Kagome晶格独特的三角与六角嵌套结构导致电子色散能够产生狄拉克能带与平带共存的奇异能带结构。其中狄拉克能带具有无质量的电子，其电子态密度低，运动速度快。体系在自旋轨道耦合的作用下，线性交叉的狄拉克能带能够打开特定大小的能隙，当其能带受到拓扑保护时，边缘能隙处能够产生输运方向与自旋方向锁定的金属态。此外，自旋非对称分布对Kagome材料拓扑输运的影响也是当前亟待研究的一个重要的科学问题。一方面，材料的内禀磁性导致能带时间反演对称性破缺，会诱导四重简并的狄拉克点劈裂为两个二重简并的外尔点，这个过程能够产生奇异的量子现象，如量子反常霍尔效应；同时，内禀磁性也能调控电子的自旋极化强度，对于设计拓扑自旋电子器件有着极大的帮助。另一方面，内禀磁性导致体系的时间反演对称性破缺会破坏体系的拓扑稳定性，从而影响到表面电子拓扑非平庸输运。与狄拉克能带不同，准平带电子不发生明显色散，呈现出高度简并，电子运动速度慢、质量大，电子间主要以库伦相互作用为主，这会产生许多强关联电子效应，如可调控的声-电耦合和高温超导等。针对Kagome晶格的线性能带色散和平带凝聚，本文主要做了以下三个方面的研究：

（1）对三维磁性Kagome半金属材料SbV₃S₅的拓扑输运进行了理论计算研究。计算结果表明，在内禀磁性的影响下，费米面附近能带电子具有相同的自旋方向，即该材料出现自旋极化输运过程。磁性导致能带时间反演对称破缺，费米面附近出现多个能带交叉的外尔点，沿高对称点路径Γ-M-K-Γ费米面上的能带具有一定的弱拓扑结构，其能带中的两个外尔点是一对手性外尔点。此外，费米面附近的能带具有两个稳定存在的第二类范霍夫奇点，其对应的能级位置出现一个大的电子态密度变化。表面电子散射与拓扑电荷输运数据表明，内禀磁性破坏了表面拓扑输运的稳定性，电子拓扑输运局域在能谷边缘，费米能级上能谷边缘的局域拓扑输运导致了一个小的反常霍尔电导平台。

（2）为探究磁性强弱对Kagome晶格拓扑输运的影响，本文提出了一种非磁二维蜂窝-Kagome晶格材料Bi₂Te₃。相关计算数据表明，在不考虑自旋轨道耦合作用时，该材料是狄拉克半金属材料，费米面上存在两个四重简并的能带线性交叉点，对应能级位置电子态密度较小。在自旋轨道耦合作用下，费米面上的能带交叉点打开了一个0.486 eV的带隙，且价带顶到导带底的轨道翻转与宇称翻转表征了能带的强拓扑结构。此外，费米面附近也存在典型的Kagome晶格能带-平带，对应能级位置处出现了相应的电子态密度突变峰值。边缘电子态与拓扑电荷输运数据展示了该材料的边缘电子自旋锁定输运，其边缘拓扑输运导致了一个长程稳定的量子自旋霍尔电导平台，自旋霍尔电导的大小与空间对称性高低呈现正相关关联。

（3）为研究Kagome晶格平带的声-电耦合特性，本文人工设计了一种Kagome-石墨烯材料与两种Kagome-石墨炔材料。理论计算表明，Kagome-石墨烯具有符合研究要求的平带结构，且声子色散没有虚频。计算数据表明，Kagome-石墨烯具有与石墨烯相同的成键方式，主要表现为sp²、p_z轨道杂化。一部分p_z轨道电子局域在Kagome晶格的三角结构当中，电子运动速度低，质量大，基本不发生散射；即费米面上的平带电子都来自于p_z轨道，其对应能级位置具有一个非常大的电子态密度。声-电耦合与超导电性计算数据表明，Kagome-石墨烯在Γ点有一个大的声子线宽峰值，且体系拥有大的声-电耦合强度和高的超导相变温度，其超导准粒子态密度大，超导能隙呈现各向同性的特征。

本文的研究工作给出了Kagome晶格的自旋电子输运与平带凝聚相关理论计算数据，揭示了Kagome晶格线性交叉能带的拓扑特征与平带的超导电性，为设计拓扑自旋电子器件与新型高温超导材料提供了丰富的理论数据。

