标题： 有限石墨烯纳米带的拓扑能级通过电场的调控 显示英文标题

标题： Topological Energy levels of Finite Graphene Nanoribbons Tuned by Electric Fields

摘要： 在本综合性研究中，我们对有限的扶手椅型石墨烯纳米带（AGNRs）和异质结构在横向电场下的拓扑态（TSs）的斯塔克位移进行了理论研究。 我们的重点集中在AGNRs的多个端部锯齿形边缘态以及$9-7-9$AGNR异质结构的界面态上。 对于正式的TSs，我们在TSs能级不与体态能级融合的范围内，观察到相对于电场的显著蓝移斯塔克位移。 相反，对于后者TSs，我们在费米能级附近观察到能级的振荡斯塔克位移。 同时，我们揭示了斯塔克效应对两种TSs的传输系数的影响。 值得注意的是，我们在多个端部锯齿形边缘态中发现了有趣的光谱。 在有限的$9-7-9$AGNR异质结构的情况下，传输系数的光谱表明，可以通过横向电场很好地控制拓扑界面态之间的耦合强度。 这项研究的结果不仅有助于更深入地理解石墨烯基材料的电子特性，还为下一代电子器件和量子技术的创新铺平了道路。 显示英文摘要

摘要： In this comprehensive study, we conduct a theoretical investigation into the Stark shift of topological states (TSs) in finite armchair graphene nanoribbons (AGNRs) and heterostructures under transverse electric fields. Our focus centers on the multiple end zigzag edge states of AGNRs and the interface states of $9-7-9$ AGNR heterostructures. For the formal TSs, we observe a distinctive blue Stark shift in energy levels relative to the electric field within a range where the energy levels of TSs do not merge into the energy levels of bulk states. Conversely, for the latter TSs, we identify an oscillatory Stark shift in energy levels around the Fermi level. Simultaneously, we reveal the impact of the Stark effect on the transmission coefficients for both types of TSs. Notably, we uncover intriguing spectra in the multiple end zigzag edge states. In the case of finite $9-7-9$ AGNR heterostructures, the spectra of transmission coefficient reveal that the coupling strength between the topological interface states can be well controlled by the transverse electric fields. The outcomes of this research not only contribute to a deeper understanding of the electronic property in graphene-based materials but also pave the way for innovations in next-generation electronic devices and quantum technologies.