1

Supplementary Material

for

“Spin Scattering and Noncollinear Spin Structure-Induced Intrinsic Anomalous Hall Effect

in Antiferromagnetic Topological Insulator MnBi2Te4”

Seng Huat Lee1,2†, Yanglin Zhu2,3†, Yu Wang1,2, Leixin Miao4, Timothy Pillsbury2, Hemian Yi2,

Susan Kempinger2, Jin Hu5, Colin A. Heikes6, P. Quarterman6, William Ratcliff6,

Julie A. Borchers6, Heda Zhang7, Xianglin Ke7, David Graf8, Nasim Alem4, Cui-Zu Chang1,2,

Nitin Samarth1,2 and Zhiqiang Mao1,2*

12D Crystal Consortium, Materials Research Institute, Pennsylvania State University, University Park,

PA 16802

2Department of Physics, Pennsylvania State University, University Park, PA16802

3Department of Physics and Engineering Physics, Tulane University, New Orleans,

LA 70118

4Department of Materials Science and Engineering, Pennsylvania State University, University Park,

PA 16802

5Department of Physics, University of Arkansas, Fayetteville, AR 72701

6NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg,

MD 20899

7Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824

8National High Magnetic Field Lab, Tallahassee, FL 32310

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1. Crystal Growth

High-quality single crystals of MnBi2Te4 were grown by a melt growth method. A

stoichiometric mixture of high purity MnTe (99.9%) and Bi2Te3 (99.999%) was sealed in an

evacuated, carbon-coated quartz tube jacked by another evacuated quartz tube. The carbon-coated

wall inside the quartz tube was finished by using the thermal decomposition of acetone and it was

used to avoid the direct reaction of manganese with the quartz tube. The prepared ampoule was

heated up to 1000 °C before ramp-down to 610 °C with the ramp-rate of 3 °C/hr for the annealing

process. The crystalline ingot was quenched in water after two days at the annealing temperature.

Platelike single crystals were obtained by cleaving along the basal plane from the resultant ingot.

Figure S1 shows the sharp (00L) X-ray diffraction peaks matched-well with the ICDD database

PDF card 04-020-8214, indicating the excellent crystallinity of the MnBi2Te4 crystal. Additionally,

we find MnBi2Te4 can be easily intergrown with Bi2Te3 and/or MnBi6Te9 and performed careful

XRD screening for all the samples we have prepared. Only pure MnBi2Te4 single crystals were

used for STEM, ARPES, magnetic, neutron scattering, resistivity, magnetoresistance, torque and

Hall measurements.

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Figure S1. XRD pattern of a high-quality MnBi2Te4 crystal. All peaks are matched-well with the

(00L) peaks of the ICDD PDF card 04-020-8214, demonstrating the excellent crystallinity of our

sample.

2. Scanning Transmission Electron Microscope (STEM)

(a) Methods

Crystal structure was studied using transmission electron microscopy (TEM). The

electron transparent TEM sample was prepared using an FEI Helios NanoLab Dual-Beam

Focused Ion Beam system. It was thinned down using 30 kV and subsequently 5 kV Ga

ion beam, and was further polished at 2 kV to remove the redeposition and amorphous

damage layer. The selected area electron diffraction pattern (SAEDP) was taken on the FEI

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Talos S/TEM at an operating voltage of 200 kV. The STEM imaging was performed on a

spherical aberration corrected FEI Titan3 S/TEM at an operating voltage of 200 kV with a

probe convergence semi-angle of 30 mrad and the point resolution of 80 pm.

(b) STEM Analysis

The HAADF detector collects the electrons that are scattered to the outer angles, and

the collected signals are roughly proportional to the atomic number Z2. It allows us to

identify the chemical identity of all the atoms in the lattice. Since the intensity collected is

formed from all the atoms in one atomic column, the Mn vacancies result in a reduction in

the intensity of Mn atomic columns in the HAADF-STEM image as shown in Fig. S2a.

The simultaneously acquired BF-STEM image in Fig. S2b shows the missing Mn atomic

columns at the same positions. Combining with the HAADF image, it provides more proof

to the existence of the Mn vacancies.

