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During the recent years, the iron-based superconductors with a topological band structure have attracted intensive attention from the science community as a new and promising platform for emerging Majorana zero modes in their vortex core. These topological iron-based superconductors possess all of the desirable properties, i.e. single material, high-Tc superconductivity, strong electron-electron correlation and topological band structure, thus successfully avoiding the difficulties suffered by previous Majorana platforms, such as intrinsic topological superconductors and multiple types of proximitized heterostructures. So far, one has observed pristine vortex Majorana zero modes in several different compounds of iron-based superconductors. The systematic studies performed on those systems show that the vortex Majorana zero modes are quite evident experimentally and very clear theoretically, leading to a bright future in applications. The vortex cores of iron-based superconductors can become one of the major candidates for exploring topological quantum computing in the future. In this review article, we will focus on Fe(Te, Se) single crystal, to introduce the original ideas and research progress of the new emerging “iron home” for Majorana zero modes. Having elabrated the basic band structures and the experimental facts of the observed vortex zero modes in Fe(Te, Se), we will systematically summarize the main observations and fundamental physics of vortex Majorana zero modes in Fe(Te, Se). First of all, with the help of the observed behavior of Majorana wavefunction and quasiparticle poisioning, we will analyze the emerging mechanism of vortex Majorana zero modes in Fe(Te, Se). Then we will elaborate the measurements on Majorana symmetry and topological nature of vortex Majorana zero modes, assisted by several existing Majorana theories. After that, we will switch our view angle from quantum physics to quantum engineering, and comprehensively analyze the fate of vortex Majorana zero modes in a real material under a real environment, which may benefit the potential engineering applications in the future. This review article follows the physical properties of vortex Majorana zero modes, and emphasizes the link between theories and experiments. Our goal is to bridge the gap between the classical Majorana theories and the new emerging Majorana platform in iron-based superconductors, and help the readers to understand the experimental observations of the newly discovered “iron home” for Majoranas.
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Keywords:
- Majorana zero modes /
- iron-based superconductors /
- superconducting vortex /
- topological matters
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图 1 铁基超导体是更优的MZM载体材料 (a)在铁基超导体中寻找MZM的原始思路; (b)铁基超导的能带结构, 其中各能带的轨道成分分别为: α (dxz); β (dyz); γ (dxy); η (dxy); δ (dxz); ω (pz); (c)铁基超导的费米面以及超导配对对称性[74,76]; (d)铁基超导体的电子-电子关联[81]; (e)ARPES测量的Fe(Te, Se)单晶Γ点的能带结构[82]
Figure 1. Fe-based superconductors (FeSCs) as a better Majorana platform: (a) The original idea for searching Majorana zero mode (MZM) in FeSCs; (b) the typical band structure of FeSCs, the orbital characters of each band are as follows: α (dxz); β (dyz); γ (dxy); η (dxy); δ (dxz); ω (pz); (c) the typical Fermi surface and superconducting order parameters of FeSCs[74,76]; (d) the mass renormalization among different compounds, indicating strong electron-electron interactions in FeSCs[81]; (e) band structure near the Γ point of Fe(Te, Se) single crystals measured by ARPES[82].
图 2 Fe(Te, Se)的拓扑能带结构与能带反转机制 (a) FeSe单晶的第一性原理计算(不含SOC), 其中红圈的尺寸表示pz轨道的成分[87]; (b) Fe(Te, Se)的晶体结构[87]; (c)拓扑能带的形成机制[87]; (d) FeTe0.5Se0.5单晶的第一性原理计算(不含SOC)[87]; (e)超高分辨率激光ARPES测得的FeTe0.55Se0.45布里渊区中心的体态能带结构[101]; (f) Fe(Te, Se)能带的拓扑相(含SOC), 其中TDS代表拓扑Dirac半金属, TI代表拓扑绝缘体[101]
Figure 2. The mechanism of topological band structure and band inversion of Fe(Te, Se): (a) First-principle calculation of band structure of FeSe (without SOC), the size of red circles represents the components of pz orbital[87]; (b) crystal structure of Fe(Te, Se)[87]; (c) band inversion mechanism and orbital overlapping in Fe(Te, Se)[87]; (d) first-principle calculation of band structure of FeTe0.5Se0.5 (without SOC)[87]; (e) experimental band structure around Γ in FeTe0.55Se0.45 measured by ultra-high resolution laser ARPES[101]; (f) realistic topological band structure in FeTe0.55Se0.45 (with SOC). TDS stands for topological Dirac semimetal, TI stands for topological insulator[101].
