Majorana zero mode and its lattice construction in iron-based superconductors

Li Geng; Ding Hong; Wang Zi-Qiang; Gao Hong-Jun

doi:10.7498/aps.73.20232022

Abstract

Majorana zero modes (MZMs) obey non-Abelian braiding statistics. The braiding of MZMs can be used to construct the basic unit − topological qubit − of the topological quantum computation, which is immune to environmental noise and can achieve fault-tolerant quantum computation. The existing MZM platforms include hybrid structures such as topological insulator/superconductor, semiconducting nanowire/superconductor and 1d magnetic atomic chain/superconductor, and single materials such as 2M-WS₂, 4Hb-TaS₂, and iron-based superconductors (IBSs). The IBSs have advantages such as easy to fabricate, pure MZMs and high surviving temperatures of MZMs. Recently, a large-scale, ordered and tunable MZM lattice has been observed in LiFeAs, which provides a promising platform to future topological quantum computation. In this paper, first, we review the experimental observations of MZMs in IBSs, focusing on FeTe_0.55Se_0.45, (Li_0.84Fe_0.16)OHFeSe, CaKFe₄As₄ and LiFeAs. Next, we introduce the critical experimental evidences of the MZMs. We also review the recent research work on the ordered and tunable MZM lattice in LiFeAs. Finally, we give conclusion and perspective on future Majorana research.

Keywords:

Authors and contacts

1.
Nanoscale Physics and Devices Laboratory, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
3.
Hefei National Laboratory, Hefei 230088, China
4.
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 201210, China
5.
Boston College, Chestnut Hill, MA 02467, USA

Corresponding author: Gao Hong-Jun, hjgao@iphy.ac.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2019YFA0308500, 2018YFA0305700), the National Natural Science Foundation of China (Grant Nos. 61888102, 51991340, 52072401), the Innovation Program of Quantum Science and Technology, China (Grant No. 2021ZD0302700), and the CAS Project for Young Scientists in Basic Research (Grant No. YSBR-003).

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图 1 (a) FeTe_0.55Se_0.45的高分辨扫描隧道显微镜(STM)图像; (b) 0.5 T下FeTe_0.55Se_0.45表面零偏压dI/dV map 图像; (c)单个磁通涡旋的零偏压dI/dV map 图像; (d)在磁通涡旋中心(红色)以及边界(黑色)处的dI/dV 谱线; (e)沿图(c)箭头方向的空间分辨dI/dV 谱线^[55]

Figure 1. (a) High-resolution scanning tunneling microscope (STM) topography of FeTe_0.55Se_0.45; (b) large-scale dI/dV map of FeTe_0.55Se_0.45 surface at 0 meV under 0.5 T; (c) dI/dV map of a typical vortex hosting MZM at 0 meV; (d) dI/dV curves taken at the center (red) and at the edge (black) of the vortex in panel (c); (e) a waterfall like plot of dI/dV line-cut along the dashed arrow in panel (c), the black curve corresponds to vortex center^[55].

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图 2 (a) DFT+DMFT计算得到的CaKFe₄As₄能带结构; (b) Γ—M方向的ARPES能谱图像; (c)对称化的EDC能谱曲线; (d) CaKFe₄As₄的STM形貌图; (e)一个涡旋附近的零偏压dI/dV map图像; (f)涡旋附近不同位置的dI/dV谱线比较; (g)不同涡旋束缚态的空间分布图案^[59]

Figure 2. (a) DFT+DMFT calculation results for the band structures of CaKFe₄As₄; (b) ARPES spectral intensity plots along the Γ–M direction on CaKFe₄As₄; (c) symmetrized EDCs at the momentum points marked by the red arrows in panel (b), the superconducting gap values of 5.9 meV is attributed to the topological surface bands; (d) STM topography of CaKFe₄As₄; (e) zero-bias conductance map around a vortex core; (f) comparison of dI/dV spectra at vortex core (P1), middle (P2), edge (P3), and without magnetic field (SC gap); (g) spatial patterns of vortex-bound states at energies corresponding to L₀ (MZM), L_–1, L_–2, and L_–3^[59].

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图 3 (a) Li(Fe, Co)As的晶格结构与布里渊区; (b) LiFeAs的能带结构示意图; (c) LiFeAs涡旋内部(红色)和外部(黑色)的大范围dI/dV 谱线; (d)跨越一个杂质辅助涡旋的大范围dI/dV谱线图, 显示出杂质的电子掺杂效应; (e)一个杂质辅助涡旋的零偏压dI/dV map图像; (f)图(e)中沿箭头方向的空间分辨dI/dV 谱线图^[60]

Figure 3. (a) Crystal structure and Brillouin zone of Li(Fe, Co)As; (b) LiFeAs band dispersion along Γ–M and Γ–Z; (c) wide range dI/dV spectra measured at an impurity assisted vortex (red curve) and on a clean surface region without impurities (black curve); (d) wide range line-cut intensity plot for an impurity assisted vortex, showing electron doping effect; (e) a zero bias conductance map around an impurity assisted vortex; (f) dI/dV intensity measured under 2.0 T along the white dashed line indicated in panel (e)^[60].