The unique triangular and honeycomb hexagonal nested structure of the Kagome lattice leads to the formation of an exotic band structure in which Dirac bands and flat bands coexist. The Dirac bands are composed of massless electrons with low electron state density and high velocity. Under the effect of spin-orbit coupling, the linearly intersecting Dirac bands can open a gap of a certain size. When the band is topologically protected, a metallic state with transport direction and spin direction locking can be generated at the edge of the gap. In addition, the spin-asymmetric distribution also has an impact on the topological transport of Kagome materials, which is currently an urgent scientific issue. On the one hand, the intrinsic magnetism of the material leads to a breaking of the time-reversal symmetry of the band structure, inducing the splitting of the four-fold degenerate Dirac point into two doubly degenerate Weyl points, which can produce unusual quantum phenomena, such as the Quantum Anomalous Hall Effect(QAHE). At the same time, the intrinsic magnetism can also regulate the spin polarization intensity of electrons, which is of great help in designing topological spintronic devices. On the other hand, the breaking of the time-reversal symmetry of the system due to the intrinsic magnetism will destroy the topological stability of the system, thus affecting the non-trivial surface electronic transport. Unlike the Dirac bands, the flat bands have no obvious dispersion, exhibiting high degeneracy, slow electron velocity, heavy mass, and mainly Coulomb interactions between electrons, which can produce many strong correlated electron effects, such as tunable phonon-electron coupling and high-temperature superconductivity. In this paper, we mainly investigate the linear band dispersion and flat band condensation of the Kagome lattice in the following three aspects:

(1) Theoretical calculations were conducted on the three-dimensional antiferromagnetic Kagome weak topological semimetal SbV₃S₅. The results show that under the influence of intrinsic magnetism, the spin directions of the band electrons near the Fermi surface are the same, indicating the occurrence of spin-polarized transport in the material. The magnetism-induced breaking of the band time-reversal symmetry results in multiple Weyl points near the Fermi surface where several bands cross. The bands along the high-symmetry point path Γ-M-K-Γ on the Fermi surface have a certain topological structure, and the two Weyl points in the band are a pair of chiral Weyl points. In addition, there are two stable Van Hove singularities near the Fermi surface, corresponding to a large electron density of states. Surface electron scattering and topological charge transport data show that intrinsic magnetism disrupts the stability of surface topological transport, and the electron topological transport is localized at the valley edge, resulting in a small anomalous Hall conductivity plateau at the Fermi level.

(2) To investigate the influence of magnetic strength on topological transport, this paper proposes a non-magnetic two-dimensional honeycomb-Kagome lattice material, Bi₂Te₃. Relevant computational data shows that when spin-orbit coupling is not considered, this material is a Dirac semimetal with two fourfold degenerate band linear crossing points on the Fermi surface, corresponding to a small electron state density in energy level positions. Under spin-orbit coupling, the band crossing points on the Fermi surface open a gap of approximately 0.486 eV, and the band inversion and parity inversion from the valence band top to the conduction band bottom characterize the topological structure of the band. In addition, there also exists a typical Kagome lattice band-flat band near the Fermi surface, corresponding to a sharp peak of electron state density at the corresponding energy level position. The edge electronic state and topological charge transport data demonstrate the edge electronic spin-locked transport of the material, and the edge topological transport results in a long-range stable quantum spin Hall conductivity platform, which exhibits a positive correlation between the size of spin Hall conductivity and the degree of spatial symmetry.

(3) To study the acoustoelectric coupling characteristics of the Kagome lattice flat band, this paper artificially designs a Kagome-graphene material and two Kagome-graphyne materials. Theoretical calculations show that the Kagome-graphene has a flat band structure that meets the research requirements, and the phonon dispersion has no imaginary frequency. Relevant computational data shows that Kagome-graphene has the same bonding mode as graphene, mainly manifested as sp² and p_z orbital hybridization. Some p_z orbital electrons are localized in the triangular structure of the Kagome lattice, with a low electron velocity and a large mass, which results in minimal scattering. That is, the flat band electrons on the Fermi surface all come from the p_z orbital, and their corresponding energy level positions have a very high electron state density. Acoustoelectric coupling and superconductivity computational data show that Kagome-graphene has a large phonon linewidth peak at the Γ point, and the system has a strong acoustoelectric coupling strength and a high superconducting transition temperature. The density of superconducting quasi-particles is large, and the superconducting gap exhibits isotropic characteristics.

The thesis provides theoretical computational data on the spin transport and flat band condensation of the Kagome lattice, revealing the topological characteristics of the linear crossing band and the superconductivity of the flat band. This provides rich theoretical data for designing topological spin electronic devices and new high-temperature superconducting materials.

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