⇦Chinese Physics Letters, 2019, Vol. 36, No. 11, Article code 117303 Experimental Evidence of Topological Surface States in Mg$_{3}$Bi$_{2}$ Films Grown by Molecular Beam Epitaxy * Tong Zhou (周侗)1,2,3,4†, Xie-Gang Zhu (朱燮刚)2,4†, Mingyu Tong (童明玉)5†, Yun Zhang (张云)2,4, Xue-Bing Luo (罗学兵)2,4, Xiangnan Xie (谢向男)1, Wei Feng (冯卫)2,4, Qiuyun Chen (陈秋云)2,4, Shiyong Tan (谭世勇)2,4, Zhen-Yu Wang (王振宇)1,3,7**, Tian Jiang (江天)1,5, Yuhua Tang (唐玉华)1**, Xin-Chun Lai (赖新春)2**, Xuejun Yang (杨学军)1,6 Affiliations 1State Key Laboratory of High Performance Computing, College of Computer, National University of Defense Technology, Changsha 410073 2Science and Technology on Surface Physics and Chemistry Laboratory, Jiangyou 621908 3National Innovation Institute of Defense Technology, Academy of Military Sciences PLA China, Beijing 100010 4Institute of Materials, China Academy of Engineering Physics, Mianyang 621700 5College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073 6Academy of Military Sciences PLA China, Beijing 100010 7Beijing Academy of Quantum Information Sciences, Beijing 100084 Received 14 August 2019, online 21 October 2019 *Supported by the Science Challenge Project under Grant No TZ2016004, the Opening Foundation of State Key Laboratory of High Performance Computing under Grant No 201601-02, the Foundation of President of CAEP under Grant No 201501040, the Natural Science Foundation of Hunan Province under Grant No 2016JJ1021, the National Basic Research Program of China under Grant Nos 2015CB921303 and 2012YQ13012508, the General Program of Beijing Academy of Quantum Information Sciences under Grant No Y18G17, and the Youth Talent Lifting Project under Grant No 17-JCJQ-QT-004.
†Tong Zhou, Xie-Gang Zhu and Mingyu Tong contributed equally to this work.
**Corresponding authors. Email: oscarwang2008@sina.com; yhtang62@163.com; laixinchun@caep.cn Citation Text: Zhou D, Zhu X G, Tong M Y, Zhang Y and Luo X B et al 2019 Chin. Phys. Lett. 36 117303    Abstract Nodal line semimetal (NLS) is a new quantum state hosting one-dimensional closed loops formed by the crossing of two bands. The so-called type-II NLS means that these two crossing bands have the same sign in their slopes along the radial direction of the loop, which requires that the crossing bands are either right-tilted or left-tilted at the same time. According to the theoretical prediction, Mg$_{3}$Bi$_{2}$ is an ideal candidate for studying the type-II NLS by tuning its spin-orbit coupling (SOC). High-quality Mg$_{3}$Bi$_{2}$ films are grown by molecular beam epitaxy (MBE). By in-situ angle resolved photoemission spectroscopy (ARPES), a pair of surface resonance bands around the $\bar{{{\it \Gamma}}}$ point are clearly seen. This shows that Mg$_{3}$Bi$_{2}$ films grown by MBE are Mg(1)-terminated by comparing the ARPES spectra with the first principles calculations results. Moreover, the temperature dependent weak anti-localization effect in Mg$_{3}$Bi$_{2}$ films is observed under magneto-transport measurements, which shows clear two-dimensional (2D) $e$–$e$ scattering characteristics by fitting with the Hikami–Larkin–Nagaoka model. Therefore, by combining with ARPES, magneto-transport measurements and the first principles calculations, this work proves that Mg$_{3}$Bi$_{2}$ is a semimetal with topological surface states. This paves the way for Mg$_{3}$Bi$_{2}$ to be used as an ideal material platform to study the exotic features of type-II nodal line semimetals and the topological phase transition by tuning its SOC. DOI:10.1088/0256-307X/36/11/117303 PACS:73.20.At, 73.20.Fz, 79.60.Bm, 81.10.Aj © 2019 Chinese Physics Society Article Text Topological surface states (TSSs) are a class of novel electronic states with great potential for topological quantum computation and spintronics applications. Materials with TSSs, such as topological insulators and topological semimetals, have emerged in the past decade as a major research focus in condensed matter physics.