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The discovery of topological materials – condensed matter systems that have nontrivial topological invariants – marked the commencement of a new era in condensed matter physics and materials science. Three dimensional topological insulators (3D TIs) are one of the first discovered and the most studied among all topological materials. The bulk material of the TIs have the characteristics of the insulator, having a complete energy gap. Their surface electronic states, on the other hand, have the characteristics of a conductor, with energy band passes continuously through the Fermi surface. The conductivity of this topological surface state (TSS) is protected by the time reversal symmetry of the bulk material. The TSS is highly spin-polarized and form a special spin-helical configuration that allows electrons with specific spin to migrate only in a specific direction on the surface. By this means, surface electrons in TIs can " bypass” the influence of local impurities, achieving a lossless transmission of spin-polarized current. The existence of TIs directly leads to a variety of novel transport, magnetic, electrical, and optical phenomena, such as non-local quantum transport, quantum spin Hall effect, etc., promising wide application prospects. Recently, several research groups have searched all 230 non-magnetic crystal space groups, exhausting all the found or undiscovered strong/weak TIs, topological crystalline insulators (TCI), and topological semimetals. This series of work marks that theoretical understanding of non-magnetic topological materials has gone through a period of one-by-one prediction and verification, and entered the stage of the large-area material screening and optimization. Parallel to non-magnetic TIs, magnetic topological materials constructed by ferromagnetic or antiferromagnetic long range orders in topological systems have always been an important direction attracting theoretical and experimental efforts. In magnetic TIs, the lack of time reversal symmetry brings about new physical phenomena. For example, when a ferromagnetic order is introduced into a three-dimensional TI, the Dirac TSS that originally intersected at one point will open a magnetic gap. When the Fermi surface is placed just in the gap, the quantum anomalous Hall effect can be implemented. At present, the research on magnetic topology systems is still in the ascendant. It is foreseeable that these systems will be the main focus and breakthrough point of topology material research in the next few years. Angle-resolved photoemission spectroscopy (ARPES) is one of the most successful experimental methods of solid state physics. Its unique k-space-resolved single-electron detection capability and simple and easy-to-read data format make it a popular choice for both theoretists and experimentalists. In the field of topological materials, ARPES has always been an important experimetnal technique. It is able to directly observe the bulk and surface band structure of crystalline materials, and in a very intuitive way. With ARPES, it is incontrovertible to conclude whether a material is topological, and which type of topological material it belongs to. This paper reviews the progress of ARPES research on TIs since 2008, focusing on the experimental energy band characteristics of each series of TIs and the general method of using ARPES to study this series of materials. Due to space limitations, this paper only discusses the research progress of ARPES for strong 3D TIs (focusing on the Bi2Se3 series) and magnetic TIs (focusing on the MnBi2Te4 series). Researches involving TCIs, topological Kondo insulators, weak 3D TIs, topological superconductors and heterostructures based on topological insulators will not be discussed. This paper assumes that the reader has the basic knowledge of ARPES, so the basic principles and system components of ARPES are not discussed. -
Keywords:
- Topological Insulator /
- Magnetic Topological Insulator /
- Angle-resolved Photoemission Spectroscopy /
- Energy Bands
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图 1 自旋分辨ARPES原理及示意图[58,71,72] (a) Mott自旋分析仪的原理示意图[58]; (b) Mott自旋分析仪的设计示意图[71]; (c) 加入两个互相垂直的VLEED自旋分析仪的一个ARPES系统(东京大学Shin-Kondo小组)[72], 其中VLEED的设计来自广岛大学Okuda小组
Figure 1. Principles and skematics of spin-resolved ARPES[58,71,72]: (a) Principle of the spin-selective exchange interaction used in a Mott spin detector[58]; (b) design skematics of a Mott spin detector[71]; (c) skematics of an ARPES system equipped with two parallelly arranged VLEED spin detectors[72], the design of which were adapted from Team of Okuda in Hiroshima University.
