  Electrical switching of N\'eel order in an antiferromagnetic insulator is
desirable as a basis for memory applications. Unlike electrically-driven
switching of ferromagnetic order via spin-orbit torques, electrical switching
of antiferromagnetic order remains poorly understood. Here we investigate the
low-field magnetic properties of 30 nm thick, c-axis oriented
$\alpha$-Fe$_2$O$_3$ Hall devices using a diamond nitrogen-vacancy (NV) center
scanning microscope. Using the canted moment of $\alpha$-Fe$_2$O$_3$ as a
magnetic handle on its N\'eel vector, we apply a saturating in-plane magnetic
field to create a known initial state before letting the state relax in low
field for magnetic imaging. We repeat this procedure for different in-plane
orientations of the initialization field. We find that the magnetic field
images are characterized by stronger magnetic textures for fields along
$[\bar{1}\bar{1}20]$ and $[11\bar{2}0]$, suggesting that despite the expected
3-fold magneto-crystalline anisotropy, our $\alpha$-Fe$_2$O$_3$ thin films have
an overall in-plane uniaxial anisotropy. We also study current-induced
switching of the magnetic order in $\alpha$-Fe$_2$O$_3$. We find that the
fraction of the device that switches depends on the current pulse duration,
amplitude and direction relative to the initialization field. Specifically, we
find that switching is most efficient when current is applied along the
direction of the initialization field.
