptibility χ. The effective moment estimated from the Curie constant is close to the theoretical value for Eu 2+ (7.94 μ ). () Temperature profile of magnetization (M) and (C) derivative of magnetization (dm/dt) at various magnetic fields. In panel (), the closed triangles denote the transition temperature to the anti-ferromagnetic phase of Eu sublattice, while the vertical arrows the spin-flop antiferromagnetic phase. All the transition temperatures were determined from the dm/dt profiles. (D) Field profiles of magnetization and (E) derivative of magnetization (dm/dh) (up to 33 T) at various temperatures. The positions of H f correspond to the large peaks in the dm/dh profiles, while those of H c are discerned as small peak or cusp structures as denoted by closed triangles.

8 Magnetic properties for EuZni 2 As shown in Fig. S5A, the magnetic susceptibility (M/H) for EuZni 2 steeply drops below T N ( 2 K) for H [1], whereas it is almost constant below T N for H [1]. This indicates that the Eu moments are oriented perpendicular to the c axis in the AFM phase at T for EuZni 2, in contrast to EuMni 2. The magnetization almost linearly increases with increasing field for both H [1] and H [1], whereas we observed a weak anomaly around 1 T in the profile for H [11] (Fig. S5), which suggests the Eu moments tend to be oriented parallel to the [11] direction in the AFM phase due to a weak anisotropy within the ab plane. To study the AFM arrangement for EuZni 2, we have measured the resonant x-ray magnetic scattering at E=6.975 kev (near the Eu L 3 absorption edge). We clearly observed the ( L) (L=odd) magnetic reflections below T N, in analogy with EuMni 2. As a typical example, Figs. S5C and S5D present the profiles of the ( 7) reflection along [1] at 5 K (below T N) and 3 K (above T N), and its energy scan at 5 K, respectively. The overall features are quite similar to those observed in EuMni 2. ased on these data, we can naturally assume that EuZni 2 hosts the same AFM stacking sequence along the c axis as EuMni 2 (type 2 in Fig. S2), which results in the AFM order shown in Fig. S5E as a plausible model at T. There the magnetic structure of the Eu sublattice is analogous to those in the spin-flop AFM phase for EuMni 2.

9 M ( /f.u.) emu/mol) A T N = 2 K H =.1 T H [1] H [1] C E=6.975 kev ( 7) s p c s E Eu T (K) 12 i T = 1.9 K D ( 7) 5 K i 3- Zn H (T) H [1] H [1] H [11] 4 5 c a b Fig. S5: Magnetization and resonant x-ray magnetic scattering near the Eu L 3 absorption edge for EuZni 2. (A) Temperature profile of magnetic susceptibility along [1] (red symbols) and [1] (blue) measured at.1 T. () Field profiles of magnetization for along [1] (red), [1] (blue) and [11] (green) at 1.9 K. (C) Profile of the ( 7) magnetic reflection along [1] at kev at 5 K (below T N) and 3 K (above T N). The inset shows schematic illustration of the measurement configuration. (D) Energy profile of the ( 7) magnetic reflection at 5 K. (E) Schematic illustration of the plausible magnetic structure for EuZni 2 at T.

10 Magnetization and transport features around 2 T for EuMni 2 As shown in Fig. S6A, the magnetization (or field derivative of magnetization) curve shows no anomaly at 1.4 K around 2 T, where clear hysteretic anomaly is discerned in the ρ xx and ρ zz curves. A 1.4 K H c H c C H c Fig. S6: Field profile of (A) derivative of magnetization dm/d(μ H), () in-plane resistivity ρ xx and (C) interlayer resistivity ρ zz at 1.4 K between 14 T and 26 T. Red and blue lines denote the field-increasing and -decreasing runs, respectively.