图 2 时间分辨ARPES原理及示意图[89,90] (a) 时间分辨ARPES原理图[89]; (b) 时间分辨ARPES的一种设计[90], 利用掺钛蓝宝石激光器发出的800 nm基频红外激光作为激发光, 再利用基于气体谐振腔的高次谐波发生器生成基频激光的高阶倍频光, 并以之作为探测光
Figure 2. Principles and skematics of time-resolved ARPES[89,90]: (a) Principles of time-resolved ARPES[89]; (b) design example of a time-resolved ARPES system[90], using the 800 nm inferred radiation from a Ti:sapphire laser as the pump beam and its high hamonics produced by a gas jet as the probe beam.
图 3 ARPES实验发现的第一类强三维拓扑绝缘体—Bi1-xSbx合金和Sb单晶[10] (a) Bi1–xSbx
$\bar \varGamma - \bar M$ 方向能带结构. 数字1−5标示5个拓扑表面态; (b) 图 (a) 中绿色虚线标示的动量分布曲线 (MDC) 的自旋极化情况. Spin up (Spin down) 表示极化方向朝向图 (a) 的前(后)方; (c) 自旋分辨ARPES测量的实验仪器布局; (d) Sb单晶$\bar \varGamma - \bar M$ 方向能带结构. (e), (f) 图 (d) 中白色虚线标示的MDC的自旋极化情况. Up/down的意义同图 (b)Figure 3. The first 3 D topological insulators, Bi1-xSbx and Sb, discovered by ARPES[10]: (a) ARPES band structure along
$\bar \varGamma - \bar M$ for Bi1-xSbx. Numbers mark the five topological surface states; (b) spin polarization along the green dashed line in Fig.(a). Up/down represents a polarization out of/into the page; (c) experimental geometry of the spin-ARPES setup; (d) ARPES band structure along$\bar \varGamma - \bar M$ for Sb; (e), (f) spin polarization along the white dashed line in Fig.(d).
图 4 Bi2Se3和Bi2Te3单晶的最初几个ARPES研究[11,12,122] (a) Bi2Se3和Bi2Te3的晶体结构[11]; (b) Bi2Se3和Bi2Te3的体和表面布里渊区[124]; (c) Bi2Te3的狄拉克锥(沿两个高对称方向)[11]. SSB: 拓扑表面态; BCB: 体导带; BVB: 体价带; (d) Bi2Se3的狄拉克锥(沿两个高对称方向)[12]; (e) Bi2Te3狄拉克锥及其形成的费米面[122]; (f) 图 (e) 能带的自旋极化情况[122]
Figure 4. The first ARPES studies on Bi2Se3 and Bi2Te3 single crystals[11,12,122]: (a) Crystal structures of Bi2Se3 and Bi2Te3[11]; (b) bulk and surface Brillouin zone of Bi2Se3 and Bi2Te3[124]; (c) Dirac cone of Bi2Te3 (along two high symmetry directions)[11]. SSB: topological surface state; BCB: bulk conduction band; BVB: bulk valence band. (d) Dirac cone of Bi2Se3 (along two high symmetry directions)[12]; (e) the Fermi surface made by the Dirac cone of Bi2Te3[122]; (f) spin polarization of the bands in Fig.(e)[122].
图 5 Bi2Se3系列TI狄拉克锥的六角变形[11,127-129] (a) (Bi1-δSnδ)2Te3材料的雪花形费米面, 上图材料为Bi2Te3 (δ = 0), 下图材料为 δ = 0.67%[11]; (b) Bi2Se3的等能面形变, 上图位于费米面, 下图位于费米面以下150 meV[127]; (c) 利用CD-ARPES导出的费米面自旋极化sz分量. sin3θ 周期性清晰可见[128]; (d) 量子阱表面态存在时自旋极化的分布[129]
Figure 5. Hexagongal warping of the Dirac cones in the Bi2Se3-class TIs[11,127-129]: (a) Snowflake-like Fermi surfaces of (Bi1-δSnδ)2Te3. Top: Bi2Te3(δ = 0); bottom: δ = 0.67%[11]; (b) constant energy contours of Bi2Se3. Top: E = EF; bottom: E = -150 meV[127]; (c) sz component of the spin polarization vector, extracted from CD-ARPES data. The sin3θ periodicity is clearly visible[128]; (d) distribution of the spin polarization vector in the presence of quantum well states[129].
