$\theta$ and $\phi$ are the the azimuth and polar angles defined the magnetization orientation, as shown in Figure \[fig1\], the sketch of Cartesian coordination is also included in the figure. E($\theta$, $\phi$) is the total energy per unit cell when the magnetization orientation is ($\theta$, $\phi$). Figure \[fig6\] summarizes the $MAE_{xz}$ $MAE_{yz}$ and $MAE_{xy}$ as a function of rotation angle for all TM-chain-GY systems. The results indicate that Fe-chain-GY and Co-chain-GY show relatively larger MAE as 2.422 meV and -2.076 meV, respectively. Fe-chain-GY and Co-chain-GY show easy-axis anisotropy and easy-plane anisotropy,respectively. V-, Cr-, and Mn-chain-GY systems show easy-axis anisotropy and their MAEs are 0.816 meV, 0.605 meV and 1.054 meV, respectively. Our results also indicate that when the quantization axis lies in the *xy* plane the MAE is smaller except for V-chain-GY which exhibits 0.522 meV in-plane anisotropy as shown in Fig.\[fig6\].\
The origin of magnetic anisotropy of TM-chain-GY is the SOC effect of the systems. The spin-conserving term of the SOC Hamilton $\hat{H}_{SO}=\lambda
\hat{L}\cdot \hat{S}$ can be defined as:[@55; @56] $$\hat{H}_{SO}^{sc}=\lambda
\hat{S}_n(\hat{L}_zcos\theta+\frac{1}{2}\hat{L}_+e^{-i\phi}sin\theta
 + \frac{1}{2}\hat{L}_-e^{i\phi}sin\theta) \label{eq7}$$ Where the term $\lambda$ is the SCO constant, $\theta$ is the azimuthal angles (as shown in Fig.\[fig1\]) of the magnetisation direction with the coordinate 