# Magneto-optical effects of an artificially-layered ferromagnetic topological
insulator

Xingyue Han  Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA  Hee Taek Yi  Department of Physics and
Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey
08854, USA  Seongshik Oh  Department of Physics and Astronomy, Rutgers, The
State University of New Jersey, Piscataway, New Jersey 08854, USA  Liang Wu
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA

###### Abstract

Abstract: Magnetic topological insulator is a fertile platform to study the
interplay between magnetism and topology. The unique electronic band structure
can induce exotic transport and optical properties. However, a comprehensive
optical study in both near-infrared frequency and terahertz frequency has been
lacking. Here, we report magneto-optical effects from a heterostructure of Cr-
incorporated topological insulator, CBST. We use 800 nm magneto-optical Kerr
effect to reveal a ferromagnetic order in the CBST film with a high transition
temperature at 160 K. We also use time-domain terahertz polarimetry to reveal
a terahertz Faraday rotation of 1.5 mrad and Kerr rotation of 5.1 mrad at 2 K.
The calculated terahertz Hall conductance is 0.42 $e^{2}/h$. Our work shows
the optical responses of an artificially layered magnetic topological
insulator, paving the way towards high-temperature quantum anomalous Hall
effect via heterostructure engineering.

key words: magnetic topological insulators, magneto-optical Kerr effect,
terahertz Faraday rotation, terahertz Kerr rotation

Topological insulators (TIs), characterised by the insulating bulk states and
conducting topological surface states, were originally proposed in time-
reversal-invariant systems Moore (2010); Hasan and Kane (2010); Qi and Zhang
(2011); Tokura  _et al._ (2019). The gapless topological surface states are
located in the bulk gap crossing at the Dirac point. Incorporating
ferromagnetism into TIs breaks time-reversal symmetry and opens an exchange
gap at the Dirac point Yu  _et al._ (2010); Chang  _et al._ (2013); Tokura
_et al._ (2019). The interplay between magnetism and topology has brought a
plethora of exotic phenomena Onoda and Nagaosa (2003); Wan  _et al._ (2011);
Nakatsuji  _et al._ (2015); Armitage  _et al._ (2018); Tokura  _et al._
(2019); Yu  _et al._ (2010), including the quantum anomalous Hall effect
(QAHE) Chang  _et al._ (2013), and the axion insulator states Liu  _et al._
(2020). When the Fermi energy lies in the exchange gap, the QAHE emerges due
to dissipationless chiral edge modes. In an ideal quantum anomalous Hall
state, the anomalous Hall resistance is quantized at $h/e^{2}$, while the
longitudinal resistivity vanishes. In order to realize a ferromagnetic TI
system, the first attempt was doping nonmagnetic TIs with magnetic impurities,
such as Cr- and V- doped (Bi,Sb)2Te3 Chang  _et al._ (2013, 2015); Checkelsky
_et al._ (2014). However, the inhomogeneity of the magnetic dopants suppresses
the effective energy gap. It results in a low observation temperature of QAHE,
usually below 100 mK. Following the magnetic impurity doping method, the
magnetic modulation doping technique increased the surface mass gap and
improved the homogeneity. The QAHE observation temperature was raised to 2 K
Mogi  _et al._ (2015, 2017a, 2017b); Ou  _et al._ (2018); Yi  _et al._ (2023).
Recently, intrinsic magnetic TIs with chemical formulas such as MnBi2Se$4$
Otrokov  _et al._ (2019); Deng  _et al._ (2020); Li  _et al._ (2019) have
received more attention with the report of 97% quantization persist up to 6.5
K in a 5 septuple-layer (SL) flakeDeng  _et al._ (2020).

The tunability of the magnetic and topological properties of magnetic TIs have
been investigated in various ways, such as Sb doping Guan  _et al._ (2022); Hu
_et al._ (2021), hydrostatic pressure Eckberg  _et al._ (2023); Chen  _et al._
(2019), and surface engineering Mazza  _et al._ (2022). Another important
method is using nonmagnetic spacer layer Yao  _et al._ (2021). For example, in
the intrinsic MnBi2Te4 family, due to the van der Waals nature, the magnitude
of interlayer antiferromagnetic strength can be suppressed by inserting Bi2Te3
quintuple layers (QL) between MnBi2Te4 SLs, with the chemical formula
MnBi2nTe3n+1. The electric band structure spontaneously changes with different
magnetic orders. For example, MnBi2Te4 (n=1) Deng  _et al._ (2020) and
MnBi4Te7 (n=2) Hu  _et al._ (2021) are reported to be antiferromagnetic TIs,
MnBi8Te13 (n=4) is predicted to be ferromagnetic TI Hu  _et al._ (2020); Zhong
_et al._ (2021). With the boost of research interest in the MnBi2nTe3n+1
family, replacing Mn with other magnetic elements has been investigated as
well. Density functional theory (DFT) calculations revealed that certain
elements can form stable XBi2Te4 SL structures, including X = Ti, V, Ni, Eu,
etc Li  _et al._ (2019). But X = Cr, Fe, and Co show instability to form SL
structures Li  _et al._ (2019).

