⇦Chinese Physics Letters, 2021, Vol. 38, No. 7, Article code 077102 Molecular Beam Epitaxy Growth and Electronic Structures of Monolayer GdTe$_{3}$ Zhilin Xu (徐智临)1, Shuai-Hua Ji (季帅华)1,2, Lin Tang (唐林)1, Jian Wu (吴健)1,2, Na Li (李娜)3*, Xinqiang Cai (蔡新强)1*, and Xi Chen (陈曦)1,2 Affiliations 1State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 2Frontier Science Center for Quantum Information, Beijing 100084, China 3Key Laboratory for the Physics and Chemistry of Nanodevices, Department of Electronics, Peking University, Beijing 100871, China Received 1 April 2021; accepted 6 May 2021; published online 3 July 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11934001 and 11874233), and the Science Challenge Project (Grant No. TZ2016004).
*Corresponding authors. Email: na-li07@pku.edu.cn; cxq15@tsinghua.org.cn Citation Text: Xu Z L, Ji S H, Tang L, Wu J, and Li N et al. 2021 Chin. Phys. Lett. 38 077102    Abstract GdTe$_{3}$ is a layered antiferromagnetic (AFM) metal with charge density wave (CDW). We grew monolayer (ML) GdTe$_{3}$ on graphene/6H-SiC(0001) substrates by molecular beam epitaxy. The electronic and magnetic structures are studied by scanning tunneling microscopy/spectroscopy, quasi-particle interference (QPI) and first-principles calculations. Strong evidence of CDW persisting at the two-dimensional (2D) limit is found. Band dispersions and partially gapped energy bands near the Fermi surface are revealed by the QPI patterns. By density functional theory $+ U$ calculations, AFM order with stripe pattern is found to be the magnetic ground state for ML GdTe$_{3}$. These results provide fundamental understanding and pave the way for further investigation of GdTe$_{3}$ at the 2D limit. DOI:10.1088/0256-307X/38/7/077102 © 2021 Chinese Physics Society Article Text Long range orders (LROs) in two-dimensional (2D) systems are usually suggested to be fragile due to thermal fluctuation. Nevertheless, experiments in recent years have convincingly demonstrated various 2D LROs, such as crystal lattice, charge density wave (CDW), and ferromagnetism and antiferromagnetism. The CDW order has been observed in 2D systems, such as transition metal dichalcogenides.[1–4] A CDW is a periodic distribution of charge-carrier density with an accompanying distortion in lattice. Nesting of the band structure near the Fermi surface (FS) and the electron-phonon coupling usually play a crucial role in the formation of a CDW. The 2D magnetic order has also been well established recently in monolayer (ML) van der Waals (vdW) materials.[5–13] Transition metal elements with 3$d$ electrons have been the essential constituents for most of the materials showing 2D CDW and magnetic phases. By contrast, 2D compounds containing rare earth metals with $4f$ electrons are rarely reported. Here we focus on the rare-earth tritellurides (RTe$_{3}$, R = La,…, Nd, Sm, and Gd,…, Tm). Bulk GdTe$_{3}$ belongs to the layered RTe$_{3}$ group with incommensurate CDW[14,15] and various magnetic orders.[16,17] The CDW and antiferromagnetic (AFM) orders are expected to persist at the 2D limit, providing a new platform to study 2D LROs. GdTe$_{3}$ crystallizes into layered orthorhombic structure (space group: Bmmb) above the CDW transition temperature. The Te atoms form a double square lattice in the ${ac}$ plane separated by double-corrugated GdTe slabs and there exists a vdW gap between the two neighboring Te sheets, as illustrated in Fig. 1(a). CDW establishes below $\sim $379 K,[18,19] and the AFM order below the Néel temperature ($T_{\rm N} = 12.0$ K,[18] the highest among RTe$_{3} $[16,17]). Under high pressure, a superconducting phase in GdTe$_{3}$ emerges.