Moreover, the septuple layers in LAADF-STEM image in Fig. S2c exhibit bright

contrast at the positions where the vacancies site. In the LAADF-STEM, the more

coherently scattered electrons with lower scattering angles are collected compared to the

HAADF detector. Due to the diffraction contrast present at lower scattering angles, the

strain associated with the defects can be observed leading to a variation in the contrast.

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Figure S2. Simultaneously acquired (a) HAADF-STEM, (b) BF-STEM, and (c) LAADF-STEM

from the [100] axis of the MnBi2Te4 crystal. The locations of the Mn vacancies are indicated by

the yellow arrows.

3. ARPES

ARPES measurements were carried out in an ultrahigh-vacuum (UHV) system with a base

pressure ~1×10-11 mbar. The photoelectrons are excited by an unpolarized He-I light (~21.218eV)

and collected by an Omicron-Scienta DA30L analyzer. Fresh surfaces were obtained by cleaving

MnBi2Te4 single crystal sample in a preparation chamber with a pressure ~2×10-9 mbar and then

the sample with the fresh surface was transferred into the main chamber for the ARPES

measurements. The angular and energy resolution of the DA30L analyzer was 0.1° and ~6 meV,

respectively.

4. Magnetization

Magnetization and magnetic susceptibility measurements were taken by using commercial

SQUID magnetometers. The field was applied perpendicular and parallel to the c-axis respectively.

Due to the small size of the crystal, the samples were surrounded in plastic wrap for mounting,

which introduces a very slight diamagnetic signal to the data. For the temperature sweep

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measurements on sample 1 (S#1), it was performed in a SQUID with a maximum field of 5T. In

these measurements, the sample was cooled from 305 K to 5 K in either a 1 T field for the

field-cooled (FC) curve or a 1.3 x 10-4 T field for the zero-field-cooled (ZFC) curve (which

corresponds to the remanence in the magnet after demagnetization). The data obtained from these

measurements are presented in Fig. S3a, from which an AFM transition is observed. The spin-easy

axis of the AFM state is along the c-axis. For the field sweep measurements on S#1, the

temperature was set at 5 K. The applied field was set to 5 T, then decreased through zero to – 5 T,

and then increased back to 5 T. The measured isothermal magnetization (Fig. S3b) shows a sharp

spin-flop transition at ~ Hc1 = 3.57T for H//c, but linear field dependence for H//ab.

Figure S3. (a) Field cooled (FC) and zero-field cooled (ZFC) temperature dependences of

magnetic susceptibility χ measured at 1 T for a MnBi2Te4 crystal (S#1) aligned with the magnetic

field parallel (H//c) and perpendicular (H⊥c) to the c-axis, respectively. Inset: Modified Curie-

Weiss plot with χo = -0.007 emu/mol Oe. The fit of susceptibility data in the 100-300K range by

the modified Curie-Weiss law yields θCW = 5.0(03)K. (b) Isothermal magnetization at 5K for S#1.

(c) Isothermal magnetization under various field orientation angles at 2K for MnBi2Te4 (S#2).

We also performed isothermal magnetization measurements up to 7 T under various field

orientations at 2 K on another MnBi2Te4 sample (S#2) using a SQUID equipped with a sample

rotator. The data obtained from these measurements are shown in Fig. S3c. The spin-flop transition

near Hc1 in this sample is broader than that in S#1, which is likely due to the different levels of

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disorders between these two samples. As shown in Fig. S3c, as the tilt angle of magnetic field

relative to the c-axis increases, M(H) shows a gradual evolution from a steep metamagnetic

transition near Hc1 for H//c (θ = 0°) to crossover transitions for 0 < θ < 90°, and finally to linear

dependence for θ ≈ 90°.

5. Neutron Diffraction

(a) Methods

Neutron scattering experiments were performed using the BT-4 triple axis

spectrometer at the NIST Center for Neutron Research. An instrument configuration of

open-pg-pg-40’-s-pg-40’-100’ was used at 14.7 meV, with two pyrolytic graphite (pg)

filters before the sample and one after the sample to eliminate higher order neutrons. For

the measurements with no applied magnetic field, the sample was mounted in a sealed

aluminum can in a He environment aligned to either the H0L or HHL zones and the sample

was cooled using a standard close cycle refrigerator. For measurements in field, the sample

was attached at the center of the rotation axis of a one-axis in-magnet custom-designed

rotation stage based on an Attocube piezoelectric rotation stage [SM1]. This assembly was

placed in the bore of a 7 T vertical field magnet in a liquid He dewar, with the sample

cooled by He exchange gas. The sample was aligned with a (100) plane normal to the tilt

axis to allow for the continuous rotation between the HK0 scattering plane and the 𝐻�̅�𝐿

scattering plane. Starting in HK0 with the field direction aligned along the c-axis, we were

able to tilt slightly with the rotation stage to reach 10𝐿 and 1̅0𝐿 reflections. This allowed