图 6 对Bi2Se3系列TI的表面掺杂引入新的量子阱表面能带[129,133] (a) 新表面能带的形成过程. 随着Fe掺杂原子的增加, 量子阱能带逐对生成. 每对新能带在两个TRIM点间增加两条通过费米面的能带[133]; (b) 钾掺杂Bi2Se3中观察到的同一现象, 及其定量计算解释[129]; (c) 解释此现象的物理图像[129]
Figure 6. Introduction of quantum well surface states to the Bi2Se3-class TIs by surface deposition[129,133]: (a) Formation of the new surface bands[133]. Pairs of quantum well states are form progressively with increasing surface Fe dosage. Each pair of new states adds two Fermi-crossing bands between two TRIMs; (b) same phenomenon as (a) observed in K-doped Bi2Se3, compared with ab initio calculation results[129]; (c) physical explanation of this phenomenon[129].
图 7 利用自能虚部研究TI中狄拉克电子的多体相互作用[140-142] (a) Bi2Se3狄拉克电子的自能虚部随束缚能的变化曲线(上), 以及可能的散射通道分析(下)[140]; (b) Bi2Se3狄拉克电子的自能虚部随束缚能的变化曲线(上), 以及MDC的半高全宽(下)[141]. (a) 和 (b) 中的拓扑表面态均沿
$\bar \varGamma - \bar K$ 方向截取; (c) Bi2Te3狄拉克电子的自能虚部随束缚能的变化曲线, 上(下)图利用MDC (EDC) 的分析得到[142]Figure 7. Study of many-body interactions of the Dirac fermions in TIs by analyzing the imaginary part of the self energy[140-142]: (a) Results of Ref. [140]. Top: Imaginary part of the self energy (ImΣ) versus binding energy (Eb) for the topological surface state (TSS) of Bi2Se3. Bottom: Analysis of possible scattering channels; (b) results of Ref. [141]. Top: ImΣ vs. Eb for the TSS of Bi2Se3. Bottom: Full width half maximum (FWHM) of the momentum distribution curves (MDCs); (c) results of Ref. [142]. Top/bottom: ImΣ vs. Eb for the TSS of Bi2Te3, obtained from MDC/EDC analysis.
图 8 TI拓扑表面态圆二色性信号和自旋极化信号的复杂性[146,148,150] (a) 两份文献中CD-ARPES和SARPES的实验配置[148,150]. 图中p和 π(s和 σ)意义相同; (b) Bi2Te3狄拉克电子的CD信号[146]. 入射光子能量从21 到55 eV变化过程中, CD信号发生了两次反转; (c) Bi2Se3自旋极化信号的三个分量随光子能量的变化[150]; (d) Bi2Se3自旋极化信号在两种不同的线偏振入射光作用下方向完全相反[148]; (e) 自旋信号随入射光偏振变化的复杂响应[148]. ± sp-pol表示右(左)倾45° 的线偏振光
Figure 8. Complexity of the CD-ARPES and SARPES signals[146,148,150]: (a) Experimental geometries in Refs. [148,150]. p and π (s and σ) are the same; (b) CD-ARPES signal of the Dirac fermions in Bi2Te3[146]. The CD signal reverses sign for two times as the photon energy goes from 21 to 55 eV; (c) the three components of the spin polarization vector (Px, Py, Pz) vs. photon energy[150]; (d) sign reversal of P under two different linearly polarized incident lights[148]; (e) complex response of P as a function of light polarization[148]. ± sp-pol: 45° tilted linearly polarized light.
