Magnetic moment of a current carrying loop $\vec{M}=N I \vec{A}$

Here $N=1, A=a^{2}+2 \pi\left(\frac{a}{2}\right)^{2}=a^{2}\left[1+\frac{\pi}{2}\right]$

From Screw law, direction of $m$ is outward or in +ve z-direction.

$\therefore \quad \vec{M}=I a^{2}\left[1+\frac{\pi}{2}\right] \hat{k}$