Figure 1: Structure and transport of CBST. a. Layered structure of CBST. The
17-tuple layer structure is repeated twice. The film terminates at BST layer.
b. Magnetic field dependence of Hall resistance of CBST measured at 2 K.

Heterostructure engineering is a powerful tool to tune the magnetic and
topological properties of Cr-incorporated TIs Jiang  _et al._ (2020); Mogi
_et al._ (2022); Yasuda  _et al._ (2016); Yao  _et al._ (2021). In our work,
we report a heterostructure of Cr-incorporated topological insulator, dubbed
as CBST. The layer-by-layer epitaxial deposition of each Sb2Te3, Bi2Te3 and
CrTe was carried out based on the Cr(Bi1/3Sb2/3)6Te10 structure (Fig.1a). We
performed magneto-optical measurements in both near-infrared (NIR) and
terahertz (THz) frequencies. Ferromagnetic order is observed using magneto-
optical Kerr effect (MOKE). The magnetic transition temperature at 160 K is
higher than any other Cr-incorporated magnetic TIs. We also performed THz
measurements in CBST which exhibits a Faraday rotation of 1.5 mrad and Kerr
rotation of 5.1 mrad under 6 T at 2 K.

In the CBST film, the Cr(Bi1/3Sb2/3)6Te10 structure repeats twice, terminating
at (Bi1/2Sb1/2)2Te3 (BST) layer (Fig.1a). The 6.5 nm-thick film is grown on a
1 cm $\times$ 1 cm Al2O3 (0001) substrate using a custom-built molecular beam
epitaxy (MBE) system (SVTA) under a base pressure of 5 $\times$ 10-10 Torr.
High-purity (99.999%) Bi,Sb, Te, and Cr were thermally evaporated using
effusion cells, and all the source fluxes were calibrated in-situ by quartz
crystal micro-balance and ex-situ by Rutherford backscattering spectroscopy.
We employ thin epitaxial Cr2O3 as a buffer layer to assist the topological
layers in adhering to the Al2O3 substrate, and naturally oxidized amorphous
CrOx as a capping layer to suppress degradation and boost ferromagnetic order
Yao  _et al._ (2021); Yi  _et al._ (2023). High-angle annular dark-field
scanning transmission electron microscope (HAADF-STEM) image has revealed that
Cr is incorporated into the neighboring Sb2Te3 layer instead of forming a van
der Waals gap between Sb2Te3 layers Yao  _et al._ (2021). The transport data
at 2 K (Fig.1b) with single coercive field inicates that surface states only
form at top and bottom surfaces, suggesting that the entire system is behaving
as a single magnetic TI layer.

Figure 2: MOKE results of CBST at 800 nm. a. Schematic of MOKE measurement
from CBST. b. Temperature dependent MOKE under field-cool. $\pm$25 mT magnetic
field is applied perpendicular to the sample surface. c. MOKE Hysteresis loop
measured at 3 K. d. MOKE hysteresis loops measured from 3 K to 160 K. e.
Temperature dependence of the field-symetrized $\theta_{K}$ at 0 T.

We firstly utilize MOKE to probe the time-reversal symmetry breaking in CBST.
A mode-locked Ti:sapphire laser with 800 nm centre wavelength, 80 MHz
repetition rate and 50 fs pulse duration is used to measure the Kerr rotation
$\theta_{K}^{NIR}$ from the CBST sample Xu  _et al._ (2022).
$\theta_{K}^{NIR}$ is the polarization plane change in the reflection beam
proportional to the net magnetization along the laser propagation direction
McCord (2015) (Fig.2a). We carried out temperature dependent MOKE measurements
under an out-of-plane magnetic field at $\pm$ 25 mT. As shown in Fig.2b,
cooling from 180 K to 10 K, $\theta_{K}^{NIR}$ emerges around 160 K and shows
opposite signs under opposite magnetic fields. Fig.2c and d show the magnetic
field dependent MOKE at 3 K and higher temperatures. $\theta_{K}^{NIR}$ at 3 K
shows a hysteresis loop with the coercive field HC at 70 mT and saturates at 1
mrad. Above the saturation field, there is no signs of spin-flip/flop
transitions up to $\pm$ 6 T, consistent with the behavior from a ferromagnetic
order. The coercive field and MOKE signal monotonically decrease with
increasing temperature, and disappears at 160 K (Fig.2d). Fig.2e shows zero-
field MOKE