图 9 虽然CD-ARPES和SARPES数据受多种参数影响, 利用SARPES研究TI表面态的自旋构型仍然是可能的[152] (a) 当入射光能量为50 eV时, 自旋信号不受入射光偏振的影响; (b) 当入射光能量为6 eV时, 自旋信号随入射光偏振的反转而反转
Figure 9. Possibility for studying the initial state spin configuration despite the complexity of CD-ARPES and SARPES signals[152]: (a) When hν = 50 eV, the spin signal is unaffected by incident light polarization; (b) when hν = 6 eV, the spin signal changes sign as the light polarization reverses.
图 10 利用trARPES和2PPE ARPES研究Bi2Se3费米面以上的电子学结构[153,154] (a) 图中涉及的三种光电效应过程[154]: (i) 电子连续吸收两个6 eV光子后发射(对应于图(b)), (ii) 电子同时吸收一个1.5 eV和一个6 eV光子后发射(对应于图 (c) 的左上两图), (iii) 普通ARPES过程, 对应于图 (b) 和图(c) 的下方小图; (b) 由6 eV + 6 eV 2 PPE过程探测到的非占据态能带[153]; (c) 由1.5 eV + 6 eV 2PPE过程探测到的能带, 其中的X型狄拉克锥是费米面以下的拓扑表面态的投影, 并非第二狄拉克锥[154]
Figure 10. Studying the unoccupied electronic states of Bi2Se3 using trARPES and 2PPE ARPES[153,154]: (a) Three relavant processes of photoemission[154]. (i) An electron photoemits after absorbing two 6 eV photons consecutively (situation in Fig. (b)); (ii) An electron photoemits after simultaneously absorbing a 1.5 eV and a 6 eV photon [two upper left panels in Fig. (c)]; (iii) Normal photoemission (two lower panels in (b) and (c)); (b) unoccupied bands revealed by Process (i)[153]; (c) bands reveals by Process (ii)[154]. The X-shaped Dirac cone is a projection of the TSS below EF, not the 2nd TSS.
图 11 利用trARPES分析TI非稳恒态的动力学弛豫过程[159,160] (a), (b) 空穴掺杂Bi2Se3的动力学弛豫过程的 (a) 实验数据和 (b) 物理图像示意[159]. 在t ≈ 0 ps时, 激发光把电子激发到很高的非占据态中; 随着时间推移, 电子逐渐占据较低能的能带. 在t = 2.5—9.0 ps这段时间内, 导带底始终作为电子库向表面态供给电子; (c) 电子掺杂Bi2Se3的动力学弛豫过程[160]. 导带底在此实验中不可见
Figure 11. Studying the relaxation process of the excited state using trARPES[159,160]: (a), (b) The trARPES result ((a)) and skematics of physical process ((b)) of dynamic relaxation process of hole-doped Bi2Se3[159]; (c) dynamic relaxation process of electron-doped Bi2Se3[160].
图 12 利用trARPES构造和研究Floquet拓扑绝缘体[166,167] (a) 时域能带结构的两种形成机制: Floquet机制和Volkov机制(见正文)[167]; (b) 对Floquet拓扑绝缘体的首次观测[166]; (c) 图 (b) 数据的高精度重复[167]; (d) p偏振(左二图)和s偏振(右二图)入射光对不同动量方向(一、三图为kx, 二、四图为ky)的Floquet能带的影响[167]. p偏振时ky方向能带产生Floquet能隙, 而s偏振时kx方向能带产生Floquet能隙(红色箭头标示能隙位置)
Figure 12. Realization of Floquet TI by trARPES[166,167]: (a) Skematics of the Floquet and Volkov mechanisms[167]; (b) the first experimental observation of a Floquet TI[166]; (c) a higher quality reproduction of data in Fig. (b)[167]; (d) influence on the Floquet bands at different momenta under differently polarized light[167]. Floquet gap (red arrows) appears along ky/kx under p/s-polarized light.
