(0,0)                       (90,90)
   Cr(AFM)        0.612             0.607             4.33               0.012                  0.007                     (0,0)                        (90,0)
   Cr(FM)         0.541             0.605             4.33               0.019                  0.005                     (0,0)                       (90,90)
     Mn           1.065             1.054             3.98               0.040                  0.005                     (0,0)                        (90,0)
     Fe           2.232             2.422             2.95               0.112                  0.089                     (0,0)                       (90,90)
     Co          -2.076            -1.750             1.99               0.230                  0.105                     (90,0)                       (0,0)

![MAE as a function of magnetization angle, from x to z axis, y to z axis and x to y axis for hard axis to easy axis of TM-chain-GY, where the Cr-chain only show the FM states.[]{data-label="fig6"}](Fig6.eps "fig:"){width="3.5in"}\

Magnetic anisotropy of TM-chain-GY
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Due to the reduced dimensions for TM chain, magnetic anisotropy can be enhanced and show unique properties. In previous study, it was indicated that 1D (chain or wire) nanostructure on metal surfaces shown lager MAE than 3D bulk material.[@52; @59; @60] Therefore, it is interesting to investigate magnetic anisotropy of 1D TM nanowire on the surface of GY. One of the most interesting issues is that : Is the easy axis parallel or perpendicular to the GY plane?\
To this end, we perform noncollinear magnetic calculation using the GGA(PBE) exchange-correction functional for TM-chain-GY. It well known that the MAE is induced by spin-orbit coupling (SOC) effect. As for MAE calculations, the SOC effect are included. 