$\theta^{NIR}_{K}(0T)=\frac{\theta^{NIR}_{K}(+0T)-\theta^{NIR}_{K}(-0T)}{2}$
(1)

under different temperatures. The high transition temperature indicates an
enhanced ferromagnetic order in the CBST film, compared to other Cr-
incorporated magnetic TIs (TABLE.1), achieved by the high Cr concentration
from the layer-by-layer growth method and an active capping layer CrOx Yao
_et al._ (2021); Yi  _et al._ (2023).

Type | Sample | TN/TC
---|---|---
intrinsic |  MnBi2Te4Gong  _et al._ (2019); Zhao  _et al._ (2021); Liu  _et al._ (2021, 2022) | 15-25 K
MnBi6Te10Yan  _et al._ (2022) | 13 K
MnBi8Te13Hu  _et al._ (2020) | 10 K
uniform doping | Cr0.15(Bi0.1Sb0.9)1.85Te3 Chang  _et al._ (2013) | 15 K
Cr0.22(Bi0.2Sb0.8)1.78Te3 Checkelsky  _et al._ (2014) | 45 K
Cr0.29Sb1.71Te3 Chang  _et al._ (2015) | 59 K
modulation doping | Cr0.57(Bi0.26Sb0.74)1.43Te3 Okada  _et al._ (2016) | 70 K
Cr0.95(Bi0.32Sb0.68)1.05Te3 Mogi  _et al._ (2015) | 80 K
| layered heterostructure (this work) | 160 K
Table 1: Magnetic transition temperatures of MnBi2nTe3n+1 family and Cr-
incorporated magnetic TIs.  Figure 3:  THz result of CBST. a. Schematics of
the polarization state of the transmitted THz waves after CBST. The incident
THz polarization is parallel to x-axis. b. Terahertz signals in time domain
for the CBST. The transmitted light is composed of the main pulse and the
first echo. They are separated in time domain due to different light path. Ex
and Ey are the transmitted signals parallel and perpendicular, respectively,
to the incident polarization at 0 T. c. Spectra of THz Faraday rotation at 2
K. d. Spectra of $\Theta^{THz}$, the sum of THz Faraday rotation
$\theta_{\mathrm{F}}^{\mathrm{THz}}$ and THz Kerr rotation, at 2 K. The shaded
area in c and d are error bars. e. Field dependence of $\Theta^{THz}$ at 2 K.
Each data point is calculated from the average value in the range 0.2 THz to
1.4 THz. f. Temperature dependence of $\Theta^{THz}$ (blue squares) and
$\theta_{K}^{NIR}$ (black curve).

Then we study the low-energy response of CBST film by transmission time-domain
terahertz (THz) polarimetry from 0.2 THz (0.8 meV) to 1.2 THz (5.0 meV) Han
_et al._ (2022). A pair of photoconductive antennas are used to generate and
probe THz radiations. A delay stage introducing time delay between THz
emission and the detection path enables the time-domain measurement. A THz
polarizer mounted on a rotation stage is used to analyze THz polarization
transmitted through the CBST film. In magnetic TIs, the exchange energy gap
opening at the TI surface is usually tens of meV Lee  _et al._ (2015); Chen
_et al._ (2010). Thus interband transition is prohibited under THz excitation.
We then measure the THz Faraday $\theta_{\mathrm{F}}^{\mathrm{THz}}$ and Kerr
rotation $\theta_{\mathrm{K}}^{\mathrm{THz}}$ from CBST film. By definition,
Faraday rotation refers to the polarization change of the transmitted light,
while Kerr rotation is about the reflected light. In the time-domain
spectroscopy, the transmitted light is composed of the main pulse and a series
of echos which are separated in time (Fig.3b). The main pulse is the directly
transmitted THz wave (Fig.3a). The first echo is firstly reflected at the
substrate/TI interface, then reflected back at the substrate surface, finally
transmitted through the film (Fig.3a). Higher orders of echos appear later in
time, and are not being considered here. Therefore, the polarization change of
the main pulse is the Faraday rotation $\theta_{\mathrm{F}}^{\mathrm{THz}}$
(Fig.3a,b). The polarization change of the first echo is the sum of Faraday
rotation and Kerr rotation
$\Theta^{THz}=\theta_{\mathrm{F}}^{\mathrm{THz}}+\theta_{\mathrm{K}}^{\mathrm{THz}}$
(Fig.3a,b). Here we use the superscript "THz" to distinguish the THz rotations
with the NIR MOKE results in Fig.2. By fixing the incident polarization (x
direction), the THz rotation angles can be calculated from the ratio of the
perpendicular component ($E_{y}$) and the parallel component ($E_{x}$).