图 13 Bi2Se3系列TI少层薄膜的拓扑相变[168,169]. 从图中可以看到, 6 QL以下的Bi2Se3和5 QL以下的Sb2Te3是拓扑平庸的 (a) Bi2Se3少层薄膜的ARPES能带结构. 完整的狄拉克锥在6 QL样品中形成[168]; (b) Sb2Te3少层薄膜的二次微分ARPES能带结构(上)和第一性原理计算结果(下)[169]. 结合实验和理论, 可以看出完整的狄拉克锥在5 QL样品中形成
Figure 13. Topological phase transition on few-layer TI films[168,169]. Bi2Se3/Sb2Te3 films thinner than 6 QL/5 QL are topologically trivial: (a) ARPES band structure on few layer Bi2Se3 films. A well-defined Dirac cone is not present for films thinner than 6 QL[168]; (b) ARPES band structure (top) and first principles calculation results (bottom) on few layer Sb2Te3 films[169]. A Dirac cone forms in 5 QL and thicker samples.
图 14 Bi2Te2Se与GeBi2Te4的ARPES能带图[183] (a)−(c) Bi2Te2Se的 (a) 晶体结构, (b) 第一性原理计算的体态和表面态, 以及 (c) 沿两个高对称方向的ARPES能带图; (d)−(f) 同 (a)−(c), 但体系为GeBi2Te4. 由图可见, Bi2Te2Se和GeBi2Te4都是具有
$\bar \varGamma $ 点单一狄拉克锥的强三维TIFigure 14. ARPES band structure of Bi2Te2Se and GeBi2Te4[183]: (a) Crystal structure, (b) bulk and surface states by ab initio calculations, and (c) ARPES bands along two high symmetry directions, of Bi2Te2Se; (d)−(f) Same as (a)−(c) but for GeBi2Te4. It is clear that both compounds are 3 D TIs with a single Dirac cone at
$\bar \varGamma $ .
图 15 TlBi(S1-δSeδ)2体系中的拓扑相变[209] (a) 沿
$\bar \varGamma - \bar M$ 方向的ARPES k-E图, 样品自左至右分别为 δ = 0, 0.2, 0.4, 0.6, 0.8和1.0(下同). 由图可见体系从一个普通绝缘体 (δ < 0.6) 转变成一个拓扑绝缘体 (δ ≥ 0.6). 普通绝缘体的“轴子角参数”θ = 0或2π, 等价于主拓扑数 (TQN)ν0 = 0; 拓扑绝缘体的“轴子角参数”θ = π, 等价于主拓扑数ν0 = 1; (b) 对应样品的费米能ARPES扫描, 箭头标示自旋取向和简并情况; (c) 对应样品的表面态和体能带色散情况(左右两小图是能量分布曲线); (d) 对应样品的三维能带色散图Figure 15. Topological phase transition in TlBi(S1-δSeδ)2[209]: (a) ARPES k-E maps along
$\bar \Gamma - \bar M$ , for δ = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 (left to right). The system evolves from a normal band insulator (δ < 0.6) to a TI (δ ≥ 0.6). The axion angular parameter θ = 0 or 2π for a normal insulator [equivalent to the topological quantum number (TQN) ν0 = 0], while θ = π for a TI; (b) ARPES maps at EF; (c) bulk and surface band dispersion (EDCs for δ = 0 and 1), and (d) 3 D band dispersion maps, for corresponding values of δ.
