Fully spin-polarized open and closed nodal lines in β-borophene by magnetic proximity effect

Recently, the experimental realization of Dirac fermions in beta(12)-and X-3-boron sheets has attracted tremendous attention and simulated a broad exploration for the novel topologically nontrivial states in two-dimensional materials. Herein, by combining first-principles and tight-binding calculations, we discover a coexistence of open nodal arc and closed nodal loop produced by three-band touching in the recently synthesized beta-boron sheet [Q. Zhong et al., Phys. Rev. Mater. 1, 021001 (2017)]. The symmetry analysis reveals that the nodal lines are protected by both time-reversal symmetry (TRS) and mirror-reflection symmetry (MRS). Intriguingly, when TRS is broken, e.g., via introducing magnetic proximity effect, a topological phase transition occurs, namely, the original spin-degenerate nodal line (loop) decays into fully spin-polarized nodal line (loop). Furthermore, when Rashba spin-orbit coupling (SOC) is turned on to break the MRS, spin-up and -down states are shifted toward opposite momentum directions, leading to the emergence of two new nodal loops around the Gamma point. In addition, inclusion of both Rashba SOC and exchange field leads to band-gap opening of the Dirac points and nodal lines. Our findings not only reveal a form of nodal line in the beta-boron sheet, but also offer an alternative approach to realize spin-polarized nodal line semimetals with promising applications in spintronic devices.