图 3 FeTe0.55Se0.45单晶中线性色散的Dirac表面态的实验观测[100] (a) ARPES光子偏振与轨道的选择性; (b) p偏振下观测到的Dirac表面态; (c) s偏振观测到的体态dxz能带; (d)Γ点能带的轨道特性分析; (e), (f)两种偏振下轨道选择性分析, 其中底部的轨道标记代表当前条件下具有ARPES选择活性的轨道, 能带示意图中的虚线部分表示被矩阵元效应禁闭掉的能带
Figure 3. Experimental observation of the linear dispersion of Dirac surface states in FeTe0.55Se0.45[100]: (a) The matrix element effect which defines the selection rule of ARPES intensity, depending on relationship between photon polarization and electron orbitals; (b) the Dirac surface states observed under p-polarization; (c) the dxz bulk bands observed under s-polarization; (d) orbital characters around Γ in FeTe0.55Se0.45; (e), (f) orbital character determined by the matrix element analysis under p- and s-polarization, respectively. The orbital characters marked at the bottom represent the active orbitals under certain polarization and momentum. The dashed parts in the band structure represent the intensity suppressed by the selection rule.
图 4 自旋轨道锁定的Dirac表面态发生各向同性超导配对 (a) FeTe0.55Se0.45中Dirac表面态的自旋轨道锁定性质[100]; (b)自旋分辨的ARPES实验结果: 测量的动量位置为(a)图标注的Cut 1[100]; (c)自旋分辨的ARPES实验结果, 测量的动量位置为(a)图标注的Cut 2[100]; (d) Dirac表面态kF处的能谱变温实验, 数据表明Dirac表面态在低温下打开超导能隙, 且该能隙关闭的温度大致与体态超导临界温度相同[100]; (e) Dirac表面态的超导能隙是各向同性的[100]; (f)总结FeTe0.5Se0.5单晶的拓扑绝缘体态能带的主要观测, 其中有拓扑能带反转、线性Dirac色散、自旋轨道锁定、大超导能隙、小费米能[102]; (g) FeTe0.55Se0.45单晶中的 Dirac表面态感受到来自体态的s ±波超导近邻效应, 满足Fu-Kane模型的要求[100]
Figure 4. Spin-momentum locking and isotropic superconducting gap on the Dirac surface state: (a) The spin-momentum locking feature in FeTe0.55Se0.45 single crystal[100]; (b), (c) spin-resolved ARPES data measured along Cut 1 and Cut 2 in panel (a), respectively[100]; (d) temperature dependent energy distribution curves measured at kF of the Dirac surface state indicating a superconducting gap of the Dirac surface state opens below 14.5 K, which is the bulk Tc[100]; (e) isotropic superconducting gap on the Dirac surface state[100]; (f) summary of the main observations of the Dirac surface state on FeTe0.55Se0.45 single crystal, i.e. topological band inversion, linear dispersion, spin-momentum locking, large superconducting gap, small Fermi energy[102]; (g) the Dirac surface state of FeTe0.55Se0.45 single crystal acquires an effective spinless pairing due to the proximity effective from s ±-wave bulk superconductivity, which satisfies all of the requirements of Fu-Kane model[100].