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图 4 (a) Fe原子沉积在FeTe_0.55Se_0.45表面的STM图像; (b)跨过一个Fe原子的空间分辨dI/dV 谱线图; (c), (d)在一个Fe原子上的dI/dV 谱线图随隧穿势垒的变化; (e), (f)在外加2 T磁场下, 一个Fe原子上的dI/dV 谱线图随隧穿势垒的变化^[96]

Figure 4. (a) STM image of FeTe_0.55Se_0.45 after atomic Fe atom deposition; (b) intensity plot of a series of spectra detected across Fe adatom; (c), (d) tunnel-barrier conductance dependence of the dI/dV spectra on a Fe atom and its intensity plots; (e), (f) the same as panel (c) and (d), but measured under a magnetic field of 6 T^[96].

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图 5 (a)跨越拓扑磁通涡旋的空间分辨dI/dV 谱线图; (b)跨越平庸磁通涡旋的空间分辨dI/dV 谱线图; (c) 35个拓扑磁通涡旋的涡旋束缚态能量统计图; (d) 26个平庸磁通涡旋的涡旋束缚态能量统计图^[81]

Figure 5. (a) Intensity plot and waterfall plot of a dI/dV linecut through a topological vortex core, showing the integer quantized vortex bond states; (b) the intensity plot and waterfall plot of a dI/dV linecut through an trivial vortex core, showing the half-odd-integer quantized vortex bond states; (c) a histogram of averaged level energies for 35 topological vortices; (d) a histogram of averaged level energies for 26 ordinary vortices^[81].

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图 6 (a)利用隧穿势垒调节耦合强度实验的示意图; (b)在不同的隧穿电导G_N下磁通涡旋中心的dI/dV谱线图; (c)不同能量下微分电导随隧穿势垒的变化的三维视觉图像; (d), (e)不同能量下微分电导随隧穿势垒变化的轮廓曲线^[101]

Figure 6. (a) Schematic of tunnel-coupling tunable experiment; Inset: dI/dV spectrum measured at vortex center under 2 T; (b) an overlapping plot of dI/dV spectra at vortex center under different G_N; (c) a three-dimensional schematic diagram depicting the variation in differential conductance values with respect to changes in energy and tunnel junction; (d) line profile of panel (c) along the dashed line at zero bias; (e) line profile of panel (c) along the dashed lines at high bias values^[101].

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图 7 (a)第一类褶皱的STM图像和高度曲线; (b)第二类褶皱的STM图像和高度曲线; (c)两类褶皱与正常区域处dI/dV谱线的比较; (d)—(f)两类褶皱与正常区域处LiFeAs的能带结构示意图^[109]

Figure 7. (a) STM topography (top) and height profile (bottom) of the first type of wrinkle; (b) STM torphology (top) and height profile (bottom) of the second type of wrinkle; (c) comparison of dI/dV spectra between the two wrinkles and the normal region; (d)–(f) schematic diagram of tuning of LiFeAs band structures by strain^[109].

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图 8 (a)双轴电荷密度波区域的STM图像; (b)图(a)的傅里叶变换: (c)不同区域的dI/dV谱线的比较; (d)图(a)中沿不同方向箭头的空间分辨dI/dV谱线图; (e) 0.5 T下, 双轴电荷密度波区域的0偏压dI/dV map图像; (f)图(e)中红色箭头方向的空间分辨dI/dV谱线图^[61]

Figure 8. (a) STM topography of large area biaxial charge density wave region; (b) corresponding Fourier transform of panel (a); (c) comparison of dI/dV spectra in different regions; (d) dI/dV intensity spectra along the arrows marked in panel (a); (e) zero bias dI/dV map of the biaxial CDW region under 0.5 T magnetic field; (f) intensity plot of the dI/dV spectra along the red arrow in panel (e)^[61].

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图 9 大面积、有序可调马约拉纳零能模阵列的形成, (上半部分)不同磁场下的马约拉纳零能模阵列, (下半部分) 6 T下微米尺度有序的马约拉纳零能模阵列^[61]

Figure 9. Formation of large-scale, ordered and tunable MZM lattice. Upper panel: Series of zero energy dI/dV maps of the MZM vortices in the biaxial CDW region under magnetic fields of 0.5, 2, 4, 5, and 6 T. Lower panel: Micrometer-sized ordered MZM lattice under 6 T^[61].

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图 10 (a)磁通涡旋的数量和间距随磁场的变化; (b)不同磁场下的平均dI/dV谱线^[61]

Figure 10. (a) Number (black) and the spacing of neighboring vortices (red) under different magnetic fields, the scanning area is 240 nm × 240 nm; (b) averaged dI/dV spectra under different magnetic fields^[61].

DownLoad: Full-Size Img PowerPoint

图 11 基于马约拉纳零能模阵列的可能编织方案, 包括利用STM针尖物理上移动涡旋和只通过测量实现编织的方案

Figure 11. Schematic of possible braiding approaches based on the MZM lattice. Upper panel: Physically move the vortices around each other using a STM tip. Lower panel: Measurement-only approach without physically moving the vortices.

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Vol. 73, Issue 3 - 2024-02-05

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