图 16 HgTe单晶的ARPES研究[217] (a) 过
$\bar \varGamma $ 点的ARPES k-E图(左)及各能带的色散情况示意(右). TSS: 狄拉克锥拓扑表面态; (b) 能带的kx–ky–E三维色散示意, 各能带颜色同图 (a); (c) TSS和体能态的kz色散, 可见TSS在kz方向无色散, 而其他能带色散非常明显; (d) SARPES测得的TSS自旋极化, 可见狄拉克锥的自旋动量锁定行为Figure 16. ARPES on single crystal HgTe[217]: (a) ARPES k-E map (left) and guides-for-the-eye band dispersion (right). TSS: topological surface state (Dirac cone); (b) skematic kx–ky–E 3 D band dispersion; (c) kz dispersion of the TSS and the bulk states. TSS has no kz dispersion, while other bands show clear out-of-plane dispersive pattern; (d) spin polarization detected with SARPES, showing the spin-momentum lock behavior.
图 17 表面掺杂HgTe单晶的ARPES研究[217] (a)−(e) 碱金属Cs当量增加时能带的变化. 显然狄拉克锥的上半部分变得可见, 且Γ8能带上下两支之间存在能隙; (f)−(j) 对应的CD-ARPES信号, 表明锥的上半部分具有典型的拓扑表面态特征; (k) 碱金属K当量增加时能带的变化; (l) 图 (k) 的能量分布曲线, 显示体能隙从290 meV增加到392 meV
Figure 17. ARPES on surface-doped HgTe single crystals[217]: (a)−(e) Band evolution as Cs dosage increases. The upper half of the Dirac cone becomes visible, and a gap exists between the upper and the lower Γ8 band; (f)−(j) corresponding CD-ARPES signal, indicating the topological nature of the upper cone; (k) band evolution as K dosage increases; (l) EDCs of Fig. (k), showing an increase of the gap, from 290 to 392 meV.
图 18 半Heusler拓扑绝缘体的ARPES研究[219,221,222] (a) 第一性原理计算给出的典型半Heusler化合物的体能隙 (EΓ6 – EΓ8) 大小与体系平均原子序数
$\left\langle Z \right\rangle $ 的联系[219]. 体能隙大于零表示体系为普通绝缘体, 体能隙小于零表示体系为拓扑绝缘体. 由图可见,$\left\langle Z \right\rangle $ 比较大(即自旋轨道耦合比较大)的体系更倾向于成为拓扑绝缘体; (b) 半Heusler TI GdPtBi的ARPES费米能扫描[221]; (c) GdPtBi的过$\bar \varGamma $ 点能带图[221]; (d) 另一个半Heusler TI LuPtBi的过$\bar \varGamma $ 点能带图[222], 左图为原始数据, 右图为二次微分后的能带. TSS: 拓扑表面态Figure 18. ARPES on half Heusler TIs[219,221,222]: (a) Bulk band gap (EΓ6 – EΓ8) of typical half Heusler compounds as a function of the system’s average atomic number
$\left\langle Z \right\rangle $ [219]. Positive/negative gap represents normal insulator/TI. Compounds with larger$\left\langle Z \right\rangle $ (larger SOC) are prone to become TIs; (b) ARPES map at EF on GdPtBi[221]; (c) ARPES band dispersion on GdPtBi[221]; (d) ARPES band dispersion on LuPtBi[222]. Left: raw data. Right: second derivative band map.
图 19 对磁性掺杂TI的首个ARPES研究[224] (a) 16% Fe体掺杂的Bi2Se3的能带. 由图可见, 狄拉克点打开了一个能隙; (b) 1% Mn体掺杂的Bi2Se3的能带. 不仅狄拉克点具有能隙, 而且样品的费米面被调节至狄拉克点中, 实现了表面态的绝缘相
Figure 19. The first ARPES study on magnetic-doped TI[224]: (a) ARPES bands for 16% Fe bulk doped Bi2Se3. A gap is visible at the TSS; (b) ARPES bands for 1% Mn bulk doped Bi2Se3. The gap not only exist but also locates right at EF, realizing an insulating phase of the surface state.