[20] Therefore, GdTe$_{3}$ at the 2D limit is also a promising material to investigate the interplay between CDW, AFM and superconducting orders, a fascinating topic in unconventional superconductivity. In principle, GdTe$_{3}$ flakes can be prepared by mechanical exfoliation. Recently, vdW GdTe$_{3}$ flakes as thin as 22 nm has been reported as an antiferromagnetic metal with rather high mobility.[18] However, ML GdTe$_{3}$ is still not achieved. Whether the CDW and AFM orders can persist at the 2D limit is unclear yet. In this work, we successfully prepare ML GdTe$_{3}$ by molecular beam epitaxy (MBE). CDW and AFM orders are then studied. We grew ML GdTe$_{3}$ via MBE on single or bilayer graphene prepared on a nitrogen-doped 6H-SiC(0001) substrate (resistivity $\sim $0.1 $\Omega$$\cdot$cm).[21] The base pressure of the MBE system is $\sim $$2\times 10^{-10}$ torr. Gadolinium foils (Alfa, 99.9%) and Te shots (Alfa, 99.9999%) were introduced into the Knudsen cells on vacuum chamber and degassed before the MBE growth. ML GdTe$_{3}$ was prepared by co-evaporating Gd at 1150 ℃ and Te at 265 ℃. The vacuum was better than $2.5 \times 10^{-9}$ torr during the growth. The nominal flux of tellurium is much higher than that of gadolinium. The growth rate is about 0.1 ML/min. The temperature window of the substrate is between 220 ℃ and 250 ℃. Higher temperature leads to the formation of other phases while lower temperature reduces the quality of the films. After the growth, samples were transferred to the low-temperature scanning tunneling microscope (STM, 4.4 K) from the growth chamber without breaking the vacuum. The crystal and electronic structures of the ML GdTe$_{3}$ were studied by STM. Before imaging, the Pr–Ir alloy tips were checked and modified on the surface of Ag (111) islands. Figure 1(b) shows the STM topography of a typical sample. The apparent height between the first layer and graphene is about 1.2 nm [see the red line in Fig. 1(c)]. Such a height is about the thickness of one ML GdTe$_{3}$ (half of a unit cell). In Fig. 1(d), the atomic resolution of ML GdTe$_{3}$ is achieved with sample bias at 0.1 V. The in-plane lattice constants are $a=c = 4.35 \pm 0.05$ Å, which agree well with the previous result.[18] CDW features with stripe pattern can be revealed in Fig. 1(d) and the inset, a strong evidence for the persistence of CDW state in GdTe$_{3}$ at 2D limit. From the fast Fourier transformation (FFT) [see the top right inset of Fig. 1(d)] and the two cross-sectional line-cuts [see Fig. 1(e)], the CDW wave vector ${\boldsymbol q}_{{\rm CDW}}$ is estimated to be $(0.27 \pm 0.01){\boldsymbol q}_{\rm atom}$, slightly smaller than those in Refs. [18,22]. It should be pointed out that the step height between the second layer and the ML GdTe$_{3}$ is about 0.9 nm, close to the thickness of one unit cell GdTe$_{2}$.[23] Thus, the second layer is no longer GdTe$_{3}$ but GdTe$_{2}$. The physical properties of ML Gd$_{2}$Te$_{5}$ (ML GdTe$_{2}$ on ML GdTe$_{3}$) will be discussed elsewhere.
cpl-38-7-077102-fig1.png
Fig. 1. (a) Crystal structure of bulk GdTe$_{3}$. (b) STM topography of ML GdTe$_{3}$ films grown on graphene (bias 3 V, current 20 pA). ML Gd$_{2}$Te$_{5}$ also forms due to the formation of ML GdTe$_{2}$ on ML GdTe$_{3}$. (c) Step height along the lines indicated in (b). (d) Atomic resolution and CDW of ML GdTe$_{3}$ (bias 0.1 V, current 200 pA). The top left is the STM image measured at bias 1 V, current 200 pA. The top right inset shows the FFT image. (e) Two cross-sectional line-cuts [indicated as the dashed lines in the right inset of (d)] of the FFT image. CDW wave vector is about ${\boldsymbol q}_{\rm CDW} \approx 0.27 {\boldsymbol q}_{\rm atom}$.