图 5 FeTe0.55Se0.45单晶中Dirac半金属态的实验观测[101] (a), (b)Γ点附近能带自旋积分和自旋分辨信号; (c)主轴对称性保护的Dirac半金属(001)面的能带结构, 自旋极化的表面态与体态混合; (d), (e)体态Dirac能带(dyz)四个代表性kF上的自旋极化ARPES信号, 实验发现体态Dirac能带具有螺旋自旋特征; (f) FeTe0.55Se0.45单晶中的拓扑非平庸能带结构, 费米能级附近为拓扑绝缘体态, 费米能级以上有Dirac半金属态; (g)Dirac半金属态的输运证据. 在高磁场下, 磁阻随磁场呈线性变化, 表明材料达到量子极限, 这是拓扑能带的证据. 磁阻测量在16 K下进行. 图中PMF和SMF分别代表在脉冲磁场和恒稳磁场实验条件下测量的结果
Figure 5. Evidence of Dirac semimetal phase in FeTe0.55Se0.45 single crystal[101]: (a), (b) The spin-integrated and spin-resolved ARPES spectrum around Γ respectively; (c) the projected band structure on the (001) surface of a C4 symmetry protected Dirac semimetal. The spin polarized surface states are mixing with the bulk bands; (d), (e) spin polarization of the dyz bulk band measured on four representative kF around the Fermi surface (as indicates in the insert). It is clear that the dyz bulk band has the helical spin texture; (f) summary of the topological band structure along the in-plane momentum. There are a strong topological insulator phase around the Fermi level and a topological Dirac semimetal phase above it; (g) the transport evidence of Dirac semimetal phase in FeTe0.55Se0.45 single crystal. The linear transverse magnetoresistance indicates the incorporation of bulk Dirac electrons. The transport experiments were carried at 16 K. The PMF and SMF represent pulse and static magnetic field respectively.
图 6 FeTe0.5Se0.5单晶超导磁通涡旋中MZM的发现 (a)理论预期MZM出现在 FeTe0.5Se0.5单晶的超导磁通涡旋中[100]; (b) STM测量的FeTe0.5Se0.5单晶表面形貌图[102]; (c)零偏压电导绘图, 可以观察到涡旋晶格[102]; (d)部分涡旋中心存在尖锐的ZBCP[102], 在(c)中红框选中的涡旋中心测量的dI/dV谱如红色曲线所示, 在该涡旋边缘测量的dI/dV谱如黑色曲线所示; (e), (f)涡旋零能束缚态的真实性检验之一: 测量到的束缚态是纯净的涡旋束缚态; (e)加场测量前, 先在零磁场下利用dI/dV谱和零偏压电导绘图的方法检验所选区域, 选择无杂质的区域进行涡旋测量, 左图为2 T下零偏压电导绘图测到磁通涡旋, 右图为0 T下在与左图同样空间位置测量的零偏压电导绘图[102]; (f)ZBCP在不同隧道结电阻条件下稳定存在[102]; (g)涡旋零能束缚态的真实性检验之二: ZBCP的半峰全宽接近系统分辨率, 是真实的单峰[102]; (h), (i)涡旋零能束缚态的真实性检验之三: 测量到的束缚态是真正的零能态, 其中(h)为STM直接测量到的I(V)曲线和lock-in输出的dI/dV(V)曲线未经校准过的原始数据[102]; (i)为(h)中红框区域的放大显示[102]
Figure 6. The discovery of vortex Majorana zero mode in FeTe0.55Se0.45 single crystal: (a) The theoretical prediction of vortex MZMs in FeTe0.55Se0.45 single crystal[100]; (b) STM topography of FeTe0.55Se0.45 single crystal[102]; (c) zero-bias conductance map which shows vortex lattice[102]; (d) a sharp zero-bias conductance peak measured at the center of a vortex. In order to make sure the observation is indeed a zero energy vortex bound state, three careful checks are listed as follows. First of all, to make sure that the signal measured is indeed from vortex bound state[102]: (e) ZBC map after and before applying a magnetic field. It shows the local environment of the vortex is clean and free of impurities[102]; (f) ZBCP is stable under different tunneling barriers. Secondly, to make sure that the observed ZBCP is truly a single peak[102]; (g) FWHM of ZBCP measured under different tunneling barriers; (h), (i) the observed ZBCP is truly a zero mode[102]. (h) is the simultaneous measured I(V) curve and dI/dV curve on the center of a vortex core[102], (i) is the enlarged display of red box area in Fig. (h).