To measure the field-dependence of $\theta_{\mathrm{F}}^{\mathrm{THz}}$ and
$\Theta^{THz}$, the magnetic field is swept to 6 T to magnetize the film
before measurements. A sequence of 6 T $\rightarrow$ 4 T $\rightarrow$ 2 T
$\rightarrow$ 0 T $\rightarrow$ -6 T $\rightarrow$ -4 T $\rightarrow$ -2 T
$\rightarrow$ 0 T is followed to apply the field. Fig.3c displays the field-
symmetrized THz Faraday and Kerr rotation spectra

$\theta_{\mathrm{F}}^{\mathrm{THz}}(B)=\frac{\theta_{\mathrm{F}}^{\mathrm{THz}}(+B)-\theta_{\mathrm{F}}^{\mathrm{THz}}(-B)}{2}$
(2)

at 2 K. A modest frequency dependence agrees with previous THz measurements in
magnetic TIs Okada  _et al._ (2016); Mogi  _et al._ (2022). Under 0 T,
$\theta_{\mathrm{F}}^{\mathrm{THz}}$ shows an average value of 0.57 mrad. It
increases monotonically with magnetic field and reaches 1.5 mrad under 6 T.
Field-symmetrized spectra of $\Theta^{THz}$ in the range 0.3 THz (1.2 meV) to
0.76 THz (3.1 meV) are displayed in Fig.3d. The more substantial frequency
dependence of $\Theta^{THz}$ at high fields comes from the temporal overlap
between the main pulse and the echo Okada  _et al._ (2016). The average value
of $\Theta^{THz}$ is 2.3 mrad under 0 T and 5.1 mrad under 6 T. The field and
temperature dependence of $\Theta^{THz}$ are displayed in Fig.3e and f. A
hysteresis loop is observed in $\Theta^{THz}$, and its temperature dependence
well agrees with $\theta_{K}^{NIR}$. These observations corroborates the
magnetic origin of the THz rotations in CBST.

Figure 4: THz anomalous Hall conductance of CBST.  $G_{xy}$ is the complex
anomalous Hall conductivity. a. and b. shows the real and imaginary part of
the 2K $G_{xy}$, respectively. Shaded area is the error bar.

THz Faraday rotation is related to THz Hall conductance $G_{xy}$ via

$G_{xy}=\theta_{\mathrm{F}}^{\mathrm{THz}}(G_{xx}+\dfrac{n_{s}+1}{Z_{0}})$ (3)

Here $G_{xx}$ is the longitudinal conductance, $n_{s}$ is the refractive index
of the substrate (3.1 for sapphire), and $Z_{0}$ ($\approx$376.7 $\Omega$) is
the vacuum impedance. We obtain a small $G_{xx}$ in the probing frequency
range from the nearly unity THz transmission $T_{xx}$, according to

$G_{xx}=\dfrac{n_{s}+1}{Z_{0}}[\dfrac{e^{i\omega(n_{s}-1)\Delta
L/c}}{T_{xx}}-1]$ (4)

Here $c$ is the speed of light, and $\Delta L$ is the thickness mismatch
between the CBST sample and reference substrate. As shown in Fig.4a., the Hall
conductance is almost constant at 0.14 $e^{2}/h$ under 0 T. Under higher
magnetic fields, $G_{xy}$ decreases at higher frequencies. Under 6 T, $G_{xy}$
reaches a maximum value of 0.69 $e^{2}/h$ at 0.32 THz, and converges to 0.33
$e^{2}/h$ at 0.7 THz. The average value of the 6 T spectrum is 0.42 $e^{2}/h$.
The frequency dependence at high field might come from cyclotron resonancesWu
_et al._ (2016, 2015). A small but nonzero value in the imaginary part of the
complex $G_{xy}$ (Fig.4b.) also indicates the emergence of dissipation.