[1–10] TSSs are immune to non-magnetic disorders. They eliminate 180$^{\circ}$ backscattering under the protection of time-reversal symmetry and they prevent a full gap forming when perturbed due to the odd number of band crossings. Moreover, due to the time-reversal symmetry, there are two closed electrons scattering paths for surface states. One path accumulates a Berry phase of $\pi$ when across single Dirac cone and induces a destructive quantum interference with the other path, leading to the enhancement of the conductivity. This is the origination of weak anti-localization (WAL) effect, which only exists under low magnetic field for the time-reversal symmetry reason.[11,12] Nodal line semimetals (NLSs) are novel proposed topological solid-state phases, which host one-dimensional closed loops or line degeneracies formed by the intersection of two bands. They are proposed as type-II NLSs when the linear spectrum at every point of the nodal line is strongly tilted and tipped over along one transverse direction, which may lead to different magnetic, optical, and transport properties compared with conventional nodal loops. To date, only K$_{4}$P$_{3}$ was theoretically predicted as a type-II NLS, but no candidate material has been experimentally reported. Therefore, it is highly desirable to experimentally realize and study topological materials with a clear type-II NLS signature.[13–15] Here we investigate Mg$_{3}$Bi$_{2}$, which is predicted as owning TSSs and is an ideal material platform for studying type-II NLSs.[14,15] Previous studies of Mg$_{3}$Bi$_{2}$ mainly focused on its good thermoelectric properties as one of Mg$_{3}$Sb$_{2}$-class materials.[16,17] From first principles calculations, Mg$_{3}$Bi$_{2}$ was predicted as a three-dimensional (3D) topological insulator with spin-orbit coupling (SOC) and a type-II NLS without SOC. A topology transition from 3D topological insulator to type-II NLS would be expected by tuning the strength of SOC effect. The previous calculation on Mg$_{3}$Bi$_{2}$ only considered the Mg- and Bi-terminated surface cases, which is insufficient while there are two kinds of Mg atoms distinguished by its structure symmetry.[14,15] Furthermore, ARPES measurements were performed on single crystal Mg$_{3}$Bi$_{2}$ to verify the theoretical calculations.[15] However due to Mg vacancies, Mg$_{3}$Bi$_{2}$ is a heavy p-type material. Although potassium surface doping has been carried out, their ARPES results do not show the conduction bands or the surface states critical crossing point. Thus, direct experimental evidence for the TSSs in Mg$_{3}$Bi$_{2}$ is still lacking.
cpl-36-11-117303-fig1.png
Fig. 1. Characterization of the crystal structure of Mg$_{3}$Bi$_{2}$ film. (a) Crystal structure of Mg$_{3}$Bi$_{2}$. The Mg(1)-Bi-Mg(2)-Bi-Mg(1) quintuple layer (QL) is also schematically shown. (b) Bulk Brillouin zone and the projected surface Brillouin zone of the (001) surface. [(c), (d)] RHEED patterns of Mg$_{3}$Bi$_{2}$ film grown on a graphene/6H-SiC(0001) substrate, with the incident electron beam along the $\bar{\it \Gamma}-\bar{K}$ and $\bar{\it \Gamma}-\bar{M}$ directions, respectively. (e) Spacing of the RHEED streaks in (c) and (d). (f) XRD patterns of the Mg$_{3}$Bi$_{2}$ film, with all the (0, 0, $1n$) diffraction peaks exclusively marked.
