  The Rashba effect, a spin splitting in electronic band structure, attracts
much attention for the potential applications in spintronics with no
requirement of external magnetic field. Realizing one-dimensional (1D) Rashba
system is a big challenge due to the difficulties of growing high-quality
heavy-metal nanowires or introducing strong spin-orbit coupling (SOC) and
broken inversion symmetry in flexible materials. Here, based on
first-principles calculations, we propose a pathway to realize the Rashba
spin-split by adsorbing Gd atom on zigzag graphene nanoribbons (Gd-ZGNR) and
further investigate the magnetic anisotropy energy (MAE). Perpendicular MAE and
unconventional MAE contributions in k-space are found in the self-assembled
Gd-ZGNR system, which present a remarkable Rashba effect (the estimated
strength is 1.89 eV {\AA}) attributed to strong SOC (~65.6 meV) and the
asymmetric adsorption site at the nanoribbons edge. Moreover, first-order MAE
is connected to the intrinsic Rashba effect beyond the traditional second-order
MAE, which is confirmed based on the analysis of electronic structures
perturbed with SOC in comparison with metastable Gd-ZGNR at central symmetric
adsorption site. The dependence of first-order MAE as well as Rashba effect of
Gd-ZGNRs on the ribbon width are also examined. This work offers new
perspective to achieve 1D Rashba system and provides fundamental understanding
on the magnetic anisotropy, which would be of great significance for searching
Majorana fermions and promoting the potential applications in spintronics.