图 20 磁性掺杂和非磁掺杂的Bi2Se3薄膜的能带自旋分析[225] (a) Mn-Bi2Se3薄膜的能带自旋. 在这个铁磁掺杂的TI样品中,
$\bar \varGamma $ 点巨大的z方向自旋分量和普通的自旋-动量锁定行为(自旋分量沿面内切线方向)共同构成了“刺猬状自旋纹理”; (b) Zn-Bi2Se3薄膜的能带自旋. 在这个非磁掺杂的TI样品中,$\bar \varGamma $ 点的z方向自旋分量为零, 体系只显示出普通的自旋-动量锁定行为Figure 20. SARPES analysis on magnetic (Mn) and non-magnetic (Zn) doped Bi2Se3 films[225]: (a) Band spins in Mn-Bi2Se3 films. In this ferromagnetically doped TI, the large sz component at
$\bar \varGamma $ and the typical spin-momentum lock behavior comprises the “hedgehog spin texture”; (b) band spins in Zn-Bi2Se3 films. In this non-magnetically doped TI, sz = 0 at$\bar \varGamma $ , only the spin-momentum locking is seen.
图 21 (MnBi2Te4)m(Bi2Te3)n (m = 1, n = 0, 1, 2, …) 系列磁性拓扑绝缘体以及普通拓扑绝缘体Bi2Te3的 (a) 晶体结构[234], (b) 理论预言的基态磁结构[237], (c) 扫描透射电子显微镜原子分辨HAADF STEM图样[237], 以及 (d) 选区电子衍射纹样[237]. 由图可见, 此系列化合物的构成单元是交替出现的Bi2Te3五重层(记为023层)和MnBi2Te4七重层(记为124层). Mn原子层位于124七重层的正中, 是体系磁性的来源. MnBi2Te4的体材料基态磁结构被中子衍射实验初步确定为面外磁矩的A型反铁磁
Figure 21. (a) Crystal structures[234], (b) predicted ground state magnetic structure[237], (c) Atomic resolution HAADF STEM images[237], and (d) Selected area electron diffraction (SAED) patterns[237], of magnetic TIs (MnBi2Te4)m(Bi2Te3)n (m = 1, n = 0, 1, 2, …) and 3 D TI Bi2Te3. This series of compounds consists of alternatively stacking Bi2Te3 QLs (023 layers) and MnBi2Te4 septuple layers (124 layers). The 0 K magnetic structure of bulk MnBi2Te4 is determined by neutron diffraction to be out-of-plane A-type antiferromagnetic.
图 22 MnBi2Te4的早期ARPES数据, 观察到较大的表面态能隙. 图中数据的光子能量和测量温度分别为 (a) 9 eV, 17 K(右图为二次微分分析结果)[242]; (b) 30和300 K, 光子能量未标示[243]; (c) 21.5 eV, 7 K(左)和47 K(右)[244]; (d) 21.5 eV, 18 K[235]
Figure 22. Early ARPES data on MnBi2Te4, observing sizable gaps at the surface state. The photon energies and measuring temperatures are (a) 9 eV, 17 K (right: 2nd derivative result)[242]; (b) 30 and 300 K, photon energy unmarked[243]; (c) 21.5 eV, 7 K (left) and 47 K (right)[245]; (d) 21.5 eV, 18 K[235].