To realize novel topological materials-based devices, such as spin transistors, topological quantum computation, or even to achieve the quantum anomalous Hall effect (QAHE), well-controlled thickness of ultrathin films are inevitable.[18–21] In this work, different from Ref. [15] we have grown Mg$_{3}$Bi$_{2}$ films by molecular beam epitaxy (MBE), which allows us to control the thickness of Mg$_{3}$Bi$_{2}$ film layer by layer. MBE also allows us to in-situ dope Sb to tune Mg$_{3}$Bi$_{2}$ SOC, which lay the foundation for studying the exotic properties of type-II NLSs. The in-situ reflection high energy electron diffraction (RHEED) and ex-situ x-ray diffraction (XRD) measurements were performed to confirm the high quality of our Mg$_{3}$Bi$_{2}$ films. First principles calculations and in-situ ARPES measurements were carried out to study the band structure of Mg$_{3}$Bi$_{2}$. Our first-principles calculations on Mg$_{3}$Bi$_{2}$ consider three types of surface terminations, i.e., Mg(1)-, Mg(2)- and Bi-termination, while the previous calculations only calculated Mg- and Bi-terminated surfaces.[15] This sufficient consideration makes us obtain different results from previous work on the termination of Mg$_{3}$Bi$_{2}$. The ARPES spectra are consistent with our first principles calculations, and we confirmed that the natural surface of our films is type-I Mg-terminated whose definition will be given in the following. To further confirm the topology of its surface states, magneto-transport measurements were performed. We have observed temperature dependence of WAL effect in Mg$_{3}$Bi$_{2}$ films under low magnetic field, which shows a clear 2D $e$–$e$ scattering characteristics by fitting with the Hikami–Larkin–Nagaoga (HLN) model. This provides auxiliary evidence on the existence of topological surface states in Mg$_{3}$Bi$_{2}$ films.
cpl-36-11-117303-fig2.png
Fig. 2. Theoretical calculations of Mg$_{3}$Bi$_{2}$. (a) Bulk band structures by first principles calculations. (b) Comparison between the bulk band structures by first principles (solid red lines) and tight-binding method (dashed-black lines). (c)–(e) Band spectra of Bi-, Mg(1)- and Mg(2)-terminated semi-infinite (001) surface slabs. The insets show the detailed electronic structures around the $\bar{\it \Gamma}$ point near Fermi level.
Mg$_{3}$Bi$_{2}$ crystalizes in a layered Kagome lattice structure with a space group of ${P\bar{{3}}m1}$ (No. 164). As shown in Fig. 1(a), the unit cell contains five atomic layers with a stacking sequence of Mg(1)-Bi-Mg(2)-Bi-Mg(1) along the (001) crystallographic orientation, which could be defined as a quintuple layer (QL), similar to that of $A_{2}$B$_{3}$-type topological insulators. Obviously, Mg$_{3}$Bi$_{2}$ owns a centrosymmetric lattice structure. Figure 1(b) depicts the bulk Brillouin zone and the projected surface Brillouin zone (SBZ) of the (001) surface with the high symmetric momentum points marked exclusively. We have grown Mg$_{3}$Bi$_{2}$ single crystalline films on graphene, which was epitaxially grown on a 6H-SiC(0001) substrate. RHEED patterns were collected simultaneously during the growing process, and the sharp streaks along $\bar{\it \Gamma}-\bar{K}$ and $\bar{\it \Gamma}-\bar{M}$ directions in the SBZ indicate the high quality of the film (Figs. 1(c) and 1(d)). XRD clearly revealed the (001), (002), (003), (004) and (005) peaks of Mg$_{3}$Bi$_{2}$ (Fig. 1(f)). The (001) peak intensity is even stronger than that of the SiC(0001) substrate signal, indicating the high crystalline quality of our as-grown samples. The in-plane lattice constant deduced from the RHEED streak spacing (Fig. 1(e)) is 4.677 Å and the out-of-plane lattice constant calculated from XRD patterns is 7.403 Å, which are both consistent with previously reported values.
cpl-36-11-117303-fig3.png
Fig. 3. ARPES measurements on Mg$_{3}$Bi$_{2}$. (a) ARPES spectrum along $\bar{\it \Gamma}-\bar{K}$ direction and (b) the corresponding calculated spectrum with Mg(1)-termination. [(c), (d)] Fermi surface contour from ARPES and calculation. The surface states contribution is marked by black dashed line. (e) ARPES spectrum near Fermi level in which the SRB and S band are more pronounced, the SRB1 and SRB2 are marked by red slash line. (f) The corresponding MDCs to (e). The SRB slope between the energies of $-0.08$ and $-0.22$ eV is obtained by fitting their peak position. The slopes of SRB1 and SRB2 are $C_{1}=4.91$ and $C_{2}=-4.91$, respectively. [(g), (h)] The corresponding calculated spectrum of Mg(1) termination and Bi termination. The slopes of SRB1 and SRB2 are $\pm$5.16 and $\pm$3.41 for Mg(1)- and Bi-terminated surface.