In the ideal QAHE state, $G_{xx}$ vanishes to zero, while $G_{xy}$ is
quantized in unit of $e^{2}/h$. The calculated THz Faraday and Kerr rotations
are $tan^{-1}(\frac{2\alpha}{n_{s}+1})=3.67$ mrad and
$tan^{-1}(\frac{4n_{s}\alpha}{n_{s}^{2}-1})=10.5$ mrad, respectively Wu  _et
al._ (2016); Okada  _et al._ (2016); Mogi  _et al._ (2022). To account for the
deviation between experimental results and theoretical calculations, different
reasons were proposed such as trivial bands involvement Li  _et al._ (2016);
Pan  _et al._ (2020) and emergence of superparamagnetic order Lachman  _et
al._ (2015); Pan  _et al._ (2020). Further investigation in CBST about the
Fermi level and possible superparamagnetic order are needed towards the
realization of QAHE . Looking forward, we hope this work generate interests in
realizing high-temperature QAHE at terahertz frequency.

Acknowledgments This project at Penn and Rutgers is mainly sponsored by the
Army Research Office and was accomplished under Grant Number W911NF-20-2-0166.
X.H. is also partially supported by the Gordon and Betty Moore Foundation’s
EPiQS Initiative, Grant GBMF9212 to L.W. and the NSF EPM program under grant
no. DMR-2213891 for manuscript writing. The work at Rutgers by H. Yi and S. O.
was also supported by NSF DMR2004125, and the center for Quantum Materials
Synthesis (cQMS), funded by the Gordon and Betty Moore Foundation’s EPiQS
initiative through grant GBMF10104.