图 7 涡旋MZM的波函数[102] (a)磁通涡旋的零偏压电导绘图; (b)沿着(a)图中黑色点线所示位置测量的dI/dV(r, V)强度分布图; (c)与(b)图对应的dI/dV谱; (d)从(c)图中选取的dI/dV谱的重叠表示; (e)ZBCP的强度(上图)以及FWHM(下图)在空间上的分布; (f)ARPES和STS数据对比, 实验测得Dirac表面态Δ0 = 1.8 meV, EF = 4.4 meV, ξ = νF/Δ0
= 123 Å; (g)实验测量和模型计算的MZM强度空间分布, 参数为(f)图的参数. Figure 7. The wavefunction of vortex Majorana zero mode[102]: (a) A zero bias conductance map of a topological vortex; (b) a dI/dV(r, V) line-cut intensity plot along the black dashed line indicated in (a); (c) a waterfall-like plot of (b) with 65 spectra; (d) an overlapping display of eight dI/dV spectra selected from (c); (e) spatial dependence of the height (top) and FWHM (bottom) of the ZBCP; (f) comparison between ARPES and STS results, Δ0
= 1.8 meV, EF = 4.4 meV, ξ = νF/Δ0 = 123 Å; (g) comparison between the measured ZBCP peak intensity with a theoretical calculation of MZM spatial profile with the parameters extracted from (f) 
图 8 涡旋MZM的准粒子中毒效应[102] (a)三个不同磁通涡旋中测量的涡旋中心谱和边缘谱, Background定义为涡旋边缘谱–1—1 meV的积分值, 显然, 超导能隙越“软”, 涡旋中心的ZBCP峰宽越大; (b) 0.55 K (左侧)和4.2 K (右侧)下测量的涡旋束缚态; (c)涡旋MZM的变温实验, 其中灰色曲线为最低温曲线的数值温度卷积; (d)涡旋MZM振幅随温度的变化, 振幅定义为ZBCP的峰谷差; (e)降低准粒子中毒有望提高MZM的存活温度, 左图: 使用C/T拟合MZM振幅随温度的变化关系, C与涡旋MZM的存活温度正相关; 右图: 对9个涡旋MZM变温实验的总结, 0.55 K下振幅越大的MZM可以在更高的温度下存活; (f)涡旋MZM随温度变化行为的合理机制, 红色曲线为涡旋MZM, 蓝色曲线为体能带催生的体态涡旋束缚态的示意图
Figure 8. Quasiparticle poisoning of vortex Majorana zero modes[102]: (a) Three vortex Majorana zero modes measured on different locations, the FWHM of ZBCP at the center of the vortex core is larger when the SC gap around the vortex core is softer; (b) a zero bias conductance map of vortex and line-cut intensity plot of Majorana zero modes measured under 0.55 K (left) and 4.2 K (right), respectively; (c) temperature evolution of ZBCPs in a vortex core. The gray curves are numerically broadened 0.55 K data at each temperature; (d) amplitude of the ZBCPs of three vortex MZMs under different temperatures. The amplitude is defined as the peak-valley difference of the ZBCP; (e) left panel: C/T fitting of amplitude of Majorana ZBCPs under different temperatures. Right panel: summary on several temperature evolution measurements; (f) schematic explanation of the temperature effect on Majorana ZBCPs. The red line is the vortex MZM and the blue line is the bound state of body votex.
图 9 三维涡线模型中的涡线拓扑相变. 第一行: 拓扑材料费米面随化学势的变化. 第二行: kz = 0处的低能涡线束缚态随化学势的变化, 其中红色曲线代表最低能的涡线束缚态. μ < |mΓ|时, 费米能级位于体态能隙内, μ = |mΓ|时费米能级位于导带底／价带顶. μ = μc时, kz = 0处的涡线束缚态发生拓扑能带反转, 涡线拓扑相变发生. 第三行: 不同化学势下涡线束缚态的kz 色散. 第四行: 当材料为拓扑绝缘体时, 表面MZM随化学势的演化. 第五行: 当材料为普通绝缘体时, 表面MZM随化学势的演化. 本图改编自文献[176], 部分内容为原创
Figure 9. Topological vortex phase transition in the three-dimensional vortex line model. The first line: Evolution of the band structure of a topological material by tuning the chemical potentials. The second line: Evolution of the low energy vortex bound state at kz = 0 under different chemical potentials. The third line: The kz dispersion of low energy vortex bound states. The fourth and fifth line: evolution of vortex Majorana zero modes under different chemical potentials in topological insulator and normal insulators, respectively. This figure is adapeted from Ref. [176], some features are added by us.
