Ghosh Sovan, Chittari Bheema Lingam
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, 741246, West Bengal, India.
Sci Rep. 2025 Apr 1;15(1):11085. doi: 10.1038/s41598-025-94773-w.
We investigate the orbital Hall conductivity in bilayer graphene (G/G) by modifying one or both the layers as Haldane type ([Formula: see text] : Graphene/Haldane and [Formula: see text] : Haldane/Haldane) with the inclusion of next nearest neighbour (NNN) hopping strength ([Formula: see text]) and flux (ϕ). It is observed that the low energy bands of [Formula: see text] and [Formula: see text] are isolated with a gap at charge neutrality with the next nearest neighbour (NNN) hopping term [Formula: see text]. The time reversal (TR) symmetry breaking with [Formula: see text] induces a large orbital magnetic moment ([Formula: see text]) for the [Formula: see text] band in [Formula: see text] and [Formula: see text] bilayers. This TR symmetry breaking, modulated by the [Formula: see text] strength, leads to the emergence of Orbital Ferromagnetism and Valley Orbital Magnetism within the BZ for the Haldane single layer as well for both [Formula: see text] and [Formula: see text]. We show that for the applied longitudinal electric fields, the intrinsic angular momentum ([Formula: see text]) gives the orbital current ([Formula: see text]) along a transverse direction and generates the orbital Hall conductivity (OHC). We further show that the orbital magnetic polarity leads the Haldane single layer to Orbital Chern Insulator. Interestingly, the orbital Hall conductivities are finite and exhibit a large plateau in the gap over the occupied bands. Moreover, the accumulation of orbital magnetic moment of the bands in Haldane graphene bilayer shows Orbital Hall Insulator and Orbital Chern Insulators with large plateaus. Similarly, we show that in the hetero-bilayers, one of the layers of the Haldane type generates the orbital magnetism and induces the OHC. We conclude that the isolated bands in Haldane graphene bilayers with external stimuli are of an orbital nature and have various orbital Hall phases.
我们通过将双层石墨烯（G/G）中的一层或两层修改为霍尔丹型（[公式：见正文]：石墨烯/霍尔丹和[公式：见正文]：霍尔丹/霍尔丹）来研究其轨道霍尔电导率，其中包含次近邻（NNN）跳跃强度（[公式：见正文]）和磁通（ϕ）。据观察，在存在次近邻（NNN）跳跃项[公式：见正文]的情况下，[公式：见正文]和[公式：见正文]的低能带在电荷中性处被一个能隙隔开。[公式：见正文]导致的时间反演（TR）对称性破缺在[公式：见正文]和[公式：见正文]双层中为[公式：见正文]能带诱导出一个大的轨道磁矩（[公式：见正文]）。这种由[公式：见正文]强度调制的TR对称性破缺，导致在布里渊区（BZ）内，对于霍尔丹单层以及[公式：见正文]和[公式：见正文]两者都出现了轨道铁磁性和谷轨道磁性。我们表明，对于施加的纵向电场，本征角动量（[公式：见正文]）会沿横向给出轨道电流（[公式：见正文]）并产生轨道霍尔电导率（OHC）。我们进一步表明，轨道磁极性使霍尔丹单层成为轨道陈绝缘体。有趣的是，轨道霍尔电导率是有限的，并且在占据能带的能隙中呈现出一个大的平台。此外，霍尔丹石墨烯双层中能带的轨道磁矩积累显示出具有大平台的轨道霍尔绝缘体和轨道陈绝缘体。同样，我们表明在异质双层中，霍尔丹型的其中一层会产生轨道磁性并诱导出OHC。我们得出结论，受外部刺激的霍尔丹石墨烯双层中的孤立能带具有轨道性质，并具有各种轨道霍尔相。