The CDW wave vector, CDW gap and band structures of ML GdTe$_{3}$ can be analyzed by the first-principles calculations, scanning tunneling spectroscopy (STS) and quasi-particle interference (QPI). The calculations based on DFT $+ U$ method were performed with the Vienna ab initio simulation package (VASP).[24,25] In the calculations, we employed a projector-augmented wave pseudopotential[26] and Perdew–Burke–Ernzerhof exchange-correlation functional (GGA).[27] The vacuum space was set as 18 Å and the energy cutoff was 350 eV. We used a $k$-mesh of $11 \times 1\times 11$ for primitive cell self-consistent calculations $4 \times 1\times 4$ for $2 \times 1\times 2$ supercell with AFM order. The convergence condition of the electronic self-consistent loop was 10$^{-5}$ eV. The strong correlation of 4$f$-orbital of Gd was treated with DFT $+ U$ formulism.[28] The effective $U$ value was chosen as 6 eV, as the same as that in the calculation of GdI$_{2}$.[29] The STS ($dI/dV$) spectra are proportional to the local density of states (LDOS) of a sample. In our experiment, the spectra were obtained through a lock-in amplifier with bias modulation of 5 mV (or 1 mV) at 913 Hz. The QPI patterns are the spatial mapping of the intensity of $dI/dV$ at fixed sample voltage (i.e., fixed energy). It is related to the interferences of all possible scattering processes for the states on a contour of constant energy (CCE).[30,31] The FFT of a QPI pattern can be used to unveil the band dispersion and the CDW gapped band.
cpl-38-7-077102-fig2.png
Fig. 2. (a) Calculated FS of ML GdTe$_{3}$. The arrow is the CDW wave vector. (b) STS of ML GdTe$_{3}$ (set point: 0.5 V, 200 pA). Inset shows the STS near Fermi energy $E_{\rm F}$ (set point: 0.1 V, 200 pA).
Not only Fermi surface (FS) nesting but also electron-phonon coupling are believed to be the mechanisms of CDW order in RTe$_{3}$. The CDW wave vector can be related to the FS topology.[14,15,32–35] As shown in Fig. 2(a), the calculated FS of ML GdTe$_{3}$ is similar to that of the bulk RTe$_{3} $.[14,32] CDW wave vector ${\boldsymbol q}_{{\rm CDW}}$ from experiment is illustrated in Fig. 2(a) by an arrow. It connects two almost parallel FS parts. The $dI/dV$ spectrum with weak gap feature is shown in Fig. 2(b). The arrows in Fig. 2(b) mark the estimation of the gap size $2\varDelta_{\rm CDW} \approx 0.42$ eV, reproduced from bulk GdTe$_{3}$.[18] Nonzero LDOS inside the CDW gap means that the FS is partially gapped. Unlike in bulk GdTe$_{3}$, a dip exists at the FS [see the inset of Fig. 2(b)]. The spatial distribution of the dip is not uniform (not shown). The origin of the dip needs further investigation.
cpl-38-7-077102-fig3.png
Fig. 3. (a)–(j) Typical QPI patterns with the corresponding FFT images below. The QPIs are recorded when scanning STM images at 200 pA with corresponding sample bias voltages, except $E_{\rm F}$. The QPI at 0 V is recorded at constant height mode when scanning. (k) CCE for $E_{\rm F} + 0.5$ eV by calculations. The black arrow indicates an observed QPI wave vector. (l) Calculated FS with estimated gapped area illustrated by shadow. The black arrows indicate possible QPI wave vectors observed around $E_{\rm F}$. (m) The band dispersion of ML GdTe$_{3}$. Blue dots with error bars are extracted from QPI patterns. The wave vectors are illustrated in (f) and (k). The red curve is the fitting result of experimental data by tight binding model. The gray curves are the band dispersion by DFT calculations.