图 10 共振Andreev反射与MZM本征量子电导 (a)半导体异质结中的经典共振隧穿[207], 隧穿电子能量与双势垒准束缚态能量一致时透射系数为1; (b)经典共振隧穿的电子波函数[105]; (c)经典共振隧穿的替代实验构型: 两针尖跨越隧穿[204], 实现经典共振隧穿的必要条件是两针尖与准束缚态之间的跃迁几率幅相等(t1 = t2); (d) MZM导致的MIRAR[204]. 与经典共振隧穿不同, MIRAR是电子和空穴希尔伯特空间的跨越共振隧穿, 因为入射电子和反射空穴在同一个物理针尖上完成, 因此空穴和电子的跃迁几率幅相等(th = te), MZM对称性保证入射和反射的隧穿耦合强度相等 (${\varGamma }_{\rm t}^{\mathrm{e}}={\varGamma }_{\rm t}^{\mathrm{h}}$ ; Γt = 2πρ0|t|2, 其中 ρ0是有关的态密度), 实现共振条件; (e) MZM参与的Andreev反射的空间波函数, 其中蓝色代表电子部分, 红色代表空穴部分[105]; (f) CdGM束缚态参与的Andreev反射的空间波函数[105]; (g) Law-Lee-Ng理论揭示Majorana模式的本征共振电导[204]; (h)理论计算的Majorana量子电导为2e2/h, 其中蓝色曲线对应(g)中超导体中存在偶数个涡旋的情况, 红色曲线是存在奇数个涡旋的情况[204]; (i)Majorana量子电导的有限温标度行为以及准粒子中毒效应对量子电导的影响
Figure 10. Resonance Andreev reflection induced Majorana quantum conductance: (a) Conventional electron resonance tunneling in a semiconductor heterostructure[207]; (b) the wavefucntion of conventional resonance tunneling[105]; (c) two tips cross-tunneling can be regarded as a replacement of semiconductor heterostructure for realizing semiconductor heterostructure under the condition of equal hopping amplitude around the two tips (t1 = t2)[204]; (d) the Majorana induced resonance Andreev reflection (MIRAR) can be regarded as a superconducting version of the conventional resonance tunneling in the particle-hole Hilbert space. Here a single electrode plays both roles of electron and hole electrode[204]. Due to the particle-hole equivalence property, Majorana modes couple to the incident electron and reflected hole with equal tunneling coupling strength, which satisfies the resonant condition ($ {\varGamma }_{\mathrm{t}}^{\mathrm{e}}={\varGamma }_{\mathrm{t}}^{\mathrm{h}} $; Γt = 2πρ0|t|2, ρ0 being the related density of states); (e), (f) the wavefucntion of Andreev reflection mediated by MZM and a conventional Andreev bound states, respectively[105]; (g) the material setup used in Law-Lee-Ng model[204]; (h) the theoretical calculated re-sonance quantum conductance of Majorana modes[204]; (i) theoretical calculated Majorana conductance under finite temperature and poisoning rate.
图 11 变耦合强度STM谱方法测量涡旋MZM电导平台[105] (a)变耦合强度STM谱方法. STM针尖反馈处于工作状态时, 隧穿电流(It)和扫描偏压(Vs)决定了隧道结大小(GN ≡ It/Vs, GN与$ {\varGamma }_{\mathrm{t}} $正相关), 当改变这两个参数时, STM针尖高度会相应变化, 改变了针尖与样品的隧穿耦合强度($ {\varGamma }_{\mathrm{t}} $); (b)涡旋中心dI/dV谱随GN的变化, MZM出现电导平台特征; (c), (d) FeTe0.55Se0.45单晶涡旋MZM电导的常见行为: 非量子化的电导平台; (e), (f) FeTe0.55Se0.45单晶涡旋MZM电导的罕见行为: 量子化的电导平台; (g)涡旋MZM电导平台值(GP)的统计分布, 样本容量为31; (h)有限能量CdGM束缚态电导行为无平台特征; (i)超导能隙外连续态电导行为无平台特征; (j)零场下零偏压电导行为无平台特征
Figure 11. Variable-tunnel-coupling STM method and the observation of conductance plateau of vortex Majorana zero modes[105]: (a) The tunnel coupling strength can be changed by the tip-sample separation distance under the effect of STM regulation loop; (b) a three-dimensional plot of tunnel coupling dependent measurement, dI/dV(E, GN), which shows a zero bias conductance plateau; (c), (d) the general phenomena observed on Majorana conductance of FeTe0.55Se0.45, i.e. non-quantized plateau; (e), (f) the rare case of Majorana conductance of FeTe0.55Se0.45, i.e. quantized plateau; (g) a histogram of the plateau conductance (GP) from 31 sets of data; (h)—(j) the conductance evolution under different tunnel couplings. It shows no plateau feature measured on finite energy CdGM states, the continuum outside the superconducting gap and zero filed superconducting state, respectively.