图 23 MnBi2Te4的最新ARPES数据, 观察到零能隙的表面态[247] (a) 一个典型的沿
$\bar K - \bar \varGamma - \bar K$ 方向的ARPES k-E图, 光子能量和测量温度为6.3 eV和10 K. 由图可见, 一个线性的X形无能隙电子态出现在导带和价带之间; (b)$\bar \varGamma $ 点处的kz色散, 光子能量范围为6−20 eV. 由图可见, 价带顶的周期性色散非常清晰; (c)−(e) 经过体对称点 Γ4, Z4, Γ5的沿$\bar M - \bar \varGamma - \bar M$ (kx) 方向的ARPES k-E图(对应光子能量分别为7.5, 10.5和13.5 eV). 由图可见, 无能隙表面态没有kz方向的色散, 而价带顶的束缚能由 Γ点的0.33 eV变成Z点的0.4 eV, 从而使体能隙由0.13 eV变成0.20 eVFigure 23. Surface and bulk electronic structure of MnBi2Te4[247]: (a) A typical ARPES k-E map along
$\bar K - \bar \varGamma - \bar K$ (ky), taken at 10 K under photon energy hν = 6.3 eV. A linear, X-shaped, gapless state exists between the valence and the conduction bands; (b) kz dispersion map at$\bar \varGamma $ , taken with 6− 20 eV photons. Periodic dispersion pattern on the VBM is seen clearly. (c)−(e) k-E maps along$\bar M - \bar \varGamma - \bar M$ (kx) taken at the Γ4, Z4 and Γ5 points marked in Fig. (b) (correspond to hν = 7.5, 10.5 and 13.5 eV, respectively). It is clear that the gapless state forming the Dirac cone has no kz dispersion, while the VBM evolves from –0.33 eV (Γ) to –0.4 eV (Z), consequently changing the bulk gap from 0.13 to 0.20 eV.
图 24 MnBi2Te4无能隙狄拉克锥的稳定性[247]. 每一小图的左边为ARPES原始数据, 右边为二次微分后的数据. 所有数据均采自6.3 eV激光ARPES系统, 沿
$\bar K - \bar \varGamma - \bar K$ 高对称方向. 每一小图测量的样品分别为 (a) 超高真空解理的样品, 测量温度为10 K(低于Neel温度); (b) 超高真空解理的样品, 测量温度为300 K(远高于Neel温度); (c) 在空气中解理然后放入真空腔的样品, 测量温度为10 K. 虽然不同的解理和测量条件导致不同的表面费米能位置和载流子浓度, 但狄拉克锥的无能隙性质保持不变, 且体能隙的大小 (150 meV) 未发生可观测的变化Figure 24. Robustness of the Dirac surface state[247]. The figure shows ARPES raw (left) and second derivative (right) k-E maps taken with 6.3 eV laser light along
$\bar K - \bar \varGamma - \bar K$ (ky) for (a) a pristine sample cleaved and measured at 10 K (below TN), (b) a pristine sample cleaved and measured at 300 K (way above TN), and (c) a sample cleaved in air at room temperature, and measured at 10 K. Despite the overall carrier doping induced by different cleaving conditions, the gapless Dirac cone is clearly seen for all cases, along with an unchanged bulk band gap sized 150 meV.表 1 MnBi2Te4的ARPES研究小结(截至2019年7月22日). 表中“能隙大小”指的是每份文献各自认为的表面态中的
$\bar \varGamma $ 点能隙大小.Table 1. Summary of ARPES studies on MnBi2Te4 (as of 22 July 2019).
文献序号 样品形态 测量温度/K 光子能量/eV 能隙大小/meV 备注 [242] 单晶 17/300 28/9 70 [243] 单晶 30/300 未提及 ~85/115 [235] 单晶 18 21.5 100 [244] (v1, v2) 单晶 18/40 7.25/9/11/13.75/15 50 在这篇arXiv文章的第三个版本(张贴于2019年7月9日)里, 作者加入了零能隙的数据. [245] 单晶 7/18/47/80 21.5/79 100 [246] 薄膜 25 21.2 0 此文献观察到了零能隙, 但作者认为测量温度不够低, 测得的是无能隙的顺磁拓扑表面态. [247] 单晶 10/300 6.3/7-40 0 张贴于2019年7月8日 [248] 单晶 7.5 7/10-22 0 张贴于2019年7月11日 [249] 单晶 10/50 13.8/47/51 0 张贴于2019年7月15日 [250] 单晶 8/60 6.36/6.7 0 张贴于2019年7月22日 
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