By QPI measurement, more evidence for the CDW gap can be provided, and band structures of ML GdTe$_{3}$ in the CDW states can be unveiled and compared with calculations. We obtain the QPI patterns with sample bias voltage between 1.0 V and $-0.5$ V at interval of 0.1 V. Five selected patterns with FFTs are shown in Figs. 3(a)–3(j). It should be noted that QPI patterns are dominated by the stationary wave vectors connecting two ${\boldsymbol k}$ points on CCE.[30,31] For sample bias outside the CDW gap, the stationary wave vector is along the $\varGamma$–$M$ direction [see black arrow in Fig. 3(k)]. Due to such surrounding wave vectors, a squared pattern forms in the FFT images [see Figs. 3(f) and 3(j) for typical results]. As illustrated in Fig. 3(k), the QPI wave vector is related to the crystal wave vector ${\boldsymbol k}$ as QPI = 2${\boldsymbol k}$. Thus, QPI wave vectors can be used to identify the band dispersion of ML GdTe$_{3}$. The experimental results are shown in Fig. 3(m) by blue dots with error bars. The $\epsilon ({\boldsymbol k})$ can be fitted based on the tight binding model as follows:[32,33] $$ \epsilon ({\boldsymbol k})= E_{0}-2t_{\rm para}\Big[\cos\Big(\frac{1}{\sqrt 2 }ka\Big)\Big].~~ \tag {1} $$ The fitting results are $E_{0} = -2.2$ eV and $t_{\rm para} = -2.0$ eV. For comparison, the calculated band dispersions (gray curves) are shown in Fig. 3(m), and agree well with the experimental results. One of the calculated bands originates only from the Te $p_{x}$- and $p_{y}$-orbitals while the other has additional contribution from the hybridization between the Te $p_{z}$-orbital and the Gd $d$-orbital. The small difference between these two dispersions cannot be distinguished by QPI. The partially gapped energy bands near $E_{\rm F}$ can be identified by QPI, expected from the features of CDW gap. As shown in Figs. 3(b)–3(d) and 3(g)–3(i), the squared pattern in FFT images outside the CDW gap disappears. The FFT of QPI patterns no longer shows $C_{4}$ rotation symmetry and such symmetry breaks into $C_{2}$. Based on the calculations without considering CDW order, the CCEs of energy bands share similar shapes [see Figs. 3(k) and 3(l) as typical examples] and corresponding stationary wave vectors. Thus, the disappearance of squared pattern and the symmetry breaking are related to the partially gapped energy band, estimated by the shadow in Fig. 3(l) for FS. Based on the estimation, the scattering processes around the wave vectors of QPI should be suppressed, and the stationary wave vectors now locate mainly around ${\boldsymbol Q}_{1}$ and ${\boldsymbol Q}_{2}$. ${\boldsymbol Q}_{1}$ can be seen clearly when bias is 0.1 V [see Figs. 3(b) and 3(g)] as three X patterns along the $\varGamma$–$Z$ direction in the FFT image. ${\boldsymbol Q}_{2}$ is clearer when the bias is $-0.1$ V [see Figs. 3(d) and 3(i)]. The asymmetric QPI patterns above and below $E_{\rm F}$ indicate that the gapped parts of the band differ in CCEs for different energies. One more notable feature is that both ${\boldsymbol Q}_{1}$ and ${\boldsymbol Q}_{2}$ are comparably weaker when the bias is set to 0 V [see Figs. 3(c) and 3(h)], implying that the density of states is more strongly suppressed on FS by CDW. Considering the difficulty of probing the AFM order of the ML GdTe$_{3}$ experimentally, spin-polarized calculations with collinear configurations were carried out. The band unfolding technique[36] was also introduced to recover an effective primitive cell picture of ML GdTe$_{3}$ band structure with different magnetic structures. We used the VASPKIT code for postprocessing of all VASP calculated data.[37]
cpl-38-7-077102-fig4.png
Fig. 4. (a)–(d) Top views of different spin configurations: the intralayer AFM (a), interlayer AFM (b), stripe AFM (c), FM (d) with only the Gd ions shown. Larger circles sit on the upper layer and the smaller on lower layer. Up/down spins are represented by arrows. (e)–(g) Unfolded bands of different AFM magnetic structures, sizes of data points corresponding to the strength of bands.