## References

  * Moore (2010) Joel E Moore, “The birth of topological insulators,” Nature 464, 194–198 (2010).
  * Hasan and Kane (2010) M Zahid Hasan and Charles L Kane, “Colloquium: topological insulators,” Reviews of modern physics 82, 3045 (2010).
  * Qi and Zhang (2011) Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
  * Tokura  _et al._ (2019) Yoshinori Tokura, Kenji Yasuda, and Atsushi Tsukazaki, “Magnetic topological insulators,” Nature Reviews Physics 1, 126–143 (2019).
  * Yu  _et al._ (2010) Rui Yu, Wei Zhang, Hai-Jun Zhang, Shou-Cheng Zhang, Xi Dai, and Zhong Fang, “Quantized anomalous hall effect in magnetic topological insulators,” science 329, 61–64 (2010).
  * Chang  _et al._ (2013) Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, _et al._ , “Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator,” Science 340, 167–170 (2013).
  * Onoda and Nagaosa (2003) Masaru Onoda and Naoto Nagaosa, “Quantized anomalous hall effect in two-dimensional ferromagnets: quantum hall effect in metals,” Physical review letters 90, 206601 (2003).
  * Wan  _et al._ (2011) Xiangang Wan, Ari M Turner, Ashvin Vishwanath, and Sergey Y Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,” Physical Review B 83, 205101 (2011).
  * Nakatsuji  _et al._ (2015) Satoru Nakatsuji, Naoki Kiyohara, and Tomoya Higo, “Large anomalous hall effect in a non-collinear antiferromagnet at room temperature,” Nature 527, 212–215 (2015).
  * Armitage  _et al._ (2018) NP Armitage, EJ Mele, and Ashvin Vishwanath, “Weyl and dirac semimetals in three-dimensional solids,” Reviews of Modern Physics 90, 015001 (2018).
  * Liu  _et al._ (2020) Chang Liu, Yongchao Wang, Hao Li, Yang Wu, Yaoxin Li, Jiaheng Li, Ke He, Yong Xu, Jinsong Zhang, and Yayu Wang, “Robust axion insulator and chern insulator phases in a two-dimensional antiferromagnetic topological insulator,” Nature materials 19, 522–527 (2020).
  * Chang  _et al._ (2015) Cui-Zu Chang, Weiwei Zhao, Duk Y Kim, Haijun Zhang, Badih A Assaf, Don Heiman, Shou-Cheng Zhang, Chaoxing Liu, Moses HW Chan, and Jagadeesh S Moodera, “High-precision realization of robust quantum anomalous hall state in a hard ferromagnetic topological insulator,” Nature materials 14, 473–477 (2015).
  * Checkelsky  _et al._ (2014) JG Checkelsky, R Yoshimi, A Tsukazaki, KS Takahashi, Y Kozuka, J Falson, M Kawasaki, and Y Tokura, “Trajectory of the anomalous hall effect towards the quantized state in a ferromagnetic topological insulator,” Nature Physics 10, 731–736 (2014).
  * Mogi  _et al._ (2015) M Mogi, R Yoshimi, A Tsukazaki, K Yasuda, Y Kozuka, KS Takahashi, M Kawasaki, and Y Tokura, “Magnetic modulation doping in topological insulators toward higher-temperature quantum anomalous hall effect,” Applied Physics Letters 107, 182401 (2015).
  * Mogi  _et al._ (2017a) M Mogi, M Kawamura, R Yoshimi, A Tsukazaki, Y Kozuka, N Shirakawa, KS Takahashi, M Kawasaki, and Y Tokura, “A magnetic heterostructure of topological insulators as a candidate for an axion insulator,” Nature materials 16, 516–521 (2017a).
  * Mogi  _et al._ (2017b) Masataka Mogi, Minoru Kawamura, Atsushi Tsukazaki, Ryutaro Yoshimi, Kei S Takahashi, Masashi Kawasaki, and Yoshinori Tokura, “Tailoring tricolor structure of magnetic topological insulator for robust axion insulator,” Science advances 3, eaao1669 (2017b).
  * Ou  _et al._ (2018) Yunbo Ou, Chang Liu, Gaoyuan Jiang, Yang Feng, Dongyang Zhao, Weixiong Wu, Xiao-Xiao Wang, Wei Li, Canli Song, Li-Li Wang, _et al._ , “Enhancing the quantum anomalous hall effect by magnetic codoping in a topological insulator,” Advanced materials 30, 1703062 (2018).
  * Yi  _et al._ (2023) Hee Taek Yi, Deepti Jain, Xiong Yao, and Seongshik Oh, “Enhanced quantum anomalous hall effect with an active capping layer,” Nano Letters 23, 5673–5679 (2023), https://doi.org/10.1021/acs.nanolett.3c01313 .
  * Otrokov  _et al._ (2019) Mikhail M Otrokov, Ilya I Klimovskikh, Hendrik Bentmann, D Estyunin, Alexander Zeugner, Ziya S Aliev, S Gaß, AUB Wolter, AV Koroleva, Alexander M Shikin, _et al._ , “Prediction and observation of an antiferromagnetic topological insulator,” Nature 576, 416–422 (2019).