We have theoretically investigated the electronic structures of Mg$_{3}$Bi$_{2}$ by adopting the projector augmented wave (PAW) method as integrated in the VASP package. The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof functional is used for exchange-correction potential. The Brillouin zone is sampled with $11\times 11\times 5 {\it \Gamma}$-centered $k$-mesh for the structural optimization and with $15\times 15\times 11 k$-mesh for self-consistent calculations. SOC effect is fully taken into consideration throughout the whole calculations. The VASP2WANNIER90 interface is used to construct the first principles tight-binding Hamiltonian. Mg $s$ and $p$ and Bi $p$ orbitals are taken into consideration and no maximizing localization procedure is performed to construct the Wannier orbitals.[22–24] The calculations of the surface state spectra and constant energy contours (CECs) are carried out by making use of the WANNIER-TOOLS packages.[25] Since our grown sample is relatively thick, the surface states and Fermi surface are calculated by the tight-binding based Green function method for the semi-infinite sample. The bulk band structures are well reproduced by the tight-binding Hamiltonian, as shown in Figs. 2(a) and 2(b). In principle, there are three different possible terminations, i.e., Bi-, Mg(1)- and Mg(2)-terminated surfaces for the (001) surface of Mg$_{3}$Bi$_{2}$, the band spectra of which are demonstrated in Figs. 2(c)–2(e), respectively. From the bulk band spectra, Mg$_{3}$Bi$_{2}$ is a semi-metal with band 'overlapping' at $\bar{\it \Gamma}$ and $\bar{M}$. The non-trivial topological nature of the surface states around $\bar{\it \Gamma}$ point near the Fermi level has been thoroughly discussed.[15] Therefore, theoretically, Mg$_{3}$Bi$_{2}$ is a semi-metal with non-trivial topological surface states, similar to the semi-metal antimony element.[26] The TSS on the Bi-terminated surface possesses a Dirac point (DP) at $\bar{\it \Gamma}$ inside the local energy gap, while the DPs for the Mg-terminated surfaces are buried in the valence bands. Apart from the TSSs, there are a pair of surface resonance bands (SRBs) around $\bar{\it \Gamma}$, with different binding energies for the three different terminations. For the Bi-terminated surface, the SRBs share the same peak point at the valence band edge, while there is a Rashba-type splitting between the SRBs for the Mg(1)- and Mg(2)-terminated surfaces. As for the Mg(1)-termination, there are a pair of clear surface states on the brink of the valence band continuum (marked as S), while it is missing for the Mg(2)-termination. The above-mentioned features would offer us clear fingerprints for our experimental confirmation of the surface termination in our Mg$_{3}$Bi$_{2}$ films.
cpl-36-11-117303-fig4.png
Fig. 4. Weak anti-localization effect in the Mg$_{3}$Bi$_{2}$ film. (a) Temperature dependence of resistivity in Mg$_{3}$Bi$_{2}$ at zero and 9 T magnetic field. Inset: detail with enlarged scale of the black box. (b) Normalized magneto-resistance at different temperatures. (c) Magneto-conductance in perpendicular magnetic field configuration measured at various temperatures and their best fits to the HLN equation. (d) The extracted coherence length $l_{\phi}$ (left-hand axis) and coefficient $A$ (right-hand axis) from the HLN fitting under different temperatures. The value of $l_{\phi}$ fits well with $l_{\phi }\propto T^{-0.51}$.