The ground state energies of ML GdTe$_{3}$ with four kinds of magnetic orders are calculated by DFT $+ U$. Figures 4(a)–4(d) show the magnetic structures of intralayer AFM, interlayer AFM, striped AFM, and FM, respectively. The ground state energy per Gd atom of each magnetic structure is shown as follows: $$\begin{alignat}{1} E_{\rm intra}=E_{0}-2J_{1}+2J_{3},~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} E_{\rm inter}=E_{0}+2J_{1}-2J_{2}+2J_{3},~~ \tag {3} \end{alignat} $$ $$\begin{alignat}{1} E_{\rm stripe}=E_{0}-2J_{3},~~ \tag {4} \end{alignat} $$ $$\begin{alignat}{1} E_{\rm FM}=E_{0}+2J_{1}+2J_{2}+2J_{3},~~ \tag {5} \end{alignat} $$ where $J_{1}$, $J_{2}$ and $J_{3}$ are the Heisenberg exchange interactions [see colored lines in Fig. 4(a)] while the energy $E_{0}$ is independent of magnetism.
Table 1. Ground energies of intralayer AFM, interlayer AFM, striped AFM, FM magnetic structure per Gd atom with different effective Coulomb interaction $U$.
Effective $U$ (eV) Energy per Gd atom (eV)
intra inter striped FM
4 $-27.1260$ $-27.1250$ $-27.1301$ $-27.1250$
5 $-27.1022$ $-27.1008$ $-27.1065$ $-27.1003$
6 $-27.0866$ $-27.0849$ $-27.0911$ $-27.0839$
7 $-27.0786$ $-27.0764$ $-27.0832$ $-27.0748$
8 $-27.0778$ $-27.0752$ $-27.0826$ $-27.0729$
The striped AFM state as the true ground state is unveiled by comparing the calculated ground state energy. As shown in Table 1, for rational $U$ varying from 4 eV to 8 eV, the striped AFM state keeps the lowest energy. The Heisenberg exchange interactions with different $U$ in Table 2 show that the third neighbor interaction $J_{3}$ is much larger than the others, leading to the lowest energy of the striped AFM state. The exchange interactions $J_{1}$, $J_{2}$ and $J_{3}$ originate mainly from interactions with itinerant electrons via the RKKY mechanism.[38] In such a situation, larger exchange interactions can locate at larger Gd–Gd distance, such as in GdPtBi.[39]
Table 2. Nearest neighbor, next nearest neighbor and third neighbor Heisenberg interactions with different Coulomb interaction $U$.
Effective $U$ (eV) Heisenberg interactions
$J_{1}$ (meV) $J_{2}$ (meV) $J_{3}$ (meV)
4 0.2486 $-0.0097$ 1.1638
5 0.3977 0.1221 1.2886
6 0.5614 0.2543 1.4106
7 0.7452 0.4040 1.5355
8 0.9494 0.5673 1.6662
We use the band unfolding technique to determine the perturbation caused by different AFM magnetic structures. Figures 4(e)–4(g) show the band structures of ML GdTe$_{3}$ with intralayer AFM, interlayer AFM, and striped AFM order. The interlayer AFM order has a four-fold symmetry, which is higher than the two-fold symmetry in intralayer AFM and striped AFM order. In that case, band structures of intralayer AFM and striped AFM order are more complicated. Compared to the band structures in Fig. 4(f), additional bands can be found in Figs. 4(e)–4(g). Those additional bands are much weaker than the “original” bands which have the same dispersions as band structures of interlayer AFM order in Fig. 4(f). Thus, the different magnetic structures make small difference in band structures, causing little impact on the CDW of ML GdTe$_{3}$. In summary, we have successfully grown ML GdTe$_{3}$ by MBE. The CDW order should persist at the 2D limit. QPI patterns show distinct features outside and inside the CDW gap, well understood by the gapped energy band near $E_{\rm F}$. The band dispersions revealed by QPI agree well