图 12 Dirac表面态导致涡旋束缚态半整数能级嬗移 (a)普通s波超导体的涡旋束缚态呈半整数能级序列分布, 其相关能带为常规的抛物线体能带[104]; (b)普通s波超导体的磁通涡旋未达到量子极限时(Texp < TQL = TcΔ/EF), 各级涡旋束缚态卷积在一起, 在空间上呈色散分布, 涡旋中心出现的巨大ZBCP, 为多个非零能束缚态的卷积[231]; (c) Fu-Kane模型的涡旋束缚态呈整数能级序列分布, 其相关能带为Dirac表面态, Dirac表面态引入了磁通涡旋束缚态的半整数能级嬗移[104]; (d)当Dirac表面态的化学势恰好位于Dirac点时称为零掺杂极限, 此时MZM是超导能隙内惟一允许的涡旋束缚态[104]; (e)磁通涡旋的相关能带为图(c)左图时, 涡旋束缚态的态密度径向空间分布, 紫色为准粒子自旋向下的组分, 绿色为准粒子自旋向上的组分[210], 插图: Fu-Kane模型(红色)和普通s波超导体(绿色)磁通涡旋束缚态最低三能级的二维自旋积分态密度. Fu-Kane模型的磁通涡旋束缚态的最低两级波函数呈实心球分布, 而普通s波超导体中只有最低能级为实心球分布, 这一态密度空间图样的差别是涡旋束缚态半整数能级嬗移的强证据[104]; (f)理论计算的趋近零掺杂极限时的涡旋束缚态, 超导能隙中允许的束缚态只有MZM, 数值模拟中磁场方向选择为垂直样品表面向下(图(a)所示为实验中实际使用的磁场方向)
Figure 12. Surface Dirac electron induced half-integer level shift of vortex bound states: (a) Half-odd-integer quantized level sequences of vortex bound states in a conventional s-wave superconductor. There are only parabolic bulk bands involved[104]; (b) the quantum limit is difficult to reach in conventional s-wave superconductors, so that a large ZBCP observed in the center of vortex core is generally due to multiple overlapping of densely packed non-zero peaks[231]; (c) integer quantized level sequences of the vortex bound state in Fu-Kane model. The intrinsic spin Berry phase carried by Dirac surface states induces the half-integer level shift[104]; (d) the zero-doping limit is defined as the chemical potential is approaching the energy of the Dirac point. In this case, a vortex MZM is the only allowed subgap bound state[104]; (e) the theoretical calculated angular momentum resolved wavefunction of BdG eigenstate, the blue and green curves are spin down and up components, respectively[210]. Insert: The calculated spin-integrated 2 D local density of states of three lowest levels of vortex bound states in the case of (c) and (a), respectively[104]; (f) theoretical calculated eigenvalue of BdG Hamiltonian near the zero chemical potential limit.