  * Deng  _et al._ (2020) Yujun Deng, Yijun Yu, Meng Zhu Shi, Zhongxun Guo, Zihan Xu, Jing Wang, Xian Hui Chen, and Yuanbo Zhang, “Quantum anomalous hall effect in intrinsic magnetic topological insulator mnbi2te4,” Science 367, 895–900 (2020).
  * Li  _et al._ (2019) Jiaheng Li, Yang Li, Shiqiao Du, Zun Wang, Bing-Lin Gu, Shou-Cheng Zhang, Ke He, Wenhui Duan, and Yong Xu, “Intrinsic magnetic topological insulators in van der waals layered mnbi2te4-family materials,” Science Advances 5, eaaw5685 (2019).
  * Guan  _et al._ (2022) Y. D. Guan, C. H. Yan, S. H. Lee, X. Gui, W. Ning, J. L. Ning, Y. L. Zhu, M. Kothakonda, C. Q. Xu, X. L. Ke, J. W. Sun, W. W. Xie, S. L. Yang, and Z. Q. Mao, “Ferromagnetic ${\mathrm{mnbi}}_{4}{\mathrm{te}}_{7}$ obtained with low-concentration sb doping: A promising platform for exploring topological quantum states,” Phys. Rev. Mater. 6, 054203 (2022).
  * Hu  _et al._ (2021) Chaowei Hu, Shang-Wei Lien, Erxi Feng, Scott Mackey, Hung-Ju Tien, Igor I Mazin, Huibo Cao, Tay-Rong Chang, and Ni Ni, “Tuning magnetism and band topology through antisite defects in sb-doped mnbi 4 te 7,” Physical Review B 104, 054422 (2021).
  * Eckberg  _et al._ (2023) Christopher Eckberg, Gang Qiu, Tao Qu, Sohee Kwon, Yuhang Liu, Lixuan Tai, David Graf, Su Kong Chong, Peng Zhang, Kin L Wong, _et al._ , “Structural tuning magnetism and topology in a magnetic topological insulator,” arXiv preprint arXiv:2301.03078 (2023).
  * Chen  _et al._ (2019) K. Y. Chen, B. S. Wang, J.-Q. Yan, D. S. Parker, J.-S. Zhou, Y. Uwatoko, and J.-G. Cheng, “Suppression of the antiferromagnetic metallic state in the pressurized $\mathrm{MnB}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{4}$ single crystal,” Phys. Rev. Mater. 3, 094201 (2019).
  * Mazza  _et al._ (2022) Alessandro R Mazza, Jason Lapano, Harry M MeyerIII, Christopher T Nelson, Tyler Smith, Yun-Yi Pai, Kyle Noordhoek, Benjamin J Lawrie, Timothy R Charlton, Robert G Moore, _et al._ , “Surface-driven evolution of the anomalous hall effect in magnetic topological insulator mnbi2te4 thin films,” Advanced Functional Materials 32, 2202234 (2022).
  * Yao  _et al._ (2021) Xiong Yao, Hee Taek Yi, Deepti Jain, Myung-Geun Han, and Seongshik Oh, “Spacer-layer-tunable magnetism and high-field topological hall effect in topological insulator heterostructures,” Nano Letters 21, 5914–5919 (2021).
  * Hu  _et al._ (2020) Chaowei Hu, Lei Ding, Kyle N Gordon, Barun Ghosh, Hung-Ju Tien, Haoxiang Li, A Garrison Linn, Shang-Wei Lien, Cheng-Yi Huang, Scott Mackey, _et al._ , “Realization of an intrinsic ferromagnetic topological state in mnbi8te13,” Science advances 6, eaba4275 (2020).
  * Zhong  _et al._ (2021) Haoyuan Zhong, Changhua Bao, Huan Wang, Jiaheng Li, Zichen Yin, Yong Xu, Wenhui Duan, Tian-Long Xia, and Shuyun Zhou, “Light-tunable surface state and hybridization gap in magnetic topological insulator mnbi8te13,” Nano Letters 21, 6080–6086 (2021).
  * Jiang  _et al._ (2020) Jue Jiang, Di Xiao, Fei Wang, Jae-Ho Shin, Domenico Andreoli, Jianxiao Zhang, Run Xiao, Yi-Fan Zhao, Morteza Kayyalha, Ling Zhang, _et al._ , “Concurrence of quantum anomalous hall and topological hall effects in magnetic topological insulator sandwich heterostructures,” Nature Materials 19, 732–737 (2020).
  * Mogi  _et al._ (2022) M Mogi, Y Okamura, M Kawamura, R Yoshimi, K Yasuda, A Tsukazaki, KS Takahashi, T Morimoto, N Nagaosa, M Kawasaki, _et al._ , “Experimental signature of the parity anomaly in a semi-magnetic topological insulator,” Nature Physics 18, 390–394 (2022).
  * Yasuda  _et al._ (2016) K Yasuda, R Wakatsuki, T Morimoto, R Yoshimi, A Tsukazaki, KS Takahashi, M Ezawa, M Kawasaki, N Nagaosa, and Y Tokura, “Geometric hall effects in topological insulator heterostructures,” Nature Physics 12, 555–559 (2016).
  * Xu  _et al._ (2022) Yishuai Xu, Zhuoliang Ni, Yizhou Liu, Brenden R Ortiz, Qinwen Deng, Stephen D Wilson, Binghai Yan, Leon Balents, and Liang Wu, “Three-state nematicity and magneto-optical kerr effect in the charge density waves in kagome superconductors,” Nature physics 18, 1470–1475 (2022).
  * McCord (2015) Jeffrey McCord, “Progress in magnetic domain observation by advanced magneto-optical microscopy,” Journal of Physics D: Applied Physics 48, 333001 (2015).