with calculations. Calculations also show striped AFM order as the ground magnetic state. We thank Jiaheng Li and Yang Li for valuable discussion. References 2D transition metal dichalcogenidesStrongly enhanced charge-density-wave order in monolayer NbSe2Gate-tunable phase transitions in thin flakes of 1T-TaS2Ultrathin Nanosheets of Vanadium Diselenide: A Metallic Two-Dimensional Material with Ferromagnetic Charge-Density-Wave BehaviorIsing-Type Magnetic Ordering in Atomically Thin FePS 3Raman spectroscopy of atomically thin two-dimensional magnetic iron phosphorus trisulfide (FePS 3 ) crystalsDiscovery of intrinsic ferromagnetism in two-dimensional van der Waals crystalsLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limitMagnon-assisted tunnelling in van der Waals heterostructures based on CrBr3Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substratesRoom Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer LimitEnhancement of interlayer exchange in an ultrathin two-dimensional magnetFermi surface nesting and charge-density wave formation in rare-earth tritelluridesFermi surface nesting and the origin of charge density waves in metalsMagnetic properties of rare-earth metal tritellurides R Te 3 ( R = C e , P r , N d , G d , D y ) Magnetic properties of the charge density wave compounds R Te 3 ( R = Y , La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm)High mobility in a van der Waals layered antiferromagnetic metalEffect of chemical pressure on the charge density wave transition in rare-earth tritellurides R Te 3 Pressure dependence of the charge-density-wave and superconducting states in GdTe 3 ,   TbTe 3 , and DyTe 3 The growth and morphology of epitaxial multilayer grapheneDivergence in the Behavior of the Charge Density Wave in RE Te 3 ( RE = Rare-Earth Element) with Temperature and RE ElementThe Nonstoichiometry of Gadolinium DitellurideAb initio molecular dynamics for liquid metalsEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setProjector augmented-wave methodGeneralized Gradient Approximation Made SimpleFirst-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA + U methodQuantum Anomalous Hall Effect in Magnetic Insulator HeterostructurePower-law decay of standing waves on the surface of topological insulatorsStationary phase approximation approach to the quasiparticle interference on the surface of a strong topological insulatorAngle-resolved photoemission study of the evolution of band structure and charge density wave properties in R Te 3 ( R = Y , La, Ce, Sm, Gd, Tb, and Dy)Theory of stripes in quasi-two-dimensional rare-earth telluridesAlternative route to charge density wave formation in multiband systemsWave-vector-dependent electron-phonon coupling and the charge-density-wave transition in TbT e 3 Unfolding First-Principles Band StructuresVASPKIT: A User-friendly Interface Facilitating High-throughput Computing and Analysis Using VASP CodeNote on the Interactions between the Spins of Magnetic Ions or Nuclei in MetalsLong-range magnetic interaction and frustration in double perovskites Sr2NiIrO6 and Sr2ZnIrO6
[1] Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V, and Kis A 2017 Nat. Rev. Mater. 2 17033
[2] Xi X, Zhao L, Wang Z, Berger H, Forró L, Shan J, and Mak K F 2015 Nat. Nanotechnol. 10 765
[3] Yu Y, Yang F, Lu X F, Yan Y J, Cho Y H, Ma L, Niu X, Kim S, Son Y W, Feng D, Li S, Chen X H, and Zhang Y 2015 Nat. Nanotechnol. 10 270