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us to reach the (1̅02) nuclear reflection and the (1̅01

2) magnetic reflection with more than

97% of the projection of the applied field along the c-axis in each case.

(b) Magnetic Structure

In zero field we measure magnetic reflections that are consistent with the 𝑘 =

(0,0,1

2) A-type antiferromagnetic (AFM) structure recently reported with all Mn spins

pointing along the c-axis, ferromagnetically aligned within a septuple layer, and

antiferromagnetically aligned between septuple layers [SM2]. As an example, Fig. S4a

shows the temperature dependence of the intensity of the (1 0 2.5) reflection, which is

consistent with TN = 22K. Field-dependent order parameters were similary obtained by

tracking the peak intensity of the (1̅01

2) and (1̅02) reflections. With the magnetic field

applied along c (as shown in Fig. 4a in main text), we observe a reduction in the scattering

intensity for the (1̅01

2) reflection (Fig. S4c) starting at 2.5 T, with a large decrease above

3.5 T. Concurrently at 3.5 T, we observe a large increase in the scattering intensity of the

(1̅02) reflection (Fig. S4b) which increases linearly to our maximal applied field. We also

still observe scattering intensity at the (1̅01

2) reflection to our maximum applied field.

Interestingly, we observe no appreciable change in the scattering intensity of the (110)

reflection with applied field.

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Figure S4. (a) Temperature dependence of the (1 0 2.5) neutron reflection at zero magnetic field.

(b) and (c) are the θ scans through the (1̅ 0 2) and (1̅ 0 1

2) neutron reflection peaks at various

fields, 4.2K. The appearance of a weak secondary peak is indicative of a crystallite with a small

offset relative to the primary crystallite.

Starting from the crystal space group 𝑅3̅𝑚, with a single 𝑘 = (0,0,1

2) magnetic

propagation vector and using the MAXMAGN and k-SUBGROUPSMAG tools from the

Bilbao Crystallographic Server we determine there is no magnetic space group that allows

for a net moment per unit cell [SM3]. By adding a second, field induced 𝑘 = (0,0,0)

propagation vector for a multi-k magnetic structure, a net moment is allowed concurrent

with scattering at the (1̅01

2) position. The magnetic field order parameters shown in Fig. 4a

illustrate 3 regimes, consistent with the torque magnetometry data. In regime I, up to 2.5 T,

the zero field magnetic structure is maintained. In regime II from 2.5 T to 3.5 T, we observe

a small reduction scattering for a 𝑘 = (0,0,1

2) reflection with no measured increase in any

𝑘 = (0,0,0) reflections. Regime III from 3.5 to 6.5 T shows a concurrent large decrease in

𝑘 = (0,0,1

2) scattering and a large increase in some but not all 𝑘 = (0,0,0) reflections. This

is consistent with the picture of a transition from an A-type AFM at low field, through to a

canted-AFM phase at intermediate field with a continuous rotation of the moments anti-

aligned to the field direction. Furthermore, this picture of alternating canted layers is also

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consistent with scattering increase at the (1̅02) reflection but not the (110) reflection.

Interestingly, any rotation of the Mn moments orthogonal to the c-axis either breaks the

3-fold rotational symmetry of the crystal resulting in a monoclinic or lower symmetry

structure, or must result in a more complicated set of magnetic propagation vectors with an

in-plane component.