  * Gong  _et al._ (2019) Yan Gong, Jingwen Guo, Jiaheng Li, Kejing Zhu, Menghan Liao, Xiaozhi Liu, Qinghua Zhang, Lin Gu, Lin Tang, Xiao Feng, _et al._ , “Experimental realization of an intrinsic magnetic topological insulator,” Chinese Physics Letters 36, 076801 (2019).
  * Zhao  _et al._ (2021) Yi-Fan Zhao, Ling-Jie Zhou, Fei Wang, Guang Wang, Tiancheng Song, Dmitry Ovchinnikov, Hemian Yi, Ruobing Mei, Ke Wang, Moses H. W. Chan, Chao-Xing Liu, Xiaodong Xu, and Cui-Zu Chang, “Even–odd layer-dependent anomalous hall effect in topological magnet mnbi2te4 thin films,” Nano Letters 21, 7691–7698 (2021), pMID: 34468149, https://doi.org/10.1021/acs.nanolett.1c02493 .
  * Liu  _et al._ (2021) Shanshan Liu, Jiexiang Yu, Enze Zhang, Zihan Li, Qiang Sun, Yong Zhang, Lun Li, Minhao Zhao, Pengliang Leng, Xiangyu Cao, _et al._ , “Gate-tunable intrinsic anomalous hall effect in epitaxial mnbi2te4 films,” arXiv preprint arXiv:2110.00540 (2021).
  * Liu  _et al._ (2022) N. Liu, S. Schreyeck, K.M. Fijalkowski, M. Kamp, K. Brunner, C. Gould, and L.W. Molenkamp, “Antiferromagnetic order in mnbi2te4 films grown on si(1 1 1) by molecular beam epitaxy,” Journal of Crystal Growth 591, 126677 (2022).
  * Yan  _et al._ (2022) Chenhui Yan, Yanglin Zhu, Leixin Miao, Sebastian Fernandez-Mulligan, Emanuel Green, Ruobing Mei, Hengxin Tan, Binghai Yan, Chao-Xing Liu, Nasim Alem, Zhiqiang Mao, and Shuolong Yang, “Delicate ferromagnetism in mnbi6te10,” Nano Letters 22, 9815–9822 (2022), pMID: 36315185, https://doi.org/10.1021/acs.nanolett.2c02500 .
  * Okada  _et al._ (2016) Ken N Okada, Youtarou Takahashi, Masataka Mogi, Ryutaro Yoshimi, Atsushi Tsukazaki, Kei S Takahashi, Naoki Ogawa, Masashi Kawasaki, and Yoshinori Tokura, “Terahertz spectroscopy on faraday and kerr rotations in a quantum anomalous hall state,” Nature communications 7, 1–6 (2016).
  * Han  _et al._ (2022) Xingyue Han, Anastasios Markou, Jonathan Stensberg, Yan Sun, Claudia Felser, and Liang Wu, “Giant intrinsic anomalous terahertz faraday rotation in the magnetic weyl semimetal ${\mathrm{co}}_{2}\mathrm{MnGa}$ at room temperature,” Phys. Rev. B 105, 174406 (2022).
  * Lee  _et al._ (2015) Inhee Lee, Chung Koo Kim, Jinho Lee, Simon JL Billinge, Ruidan Zhong, John A Schneeloch, Tiansheng Liu, Tonica Valla, John M Tranquada, Genda Gu, _et al._ , “Imaging dirac-mass disorder from magnetic dopant atoms in the ferromagnetic topological insulator crx (bi0. 1sb0. 9) 2-xte3,” Proceedings of the National Academy of Sciences 112, 1316–1321 (2015).
  * Chen  _et al._ (2010) YL Chen, J-H Chu, JG Analytis, ZK Liu, Kyushiro Igarashi, H-H Kuo, XL Qi, Sung-Kwan Mo, RG Moore, DH Lu, _et al._ , “Massive dirac fermion on the surface of a magnetically doped topological insulator,” Science 329, 659–662 (2010).
  * Wu  _et al._ (2016) Liang Wu, M Salehi, N Koirala, J Moon, S Oh, and NP Armitage, “Quantized faraday and kerr rotation and axion electrodynamics of a 3d topological insulator,” Science 354, 1124–1127 (2016).
  * Wu  _et al._ (2015) Liang Wu, Wang-Kong Tse, M Brahlek, CM Morris, R Valdés Aguilar, N Koirala, S Oh, and NP Armitage, “High-resolution faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator cu 0.02 bi 2 se 3,” Physical review letters 115, 217602 (2015).
  * Li  _et al._ (2016) W Li, M Claassen, Cui-Zu Chang, B Moritz, T Jia, C Zhang, S Rebec, JJ Lee, M Hashimoto, D-H Lu, _et al._ , “Origin of the low critical observing temperature of the quantum anomalous hall effect in v-doped (bi, sb) 2te3 film,” Scientific reports 6, 32732 (2016).
  * Pan  _et al._ (2020) Lei Pan, Xiaoyang Liu, Qing Lin He, Alexander Stern, Gen Yin, Xiaoyu Che, Qiming Shao, Peng Zhang, Peng Deng, Chao-Yao Yang, _et al._ , “Probing the low-temperature limit of the quantum anomalous hall effect,” Science advances 6, eaaz3595 (2020).
  * Lachman  _et al._ (2015) Ella O Lachman, Andrea F Young, Anthony Richardella, Jo Cuppens, HR Naren, Yonathan Anahory, Alexander Y Meltzer, Abhinav Kandala, Susan Kempinger, Yuri Myasoedov, _et al._ , “Visualization of superparamagnetic dynamics in magnetic topological insulators,” Science advances 1, e1500740 (2015).