To directly confirm the TSSs of Mg$_{3}$Bi$_{2}$ films, we performed in-situ ARPES measurements at 12 K. Figure 3(a) shows the experimental band structure in the $\bar{K}-\bar{\it \Gamma}-\bar{K}$ direction. A Rashba-type splitting structure could be clearly identified, as shown by the guidelines in Fig. 3(a), indicating an Mg-termination sample surface. What is more, the sharply dispersive bands marked as S matches well with the theoretical calculation for Mg(1)-termination, as shown in Fig. 3(b). The constant energy contour (CEC) at Fermi energy for the experimental and theoretical calculation are compared in Figs. 3(c) and 3(d), in which they match quite well with each other, despite the fact that the bulk continuum of the valence bands dominates the spectra weight in Fig. 3(d). Therefore, we have preliminarily confirmed that our Mg$_{3}$Bi$_{2}$ films have a Mg(1)-termination, which is different from the reported Bi-termination for single crystal Mg$_{3}$Bi$_{2}$.[15] The ARPES spectrum near Fermi level is shown in Fig. 3(e), in which the SRB and S band are more pronounced. Figure 3(f) depicts its corresponding momentum distribution curves (MDCs). A Lorentzian-type peak fit procedure was applied to each curve between the energies $-0.08$ and $-0.22$ eV to find the peak position of SRB1 and SRB2. Their peak position was forward linear fitted and thus we obtained the slopes of SRB1 and SRB2 to be $C_{1}=4.91$ and $C_{2}=-4.91$, respectively. The detailed fitting result is shown in Fig. S1 in the Supplementary Material. Figures 3(g) and 3(h) show the corresponding calculated spectrum of Mg(1) termination and Bi termination. The slopes of SRB1 and SRB2 are $\pm$5.16 and $\pm$3.41 for Mg(1)- and Bi-terminated surface. Obviously our measured SRB slope is consistent with Mg(1) termination. Thus by comparing the slope of SRB from the ARPES data with the calculated results, we further confirm that our Mg$_{3}$Bi$_{2}$ film is Mg(1)-terminated. Magneto-transport is another way to study the TSSs. For transport measurements, we grew Mg$_{3}$Bi$_{2}$ films on Al$_{2}$O$_{3}$(0001) substrates to eliminate the substrate effect. The characterization of Mg$_{3}$Bi$_{2}$ films on Al$_{2}$O$_{3}$ is shown in Fig. S2 in the Supplementary Material. The temperature dependence of longitudinal resistivity $\rho_{xx}$ in the Mg$_{3}$Bi$_{2}$ film is shown in Fig. 4(a), in which the applied magnetic field (9 T) is perpendicular to the electrical field. Here, $\rho_{xx}$ keeps decreasing from room temperature to 8 K, which is consistent with the behavior of semimetals. As the temperature drops below 8 K, $\rho_{xx}$ experiences an upturn (inset). This phenomenon was also observed in other topological materials and was explained as the freezing out of the bulk carriers, which may be caused by the presence of Coulomb interaction or $e$–$e$ correlation among surface states on the surface of Mg$_{3}$Bi$_{2}$ due to disorder.[27] Figure 4(b) demonstrates the temperature-dependent evolution of the normalized magneto-resistance (MR), defined as $MR=\frac{R(B)-R(0)} {R(0)} \times 100\%$, where $R(B)$ and $R(0)$ are the resistances measured under magnetic field of $B$ and zero field, respectively. A characteristic dip that hallmarked the existence of week anti-localization (WAL) exists at low temperature. Theoretically, the WAL phenomena for perpendicular magnetic field could be explained by the Hikami–Larkin–Nagaoka (HLN) model.[28] To confirm the magneto-transport behaviors under perpendicular field, we adopt the HLN model to fit the transport data. The HLN model can be expressed as $$\begin{align} \Delta G_{xx} (B)\equiv \,&G(B)-G(0)\\ \cong\,& A\frac{e^{2}}{2\pi^{2}\hslash }\Big[{\it \Psi}\Big(\frac{1}{2}+\frac{B_{\phi } }{B}\Big)-\ln\Big(\frac{B_{\phi } }{B}\Big) \Big], \end{align} $$ where $G(B)$ is the magneto-conductance (i.e., inverse of $R(B)$), prefactor $A$ is the coefficient of WAL, $e$ is the electronic charge, $\hslash$ is the reduced Planck constant, ${\it \Psi}$ is the Digamma function, $B_{\phi} =\hslash/(4el_{\phi})$ is a characteristic magnetic field, and $l_{\phi}$ is the phase coherence length. Figure 4(c) depicts the evolution of $\Delta G_{xx}$ versus temperature, together with the fitting results by the HLN model. The extracted coherence length $l_{\phi}$ as a function of $T$ is shown in Fig. 4(d). Here $l_{\phi}$ decreases from 210.9 nm at 2 K to 62.5 nm at 15 K, following a power law dependence on $T$ as $l_{\phi }\propto T^{-0.51}$. This monotonic decrease behavior was also observed in other topological materials,[29] where $l_{\phi }\propto T^{-0.5}$ and $l_{\phi }\propto T^{-0.75}$ for 2D and 3D systems, respectively. The power factor of $-0.51$ for our Mg$_{3}$Bi$_{2}$ system indicates that the WAL mainly originated from the 2D topological surface states at low temperature. The WAL coefficient $A$ at different temperatures is also shown in Fig. 4(d) (right-hand axis), where $A=N\alpha$, $\alpha$ equals $-0.5$ for the TSSs of symplectic universality class, and $N$ is the number of independent conducting channels.[28,30] Multiple channels could be involved in the transport due to the existence of different surface states, as well as possible conducting channels from bulk states.[31–33] The centrosymmetric lattice structure of Mg$_{3}$Bi$_{2}$ helps us to exclude that WAL origins from strong SOC. The value $A$ is between $-$0.87 and $-$0.47 for different temperatures, which means that $N=1$ or 2, indicating that one or two TSSs contribute as the conducting channel. In summary, we have reported the growth of high-quality Mg$_{3}$Bi$_{2}$ crystalline films on graphene/6H-SiC(0001) substrate. The existence of topological surface states in Mg$_{3}$Bi$_{2}$ is confirmed by first principles calculations, ARPES and magneto-transport measurements. The magneto-transport measurements exhibit 2D WAL effect in the low magnetic field regime, which fits well with the HLN model. The coherence lengths show a power law dependent on $T$, specifically in a relationship of $l_{\phi }\propto T^{-0.51}$. This suggests that the main contribution is from TSSs which provides auxiliary experimental evidence for the TSSs. It is predicted that replacing Bi by Sb or As would tune the SOC strength of Mg$_{3}$Bi$_{2}$ and result in a reduced spin-orbit band gap in Mg$_{3}$Bi$_{2-x}$Sb(As)$_{x}$ alloys,[15] which offers the possibilities of studying the exotic topological nature of type-II nodal line fermions. MBE techniques that we have adopted in our current work would offer a solution to the above-mentioned idea of realizing a possible type-II NLS. In the Supplementary Material, we present the methods on sample preparation, measurements detail, the detailed fitting procedure on SRB slope and characterization of Mg$_{3}$Bi$_{2}$ on Al$_{2}$O$_{3}$. We thank Quan-Sheng Wu and Chang-Ming Yue for helpful discussions. References Topological insulators and superconductorsColloquium : Topological insulatorsTopological Crystalline Insulators and Topological Superconductors: From Concepts to MaterialsTopological Insulators in Three DimensionsWeyl and Dirac semimetals in three-dimensional solidsSynthesis of semimetal A3Bi (A=Na, K) thin films by molecular beam epitaxyThe rise of grapheneTransmission of topological surface states through surface barriersEvolution of Topological Surface States in Antimony Ultra-Thin FilmsThree Dirac points on the (110) surface of the topological insulator Bi 1− x Sb x Weak-antilocalization and surface dominated transport in topological insulator Bi 2 Se 2 TeTransport evidence for the three-dimensional Dirac semimetal phase in ZrT e 5 Type-II nodal loops: Theory and material realizationTopological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg 3 Bi 2Realization of a Type-II Nodal-Line Semimetal in Mg 3 Bi 2Isotropic Conduction Network and Defect Chemistry in Mg 3+ δ Sb 2 -Based Layered Zintl Compounds with High Thermoelectric PerformanceGrowth and transport properties of Mg3X2 (X = Sb, Bi) single crystalsExperimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological InsulatorHydrogenated antimonene as quantum spin Hall insulator: A first-principles studyGraphene-like Be 3 X 2 ( X = C, Si, Ge, Sn): A new family of two-dimensional topological insulatorsInteger Quantum Hall Effect in a Two-Orbital Square Lattice with Chern Number C = 2Maximally localized generalized Wannier functions for composite energy bandsMaximally localized Wannier functions for entangled energy bandswannier90: A tool for obtaining maximally-localised Wannier functionsWannierTools: An open-source software package for novel topological materialsTopological surface states scattering in antimonyEffective n-type doping of Mg3Sb2 with group-3 elementsSpin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random SystemWeak Anti-localization and Quantum Oscillations of Surface States in Topological Insulator Bi2Se2TeArticle identifier not recognizedElectron interaction-driven insulating ground state in Bi 2 Se 3 topological insulators in the two-dimensional limitWeak antilocalization and spin-orbit interaction in a two-dimensional electron gasObservation of Anderson Localization in Ultrathin Films of Three-Dimensional Topological Insulators
[1] Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[2] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[3] Ando Y and Fu L 2015 Annu. Rev. Condens. Matter Phys. 6 361
[4] Fu L, Kane C L and Mele E J 2007 Phys. Rev. Lett. 98 106803
[5] Armitage N P, Mele E J and Vishwanath A 2018 Rev. Mod. Phys. 90 015001
[6] Wen J, Guo H, Yan C H, Wang Z Y, Chang K, Deng P, Zhang T, Zhang Z D, Ji S H, Wang L L, He K, Ma X C, Chen X and Xue Q K 2015 Appl. Surf. Sci. 327 213
[7] Geim A K and Novoselov K S 2007 Nat. Mater. 6 183
[8] Seo J, Roushan P, Beidenkopf H, Hor Y S, Cava R J and Yazdani A 2010 Nature 466 343
[9] Yao G, Luo Z, Pan F, Xu W, Feng Y P and Wang X 2013 Sci. Rep. 3 2010
[10] Zhu X G, Stensgaard M, Barreto L, Silva W S E, Søren U, Michiardi M, Bianchi M, Dendzik M and Hofmann P 2013 New J. Phys. 15 103011
[11] Gopal R K, Singh S, Chandra R and Mitra C 2015 AIP Adv. 5 047111
[12] Zheng G, Lu J, Zhu X, Ning W, Han Y, Zhang H, Zhang J, Xi C, Yang J, Du H, Yang K, Zhang Y and Tian M 2016 Phys. Rev. B 93 115414
[13] Li S, Yu Z M, Liu Y, Guan S, Wang S S, Zhang X, Yao Y and Yang S A 2017 Phys. Rev. B 96 081106
[14] Zhang X, Jin L, Dai X and Liu G 2017 J. Phys. Chem. Lett. 8 4814
[15] Chang T R, Pletikosic I, Kong T, Bian G, Huang A, Denlinger J, Kushwaha S K, Sinkovic B, Jeng H T, Valla T, Xie W and Cava R J 2019 Adv. Sci. 6 1800897
[16] Tamaki H, Sato H K and Kanno T 2016 Adv. Mater. 28 10182
[17] Xin J, Li G, Auffermann G, Borrmann H, Schnelle W, Gooth J, Zhao X, Zhu T, Felser C and Fu C 2018 Mater. Today Phys. 7 61
[18] Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C and Xue Q K 2013 Science 340 167
[19] He X and Li J B 2019 Chin. Phys. B 28 037301
[20] Song L L, Zhang L Z, Guan Y R, Lu J C, Yan C X and Cai J M 2019 Chin. Phys. B 28 037101
[21] Yu H L, Zhai Z Y and Bian X T 2016 Chin. Phys. Lett. 33 117305
[22] Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847
[23] Souza I, Marzari N and Vanderbilt D 2001 Phys. Rev. B 65 035109
[24] Mostofi A A, Yates J R, Lee Y S, Souza I, Vanderbilt D and Marzari N 2008 Comput. Phys. Commun. 178 685
[25] Wu Q S, Zhang S N, Song H F, Troyer M and Soluyanov A A 2018 Comput. Phys. Commun. 224 405
[26] Narayan A, Rungger I and Sanvito S 2012 Phys. Rev. B 86 201402
[27] Gorai P, Toberer E S and Stevanovic V 2019 J. Appl. Phys. 125 025105
[28] Hikami S, Larkin A I and Nagaoka Y 1980 Prog. Theor. Phys. 63 707
[29] Bao L, He L, Meyer N, Kou X, Zhang P, Chen Z G, Fedorov A V, Zou J, Riedemann T M, Lograsso T A, Wang K L, Tuttle G and Xiu F 2012 Sci. Rep. 2 726
[30] Dybko K, Mazur G P, Wolkanowicz W W, Szot M, Dziawa P, Domagala J Z, Wiater M, Wojtowicz T, Grabecki G and Story T 2018 arXiv:1812.08711 [cond-mat.mes-hall]
[31] Liu M, Chang C Z, Zhang Z, Zhang Y, Ruan W, He K, Wang L L, Chen X, Jia J F, Zhang S C, Xue Q K, Ma X and Wang Y 2011 Phys. Rev. B 83 165440
[32] Spirito D, Di Gaspare L, Evangelisti F, Di Gaspare A, Giovine E and Notargiacomo A 2012 Phys. Rev. B 85 235314
[33] Liao J, Ou Y, Feng X, Yang S, Lin C, Yang W, Wu K, He K, Ma X, Xue Q K and Li Y 2015 Phys. Rev. Lett. 114 216601