[4] Xu K, Chen P, Li X, Wu C, Guo Y, Zhao J, Wu X, and Xie Y 2013 Angew. Chem. 125 10671
[5] Lee J U, Lee S, Ryoo J H, Kang S, Kim T Y, Kim P, Park C H, Park J G, and Cheong H 2016 Nano Lett. 16 7433
[6] Wang X, Du K, L, Y Y F, Hu P, Zhang J, Zhang Q, Owen M H S, Lu X, Gan C K, Sengupta P, Kloc C, and Xiong Q 2016 2D Mater. 3 031009
[7] Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, and Zhang X 2017 Nature 546 265
[8] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P, and Xu X 2017 Nature 546 270
[9] Ghazaryan D, Greenaway M T, Wang Z et al. 2018 Nat. Electron. 1 344
[10] Deng Y, Yu Y, Song Y, Zhang J, Wang N Z, Sun Z, Yi Y, Wu Y Z, Wu S, Zhu J, Wang J, Chen X H, and Zhang Y 2018 Nature 563 94
[11] Bonilla M, Kolekar S, Ma Y, Diaz H C, Kalappattil V, Das R, Eggers T, Gutierrez H R, Phan M H, and Batzill M 2018 Nat. Nanotechnol. 13 289
[12] O'Hara D J, Zhu T, Trout A H, Ahmed A S, Luo Y K, Lee C H, Brenner M R, Rajan S, Gupta J A, McComb D W, and Kawakami R K 2018 Nano Lett. 18 3125
[13] Klein D R, MacNeill D, Song Q, Larson D T, Fang S, Xu M, Ribeiro R A, Canfield P C, Kaxiras E, Comin R, and Jarillo-Herrero P 2019 Nat. Phys. 15 1255
[14] Laverock J, Dugdale S B, Major Z, Alam M A, Ru N, Fisher I R, Santi G, and Bruno E 2005 Phys. Rev. B 71 085114
[15] Johannes M D and Mazin I I 2008 Phys. Rev. B 77 165135
[16] Iyeiri Y, Okumura T, Michioka C, and Suzuki K 2003 Phys. Rev. B 67 144417
[17] Ru N, Chu J H, and Fisher I R 2008 Phys. Rev. B 78 012410
[18] Lei S, Lin J, Jia Y et al. 2020 Sci. Adv. 6 eaay6407
[19] Ru N, Condron C L, Margulis G Y, Shin K Y, Laverock J, Dugdale S B, Toney M F, and Fisher I R 2008 Phys. Rev. B 77 035114
[20] Zocco D A, Hamlin J J, Grube K, Chu J H, Kuo H H, Fisher I R, and Maple M B 2015 Phys. Rev. B 91 205114
[21] Hass J, de Heer W A, and Conrad E H 2008 J. Phys.: Condens. Matter 20 323202
[22] Malliakas C D and Kanatzidis M G 2006 J. Am. Chem. Soc. 128 12612
[23] Niihara K, Shishido T, and Yajima S 1972 Bull. Chem. Soc. Jpn. 45 1239
[24] Kresse G and Hafner J 1993 Phys. Rev. B 47 558(R)
[25] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[26] Blöchl P E 1994 Phys. Rev. B 50 17953
[27] Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[28] Anisimov V I, Aryasetiawan F, and Lichtenstein A I 1997 J. Phys.: Condens. Matter 9 767
[29] Xu G, Wang J, Felser C, Qi X L, and Zhang S C 2015 Nano Lett. 15 2019
[30] Wang J, Li W, Cheng P, Song C, Zhang T, Deng P, Chen X, Ma X, He K, Jia J F, Xue Q K, and Zhu B F 2011 Phys. Rev. B 84 235447
[31] Liu Q, Qi X L, and Zhang S C 2012 Phys. Rev. B 85 125314
[32] Brouet V, Yang W L, Zhou X J, Hussain Z, Moore R G, He R, Lu D H, Shen Z X, Laverock J, Dugdale S B, Ru N, and Fisher I R 2008 Phys. Rev. B 77 235104
[33] Yao H, Robertson J A, Kim E A, and Kivelson S A 2006 Phys. Rev. B 74 245126
[34] Eiter H M, Lavagnini M, Hackl R, Nowadnick E A, Kemper A F, Devereaux T P, Chu J H, Analytis J G, Fisher I R, and Degiorgi L 2013 Proc. Natl. Acad. Sci. USA 110 64
[35] Maschek M, Rosenkranz S, Heid R, Said A H, Giraldo-Gallo P, Fisher I R, and Weber F 2015 Phys. Rev. B 91 235146
[36] Ku W, Berlijn T, and Lee C C 2010 Phys. Rev. Lett. 104 216401
[37] Wang V, Xu N, Liu J C, Tang G, and Geng W T 2019 arXiv:1908.08269 [cond-mat.mtrl-sci]
[38] Van Vleck J H 1962 Rev. Mod. Phys. 34 681
[39] Ou X, Li Z, Fan F, Wang H, and Wu H 2014 Sci. Rep. 4 7542