INELASTIC LIGHT SCATTERING IN LOW DIMENSIONAL QUANTUM SPIN SYSTEMS BY ADRIAN
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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).