图 13 实验观测Dirac表面态导致的整数量子化涡旋束缚态[104] (a)拓扑涡旋#1的dI/dV(r, V)强度分布图, 整数量子化的涡旋束缚态能量呈分立分布, 且不随空间位置改变; (b)采用简单高斯拟合提取(a)中实验数据的束缚态峰位; (c)数值计算的#1磁通涡旋的束缚态能量与实验观测值的比较. 右侧坐标轴为使用束缚态能级间距归一化的束缚态能量, 能级序列具有明显的整数量子化特征; (d)拓扑涡旋#11的dI/dV(r, V)强度分布图, 这个涡旋靠近零掺杂极限; (e)图(d)的dI/dV谱, 在超导能隙附近具有量子化的高能涡旋涡旋束缚态; (f)与(c)相同, 是#11的情况; (g)整数量子化(红蓝圆圈)和近零掺杂极限下(黑星)的MZM强度空间分布; (h) Fu-Kane模型计算的不同化学势下的MZM波函数, 化学势越小MZM波函数空间分布越长; (i) 35个具有整数量子化行为的拓扑涡旋的直方图统计, 涡旋束缚态能量被能级间距归一化. 插图: 这35个涡旋所有涡旋束缚态归一化能量的分布图, 涡旋束缚态能量呈整数能级序列; (j)实验测量的整数量子化的拓扑涡旋最低能的三个涡旋束缚态的态密度空间分布, 与图12(e)中Dirac表面态演生涡旋束缚态的理论计算一致. 实验中磁场垂直样品表面向下
Figure 13. Observation of integer quantized vortex bound states[104]: (a) A dI/dV(r, V) line-cut intensity plot measured on a topological vortex #1. Integer quantized vortex bound states are clearly observed; (b) peak positions extracted from (a); (c) the comparison between experimental observed and theoretical calculated level energy in topological vortex #1; (d) same as (a), but measured on vortex #11, which is close to the zero chemical potential limit; (e) overlapping display of dI/dV spectra selected from (d); (f) same as (c), but shows the case of vortex #11; (g) the comparison of observed MZM line profile in topological vortices under integer quantization (open circles) and near the zero chemical potential limit (dark stars); (h) the calculated MZM wavefuction under different chemical potential by Fu-Kane model; (i) a histogram of averaged level energies normalized by the first level spacing, i.e. the ratio EL/ΔE. The statistical analysis is performed among all the 35 topological vortices which show integer quantized CdGMs levels; (j) experimentally observed spatial pattern of the lowest three levels of vortex bound state in a topological vortex.
图 14 材料不均性帮助平庸涡旋共存 (a)平庸涡旋#8的dI/dV(r, V)强度分布图, 半整数量子化的涡旋束缚态能量呈分立分布, 且不随空间位置改变[104]; (b)数值计算的#8磁通涡旋的束缚态能量与实验观测值的比较, 右侧坐标轴为使用束缚态能级间距归一化的束缚态能量, 能级序列具有明显的半整数量子化特征[104]; (c) 26个具有半整数量子化行为的平庸涡旋的直方图统计, 涡旋束缚态能量被能级间距归一化, 插图: 26个涡旋所有涡旋束缚态归一化能量的分布图, 涡旋束缚态能量呈半整数能级序列[104]; (d)无序可以将强拓扑绝缘体变为普通绝缘体, 从左到右非磁性散射势逐渐增强[262]; (e)掺杂可以将强拓扑绝缘体变为弱拓扑绝缘体, 从左到右Te含量依此变大, 三个状态分别为普通绝缘体、强拓扑绝缘体、弱拓扑绝缘体. 绿色能带为奇宇称的pz轨道, 红色能带为偶宇称的dyz和dxz[201]
Figure 14. The inhomogeneity of material helps coexisting ordinary and topological vortices: (a) A dI/dV(r, V) line-cut intensity plot measured on ordinary vortex #8. Half-odd-integer quantized vortex bound states are clearly observed[104]; (b) the comparison between experimental observed and theoretical calculated level energy in ordinary vortex #8[104]; (c) a histogram of averaged level energies normalized by the first level spacing, i.e. the ratio EL/ΔE. The statistical analysis is performed among all the 26 ordinary vortices which show half-odd-integer quantized CdGM levels[104]; (d) surface disorder transforms the strong topological insulator to a normal insulator. The scattering potentials are gradually larger from left to right[262]; (e) concentration of the dopants could drive a strong topological insulator to be a normal insulator or weak topological insulator in Fe(Te, Se). The bands in green (red) represent pz (dxz/dyz) orbital with odd (even) parity[201].
图 15 两种涡旋的空间分布[104] (a), (c), (e)在极低温(40 mK)和弱磁场(2 T)条件下测量的零偏压电导绘图. 随机挑选的三个区域相隔很远. 对这三个区域出现的磁通涡旋进行无差别dI/dV(r, V)测量, 用来鉴别各涡旋的束缚态行为. 其中黄色圆圈标记的涡旋代表存在涡旋MZM的拓扑涡旋, 蓝色圆圈标记的为不存在涡旋MZM的平庸涡旋, 实(虚)线圆圈代表(不)符合整数/半整数量子化行为, 绿色点线将同类涡旋围在一起, 可见同类涡旋总是成群出现, 这表明不均性导致样品表面某些区域Dirac表面态消失, 而在其他区域Dirac表面态保持完好; (b), (d), (f)三个区域中不同类型涡旋的统计数据, 数据测量条件为40 mK, 2.0 T
Figure 15. Spatial distribution of the two classes of vortices[104]: (a), (c), (e) Zero-bias conductance maps of three well-separated regions. The yellow solid circles mark the vortices with ZBCPs and integer quantized CdGM levels, yellow dashed circles mark the vortices with ZBCPs but its CdGM level sequences can not be fitted to integer quantization, blue solid circles mark the vortices without ZBCPs and half-integer quantized CdGM levels, and blue dashed circles mark the vortices without ZBCPs or half-integer quantized CBS levels. The green dashed lines encircle the same class of vortices. Topological vortices and ordinary vortices usually group together, which indicates topological region and trivial region coexist on the sample surface due to spatial inhomogeneity; (b), (d), (f) summary of the ratio of different types of vortices in the three regions, respectively. The data in the three regions are measured under 40 mK and 2.0 T.