6. Torque Simulation

Since the magnetic torque τ measurements have to be performed with the magnetic field

being misaligned relative to the c-axis of the sample (Fig. S5a) as explained below, the measured

τ is not directly proportional to magnetization, instead dependent on the components of the

magnetization and magnetic field along the c-axis and in-plane directions. From the simple

simulations based on the measured magnetization and the field tilt angle, we can qualitatively

capture the features seen in the measured τ (Fig. S5b and S5c), as presented below. τ is a vector,

which can be expressed as

τ = M × μoH (1)

where M is the magnetization and H is the applied magnetic field. From this expression, it can be

seen that τ = 0 when M//H. Therefore, the direction of the sample’s magnetization must deviate

from the field direction to observe τ, as shown in Fig. S5a. For simplicity, M and H are assumed

within the z-y plane. According to Fig. S5a, τ can be expressed as:

τ = μo(MyHz – MzHy) (2)

where Mz, My, Hz and Hy are the z- and y-direction components of M and H respectively.

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Figure S5. (a) Schematic of magnetization induced by a tilted magnetic field. (b) Components of

MyHz and MzHy in eq. (2) and their difference, simulated based on the measured and extrapolated

magnetization and a magnetic field tilt angle of 6°. (c) Zoom-in of MyHz - MzHy in (b).

In actual experiments, there is always an inevitable sample misalignment, so both MyHz

and MzHy components in eq. (2) are finite. Hence, τ is determined by the difference of MyHz and

MzHy rather than proportional to the magnetization. Fig. S6a and S6b show the out-of-plane (H//c)

and in-plane (H//ab) magnetization of MnBi2Te4 respectively, measured by a SQUID

magnetometer. Given the available maximum field of the SQUID is 7T, less than the critical field

Hc2 ~ 7.7T, we have to extropalate the magnetization data to higher field to produce the torque

data beyond Hc2 using eq. (2). For H//ab, since M is linearly dependent on H up to 7T, a linear

extrapolation beyond Hc2 (i.e. red dashed line in Fig. S6b) is made. For H//c, the extrapolation

beyond Hc2 is not staightfoward, since the slope change of M(H) across Hc2 is unknown. We tried

many different extrapolations and find the extrapolation shown in Fig. S6a can yield the torque

data (Fig S5c) which look similar to the experimentally measured torque (Fig. 4a in the main text).

(a) (b) (c)

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Figure S6. The measured magnetization data and its extrapolation beyond Hc2for (a) H//c and

(b) H//ab at 2K for MnBi2Te4

With the extrapolated magnetization shown in Fig. S6 and the actual field tilt angle of 6°

from c-axis in the experiment, magnetic toque is simulated using eq. (2). The field components on

the y- and z-axis lead to My and Mz components. In Fig. S5b, the simulated MzHy, MyHz, as well

as MzHy - MyHz are shown. As seen in Fig. S5c, the simulated MzHy - MyHz, which is proportional

to the magnetic torque (eq. (2)), captures the main features of the measured toque shown in Fig. 4a

in the main text, including the peak around Hc1 (~ 3.57 T), the kink near Hc2 (~7.7T) and weak

field dependence above Hc2.

7. Magnetotransport Measurements up to 35 T

The magnetotransport and magnetic torque measurements were performed at the National

High Magnetic Field Laboratory in Tallahassee. The standard four-probe technique with silver

epoxy cured at 160C in argon enviroment for the contacts was employed for the in-plane

resistivity ρxx, out-of-plane ρzz, and Hall resistivity ρxy measurements. A small dc current of 1 mA

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was applied for all the transport measurements. Figure S7 shows the magnetotransport

measurements for both H//c and H⊥c up to 35T at various temperatures. Both ρxx and ρzz show

clear increase above 15T for H//C, as shown in Fig. S7a and S7b.

Figure S7. (a) and (c) show the in-plane resistivity ρxx for H//c and H⊥c, respectively. (b) and (d)

show the out-of-plane resistivity ρzz for H//c and H⊥c, respectively. The schematics illustrate the

setup of the magnetotransport experiments.

References

[SM1] Certain commercial equipment, instruments, or materials (or suppliers, or software, ...) are

identified in this paper to foster understanding. Such identification does not imply

recommendation or endorsement by the National Institute of Standards and Technology,

nor does it imply that the materials or equipment identified are necessarily the best

available for the purpose.

[SM2] J. Q. Yan, Q. Zhang, T. Heitmann, Z. L. Huang, W. D. Wu, D. Vaknin, B. C. Sales, and

R. J. McQueeney, arXiv:1902.10110 (2019).

[SM3] M. I. Aroyo et al., Zeitschrift Für Kristallographie - Crystalline Materials 221, 15 (2006).