11 Hall angle and step size between the consecutive 1/ρyx plateaus for EuMni 2 ased on semiclassical transport theories, ω cτ is given by Hall angle ρ yx/ρ xx (Fig. S7A), where ω c is the cyclotron frequency and τ the scattering time. As shown in Figs.S7A and S7, the ρ yx/ρ xx peaks nicely correspond to the plateaus of inverse Hall resistivity 1/ρ yx at 1.4 K for sample #1. The value of ρ yx/ρ xx = ω cτ is larger than unity in the entire field range of the spin-flop phase and reaches 5 at around F/=1.5. In Fig. S7, we plot the inverse Hall resistivity 1/ρ yx as a function of F/. The step size between the successive 1/ρ yx plateaus (1/ρ yx ) are determined from the plateau values at F/=1.5, 2.5, and 3.5, as shown in Fig. S7. For sample #1, for instance, the value of 1/ρ yx is estimated to be Ω 1 cm 1 (i.e. ρ yx Ωcm). The present compound can be regarded as a multilayer quantum Hall system, consisting of a quasi 2D Dirac fermion conducting layer of i square net and a blocking layer of Eu-Mn-i layer. For the sample with thickness t, the Hall resistance in the quantum Hall regime is given by R yx = h/(zνe 2 ) where Z =Z * t is the total number of the i stacking layers along the c axis, Z * the number of the i stacking layers per unit thickness, and ν = s (n + 1 γ) the filling factor for the 2 single i layer. The Hall resistivity is hence expressed as ρ yx = R yx t = h (Z νe 2 ) which is independent of thickness of the sample. This expression yields the inverse Hall resistivity as 1 = Z s (n + 1 e2 γ) ρ yx 2 h which results in the step size between the successive 1/ρ yx plateaus expressed by 1 ρ = Z s e2 yx h y utilizing the 1/ρ yx values obtained in experiments, we are able to estimate the values of degeneracy factor s for sample #1 and sample #2, which are summarized in Table 1 in the main text.

12 A 8 6 H=H c 1.4 K sample #1 yx / xx 4 2 1/ yx (1 3-1 cm -1 ) K sample #1 1/ yx 1.9x1 3-1 cm F / Fig. S7: (A) Hall angle ρ yx/ρ xx (corresponding to ω cτ) and () inverse Hall resistivity 1/ρ yx versus F/ at 1.4 K for sample #1, where F is the frequency of SdH oscillation and = μ (H +M) the magnetic induction. The data were taken in a field-decreasing run. Horizontal lines in () denote the positions of 1/ρ yx plateau, from which we have estimated 1/ρ yx, the step size between the consecutive plateaus.

13 I (a.u.) I (a.u.) I (a.u.) ( 4) ( 6) ( 4) (1 3) ( 6) ( 8) (1 9) (1 1 8) (1 1 2) (1 1 ) (2 ) (2 6) (2 1 5) (1 1 1) (2 8) (2 2 ) (2 1 9) (2 2 6) (2 12) (3 1 2) (1 1 4) (1 1 2) ( 1) (2 ) (1 1 8) (1 1 1) (2 8) (1 1 12) (2 2 ) (1 1 14) ( 16) (3 1 4) ( 6) (1 1 4) ( 8) (1 1 1) (2 8) (1 1 ) (1 1 2) (1 1 4) ( 8) (2 ) ( 1) (1 1 12) (2 2 ) (2 1 9) (2 2 6) (1 1 14) ( 16) (3 1 4) Powder x-ray diffraction profile for each compound and typical geometry of the samples and electrodes A * * (1 9) (1 1 8) (2 6) ( 12) * EuMni 2 * (2 6) ( 12) SrMni 2 C EuZni 2 D * * (deg.) sample #1 ( xx, yx ) sample #3 ( zz ) E 7 1 mm 1 mm Fig. S8: (A-C) Profiles of the 2θ-θ powder x-ray diffraction for (A) EuMni 2, () SrMni 2, and (C) EuZni 2 measured by Cu-Kα radiation at room temperature. The numbers in each panel are Miller indices based on the space group I4/mmm. * denotes the reflections from the i flux stuck to the crystal surfaces. (D, E) Geometry of the samples and electrodes.