图 16 Fe(Te, Se)涡旋中有无MZM的微观机制[104] (a)材料不均性导致的两种表面共存, 其中棕色表面无Dirac表面态, 对应的体态为普通绝缘体或弱拓扑态; (b)在有Dirac表面态的表面区域上涡旋MZM的经验相图; (c)在没有Dirac表面态的表面区域上涡旋MZM的经验相图, 其中相图中蓝色越深的区域代表涡旋MZM越强, 更易被STM实验观测; 颜色越淡代表涡旋MZM强度越弱, 不易被观测; 横轴代表驱动涡线量子相变的量子参数, 比如化学势、Zeeman能等; 纵轴是MZM的有效温度, 包括温度展宽、仪器展宽、基础准粒子中毒展宽等; 红色点线表示估计的目前实验可以覆盖的相区
Figure 16. Mechanism of the presence or absence of MZMs in Fe(Te, Se)[104]: (a) Fe(Te, Se) single crystals are intrinsically inhomogeneous. Disappearance of Dirac surface states is possible in some regions of the (001) surface (brown color). In the conventional regions, the corresponding bulk states can be normal insulators or weak topological insulators. Consequently, the Dirac surface state moves deeper into the bulk and go around the conventional region, as indicated by the gray boundary inside the crystal. In other topological regions (gray color), where Dirac surface states remain intact, the corresponding bulk states are still in the strong topological insulating phase; (b) a schematic phase diagram of vortex MZMs appearing in topological regions (topological vortices). The gradient blue areas in (b) and (c) indicate the phase sector that MZMs can be detected by STM/S experiments. In the dark blue sector, the Majorana wave function is more localized on the sample surface, while in brighter positions, the Majorana wave function strongly hybridizes with bulk quasiparticles and moves deeper beneath the surface, leading to weak ZBCP signal measured by STM/S. The vertical axis demonstrates MZMs evolution as a function of effective temperature which can be represented by extrinsic broadening of the observed ZBCPs. The horizontal axis demonstrates the MZMs evolution as a function of quantum parameters, e.g., chemical potential (μ) measured from the Dirac point. The black dots with an arrow indicate the quantum critical points in which a vortex phase transition happens. Across the critical point, the vortex line turns to be topological trivial and MZMs disappear in the topological region. The red dashed line indicates the achievable region in experiments; (c) a schematic phase diagram of vortex MZMs appearing in conventional regions (ordinary vortices). There are no MZMs in our measurements in those vortices. The observable MZMs can only exist above the critical points when the vortex phase transition turns the trivial vortex line into a 1D topological superconductor in the conventional region.
图 17 编织涡旋MZM, 探索拓扑量子计算. 左上: 拓扑铁基超导体中体超导k-近邻效应诱导的表面等效无自旋手性p波配对[84]; 左下: Fe(Te, Se)单晶中纯净的涡旋MZM[103]; 中间: 利用STM针尖操纵铁基超导表面的涡旋MZM[102]; 右图: 涡旋MZM编织操作与拓扑量子比特[3] (本图部分为原创)
Figure 17. Braiding vortex MZMs and topological quantum computing. Left-top panel: Surface effective spinless p-wave pairing induce by k-proximity effect from bulk bands in Fe(Te, Se)[84]. Left-bottom panel: The pristine vortex MZM observed in Fe(Te, Se)[103]. Middle panel: It is possible to use a STM tip to manipulate vortex MZMs on the surface of Fe(Te, Se)[102]. Right panel: Topological qubit built by braiding four vortex MZMs[3].
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